Singular Solutions of Nonlinear Elliptic and Parabolic Equations 9783110332247, 9783110315486

Table of contents :
Contents
Foreword
I. Nonlinear elliptic equations with L1-data
1. Nonlinear elliptic equations of the second order with L1-data
1.1 Introduction
1.2 General conditions for the limiting summability of solutions
1.3 The cases where the right-hand side of the equation belongs to logarithmic classes
1.4 On the integrally logarithmic conditions for the limiting summability of solutions
1.5 The case where the right-hand side of equation belongs to Lebesgue spaces close to L1(O)
1.6 On the convergence of functions from satisfying special integral inequalities
1.7 On the existence of entropy solutions for the equations with degenerate coercivity and L1-data
1.8 A priori properties of the entropy solutions of equations with degenerate coercivity and L1-data
2. Nonlinear equations of the fourth order with strengthened coercivity and L1-data
2.1 Introduction
2.2 Set of functions
2.3 Definition and some properties of entropy solutions
2.4 One a priori estimate for the entropy solutions
2.5 Notion of H-solution
2.6 On uniqueness of the entropy solution
2.7 Theorems on existence
2.8 Entropy solutions as elements of the Sobolev spaces and the existence of W-solutions
2.9 On the summability of functions from satisfying certain integral inequalities
2.10 Improvement of the properties of summability for the solutions of problem (2.1.6), (2.1.7)
2.11 Some characteristics of the set of functions
2.12 Set of functions
2.13 Definition and a priori estimates of the proper entropy solutions
2.14 Existence of the proper entropy solutions
2.15 Relationship with the entropy solutions and the theorem on uniqueness
2.16 Relationship with H-solutions and W-solutions
2.17 Properties of summability of the proper entropy solutions
2.18 Relationship with generalized solutions
2.19 Examples of coefficients and the right-hand sides of Eq. (2.1.6)
II. Removability of singularities of the solutions of quasilinear elliptic and parabolic equations of the second order
3. Removability of singularities of the solutions of quasilinear elliptic equations
3.1 Introduction
3.2 Removability of isolated singularities
3.2.1 Formulation of assumptions and principal results
3.2.2 Integral estimates of the solutions for 1 < p < n
3.2.3 Pointwise estimates of the solutions for 1 < p < n
3.3 Removability of singularities of the solutions of elliptic equations on manifolds
3.3.1 Formulation of assumptions and main results
3.3.2 Integral estimates for the gradient of solution in the case 1 < p < n - s
3.3.3 Pointwise integral estimates for the solution in the case 1 < p < n - s
3.4 Removability of isolated singularities of the solutions of elliptic equations with absorption
3.4.1 Formulation of the assumptions and main results
3.4.2 Proof of Theorem 3:4:2
3.4.3 Integral estimates for the gradient of the solution
3.4.4 Proof of Theorem 3:4:1
4. Removability of singularities of the solutions of quasilinear parabolic equations
4.1 Introduction
4.2 Removability of isolated singularities
4.2.1 Formulation of assumptions and main results
4.2.2 Integral estimates for the solution
4.2.3 Pointwise estimates of the solution
4.3 Removability of isolated singularities for the solutions of quasilinear parabolic equations with absorption
4.3.1 Formulation of the assumptions and main results
4.3.2 Proof of Theorem 4:3:2
4.3.3 Integral estimates for the solution
4.3.4 Proof of Theorem 4:3:1
5. Quasilinear elliptic equations with coefficients from the Kato class
5.1 Introduction
5.2 Harnack’s inequality
5.2.1 Formulation of assumptions and main results
5.2.2 Proof of Theorem 5:2:1
5.2.3 Proof of Theorem 5:2:2
5.3 Removability of isolated singularities
5.3.1 Statement of propositions and main results
5.3.2 Proof of Theorem 5:3:2
5.3.3 Proof of Theorem 5:3:1
5.4 Removability of isolated singularities for the solutions of quasilinear elliptic equations with absorption
5.4.1 Formulation of assumptions and main results
5.4.2 Proof of Theorem 5:4:2
5.4.3 Integral and pointwise estimates for the gradient of the solution
III. Boundary regimes with peaking for quasilinear parabolic equations
6. Energy methods for the investigation of localized regimes with peaking for parabolic second-order equations
6.1 Introduction: localized and nonlocalized singular boundary regimes
6.2 Sufficient conditions for the localization of boundary regimes with peaking
6.3 Sharp conditions for the effective localization of boundary regimes: the case of slow diffusion p > q
6.4 Effective localization of singular boundary regimes for quasihomogeneous parabolic equations
6.5 Effective localization of singular boundary regimes for the equations of nonstationary fast-diffusion type
7. Method of functional inequalities in peaking regimes for parabolic equations of higher orders
7.1 Boundary peaking regimes for quasilinear parabolic equations of higher orders
7.2 Energy functions of the solutions and the main system of functional inequalities
7.3 Localized singular boundary regimes: the case of slow diffusion p > q
7.4 Localized boundary regimes: the case p = q
8. Nonlocalized regimes with singular peaking
8.1 Propagation of blow-up waves
8.2 Estimates for the blow-up wave in the equation of slow-diffusion type
8.3 Blow-up waves in quasihomogeneous parabolic equations
9. Appendix: Formulations and proofs of the auxiliary results
9.1 Interpolation inequalities
9.2 Systems of differential inequalities
9.3 Functional inequalities
Bibliography

Citation preview

Alexander A. Kovalevsky, Igor I. Skrypnik, Andrey E. Shishkov Singular Solutions of Nonlinear Elliptic and Parabolic Equations

De Gruyter Series in Nonlinear Analysis and Applications

Editor in Chief Jürgen Appell, Würzburg, Germany Editors Catherine Bandle, Basel, Switzerland Alain Bensoussan, Richardson, Texas, USA Avner Friedman, Columbus, Ohio, USA Mikio Kato, Nagano, Japan Wojciech Kryszewski, Torun, Poland Umberto Mosco, Worcester, Massachusetts, USA Louis Nirenberg, New York, USA Simeon Reich, Haifa, Israel Alfonso Vignoli, Rome, Italy Katrin Wendland, Freiburg, Germany

Volume 24

Alexander A. Kovalevsky, Igor I. Skrypnik, Andrey E. Shishkov

Singular Solutions of Nonlinear Elliptic and Parabolic Equations

Mathematics Subject Classification 2010 35A20, 35B40, 35B44, 35B45, 35J25, 35J40, 35J60, 35K20, 35K35, 35K55, 35R05 Authors Prof. Alexander A. Kovalevsky Institute of Mathematics and Mechanics Ural Branch of the Russian Academy of Sciences ul. S. Kovalevskoi 16 Yekaterinburg, 620990 Russia and Institute of Mathematics and Computer Science Ural Federal University pr. Lenina 51 Yekaterinburg, 620000 Russia [email protected] Dr. Igor I. Skrypnik Institute of Mathematics National Academy of Sciences of Ukraine vul.Tereschenkivska 3 Kiev, 01601 Ukraine [email protected]

Prof. Andrey E. Shishkov Institute of Applied Mathematics and Mechanics National Academy of Sciences of Ukraine ul. Dobrovol‘skogo 1 Slavyansk, Donetsk region, 84116 Ukraine [email protected]

ISBN 978-3-11-031548-6 e-ISBN (PDF) 978-3-11-033224-7 e-ISBN (EPUB) 978-3-11-039008-7 Set-ISBN 978-3-11-033225-4 ISSN 0941-813X Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. ©2016 Walter de Gruyter GmbH, Berlin/Boston Printing and binding: CPI books GmbH, Leck ♾Printed on acid-free paper Printed in Germany www.degruyter.com

Foreword

We present the results of our investigations of some actual fields of the contemporary theory of nonlinear partial differential equations. Briefly speaking, these fields can be described as follows: the investigation of the existence and properties of solutions of the equations with weakly integrable and irregular data, the analysis of conditions required for the removability of singularities of the solutions, and the study of the boundary regimes with singular peaking. As a common specific feature of the indicated fields, we can mention the fact that the results obtained in these directions and the methods used for their investigation reflect, to a significant extent, the singular character of solutions of the analyzed equations. The first direction includes the investigation of divergent nonlinear equations with L1 -right-hand sides and measures as the right-hand sides. This field has been extensively developed since the end of the 1980s. At present, the theory of nonlinear second-order elliptic equations with L1 -data or data measures is, for the most part, constructed. Within the framework of this theory, the notions of weak, entropy, and renormalized solutions of the investigated equations were introduced, the theorems on existence and uniqueness of these solutions were proved, and their properties of summability were described. Among the researchers who made significant contributions to the development of the investigated theory, we mention, e.g., Ph. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, F. Murat, M. Pierre, and J. L. Vázquez. Some problems related to this direction are considered in Chapters 1 and 2 (Part I of the present book). This part is mainly based on the results obtained by A. A. Kovalevsky. In Chapter 1, we consider the nonlinear second-order elliptic equations with L1 -data. We present the conditions that should be imposed on the right-hand side of the equation in order to guarantee the limiting summability of the entropy solution of the corresponding Dirichlet problem, describe the improvement of the properties of summability of the entropy solution depending on the increase in the integrability of the right-hand side of the equation, and present the detailed description of the results on the existence and the a priori properties of the entropy solutions of the Dirichlet problem for the equations with degenerate coercivity and L1 -right-hand sides. Chapter 2 is devoted to the investigation of nonlinear fourth-order elliptic equations with strengthened coercivity and L1 -data. In this chapter, we analyze the notions of entropy and regular entropy solutions of the Dirichlet problem for the indicated equations. We present the a priori estimates, establish

vi the theorems on existence and uniqueness of these solutions, and describe their summability properties. Some other types of solutions of the same problem are also considered. The second direction is connected with the study of the removability of singularities of the solutions of quasilinear elliptic and parabolic equations. The researches in this direction are extensively developed since the 1960s. The exact conditions of removability of the singularities are established for the solutions of quasilinear elliptic and parabolic equations with coefficients from the Lebesgue and Kato classes. Among the researchers who made significant contributions to the development of this direction, we can mention, e.g., D. G. Aronson, H. Brézis, A. Friedman, S. Kamin, L. A. Peletier, V. A. Kondrat’ev, E. M. Landis, J. Serrin, and L. Véron. Part II of the present book is devoted to the investigation of some problems encountered in this field. This part includes Chapters 3–5 and mainly contains the results obtained by V. Liskevich, I. I. Skrypnik, and I. V. Skrypnik. In Chapter 3, we describe the exact conditions of removability of the isolated singularities and the singularities on manifolds of the solutions of quasilinear elliptic equations with coefficients from the Lebesgue classes. In Chapter 4, we present the exact conditions of removability of the isolated singularities of solutions of the quasilinear parabolic equations with coefficients from the Lebesgue classes. Finally, in Chapter 5, we consider the quasilinear elliptic equations with coefficients from the Kato classes. In this chapter, we prove Harnack’s inequality for nonnegative solutions and describe the exact conditions of removability of the isolated singularities of solutions of the analyzed equations. The third direction is connected with the investigation of the asymptotic properties of solutions of parabolic equations with boundary data singularly peaking at a certain time. These studies were stimulated by the analysis of various new physicomathematical models, including the models connected with the problems of controlled thermonuclear synthesis extensively developed since the beginning of the 1960s. For the first time, the effect of spatial localization of the blow-up set of the solution or, in other words, of localization of the singular boundary regime was discovered by A. A. Samarskii and I. M. Sobol’ (1963) for a model quasilinear heat equation. Later, for various classes of model parabolic equations admitting the application of the method of sub- and supersolutions, the conditions of localization and nonlocalization of boundary regimes and various estimates of the behavior of solutions in the vicinity of the time of peaking of the boundary regime were established by A. A. Samarskii, S. P. Kurdyumov, V. A. Galaktionov, B. H. Gilding, M. A. Herrero, C. Cortazar, M. Elgueta, and other researcheres. At the end of the 1990s, A. E. Shishkov developed an alternative method for the investigation of singular boundary regimes for general quasilinear second-order parabolic equations not connected with the barrier

vii technique and obtained as a modification of the method of local energy estimates used earlier in the study of the supports of solutions of degenerate parabolic equations. In our book, the methods and results obtained in this direction are presented in Part III including Chapters 6–9. The results presented in these chapters were, for the most part, obtained by A. E. Shishkov. In Chapter 6, on the basis of the proposed modified method of local energy estimates, we describe the exact sufficient conditions of localization of the singular boundary regimes for general divergent “doubly” nonlinear second-order parabolic equations of the types of slow, fast, and “neutral” diffusion. Note that, in recent years, A. E. Shishkov and V. A. Galaktionov have developed a method of local integral estimates for the case of general quasilinear parabolic equations of arbitrary order. In Chapter 7, we present a modified proof of the sufficient conditions of localization of the singular regimes for various classes of higher-order equations. In Chapter 8, we study the nonlocalized regimes and establish the exact upper bounds for the speed of propagation of the singularity wave. Finally, Chapter 9 contains some technical elements of the proofs of the results presented in Chapters 6–8. Authors, Donetsk, Ukraine.

Contents

Foreword

v

I

Nonlinear elliptic equations with L1 -data

1

Nonlinear elliptic equations of the second order with L1 -data

3

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2 General conditions for the limiting summability of solutions . . . . 16 1.3 The cases where the right-hand side of the equation belongs to logarithmic classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.4 On the integrally logarithmic conditions for the limiting summability of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.5 The case where the right-hand side of equation belongs to Lebesgue spaces close to L1 (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ◦

1.6 On the convergence of functions from W 1,p (Ω) satisfying special integral inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 1.7 On the existence of entropy solutions for the equations with degenerate coercivity and L1 -data . . . . . . . . . . . . . . . . . . . . . . . . . . 61 1.8 A priori properties of the entropy solutions of equations with degenerate coercivity and L1 -data . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2

Nonlinear equations of the fourth order with strengthened coercivity and L1 -data

83

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 ◦

2.2 Set of functions H 1,q 2,p (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.3 Definition and some properties of entropy solutions . . . . . . . . . . . 94 2.4 One a priori estimate for the entropy solutions . . . . . . . . . . . . . . . 98 2.5 Notion of H-solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 2.6 On uniqueness of the entropy solution . . . . . . . . . . . . . . . . . . . . . . . 107 2.7 Theorems on existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 2.8 Entropy solutions as elements of the Sobolev spaces and the existence of W -solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

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2.9 On the summability of functions from H 1,q 2,p (Ω) satisfying certain integral inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 2.10 Improvement of the properties of summability for the solutions of problem (2.1.6), (2.1.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 ◦

2.11 Some characteristics of the set of functions H 1,q 2,p (Ω) . . . . . . . . . . 150 b 1,q (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 2.12 Set of functions H 2,p 2.13 Definition and a priori estimates of the proper entropy solutions 157 2.14 Existence of the proper entropy solutions . . . . . . . . . . . . . . . . . . . . 164 2.15 Relationship with the entropy solutions and the theorem on uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 2.16 Relationship with H-solutions and W -solutions . . . . . . . . . . . . . . 171 2.17 Properties of summability of the proper entropy solutions . . . . . . 175 2.18 Relationship with generalized solutions . . . . . . . . . . . . . . . . . . . . . . 180 2.19 Examples of coefficients and the right-hand sides of Eq. (2.1.6) . . 184

II Removability of singularities of the solutions of quasilinear elliptic and parabolic equations of the second order 3

Removability of singularities of the solutions of quasilinear elliptic equations

191

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 3.2 Removability of isolated singularities . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Formulation of assumptions and principal results . . . . . . . 3.2.2 Integral estimates of the solutions for 1 < p < n . . . . . . . 3.2.3 Pointwise estimates of the solutions for 1 < p < n . . . . . . 3.3 Removability of singularities of the solutions of elliptic equations on manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Formulation of assumptions and main results . . . . . . . . . . 3.3.2 Integral estimates for the gradient of solution in the case 1 < p < n − s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Pointwise integral estimates for the solution in the case 1 < p < n − s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

192 192 194 198 200 200 201 210

3.4 Removability of isolated singularities of the solutions of elliptic equations with absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 3.4.1 Formulation of the assumptions and main results . . . . . . . 217 3.4.2 Proof of Theorem 3.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

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3.4.3 3.4.4 4

Integral estimates for the gradient of the solution . . . . . . . 219 Proof of Theorem 3.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

Removability of singularities of the solutions of quasilinear parabolic equations

226

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

5

4.2 Removability of isolated singularities . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Formulation of assumptions and main results . . . . . . . . . . 4.2.2 Integral estimates for the solution . . . . . . . . . . . . . . . . . . . . 4.2.3 Pointwise estimates of the solution . . . . . . . . . . . . . . . . . . .

227 227 229 233

4.3 Removability of isolated singularities for the solutions of quasilinear parabolic equations with absorption . . . . . . . . . . . . . . 4.3.1 Formulation of the assumptions and main results . . . . . . . 4.3.2 Proof of Theorem 4.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Integral estimates for the solution . . . . . . . . . . . . . . . . . . . . 4.3.4 Proof of Theorem 4.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

237 237 238 240 247

Quasilinear elliptic equations with coefficients from the Kato class

250

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 5.2 Harnack’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Formulation of assumptions and main results . . . . . . . . . . 5.2.2 Proof of Theorem 5.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Proof of Theorem 5.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

251 251 253 262

5.3 Removability of isolated singularities . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Statement of propositions and main results . . . . . . . . . . . . 5.3.2 Proof of Theorem 5.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Proof of Theorem 5.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

263 263 264 270

5.4 Removability of isolated singularities for the solutions of quasilinear elliptic equations with absorption . . . . . . . . . . . . . . . . . 5.4.1 Formulation of assumptions and main results . . . . . . . . . . 5.4.2 Proof of Theorem 5.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Integral and pointwise estimates for the gradient of the solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

272 272 273 275

III Boundary regimes with peaking for quasilinear parabolic equations 6

Energy methods for the investigation of localized regimes with peaking for parabolic second-order equations

283

6.1 Introduction: localized and nonlocalized singular boundary regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

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6.2 Sufficient conditions for the localization of boundary regimes with peaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 6.3 Sharp conditions for the effective localization of boundary regimes: the case of slow diffusion p > q . . . . . . . . . . . . . . . . . . . . 304 6.4 Effective localization of singular boundary regimes for quasihomogeneous parabolic equations . . . . . . . . . . . . . . . . . . . . . . 317 6.5 Effective localization of singular boundary regimes for the equations of nonstationary fast-diffusion type . . . . . . . . . . . . . . . . 327 7

Method of functional inequalities in peaking regimes for parabolic equations of higher orders

333

7.1 Boundary peaking regimes for quasilinear parabolic equations of higher orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 7.2 Energy functions of the solutions and the main system of functional inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 7.3 Localized singular boundary regimes: the case of slow diffusion p > q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 7.4 Localized boundary regimes: the case p = q . . . . . . . . . . . . . . . . . 361 8

Nonlocalized regimes with singular peaking

377

8.1 Propagation of blow-up waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 8.2 Estimates for the blow-up wave in the equation of slow-diffusion type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 8.3 Blow-up waves in quasihomogeneous parabolic equations . . . . . . 400 9

Appendix: Formulations and proofs of the auxiliary results

412

9.1 Interpolation inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 9.2 Systems of differential inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 413 9.3 Functional inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 Bibliography

425

Part I

Nonlinear elliptic equations with L1-data

Chapter 1

Nonlinear elliptic equations of the second order with L1-data 1.1

Introduction

Let n ∈ N, n > 2, Ω be a bounded open set in Rn and let p ∈ (1, n). Also let c1 , c2 > 0, g ∈ Lp/(p−1) (Ω), g > 0 in Ω, and let, for any i ∈ {1, . . . , n}, ai be a Carathéodory function on Ω × Rn . We assume that the following inequalities hold for almost all x ∈ Ω and any ξ ∈ Rn : n X

|ai (x, ξ)| 6 c1 |ξ|p−1 + g(x),

(1.1.1)

i=1 n X

ai (x, ξ)ξi > c2 |ξ|p .

(1.1.2)

i=1

In addition, it is assumed that the following inequality is true for almost all x ∈ Ω and any ξ, ξ 0 ∈ Rn , ξ 6= ξ 0 : n X

[ai (x, ξ) − ai (x, ξ 0 )](ξi − ξi0 ) > 0.

(1.1.3)

i=1

Let f ∈

L1 (Ω).

We consider the following Dirichlet problem: −

n X ∂ ai (x, ∇u) = f ∂xi

in Ω,

(1.1.4)

i=1

u = 0 on ∂Ω.

(1.1.5)

The existence and properties of solutions of this and similar problems for the second-order equations with right-hand sides from the space L1 (Ω) or from the class of Radon measures were studied in numerous works (see, e.g., [4, 5, 13, 18, 19, 22–28, 35, 65, 68, 69, 71–73, 97, 105, 107, 108, 111]), and the notions of weak, entropy, and renormalized solutions were introduced and investigated. ◦

The notion of weak solution (solution from W 1,1 (Ω) in a sense of the integral identity for smooth functions) is a natural analog of the ordinary notion of generalized solution, which is, generally speaking, meaningless in the case of L1 -data. The theorems on existence of weak solutions were proved, e.g., in [23–26]. Note that a weak solution exists, generally speaking, not for all admissible values of the parameter p. In what follows, we consider this problems in more detail. In addition, in the case where p = 2, there exists an example of nonuniqueness of weak solutions (see Remark 8 in [23]).

4

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

An efficient approach to the investigation of the solvability of problems with L1 -data similar to problem (1.1.4), (1.1.5) was proposed in [13]. It is based on the introduction of the notion of entropy solutions of the analyzed problems. Generally speaking, the entropy solutions are elements of a set of functions, which is much wider than the corresponding energy Sobolev space. It turns out that the entropy solution, e.g., of problem (1.1.4), (1.1.5), exists without additional restrictions imposed on the parameter p and is unique. As applied to problem (1.1.4), (1.1.5), the idea of the approach proposed in [13] can be described as follows: Parallel with the initial problem, it is necessary to consider a sequence of similar problems with the data fl ∈ C0∞ (Ω) approximating the function f in L1 (Ω) and study the corresponding sequence of generalized solutions ul ∈ ◦

W 1,p (Ω) of these problems. In this case, (i) some estimates uniform in l are established for the measures of the sets {|ul | > k} and {|∇ul | > k}, k > 0; (ii) these estimates are used to prove that, for some increasing sequence {lj } ⊂ N, the sequences {ulj } and {Di ulj }, i = 1, . . . , n, converge almost everywhere on Ω to certain functions u : Ω → R and vi : Ω → R, i = 1, . . . , n, respectively; (iii) the established convergences are used to pass to the limit in the integral identity corresponding to the approximating problems. As a result, we get the family of integral inequalities specifying the function u as an entropy solution of problem (1.1.4), (1.1.5). Note that the functions vi are uniquely determined by the function u and can be regarded as derivatives of u of a certain kind. Parallel with [13], the problems of existence and uniqueness of the entropy solutions of the Dirichlet problem for nonlinear second-order elliptic equations with L1 -data (or measures used as data) were studied, e.g., in [27, 111]. As for the renormalized solutions, they are, briefly speaking, elements of the same set of functions as the entropy solutions. However, unlike the entropy solutions, they satisfy a different family of integral relations. In numerous cases, the notions of entropy and renormalized solutions are equivalent. This is true, e.g., for problem (1.1.4), (1.1.5). The existence and uniqueness of renormalized solutions were studied, e.g., in [19, 35, 97, 108]. We now present some definitions and results obtained for problem (1.1.4), (1.1.5) and the corresponding classes of functions frequently used in what follows. Definition 1.1.1. A weak solution of problem (1.1.4), (1.1.5) is defined as ◦

a function u ∈ W 1,1 (Ω) for which the following conditions are satisfied: (i) for any i ∈ {1, . . . , n}, we have ai (x, ∇u) ∈ L1 (Ω);

Section 1.1

5

Introduction

(ii) for any function ϕ ∈ C0∞ (Ω),  Z X Z n f ϕ dx. ai (x, ∇u)Di ϕ dx = Ω

Ω

i=1

For this definition, see, e.g., [23, 24]. If p > 2 − 1/n, then, according to Theorem 1 in [24], there exists a weak solution of problem (1.1.4), (1.1.5) from ◦

W 1,λ (Ω) for any λ, 1 6 λ < n(p − 1)/(n − 1). For any k > 0, assume that Tk is a function on R such that ( s for |s| 6 k, Tk (s) = k sign s for |s| > k. ◦

◦

It is known that if λ > 1, u ∈ W 1,λ (Ω), and k > 0, then Tk (u) ∈ W 1,λ (Ω), and, for any i ∈ {1, . . . , n}, Di Tk (u) = Di u · 1{|u| 0. It is clear that

◦

◦

W 1,p (Ω) ⊂ T 1,p (Ω).

(1.1.7)

◦

At the same time, the set T 1,p (Ω) contains functions that do not belong to L1 (Ω). Thus, let λ > n, y ∈ Ω, ρ > 0, let B1 and B2 be closed balls in Rn centered at the point y with radii ρ and ρ/2, respectively, let B1 ⊂ Ω, let ϕ ∈ C 1 (Ω), where 0 6 ϕ 6 1 in Ω, ϕ = 1 in B2 , and ϕ = 0 in Ω \ B1 , and let w be a function on Ω such that ( |x − y|−λ ϕ(x) if x ∈ Ω \ {y}, w(x) = 0 if x = y. ◦

◦

Then w ∈ T 1,p (Ω) \ L1 (Ω). Hence, the set T 1,p (Ω) is wider than the space ◦

W 1,p (Ω). ◦ Note that the elements of the set T 1,p (Ω) are measurable functions. Indeed, ◦

if u ∈ T 1,p (Ω), then the measurability of the function u follows from the measurability of the functions Tk (u), k ∈ N, and pointwise convergence of the sequence {Tk (u)} to u. ◦

The set of functions T 1,p (Ω) was introduced in [13]. For any u : Ω → R and x ∈ Ω, we set k(u, x) = min{l ∈ N : |u(x)| 6 l}.

6

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data ◦

Definition 1.1.2. Let u ∈ T 1,p (Ω) and i ∈ {1, . . . , n}. Then δi u is a function on Ω such that, for any x ∈ Ω, δi u(x) = Di Tk(u,x) (u)(x).

(1.1.8)

◦

Proposition 1.1.1. Let u ∈ T 1,p (Ω) and i ∈ {1, . . . , n}. Then, for any k > 0, Di Tk (u) = δi u · 1{|u| 0 and show that Di Tk (u) = δi u a.e. on {|u| < k}.

(1.1.10)

We set l0 = min{l ∈ N : k 6 l}. Since Tl0 (u) = Tk (u) on {|u| < k}, there exists a set El0 ⊂ Ω of measure zero such that, for any x ∈ {|u| < k} \ El0 , Di Tl0 (u)(x) = Di Tk (u)(x).

(1.1.11)

If l0 = 1, then, for any x ∈ {|u| < k}, we get k(u, x) = l0 . Thus, relation (1.1.10) follows from (1.1.11). Now let l0 > 1. It is easy to see that if l ∈ N, l < l0 , then there exists a set El ⊂ Ω of measure zero such that, for any x ∈ {|u| 6 l} \ El , Di Tl (u)(x) = Di Tk (u)(x).

(1.1.12)

Further, let x ∈ {|u| < k} \

l0 [

El .

l=1

Setting l1 = k(u, x), we conclude that l1 6 l0 . If l1 < l0 , then x ∈ {|u| 6 l1 } \ El1 and, by virtue of (1.1.12), Di Tk (u)(x) = δi u(x). If l1 = l0 , then the values of the functions Di Tk (u) and δi u at the point x are equal by virtue of relation (1.1.11). Thus, we conclude that (1.1.10) is also true for l0 > 1. It is clear that if meas {|u| > k} > 0, then Di Tk (u) = 0 a.e. on {|u| > k}. This result and relation (1.1.10) yield (1.1.9).  Note that Proposition 1.1.1 was proved in [69]. We also note that the equality of the form (1.1.9) in [13] is used to define a “gradient” of elements of ◦

a certain class of functions containing the set T 1,p (Ω). The direct definition of ◦

the functions δi u with u ∈ T 1,p (Ω) by formula (1.1.8) was given in [69]. ◦

Relations (1.1.6) and (1.1.7) and Proposition 1.1.1 imply that if u ∈ W 1,p (Ω), then, for any i ∈ {1, . . . , n}, we have δi u = Di u a.e. on Ω. Moreover, it follows ◦

◦

from (1.1.6) and Proposition 1.1.1 that if u ∈ T 1,p (Ω) ∩ W 1,1 (Ω), then, for any i ∈ {1, . . . , n}, we get δi u = Di u a.e. on Ω.

Section 1.1

7

Introduction ◦

Finally, Proposition 1.1.1 implies that if u ∈ T 1,p (Ω), then, for any i ∈ {1, . . . , n}, we find Di Tk (u) → δi u a.e. on Ω. This enables us to conclude that ◦

if u ∈ T 1,p (Ω), then the functions δi u, i = 1, . . . , n, are measurable. ◦

Definition 1.1.3. If u ∈ T 1,p (Ω), then δu is a mapping from Ω into Rn such that, for any x ∈ Ω and any i ∈ {1, . . . , n}, we have (δu(x))i = δi u(x). For any λ ∈ [1, n), we set λ∗ = nλ/(n − λ). ◦

∗

Recall that (see, e.g., [48, Chap. 7 ]) if λ ∈ [1, n), then W 1,λ (Ω) ⊂ Lλ (Ω), and there exists a positive constant cn,λ depending only on n and λ and such ◦

that, for any function u ∈ W 1,λ (Ω), Z 1/λ∗ Z 1/λ λ∗ λ |u| dx 6 cn,λ |∇u| dx . Ω

(1.1.13)

Ω ◦

Proposition 1.1.2. Let u ∈ T 1,p (Ω), λ ∈ [1, p ], and let |δu| ∈ Lλ (Ω). Then ◦

u ∈ W 1,λ (Ω) and Di u = δi u a.e. on Ω for any i ∈ {1, . . . , n}. Proof. Under the imposed conditions, by virtue of Proposition 1.1.1 and inequality (1.1.13), for the sequence of functions Tk (u), k ∈ N, we obtain ◦

Tk (u) ∈ W 1,λ (Ω), Tk (u) → u strongly in Lλ (Ω), and Di Tk (u) → δi u strongly in Lλ (Ω), i = 1, . . . , n. Thus, for any i ∈ {1, . . . , n}, there exists the weak derivative Di u, Di u = δi u a.e. on Ω, and it is possible to conclude that ◦

u ∈ W 1,λ (Ω) and Tk (u) → u strongly in W 1,λ (Ω). Hence, u ∈ W 1,λ (Ω).



This simple result can be used to study the summability properties of the solutions to problems of the form (1.1.4), (1.1.5). It was formulated and applied, e.g., in [65, 69, 71, 73]. ◦

◦

Proposition 1.1.3. Let u ∈ T 1,p (Ω) and let v ∈ W 1,p (Ω) ∩ L∞ (Ω). Then ◦

(i) u − v ∈ T 1,p (Ω); (ii) if k > 0 and i ∈ {1, . . . , n}, then Di Tk (u−v) = δi u−δi v a.e. on {|u−v| < k}. Proof. It is clear that there exists a set E ⊂ Ω of measure zero such that, for any x ∈ Ω \ E, we have |v(x)| 6 ||v||L∞ (Ω) .

(1.1.14)

We fix any k > 0 and set k1 = k + ||v||L∞ (Ω) . For any j ∈ N, we also set uj = Tj (u) − v.

(1.1.15)

8

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

Now let j ∈ N, j > k1 and let x ∈ {|uj | < k} \ E. Then |Tj (u(x)) − v(x)| < k.

(1.1.16)

If |u(x)| > j, then, by virtue of (1.1.14) and (1.1.16), we find j = |Tj (u(x))| 6 |Tj (u(x)) − v(x)| + |v(x)| < k1 . Hence, j < k1 , which contradicts the initial assumption on j. Therefore, |u(x)| 6 j. Thus, relation (1.1.16) implies that |u(x) − v(x)| < k. This and inequality (1.1.14) imply the inequality |u(x)| < k1 . Hence, in view of (1.1.15), we get uj (x) = Tj (u(x)) − v(x) = u(x) − v(x) = Tk1 (u(x)) − v(x). Thus, uj = Tk1 (u) − v a.e. on {|uj | < k}. Therefore, for any i ∈ {1, . . . , n}, we obtain Di uj = Di Tk1 (u) − Di v a.e. on {|uj | < k}. ◦

It is now possible to conclude that the sequence {Tk (uj )} is bounded in

W 1,p (Ω). Hence, in view of the fact that Tk (uj ) → Tk (u−v) strongly in Lp (Ω), ◦

we obtain Tk (u − v) ∈ W 1,p (Ω). Thus, in view of the arbitrariness of k > 0, ◦

we get u − v ∈ T 1,p (Ω). This proves assertion (i). Let k > 0 and let i ∈ {1, . . . , n}. We set k1 = k + ||v||L∞ (Ω) . By virtue of (1.1.14), we find {|u − v| < k} \ E ⊂ {|u| < k1 }. (1.1.17) Then Tk (u − v) = Tk1 (u) − v a.e. on {|u − v| < k} and, hence, Di Tk (u − v) = Di Tk1 (u) − Di v

a.e. on {|u − v| < k}.

(1.1.18)

In addition, by virtue of Proposition 1.1.1 and (1.1.17), we conclude that Di Tk1 (u) = δi u a.e. on {|u − v| < k}. Hence, in view of (1.1.18), we obtain Di Tk (u − v) = δi u − δi v

a.e. on {|u − v| < k}. 

This proves the validity of assertion (ii). The presented result is taken from [73]. ◦

◦

Further, we note that if u ∈ T 1,p (Ω), ϕ ∈ W 1,p (Ω) ∩ L∞ (Ω), k > 0, and i ∈ {1, . . . , n}, then the function ai (x, δu)(δi u − δi ϕ) is summable on the set {|u − ϕ| < k}. This follows from (1.1.1) and Proposition 1.1.1.

Section 1.1

9

Introduction

Definition 1.1.4. An entropy solution of problem (1.1.4), (1.1.5) is de◦

fined as a function u ∈ T 1,p (Ω) such that the following inequality is true for any ϕ ∈ C0∞ (Ω) and any k > 0: X  Z Z n f Tk (u − ϕ) dx. (1.1.19) ai (x, δu)(δi u − δi ϕ) dx 6 {|u−ϕ| 0. The notion of entropy solution to problem (1.1.4), (1.1.5) was introduced and studied in detail in [13]. By virtue of the assumptions concerning the functions ai made above and Theorem 6.1 in [13], the following assertion is true: Theorem 1.1.1. There exists a unique entropy solution of problem (1.1.4), (1.1.5). Note that condition (1.1.3) is used not only in the proof of uniqueness of the entropy solution but also in the proof of its existence. We also note that, as a result of application of the approach proposed in [13], the entropy solution u of problem (1.1.4), (1.1.5) is obtained as a pointwise limit ◦

of the sequence of solutions ul ∈ W 1,p (Ω) of the Dirichlet problems with smooth data fl approximating the function f in L1 (Ω). Some additional constructions enable us to show that, for any k > 0, Tk (ul ) → Tk (u) strongly in W 1,p (Ω). This, in turn, enables us to conclude that the following equality holds for any ◦

ϕ ∈ W 1,p (Ω) ∩ L∞ (Ω) and k > 0:  X Z Z n ai (x, δu)(δi u − δi ϕ) dx = f Tk (u − ϕ) dx. {|u−ϕ| 2 − 1/n and let u be an entropy solution of problem (1.1.4), (1.1.5). Then u is a weak solution of problem (1.1.4), (1.1.5) ◦

and u ∈ W 1,λ (Ω) for any λ ∈ [1, r). Finally, Theorems 1.1.1 and 1.1.4 yield the following result: Theorem 1.1.5. Let p > 2 − 1/n. Then there exists a weak solution of ◦

problem (1.1.4), (1.1.5) from W 1,λ (Ω) for any λ ∈ [1, r). As already indicated, this result was obtained in [24]. Remark 1.1.1. If p 6 2 − 1/n, then problem (1.1.4), (1.1.5) cannot have weak solutions (see the corresponding example in [13]). The construction of examples of this kind is based on the application of the well-known principle of uniform boundedness [40, Chap. 2 ]. Remark 1.1.2. Generally speaking, the weak solutions of problem (1.1.4), ◦

(1.1.5) cannot belong to the space W 1,r (Ω). This fact was mentioned in [24], although the corresponding examples were not presented in the cited work. Some examples showing that the entropy and weak solutions of problem (1.1.4), ◦

(1.1.5) cannot belong to W 1,r (Ω) can be found in [68, 69]. These examples are considered in Sections 1.3 and 1.4. In connection with the last remark, it is reasonable to study additional conditions that should be imposed on the function f to guarantee that the analyzed ◦

types of solutions of problem (1.1.4), (1.1.5) belong to the limit space W 1,r (Ω). Thus, in [24], the existence of a weak solution of problem (1.1.4), (1.1.5) from ◦

the space W 1,r (Ω) was proved under the conditions p > 2 − 1/n and f ln(1 + |f |) ∈ L1 (Ω). In [65, 68, 69, 71], this was done in a different way under weaker assumptions about p and f. The presentation of the results obtained in these works constitutes a significant part of the present chapter (Sections 1.2–1.5). In addition, in this chapter, one can find a detailed presentation of the results obtained in [72, 73] concerning the existence and a priori properties of the entropy solutions of the Dirichlet problem for nonlinear second-order elliptic equations with degenerate coercivity and L1 -right-hand sides (Sections 1.6 –1.8). The main structural difference between these equations and Eq. (1.1.4) is that their coefficients a¯ i also depend on the variable corresponding to the unknown

Section 1.1

11

Introduction

function and it is assumed that a condition of the form n X

a¯ i (x, s, ξ)ξi >

i=1

c|ξ| ¯ p , (1 + |s|)p1

p1 > 0,

is satisfied instead of condition (1.1.2). Among the earlier works devoted to the investigation of existence of the solutions of elliptic equations with degenerate coercivity, we can especially mention, e.g., [4, 22]. In [4], in particular, a theorem on existence of the entropy solution of the Dirichlet problem for nonlinear elliptic equations with degenerate coercivity and L1 -data was proved. Regarding to the conditions of growth of the coefficients of the investigated equations with respect to s, a more general result than the indicated theorem was obtained in [72]. We now present a more detailed description of the structure of the present chapter. In Sec. 1.2, we discuss the results obtained in [71] concerning the general conditions under which the solutions of problem (1.1.4), (1.1.5) belong to ◦

the limit spaces Lq (Ω) and W 1,r (Ω). The indicated conditions are formulated in terms of the summability on [1, +∞) of some numerical functions depending on f. The main structural component of these functions is a function whose values are equal to the integrals of the function |f | over the sets {|f | > s}, s > 0. In Sections 1.3 and 1.4, we present the results obtained as corollaries of the main theorems from Sec. 1.2. In Sec. 1.3, we consider the cases where the right-hand side of Eq. (1.1.4) belongs to the logarithmic classes. In Sec. 1.4, we obtain some integrally logarithmic conditions that should be imposed on the function f to guarantee that the entropy solution of problem (1.1.4), (1.1.5) ◦

belongs to the spaces Lq (Ω) and W 1,r (Ω). Moreover, in Sec. 1.4, we present an example demonstrating a certain degree of accuracy of one of these conditions. The direct proof of the main results presented in Sections 1.3 and 1.4 was obtained in [65, 68, 69]. In Sec. 1.5, we consider the case where the function f belongs to the space Lm (Ω) with m satisfying the inequality 1 < m < p∗ /(p∗ − 1) and describe the dependence of the summability properties of the entropy solution of problem (1.1.4), (1.1.5) on the parameter m. The main results of this section were published in [65]. Note that, under the indicated condition imposed on f and p > 2 − 1/n, the existence of a weak solution of problem (1.1.4), (1.1.5) from the Sobolev space with the same dependence on m as described in Sec. 1.5 for the entropy and weak solutions of the problem under a more general condition m∗ (p − 1) > 1 was established in [24]. In Sec. 1.6, we present the results on convergence of the functions in the space ◦

W 1,p (Ω) closely connected with the outlined approach to the investigation of solvability of the Dirichlet problem with L1 -data [13]. The realizations of this

12

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

approach in various cases depending on conditions imposed on the coefficients of equations lead to certain basic situations whose generalizations yield the proposed results. On one hand, the presented assertions are of significant independent interest. On the other hand, they enable us to simplify the proof of the existence of solutions of the nonlinear problems with L1 -data within the framework of the approach proposed in [13]. The application of the results given in Sec. 1.6 to the proof of the existence of the entropy solution of the Dirichlet problem for a nonlinear elliptic equation of the second order with degenerate coercivity, zero lower-order coefficient, and the L1 -right-hand side can be found in Sec. 1.7. Generally speaking, in Sections 1.6 and 1.7, we present the main results of the work [72]. Finally, in Sec. 1.8, we study the Dirichlet problem for a nonlinear secondorder elliptic equation of the general divergent form with L1 -right-hand side. It is assumed that the conditions imposed on the leading coefficients of this equation include the condition of degenerate coercivity. The main results of the section deal with the a priori properties of summability and the estimates for the entropy solutions of the analyzed problem. In this case, no restrictions are imposed on the growth and sign of the lower-order coefficient of the equation. Certain conditions on the growth of the lower-order coefficient in the variables corresponding to the unknown function and its gradient are imposed only in the theorem on existence of the entropy solution formulated at the end of the section. In Sec. 1.8, we present the main results obtained in [73]. Note that the conditions for the coefficients of equations used in [4, 5, 13, 18, 19, 22–25, 27, 28, 35, 65, 68, 69, 71, 72, 97, 105, 107, 108, 111], and other works are less general than the conditions proposed in [73]. Moreover, to the best of our knowledge, the main results presented in [73] are new not only in the analyzed general statement but also in the particular cases considered earlier. To make our presentation more complete, at the end of the introductory section, we present the analysis of the above-mentioned notion of renormalized solution to problem (1.1.4), (1.1.5). Definition 1.1.5. A renormalized solution to problem (1.1.4), (1.1.5) is ◦

defined as a function u ∈ T 1,p (Ω) such that Z 1 |δu|p dx = 0; (i) lim m→∞ m {m6|u|62m} ◦

(ii) for any h ∈ C01 (R) and ϕ ∈ W 1,p (Ω) ∩ L∞ (Ω),   Z X Z X n n ai (x, δu)Di ϕ h(u) dx + ai (x, δu)δi u h0 (u)ϕdx Ω i=1 Ω i=1 Z = f h(u)ϕdx. Ω

For this definition, see, e.g., [97].

Section 1.1

13

Introduction

Theorem 1.1.6. The function u : Ω → R is a renormalized solution of problem (1.1.4), (1.1.5) iff this function is an entropy solution of problem (1.1.4), (1.1.5). Proof. Let u be a renormalized solution of problem (1.1.4), (1.1.5). It is ◦

clear that u ∈ T 1,p (Ω). We fix any ϕ ∈ C0∞ (Ω) and k > 0. By virtue of Propositions 1.1.3 and 1.1.1, we find ◦

Tk (u − ϕ) ∈ W 1,p (Ω) ∩ L∞ (Ω)

(1.1.20)

and, for any i ∈ {1, . . . , n}, Di Tk (u − ϕ) = (δi u − δi ϕ) · 1{|u−ϕ| 2. Now let, for any m ∈ N, σm be a function on R such that σm (s) = σ(s/m), s ∈ R. It is clear that {σm } ⊂ C01 (R). Thus, in view of (1.1.20), by virtue of the condition (ii) in Definition 1.1.5 for any m ∈ N, we find  Z X n ai (x, δu)Di Tk (u − ϕ) σm (u)dx Ω

i=1

+

Z X n Ω

 Z 0 ai (x, δu)δi u Tk (u − ϕ)σm (u) dx = f Tk (u − ϕ)σm (u) dx. Ω

i=1

(1.1.22) We set σ¯ = max |σ 0 |. By using (1.1.1) and the properties of the function σ, [−2,2]

we conclude that, for any m ∈ N, Z  n  X 0 ai (x, δu)δi u Tk (u − ϕ)σm (u) dx Ω

i=1

σk ¯ (c1 + 1) 6 m

Z

σk ¯ |δu| dx + m {m6|u|62m} p

Z

g p/(p−1) dx.

Ω

Thus, by virtue of the condition (i) of Definition 1.1.5, we get  Z X n 0 lim ai (x, δu)δi u Tk (u − ϕ)σm (u) dx = 0. m→∞ Ω

i=1

Moreover, since σm (u) → 1 on Ω, we obtain  Z X n lim ai (x, δu)Di Tk (u − ϕ) σm (u) dx m→∞ Ω

i=1

(1.1.23)

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

14

=

Z X n Ω

 ai (x, δu)Di Tk (u − ϕ) dx, Z

Z lim

m→∞ Ω

f Tk (u − ϕ) dx.

f Tk (u − ϕ)σm (u) dx =

(1.1.25)

Ω

Relations (1.1.21)–(1.1.25) imply that X  Z Z n ai (x, δu)(δi u − δi ϕ) dx = f Tk (u − ϕ) dx. {|u−ϕ| 0. Since Tm (u) ∈ W 1,p (Ω) ∩ L∞ (Ω), in view of (1.1.19), we conclude that X  Z Z n ai (x, δu)(δi u − Di Tm (u)) dx 6 f Tk (u − Tm (u)) dx. {|u−Tm (u)| c2 |δu|p dx. {m6|u| M1 ||ϕ||L∞ (Ω) . Also let m ∈ N, m > M. We set ϕm = Tm (u) − h(Tm (u))ϕ. (1.1.30) ◦

We get ϕm ∈ W 1,p (Ω) ∩ L∞ (Ω) and, for any i ∈ {1, . . . , n}, δi ϕm = δi u · 1{|u| ln k0 , N ∈ N, N > k1 , and k ∈ N, k1 6 k 6 N. In view of the inequality ek > k0 and (1.2.2), we obtain Z |v|α dx 6 e(k+1)α meas {|v| > ek } 6 eα ϕ(ek ). (1.2.3) {ek 6|v| 0. Then Z |δu|p dx 6 c3 [ k γp/(p−1) + k f˜(k γ )], (1.2.4) {|u| k} 6 c4 k −q [ k γp/(p−1)−1 + f˜(k γ )]p

.

Proof. Definition 1.1.4 and inequality (1.1.2) imply that Z Z c2 |δu|p dx 6 f Tk (u) dx. {|u| k} 6 k −q/γ ϕ(k) for any k > 1. Thus, by Lemma 1.2.1, we get |u|γ ∈ Lq/γ (Ω) and, hence, u ∈ Lq (Ω). 

Section 1.2

19

General conditions for the limiting summability of solutions

Theorem 1.2.2. Let p > 2 − 1/n and let inequality (1.2.9) be true. Then there exists a weak solution of problem (1.1.4), (1.1.5) that belongs to Lq (Ω). This result is a consequence of Theorems 1.1.1, 1.1.4, and 1.2.1. Theorem 1.2.3. Let Z 1

+∞

1 ˜ n/(n−1) ds < +∞, [f (s)] s

(1.2.11)

and let u be an entropy solution of problem (1.1.4), (1.1.5). Then |δu| ∈ Lr (Ω). Proof. Let γ be a number defined by relation (1.2.10) and let β=

(p − 1)(n − p) . p(2n − 1)

Assume that ψ is a function on [1, +∞) such that, for any s ∈ [1, +∞), Z s f˜(t)dt. ψ(s) = s−1

The function ψ is continuous on [1, +∞) and, since the function f˜ is nonincreasing, we conclude that f˜(s) 6 ψ(s) 6 f˜(s − 1)

(1.2.12)

for any s > 1. Now let ψ1 be a function on [1, +∞) such that, for any s ∈ [1, +∞), ψ1 (s) = sp(n−1)/(n−p) [ s−1/2 + ψ(sγ )]−p/(n−p) . It is clear that ψ1 ∈ C([1, +∞)),

(1.2.13)

ψ1 (s) → +∞ as s → +∞.

(1.2.14)

We fix any number k > 1. Note that ψ1 (1) 6 k p . Thus, in view of properties (1.2.13) and (1.2.14), there exists k1 > 1 such that ψ1 (k1 ) = k p . This yields the equalities −1/2

k1 = k (n−p)/(n−1) [ k1 −1/2

k −p k1 [ k1

+ ψ(k1γ )]1/(n−1) ,

−1/2

+ ψ(k1γ )] = k1−q [ k1

+ ψ(k1γ )]n/(n−p) .

(1.2.15) (1.2.16)

We set G = {|u| < k1 , |δu| > k}. It is clear that meas {|δu| > k} 6 meas {|u| > k1 } + meas G.

(1.2.17)

20

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

By virtue of Lemma 1.2.2, we obtain −1/2

meas {|u| > k1 } 6 c4 k1−q [ k1

+ f˜(k1γ )]n/(n−p) .

(1.2.18)

Since k p 6 |δu|p on G, Lemma 1.2.2 now implies that Z 1/2 |δu|p dx 6 c3 [ k1 + k1 f˜(k1γ )]. k p meas G 6

(1.2.19)

{|u| k} 6 (c3 + c4 )k −p k1 [ k1

+ ψ(k1γ )].

(1.2.20)

Using (1.2.15) and (1.2.12), we get −1/2

k −p k1 [ k1

−1/2

+ ψ(k1γ )] = k −r [ k1

+ ψ(k1γ )]n/(n−1)

−1/2

+ f˜(k1γ − 1)]n/(n−1) 6 k −r [ k −βp/(p−1) + f˜(k β − 1)]n/(n−1) . 6 k −r [ k1

In view of (1.2.20), for any k > 1, this yields meas {|δu| > k} 6 (c3 + c4 )k −r [ k −βp/(p−1) + f˜(k β − 1)]n/(n−1) .

(1.2.21)

Let ϕ be a function on [1, +∞) such that, for any s ∈ [1, +∞), ϕ(s) = (c3 + c4 )[ s−p/(p−1) + f˜(s)]n/(n−1) . The function ϕ is positive, nonincreasing, and measurable. Moreover, by virtue of (1.2.11), it satisfies inequality (1.2.1). In view of the definition of the function ϕ, it follows from (1.2.21) that, for any k > 1,   |δu|β meas > k 6 k −r/β ϕ(k). 2 Thus, by Lemma 1.2.1, |δu|β ∈ Lr/β (Ω). Hence, |δu| ∈ Lr (Ω).



Theorem 1.2.4. Let p > 2 − 1/n and let inequality (1.2.11) be true. Assume ◦

that u is an entropy solution of problem (1.1.4), (1.1.5). Then u ∈ W 1,r (Ω). This result is a consequence of Theorem 1.2.3 and Proposition 1.1.2. Theorem 1.2.5. Let p > 2 − 1/n and let inequality (1.2.11) be true. Then ◦

there exists a weak solution of problem (1.1.4), (1.1.5) from W 1,r (Ω). This assertion is a consequence of Theorems 1.1.1, 1.1.2, 1.2.3, and 1.2.4. In conclusion, we note that all results presented in this section were obtained in [71].

Section 1.3

1.3

The cases where the right-hand side of the equation belongs to

21

The cases where the right-hand side of the equation belongs to logarithmic classes

We now consider several assertions following from the theorems presented in the previous section. In order to prove the first group of these assertions, we need the following simple auxiliary assertion: Lemma 1.3.1. Let σ > 0, and let f [ ln(1 + |f |)]σ ∈ L1 (Ω). Then, for any λ > 1/σ, Z +∞ 1 ˜ λ (1.3.1) [f (s)] ds < +∞. s 1 Proof. Let λ > 1/σ. We set Φ = |f |[ ln(1 + |f |)]σ ,

Z M=

λ Φdx .

Ω

Let s > e. Assume that {|f | > s} 6= ∅. It is easy to see that |f | 6 (ln s)−σ Φ on {|f | > s}. Then

Z

|f | dx 6 M 1/λ (ln s)−σ .

{|f |>s}

It is clear that this inequality also holds in the case where {|f | > s} = ∅. Thus, in view of the definition of the function f˜, for any s > e, we find f˜(s) 6 M 1/λ (ln s)−σ . Therefore, by using the inequality σλ > 1, we conclude that, for any N > e, Z N 1 ˜ λ M [f (s)] ds 6 . s σλ − 1 e This implies that inequality (1.3.1) is true.



Theorem 1.3.1. Let σ > (n − p)/n and let f [ ln(1 + |f |)]σ ∈ L1 (Ω). Assume that uis an entropy solution of problem (1.1.4), (1.1.5). Then u ∈ Lq (Ω). Theorem 1.3.2. Let p > 2−1/n, let σ > (n−p)/n, and let f [ ln(1+|f |)]σ ∈ Then there exists a weak solution of problem (1.1.4), (1.1.5) from Lq (Ω).

L1 (Ω).

Theorem 1.3.1 is a consequence of Lemma 1.3.1 and Theorem 1.2.1, whereas Theorem 1.3.2 follows from Lemma 1.3.1 and Theorem 1.2.2.

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

22

Theorem 1.3.3. Let p > 2−1/n, let σ > (n−1)/n, and let f [ ln(1+|f |)]σ ∈ Assume that u is an entropy solution of problem (1.1.4), (1.1.5). Then

L1 (Ω). ◦

u ∈ W 1,r (Ω). Theorem 1.3.4. Let p > 2−1/n, let σ > (n−1)/n, and let f [ ln(1+|f |)]σ ∈ ◦

L1 (Ω).Then there exists a weak solution of problem (1.1.4),(1.1.5) from W 1,r (Ω). Theorem 1.3.3 is a consequence of Lemma 1.3.1 and Theorem 1.2.4, while Theorem 1.3.4 follows from Lemma 1.3.1 and Theorem 1.2.5. Note that Theorems 1.3.3 and 1.3.4 were, in fact, established in [65]. Theorems 1.3.1 and 1.3.2 have not been published earlier. We now consider more general assertions following from the theorems presented in Sec. 1.2. To this end, we need successive superpositions of the logarithmic function and a proper analog of Lemma 1.3.1. We define a sequence of numbers sj as follows: s1 = 1,

sj = esj−1 ,

j = 2, 3, . . . .

Now let, for any j ∈ N, bj : [sj , +∞) → [0, +∞) be a function such that bj (s) = ln . ln ln} s, | . .{z

s ∈ [sj , +∞).

j

Note that if j ∈ N and s > sj , then bj (s) > 0. Lemma 1.3.2. Let m ∈ N, σ > 0, λ > 1/σ, and let Y 1/λ m f bj (sj + |f |) [ bm+1 (sm+1 + |f |)]σ ∈ L1 (Ω). j=1

Then

Z 1

+∞

1 ˜ λ [f (s)] ds < +∞. s

(1.3.2)

Proof. We set Y 1/λ m [ bm+1 (sm+1 + |f |)]σ . Φ = |f | bj (sj + |f |) j=1

It is clear that Φ ∈ L1 (Ω). By M we denote the integral of the function Φ over Ω. Let s > sm+2 . Assume that {|f | > s} 6= ∅. It is easy to see that Y −1/λ m |f | 6 bj (s) [ bm+1 (s)]−σ Φ on {|f | > s}. j=1

Section 1.3

23

The cases where the right-hand side of the equation belongs to

Then Z |f | dx 6 M {|f |>s}

Y m

−1/λ bj (s) [ bm+1 (s)]−σ .

j=1

It is clear that this inequality also holds in the case where {|f | > s} = ∅. Thus, by the definition of the function f˜, we conclude that, for any s > sm+2 , Y −1/λ m ˜ f (s) 6 M bj (s) [ bm+1 (s)]−σ . (1.3.3) j=1

Let N > sm+2 . In view of (1.3.3) and the inequality σλ > 1, we obtain m −1 Z N Z N 1 Y 1 ˜ λ λ [f (s)] ds 6 M bj (s) [ bm+1 (s)]−σλ ds sm+2 s sm+2 s j=1

Mλ

[ bm+1 (s)]1−σλ |N sm+2 1 − σλ Mλ [ bm+1 (sm+2 )]1−σλ . 6 σλ − 1

=

This implies that inequality (1.3.2) holds. Theorem 1.3.5. Let m ∈ N, σ > (n − p)/n, and let Y (n−p)/n m f bj (sj + |f |) [ bm+1 (sm+1 + |f |)]σ ∈ L1 (Ω).



(1.3.4)

j=1

Assume that u is an entropy solution of problem (1.1.4), (1.1.5).Then u ∈ Lq (Ω). Theorem 1.3.6. Let p > 2 − 1/n, m ∈ N, σ > (n − p)/n, and let inclusion (1.3.4) be true. Then there exists a weak solution of problem (1.1.4), (1.1.5) from Lq (Ω). Theorem 1.3.5 is a consequence of Lemma 1.3.2 and Theorem 1.2.1, while Theorem 1.3.6 follows from Lemma 1.3.2 and Theorem 1.2.2. Theorem 1.3.7. Let p > 2 − 1/n, m ∈ N, σ > (n − 1)/n, and let Y (n−1)/n m [ bm+1 (sm+1 + |f |)]σ ∈ L1 (Ω). f bj (sj + |f |)

(1.3.5)

j=1

Assume that u is an entropy solution of problem (1.1.4), (1.1.5). Then u ∈ ◦

W 1,r (Ω). Theorem 1.3.8. Let p > 2 − 1/n, m ∈ N, σ > (n − 1)/n, and let inclusion (1.3.5) be valid. Then there exists a weak solution of problem (1.1.4), (1.1.5) ◦

from W 1,r (Ω).

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

24

Theorem 1.3.7 is a consequence of Lemma 1.3.2 and Theorem 1.2.4. At the same time, Theorem 1.3.8 follows from Lemma 1.3.2 and Theorem 1.2.5. Note that (1.3.5) is only a sufficient condition. Here, we restrict ourselves to the substantiation of this assertion solely for weak solutions. By using the Laplace operator as an example (this corresponds to the case where p = 2), we now show that a weak solution of problem (1.1.4), (1.1.5) from the space ◦

W 1,r (Ω) may also exist under weaker requirements imposed on the right-hand side of the equation as compared with condition (1.3.5). Example 1.3.1. Let Ω = {x ∈ Rn : |x| < 1} and let g1 be a function from the class C 2 ((0, +∞)) such that g1 = 0 on [1/2, +∞) and, for any s ∈ (0, e−e ),  1 −(n−1)/n  1 −(2n−1)/n ln ln . g1 (s) = ln s s Also let u1 and f1 be functions on Ω such that, for any x ∈ Ω \ {0}, u1 (x) = |x|2−n g1 (|x|), f1 (x) = (n − 3)|x|1−n g10 (|x|) − |x|2−n g100 (|x|) . It is easy to see that the function u1 is a weak solution of problem −∆u = f1

in Ω,

u = 0 on ∂Ω.

◦

In this case, u1 ∈ W 1,r (Ω) and, for any σ ∈ (0, (n − 1)/n), f1 [ ln(1 + |f1 |)](n−1)/n [ ln ln(e + |f1 |)]σ ∈ L1 (Ω). However, f1 [ ln(1 + |f1 |)](n−1)/n [ln ln(e + |f1 |)](n−1)/n ∈ / L1 (Ω), and, hence, the function f1 does not satisfy inclusion (1.3.5) for any m ∈ N and σ > (n − 1)/n. In conclusion, we present an example in which the right-hand side of the equation has a certain “logarithmic” summability but the weak solution of the ◦

corresponding Dirichlet problem does not belong to W 1,r (Ω). Example 1.3.2. Let Ω = {x ∈ Rn : |x| < 1} and let g2 be a function from the class C 2 ((0, +∞)) such that g2 = 0 on [1/2, +∞) and, for any s ∈ (0, e−1 ),  1 −(n−1)/n g2 (s) = ln . s Also let u2 and f2 be functions on Ω such that, for any x ∈ Ω \ {0}, u2 (x) = |x|2−n g2 (|x|), f2 (x) = (n − 3)|x|1−n g20 (|x|) − |x|2−n g200 (|x|).

Section 1.4 On the integrally logarithmic conditions for the limiting summability 25

Then the function u2 is a weak solution of the problem −∆u = f2

in Ω,

u = 0 on ∂Ω. ◦

In this case, for any λ ∈ [1, r), we get u2 ∈ W 1,λ (Ω) but |∇u2 | ∈ / Lr (Ω) ◦

and, hence, u2 ∈ / W 1,r (Ω). In addition, for any σ ∈ (0, (n − 1)/n), we conclude that f2 [ ln(1 + |f2 |)]σ ∈ L1 (Ω) but f2 [ ln(1 + |f2 |)](n−1)/n ∈ / L1 (Ω). Note that the direct proof of Theorem 1.3.8 was given in [68]. Examples 1.3.1 and 1.3.2 are taken from the cited work. Theorems 1.3.5, 1.3.6, and 1.3.7 have not been published earlier.

1.4

On the integrally logarithmic conditions for the limiting summability of solutions

We now consider some corollaries of the theorems presented in Sec. 1.2 obtained for more general conditions imposed on f as compared with requirements (1.3.4) and (1.3.5). These conditions are called integrally logarithmic. On one hand, they contain integrals of the function f over sets of the form {|f | > k}, k > 0. On the other hand, they contain the expressions that depend on successive superpositions of the logarithmic function. In fact, these conditions can be regarded as certain qualified upper bounds of the values of f˜(k) for sufficiently large k. Theorem 1.4.1. Let c > 0, m ∈ N, σ > (n − p)/n, and let, for any k > sm+1 , Z {|f |>k}

Y −(n−p)/n m [ bm+1 (k)]−σ . |f |dx 6 c bj (k)

(1.4.1)

j=1

Assume that u is an entropy solution of problem (1.1.4), (1.1.5). Then u ∈ Lq (Ω). Proof. By virtue of (1.4.1) and the definition of the function f˜, for any s > sm+1 , we get [f˜(s)]n/(n−p) 6 cn/(n−p)

Y m

−1 bj (s) [ bm+1 (s)]−σn/(n−p) .

j=1

This implies that inequality (1.2.9) is true. Hence, by Theorem 1.2.1, we conclude that u ∈ Lq (Ω). 

26

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

Theorem 1.4.2. Let p > 2 − 1/n. Let c > 0, m ∈ N, σ > (n − p)/n, and let inequality (1.4.1) be true for any k > sm+1 ,. Then there exists a weak solution of problem (1.1.4), (1.1.5) from Lq (Ω). Proof. The validity of inequality (1.2.9) follows from the proof of the previous theorem. Thus, Theorem 1.2.2 yields the required assertion.  Theorem 1.4.3. Let p > 2 − 1/n. Let c > 0, m ∈ N, σ > (n − 1)/n, and let Y −(n−1)/n Z m |f |dx 6 c bj (k) [ bm+1 (k)]−σ (1.4.2) {|f |>k}

j=1

for any k > sm+1 . Assume that u is an entropy solution of problem (1.1.4), ◦

(1.1.5). Then u ∈ W 1,r (Ω). Proof. By virtue of (1.4.2) and the definition of the function f˜, for any s > sm+1 , we can write [f˜(s)]n/(n−1) 6 cn/(n−1)

Y m

−1 bj (s) [ bm+1 (s)]−σn/(n−1) .

j=1 ◦

This means that inequality (1.2.11) is true. Thus, u ∈ W 1,r (Ω) by Theorem 1.2.4.  Theorem 1.4.4. Let p > 2 − 1/n. Let c > 0, m ∈ N, σ > (n − 1)/n, and let inequality (1.4.2) hold for any k > sm+1 . Then there exists a weak solution ◦

of problem (1.1.4), (1.1.5) from W 1,r (Ω). Proof. Inequality (1.2.11) follows from the proof of Theorem 1.4.3. Therefore, Theorem 1.2.5 yields the required assertion.  Note that the direct proof of Theorem 1.4.3 was given in [69]. Theorem 1.4.4 was also established in the cited work. Theorems 1.4.1 and 1.4.2 were not published earlier. We now consider two assertions concerning the necessary and sufficient conditions for the function f to satisfy the condition of Theorem 1.4.3. Proposition 1.4.1. Let c > 0, m ∈ N, σ > (n − 1)/n, and let inequality (1.4.2) be true for any k > sm+1 . Then f [ ln(1 + |f |)]t ∈ L1 (Ω) for any t ∈ (0, (n − 1)/n).

Section 1.4 On the integrally logarithmic conditions for the limiting summability 27

Proof. Let t ∈ (0, (n − 1)/n). We fix t1 such that  −1 n−1 t1 > −t , n

(1.4.3)

1/t

1 and k0 ∈ N such that k0 > sm+1 . For any k ∈ N, k > k0 , we set

Gk = { ek

t1

t1

t1

G0k = { |f | > ek }.

< |f | 6 e(k+1) },

Note that G0k0 =

∞ [

Gk .

(1.4.4)

k=k0 t

We fix any k ∈ N, k > k0 . Since k t1 > sm+1 , we get bj (ek 1 ) > 1 for any j ∈ {1, . . . , m + 1}. Thus, by the condition of proposition, we find Z |f | dx 6 ck −t1 (n−1)/n . (1.4.5) G0k

Moreover, by the definition of the set Gk , we get [ ln(1 + |f |)]t 6 2(1+t1 )t k t1 t

on Gk .

(1.4.6)

Relations (1.4.5) and (1.4.6) now yield Z |f |[ ln(1 + |f |)]t dx 6 2(1+t1 )t ck −t1 [(n−1)/n−t] . Gk

The obtained inequality and relation (1.4.3) enable us to conclude that ∞ Z X |f |[ ln(1 + |f |)]t dx < ∞. k=k0

Gk

In view of relation (1.4.4) and the fact that the sets Gk are pairwise disjoint, we conclude that the function f [ ln (1 + |f |) ]t is summable on G0k0 . Hence, this function is summable on Ω.  Note that the assertion converse to Proposition 1.4.1 is not true. This is demonstrated by Example 1.4.3 (presented in what follows). Proposition 1.4.2. Let m ∈ N, let σ > (n−1)/n, and let inclusion (1.3.5) be true. Then there exists c > 0 such that inequality (1.4.2) is true for any k > sm+1 . Proof. We set Φ=

Y m j=1

(n−1)/n bj (sj + |f |)

[ bm+1 (sm+1 + |f |) ]σ .

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

28

By the condition of proposition, we have f Φ ∈ L1 (Ω). We set Z |f | Φ dx + 1. c=

(1.4.7)

Ω

We fix any k > sm+1 . It is easy to see that, for any x ∈ { |f | > k }, Y (n−1)/n m bj (k) [ bm+1 (k) ]σ < Φ(x). j=1

Then Y m

Z |f | dx 6 {|f |>k}

−(n−1)/n bj (k)

−σ

Z |f | Φ dx.

[ bm+1 (k) ]

Ω

j=1

This relation and (1.4.7) yield inequality (1.4.2).



In connection with the presented propositions, we consider the following illustrative example. By B we denote the open unit ball in Rn centered at the origin. Example 1.4.1. Let σ ∈ ((n−1)/n, 1) and let f1 be a function on B such that       1 ln 1 −(2n−1)/n ln ln 1 −σ if 0 < |x| 6 e−e , n |x| |x| f1 (x) = |x|  0 if |x| = 0 or |x| > e−e . It is easy to see that the function f1 is summable on B and Z f1 dx 6 2κn ,

(1.4.8)

B

where κn is the surface area of the unit sphere in Rn . We now show that the following assertions are true: (i) for any k > e, the inequality Z |f1 |dx 6 2κn n2 e (ln k)−(n−1)/n (ln ln k)−σ {|f1 |>k}

holds; (ii) the function f1 [ ln(1 + |f1 |)](n−1)/n does not belong to L1 (B). Indeed, we fix an arbitrary k > e. First, we assume that k 6 ene . Then 1 6 ne(ln k)−(n−1)/n and 1 6 n(ln ln k)−σ . Hence, in view of (1.4.8), we obtain Z |f1 |dx 6 2κn n2 e(ln k)−(n−1)/n (ln ln k)−σ . {|f1 |>k}

Now let k >

ene .

We set Bk = {x ∈ Rn : |x| 6 k −1/n }.

Section 1.4 On the integrally logarithmic conditions for the limiting summability 29

Assume that (i)

Bk = {x ∈ Rn : i−1 6 |x| 6 k −1/n } (i)

for any i ∈ N, i > k 1/n . We fix i ∈ N, i > k 1/n . For any x ∈ Bk , we get ln ln k 6 n ln ln(1/|x|) and, hence, |f1 (x)| 6 n(ln ln k)−σ

1 |x|n

 ln

1 |x|

−(2n−1)/n .

Then Z

−σ

(i)

Z

1/k1/n

|f1 |dx 6 κn n(ln ln k)

Bk

1/i 2

−(n−1)/n

6 2κn n (ln k)

  1 1 −(2n−1)/n ln dρ ρ ρ

(ln ln k)−σ .

This yields Z

|f1 |dx 6 2κn n2 (ln k)−(n−1)/n (ln ln k)−σ .

Bk

Hence, in view of the inclusion { |f1 | > k } ⊂ Bk , we arrive at the required estimate for the integral of the function |f1 | over the set { |f1 | > k }. Thus, we conclude that assertion (i) is true. We set    n − 1 4 −e α = min . , e n+1 Let Gi = {x ∈ Rn : i−1 6 |x| 6 α} for any i ∈ N, i > 1/α. We fix i ∈ N, i > 1/α, and take x ∈ Gi . It is clear that 1 n−1 1 ln + ln > 0. 4 |x| n+1

(1.4.9)

  1 n + 1 1 (n−1)/(n+1) 1 < ln < . ln ln |x| |x| n − 1 |x|

(1.4.10)

Moreover, we find

In view of (1.4.10), we obtain  |f1 (x)| >

n−1 n+1

3

1 . |x|

By virtue of (1.4.9), this yields ln(1 + |f1 (x)|) >

1 1 ln 4 |x|

and, hence, (n−1)/n

|f1 (x)|[ ln(1 + |f1 (x)|)]

    1 −1 1 −σ 1 ln ln ln . > 4|x|n |x| |x|

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

30 Thus, Z

(n−1)/n

|f1 |[ ln(1 + |f1 |)] Gi

    1 1 −1 1 −σ ln ln ln dρ ρ ρ 1/i ρ     1 1−σ κn 1−σ − ln ln = (ln ln i) . 4(1 − σ) α

κn dx > 4

Z

α

This yields Z

|f1 |[ ln(1 + |f1 |)](n−1)/n dx → ∞ as i → ∞.

Gi

Hence, the function f1 [ ln (1+|f1 |) ](n−1)/n does not belong to L1 (B). Therefore, assertion (ii) is true. Remark 1.4.1. By virtue of Proposition 1.4.2, the validity of the condition of Theorem 1.3.7 for the function f implies the validity of the condition of Theorem 1.4.3 for f. The converse assertion is not true, as follows from assertions (i) and (ii) in Example 1.4.1. Moreover, these assertions show that, generally speaking, the exponent t = (n − 1)/n is not attained in the final inclusion of Proposition 1.4.1. Consider one more proposition. It proves to be useful for the clarification of the properties of entropy and weak solutions in model situations. In particular, by using this proposition, we can establish a certain kind of accuracy of the condition of Theorem 1.4.3 for the function f (see Example 1.4.3 in what follows). For any i ∈ {1, . . . , n}, we define a function Ai : Rn → R as follows: ( |ξ|p−2 ξi for ξ 6= 0, Ai (ξ) = 0 for ξ = 0. Proposition 1.4.3. Let p > 2 − 1/n. Let h ∈ C 2 ((0, +∞)), α ∈ (0, 1), and let the following conditions be satisfied: 1) h > 0 on (0, α) and h = 0 on [ α, +∞); 2) h(s) → 0 as s → 0; n−p h(s) − sh0 (s) > 0 for any s ∈ (0, α); 3) p−1 4) there exist M > 0, γ ∈ (0, (n − p)/(p − 1)), and β ∈ (0, α) such that h(s) > M sγ for any s ∈ (0, β); n−p M1 h(s) 5) there exist M1 ∈ (0, 1) and β1 ∈ (0, α) such that s|h0 (s)| 6 p −1 for any s ∈ (0, β1 ); p−2 Z α n−p 0 h(s) − sh (s) [ |h0 (s)| + s|h00 (s)| ] ds < +∞. 6) p−1 0

Section 1.4 On the integrally logarithmic conditions for the limiting summability 31

Let u be a function on B such that, for any x ∈ B\{0}, u(x) = |x|−(n−p)/(p−1) h(|x|). Assume that F is a function on B such that, for any x ∈ B, 0 < |x| < α,  p−2 0 1−n n − p F (x) = |x| h(|x|) − |x|h (|x|) p−1 h i × (n − 2p + 1)h0 (|x|) − (p − 1)|x|h00 (|x|) , and F (x) = 0 for any x ∈ B, ◦

|x| > α.

Then F ∈ L1 (B), u ∈ W 1,1 (B), and the following assertions are true: (∗1 ) |∇u| ∈ Lλ (B) for any λ ∈ [1, r );  Z X Z n (∗2 ) for any function ϕ ∈ C0∞ (B), Ai (∇u)Di ϕ dx = F ϕ dx; B

(∗3 ) Tk (u − ϕ) ∈

◦

W 1,p (B)

for any ϕ ∈

B

i=1

C0∞ (B)

and k > 0; n X (∗4 ) for any ϕ ∈ C0∞ (B) and k > 0, the function Ai (∇u)Di Tk (u−ϕ) is summable on B and

Z X n B

Z

α

(∗5 ) if Z0 α (∗6 ) if 0

i=1

 Z Ai (∇u)Di Tk (u−ϕ) dx = F Tk (u−ϕ) dx;

i=1

B

1 r h (s)ds < +∞, then |∇u| ∈ Lr (B); s 1 r h (s)ds = +∞, then |∇u| 6∈ Lr (B). s

Proof. The inclusion F ∈ L1 (B) is valid by virtue of condition 6). ◦

We now show that u ∈ W 1,1 (B). First, we note that, by virtue of conditions 2) and 5), there exists a positive number M2 such that, for any s ∈ (0, α), n−p h(s) + s|h0 (s)| 6 M2 . p−1

(1.4.11)

Let w be a function on B such that, for any x ∈ B\{0}, w(x) = |x|−(n−1)/(p−1) . Since p > 2 − 1/n, we have n−1 < n, p−1 and, hence, the function w is summable on B.

(1.4.12)

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

32

Note that, by virtue of condition 1) and (1.4.11), |u| 6

p−1 M2 w n−p

on B\{0}.

Thus, we conclude that u ∈ L1 (B). Let F0 be a function on B such that, for any x ∈ B\{0}, F0 (x) =

n−p h(|x|) − |x|h0 (|x|). p−1

Conditions 1) and 3) imply that F0 (x) > 0 for any x ∈ B\{0}. By v we denote the restriction of the function u to B\{0}. We have v ∈ C 1 (B\{0}) and, hence, for any i ∈ {1, . . . , n} and x ∈ B\{0}, Di v (x) = −xi |x|−1 w(x)F0 (x). For any j ∈ N, we set   1 n < |x| < 1 , Kj = x ∈ R : 1+j

 Sj =

(1.4.13)

 1 . x ∈ R : |x| = 1+j n

We now fix i ∈ {1, . . . , n} and denote an extension of Di v to B by wi . By virtue of condition 1) and relations (1.4.11), and (1.4.13), we conclude that |wi | 6 M2 w on B\{0} and, hence, wi ∈ L1 (B). Let ϕ ∈ C0∞ (B) and j ∈ N. In view of the smoothness of the function v, by using the formula of integration by parts, we obtain Z Z Z uDi ϕ dx = − wi ϕ dx + uϕ cos(ν, ei ) dSj , (1.4.14) Kj

Kj

Sj

where ν is the unit vector of inner normal to Sj and ei is the unit vector of the ith axis. By the definition of the function u, we get Z n−(n−1)/(p−1)    1 1 uϕ cos(ν, ei ) dSj 6 κn max |ϕ(x)|. h 1+j 1 + j x∈B Sj

In view of condition 2) and inequality (1.4.12), this enables us to conclude that Z lim u ϕ cos(ν, ei ) dSj = 0. j→∞

Sj

Thus, in view of the summability of the functions u and wi on B, we can pass to the limit in (1.4.14) as j → ∞ and obtain Z Z uDi ϕ dx = − wi ϕ dx. B

B

Hence, there exists the weak derivative Di u and Di u = wi a.e. on B. Thus, by virtue of (1.4.13), for almost all x ∈ B\{0}, Di u (x) = −xi |x|−1 w(x)F0 (x).

(1.4.15)

Section 1.4 On the integrally logarithmic conditions for the limiting summability 33

We can now conclude that u ∈ W 1,1 (B). Since supp u ⊂ B by virtue of ◦

condition 1), we find u ∈ W 1,1 (B). Note that, in view of (1.4.15), |∇u| = wF0

a.e. on B.

(1.4.16)

We now prove the validity of assertion (∗1 ). Let λ ∈ [1, r). In view of (1.4.16), (1.4.11), and condition 1), we conclude that |∇u|λ 6 M2λ |w|λ a.e. on B. Since λ(n − 1)/(p − 1) < n and, hence, w ∈ Lλ (B), this yields |∇u| ∈ Lλ (B). Thus, the validity of assertion (∗1 ) is proved. For any i ∈ {1, . . . , n, }, let vi be a function on B such that, for any x ∈ B\{0}, vi (x) = −xi |x|−n [F0 (x)]p−1 . By virtue of (1.4.11) and condition 1), for any i ∈ {1, . . . , n, }, we get vi ∈ L1 (B). Moreover, for any i ∈ {1, . . . , n} and j ∈ N, we can write |vi | 6 M2p−1 (1 + j)n−1    n−p 1 1 |vi | 6 (1+j)n−1 h + p−1 1+j 1+j

on Kj , (1.4.17)   p−1 0 1 h 1+j

on Sj .

(1.4.18)

We set Bα = {x ∈ Rn : 0 < |x| < α}. According to to condition 3), for any i ∈ {1, . . . , n, }, we find vi |Bα ∈ C 1 (Bα ). (1.4.19) Furthermore, in view of condition 3) and the definition of the function F, we obtain n X − Di (vi |Bα ) = F |Bα . (1.4.20) i=1 ◦

We now show that, for any function ϕ ∈ W 1,p (B) ∩ L∞ (B),  Z X Z n lim vi Di ϕ dx = F ϕ dx. j→∞

Kj

(1.4.21)

B

i=1

◦

Indeed, let ϕ ∈ W 1,p (B) ∩ L∞ (B) and let {ϕm } be a sequence of functions from C0∞ (B) such that ϕm → ϕ strongly in W 1,p (B),

(1.4.22)

ϕm → ϕ a.e. on B,

(1.4.23)

∀ m ∈ N,

|ϕm | 6 kϕkL∞ (B) + 1 on B.

For any j ∈ N, we set    1 1 n−p h + qj = p−1 1+j 1+j

  p−1 0 1 h . 1+j

(1.4.24)

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

34

By virtue of conditions 2) and 5), we get lim qj = 0.

(1.4.25)

j→∞

We now fix j ∈ N. Let m ∈ N. Using (1.4.19), the formula of integration by parts, relation (1.4.20), and condition 1), we obtain   Z Z X Z X n n F ϕm dx + vi cos(ν, ei ) ϕm dSj . (1.4.26) vi Di ϕm dx = Kj

Kj

i=1

Sj

i=1

By virtue of relations (1.4.18) and (1.4.24), we get Z  n  X vi cos(ν, ei ) ϕm dSj 6 (kϕkL∞ (B) + 1)nκn qj . Sj

i=1

It follows from this inequality and (1.4.26) that Z X  Z n F ϕm dx 6 (kϕkL∞ (B) + 1)nκn qj . vi Di ϕm dx − Kj

(1.4.27)

Kj

i=1

By using (1.4.17), we conclude that Z X   Z X n n vi Di ϕm dx − vi Di ϕ dx Kj

Kj

i=1

i=1

6 M2p−1 (1 + j)n−1

n Z X i=1

|Di (ϕm − ϕ)|dx.

(1.4.28)

B

Moreover, as a result of (1.4.24), we find Z Z F ϕ dx − F ϕ dx m Kj B Z Z
6 |F ||ϕm − ϕ|dx + kϕkL∞ (B) + 1 B

|F |dx.

(1.4.29)

B\Kj

Relations (1.4.27)–(1.4.29) now yield Z X  Z n vi Di ϕ dx − F ϕ dx Kj

i=1

B

6 M2p−1 (1 + j)n−1

n Z X i=1

Z |Di (ϕm − ϕ)|dx +

B

|F ||ϕm − ϕ|dx B

 Z + (kϕkL∞ (B) + 1) nκn qj + B\Kj

 |F |dx .

Section 1.4 On the integrally logarithmic conditions for the limiting summability 35

Thus, by using (1.4.22)–(1.4.25) and the fact that meas (B\Kj ) → 0 as j → ∞, we arrive at (1.4.21). Note that, by virtue of relations (1.4.15) and (1.4.16) and conditions 1) and 3), for any i ∈ {1, . . . , n}, Ai (∇u) = vi a.e. on B. (1.4.30) In view of relation (1.4.21), this enables us to conclude that assertion (∗2 ) is true. We now prove the validity of assertion (∗3 ). Let ϕ ∈ C0∞ (B) and k > 0. Also let hk be a function from C 1 (R) with the following properties: hk (s) = s, 3 for |s| 6 k ; hk (s) = k sign s for |s| > 2k , and 0 6 h0k 6 1 on R. We define 2 the function uk : B → R as follows:  hk (u(x) − ϕ(x)) for x ∈ B\{0}, uk (x) = 3  k for x = 0. 2 Under condition 4), we have uk ∈ C 1 (B). Moreover, in view of condition 1), we ◦

get supp uk ⊂ Bα ∪ supp ϕ. Therefore, uk ∈ C01 (B) and, hence, uk ∈ W 1,p (B). ◦

However, in this case, we also have Tk (uk ) ∈ W 1,p (B). Thus, with regard for ◦

the fact that Tk (u − ϕ) = Tk (uk ) on B\{0}, we obtain Tk (u − ϕ) ∈ W 1,p (B). This means that the validity of assertion (∗3 ) is proved. We now prove the validity of assertion (∗4 ). Let ϕ ∈ C0∞ (B) and k > 0. We set −1   γ1   n−p M −γ , k1 = min β, . γ1 = p−1 k + max |ϕ(x)| x∈B

B0

Rn

By we denote the open ball in of radius k1 centered at the origin. By virtue of condition 4), we get B 0 \{0} ⊂ { |u − ϕ| > k}. This inclusion and assertion (∗3 ) imply that, for any i ∈ {1, . . . , n}, Ai (∇u)Di Tk (u − ϕ) = 0 a.e. on B 0 .

(1.4.31)

In addition, by virtue of (1.4.11), (1.4.30), and condition 1), for any i ∈ {1, . . . , n}, we obtain |Ai (∇u)Di Tk (u − ϕ)| 6 M2p−1 k11−n |Di Tk (u − ϕ)| a.e. on B\B 0 .

(1.4.32)

Relations (1.4.31) and (1.4.32) imply that, for any i ∈ {1, . . . , n}, the function Ai (∇u)Di Tk (u − ϕ) is summable on B. Using this fact, assertion (∗3 ), and relations (1.4.21) and (1.4.30), we obtain  Z X Z n Ai (∇u)Di Tk (u − ϕ) dx = F Tk (u − ϕ) dx. B

i=1

Hence, assertion (∗4 ) is true.

B

36

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

We now prove the validity of assertion (∗5 ). Let Z α 1 r h (s)ds < +∞. 0 s

(1.4.33)

We set G = {x ∈ Rn : |x| 6 β1 }. Assume that, for any j ∈ N,   β1 6 |x| 6 β1 . G(j) = x ∈ Rn : 1+j By virtue of (1.4.16) and condition 5), for almost all x ∈ G\{0}, we get |∇u|r (x) 6 (2n)r |x|−n hr (|x|). Thus, for any j ∈ N, Z G(j)

|∇u|r dx 6 (2n)r κn

Z 0

α

1 r h (s)ds. s

In view of (1.4.33), this implies that the function |∇u|r is summable on G. We also note that, by virtue of relations (1.4.16) and (1.4.11) and condition 1), |∇u| < M2 β1−n a.e. on B\G and, hence, the function |∇u|r is summable on B\G. This enables is to conclude that |∇u| ∈ Lr (B). Thus, the validity of assertion (∗5 ) is proved. Finally, we prove the validity of assertion (∗6 ). Assume that Z α 1 r (1.4.34) h (s)ds = +∞. 0 s Let G and G(j) , j ∈ N, be the sets introduced in the proof of the validity of assertion (∗5 ). By virtue of condition 5), for any x ∈ G\{0}, we get F0 (x) > (1 − M1 )

n−p h(|x|). p−1

In view of (1.4.16), this implies that, for almost all x ∈ G\{0},  r r r n−p |∇u| (x) > (1 − M1 ) |x|−n hr (|x|). p−1 Thus, for any j ∈ N, we find r Z β1  Z 1 r r r n−p h (s)ds. |∇u| dx > κn (1 − M1 ) p−1 β1 /(1+j) s G(j) Hence, in view of (1.4.34), we obtain Z lim |∇u|r dx = ∞. j→∞

Therefore, |∇u| 6∈ proved.

Lr (B).

G(j)

This means that the validity of assertion (∗6 ) is 

Section 1.4 On the integrally logarithmic conditions for the limiting summability 37

In using Proposition 1.4.3, we first present a fairly simple example. It is not directly connected with the integrally logarithmic conditions studied in the present section. However, it is of interest in the general context of investigation of the limit summability of solutions. In this example, the right-hand side of Eq. (1.1.4) is characterized by a certain “logarithmic” summability but the entropy (and, hence, weak) solution of problem (1.1.4), (1.1.5) does not belong ◦

to the space W 1,r (Ω) and may also lie outside the space Lq (Ω). Example 1.4.2. Let p > 2 − 1/n, σ1 ∈ (0, 1/r). We set α1 = e−2σ1 (p−1)/(n−p) . Assume that ψ1 is a function from C 2 ((0, +∞)) such that ψ1 = 0 on (0, α1 /3 ], ψ1 = 1 on [ 2α1 /3, +∞), and ψ1 does not decrease on (α1 /3, 2α1 /3). Let h1 be a function on (0, +∞) such that  1 −σ1   1 − ψ1 (s) + ψ1 (s)(α1 − s)3 h1 (s) = ln s for any s ∈ (0, α1 ) and h1 (s) = 0 for any s > α1 . We have h1 ∈ C 2 ((0, +∞)), α1 ∈ (0, 1), and conditions 1)– 6) of Proposition 1.4.3 are satisfied for the function h1 and the number α1 . In addition, by virtue of the inequality σ1 < 1/r, we obtain Z α1 1 r (1.4.35) h (s)ds = +∞. s 1 0 Now let F1 be a function on B such that  p−2 n−p h1 (|x|) − |x|h01 (|x|) F1 (x) = |x|1−n p−1   × (n − 2p + 1)h01 (|x|) − (p − 1)|x|h001 (|x|) for any x ∈ B, 0 < |x| < α1 , and F1 (x) = 0 for any x ∈ B, |x| > α1 . Finally, let u be a function on B such that, for any x ∈ B\{0}, u(x) = |x|−(n−p)/(p−1) h1 (|x|). By virtue of Proposition 1.4.3 and relation (1.4.35), we obtain F1 ∈ L1 (B), ◦

u ∈ W 1,1 (B), and the following assertions are true: (i) |∇u| ∈ Lλ (B) for any λ ∈ [1, r); (ii) |∇u| 6∈ Lr (B); (iii) for any function ϕ ∈ C0∞ (B),  Z X Z n Ai (∇u)Di ϕ dx = F1 ϕ dx; B

i=1

B

38

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data ◦

(iv) Tk (u − ϕ) ∈ W 1,p (B) for any ϕ ∈ C0∞ (B) and k > 0; the function n P Ai (∇u)Di Tk (u − ϕ) is summable on B, and i=1  Z X Z n Ai (∇u)Di Tk (u − ϕ) dx = F1 Tk (u − ϕ) dx. B

B

i=1

Now let Ω = B and let the coefficients ai , i = 1, . . . , n, and the right-hand side f in Eq. (1.1.4) be such that 1) ai (x, ξ) = Ai (ξ) for any i ∈ {1, . . . , n} and (x, ξ) ∈ Ω × Rn ; 2) f = F1 . By virtue of assertions (i) and (iii), the function u is a weak solution of problem (1.1.4), (1.1.5). According to assertion (iv) and relation (1.1.6), the same function is an entropy solution of problem (1.1.4), (1.1.5). At the same time, ◦

the function u does not belong to the space W 1,r (Ω) by assertion (ii). We also note that the inclusion f [ ln(1 + |f |)]t ∈ L1 (Ω) is true for any t, 0 < t < σ1 min{1, p − 1}. In this case, by virtue of the assumption σ1 < 1/r, we get the inequality σ1 min{1, p − 1} < (n − 1)/n. Finally, it is easy to see that if σ1 < 1/q, then u ∈ / Lq (Ω). Prior to passing to the last example in this section, we introduce some functions used in this example. For any m ∈ N, let gm be a function on (0, 1/sm+1 ) such that, for any s ∈ (0, 1/sm+1 ), m 1  Y gm (s) = bj . s j=1

For any m ∈ N, we have gm ∈ C 2 ((0, 1/sm+1 )) and gm > 1 on (0, 1/sm+1 ). We fix a sequence of numbers αm such that, for any m ∈ N,  2 0 < αm < min 1/sm+1 , e−mn /(n−p) . Assume that, for any m ∈ N, ψm is a function from C 2 ((0, +∞)) such that ψm = 0 on (0, αm /3 ], ψm = 1 on [ 2αm /3, +∞), and ψm is nondecreasing on [ αm /3, 2αm /3 ]. For any m ∈ N, we set τm = 1 + max

s∈(0,+∞)

0 ψm (s) + max

s∈(0,+∞)

00 |ψm (s)|.

For any m ∈ N, let hm be a function on (0, +∞) such that   hm (s) = [ gm (s)]−1/r 1 − ψm (s) + ψm (s)(αm − s)3 for any s ∈ (0, αm ) and hm (s) = 0 for any s > αm .

Section 1.4 On the integrally logarithmic conditions for the limiting summability 39

We have {hm } ⊂ C 2 ((0, +∞)). In addition, if m ∈ N and s ∈ (0, αm ), then n−p hm (s) − sh0m (s) > 0. p−1

(1.4.36)

We now formulate several useful lemmas about properties of the functions hm . The detailed proofs of these auxiliary statements can be found in [69]. Lemma 1.4.1. Let m ∈ N and s ∈ (0, αm /3 ]. Then   n − p m/r (n−p)/r n−p s , s|h0m (s)| 6 hm (s). hm (s) > m 2(p − 1) Assume that, for any m ∈ N, Φm is a function on (0, αm ) such that, for any s ∈ (0, αm ), p−2  n−p hm (s) − sh0m (s) [ |h0m (s)| + s|h00m (s)| ]. Φm (s) = p−1 Lemma 1.4.2. Let p > 2 − 1/n, m ∈ N, and s ∈ (0, αm /3 ]. Then    1 −1 3m n − p p−2 [ gm (s)]−(n−1)/n ln . Φm (s) 6 rs p − 1 s Lemma 1.4.3. Let p > 2 − 1/n m ∈ N, and s ∈ [ αm /3, αm ). Then p Φm (s) 6 ( 3/αm )p+2 τm .

Lemma 1.4.4. Let p > 2 − 1/n and m ∈ N. Then Z αm p Φm (s)ds 6 ( 3/αm )p+2 τm . 0

Lemma 1.4.5. Let m ∈ N. Then Z αm 1 r h (s)ds = +∞. s m 0 Example 1.4.3. Let p > 2 − 1/n, c > 0, and m ∈ N. We fix a positive number µ such that c p+1 µp−1 6 α , (1.4.37) κn m+1   sm+1 αm+1 n+p+1 p−1 . (1.4.38) µ 6 p 3 nτm+1

40

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

Let Ω = B and let F be a function on Ω such that  p−2 n−p F (x) = µp−1 |x|1−n hm+1 (|x|) − |x|h0m+1 (|x|) p−1   × (n − 2p + 1)h0m+1 (|x|) − (p − 1)|x|h00m+1 (|x|) for any x ∈ Ω, 0 < |x| < αm+1 , and F (x) = 0 for any x ∈ Ω, |x| > αm+1 . By the definition of the function Φm+1 , for any x ∈ Ω, 0 < |x| < αm+1 , we conclude that |F (x)| 6 nµp−1 |x|1−n Φm+1 (|x|). (1.4.39) This and Lemma 1.4.4 imply the inclusion F ∈ L1 (Ω). Assume that the coefficients ai , i = 1, . . . , n, and the right-hand side f of Eq. (1.1.4) are such that: 1) ai (x, ξ) = Ai (ξ) for any i ∈ {1, . . . , n} and (x, ξ) ∈ Ω × Rn ; 2) f = F. Let u be a function on Ω such that, for any x ∈ Ω\{0}, u(x) = µ|x|−(n−p)/(p−1) hm+1 (|x|). We now show that the following propositions are true: (i) the function u is a weak solution of problem (1.1.4), (1.1.5); (ii) the function u is an entropy solution of problem (1.1.4), (1.1.5); ◦

(iii) the function u does not belong to the space W 1,r (Ω); Z (iv) for any k > sm+1 ,

 m+1 −(n−1)/n Y bj (k) |f |dx 6 c .

{|f |>k}

j=1

Indeed, by the definition of the function hm+1 , relation (1.4.36), and Lemmas 1.4.1 and 1.4.4, conditions 1)– 6) of Proposition 1.4.3 are satisfied for the function µhm+1 and the number αm+1 . Due to Proposition 1.4.3, Lemma 1.4.5, ◦

and assumptions 1) and 2) of the analyzed example, we have u ∈ W 1,1 (Ω) and the following assertions are true: (i 0 ) |∇u| ∈ Lλ (Ω) for any λ ∈ [1, r);  Z Z X n (ii 0 ) for any function ϕ ∈ C0∞ (Ω), ai (x, ∇u)Di ϕ dx = f ϕ dx; Ω ◦

i=1

(iii 0 ) Tk (u − ϕ) ∈ W 1,p (Ω) for any ϕ ∈ C0∞ (Ω) and k > 0;

Ω

Section 1.4 On the integrally logarithmic conditions for the limiting summability 41

(iv 0 ) for any ϕ ∈ C0∞ (Ω) and k > 0, the function

n P

ai (x, ∇u)Di Tk (u − ϕ)

i=1

is summable on Ω and  Z X Z n f Tk (u − ϕ) dx; ai (x, ∇u)Di Tk (u − ϕ) dx = Ω

Ω

i=1

◦

(v 0 ) the function u does not belong to the space W 1,r (Ω). Assertions (i 0 ) and (ii 0 ) imply that the function u is a weak solution of problem (1.1.4), (1.1.5). Hence, proposition (i) is true. ◦

◦

◦

By virtue of assertion (iii 0 ), we have u ∈ T 1,p (Ω). Thus, u ∈ T 1,p (Ω)∩ W 1,1 (Ω) and, hence, ∇u = δu a.e. on Ω. In view of this fact and relation (1.1.6), we conclude that if ϕ ∈ C0∞ (Ω) and k > 0, then, for any i ∈ {1, . . . , n}, Di Tk (u − ϕ) = (δi u − δi ϕ) · 1{|u−ϕ| sm+1 . We set B˜ = { x ∈ Rn : |x| 6 αm+1 /3 },

B˜ k = { x ∈ Rn : |x| 6 k −1/2n }.

Assume that H is a function on B˜ k such that, for any x ∈ B˜ k \{0},   1 −2+1/n 1 ln . H(x) = |x|n |x| It is easy to see that the function H is summable on B˜ k and Z Hdx 6 3nκn (ln k)−(n−1)/n .

(1.4.40)

B˜ k

We now estimate the integral of the function |f | over the set B˜ ∩ B˜ k . Let ˜ B˜ k )\{0}. Hence, 0 < |x| 6 αm+1 /3. Thus, by virtue of relation (1.4.39) x ∈ (B∩ and assumption 2), we get |f (x)| 6 nµp−1 |x|1−n Φm+1 (|x|). Moreover, according to Lemma 1.4.2,     1 −1 3(m + 1) n − p p−2 −(n−1)/n [ gm+1 (|x|)] . ln Φm+1 (|x|) 6 r|x| p−1 |x| These inequalities imply that |f (x)| 6 3n(m + 1)µ

p−1

p−2

(n − p)

−n

|x|

−(n−1)/n

[ gm+1 (|x|)]

  1 −1 . ln |x| (1.4.41)

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

42

We now establish a suitable estimate for gm+1 (|x|). Since x ∈ B˜ k , we find ln ln(1/|x|) > ln ln k − ln(2n). In addition, ln(1/αm+1 ) > 2n. Since |x| < αm+1 , this fact and the previous inequality imply that b2 (1/|x|) > b2 (k)/2. Using this inequality and the initial estimate for αm+1 and proceeding by induction on j, we conclude that bj (1/|x|) > bj (k)/2 for any j ∈ {2, . . . , m + 1}. Hence, gm+1 (|x|) > 2−m

 m+1 Y j=2

 1 bj (k) ln . |x|

This inequality and relation (1.4.41) yield the following inequality: m p−1

|f (x)| 6 3n(m + 1)2 µ

p−2

(n − p)

 m+1 Y

−(n−1)/n bj (k) H(x),

j=2

whence, in view of inequality (1.4.40), we conclude that Z ˜ B˜ k B∩

2

m p−1

|f |dx 6 9n κn (m+1)2 µ

p−2

(n−p)

 m+1 Y

−(n−1)/n bj (k) .

(1.4.42)

j=1

We now show that {|f | > k} ⊂ B˜ ∩ B˜ k .

(1.4.43)

Indeed, let x ∈ {|f | > k}. Then k < |f (x)|.

(1.4.44)

If x = 0, then it is clear that x ∈ B˜ ∩ B˜ k . Let x 6= 0. It is obvious that |x| < αm+1 . Assume that |x| > αm+1 /3. Then, by virtue of Lemma 1.4.3, we get p Φm+1 (|x|) 6 (3/αm+1 )p+2 τm+1 . This relation and (1.4.39) imply that p |f (x)| 6 nµp−1 (3/αm+1 )n+p+1 τm+1 .

Hence, in view of (1.4.38) and the inequality sm+1 < k , we obtain |f (x)| < k. Therefore, we arrive at a contradiction with (1.4.44). Hence, |x| 6 αm+1 /3, ˜ In addition, by virtue of Lemma 1.4.2 and the inequality gm+1 (|x|) > 1, and x ∈ B. we find   1 −1 p−2 −1 . Φm+1 (|x|) 6 3(m + 1)(n − p) |x| ln |x|

Section 1.4 On the integrally logarithmic conditions for the limiting summability 43

This inequality and (1.4.39) imply that   1 −1 |f (x)| 6 3n(m + 1)µp−1 (n − p)p−2 |x|−n ln . |x|

(1.4.45)

Note that, by virtue of the initial estimate for αm+1 , we get ln(1/|x|) > n2 (m + 1)/(n − p). In view of this inequality, relations (1.4.44) and (1.4.45) yield k < 3µp−1 np−2 |x|−n .

(1.4.46)

Moreover, since αm+1 < min{1/n, 1/sm+1 }, by using (1.4.38), we obtain 4(µn)2(p−1) < sm+1 < k. Thus, it follows from (1.4.46) that |x|n < k −1/2 . Hence, x ∈ B˜ k and, finally, we get x ∈ B˜ ∩ B˜ k . Thus, the inclusion (1.4.43) is proved. Note that 9n2 κn (m + 1)2m µp−1 (n − p)p−2 < c. (1.4.47) This inequality is true in view of the initial estimate for αm+1 , relation (1.4.37), and the inequality p > 3/2. Relations (1.4.42), (1.4.43), and (1.4.47) imply the upper bound for the integral of the function |f | over the set {|f | > k}, which enables us to conclude that proposition (iv) is true. Finally, without any details, we note that the following proposition is true: (v) f [ ln(1 + |f |)]t ∈ L1 (Ω) for any t ∈ (0, (n − 1)/n). Remark 1.4.2. Propositions (ii)–(iv) in Example 1.4.3 imply that the condition of Theorem 1.4.3 for the function f cannot, generally speaking, be weakened without violating the condition that the entropy solution of prob◦

lem (1.1.4), (1.1.5) belongs to the space W 1,r (Ω). As shown in Example 1.4.3, the minimal possible weakening of the condition of Theorem 1.4.3 for the function f (i.e., the assumption that σ = (n − 1)/n) leads to situations in which the entropy solution of problem (1.1.4), (1.1.5) (by virtue of Theorem 1.1.1, ◦

it is unique) does not belong to W 1,r (Ω) and there exists a weak solution of ◦

problem (1.1.4), (1.1.5) which also does not belong to W 1,r (Ω). Remark 1.4.3. By virtue of Proposition 1.4.1, the validity of the condition of Theorem 1.4.3 for the function f implies the validity of the following property: For any t ∈ (0, (n − 1)/n), the function f [ ln(1 + |f |)]t belongs to L1 (Ω). As follows from propositions (ii), (iii), and (v) in Example 1.4.3 and Theorem 1.4.3, the converse assertion is, generally speaking, not true. In conclusion, we note that Propositions 1.4.1–1.4.3 and Examples 1.4.1–1.4.3 were published in [69].

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

44

1.5

The case where the right-hand side of equation belongs to Lebesgue spaces close to L1 (Ω)

First, we consider a series of auxiliary results frequently used in this section. Lemma 1.5.1. Let v be a measurable function on Ω, let α > 0, and let M > 0. Assume that the inequality meas {|v| > k} 6 M k −α

(1.5.1)

is true for any k > 0. Then v ∈ Lλ (Ω) for any λ ∈ (0, α). Proof. We fix any λ ∈ (0, α). Let ϕ be a function on [1, +∞) such that ϕ(s) = M sλ−α for any s ∈ [1, +∞). It is clear that the function ϕ is nonnegative, nonincreasing, and measurable. Moreover, it satisfies inequality (1.2.1). Finally, by virtue of (1.5.1), for any k > 1, we find meas {|v| > k} 6 k −λ ϕ(k). Applying Lemma 1.2.1, we conclude that v ∈ Lλ (Ω).  ◦

Lemma 1.5.2. Let u ∈ T 1,p (Ω) and k > 0. Then Z Z p |∇Tk (u)| dx = |δu|p dx. {|u| 0. Assume that, for any l ∈ N, Z |∇Tl (u)|λ dx 6 c. (1.5.2) Then u ∈

Ω

◦

W 1,λ (Ω).

Proof. By virtue of Proposition 1.1.1, |∇Tl (u)| → |δu| a.e. on Ω. Thus, taking into account (1.5.2) and using the Fatou lemma, we obtain |δu| ∈ Lλ (Ω). ◦

It follows from Proposition 1.1.2 that u ∈ W 1,λ (Ω).



◦

For any u ∈ T 1,p (Ω) and k > 0, we set Z I(u, k) = |∇Tk (u)|p dx. Ω ◦

Lemma 1.5.4. Let u ∈ T 1,p (Ω) and k > 0. Then ∗

∗

meas {|u| > k} 6 cpn,p k −p [I(u, k)]p

∗ /p

.

(1.5.3)

Proof. By using the definition of the function Tk , i.e., the fact that |Tk (s)| = k for |s| > k, we arrive at the inequality Z ∗ ∗ (1.5.4) k p meas {|u| > k} 6 |Tk (u)|p dx. Ω

Section 1.5

The case where the right-hand side of equation belongs to

45

◦

Since Tk (u) ∈ W 1,p (Ω), by virtue of (1.1.13), we find Z ∗ ∗ ∗ |Tk (u)|p dx 6 cpn,p [I(u, k)]p /p . Ω



Thus, in view of (1.5.4), we obtain (1.5.3). ◦

Lemma 1.5.5. Let u ∈ T 1,p (Ω) and let k, k1 > 0. Then ∗

∗

meas {|δu| > k} 6 cpn,p k1−p [I(u, k1 )]p

∗ /p

+ k −p I(u, k1 ).

(1.5.5)

Proof. We set G = {|u| < k1 , |δu| > k}. It is clear that meas {|δu| > k} 6 meas {|u| > k1 } + meas G,

(1.5.6)

and, by virtue of Lemma 1.5.4, ∗

∗

meas {|u| > k1 } 6 cpn,p k1−p [I(u, k1 )]p

∗ /p

.

(1.5.7)

We now estimate the measure of the set G. Assume that meas G > 0. In view of Proposition 1.1.1, we get |∇Tk1 (u)| = |δu| a.e. on {|u| < k1 }. Then k 6 |∇Tk1 (u)| a.e. on G and, hence, meas G 6 k −p I(u, k1 ).

(1.5.8)

It is clear that this inequality remains valid in the case meas G = 0. Relations (1.5.6)–(1.5.8) yield (1.5.5).  Lemma 1.5.6. Let u be an entropy solution of problem (1.1.4), (1.1.5). Then, for any k > 0, Z c2 I(u, k) 6

f Tk (u) dx. Ω

This result is a simple consequence of Definition 1.1.4, inequality (1.1.2), and Lemma 1.5.2. Remark 1.5.1. If u is an entropy solution of problem (1.1.4), (1.1.5), then, by virtue of Lemmas 1.5.4–1.5.6, for any k > 0, we obtain ∗

∗

meas {|u| > k} 6 cpn,p (||f ||L1 (Ω) /c2 )p /p k −q ,  ∗  ∗ meas {|δu| > k} 6 cpn,p (||f ||L1 (Ω) /c2 )p /p + ||f ||L1 (Ω) /c2 k −r .

(1.5.9) (1.5.10)

These estimates and Lemma 1.5.1 yield Theorem 1.1.3. Inequalities (1.5.9) and (1.5.10) were established in [13]. In what follows, it is shown that, in the presence of certain additional information about the summability of the function f, the entropy solution of problem (1.1.4), (1.1.5) may satisfy stronger estimates than (1.5.9) and (1.5.10).

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

46

As a result, this solution may have better properties of summability as compared with the properties described in Theorem 1.1.3. An essential role in the substantiation and correction of the assertions made above is played by the following two lemmas: Lemma 1.5.7. Let 1 < m < p∗ /(p∗ − 1) and let ψ ∈ Lm (Ω), ψ > 0 on Ω. ◦

Assume that u ∈ T 1,p (Ω) and that the inequality Z ψ|Tk (u)|dx I(u, k) 6

(1.5.11)

Ω

is true for any k > 0. Then there exist positive numbers M1 and M2 depending only on n, p, m, and ||ψ||Lm (Ω) such that, for any k > 0, meas {|u| > k} 6 M1 k −nm(p−1)/(n−mp) meas {|δu| > k} 6 M2 k

−m∗ (p−1)

.

(1.5.12) (1.5.13)

Proof. First, we note that, by virtue of the inequality m < p∗ /(p∗ − 1), we have mp < n and p∗ (m − 1) < m. We set m1 = 1 −

p∗ (m − 1) n − mp nm(p − 1) m−1 ∗ p , m2 = , m3 = , θ= . m mp n−m n − mp

It is clear that 0 < m1 < 1 and 0 < m2 < 1. In addition, we obtain   m1 m1 m3 ∗ p 1− . = θ, θm3 = p − p(1 − m2 ) 1 − m2

(1.5.14)

By c0i , i = 1, 2, . . . , we denote positive constants depending only on n, p, m, and ||ψ||Lm (Ω) . We fix k > 0. By using (1.5.11), the Hölder inequality, and (1.1.13), we get Z m1 I(u, k) 6 k ψ|Tk (u)|1−m1 dx Ω Z (m−1)/m ∗ 6 k m1 ||ψ||Lm (Ω) |Tk (u)|p dx 6 c01 k m1 [I(u, k)]m2 . Ω

It is now possible to conclude that I(u, k) 6 c02 k m1 /(1−m2 ) .

(1.5.15)

Thus, in view of Lemma 1.5.4 and the first equality in (1.5.14), we find meas {|u| > k} 6 c03 k −θ . We set k1 = k m3 . By analogy with (1.5.15), we conclude that m1 /(1−m2 )

I(u, k1 ) 6 c02 k1

.

Hence, I(u, k1 ) 6 c02 k m1 m3 /(1−m2 ) .

(1.5.16)

Section 1.5

The case where the right-hand side of equation belongs to

47

Thus, in view of Lemma 1.5.5 and equalities (1.5.14), we obtain meas {|δu| > k} 6 c04 k −θm3 .

(1.5.17)

Inequalities (1.5.16) and (1.5.17) yield the assertion of the lemma.



Lemma 1.5.8. Let b0 , b00 > 0, let 1 < m < p∗ /(p∗ − 1), let ψ ∈ Lm (Ω), ◦

ψ > 0, on Ω, and let m∗ (p − 1) > 1. Assume that u ∈ T 1,p (Ω) and that the following inequalities are true for any k > 0 : meas {|u| > k} 6 b0 k −nm(p−1)/(n−mp) , Z p 00 |∇Tk (u)| dx 6 b ψ dx.

Z

{k−16|u|k−1}

(Ω).

Proof. We set λ = m∗ (p − 1). By the conditions of the lemma imposed on m, we have λ ∈ [1, p). We also note that nm(p − 1) λ∗  λ  1 λ∗ = , 1− = . (1.5.20) n − mp λ mp p We set λ1 = λ∗ (p − λ)/λ. We have λ1 ∈ (0, 1) and (1 − λ1 )

m = λ∗ . m−1

(1.5.21)

By c0i , i = 1, 2, . . . , we denote positive constants depending only on n, p, b00 , m, ||ψ||Lm (Ω) , and meas Ω. We fix any N ∈ N. By using the Hölder inequality, (1.5.18), the first equality in (1.5.20), and (1.5.21), we obtain Z
(m−1)/m N 1−λ1 ψ dx 6 N 1−λ1 meas { |u| > N } ||ψ||Lm (Ω)

b0 ,

{|u|>N }

6 (b0 + 1)N 1−λ1 −λ

∗ (m−1)/m

||ψ||Lm (Ω) = c01 .

(1.5.22)

Further, we set Z IN =

|∇TN (u)|λ dx,

Ω

Z JN = Ω

|∇TN (u)|p dx. ( 1 + |TN (u)| )λ1

By virtue of the Hölder inequality and the definition of λ1 , we find Z |∇TN (u)|λ IN = (1 + |TN (u)|)λ1 λ/p dx λ1 λ/p Ω (1 + |TN (u)|) Z (p−λ)/p λ/p λ∗ . 6 JN (1 + |TN (u)|) dx Ω

(1.5.23)

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

48

We now estimate JN . By using Proposition 1.1.1 and (1.5.19), we obtain JN 6

Z Z N N X X 1 1 p 00 |∇T (u)| dx 6 b ψ dx. (1.5.24) k k λ1 {k−16|u|k−1} k=1

k=1

In view of (1.5.21) and (1.5.22), the sum on the right-hand side of inequality (1.5.24) can be estimated as follows: Z N X 1 ψ dx k λ1 {|u|>k−1} k=1 X Z N N N Z X 1 X 1 = ψ dx + ψ dx k λ1 k λ1 {|u|>N } k=1 k=1 l=k {l−16|u|N } {l−16|u| 0, meas {|u| > k} 6 C1 k −nm(p−1)/(n−mp) , meas {|δu| > k} 6 C2 k −m

∗ (p−1)

.

This result is obtained as a consequence of Lemmas 1.5.6 and 1.5.7. Theorem 1.5.2. Let 1 < m < p∗ /(p∗ − 1) and let f ∈ Lm (Ω). Assume that u is an entropy solution of problem (1.1.4), (1.1.5). Then the following assertions are true: 1) u ∈ Lλ (Ω) for any λ ∈ (0, nm(p − 1)/(n − mp)); 2) |δu| ∈ Lλ (Ω) for any λ ∈ (0, m∗ (p − 1)). This result is a consequence of Theorem 1.5.1 and Lemma 1.5.1. Theorem 1.5.3. Let 1 < m < p∗ /(p∗ − 1), let m∗ (p − 1) > 1 and let f ∈ Lm (Ω). Assume that u is an entropy solution of problem (1.1.4), (1.1.5). ◦

Then u ∈ W 1,m

∗ (p−1)

(Ω).

Proof. By Theorem 1.5.1, there exists b0 > 0 such that, for any k > 0, meas {|u| > k} 6 b0 k −nm(p−1)/(n−mp) .

(1.5.27)

Moreover, by virtue of the arguments used in the proof of Theorem 1.1.6 for any k, l > 0, we find Z Z c2 |δu|p dx 6 l |f |dx. {k6|u|k}

Hence, in view of Proposition 1.1.1 and Lemma 1.5.6, we conclude that, for any k > 0, Z Z 1 p |∇Tk (u)| dx 6 |f |dx. (1.5.28) c2 {|u|>k−1} {k−16|u| 1, and let f ∈ Lm (Ω). Then there exists a weak solution of problem (1.1.4), (1.1.5) from ◦

W 1,m

∗ (p−1)

(Ω).

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

50

Proof. By virtue of Theorem 1.1.1, there exists an entropy solution of problem (1.1.4), (1.1.5). We denote this solution by u. By Theorem 1.5.3, we get ◦

∗

u ∈ W 1,m (p−1) (Ω). Hence, |δu| ∈ L1 (Ω). This inclusion and Theorem 1.1.2 imply that u is a weak solution of problem (1.1.4), (1.1.5).  Note that if p > 2 − 1/n and m ∈ [1, n), then m∗ (p − 1) > 1. By using this inequality and Theorem 1.5.4, we arrive at the following assertion: Theorem 1.5.5. Let p > 2 − 1/n, let 1 < m < p∗ /(p∗ − 1), and let f ∈ Lm (Ω). Then there exists a weak solution of problem (1.1.4), (1.1.5) from ◦

W 1,m

∗ (p−1)

(Ω).

The same result was obtained in [24] (Theorem 3) with the following difference: instead of the condition p > 2 − 1/n, it was assumed that p > 2 − 1/n. Parallel with Theorem 1.5.5 (somewhat sharpening the indicated result from [24]), Theorem 1.5.4 also yields an assertion on the existence of a weak solution of problem (1.1.4), (1.1.5) also in the case where p < 2 − 1/n. In this connection, we first note that n/(np − n + 1) < p∗ /(p∗ − 1) and the following assertion is true: If p < 2 − 1/n, then 1 < n/(np − n + 1). Theorem 1.5.6. Let p < 2 − 1/n, let n/(np − n + 1) 6 m < p∗ /(p∗ − 1), and let f ∈ Lm (Ω). Then there exists a weak solution of problem (1.1.4), (1.1.5) ◦

from W 1,m

∗ (p−1)

(Ω).

Proof. It is clear that 1 < m < p∗ /(p∗ − 1). Moreover, in view of the inequality n/(np − n + 1) 6 m, we have m∗ (p − 1) > 1. Theorem 1.5.4 now yields the required assertion.  Note that Lemmas 1.5.7 and 1.5.8 were proved in [65]. Moreover, in proving Lemma 1.5.8, the procedure proposed in the proof of Theorem 3 in [24] was somewhat modified. Theorems 1.5.2 and 1.5.3 and, in fact, Theorem 1.5.6, were also established in [65]. We also note that analogs of Theorems 1.5.1 and 1.5.3 for the entropy solutions of the Dirichlet problem for nonlinear equations of the second order with degenerate coercivity were obtained in [4] with the use of a different technique. At the end of the section, we consider the case where the right-hand side of Eq. (1.1.4) belongs to the Lebesgue space with exponent p∗ /(p∗ − 1). In this case, as shown in what follows, the entropy solution of problem (1.1.4), (1.1.5) ◦

belongs to the space W 1,p (Ω). This property is stronger than the property of the same solution under the conditions of Theorem 1.5.3 because the inequality 1 < m < p∗ /(p∗ − 1) yields the inequality m∗ (p − 1) < p.

◦

Section 1.6

On the convergence of functions from W 1,p (Ω) ∗

51

∗

Theorem 1.5.7. Let f ∈ Lp /(p −1) (Ω) and let u be an entropy solution of problem (1.1.4), (1.1.5). Then the following assertions are true: ◦

1) u ∈ W 1,p (Ω); 2) u is a weak solution of problem (1.1.4), (1.1.5); 3) u is a weak solution of problem (1.1.4), (1.1.5). ∗

∗

Proof. By b we denote the norm of the function f in Lp /(p −1) (Ω). Let k > 0. By using Lemma 1.5.6, the Hölder inequality, and (1.1.13), we obtain Z 1/p∗ Z p∗ c2 I(u, k) 6 f Tk (u) dx 6 b |Tk (u)| dx 6 b cn,p [ I(u, k)]1/p . Ω

Ω

Hence, I(u, k) 6 (b cn,p /c2 )p/(p−1) . ◦

This result and Lemma 1.5.3 imply that u ∈ W 1,p (Ω). Then |δu| ∈ L1 (Ω). Therefore, by virtue of Theorem 1.1.2, u is a weak solution of problem (1.1.4), (1.1.5). Hence,  Z X Z n ∞ ∀ ϕ ∈ C0 (Ω), ai (x, ∇u)Di ϕ dx = f ϕ dx. (1.5.29) Ω

Ω

i=1

Note that, in view of (1.1.1) and the inclusion |∇u| ∈ Lp (Ω), we have ai (x, ∇u) ∈ Lp/(p−1) (Ω) for any i ∈ {1, . . . , n}. Thus, it follows from (1.5.29) ◦

that, for any function ϕ ∈ W 1,p (Ω),  Z X Z n ai (x, ∇u)Di ϕ dx = f ϕ dx. Ω

i=1

Ω

Hence, u is a weak solution of problem (1.1.4), (1.1.5).



Finally, we note that, for f ∈ Lm (Ω) with m > p∗ /(p∗ − 1), the improvement of the summability of the weak solution of problem (1.1.4), (1.1.5) can be established by using Stampacchia’s method (see, e.g., [29, 60, 126]). ◦

1.6

On the convergence of functions from W 1,p (Ω) satisfying special integral inequalities

We now consider a series of results frequently used in Sections 1.7 and 1.8. Lemma 1.6.1. Let u be a measurable function on Ω, let M > 0, let γ > 0, and let, for any k ∈ N, meas {|u| > k} 6 M k −γ . (1.6.1)

52

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

Then u ∈ Lλ (Ω) for any λ ∈ (0, γ) and Z |u|λ dx 6 2γ+(γ+λ)/(γ−λ) M + meas Ω.

(1.6.2)

Ω

Proof. We fix λ ∈ (0, γ) and set λ1 = 2/(γ − λ). By virtue of (1.6.1), for any k ∈ N, we get Z |u|λ dx 6 2γ+λ1 λ M k −2 . {kλ1 6|u| 1, and 0 < θ < p. Assume that the following inequality holds for any k > 1 : Z |δu|p dx 6 M k θ . (1.6.3) {|u| 1, ∗

meas {|u| > k} 6 cpn,p M n/(n−p) k −n(p−θ)/(n−p) , ∗

meas {|δu| > k} 6 (cpn,p + 1)M n/(n−θ) k −n(p−θ)/(n−θ) .

(1.6.4) (1.6.5)

Proof. Let k > 1. By virtue of (1.6.3) and Lemma 1.5.2, we get I(u, k) 6 M k θ . This inequality and Lemma 1.5.4 imply inequality (1.6.4). We now set k1 = M 1/(n−θ) k (n−p)/(n−θ) . By virtue of (1.6.3) and Lemma 1.5.2, we obtain I(u, k1 ) 6 M k1θ . Thus, in view of Lemma 1.5.5, we arrive at inequality (1.6.5).  Lemmas 1.6.1 and 1.6.2 yield the following assertion: ◦

Lemma 1.6.3. Let u ∈ T 1,p (Ω), M > 1, and 0 < θ < p. Assume that, for any k > 1, the following inequality holds: Z |δu|p dx 6 M k θ . {|u| 1, 0 < θ < p, and {uj } ⊂ W 1,p (Ω). Assume that the following inequality holds for any k > 1 and j ∈ N: Z |∇uj |p dx 6 M k θ . (1.6.6) {|uj | 0,

(1.6.8)

◦

Proof. Let k > 0 and j ∈ N. We have Tk (uj ) ∈ W 1,p (Ω). Hence, by using (1.6.6), we obtain Z p p kTk (uj )k ◦ 6 k meas Ω + n |∇ uj |p dx W 1,p (Ω)

{|uj | 0, the sequence {Tk (uj )} is bounded in W 1,p (Ω). 0 } ⊂ N and a sequence {w } ⊂ Thus, there exist an increasing sequence {jm k ◦

◦

1,p (Ω). 0 ) → wk weakly in W W 1,p (Ω) such that, for any k ∈ N, we have Tk (ujm For any k ∈ N, this yields 0 ) → wk Tk (ujm

strongly in L1 (Ω).

(1.6.9)

Therefore, by virtue of (1.6.6) and Lemma 1.6.2, for any k > 1 and j ∈ N, we obtain ∗ meas {|uj | > k} 6 cpn,p M n/(n−p) k −n(p−θ)/(n−p) . (1.6.10) We now fix t > 0 and ε > 0. Let k ∈ N and ∗

cpn,p M n/(n−p) k −n(p−θ)/(n−p) 6 ε/4.

(1.6.11)

By virtue of (1.6.9), there exists m0 ∈ N such that, for any m ∈ N, m > m0 , we can write Z 0 ) − wk |dx 6 εt/4. |Tk (ujm (1.6.12) Ω

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

54

We now fix m, l ∈ N, m, l > m0 . It is clear that 0 − uj 0 | > t} 6 meas {|uj 0 | > k} + meas {|uj 0 | > k} meas {|ujm m l l 0 ) − Tk (uj 0 )| > t}. + meas {|Tk (ujm l

(1.6.13)

In view of (1.6.10) and (1.6.11), we get 0 | > k} + meas {|uj 0 | > k} 6 ε/2. meas {|ujm l

(1.6.14)

Moreover, by using (1.6.12), we find 1 0 ) − Tk (uj 0 )| > t} 6 meas {|Tk (ujm l t

Z Ω

0 ) − Tk (uj 0 )|dx 6 ε/2. |Tk (ujm l

This inequality and inequalities (1.6.13) and (1.6.14) now imply that 0 − uj 0 | > t} 6 ε. meas {|ujm l 0 } is fundamental in measure. Thus, by virtue of Hence, the sequence {ujm the Riesz theorem, there exists a measurable function u : Ω → R such that 0 → u in measure. Therefore, there exists an increasing sequence {jl } ⊂ N ujm for which assertion (1.6.7) is true.

◦

Let k > 0. Since the sequence {Tk (uj )} is bounded in W 1,p (Ω) and, by virtue of (1.6.7), Tk (ujl ) → Tk (u) strongly in Lp (Ω), we conclude that Tk (u) ∈ ◦

◦

◦

W 1,p (Ω) and Tk (ujl ) → Tk (u) weakly in W 1,p (Ω). Hence, u ∈ T 1,p (Ω) and assertion (1.6.8) is true.  Note that the main ideas of the proof of Theorem 1.6.1 are contained in [13, Theorem 6.1]. Prior to the analysis of the second main result of the present section, we prove a simple auxiliary assertion. Lemma 1.6.4. Let µ ∈ L1 (Ω), µ > 0 a.e. on Ω. Also let Ej , for any j ∈ N, be a measurable set in Rn such that Ej ⊂ Ω and let Z lim µ dx = 0. (1.6.15) j→∞

Ej

Then lim meas Ej = 0.

j→∞

(1.6.16)

Proof. For any t > 0, we set Gt = {µ 6 t}. We now show that lim meas Gt = 0.

t→0

(1.6.17)

Indeed, assume that the family meas Gt , t > 0, does not converge to zero as t → 0. Then there exist ε0 > 0 and a decreasing sequence {ti }, ti → 0, such that ∀ i ∈ N, meas Gti > ε0 . (1.6.18)

◦

Section 1.6

On the convergence of functions from W 1,p (Ω)

We set G=

∞ \

55

Gti .

i=1

Note that Gti+1 ⊂ Gti for any i ∈ N. Therefore, meas Gti → meas G. Thus, in view of (1.6.18), we conclude that meas G > 0. Since µ > 0 a.e. on Ω, there exists a measurable set E ⊂ Ω of measure zero such that µ(x) > 0 for any x ∈ Ω \ E. Let x ∈ G \ E. Then µ(x) > 0. On the other hand, by the definition of the sets G and Gti , for any i ∈ N, we obtain µ(x) 6 ti . Thus, in view of ti → 0, we arrive at the inequality µ(x) 6 0. This contradiction proves relation (1.6.17). Further, we fix any ε > 0. By virtue of (1.6.17), there exists t > 0 such that meas Gt 6 ε/2.

(1.6.19)

According to (1.6.15), there exists j0 ∈ N such that, for any j ∈ N, j > j0 , we get Z µ dx 6 εt/2. (1.6.20) Ej

We fix j ∈ N, j > j0 . It is clear that meas Ej = meas (Ej ∩ {µ 6 t}) + meas (Ej ∩ {µ > t}) 6 meas Gt + meas (Ej ∩ {µ > t}).

(1.6.21)

Since µ > 0 a.e. on Ω, we obtain Z Z µ dx > µ dx > t meas (Ej ∩ {µ > t}). Ej

Ej ∩{µ>t}

This inequality and (1.6.19)–(1.6.21) imply that meas Ej 6 ε. We can now conclude that (1.6.16) is true.  Theorem 1.6.2. Let Φ be a Carathéodory function on Ω × R × Rn × Rn , and let the following conditions be satisfied: (i) for any k > 0, there exist c˜k > 0 and g˜k ∈ L1 (Ω), g˜k > 0 on Ω, such that, for almost all x ∈ Ω and any s ∈ R, |s| 6 k, and ξ, ξ 0 ∈ Rn , the following inequalities are true: 0 6 Φ(x, s, ξ, ξ 0 ) 6 c˜k (|ξ|p + |ξ 0 |p ) + g˜k (x); (ii) for almost all x ∈ Ω and all s ∈ R and ξ, ξ 0 ∈ Rn , ξ 6= ξ 0 , the following inequality is true: Φ(x, s, ξ, ξ 0 ) > 0. ◦

◦

Let {uj } ⊂ W 1,p (Ω), let u ∈ T 1,p (Ω), let uj → u in measure, and let lim sup meas {|uj | > m} = 0,

(1.6.22)

lim sup meas {|∇uj | > m} = 0.

(1.6.23)

m→∞ j∈N m→∞ j∈N

56

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

Finally, assume that there exist functions ν1 , ν2 : (0, +∞) → (0, +∞) such that ν2 (s) → 0 as s → 0 and, for any m > 1 and k ∈ (0, ν1 (m)], the following inequality is true: Z lim Φ(x, uj , ∇uj , δu) dx 6 ν2 (k). (1.6.24) j→∞ {|u | t, we now set  Gt,m = (s, ξ, ξ 0 ) ∈ R×Rn ×Rn : |s| 6 m, |ξ| 6 m, |ξ 0 | 6 m, |ξ −ξ 0 | > t . It is clear that, for any t > 0 and m > t, the set Gt,m is nonempty, closed, and bounded. For any t > 0 and m > t, let µt,m be a function on Ω such that   min Φx for x ∈ Ω \ E, µt,m (x) = Gt,m 0 for x ∈ E. We fix t > 0 and m > t. It is easy to see that µt,m > 0 on Ω \ E.

(1.6.25)

In addition, by virtue of condition (i), we get µt,m 6 2c˜m mp + g˜m

a.e. on Ω.

(1.6.26)

We now prove that the function µt,m is measurable. Let R be the set of all (s, ξ, ξ 0 ) ∈ R × Rn × Rn such that s is rational and, for any i ∈ {1, . . . , n}, (0) (0) ξi and ξi0 are also rational. We set Gt,m = Gt,m ∩ R. Thus, Gt,m 6= ∅ and, (0)

moreover, Gt,m is dense in Gt,m . Indeed, let (s, ξ, ξ 0 ) ∈ Gt,m and ε > 0. Assume that |ξ − ξ 0 | > t. Thus, taking λ such that   t ε , t, this yields the inequality (ξ, ξ 0 ) > 0. The same inequality is true if |ξ 0 | = m. Hence, there exists j ∈ {1, . . . , n} for which ξj ξj0 > 0. We now take λ such that   |ξj | |ξj0 | ε ε , , , . 0 < λ < min 1, 2 2 2m 4 Let s0 = (1 − λ)s and let η, η 0 ∈ Rn . Hence,   ηj = ξj − λ 2(1−sign ξj )/2 sign ξj − min(0, sign(ξj − ξj0 )) ,   0 ηj0 = ξj0 − λ 2(1+sign ξj )/2 sign ξj0 + min(0, sign(ξj − ξj0 )) . For any i ∈ {1, . . . , n}, i 6= j, we have ηi = ξi and ηi0 = ξi0 . Therefore, inequalities (1.6.27) and (1.6.28) are satisfied. Thus, in any case, there exists a triple (s0 , η, η 0 ) ∈ R × Rn × Rn for which inequalities (1.6.27) and (1.6.28) are true. Now let s¯ be a rational number such that |s¯ − s0 | < min(m − |s0 |, ε/2), ¯ ξ¯0 ∈ Rn . Moreover, for any i ∈ {1, . . . , n}, the numbers ξ¯i and ξ¯0 and let ξ, i are rational and, in addition, 1 |ξ¯i − ηi | < min(m − |η|, |η − η 0 | − t, ε), 2n 1 |ξ¯i0 − ηi0 | < min(m − |η 0 |, |η − η 0 | − t, ε). 2n 2 +|ξ−ξ| 2 +|ξ¯0 −ξ 0 |2 < 4ε2 . ¯ ξ¯0 ) ∈ G(0) . Using (1.6.28), we obtain |s−s| ¯ Then (s, ¯ ξ, ¯ t,m

(0)

This enables us to conclude that the set Gt,m is dense in Gt,m . Further, we fix λ ∈ R. We now show that [ {x ∈ Ω \ E : µt,m (x) < λ} = {x ∈ Ω \ E : Φ(x, s, ξ, ξ 0 ) < λ}. (1.6.29) (0)

(s, ξ, ξ 0 )∈Gt,m

58

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

Indeed, let y belong to the left-hand side of equality (1.6.29). We have y ∈ Ω\E and µt,m (y) < λ. Thus, by virtue of the definition of the function µt,m , there exists a triple (s, ξ, ξ 0 ) ∈ Gt,m such that Φy (s, ξ, ξ 0 ) < λ. Hence, in view of the continuity of the function Φy on R × Rn × Rn , there exists ε > 0 such ˜ ξ˜0 ) < λ is true for any triple ( s, ˜ ξ˜0 ) ∈ R × Rn × Rn that the inequality Φy (s, ˜ ξ, ˜ ξ, satisfying the inequality |s˜ − s|2 + |ξ˜ − ξ|2 + |ξ˜0 − ξ 0 |2 < ε2 . Thus, in view of the (0) (0) fact that set Gt,m is dense in Gt,m , there exists a triple (s0 , η, η 0 ) ∈ Gt,m for 0 0 which Φy (s , η, η ) < λ. In view of the definition of the function Φy , the last inequality means that Φ(y, s0 , η, η 0 ) < λ. Hence, the point y belongs to the right-hand side of equality (1.6.29). Conversely, let y belong to the right-hand side of equality (1.6.29). Thus, there exists a triple (s, ξ, ξ 0 ) ∈ Gt,m such that y ∈ Ω \ E and Φ(y, s, ξ, ξ 0 ) < λ. Therefore, by virtue of the definition of the function µt,m , we conclude that µt,m (y) 6 Φ(y, s, ξ, ξ 0 ) < λ. This means that y belongs to the left-hand side of equality (1.6.29). This reasoning proves the validity of relation (1.6.29). Since Φ is a Carathéodory function on Ω×R×Rn×Rn, the set {Φ(·,s, ξ, ξ 0 ) < λ} is measurable for any triple (s, ξ, ξ 0 ) ∈ R × Rn × Rn . This implies that, for any (0) triple (s, ξ, ξ 0 ) ∈ Gt,m , the set {x ∈ Ω\E : Φ(x, s, ξ, ξ 0 ) < λ} is measurable. Hence, (0)

since set Gt,m is countable, equality (1.6.29) implies that the set {µt,m < λ} is measurable and, therefore, the function µt,m is measurable. In view of (1.6.25) and (1.6.26), we conclude that, for any t > 0 and m > t, the function µt,m is summable on Ω. Further, we note that lim meas {|u| > m} = 0,

m→∞

lim meas {|δ u| > m} = 0.

m→∞

(1.6.30)

Indeed, the union of all sets {|u| < m}, m ∈ N, is Ω. Moreover, {|u| < m} ⊂ {|u| < m + 1} if m ∈ N. Hence, meas {|u| < m} → meas Ω. This yields the first equality in (1.6.30). The validity of the second equality is proved similarly. We now proceed to the direct proof of the assertion of the theorem. Fix t > 0 and ε > 0. By virtue of (1.6.22), (1.6.23), and (1.6.30), there exists m > max(1, t) such that sup meas {|uj | > m} 6 ε,

sup meas {|∇uj | > m} 6 ε,

j∈N

j∈N

meas {|u| > m} 6 ε,

meas {|δu| > m} 6 ε.

(1.6.31) (1.6.32)

For any k, j ∈ N, we set n E(k, j) = |uj | 6 m, |u| 6 m, |∇uj | 6 m, o |δu| 6 m, |uj − u| < 1/k, |∇uj − δu| > t . Let k, j ∈ N and let x ∈ E(k, j) \ E. We have |uj (x)| 6 m,

|∇uj (x)| 6 m,

|δu(x)| 6 m,

|∇uj (x) − δu(x)| > t,

◦

Section 1.6

On the convergence of functions from W 1,p (Ω)

59

and, hence, (uj (x), ∇uj (x), δu(x)) ∈ Gt,m . Thus, by the definition of the functions µt,m and Φx , we get µt,m (x) 6 Φ(x, uj (x), ∇uj (x), δu(x)). It is now possible to conclude that, for any k, j ∈ N, Z Z µt,m dx 6 Φ(x, uj , ∇uj , δu) dx. E(k,j)

(1.6.33)

E(k,j)

Note that, by virtue of condition (i), the inequality Φ(x, s, ξ, ξ 0 ) > 0 is true for almost all x ∈ Ω and all s ∈ R and ξ, ξ 0 ∈ Rn . Thus, by virtue of condition (1.6.24), there exists a sequence {jk } ⊂ N such that, for any k ∈ N, k > 1/ν1 (m), and j ∈ N, j > jk , we obtain Z Φ(x, uj , ∇uj , δu) dx 6 2ν2 (1/k). (1.6.34) E(k,j)

Relations (1.6.33) and (1.6.34) imply that, for any k ∈ N, k > 1/ν1 (m), and j ∈ N, j > jk , the following inequality is true: Z µt,m dx 6 2ν2 (1/k). (1.6.35) E(k,j)

We now show that sup meas E(k, j) → 0 as k → ∞.

(1.6.36)

j>jk

Indeed, assume that assertion (1.6.36) is not true. Then there exist τ > 0 and an increasing sequence {kl } ⊂ N ∩ [1/ν1 (m), +∞) such that, for any l ∈ N, we get sup meas E(kl , j) > τ. j>jkl

This implies that, for any l ∈ N, there exists tl ∈ N, tl > jkl , such that meas E(kl , tl ) > τ /2.

(1.6.37)

By virtue of (1.6.35) for any l ∈ N, we can write Z µt,m dx 6 2ν2 (1/kl ) E(kl ,tl )

and, hence, Z µt,m dx = 0.

lim

l→∞

E(kl ,tl )

Note that µt,m ∈ L1 (Ω) and µt,m > 0 a.e. on Ω. Using Lemma 1.6.4, we conclude that meas E(kl , tl ) → 0, which contradicts (1.6.37). This contradiction proves that assertion (1.6.36) is true.

60

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

It follows from this assertion that there exists k ∈ N such that sup meas E(k, j) 6 ε.

(1.6.38)

j>jk

In view of the convergence of {uj } to u in measure, there exists jk0 ∈ N such that, for any j ∈ N, j > jk0 , meas {|uj − u| > 1/k} 6 ε.

(1.6.39)

We fix j ∈ N, j > max(jk , jk0 ). It is clear that meas {|∇uj − δu| > t} 6 meas {|uj | > m} + meas {|u| > m} + meas {|∇uj | > m} + meas {|δu| > m} + meas {|uj − u| > 1/k} + meas E(k, j). From this relation and inequalities (1.6.31), (1.6.32), (1.6.38), and (1.6.39), we obtain meas {|∇uj − δu| > t} 6 6ε. Hence, for any i ∈ {1, . . . , n}, we conclude that Di uj → δi u in measure.  Note that some ideas taken from the works [13] and [64] were used in the proof of Theorem 1.6.2. Corollary 1.6.1. Let 0 < p˜ < p∗ , c˜ > 0, g˜ ∈ L1 (Ω), g˜ > 0 on Ω, and let Φ be a Carathéodory function on Ω × R × Rn × Rn . Assume that the following conditions are satisfied: (i) for almost all x ∈ Ω and all s ∈ R and ξ, ξ 0 ∈ Rn , the following inequality is true: 0 6 Φ(x, s, ξ, ξ 0 ) 6 c(|s| ˜ p˜ + |ξ|p + |ξ 0 |p ) + g(x); ˜ (ii) the inequality Φ(x, s, ξ, ξ 0 ) > 0 is true for almost all x ∈ Ω and all s ∈ R and ξ, ξ 0 ∈ Rn , ξ 6= ξ 0 . ◦

◦

◦

Also let {uj } ⊂ W 1,p (Ω), let u ∈ W 1,p (Ω), let uj → u weakly in W 1,p (Ω), and let there exist k0 > 0 such that Z lim Φ(x, uj , ∇uj , ∇u) dx = 0. (1.6.40) j→∞

{|uj −u| 0 such that, for any j ∈ N, we get Z Z |∇uj |dx 6 M. (1.6.41) |uj |dx 6 M, Ω

Ω

By the Chebyshev inequality, for any m > 0 and j ∈ N, we find Z 1 |uj |dx. meas {|uj | > m} 6 m Ω Thus, in view of the first inequality in (1.6.41), we obtain (1.6.22). Similarly, by using the Chebyshev inequality for the functions |∇uj | and the second inequality in (1.6.41), we establish (1.6.23). Let ν1 , ν2 : (0, +∞) → (0, +∞) be functions such that ν1 (s) = k0 and ν2 (s) = s for any s ∈ (0, +∞). Note that Φ(x, s, ξ, ξ 0 ) > 0 for almost all x ∈ Ω and all s ∈ R and ξ, ξ 0 ∈ Rn . Thus, relation (1.6.40) implies that Z lim Φ(x, uj , ∇uj , ∇u) dx 6 ν2 (k) j→∞ {|u | 0 and g¯k ∈ L1 (Ω), g¯k > 0 on Ω, such that, for almost all x ∈ Ω and all s ∈ R, |s| 6 k, and ξ ∈ Rn ,

62

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

the inequality

n X

|ai (x, s, ξ)|p/(p−1) 6 c¯k |ξ|p + g¯k (x)

(1.7.1)

i=1

holds. Moreover, we suppose that there exist p1 ∈ [ 0, p − 1) and c1 > 0 such that, for almost all x ∈ Ω and all s ∈ R and ξ ∈ Rn , the following inequality is true: n X

ai (x, s, ξ)ξi >

i=1

c1 |ξ|p . (1 + |s|)p1

(1.7.2)

Finally, we assume that the following inequality is valid for almost all x ∈ Ω and all s ∈ R and ξ, ξ 0 ∈ Rn , ξ 6= ξ 0 : n X

[ ai (x, s, ξ) − ai (x, s, ξ 0 )](ξi − ξi0 ) > 0.

(1.7.3)

i=1

Let f ∈ L1 (Ω). We now consider the following Dirichlet problem: −

n X ∂ ai (x, u, ∇u) = f ∂xi

in Ω,

(1.7.4)

i=1

u = 0 on ∂Ω. ◦

(1.7.5)

◦

Note that if u ∈ T 1,p (Ω), v ∈ W 1,p (Ω) ∩ L∞ (Ω), k > 0, and i ∈ {1, . . . , n}, then the function ai (x, u, δu)(δi u − δi v) is summable on the set {|u − v| < k}. This follows from Proposition 1.1.1 and inequality (1.7.1). Definition 1.7.1. The entropy solution of problem (1.7.4), (1.7.5) is de◦

fined as a function u ∈ T 1,p (Ω) such that, for any v ∈ C0∞ (Ω) and k > 0, the following inequality is true: X  Z Z n ai (x, u, δu)(δi u − δi v) dx 6 f Tk (u − v) dx. {|u−v| 1 and j ∈ N. Setting ϕ = Tk (uj ) in (1.7.6) and using (1.1.6) and (1.7.2), we obtain  p1  Z 2 p |∇uj | dx 6 kf kL1 (Ω) k 1+p1 . (1.7.7) c1 {|uj | 1 and k > 0. Assume that zm is a function on R such that zm (s) = z(s/m) for any s ∈ R. We have zm ∈ C 1 (R), 0 6 zm 6 1 on R,

zm = 1 on [−m, m],

zm = 0 on R \ (−2m, 2m),

0 and |zm | 6 c/m on R.

For any j ∈ N, we set wj = T2m (uj ) − T2m (u). Now let j ∈ N, j > 2m. Setting ϕj = zm (uj )zm (u)Tk (wj ), ◦

we find ϕj ∈ W 1,p (Ω) and, for any i ∈ {1, . . . , n}, 0 Di ϕj = zm (uj )zm (u)Di Tk (wj ) + zm (uj )zm (u)Tk (wj )Di uj 0 + zm (uj )zm (u)Tk (wj )Di T2m (u) a.e. on Ω. (1.7.10)

By virtue of (1.7.6), Z X n Ω

i=1

 Z ai (x, Tj (uj ), ∇uj )Di ϕj dx = Tj (f )ϕj dx. Ω

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

64

In view of this equality, equality (1.7.10), the properties of the function zm , and relations (1.7.1), (1.7.7), and (1.7.8), we obtain Z X n Ω

 ai (x, Tj (uj ), ∇uj )Di Tk (wj ) zm (uj )zm (u) dx 6 σ(m)k,

i=1

where σ(m) is a positive constant depending only on n, p, c1 , c, kf kL1 (Ω) , and m. Thus, by using Proposition 1.1.1, inequality (1.7.3), the properties of the function zm , and the definition of the function Φ, we conclude that, for any j ∈ N, j > 2m, Z Φ(x, uj , ∇uj , δu) dx {|uj | 0.

Section 1.7

65

On the existence of entropy solutions for the equations

Also, for any m ∈ N, let τm be a function on R such that, for any s ∈ R,   1 τ (m(|s| − k)) + k sign s. τm (s) = m ◦

We fix any m ∈ N, m > 1/k, and take j ∈ N. Thus, τm (uj − ϕ) ∈ W 1,p (Ω). Hence, by virtue of (1.7.6),  Z X Z n 0 Tj (f )τm (uj −ϕ) dx. ai (x, Tj (uj ), ∇uj )Di (uj −ϕ) τm (uj −ϕ) dx = Ω

Ω

i=1

By using this equality, relations (1.7.1), (1.7.2), (1.7.7)–(1.7.9), and the Fatou lemma, we obtain  Z X n 0 ai (x, u, δu)δi u τm (u − ϕ) dx Ω

i=1

6

Z X n Ω

 Z 0 ai (x, u, δu)Di ϕ τm (u − ϕ) dx + f τm (u − ϕ) dx. Ω

i=1

Thus, passing to the limit as m → ∞, we arrive at the inequality X  Z Z n ai (x, u, δu)(δi u − δi ϕ) dx 6 f Tk (u − ϕ) dx. {|u−ϕ| 0, there exist c¯k > 0 and g¯k ∈ L1 (Ω), g¯k > 0 on Ω, such that the following inequality is true for almost all x ∈ Ω and all s ∈ R, |s| 6 k, and ξ ∈ Rn : n X

|ai (x, s, ξ)|p/(p−1) 6 c¯k |ξ|p + g¯k (x).

(1.8.1)

i=1

Moreover, it is assumed that there exist p1 ∈ [ 0, p − 1), p2 ∈ [ 0, p − p1 ), c1 , c2 > 0, and g1 ∈ L1 (Ω), g1 > 0 on Ω, such that the following inequality holds for almost all x ∈ Ω and all s ∈ R and ξ ∈ Rn : n X

ai (x, s, ξ)ξi >

i=1

c1 |ξ|p − c2 p2 (1 + |s|)p2 − g1 (x). (1 + |s|)p1

(1.8.2)

Finally, we assume that the following inequality holds for almost all x ∈ Ω and all s ∈ R and ξ, ξ 0 ∈ Rn , ξ 6= ξ 0 : n X

[ ai (x, s, ξ) − ai (x, s, ξ 0 )](ξi − ξi0 ) > 0.

(1.8.3)

i=1

Let a0 be a Carathéodory function on Ω × R × Rn and let f ∈ L1 (Ω). We consider the following Dirichlet problem: −

n X ∂ ai (x, u, ∇u) + a0 (x, u, ∇u) = f ∂xi

in Ω,

(1.8.4)

i=1

u = 0 on ∂Ω.

(1.8.5)

We now define several types of solutions of this problem and describe the relationships between them. Definition 1.8.1. A weak solution of problem (1.8.4), (1.8.5) is defined as ◦

a function u ∈ W 1,1 (Ω) satisfying the conditions: 1) ai (x, u, ∇u) ∈ L1 (Ω) for any i ∈ {1, . . . , n, }; 2) a0 (x, u, ∇u) ∈ L1 (Ω);

Section 1.8

A priori properties of the entropy solutions of equations

67

3) for any function v ∈ C0∞ (Ω),  Z X Z n f v dx. ai (x, u, ∇u)Di v + a0 (x, u, ∇u)v dx = Ω

Ω

i=1

Definition 1.8.2. A T -solution of problem (1.8.4), (1.8.5) is defined as ◦

a function u ∈ T 1,p (Ω) satisfying the conditions: 1) ai (x, u, δu) ∈ L1 (Ω) for any i ∈ {1, . . . , n, }; 2) a0 (x, u, δu) ∈ L1 (Ω); 3) for any function v ∈ C0∞ (Ω),  Z X Z n ai (x, u, δu)Di v + a0 (x, u, δu)v dx = f v dx. Ω

Ω

i=1

Proposition 1.8.1. Let u be a T -solution of problem (1.8.4), (1.8.5) and let |δu| ∈ L1 (Ω). Then u is a weak solution of problem (1.8.4), (1.8.5). ◦

Proof. Since u ∈ T 1,p (Ω) and |δu| ∈ L1 (Ω), by virtue of Proposition 1.1.2, ◦

we conclude that u ∈ W 1,1 (Ω) and, for any i ∈ {1, . . . , n}, Di u = δi u a.e. on Ω. This fact and Definition 1.8.2 imply that conditions 1)–3) of Definition 1.8.1 are satisfied. Hence, u is a weak solution of problem (1.8.4), (1.8.5).  ◦

◦

Thus, we note that if u ∈ T 1,p (Ω), v ∈ W 1,p (Ω) ∩ L∞ (Ω), k > 0, and i ∈ {1, . . . , n}, then the function ai (x, u, δu)(δi u − δi v) is summable on the set {|u − v| < k}. This follows from Proposition 1.1.1 and (1.8.1). Definition 1.8.3. An entropy solution of problem (1.8.4), (1.8.5) is defined ◦

as a function u ∈ T 1,p (Ω) satisfying the conditions: 1) a0 (x, u, δu) ∈ L1 (Ω); 2) for any v ∈ C0∞ (Ω) and k > 0, X  Z n ai (x, u, δu)(δi u − δi v) dx {|u−v| 0.

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

68

◦

The proof of the lemma is based on the approximation of a function from

W 1,p (Ω) ∩ L∞ (Ω) by a sequence of smooth functions uniformly bounded in Ω, the use of the inequality from condition 2) of Definition 1.8.3 for these approximating functions, and the possibility to pass to the limit with the use of inequality (1.8.1) and Propositions 1.1.1 and 1.1.3. Proposition 1.8.2. Let u be an entropy solution of problem (1.8.4), (1.8.5). Then the following assertions are true: 1) for any λ, 0 < λ < n(p − 1 − p1 )/(n − p), the function |u|λ is summable on Ω; 2) for any λ, 0 < λ < n(p − 1 − p1 )/(n − 1 − p1 ), the function |δu|λ is summable on Ω. The proof of this proposition is based on the use of the inequalities 0 6 p2 < p − p1 , relations (1.8.2) and (1.1.13), Proposition 1.1.1, and Lemma 1.6.3. However, the proof of Proposition 1.8.2 is much more complicated than the proof of a similar result for the entropy solution of problem (1.7.4), (1.7.5) presented in Remark 1.7.2. This is explained by a higher complexity of the structure of condition (1.8.2) as compared with the structure of condition (1.7.2) (for more detail, see [73]). Proposition 1.8.3. Let u be an entropy solution of problem (1.8.4), (1.8.5). Assume that the function (1 + |u|)p2 is summable on Ω and that ai (x, u, δu) ∈ L1 (Ω) for any i ∈ {1, . . . , n, }. Then u is a T -solution of problem (1.8.4), (1.8.5). ◦

Proof. By virtue of the conditions of the proposition, we get u ∈ T 1,p (Ω), and conditions 1) and 2) of Definition 1.8.2 are satisfied. We now show that condition 3) of Definition 1.8.2 is satisfied. Let v ∈ C0∞ (Ω). We fix k > max |v| and set Em = { |u−Tm (u)+v| < k} for any m ∈ N. Ω

◦

Let m ∈ N. Since Tm (u) − v ∈ W 1,p (Ω) ∩ L∞ (Ω), by virtue of Lemma 1.8.1, we obtain  Z X n ai (x, u, δu)(δi u − Di Tm (u) + Di v) dx Em

i=1

Z [f − a0 (x, u, δu)] Tk (u − Tm (u) + v) dx.

6 Ω

By using Proposition 1.1.1 and inequality (1.8.2), we find  Z X n ai (x, u, δu)(δi u − Di Tm (u) + Di v) dx Em

i=1

(1.8.6)

Section 1.8

X n

Z = Em

Z > Em

69

A priori properties of the entropy solutions of equations

Em ∩{|u|>m}

i=1

X n

X n

 Z ai (x, u, δu)Di v dx +  Z ai (x, u, δu)Di v dx − c2 p2

 ai (x, u, δu)δi u dx

i=1 p2

Z

(1+|u|) dx −

g1 dx. {|u|>m}

{|u|>m}

i=1

Thus, in view of (1.8.6), we conclude that, for any m ∈ N,  Z X n ai (x, u, δu)Di v dx Em

i=1

Z 6

[f − a0 (x, u, δu)] Tk (u − Tm (u) + v) dx Z Z p2 + c2 p2 (1 + |u|) dx + g1 dx. Ω

{|u|>m}

It is clear that

∞ S

(1.8.7)

{|u|>m}

Em = Ω. Furthermore, for any m ∈ N, we find Em ⊂ Em+1 .

m=1

Therefore, meas (Ω \ Em ) → 0. Since the functions ai (x, u, δu) are summable on Ω by the condition of the proposition, we obtain   Z X Z X n n ai (x, u, δu)Di v dx → ai (x, u, δu)Di v dx. (1.8.8) Em

Ω

i=1

i=1

Further, since meas {|u| > m} → 0, in view of the summability of the functions (1 + |u|)p2 and g1 , we conclude that Z Z p2 (1 + |u|) dx → 0, g1 dx → 0. (1.8.9) {|u|>m}

{|u|>m}

Finally, we get Z Z [f −a0 (x, u, δu)] Tk (u−Tm (u)+v) dx → [f −a0 (x, u, δu)]v dx. Ω

(1.8.10)

Ω

This follows from the facts that Tk (u − Tm (u) + v) → v on Ω and that the functions f and a0 (x, u, δu) are summable on Ω. Relations (1.8.7)–(1.8.10) imply that, for any function v ∈ C0∞ (Ω),  Z X Z n ai (x, u, δu)Di v dx 6 [f − a0 (x, u, δu)]v dx. Ω

Ω

i=1

C0∞ (Ω),

Hence, for any function v ∈ we get  Z Z X n ai (x, u, δu)Di v dx = [f − a0 (x, u, δu)]v dx. Ω

i=1

Ω

Therefore, condition 3) in Definition 1.8.2 is satisfied and it is possible to conclude that u is a T -solution of problem (1.8.4), (1.8.5). 

70

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

Corollary 1.8.1. Let u be an entropy solution of problem (1.8.4), (1.8.5). Assume that the functions (1 + |u|)p2 and |δu| are summable on Ω and that ai (x, u, δu) ∈ L1 (Ω) for any i ∈ {1, . . . , n}. Then u is a weak solution of problem (1.8.4), (1.8.5). This result follows from Propositions 1.8.3 and 1.8.1. In connection with the next assertion, we make the following useful remark: Remark 1.8.1. If p1 < (p − 1)/(n − p + 1), then 1
0 on Ω. Assume that 1 )/(p−1), c¯ > 0, and g¯ ∈ the following inequality holds for almost all x ∈ Ω and all s ∈ R and ξ ∈ Rn : n X

|ai (x, s, ξ)|p/(p−1) 6 c¯ (|s|p¯ + |ξ|p ) + g(x). ¯

(1.8.13)

i=1

Let u be an entropy solution of problem (1.8.4), (1.8.5). Then the following assertions are true: (i) ai (x, u, δu) ∈ Lλ (Ω) for any number λ satisfying the inequality   ∗ n(p − 1 − p1 ) p (p − 1 − p1 ) , (1.8.14) 1 6 λ < min p(p ¯ − 1) (n − 1 − p1 )(p − 1) and any i ∈ {1, . . . , n, }; (ii) u is a T -solution of problem (1.8.4), (1.8.5). Proof. First, we note that inequality (1.8.11) holds by virtue of the first inequality in (1.8.12) and Remark 1.8.1. This inequality and the condition of the proposition imposed on p¯ imply that the set of numbers λ satisfying inequality (1.8.14) is nonempty. Let the number λ satisfy inequality (1.8.14) and let i ∈ {1, . . . , n}. By virtue of inequality (1.8.13) and the inequality λ(p − 1) < p, we obtain  λ(p−1)p/p  ¯ |ai (x, u, δu)|λ 6 (c+1) ¯ |u| +|δu|λ(p−1) + g¯ + 1 a.e. on Ω. (1.8.15) Inequality (1.8.14) yields λ(p−1)p/p ¯ < n(p−1−p1 )/(n−p),

λ(p−1) < n(p−1−p1 )/(n−1−p1 ).

¯ Thus, by virtue of Proposition 1.8.2, the functions |u|λ(p−1)p/p and |δu|λ(p−1) are summable on Ω. Hence, in view of (1.8.15), we get ai (x, u, δu) ∈ Lλ (Ω).

Section 1.8

A priori properties of the entropy solutions of equations

71

Thus, the validity of assertion (i) is established. This assertion implies that, for any i ∈ {1, . . . , n}, we have ai (x, u, δu) ∈ L1 (Ω). In addition, by virtue of the second inequality in (1.8.12) and Proposition 1.8.2, the function (1 + |u|)p2 is summable on Ω. Thus, all conditions of Proposition 1.8.3 are satisfied and, hence, assertion (ii) is true.  Corollary 1.8.2. Let p > 2 − 1/n,   n  1 p−1 p1 < min p−2+ , , n−1 n n−p+1

(1.8.16)

p2 < n(p−1−p1 )/(n−p), 0 < p¯ < p∗ (p−1−p1 )/(p−1), c¯ > 0, and g¯ ∈ L1 (Ω), g¯ > 0 on Ω. Assume that the following inequality is true for almost all x ∈ Ω and all s ∈ R and ξ ∈ Rn : n X

|ai (x, s, ξ)|p/(p−1) 6 c(|s| ¯ p¯ + |ξ|p ) + g(x). ¯

i=1

Let u be an entropy solution of problem (1.8.4), (1.8.5). Then u is a weak solution of problem (1.8.4), (1.8.5). Proof. Since all conditions of Proposition 1.8.4 are satisfied, this proposition implies that u is a T -solution of problem (1.8.4), (1.8.5). In view of (1.8.16), we get p1 < n(p − 2 + 1/n)/(n − 1). Hence, 1 < n(p − 1 − p1 )/(n − 1 − p1 ). This fact and assertion 2) of Proposition 1.8.2 imply that |δu| ∈ L1 (Ω). Thus, it follows from Proposition 1.8.1 that u is a weak solution of problem (1.8.4), (1.8.5).  In what follows, we focus on the properties of summability of the entropy solutions of problem (1.8.4), (1.8.5). Some of these properties have been already described in Proposition 1.8.2. The improvement of the properties of summability of the entropy solutions established in assertions 1) and 2) of this proposition under an additional condition imposed on p2 is one of the problems discussed at the end of the present section First, we present one auxiliary result. Lemma 1.8.2. Let h ∈ C 1 (R) and let h(0) = 0. Suppose that ◦

u ∈ W 1,p (Ω) ∩ L∞ (Ω). ◦

Then h(u) ∈ W 1,p (Ω) ∩ L∞ (Ω) and, for any i ∈ {1, . . . , n, }, Di h(u) = h0 (u)Di u a.e. on Ω. Proof. By the continuity of the function h on R and the inclusion u ∈ L∞ (Ω), we get h(u) ∈ L∞ (Ω).

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

72

We set m = kukL∞ (Ω) + 1. Let {uj } be a sequence of functions from C0∞ (Ω) such that uj → u strongly in W 1,p (Ω), (1.8.17) uj → u a.e. on Ω, ∀ j ∈ N,

(1.8.18)

|uj | 6 m on Ω.

(1.8.19)

Since h ∈ C 1 (R) and h(0) = 0, we have {h(uj )} ⊂ C01 (Ω).

(1.8.20)

Properties (1.8.18) and (1.8.19) and the inclusion h(u) ∈ L∞ (Ω) imply that h(uj ) → h(u) strongly in Lp (Ω).

(1.8.21)

We fix i ∈ {1, . . . , n}. Since u ∈ L∞ (Ω), Di u ∈ Lp (Ω), and the function h0 is continuous on R, we conclude that h0 (u)Di u ∈ Lp (Ω). Taking into account (1.8.19), for any j ∈ N, we obtain Z Z 0 p p |Di h(uj ) − h (u)Di u| dx 6 2 |h0 (uj ) − h0 (u)|p |Di u|p dx Ω

Ω


p + 2 max |h0 |

Z

[−m,m]

|Di uj − Di u|p dx.

Ω

Hence, in view of properties (1.8.17)–(1.8.19), we get Di h(uj ) → h0 (u)Di u strongly in Lp (Ω).

(1.8.22)

Relations (1.8.20)–(1.8.22) imply that there exists the weak derivative Di h(u) and Di h(u) = h0 (u)Di u a.e. on Ω. We can now conclude that h(u) ∈ W 1,p (Ω). In addition, by virtue of (1.8.21) and (1.8.22), we get h(uj ) → h(u) strongly in W 1,p (Ω). This fact and (1.8.20) ◦

imply that h(u) ∈ W 1,p (Ω).



Proposition 1.8.5. Let p2 = 0 and g1 = 0 on Ω. Let u be an entropy solution of problem (1.8.4), (1.8.5), let h ∈ C 1 (R), h(0) = 0, let h be bounded on R, and let h0 > 0 on R. Then the function |δu|p h0 (u)/(1+|u|)p1 is summable on Ω and the following inequality holds: Z Z 1 |δu|p 0 h (u) dx 6 (1.8.23) [f − a0 (x, u, δu)] h(u) dx. p1 c1 Ω Ω (1 + |u|) Proof. We fix k > sup |h|. For any m ∈ N, we set R

vm = Tm (u) − h(Tm (u)),

(1.8.24)

Em = { |u − Tm (u) + h(Tm (u))| < k }.

(1.8.25)

Section 1.8

73

A priori properties of the entropy solutions of equations ◦

◦

Let m ∈ N. Since u ∈ T 1,p (Ω), we get Tm (u) ∈ W 1,p (Ω) ∩ L∞ (Ω). Thus, ◦

by virtue of Lemma 1.8.2, h(Tm (u)) ∈ W 1,p (Ω) ∩ L∞ (Ω) and, for any i ∈ {1, . . . , n}, we conclude that Di h(Tm (u)) = h0 (Tm (u))Di Tm (u) a.e. on Ω.

(1.8.26)

◦

It is clear that vm ∈ W 1,p (Ω) ∩ L∞ (Ω). Thus, by Lemma 1.8.1,  Z X Z n ai (x, u, δu)(δi u − δi vm ) dx 6 [f − a0 (x, u, δu)] Tk (u − vm ) dx. Em

Ω

i=1

(1.8.27)

By using Proposition 1.1.1, relations (1.8.2) and (1.8.26), the inclusion {|u| < m} ⊂ Em , and the fact that the function h0 is nonnegative on R, we obtain  Z X n ai (x, u, δu)(δi u − δi vm ) dx Em

i=1

X n

Z = Em

 ai (x, u, δu)(δi u − Di Tm (u)) dx

i=1

X n

Z + Em

i=1

X n

Z > Em

= Em

 ai (x, u, δu)Di h(Tm (u)) dx

i=1

X n

Z

 ai (x, u, δu)Di h(Tm (u)) dx

 ai (x, u, δu)Di Tm (u) h0 (Tm (u)) dx

i=1

X n

Z = {|u| 0 on R, and let Z +∞ (1 + |t|)p1 h(t)dt < +∞. −∞

Assume that h1 is a function on R such that, for any s ∈ R, Z s h1 (s) = (1 + |t|)p1 h(t)dt. 0

Then h1 ∈ C 1 (R), h1 (0) = 0, h1 is bounded on R and, for any s ∈ R, we have h01 (s) = (1 + |s|)p1 h(s). This fact and Proposition 1.8.5 imply that the function |δu|p h(u) is summable on Ω. Hence, assertion (i) is true. Now let h ∈ C 1 (R), h(0) = 0, and Z +∞ (1 + |t|)p1 |h0 (t)|p dt < +∞. −∞ ◦

We fix any k ∈ N. By virtue of Lemma 1.8.2, we get h(Tk (u)) ∈ W 1,p (Ω) and k∇h(Tk (u))| = |h0 (Tk (u))| |∇Tk (u)| a.e. on Ω. Thus, by using (1.1.13) and Proposition 1.1.1, we obtain Z

p∗

|h(Tk (u))| dx 6 Ω

∗ cpn,p

Z

0

p

p

|h (u)| |δu| dx

p∗ /p .

(1.8.29)

{|u| 0 on R. Then the function |δu|p h0 (u)/(1 + |u|)p1 is summable on Ω and the following inequality holds: Z Z |δu|p 1 0 [f − a0 (x, u, δu)]h(u) dx h (u) dx 6 p1 c1 Ω Ω (1 + |u|) Z 1 (1.8.30) + [ c2 p2 (1 + |u|)p2 + g1 ]h0 (u) dx. c1 Ω Proof. We fix k > supR |h| and, for any m ∈ N, define the function vm and the set Em by relations (1.8.24) and (1.8.25). Let m ∈ N. By analogy with the proof of Proposition 1.8.5, we conclude ◦

◦

that h(Tm (u)) ∈ W 1,p (Ω), vm ∈ W 1,p (Ω) ∩ L∞ (Ω), and relations (1.8.26) and (1.8.27) are true. It follows from the condition of the proposition imposed on the number p2 and Proposition 1.8.2 that the function (1 + |u|)p2 is summable on Ω. By using this fact, Proposition 1.1.1, relations (1.8.2) and (1.8.26), the inclusion {|u| < m} ⊂ Em , and the fact that the function h0 is nonnegative and bounded on R, we get  Z X n ai (x, u, δu)(δi u−δi vm ) dx Em

i=1

X n

Z = {|u|m}

Z > c1 {|u| 0 on R, and Z +∞ (1 + |t|)p1 h(t)dt < +∞, −∞

sup (1 + |t|)p1 h(t) < +∞,

(1.8.32)

t∈R

then the function |δu|p h(u) is summable on Ω; (ii) if h ∈ C 1 (R), h(0) = 0, and Z +∞ (1+|t|)p1 |h0 (t)|p dt < +∞, sup (1+|t|)p1 |h0 (t)|p < +∞, (1.8.33) −∞

t∈R p∗

then h(u) ∈ L (Ω). Proof. Let h ∈ C(R), h > 0 on R, and let inequalities (1.8.32) be true. We now define the function h1 in the same way as in the proof of Corollary 1.8.3. Thus, we have h1 ∈ C 1 (R), h1 (0) = 0, the functions h1 and h01 are bounded on R, and h01 > 0 on R. Hence, by virtue of Proposition 1.8.6, the function |δu|p h01 (u)/(1 + |u|)p1 is summable on Ω. Thus, in view of the fact that h01 (s) = (1 + |s|)p1 h(s) for any s ∈ R, the function |δu|p h(u) is summable on Ω. Therefore, assertion (i) is true. Now let h ∈ C 1 (R), h(0) = 0, and let inequalities (1.8.33) be true. Then, for any k ∈ N, inequality (1.8.29) is true and the function |h0 (u)|p |δu|p is summable on Ω in view of assertion (i). Thus, by virtue of the Fatou lemma, ∗ we conclude that h(u) ∈ Lp (Ω). Hence, assertion (ii) is true.  Corollary 1.8.5. Let p2 < n(p−1−p1 )/(n−p), let u be an entropy solution of problem (1.8.4), (1.8.5), and let β > 1. Then the function (1 +

|u|)p1 +1 [ ln(2

|δu|p + |u|)] [ ln ln(3 + |u|)]β

is summable on Ω. Proof. Assume that h is a function on R such that, for any s ∈ R, h(s) =

(1 +

|s|)p1 +1 [ ln(2

1 . + |s|)] [ ln ln(3 + |s|)]β

Hence, h ∈ C(R), h > 0 on R, and inequalities (1.8.32) hold. Thus, by virtue of the assertion (i) of Corollary 1.8.4, the function |δu|p h(u) is summable on Ω. Therefore, the required assertion is true. 

Section 1.8

77

A priori properties of the entropy solutions of equations

Corollary 1.8.6. Let p2 < n(p − 1 − p1 )/(n − p), let u be an entropy solution of problem (1.8.4), (1.8.5). and let h ∈ C(R). Assume that the following conditions are satisfied: The function h is even, h > 0 on R, the function h is nonincreasing on [ 0, +∞), and Z +∞ 1 [ h(t)](n−p)/n dt < +∞. (1.8.34) t 1 Then the functions |u|n(p−1−p1 )/(n−p) h(u),

|δu|n(p−1−p1 )/(n−1−p1 ) [ h(u)](n−p)/(n−1−p1 )

are summable on Ω. Proof. Let h1 be a function on R such that, for any s ∈ R, Z s ∗ [ h(t)]1/p h1 (s) = dt. (p1 +1)/p 0 (1 + |t|) We have h1 ∈ C 1 (R),

h1 (0) = 0,

(1 + |t|)p1 |h01 (t)|p =

∀ t ∈ R,

[ h(t)](n−p)/n . 1 + |t|

Relations (1.8.34) and (1.8.36) yield Z +∞ (1 + |t|)p1 |h01 (t)|p dt < +∞.

(1.8.35) (1.8.36)

(1.8.37)

−∞

Since the function h is even and nonincreasing on [ 0, +∞), we conclude that ∀ t ∈ R,

h(t) 6 h(0).

(1.8.38)

Hence, it follows from (1.8.36) that sup (1 + |t|)p1 |h01 (t)|p < +∞.

(1.8.39)

t∈R

In view of (1.8.35), (1.8.37), and (1.8.39), assertion (ii) of Corollary 1.8.4 implies that ∗ h1 (u) ∈ Lp (Ω). (1.8.40) Let s ∈ R. In view of the fact that the function h is even and nonincreasing on [ 0, +∞), we obtain ∗

|h1 (s)| >

[ h(s)]1/p |s| . (1 + |s|)(p1 +1)/p

(1.8.41)

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

78

If |s| > 1, then, by using (1.8.41), we find ∗

|s|1−(p1 +1)/p [ h(s)]1/p 6 2|h1 (s)|.

(1.8.42)

At the same time, if |s| < 1, then, in view of (1.8.38), we get ∗

∗

|s|1−(p1 +1)/p [ h(s)]1/p 6 [ h(0)]1/p .

(1.8.43)

Relations (1.8.42) and (1.8.43) imply that, for any s ∈ R, ∗

∗

|s|1−(p1 +1)/p [ h(s)]1/p 6 2|h1 (s)| + [ h(0)]1/p . Then ∗

∗

∗

|u|n(p−1−p1 )/(n−p) h(u) 6 4p |h1 (u)|p + 2p h(0) on Ω. Hence, in view of (1.8.40), we conclude that the function |u|n(p−1−p1 )/(n−p) h(u) is summable on Ω. We now set n−p n(p − 1 − p1 ) , λ= . λ1 = n − 1 − p1 n − 1 − p1 Note that λ1 < p and λ1 (1 + p1 ) n(p − 1 − p1 ) = , p − λ1 n−p

(λ − 1)

n−p p +1= . λ1 n

(1.8.44)

Let s ∈ R and ξ ∈ Rn . By virtue of the Young inequality with the exponents p/λ1 and p/(p − λ1 ) and relations (1.8.44), we obtain |ξ|λ1 [ h(s)]λ = 6

|ξ|λ1 [ h(s)]λ−1+λ1 /p × (1 + |s|)(p1 +1)λ1 /p [ h(s)](p−λ1 )/p (1 + |s|)(p1 +1)λ1 /p |ξ|p [ h(s)](n−p)/n + (1 + |s|)n(p−1−p1 )/(n−p) h(s). (1 + |s|)p1 +1

Thus, |δu|n(p−1−p1 )/(n−1−p1 ) [ h(u)](n−p)/(n−1−p1 ) 6

|δu|p [ h(u)](n−p)/n + (1 + |u|)n(p−1−p1 )/(n−p) h(u) on Ω. (1 + |u|)p1 +1

(1.8.45)

According to the properties of the function h and assertion (i) of Corollary 1.8.4, the function |δu|p [ h(u)](n−p)/n /(1 + |u|)p1 +1 is summable on Ω. This fact, the property of summability of the function |u|n(p−1−p1 )/(n−p) h(u) established above, and properties (1.8.38) and (1.8.45) imply that the function |δu|n(p−1−p1 )/(n−1−p1 ) [ h(u)](n−p)/(n−1−p1 ) is summable on Ω. 

Section 1.8

79

A priori properties of the entropy solutions of equations

Corollary 1.8.7. Let p2 < n(p−1−p1 )/(n−p), let u be an entropy solution of problem (1.8.4), (1.8.5), and let β > n/(n − p). Then the functions |u|n(p−1−p1 )/(n−p) , [ ln(2 + |u|)]n/(n−p) [ ln ln(3 + |u|)]β |δu|n(p−1−p1 )/(n−1−p1 ) [ ln(2 + |u|)]n/(n−1−p1 ) [ ln ln(3 + |u|)]β(n−p)/(n−1−p1 ) are summable on Ω. Proof. Let h be a function on R such that 1 h(s) = [ ln(2 + |s|)]n/(n−p) [ ln ln(3 + |s|)]β for any s ∈ R. The function h satisfies the conditions of Corollary 1.8.6, which means that the required assertion is true.  Corollary 1.8.7 yields the following proposition: Corollary 1.8.8. Let p2 < n(p−1−p1 )/(n−p), let u be an entropy solution of problem (1.8.4), (1.8.5), and let β > 1/(p − 1 − p1 ). Then |u| [ ln(2 +

|u|)]1/(p−1−p1 ) [ ln ln(3

[ ln(2 +

|u|)]1/(p−1−p1 ) [ ln ln(3

+ |u|)]β

|δu| + |u|)]β

∈ Ln(p−1−p1 )/(n−p) (Ω), ∈ Ln(p−1−p1 )/(n−1−p1 ) (Ω).

It is clear that, for p2 < n(p − 1 − p1 )/(n − p), Corollary 1.8.8 gives more exact information about the properties of summability of the entropy solution of problem (1.8.4), (1.8.5) as compared with Proposition 1.8.2. Moreover, if we trace the chain of results leading to Corollary 1.8.8, then we can see that this chain includes Proposition 1.8.2. We now formulate one of the theorems on existence of the entropy solution of problem (1.8.4), (1.8.5). Theorem 1.8.1. Let p2 = 0 and g1 = 0 on Ω . Also let c > 0 , 0 < σ < p − 1 − p1 , g ∈ L1 (Ω) , g > 0 on Ω , ϕ ∈ C(R) , ϕ > 0 on R, and Z +∞ (1 + |t|)p1 ϕ(t)dt < +∞. −∞

Assume that the following inequality holds for almost all x ∈ Ω and all s ∈ R and ξ ∈ Rn : |a0 (x, s, ξ)| 6 c(|s|σ + |ξ|σ ) + |ξ|p ϕ(s) + g(x). Then the entropy solution of problem (1.8.4), (1.8.5) exists.

(1.8.46)

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

80

The same result was formulated in [73] under the additional condition p1 < (p − 1)/(n − p + 1). It turns out that this condition is unnecessary. The proof of Theorem 1.8.1 is similar to the proof of Theorem 1.7.1 but is more complicated due to the presence of the lower-order coefficient a0 in Eq. (1.8.4). Omitting some technical details, we restrict ourselves to the description of the main stages of the proof of Theorem 1.8.1. They can be described as follows: First, we consider the sequence of generalized solutions ◦

uj ∈ W 1,p (Ω) ∩ L∞ (Ω) of the approximating problems −

n X ∂ (j) (j) A (x, u, ∇u) + A0 (x, u, ∇u) = fj ∂xi i

in Ω,

(1.8.47)

i=1

u = 0 on ∂Ω, (j)

where fj = Tj (f ) and Ai

(j)

and A0

(j)

(1.8.48)

are functions on Ω × R × Rn such that (j)

Ai (x, s, ξ) = ai (x, Tj (s), ξ) and A0 (x, s, ξ) = Tj (a0 (x, s, ξ)) for any triple (x, s, ξ) ∈ Ω × R × Rn . If we substitute the function h(uj ) with any bounded function h ∈ C 1 (R) satisfying the conditions h(0) = 0 and h0 > 0 on R in the integral identity corresponding to the approximating problem with the right-hand side fj , then we obtain [in view of Lemma 1.8.2 and inequalities (1.8.2) and (1.8.46)] an integral inequality that can be used to deduce a series of uniform integral estimates for the functions uj . In particular, this enables us to prove the existence of an increasing sequence ◦

{jl } ⊂ N and a function u ∈ T 1,p (Ω) such that ujl → u a.e. on Ω, ∀ k > 0,

(1.8.49) ◦

Tk (ujl ) → Tk (u) weakly in W 1,p (Ω).

(1.8.50)

Thus, by using the integral identities corresponding to the approximating problems (1.8.47) and (1.8.48), inequalities (1.8.1) – (1.8.3), and properties (1.8.49) and (1.8.50), we conclude that ∀ i ∈ {1, . . . , n}, ∀ k > 0,

Di ujl → δi u in measure,

Tk (ujl ) → Tk (u) strongly in W

1,p

(Ω).

(1.8.51) (1.8.52)

The above-mentioned uniform integral estimates for the functions uj (one of them is a uniform estimate for the integrals of the functions |∇uj |p ϕ(uj ) over Ω), inequalitiy (1.8.46), and properties (1.8.49) and (1.8.51) yield the inclusion a0 (x, u, δu) ∈ L1 (Ω).

Section 1.8

A priori properties of the entropy solutions of equations

81

By using the same uniform integral estimates for the functions uj and properties (1.8.49), (1.8.51), and (1.8.52), we pass to the limit in the integral identities corresponding to the approximating problems (1.8.47) and (1.8.48). This enables us to conclude that u is an entropy solution of problem (1.8.4), (1.8.5). Note that a result similar to Theorem 1.8.1 remains valid for p2 6= 0. In this case, it is necessary to additionally require that the function ϕ must satisfy the condition sup (1 + |t|)p1 ϕ(t) < +∞. t∈R

In conclusion, we present some examples in which inequalities (1.8.1)–(1.8.3) and (1.8.46) are satisfied. Example 1.8.1. Let p > 2, p1 ∈ [ 0, p − 1), γ ∈ (0, p − 1 − p1 ), p2 = (γp + p1 )/(p − 1), β ∈ R, let ψ be a nonnegative continuous function on R, and let g0 ∈ Lp/(p−2) (Ω), g0 > 0 on Ω. Assume that, for any i ∈ {1, . . . , n, }, ai is a function on Ω × R × Rn such that ai (x, s, ξ) =

|ξ|p−2 ξi + β|s|γ + g0 (x)ψ(s)ξi (1 + |s|)p1

for any triple (x, s, ξ) ∈ Ω × R × Rn . Then inequalities (1.8.1)–(1.8.3) hold and, in addition, c¯k = 6p/(p−1) n, g¯k = (3|β|k γ )p/(p−1) n + 3p/(p−1) n

h

ip/(p−2) p/(p−2) max ψ(t) g0 ,

t∈[−k,k]

c1 = (p − 1)/p, c2 > 0, c2 depends only on n, p, p1 , γ, and β, and g1 = 0 on Ω. Example 1.8.2. Let β > 1, β1 , β2 ∈ R, 0 < σ < p − 1 − p1 , let g ∈ L1 (Ω), g > 0 on Ω, and let ϕ be a function on R such that, for any s ∈ R, ϕ(s) =

1 (1 +

|s|)p1 +1 [ ln(2

+ |s|)]β

.

Assume that a0 is a function on Ω × R × Rn such that a0 (x, s, ξ) = β1 |s|σ + β2 |ξ|σ + |ξ|p ϕ(s) + g(x) for any triple (x, s, ξ) ∈ Ω×R×Rn . Then the function ϕ satisfies the conditions of Theorem 1.8.1 and the function a0 satisfies inequality (1.8.46). Finally, we make some remarks concerning the results presented in this section. Propositions 1.8.3 and 1.8.4 and Corollary 1.8.2 generalize the results obtained in [13] to the case where p1 = 0, p2 = 0, and g1 ≡ 0. Even in this case,

82

Chapter 1 Nonlinear elliptic equations of the second order with L1 -data

Corollary 1.8.8 gives stronger results than the results obtained in [25] for weak solutions. Estimates (1.8.23) and (1.8.30) play an important role in the study of the properties of summability of the entropy solutions under restrictions imposed on a function a0 of the form (1.8.46) and in the improvement of the properties of summability of the function f. In fact, this is a fairly general problem that requires separate investigations. In the case where the leading coefficients of Eq. (1.8.4) are independent of u, have the order of growth p − 1 in ∇u, and satisfy the standard coercivity condition (p1 = 0, p2 = 0, and g1 ≡ 0) and the lower-order coefficient a0 is independent of ∇u, has an arbitrary order of growth in u, and is a nondecreasing function of u, the existence of an entropy solution of problem (1.8.4), (1.8.5) was proved in [13]. In the case where the leading coefficients of Eq. (1.8.4) may have the order of growth in u not greater than p − 1, have the order of growth in ∇u equal to p − 1, and satisfy the standard condition of coercivity (p1 = 0, p2 = 0, and g1 ≡ 0), the lower-order coefficient a0 satisfies inequality (1.8.46) with c = 0 and ϕ ∈ L1 (R), and the right-hand side of the equation is a bounded Radon measure on Ω, the existence of a T -solution of the corresponding Dirichlet problem was established in [105]. The cited work contains useful procedures of construction of the integral estimates of solutions of the approximating problems close, in a certain sense, to our methods, as well as the reasoning used to prove strong convergence of the truncation of these solutions in W 1,p (Ω), which was applied in the proof of Theorem 1.8.1. In the case where the leading coefficients of Eq. (1.8.4) satisfy the same conditions as in [105], the behavior of the lower-order coefficient as a function of ∇u is also similar to its behavior in [105], and the right-hand side of the equation belongs to L1 (Ω), the existence of an entropy solution of the corresponding Dirichlet problem was proved in [111].

Chapter 2

Nonlinear equations of the fourth order with strengthened coercivity and L1-data 2.1

Introduction

As mentioned in Chapter 1, an efficient approach to the solution of nonlinear second-order elliptic equations with L1 -right-hand sides was proposed in [13]. At the same time, for higher-order elliptic equations with L1 -data, an approach similar to the approach proposed in [13] can be realized under the condition of strengthened coercivity for the coefficients of equations. For the first time, this was demonstrated for nonlinear equations of the fourth order in [63, 64]. Later, the ideas advanced in the cited works were developed in [66, 74–77, 80] in the investigation of the solvability of nonlinear elliptic equations of the fourth and arbitrary even orders with strengthened coercivity and L1 -data. Moreover, the equations degenerate in the space variables and equations with anisotropy were also studied. In the second chapter, we present the results obtained in [63, 64, 67, 74] concerning the existence and properties of solutions of the Dirichlet problem for nonlinear elliptic equations of the fourth order with strengthened coercivity and L1 -right-hand sides. We now formulate basic assumptions used in the present chapter and introduce the required notation. Let n ∈ N, n > 2, and let Ω be a bounded open set in Rn . By Λ we denote the set of all n-dimensional multiindices α such that |α| = 1 or |α| = 2. By Rn,2 we denote the space of all mappings ξ : Λ → R. Accordingly, if u ∈ W 2,1 (Ω), then ∇2 u : Ω → Rn,2 and (∇2 u(x))α = Dα u(x) for any x ∈ Ω and α ∈ Λ. Let p, q ∈ R and 1 < p < n/2,

(2.1.1)

2p < q < n.

(2.1.2)

Let c1 , c2 > 0, let g1 and g2 be nonnegative functions on Ω, g1 , g2 ∈ L1 (Ω), and let, for any α ∈ Λ, Aα : Ω×Rn,2 → R be a Carathéodory function. Assume that the following inequalities are true for almost all x ∈ Ω and all ξ ∈ Rn,2 : X X |Aα (x, ξ)|q/(q−1) + |Aα (x, ξ)|p/(p−1) |α|=1

|α|=2

84

Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity

6 c1

X

|ξα |q +

|α|=1

X

Aα (x, ξ)ξα > c2

α∈Λ

X

X

|ξα |p

 + g1 (x),

(2.1.3)

|α|=2 q

|ξα | +

|α|=1

X

p

|ξα |

 − g2 (x).

(2.1.4)

|α|=2

Moreover, we assume that the following inequality holds for almost all x ∈ Ω and all ξ, ξ 0 ∈ Rn,2 , ξ 6= ξ 0 : X [Aα (x, ξ) − Aα (x, ξ 0 )](ξα − ξα0 ) > 0. (2.1.5) α∈Λ

Also let F : Ω × R → R be a Carathéodory function. Consider the following Dirichlet problem: X (−1)|α| Dα Aα (x, ∇2 u) = F (x, u) in Ω,

(2.1.6)

α∈Λ

Dα u = 0,

|α| = 0, 1,

on ∂Ω.

(2.1.7)

The exact definitions of solutions of this problem are given in what follows. In the theorems on existence and uniqueness of solutions, it is necessary to impose certain additional conditions on the function F . They admit any order of growth of the function F (x, u) in the second variable variable and, for a fixed value of this variable, require that the corresponding function must belong to the class L1 (Ω). Note that the realization of the ideas used in the approach proposed for the second-order equations in [13] has numerous specific features in the case of higher-order equations with L1 -data. First, the standard cutoff functions Tk cannot be used in this case, as in [13], in order to obtain the required estimates of solutions of the approximating problems with regular data. Second, the application of the other (smooth) functions instead of the standard cutoff functions in the corresponding integral identities is connected with the necessity of quite delicate handling of the terms containing high-order derivatives of suitable test functions. Actually, these circumstances determine the structure of high-order equations with L1 -data and the energy space for approximating problems required to guarantee the possibility of realization of the ideas of the proposed approach. Thus, just the condition of strengthened coercivity (2.1.4) ◦

and the corresponding Sobolev space W 1,q 2,p (Ω) (see Sec. 2.2) prove to be suitable for problem (2.1.6), (2.1.7) if the function F satisfies the above-mentioned additional conditions. Note that the high-order equations with conditions of the form (2.1.3) and (2.1.4) imposed on the coefficients but with sufficiently regular data were introduced in [123]. Moreover, the regularity of the data assumed in the cited work

Section 2.1

85

Introduction

admits, unlike the case of L1 -data, the investigation of the solvability of equations within the framework of the ordinary theory of monotonic operators [89]. The structure of the present chapter is as follows. In Sec. 2.2, we introduce a ◦

space W 1,q 2,p (Ω) playing (in what follows) the role of energy space for the problems that approximate problem (2.1.6), (2.1.7). Then, with the help of a spe◦

cial sequence of functions hk ∈ C 2 (R), we introduce a set of functions H 1,q 2,p (Ω) ◦

which is broader than the space W 1,q 2,p (Ω) and contains even nonsummable func◦

α tions. For the functions u ∈ H 1,q 2,p (Ω), we define their “derivatives” δ u of the first and second orders that coincide with the ordinary weak derivatives for ◦

u ∈ W 1,q 2,p (Ω). In Sec. 2.3, we introduce the notion of entropy solution of problem (2.1.6), (2.1.7) and establish some important properties of solutions of this kind. By def◦

inition, the entropy solution is an element of the set H 1,q 2,p (Ω). Note that the proposed definition of the entropy solution is connected, just as the definition ◦

of the set H 1,q 2,p (Ω), with the use of the sequence of functions hk . This sequence plays a role similar to the role played by the standard cutoff functions Tk in ◦

the case of the second-order equations with L1 -data and the set H 1,q 2,p (Ω) is ◦

an analog of the set T 1,q (Ω) traditionally used in the indicated case. We also note that the specific feature of the definition of entropy solutions for the analyzed problem, by analogy with the corresponding definition given in [13] for the case of second-order equations, is connected with the requirement of validity, for the function regarded as an entropy solution, of a certain family of integral inequalities connected with Eq. (2.1.6) and depending on arbitrary ϕ ∈ C0∞ (Ω) and k ∈ N (see Definition 2.3.1). This family serves as a basis for the investigation of the properties of entropy solutions. Moreover, we note that one of distinctions of the proposed definition of the entropy solution from the corresponding definition in [13] is connected with the presence of an additional term on the right-hand side of the required integral inequalities. This term depends on the norm of the function ϕ ∈ C0∞ (Ω) in a certain Sobolev space and on the power of the number k ∈ N with negative exponent. Briefly speaking, the presence of this additional term is connected with the problems of existence and uniqueness of solutions and reflects essential specific features of the high-order equations as compared with the second-order equations. Section 2.4 is devoted to the construction of an a priori integral estimate for the “derivatives” of the entropy solution. This estimate plays an important role in the proof of uniqueness of this solution. In Sec. 2.5, we introduce the notion of H-solution of problem (2.1.6), (2.1.7) and show that the entropy solution of this problem is also an H-solution. The theorem on uniqueness

86

Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity

of the entropy solution is proved in Sec. 2.6. The theorems on existence of the H-solution and entropy solution are proved in Sec. 2.7. The problems of belonging of the entropy solutions to Sobolev spaces and the existence of the so-called W -solutions of problem (2.1.6), (2.1.7) are studied in Sec. 2.8. Note that the H- and W -solutions are, in fact, solutions in a sense of the integral identity corresponding to problem (2.1.6), (2.1.7). In Sec. 2.9, we establish some properties of integrability for functions from ◦

the set H 1,q 2,p (Ω) satisfying special integral inequalities. In this case, we use methods similar to the methods applied in Sections 1.2 and 1.5. In Sec. 2.10, we show that the entropy solutions of problem (2.1.6), (2.1.7) satisfy the integral inequalities from the previous section. This fact, together with suitable conditions imposed on the right-hand side of Eq. (2.1.6) and the results from Sec. 2.9, enable us to get stronger properties of summability of the entropy solutions of problem (2.1.6), (2.1.7), as compared with the properties described in Sections 2.3 and 2.8. In the subsequent sections of the chapter, we present the results connected with the investigation of a certain modification of the notion of entropy solution of the analyzed problem. This modification is called a proper entropy solution (see Definition 2.13.1). The presented results demonstrate that the definition of proper entropy solution is more natural than the definition of entropy solution. In addition, the form of integral inequalities in the definition of proper entropy solution agrees with one of the equivalent formulations of the entropy condition in the case of the second-order equations with L1 -data (as for this formulation, see Lemma 3.2 in [13]). ◦

In Sec. 2.11, we consider some useful characteristics of the set H 1,q 2,p (Ω). In particular, it is shown that this set is indeed independent of the sequence {hk }. Actually, it is defined by a sufficiently broad class H ⊂ C 2 (R) containing this sequence. ◦

In Sec. 2.12, we introduce a subset of functions from H 1,q 2,p (Ω) whose “derivatives” of the first and second orders satisfy certain uniform integral estimates. b 1,q (Ω) and plays an important role in our subseThis subset is denoted by H 2,p quent presentation. In this section, we also prove some estimates for the meab 1,q (Ω) and describe the properties sures of sets connected with functions from H 2,p of summability of these functions and their “derivatives”. In Sec. 2.13, we present the definition of proper entropy solution for problem (2.1.6), (2.1.7) and establish two a priori estimates for solutions of this kind. b 1,q (Ω). Note that the proper entropy solution is defined as a function from H 2,p Unlike the definition of entropy solution, the main condition for a function required to be a proper entropy solution is connected with the entire set of functions from H with nonnegative derivatives of the first order but not with a single sequence {hk } from this set.

Section 2.1

87

Introduction

In Sec. 2.14, we prove the theorem on existence of a proper entropy solution. Note that the formulation of this result (Theorem 2.14.1) does not contain the condition q > p1 := (3n−2)p/(n+p−1) required in the formulation of the theorem on existence of an entropy solution (Theorem 2.7.2). In Sec. 2.15, we show that, under the condition q > p1 , the proper entropy solution is an entropy solution. In addition, the results obtained in this section demonstrate that, for the same restriction on q and the same requirements to the function F as in the conditions of Theorem 2.14.1, the notions of proper entropy solution and entropy solution are equivalent. Here, we also prove the theorem on uniqueness of the proper entropy solution (Theorem 2.15.3). Note that, unlike the formulation of the result on uniqueness of the entropy solution (Theorem 2.6.1), the statement of Theorem 2.15.3 contains the condition q > p1 . The question whether the theorem on uniqueness of the proper entropy solution is true without this condition remains open. In Sec. 2.16, we prove that the proper entropy solution of problem (2.1.6), (2.1.7) is an H-solution and, under the condition q > p2 := np/(np − n + 1), it is a W -solution of this problem. In Sec. 2.17, we study the properties of summability of the proper entropy solutions. The main results of this section are formulated in the form of Proposition 2.17.3. The functional multipliers belonging, together with the proper entropy solutions and their “derivatives,” to certain limiting Lebesgue spaces are described in this proposition. In Sec. 2.18, we give a standard definition of the generalized solution of problem (2.1.6), (2.1.7) in the case of sufficient regularity of the right-hand side of the equation of this problem. It is easy to see that the generalized solution is also a proper entropy solution. In this section, we also establish the conditions that should be imposed on the function F to guarantee the validity of the converse assertion. Finally, in Sec. 2.19, we consider some examples of validity of inequalities (2.1.3)–(2.1.5) and the conditions imposed on the function F in the theorems on existence and uniqueness of solutions of problem (2.1.6), (2.1.7) presented in Chapter 2. Note that the contents of Sections 2.1–2.8 include the results obtained in [63, 64]. Sections 2.9 and 2.10 reflect results obtained in [67] and the results established in [74] are presented in Sections 2.11–2.18 At the end of the introductory section, we recall that, in this chapter, it is always assumed that 1 < p < n/2 [see (2.1.1)]. For p > n/2, the solvability of problems of the form (2.1.6), (2.1.7) with L1 -data under the standard conditions of growth and coercivity imposed on the coefficients Aα can be es◦

tablished in the space W 2,p (Ω) on the basis of the well-known results from the theory of monotonic operators [89]. This follows from the boundedness

88

Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity ◦

of the embedding W 2,p (Ω) in C(Ω) for n < 2p. The case p = n/2 requires separate investigations. ◦

Set of functions H 1,q 2,p (Ω)

2.2

1,q By W2,p (Ω) we denote the set of all functions from W 1,q (Ω) with weak deriva1,q tives of the second order from Lp (Ω). The set W2,p (Ω) is a Banach space with the norm X Z 1/p |Dα u|p dx kuk = kukW 1,q (Ω) + . |α|=2 Ω ◦

1,q ∞ By W 1,q 2,p (Ω) we denote the closure of the set C0 (Ω) in W2,p (Ω). In addition, we use the following notation: If t ∈ [1, +∞], then | · |t is the nt norm in Lt (Ω); moreover, if t ∈ [1, n), then t∗ = n−t . ◦

∗

It is known (see, e.g., [48, Chap. 7]) that W 1,q (Ω) ⊂ Lq (Ω) and there exists a positive constant c0 depending only on n and q and such that X |u|q∗ 6 c0 |Dα u|q (2.2.1) |α|=1 ◦

for any function u ∈ W 1,q (Ω). For any k ∈ N, let ψk be a function defined on R such that k + 1 k+3 ψk (s) = s − sk+2 + s , s ∈ R. k+3 For any k ∈ N, we now define a function hk : R → R by setting  s if |s| 6 k,     h   i  ψk |s|−k + 1 k sign s if k < |s| < 2k, hk (s) = k     2k k + 2 sign s if |s| > 2k. k+3 For any k ∈ N, we have hk ∈ C 2 (R) and, in addition, |hk | 6 2k

on R,

(2.2.2)

0 6 h0k 6 1 on R,

(2.2.3)

|h00k | 6 3 on R.

(2.2.4)

Lemma 2.2.1. Let k, j ∈ N and j > 2k. Then, for any s ∈ R, hk (hj (s)) = hk (s).

(2.2.5)

◦

Section 2.2

Set of functions H 1,q 2,p (Ω)

89

Proof. Let s ∈ R. If |s| 6 j, then, by virtue of the definition of the function hj , we find hj (s) = s and, hence, equality (2.2.5) is true. However, if |s| > j, then, by virtue of the properties of the function hj , we get |hj (s)| > 2k. Hence, by the definition of the function hk , the following equality holds: hk (hj (s)) = 2k

k+2 sign hj (s). k+3

Together with the equalities hk (s) = 2k

k+2 sign s and sign hj (s) = sign s, k+3 

this yields (2.2.5). ◦

◦

1,q Lemma 2.2.2. Let u ∈ W 1,q 2,p (Ω), k ∈ N. Then hk (u) ∈ W 2,p (Ω) and the following propositions are true:

1) for any n-dimensional multiindex α, |α| = 1, Dα hk (u) = h0k (u)Dα u a.e. on Ω; 2) for any n-dimensional multiindex α, |α| = 2, X |Dβ u|2 |Dα hk (u) − h0k (u)Dα u| 6 |h00k (u)|

a.e. on Ω.

|β|=1

Proof. We set wα = h0k (u)Dα u

(2.2.6)

for α with |α| = 1 and 0

00

wα = h0k (u)Dα u + h00k (u)Dα uDα u

(2.2.7)

for α with |α| = 2, where |α0 | = |α00 | = 1, α0 + α00 = α. It is clear that wα ∈ Lq (Ω),

|α| = 1,

and wα ∈ Lp (Ω),

|α| = 2.

(2.2.8)

1,q Consider a sequence {ui } ⊂ C0∞ (Ω) such that ui → u strongly in W2,p (Ω) 2 and ui → u a.e. on Ω. Since {hk (ui )} ⊂ C0 (Ω), we obtain

Dα hk (ui ) = h0k (ui )Dα ui for α with |α| = 1 and 0

00

Dα hk (ui ) = h0k (ui )Dα ui + h00k (ui )Dα ui Dα ui for α with |α| = 2. In view of the properties of the functions hk , we obtain lim |hk (ui ) − hk (u)|q = 0, X |Dα hk (ui )−wα |q = 0, lim |Dα hk (ui )−wα |p = 0. i→∞

lim

i→∞

X |α|=1

i→∞

|α|=2

(2.2.9) (2.2.10)

90

Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity

By using these limit relations, we can show, in the ordinary way that, for any α ∈ Λ, there exists the weak derivative Dα hk (u), Dα hk (u) = wα a.e. on Ω. ◦

Thus, by virtue of (2.2.8)–(2.2.10), we conclude that hk (u) ∈ W 1,q 2,p (Ω). Moreover, in view of (2.2.6) and (2.2.7), assertions 1) and 2) are true.  ◦

By H 1,q 2,p (Ω) we denote the set of all functions u : Ω → R satisfying the ◦

condition: ∀ k ∈ N, hk (u) ∈ W 1,q 2,p (Ω). ◦

Any function u from the set H 1,q 2,p (Ω) is measurable. This follows from the measurability of the functions hk (u), k ∈ N, and the pointwise convergence of {hk (u)} to u. The following properties hold: ◦

◦

1,q W 1,q 2,p (Ω) ⊂ H 2,p (Ω),

(2.2.11)

◦

◦

1,q ∞ ∞ H 1,q 2,p (Ω) ∩ L (Ω) = W 2,p (Ω) ∩ L (Ω).

(2.2.12)

Property (2.2.11) directly follows from Lemma 2.2.2. Property (2.2.12) is a consequence of (2.2.11) and the fact that if u ∈ L∞ (Ω) and k ∈ N, k > |u|∞ , then hk (u) = u a.e. on Ω. ◦

However, we note that, among elements of the set H 1,q 2,p (Ω), there are functions that do not have even the property of local summability. Indeed, if B is a closed ball, B ⊂ Ω, x0 is the center of the ball B, ϕ is a function from the class C0∞ (Ω), ϕ = 1 on B, and v is a function on Ω such that, ∀ x ∈ Ω\{x0 }, ◦

1 v(x) = |x − x0 |−n ϕ, then v ∈ H 1,q 2,p (Ω)\Lloc (Ω). ◦

Lemma 2.2.3. Let u ∈ H 1,q 2,p (Ω). Then, for any k, j ∈ N, j > 2k, and α ∈ Λ, Dα hk (u) = Dα hj (u) a.e. on {|u| 6 k},

(2.2.13)

Dα hk (u) = 0 a.e. on {|u| > j}.

(2.2.14)

Proof. Let k, j ∈ N, j > 2k, and α ∈ Λ. Thus, by Lemma 2.2.1, we get hk (u) = hk (hj (u)).

(2.2.15)

Assume that |α| = 1. Then it follows from relation (2.2.15) and assertion 1) of Lemma 2.2.2 that Dα hk (u) = h0k (hj (u))Dα hj (u) a.e. on Ω. Hence, in view of the relations h0k (hj (u)) = 1 on {|u| 6 k},

(2.2.16)

h0k (hj (u))

(2.2.17)

we arrive at (2.2.13) and (2.2.14).

= 0 on {|u| > j},

◦

Section 2.2

Set of functions H 1,q 2,p (Ω)

91

Now let |α| = 2. By virtue of (2.2.15) and assertion 2) of Lemma 2.2.2, we find X |Dβ hj (u)|2 a.e. on Ω. |Dα hk (u) − h0k (hj (u))Dα hj (u)| 6 |h00k (hj (u))| |β|=1

Thus, according to properties (2.2.16) and (2.2.17) and the fact that h00k (hj (u)) = 0 on {|u| 6 k} ∪ {|u| > j}, 

we again obtain (2.2.13) and (2.2.14). ◦

α Definition 2.2.1. If u ∈ H 1,q 2,p (Ω) and α ∈ Λ, then δ u is a function defined on Ω such that δ α u = Dα h1 (u) on {|u| 6 1}

and, ∀ k ∈ N,

δ α u = Dα h2k (u) on {2k−1 < |u| 6 2k }. ◦

Lemma 2.2.4. Let u ∈ H 1,q 2,p (Ω). Then, for any α ∈ Λ and k ∈ N, δ α u = Dα hk (u) a.e. on {|u| 6 k}.

(2.2.18)

Proof. Let α ∈ Λ and k ∈ N. We fix j ∈ N such that 2j−1 > k and show that δ α u = Dα h2j (u) a.e. on {|u| 6 2j }. (2.2.19) Indeed, by virtue of Lemma 2.2.3, for any l ∈ {0, 1, . . . , j}, we can write Dα h2l (u) = Dα h2j (u) a.e. on {|u| 6 2l }. Hence, in view of the definition δ α u and the fact that j [ j {|u| 6 2 } = {|u| 6 1} ∪ {2l−1 < |u| 6 2l }, l=1

we obtain (2.2.19). Since 2j−1 > k, we get Dα hk (u) = Dα h2j (u) a.e. on {|u| 6 k} by Lemma 2.2.3. Relations (2.2.19) and (2.2.20) yield (2.2.18). ◦

(2.2.20) 

It follows from Lemma 2.2.4 that if u ∈ H 1,q 2,p (Ω) and k ∈ N, then there exist finite integrals of the functions |δ α u|q , |α| = 1, and |δ α u|p , |α| = 2, over the set {|u| 6 k}. ◦ In addition, Lemmas 2.2.4 and 2.2.2 imply that if u ∈ W 1,q 2,p (Ω) and α ∈ Λ, then δ α u = Dα u a.e. on Ω.

92

Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity

We set r=

n(q − 1) , n−1

c00 = 2q

∗ /q

∗

(c0 n)q + 2nq .

◦

Lemma 2.2.5. Let u ∈ H 1,q 2,p (Ω), M > 1, and let, for any k ∈ N, the following inequality be true:  X Z X (2.2.21) |δ α u|p dx 6 M k. |δ α u|q + {|u| k} 6 c00 M n/(n−q) k −r , X  α meas |δ u| > k 6 c00 M n/(n−1) k −r ,

(2.2.22) (2.2.23)

|α|=1

 X α |δ u| > k 6 c00 M n/(n−1) k −pr/q . meas

(2.2.24)

|α|=2

Proof. Let k ∈ N. By virtue of relation (2.2.1), we find X |hk (u)|q∗ 6 c0 |Dα hk (u)|q .

(2.2.25)

|α|=1

Using Lemmas 2.2.1–2.2.4, property (2.2.3), and inequality (2.2.21), for any α with |α| = 1, we obtain Z |Dα hk (u)|qq = |Dα hk (u)|q dx {|u| k} 6 |hk (u)|qq∗ holds. Hence, in view of inequality (2.2.27), we obtain ∗

meas{|u| > k} 6 (c00 − 2nq )M n/(n−q) k −r . Therefore, inequality (2.2.22) holds.

(2.2.28)

◦

Section 2.2

Set of functions H 1,q 2,p (Ω)

93

Now let k1 , k2 ∈ N, and ∗

∗

∗ k21+r

1+r∗

M q/(n−q) k q 6 k11+r 6 21+r M q/(n−q) k q , M

q/(n−q) p

k 6

62

M

q/(n−q) p

(2.2.29)

k .

(2.2.30)

i = 1, 2.

(2.2.31)

By analogy with (2.2.28), we find ∗

meas{|u| > ki } 6 (c00 − 2nq )M n/(n−q) ki−r , We set B1 =

X |α|=1

B2 =

X

kq |δ u| > q , n α



q

|δ α u|p >

|α|=2

|u| < k1 ,

kp , n2p

 |u| < k2 .

By virtue of (2.2.21), kq meas B1 6 nq

Z

kp meas B2 6 n2p

Z

{|u| q n |α|=1

|α|=1

6 meas{|u| > k1 } + meas B1 6 c00 M n/(n−1) k −r . Similarly, by using (2.2.31), (2.2.33), and (2.2.30),we get  X  X kp α α p meas |δ u| > k 6 meas |δ u| > 2p n |α|=2

|α|=2

6 meas{|u| > k2 } + meas B2 6 c00 M n/(n−1) k −pr/q . Hence, inequalities (2.2.23) and (2.2.24) hold.



Lemma 2.2.6. Let u be a measurable function on Ω, M > 0, γ > 0, and let meas {|u| > k} 6 M k −γ for any k ∈ N. Then u ∈ Lλ (Ω) and |u|λ 6 41/(γ−λ) [meas Ω + 2γ M ]1/λ for any λ ∈ (0, γ). In fact, this result coincides with Lemma 1.6.1. In this section, it is presented in the original form (see [64]). Lemmas 2.2.5 and 2.2.6 yield certain properties of summability for functions ◦

from the set H 1,q 2,p (Ω) satisfying inequality (2.2.21) for any k ∈ N.

94

Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity

2.3

Definition and some properties of entropy solutions ◦

n,2 and, We introduce the following notation: if u ∈ H 1,q 2,p (Ω), then δ2 u : Ω → R α for any x ∈ Ω and α ∈ Λ, (δ2 u(x))α = δ u(x). ◦

◦

1,q ∞ We note that if u ∈ H 1,q 2,p (Ω), ϕ ∈ W 2,p (Ω) ∩ L (Ω), k ∈ N, and α ∈ Λ, then, by virtue of (2.1.3) and Lemma 2.2.4, there exist finite integrals of the functions Aα (x, δ2 u)δ α u and Aα (x, δ2 u)δ α ϕ over the set {|u − ϕ| < 2k}.

Definition 2.3.1. An entropy solution of problem (2.1.6), (2.1.7) is defined ◦

as a function u ∈ H 1,q 2,p (Ω) satisfying the conditions: 1) F (x, u) ∈ L1 (Ω); 2) there exist c > 0, b ∈ (1, r), and γ > 0 such that, for any ϕ ∈ C0∞ (Ω) and  Z k ∈ N,  X α α Aα (x, δ2 u)(δ u − δ ϕ) h0k (u − ϕ)dx {|u−ϕ| 0 such that, for any k ∈ N, X  Z X |δ α u|q + |δ α u|p dx 6 c3 [ |F (x, u)|1 + c + 1 ]k. (2.3.1) {|u|6k}

|α|=1

|α|=2

Proof. By virtue of Definition 2.3.1, there exists c > 0 such that, for any k ∈ N, X  Z Z Aα (x, δ2 u)δ α u h0k (u)dx 6 F (x, u)hk (u)dx + c. (2.3.2) {|u| 1 and q(p−1) yield assertion 6). 

Proposition 2.3.1. Let u be an entropy solution of problem (2.1.6), (2.1.7). Then, for any function ϕ ∈ C0∞ (Ω),  Z X α α lim Aα (x, δ2 u)(δ u − δ ϕ) dx k→∞

{|u−ϕ| 0, b ∈ (1, r), and γ > 0 such that, for any ϕ ∈ C0∞ (Ω) and k ∈ N, X  Z α α Aα (x, δ2 u)(δ u − δ ϕ) h0k (u − ϕ)dx {|u−ϕ| 2|ϕ|∞ , we have Ek ⊂ {|u| > k/2}. Thus, Lemmas 2.3.1 and 2.2.5 imply that lim meas Ek = 0. (2.3.7) k→∞

We fix k ∈ N. In view of relation (2.2.3) and the fact that h0k (u − ϕ) = 1 on {|u − ϕ| < k}, relation (2.3.6) yields the inequality X  Z Z α α Aα (x, δ2 u)(δ u−δ ϕ) dx − F (x, u)hk (u−ϕ)dx + Ik {|u−ϕ| − g2 dx. Ek

This inequality and inequality (2.3.8) imply that X  Z Z α α Aα (x, δ2 u)(δ u − δ ϕ) dx − F (x, u)hk (u − ϕ)dx {|u−ϕ| 0, b ∈ (1, r), and γ > 0 such that, for any ϕ ∈ W 1,q 2,p (Ω) ∩ L∞ (Ω) and k ∈ N, X  Z Aα (x, δ2 u)(δ α u − δ α ϕ) h0k (u − ϕ)dx {|u−ϕ| 0 such that h ib Ik (ϕ) 6 Jk (ϕ) + c 1 + kϕkW 1,b (Ω) k −γ . (2.3.10) ◦

∞ for any ϕ ∈ C0∞ (Ω) and k ∈ N. We now fix any ϕ ∈ W 1,q 2,p (Ω) ∩ L (Ω) ∞ and k ∈ N. It is easy to see that there exists a sequence {ϕj } ⊂ C0 (Ω) with the following properties: 1,q (Ω), ϕj → ϕ strongly in W2,p

(2.3.11)

ϕj → ϕ a.e. on Ω,

(2.3.12)

∀ j ∈ N,

|ϕj | 6 |ϕ|∞ + 1 on Ω.

By virtue of (2.3.10), for any j ∈ N, we find h ib Ik (ϕj ) 6 Jk (ϕj ) + c 1 + kϕj kW 1,b (Ω) k −γ .

(2.3.13)

(2.3.14)

Properties (2.3.11) and (2.3.12) enable us to conclude that lim Jk (ϕj ) = Jk (ϕ),

(2.3.15)

lim kϕj kW 1,b (Ω) = kϕkW 1,b (Ω) .

(2.3.16)

lim Ik (ϕj ) = Ik (ϕ).

(2.3.17)

j→∞ j→∞

We now show that j→∞

We set Ek = {|u| 6 2k + |ϕ|∞ + 1},

Φ=

X α∈Λ

Aα (x, δ2 u)(δ α u − δ α ϕ).

98

Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity

Assume that, for any j ∈ N, X Φj = Aα (x, δ2 u)(δ α ϕ − δ α ϕj ). α∈Λ

By virtue of (2.1.3) and Lemma 2.2.4, the functions Φ and Φj are integrable in the set Ek . Moreover, propery (2.3.11) implies that Z |Φj |dx = 0. (2.3.18) lim j→∞ E k

By using (2.3.13) and the properties of the function hk , we conclude that, for any j ∈ N, Z Z |Ik (ϕj ) − Ik (ϕ)| 6 |Φ||h0k (u − ϕj ) − h0k (u − ϕ)|dx + |Φj |dx. Ek

Ek

Hence, in view of (2.3.12) and (2.3.18), we obtain (2.3.17). In turn, relations (2.3.14)–(2.3.17) imply that h ib Ik (ϕ) 6 Jk (ϕ) + c 1 + kϕkW 1,b (Ω) k −γ . Therefore, inequality (2.3.9) holds.

2.4



One a priori estimate for the entropy solutions

Lemma 2.4.1. Let k ∈ N, τ ∈ (0, 1]. Then 1) |h00k (s)| 6 6kτ k for any s ∈ R, |s| 6 k(1 + τ ); 2) |h00k (s)| 6 3τ −1 (1 − h0k (s)) for any s ∈ R, |s| > k(1 + τ ). Proof. By the definition of the function ψk , we get ψk00 (s) = (k + 1)(k + 2)(s − 1)sk for any s ∈ R. Hence, for any s ∈ [0, τ ], |ψk00 (s)| 6 6k 2 τ k . This relation and the definition of the function hk imply the validity of assertion 1). Thus, by the definition of the function ψk , for any s ∈ R, we obtain 1 00 |ψ (s)| − 3τ −1 (1 − ψk0 (s)) k k 6 3(k + 1)|s − 1||s|k − 3τ −1 sk+1 [(k + 1)(1 − s) + 1]. Hence, for any s ∈ [τ, 1], 1 00 |ψ (s)| 6 3τ −1 (1 − ψk0 (s)). k k

Section 2.4

One a priori estimate for the entropy solutions

99

This inequality and the definition of the function hk imply the validity of assertion 2).  Lemma 2.4.2. Let u be an entropy solution of problem (2.1.6), (2.1.7). Then there exist c > 0, b ∈ (1, r), and γ > 0 such that, for any k ∈ N, k > 2, and m ∈ N, the following estimate holds:  X Z X α p α q |δ u| + |δ u| dx c4 {k6|u|6k+m}

|α|=2

|α|=1

Z



6m

|F (x, u)| + g1 + g2 dx

{|u|>k/4}

 n/(n−q) −1 + |F (x, u)|1 + c + 1 k h ib X + c 1 + |u|b + |δ α u|b m−γ .

(2.4.1)

|α|=1

Proof. By virtue of Lemma 2.3.3, there exist c > 0, b ∈ (1, r), and γ > 0 ◦

∞ such that, for any ϕ ∈ W 1,q 2,p (Ω) ∩ L (Ω) and m ∈ N, X  Z Aα (x, δ2 u)(δ α u − δ α ϕ) h0m (u − ϕ)dx {|u−ϕ| 4. Moreover, we set ψ = hk (u). It is clear that ◦ ∞ (2.4.3) ψ ∈ W 1,q 2,p (Ω) ∩ L (Ω). In view of (2.2.2) and the fact that ψ = u on {|u| 6 k}, we obtain Z Z F (x, u)hm (u − ψ)dx 6 2m |F (x, u)|dx.

(2.4.4)

{|u|>k}

Ω

In addition, by virtue of (2.2.3) and Lemmas 2.2.1–2.2.4, we get X kψkW 1,b (Ω) 6 |u|b + |δ α u|b .

(2.4.5)

|α|=1

According to (2.4.3)–(2.4.5), it follows from (2.4.2) that X  Z Aα (x, δ2 u)(δ α u − δ α ψ) h0m (u − ψ)dx {|u−ψ|k}

h ib X |F (x, u)|dx + c 1 + |u|b + |δ α u|b m−γ . |α|=1

(2.4.6)

100 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity We also note that, by virtue of relations (2.4.2), (2.1.4), (2.2.2), and (2.2.3) and Lemma 2.2.5, we arrive at the following inequalities:  X Z X α p α q (2.4.7) |δ u| dx 6 c5 [ |F (x, u)|1 + c + 1 ]k, |δ u| + {|u|62k}

|α|=2

|α|=1

∗

meas{|u| > k} 6 c6 [ |F (x, u)|1 + c + 1 ]n/(n−q) k −r . We set σ = (r∗ − 1)

q − 2p , qp

θ1 = h0k (u),

(2.4.8)

θ2 = |h00k (u)|.

It is clear that θ2 = 0 on {|u| 6 k}.

θ1 = 1,

(2.4.9)

Lemma 2.4.1 implies that on {k < |u| < k(1 + k −σ )},

θ2 6 6k 1−σk σ

θ2 6 3k (1 − θ1 ) on {|u| > k(1 + k

−σ

)}.

(2.4.10) (2.4.11)

Moreover, by virtue of (2.2.2) and Lemmas 2.2.1, 2.2.2, and 2.2.4, for any α ∈ Λ, we find X |δ α ψ − θ1 δ α u| 6 (|α| − 1)θ2 |δ β u|2 a.e. on {|u − ψ| < 2m}. (2.4.12) |β|=1

By Ik,m we denote the left-hand side of inequality (2.4.6). We set 0 = {|u − ψ| < 2m} ∩ {|u| > k(1 + k −σ )}, Ek,m

0 Ik,m

00 Ik,m

00 = {|u − ψ| < 2m} ∩ {|u| > k}, Ek,m X  Z X α q α p = |δ u| + |δ u| (1 − θ1 )h0m (u − ψ)dx, 0 Ek,m

Z = 00 Ek,m

|α|=1

X

|α|=2

 X  α 2 |Aα (x, δ2 u)| |δ u| θ2 h0m (u − ψ)dx.

|α|=2

|α|=1

Using (2.1.4), (2.2.3), (2.4.9), and (2.4.12), we obtain X  Z α Ik,m = Aα (x, δ2 u)δ u (1 − θ1 )h0m (u − ψ)dx {|u−ψ|k}

(2.4.18) We now note that {2k 6 |u| 6 2k + m/2} ⊂ {|u − ψ| < m}.

(2.4.19)

This can easily be proved if we take into account the fact that m > 4 and, by the definition of the function hk , we have hk (s) = 2k for s ∈ R, |s| > 2k.

k+2 sign s k+3

103

Notion of H-solution

Section 2.5

Inclusion (2.4.19) implies that 0 {2k 6 |u| 6 2k + m/2} ⊂ Ek,m

and (1 − θ1 )h0m (u − ψ) = 1 on {2k 6 |u| 6 2k + m/2}. Then 0 Ik,m >

X

Z {2k6|u|62k+m/2}

|δ α u|q +

X

 |δ α u|p dx.

(2.4.20)

|α|=2

|α|=1

Inequalities (2.4.20), (2.4.18), and (2.4.6) yield the inequality X  Z X c2 |δ α u|q + |δ α u|p dx {2k6|u|62k+m/2}

|α|=1

|α|=2

Z {|F (x, u)| + g1 + g2 }dx

6 (c12 + 4)m {|u|>k}

n/(n−q) −1

+ c13 [ |F (x, u)|1 +c+1 ]

k

 + 2c 1+|u|b +

X

b

|δ u|b m−γ . α

|α|=1

Since this inequality is true for any k ∈ N and m ∈ N, m > 4, it is easy to see that inequality (2.4.1) is true for any k ∈ N, k > 2, and m ∈ N. 

2.5

Notion of H-solution

Definition 2.5.1. An H-solution of problem (2.1.6), (2.1.7) is defined as a func◦

tion u ∈ H 1,q 2,p (Ω) satisfying the following conditions: 1) F (x, u) ∈ L1 (Ω); 2) Aα (x, δ2 u) ∈ L1 (Ω) for any α ∈ Λ; 3) for any function ϕ ∈ C0∞ (Ω),  Z X Z α Aα (x, δ2 u)δ ϕ dx = F (x, u)ϕ dx. Ω

Ω

α∈Λ

Theorem 2.5.1. Let u be an entropy solution of problem (2.1.6), (2.1.7). Then u is an H-solution of this problem. ◦

1 Proof. By Definition 2.3.1, we have u ∈ H 1,q 2,p (Ω) and F (x, u) ∈ L (Ω). Moreover, by virtue of assertion 6) of Lemma 2.3.2, Aα (x, δ2 u) ∈ L1 (Ω) for any α ∈ Λ. In addition, according to Lemma 2.3.3, there exist c > 0, b ∈ (1, r),

104 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity and γ > 0 such that X Z {|u−ψ| |ϕ|∞ and set ϕk = hk (u) − ϕ

for any k ∈ N. Further, we fix k ∈ N. By virtue of (2.5.1), this yields X  Z α α Aα (x, δ2 u)(δ u − δ ϕk ) h0m (u − ϕk ) dx {|u−ϕk | k(1 + k −σ )}, 00 Ek,m = {|u − ϕk | < 2m} ∩ {|u| > k}, X  Z α Jk,m = Aα (x, δ2 u)δ ϕ h0m (u − ϕk )dx, {|u−ϕk |k}

00 . By virtue of (2.4.10) and (2.4.11), we get We now estimate Ik,m 00 Ik,m

6 6k

1−σk

X

Z {|u|62k}

+ 3k σ

|α|=2

X

Z 0 Ek,m

 X  α 2 |Aα (x, δ2 u)| |δ u| dx |α|=1

|Aα (x, δ2 u)|

|α|=2

 X

 |δ α u|2 (1−θ1 )h0m (u−ϕk )dx.

|α|=1

(2.5.5)

Using (2.4.15), the Hölder inequality, (2.1.3), and (2.4.7), we find X  X  Z α 2 |Aα (x, δ2 u)| |δ u| dx 6 c8 [ |F (x, u)|1 +c+1 ]k. {|u|62k}

|α|=2

|α|=1

(2.5.6)

By analogy with (2.4.17), we conclude that X  X  Z α 2 σ |Aα (x, δ2 u)| |δ u| (1 − θ1 )h0m (u − ϕk )dx k 0 Ek,m

6

|α|=2

c2 0 I + (c2 /c1 ) 6 k,m

|α|=1

Z

g1 dx + c6 c10 [ |F (x, u)|1 +c+1 ]n/(n−q) k −1 . {|u|>k} (2.5.7)

Relations (2.5.5)–(2.5.7) yield Z c2 0 00 + 3(c2 /c1 ) Ik,m 6 Ik,m g1 dx + c14 [ |F (x, u)|1 + c + 1 ]n/(n−q) k −1 . 2 {|u|>k} Thus, relations (2.5.4) and (2.5.2) imply the inequality X  Z α Aα (x, δ2 u)δ ϕ h0m (u − ϕk )dx {|u−ϕk |k}

where C(u) = c14 [ |F (x, u)|1 + c + 1 ]n/(n−q) ,  b X α C(u, ϕ) = c 1 + kϕkW 1,b (Ω) + |u|b + |δ u|b . |α|=1

In turn, it follows from relation (2.5.8) that  Z X α Aα (x, δ2 u)δ ϕ dx Ω

α∈Λ

Z

X

Z F (x, u)hm (u − ϕk )dx +

6

{|u−ϕk |>2m}

Ω

+

Z X Ω

 Aα (x, δ2 u)δ α ϕ dx

α∈Λ

 α |Aα (x, δ2 u)||δ ϕ| {1 − h0m (u − ϕk )}dx

α∈Λ

Z

(g1 + g2 )dx + C(u)k −1 + C(u, ϕ)m−γ .

+ c15

(2.5.9)

{|u|>k}

Thus inequality (2.5.9) holds for any k ∈ N. Consider the behavior of different terms on the right-hand side of this inequality as k → ∞. Since m > |ϕ|∞ and u − ϕk → ϕ on Ω, we get hm (u − ϕk ) → ϕ, h0m (u − ϕk ) → 1 on Ω. Therefore, Z Z lim

k→∞ Ω

lim

Z X

k→∞ Ω

F (x, u)hm (u − ϕk )dx =

F (x, u)ϕdx,

(2.5.10)

Ω

  |Aα (x, δ2 u)||δ α ϕ| 1 − h0m (u − ϕk ) dx = 0.

(2.5.11)

α∈Λ

Moreover, by virtue of (2.4.8), we obtain Z lim (g1 + g2 )dx = 0. k→∞ {|u|>k}

(2.5.12)

We now show that lim meas {|u − ϕk | > 2m} = 0.

k→∞

(2.5.13)

Indeed, since |ϕ|∞ 6 m, we conclude that, for any k ∈ N, {|u − ϕk | > 2m} ⊂ {|u| > k}.

(2.5.14)

Section 2.6

107

On uniqueness of the entropy solution

It is clear that meas {|u| > k} → 0 as k → ∞. Hence, in view of (2.5.14), we obtain (2.5.13). Then X  Z α lim Aα (x, δ2 u)δ ϕ dx = 0. (2.5.15) k→∞ {|u−ϕk |>2m}

α∈Λ

Using (2.5.10)–(2.5.12) and (2.5.15) and passing to the limit in inequality (2.5.9) as k → ∞, we get  Z X Z F (x, u)ϕdx + C(u, ϕ)m−γ . Aα (x, δ2 u)δ α ϕ dx 6 Ω

Ω

α∈Λ

Passing to the limit in this inequality as m → ∞, we find  Z Z X Aα (x, δ2 u)δ α ϕ dx 6 F (x, u)ϕdx. Ω

α∈Λ

Ω

Note that ϕ in this inequality is an arbitrary function from C0∞ (Ω). Thus, for any function ϕ ∈ C0∞ (Ω), we obtain  Z Z X α F (x, u)ϕdx. Aα (x, δ2 u)δ ϕ dx = Ω

α∈Λ

Ω

Hence, the function u possesses all properties required to assert that it is an H-solution of problem (2.1.6), (2.1.7). 

2.6

On uniqueness of the entropy solution

Prior to proving the theorem of uniqueness of the entropy solution of problem (2.1.6), (2.1.7), we present a series of auxiliary results. Lemma 2.6.1. Let u and v be entropy solutions of problem (2.1.6), (2.1.7) and let δ α u = δ α v a.e. on Ω for any α, |α| = 1. Then u = v a.e. on Ω. Proof. We set Φ = |F (x, u)| + |F (x, v)| + g1 + g2 , C0 = |F (x, u)|1 + |F (x, v)|1 + 1. By virtue of Lemma 2.4.2, there exist c > 0, b ∈ (1, r), and γ > 0 such that, for any m ∈ N and k ∈ N, k > 5m, the following inequality is true: X  X  Z Z c4 |δ α u|q dx + c4 |δ α v|q dx {k−m6|u|6k}

|α|=1

{k−m6|v|6k}

|α|=1

108 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity Z

Z

Φ dx + 3[C0 + c]n/(n−q) k −1

Φ dx + m

6m {|u|>k/5}

{|v|>k/5}

b  X (|δ α u|b + |δ α v|b ) m−γ . (2.6.1) + 2c(1 + meas Ω)b 1 + |u|b + |v|b + |α|=1

For any k ∈ N, let Tk be a function on R such that ( s if |s| 6 k, Tk (s) = k sign s if |s| > k. ◦

◦

1,q (Ω) for any k ∈ N. Thus, in view of Since u ∈ H 1,q 2,p (Ω), we have hk (u) ∈ W the fact that Tk (s) = Tk (hk (s)) for k ∈ N and s ∈ R, we get

∀ k ∈ N,

◦

Tk (u) ∈ W 1,q (Ω).

A similar assertion also holds for the function v. We now fix m ∈ N and k ∈ N, k > 5m. Note that Tk (u) = hk (u) on {|u| 6 k}. Therefore, by virtue of Lemma 2.2.4 for any α, |α| = 1, we find Dα Tk (u) = δ α u a.e. on {|u| 6 k}

(2.6.2)

Dα Tk (v) = δ α v

(2.6.3)

and, similarly, a.e. on {|v| 6 k}.

It is also clear that, for any α, |α| = 1, Dα Tk (u) = 0 a.e. on {|u| > k},

(2.6.4)

Dα Tk (v) = 0 a.e. on {|v| > k}.

(2.6.5)

We set w = Tm (Tk (u) − Tk (v)). ◦

Since w ∈ W 1,q (Ω), by virtue of (2.2.1), we have X Z q∗ /q Z ∗ ∗ |w|q dx 6 (c0 n)q |Dα w|q dx . Ω

(2.6.6)

|α|=1 Ω

We now establish a lower bound for the left-hand side of this inequality and an upper bound for its right-hand side. To this end, we introduce the sets (1)

Em,k = {|u − v| 6 m, |u| 6 k, |v| 6 k}, (2)

Em,k = {|Tk (u) − Tk (v)| 6 m, |u| > k, |v| > k},

Section 2.6

109

On uniqueness of the entropy solution (3)

Em,k = {|Tk (u) − Tk (v)| 6 m, |u| 6 k, |v| > k}, (4)

Em,k = {|Tk (u) − Tk (v)| 6 m, |u| > k, |v| 6 k}. It is obvious that {|Tk (u) − Tk (v)| 6 m} =

4 [

(i)

Em,k

i=1

and

Z

∗

(1) Em,k

|u − v|q dx 6

Z

∗

|w|q dx.

(2.6.7)

Ω

We also note that (3)

(2.6.8)

(4)

(2.6.9)

Em,k ⊂ {k − m 6 |u| 6 k}, Em,k ⊂ {k − m 6 |v| 6 k}. Further, we fix α, |α| = 1 and obtain Z

|Dα w|q dx =

Ω

4 Z X

|Dα Tk (u) − Dα Tk (v)|q dx.

(i)

Em,k

i=1

Thus, by using the condition of the lemma and relations (2.6.2)–(2.6.5), (2.6.8), and (2.6.9), we get Z Z Z |Dα w|q dx 6 |δ α u|q dx + |δ α v|q dx. {k−m6|u|6k}

Ω

{k−m6|v|6k}

This inequality together with inequalities (2.6.6) and (2.6.7) yield Z X  Z q∗ 0 q∗ α q |u − v| dx 6 (c n) |δ u| dx (1)

{k−m6|u|6k}

Em,k

|α|=1

X

Z + {k−m6|v|6k}

 q∗ /q |δ α v|q dx .

(2.6.10)

|α|=1

We now set n/(n−q)

b



+ 2c(1 + meas Ω) 1 + |u|b + |v|b +

C1 = 3[C0 + c]

X

b (|δ u|b + |δ v|b ) . α

α

|α|=1

It follows from (2.6.10) and (2.6.1) that Z ∗ |u − v|q dx {|u−v|6m,|u|6k,|v|6k}

 Z 6 c16 m {|u|>k/5}

Z Φ dx + m {|v|>k/5}

Φ dx + C1 (k

−1

+m

−γ

q∗ /q ) .

110 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity This inequality holds for all k ∈ N, k > 5m. Thus, passing to the limit in it as k → ∞, we find Z  q∗ /q ∗ |u − v|q dx 6 c16 C1 m−γ . (2.6.11) {|u−v|6m}

In performing this limiting transition, we have used the fact that, for the entropy solutions u and v, meas {|u| > k} → 0 and meas {|v| > k} → 0 as k → ∞ by virtue of Lemmas 2.3.1 and 2.2.5. Finally, passing to the limit as m → ∞ in inequality (2.6.11), we conclude that u = v a.e. on Ω.  In what follows, we need some elementary inequalities for the functions hk and some other auxiliary functions. Lemma 2.6.2. Let k, m ∈ N, and k > 4m. Then, for any s ∈ R, |s| > k + 2m, the following inequality is true: |hk (s)| > k + m.

(2.6.12)

Proof. Let s ∈ R, |s| > k + 2m. If |s| > 2k, then, by virtue of the definition of the function hk and the inequality k > 4m, we have k+2 > k + m. |hk (s)| = 2k k+3 Now let |s| < 2k. We set s1 = (|s| − k)/k. Thus, in view of the definition of the function hk , we get   2m + k. (2.6.13) |hk (s)| = kψk (s1 ) + k > kψk k However, by the definition of the function ψk and the inequality k > 4m, we find     2m 2m k+2 m 2m > − > . ψk k k k k Thus, in view of (2.6.13), we obtain (2.6.12).  Let a be a nondecreasing function from the class C ∞ (R) such that a(s) = s for s 6 −1 and a(s) = 0 for s > 0 and let c0 be a positive constant such that a0 6 c0 , |a00 | 6 c0 on R. We now define a function ak,m : R → R for any k, m ∈ N, by setting     |s| − k − m + k + m sign s, s ∈ R. ak,m (s) = ma m Lemma 2.6.3. Let k, m ∈ N. Then ak,m ∈ C ∞ (R) and this function has the following properties: if |s| 6 k,

then ak,m (s) = s,

(2.6.14)

Section 2.6

111

On uniqueness of the entropy solution

if |s| > k + m,

then ak,m (s) = (k + m) sign s, k+m if k 6 |s| 6 k + m, then |s| 6 |ak,m (s)|, k |ak,m | 6 k + m on R, 0 6 a0k,m 6 c0 on R, c0 |a00k,m | 6 on R. m

(2.6.15) (2.6.16) (2.6.17) (2.6.18) (2.6.19)

All properties listed in the lemma are simple consequences of the definition of the functions ak,m . Lemma 2.6.4. Let k, m ∈ N, and k > 5m. Also let s, t ∈ R and let the following inequalities be true: |t| > k,

|hk (t)| 6 k + m,

(2.6.20)

|s − ak,m (hk (t))| 6 2m.

(2.6.21)

|t| 6 k + 2m,

(2.6.22)

k − 4m 6 |s| 6 k + 3m.

(2.6.23)

Then

Proof. Inequality (2.6.22) follows from Lemma 2.6.2 and the second inequality in (2.6.20). By virtue of (2.6.21) and (2.6.17), we get |s| 6 |s − ak,m (hk (t))| + |ak,m (hk (t))| 6 k + 3m.

(2.6.24)

Finally, by using relations (2.6.20), (2.6.16), (2.6.21), and (2.6.24) and the inequality k > 5m, we find k+m k+m |ak,m (hk (t))| 6 (2m + |s|) k k k+m m m 6 |s| + |s| + 2m 6 |s| + 3m + 5m 6 |s| + 4m. k k k

k 6 |hk (t)| 6

Hence, we have |s| > k − 4m and, therefore, in view of (2.6.24), we obtain (2.6.23).  Theorem 2.6.1. Assume that a function F (x, ·) is nonincreasing on R for almost all x ∈ Ω. Let u and v be entropy solutions of problem (2.1.6), (2.1.7). Then u = v a.e. on Ω. Proof. For any k, m ∈ N, we set fk,m = ak,m (hk (u)),

gk,m = ak,m (hk (v)),

112 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity X

Z Ik,m = {k−4m6|u|6k+3m}

|δ α u|q +

X

+ {k−4m6|v|6k+3m}

 |δ α u|p dx

|α|=2

|α|=1

Z

X

α

q

|δ v| +

X

α

p

|δ v|

 dx.

|α|=2

|α|=1

By virtue of Lemmas 2.3.3 and 2.4.2, there exist c > 0, γ > 0, and a sequence {σk } of positive numbers that converges to zero and is such that, for any m ∈ N and k ∈ N, k > 5m, we have X  Z Aα (x, δ2 u)(δ α u−δ α gk,m ) h0m (u−gk,m )dx {|u−gk,m |k}∪{|v|>k}

Further, we set Ek,m = {|u − v| < 2m, |u| 6 k, |v| 6 k}, 0 = {|u − gk,m | < 2m, |v| > k}, Ek,m 00 = {|u − gk,m | < 2m, |v| 6 k, |u| > k}. Ek,m

Using (2.1.4), (2.2.3), and the fact that gk,m = v on Ek,m , δ α gk,m = δ α v a.e. on Ek,m ∀ α ∈ Λ, we arrive at the inequality  X Z α α 0 Aα (x, δ2 u)(δ u − δ v) h0m (u − v)dx Ik,m > Ek,m

α∈Λ

 Z |Aα (x, δ2 u)||δ α gk,m | dx −

X

Z − 0 00 Ek,m ∪Ek,m

g2 dx.

{|u|>k}∪{|v|>k}

α∈Λ

(2.6.32) We now estimate the second integral on the right-hand side of this inequality. We set 0 000 ∩ {|hk (v)| 6 k + m}, = Ek,m Ek,m 0 Bk,m = {k − 4m 6 |u| 6 k + 3m},

00 Bk,m = {k − 4m 6 |v| 6 k + 3m}.

000 ⊂ B 0 00 By virtue of Lemma 2.6.4, we conclude that Ek,m k,m ∩Bk,m . Using (2.6.14), 00 0 00 we also obtain Ek,m ⊂ Bk,m ∩ Bk,m . Further, we note that δ α gk,m = 0 a.e. on {|hk (v)| > k + m} for any α ∈ Λ. By using these properties and the numerical and integral Hölder inequalities, we obtain X  Z |Aα (x, δ2 u)||δ α gk,m | dx 0 00 Ek,m ∪Ek,m

α∈Λ

X

Z = 000 ∪E 00 Ek,m k,m

 |Aα (x, δ2 u)||δ gk,m | dx α

|α|=1

Z + 000 ∪E 00 Ek,m k,m

X |α|=2

 |Aα (x, δ2 u)||δ α gk,m | dx

114 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity Z 6

X

0 Bk,m

q/(q−1)

(q−1)/q



|Aα (x, δ2 u)|

dx

|α|=1

Z

X

× 00 Bk,m

Z

|α|=1

X

+ 0 Bk,m

 1/q |δ α gk,m |q dx

|Aα (x, δ2 u)|

p/(p−1)

(p−1)/p

 dx

|α|=2

Z

X

× 00 Bk,m

 1/p |δ α gk,m |p dx .

(2.6.33)

|α|=2

By virtue of (2.1.3), we get X  Z X q/(q−1) p/(p−1) |Aα (x, δ2 u)| + |Aα (x, δ2 u)| dx 0 Bk,m

|α|=1

|α|=2

Z 6 c1 Ik,m +

g1 dx. (2.6.34) {|u|>k/5}

By virtue of relations (2.2.3), (2.2.4), (2.6.18), and (2.6.19) and Lemmas 2.2.1, 2.2.2, and 2.2.4, we conclude that, for any α, |α| = 1, |δ α gk,m | 6 c0 |δ α v| a.e. on {|v| 6 2k} and, for any α, |α| = 2, |δ α gk,m | 6 c0 |δ α v| + 4c0

X

|δ β v|2

a.e. on {|v| 6 2k}.

|β|=1

Then X

Z 00 Bk,m

Z

X

00 Bk,m

 |δ α gk,m |q dx 6 cq0 Ik,m ,

(2.6.35)

|α|=1

 2p/q |δ α gk,m |p dx 6 (2c0 )p Ik,m + c17 cp0 Ik,m .

(2.6.36)

|α|=2

Relations (2.6.33)–(2.6.36) imply that X  Z α |Aα (x, δ2 u)||δ gk,m | dx 0 00 Ek,m ∪Ek,m

α∈Λ

6

c18 cq0 [Ik,m

+

2p/q Ik,m ]

Thus, in view of (2.6.32), we obtain X  Z α α Aα (x, δ2 u)(δ u − δ v) h0m (u − v)dx Ek,m

α∈Λ

Z +2

g1 dx. {|u|>k/5}

Section 2.7

115

Theorems on existence

 2p/q  0 6 Ik,m + c18 cq0 Ik,m +Ik,m Z Z g1 dx + +2 {|u|>k/5}

g2 dx.

(2.6.37)

{|u|>k}∪{|v|>k}

Since u and v are equal in rights, by analogy with (2.6.37), we get X  Z α α Aα (x, δ2 v)(δ v − δ u) h0m (v − u)dx Ek,m α∈Λ

 2p/q  00 6 Ik,m + c18 cq0 Ik,m +Ik,m Z Z +2 g1 dx + {|v|>k/5}

g2 dx.

(2.6.38)

{|u|>k}∪{|v|>k}

Adding inequalities (2.6.37) and (2.6.38), in view of the fact that the function hm is odd and relations (2.1.5), (2.6.27), (2.6.29), and (2.6.31), we find  X Z α α [Aα (x, δ2 u) − Aα (x, δ2 v)](δ u − δ v) dx {|u−v|6m,|u|6k,|v|6k}

α∈Λ

Z [ |F (x, u)| + |F (x, v)| + g1 + g2 ]dx

6 4m {|u|>k/5}∪{|v|>k/5}

i h + 2c18 cq0 mσk + cm−γ + (mσk + cm−γ )2p/q + 2cm−γ . In this inequality, we first pass to the limit as k → ∞ and then to the limit as m → ∞. As a result, taking into account (2.1.5), we obtain X [ Aα (x, δ2 u) − Aα (x, δ2 v) ](δ α u − δ α v) = 0 a.e. on Ω. α∈Λ

Due to (2.1.5), this yields δ2 u = δ2 v a.e. on Ω. Thus, by virtue of Lemma 2.6.1, u = v a.e. on Ω. 

2.7

Theorems on existence

In this section, we prove the following fundamental results: Theorem 2.7.1. Assume that the following conditions are satisfied: 1) for almost all x ∈ Ω, the function F (x, ·) is nonincreasing on R; 2) for any s ∈ R, the function F (·, s) belongs to L1 (Ω). Then there exists an H-solution of problem (2.1.6), (2.1.7). We set

3n − 2 p. n+p−1 By virtue of (2.1.1), we conclude that p1 ∈ (2p, n). p1 =

116 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity Theorem 2.7.2. Let conditions 1) and 2) of Theorem 2.7.1 be satisfied and let q > p1 . Then there exists an entropy solution of problem (2.1.6), (2.1.7). We start from the common part of the proof of Theorems 2.7.1 and 2.7.2. It is connected with the estimation of the solutions of approximating problems for problem (2.1.6), (2.1.7). To this end, we introduce a special sequence of functions. We set a1 = 4 −

14 −1/2 e , 3

a2 = 4 − 5e−1/2 ,

and define a function χ : R → R by setting ( s − a1 s3 + a2 s4 χ(s) = a3 − e−s

a3 =

1 61 −1/2 + e 4 48

for s 6 1/2, for s > 1/2.

For any k ∈ N, we now define a function χk : R → R by setting  s for |s| 6 k, i χk (s) = h  |s|−k   χ + 1 k sign s for |s| > k. k Thus, for any k ∈ N, we have χk ∈ C 2 (R), |χk | 6 3k

on R,

(2.7.1)

6 1 on R, 8 |χ00k | 6 χ0k on R. k

(2.7.2)

0
k} ◦

Proof. We fix any k, l ∈ N. We have χk (ul ) ∈ W 1,q 2,p (Ω) and, in addition, Dα χk (ul ) = χ0k (ul )Dα ul

(2.7.10)

for α with |α| = 1 a.e. on Ω. Moreover, for α with |α| = 2 a.e. on Ω, X |Dα χk (ul ) − χ0k (ul )Dα ul | 6 |χ00k (ul )| |Dβ ul |2 . (2.7.11) |β|=1

Substituting the function χk (ul ) in (2.7.7) instead of v, we obtain  Z X α Aα (x, ∇2 ul )D χk (ul ) dx Ω

α∈Λ

Z +

Z Fl (x, ul )χk (ul )dx =

Ω

fl χk (ul )dx.

(2.7.12)

Ω

By Ik,l we denote the first integral on the left-hand side of inequality (2.7.12). We set  Z X X α q α p Jk,l = |D ul | + |D ul | χ0k (ul )dx. Ω

|α|=1

|α|=2

Using (2.1.4), (2.7.10), and (2.7.11), we get Z Ik,l > c2 Jk,l − g2 χ0k (ul )dx  X  Z  XΩ − |Aα (x, ∇2 ul )| |Dα ul |2 |χ00k (ul )|dx. Ω

|α|=2

|α|=1

(2.7.13)

118 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity We now establish a suitable estimate for the last integral on the right-hand 0 and set side of inequality (2.7.13). We denote this integral by Jk,l c21 =

i−1 c2 h 2p/(p−1) c1 n + nq/2 . 16

By using (2.4.15), the Young inequality, and (2.1.3), we obtain  X  X α 2 |Aα (x, ∇2 ul )| |D ul | |α|=2

|α|=1

c2 6 16

X

α

q

|D ul | +

|α|=1

X

α

p

|D ul |

 +

|α|=2

c2 g1 + c22 16c1

a.e. on Ω.

Thus, in view of (2.7.2) and (2.7.3), we find 0 6 Jk,l

c2 c2 Jk,l + |g1 |1 + c22 meas Ω. 2 2c1

(2.7.14)

Relations (2.7.13) and (2.7.14) yield the inequality Ik,l >

c2 Jk,l − c23 . 2

Hence, in view of (2.7.12) and (2.7.1), we arrive at the inequality Z c2 Jk,l + Fl (x, ul )χk (ul )dx 6 3k|fl |1 + c23 . 2 Ω

(2.7.15)

By virtue of (2.7.4) and the properties of the function χk , we conclude that Fl (x, ul )χk (ul ) > 0 a.e. on Ω, Fl (x, ul )χk (ul ) > k|Fl (x, ul )| a.e. on {|ul | > k}. Further, in view of relations (2.7.6) and (2.7.15), this yields Z c2 Jk,l + k |Fl (x, ul )|dx 6 3k(|f |1 + 1) + c23 . 2 {|ul |>k} Then  Z X X c2 α q α p |D ul | + |D ul | χ0k (ul )dx 6 3k(|f |1 + 1) + c23 , (2.7.16) 2 Ω |α|=1 |α|=2 Z k |Fl (x, ul )|dx 6 3k(|f |1 + 1) + c23 . (2.7.17) {|ul |>k}

Since χ0k = 1 on [−k, k], inequality (2.7.8) follows from (2.7.16). Relation (2.7.17) implies inequality (2.7.9). 

Section 2.7

119

Theorems on existence

Lemmas 2.7.1 and 2.2.5 yield the following assertion: Lemma 2.7.2. For any k, l ∈ N, the following inequalities are true: ∗

meas {|ul | > k} 6 c24 [ |f |1 + 1 ]n/(n−q) k −r , X  α meas |D ul | > k 6 c25 [ |f |1 + 1 ]n/(n−1) k −r ,

(2.7.18) (2.7.19)

|α|=1

meas

X

α



|D ul | > k

6 c25 [ |f |1 + 1 ]n/(n−1) k −pr/q .

(2.7.20)

|α|=2

Lemma 2.7.3. For any k, l ∈ N, the following estimate holds: Z Z c26 |Fl (x, ul )|dx 6 |f |dx + [ |f |1 +1 ]k −1 + |fl −f |1 . (2.7.21) {|ul |>2k}

{|ul |>k}

Proof. Let z be a function from the class C 2 (R) such that 0 6 z 6 1 on R, z = 0 on [−1, 1], z = 1 on (−∞, −2] ∪ [2, +∞), and z 0 (s) sign s > 0 for any s ∈ R. We set  c˜ = max max |z 0 |, max |z 00 | . R

R

We now fix any k, l ∈ N and define a function zk : R → R, by setting s  s  z zk (s) = h1 k k for any s ∈ R. It is clear that zk ∈ C 2 (R). By virtue of (2.2.2)–(2.2.4) and the properties of the function z, for any s ∈ R, we find |zk (s)| 6 2,

(2.7.22)

0 6 zk0 (s) 6 (1 + 2c)k ˜ −1 ,

(2.7.23)

|zk00 (s)| 6 (3 + 4c)k ˜ −2 .

(2.7.24)

Moreover, ∀ s ∈ R, |s| 6 k,

zk (s) = 0;

(2.7.25)

∀ s ∈ R, |s| > 2k,

|zk (s)| > 1;

(2.7.26)

∀ s ∈ R, |s| > 2k,

zk00 (s) = 0.

(2.7.27)

◦

By properties of the function zk , we get zk (ul ) ∈ W 1,q 2,p (Ω). Moreover, for α with |α| = 1 a.e. on Ω, we find Dα zk (ul ) = zk0 (ul )Dα ul ,

(2.7.28)

120 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity and, for α with |α| = 2 a.e. on Ω, |Dα zk (ul ) − zk0 (ul )Dα ul | 6 |zk00 (ul )|

X

|Dβ ul |2 .

(2.7.29)

|β|=1

Substituting the function zk (ul ) in (2.7.7) instead of v and using (2.7.22) and (2.7.25), we obtain  Z X α Aα (x, ∇2 ul )D zk (ul ) dx Ω

α∈Λ

Z

Z

+

|f |dx + 2|fl − f |1 .

Fl (x, ul )zk (ul )dx 6 2

(2.7.30)

{|ul |>k}

Ω

0 We denote the first integral on the left-hand side of inequality (2.7.30) by Ik,l and set  X  Z X α 2 00 |Aα (x, ∇2 ul )| |D ul | |zk00 (ul )|dx. Ik,l = Ω

|α|=2

|α|=1

Using (2.1.4), (2.7.23), (2.7.25), (2.7.28), and (2.7.29), we get 00 0 − (1 + 2c)|g ˜ 2 |1 k −1 . Ik,l > −Ik,l

This relation and (2.7.30) imply that Z Fl (x, ul )zk (ul )dx Ω Z 00 62 |f |dx + (1 + 2c)|g ˜ 2 |1 k −1 + 2|fl − f |1 + Ik,l . (2.7.31) {|ul |>k}

By virtue of (2.7.4) and the definition of the function zk , we find Fl (x, ul )zk (ul ) > 0 a.e. on Ω, and, by virtue of (2.7.26), Fl (x, ul )zk (ul ) > |Fl (x, ul )| a.e. on {|ul | > 2k}. Thus, Z

Z |Fl (x, ul )|dx.

Fl (x, ul )zk (ul )dx > Ω

{|ul |>2k}

This inequality and (2.7.31) yield the inequality Z |Fl (x, ul )|dx {|ul |>2k} Z 00 62 |f |dx + (1 + 2c)|g ˜ 2 |1 k −1 + 2|fl − f |1 + Ik,l . {|ul |>k}

(2.7.32)

Section 2.7

121

Theorems on existence

00 . By virtue of (2.7.27) and (2.7.24), It remains to estimate the integral Ik,l we get  X  X Z α 2 00 −2 |Aα (x, ∇2 ul )| |D ul | dx. (2.7.33) Ik,l 6 (3 + 4c)k ˜ {|ul |62k}

|α|=2

|α|=1

000 . Using Denote the integral on the right-hand side of inequality (2.7.33) by Ik,l (2.4.15), the Young inequality, and (2.1.3), we obtain X  Z X 000 α q α p Ik,l 6 c27 |D ul | + |D ul | dx + c28 . {|ul |62k}

|α|=1

|α|=2

000 6 c [ |f | + 1 ]k. This inequality Hence, in view of Lemma 2.7.1, we find Ik,l 29 1 and (2.7.33) enable us to conclude that 00 6 c30 [ |f |1 + 1 ]k −1 . Ik,l

By using this estimate and (2.7.32), we arrive at (2.7.21).



For any t > 0 and i, j ∈ N, we set Mt (i, j) = meas{|ui − uj | > t},  X α α |D ui − D uj | > t . Nt (i, j) = meas α∈Λ

We now establish some estimates for the introduced quantities. They play an important role in the proof of existence of solutions to problem (2.1.6), (2.1.7). The required estimate for Mt (i, j) can be readily obtained. This is done in the following lemma: Lemma 2.7.4. Let t > 0, i, j, k ∈ N. Then Z ∗ Mt (i, j) 6 2c24 [ |f |1 +1 ]n/(n−q) k −r + t−q |hk (ui )−hk (uj )|q dx.

(2.7.34)

Ω

Proof. We set B = {|hk (ui ) − hk (uj )| > t}. Since {|ui − uj | > t} ⊂ {|ui | > k} ∪ {|uj | > k} ∪ B, we get Mt (i, j) 6 meas {|ui | > k} + meas {|uj | > k} + meas B. This inequality, relation (2.7.18), and the fact that Z −q meas B 6 t |hk (ui ) − hk (uj )|q dx Ω

yield (2.7.34).



122 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity We now establish the required estimate for Nt (i, j). For any t > 0 and m, k, i, j ∈ N, we set X X Et,m,k (i, j) = |Dα ui −Dα uj | > t, |Dα ui | 6 m, α∈Λ

α∈Λ

X

α

|D uj | 6 m,

α∈Λ

 1 |ui | 6 k, |uj | 6 k, |ui − uj | 6 . k

Lemma 2.7.5. Let t > 0 and m, k, i, j ∈ N. Then ∗

Nt (i, j) 6 c31 [ |f |1 + 1 ]n/(n−q) (m−pr/q + k −r ) + M1/k (i, j) + meas Et,m,k (i, j).

(2.7.35)

Proof. We have   X  X X |Dα uj | > m |Dα ui | > m ∪ |Dα ui − Dα uj | > t ⊂ α∈Λ

α∈Λ

α∈Λ



∪ |ui | > k} ∪ {|uj | > k  ∪ |ui − uj | > 1/k ∪ Et,m,k (i, j). Hence, Nt (i, j) 6 meas

X

  X |Dα uj | > m |Dα ui | > m + meas α∈Λ

α∈Λ

+ meas {|ui | > k} + meas {|uj | > k} + M1/k (i, j) + meas Et,m,k (i, j). Thus, by using (2.7.18)–(2.7.20), we arrive at inequality (2.7.35).



In order to be able to efficiently use estimate (2.7.35), we now establish one more estimate for some integrals connected with the sets Et,m,k (i, j). In this connection, we introduce several auxiliary functions and sets. Definition 2.7.1. Let x ∈ Ω. Then Ax is a function on Rn,2 × Rn,2 such that, for any couple (ξ, ξ 0 ) ∈ Rn,2 × Rn,2 , X Ax (ξ, ξ 0 ) = [Aα (x, ξ) − Aα (x, ξ 0 )](ξα − ξα0 ). α∈Λ

Since Aα , α ∈ Λ, are Carathéodory functions and inequality (2.1.5) is true for almost all x ∈ Ω and all ξ, ξ 0 ∈ Rn,2 , ξ 6= ξ 0 , there exists a set E ⊂ Ω of measure zero such that (i) for any x ∈ Ω\E, the function Ax is continuous on Rn,2 × Rn,2 ; (ii) for any x ∈ Ω\E and ξ, ξ 0 ∈ Rn,2 , ξ 6= ξ 0 , the inequality Ax (ξ, ξ 0 ) > 0 is valid.

Section 2.7

123

Theorems on existence

For any t > 0 and m ∈ N, m > t, we set   X X X Gt,m = (ξ, ξ 0 ) ∈ Rn,2 ×Rn,2 : |ξα −ξα0 | > t, |ξα | 6 m, |ξα0 | 6 m . α∈Λ

α∈Λ

α∈Λ

It is easy to see that, for any t > 0 and m ∈ N, m > t, the set Gt,m is nonempty. Definition 2.7.2. Let t > 0 and m ∈ N, m > t. Then µt,m is a function on Ω such that   min Ax , if x ∈ Ω\E, µt,m (x) = Gt,m 0, if x ∈ E. By using assertions (i) and (ii), relation (2.1.3), and the fact that Aα , α ∈ Λ, are Carathéodory functions, we can make the following conclusion: If t > 0 and m ∈ N, m > t, then µt,m > 0 on Ω,

µt,m > 0 a.e. on Ω,

µt,m ∈ L1 (Ω).

(2.7.36) (2.7.37)

For any i, j ∈ N, we set ρ(i, j) = |fi − fj |1 + |Fi (x, ui ) − Fj (x, uj )|1 . Lemma 2.7.6. Let t > 0, m, k, i, j ∈ N, and m > t. Then Z µt,m dx 6 2ρ(i, j)k −1 + c32 [ |f |1 + 1 ]k −(q−2p)/qp Et,m,k (i,j)

+ c32 [ |f |1 + 1 ] [ M1/k (i, j) ](q−2p)/qp k 2 . Proof. Let z and c˜ be, respectively, the function and number introduced in the proof of Lemma 2.7.3. We now define a function τk : R → R by setting, for any s ∈ R, s . τk (s) = 1 − z k We also define one more function σk : R → R by setting, for any s ∈ R, σk (s) =

1 h1 (ks). k

Further, we set wk = σk (ui − uj )τk (ui )τk (uj ). ◦

By the properties of the functions z and h1 and the fact that ui , uj ∈ W 1,q 2,p (Ω), ◦

we conclude that wk ∈ W 1,q 2,p (Ω) and the following assertions are true: a) if α ∈ Λ, then Dα wk = 0 a.e. on {|ui | > 2k} ∪ {|uj | > 2k};

124 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity b) if |α| = 1, then |Dα wk − σk0 (ui −uj )τk (ui )τk (uj )Dα (ui −uj )| 6 2ck ˜ −2 (|Dα ui | + |Dα uj |) a.e. on Ω; c) if |α| = 2, then |Dα wk − σk0 (ui − uj )τk (ui )τk (uj )Dα (ui − uj )| 6 2ck ˜ −2 (|Dα ui | + |Dα uj |) + 2(4c˜2 k −1 + |σk00 (ui − uj )|) X  X β 2 β 2 × a.e. on Ω. |D ui | + |D uj | |β|=1

|β|=1

By virtue of (2.7.7),  Z Z X Aα (x, ∇2 ui )Dα wk + Fi (x, ui )wk dx = fi wk dx, Ω

Ω



Aα (x, ∇2 uj )Dα wk + Fj (x, uj )wk dx =

α∈Λ

Z fj wk dx. Ω

α∈Λ

Hence, Z X Ω

Ω

α∈Λ

Z X

 2 [ Aα (x, ∇2 ui ) − Aα (x, ∇2 uj ) ]D wk dx 6 ρ(i, j). k α

(2.7.38)

By using propositions a)–c), the properties of the functions z and h1 , and relations (2.1.5) and (2.7.38), we obtain  X Z α [Aα (x, ∇2 ui ) − Aα (x, ∇2 uj )]D (ui − uj ) dx Et,m,k (i,j)

α∈Λ

2 6 ρ(i, j) + 2ck ˜ −2 k

+ 8c˜2 k −1

Z

X

Z {|ui | τ.

(2.7.55)

For any s ∈ N, we set Gs = Et,m,ks (lis , ljs ). By virtue of (2.7.52) and (2.7.54), Z lim s→∞

µt,m dx = 0.

Gs

In view of this relation, (2.7.36), and (2.7.37), we obtain lim meas Gs = 0.

s→∞

However, this contradicts (2.7.55). The obtained contradiction proves (2.7.53). Relations (2.7.51) and (2.7.53) imply that there exists k ∈ N such that bk 6 ε. Thus, for any i, j ∈ N, i, j > nk , we get Nt (li , lj ) 6 ε. This result enables us to conclude that, for any α ∈ Λ, the sequence {Dα uli } is fundamental in measure.  By virtue of (2.7.47), Lemma 2.7.7, and the Riesz theorem, there exist measurable functions u : Ω → R and v (α) : Ω → R, α ∈ Λ, such that the sequence {uli } converges to u in measure and, for any α ∈ Λ, the sequence {Dα uli } converges to v (α) in measure. It is well known that the analyzed sequences contain subsequences convergent to the corresponding functions almost everywhere on Ω. Thus, without loss of generality, we can assume that uli → u a.e. on Ω, ∀ α ∈ Λ,

α

D uli → v

(α)

(2.7.56)

a.e. on Ω.

(2.7.57) ◦

By using (2.7.46) and (2.7.56), we conclude that hk (u) ∈ W 1,q 2,p (Ω) for any k ∈ N and ◦ hk (uli ) → hk (u) weakly in W 1,q (2.7.58) 2,p (Ω). ◦

Thus, we get u ∈ H 1,q 2,p (Ω).

128 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity Lemma 2.7.8. For any α ∈ Λ, the equality δ α u = v (α) is true a.e. on Ω. Proof. Let α ∈ Λ. We fix any k ∈ N. By virtue of Lemma 2.2.2 for any i ∈ N, we find Dα hk (uli ) = Dα uli a.e. on {|uli | 6 k}. Thus, in view of relations (2.7.56) and (2.7.57), we obtain Dα hk (uli ) → v (α)

a.e. on {|u| < k}.

Hence, by using (2.7.44) and (2.7.45), we conclude that the function v (α) is integrable on the set {|u| < k} and, for any function ϕ ∈ L∞ (Ω), Z lim [Dα hk (uli ) − v (α) ]ϕ dx = 0. (2.7.59) i→∞ {|u| 0 be a given number. It is clear that there exists ε1 > 0 such that, for any measurable set G ⊂ Ω satisfying the inequality meas G 6 ε1 , we have Z (|f | + |F (x, u)|)dx 6 ε. G

We fix k ∈ N such that the following inequalities are satisfied: [ |f |1 + 1 ]k −1 6 ε, c24 [ |f |1 + 1 ]n/(n−q) k

−r∗

(2.7.65) 6 ε1 .

(2.7.66)

According to condition 2) of Theorem 2.7.1, the functions F (·,−2k) and F (·, 2k) belong to L1 (Ω). Hence, there exists ε2 > 0 such that, for any measurable set G ⊂ Ω satisfying the inequality meas G 6 ε2 , we get Z ( |F (·, −2k)| + |F (·, 2k)| )dx 6 ε. G

By virtue of (2.7.64), there exists a measurable set Ω1 ⊂ Ω such that meas (Ω\Ω1 ) 6 min (ε1 , ε2 )

(2.7.67)

and Fli (x, uli ) → f − F (x, u) uniformly on Ω1 . Then there exists i1 ∈ N such that, for any i ∈ N, i > i1 , Z (2.7.68) |Fli (x, uli ) − (f − F (x, u))|dx 6 ε. Ω1

In addition, by virtue of (2.7.5), there exists i2 ∈ N such that, for any i ∈ N, i > i2 , |fli − f |1 6 ε. (2.7.69) We fix i ∈ N, i > max(i1 , i2 ). Using (2.7.67) and (2.7.68), we obtain |Fli (x, uli ) − (f − F (x, u))|1 Z Z |Fli (x, uli )|dx + 6 {|uli |>2k}

|Fli (x, uli )|dx + 2ε. (2.7.70)

(Ω\Ω1 )∩{|uli | k} 6 ε1 . Then Z |f |dx 6 ε. (2.7.71) {|uli |>k}

Relations (2.7.21), (2.7.65), (2.7.69), and (2.7.71) imply that Z |Fli (x, uli )| dx 6 3c−1 26 ε. {|uli |>2k}

(2.7.72)

130 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity We note that, in view of condition 1) of Theorem 2.7.1, |Fli (x, uli )| 6 2(|F (·, −2k)| + |F (·, 2k)|) a.e. on {|uli | < 2k}. This fact and the inequality Z ( |F (·, −2k)| + |F (·, 2k)| )dx 6 ε Ω\Ω1

[valid by virtue of (2.7.67)] yield the inequality Z |Fli (x, uli )| dx 6 2ε.

(2.7.73)

(Ω\Ω1 )∩{|uli | 0, it is clear that there exists ε1 > 0 such that, for any measurable set G ⊂ Ω satisfying the inequality meas G 6 ε1 , we get Z |Aα (x, δ2 u)|dx 6 ε/4. G

By virtue of (2.7.74), there exists a measurable set Ω1 ⊂ Ω such that  −n/(n−1) λ/(λ−1) ) meas (Ω \ Ω1 ) 6 min ε1 , (4−r/(r−λ1 ) ε c−1 42 [ |f |1 + 1 ] and Aα (x, ∇2 uli ) → Aα (x, δ2 u) uniformly on Ω1 . Then there exists i1 ∈ N such that, for any i ∈ N, i > i1 , Z (2.7.76) |Aα (x, ∇2 uli ) − Aα (x, δ2 u)|dx 6 ε/2. Ω1

We fix i ∈ N, i > i1 . Using the inequality for meas(Ω\Ω1 ) and relations (2.7.75) and (2.7.76), we obtain |Aα (x, ∇2 uli ) − Aα (x, δ2 u)|1 Z |Aα (x, ∇2 uli )|dx + 3ε/4 6 Ω\Ω1

 (λ−1)/λ 6 meas(Ω\Ω1 )

Z

λ

|Aα (x, ∇2 uli )| dx

1/λ + 3ε/4 6 ε.

Ω

Hence, Aα (x, ∇2 uli ) → Aα (x, δ2 u) strongly in L1 (Ω).



Relations (2.7.7) and (2.7.5) and Lemmas 2.7.9 and 2.7.10 imply that, for any function ϕ ∈ C0∞ (Ω),  Z X Z α Aα (x, δ2 u)δ ϕ dx = F (x, u)ϕdx. Ω

α∈Λ

Ω

The established properties of the function u enable us to conclude that u is an H-solution of problem (2.1.6), (2.1.7).

132 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity Lemma 2.7.11. Let q > p1 . Then there exist c > 0, b ∈ (1, r), and γ > 0 such that, for any ϕ ∈ C0∞ (Ω) and k ∈ N, X  Z α α Aα (x, δ2 u)(δ u − δ ϕ) h0k (u − ϕ)dx {|u−ϕ| p1 , we have   1 p−1 q + 2 < 1. r p We fix a number σ such that   1 p−1 q + 2 < σ < 1, r p and set c = 4r

(2.7.77)

∗ +3+1/(1−σ)

n4 (c1 +1)(1+c24 +c25 +|g1 |1 +meas Ω) [ |f |1 +1 ]n/(n−q) ,   p−1 prσ b = rσ, γ = rσ − q + 2 , r1 = . p q(p − 1)

By virtue of (2.7.77), we get b ∈ (1, r) and γ > 0. It is clear that 2 γ 1 + + = 1. r1 b b

(2.7.78)

Further, we fix ϕ ∈ C0∞ (Ω) and k ∈ N and set, for any l ∈ N,  Z X (1) α Il = Aα (x, ∇2 ul )D ϕ h0k (ul − ϕ)dx, Ω (2) Il

α∈Λ

Z

{fl − Fl (x, ul )}hk (ul − ϕ)dx,  X  |Aα (x, ∇2 ul )| |Dα (ul − ϕ)|2 |h00k (ul − ϕ)|dx. =

Ω

Il =

Z X Ω

|α|=2

|α|=1

Let l ∈ N. We set vl = hk (ul − ϕ). By virtue of Lemma 2.2.2, we have ◦

vl ∈ W 1,q 2,p (Ω), and the following assertions are true: a) for any n-dimensional multiindex α, |α| = 1, Dα vl = h0k (ul − ϕ)Dα (ul − ϕ) a.e. on Ω; b) for any n-dimensional multiindex α, |α| = 2, X |Dα vl − h0k (ul −ϕ)Dα (ul −ϕ)| 6 |h00k (ul −ϕ)| |Dβ (ul −ϕ)|2 a.e. on Ω. |β|=1

Section 2.7

133

Theorems on existence

Moreover, according to (2.7.7),  Z X (2) Aα (x, ∇2 ul )Dα vl dx = Il . Ω

α∈Λ

In view of the assertions a) and b), this yields the inequality  Z X (1) (2) α Aα (x, ∇2 ul )D ul h0k (ul − ϕ)dx 6 Il + Il + Il . Ω

(2.7.79)

α∈Λ

We now estimate the integral Il . To this end, we set X 2 X 2 X α α Φl = |Aα (x, ∇2 ul )|, Φ1,l = |D ul | , Φ = |D ϕ| , |α|=2

|α|=1

|α|=1

El = {|ul − ϕ| > k}. In view of (2.2.4) and the fact that h00k (s) = 0, for s ∈ [ −k, k ], we obtain Z Z Il 6 6 Φl Φ1,l dx + 6 Φl Φ dx. (2.7.80) El

El

By virtue of relations (2.1.3), (2.7.19), and (2.7.20) and Lemma 2.2.6, we get  c 1/r1 , (2.7.81) Φl ∈ Lr1 (Ω), |Φl |r1 6 n2 16n4  c 2/b Φ1,l ∈ Lb/2 (Ω), |Φ1,l |b/2 6 . (2.7.82) 16n4 Further, by setting El0 = El ∩ {|ϕ| 6 k/2},

El00 = El ∩ {|ϕ| > k/2},

we find meas El = meas El0 + meas El00 .

(2.7.83)

It is easy to see that meas El0 6 meas {|ul | > k/2}. In view of (2.7.18), this yields meas El0 6

c ∗ k −r . 4 32n

For any x ∈ El00 , we obtain 1 < 2k −1 |ϕ(x)|. Therefore, Z meas El00 6 2b k −b |ϕ|b dx.

(2.7.84)

(2.7.85)

Ω

Relations (2.7.83)–(2.7.85) yield meas El 6

c (1 + |ϕ|b )b k −b . 16n4

(2.7.86)

134 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity Taking into account relation (2.7.78) and using the Hölder inequality and relations (2.7.80), (2.7.81), (2.7.82), and (2.7.86), we get Il 6 6|Φl |r1 |Φ1,l |b/2 (meas El )γ/b + 6|Φl |r1 |Φ|b/2 (meas El )γ/b 6 c(1 + kϕkW 1,b (Ω) )b k −γ . Thus, it follows from (2.7.79) that, for any l ∈ N,  Z X (1) (2) Aα (x, ∇2 ul )Dα ul h0k (ul −ϕ)dx 6 Il + Il + c(1+kϕkW 1,b (Ω) )b k −γ . Ω

α∈Λ

(2.7.87)

By virtue of relations (2.2.2), (2.2.3), (2.7.5), and (2.7.56) and Lemmas 2.7.9 and 2.7.10, we find  Z X (1) α (2.7.88) lim Ili = Aα (x, δ2 u)D ϕ h0k (u − ϕ)dx, i→∞

Ω

α∈Λ (2)

lim Ili =

i→∞

Z F (x, u)hk (u − ϕ)dx.

(2.7.89)

Ω

Hence, in view of relations (2.1.4), (2.2.3), (2.7.56), (2.7.61), and (2.7.74) and the Fatou lemma, relations (2.7.87)–(2.7.89) imply that  Z X α Aα (x, δ2 u)δ u h0k (u − ϕ)dx Ω

α∈Λ

6

Z X Ω

 Aα (x, δ2 u)Dα ϕ h0k (u − ϕ)dx

α∈Λ

Z

F (x, u)hk (u − ϕ)dx + c(1 + kϕkW 1,b (Ω) )b k −γ .

+



Ω

Thus, by using conditions 1) and 2) of Theorem 2.7.1, we have shown that u is an H-solution of problem (2.1.6), (2.1.7). Under the assumption that q > p1 , by using the above-mentioned conditions, relation (2.7.62), and Lemma 2.7.11, we conclude that u is an entropy solution of problem (2.1.6), (2.1.7). Thus, Theorems 2.7.1 and 2.7.2 are proved.

2.8

Entropy solutions as elements of the Sobolev spaces and the existence of W -solutions ◦

1 Lemma 2.8.1. Let u ∈ H 1,q 2,p (Ω)∩L (Ω), let α be an n-dimensional multiindex such that |α| = 1, and let δ α u ∈ L1 (Ω). Then there exists the weak derivative Dα u and Dα u = δ α u a.e. on Ω.

Section 2.8

Entropy solutions as elements of the Sobolev spaces

135

Proof. We fix an arbitrary function w ∈ C0∞ (Ω). Thus, for any k ∈ N, we obtain Z Z   (uDα w + wδ α u)dx 6 max |Dα w| |hk (u) − u| dx Ω Ω Ω Z   + max |w| |Dα hk (u) − δ α u| dx. (2.8.1) Ω

Ω

Note that hk (u) → u pointwise on Ω and, for any k ∈ N, the inequality |hk (u) − u| 6 2|u| is true. Hence, by virtue of the Lebesgue theorem on the limit transition under the integral sign, we obtain Z lim |hk (u) − u|dx = 0. (2.8.2) k→∞

Ω

By using Lemmas 2.2.1, 2.2.2, and 2.2.4 and relation (2.2.3), we conclude, for any k ∈ N, that Z Z α α |D hk (u) − δ u|dx 6 2 |δ α u|dx. {|u|>k}

Ω

This relation and the fact that meas {|u| > k} → 0 as k → ∞ (by virtue of the inclusion u ∈ L1 (Ω)) yield the relation Z lim |Dα hk (u) − δ α u|dx = 0. (2.8.3) k→∞

Ω

Relations (2.8.1)–(2.8.3) imply that Z Z uDα w dx = − wδ α u dx. Ω

Ω

In view of the arbitrariness of w, this equality implies that there exists the weak derivative Dα u and, moreover, Dα u = δ α u a.e. on Ω.  ◦

1 Lemma 2.8.2. Let u ∈ H 1,q 2,p (Ω) ∩ L (Ω), let α be an n-dimensional multiindex such that |α| = 2, and let δ α u ∈ L1 (Ω). Moreover, assume that Z lim |Dα hk (u)|dx = 0. (2.8.4) k→∞ {|u|>k}

Then there exists the weak derivative Dα u and Dα u = δ α u a.e. on Ω. Proof. We fix an arbitrary function w ∈ C0∞ (Ω) and conclude, for any k ∈ N, that Z Z   (uDα w − wδ α u)dx 6 max |Dα w| |hk (u) − u| dx Ω Ω Ω Z   + max |w| |Dα hk (u) − δ α u| dx. (2.8.5) Ω

Ω

136 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity By virtue of Lemma 2.2.4, for any k ∈ N, we obtain Z Z Z α α α |D hk (u)|dx + |D hk (u) − δ u|dx 6

|δ α u| dx.

{|u|>k}

{|u|>k}

Ω

Using this relation, the inclusions u ∈ L1 (Ω) and δ α u ∈ L1 (Ω), and condition (2.8.4), we conclude that Dα hk (u) → δ α u strongly in L1 (Ω). It is also clear that hk (u) → u strongly in L1 (Ω). Thus, relation (2.8.5) yields the equality Z Z wδ α udx. uDα wdx = Ω

Ω

In view of the arbitrariness of w, this equality implies that there exists the weak derivative Dα u and Dα u = δ α u a.e. on Ω.  Theorem 2.8.1. Let u be an entropy solution of problem (2.1.6), (2.1.7). ◦

Then u ∈ W 1,λ (Ω) for any λ ∈ (1, r). Proof. Let λ ∈ (1, r). By virtue of Lemmas 2.3.2 and 2.8.1, we have u ∈ Lλ (Ω) and, for any α, |α| = 1, there exists the weak derivative Dα u, Dα u ∈ Lλ (Ω). Hence, u ∈ W 1,λ (Ω). ◦

Let k ∈ N. We have hk (u) ∈ W 1,λ (Ω). Lemmas 2.2.1, 2.2.2, and 2.2.4, and relation (2.2.3) imply that, for any α, |α| = 1, |Dα hk (u)| 6 |Dα u| a.e. on Ω. This inequality yields the following inequality: khk (u)kW 1,λ (Ω) 6 kukW 1,λ (Ω) . It is now clear that there exist an increasing sequence {ki } ⊂ N and a function ◦

◦

v ∈ W 1,λ (Ω) such that hki (u) → v weakly in W 1,λ (Ω). Thus, hki (u) → v strongly in Lλ (Ω). On the other hand, hk (u) → u strongly in Lλ (Ω). Hence, ◦

u = v a.e. on Ω and, therefore,u ∈ W 1,λ (Ω). We set p2 =



np . n(p − 1) + 1

Note that, by virtue of (2.1.1), p2 ∈ (1, n). We also note that the inequalities rp/q > 1 and p2 > 2p are equivalent to the inequalities q > p2 and p < 3/2 − 1/n, respectively. Theorem 2.8.2. Let q > p2 , and let u be an entropy solution of problem (2.1.6), (2.1.7). Then, for any n-dimensional multiindex α, |α| = 2, there exists the weak derivative Dα u and, moreover, Dα u = δ α u a.e. on Ω.

Section 2.8

Entropy solutions as elements of the Sobolev spaces

137

Proof. Let α be any n-dimensional multiindex such that |α| = 2. We now show that Z |Dα hk (u)|dx → 0 as k → ∞. (2.8.6) {|u|>k}

First, by virtue of Lemmas 2.3.1 and 2.2.5, we conclude that there exists M > 0 such that, for any k ∈ N, ∗

meas{|u| > k} 6 M k −r , X  X |δ β u|q + |δ β u|p dx 6 M k.

Z {|u|62k}

|β|=1

(2.8.7) (2.8.8)

|β|=2

We fix k ∈ N. It is clear that Z |Dα hk (u)|dx 6 [meas{|u| > k}](p−1)/p |Dα hk (u)|p .

(2.8.9)

{|u|>k}

Note that, by virtue of Lemmas 2.2.1, 2.2.2, and 2.2.4 and relations (2.2.3) and (2.2.4), we get X |Dα hk (u)| 6 |δ α u| + 3 |δ β u|2 a.e. on {|u| 6 2k}. |β|=1

Moreover, by virtue of Lemma 2.2.3, Dα hk (u) = 0 a.e. on {|u| > 2k}. In view of these properties and (2.8.8), we obtain |Dα hk (u)|p 6 6n(M + n meas Ω)1/p k 1/p .

(2.8.10)

Relations (2.8.7), (2.8.9), and (2.8.10) imply that Z ∗ |Dα hk (u)|dx 6 6n(M + n meas Ω)k −r (p−1)/p+1/p . {|u|>k}

By using this relation and the fact that the inequality −r∗ (p − 1)/p +1/p < 0 holds for q > p2 , we arrive at property (2.8.6). We also note that the condition q > p2 and Lemma 2.3.2 yield the inclusion δ α u ∈ L1 (Ω). By virtue of Lemma 2.3.2, we also have u ∈ L1 (Ω). Further, by using Lemma 2.8.2, we can show that there exists the weak derivative Dα u and, moreover, Dα u = δ α u a.e. on Ω.  Remark 2.8.1. Theorem 2.8.2 and Lemma 2.3.2 imply that if q > p2 and u is an entropy solution of problem (2.1.6), (2.1.7), then Dα u ∈ Lλ (Ω) for any α, |α| = 2, and λ ∈ [1, rp/q). Theorem 2.8.3. Let q > p2 and let u be an entropy solution of prob◦

lem (2.1.6), (2.1.7). Then u ∈ W 2,1 (Ω). Proof. Theorems 2.8.1 and 2.8.2 and Lemma 2.3.2 imply that u ∈ W 2,1 (Ω).

138 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity ◦

We fix k ∈ N. It is clear that hk (u) ∈ W 2,1 (Ω). Using Lemmas 2.2.4, 2.3.2, and 2.8.1 and Theorem 2.8.2, we obtain XZ |Dα hk (u)|dx khk (u) − ukW 2,1 (Ω) 6 |hk (u) − u|1 + α∈Λ {|u|>k}

+

XZ

|Dα u|dx.

(2.8.11)

α∈Λ {|u|>k}

By the proofs of Theorems 2.8.1 and 2.8.2, for any α, |α| = 1, we obtain |Dα hk (u)| 6 |Dα u| a.e. on Ω. Moreover, for any α, |α| = 2, we get property (2.8.6). Hence, it follows from (2.8.11) that khk (u) − ukW 2,1 (Ω) → 0 ◦

as k → ∞. Therefore, u ∈ W 2,1 (Ω).



Definition 2.8.1. The W -solution of problem (2.1.6), (2.1.7) is defined as ◦

a function u ∈ W 2,1 (Ω) satisfying the conditions: 1) F (x, u) ∈ L1 (Ω); 2) Aα (x, ∇2 u) ∈ L1 (Ω) for any α ∈ Λ; 3) for any function ϕ ∈ C0∞ (Ω),  Z Z X Aα (x, ∇2 u)Dα ϕ dx = F (x, u)ϕ dx. Ω

Ω

α∈Λ

Theorem 2.8.4. Let q > p2 and let u be an entropy solution of problem (2.1.6), (2.1.7). Then u is a W -solution of problem (2.1.6), (2.1.7). ◦

Proof. By Theorem 2.8.3, we have u ∈ W 2,1 (Ω). By virtue of Theorem 2.5.1, u is an H-solution of problem (2.1.6), (2.1.7) and, hence, F (x, u) ∈ L1 (Ω), Aα (x, δ2 u) ∈ L1 (Ω) for any α ∈ Λ, and  Z X Z α Aα (x, δ2 u)δ ϕ dx = F (x, u)ϕ dx Ω

α∈Λ

Ω

for any function ϕ ∈ C0∞ (Ω). By using these properties and the fact that, in view of Lemmas 2.3.2 and 2.8.1 and Theorem 2.8.2, ∇2 u = δ2 u a.e. on Ω, we conclude that u is a W -solution of problem (2.1.6), (2.1.7).  Theorem 2.8.5. Let conditions 1) and 2) of Theorem 2.7.1 be satisfied and let q > p2 . Then there exists a W -solution of problem (2.1.6), (2.1.7). Proof. It follows from the results presented in Sec. 2.7 and conditions 1) ◦

and 2) of Theorem 2.7.1 that there exists a function u ∈ H 1,q 2,p (Ω), which is an H-solution of problem (2.1.6), (2.1.7) and has the following properties: a) u ∈ L1 (Ω);

◦

Section 2.9

On the summability of functions from H 1,q 2,p (Ω)

139

b) δ α u ∈ L1 (Ω) for any α, |α| = 1; c) δ α u ∈ Lλ (Ω) for any α, |α| = 2, and λ ∈ (0, pr/q), Z ∗ |Dα hk (u)|dx < +∞. d) for any α, |α| = 2, sup k r (p−1)/p−1/p {|u|>k}

k∈N

By using properties a)–d), Lemmas 2.8.1 and 2.8.2, and the condition q > p2 , we conclude that u ∈ W 2,1 (Ω) and ∇2 u = δ2 u a.e. on Ω. ◦

(2.8.12)

◦

2,1 (Ω). In view of relation (2.8.12), Since u ∈ H 1,q 2,p (Ω), we get {hk (u)} ⊂ W Lemmas 2.2.1–2.2.4, and property (2.2.3), for any k ∈ N, we can write XZ XZ α khk (u) − ukW 2,1 (Ω) 6 2 |D u| dx + |Dα hk (u)| dx. |α|62 {|u|>k}

|α|=2 {|u|>k}

Thus, in view of the fact that meas{|u| > k} → 0 as k → ∞, property d), and the condition q > p2 , we find khk (u) − ukW 2,1 (Ω) → 0 as k → ∞. ◦

Hence, u ∈ W 2,1 (Ω). Using (2.8.12) and that fact that u is an H-solution of problem (2.1.6), (2.1.7), we conclude that u is a W -solution of problem (2.1.6), (2.1.7).  As already indicated, the inequalities p2 > 2p and p < 3/2 − 1/n are equivalent. Hence, in view of the inequality q > 2p and Theorems 2.8.4 and 2.8.5, we arrive at the following assertions: Theorem 2.8.6. Let p > 3/2 − 1/n and let u be an entropy solution of problem (2.1.6), (2.1.7). Then u is a W -solution of problem (2.1.6), (2.1.7). Theorem 2.8.7. Let p > 3/2 − 1/n and let conditions 1) and 2) of Theorem 2.7.1 be satisfied. Then there exists a W -solution of problem (2.1.6), (2.1.7). ◦

2.9

On the summability of functions from H 1,q 2,p (Ω) satisfying certain integral inequalities ◦

Lemma 2.9.1. Let u ∈ H 1,q 2,p (Ω), k ∈ N. Then 1) if α is an n-dimensional multiindex, |α| = 1, then Dα hk (u) = h0k (u)δ α u a.e. on Ω;

140 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity 2) if α is an n-dimensional multiindex, |α| = 2, then X |δ β u|2 a.e. on Ω. |Dα hk (u)| 6 |δ α u| + 3 |β|=1

Proof. We have hk (u) = hk (h2k (u)).

(2.9.1)

Let α be an n-dimensional multiindex with |α| = 1. By virtue of (2.9.1), Dα hk (u) = h0k (h2k (u))Dα h2k (u) a.e. on Ω. Hence, by using properties of the functions hk and h2k and Lemma 2.2.4, we conclude that Dα hk (u) = h0k (u)δ α u a.e. on Ω. Thus, assertion 1) is proved. Now let α be an n-dimensional multiindex with |α| = 2. Using (2.9.1), we get X |Dβ h2k (u)|2 a.e. on Ω. |Dα hk (u)−h0k (h2k (u))Dα h2k (u)| 6 |h00k (h2k (u))| |β|=1

Thus, in view of relations (2.2.3) and (2.2.4) and Lemma 2.2.4, we find X |Dα hk (u)| 6 |δ α u| + 3 |δ β u|2 a.e. on Ω. |β|=1



Hence, assertion 2) is proved. ◦

Proposition 2.9.1. Let u ∈ H 1,q 2,p (Ω), let λ ∈ [1, q], and let, for any n-dimensional multiindex α, |α| = 1, the inclusion δ α u ∈ Lλ (Ω) be true. Then ◦

u ∈ W 1,λ (Ω). ◦

Proof. It is clear that {hk (u)} ⊂ W 1,λ (Ω). Thus, by virtue of relations (1.1.13) and (2.2.3) and Lemma 2.9.1, for any k ∈ N, we arrive at the inequality Z 1/λ∗ XZ 1/λ λ∗ α λ |hk (u)| dx |δ u| dx 6 ncn,λ . Ω

|α|=1 Ω

Hence, in view of the fact that hk (u) → u pointwise in Ω, we obtain u ∈ Lλ (Ω). Since |hk (u)| 6 |u| for any k ∈ N, we conclude that hk (u) → u strongly in Lλ (Ω). In addition, by virtue of Lemmas 2.2.4 and 2.9.1 and property (2.2.3), for any n-dimensional multiindex α, |α| = 1, we see that Dα hk (u) → δ α u strongly in Lλ (Ω). It is now possible to conclude that u ∈ W 1,λ (Ω) and ◦

hk (u) → u strongly in W 1,λ (Ω). Hence, u ∈ W 1,λ (Ω). ◦



Proposition 2.9.2. Let u ∈ H 1,q 2,p (Ω), let λ ∈ [ 1, p ], and let, for any n-dimensional multiindex α, |α| = 1, the inclusion δ α u ∈ L2λ (Ω) be true. Assume that the inclusion δ α u ∈ Lλ (Ω) is true for any n-dimensional multiindex α, ◦

|α| = 2. Then u ∈ W 2,λ (Ω).

◦

Section 2.9

On the summability of functions from H 1,q 2,p (Ω)

141

◦

Proof. It is clear that {hk (u)} ⊂ W 2,λ (Ω). By the proof of Proposition 2.9.1, we have u ∈ W 1,λ (Ω) and hk (u) → u strongly in W 1,λ (Ω). Moreover, for any n-dimensional multiindex α, |α| = 2, by virtue of Lemmas 2.2.4 and 2.9.1, we conclude that Dα hk (u) → δ α u strongly in Lλ (Ω). Thus, it is also possible to conclude that u ∈ W 2,λ (Ω) and hk (u) → u strongly in W 2,λ (Ω). Hence, ◦

u ∈ W 2,λ (Ω).

 ◦

For any u ∈ H 1,q 2,p (Ω) and k ∈ N, we set  Z X X I(u, k) = |δ α u|q + |δ α u|p h0k (u)dx. Ω

|α|=1

|α|=2

◦

Lemma 2.9.2. Let u ∈ H 1,q 2,p (Ω) and k ∈ N. Then ∗

∗

meas{|u| > k} 6 (ncn,q )q k −q [ I(u, k) ]q

∗ /q

.

(2.9.2)

Proof. Since |hk (u)| > k on {|u| > k}, we have Z ∗ ∗ k q meas{|u| > k} 6 |hk (u)|q dx.

(2.9.3)

Ω

We estimate the right-hand side of this inequality by using inequality (1.1.13), Lemma 2.9.1, and property (2.2.3). This yields X Z q∗/q Z ∗ ∗ q∗ q∗ α q |hk (u)| dx 6 (ncn,q ) |D hk (u)| dx 6 (ncn,q )q [ I(u, k) ]q /q . Ω

|α|=1 Ω



Hence, in view of (2.9.3), we arrive at (2.9.2). ◦

Lemma 2.9.3. Let u ∈ H 1,q 2,p (Ω), let α be an n-dimensional multiindex with |α| = 1, and let k, k1 ∈ N. Then ∗

∗

meas{|δ α u| > k} 6 (ncn,q )q k1−q [ I(u, k1 ) ]q

∗ /q

+ k −q I(u, k1 ).

(2.9.4)

Proof. Setting G = { |u| < k1 , |δ α u| > k }, we find meas{|δ α u| > k} 6 meas{|u| > k1 } + meas G,

(2.9.5)

and, by virtue of Lemma 2.9.2, ∗

∗

meas{|u| > k1 } 6 (ncn,q )q k1−q [ I(u, k1 ) ]q

∗ /q

.

We now estimate meas G. Since k 6 |δ α u|h0k1 (u) on G, we get Z k q meas G 6 |δ α u|q h0k1 (u) dx 6 I(u, k1 ). Ω

(2.9.6)

142 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity Hence, meas G 6 k −q I(u, k1 ). Thus, in view of relations (2.9.5) and (2.9.6), we obtain (2.9.4).  ◦

Lemma 2.9.4. Let u ∈ H 1,q 2,p (Ω), let α be an n-dimensional multiindex with |α| = 2, and let k, k1 ∈ N. Then ∗

∗

meas{|δ α u| > k} 6 (ncn,q )q k1−q [ I(u, k1 ) ]q

∗ /q

+ k −p I(u, k1 ).

The proof of this lemma is similar to the proof of Lemma 2.9.3. Proposition 2.9.3. Let µ1 , µ2 > 0, let (n − 1)/n < σ < 1, and let Φ be a nonnegative measurable function on Ω such that Φ[ ln(1 + Φ)]σ ∈ L1 (Ω). Also ◦

let u ∈ H 1,q 2,p (Ω). Assume that the following inequality is true for any k ∈ N : Z I(u, k) 6 µ1 Φ|hk (u)| dx + µ2 . (2.9.7) Ω

Then ◦

1) u ∈ W 1,r (Ω); 2) δ α u ∈ Lrp/q (Ω) for any n-dimensional multiindex α, |α| = 2; ◦

3) if rp/q > 1, then u ∈ W 2,rp/q (Ω). Proof. By µi , i = 3, 4, . . . , we denote positive constants depending only on n, q, µ1 , µ2 , σ, meas Ω, and the norm of the function Φ[ ln(1+Φ)]σ in L1 (Ω). We set q1 = q−1 2q and fix k ∈ N, k > e. By virtue of (2.9.7), we obtain Z Z I(u, k) 6 µ1 Φ|hk (u)| dx + µ1 Φ|hk (u)| dx + µ2 . (2.9.8) {Φ6kq1 }

{Φ>kq1 }

By using the Hölder inequality, inequality (1.1.13), property (2.2.3), and applying Lemma 2.9.1, we get Z Z Φ|hk (u)| dx 6 k q1 |hk (u)| dx {Φ6kq1 }

Ω

1/q∗ ∗ |hk (u)|q dx Ω X Z 1/q ∗ q1 (q −1)/q ∗ α q 6 n cn,q k (meas Ω) |D hk (u)| dx 6 k q1 (meas Ω)(q

Z

∗ −1)/q ∗

|α|=1 Ω

6 µ3 k q1 [ I(u, k) ]1/q .

(2.9.9)

In view of (2.2.2) and the fact that Φ < q1−σ ( ln k)−σ Φ[ ln(1 + Φ)]σ

on {Φ > k q1 },

◦

Section 2.9

On the summability of functions from H 1,q 2,p (Ω)

143

we can estimate the second integral on the right-hand side of (2.9.8) as follows: Z Z Z −σ −σ Φ[ ln(1+Φ)]σ dx Φ dx 6 2kq1 (ln k) Φ|hk (u)| dx 6 2k {Φ>kq1 } −σ

{Φ>kq1 }

6 µ4 k ( ln k)

Ω

.

(2.9.10)

By the Young inequality, it follows from (2.9.8)–(2.9.10) that I(u, k) 6 µ1 µ3 k q1 [ I(u, k) ]1/q + µ1 µ4 k (ln k)−σ + µ2 1 q−1 6 I(u, k) + µ5 [ k 1/2 + k (ln k)−σ ]. q q Thus, we conclude that the following inequality is true for any k ∈ N, k > e: I(u, k) 6 µ6 k ( ln k)−σ .

(2.9.11)

Let α be any n-dimensional multiindex such that |α| = 1. We fix k ∈ N, k > e(n−1)/(n−q) . It is clear that there exists η > e such that η n−1 (ln η)σ = k n−q . This equality implies that ∗

η −r ( ln η)−σn/(n−q) = k −q η ( ln η)−σ = k −r ( ln η)−σn/(n−1) ,

(2.9.12)

n−q

(2.9.13)

k

n

6η .

Let k1 be the minimum natural number larger than η. Relations (2.9.12) and (2.9.13) yield ∗

k1−r ( ln k1 )−σn/(n−q) 6 µ7 k −r ( ln k)−σn/(n−1) ,

(2.9.14)

k −q k1 ( ln k1 )−σ 6 µ8 k −r ( ln k)−σn/(n−1) .

(2.9.15)

In addition, since k1 > e, by virtue of (2.9.11), we conclude that I(u, k1 ) 6 µ6 k1 ( ln k1 )−σ .

(2.9.16)

Using Lemma 2.9.3 and inequalities (2.9.14)–(2.9.16), we obtain meas{|δ α u| > k} 6 µ9 k −r ( ln k)−σn/(n−1) . Hence, by virtue of the inequality σ > (n − 1)/n and Lemma 1.2.1, we get δ α u ∈ Lr (Ω). Thus, for any n-dimensional multiindex α, |α| = 1, we find δ α u ∈ Lr (Ω). ◦

In view of Proposition 2.9.1, this yields u ∈ W 1,r (Ω). Hence, assertion 1) is proved. We now prove assertion 2). Let α be any n-dimensional multiindex such that |α| = 2. We fix k ∈ N, k > eq(n−1)/p(n−q) . It is clear that there exists y > e for which y n−1 ( ln y)σ = k p(n−q)/q .

144 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity This equality implies that ∗

y −r ( ln y)−σn/(n−q) = k −p y( ln y)−σ = k −rp/q ( ln y)−σn/(n−1) ,

(2.9.17)

p(n−q)

(2.9.18)

k

qn

6y .

Let k2 be the minimum natural number greater than y. Relations (2.9.17) and (2.9.18) enable us to conclude that ∗

k2−r ( ln k2 )−σn/(n−q) 6 µ10 k −rp/q ( ln k)−σn/(n−1) ,

(2.9.19)

k −p k2 ( ln k2 )−σ 6 µ11 k −rp/q ( ln k)−σn/(n−1) .

(2.9.20)

In addition, since k2 > e, by virtue of (2.9.11), we obtain I(u, k2 ) 6 µ6 k2 ( ln k2 )−σ .

(2.9.21)

Using Lemma 2.9.4 and inequalities (2.9.19)–(2.9.21), we get meas{|δ α u| > k} 6 µ12 k −rp/q ( ln k)−σn/(n−1) . Thus, by virtue of the inequality σ > (n − 1)/n and Lemma 1.2.1, we find δ α u ∈ Lrp/q (Ω). Hence, the validity of assertion 2) is proved. It remains to establish the validity of assertion 3). Assume that rp/q > 1. Then rp/q ∈ [1, p]. Since 2p < q, we have 2rp/q < r. Thus, it follows from the proof of assertion 1) that δ α u ∈ L2rp/q (Ω) for any n-dimensional multiindex α, |α| = 1. Further, in view of assertion 2) and Proposition 2.9.2, we conclude that ◦

u ∈ W 2,rp/q (Ω). Hence, assertion 3) is true.



Proposition 2.9.4. Let µ0 , µ00 > 0, 1 6 m < nq/(nq − n + q), and ◦

Φ ∈ Lm (Ω), Φ > 0 on Ω. Also let u ∈ H 1,q 2,p (Ω) and let, for any k ∈ N, the following inequality be true : Z 0 I(u, k) 6 µ Φ|hk (u)|dx + µ00 . (2.9.22) Ω

Under these conditions, ◦

1) if λ ∈ [ 1, (q − 1)m∗ ), then u ∈ W 1,λ (Ω);
2) if α is an n-dimensional multiindex, |α| = 2, and λ ∈ 0, pq (q − 1)m∗ , then δ α u ∈ Lλ (Ω); 3) if

p q (q

◦ 
− 1)m∗ > 1 and λ ∈ 1, pq (q − 1)m∗ , then u ∈ W 2,λ (Ω).

Proof. First, by the condition imposed on the number m, we get mq < n and (m − 1)q ∗ < m.

◦

Section 2.9

On the summability of functions from H 1,q 2,p (Ω)

145

We set m1 = 1 −

m−1 ∗ q , m

(m − 1)q ∗ , mq nm(q − 1) θ= . n − mq m2 =

m3 =

n − mq , n−m

It is clear that 0 < m1 6 1 and 0 6 m2 < 1. Moreover, we get   m1 = θ, q∗ 1 − (1 − m2 )q m1 m3 . m3 θ = q − 1 − m2

(2.9.23) (2.9.24)

By µi , i = 13, 14, . . . , we denote positive constants depending only on n, q, µ0 , µ00 , m, meas Ω, and the norm of the function Φ in Lm (Ω). We fix any k ∈ N. Assume that m > 1. Then m1 < 1. By using (2.9.22), (2.2.2), and the Hölder inequality, we obtain Z 0 m1 I(u, k) 6 µ (2k) Φ|hk (u)|1−m1 dx + µ00 Ω 0

m1

6 µ (2k)

Z kΦkLm (Ω)

(m−1)/m |hk (u)| dx + µ00 . q∗

Ω

This result, inequality (1.1.13), property (2.2.3), and Lemma 2.9.1 imply that I(u, k) 6 µ13 k m1 [I(u, k)]m2 + µ00 . Hence, I(u, k) 6 µ14 k m1 /(1−m2 ) .

(2.9.25)

It is easy to see that this inequality is also valid for m = 1. Thus, we have proved that inequality (2.9.25) is true for any k ∈ N. Let λ ∈ [1,(q−1)m∗ ) and let α be an n-dimensional multiindex with |α| = 1. We fix k ∈ N and k1 ∈ N such that k m3 < k1 6 2k m3 .

(2.9.26)

By virtue of (2.9.25) and (2.9.26), we find I(u, k1 ) 6 µ15 k m1 m3 /(1−m2 ) . In view of this inequality, Lemma 2.9.3, and relations (2.9.26), (2.9.23), and (2.9.24), we obtain meas{|δ α u| > k} 6 µ16 k −m3 θ . Thus, by virtue of Lemma 2.2.6, δ α u ∈ Lλ (Ω). Hence, by using the inequality ◦

(q − 1)m∗ < q and Proposition 2.9.1, we get u ∈ W 1,λ (Ω). This means that assertion 1) is proved.

146 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity We now prove assertion 2). Let
α be an n-dimensional multiindex with |α| = 2 and let λ ∈ 0, pq (q − 1)m∗ . We fix k ∈ N and k2 ∈ N such that k pm3 /q < k2 6 2k pm3 /q .

(2.9.27)

By virtue of (2.9.25) and (2.9.27), we find I(u, k2 ) 6 µ17 k pm1 m3 /q(1−m2 ) . Using this inequality, Lemma 2.9.4, and relations (2.9.27), (2.9.23), and (2.9.24), we obtain meas{ |δ α u| > k } 6 µ18 k −pm3 θ/q . In view of Lemma 2.2.6, we conclude that δ α u ∈ Lλ (Ω). Thus, assertion 2) is proved. It remains to establish the validity of assertion 3). Let pq (q − 1)m∗ > 1 and ◦
 let λ ∈ 1, pq (q − 1)m∗ . We have u ∈ H 1,q 2,p (Ω) and λ ∈ [1, p]. Moreover, 2λ ∈ ∗ [ 1, (q−1)m ). Thus, by the proof of assertion 1), we have δ α u ∈ L2λ (Ω) for any n-dimensional multiindex α with |α| = 1. Finally, according to assertion 2), we conclude that δ α u ∈ Lλ (Ω) for any n-dimensional multiindex α with |α| = 2. ◦

Thus, Proposition 2.9.2 enables us to conclude that u ∈ W 2,λ (Ω). Hence, assertion 3) is true. 

2.10

Improvement of the properties of summability for the solutions of problem (2.1.6), (2.1.7)

Proposition 2.10.1. Let the following conditions be satisfied: 1) for almost all x ∈ Ω, the function F (x, ·) is nonincreasing on R; 2) F (·, 0) ∈ L1 (Ω). Assume that u is an entropy solution of problem (2.1.6), (2.1.7). Then there exists c > 0 such that, for any k ∈ N, Z c2 I(u, k) 6 F (·, 0)hk (u)dx + |g2 |1 + c. (2.10.1) Ω

Proof. By virtue of Definition 2.3.1, F (x, u) ∈ L1 (Ω) and there exists c > 0 such that, for any k ∈ N, X  Z Z α 0 Aα (x, δ2 u)δ u hk (u)dx 6 F (x, u)hk (u)dx + c. (2.10.2) {|u| 3/2 − 1/n and let the conditions of Theo◦

◦

rem 2.10.1 be satisfied. Then u ∈ W 1,r (Ω) ∩ W 2,rp/q (Ω). To prove this assertion, it suffices to use Theorem 2.10.1, take into account the inequality q > 2p, and note that p2 6 2p by virtue of the inequality p > 3/2 − 1/n. Theorem 2.10.2. Assume that, for almost all x ∈ Ω, the function F (x, ·) is nonincreasing on R, 1 6 m < nq/(nq − n + q), and F (·, 0) ∈ Lm (Ω). Let u be an entropy solution of problem (2.1.6), (2.1.7). Then ◦

1) if λ ∈ [1, (q − 1)m∗ ), then u ∈ W 1,λ (Ω); 2) if α is an n-dimensional multiindex, |α| = 2, and λ ∈ (0, pq (q − 1)m∗ ), then δ α u ∈ Lλ (Ω); 3) if

p q (q

◦

− 1)m∗ > 1 and λ ∈ [1, pq (q − 1)m∗ ), then u ∈ W 2,λ (Ω).

This result follows from Propositions 2.10.1 and 2.9.4. Corollary 2.10.2. Let q > p2 , and let the conditions of Theorem 2.10.2. ◦

◦

be satisfied. Then pq (q − 1)m∗ > 1 and u ∈ W 1,2λ (Ω) ∩ W 2,λ (Ω) for any λ ∈ [1, pq (q − 1)m∗ ).

148 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity To prove this assertion, it suffices to use Theorem 2.10.2 and the fact that the inequality q > p2 yields the inequality pq (q − 1)m∗ > 1. Remark 2.10.1. It is easy to see that if q 6 p2 , then nq/(npq − np + q) > 1. Furthermore, the inequality nq/(npq − np + q) < m < n yields the inequality p nm (q − 1) > 1. q n−m Corollary 2.10.3. Let q 6 p2 , let nq/(npq − np + q) < m < nq/(nq − n + q), let, for almost all x ∈ Ω, the function F (x, ·) be nonincreasing on R, and let F (·, 0) ∈ Lm (Ω). Assume that u is an entropy solution of problem (2.1.6), (2.1.7). ◦

◦

Then pq (q −1)m∗ > 1 and u ∈ W 1,2λ (Ω)∩ W 2,λ (Ω) for any λ ∈ [1, pq (q −1)m∗ ). This corollary follows from Theorem 2.10.2 and Remark 2.10.1. We now present several assertions concerning the W -solutions of problem (2.1.6), (2.1.7). For this purpose, we use the following two propositions: Proposition 2.10.2. Let the following conditions be satisfied: 1) for almost all x ∈ Ω, the function F (x, ·) is nonincreasing on R; 2) for any s ∈ R, the function F (·, s) belongs to L1 (Ω). ◦

Then there exists a function u ∈ H 1,q 2,p (Ω) such that (i) u is an H-solution of problem (2.1.6), (2.1.7); Z (ii) for any k ∈ N, c2 I(u, k) 6 12 |F (·, 0)||hk (u)|dx + 2c23 . Ω

Proof. Let {Fl }, {fl }, and {ul } be the same sequences of functions as in Sec. 2.7. As follows from Sec. 2.7, there exist an increasing sequence {li } ⊂ N ◦

and a function u ∈ H 1,q 2,p (Ω) such that u is an H-solution of problem (2.1.6), (2.1.7), uli → u a.e. on Ω, (2.10.4) and, ∀ α ∈ Λ,

Dα uli → δ α u a.e. on Ω.

(2.10.5)

Let k, l ∈ N. By virtue of (2.7.12), (2.7.4), and properties of the function χk , we obtain  Z X Z α Aα (x, ∇2 ul )D χk (ul ) dx 6 fl χk (ul )dx. (2.10.6) Ω

α∈Λ

Ω

149

Section 2.10 Improvement of the properties of summability for the solutions

By the proof of Lemma 2.7.1, the following inequality is true:  Z X X c2 α q α p |D ul | + |D ul | χ0k (ul )dx 2 Ω |α|=1 |α|=2  Z X α 6 Aα (x, ∇2 ul )D χk (ul ) dx + c23 . Ω

α∈Λ

This inequality and (2.10.6) imply that  Z X Z X c2 α q α p 0 |D ul | + |D ul | χk (ul )dx 6 fl χk (ul )dx + c23 . 2 Ω Ω |α|=1

|α|=2

Thus, in view of (2.10.4), (2.10.5), the strong convergence of {fl } to F (·, 0) in L1 (Ω), and the properties of the functions χk , by using the Fatou lemma, we conclude that, for any k ∈ N, X  Z Z X c2 α q α p |δ u| + |δ u| dx 6 |F (·, 0)| |χk (u)|dx + c23 . 2 {|u| 2k, by using (2.2.3) and (2.10.7), we get X  Z X α q α p c2 I(u, k) 6 c2 |δ u| + |δ u| dx {|u| p2 and let the conditions of Proposition 2.10.2 ◦

◦

2,1 (Ω) such that be satisfied. Then there exists a function u ∈ H 1,q 2,p (Ω)∩ W

(i) u is a W -solution of problem (2.1.6), (2.1.7); Z (ii) for any k ∈ N, c2 I(u, k) 6 12 |F (·, 0)| |hk (u)|dx + 2c23 . Ω ◦

Proof. By virtue of Proposition 2.10.2, there exists a function u ∈ H 1,q 2,p (Ω) such that u is an H-solution of problem (2.1.6), (2.1.7). For any k ∈ N, we get Z |F (·, 0)| |hk (u)|dx + 2c23 .

c2 I(u, k) 6 12 Ω

150 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity Thus, by using the properties of the functions hk and Lemmas 2.2.5 and 2.2.6, we obtain δ α u ∈ Lλ (Ω) for any α, |α| = 1, and λ ∈ (0, r), α

λ

δ u ∈ L (Ω) for any α, |α| = 2, and λ ∈ (0, rp/q).

(2.10.9) (2.10.10)

Since q > p2 , we find rp/q > 1. Thus, assertions (2.10.9) and (2.10.10) and ◦

Proposition 2.9.2 imply that u ∈ W 2,1 (Ω). In addition, by virtue of assertions (2.10.9) and (2.10.10) and Lemmas 2.8.1 and 2.8.2, we have ∇2 u = δ2 u a.e. on Ω. Therefore, in view of the fact that u is an H-solution of problem (2.1.6), (2.1.7), we conclude that u is a W -solution of problem (2.1.6), (2.1.7).  Propositions 2.9.3, 2.9.4, and 2.10.3 yield the following results: Theorem 2.10.3. Let q > p2 , let the conditions of Proposition 2.10.2 be satisfied, let (n − 1)/n < σ < 1, and let F (·, 0)[ ln(1 + |F (·, 0)|)]σ ∈ L1 (Ω). Then there exists a W -solution u of problem (2.1.6), (2.1.7) such that ◦

◦

u ∈ W 1,r (Ω) ∩ W 2,rp/q (Ω). Theorem 2.10.4. Let q > p2 , let the conditions of Proposition 2.10.2 be satisfied, let 1 6 m < nq/(nq−n+q), and let F (·, 0) ∈ Lm (Ω). Then there exists a W -solution u of problem (2.1.6), (2.1.7) such that, for any λ ∈ [1, pq (q−1)m∗ ), the following inclusion is true: ◦

◦

u ∈ W 1,2λ (Ω) ∩ W 2,λ (Ω). Note that, in [70, 78, 79, 81], the analogs of the main results presented in this section were obtained, on the basis of the ideas advanced in [65], for the solutions of nondegenerate and degenerate nonlinear elliptic equations of the fourth and higher orders with strengthened coercivity.

2.11

Some characteristics of the set of ◦

functions H 1,q 2,p (Ω) We first present the following auxiliary assertion: Lemma 2.11.1. Let h ∈ C 2 (R), h(0) = 0, and let the functions h, h0 , ◦

◦

1,q and h00 be bounded on R. Assume that u ∈ W 1,q 2,p (Ω). Then h(u) ∈ W 2,p (Ω) and the following assertions are valid:

(i) for any n-dimensional multiindex α, |α| = 1, Dα h(u) = h0 (u)Dα u a.e. on Ω;

◦

Section 2.11 Some characteristics of the set of functions H 1,q 2,p (Ω)

151

(ii) for any n-dimensional multiindices β and γ such that |β| = |γ| = 1, Dβ+γ h(u) = h0 (u)Dβ+γ u + h00 (u)Dβ u · Dγ u a.e. on Ω. The proof of the lemma is similar to the proof of Lemma 2.2.2. By H we denote the set of all functions h ∈ C 2 (R) satisfying the following conditions: h(0) = 0 and there exists µ > 0 such that, for any s ∈ R, |s| > µ, the following equality holds: h0 (s) = 0. It is clear that {hk } ⊂ H. Proposition 2.11.1. Assume that {ϕk } ⊂ C 2 (R) and there exists a sequence {mk } of positive numbers such that mk → +∞ as k → +∞ and, for any k ∈ N, ϕk (s) = s if |s| < mk , |ϕk (s)| > mk

(2.11.1)

if |s| > mk .

(2.11.2)

◦

Let u : Ω → R be a function such that ϕk (u) ∈ W 1,q 2,p (Ω) for any k ∈ N. Then ◦

(i) h(u) ∈ W 1,q 2,p (Ω) for any h ∈ H; ◦

(ii) u ∈ H 1,q 2,p (Ω). Proof. We fix h ∈ H and, for any k ∈ N, define vk = h(ϕk (u)). In view of the ◦

properties of the function h, the inclusion {ϕk (u)} ⊂ W 1,q 2,p (Ω), Lemma 2.11.1, ◦

the convergence of {mk } to +∞, and relation (2.11.1), we get {vk } ⊂ W 1,q 2,p (Ω), h(u) ∈ Lq (Ω), and vk → h(u) strongly in Lq (Ω). (2.11.3) Since h ∈ H, there exists µ > 0 such that, for any s ∈ R, |s| > µ, we have = 0. Therefore,

h0 (s)

∀ s ∈ R, |s| > µ,

h(s) ∈ {h(−µ), h(µ)}.

(2.11.4)

It is clear that there exists l ∈ N such that ∀ k ∈ N, k > l,

mk > µ.

(2.11.5)

We fix k ∈ N, k > l. Relations (2.11.1), (2.11.2), (2.11.4), and (2.11.5) imply that if x ∈ {|u| < µ}, then vk (x) = vl (x), if x ∈ {|u| > µ}, then vk (x) ∈ {h(−µ), h(µ)}. This enables us to conclude that |Dα vk | 6 |Dα vl | a.e. on Ω for any α ∈ Λ.

152 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity Hence, in view of the boundedness of the function h, we conclude that {vk } ◦

is bounded in W 1,q 2,p (Ω). Together with (2.11.3), this yields the inclusion h(u) ∈ ◦

W 1,q 2,p (Ω). Thus, assertion (i) of the proposition is true. Hence, in view of the fact that ◦

{hk } ⊂ H, we arrive at the inclusion u ∈ H 1,q 2,p (Ω).



Proposition 2.11.2. Assume that all conditions of Proposition 2.11.1 are satisfied. Then, for any α ∈ Λ and k ∈ N, Dα ϕk (u) = δ α u a.e. on {|u| < mk }.

(2.11.6)

Proof. Let α ∈ Λ and k ∈ N. We fix l ∈ N, l > mk . By the definition of the function hl and relation (2.11.1), we get ϕk (u) = hl (u) on {|u| < mk }. Further, ◦

in view of the fact that ϕk (u) ∈ W 1,q 2,p (Ω) by the condition of Proposition 2.11.1 ◦

and hl (u) ∈ W 1,q 2,p (Ω) by the assertion (ii) of the same proposition, we conclude that Dα ϕk (u) = Dα hl (u) a.e. on {|u| < mk }. This fact, Lemma 2.2.4, and the inequality l > mk imply the validity of (2.11.6).  ◦

◦

1,q Proposition 2.11.3. Let u : Ω → R. Then u ∈ H 1,q 2,p (Ω) iff h(u) ∈ W 2,p (Ω) for any h ∈ H. ◦

◦

1,q Proof. Assume that u ∈ H 1,q 2,p (Ω). Hence, hk (u) ∈ W 2,p (Ω) for any k ∈ N. We now set ϕk = hk and mk = k for any k ∈ N. It is clear that all conditions of Proposition 2.11.1 are satisfied. Thus, by virtue of this proposition for ◦

any h ∈ H, we obtain h(u) ∈ W 1,q 2,p (Ω). ◦

Conversely, let h(u) ∈ W 1,q 2,p (Ω) for any h ∈ H. Thus, in view of the fact ◦

that {hk } ⊂ H, for any k ∈ N, we obtain hk (u) ∈ W 1,q 2,p (Ω) and, therefore, ◦

u ∈ H 1,q 2,p (Ω).

 ◦

◦

1,q ∞ Proposition 2.11.4. Let u ∈ H 1,q 2,p (Ω), let w ∈ W 2,p (Ω) ∩ L (Ω), and let ◦

∞ h ∈ H. Then h(u − w) ∈ W 1,q 2,p (Ω) ∩ L (Ω) and the following assertions are true :

(i) for any n-dimensional multiindex α, |α| = 1, Dα h(u − w) = h0 (u − w)(δ α u − δ α w) a.e. on Ω; (ii) for any n-dimensional multiindices β and γ such that |β| = |γ| = 1, Dβ+γ h(u − w) = h0 (u − w)(δ β+γ u − δ β+γ w) + h00 (u − w)(δ β u − δ β w)(δ γ u − δ γ w) a.e. on Ω.

◦

Section 2.11 Some characteristics of the set of functions H 1,q 2,p (Ω)

153

Proof. Since h ∈ H, there exists µ > 0 such that ∀ s ∈ R, |s| > µ,

h0 (s) = 0.

(2.11.7)

Then ∀ s 6 −µ, h(s) = h(−µ),

(2.11.8)

∀ s > µ, h(s) = h(µ).

(2.11.9)

It is clear that the function h is bounded on R. Hence, h(u − w) ∈ L∞ (Ω). It is obvious that there exists a set E ⊂ Ω of measure zero for which ∀ x ∈ Ω \ E,

|w(x)| 6 |w|∞ .

(2.11.10)

We fix k ∈ N such that k > µ + |w|∞ .

(2.11.11)

Let x ∈ Ω \ E and let |u(x)| > k. By using relations (2.11.10) and (2.11.11) and the properties of the function hk , we obtain |u(x) − w(x)| > µ,

|hk (u(x)) − w(x)| > µ,

(2.11.12)

sign(u(x) − w(x)) = sign(hk (u(x)) − w(x)). Together with (2.11.8) and (2.11.9), this relation yields h(u − w) = h(hk (u) − w) a.e. on Ω.

(2.11.13)

Moreover, in view of (2.11.7) and (2.11.12), we conclude that if x ∈ Ω \ E

and |u(x)| > k,

0

then h (u − w)(x) = 0,

h0 (hk (u) − w)(x) = 0,

h00 (u − w)(x) = 0,

h00 (hk (u) − w)(x) = 0.

(2.11.14)

◦

◦

It is clear that hk (u)−w ∈ W 1,q 2,p (Ω). Thus, by Lemma 2.11.1, h(hk (u)−w) ∈

W 1,q 2,p (Ω), and the following assertions are true: (i 0 ) for any n-dimensional multiindex α, |α| = 1, Dα h(hk (u) − w) = h0 (hk (u) − w)Dα (hk (u) − w) a.e. on Ω; (ii 0 ) for any n-dimensional multiindices β and γ such that |β| = |γ| = 1, Dβ+γ h(hk (u) − w) = h0 (hk (u) − w)Dβ+γ (hk (u) − w) + h00 (hk (u) − w)Dβ (hk (u) − w)·Dγ (hk (u) − w) a.e. on Ω.

154 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity Together with (2.11.13), (2.11.14), and Lemma 2.2.4, these results imply the validity of the analyzed proposition.  ◦

◦

1,q ∞ Corollary 2.11.1. Let u ∈ H 1,q 2,p (Ω) and let w ∈ W 2,p (Ω) ∩ L (Ω). Then ◦

u − w ∈ H 1,q 2,p (Ω). ◦

Proof. Let k ∈ N. Since hk ∈ H, we conclude that hk (u − w) ∈ W 1,q 2,p (Ω) ◦

by virtue of Proposition 2.11.4. Hence, u − w ∈ H 1,q 2,p (Ω).

2.12



c1,q (Ω) Set of functions H 2,p ◦

For any function u ∈ H 1,q 2,p (Ω), we set X X Φu = |δ α u|q + |δ α u|p . |α|=1

Lemma 2.2.4 implies that if u ∈ summable on {|u| < k}.

|α|=2 ◦

H 1,q 2,p (Ω)

and k ∈ N, then the function Φu is

◦

For any function u ∈ H 1,q 2,p (Ω), we set Z 1 Φu dx. Mu = sup k∈N k {|u| k. Hence, Z ∗ q∗ k meas{|u| > k} 6 |hk (u)|q dx.

(2.12.2)

(2.12.3)

Ω

We now estimate the integral on the right-hand side of this inequality. Since ◦

hk (u) ∈ W 1,q (Ω), by virtue of (2.2.1), we find X |hk (u)|q∗ 6 c0 |Dα hk (u)|q .

(2.12.4)

|α|=1

Let α be any n-dimensional multiindex with |α| = 1. By virtue of Proposition 2.11.4, we get Dα hk (u) = h0k (u)δ α u a.e. on Ω. Thus, in view of the

b 1,q (Ω) Section 2.12 Set of functions H 2,p

155

equality h0k (s) = 0, for |s| > 2k, by using (2.2.3) and (2.12.1), we obtain Z |Dα hk (u)|q dx 6 2Mu k. Ω

Hence, by virtue (2.12.4), we conclude that Z ∗ ∗ ∗ |hk (u)|q dx 6 (c0 n)q (2Mu k)q /q . Ω



This inequality and (2.12.3) yield (2.12.2). We set r¯ =

n(q − 1) q(n − 1)

and c¯ = 2q

∗ /q

∗

(c0 n)q + 1.

b 1,q (Ω). Then, for any k ∈ N, Proposition 2.12.2. Let u ∈ H 2,p meas{Φu > k} 6 3c(M ¯ u + 1)n/(n−1) k −r¯ .

(2.12.5)

Proof. Let k ∈ N. We set ∗

k1 = (Mu + 1)1/(n−1) (ck) ¯ 1/(1+r ) . Also let l ∈ N, k1 6 l < k1 + 1.

(2.12.6)

It is clear that meas{Φu > k} 6 meas{Φu > k, |u| < l} + meas{|u| > l} and

(2.12.7)

Z k meas{Φu > k, |u| < l} 6

Φu dx. {|u| k, |u| < l} 6 Mu lk −1 .

(2.12.8)

Moreover, by virtue of Proposition 2.12.1, meas{|u| > l} 6 cM ¯ uq

∗ /q

∗

l−r .

(2.12.9)

In view of the definition of the number k1 , relations (2.12.6)–(2.12.9) yield (2.12.5).  Propositions 2.12.1 and 2.12.2 and Lemma 2.2.6 imply the following assertion: b 1,q (Ω). Then Corollary 2.12.1. Let u ∈ H 2,p (i) u ∈ Lλ (Ω) for any λ ∈ (0, r∗ ); (ii) Φu ∈ Lλ (Ω) for any λ ∈ (0, r). ¯

156 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity In turn, by using assertion (ii) of Corollary 2.12.1, we arrive at the following corollary: b 1,q (Ω). Then Corollary 2.12.2. Let u ∈ H 2,p (i) δ α u ∈ Lλ (Ω) for any λ ∈ (0, r) and any n-dimensional multiindex α, |α| = 1; (ii) δ α u ∈ Lλ (Ω) for any λ ∈ (0, rp/q) and any n-dimensional multiindex α, |α| = 2. Corollary 2.12.2 and Propositions 2.9.1 and 2.9.2 yield the following assertions: ◦

b 1,q (Ω) ⊂ W 1,λ (Ω) for any λ ∈ [1, r). Corollary 2.12.3. H 2,p ◦

b 1,q (Ω) ⊂ W 2,λ (Ω) for any λ ∈ Corollary 2.12.4. Let q > p2 . Then H 2,p [1, rp/q). We can also prove the following assertion: b 1,q (Ω). Then Aα (x, δ2 u) ∈ L1 (Ω) for any Corollary 2.12.5. Let u ∈ H 2,p α ∈ Λ. Proof. We set Φ = c1 Φu + g1 + 1. Since (q − 1)/q ∈ (0, r), by virtue of the assertion (ii) of Corollary 2.12.1, we get Φu ∈ L(q−1)/q (Ω). This fact and the inclusion g1 ∈ L1 (Ω) imply that Φ ∈ L(q−1)/q (Ω).

(2.12.10)

Let α ∈ Λ. As a consequence of (2.1.3), we obtain |Aα (x, δ2 u)| 6 Φ(q−1)/q a.e. on Ω. Thus, in view of (2.12.10), we find Aα (x, δ2 u) ∈ L1 (Ω).  Further, for any k ∈ N, we now define a function h¯ k : R → R as follows: h¯ k (s) = kh1 (s/k),

s ∈ R.

It is clear that, for any k ∈ N, we have h¯ k ∈ C 2 (R) and h¯ k (s) = s for |s| 6 k, h¯ 0k (s) = 0 for |s| > 2k. In addition, properties (2.2.2)–(2.2.4) imply that, for any k ∈ N, |h¯ k | 6 2k on R, 0 6 h¯ 0k 6 1 on R, 3 on R. |h¯ 00k | 6 k Note that, by virtue of (2.12.11) and (2.12.12), {h¯ k } ⊂ H.

(2.12.11) (2.12.12) (2.12.13) (2.12.14) (2.12.15)

Section 2.13 Definition and a priori estimates of the proper entropy solutions

157

Remark 2.12.1. If u : Ω → R, then the following assertions are equivalent: ◦

(i) u ∈ H 1,q 2,p (Ω); ◦

(ii) h¯ k (u) ∈ W 1,q 2,p (Ω) for any k ∈ N. This result follows from Propositions 2.11.1 and 2.11.3. The functions h¯ k are essentially used in what follows. At the end of this section, we present the following auxiliary result obtained for these functions: b 1,q (Ω). Then, for any k ∈ N, Lemma 2.12.1. Let u ∈ H 2,p  X  Z X |Aα (x, δ2 u)| |δ β u|2 |h¯ 00k (u)|dx Ω

|α|=2

|β|=1 3

6 6n (c1 +1)(1+|g1 |1 +meas Ω)(Mu +1)k 2/q−1/p .

(2.12.16)

Proof. We set Φ=

X

 X  β 2 |Aα (x, δ2 u)| |δ u| .

|α|=2

|β|=1

By virtue of (2.1.3), we get Φ 6 (c1 + 1)n3 (Φu + g1 )1+2/q−1/p

a.e. on Ω.

(2.12.17)

Let k ∈ N. Using (2.12.12), (2.12.15), and (2.12.17) and the Hölder inequality, we obtain Z Z (Φu + g1 )1+2/q−1/p dx Φ|h¯ 00k (u)|dx 6 3(c1 + 1)n3 k −1 {|u| 0, and let the following conditions be satisfied: h(0) = 0, the functions h and h0 are bounded on R, h0 > 0 on R, and |h00 | 6 M h0 Then Z

Φu h0 (u)dx 6

Ω

2 c2

(2.13.1)

on R. Z

Z F (x, u)h(u)dx + C

(g1 +g2 +1)h0 (u)dx,

(2.13.2)

Ω

Ω

where C is a positive constant depending only on n, p, q, c1 , c2 , and M. Proof. We set M0 = sup (|h(s)| + h0 (s)),

M1 = [M (c1 + 1)(2/c2 + 1)n3 ]qp/(q−2p) ,

s∈R

Φ=

X

 X  β 2 |Aα (x, δ2 u)| |δ u| .

|α|=2

|β|=1

For any k ∈ N, we define ϕk = h ◦ h¯ k . It is clear that {ϕk } ⊂ H+ , ϕk (u) → h(u), ∀ k ∈ N,

ϕ0k (u) → h0 (u) on Ω,

|ϕk (u)| +

ϕ0k (u)

6 M0

on Ω.

We fix k ∈ N. By virtue of Definition 2.13.1,  Z X Z Aα (x, δ2 u)Dα ϕk (u) dx 6 F (x, u)ϕk (u)dx. Ω

α∈Λ

(2.13.3) (2.13.4)

(2.13.5)

Ω

Moreover, by Proposition 2.11.4, the following relations are true: Dα ϕk (u) = ϕ0k (u)δ α u a.e. on Ω for any n-dimensional multiindex α, |α| = 1, and X |Dα ϕk (u) − ϕ0k (u)δ α u| 6 |ϕ00k (u)| |δ β u|2

a.e. on Ω

|β|=1

for any n-dimensional multiindex α with |α| = 2. By using these relations,

Section 2.13 Definition and a priori estimates of the proper entropy solutions

we obtain Z X Ω

159

 Aα (x, δ2 u)δ u ϕ0k (u)dx α

α∈Λ

6

Z X Ω

 Z Φ|ϕ00k (u)|dx. Aα (x, δ2 u)D ϕk (u) dx + α

(2.13.6)

Ω

α∈Λ

In view (2.1.4), we now get  Z Z X Z c2 Φu ϕ0k (u)dx 6 Aα (x, δ2 u)δ α u ϕ0k (u)dx + g2 ϕ0k (u)dx. Ω

Ω

Ω

α∈Λ

Together with (2.13.5) and (2.13.6), this inequality implies that Z Z Z Z Φ|ϕ00k (u)|dx. c2 F (x, u)ϕk (u)dx + g2 ϕ0k (u)dx + Φu ϕ0k (u)dx 6 Ω

Ω

Ω

Ω

(2.13.7) We now estimate the last integral on the right-hand side of (2.13.7). First, we note that, by virtue of (2.12.14) and (2.13.1), |ϕ00k | 6 M ϕ0k + M0 |h¯ 00k | on R. Therefore, Z

Φ|ϕ00k (u)|dx

Z

Φϕ0k (u)dx

6M

Z + M0

Φ|h¯ 00k (u)|dx.

(2.13.8)

Ω

Ω

Ω

Using relation (2.12.17) and the Young inequality, we find MΦ 6

c2 (Φu + g1 ) + M1 2

a.e. on Ω.

Hence, Z M Ω

Φϕ0k (u)dx 6

c2 2

Z Ω

Φu ϕ0k (u)dx +

Z  Ω

 c2 g1 + M1 ϕ0k (u)dx. 2

This inequality and relations (2.13.7) and (2.13.8) imply that Z Z 2 Φu ϕ0k (u)dx 6 F (x, u)ϕk (u)dx c2 Ω Ω Z  2M1 2 + +1 (g1 + g2 + 1)ϕ0k (u)dx + c2 c2 Ω Z 2M0 Φ|h¯ 00k (u)|dx. (2.13.9) + c2 Ω

160 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity Therefore, in view of (2.13.3) and (2.13.4), we get Z Z F (x, u)ϕk (u)dx → F (x, u)h(u)dx, Ω Ω Z Z (g1 + g2 + 1)ϕ0k (u)dx → (g1 + g2 + 1)h0 (u)dx.

(2.13.10) (2.13.11)

Ω

Ω

Moreover, by virtue of Lemma 2.12.1, we obtain Z Φ|h¯ 00k (u)|dx → 0.

(2.13.12)

Ω

In view of (2.13.3), (2.13.10)–(2.13.12) and the Fatou lemma, relation (2.13.9) yields inequality (2.13.2) with C = 2(1 + M1 )/c2 + 1.  Proposition 2.13.2. Let u be a proper entropy solution of problem (2.1.6), (2.1.7). Then there exists C1 > 0 such that, for any m ∈ N, m > 2, and k ∈ N, k > 2m, Z C1 Φu dx {2k6|u|62k+m} Z Z 6m |F (x, u)|dx + (g1 + g2 )dx + k 2/q−1/p . {|u|>k}

{|u|>k}

Proof. By c0i , i = 1, 2, . . . , we denote positive constants depending only on n, p, q, c1 , c2 , meas Ω, |g1 |1 , and Mu . We set  X  X β 2 |Aα (x, δ2 u)| |δ u| . Φ= |α|=2

|β|=1

Let m ∈ N, m > 2, and let k ∈ N, k > 2m. We also set w = hk (u),

h = χ2m ◦ h¯ k .

◦

∞ It is clear that w ∈ W 1,q 2,p (Ω) ∩ L (Ω) and h ∈ H+ . Thus, by Definition 2.13.1,  Z X Z Aα (x, δ2 u)Dα h(u − w) dx 6 F (x, u)h(u − w)dx. (2.13.13) Ω

Ω

α∈Λ

According to Proposition 2.11.4 and (2.2.3), we obtain X  Aα (x, δ2 u)δ α u (1 − h0k (u))h0 (u − w) α∈Λ

6

X

Aα (x, δ2 u)Dα h(u − w) + Φ(1 − h0k (u))|h00 (u − w)|

α∈Λ

+ Φ|h00k (u)|h0 (u − w) a.e. on Ω.

(2.13.14)

Section 2.13 Definition and a priori estimates of the proper entropy solutions

Moreover, by virtue of (2.1.4), X c2 Φu 6 Aα (x, δ2 u)δ α u + g2

a.e. on Ω.

161

(2.13.15)

α∈Λ

Relations (2.13.13)–(2.13.15) imply that Z c2 Φu (1 − h0k (u))h0 (u − w)dx Ω Z Z g2 (1 − h0k (u))h0 (u − w)dx F (x, u)h(u − w)dx + 6 Ω ΩZ Z 00 0 Φ|h00k (u)|h0 (u − w)dx. (2.13.16) + Φ(1 − hk (u))|h (u − w)|dx + Ω

Ω

We now estimate the integrals on the right-hand side of this inequality. As a result of (2.7.1), the equality h(0) = 0, and the fact that hk (s) = s, for |s| 6 k, we get Z Z F (x, u)h(u − w)dx 6 6m |F (x, u)|dx. (2.13.17) {|u|>k}

Ω

Using (2.2.3), (2.12.14), (2.7.2) and the fact that h0k (s) = 1 for |s| 6 k, we obtain Z Z 0 0 g2 dx. (2.13.18) g2 (1 − hk (u))h (u − w)dx 6 {|u|>k}

Ω

Thus, it follows from (2.12.14), (2.7.2), and (2.7.3) that, ∀ s ∈ R,

|h00 (s)| 6 2h0 (s) + |h¯ 00k (s)|.

Therefore, Z Z 0 00 Φ(1 − hk (u))|h (u − w)|dx 6 2 Φ(1 − h0k (u))h0 (u − w)dx Ω ZΩ + Φ(1 − h0k (u))|h¯ 00k (u − w)|dx. (2.13.19) Ω

Moreover, by using (2.12.12), (2.12.15), (2.2.3), and (2.2.2), we obtain Z Z 3 Φ dx. (2.13.20) Φ(1 − h0k (u))|h¯ 00k (u − w)|dx 6 k {|u| k(1 + k −σ ), we have |h00k (s)| 6 3k σ (1 − h0k (s)). Using these facts and relations (2.12.14), (2.7.2), and (2.2.3), we arrive at the inequalities Z Z 0 1−σk 00 Φ dx, (2.13.22) Φ|hk (u)|h (u − w)dx 6 6k {|u|k} Z g2 dx + c04 k 2/q−1/p . (2.13.27) + {|u|>k}

Section 2.13 Definition and a priori estimates of the proper entropy solutions

163

We now establish the lower bound of the integral on the left-hand side of this inequality. It is clear that Z Z Φu h0 (u − w)dx. (2.13.28) Φu (1 − h0k (u))h0 (u − w)dx > {2k6|u|62k+m}

Ω

Let x ∈ {2k 6 |u| 6 2k + m}. By the definition of the function hk , we have hk (u(x)) = 2k

k+2 sign u(x). k+3

Thus, k+2 2k 6 + m. k+3 k+3 Hence, for m > 2, we arrive at the inequality |u(x) − w(x)| < 2m. This enables us to conclude that |u(x) − w(x)| = |u(x)| − 2k

{2k 6 |u| 6 2k + m} ⊂ {|u − w| < 2m}.

(2.13.29)

Furthermore, by virtue of the inequality k > 2m, (2.12.11), and the definition of the function χ2m , we obtain h0 (u − w) = 1 for {|u − w| < 2m}. Together with (2.13.29), this yields the equality Z Z Φu h0 (u − w)dx = Φu dx. {2k6|u|62k+m}

{2k6|u|62k+m}

Thus, in view of (2.13.28), we get Z Z Φu (1 − h0k (u))h0 (u − w)dx >

Φu dx.

{2k6|u|62k+m}

Ω

This inequality and (2.13.27) enable us to conclude that Z 0 c5 Φu dx {2k6|u|62k+m} Z Z 6m |F (x, u)|dx + (g1 +g2 )dx + k 2/q−1/p . {|u|>k}



{|u|>k}

Remark 2.13.1. The a priori estimate established in Proposition 2.13.1 serves as a basis for getting the properties of summability of the proper entropy solutions of problem (2.1.6), (2.1.7). Some of these properties are considered in Sections 2.17 and 2.18. Note that the estimate under consideration is an analog of the a priori estimates obtained in [73] for the entropy solutions of the Dirichlet problem for nonlinear second-order elliptic equations with degenerate coercivity and L1 -data. The a priori estimate obtained in Proposition 2.13.2

164 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity is similar to the estimate obtained in Lemma 2.4.2 for the entropy solutions of problem (2.1.6), (2.1.7) and to the estimate established in [13] for the entropy solutions of the Dirichlet problem for nonlinear elliptic equations of the second order with L1 -data. These estimates prove to be useful for the investigation of uniqueness of the entropy solutions and the properties of their summability.

2.14

Existence of the proper entropy solutions

Theorem 2.14.1. Assume that the following conditions are satisfied: 1) for almost all x ∈ Ω, the function F (x, ·) is nonincreasing on R; 2) for any s ∈ R, the function F (·, s) belongs to L1 (Ω). Then there exists a proper entropy solution of problem (2.1.6), (2.1.7). Proof. We set f = F (·, 0) and, for any l ∈ N, define a function Fl : Ω×R → R as follows: Fl (x, s) = hl (f (x) − F (x, s)), (x, s) ∈ Ω × R. By virtue of condition 1), if l ∈ N, then, Fl (x, ·) is nondecreasing on R for almost all x ∈ Ω.

(2.14.1)

In addition, as follows from condition 2), f ∈ L1 (Ω). Hence, there exists {fl } ⊂ C0∞ (Ω) such that lim |fl − f |1 = 0,

(2.14.2)

l→∞

∀ l ∈ N,

|fl |1 6 |f |1 + 1.

In view of relations (2.1.3)–(2.1.5) and (2.14.1) and the results on solvability of the equations with monotonic operators obtained in [89], we conclude that ◦

◦

1,q if l ∈ N, then there exists a function ul ∈ W 1,q 2,p (Ω) such that, ∀ v ∈ W 2,p (Ω),  Z X Z Aα (x, ∇2 ul )Dα v + Fl (x, ul )v dx = fl v dx. (2.14.3) Ω

Ω

α∈Λ

In turn, the results presented in Sec. 2.7 imply that there exist an increasing ◦

sequence {li } ⊂ N and a function u ∈ H 1,q 2,p (Ω) such that F (x, u) ∈ L1 (Ω),

(2.14.4)

uli → u a.e. on Ω,

(2.14.5) 1

(2.14.6)

D uli → δ u a.e. on Ω.

(2.14.7)

Fli (x, uli ) → f − F (x, u) strongly in L (Ω), ∀ α ∈ Λ,

α

α

165

Section 2.14 Existence of the proper entropy solutions

Moreover, it follows from the proof of Lemma 2.7.1 that, for any k, l ∈ N,  Z X X (2.14.8) |Dα ul |p χ0k (ul )dx 6 C2 k, |Dα ul |q + Ω

|α|=2

|α|=1

where C2 is a positive constant depending only on n, p, q, c1 , c2 , meas Ω, and the norms of the functions g1 , g2 , and f in L1 (Ω). This result, together with relations (2.7.2), (2.14.5), and (2.14.7) and the Fatou lemma, implies that, for any k ∈ N, Z Φu χ0k (u)dx 6 C2 k. Ω

This enables us to conclude that, for any k ∈ N, Z Φu dx 6 C2 k. {|u| 0 in R. It is clear that there exists µ > 0 such that ∀ s ∈ R, |s| > µ, h0 (s) = 0. (2.14.13) Hence, there exists µ1 > 0 for which h0 (s) + |h00 (s)| 6 µ1 .

∀ s ∈ R,

(2.14.14)

We set µ2 = µ + |w|∞ . Now let α ∈ Λ. We set v = Aα (x, δ2 u)Dα w · h0 (u − w). Lemma 2.2.4 and relations (2.1.3), (2.14.13), and (2.14.14) imply that the function v is summable on Ω. For any i ∈ N, we set vi = Aα (x, ∇2 uli )Dα w · h0 (uli − w). In view of (2.14.5) and (2.14.7), we obtain vi → v

a.e. on Ω.

(2.14.15)

We fix ε ∈ (0, 1). By the absolute continuity of the Lebesgue integral, relation (2.14.15), and the Egorov theorem, there exists a measurable set Ω0 ⊂ Ω such that X  Z X |Dβ w|q + |Dβ w|p + |v| dx 6 εq , (2.14.16) Ω\Ω0

|β|=1

|β|=2

vi → v

uniformly on Ω0 .

According to the last property, there exists i0 ∈ N such that, for any i ∈ N, i > i0 , we have Z |vi − v|dx 6 ε. (2.14.17) Ω0

We fix i ∈ N, i > i0 . Relations (2.14.16) and (2.14.17) imply that Z Z |vi − v|dx 6 2ε + |vi |dx. Ω

(2.14.18)

Ω\Ω0

Using relations (2.14.13), (2.14.14), (2.1.3), (2.14.16), and (2.14.8) and the Hölder inequality, we arrive at the estimate Z Z |vi |dx 6 µ1 |Aα (x, ∇2 uli )||Dα w|dx Ω\Ω0

(Ω\Ω0 )∩{|uli | i1 . Relations (2.14.24) and (2.14.25) imply that Z Z |zi − z|dx 6 2ε + |zi |dx.

(2.14.26)

Ω\Ω00

Ω

By using properties (2.14.21), (2.14.13), (2.14.14), (2.1.3), (2.14.23), (2.14.24), and (2.14.8) and the Hölder inequality, we arrive at the estimate Z Z |Aα (x,∇2 uli )||Dβ uli −Dβ w||Dγ uli −Dγ w|dx |zi |dx 6 µ1 Ω\Ω00

(Ω\Ω00 )∩{|uli | p1 , we have

 1p − 1 q + 2 < 1. r p We fix a number σ such that  1p − 1 q+2 k})γ/b . Ω

We now estimate the measure of the set {|u − w| > k}. To this end, we set G1 = {|u − w| > k} ∩ {|w| 6 k/2},

G2 = {|u − w| > k} ∩ {|w| > k/2}.

It is clear that meas{|u − w| > k} = meas G1 + meas G2 .

(2.15.8)

Note that G1 ⊂ {|u| > k/2}. By the first inclusion in (2.15.1) and Proposition 2.12.1, we get meas G1 6 4b k −b cσ ¯ 2. (2.15.9) Since G2 ⊂ {|w| > k/2}, we have meas G2 6 2b k −b

Z

|w|b dx.

Ω

This inequality, (2.15.8), and (2.15.9) imply hat meas{|u − w| > k} 6 4b k −b cσ ¯ 2 (1 + kwkW 1,b (Ω) )b . Relations (2.15.7), (2.15.5), and (2.15.10) now yield Z ϕv|h00k (u − w)|dx 6 c(1 + kwkW 1,b (Ω) )b k −γ . Ω

(2.15.10)

(2.15.11)

171

Section 2.16 Relationship with H-solutions and W -solutions

Note that h0k (s) = 0 for |s| > 2k. Thus, it follows from relations (2.15.6) and (2.15.11) that X  Z Aα (x, δ2 u)(δ α u − δ α w) h0k (u − w)dx {|u−w| p1 and the following conditions are satisfied : 1) for almost all x ∈ Ω, the function F (x, ·) is nonincreasing on R; 2) for any s ∈ R, the function F (·, s) belongs to L1 (Ω). Let u be an entropy solution of problem (2.1.6), (2.1.7). Then u is a proper entropy solution of problem (2.1.6), (2.1.7). Proof. By virtue of Theorem 2.14.1, there exists a proper entropy solution of problem (2.1.6), (2.1.7). We denote this solution by v. Theorem 2.15.1 implies that v is an entropy solution of problem (2.1.6), (2.1.7). Thus, by virtue of Theorem 2.6.1, u = v a.e. on Ω. Therefore, in view of the fact that v is a proper entropy solution of problem (2.1.6), (2.1.7), we conclude that u is a proper entropy solution of problem (2.1.6), (2.1.7).  Theorems 2.15.1 and 2.6.1 yield the following assertion: Theorem 2.15.3. Assume that q > p1 and, for almost all x ∈ Ω, the function F (x, ·) is nonincreasing on R. Let u and v be proper entropy solutions of problem (2.1.6), (2.1.7). Then u = v a.e. on Ω. Note that the problem of uniqueness of the proper entropy solution of problem (2.1.6), (2.1.7) in the absence of assumption q > p1 remains open.

2.16

Relationship with H-solutions and W -solutions

Theorem 2.16.1. Let u be a proper entropy solution of problem (2.1.6), (2.1.7). Then u is an H-solution of problem (2.1.6), (2.1.7). Proof. It is clear that b 1,q (Ω), u∈H 2,p

F (x, u) ∈ L1 (Ω).

(2.16.1)

172 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity The first of these inclusions and Corollary 2.12.5 imply that Aα (x, δ2 u) ∈ L1 (Ω).

∀ α ∈ Λ,

(2.16.2)

We set 000

3

qp/(q−2p)

c = [2(c1 +1)(c2 +2)n /c2 ]

, Φ=

X

  X β 2 |Aα (x, δ2 u)| |δ u| .

|α|=2

|β|=1

Using (2.12.17) and the Young inequality, we obtain c2 2Φ 6 (Φu + g1 ) + c000 a.e. on Ω. 2

(2.16.3)

Moreover, in view of (2.12.17), the Hölder inequality, and relation (2.12.1), we get Z 1 lim (2.16.4) Φ dx = 0. k→∞ k {|u| 8 + max |w| and set, for any k ∈ N, Ω

ϕk = χm ◦h¯ k ,

vk = u − h¯ k (u) + w.

In view of the properties of the function χm and the functions h¯ k , for any k ∈ N, we conclude that ϕk ∈ H+ , |ϕk | 6 3m on R, 06

ϕ0k

(2.16.6) (2.16.7)

6 1 on R,

(2.16.8)

|ϕ00k | 6 ϕ0k + |h¯ 00k | on R.

(2.16.9)

Moreover, ϕk (vk ) → w,

ϕ0k (vk ) → 1,

ϕ00k (vk ) → 0 on Ω.

(2.16.10)

We now fix k ∈ N, k > m. In view of Remark 2.12.1 and property (2.12.13), ◦ ∞ we find h¯ k (u) − w ∈ W 1,q 2,p (Ω) ∩ L (Ω). Thus, by Definition 2.13.1 and relation (2.16.6), we get  Z X Z α Aα (x, δ2 u)D ϕk (vk ) dx 6 F (x, u)ϕk (vk )dx. (2.16.11) Ω

α∈Λ

Ω

173

Section 2.16 Relationship with H-solutions and W -solutions

By virtue of Proposition 2.11.4 and relations (2.12.14), (2.16.8), and (2.16.9), we obtain Dα ϕk (vk ) = (1 − h¯ 0k (u))ϕ0k (vk )δ α u + ϕ0k (vk )Dα w

a.e. on Ω

for any n-dimensional multiindex α, |α| = 1, and |Dα ϕk (vk ) − (1 − h¯ 0k (u))ϕ0k (vk )δ α u − ϕ0k (vk )Dα w| X X |δ β u|2 |δ β u|2 + |h¯ 00k (u)| 6 2(1 − h¯ 0k (u))ϕ0k (vk ) |β|=1

|β|=1

+

2|h¯ 00k (vk )|

X

β

2

|δ u| +

|β|=1

X

|Dβ w|2 2|ϕ00k (vk )| |β|=1

a.e. on Ω

for any n-dimensional multiindex α, |α| = 2. Hence,  X  X
Aα (x, δ2 u)Dα w ϕ0k (vk ) Aα (x, δ2 u)δ α u 1 − h¯ 0k (u) ϕ0k (vk ) + α∈Λ

α∈Λ

6

X


Aα (x, δ2 u)D ϕk (vk ) + 2Φ 1 − h¯ 0k (u) ϕ0k (vk ) α

α∈Λ

¯ 00k (vk )| a.e. on Ω. + Φ|h¯ 00k (u)| + 2Φ|h¯ 00k (vk )| + 2Φ|ϕ By using relations (2.1.4), (2.12.14), (2.16.3), and (2.16.8), we get X  α Aα (x, δ2 u)D w ϕ0k (vk ) α∈Λ

6

X

Aα (x, δ2 u)Dα ϕk (vk ) + (c000+c2 +1)(g1 +g2 +1)(1− h¯ 0k (u))

α∈Λ

¯ 00k (vk )| a.e. on Ω, + Φ|h¯ 00k (u)| + 2Φ|h¯ 00k (vk )| + 2Φ|ϕ whence, in view of relations (2.16.11), (2.12.12), (2.12.13), and (2.12.15) and the inequality k > m, we conclude that  Z X α Aα (x, δ2 u)D w ϕ0k (vk )dx Ω

α∈Λ

Z

Z 6 F (x, u)ϕk (vk )dx + (c +c2 +1) (g1 +g2 +1)(1− h¯ 0k (u))dx Ω Ω Z Z 9 ¯ 00k (vk )|dx. Φ dx + 2 Φ|ϕ + k {|u| p2 . Let u be a proper entropy solution of problem (2.1.6), (2.1.7). Then u is a W -solution of problem (2.1.6), (2.1.7). Proof. By virtue of Theorem 2.16.1, u is an H-solution of problem (2.1.6), (2.1.7). Moreover, in view of Corollary 2.12.2 and the inequality q > p2 , we get δ α u ∈ L2 (Ω) for any n-dimensional multiindex α, |α| = 1, and δ α u ∈ L1 (Ω) for any n-dimensional multiindex α, |α| = 2. Thus, by virtue of Proposition 2.16.1, u is a W -solution of problem (2.1.6), (2.1.7). 

Section 2.17 Properties of summability of the proper entropy solutions

2.17

175

Properties of summability of the proper entropy solutions

Since the proper entropy solution of problem (2.1.6), (2.1.7) belongs to the set b 1,q (Ω), Corollaries 2.12.1–2.12.4 yield the following result: of functions H 2,p Proposition 2.17.1. Let u be a proper entropy solution of problem (2.1.6), (2.1.7). Then (i) u ∈ Lλ (Ω) for any λ ∈ (0, r∗ ); (ii) Φu ∈ Lλ (Ω) for any λ ∈ (0, r); ¯ (iii) δ α u ∈ Lλ (Ω) for any λ ∈ (0, r) and any n-dimensional multiindex α, |α| = 1; (iv) δ α u ∈ Lλ (Ω) for any λ ∈ (0, rp/q) and any n-dimensional multiindex α, |α| = 2; ◦

(v) u ∈ W 1,λ (Ω) for any λ ∈ [1, r); ◦

(vi) if q > p2 , then u ∈ W 2,λ (Ω) for any λ ∈ [1, rp/q). Byusing Proposition 2.13.1,we now establish more exact properties of summability of the proper entropy solutions of problem (2.1.6), (2.1.7) as compared with the properties described by assertions (i)–(iv) in Proposition 2.17.1. Proposition 2.17.2. Assume that u is a proper entropy solution of problem (2.1.6), (2.1.7). Let h ∈ C 1 (R), let M > 0, and let the following conditions be satisfied: (i) the function h is nonnegative and bounded on R; (ii) |h0 | 6 M h on R; Z +∞ (iii) h(τ )dτ < +∞. −∞

Then Φu h(u) ∈ L1 (Ω). Proof. We define a function ϕ : R → R as follows: Z s ϕ(s) = h(τ )dτ, s ∈ R. 0 1 C (R)

In view of the inclusion h ∈ and conditions (i)–(iii), we conclude that ϕ ∈ C 2 (R), the functions ϕ and ϕ0 are bounded on R, ϕ0 > 0 on R, and |ϕ00 | 6 M ϕ0 on R. Moreover, ϕ(0) = 0 and ϕ0 (s) = h(s) for any s ∈ R. Thus, by virtue of Proposition 2.13.1, the function Φu h(u) is summable on Ω. 

176 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity Further, we fix a function a¯ ∈ C 1 (R) with the following properties: a¯ (0) = 0, a¯ is even, a¯ is nonnegative on R, a¯ is nondecreasing on [0, +∞), and ∀ s ∈ R, |s| > 1,

a¯ (s) = |s|.

(2.17.1)

Corollary 2.17.1. Let u be a proper entropy solution of problem (2.1.6), (2.1.7). Then, for any λ > 1, Φu (1 + |u|)−1 [ ln(2 + |u|)]−λ ∈ L1 (Ω).

(2.17.2)

Proof. Let λ > 1. We set M = (1 + λ/ ln 2) max |¯a0 | [−1,1]

and define a function h : R → R as follows: h(s) = (1 + a¯ (s))−1 [ ln(2 + a¯ (s))]−λ ,

s ∈ R.

It follows from the properties of the function a¯ that h ∈ C 1 (R) and that the function h satisfies conditions (i)–(iii) of Proposition 2.17.2. Hence, by virtue of this proposition, Φu h(u) ∈ L1 (Ω). Thus, in view of the fact that 1 + a¯ (s) 6 2(1 + |s|) for any s ∈ R, we arrive at inclusion (2.17.2).  For any k > 0, let Tk : R → R be a function such that ( s if |s| 6 k, Tk (s) = k sign s if |s| > k. ◦

◦

1,q (Ω) Lemma 2.17.1. Let u ∈ H 1,q 2,p (Ω) and k ∈ N. Then Tk (u) ∈ W α α and D Tk (u) = δ u · 1{|u| 0 on R and χ0k (s) = 1 for |s| 6 k, we find Z Z Φu dx 6 Φu χ0k (u)dx. {|u| 0 a.e. on Ω.

(2.18.3)

Finally, in view of (2.7.1), we conclude that |χk (u)| 6 3|Tk (u)| on Ω. Hence, by virtue of (2.18.1)–(2.18.3), we get Z Z 6 Φu dx 6 |F (·, 0)||Tk (u)|dx + c01 . (2.18.4) c 2 {|u| 0 on Ω, such that, for

∗ −1

+ g(x).

Then there exists a generalized solution of problem (2.1.6), (2.1.7).

2.19

Examples of coefficients and the right-hand sides of Eq. (2.1.6)

Consider some examples in which inequalities (2.1.3)–(2.1.5) are satisfied. Example 2.19.1. Let c1 = 1, c2 = 1, g1 = 0 on Ω, and g2 = 0 on Ω. Assume that, for any n-dimensional multiindex α, |α| = 1, the function Aα is defined on Ω × Rn,2 as follows: Aα (x, ξ) = |ξα |q−1 sign ξα ,

(x, ξ) ∈ Ω × Rn,2 .

Moreover, for any n-dimensional multiindex α, |α| = 2, the function Aα is defined on Ω × Rn,2 as follows: Aα (x, ξ) = |ξα |p−1 sign ξα ,

(x, ξ) ∈ Ω × Rn,2 .

It is clear that, for any α ∈ Λ, the function Aα satisfies the Carathéodory conditions. In addition, it is easy to see that, for any x ∈ Ω and ξ ∈ Rn,2 , inequalities (2.1.3) and (2.1.4) are true. Finally, we note that if λ > 1 and s1 , s2 ∈ R, s1 6= s2 , then (|s1 |λ−1 sign s1 − |s2 |λ−1 sign s2 )(s1 − s2 ) > 0. This implies that, for any x ∈ Ω and ξ, ξ 0 ∈ Rn,2 , ξ 6= ξ 0 , inequality (2.1.5) holds.

Section 2.19 Examples of coefficients and the right-hand sides of Eq. (2.1.6)

185

Example 2.19.2. Assume that p > 2. Let c1 = nq , c2 = 1, g1 = 0 on Ω, and g2 = 0 on Ω. Suppose that, for any n-dimensional multiindex α, |α| = 1, the function Aα is defined on Ω × Rn,2 as follows: X (q−2)/2 Aα (x, ξ) = |ξβ |2 ξα , (x, ξ) ∈ Ω × Rn,2 , |β|=1

and that, for any n-dimensional multiindex α, |α| = 2, the function Aα is defined on Ω × Rn,2 as (p−2)/2 X |ξβ |2 ξα , (x, ξ) ∈ Ω × Rn,2 . Aα (x, ξ) = |β|=2

It is clear that, for any α ∈ Λ, the function Aα satisfies the Carathéodory conditions. In addition, it is easy to see that, for any x ∈ Ω and ξ ∈ Rn,2 , inequalities (2.1.3) and (2.1.4) are true. We now show that, for any x ∈ Ω and ξ, ξ 0 ∈ Rn,2 , ξ 6= ξ 0 , inequality (2.1.5) is valid. Indeed, let x ∈ Ω and let ξ, ξ 0 ∈ Rn,2 , ξ 6= ξ 0 . We set X Si = [Aα (x, ξ) − Aα (x, ξ 0 )](ξα − ξα0 ), i = 1, 2, |α|=i

ηi =

X

|ξβ |

2

1/2 ,

µi =

|β|=i

X

|ξβ0 |2

1/2 ,

i = 1, 2.

|β|=i

Therefore, X X q−2 q q−2 0 0 ξα ξα0 + µq1 . +µq−2 S1 = [η1 ξα −µq−2 1 ) 1 ξα ](ξα −ξα ) = η1 − (η1 |α|=1

|α|=1

(2.19.1) By the Cauchy–Buniakowski inequality, we get X 1/2  X 1/2 X = η 1 µ1 . ξα ξα0 6 |ξα |2 |ξα0 |2 |α|=1

|α|=1

|α|=1

Thus, in view of (2.19.1), we obtain q q−1 S1 > η1q − (η1q−2 + µq−2 − µq−1 1 )η1 µ1 + µ1 = (η1 1 )(η1 − µ1 ).

(2.19.2)

Similarly, S2 > (η2p−1 − µp−1 2 )(η2 − µ2 ). It is clear that S1 > 0 and S2 > 0.

(2.19.3)

186 Chapter 2 Nonlinear equations of the fourth order with strengthened coercivity Since ξ 6= ξ 0 , there exists an n-dimensional multiindex γ ∈ Λ such that ξγ 6= ξγ0 . Assume that |γ| = 1. If η1 6= µ1 , then S1 > 0 by virtue of (2.19.2). However, if η1 = µ1 , then, in view of the first equality in (2.19.1), X S1 = η1q−2 (ξα − ξα0 )2 > η1q−2 (ξγ − ξγ0 )2 > 0. |α|=1

Thus, if |γ| = 1, then S1 > 0. Similarly, by virtue of (2.19.3), we conclude that if |γ| = 2, then S2 > 0. It is now possible to assert that, in any case, S1 + S2 > 0. Hence, inequality (2.1.5) is true. Example 2.19.3. Assume that p = 2 and that, for any n-dimensional multiindex α, |α| = 2, we have σα > 0. Let n  2 o n o c1 = max nq , max σα , c2 = min 1, min σα , |α|=2

|α|=2

g1 = 0 on Ω and g2 = 0 on Ω. Suppose that, for any n-dimensional multiindex α, |α| = 1, the function Aα is defined on Ω × Rn,2 in the same way as in Example 2.19.2 and that, for any n-dimensional multiindex α, |α| = 2, the function Aα is defined on Ω × Rn,2 as follows: Aα (x, ξ) = σα ξα ,

(x, ξ) ∈ Ω × Rn,2 .

It is clear that, for any α ∈ Λ, the function Aα satisfies the Carathéodory conditions. In addition, it is easy to see that, for any x ∈ Ω and ξ ∈ Rn,2 , inequalities (2.1.3) and (2.1.4) are true. By analogy with Example 2.19.2, we finally conclude that, for any x ∈ Ω and ξ, ξ 0 ∈ Rn,2 , ξ 6= ξ 0 , inequality (2.1.5) is true. Within the framework of this example, we also note that if the following conditions are satisfied: σα = 1 if there exists i ∈ {1, . . . , n} such that αi = 2, σα = 2 if there is no i ∈ {1, . . . , n} such that αi = 2, then Eq. (2.1.6) turns into the equation −∆q u + ∆2 u = F (x, u), where ∆q is the q-Laplacian and ∆2 is a biharmonic operator. In conclusion, we present a simple example of validity of the conditions imposed on the function F in the section devoted to the theorems on existence and uniqueness of the solutions of problem (2.1.6), (2.1.7).

Section 2.19 Examples of coefficients and the right-hand sides of Eq. (2.1.6)

187

Example 2.19.4. Let µ, ν ∈ L1 (Ω), ν > 0 on Ω, let ϕ be a continuous nondecreasing function on R, and let a function F be defined on Ω × R as follows: F (x, s) = µ(x) − ν(x)ϕ(s), (x, s) ∈ Ω × R. It is clear that the function F satisfies the Carathéodory conditions. Moreover, for any x ∈ Ω, the function F (x, ·) is nonincreasing on R and, for any s ∈ R, the function F (·, s) belongs to L1 (Ω). Hence, the conditions of Theorems 2.7.1 and 2.14.1 imposed on the function F are satisfied. In this case, the function ϕ may obviously have any oder of growth. Thus, we can take ϕ such that, for any s ∈ R, ϕ(s) = |s|σ sign s, where σ > 0, or such that, for any s ∈ R, ϕ(s) = es . ∗ ∗ Finally, we note that if µ ∈ Lq /(q −1) (Ω), ν0 > 0, 0 < σ 6 q ∗ − 1, ν(x) 6 ν0 σ for any x ∈ Ω, and ϕ(s) = |s| sign s for any s ∈ R, then the function F satisfies the conditions of Theorem 2.18.1.

Part II

Removability of singularities of the solutions of quasilinear elliptic and parabolic equations of the second order

Chapter 3

Removability of singularities of the solutions of quasilinear elliptic equations 3.1

Introduction

In the present chapter, we study the problem of removability of the singularities of solutions of quasilinear elliptic equations. Let n > 3 and let Ω be an open set in Rn . Consider the following model elliptic equations: −∆p u + gu|u|p−2 = f q−1

−∆p u + u|u|

in Ω \ {x0 },

= 0 in Ω \ {x0 },

p > 1,

(3.1.1)

q > p − 1.

(3.1.2)

For harmonic functions, the exact condition of removability of isolated singularities is well known. This condition can be formulated in the form  o(|x − x0 |2−n ) as x → x0 , n > 2,  u(x) =   ln 1 as x → x0 , n = 2. |x−x0 | The problem of removability of singularities for general equations of the form (3.1.1) was studied by Serrin in [112, 113] for g, f ∈ Lq (Ω), q > n/p. He established the following sufficient condition of removability of the isolated singularity:    n−p O |x − x0 |− p−1 +ε , 1 < p < n, ε > 0,   u(x) = O ln1−ε 1 p = n, ε > 0. |x−x0 | , For the equation −∆u + uq = 0, Brézis and Véron [32] proved that the isolated singularity is always removable for q > n/(n − 2). Numerous authors studied the problem of removability of isolated singularities for elliptic equations of the form (3.1.1) and (3.1.2) (see, e.g., [12, 32, 46, 47, 62, 128, 132–134]). Most of these authors considered equations of special types. For a survey of these results, see the monograph by Véron [134]. The exact conditions for the removability of singularities were obtained for general equations of the form (3.1.1) and (3.1.2) in [99, 119, 120]. A significant part of Chapter 3 is devoted to the presentation of these results. The proof is based on new a priori estimates for the solutions with singularities. The method used for getting estimates of this kind is developed in [123].

192

Chapter 3 Removability of singularities of the solutions

The chapter is organized as follows: In Sec. 3.2, we present the results obtained in [99] and establish the exact condition for the removability of isolated singularities of the solutions of quasilinear elliptic equations of the form (3.1.1). In Sec. 3.3, we present the results obtained in [119] and prove the exact condition of removability of singularities on a manifold for the solutions of general quasilinear elliptic equations of the form (3.1.1). Finally, in Sec. 3.4, we present the results obtained in [120] and establish the exact condition of removability of singularities for the solutions of general quasilinear elliptic equations of the form (3.1.2).

3.2 3.2.1

Removability of isolated singularities Formulation of assumptions and principal results

Let Ω be an open bounded set in Rn , 1 < p 6 n. Let x0 ∈ Ω. Consider the equation     n X ∂u d ∂u ai x, u, = a0 x, u, , x ∈ Ω \ {x0 }. − (3.2.1) dxi ∂x ∂x i=1

Assume that the functions ai (x, u, ξ), i = 0, . . . , n, are defined for (x, u, ξ) ∈ Ω × R × Rn , satisfy the Carathéodory conditions and, with some positive constants c1 and c2 , the inequalities n X

ai (x, u, ξ)ξi > c1 |ξ|p − g1 (x)|u|p − f1 (x),

(3.2.2)

i=1

|ai (x, u, ξ)| 6 c2 |ξ|p−1 + g2 (x)|u|p−1 + f2 (x),

i = 1, . . . , n,

|a0 (x, u, ξ)| 6 h(x)|ξ|p−1 + g3 (x)|u|p−1 + f3 (x),

(3.2.3) (3.2.4)

where h(x), gi (x), fi (x), i = 1, 2, 3, are nonnegative functions such that hp (x) ∈ Ln/(p−δ) (Ω),

fi (x), gi (x) ∈ Lpi (Ω),

(3.2.5)

p1 = p3 = n/(p − δ), p2 = n/(p−1) for p < n,

and p2 = n/(n−1−δ) for p = n

with δ ∈ (0, 1/2 min(p−1, 1)). Let ψ(t) ∈ C ∞ (R), ψ(t) = 0 for t 6 1, ψ(t) = 1 for t > 2, and − 2 6 dψ/dt 6 0 for t ∈ [1, 2].
We define a function ψr (x) = ψ |x − x0 |/r , r > 0, and say that u(x) is a solution of Eq. (3.2.1) in Ω \ {x0 } if u(x)ψr (x) ∈ W 1,p (Ω) for any r > 0 and

Section 3.2

193

Removability of isolated singularities

the integral identity     Z n Z X ∂u ∂u ∂ ai x, u, [ϕψr ]dx − a0 x, u, ϕψr dx = 0 ∂x ∂xi ∂x Ω Ω

(3.2.6)

i=1

◦

holds for any function ϕ ∈ W 1,p (Ω). Let R0 be an arbitrary number satisfying the condition 0 < R0 < min{dist({x0 }, ∂Ω), 1}. For 0 < r 6 R0 , we denote M (r) = max{|u(x)| : r 6 |x − x0 | 6 R0 }.

(3.2.7)

Note that M (r) < ∞ (see, e.g., [88]). Definition 3.2.1. We say that a solution u(x) possesses a removable singularity at a point x0 if u(x) can be defined at x0 in such a way that the function u e(x) obtained as a result is a solution of Eq. (3.2.1) in the entire domain Ω and u e(x) ∈ W 1,p (Ω). The following theorems are main results of the present section: Theorem 3.2.1. Let conditions (3.2.2)–(3.2.5) be satisfied and let u(x) be a solution of Eq. (3.2.1) in Ω \ {x0 }. Suppose that lim M (r)r(n−p)/(p−1) = 0

r→0

lim M (r)| ln r|−1 = 0

r→0

for for

p < n, p = n.

(3.2.8) (3.2.9)

Then the singularity at the point x0 is removable and, moreover, r(n−p)/(p−1) is the best function for the removability of singularity at the point x0 for 1 < p < n and | ln r|−1 is the best function for the removability of singularity at the point x0 for p = n. Theorem 3.2.1 is a consequence of the Serrin results from [113] and Theorem 3.2.2. Theorem 3.2.2. Let all conditions of Theorem 3.2.1 be satisfied. Then there exist positive constants K1 and a depending only on n, p, and the norms ||hp ||Ln/(p−δ) (Ω) , ||fi ||Lpi (Ω) , and ||gi ||Lpi (Ω) and such that the inequalities    (n−p)/(p−1) r + ρa−(n−p)/(p−1) for p < n, M (ρ) 6 K1 M (r) (3.2.10) ρ     1 1−a ln ρ + ln for p = n, (3.2.11) M (ρ) 6 K1 M (r) ln r ρ hold for any 0 < r < ρ < R0 .

194

Chapter 3 Removability of singularities of the solutions

3.2.2

Integral estimates of the solutions for 1 < p < n

In what follows, we assume that lim M (r) = ∞

r→0

(3.2.12)

because, otherwise, Theorem 3.2.1 follows from the Serrin results obtained in [36]. By γ we denote all possible constants that depend only on n, p, and the norms ||hp ||Ln/(p−δ) (Ω) , ||fi ||Lpi (Ω) , and ||gi ||Lpi (Ω) . We fix a number R1 ∈ (0, R0 ) such that M (R1 ) > 1. Let B(x0 , R) = {|x − x0 | < R},

uR (x) = max{u(x) − M (R), 0},

and E(R) = {x ∈ B(x0 , R) \ {x0 } : u(x) > M (R)}. Lemma 3.2.1. Assume that all conditions of Theorem 3.2.1 are satisfied. Then p Z ∂u p ψr (x)dx 6 γ{M p (r)rn−p + M p (R)} (3.2.13) E(R) ∂x for 0 < r < R < R1 . Proof. We substitute the function ϕ(x) = uR (x)ψrp−1 (x) in the integral identity (3.2.6). By using inequalities (3.2.2)–(3.2.4), we obtain p Z Z ∂u p ψr (x)dx 6 γ H(x)upR (x)ψrp (x)dx ∂x E(R) E(R) Z + γM p (R) [g1 (x) + g2 (x)]ψrp (x)dx E(R) p   Z ∂ψr (x) p ∂ψr (x) dx + g2 (x) +γ |u(x)| ∂x ∂x K(r) Z +γ F (x)dx, (3.2.14) E(R)

where K(r) = {r < |x − x0 | < 2r},

H(x) = hp (x) + g1 (x) + g3 (x) + f3 (x), p/(p−1)

F (x) = f1 (x) + f2

(x) + f3 (x).

Section 3.2

195

Removability of isolated singularities

We now estimate the terms on the right-hand side of (3.2.14). By using conditions (3.2.5), the Hölder inequality, and the imbedding theorem, we obtain Z H(x)uPR (x)ψrp (x)dx E(R)

Z

H(x)uPR (x)ψrp (x)dx

6

Z +

H(x)uPR (x)ψrp (x)dx

K(r)

E(R)\K(r)

p  ∂u p n−p+δ n 6 γkHkL p−δ , (Ω) R dx + M (r)r E(R)\K(r) ∂x   Z ∂ψr (x) p + g2 (x) ∂ψr (x) dx 6 γM p (r)rn−p . |u(x)|p ∂x ∂x K(r) 

δ

Z

(3.2.15)

(3.2.16)

The second and fourth terms on the right-hand side of (3.2.14) re estimated by using (3.2.5) and choosing R1 from the condition γkHkLn/(p−δ) (Ω) R1δ = 1/2. In view of (3.2.14)–(3.2.17), we obtain (3.2.13).

(3.2.17) 

Let 0 < r < ρ < R1 ,

n
o M (ρ) > max 2M (R), M R/2 .

(3.2.18)

We define E(ρ, R) = {x ∈ Ω : 0 < uR (x) < M (ρ) − M (R)}.

(3.2.19)

Lemma 3.1.2. Suppose that all conditions of Theorem 3.2.1 are satisfied. Then, for all r, ρ, and R satisfying (3.2.18), the inequality p Z ∂u p ψr (x)dx E(ρ,R) ∂x  6 γ[M (ρ)−M (R)] M p−1 (r)rn−p Z + E(ρ)



p−1   ∂u p−1 p h(x) + g3 (x)u (x) ψr (x)dx + γM p (R) ∂x

holds. Proof. We substitute the function ϕ(x) = min{uR (x), M (ρ) − M (R)}ψrp−1 (x)

(3.2.20)

196

Chapter 3 Removability of singularities of the solutions

in the integral identity (3.2.6). By using inequalities (3.2.2)–(3.2.4), we obtain p Z ∂u p ψr (x) E(ρ,R) ∂x Z H(x)upR (x)ψrp (x)dx + γ[M (ρ)−M (R)] 6γ E(ρ,R)

 p−1  ∂u ∂ψr p−1 p−1 + g2 (x)u × ∂x ∂x ψr dx K(r)∩E(R) p−1   Z ∂u h(x) + g3 (x)up−1 ψrp (x)dx + γ[M (ρ)−M (R)] ∂x E(ρ) Z + γM p (R) F1 (x)dx, (3.2.21) Z

E(R)

where F1 (x) = g1 (x) + g3 (x) + f1 (x) + [f2 (x)]p/p−1 + f3 (x) and H(x) is defined in Lemma 3.2.1. We estimate the first term on the right-hand side of (3.2.21) by analogy with (3.2.15). Indeed, we have Z H(x)upR (x)ψrp (x)dx E(ρ,R) p  Z  ∂u
dx + M (ρ)−M (R) rn−p+δ . 6 γkHkLn/(p−δ) (Ω) R δ E(ρ,R) ∂x (3.2.22) By using the Hölder inequality and Lemma 3.2.1, we obtain the following relation: p−1   Z ∂u ∂ψr p−1 ψr (x)dx 6 γ M p−1 (r)rn−p + r(n−p)/p M p−1 (R) . ∂x ∂x K(r)∩E(R)

(3.2.23) By using the definition of M (r) and the Hölder inequality, we obtain Z p−1 ∂ψr (x) dx 6 γM p−1 (r)rn−p . g2 (x)u (3.2.24) ∂x K(r)∩E(R) Thus, choosing R1 from the condition γR1δ kHkLn/(p−δ) (Ω) = 1/4,

(3.2.25)

we obtain inequality (3.2.20) from (3.2.21)–(3.2.25).



Section 3.2

197

Removability of isolated singularities

Lemma 3.2.3. Suppose that all conditions of Theorem 3.2.1 are satisfied. Then p Z −q ∂u uR ψrp (x)dx 6 γ[M p−1 (r)rn−p + ργ M p−1 (R)], (3.2.26) ∂x E(ρ) where q = 1 + δ/(n − p) and 0 < R < R1 . Proof. Substituting the function   1−q 1−q ψrp−1 (x) ϕ(x) = [M (ρ) − M (R)] − [max(uR (x), M (ρ) − M (R))] in the integral identity (3.2.6), we get p Z −q ∂u uR ψrp (x)dx ∂x E(ρ) Z p p u−q 6γ R (x)[g1 (x)u + f1 (x)]ψr (x)dx E(ρ)

 p−1  ∂u ∂ψr p−1 p−1 + ∂x + g2 (x)u (x) + f2 (x) ∂x ψr (x)dx K(r)∩E(ρ) Z +γ [uq(p−1) hp (x) + g3 (x)up−1 + f3 (x)]ψrp (x)dx. (3.2.27) Z

E(ρ)

By virtue of the Hölder inequality and the imbedding theorem, we find Z p u−q R (x)g1 (x)u dx E(ρ)\K(r) Z p p 6γ g1 (x)u−q R (x)[(u(x) − M (ρ)) + M (ρ)]dx E(ρ)\K(r) p Z  ∂u −q δ n−p p−q uR (x) dx + M 6 γkg1 kLn/(p−δ) (Ω) ρ (ρ)ρ , ∂x E(ρ)\K(r) (3.2.28) Z p p−q u−q (r)rn−p+δ . (3.2.29) R (x)|u(x)| g1 (x)dx 6 γkg1 kLn/(p−δ) (Ω) M K(r)

In view of condition (3.2.8), we get Z

q(p−1) p

u E(ρ)

h

(x)ψrp (x)dx

Z

p

6 γkh kLn/(p−δ) (Ω) Z

ρ

6γ

[M (t)]

|u|

q(p−1)n n−p(1−δ)

n−p(1−δ) n dx

K(r,ρ)∩E(ρ)

q(p−1)n n−p(1−δ)

t

n−1

n−p(1−δ) n dt

r

Z 6γ

ρ

qn(n−p)

t r

n−1− n−p(1−δ)

n−p(1−δ) n 6 γρδ(p−1) . dt

198

Chapter 3 Removability of singularities of the solutions

Similarly, we have Z

g3 (x)up−1 ψrp (x)dx 6 γρδ .

(3.2.30)

E(ρ)

Moreover, by using (3.2.5), we obtain Z [f1 (x) + f3 (x)]dx 6 γρn−p+δ , E(ρ) Z f2 (x)dx 6 γrn−p+1 .

(3.2.31) (3.2.32)

K(r)

We choose R1 from the condition γkg1 kLn/(p−δ)(Ω) R1δ = 1/4.

(3.2.33) 

In view of (3.2.27)–(3.2.33), this yields inequality (3.2.26). Lemma 3.2.4. The following estimate is true: p Z ∂u p ψr (x)dx E(ρ,R1 ) ∂x  6 γ (M (ρ) − M (R1 ))[M p−1 (r)rn−p + ργ ] + 1

(3.2.34)

Inequality (3.2.34) follows from Lemmas 3.2.2 and 3.2.3.

Pointwise estimates of the solutions for 1 < p < n

3.2.3

We define numerical sequences (1)

ρi

=

ρ (1 + 2−i ), 2

(2)

ρi

= |xi − x0 |,

and

(2)

di =

ρ ρ (2) for ρi 6 2ρ or di = i 8 · 2i 4

(2)

for ρi

> 2ρ,

i = 1, 2, . . . ,

where xi are points satisfying the condition (1)

ρi

6 |xi − x0 | 6

R1 , 2

(1)

|u(xi )| = M (ρi ),

i = 1, 2, . . . .

(3.2.35)

We also consider the sets   (1) (2) (1) Gi = x ∈ Rn : ρi+1 6 |x − x0 | 6 2ρi − ρi+1 . By using the iterative Moser method (see, e.g., [124], Chap. 8), we obtain Z (1) −n p [M (ρi ) − M (R)] 6 γdi [uR (x) + 1]p dx. (3.2.36) Gi

Removability of isolated singularities

199

By virtue of the Poincaré inequality, we get Z Z (2) upR (x)dx 6 γ[ρi ]p

(3.2.37)

Section 3.2

p ∂u dx, Di ∂x

Gi

where (1)

Di = E(R) ∩ {ρi+1 6 |x − x0 | 6 R}. Note that

  (1) uR (x) 6 M ρi+1 − M (R) for x ∈ Di .

According to Lemma 3.2.4, we find Z p nh   i o ∂u dx 6 γ M ρ(1) − M (R) [M p−1 (r)rn−p + ργ ] + 1 . i+1 Di ∂x

(3.2.38)

It follows from (3.2.36)–(3.2.38) that h   ip (1) M ρi −M (R) 6 γ2(n−p)i ρn−p n h   i o (1) × 1+ M ρi+1 −M (R) [M p−1 (r)rn−p +ργ ] . (3.2.39) If, for some i = i0 > 1, the inequality h   i (1) M ρi0 +1 − M (R) [M p−1 (r)rn−p + ργ ] 6 1

(3.2.40)

is true, then, in view of the fact that M (ρ) is a nonincreasing function in ρ, we get the following relation from (3.2.39) and (3.2.40): M (ρ) 6 M (ρ(1) ) 6 γρ(p−n)/p .

(3.2.41)

If, for all i > 1, the inequality inverse to (3.2.40) is true, then, according to (3.2.39), we find (1)

M (ρi ) − M (R) 6 γ2(n−p)i ρn−p h   i (1) × M ρi+1 − M (R) [M p−1 (r)rn−p + ργ ].

(3.2.42)

Iterating inequality (3.2.42), we obtain (1) M (ρ1 )

 (n−p)/(p−1)  r γ−(n−p)/(p−1) 6 M (R) + γ M (r) +ρ . ρ 

In both cases, we arrive at the required inequality (3.2.10). For p = n, Theorem 3.2.2 is proved in a similar way.

(3.2.43)

200

Chapter 3 Removability of singularities of the solutions

3.3

Removability of singularities of the solutions of elliptic equations on manifolds

3.3.1

Formulation of assumptions and main results

Let be a Γ-manifold of the class C 1 without boundary of dimension s contained in Ω. We say that u(x) is a solution of Eq. (3.2.1) in Ω \ Γ if, for any function ψ(x) ∈ C 1 (Ω \ Γ) equal to zero near Γ, the inclusion u(x)ψ(x) ∈ W 1,p (Ω) is true and the integral identity     Z n Z X ∂u ∂u ∂ ai x, u, (3.3.1) [ϕψ]dx + a0 x, u, ϕψdx = 0 ∂x ∂xi ∂x Ω Ω i=1

◦

holds for any function ϕ(x) ∈ W 1,p (Ω). We set r0 = min{d(Γ, ∂Ω), 1}, where d(Γ, ∂Ω) is the distance between Γ and ∂Ω. For r ∈ (0, r0 ], we define M (r) = max{|u(x)| : r 6 d(x, Γ) 6 r0 }.

(3.3.2)

Definition 3.3.1. We say that a solution u(x) of Eq. (3.2.1) has a removable singularity on the manifold Γ if u(x) can be extended to Γ so that the obtained extension u e(x) of the function u(x) satisfies Eq. (3.2.1) in Ω and u e(x) ∈ W 1,p (Ω). The following theorems are the main results of this section: Theorem 3.3.1. Assume that, for 1 < p < n − s and 1 6 s 6 n − 2, conditions (3.2.2)–(3.2.5) are satisfied and that u(x) is a solution of Eq. (3.2.1) in Ω \ Γ. Suppose that lim M (r)r(n−p−s)/(p−1) = 0.

r→0

(3.3.3)

Then the singularity of u(x) on Γ is removable and, moreover, r(n−p−s)/(p−1) is the best function for the removability of singularity on the manifold Γ. Theorem 3.3.2. Let, for p = n − s and 1 6 s 6 n − 2, conditions (3.2.2)– (3.2.5) be satisfied and let u(x) be a solution of Eq. (3.2.1) in Ω \ Γ. Suppose that lim M (r)| ln r|−1 = 0.

r→0

(3.3.4)

Then the singularity on the manifold Γ is removable and | ln r|−1 is the best function for the removability of singularity on the manifold Γ.

Section 3.3

201

Removability of singularities of the solutions of elliptic equations

Theorem 3.3.1 follows from the Serrin results presented in [113] and the following pointwise estimate: n λ/(p+λ−1)  p−1 1/(p+λ−1) M (ρ) 6 Kρ−(n−p−s)/(p−1) κ e(r) M (r)rn−s−p 
τ o +κ e(r) ρτ1 + r/ρ 2 , (3.3.5) e 6 r0 with some positive which is proved in what follows for 0 < r < θρ e constants K, λ, θ, τ1 , and τ2 and a function κ e(r) = max{M (t)t(n−s−p)/(p−1) : r 6 t 6 r0 }.

3.3.2

(3.3.6)

Integral estimates for the gradient of solution in the case 1 < p < n − s

We fix a covering of the manifold Γ with local neighborhoods U (i) , i = 1, . . . , I, (i) (i) such that the following inclusion is true in local coordinates y (i) = (y1 ), . . . , yn corresponding to a neighborhood U (i) :  (i) (i) Γ ∩ C (i) (R0 , H0 ) ⊂ y1 = . . . = yn−s = 0 ; here, n o C (i) (R0 , H0 ) = y (i) : |[y (i) ]0 | 6 R0 , |[y (i) ]00 | 6 H0 , (i)

(i)

[y (i) ]0 = (y1 , . . . , yn−s ),

(i)

[y (i) ]00 = (yn−s+1 , . . . , yn(i) ).

It is possible to assume that R0 and H0 are independent of i and such that C (i) (R0 , H0 ) ⊂ Ω,

I [

C (i) (R0 , H0 /2) ⊃ Γ.

i=1

For the solution u(x) of Eq. (3.3.1) rewritten in the form u(i) (y (i) ) in terms of the coordinates y (i) , we define n o m(i) (r) = max |u(i) (y (i) )| : y (i) ∈ C (i) (R0 , H0 /2) \ C (i) (r, H0 /2) and set m(r) = max{m(i) (r), i = 1, . . . , I}.

(3.3.7)

Our subsequent analysis is performed in local coordinates. For simplicity, we omit the index i. Thus, it suffices to consider the case C(R0 , H0 ) ∩ Γ ⊂ {x1 = . . . = xn−s = 0},

(3.3.8)

where C(R0 , H0 ) = {x ∈ Rn : |x0 | < R0 , |x00 | < H0 } and x0 = (x1 , . . . , xn−s ), x00 = (xn−s+1 , . . . , xn ). We can also assume that, for 0 < r 6 R0 , the inequality max{|u(x)| : x ∈ C(R0 , H0 ) \ C(r, H0 )} 6 m(C0 r)

(3.3.9)

is true with a fixed constant C0 and a function m(r) given by equality (3.3.7).

202

Chapter 3 Removability of singularities of the solutions

For 0 < r 6 R0 , we set E(r) = {x ∈ C(R0 , H0 ) : u(x) > m(r)}

(3.3.10)

and note that (3.3.9) implies the inclusion E(r) ⊂ C(r/C0 , H0 ). We now define a function ur (x) = max{u(x) − m(r); 0},

x ∈ C(R0 , H0 ) \ Γ.

(3.3.11)

We fix a function τ : R → R from class C ∞ (R) such that dτ (t) 6 2 for t ∈ R. dt For r > 0 and h > 0, we define the following functions: τ (t) = 0 for t 6 1,

τ (t) = 1 for t > 2,

06

ψr (x) = τ (|x0 |/r) and χh (x) = 1 − τ (|(x00 − ξ 00 )/h|),

(3.3.12)

where ξ 00 = (ξN −s+1 , . . . , ξN ) is a fixed point such that |ξ 00 | 6 H/2. For 0 < t 6 R0 , we also define a function κ(t) by the equality   κp−1 (t) = max mp−1 (r)rn−p−s : t 6 r 6 R0 . (3.3.13) The boundedness of the function κ(t) follows from condition (3.3.3). In what follows, it is assumed that this condition is satisfied. Lemma 3.3.1. Assume that all conditions of Theorem 3.3.1 are satisfied. Then there exist positive constants K1 and R1 depending only on known parameters and such that, for 0 < r < ρ 6 R1 ,

0 < h 6 H0 /2,

ρ 6 h,

(3.3.14)

the estimate  Z ∂u p  (n−p+1)/n p p ψr (x)χh (x)dx 6 K1 mp (C0 r)rn−p−s + mp (ρ) ρn−s hs E(ρ) ∂x  ∂u p h ρ ip Z p p + (x)dx (3.3.15) ψ (x)χ r 2h h E(ρ) ∂x is true with functions ψr (x) and χh (x) given by equalities (3.3.12). Proof. By virtue of the Serrin theorem [113] on the removability of singularities, under condition (3.3.3), it suffices to prove Theorem 3.3.1 only in the case where m(r) → ∞ as r → 0. Thus, we can impose the following conditions on R1 : R1 6 R0 ,

m(R1 ) > 1.

We substitute the functions ϕ(x) = uρ (x)ψrp (x)χph (x),

ψ(x) = ψr/2 (x)χ2h (x),

(3.3.16)

Section 3.3

Removability of singularities of the solutions of elliptic equations

203

in the integral identity (3.3.1) and assume that 0 < r < ρ < R0 /2 and 0 < h 6 H0 /2. In view of conditions (3.2.2)–(3.2.5) and the Young inequality, we arrive at the estimate Z 5 ∂u p X p p ψ (x)χ (x)dx 6 γ Ii , (3.3.17) r h E(ρ) ∂x i=1

where Z I1 = E(ρ)

h1 (x)upρ (x)ψrp (x)χph (x)dx,

Z

∂ψ p r p upρ (x) χh (x)dx, ∂x E(ρ) Z ∂χ p h upρ (x)ψrp (x) I3 = dx, ∂x E(ρ) I2 =

Z I4 =

up (x)[f2 (x) + g2 (x)]p/(p−1) χph (x)dx, Z p I5 = m (ρ) h2 (x)ψrp (x)χph (x)dx,

E(ρ)∩K 0 (r)

E(ρ)

 p/(p−1) h1 (x) = f1 (x) + g1 (x) + f2 (x) + g2 (x) + f3 (x) + g3 (x) + hp (x),  p/(p−1) h2 (x) = f1 (x) + g1 (x) + f2 (x) + g2 (x) + f3 (x) + g3 (x), K 0 (r) = {x ∈ Rn : r 6 |x0 | 6 2r}. Further, we estimate the terms on the right-hand side of (3.3.17). By using conditions (3.2.2)–(3.2.4), the Hölder inequality, and the imbedding theorem, we obtain Z  ∂u p  δ/n p p p n−p−s s I1 6 γ meas E(ρ) h + I3 . ψr (x)χh (x)dx + m (C0 r)r E(ρ) ∂x (3.3.18) The following estimate directly follows from the definition of m(r): I2 6 γmp (C0 r)rn−p−s hs . The integral I3 can be represented in the form Z Z ∂χ p ∂χ p h h upρ (x) I3 = upρ (x)ψrp (x) dx + dx. ∂x ∂x E(ρ)∩K 0 (r) E(ρ)\K 0 (r)

(3.3.19)

(3.3.20)

In the first integral in (3.3.20), we estimate uρ (x) in terms of m(C0 r). To estimate the second integral, we use the Poincaré inequality in the variable x0 .

204

Chapter 3 Removability of singularities of the solutions

This yields   ∂u p  ρ p Z p p I3 6 γ mp (C0 r)rn−s hs−p + ψr (x)χ2h (x)dx . h E(ρ) ∂x

(3.3.21)

To estimate the integral I4 , we majorize uρ (x) by m(C0 r) and apply the Hölder inequality. As a result, we arrive at the inequality  (n−p+δ)/n I4 6 γmp (C0 r) rn−s hs .

(3.3.22)

Finally, by using the Hölder inequality, we get  (n−p+δ)/n I5 6 γmp (ρ) ρn−s hs .

(3.3.23)

In addition to (3.3.16), we impose the following condition on R1 : γ [meas C(R1 /C0 , H0 )]δ/n < 1/2.

(3.3.24)

In view of the fact that, for ρ > R1 , the inclusion E(ρ) ⊂ C(R1 /C0 , H0 ) is true, relations (3.3.17)–(3.3.24) yield estimate (3.3.15), which completes the proof of Lemma 3.3.1.  We need a global analog of estimates (3.3.15). It is possible to construct a function de : Ω → R from the class C 1 such that, for x ∈ Ω, the inequality e 6 k2 d(x, Γ) k1 d(x, Γ) 6 d(x)

(3.3.26)

holds with nonnegative constants k1 and k2 . In what follows, we include these e we can assume constants in the set of known parameters. For the function d, 0 e = |x | for x ∈ C(R0 , H0 ). that d(x) We define e < r}, e C(r) = {x ∈ Ω : d(x) e m(r) e = max{|u(x)| : x ∈ Ω \ C(r)}, e 0}, u er (x) = max{u(x) − m(r),

e ψer (x) = τ (d(x)/r),

e e E(r) = {x ∈ Ω : u(x) > m(r)},

(3.3.27) (3.3.28)

where τ is the function used in equalities (3.3.12). Substituting the functions ϕ(x) = u ep (x)ψerp−1 (x),

ψ(x) = ψer (x)

in (3.3.1) and estimating the integrals in the same way as in the proof of Lemma 3.3.1, we arrive at the following result:

Section 3.3

Removability of singularities of the solutions of elliptic equations

205

Lemma 3.3.2. Suppose that all conditions of Theorem 3.3.1 are satisfied. Then there exist positive constants K2 and R2 depending only on known parameters and such that the inequality Z ∂u p n o (n−s)(n−p+δ) ep n e p (e r)e rn−p−s + m e p (e ρ)e ρ (3.3.28) ψre(x)dx 6 K2 m e ρ) ∂x E(e holds for 0 < re < ρe < R2 . Note that the inequality m(k e 3 r) 6 m(r) 6 m(k e 4 r)

(3.3.29)

is true for some positive constants k3 and k4 and 0 < r 6 R0 . Lemma 3.3.3. Assume that all conditions of Theorem 3.3.1 are satisfied. Then there exist positive constants K3 and R3 depending solely on the known parameters and such that, for 0 < r < ρ 6 R3 ,

0 < h 6 H0 /2,

ρ 6 γ1 h,

the estimate Z ∂u p p p ψr (x)χh dx ∂x E(ρ) n  (n−p+δ)/n o 6 K3 mp (C0 r)rn−p−s hs + mp (ρ) ρn−s hs

(3.3.30)

(3.3.31)

is true. Proof. By using inequality (3.3.28) with re = k3 and ρe = k3 ρ and inequality (3.3.29) for 0 < r < ρ 6 R2 /k3 , we arrive at the estimate Z ∂u p p p ψr (x)χH0 /2 (x)dx ∂x E(ρ) o n (n−s)(n−p+δ) n . (3.3.32) 6 γ mp (C0 r)rn−p−s hs + mp (ρ)ρ Hence, inequality (3.3.31) is true for h = H0 /2 with a constant K3 (H0 ) depending on H0 . Further, we prove (3.3.31) by induction on j for h = hj = H0 /2j . Assume that inequality (3.3.31) is true for some j > 1 with a constant K3 (hj ). It is necessary to prove that this inequality holds for j + 1. By using (3.3.15) with h = hj+1 and (3.3.31) with h = hj , we arrive at the estimate Z ∂u p p p ψr (x)χhj+1 (x)dx E(ρ) ∂x   h ρ ip K3 (hj ) 6 2s K1 1 + hj+1 n  (n−p+δ) o × mp (C0 r)rn−p−s hsj+1 + mp (ρ) ρn−s hsj+1 n . (3.3.33)

206

Chapter 3 Removability of singularities of the solutions

Thus, we get an inequality of the form (3.3.31) for h = hj+1 with   h ρ ip K3 (hj ) , j = 1, 2, . . . . K3 (hj+1 ) = 2s K1 1 + hj+1

(3.3.34)

To complete the proof of Lemma 3.3.3, it suffices to show that, by the proper choice of the number γ1 , we can guarantee the validity of the estimate K3 (hj ) 6 γ for j = 1, 2, ..., J, where J is determined from the condition γ1

H0 H0 < ρ 6 γ1 J−1 . J 2 2

Relation (3.3.34) immediately implies that the required uniform estimate K3 (hj ) is true for γ1 obtained from the condition 2s K1 γ1p = 1/2.

(3.3.35)

Thus, we arrive at estimate (3.3.31) for the specified value of γ1 and R3 = min(R1 , R2 /K3 ).  For 0 < ρ0 < ρ 6 R0 /2, we define a set  E(ρ0 , ρ) = x ∈ E(ρ) : u(x) < m(ρ0 ) . Moreover, for ρ > 0, we define a function  u(ρ) (x) = min u(x), m(ρ) ,

x ∈ C(R0 , H0 ).

(3.3.36)

(3.3.37)

Lemma 3.3.4.  Suppose that all conditions of Theorem 3.3.1 are satisfied. Let θ ∈ 0, 1/2 and λ ∈ (0, 1) be some numbers that depend only on known parameters. Then there exist positive constants K4 and R4 depending only on known parameters and such that, for ρ > R4 , 0 < r < θρ/2 < ρ0 6 θρ,

ρ 6 h,

the estimate Z ∂u p uλ−1 (x) ψrp (x)χph (x)dx ∂x E(ρ0 ,ρ)  6 K4 J (1) +J (2) +J (3) +R(1) (C0 r, ρ0 , h)

(3.3.38)

(3.3.39)

is true. Here, we have used the following notation:   p J (1) = mλ (ρ0 ) − mλ (ρ) p−1 Z ∂u p  λ − p × u (x) − mλ (ρ) p−1 uλ−1 (x) ψrp (x)χph (x)dx, ∂x 0 E(ρ )

Section 3.3

Removability of singularities of the solutions of elliptic equations

J (2) =

1 hp

Z E(ρ)

J

(3)

207

 λ p u (x) − mλ (ρ) [u(x)](1−λ)(p−1) ψrp (x)χp2h (x)dx, Z



= E(ρ0 )

p+λ−1 u(x) h1 (x)ψrp (x)χph (x)dx,

R(1) (r, ρ, h) = mλ (ρ)mp−1 (r)rn−p−s hs    n−s s  n−p+δ h 1 p−1 in−s−p s p+λ−1 n + rpρ p +m (ρ) ρ h h .

(3.3.40)

Proof. We substitute the functions ψ(x) = ψ r2 (x)χ2h (x) and    0 ϕ(x) = max [uρ (x)]λ − mλ (ρ), 0 ψrp (x)χph (x) 1 − ψp (x) in the integral identity (3.3.1) under the assumption that condition (3.3.38) is satisfied and 2ρ 6 min(R1 , R2 ). By using conditions (3.2.2)–(3.2.5) and the Young inequality, we arrive at the estimate Z 11 ∂u p X uλ−1 (x) ψrp (x)χph (x)dx 6 γ Ii , (3.3.41) ∂x E(ρ0 ,ρ) i=6

where

Z I6 =

E(ρ0 ,ρ)

h1 (x)up+λ−1 (x)ψrp (x)χph (x)dx,

Z

∂ψ p r p up+λ−1 (x) χh (x)dx, ∂x 0 E(ρ ,ρ) Z  p+λ−1 p  (1−λ)(p−1) p ∂χh p I8 = u (x) − mλ (ρ) u(x) ψr (x) dx, ∂x E(ρ0 ,ρ)  Z ∂u p−1  ∂χ   h p−1 p I9 = mλ (ρ0 ) − mλ (ρ) χh (x) + h(x)χh (x) ψrp (x)dx, ∂x E(ρ0 ) ∂x   Z   ∂ψr p−1 ∂u p−1 p I10 = mλ (ρ0 ) − mλ (ρ) + (f2 +g2 )up−1 ψr (x)χh (x), ∂x ∂x 0 E(ρ ) I7 =

  I11 = mλ (ρ0 ) − mλ (ρ) Z ∂χ h i h p−1 p × up−1 (x) (f2 + g2 ) χh (x) + (f3 + g3 )χh (x) ψrp (x)dx. ∂x E(ρ0 ) We now estimate the terms on the right-hand side of (3.3.41). By using conditions (3.2.2)–(3.2.5), the Hölder inequality, and the imbedding theorem, we obtain Z   I6 6 γ uλ−1 (x) upρ + mp (ρ) h1 (x)ψrp (x)χph (x)dx E(ρ0 ,ρ)

208

Chapter 3 Removability of singularities of the solutions

Z

∂  p F (1) (u)ψr (x)χh (x) dx E(ρ) ∂x  n−s s (n−p+δ)/n  p+λ−1 +γ ρ h m(ρ) ,

 δ/n 6 γ ρn−s hs

where


 0 (λ−1)/p  (ρ0 )  F (1) u(x) = u(ρ ) (x) u (x) − m(ρ) + .

(3.3.42) (3.2.43)

Further, we estimate the integral on the right-hand side of (3.3.42) as follows: Z ∂  p F (1) (u)ψr (x)χh (x) dx E(ρ) ∂x   p+λ−1  n−s s−p n−s−p s  6 γ m(ρ0 ) ρ h +r h  Z ∂u p + uλ−1 (x) ψrp (x)χph (x)dx . (3.3.44) ∂x E(ρ0 ,ρ) The estimate for I7 directly follows from the definition of the set E(ρ0 , ρ):  p+λ−1 n−s−p s I7 6 γ m(ρ0 ) r h . (3.3.45) By using the Young inequality, we get the estimate
I8 + I11 6 γ J (2) + J (3) , where J (2) and J (3) are the same quantities as in (3.3.39) By using the Young inequality, we estimate I9 as follows:
I9 6 γ J (1) + J (2) + J (3) .

(3.3.46)

(3.3.47)

Finally, in view of conditions (3.2.2)–(3.2.5), the Hölder inequality, and Lemma 3.3.3, we arrive at the following estimate for I10 :  Z  p−1 ∂u p p  1  λ 0  p p λ rn−s−p hs p I10 6 γ m (ρ )−m (ρ) ψr (x)χh (x)dx E(ρ0 ) ∂x  mp−1 (C0 r)  n−s s  n−p+1+δ n · r h + r  λ 0 n p−1 λ 6 γ m (ρ )−m (ρ) m (C0 r)rn−s−p hs o n−s−p (n−s−p) p−1 p hs . (3.3.48) + mp−1 (ρ0 )r p ρ We choose sufficiently small R4 . In this case, inequalities (3.3.41), (3.3.43)– (3.3.48) imply estimate (3.3.39).  In what follows, the estimates are obtained under the assumption that the following condition is satisfied for the considered values of ρ0 and ρ: m(ρ0 ) − m(ρ) > 0.

(3.3.49)

Section 3.3

Removability of singularities of the solutions of elliptic equations

209

Lemma 3.3.5. Suppose that the conditions of Lemma 3.3.4 are satisfied and inequality (3.3.49) is true. Then there exists a positive constant K5 depending only on known parameters and such that the following estimate is true: n h J (1) 6 K5 mλ (ρ0 ) mp−1 (C0 r)rn−s−p hs o  n−s−p i + mp−1 (ρ0 ) r1/p ρ(p−1)/p hs + J (2) + J (3) , (3.3.50) where J (1) , J (2) , and J (3) are given by equalities (3.3.40). Proof. We substitute the functions ψ(x) = ψ r2 (x)χ2h (x) and n − 1  − 1 o ϕ(x) = mλ (ρ0 )−mλ (ρ) p−1 − (uρ0 (x)+m(ρ0 ))λ −mλ (ρ) p−1 ψrp (x)χph (x) in the integral identity (3.3.1). By using conditions (3.3.2)–(3.3.5) and the Young inequalities, we arrive at the estimate  J (1) 6 γ J (2) + J (3) + I10 , (3.3.51) Estimate (3.3.50) directly follows from inequalities (3.3.39) and (3.3.48). We introduce the notation  p  (1−λ)(p−1) Fρ (u(x)) = uλ (x) − mλ (ρ) u(x)



(3.3.52)

for x ∈ E(ρ) and ρ and λ satisfying the conditions of Lemma 3.3.4. Theorem 3.3.3. Let the conditions of Theorem 3.3.1 be satisfied and let r, ρ, ρ0 , h, and λ satisfy the conditions of Lemma 3.3.4. Then there exists a positive constant K6 depending only on known parameters and such that the estimate Z ∂u p uλ−1 (x) ψrp (x)χph (x)dx ∂x 0 E(ρ ,ρ)  6 K6 R(1) (C0 r, ρ0 , h)  Z 
 1 (3.3.53) + Fρ u(x) ψrp (x) h1 (x)χph (x) + p χp2h (x) dx h E(ρ) is true. Proof. Assume that condition (3.3.49) is satisfied because, otherwise, inequality (3.3.53) is obvious.The estimate for the term J (1) on the right-hand side of inequality (3.3.39) is obtained in Lemma 3.3.5. By virtue of the inequality 
 p+λ−1 up+λ−1 (x) 6 γ Fρ u(x) + m(ρ) for x ∈ E(ρ) and the Young inequality, we deduce the estimate for J (3) . By using the established estimate for J (3) and relations (3.3.39) and (3.3.50), we arrive at inequality (3.3.53). 

210

3.3.3

Chapter 3 Removability of singularities of the solutions

Pointwise integral estimates for the solution in the case 1 < p < n − s

We introduce the notation Z (n−p+δ)/n   n  n−s s (p−δ)/n I(ρ, h) = Fρ (u(x))ψrp (x)χph (x) n−p+δ dx ρ h , E(ρ) (3.3.54) n−s sp n−s−p +p , β1 = s + , α1 = −λ p−1 n n n−s−p n−s s α2 = −λ +p+δ , β2 = s + δ , p−1 n n δ n − s sp n−s−p  + p− , β3 = s + , (3.3.55) α3 = −λ p−1 2 n n where Fρ (u(x)) is given by equality (3.3.52). We can assume that δ is sufficiently small. In particular, we assume that the inequality n−s−p δ(n − s) > p 2n is true. Lemma 3.3.6. Suppose that the conditions of Lemma 3.3.4 are satisfied. Then there exist constants K6 and R4 depending only on known parameters and such that R4 6 R3 and the following estimate is true:  ρ (n−s) p (n−s)(p−δ) n n I(θρ, h) + K6 I(ρ, 2h) + R(2) (r, ρ, h), I(ρ, h) 6 2p−1 θ− h (3.3.56) for ρ 6 R4 , where n  λ R(2) (r, ρ, h) = K6 ρα1 hβ1 κ(C0 r) mp−1 (C0 r)rn−np−s h i p+λ−1 o δ(n−s)  + ρα2 hβ2 + ρα3 hβ3 r 2n κ(C0 r) and the function κ(r) is given by equality (3.3.13). Proof. By χ(E(θρ, ρ)) and χ(E(θρ)) we denote the characteristic functions of the sets E(θρ, ρ) and E(θ, ρ), respectively. We now estimate I(ρ, h). To this end, we use the following inequality:  λ p u (x)−mλ (ρ)  p 6 uλ (x)−mλ (ρ) χ(E(θρ, ρ))  + 2p−1 [uλ (x)−mλ (θρ)]p + [mλ (θρ)−mλ (ρ)]p χ(E(θρ, ρ))  p 6 2p−1 [uλ (x)−mλ (θρ)]p χ(E(θρ))+2p−1 [u(θρ) (x)]λ −mλ (ρ) for x ∈ E(ρ).

Section 3.3

Removability of singularities of the solutions of elliptic equations

211

We arrive at the estimate  p/n I(ρ, h) 6 2p−1 θ(n−s)(p−δ)/n I(θρ, h) + 2p−1 pn−s hs I12 ,

(3.3.57)

where Z I12 = E(ρ)

n−p n o n n p  (1−λ)(p−1) p n−p dx (u(θρ) (x))λ −mλ (ρ) u(x) ψr (x)χph (x) .

Estimating the last integral by the imbedding theorem, we get I12 6 γ(I13 + I14 + I15 ),

(3.3.58)

where Z

 λ−1 ∂u p p p u(x) ψr (x)χh (x)dx, ∂x E(θρ,ρ) Z  (λ−1)(p−1) ∂ψr p p λp I14 = m (θρ) u(x) χh (x)dx, ∂x E(ρ)∩K(r) Z  (θρ) p  (λ−1)(p−1) p ∂χh p = (u (x))λ − mλ (ρ) u(x) ψr (x) dx. ∂x E(ρ) I13 =

I15

(3.3.59)

We now estimate the remaining integrals in (3.3.58). Thus, the integral I13 is estimated by Theorem 3.3.3 with subsequent application of the Hölder inequality. We get   (δ−p)/n 1 (3.3.60) I13 6 γ R(1) (C0 r, θρ, h) + I(ρ, h) ρn−s hs + p I(ρ, 2h) . h In estimating I14 , we majorize u(x) on E(ρ) ∩ K(r) by m(C0 r) with the constant C0 introduced in inequality (3.3.9). We find  (1−λ)(p−1) n−p−s s I14 6 γmλp (θρ) m(C0 r) r h . (3.3.61) To estimate the integral I15 , we use the Hölder inequality. This yields Z
γ γ I15 6 p Fρ u(x) ψrp (x)χp2h (x)dx 6 p I(ρ, 2h), (3.3.62) h E(ρ) h Finally, by using (3.3.13), we estimate mp−1 (C0 t) 6 κp−1 (C0 r)(C0 t)−(n−p−s) for r 6 t 6 R0 and arrive at the following inequality:  p R(1) (C0 r, θρ, h) ρn−s hs n n  λ  p−1 n−p−s 6 γ ρα1 hβ1 κ(C0 r) m(C0 r) r p+λ−1 o  δ(n−s)  κ(C0 r) , + ρα2 hβ2 +ρα3 hβ3 r 2n where αi and βi are given by equalities (3.3.55).

(3.3.63)

(3.3.64)

212

Chapter 3 Removability of singularities of the solutions

Estimate (3.3.56) now follows from inequalities (3.3.57)–(3.3.64) for sufficiently small ρ.  Further, we define the numbers λ, θ, and γ2 by the equalities   p−1 δ(n − s) , , λ = min 4n n−s−p 1 4pn (n−s)p/n ln = = 2−2(n+1) , ln 2, γ2 θ δ(n − s)

(3.3.65) (3.3.66)

where K6 is the constant from inequality (3.3.56). Theorem 3.3.4. Suppose that the conditions of Lemma 3.3.4 are satisfied and λ, θ, and γ2 are given by conditions (3.3.65) and (3.3.66). Then there exists a constant K7 depending only on known parameters and such that the estimate    r  n−s−p p+λ−1 p−1 n−s s I(ρ, h) 6 K7 ρ h m(C0 r) ρ   p+λ−1 p−λ n−p−s p−1 κ(C0 r) + R(2) (r, ρ, h) (3.3.67) + hn ρ holds for ρ 6 R4 and ρ 6 γ2 h. Proof. We define A = 2p−1 θ−

(n−s)(p−δ) n

,

p (n−s) n

B = K6 γ2

,

(3.3.68)

and the numbers M1 and M2 by the conditions 2r < ρθM1 6 2r/θ,

H0 /2 6 2M2 h < H0 .

(3.3.69)

In view of this notation, inequality (3.3.56) can be rewritten in the form I(ρ, h) 6 AI(θρ, h) + BI(ρ, 2h) + R(2) (r, ρ, h).

(3.3.70)

Iterating the last inequality, by induction on the number of iterations, we arrive at the following estimate: I(ρ, h) 6

M 2 −1 X

2

M1 +m

A

m=0 M 1 −1 M 2 −1 X X

+

i=0

M1

m

m

B I(2r, 2 h) +

M 1 −1 X

An B M2 I(θi ρ, H0 )

i=0

An B m R(2) (r, θi ρ, 2m h).

(3.3.71)

m=0

Further, we estimate the values I(2r, 2m h) and I(θi ρ, H0 ) and the first two sums in (3.3.71). By the definition of the function ψr (x) and inequality (3.3.9), we find  p+λ−1 n−s s I(2r, 2m h) 6 γ2ms m(C0 r) r h . (3.3.72)

Section 3.3

Removability of singularities of the solutions of elliptic equations

213

By virtue of this estimate and the definition of A and B, we obtain the following estimate for the first term on the right-hand side of (3.3.71): M 2 −1 X

2M1 +m AM1 B m I(2r, 2m h)

m=0 2 −1  p+λ−1 n−s s −M1 (n−s)(p−δ) −p ln 2 MX  s+1 m n ln θ 6 γ m(C0 r) r h [θ ] 2 B

m=0 2 −1 ln 2 M (n−s)p  X −p ln  s+1  p+λ−1 n−s s  ρ  (n−s)(p−δ) n θ m 2 K6 γ2 n 6 γ m(C0 r) r h r m=0  p+λ−1 n−s−p r p−1 . (3.3.73) 6 γρn−s hs m(C0 r) ρ

In view of (3.3.65), we arrive at the inequality )n−p+δ (Z θi ρ n   C0 − n(n−p−s)(p+λ−1) +n−s−1 (n−s)(p−δ) p+λ−1 n dt I(θi ρ, H0 ) 6 γ κ(C0 r) [θi ρ] t (p−1)(n−p+δ) r

p+λ−1  p−λ n−p−s  p−1 κ(C0 r) . 6 γ θi ρ

(3.3.74)

By using estimate (3.3.74) and the values λ, θ, and γ2 , we get the following estimate for the second sum in (3.3.71): M 1 −1 X

2M2 +i Ai B M2 I(θi ρ, H0 )

i=0

6γ

h

p iM2 (n−s) n 2K6 γ2

M 1 −1 X

p −

2 θ

i  p+λ−1 p−λ n−p−s p−1 ρ κ(C0 r)

(n−s)(p−δ) +p−λ n−p−s n p−1

i=0 p−λ n−p−s p−1

6 γρ

 p+λ−1 hn κ(C0 r) .

(3.3.75)

The third sum in (3.3.71) can be estimated as follows: M 1 −1 M 2 −1 X X i=0

Ai B m R(2) (C0 r, θi ρ, 2m h)

m=0

 λ 6 γ κ(C0 r) mp−1 (C0 r)rh−p−s ρα1 hβ1 ×

M 1 −1 X



2p−1 θ−

(n−s)(p−δ) +α1 n

2 −1 p i MX  β1 (n−s) n m 2 K 6 γ2

m=0

i=0

p+λ−1 α2 β2 + γ κ(C0 r) ρ h 

×

M 1 −1 X i=0



2p−1 θ−

(n−s)(p−δ) +α2 n

2 −1 p i MX  β2 (n−s) n m 2 K 6 γ2

m=0

214

Chapter 3 Removability of singularities of the solutions

 p+λ−1 δ(n−s) α3 β3 + γ κ(C0 r) r 2n ρ h ×

M 1 −1 X

2 −1 p  β3  p−1 − (n−s)(p−δ) +α2 i MX (n−s) n m n 2 K6 γ2 2 θ

m=0

i=0

6 γR

(2)

(r, ρ, h).

(3.3.76)

Gathering estimates (3.3.70)–(3.3.76), we arrive at the required inequality (3.3.67). Thus, Theorem 3.3.4 is proved.  We fix a function ω ∈ C ∞ (R) equal to one in the interval (3/4, 5/4), equal to zero outside the interval (1/2, 3/2) , and such that dω(t) 6 γ. 0 6 ω(t) 6 1 and dt We also define the following functions:  0  |x | η(x) = η1 (x)η2 (x), η1 (x) = ω , ρ

 and η2 (x) = ω

γ|x00 − ξ 00 | ρ

 .

Moreover, let G = {x : η(x) = 1}. We substitute the functions ϕ(x) = uk (x)u2ρ (x)η l+1 (x) and ψ(x) = η p−1 (x),

(3.3.77)

where k and l are arbitrary nonnegative numbers, in the integral identity (3.3.1). By conditions (3.3.2)–(3.3.5) and the Young inequality, we find Z ∂u p uk (x) η l+p (x)dx ∂x E(2ρ) Z  k+p p 6 γ(k + l + 1) u (x)h1 (x)η l+p (x) + ρ−p uk+p (x)η l (x) dx, E(2ρ)

(3.3.78) where the function h1 (x) is defined in Lemma 3.3.1. Estimating the integral on the right-hand side of (3.3.78) with the help of the Hölder inequality, we obtain Z uk+p (x)h1 (x)η l+p (x)dx E(2ρ)

Z



6γ

k+p

u

(x)η

l+p

 n (x) n−p+δ dx

n−p+δ n

,

(3.3.79)

E(2ρ)

Z

k+p

u E(2ρ)

l

p−δ

Z

(x)η (x)dx 6 γρ

E(2ρ)

 k+p  n u (x)η l (x) n−p+δ dx

n−p+δ n

. (3.3.80)

Section 3.3

Removability of singularities of the solutions of elliptic equations

215

Inequalities (3.3.79) and (3.3.80) yield the following estimate: Z ∂u p uk (x) η l+p (x)dx ∂x E(2ρ) p

6 γ(k + l + 1) · ρ

−δ

Z

n−p+δ n  k+p  n l . u (x)η (x) n−p+δ dx

(3.3.81)

E(2ρ)

 The obtained inequality enables us to estimate max u(x) : x ∈ G by using the iterative Moser method. We omit the reasoning standard for this procedure (see, e.g., Chap. 8 in [124]). The estimate obtained as a result takes the following form: Z  p+λ−1 −n [u(x)]p+λ−1 η p (x)dx. (3.3.82) max [u(x)] : x ∈ G 6 γρ E(2ρ)

In what follows, we assume that the inequality m(2ρ) − m(4ρ) > µm(2ρ)

(3.3.83)

holds with a constant µ specified by the equality µ = 1 − 2−(n−p−s)/p .

(3.3.84)

In this case, we arrive at the estimate u(x) >

1 m(4ρ) for x ∈ E(2ρ). 1−µ

This yields the inequality Z [u(x)]p+λ−1 η p (x)dx E(2ρ) Z 6γ E(4ρ)

 p+λ−1 p u(x) − m(4ρ) ψr (x)χph (x)dx

(3.3.85)

with functions ψr (x) and χh (x) given by equalities (3.3.12) with h = 4ρ/γ2 . By using the Hölder inequality and (3.3.66), we get the following estimate from (3.3.82) and (3.3.85): 

p+λ−1 m(ρ) 6γ

 λ  r N −s−p κp−1 (C0 r) p−1 p−1 m (C r) 0 ρn−p−s ρ  p+λ−1   p−1  r δ(n−s)  κ (C0 r) p−1 2n δ , + ρ + ρn−p−s ρ



(3.3.86)

Note that this inequality is established under the assumption that inequality (3.3.83) is true.

216

Chapter 3 Removability of singularities of the solutions

However, if inequality (3.3.83) is not true, then m(2ρ) 6

1 m(4ρ). 1−µ

(3.3.87)

Thus, in view of (3.3.86) and (3.3.87), we arrive at the following estimate independent of the validity of inequality (3.3.83):    1 − n−p−s p−1 m(2ρ) 6 γρ κ(C0 r)rn−s−p p+λ−1   r  δ(n−s)  δ0 1 2n(p+λ−1) +κ(C0 r) ρ p+λ−1 + + m(4ρ), ρ 1−µ h i where δ 0 = min (n − s − p)/2, δ .

(3.3.88)

We define an integer I(ρ) by the condition R0 /2 6 2I(ρ)ρ < R0 . Iterating inequality (3.3.88), we obtain the estimate m(2ρ) 6 γ

I(ρ) X

2

−

i(n−p−s−δ 0 ) p−1

R(3) (r, ρ) +

i=0



1 I(ρ)−1 m(2I(ρ) ρ), 1−µ

(3.3.89)

where R

(3)

(r, ρ) = ρ

− n−p−s p−1



  λ   1 κ(C0 r) p+λ−1 mp−1 (C0 r)rn−s−p p+λ−1 i h δ0 δ(n−s) 2n(p+λ−1) p+λ−1 + (r/ρ) . + κ(C0 r) ρ

(3.3.90)

By the choice of the number µ, we find 

1 1−µ

I(ρ)

(n−p−s) n−p−s  − ln(1−µ) I(ρ)· ln 2 p 6 (R0 /ρ) p . = 2I(ρ) =2

The last inequality and (3.3.89) yields the estimate   − n−p−s p m (R0 /2) . m(2ρ) 6 γ R(3) (r, ρ) + ρ

(3.3.91)

Passing to the limit as r → 0 in inequality (3.3.91), we arrive at the estimate a− n−p−s p−1

m(2ρ) 6 γρ

with some a > 0. Thus, Theorem 3.3.1 now follows from (3.3.92) and the Serrin results obtained in [113]. For p = n, the proof of Theorem 3.3.2 is similar. 

Section 3.4

3.4 3.4.1

217

Removability of isolated singularities of the solutions

Removability of isolated singularities of the solutions of elliptic equations with absorption Formulation of the assumptions and main results

Consider an elliptic equation of the form n  X ∂u  d  ∂u  ai x, u, + g(x, u) = a0 x, u, , − dxi ∂x ∂x

x ∈ Ω \ {x0 }.

(3.4.1)

i=1

Assume that the functions g(x, u) and ai (x, u, ξ), i = 0, . . . , n, satisfy the Carathéodory conditions, the functions ai (x, u, ξ), i = 0, . . . , n, satisfy inequalities (3.2.2)–(3.2.4) and conditions (3.2.5) and, in addition, g(x, u) sign u > c1 |u|q − f4 (x), f4 (x) > 0,

|g(x, u)| 6 c2 |u|q + f4 (x),  1  f4 (x) ∈ Ln/(p−δ) (Ω), δ ∈ 0, min(1, p − 1) . 2

(3.4.1a)

Definition 3.4.1. We say that u(x) is a solution of Eq. (3.4.1) in Ω \ {x0 } if, for an any function ξ(x) ∈ C ∞ (Ω) equal to zero in a certain neighborhood ◦

of {x0 } and any function ψ ∈ W 1,p (Ω) ∩ Lq+1 (Ω), the inclusion u(x)ξ(x) ∈ W 1,p (Ω) ∩ Lq+1 (Ω) is true and the equality  Z X n   ∂u  ∂ϕ ∂u  ai x, u, (3.4.2) − a0 x, u, ϕ + g(x, u)ϕ dx = 0 ∂x ∂xi ∂x Ω i=1

holds for ϕ(x) = ξ(x)ψ(x). Definition 3.4.2. We say that a solution u(x) of Eq. (3.4.1) has a removable singularity at a point {x0 } if u(x) ∈ W 1,p (Ω) ∩ Lq+1 (Ω) and equality (3.4.2) ◦

is true for any function ϕ ∈ W 1,p (Ω) ∩ Lq+1 (Ω). The following theorems are the main results of the present section: Theorem 3.4.1. Let inequalities (3.2.2)–(3.2.4) be true, let conditions (3.2.5) be satisfied, and let u(x) be a solution of Eq. (3.4.1) in Ω \ {x0 }. Also let the inequality q>

n(p − 1) n−p

(3.4.3)

hold. Then the singularity of u(x) at the point {x0 } is removable. Theorem 3.4.2. Assume that all conditions of Theorem 3.4.1 are satisfied. Then p u(x) 6 γ|x − x0 |− q−p+1 , |x − x0 | > 0. (3.4.4)

218

Chapter 3 Removability of singularities of the solutions

Proof of Theorem 3.4.2

3.4.2

In what follows, we assume that lim M (r) = ∞.

r→0

(3.4.5)

Otherwise, Theorem 3.4.1 follows from the Serrin results presented in [113]. We fix R1 > 0 such that M (R1 ) > 1.

(3.4.6)

For ρ > 0 and σ ∈ (0, 1), we fix a function ϕρ,σ (x) ∈ C ∞ (Rn ) satisfying the conditions: ϕρ,σ (x) = 1 for ρ 6 |x − x0 | 6 2ρ,  ϕρ,σ (x) = 0 outside the set (1 − σ)ρ < |x − x0 | < (2 + σ)ρ , ∂ϕ (x) 2 ρ,σ , and 0 6 ϕρ,σ (x) 6 1. 6 ∂x σρ Substituting the function ϕ = uϕpρ,σ in the integral identity (3.4.2) and using the Hölder and Young inequalities and the definition of M (ρ), we get Lemma 3.4.1. The inequality  Z  ∂u p q+1 ϕpρ,σ (x)dx 6 γσ −p ρn−p M p (ρ − σp) + |u| ∂x Ω is true.

We fix a number R2 ∈ 0, R1 /2 such that M (R2 ) > M R1 /2 and assume that ρ is an arbitrary number from the interval (0, R2 ). Further, we consider numerical sequences {ρj } and {σj } defined by the equalities 1  ρj = + 2−j ρ, σj = 2−(j+1) , j = 1, 2, . . . . 2 We choose a point xj such that M (ρj ) = u(xj ) and ρj 6 |xj − x0 | 6 R2 , and denote ρ0j = |xj − x0 |. We also define ϕj (x) = ϕρ0 j , σj (x). We substitute the function
k ϕ(x) = 1 + |u(x)| u(x)ϕl+p k, l > 0, j (x), in the integral identity (3.4.2). By the iterative Moser method (see, e.g., [124], Chap. 8), we find Z q+1  q+1
q+1 p jp = u(xj ) ϕj (x)dx. (3.4.7) M (ρj ) 6 γ2 δ ρ−n 1 + |u(x)| Ω

Section 3.4

Removability of isolated singularities of the solutions

219

We now estimate the integral on the right-hand side of (3.4.7). In view of Lemma 3.4.1, we conclude that p  q+1 1  M (ρj ) 6 γρ−p 2jp(1+ δ ) M (ρ0j − σj ρ0j )  p 1 = γ2jp(1+ δ ) ρ−p M (ρj+1 ) . (3.4.8) Iterating inequality (3.4.8), we obtain (3.4.4), which completes the proof of Theorem 3.4.2.

3.4.3

Integral estimates for the gradient of the solution


For r ∈ (0, R2 ), we define a function ψr (x) : Rn → R, ψr (x) = ψer |x| , where ψer (x) : R → R is a function specified by the equalities ψer (t) = 0 for t 6 r, ψer (t) =

1 (1 − θ) ln ln 1r

ψer (t) = 1 for t > R(r), Z

t

r

dz z ln z1

for r 6 t 6 R(r),

(3.4.10)

θ is a number from the interval (0, 1) determined in what follows, and R(r) is the number defined by the equality  1 θ 1 ln = ln . (3.4.11) R(r) r For ρ ∈ (0, R0 ), we define a function uρ (x) and a set E(ρ) by the equalities   f(ρ) uρ (x) = u(x) − M for x ∈ B(ρ) \ {x0 }, + for x ∈ Ω \ B(ρ), f(ρ) , E(ρ) = x ∈ B(ρ) \ {x0 } : u(x) > M f(ρ) = M (ρ) + ln 1 . M ρ

uρ (x) = 0



Further, we define a number p and a function F1 (r, ρ) as follows: p=

F1 (r, ρ) =

  ρn−p    h i2− n   p   ln 1r   ln ln 1r   h i2− n   p  ln 1 ρ

qp , q−p+1 for q >

n(p−1) n−p ,

for q =

n(p−1) n−p ,

n < 2p,

for q =

n(p−1) n−p ,

n = 2p,

for q =

n(p−1) n−p ,

n > 2p.

(3.4.12)

220

Chapter 3 Removability of singularities of the solutions

Lemma 3.4.2. Assume that all conditions of Theorem 3.4.1 are satisfied and, in addition, that the solution u(x) satisfies the inequality n−p−λ u(x) 6 γ|x − x0 |− p−1 for 0 < |x| 6 R1 (3.4.13) with λ ∈ [0, n − p). Then, for r ∈ (0, R2 ) and R(r) < ρ < R2 , the estimate h  Z 1 i−p 1 ∂u p p 1 λ+δ F1 (r, ρ) + ρ (3.4.14) ln ψr (x)dx 6 γ ln ln r ρ E(ρ) u(x) ∂x is satisfied with p and F1 (r, ρ) given by equalities (3.4.12) and δ > 0 depending only on known parameters. Proof. We substitute the function   u(x) ψrp (x) ϕ(x) = ln f M (ρ) + in the integral identity (3.4.2). By virtue of inequalities (3.2.2)–(3.2.4), we obtain Z Z 1 ∂u p p f(ρ))ψrp (x)dx uq ln(u/M ψr (x)dx + u ∂x E(ρ) E(ρ) Z   p−1 γ 6γ u g1 (x) + u−1 f1 (x) ψrp (x)dx + ln ln(1/r) E(ρ)   p−1 Z f(ρ))]+ ∂u [ln(u/M p−1 × + + u g2 (x)+f2 (x) ψrp−1 (x)dx 1 ∂x K(r) |x−x0 | ln |x−x0 |   Z ∂u p−1 u +γ ln h(x) + g3 (x)up−1 +f3 (x) ψrp (x)dx, (3.4.15) f(ρ) ∂x M E(ρ)  where K(r) = r < |x − x0 | < R(r) . In view of (3.4.6) and (3.4.13), we get Z    p−1  (3.4.16) u g1 (x) + u−1 f1 (x) ψrp (x)dx 6 γ ρλ+δ + ρn−p+δ . E(ρ)

According to the Young inequality, we find   Z ∂u p−1 1 u 1 |x − x0 |−1 ln−1 ln ψrp−1 dx 1 f |x − x | ln ln r K(r) 0 ∂x M (ρ) + Z Z 1 1 1 ∂u p p u ψ p dx 6 uq ln ψr dx + f(ρ) r 4 E(ρ) u ∂x 4 E(ρ) M p(1− p−1 )   −p Z qp 1 u 1 −p + γ ln dx. ln |x−x0 | p f(ρ) + |x−x0 | M K(r) ln ln 1r (3.4.17)

Section 3.4

Removability of isolated singularities of the solutions

221

We now estimate the last term on the right-hand side of (3.4.17). By using condition (3.4.13), we obtain Z

1 

ln ln 1r

p

K(r)

−p p(1− p−1 )  qp 1 u ln dx f(ρ) + |x − x0 | M − p−1 Z ρ q−p+1 1 1 ln |x − x0 |n−1−p d|x − x0 | 6 γ  p 1 |x − x | 0 r ln ln

 |x − x0 |−p ln

r

1

p F1 (r, ρ).

(3.4.18)

Similarly, in view of (3.4.13), we conclude that Z u ln f
γ M up−1 g2 (x) + f2 (x) dx 1 1 ln ln r K(r) ln |x−x0 | · |x − x0 |
1 n−p + ρλ . 6γ 1 ρ ln ln r In addition, by virtue of (3.4.13), we get Z    1 u  g3 (x)up−1 +f3 (x) ψrp (x)dx 6 γ ρn−p+δ +ρλ+δ ln . ln f(ρ) ρ M E(ρ)

(3.4.19)

6 γ

Finally, we find Z h(x) ln E(ρ)

ln ln 1r

u ∂u p−1 p ψr dx f(ρ) ∂x M Z 1 1 ∂u p p 1 ψ dx + γρδ+λ lnp . 6 4 E(ρ) u ∂x r ρ

(3.4.20)

(3.4.21)

Combining estimates (3.4.15)–(3.4.21), we arrive at the required inequality (3.4.14).  We define a function u(ρ) (x) and a set E(ρ, 4ρ) as follows:  u(ρ) (x) = min [u(x) − M (4ρ)]+ , M (ρ) − M (4ρ) ,  E(ρ, 4ρ) = x : M (4ρ) < u(x) < M (ρ) . Lemma 3.4.3. Suppose that the conditions of Theorem 3.4.1 are satisfied and inequality (3.4.13) is true. Then the estimate Z ∂u p (3.4.22) dx 6 γM (ρ)ρλ+δ E(ρ,4ρ) ∂x is true for 0 < ρ < R2 .

222

Chapter 3 Removability of singularities of the solutions

Proof. We substitute the function ϕ(x) = u(ρ) (x)ψrp (x) for r ∈ (0, e−1 ),

R(r) < ρ < R2 ,

in the integral identity (3.3.1). By using inequalities (3.2.2)–(3.2.4), we obtain Z Z ∂u p p uq (x)u(ρ) (x)ψrp (x)dx ψr (x)dx + E(4ρ) E(ρ,4ρ) ∂x Z 6 γM (ρ) up−1 (x)H1 (x)ψrp (x)dx E(4ρ)

Z

∂ψ r p−1 up−1 (x)H2 (x) ψ (x)dx ∂x r E(4ρ) Z ∂u p−1 ∂ψ r p−1 +γ u(ρ) (x) ψ (x)dx, ∂x ∂x r E(4ρ)

+ γM (ρ)

(3.4.23)

where H1 (x) = g1 (x)+f1 (x)+hp (x)+g3 (x)+f3 (x) and H2 (x) = g2 (x)+f2 (x). By using conditions (3.4.5) and inequality (3.4.13), we conclude that Z M (ρ) up−1 (x)H1 (x)ψrp (x)dx E(4ρ)

Z 6 γM (ρ)

 (p−1) p  n u ψr (x) n−p+δ dx

 n−p+δ n

E(4ρ)

Z 6 M (ρ)

4ρ

|x − x0 |

n +n−1 −(n−p−λ) n−p+δ

 n−p+δ n

d|x − x0 |

r

6 M (ρ)ρλ+δ .

(3.4.24)

Further, we get Z ∂ψ r p−1 M (ρ) up−1 (x)H2 (x) ψ (x)dx ∂x r E(4ρ) Z  n−p+1+δ  n  ∂ψ n−p+1+δ n r p−1 p−1 6 γM (ρ) dx u (x) ψr (x)dx ∂x E(4ρ)  n  Z R(r)  n−p+1+δ 1 1 −(n−p+1−λ) 6γ M (ρ) |x − x | 0 1 ln ln 1r ln |x−x r 0|  n−p+1+δ n n−1 × |x − x0 | d|x − x0 |  λ+δ , 6 γM (ρ) R(r)

(3.4.25)

Section 3.4

Removability of isolated singularities of the solutions

223

Z

∂u p−1 ∂ψ r p−1 u(ρ) (x) ψ (x)dx ∂x ∂x r E(4ρ) Z 1 6 u(ρ) (x)uq (x)ψrp (x)dx 2γ E(4ρ)

q−p+1  q(p−1) Z ∂ψ p qp−p+1 qp−p+1 1 ∂u p p r ρ +γ u (x) . ψr (x)dx dx ∂x E(4ρ) u ∂x E(4ρ) (3.4.26) By the definition of ψr (x), we find (Z ) q−p+1 Z qp−p+1 ∂ψ p R(r) n−1 d|x−x | |x−x | r 0 0 p |x−x | u(ρ) (x) dx 6 γM (ρ)
0 p ∂x E(4ρ) r ln 1

Z

|x−x0 |

6 γM (ρ)F2 (r), where

 q−p+1    R(r) (n−p) qp−p+1 F2 (r) = h i−(n−p) q−p+1 qp−p+1   ln 1 R(r)

(3.4.27) if q >

n(p−1) n−p ,

if q =

n(p−1) n−p .

(3.4.28)

Combining estimates (3.4.23)–(3.4.27) and using Lemma 3.4.2, we obtain Z ∂u p  λ+δ p ψr (x)dx 6 γM (ρ)ρλ+δ + γM (ρ) R(r) E(ρ,4ρ) ∂x  q(p−1)  qp−p+1 1 −p λ+δ p 1 F1 (r, ρ) + ρ + γM (ρ)F2 (r) ln ln ln , r ρ  q(p−1)  q(p−1) q−p+1 h qp−p+1 1 i−(n−1) qp−p+1 h 1 i(2− np ) qp−p+1 1 −p F1 (r, ρ) ln 6 ln F2 (r) ln ln r R(r) r −θ(n−1)(q−p+1)+(2− n )(p−1) p h 1i qp−p+1 = ln . (3.4.29) r The last equality in (3.4.29) is deduced with regard for the definition of R(r). We now choose θ ∈ (0, 1) from the conditions n n(p − 1) −θ(n − 1)(q − p + 1) + (2 − )q(p − 1) < 0 for q = , p n−p 1 n(p − 1) θ= for q > . 2 n−p The possibility of the indicated choice of θ is confirmed, for q = n(p−1) n−p and p < n, by the following inequality: n n(p − 1)2 (2 − )q(p − 1) < q(p − 1) = q = p n−p (n − 1)p(p − 1) , p−1 p

(3.4.32)

for

n−p−λ δ 6 . p−1 p

(3.4.33)

It follows from (3.4.32) that M (ρ) 6 M

R  2

2

+γ

 1  n−p−λ − δ p−1

p

ρ

for

n−p−λ δ > . p−1 p

(3.4.34)

Further, it follows from (3.4.33) that M (ρ) 6 M

R  2

2

+ ln

1 ρ

for

n−p−λ δ 6 . p−1 p

(3.4.35)

Section 3.4

Removability of isolated singularities of the solutions

225

Theorem 3.4.2 now implies that inequality (3.4.13) is true with λ=0 λ=

n−p p − >0 p−1 q−p+1

for q =

n(p − 1) , n−p

for q >

n(p − 1) . n−p

(3.4.36)

Inequalities (3.4.34) and (3.4.35) guarantee successive improvements of estimate (3.4.13) as λ increases by δ/p in each step, as long as n − p − λ is positive. Thus, starting from inequality (3.4.13) with the value of λ specified by equalities (3.4.36), after finitely many steps, by virtue of (3.4.34) and (3.4.35), we obtain u(x) 6 γ ln

1 |x − x0 |

for 0 < |x − x0 | < R2 /2.

(3.4.37)

We now rewrite Eq. (3.4.1) in the form n  X ∂u  d  ∂u  ai x, u, +e a0 x, u, = 0, − dxi ∂x ∂x

(3.4.38)

i=1

where e a0 (x, u, ξ) = a0 (x, u, ξ) + ge(x) and ge(x) = g(x, u(x)). The coefficients of Eq. (3.4.38) satisfy conditions (3.2.2)–(3.2.5). Thus, Theorem 3.4.1 follows from Theorem 3.2.1. 

Chapter 4

Removability of singularities of the solutions of quasilinear parabolic equations 4.1

Introduction

In the present chapter, we study the problem of removability of isolated singularities for the solutions of quasilinear parabolic equations. Let n > 3, let Ω be an open set in Rn , let ΩT = Ω × (0, T ), and let ◦ ΩT = ΩT \ {(0, 0)}. Consider the following model parabolic equations: ∂u − ∆p u + gu|u|p−2 = f in Ω◦T , p > 2, ∂t ∂u − ∆p u + u|u|q−1 = f in Ω◦T , q > p − 1. ∂t In addition, we assume that u(x, 0) = 0 for x ∈ Ω \ {0}.

(4.1.1) (4.1.2)

(4.1.3)

The problem of removability of singularities for the solutions of general parabolic equations of the form (4.1.1) with p = 2 satisfying the initial condition (4.1.3) was studied by Aronson [7, 8] and Aronson and Serrin [9] for g, f ∈ Lq,r (ΩT ), nq + 2r < 2. They obtained the following sufficient condition for the removability of an isolated singularity: u(x, t) = o((|x| + t1/2 )−n ). For the equation ut − ∆u + uq = 0, Brézis and Friedman proved [30] that an isolated singularity is always removable for q > 1 + 2/n. Numerous authors studied the problem of removability of isolated singularities for parabolic equations of the form (4.1.1), (4.1.2) (see, e.g., [31, 51, 55–58, 95, 96]). Most of these researchers considered equations of special form. For a survey of these results, see the monograph by Véron [134]. The exact conditions of removability of isolated singularities for general equations of the form (4.1.1), (4.1.2) were obtained in [100, 122]. The presentation of these results constitutes a significant part of this chapter. In Sec. 4.2, we give the results obtained in [100] and establish the exact condition of removability of isolated singularities for the solutions of general quasilinear parabolic equations of the form (4.1.1). In Sec. 4.3, we give the results obtained in [122] and prove the exact condition of removability of isolated singularities for the solutions of general quasilinear parabolic equations of the form (4.1.2).

Section 4.2

4.2 4.2.1

227

Removability of isolated singularities

Removability of isolated singularities Formulation of assumptions and main results

Let Ω be an open bounded set in Rn , let ΩT = Ω × [0, T ), let {0} ∈ Ω, and let t0 ∈ [0, T ). Consider an equation n

 ∂u X d  ∂u  ∂u  − ai x, t, u, = a0 x, t, u, , ∂t dxi ∂x ∂x

(4.2.1)

i=1

(x, t) ∈ ΩT \ {(0, t0 )}. We study the following two cases: t0 = 0 or t0 > 0. In the first case, we additionally assume that u(x, 0) = 0,

x ∈ Ω \ {0}.

(4.2.2)

We also suppose that the functions ai (x, t, u, ξ) i = 0, 1, . . . , n, are defined for (x, t, u, ξ) ∈ Ω × (0, T ) × R × Rn and satisfy the Carathéodory condition and that the inequalities n X

ai (x, t, u, ξ)ξi > c1 |ξ|p − g1 (x, t)|u|p − f1 (x, t),

p > 2,

(4.2.3)

i = 1, . . . , n,

(4.2.4)

i=1

|ai (x, t, u, ξ)| 6 c2 |ξ|p−1 + g2 (x, t)|u|p−1 + f2 (x, t), p−1

|a0 (x, t, u, ξ)| 6 h(x, t)|ξ|

p−1

+ g3 (x, t)|u|

+ f3 (x, t)

(4.2.5)

hold with some positive constants c1 and c2 and nonnegative functions h(x, t), gi (x, t), fi (x, t), i = 1, 2, 3, such that H(x, t) ∈ Lr0 (0, T ; Lq0 (Ω)), n p + n(p − 2) + = p(1 − δ), r0 q0

r0 > 1, q0 > 1,

H(x, t) = 1 + hp (x, t) + f1 (x, t) + [f2 (x, t)]p/(p−1) + f3 (x, t) + g1 (x, t) + [g2 (x, t)]p/(p−1) + g3 (x, t).

(4.2.6)

A solution of problem (4.2.1), (4.2.2) is defined as a function u(x, t) such that u(x, t)ϕ(x, t) ∈ V 2,p (ΩT ) = C(0, T ; L2 (Ω)) ∩ Lp (0, T ; W 1,p (Ω)) and the following integral identity is true: Z Z τZ  n ∂u  ∂ϕ ∂ϕ X  + I(u, ϕ) ≡ u(x, τ )ϕ(x, τ )dx + −u ai x, t, u, ∂t ∂x ∂xi Ω 0 Ω i=1    ∂u − a0 x, t, u, ϕ dxdt = 0, (4.2.7) ∂x

228

Chapter 4 Removability of singularities of the solutions

where ϕ(x, t) ∈ V 2,p (ΩT ) is an arbitrary function such that ∂ϕ/∂t ∈ L2 (ΩT ),

ϕ(x, t) ≡ 0 for (x, t) ∈ {(0, 0)} ∪ {∂Ω × (0, T )},

and τ is an arbitrary number, τ ∈ (0, T ]. In what follows, without loss of generality, we assume that ∂u/∂t ∈ L2 (ΩT ) because, otherwise, by analogy with [38], it is possible to consider the Steklov mean. We set   p  t |x| + p+n(p−2) 6 1 . D(r) = (x, t) : r r Let R0 be a sufficiently small number such that D(R0 ) ⊂ ΩT and M (r) = sup{|u(x, t)| : (x, t) ∈ D(R0 ) \ D(r)}. The following theorem is the main result of the present section: Theorem 4.2.1. Let conditions (4.2.3)–(4.2.6) be satisfied and let u(x, t) be a solution of Eq. (4.2.1) satisfying the initial condition (4.2.2). Moreover, assume that the following equality holds: lim M (r)rn = 0.

r→0

(4.2.8)

Then the singularity at the point {(0, 0)} is removable. Theorem 4.2.1 is obtained as consequence of the following result: Theorem 4.2.2. Suppose that all conditions of Theorem 4.2.1 are satisfied. Then the following inequality is true : M (r) 6 γrα−n

with some α > 0.

(4.2.9)

In the case where (0, t0 ) is an inner point of ΩT , we consider   p  |x| |t − t0 | e D(r) = (x, t) : + p+n(p−2) 6 1 , r r e 0 ) \ D(r)}. e f M (r) = sup{|u(x, t)|, (x, t) ∈ D(R Theorem 4.2.3. Suppose that all conditions of Theorem 4.2.1 are satisfied and f(r)rn = 0. lim M (4.2.10) r→0

Then the singularity at the point (x0 , t0 ) ∈ ΩT is removable. Note that Theorem 4.2.2 yields the inequality   R  0 6 γ. sup |u(x, t)|, (x, t) ∈ D 2

Section 4.2

Removability of isolated singularities

229

By using this inequality, as a result of the standard reasoning (see, e.g., [100]), we obtain Theorem 4.2.1 from Theorem 4.2.2.

4.2.2

Integral estimates for the solution

Let ω(s) ∈ C ∞ (R), ω(s) = 0 for s 6 1, ω(s) = 1 for s > 2p , 0 6 dω(s)/ds 6 γ, and 0 6 ω(s) 6 1 for s ∈ R. We set  p  t |x| + p+n(p−2) . ηr (x, t) = ω r r In what follows, we assume that lim M (r) = ∞.

r→0

Further, we fix a number R1 from the interval (0, R0 ) such that M (R1 ) > 1 and, for r ∈ (0, R1 ], set  1 M ∗ (r) = n max M (ρ)ρn : r 6 ρ 6 R1 + 1. r A function uR (x, t) for R ∈ (0, R1 ) and a set E(R) are defined by the equalities
uR (x, t) = u(x, t) − M (R) + , (x, t) ∈ D(R), uR (x, t) = 0 for (x, t) ∈ ΩT \ D(R),  E(R) = (x, t) ∈ D(R) : u(t, x) > M (R) . Lemma 4.2.1. Assume that all conditions of Theorem 4.2.1 are satisfied. Then Z ZZ ∂u p p sup u2R (x, t)ηrp (x, t)dx + ηr (x, t)dxdt 0 p − 1 + p/n, (4.3.36) for q = p − 1 + p/n.

Further, by using the Young and Hölder inequalities, (4.3.18), in view of the choice of κ, we obtain p ZZ ∂u p 1 ψ (x, t) dx dt + γ M ˜ (ρ)ρε(p−1)+pδ I10 6 8 E(ρ,4ρ) ∂x r )(p−1)/p (Z Z p ∂u p −κ ˜ (ρ) ˜ (4ρ)] ψr (x, t) dx dt +γ M [u− M ∂x E(ρ,4ρ) ×ρ(ε(p−1)+3/4δp)1/p , ˜ (ρ){ρε(p−1)+pδ + F4 (r)}, I11 6 γ M

(4.3.37) (4.3.38)

By virtue of Lemmas 4.3.3 and 4.3.4 and the Hölder inequality, we arrive at the following estimate: p ZZ ∂u dx dt E(ρ,4ρ) ∂x  ˜ 6 γ M (ρ) ρε(p−1)+(pδ)/2 + F3 (r) + F4 (r)

246

Chapter 4 Removability of singularities of the solutions

n o q(p−1) qp−p+1 + [F5 (r)](q−p+1)/(qp−p+1) ρε(p−1)+pδ lnρ 1/ρ  (4.3.39) + F6 (r)ρ(ε(p−1)+δp)1/p , where ) q(p−1)  −p −q0 qp−p+1 q−p+1 1 F1 (r) + ln ln p+n(p−2) F2 (r) F6 (r) = ln ln p+n(p−2) [F5 (r)] qp−p+1 r r ( )p−1 −p −q0  p 1 1 + ln ln p+n(p−2) F1 (r) + ln ln p+n(p−2) F2 (r) [F4 (r)]1/p . r r (

1

We now substantiate the possibility of the limit transition as r → 0 in inequality (4.3.39). The equality lim F3 (r) = lim F4 (r) = lim F5 (r) = 0

r→0

r→0

r→0

(4.3.40)

directly follows from the definition of F3 (r), F4 (r), and F5 (r). By the definition of F1 (r) and F2 (r), we get lim F1 (r) = 0 for q > p − 1 + p/n;

p < n(p − 1),

(4.3.41)

lim F2 (r) = 0 for q > p − 1 + p/n;

q 0 < 2.

(4.3.42)

t→0 t→0

Moreover, the following equalities are true:  −p 1 lim ln ln p+n(p−2) F1 (r) = 0 for q > p−1+p/n, p = n(p−1), r→0 r  −p 1 lim ln ln p+n(p−2) F2 (r) = 0 for q = p−1+p/n, q 0 = 2. r→0 r

(4.3.43) (4.3.44)

Further, for q > p − 1 + p/n and p > n(p − 1), we have [F1 (r)]−q(p−1) [F2 (r)]q−p+1

 −p(p−1)(n+1) θ+[1− n(p−1) ](p−1+p/n)(p−1) n p 1 6 γ ln , (4.3.45) r  n(p−1)  1 [1− p −θ](p−1) 6 γ ln . (4.3.46) r 

[F1 (r)]p−1 [F4 (r)]q−p+1

The right-hand sides of inequalities (4.3.45) and (4.3.46) tend to zero as r → 0 provided that   p + n(p − 1) 1 , (p − (n − 1/2)(p − 1))+ . θ > max (4.3.47) p(n + 1) p

Section 4.3

247

Removability of isolated singularities for the solutions

Similarly, for q = p − 1 + p/n and q 0 > 2, we obtain [F2 (r)]q(p−1) [F5 (r)]q−p+1  −p(p−1)(n+1) θ+[1− 1 ](p−1)(p−1+p/n)  n p−2+p/n 1 6 γ ln , r 1   1 (1− p−2+p/n −θ)(p−1) p−1 [F2 (r)] [F4 (r)] 6 γ ln . r

(4.3.48)

(4.3.49)

The right-hand sides of inequalities (4.3.48) and (4.3.49) tend to zero as r → 0 provided that   1 p + n(p − 1) ,1 − θ > max . (4.3.50) p(n + 1) 2(p − 2 + p/n) Finally, we choose θ equal to the maximum value of the right-hand sides of inequalities (4.3.48) and (4.3.49) and obtain lim F6 (r) = 0.

(4.3.51)

r→0

Thus, using obtained results and passing to the limit as r → 0 in inequality (4.3.39), we arrive at (4.3.32). 

4.3.4

Proof of Theorem 4.3.1

First, we prove an auxiliary estimate. Lemma 4.3.6. Suppose that all conditions of Theorem 4.3.1 are satisfied. Then (n+p)r 0  n p+n(p−2) ρ  ρ  p(n+p+2r00 ) p+n(p−2) − 0 0 r0 ˜2 ˜ (ρ) − M ˜ (2ρ) 6 γ ρ r0 ˜ρ M M + ρ r0 M 2 2 

Z Z

up2ρ (x, t)ξρp (x, t) dx dt

×

0 n+p−nr0 0) p(n+p+2r0

ΩT

where ξρ (x, t) = ξρ (|x|p+n(p−2) + t) and ξρ (s) : R → R is a function from the class C ∞ such that ξρ (s) = 1 for s > ρp+n(p−2) , ξρ (s) = 0 for s 6 (ρ/2)p+n(p−2) , 0 6 ξρ (s) 6 1,

and |dξρ /ds| 6 γ/ρp+n(p−2) .

.

(4.3.52)

248

Chapter 4 Removability of singularities of the solutions

Proof. We substitute the function ϕ(x, t) = [u2ρ (x, t)]t ξpk (x, t),

l, k > 0,

in (4.3.3) and perform integration by parts. By analogy with (4.2.31), we arrive at the estimate ZZ ul2ρ (x, t)ξρk (x, t) dx dt ΩT

i(n+p)/n h n − p+n(p−2) p+n(p−2) n 0 r0 ˜ 2 (ρ/2) + ρ r0 − r0 M ˜ p (ρ/2) M 6 γ(l + k)ρ(p/n+2) ρ r0 1/θ

Z Z

0 lθ−l0 u2ρ (x, t)ξρkθ−k (x, t) dx dt

×

.

(4.3.53)

ΩT

where θ=

nr00 , n+p

l0 =

p(n + 2) 0 r , n+p 0

k 0 = pr00 ,

l and k are arbitrary positive numbers such that lθ − l0 > p and kθ − k 0 > p. In view of inequality (4.3.33), we can apply the iterative Moser method. This yields (4.3.52).  Lemma 4.3.7. The following estimate is true with a constant a depending only on the known quantities : ˜ (ρ) − M ˜ (2ρ) 6 γρa−n . M

(4.3.54)

Proof. We estimate the integral on the right-hand side of (4.3.52). By the Poncaré inequality and Lemma 4.3.5, we find ZZ ZZ up2ρ (x, t)ξρp (x, t) dx dt 6 uρ/2 (x, t) dx dt ΩT

ΩT

6 γρρ

ZZ

p ∂u dx dt E(ρ/2,2ρ) ∂x

˜ (ρ)ρp+ε(p−1)+δp/2 . 6 γM

(4.3.55)

In view of (4.3.52) and (4.3.55), we get the inequality   n p+n(p−2) p+n(p−2) p+n(p−2) p+n(p−2) − −2 q−1 0 −p q−1 r0 r0 r ˜ ˜ +ρ 0 M (ρ) − M (2ρ) 6 γ ρ   n+p−nr000 p+n(p−2) p(n+p+2r0 ) − q−1 +p+ε(p−1)+ δp 2 × ρ .

(4.3.56)

Section 4.3

249

Removability of isolated singularities for the solutions

We have n p+n(p−2) p+n(p−2) p+n(p−2) p+n(p−2) − −2 6 −p , 0 r0 r0 q−1 r0 q−1   n p+n(p−2) p+n(p−2) (n+p)r00 − − 2 r0 r00 q−1 p(n+p+2r00 )   p+n(p−2) n+p−nr00 + p− > −n. q−1 p(n+p+2r00 ) The last two inequalities and inequality (4.3.55) give estimate (4.3.54).



˜ j }j∈N by the equalities We define sequences {ρj }j∈N and {M ρj = R˜ 0 /2j

˜j = M ˜ (ρj ). and M

By using (4.3.54), we get ˜j −M ˜ j−1 6 γ2j(n−a) , M

j = 0, 1, 2, . . . .

(4.3.57)

Finding the sum of inequalities (4.3.57), we obtain ˜j −M ˜ 1 6 γ2j(n−a) . M This yields the estimate ˜ (ρ) 6 γ(M ˜ 1 + ρa−n ), M which, in turn, implies the estimate h  ia−n 1/(p+n(p−2)) ˜ |u(x, t)| 6 γ |x| + |t| + M1 .

(4.3.58)

This inequality guarantees the removability of singularity for the solution u(x, t) at {(0, 0)}. Theorem 4.3.1 is thus proved.

Chapter 5

Quasilinear elliptic equations with coefficients from the Kato class 5.1

Introduction

In the present chapter, we study some qualitative properties of the solutions of quasilinear elliptic equations with coefficients from the Kato classes. After the outstanding works by De Giorgi [37], Nash [98], and Moser [93], numerous authors studied the Hölder property and Harnack’s inequality for the solutions of elliptic equations. Ladyzhenskaya and Ural’tseva [88] and Serrin [112] established the local Hölder property and Harnack’s inequality for the solutions of general divergent-type quasilinear elliptic equations under the condition that the coefficients of equations belong to the corresponding Lq -classes. In [1], Aizenman and Simon proved Harnack’s inequality for the nonnegative solutions of the equation −∆u + V (x)u = 0 under the condition that V (x) belongs to the Kato class. In [33], Chiarenza, Fabes, and Garofalo proved a similar result for a linear divergent elliptic equation with variable coefficients and a potential from the Kato class. In [83], by using the Chiarenza–Fabes–Garofalo method, Kurata proved Harnack’s inequality for the equation −

n n X ∂u
X ∂u ∂ bi (x) aij (x) + + V (x)u = 0 ∂xi ∂xj ∂xi

i,j=1

i=1

Pn

under the condition that i=1 b2i (x) and V (x) belong to the Kato class. Later, Biroli [20] introduced the nonlinear Kato class and proved Harnack’s inequality for the nonnegative solutions of an elliptic equation of the form −∆p u + V (x)up−1 = 0. For a general divergent-type quasilinear elliptic equation with coefficients from the Kato classes, Harnack’s inequality was obtained in [121]. The problem of removability of isolated singularities for the equation −∆u + V (x)u = 0, where V (x) belongs to the Kato class, was considered in [130]. In [90, 91], the exact conditions of removability of isolated singularities were established for general quasilinear elliptic equations of divergent type with coefficients from the Kato class.

Section 5.2

251

Harnack’s inequality

A significant part of this chapter is devoted to the presentation of the results obtained in [90, 91, 121]. In Sec. 5.2, we present the results from [121] and prove Harnack’s inequality for the nonnegative solutions of quasilinear elliptic equations with coefficients from the Kato class. In Sec. 5.3, we present the results obtained in [90] and establish the exact condition of removability of isolated singularities for the solutions of quasilinear elliptic equations with coefficients from the Kato classes. Finally, in Sec. 5.4, we give the results obtained in [91] and prove the exact condition of removability of isolated singularities for the solutions of quasilinear elliptic equations with absorption and coefficients from the Kato classes.

5.2 5.2.1

Harnack’s inequality Formulation of assumptions and main results

Consider nonnegative solutions of the equation     n X ∂u d ∂u ai x, u, = a0 x, u, , x ∈ B1 = { |x| < 1 }. dx ∂x ∂x

(5.2.1)

i=1

It is assumed that the functions ai (x, u, ξ),

i = 0, 1, . . . , n,

x ∈ B1 , u ∈ R, ξ ∈ Rn ,

satisfy the Carathéodory conditions and the following inequalities are valid: n X

ai (x, u, ξ)ξi > c1 |ξ|p − g1 (x)|u|p − f1 (x),

1 < p < n,

(5.2.2)

i = 1, . . . , n,

(5.2.3)

i=1

|ai (x, u, ξ)| 6 c2 |ξ|p−1 + g2 (x)|u|p−1 + f2 (x),

|a0 (x, u, ξ)| 6 h(x)|ξ|p−1 + g3 (x)|u|p−1 + f3 (x),

(5.2.4)

where the functions fi (x), gi (x), and h(x) are nonnegative. To find conditions on the functions h(x), gi (x), fi (x), i = 1, 2, 3, we introduce classes Kp , K˜ p , and K2,p generalizing the well-known Kato class for p 6= 2. We say that a nonnegative function g(x) ∈ L1 (B1 ) belongs to Kp if 1/(p−1) Z R Z 1 dr lim sup (5.2.5) g(z)dz = 0. R→0 x∈B1 0 rn−p B(x,r) r We say that a nonnegative function g(x) ∈ L1 (B1 ) belongs to K˜ p if 1/p Z R Z 1 dr lim sup g(z)dz = 0. R→0 x∈B1 0 rn−p B(x,r) r

(5.2.6)

252

Chapter 5 Quasilinear elliptic equations with coefficients from the Kato class

We say that a nonnegative function g(x) ∈ L1 (B1 ) belongs to K2,p if 1  Z Z R max(p,2)−1 dr 1 g(z)dz = 0. lim sup R→0 x∈B1 0 rn−p B(x,r) r

(5.2.7)

In what follows, we assume that p/(p−1)

g1 (x), f1 (x), g2

p/(p−1)

(x), f2

(x) ∈ K˜ p ,

hp (x), g3 (x), f3 (x) ∈ Kp .

(5.2.8)

1,p (B1 ) satisfying A solution of Eq. (5.2.1) is defined as a function u(x) ∈ Wloc the integral identity      Z X n ∂u ∂u ∂ϕ + a0 x, u, ϕ dx = 0 (5.2.9) ai x, u, ∂x ∂xi ∂x B1 i=1

◦

for any function ϕ(x) ∈ W 1,p loc (B1 ). To formulate the results, we introduce the following notation: Z R  1/p Z 1 dr p.(p−1) [ f1 (z) + f2 f (R) = sup sup (z) ]dz n−p r r x0 ∈B1 x∈B(x0 ,R) B(x,r) 0 1/(p−1)  Z R Z 1 dr + f3 (z)dz , (5.2.10) n−p r r 0 B(x,r) Z R  1/p Z 1 dr p/(p−1) sup (z) ]dz g(R) = sup [g1 (z) + g2 n−p r r x0 ∈B1 x∈B(x0 ,R) 0 B(x,r)    Z Z R 1/(p−1) dr 1 g (z)dz , (5.2.11) + 3 n−p r r B(x,r) 0 1/(p−1) Z R Z 1 dr p sup h(R) = sup h (z)dz . (5.2.12) n−p r r x0 ∈B1 x∈B(x0 ,R) 0 B(x,r) Theorem 5.2.1. Let u(x) be a nonnegative solution of Eq. (5.2.1). Suppose that inequalities (5.2.2)–(5.2.4) are true and conditions (5.2.8) are satisfied. Also let v0 be a constant such that f (R) + g(R) + h(R) 6 v0 for 0 < R < 1/4. (5.2.13)
Then, for any q ∈ 0, n(p − 1)/(n − p) , there exist constants K1 and α depending only on n, p, c1 , c2 , v0 , and q and such that the estimate Z [u(x) + αf (R) ]q dx 6 K1 Rn inf [ u(x) + αf (R) ]q (5.2.14) B(x0 ,R/2)

B(x0 ,R)

is true for any point x0 ∈ B1 such that B(x0 , 4R) ⊂ B1 .

Section 5.2

253

Harnack’s inequality

To prove Theorem 5.2.1, we use the following lemma: Lemma 5.2.1. For any ε > 0, there exist R0 = R0 (ε) < 1 and τ (ε) depending only on n, p, and ε and such that the inequality 1/(p−1) Z R0  Z 1 dr sup H(z)dz 6 τ (ε) (5.2.15) n−p r r x∈B1 0 B(x,r) implies the estimate Z

∂ϕ(x) p H(x)|ϕ(x)| dx 6 ε ∂x dx B(x0 ,R) B(x0 ,R) Z

p

(5.2.16)

◦

for any function ϕ(x) ∈ W 1,p (B(x0 , R)), whenever R 6 R0 and B(x0 , 4R0 ) ⊂ B1 . The proof of the lemma can be found in [20]. Theorem 5.2.2. Let u(x) be a nonnegative solution of Eq. (5.2.1). Suppose that inequalities (5.2.2)–(5.2.4) are true and conditions (5.2.8) are satisfied. Then there exist positive constants K2 and τ∗ , depending only on n, p, c1 , c2 , and v0 such that the estimate max

[ u(x) + αf (R) ] 6 K2

x∈B(x0,R)

[ u(x) + αf (R) ]

min

(5.2.17)

x∈B(x0 ,R)

holds for R and α satisfying the inequality g(R) + h(R) 6 τ∗ ,

α > τ∗−1 ,

(5.2.18)

and any point x0 ∈ B1 such that B(x0 , 4R) ⊂ B1 ,

5.2.2

Proof of Theorem 5.2.1

Let x0 be a point from B1 and let R < (1 − |x0 |)/4. We fix x1 ∈ B(x0 , R/2). For r < R, we define a function ξ(x) ∈ C0∞ (B(x1 , r)) equal to 1 in B(x1 , r/2) and such that ∂ξ(x) 6 2/r. 0 6 ξ(x) 6 1 and ∂x For positive α, l, and δ, we also define v(x) = [ u(x) + αf (R) ]−1 , λ=

(p − 1)(n − 1) , p(n − p − 1)

α > 0,

L = B(x0 , r) ∩ { v(x) > l },

k =p+

(p2 − 1)(1 + λ) . p−1−λ

(5.2.19)

Lemma 5.2.2. Assume that conditions (5.2.2)–(5.2.4) are satisfied. Then the following estimate is true:     Z  Z v(x) v(x) − l −1−λ ∂v p k s − l −1−λ ds + v(x) 1 + 1+ ∂x ξ (x) dx δ δ L l

254

Chapter 5 Quasilinear elliptic equations with coefficients from the Kato class

  v(x) − l) (1+λ)(p+1) k−p u(x) 1 + ξ (x)dx 6 γδ r δ L  Z v(x)  Z s − l −1−λ p dsH1 (x)ξ k (x)dx 1+ v (x) +γ δ l L   Z v(x) − l −1−λ H2 (x)ξ k (x)dx, +γ v p+1 (x) 1 + δ L p −p

Z

(5.2.20)

where H1 (x) = g1 (x) + g3 (x) + hp (x) + f1 (x)f −p (R)α−p + f3 (x)f −1−p (R)α1−p , p p+1

H2 (x) = g1 (x) + g2

p p+1

(x) + [f1 (x) + f2

(5.2.21)

(x) ]f −p (R)α−p .

Proof. We substitute the function  Z v(x)    s − l −1−λ ds ξ k (x), ϕ(x) = v 2p−1 (x) 1+ δ l +

(5.2.22)

k > 0,

in the integral identity (5.2.9). This yields  Z Z v(x)  s − l −1−λ ∂u p k ds ξ (x)g k (x)dx v 2p (x) 1+ δ ∂x L l   Z 5 X v(x) − l −1−λ ∂u p k 2p+1 v (x) 1 + + 6 γ Ij , ξ (x)dx ∂x δ L

(5.2.23)

j=1

where  s − l −1−λ ds [g1 (x)up (x) + f1 (x)]ξ k (x)dx, δ L l   Z v(x) − l −1−λ 2p+1 = v (x) 1 + [g1 (x)up (x) + f1 (x)]ξ k (x)dx, δ L  Z Z v(x)  s − l −1−λ 1 v 2p+1 (x) 1+ ds = r L δ l  p−1  ∂u p−1 × + g2 (x)u (x) + f2 (x) ξ k−1 (x)dx, ∂x  Z Z v(x)  s − l −1−λ ∂u p−1 k 2p−1 = h(x)v (x) 1+ ds ξ (x)dx, δ ∂x L 1 −1−λ Z Z v(x)  s−l ds[g3 (x)up−1 (x) + f3 (x)]ξ k (x)dx. = v 2p−1 (x) 1+ δ L l Z

I1 = I2 I3

I4 I5

v 2p (x)

Z

v(x) 

1+

Section 5.2

255

Harnack’s inequality

In what follows, we use the following easily verifiable inequality:  Z v(x)  s − l −1−λ ds 6 γδ. 1+ δ l

(5.2.24)

By using the definition of the function v(x) and the Young inequality, we obtain  Z Z v(x)  s − l −1−λ p ds I1 6 v (x) 1+ δ L l × [g1 (x)up (x) + f1 (x)f −p (R)α−p ]ξ k (x)dx,   Z v(x) − l −1−λ I2 6 γ v p+1 (x) 1 + δ L × [g1 (x) + f1 (x)f −p (R)α−p ]ξ k (x)dx,  Z Z v(x)  1 v(x) − l −1−λ ∂u p k 2p+1 γI3 6 v (x) 1+ ∂x g (x)dx 4 L δ l   Z δp v(x) − l (1+λ)(p−1) k−p ξ (x)dx +γ p v(x) 1 + r L δ   Z v(x) − l −1−λ p+1 +γ v (x) 1 + δ L

(5.2.25)

(5.2.26)

p

p

× [g2p−1 (x) + f2p−1 (x)f −p (R)α−p ]ξ k (x)dx,  Z Z v(x)  1 s − l −1−λ ∂u p k 2p γI4 6 v (x) 1+ ds ξ (x)dx 4 L δ ∂x l −1−λ Z Z v(x)  s−l ds hp (x)ξ k (x)dx, +γ v p (x) 1+ δ L l  Z Z v(x)  s − l −1−λ p ds I5 6 γ v (x) 1+ δ L l × [g3 (x) + f3 (x)f 1−p (R)α1−p ]ξ k (x)dx.

(5.2.27)

(5.2.28)

(5.2.29)

Estimate (5.2.20) is obtained as a direct corollary of inequalities (5.2.23) and (5.2.25)–(5.2.29).  Denote 1 Φ(x) = δ

Z

v(x) 1/p

s l

   s − l −(1+λ)/p 1+ ds . δ

(5.2.30)

Lemma 5.2.3. Let conditions (5.2.2)–(5.2.4) and (5.2.8) be satisfied. Then there exists a number θ1 ∈ (0, 1) that depends only on n, p, c1 , c2 , and v0 such

256

Chapter 5 Quasilinear elliptic equations with coefficients from the Kato class

that the estimate Z ∂Φ(x) p k ∂x ξ (x)dx L   Z v(x) − 1 (1+λ)(p−1) k−p −p v(x) 1 + 6 γr ξ (x)dx δ L Z Z + γlp δ 1−p H1 (x)ξ k (x)dx + γlp+1 δ −p H2 (x)ξ k (x)dx

(5.2.31)

L

L

is true whenever R and α satisfy the inequality g(R) + h(R) 6 θ1 ,

α > θ1−1 .

(5.2.32)

Proof. In view of inequalities (5.2.2) and (5.2.6) and the definition of the function Φ(x), we conclude that, in order to prove the lemma, it suffices to establish the following estimates: 1 1 2p γI9 6 I6 + γI7 , (5.2.33) 2p−1 γI8 6 I6 + γI7 , 4 4 where  Z  Z v(x)  s − l −1−λ I6 = 1+ ds δ L l    v(x) − l −1−λ ∂v p k + v(x) 1 + ∂x ξ (x)dx, δ   Z v(x) − l (1+λ)(p−1) k−p p −p ξ (x)dx, I7 = δ r v(x) 1 + δ L  Z Z v(x)  s − l −1−λ p ds H1 (x)ξ k (x)dx, I8 = (v(x) − l) 1+ δ L l   Z v(x) − l −1−λ I9 = (v(x) − l)p+1 1 + H2 (x)ξ k (x)dx. δ L Let us prove that the second inequality in (5.2.33) follows from Lemma 5.2.1. Condition (5.2.15) for the function H2 follows from (5.2.32) for a sufficiently small θ1 since 1/(p−1) Z Z R dr 1 H (z)dz 6 γ[g(R) + αp/(p−1) ]. (5.2.34) 2 n−p r r Br (x) 0 Hence, for γ[g(R) + αp/(p−1) ] 6 τ (ε), we get     Z ∂ v(x) − l −(1+λ)/p k/p (p+1)/p ξ (x) dx. I9 6 ε 1+ ∂x (v(x) − l) δ Br (x) By estimating the last integral for a properly chosen ε, we arrive at inequality (5.2.33) for I9 .

Section 5.2

257

Harnack’s inequality

To prove the first inequality in (5.2.33), we introduce an auxiliary function w(x) as a solution of the problem ◦

w(x) ∈ W 1,p B(x0 , R),

−∆p w = H1 (x),

(5.2.35)

where ∆p is the p-Laplacian. Condition (5.2.8) guarantees the inclusion ◦

H1 (x) ∈ [ W 1,p (BR )(x0 ) ]∗ . Hence, the existence of w(x) immediately follows from the theory of monotone operators. For w(x), the estimate |w(x)| 6 γ[g(R) + h(R) + α

p − (p−1)

+ α−1 ]

(5.2.36)

follows from the results established in [59]. By using the definition of a solution of problem (5.2.35), we obtain n Z X ∂w p−2 ∂w I8 = ∂x ∂xi L i=1    Z v(x)  ∂ s − l −1−λ × 1+ (v(x) − l)p ds ξ k (x) dx. (5.2.37) ∂xi δ l Differentiating the expression in the braces and using Young’s inequality, we arrive at the estimate 1 (5.2.38) 2p−1 γI8 6 I6 + γ(I7 + I10 ), 8 where  Z Z v(x)  s − l −1−λ ∂w p k p I10 = (v(x) − l) 1+ ds ξ (x)dx. δ ∂x L l We now use the integral identity for the solution of problem (5.2.35) once again. This yields  Z Z v(x) s − l −1−λ p dsH1 (x)ξ k (x)dx I10 = w(x)(v − l) 1+ δ L l    Z v(x) n Z X ∂w p−2 ∂w ∂ s − l −1−λ p k (v − l) dsξ (x) dx. − 1 + ∂x ∂xi ∂xi δ i=1

L

l

Estimating the last integral by analogy with the right-hand side of (5.2.37), we get the inequality I10 6 γ max {|w(x)|x ∈ BR (x0 )}˙[I8 + I6 + I7 + I10 ].

(5.2.39)

By virtue of estimate (5.2.36), we can choose sufficiently small θ1 such that the first inequality in (5.2.33) follows from (5.2.32), (5.2.38), and (5.2.39). 

258

Chapter 5 Quasilinear elliptic equations with coefficients from the Kato class

Consider a sequence of numbers Rj = R2−j , j = 0, 1, 2, . . . , and a sequence of functions ξj (x) ∈ C0∞ (Bj ), j = 0, 1, 2, . . . , such that ξj (x) = 1 in Bj+1 = B(x1 , Rj+1 ), ξj (x) = 0 outside Bj , ∂ξj 6 2R−1 . 0 6 ξj (x) 6 1, and j ∂x Let l0 = 0. For any j > 1, we define   Z v(x) v(x) − lj (1+λ)(p−1) k−p 1 ξj (x)dx, κ= n Rj Lj lj+1 lj+1 − lj

(5.2.40)

Lj = Bj ∩ {v(x) > lj }, and k > p is a fixed number determined in what follows. Equality (5.2.40) specifies lj+1 if lj is known. Thus, if l0 s known, then we can determine the entire sequence {lj }. We set     Z v(x) s − lj −(1+λ)/p 1/p ds . s 1+ δj = lj+1 − lj , Φj (x) = δj lj + Lemma 5.2.4. Let conditions (5.2.2)–(5.2.4) and (5.2.8) and inequality (5.2.32) be satisfied. Then there exists κ that depends only on n, p, c1 , c2 , and v0 such that the estimate 1/(p−1)  Z 1 p−n H1 (x)ξj (x)dx δj 6 δj−1 + γlj Rj 2 Bj  1/p Z + γ Rjp−n ljp−1 H2 (x)ξj (x)dx (5.2.41) Bj

holds for any j > 1. Proof. We assume that δj > δj−1 /2. Otherwise, inequality (5.2.41) is obvious. Let us represent Lj in the form   v(x) − lj 0 00 0 6 ε ∩ Lj , L00j = Lj \L0j , Lj = Lj ∪ Lj , Lj = δj where ε ∈ (0, 1) will be chosen later. We have   Z v(x) − lj (1+λ)(p−1) k−p 1 v(x) ξj (x)dx Rj00 L0j δj Z 2n (1+λ)(p−1) 6 n ε v(x)ξjk−p (x)dx Rj−1 Lj−1 ∩{v>lj }

Section 5.2

259

Harnack’s inequality

2n 6 n ε(1+λ)(p−1) Rj−1

  v(x) − lj−1 (1+λ)(p−1) k−p v(x) ξj−1 (x)dx δj−1 Lj−1

Z

6 2n ε(1+λ)(p−1) κlj .

(5.2.42)

Further, we use the following evident inequality:   1 v(x) − lj 1−(1+λ)/p Φj (x) > γ(ε)v p (x) δj

(5.2.43)

for v(x) − lj > δj ε, where the constant γ(ε) depends on the known parameters and ε. By the choice of λ, we can use the imbedding theorem and obtain   Z v(x) − lj (1+λ)(p−1) k−p 1 ξj (x)dx v(x) Rj00 L00j δj Z pλ (1+λ)(p−1) (k−p) p p−1−λ −n 6 γ(ε)lj [Φj (x)] p−1−λ ξj Rj (x)dx Lj

p−1−λ ∂Φj (x) p (k−p) (p−1)(1+λ) ξ dx ∂x j Lj  (1+λ)(p−1) Z p−1−λ p−1−λ −p (k−p) (p−1)(1+λ) p −n . dx + Rj Φj (x)ξj

pλ p−1−λ

6 γ(ε)lj

 Z Rjp−n

(5.2.44)

Lj

By using Lemma 5.2.3 and choosing k from the condition (k − p)

p−1−λ = p + 1, (p − 1)(1 + λ)

we obtain the following relation from (5.2.44):   Z v(x) − lj (1+λ)(p−1) k−p −n Rj v(x) ξj (x)dx δj L00 j    Z pλ v(x) − lj (1+λ)(p−1) p−1−λ −n ξj (x)dx 6 γ(ε)lj Rl v(x) 1 + δj Lj Z + Rjp−n δj1−p ljp H1 (x)ξj (x)dx Bj

+

Rjp−n δj−p ljp+1

 (1+λ)(p−1)

Z

p−1−λ

H2 (x)ξj (x)dx

.

(5.2.45)

Bj

Here, we also use the following evident inequality:   1 v(x) − lj 1−(1+λ)/p p Φj (x) 6 γv (x) 1 + for x ∈ Lj . δj

(5.2.46)

260

Chapter 5 Quasilinear elliptic equations with coefficients from the Kato class

By analogy with (5.2.42), by virtue of the inequality δj > δj−1 /2, we obtain   Z v(x) − lj (1+λ)(p−1) Rj−n ξj (x)dx v(x) 1 + δj Lj   Z v(x) − lj−1 (1+λ)(p−1) k−p n −n 6 2 Rj−1 v(x) 3 ξj−1 (x) δj−1 Lj−1 ∩{v>lj } 6 2n 3(1+λ)(p−1) κlj .

(5.2.47)

By using equality (5.2.30) and estimates (5.2.42), (5.2.45), and (5.2.47), we get (1+λ)(p−1)

κ 6 2n ε(1+λ)(p−1) κ + γ(ε)κ p−1−λ  (1+λ)(p−1)  Z p−1−λ . + γ(ε) δj−p lj−1 Rjp−n [δj ljp H1 (x) + ljp+1 H2 (x)]ξj (x)dx Bj

(5.2.48) We now first choose ε and then κ to guarantee the validity of the equalities 2n ε(1+λ)(p−1) = 1/4 and γ(ε)κ λp/(p−1−λ) = 1/4. Thus, it follows from (5.2.48) that at least one of the inequalities   p−1−λ Z (1+λ)(p−1) κ 1 1−p p−1 p−n δj lj Rj H1 (x)ξj (x)dx > , 2 4γ(ε) Bj   p−1−λ Z (1+λ)(p−1) 1 κ −p p p−n H2 (x)ξj (x)dx > δj lj Rj 2 4γ(ε) Bj is satisfied. This implies that δj satisfies inequality (5.2.41).



Lemma 5.2.5. Let conditions (5.2.2)–(5.2.4) and (5.2.8) be satisfied. Then there exists θ2 ∈ (0, θ1 ) that depends only on n, p, c1 , c2 , and v0 such that the estimate   1 Z p+λ(p−1) −n p+λ(p−1) max v(x) 6 γ R v (x)dx (5.2.49) B(x0 , R ) 2

B(x0 ,R)

holds for g(R) + h(R) < θ2 and α > θ2−1 . Proof. We sum (5.2.41) over j = 1, 2, . . . , J. By using inequality (5.2.34) and a similar estimate for H1 , we obtain p

lJ+1 6 δ0 + γ[g(R) + h(R) + α p−1 ]lJ .

(5.2.50)

Choosing θ2 ∈ (0, θ1 ) such that p

γ(θ2 + α p−1 ) < 1/2

(5.2.51)

Section 5.2

261

Harnack’s inequality

and passing to the limit as J → ∞, we obtain (5.2.49) by virtue of the definition of δ0 .  Lemma 5.2.6. Let conditions (5.2.2)–(5.2.4) and (5.2.8) be satisfied and let ξ(x) be the cutoff function defined at the beginning of the section. Then, for any K > p, there exist positive constants τ (K) and γ(K) that depend only on the known parameters and K such that the inequality p Z Z ∂u k s+1 −p v (x) ξ (x)dx 6 γ(K)r v s+1−p (x)ξ k−p (x)dx (5.2.52) ∂x B(x1 ,r)

B(x1 ,r)

holds for 0 < s 6 K and p 6 k 6 K, whenever g(R) + h(R) 6 τ (K),

α > τ −1 (K),

and r 6 R.

Proof. We substitute the function ϕ(x) = v s (x)ξ k (x),

s, k > 0,

in the integral identity (5.2.9). This yields p Z ∂u s+1 v (x) ξ k (x)dx ∂x B(x1 ,r) Z −p 6 γ(K)r v s+1−p (x)ξ k−p (x)dx B(x1 ,r) Z + γ(K) v s+1−p (x)(H1 (x) + H2 (x))ξ k (x)dx,

(5.2.53)

B(x1 ,r)

where H1 (x) and H2 (x) are defined by (5.2.21) and (5.2.22). For s = p − 1 and k = p, by virtue of condition (5.2.8) and (5.2.53), we arrive at the inequality p Z ∂ n−p , ∂x ln v(x) dx 6 γr B(x1 ,r) This inequality guarantees a possibility to use the John–Nirenberg lemma for the function ln v(x). By virtue of Lemma 5.2.1, for any s > p − 1, we obtain Z γ(K) v s+1−p (x)(H1 (x) + H2 (x))ξ k (x)dx B(x1 ,r) p  Z Z ∂u k 1 −p s+1−p k−p s+1 6 v (x)ξ (x)dx v (x) ξ (x)dx + r 2 ∂x B(x1 ,r) B(x1 ,r) if τ (K) is sufficiently small. This and (5.2.53) imply (5.2.52).



By using Lemmas 5.2.4 and 5.2.5 and the John–Nirenberg lemma, after finitely many iterations of the Moser type, we obtain Theorem 5.2.1.

262

5.2.3

Chapter 5 Quasilinear elliptic equations with coefficients from the Kato class

Proof of Theorem 5.2.2

Let α, l, δ, λ, k, be the same numbers and let ξ(x) be the same function as in the proof of Lemma 5.2.2. We denote E = Br (x0 ) ∩ {u(x) > l}.

u(x) = u(x) + αf (R),

Lemma 5.2.7. Assume that conditions (5.2.2)–(5.2.4) are satisfied. Then the estimate     Z  Z u(x)  s − l −1−λ u(x) − l −1−λ ∂u p k 1+ ds + u(x) 1 + ∂x ξ (x)dx δ δ l E   p Z  u(x) − l (1+λ)(p−1) k−p δ 6 ξ (x)dx u(x) 1 + r δ E  Z Z u(x)  s − l 1−λ 1+ +γ up (x) dsH1 (x)ξ k (x)dx δ l E   Z u(x) − l −1−λ p+1 H2 (x)ξ k (x)dx (5.2.54) u (x) 1 + +γ δ E holds with functions H1 (x) and H2 (x) given by equalities (5.2.21) and (5.2.22). To prove the lemma, it suffices to substitute the function u(x) 

Z ϕ(x) = u(x)

s−l 1+ δ

l

−1−λ



ds ξ k (x)

in the integral identity and estimate the integrals by analogy with the proof of Lemma 5.2.2. Note that inequality (5.2.54) follows from (5.2.20) with v(x) and L replaced by u(x) and E, respectively. By analogy with Lemmas 5.2.3 and 5.2.4, we obtain the following assertion: Lemma 5.2.8. Suppose that conditions (5.2.2)–(5.2.4) and (5.2.8) are satisfied. Then the estimate  max u(x) 6 γ R

B(x0 , R ) 2

−n

Z

p+λ(p−1)

u



1 p+λ(p−1)

(x)dx

(5.2.55)

B(x0 ,R)

is true for R and α satisfying the condition of Lemma 5.2.5. It is easy to see that estimate (5.2.17) is a direct corollary of inequalities (5.2.13) and (5.2.55). This completes the proof of Theorem 5.2.1.

Section 5.3

5.3 5.3.1

263

Removability of isolated singularities

Removability of isolated singularities Statement of propositions and main results

Without loss of generality, we can assume that {0} ∈ Ω and {0} is a singular point of the solution u(x) of the equation −

n X d ∂u
ai x, u, = a0 (x, u), dxi ∂x

x ∈ Ω\{0}.

(5.3.1)

i=1

We assume that the functions a0 (x, u), ai (x, u, ξ), i = 1, 2, . . . , are defined for x, u ∈ R, ξ ∈ Rn and satisfy the Carathéodory conditions. Moreover, it is assumed that inequalities (5.2.2), (5.2.3), and |a0 (x, u)| 6 g3 (x)|u|p−1 + f3 (x)

(5.3.2)

are true. In addition, suppose that p/(p−1)

g1 , f1 , g1 g3 , f3 ∈ Kp for p > 2

and

p/(p−1)

∈ K˜ p ,

(5.3.3)

g3 , f3 ∈ K2,p for 1 < p < 2.

(5.3.4)

, f2

We say that u(x) is a solution of Eq. (5.3.1) in Ω\{0} if, for any function ϕ(x) ∈ W 1,p (Ω) equal to zero near the boundary of the set Ω\{0}, we have u(x)ϕ(x) ∈ W 1,p (Ω), and the following equality is true:    Z X n ∂u ∂ϕ − a0 (x, u)ϕ dx = 0. ai x, u, (5.3.5) ∂x ∂xi Ω i=1

Let R0 be an arbitrary number satisfying the inequality 0 < R0 < min {1, dist ({0}, ∂Ω)} For 0 < r 6 R0 , we denote M (r) = max {|u(x)| : r < |x| 6 R0 }.

(5.3.6)

It follows from the results of the previous section that the function u(x) is continuous for x ∈ {r 6 |x| 6 R0 }. Hence, M (r) < ∞. The following theorem is the main result of the present section: Theorem 5.3.1. Let u(x) be a solution of Eq. (5.3.1) in Ω\{0}. Suppose that inequalities (5.2.2), (5.2.3), and (5.3.2) are true and conditions (5.3.3) and (5.3.4) satisfied. In this case, if lim M (r)r(n−p)/(p−1) = 0,

r→0

then the singularity at {0} for a solution of Eq. (5.3.1) is removable.

(5.3.7)

264

Chapter 5 Quasilinear elliptic equations with coefficients from the Kato class

Example 5.3.1. The function u(x) = |x|2−n ln−α

1 |x|

is a solution of the equation −∆u + gu = 0 in B(0, 1)\{0}, where g(x) = α(2 − n)|x|−2 ln−1

1 1 + α(α + 1)|x|−2 . |x| |x|

The solution u(x) has an unremovable singularity at {0}. At the same time, 1 g∈ / K2 despite the fact that |x|−2 ln−1−ε |x| ∈ K2 for any ε > 0. Similarly, the function p−n

u(x) = |x| p−1 ln−α

1 |x|

satisfies the equation −∆p u + gup−1 = 0 in B(0, 1)\{0}, 1 where g(x) behaves as c|x|−p ln−1 |x| in the neighborhood of zero. At the same −p 1 −p time, g(x) ˜ = c|x| ln |x| belongs to Kp for β > p − 1 and to K2,p for β > 1. This means that our conditions are optimal for 1 < p 6 2. Theorem 5.3.1 is a consequence of certain pointwise estimates similar to those obtained in Chap. 3 and the following theorem:

Theorem 5.3.2. Let u(x) be a solution of Eq. (5.3.1) in Ω\{0}. Suppose that the conditions of Theorem 5.3.2 are satisfied and, in addition, M (r) 6 γr−a+(n−p)/(p−1)

(5.3.8)

max {|u(x)| : x ∈ Ω} 6 K,

(5.3.9)

for some a > 0. Then

for a constant K that depends only on n, p, c1 , c2 , v0 , and a.

5.3.2

Proof of Theorem 5.3.2

Let x0 be an arbitrary point of Ω, 0 < R1 < 21 min{1, dist ({x0 }, ∂Ω)},   1 p − 1 (p − 1)(n − 1) , a , λ = min 2 1 + p−1 p(n − p + 1) p k =p+

(p2 − 1)(1 + λ) , p−1−λ

δ, l > 0.

(5.3.10)

Section 5.3

265

Removability of isolated singularities

We fix x1 ∈ B(x0 , R1 ). For R 6 R1 , we fix a function ξ(x) ∈ C0∞ (B(x1 , R)) equal to 1 in B(x1 , R2 ) and such that | Also let ψr (x) = ψ conditions

|x|
r ,

4 ∂ξ |6 . ∂x R

r > 0, where the function ψ(t) ∈ C ∞ (R) satisfies the ψ(t) = 0 for t 6 1, ψ(t) = 1 for t > 2, −2 6

dψ 6 0 for t ∈ [1, 2]. dt

We set v(x) = u(x) + θf (R1 ), where θ is a number that depends only on the quantities n, p, c1 , and c2 , which are determined in what follows. Lemma 5.3.1. Let the conditions of Theorem 5.3.2 be satisfied. Then, for any α and β > 0, the following relation is true:     Z  Z vβ (x) v β − l −1−λ ∂v p k k s − l −1−λ ds + v α+β−1 1 + v α−1 1+ ∂x ξ ψr dx δ δ L l   Z v β (x) − 1 (1+λ)(p−1) k−p k −p p α−(β−1)(p−1) 6 γR δ v 1+ ξ ψr dx δ L   Z v β (x) − l (1+λ)(p−1) k k−p α−(β−1)(p−1) −p p v 1+ + γr δ ξ ψr dx δ L∩K(r)  Z Z vβ (x)  δ s − l −1−λ + vα 1+ ds r L∩K(r) δ l × [g2 (x)up−1 (x) + f2 (x)]ξ k ψrk−1 dx  Z Z vβ (x)  s − l −1−λ p−1+α +γ v 1+ ds H1 (x)ξ k ψrk dx δ L l   Z v β (x) − l −1−λ H2 (x)ξ k ψrk dx, +γ v p−1+α+β 1 + δ L

(5.3.11)

where L = BR (x1 ) ∩ {v β (x) > l},

K(r) = {r 6 |x| 6 2r},

H1 (x) = g1 (x) + g3 (x) + f1 (x)f −p (R)θ−p + f3 (x)f 1−p (R)θ1−p , p/(p−1)

H2 (x) = g1 (x) + g2

p/(p−1)

(x) + [f1 (x) + f2

(x)]f −p (R)θ−p .

(5.3.12) (5.3.13)

266

Chapter 5 Quasilinear elliptic equations with coefficients from the Kato class

Proof. We substitute the function    Z vβ (x)  s − l −1−λ α ds ξ k (x)ψrk (x) ϕ(x) = v (x) 1+ δ l + in the integral identity (5.3.5). After this, reasoning as in Lemma 5.2.2, we obtain (5.3.11).  Denote 1 Φ(x) = δ

Z

v β (x)

s

α−(β−1)(p−1) βp

l

   s − l −(1+λ)/p 1+ ds . δ +

(5.3.14)

Lemma 5.3.2. Let the conditions of Theorem 5.3.2 be satisfied and let α + p − 1 − βp > 0. Then there exists a number v1 ∈ (0, 1) that depends only on n, p, c1 , c2 , and v0 , such that the estimate Z ∂Φ(x) p k ∂x ξ (x)dx L   Z v β (x) − l (1+λ)(p−1) k−p α−(β−1)(p−1) −p v 1+ ξ (x)ψrk (x)dx 6 γR δ l   Z v β (x) − l (1+λ)(p−1) k α−(β−1)(p−1) −p v 1+ + γr ξ (x)ψrk−p (x)dx δ L∩K(r)  Z Z vβ (x)  s − l −1−λ + γδ 1−p r−1 vα 1+ ds δ L∩K(r) l × [g2 (x)up−1 (x) + f2 (x)]dx + γδ 1−p l

p−1+α β

Z

H1 (x)ξ k (x)ψrk (x)dx

L

+ γδ

−p

l

p−1+α+β β

Z

H2 (x)ξ k (x)ψrk (x)dx

(5.3.15)

L

holds whenever θ and R satisfy the inequalities g(R) < v1 ,

θ > v1−1 .

(5.3.16)

Proof. We have   Z v β (x) − l −1−λ v p−1+α−β 1 + H2 (x)ξ k (x)ψrk (x)dx δ L   Z β (x) − l −1−λ p−1+α−β v β H2 (x)ξ k (x)ψrk (x)dx 1+ 6 γ (v − l) β δ L Z p−1+α−β H2 (x)ξ k (x)ψrk (x)dx. (5.3.17) + γl β L

Section 5.3

Removability of isolated singularities

In view of the definition of H2 (x) and inequality (5.3.16), we find  1/(p−1)  Z R Z 1 dr −p/(p−1) H (z)dz 6 γ g(R) + θ 1 rn−p Br (x) r 0   p/(p−1) 6 γ v1 + v1 .

267

(5.3.18)

Hence, it follows from Lemma 5.3.1 and the condition α + p − 1 − βp > 0 that   Z β (x) − l −1−λ p−1+α+β v β H2 (x)ξ k (x)ψrk (x)dx (v − l) β 1+ δ L   Z 1 v β (x) − l −1−λ ∂v(x) p k α+β−1 k 6 v 1+ ∂x ξ (x)ψr (x)dx 8 L δ   Z v β − l p−1−λ p α−(β−1)(p−1) + γδ v 1+ δ L p ∂(ξ(x)ψr (x)) k−p ξ (x)ψrk−p (x)dx. (5.3.19) × ∂x Moreover, Z Z v α+p−1 L

v β (x) 

 s − l −1−λ dsH1 (x)ξ k (x)ψrk (x)dx δ l  Z Z vβ (x)  p−1+α s − l −1−λ 6 γ (v β − l) β 1+ dsH1 (x)ξ k (x)ψrk (x)dx δ L l Z p−1+α H1 (x)ξ k (x)ψrk (x)dx. (5.3.20) + γδl β 1+

L

Consider problem (5.2.35). According to the definition of solution, we can write  Z vβ (x) Z p−1+α s−l −1−λ β β 1+ dsH1 (x)ξ k (x)ψrk (x)dx I1 = (v − l) δ l L n Z X ∂w p−2 ∂w = ∂x ∂xi i=1

L

   Z vβ (x) p−1+α ∂ s−l −1−λ β k k−p β × dxξ (x)ψr (x) . 1+ (v − l) ∂xi δ l

(5.3.21)

Differentiating the expression in braces and using the Young inequality, we arrive at the estimate  Z  Z vβ (x)  1 s − l −1−λ I1 6 v α−1 1+ ds 8 L δ l    v β (x) − l −1−λ ∂v p k k α+β−1 +v 1+ ∂x ξ (x)ψr (x)dx δ

268

Chapter 5 Quasilinear elliptic equations with coefficients from the Kato class

 v β (x) − l p−1 + γδ v δ L p k−p ∂ (x)ψrk−p (x)dx + I2 + I3 , × (ξ(x)ψr (x)) ξ ∂x p

Z

α−(β−1)(p−1)



(5.3.21)

where v β (x) 

 s − l −1−λ ∂w p k k ds ξ (x)ψr (x)dx, δ ∂x l L   Z β (x) − l −1−λ ∂w p k p+α v β β−1 ξ (x)ψrk (x)dx. ds I3 = γ (v − l) β v 1+ δ ∂x L Z

I2 = γ

(v β − l)

p−1+α β

Z

1+

We again apply the integral identity for the solution of problem (5.2.35). This yields  Z Z vβ (x) p−1+α s−l −1−λ I2 = γ H1 (x)w(x)(v β − l) β 1+ ds ξ k (x)ψrk (x)dx δ L l n Z X ∂w p−2 ∂w −γ w ∂x ∂xi i=1

L



p−1+α ∂ × (v β − l) β ∂xi

Z l

v β (x)

s−l 1+ δ

−1−λ dsξ

k



(x)ψrk (x)

dx. (5.3.22)

Therefore,  Z  ∂ I2 6 γ wH1 (x) + (ξ(x)ψr (x)) wp ∂x L  Z vβ (x)  p−1+α 1 + s − l −1−λ β β × (v − l) ds ξ k−p (x)ψrk−p dx δ l   Z Z vβ (x)  s − l −1−λ p α−1 +γ w v 1+ ds δ L l    v β − l −1−λ ∂v p k k α+β−1 (5.3.23) +v 1+ ∂x ξ ψr dx + I3 . δ Further, by using the same integral identity for problem (5.2.35), we obtain   Z p+α v β (x) − l −1−λ k β−1 β β 1+ I3 = γ H1 (x)w(v − l) v ξ (x)ψrk (x)dx δ L n Z X ∂w p−1 ∂w −γ w ∂x ∂xi i=1

L

    p+α ∂ v β − l −1−λ k β k β+1 β × (v − l) v ξ (x)ψr (x) dx, (5.3.24) 1+ ∂xi δ

Section 5.3

269

Removability of isolated singularities

whence it follows that   Z p+α v β (x) − l −1−λ k ξ (x)ψrk (x)dx wH1 (x)(v β − l) β v β−1 1 + I3 6 γ δ L p−1  β Z ∂ k−1 p p α−(β−1)(p−1) v − l ξ + γδ w v (ξ(x)ψ (x) (x)ψrk−1 (x)dx. r ∂x δ L (5.3.25) We estimate the first integral on the right-hand side of (5.3.25) by analogy with (5.3.19). In view of estimate (5.2.36) for w(x), we obtain (5.3.15) from relations (5.3.17)–(5.3.25).  Consider sequences of numbers Rj = 2−j R, j = 0, 1, 2, . . . , and functions ξj (x) ∈ C0∞ (Bj ) such that ξj (x) = 1 in Bj+1 = B(x1 , Rj+1 ), ξj (x) = 0 outside Bj , 0 6 ξj (x) 6 1, and ∂ξj /∂x 6 γR−1 . j

Let l0 = 0. For any j > 1, we define 1 κ= n Rj



Z Lj

v(x)

α−(β−1)(p−1) 

v β (x) − lj lj+1 − lj

1/β lj+1

(1+λ)(p−1)

ξjk−p (x)dx,

(5.3.26)

where 1/β

Lj = BJ ∩ {v(x) > lj

},

β = (p − 1)/p,

and α = λ.

By virtue of (5.3.8) and the conditions imposed on α, β, and λ, the integral on the right-hand side of (5.3.26) is finite. We set δj = lj+1 − lj . By analogy with (5.2.49), we arrive at the estimate κ 6 2n ε(1+λ)(p−1) κ  Z + γ(ε) κ + γδj1−p ljp−1

L1

+ γr

−p



Z

v(x) 1/β

lj

Lj ∩K(r)

H1 (x)dx + κ + γδj1−p ljp−1

Z H1 (x)dx Lj

 α−(β−1)(p−1)  v β (x) − l (1+λ)(p−1) 1+ dx δj

α−(β−1)(p−1) β

+ γδj1−p lj Z ×

Lj ∩K(r)

r−1

α

p−1

v (x)[g2 (x)u

(1+λ)(p−1) p−1−λ . (x) + f2 (x)]dx

(5.3.27)

270

Chapter 5 Quasilinear elliptic equations with coefficients from the Kato class

By the choice of α, β, and λ and inequality (5.3.8), we get     Z v β (x) − lj (1+λ)(p−1) v(x) α−(β−1)(p−1) −p 1+ lim r dx = 0, (5.3.28) 1/β r→0 δj Lj ∩K(r) lj Z α−(β−1)(p−1)   β r−1 v α (x) g2 (x)up−1 + f2 (x) dx = 0. (5.3.29) lim δj1−p lj r→0

Lj ∩K(r)

By analogy with Lemma 5.2.5, we obtain the following relation from (5.3.27)– (5.3.29): 1   Z p−1+α+βλ(p−1) −n p−1+α+βλ(p−1) max v(x) 6 γ R . (5.3.30) (x)dx v B(x0 , R ) 2

B(x0 ,R)

The right-hand side of inequality (5.3.30) is finite by virtue of (5.3.8) and the choice of λ, α, and β. This completes the proof of Theorem 5.3.2.

5.3.3

Proof of Theorem 5.3.1

Lemma 5.3.3. Assume that all conditions of Theorem 5.3.1 be satisfied. Then p Z ∂u p ψr (x)dx 6 γ {M p (r)rn−p + 1}, (5.3.31) ∂x E(R)

where  E(R) = x ∈ B(R)\{0} : u(x) > M (R) and the function ψr (x) is defined in Sec. 5.3.2. This lemma is proved similarly to Lemma 3.2.1. In what follows, we assume that r and ρ satisfy the conditions 
0 < r < p < R < R0 , M (ρ) > max 2M (R), M R/2 .

(5.3.32)

Lemma 5.3.4. Assume that all conditions of Theorem 5.3.1 are satisfied. Then p Z ∂u p ψr (x)dx 6 γ [M (ρ) − M (R)] {M p−1 (r)rn−p + v0 } + γ, (5.3.33) ∂x E(ρ,R)

where E(ρ, R) = {x ∈ Ω : 0 < uR (x) < M (ρ) − M (R)} and v0 is defined in (5.2.13). Proof. Substituting the function  ϕ(x) = min uR (x), M (ρ) − M (R) ψrp (x)

Section 5.3

271

Removability of isolated singularities

in the integral identity (5.3.5), we obtain p Z ∂u p ψr (x)dx E(ρ,R) ∂x 6 γ[M (ρ) − M (R)] {M p−1 (r)rn−p + v0p−1 } Z g3 (x)up−1 (x)ψrp (x)dx + γ. + γ[M (ρ) − M (R)]

(5.3.34)

E(ρ)

By virtue of (5.3.7) and (5.2.13), we get Z Z g3 (x)up−1 (x)ψrp (x)dx 6 γ g3 (x)|x|p−n ψrp (x)dx E(ρ)

E(ρ)

6γ

∞ X i=0

ρp−n i

Z g3 (x)dx 6 γv0 ,

(5.3.35)

K(ρi )

where ρi = 2−i ρ and K(ρi ) = {ρi < |x| < ρi−1 }. By using this result and inequality (5.3.34), we arrive at the required inequality (5.3.33).  By analogy with (5.3.30), we get the estimate Z (upR (x) + 1)dx, [M (ρi ) − M (R)]p 6 γi ρ−n

(5.3.36)

Gi

where 

 ρ ρ (1) −i −i Gi = ρi − (1 − 2 ) < |x| < ρi + (1 − 2 ) , 2 2 ρ (1) ρi = (1 + 2−i ), ρi = |xi |, i = 1, 2, . . . , 2 and xi is a point such that ρi 6 |xi | 6 R0 /2,

|u(xi )| = M (ρi ).

In view of (5.3.33) and inequality (5.3.36), we arrive at the following inequality: (M (ρi )−M (R))p    n−p r p−n p−1 + v0 ρ + γ i ρp−n . 6 γ [M (ρi+1 )−M (R)] M (r) ρ i

Iterating the last inequality, we obtain   n−p 1 r p−1 a− n−p − n−p M (ρ) 6 γM (r) + γρ p−1 + γv0p−1 ρ p−1 , ρ n−p . a= p(p − 1)

(5.3.37)

(5.3.38)

272

Chapter 5 Quasilinear elliptic equations with coefficients from the Kato class

Let r → 0. Thus, by virtue of (5.3.7), we find 1

a− n−p p−1

+ γv0p−1 ρ

M (ρ) 6 γρ

− n−p p−1

(5.3.39)

for any 0 < ρ < R1 . It is possible to assume that ρa 6 v 0

(5.3.40)

because, otherwise, by virtue of Theorem 5.3.2, the function u(x) is bounded in Ω. Thus, we get the estimate M (ρ) 6 γv0 ρ

− n−p p−1

.

(5.3.41)

Repeating the previous reasoning, we arrive at the estimate a− n−p p−1

M (ρ) 6 γρ

+ γ I v0I ρ

− n−p p−1

on the I th step. After this, if we choose I from the condition   a log ρ−1 I= + 1, log (γv0 )−1

(5.3.42)

(5.3.43)

then we conclude that a− n−p p−1

M (ρ) 6 γρ

,

0 < ρ < R1 .

(5.3.44)

By virtue of Theorem 6.2.2, this estimate yields the property of boundedness of the solution u(x) in Ω. The remaining reasoning in the proof of Theorem 5.3.1 is similar to the reasoning presented in [113].

5.4

5.4.1

Removability of isolated singularities for the solutions of quasilinear elliptic equations with absorption Formulation of assumptions and main results

Consider the following equation: n X d ∂u
− ai x, u, + g(x, u) = a0 (x, u), dxi ∂x

x ∈ Ω\{0}.

(5.4.1)

i=1

Assume that the functions a0 (x, u), g(x, u), ai (x, u, ξ), i = 1, . . . , n, satisfy the Carathéodory condition, the functions a0 (x, u), ai (x, u, ξ), i = 1, . . . , n, satisfy inequalities (5.2.2), (5.2.3), (5.3.2), and, in addition, |g(x, u)| 6 c2 |u|q + f4 (x),

g(x, u)sign u > c1 |u|q − f4 (x),

where f4 (x) > 0, f4 (x) ∈ Kp for p > 2, f4 (x) ∈ K2,p for 1 < p < 2.

(5.4.2)

Section 5.4

Removability of isolated singularities for the solutions

273

Definition 5.4.1. We say that u(x) is a solution of Eq. (5.4.1) in Ω\{0} if, for any function ϕ(x) ∈ W 1,p (Ω) equal to zero near the boundary of the set Ω\{0}, we have u(x)ϕ(x) ∈ W 1,p (Ω) and the following identity holds:      Z X n ∂u ∂ϕ ∂u ai x, u, (5.4.3) − a0 x, u, ϕ + g(x, u)ϕ dx = 0. ∂x ∂xi ∂x Ω i=1

Definition 5.4.2. We say that a solution u(x) of Eq. (5.4.1) has a removable singularity at the point {0} if u(x) ∈ W 1,p (Ω) ∩ Lq+1 (Ω) and identity (5.4.3) ◦

is true for any function ϕ(x) ∈ W 1,p (Ω) ∩ Lq (Ω). The following theorem is the main result of the present section: Theorem 5.4.1. Let inequalities (5.2.2), (5.2.3), (5.3.2), and (5.4.2) be satisfied and let u(x) be a solution of Eq. (5.4.1) in Ω\{0}. Suppose that the inequality q>

n(p − 1) n−p

(5.4.4)

holds. Then the singularity of u(x) at the point {0} is removable. First, we prove the following statement: Theorem 5.4.2. Assume that the conditions of Theorem 5.4.1 are satisfied. Then the following estimate is true: |u(x)| 6 γ|x|−p/(q−p+1) ,

5.4.2

0 < |x| 6 R0 .

(5.4.5)

Proof of Theorem 5.4.2

We assume that lim M (r) = ∞

r→0

(5.4.6)

because, otherwise, Theorem 5.4.1 follows from the results obtained in the previous section. We fix a number R1 ∈ (0, R0 ) such that M (R1 ) > 1.

(5.4.7)

For ρ > 0 and σ ∈ (0, 1), we fix a function ϕρ,σ (x) with the following properties: ϕρ,σ (x) = 1 for ρ 6 |x| 6 2ρ, ϕρ,σ (x) = 0 outside the set {(1 − σ)ρ < |x| < (2 + σ)ρ},

274

Chapter 5 Quasilinear elliptic equations with coefficients from the Kato class

and the inequality ∂ϕρ,σ (x)/∂x 6 2/σρ is true. Lemma 5.4.1. Assume that the conditions of Theorem 5.4.2 are satisfied. Then the estimate  Z  p ∂u + |u|q+1 ϕpρ,σ (x)dx 6 γρn−p M p (ρ − σρ) (5.4.8) ∂x Ω is true for 0 < σ < 1 and 0 < ρ < R1 /3. Proof. We substitute the test function ϕ(x) = u(x)ϕpρ,σ (x) in the integral identity (5.4.3). By using inequalities (5.2.2)–(5.2.3), (5.3.2), and (5.4.2) and the Young inequality, we obtain  Z  p ∂u + |u|q+1 ϕpρ,σ (x)dx ∂x Ω p Z Z p p p ∂ϕρ,σ dx 6γ |u(x)| H(x)ϕρ,σ (x)dx + |u(x)| ∂x Ω Ω Z + γ [f1 (x) + f3 (x) + f4 (x)]ϕpρ,σ (x)dx, (5.4.9) Ω

where H(x) = 1 + g1 (x) + [g2 (x)]p/(p−1) + [f2 (x)]p/(p−1) + g3 (x) + f3 (x) + f4 (x) + hp (x). To estimate the integrals on the right-hand side of (5.4.9), we use the Hölder inequality and take into account the properties of the function ϕρ,σ (x) and the definition of the function M (r). This yields p Z p Z Z ∂u p 1 p p p ∂ϕρ,σ ϕ (x)dx + γ |u(x)| |u(x)| H(x)ϕρ,σ (x)dx 6 ρ,σ ∂x dx 8 Ω ∂x Ω Ω p Z p ∂ϕρ,σ dx 6 γσ −p ρn−p M p (ρ − σρ), |u(x)| ∂x Ω Z [f1 (x) + f3 (x) + f4 (x)] ϕpρ,σ (x)dx 6 γρn . Ω

Estimate (5.4.9) and the last three inequalities imply estimate (5.4.8).



Section 5.4

Removability of isolated singularities for the solutions

275



We fix a number R2 from the interval 0, R1 /2 for which M (R2 ) > M R1 /2 . Let ρ be an arbitrary number from the interval (0, R2 ). We introduce numerical sequences {ρj } and {σj } by the equalities ρj = (2−1 + 2−j )ρ,

σj = 2−(j+1) ,

j = 1, 2, . . . .

We choose a point xj such that M (ρj ) = |u(xj )|, ρj 6 |xj | 6 R2 , and denote ρ0j = |xj |. We also define ϕj (x) = ϕρ0j ,σj (x). By analogy with Lemma 5.2.5, we arrive at the estimate Z p+1 p+1 jp/σ −n [M (ρj )] (1 + |u(x)|)q+1 ϕpj (x)dx. = |u(xj )| 6 γ2 ρ (5.4.10) Ω

Further, we estimate the integral in (5.4.10) by using inequality (5.4.8). As a result, we obtain [M (ρj )]q+1 6 γρ−p aj [M (ρ0j − σj ρ0j )]p 6 γaj ρ−p [M (ρj+1 )]p ,

(5.4.11)

where a = 4p . Successively applying estimate (5.4.11), we arrive at the inequality
j j
j
j j j p 1 P p p
j+1 1 P 1 P (j+1) q+1 q+1 q+1 p q+1 q+1 q+1 j=0 a j=0 M (ρj ) 6 C8 . ρ j=0 [M (ρj+1 )] q+1 (5.4.12) Passing to the limit as j → ∞ and using the property of boundedness of the sequence M (ρ) and inequality (5.4.12), we get p

M (ρ) 6 γρ q−p+1 . This completes the proof of Theorem 5.4.2.

5.4.3

Integral and pointwise estimates for the gradient of the solution

For r ∈ (0, 1), we define a function ψr : RN → R by the equality ψr (x) = ψ˜ r (|x|), where ψ˜ r : R → R is a function given by the conditions ψ˜ r (t) ≡ 0

for t 6 r,

ψ˜ r (t) ≡ 1 ψ˜ r (t) =

1 (1 − θ) ln ln 1r

for t > R(r), Z r

t

dt t ln 1t

for r 6 t 6 R(r),

(5.4.13)

θ is a number from the interval (0, 1), and R(r) is a number specified by the equality   1 θ 1 = ln . (5.4.14) ln R(r) r

276

Chapter 5 Quasilinear elliptic equations with coefficients from the Kato class

For ρ ∈ (0, R0 ), we define functions uρ (x), F1 (r, ρ), and F2 (r, ρ) and a set E(ρ) by the following equalities: uρ (x) = [u(x) − M (ρ)]+ , E(ρ) = {x ∈ B(ρ)\{x0 )} : u(x) > M (ρ)},   qp p n−p p= , λ= − (p − 1) > 0, q−p+1 q−p+1 p−1   ρn−p for q > n(p−1)  n−p ,   h i2− n   p   ln 1r for q = n(p−1) n−p , n < 2p, F1 (r, ρ) =   for q = n(p−1) ln ln 1r  n−p , n = 2p,   n h i  2− p   ln 1 for q = n(p−1) ρ np , n > 2p.   p −1) ρλ [g(ρ) + f (ρ) + h(ρ)]p−1 ln 1 for q > n(P ρ n−p , F2 (r, ρ) = ln 1 [1 + g(ρ) + f (ρ)]p−1 for q = n(p−1) r n−p .

(5.4.15) (5.4.16)

(5.4.17)

(5.4.18)

Lemma 5.4.2. Assume that the conditions of Theorem 5.4.1 are satisfied. Then the estimate    Z 1 ∂u p p 1 p ln ln (5.4.19) F1 (r, ρ) + F2 (r, ρ) ψr (x)dx 6 γ r E(ρ) u(x) ∂x is true for r ∈ (0, 1) and R(r) < ρ < R2 . Proof. We substitute the test function ϕ(x) = ln+

u(x) p ψ (x) M (ρ) r

in the integral identity (5.4.3). After evident transformations, we get   Z u 1 ∂u p q + u ln ψrp (x)dx 6 γ(I1 + I2 ), (5.4.20) ∂x u(x) M (ρ) E(ρ) where p−1 ∂u ∂ψr p−1 u · I1 = ln ∂x ψr (x)dx, M (ρ) ∂x E(ρ)  Z ∂ψr u I2 = up−1 (x) f1 (x) + g1 (x) + (f2 (x) + g2 (x)) ln · M (ρ) ∂x E(ρ)  u ψrp−1 (x)dx. + (f3 (x) + g3 (x) + f4 (x)) ln M (ρ) Z

Section 5.4

Removability of isolated singularities for the solutions

By the Young inequality, we conclude that  p−1  Z 1 u 1 ∂u q γI1 6 + u ln ψrp−1 (x)dx + γI3 , 2 E(ρ) u ∂x M (ρ) where 

Z I3 = E(ρ)

u ln M (ρ)

277

(5.4.21)

(1−(p−1)/qp)p ∂ψr p ∂x dx.

By using Theorem 5.4.2, we find   1 −p I3 6 γ ln ln F1 (r, ρ). r

(5.4.22)

Further, we get ∞

I2 6 γ ln

1X r i=1



ρ 2i

p−n Z H(x)dx 6 γF2 (r, ρ)

(5.4.23)

K(ρi )

for q =

n(p − 1) n−p

and I2 6 γ

∞  p−n+λZ X ρ i=1

2i

  1 f1 (x) + g1 (x) + (f3 (x) + g3 (x) + f4 (x)) ln |x| K(ρi )

6 γF2 (r, ρ)

for q >

n(p − 1) . n−p

ρ . 2i Combining estimates (5.4.20)–(5.4.24), we conclude that (5.4.19).

(5.4.24)

Here, K(ρi ) = {ρi < |x| < ρi−1 }, ρi =



We now define a function u(ρ) (x) and a set E(ρ, 4ρ) as follows: u(ρ) (x) = min{[u(x) − M (4ρ)]+ , M (ρ) − M (4ρ)}, E(ρ, 4ρ) = {x ∈ BR0 (x0 ) : M (4ρ) < u(x) < M (ρ)}. Lemma 5.4.3. Assume that the conditions of Theorem 5.4.1 are satisfied. Then the following estimate is true: p Z ∂u dx 6 v0 (M (ρ) − M (4ρ))ρλ , (5.4.25) E(ρ,4ρ) ∂x where v0 is defined in (5.4.13). Proof. We substitute the function ϕ(x) = u(ρ) (x)ψrp (x) for r ∈ (0, 1),

R(r) < ρ < R1 ,

278

Chapter 5 Quasilinear elliptic equations with coefficients from the Kato class

in the integral identity (5.4.3). By using inequalities (5.2.2)–(5.2.3), (5.3.2), and (5.4.2), we obtain p Z Z 6 X ∂u p q (ρ) p ψr (x)dx + u (x)u (x)ψ (x)dx 6 γ Ii , (5.4.26) r E(4ρ) E(ρ,4ρ) ∂x i=4

where Z

up−1 (x)H(x)ψrp (x)dx,

I4 =(M (ρ) − M (4ρ))

(5.4.27)

E(4ρ) p−1

∂ψr p−1 I5 = u ∂x ψr (x)dx, E(4ρ) Z ∂ψr p−1 (ρ) p−1 ψ (x)dx. I6 = u (x)u (x) [f2 (x) + g2 (x)] ∂x r E(4ρ) Z

(ρ)

∂u (x) ∂x

(5.4.28) (5.4.29)

We have I4 6 γ(M (ρ) − M (4ρ))

∞  p−n+λZ X ρ i=1 λ

2i

H(x)dx

K(ρi )

6 γv0 (M (ρ) − M (ρ))ρ Z 1 u(ρ) (x)uq (x)ψrp (x)dx + γI7 I8 , γI5 6 2 E(4ρ) where

(5.4.30) (5.4.31)

 q(p−1) qp−p+1 1 ∂u p p ψr (x)dx , E(4ρ) u ∂x p  q−p+1 Z
qp−p+1 1+ p−1 p ∂ψr (ρ) pq dx I8 = [u (x)] . ∂x E(4ρ)

Z I7 =

Further, in view of the form of the function ψr (x), we obtain  qp  Z R(r)  q−p+1  |x|n−1 d|x| qp−p+1 1 qp−p+1 I8 6 γ(M (ρ) − M (4ρ)) ln ln  1 p r r ln |x|p |x|

6 γ(M (ρ) − M (4ρ))F3 (r), where

(5.4.32)

 q−p+1 (n−p) qp−p+1  [R(r)] F3 (r) = h i−p q−p+1 qp−p+1   ln 1 R(r)

if q >

n(p−1) n−p ,

if q =

n(p−1) n−p ,

and R(r) is given by equality (5.4.15). Moreover, we get Z  p−1 p p p−1 p−1 I6 6 γ[M (ρ) − M (4ρ)] u (x)(f2 (x) + g2 (x)) dx r 0} ∀ t ∈ [0, T ).

(6.1.4)

This function is continuous and monotonically increasing (see [53], [103], [143]). The boundary condition (6.1.3) specifies the so-called boundary peaking regime with peaking time t = T. The presence of this regime explains the necessity of investigation of the limit behavior of the solution u(t, x) and the free boundary ζ(t) as t → T. The investigations in this direction were originated

284

Chapter 6 Energy methods for the investigation of localized regimes

in the early 1960s (see [110], [84], [86], [109], [137], and the references therein), in particular, in connection with the study of the conditions of localization of boundary regimes with peaking and with the description of the corresponding sets of intensification. Definition 6.1.1. A boundary regime ψ(t) in problem (6.1.1)–(6.1.3) is called localized if the corresponding solution u(t, x) satisfies the condition lim ζu (t) := lim ζ(t) < ∞,

t→T

t→T

ζ(t) is defined in (6.1.4),

(6.1.5)

for any continuous bounded nonnegative initial function u0 with compact support supp u0 . Definition 6.1.2. For any solution u(t, x) of problem (6.1.1)–(6.1.2), the sets of intensification (or “blow-up” sets) are defined as follows: Ωs = Ωs (u) = {x : lim sup u(t, x) = ∞}, t→T Z ∞ ∗ ∗ u(t, y) dy = ∞}. Ωs = Ωs (u) = {x : lim sup t→T

(6.1.6) (6.1.7)

x

By using the notion of localization proposed in [141], [109], [50], we also introduce the following definition: Definition 6.1.3. We say that the solution u(t, x) of a problem with boundary peaking regime at t = T is effectively localized if ω := sup{x ∈ Ωs (u)} < ∞. We also say that this solution is metastable localized if ω ∗ := sup {x ∈ Ω∗s (u)} < ∞, where Ωs (u) and Ω∗s (u) are introduced in Definition 6.1.2. The first general exact results on the localization of solutions of problem (6.1.1)–(6.1.3) were obtained by Galaktionov and Samarskii [42]. For the class of singularly peaking boundary regimes ψ(t) such that ψ ∈ C 2 (0, T ),

ψ 0 (t) > 0 ∀ t ∈ (0, T ),

(6.1.8)

and the limit lim (ψ/ψ 0 )0 (t) := l

t→T

(6.1.9)

exists, it was shown that a necessary and sufficient condition of localization has the following form: l = −∞ or ψ m /ψ 0 ∈ L∞ (0, T ).

(6.1.10)

Section 6.1 Introduction: localized and nonlocalized singular boundary regimes 285

Moreover, the size of the domain of intensification satisfies the equality  0 for l = −∞, 1 ω= (6.1.11) C lim sup(ψ m (t)/ψ 0 (t)) 2 for l > −∞, t→T

where C is a positive constant that depends only on l. In particular, we have 1

C = (2m(m + 1)) 2 /(m − 1) for l = −(m − 1). This result was established in [137] for the model boundary regime 1

ψ(t) = ψ0 (T − t)− m−1

∀ t < T,

ψ0 = const > 0,

(6.1.12)

which can be regarded as the limiting case from the viewpoint of validity of condition (6.1.10). In [50], Gilding and Herrero showed that, for any continuous monotone regime with peaking ψ(t), a necessary and sufficient condition of localization can be formulated in the form of a single inequality as follows:   12 Z t −1 m ρ := lim sup ψ(t) · ψ (s) ds < ∞. (6.1.13) t→T

0

Moreover, in [50] it was also shown that (1) in the presence of localization, 1

lim sup(T − t) m−1 u(t, x) < ∞ ∀ x > 0, t→T

(2) if the absence of localization, 1

lim sup(T − t) m−1 u(t, x) = ∞ ∀ x > 0, (3) for any x ∈ Ωs (u), lim inf u(t, x) = ∞ t→T

and Ωs (u) is a connected interval that contains the point x = 0, (4) the size ω of the domain of intensification Ωs satisfies the estimates C1 ρ 6 ω 6 C2 ρ, where ρ is defined in (6.1.13), 1  (m + 1)(2m + 1) 2 C1 = , m2

C2 =

(6.1.14)

2m

2m+1 m

1

(m + 1) m (m − 1)2

!1

2

,

(5) in the class of “smooth” peaking regimes specified by conditions (6.1.8) and (6.1.9), the localization criteria (6.1.10) and (6.1.13) are equivalent.

286

Chapter 6 Energy methods for the investigation of localized regimes

In [34], for a class of continuous singular boundary regimes ψ(t) wider than the class considered in [50] since the monotonicity condition is not assumed, Cortázar and Elgueta established the following criterion of localization: Z t 1 ∗ m−1 ψ m (s) ds < ∞ (6.1.15) ρ := lim sup(T − t) t→T

0

equivalent to (6.1.13) on the subclass of monotone regimes. Moreover, in [34], with the same level of completeness as for the Dirichlet problem, the problem of localization of solutions was studied for the Neumann problem for Eq. (6.1.1) with singularly peaking boundary data (um )x (t, 0) = −h(t),

h(t) → ∞ as t → T

(6.1.16)

In particular, it was shown that the following condition plays the role of a criterion of localization in this regime: Z t 1 h(s) ds < ∞. σ ∗ := lim sup(T − t) m−1 (6.1.17) t→T

0

We now consider the dependence of the characteristics of localization of boundary peaking regimes on the structure of the equation. First, we consider Eq. (6.1.1) with m = 1. In this case, problem (6.1.1)–(6.1.3) is explicitly solvable. For u0 = 0, we have   Z t x x2 − 32 ψ(s) ds. (6.1.18) u(t, x) = (t − s) · exp − 1 4(t − s) 2π 2 0 It is clear that this solution has no thermal front. Hence, the localization in a sense of Definition 6.1.1 is absent for any regime ψ(t). Nevertheless, as shown in [141], a metastable localization is possible (see Definition 6.1.3). If ψ(t) = (T − t)α exp(γ(T − t)−β ) ∀ t < T, α > 0, γ > 0, β > 0,

(6.1.19)

then the metastable localization with ω = 0 takes place for β < 1 (LS -regime in terms of [109]). For β = 1, we observe the metastable localization with 1 1 ω = 2γ 2 (S -regime), where u(t, x) → ∞ for all x < 2γ 2 as t → T and 1 u(t, 2γ 2 ) → ∞ as t → T for α 6 2−1 . For β > 1, we have u(t, x) → ∞ as t → T for all x > 0 (HS –regime). It is quite natural to expect that the localization of boundary peaking regimes occurs for a broad class of equations whose structure differs from the structure of Eq. (6.1.1) or is more general (see [109]). Thus, in [109], for the nonlinear heat-conduction equation with gradient nonlinearity ut = (|ux |p−1 ux )x ,

p > 1,

(6.1.20)

Section 6.1 Introduction: localized and nonlocalized singular boundary regimes 287

it was shown that there exist self-similar solutions corresponding to the Dirichlet boundary condition (6.1.3) with ψ(t) = ψ0 (T − t)−l ,

l > 0,

ψ0 > 0.

(6.1.21)

The structure of this solution enables us to conclude that the limiting localized boundary regime (S -regime) corresponds to the value l = (p − 1)−1 .

(6.1.22)

In [49], the problem of localization of solutions was studied for the axially symmetric Dirichlet and Neumann problems with singularly peaking boundary data in the case of n-dimensional nonlinear heat-conduction equation, i.e., in fact, for the following “one-dimensional” equation with variable coefficient: ut = r1−n (rn−1 (um )r )r ,

u = u(t, r), r ∈ (1, ∞), n ∈ N.

(6.1.23)

It was shown that the effective and metastable localizations are equivalent for this equation. Moreover, the criteria of localization of boundary peaking regimes similar to (6.1.15) and (6.1.17) were established and the optimal upper and lower bounds were determined for the domain of intensification. We now briefly dwell upon the existing methods aimed at the investigation of localization of the boundary regimes with peaking. Starting from the appearance of the work [110] till the end of the 1990s, all investigations in this field were based on various versions of the barrier technique, the comparison of the obtained solutions with different special self-similar solutions, and finding the so-called approximate self-similar solutions (see [42], [109], [49]). An especially important role was played by the following two classes of solutions: (1) The solutions of Eq. (6.1.1) proposed by Kalashnikov [54]:  1 2 2  Aa m−1 (τ −t)− m−1 (1− xa ) m−1 ∀ x < a, ∀ t < τ,  1  U (t, x, τ, a) = m−1  m−1 0 ∀ x > a, ∀ t < τ ; A := , 2m(m+1)

(6.1.24)

where a > 0 and τ 6 T are arbitrary parameters; these solutions illustrate the effect of waiting time in the propagation of supports (analogs of these solutions for Eq. (6.1.23) were constructed in [49]). (2) The so-called Barenblatt (or Barenblatt–Pattle) solutions (see [135], [145], [10], [11], [17]); for the first time, these solutions were obtained by Zel’dovich and Kompaneets in [139] for the special cases of Eq.(6.1.23) with n = 1 and n = 3. An absolutely new method was proposed in [116], [118] without using the barrier technique. This method made it possible to study the localization of boundary regimes with singular peaking for general quasilinear divergent parabolic

288

Chapter 6 Energy methods for the investigation of localized regimes

equations of the second order in arbitrary multidimensional domains. The proposed method combines the ideas of the method of local energy estimates (see [39], [6]), the method of a priori estimates similar to the Saint-Venant principle in the investigation of the asymptotic properties and uniqueness of the generalized energy solutions of linear elliptic and parabolic equations (see [127], [61], [102], [104], [142]), and a nonlinear version of the method of estimates similar to the Saint-Venant principle (see [115], [2]). It consists in the effective estimation of the energy flows connected with the analyzed solution in an infinite family of subdomains (strips) accumulated in the vicinity of peaking time in the boundary regime. In the present chapter (Sections 6.2 and 6.3), we give the proofs of sufficient conditions for the localization of Dirichlet boundary peaking regimes for general quasilinear parabolic diffusion equations of slow Newton–non-Newton type (heat conduction) [a model example is given by Eq. (6.1.1) with m > 1]. These proofs are improved versions of the proofs presented in [116] and [118]. In Sec. 6.4, these methods are adapted for the general equations of the type of “neutral” (constant) diffusion (heat conduction) [a model example is given by Eq. (6.1.1) with m = 10]. In Sec. 6.5, the characteristics of localization are established for equations of the type of “fast” diffusion [Eq. (6.1.1) with m < 1]. Let Ω be a bounded domain in Rn , n > 1, with C 1 -smooth boundary ∂Ω = ∂0 Ω ∪ ∂1 Ω, where ∂0 Ω = {x : |x| = 1}, ∂1 Ω ⊂ {x ∈ Rn : |x| > R1 },

R1 = const > 1.

(6.1.25)

In a cylindrical domain Q = Q(T ) = (0, T ) × Ω, 0 < T < ∞, we consider the initial boundary-value problem (|u|q−1 u)t −

n X

(ai (t, x, u, Dx u))xi = 0 in Q, q = const > 0;

(6.1.26)

i=1

u(t, x) = f (t, x) on (0, T ) × ∂0 Ω; n X ∂u := ai (t, . . . , Dx u)νi = 0 on (0, T ) × ∂1 Ω; ∂N

(6.1.27)

u(0, x) = u0 ∈ Lq+1 (Ω).

(6.1.29)

(6.1.28)

i=1

Here, ν = (ν1 , . . . , νn ) = ν(x) is the unit vector of outer normal to ∂1 Ω at the point x and ai (t, x, s, ξ) are continuous functions of all arguments satisfying the following conditions of coercivity and growth: d0 |ξ|p+1 6

n X i=1

ai (t, x, s, ξ)ξi ∀ (t, x, s, ξ) ∈ Q × R1 × Rn , p = const > 0,

(6.1.30)

Section 6.1 Introduction: localized and nonlocalized singular boundary regimes 289

ai (t, . . . , ξ) 6 d1 |ξ|p

∀ (t, . . . , ξ) ∈ Q × R1 × Rn , i = 1, . . . , n,

(6.1.31)

where the constants d0 > 0 and d1 < ∞ are independent of (t, x, s, ξ). We assume that the function f (t, x) specifying the boundary regime (6.1.27) is the trace of the function f¯(t, x) on (0, T ) × ∂0 Ω and f¯(t, x) satisfies the following conditions of smoothness: 1 f¯(t, ·) ∈ C loc ([0, T ); Lq+1 (Ω)) ∩ Lp+1, loc ([0, T ); Wp+1 (Ω));

f¯t (t, ·) ∈ L1,loc ([0, T ), Lq+1 (Ω)) ∩ L

p+1 ,loc p−q+1

([0, T ), L

p+1 p−q+1

(Ω)).

(6.1.32) (6.1.33)

1 (Ω, S) we denote the closure (in the norm of the Sobolev As usual, by Wm 1 space Wm (Ω)) of the set of functions from C ∞ (Ω) vanishing in the vicinity of the set S ⊂ ∂Ω. Moreover, let h·, ·i denote the operation of pairing of elements 1 (Ω, ∂ Ω))∗ and W 1 (Ω, ∂ Ω). By using the function of the spaces (Wp+1 0 0 p+1 Z Z tZ F (t) := sup |f¯(τ, x)|q+1 dx + |Dx f¯(τ, x)|p+1 dx dτ 0 0 is independent of (t, x, s, ξ, η), there exists a generalized (in a sense of Definition 6.1.4) solution uj (t, x) of problem (6.1.26)–(6.1.31) in the domain Qj := (0, Tj ) × Ω with arbitrary Tj < T. Note that, for the functions ai ( . . . ) in (6.1.30) and (6.1.31), condition (M) may be satisfied only for p > 1. Remark 6.1.2. The domain Ω in problem (6.1.26)–(6.1.29) was chosen as a domain with model “internal” boundary ∂0 Ω = {x ∈ Rn : |x| = 1} solely to simplify the notation and subsequent constructions. In what follows, we show that the original domain Ω can be chosen in the form of any domain with C 1 -smooth “internal” boundary ∂0 Ω, which is the boundary of some convex bounded domain in Rn . Remark 6.1.3. Nonlinear heat-conduction equations (6.1.1), (6.1.20), and (6.1.23) presented above are special cases of the general equation (6.1.26). For instance, Eq. (6.1.1) rewritten in terms of the new unknown function v := |u|m−1 u corresponds to Eq. (6.1.26) with p = 1, q = m−1 , ai (t, x, s, ξ) = ξi , and n = 1. It is well known (see, e.g., [6], [39]) that, in the case of slow diffusion, i.e., for 0 < q < p,

(6.1.36)

Eq. (6.1.26) has the property of finite propagation speed of the support of any energy solution u(t, x). In particular, if supp u0 ⊂ B0 (R) := {x : |x| < R},

1 < R,

Ω \ B0 (R) 6= ∅,

(6.1.37)

then there exists a continuous nondecreasing function ζ(t), ζ(0) = R, such that supp u(t, ·) ⊂ B0 (ζ(t)) ∀ t < T.

(6.1.38)

Hence, it is natural to introduce a definition similar to Definition 6.1.1.

Section 6.2

Sufficient conditions for the localization of boundary regimes

291

Definition 6.1.5. A solution u of problem (6.1.26)–(6.1.29),(6.1.36),(6.1.37) [or the corresponding boundary regime f (or F )] is localized if there exists ¯ 6= ∅ and ζu (t) := ζ(t) < R¯ for all t < T. R¯ > R such that Ω \ B0 (R) We now present the definition of the set of intensification of a solution u of problem (6.1.26)–(6.1.29), (6.1.30)–(6.1.34) that can be used with the integral characteristics of the behavior of solutions in the vicinity of the peaking time t = T . Definition 6.1.6. The set of intensification (blow-up set) Ωs (u) of a generalized solution u of problem (6.1.26)–(6.1.29) is defined as the set of points x ∈ Ω such that Z sup |u(t, y)|q+1 dy t 0, A is specified in (6.1.24). (6.2.5) A Here, U (t, x, τ, a) is a solution given by (6.1.24). According to (6.1.11)–(6.1.14), the size ω of the domain of intensification Ωs (u) satisfies the estimate m−1 2

ω 6 C2 (m − 1)1/2 · ψ0

.

(6.2.6)

Note that the new unknown function v(t, x) = problem (6.1.26), (6.1.27), (6.1.29) in the case where q = m−1 ,

p = 1,

n = 1,

u(t, x)m

Ω = (0, ∞),

is a solution of

u0 = 0,

(6.2.7)

¯ x)m , f (t, x) = ψ(t,

and, moreover, the corresponding function F (t) defined in (6.1.34) can be represented as follows: F (t) = F1 (t) + F2 (t) + F3 (t) Z Z tZ m+1 ¯ := ψ(t, x) dx + (ψ¯ m (τ, x))2x dx dτ 0

Ω

Z t Z + 0

Ω

Ω

m+1 |(ψ¯ m (τ, x))τ | m dx



 m+1

m m+1

m

dτ

.

Section 6.2

Sufficient conditions for the localization of boundary regimes

293

¯ x) specified by relation (6.2.5), By using the explicit form of the function ψ(t, after simple calculations, we obtain 3m+1 2

Fi (t) 6 ci ψ0

m+1

(T − t)− m−1

∀ t < T,

i = 1, 2, 3,

(6.2.8)

where ci are constants determined in the explicit form and depending only q+1 m+1 on m. Note that p−q = m−1 for p and q defined in (6.2.7). Hence, estimate 3m+1

(6.2.8) corresponds to estimate (6.2.1) if we set ω0 = cψ0 2 and c = c1 +c2 +c3 . Therefore, by virtue of (6.2.7), estimate (6.2.4) can be rewritten in the form m−1

(3m+1)(m−1) 2(3m+1)

R∗ 6 L2 c 3m+1 ψ0

m−1

m−1 2

= L2 c 3m+1 ψ0

,

which corresponds to the exact relation (6.2.6). Remark 6.2.2. If uj (t, x) is a sequence of solutions of problem (6.1.26)– (6.1.31) with nonpeaking boundary data in the domains (0, Tj ) × Ω, where Tj → T as j → ∞ (this sequence has already been mentioned in Remark 6.1.1), then Theorem 6.2.1 guarantees that the supports of these solutions are bounded uniformly in j : supp uj (t, ·) ⊂ {|x| 6 L1 hb01 + L2 w0b2 } ∀ t < T,

∀ j ∈ N.

Prior to proving Theorem 6.2.1, we establish a series of auxiliary statements. First, in the layers Q(τ ) := Q ∩ {0 < t < τ } ∀ τ < T, we deduce a global integral estimate for a solution u of problem (6.1.26)–(6.1.29), (6.1.30)–(6.1.34) in terms of the boundary regime function F (τ ) under the condition that p > 0 and q > 0 are arbitrary. Lemma 6.2.1. For any energy solution of problem (6.1.26)–(6.1.29), the following a priori integral estimate is true: Z Z τZ |u(τ, x)|q+1 dx + |Dx u|p+1 dx dt Ω 0 Ω Z 6 C1 |u0 |q+1 dx + C2 F (τ ) ∀ τ ∈ (0, T ), (6.2.9) Ω

where the constants C1 and C2 depend only on known parameters of problem (6.1.26)–(6.1.29) and do not depend on the solution u. Proof. Integrating by parts (see [3]), we obtain Z τ h(|u|q−1 u)t , u − f i dt 0  Z  q q+1 q+1 |u(τ, x)| = − |u(0, x)| dx q+1 Ω

294

Chapter 6 Energy methods for the investigation of localized regimes

 Z τZ  |u(t, x)|q−1 u(t, x) − |u(0, x)|q−1 u(0, x) ft0 (t, x) dx dt

+

0

Ω

 Z  |u(τ, x)|q−1 u(τ, x) − |u(0, x)|q−1 u(0, x) f (τ, x) dx. −

(6.2.10)

Ω

We now substitute u(t, x) − f (t, x) (as a test function) in integral identity (6.1.35) and take into account (6.2.10) and the structural conditions (6.1.30) and (6.1.31). This yields Z Z τZ q |u(τ, x)|q+1 dx + d0 |Dx u|p+1 dx dt q+1 Ω 0 Ω Z τZ Z q q+1 6 dx + d1 |Dx u|p |Dx f | dx dt |u0 | q+1 Ω 0 Ω Z + (|u(τ, x)|q + |u0 |q )|f (τ, x)| dx Ω

 Z τZ  q q + |u(t, x)| + |u0 (x)| |ft0 (t, x)| dx dt. 0

(6.2.11)

Ω

Further, let us estimate the terms on the right-hand side of (6.2.11) from above. By virtue of the Young inequality with ε, we get Z τZ |Dx u|p · |Dx f | dx dt 0

Ω

Z τZ

|Dx u|p+1 dx dt + c(ε)

6ε 0

Z τZ 0

Ω

|Dx f |p+1 dx dt ∀ ε > 0;

Ω

 Z  q q |u(τ, x)| + |u0 (x)| |f (τ, x)| dx Ω

Z |u(τ, x)|

6ε

q+1

Z |u0 |

dx +

Ω

Z τZ 0

q+1



Ω

|u0 (x)|q |ft (t, x)| dx dt ku0 kqLq+1 (Ω)

Z

τ

Z |ft (t, x)|

0

τ

0

Z τZ

q+1

 1 q+1 dx dt

Ω

Z 6 εku0 kq+1 + c(ε) Lq+1 (Ω)

Z

|ft (t, x)|q+1 dx Z

q

0 0;

Section 6.2

295

Sufficient conditions for the localization of boundary regimes

Z

|u(t, x)|

6 ε sup

0 s + δ, ∀ r : s < r < s + δ.

(6.2.16)

We now substitute the test function η(t, x) = u(t, x)ηs,δ (|x|) in the integral identity (6.1.35). First, we write this identity for T = b, then for T = a, and finally, we subtract the equalities obtained as a result. By using the formula of integration by parts [3], we arrive at the equality Z Z bZ n X q |u(b, x)|q+1 ηs,δ dx + ai (t, . . . , Dx u)uxi ηs,δ dx dt q + 1 Ω(s) a Ω(s) i=1 Z q = |u(a, x)|q+1 ηs,δ (|x|) dx q + 1 Ω(s) Z bZ n X − ai (. . . , Dx u)u · (ηs,δ (|x|))xi dx dt. a

Ω(s)\Ω(s+δ) i=1

Passing to the limit as δ → 0 (see [39]), we conclude that the integral Z bZ n X ai (. . .)uνi dσ dt, a

∂0 Ω(s) i=1

exists for almost every s and the inequality Z Z bZ n X q q+1 |u(b, x)| dx + ai (t, x, u, Dx u)uxi dx dt q + 1 Ω(s) a Ω(s) i=1 Z Z bZ n X q 6 |u(a, x)|q+1 dx + ai (. . .)uνi dσ dt q + 1 Ω(s) a ∂0 Ω(s)

(6.2.17)

i=1

is true for almost every s > 1. In view of the structural conditions (6.1.30) and (6.1.31) and the Hölder inequality, inequality (6.2.17) implies that Z Z Z d0 (q + 1) b q+1 |u(b, x)| dx + |Dx u|p+1 dx dt q Ω(s) a Ω(s)  Z bZ  p Z p+1 (q + 1)d1 p+1 q+1 |Dx u| dσ dt |u(a, x)| dx + 6 q a ∂0 Ω(s) Ω(s)  Z bZ  1 p+1 × |u|p+1 dσ dt . (6.2.18) a

∂0 Ω(s)

298

Chapter 6 Energy methods for the investigation of localized regimes

This inequality is true for any p > 0 and q > 0. It is used as a starting point of our subsequent presentation. By virtue of (6.1.36) and Proposition 9.1.1, we obtain Z |u(t, x)|p+1 dσ ∂0 Ω(s)

6

k1p+1

Z |Dx u(t, x)|

p+1

θ  Z

|u(t, x)|

dx

+ k2p+1

(1−θ)(p+1) q+1

dx

Ω(s)

Ω(s)

Z

q+1

|u(t, x)|q+1 dx

 p+1 q+1

.

(6.2.19)

Ω(s)

We integrate this inequality with respect to t, substitute the result in (6.2.18), and arrive at relation (6.2.15) with C3 = k1 d1 q −1 (q + 1) and C4 = k2 d1 q −1 (q + 1).



Remark 6.2.3. As a result of the standard analysis, we conclude that inequality (6.2.19) is true for k2 = 0 if ∂1 Ω = ∅ or supp u(t, ·) ∩ ∂1 Ω = ∅. Thus, inequality (6.2.15) is true for C4 = 0 provided that supp u(t, ·) ∩ ∂1 Ω = ∅ ∀ t ∈ (a, b). We now introduce a monotonically increasing sequence {tj }, j = 1, 2, . . . , where t0 = 0, tj → T as j → ∞, and ∆j := tj −tj−1 . This sequence is specified in what follows. Further, we define two families of energy functions connected with the analyzed solution u(t, x): Z tj Z |Dx u(t, x)|p+1 dx dt ∀ s > 1, j = 1, 2, . . . , Ej (s) = tj−1 Ω(s) Z hj (s) = sup |u(t, x)|q+1 dx ∀ s > 1, j = 1, 2, . . . , (6.2.20) t∈(tj−1 ,tj ] Ω(s)

Z h0 (s) =

|u0 (x)|q+1 dx.

Ω(s)

Lemma 6.2.3. Let u(t, x) be an arbitrary energy solution of problem (6.1.26)– (6.1.29), (6.1.30)–(6.1.34), (6.1.36). Then the corresponding energy functions from (6.2.20) satisfy the following system of differential inequalities: hj (s) + Ej (s) 6 C5 hj−1 (s) + C6 ∆νj 1 (−Ej0 (s))1+µ1 + C7 ∆νj 2 (−Ej0 (s))1+µ2 , (6.2.21) hj (s) 6 (1 + γ)hj−1 + C8 ∆νj 1 (−Ej0 (s))1+µ1 + C9 ∆ν2 (−Ej0 (s))1+µ2

∀ γ > 0, (6.2.22)

Section 6.2

Sufficient conditions for the localization of boundary regimes

where ν1 =

299

(1 − θ)(p − q) (1 − θ)(q + 1) < 1, µ1 = , q(p + 1) + θ(p − q) q(p + 1) + θ(p − q) p−q q+1 , µ2 = , ν2 = q(p + 1) q(p + 1)

C8 = C8 (γ) → ∞ and C9 = C9 (γ) → ∞ as γ → 0, and θ is defined in the statement of Lemma 6.2.2. Proof. In a standard way (as in [39]), we can easily show that the function Ej (s) is differentiable almost everywhere and that the following equality is true for almost every s: Z tj Z d − Ej (s) = |Dx u(t, x)|p+1 dσ dt ∀ j ∈ N. (6.2.23) ds tj−1 ∂0 Ω(s) Setting a = tj−1 and b = tj in inequality (6.2.15) and using (6.2.23), we obtain   Z tj  Z  p+1  1 q+1 p+1 p 0 q+1 p+1 Ej (s) 6 c0 (−Ej (s)) C4 |u| dx dt tj−1 θ p+1

Z

tj

Ω(s)

Z

|u| + C3 Ej (s) tj−1 Ω(s) Z + c0 |u(tj−1 , x)|q+1 dx

q+1

 p+1  1−θ  q+1 p+1 dx dt

Ω(s) 1−θ
p 1−θ θ 6 c0 hj−1 (s) + c0 C3 Ej (s) p+1 − Ej0 (s) p+1 hj (s) q+1 ∆jp+1 1
p 1 + c0 C4 − Ej0 (s) p+1 hj (s) q+1 ∆jp+1 ,

(6.2.24)

where c0 = q(q + 1)−1 d−1 0 . We set a = tj−1 and b = tδ ∈ (tj−1 , tj ) in inequality (6.2.15). Thus, Z (1 − δ)hj (s) = |u(tδ , x)|q+1 dx. Ω(s)

This yields (1 − δ)hj (s) + c−1 0 E j (s) θ

p

0

1−θ

1−θ

6 hj−1 (s) + C3 E j (s) p+1 (−E j (s)) p+1 hj (s) q+1 ∆jp+1 0

p

1

1

+ C4 (−E j (s)) p+1 hj (s) q+1 ∆jp+1 , Z tδ Z E j (s) := |Dx u(t, x)|p+1 dx dt. tj−1

Ω(s)

(6.2.25)

300

Chapter 6 Energy methods for the investigation of localized regimes

Consider the sum of inequalities (6.2.24) and (6.2.25) for sufficiently small δ 0 and note that E j 6 Ej and −E j (s) < −Ej0 (s). Hence, by using the Young inequality with ε, we obtain hj (s) + Ej (s) 6 (1 + c0 )hj−1 + ε(1 + c0 )C3 Ej (s) + ε(1 + c0 )C3 hj (s) + (1 + c0 )C3 c1 (ε)∆νj 1 (−Ej0 (s))1+µ1 + (1 + c0 )C4 εhj (s) + (1 + c0 )C4 c2 (ε)∆νj 2 (−Ej0 (s))1+µ2

∀ ε > 0,

(6.2.26)

where c1 (ε) → ∞ and c2 (ε) → ∞ as ε → 0. Setting ε = ε := 2−1 (1 + c0 )−1 (C3 +C4 )−1 in (6.2.26), we arrive at relation (6.2.21) with C5 = 2(1 + c0 ),

C6 = 2(1 + c0 )C3 c1 (ε),

C7 = 2(1 + c0 )C4 c2 (ε).

(6.2.27)

In order to deduce estimate (6.2.22), we apply the Young inequality with ε to the terms on the right-hand side of inequality (6.2.25): pt(1 − δ)hj (s) + c−1 0 E j (s) 0

6 hj−1 (s) + εC3 E j (s) + εC3 hj (s) + C3 c1 (ε)∆νj 1 (−E j (s))1+µ1 0

+ εC4 hj (s) + c2 (ε)C4 ∆νj 2 (−E j (s))1+µ2

∀ ε > 0.

Setting δ = ε, we get (1 − ε − εC3 − εC4 )hj (s) + (c−1 0 − εC3 )E j (s) 0

0

6 hj−1 (s) + C3 c1 (ε)∆νj 1 (−E j (s))1+µ1 + C4 c2 (ε)∆νj 2 (−E j (s))1+µ2 . Finally, setting ε = ε0 := γ(1 + γ)−1 (1 + C3 + C4 )−1 , we arrive at (6.2.22) with C8 = C3 c1 (ε0 ),

C9 = C4 c2 (ε0 ).

(6.2.28) 

The lemma is proved.

Proof of Theorem 6.2.1. To prove the theorem, we analyze the properties of solutions of the infinite system of differential inequalities (6.2.21), (6.2.22). This system contains infinitely many free parameters that could be controlled. First, we set certain values of the parameters ∆i , i = 1, 2, . . . . To this end, we fix a number ξ, 0 < ξ < 1, and define ∆1 = T (1 − ξ), P∞ It is clear that i=1 ∆i = T and T − tj =

∞ X i=j+1

∆i =

∆j = ξ∆j−1

ξ ∆j , 1−ξ

∀ j > 2.

j = 1, 2, . . . .

(6.2.29)

(6.2.30)

Section 6.2

301

Sufficient conditions for the localization of boundary regimes

We now introduce normalized energy functions q+1

q+1

Aj (s) := ∆jp−q Ej (s) and Hj (s) := ∆jp−q hj (s),

j = 1, 2, . . . .

(6.2.31)

It is easy to see that the parameters of system (6.2.21), (6.2.22) satisfy the relations ν1 ν2 q + 1 (q + 1) (q + 1) q+1 = = ⇐⇒ ν1 −(1 + µ1 ) = ν2 −(1 + µ2 ) =− . µ1 µ2 p − q (p − q) (p − q) p−q In terms of the new unknown functions Aj (s) and Hj (s), system (6.2.21), (6.2.22) can be rewritten in the form Hj (s) + Aj (s) 6 C 5 Hj−1 (s) + C6 (−A0j (s))1+µ1 + C7 (−A0j (s))1+µ2 , (6.2.32) q+1

Hj (s) 6 (1 + γ)ξ p−q Hj−1 (s) + C8 (−A0j (s))1+µ1 + C9 (−A0j (s))1+µ2 , (6.2.33) j = 1, 2, . . . , where  H0 (s) =

T (1 − ξ) ξ

 q+1

p−q

q+1

h0 (s) and C 5 = C5 ξ p−q .

We now specify the parameter γ in (6.2.33). First, we fix a number λ: q+1

ξ p−q < λ < 1 with ξ from (6.2.29).

(6.2.34)

Then we set 0 < γ = γ(λ) := λξ

q+1 − p−q

−1

⇐⇒

q+1

(1 + γ)ξ p−q = λ < 1.

(6.2.35)

We now use additional conditions from Theorem 6.2.1, namely, conditions (6.1.36) and (6.1.37). It is well known (see, e.g., [39]) that, under these conditions, the support of the solution u(t, ·) continuously expands. Hence, there exists T 6 T such that supp u(t, ·) ∩ ∂1 Ω = ∅ ∀ t < T 6 T.

(6.2.36)

Thus, by virtue of Lemma 6.2.2, we have C4 = 0 for b < T in (6.2.15). Therefore, in view of (6.2.27) and (6.2.28), system (6.2.32), (6.2.33) takes the the following “truncated” form: Hj (s) + Aj (s) 6 C 5 Hj−1 (s) + C6 (−A0j (s))1+µ1 Hj (s) 6 λHj−1 (s) + C8 (−A0j (s))1+µ1 , where j := max{j : ∆1

1 − ξj 6 T }. 1−ξ

∀ j 6 j,

(6.2.37) (6.2.38)

302

Chapter 6 Energy methods for the investigation of localized regimes

Thus, it is clear that, in order to prove Theorem 6.2.1, it suffices to show that if the boundary regime satisfies restriction (6.2.1), then the supports of all solutions of system (6.2.37), (6.2.38) are bounded uniformly in j, i.e., sup{s : Aj (s) > 0} 6 R1 ,

sup{s : Hj (s) > 0} 6 R1 ,

∀ j ∈ N,

(6.2.39)

where R1 is specified in (6.1.25). We now prove (6.2.39). Iterating relation (6.2.38), we obtain Hj (s) + Aj (s) 6 C¯ 5 Hj−1 (s) + C6 (−A0j (s))1+µ1 6 C¯ 5 λHj−2 (s) + C¯ 5 C8 (−A0j−1 (s))1+µ1 + C6 (−A0j (s))1+µ1 6 C¯ 5 λ2 Hj−3 + C¯ 5 C8 λ(−A0j−2 (s))1+µ1 + C¯ 5 C8 (−A0j−1 (s))1+µ1 + C6 (−A0j (s))1+µ1 6 . . . 6 C¯ 5 λj−1 H0 (s) + C10

j X

λj−i (−A0i (s))1+µ1

∀ j 6 j,

i=1

C10 = λ−1 max(C6 λ, C¯ 5 C8 ). Since the function g(s) = s1+µ1 , µ1 > 0 is convex, the last inequality implies that Hj (s) + Aj (s) 6 C11 λ

j−1

h0 (s) + C10

X j

λ

j−i 1+µ1

1+µ1

(−A0i (s))

∀ j 6 j,

(6.2.40)

i=1

where

q+1

C11 = C 5 (T (1 − ξ)ξ −1 ) p−q . We now introduce the following family of energy functions: Uj (s) :=

j X

j−i

λ 1+µ1 Ai (s),

j = 1, 2, . . . ,

U0 (s) = 0.

(6.2.41)

i=1

Clearly, we have 1

Uj (s) − λ 1+µ1 Uj−1 (s) = Aj (s) ∀ j > 1. Therefore, relations (6.2.40) can be rewritten in the form 1

Uj (s) 6 λ 1+µ1 Uj−1 (s) + C10 (−Uj0 (s))1+µ1 + C11 λj−1 h0 (s) ∀ j : 1 6 j 6 j. Further, we estimate the initial value Uj (1) from above. We have Uj (1) =

j X i=1

j−i

λ 1+µ1 Ai (1) =

j X i=1

j−i

q+1

λ 1+µ1 ∆ip−q Ei (1) ∀ j ∈ N.

(6.2.42)

Section 6.2

Sufficient conditions for the localization of boundary regimes

303

According to Lemma 6.2.1, we find Ei (1) 6 C1 h0 (1) + C2 F (ti ) 6 C1 h0 (1) + C2 ω0 (T − ti )−α0 .

(6.2.43)

By using (6.2.30), we conclude that Uj (1) 6 C1 h0 (1)

j X

λ

j−i 1+µ1

q+1 p−q

∆i

 + C 2 ω0

i=1

6 C12 λ

j 1+µ1

1−ξ ξ

 q+1 X j p−q

j−i

λ 1+µ1

i=1

h0 (1) + C13 ω0 ,

where

q+1

C12 = C1 ∆1p−q λ

C13 = C2

µ1

1 − 1+µ

and 

(6.2.44)

1−ξ ξ

1

 q+1

p−q

(1 − λ 1+µ1 )−1 1

(1 − λ 1+µ1 )−1 .

Thus, in view of condition (6.1.37), we conclude that the functions Uj (s) solve the Cauchy problem for the following infinite system of differential inequalities: Uj (s) 6 λ1 Uj−1 (s) + C10 (−Uj0 (s))1+µ1 Uj (1) 6 C12 λj1 h0 (1) + C13 ω0 ,

λ1 = λ

∀ s : 1 < s < R1 , 1 1+µ1

(6.2.45) < 1,

1 6 j 6 j.

Finally, applying Lemma 9.2.1 to system (6.2.45), we obtain Theorem 6.2.1 with 1 1+µ1

L1 = a2 C10

µ1 1+µ1

C12

,

1 1+µ1

L2 = a2 C10

µ1 1+µ1

C13

.



Remark 6.2.4. The specific form of the boundary ∂0 Ω whose part contains peaking boundary data has been used solely to exhaust the family of subdomains Ω(s) from (6.2.14). It is obvious that this family can be specified by the following general formula: Ω(s) := {x ∈ Ω : ρ(x, ∂0 Ω) > s − 1} ∀ s > 1, where ρ(x, S) is the distance from a point x to a compact set S. Clearly, this formula for the family of subdomains Ω(s) suitable for our purposes is true for a class of domains much broader than in the case of relation (6.1.25). In particular, it is easy to see that these families of subdomains are well defined for the domains mentioned in Remark 6.1.2. Moreover, the proof presented above shows that the problem of localization of the support of solution (and also the effective localization studied in what follows) can be analyzed in exactly the same way also in the case of nonhomogeneous equation whose finite righthand side is singularly peaking in a compact subset S ⊂ Ω.

304

Chapter 6 Energy methods for the investigation of localized regimes

6.3

Sharp conditions for the effective localization of boundary regimes: the case of slow diffusion p > q

Theorem 6.3.1 ([116], [118]). Let u(t, x) be an arbitrary energy solution of problem (6.1.26)–(6.1.29), (6.1.36). Assume that the boundary function f is such that the function F (t) defined in (6.1.34) satisfies the estimate F (t) 6 ω(T − t)−α0

∀ t < T,

α0 =

q+1 , p−q

ω = const > 0.

(6.3.1)

Then there exists a constant δ = δ(ω), δ(ω) → 0 as ω → 0, which also depends on the other known parameters of the original problem, such that the following inclusion holds for the set of intensification: Ωs = Ωs (u) ⊂ Ω ∩ {x : |x| 6 1 + δ}.

(6.3.2)

This means that the boundary regime (6.3.1) and the corresponding solution u are effectively localized if ω < ω0 , where ω0 is defined by the equality 1 + δ(ω0 ) = sup{|x| : x ∈ ∂1 Ω}. Corollary 6.3.1. Under the conditions of Theorem 6.3.1, assume that the boundary regime is such that the following estimate, which is stronger than (6.3.1), is true : F (t) 6 ω(t)(T − t)−α0

∀ t < T,

ω(t) → 0 as t → T,

(6.3.3)

Then Ωs (u) ⊂ ∂0 Ω = {|x| = 1} for any solution u, i.e., the boundary regime (6.3.3) is a localized LS-regime. First, we prove the following assertion: Lemma 6.3.1. Under the conditions of Theorem 6.3.1, for any α1 such that p−1 < α1 < α0 =

q+1 , p−q

there exists a value s0 = s0 (ω) > 0, s0 (ω) → 0 as ω → 0, for which any solution u satisfies the following estimate : Z Z tZ q+1 max |u(τ, x)| dx + |Dx u(t, x)|p+1 dx dt 6 C(T − t)−α1 (6.3.4) τ 6t

Ω(s)

0

Ω(s)

∀ t < T,

∀ s > 1 + s0 (ω),

where C = C(α1 ) is independent of s.

Section 6.3 Sharp conditions for the effective localization of boundary regimes

305

Proof. Consider the differential system (6.2.20), (6.2.21) considered in Lemma 6.2.3. In this system, we use the same shifts ∆i as in (6.2.29). As new auxiliary energy functions, we take functions different from (6.2.30). Indeed, we set Aj (s) := ∆αj 1 Ej (s),

Hj (s) := ∆αj 1 hj (s),

j = 1, 2, . . . ,

(6.3.5)

where α1 > 0 is an arbitrary number from the interval ν2 ν1 1 q+1 = < α1 < α0 := = . 1 + µ2 p µ1 p−q

(6.3.6)

In terms of the new unknown functions (6.3.5), system (6.2.20), (6.2.21) can be rewritten in the form e5 Hj−1 (s) + C6 ∆ν1 −α1 µ1 (−A0j (s))1+µ1 Hj (s) + Aj (s) 6 C j

+

C7 ∆νj 2 −α1 µ2 (−A0j (s))1+µ2 ,

(6.3.7)

Hj (s) 6 (1 + γ)ξ α1 Hj−1 (s) + C8 ∆νj 1 −α1 µ1 (−A0j (s))1+µ1 + C9 ∆νj 2 −α1 µ2 (−A0j (s))1+µ2 . where



e5 = C5 · ξ α1 , C

H0 (s) =

T (1 − ξ) ξ

(6.3.8)  α1 h0 (s),

and ξ is specified in (6.2.29). We now impose the first restriction on the choice of the parameters γ and ξ. Namely, we set (1 + γ)ξ α1 := λ < 1. (6.3.9) Iterating estimate (6.3.7) and using (6.3.8), we obtain e5 λj−1 H0 (s) + C e10 Hj (s) + Aj (s) 6 C

j X

λj−i ∆νi 1 −α1 µ1 (−A0i (s))1+µ1

i=1

+ C12

j X

λj−i ∆νi 2 −α1 µ2 (−A0i (s))1+µ2 ,

(6.3.10)

i=1

e5 C8 ) and C12 = λ−1 max(C7 λ, C e5 C9 ). This ime10 = λ−1 max(C6 λ, C where C mediately implies the inequality Hj (s) + Aj (s) e5 λ 6C

j−1

e10 H0 (s) + C

X j

λ

j−i 1+µ1

i=1

+ C12

X j i=1

λ

j−i 1+µ2



∆i ∆j

 ν2 −α1 µ2 1+µ2



∆i ∆j

 ν1 −α1 µ1 1+µ1

1+µ1 (−A0i (s)) ∆νj 1 −α1 µ1

1+µ2

(−A0i (s))

∆νj 2 −α1 µ2 ,

j = 1, 2, . . . . (6.3.11)

306

Chapter 6 Energy methods for the investigation of localized regimes

Further, we introduce the following two families of functions:   ν1 −α1 µ1 j X j−i 1+µ1 ∆i (1) (Ai (s) + Hi (s)), j = 1, 2, . . . , Uj (s) = λ 1+µ1 ∆j i=1   ν2 −α1 µ2 j X j−i 1+µ2 ∆i (2) 1+µ 2 (Ai (s) + Hi (s)), j = 1, 2, . . . . Uj (s) = λ ∆j

(6.3.12)

i=1

The following relations are obvious: (1)

(1)

j = 1, 2, . . . ,

(2)

(2)

j = 1, 2, . . . ,

Uj (s) − Aj (s) − Hj (s) = Uj−1 (s) · θ1 ,

(6.3.13)

Uj (s) − Aj (s) − Hj (s) = Uj−1 (s) · θ2 , where θ1 = λ θ2 = λ

1 1+µ1



1 1+µ2



∆j−1 ∆j ∆j−1 ∆j

 ν1 −α1 µ1 1+µ1

1

= (1 + γ) 1+µ1 · ξ

 ν2 −α1 µ2 1+µ2

1

= (1 + γ) 1+µ2 · ξ

ν

1 α1 − 1+µ

1

,

ν

2 α1 − 1+µ

2

.

Here, we have also used equality (6.3.9). We now impose the following restrictions [more severe than (6.3.9)] on the choice of the parameters γ and ξ : ν

1

θ1 := (1 + γ) 1+µ1 · ξ θ2 := (1 + γ)

1 1+µ2

·ξ

1 α1 − 1+µ

1

< 1,

ν2 α1 − 1+µ 2

< 1.

(6.3.14)

Note that, under condition (6.3.6), we have ν2 ν1 > 0 and α1 − > 0. α1 − 1 + µ1 1 + µ2 Thus, for any ξ < 1, inequalities (6.3.14) are true for all γ such that 0 < γ < γ0 = γ0 (ξ). It is clear that all Hj (s), j = 1, 2, . . . , are absolutely continuous decreasing functions. Therefore, by virtue of equalities (6.3.13), inequality (6.3.11) yields 1+µ1  d (1) (1) (1) j−1 e e ∆νj 1 −α1 µ1 Uj (s) 6 C5 λ H0 (s) + θ1 Uj−1 (s) + C10 − Uj (s) ds  1+µ2 d (2) + C12 − Uj (s) (6.3.15) ∆νj 2 −α1 µ2 , j = 1, 2, . . . , ds 1+µ1  (2) e5 λj−1 H0 (s) + θ2 U (2) (s) + C e10 − d U (2) (s) · ∆νj 1 −α1 µ1 Uj (s) 6 C j−1 ds j  1+µ2 d (2) + C12 − Uj (s) · ∆νj 2 −α1 µ2 , j = 1, 2, . . . . (6.3.16) ds

Section 6.3 Sharp conditions for the effective localization of boundary regimes (1)

307

(2)

We now estimate the quantities Uj (1) and Uj (1) from above. It is clear that (1)

Uj (1) =

j X

j X

θ1j−i (Ai (1) + Hi (1)) =

i=1

θ1j−i ∆αi 1 (Ei (1) + hi (1)).

i=1

In view of Lemma 6.2.1, we conclude that hi (1) + Ei (1) 6 C1 h0 (1) + C2 F (ti ) 6 C1 h0 (1) + C2 ω(T − ti )−α0 −α0  ξ 0 ω∆−α . 6 C1 h0 (1) + C2 i 1−ξ Therefore,   j j X 1 − ξ α0 X j−i −(α0 −α1 ) (1) j−i α1 Uj (1) 6 C1 h0 (1) θ1 ∆ i + C2 ω θ1 ∆i ξ i=1

i=1

e2 ω∆−(α0 −α1 ) , e1 θj h0 (1) + C 6C 1 j where

α1 e1 = C1 ∆1 , C θ1 − ξ α1

e2 = C

(6.3.17)

C2 (1 − ξ)α0 . α ξ 0 (1 − θ1 ξ α0 −α1 )

Similarly, we get (2)

Uj (1) 6

j X

−(α0 −α1 )

b1 θj h0 (1) + C b2 ω∆ θ2j−i (Ai (1) + Hi (1)) 6 C 2 j

,

(6.3.18)

i=1

where

α1 b1 = C1 ∆1 C θ2 − ξ α1

b2 = and C

C2 (1 − ξ)α0 . ξ α0 (1 − θ2 ξ α0 −α1 )

By using relations (6.3.15)–(6.3.18), we arrive at the following “combined” system: (1) (2) e5 λj−1 H0 (s) + θUj−1 (s) Uj (s) := Uj (s) + Uj (s) 6 2C   1+µ1 d (α0 −α1 )µ1 e + max C10 ∆j − Uj (s) , ds 1+µ2   d µ2 (α0 −α1 ) , (6.3.19) C12 ∆j − Uj (s) ds b1 )θj h0 (1) + (C e2 + C b2 )ω∆−(α0 −α1 ) , j = 1, 2, . . . , e1 + C Uj (1) 6 (C (6.3.20) j

where θ = max(θ1 , θ2 ) < 1. Denote e5 H0 (1), (C e1 + C b1 )h0 (1)}. b := max{(1 − θ)−1 2C

308

Chapter 6 Energy methods for the investigation of localized regimes

By using relations (6.3.19) and (6.3.20), we readily prove that U j (s) := Uj (s) − b 6 θ U j−1 (s)   0 0 + max kj,1 (−U j (s))1+µ1 , kj,2 (−U j (s))1+µ2 , e2 + C b2 )∆−(α0 −α1 ) ω, U j (1) 6 Kj := (C j

(6.3.21)

e10 ∆µ1 (α0 −α1 ) and kj,2 = C12 ∆µ2 (α0 −α1 ) . where kj,1 = C j j The next step in the proof of the lemma is the analysis of system (6.3.21) and getting the following upper bound for the functions U j (s) satisfying this system: U j (s) 6 M j (s − 1) := max{M j,1 (s − 1), M j,2 (s − 1)} ∀ s > 1,

(6.3.22)

where M j,1 (s) and M j,2 (s) are functions M 1 (s) and M 2 (s) from (9.2.14) obtained for the following values of the parameters: k1 =

kj,1 kj,2 , k2 = , K = Kj , γ1 = µ1 , 1−θ 1−θ

and γ2 = µ2 .

We prove the required assertion by induction. For j = 1, estimate (6.3.22) can be directly verified by applying Lemma 9.2.2. Suppose that estimate (6.3.22) is true for all j 6 l − 1. It is necessary to prove that it is true for j = l. First, it is necessary to show that the sequence of functions M j (s) is nondecreasing. It is clear that e2 + C b2 )µ2 ω µ2 k2 K γ2 = (1 − θ)−1 kj,2 Kjµ2 = (1 − θ)−1 C12 (C does not depend on j, e10 (C e2 + C b2 )µ1 ω µ1 k1 K γ1 = (1 − θ)−1 kj,1 Kjµ1 = (1 − θ)−1 C does not depend on j, and   1  µ1  µ1 1+µ1  (1) kj,1 Kj  1+µ1 a2 = g ω 1  1−θ       1   µ  1 µ s1 = sj,1 = a(2) kj,2 Kj 2 1+µ2 + µ2 −µ1 kj,12 µ2 −µ1 µ1  2 µ1 µ2 kj,2  1−θ     µ2    = g2 ω 1+µ2 + g4 

for ω 6 g3 , (6.3.23) for ω > g3 ,

where 1 + γ1 g1 = γ1

1 e e b2 )µ1  1+µ1 C10 (C2 + C , 1−θ

 e 1+µ2  1 µ2 −µ1 −1 C10 e b g3 = (C2 + C2 ) , 1+µ1 C12

Section 6.3 Sharp conditions for the effective localization of boundary regimes

1 + µ2 g2 = µ2



e2 + C b2 )µ2 C12 (C 1−θ



1 1+µ2

309

 e µ2  1 µ2 −µ1 (µ2 − µ1 ) C 10 g4 = , µ1 µ 2 µ1 C12

,

i.e., the quantities sj,1 also do not depend on j . The required property of monotonicity of the sequences of functions {M j,1 (s)} and {M j,2 (s)} follows from the monotonicity of the sequences {kj,1 } and {kj,2 }. We now return to the proof of estimate (6.3.22) for j = l. Assume that this estimate is not true, i.e., there exists an interval (s1 , s2 ) such that U l (s) > M l (s − 1) ∀ s ∈ (s1 , s2 ),

U l (s1 ) = M l (s1 ),

s1 > 1.

(6.3.24)

For any s ∈ (s1 , s2 ), we get (1 − θ)U l−1 (s) 6 (1 − θ)M l−1 (s − 1) 6 (1 − θ)M l (s − 1) 6 (1 − θ)Ul (s), By virtue of (6.3.21), we find 0

0

U l (s) 6 (1 − θ)U l (s) + max{kj,1 (−U j (s))1+µ1 , kj,2 (−U j (s))1+µ2 }, U l (s1 ) = M l (s1 − 1), whence, in view of Lemma 9.2.2, we arrive at the inequality U l (s) 6 M l (s − 1) ∀ s ∈ (s1 , s2 ), which contradicts assumption (6.3.24). Hence, estimate (6.3.22) is true. Note that, by virtue of (6.3.23),  µ1 µ2  supp M j ⊂ [0, s1 ], s1 6 g5 max ω 1+µ1 , ω 1+µ2 = s1,0 ∀ j ∈ N, (6.3.25) µ2   − 1+µ 2 . g5 = max g1 , g2 + g4 g3 Therefore, in view of (6.3.21), we obtain e5 H0 (1), (C e1 + C b1 )h0 (1)} Uj (s) 6 b = max{(1−θ)−1 2C ∀ j ∈ N,

∀ s > s1,0 +1,

(6.3.26)

where s1,0 is specified in (6.3.25). In view of the definition (6.3.19) of functions Uj (s) and (6.3.26), we get the following estimate: 1 hj (s) + Ej (s) 6 b∆−α j

∀ s > 1 + s1,0 ,

∀ j ∈ N,

(6.3.27)

where α1 is specified in (6.3.5) and (6.3.6). We now sum inequalities (6.3.27) over j from j = 1 to j = i and take into account (6.2.28) and (6.2.29). This yields Z Z tiZ i X q+1 1 hti (s)+E(ti , s) := max |u(t, x)| dx + |Dx u|p+1 dx dt 6 b ∆−α j t6ti

6

Ω(s)

b ξ α1 (1 − ξ)α1 (1 − ξ α1 )

0

Ω(s)

(T − ti )−α1

∀ i ∈ N.

j=1

(6.3.28)

310

Chapter 6 Energy methods for the investigation of localized regimes

By virtue of (6.2.30), this yields the estimate ht (s) + E(t, s) 6 b1 (T − t)−α1

∀ t < T,  µ1 µ2  ∀ s > 1 + g5 max ω 1+µ1 , ω 1+µ2 := 1 + s0 (ω),

where b1 = b(1 − ξ)−α1 (1 − ξ α1 )−1 . Thus, the lemma is proved.

(6.3.29) 

We now proceed directly to the proof of Theorem 6.3.1. To do this, we consider system (6.2.20), (6.2.21) for s > 1 + s0 (ω). However, in this case, the choice of the shifts ∆i at any point s > 1 + s0 (ω) is determined by the investigated solution u (to be more precise, by its energy functions E(t, s) and ht (s)). Thus, we fix numbers ξ and η as follows: 0 < ξ < 1;

α1 < η < α 0 =

q+1 , p−q

(6.3.30)

where α1 is specified in (6.3.6). Further, we fix an arbitrary point s > 1+s0 (ω) at which the inequality E(T, s) + sup h(t, s) > b1 T α1 ξ −α1

(6.3.31)

0 s. Inequality (6.3.40) determines the structure of the new energy functions  α1 − ν1 j X j−i 1+µ1 ∆j (1) (Ai (s)+Hi (s)), j = 1, 2, . . . , Uj (s) = (1+γ) 1+µ1 ∆i i=1 (6.3.41)  α1 − ν2 j X j−i 1+µ2 ∆ j (2) Uj (s) = (1+γ) 1+µ2 (Ai (s)+Hi (s)), j = 1, 2, . . . . ∆i i=1

The following relations are obvious: (1)

(j)

(1)

j = 1, 2, . . . ;

U0 (s) = 0,

(2)

(j)

(2)

j = 1, 2, . . . ;

U0 (s) = 0,

Uj (s) − Aj (s) − Hj (s) = θ1 Uj−1 (s),

(1)

(6.3.42) Uj (s) − Aj (s) − Hj (s) = θ2 Uj−1 (s),

(2)

where ν1  ν 1 ∆j α1 − 1+µ1 α − 1 6 (1+γ) 1+µ1 · ξ 1 1+µ1 := θ1 , (1+γ) · ∆j−1 ν2   ν 1 1 ∆j α1 − 1+µ2 α − 2 (j) 1+µ 6 (1+γ) 1+µ2 · ξ 1 1+µ2 := θ2 . θ2 := (1+γ) 2 · ∆j−1

(j) θ1 :=

1 1+µ1



(6.3.43)

Section 6.3 Sharp conditions for the effective localization of boundary regimes

313

Note that the quantities θ1 and θ2 in (6.3.43) coincide with the quantities θ1 and θ2 from relations (6.3.13). We now impose restrictions (6.3.14) on the parameters γ and ξ . Thus, by using inequalities (6.3.40) and relations (6.3.42) and (6.3.43), we can prove that relations (6.3.15) and (6.3.16) are true for all s > s. Clearly, in this case, the estimates obtained for the initial values (2) (1) Uj (s) and Uj (s) differ from inequalities (6.3.17) and (6.3.18). Namely, by using definition (6.3.33), we find  α1 − ν1 j X j−i 1+µ1 ∆j (1) ∆αi 1 (Ei (s) + hi (s)) Uj (s) = (1 + γ) 1+µ1 ∆i i=1  α1 − ν1 j X j−i 1+µ1 ∆j −η η−α1 1+µ ∆αi 1 −η (1 + γ) 1 = b1 ξ T ∆i i=1  η− ν1 j X j−i 1+µ1 ∆j −η η−α1 −(η−α1 ) = b1 ξ T ∆j (1 + γ) 1+µ1 ∆i i=1

−(η−α1

6 b1 ξ −η T η−α1 ∆j

j X )

−(η−α1 )

= b1 ξ −η T η−α1 ∆j 6 b1 ξ

−η

T

η−α1

(1 −

i=1 j X

j−i

(1 + γ) 1+µ1 ξ

ν

1 ) (j−i)(α1 − 1+µ 1

θ1j−i

i=1 −1 −(η−α1 ) θ1 ) ∆j

∀ j < i∞ .

(6.3.44)

∀ j < i∞ .

(6.3.45)

Similarly, we get (2)

−(η−α1 )

Uj (s) 6 b1 ξ −η T η−α1 (1 − θ2 )−1 ∆j

We now sum the inequalities obtained as a result and conclude that, for all (1) (2) s > s, the energy functions Uj (s) = Uj (s) + Uj (s) satisfy the differential inequality (6.3.19) and the initial condition −(η−α1 )

Uj (s) 6 2(1 − θ)−1 b1 ξ −η T η−α1 ∆j

∀ j < i∞ ,

(6.3.46)

θ = max (θ1 , θ2 ). e5 λj−1 H0 (s). We arrive at the system Denote b(j) = (1 − θ)−1 2C ej (s) := Uj (s) − b(j) U ej−1 (s) + max{kj,1 (−U e 0 (s))1+µ1 , kj,2 (−U e 0 (s))1+µ2 } ∀ s > s, 6 θU j j ej (s) 6 Kj := b2 ∆−(η−α1 ) U j

∀ j < i∞ , b2 = 2(1 − θ)−1 b1 ξ −η T η−α1 , (6.3.47)

where kj,1 and kj,2 are the same as in (6.3.21) but with new values of ∆j specified by (6.3.33). Using Lemmas 9.2.5 and 9.2.6 for the analysis of system (6.3.47), we get the following uniform estimate: ej (s) 6 max{D1 U 0 (s − s), D2 } ∀ s > s, ∀ j < i∞ , U (6.3.48)

314

Chapter 6 Energy methods for the investigation of localized regimes

where −

U 0 (s) = s

(1+µ1 )(η−α1 ) µ1 (α0 −η)

,

the constants D1 and D2 have the form  1+µ1  1+µ µ1 α0 − η (1) (1) µ 1 1 D1 = 2ba1 (a2 ) α0 − α1  (1+µ1 )(η−α1 )  µ1 (α0−η) η − α1 ξ −η T η−α1 (1−θ)−1 (1−ξ)−α1 (1−ξ α1 )−1 , × α0 − α1 α0 −α1 1+µ1 η−α1  1+µ1  (1) (1) (α0 −η)(µ2 −µ1 ) (2b) α0 −η a1 (a2 ) µ1 C12 D2 = , α1 (α0 −α1 ) α0 −α1 e 1+µ2 (1 − θ)(1 − ξ) α0 −η (1 − ξ α1 ) α0 −η C10 and the constant b is determined from (6.3.26). It follows from (6.3.48) that Uj (s) 6 max{D1 U 0 (s − s), D2 + b(1) } ∀ j < i∞ ,

(6.3.49)

b(1) = (1 − θ)−1 2C5 H0 (s). We define the number δ0 > 0 by the equality U 0 (δ0 ) = (D2 + b(1) )D1−1 .

(6.3.50)

Then it is clear that (6.3.49) leads to the inequality Uj (s) 6 D1 U 0 (s − s) ∀ s : s < s < s + δ0 ,

∀ j < i∞ .

(6.3.51)

In view of the definition of the functions Uj (s), the choice of the shifts ∆j , and relation (6.3.33), we obtain the following inequality from (6.3.51): 1 hj (s) + Ej (s) 6 D1 U 0 (s − s)∆−α j

= D3 U 0 (s − s)(hj (s) + Ej (s)) where

−

D3 = D1 ξ α1 b1

α1 η

T

−

α1 η

α1 (η−α1 ) η

∀ j < i∞ ,

(6.3.52)

.

Note that relation (6.3.52) is true for any s and s such that 1 + s0 < s < s < s + δ0 , where s0 = s0 (ω) is specified in (6.3.29). (6.3.53) We now fix an arbitrary i < i∞ and sum inequalities (6.3.52) over j from 1 to i. By using, in addition, property (6.3.35) of the sequence {∆j }, we get hti (s) + E(ti , s) 6 D1 U 0 (s − s)

i X

1 1 ∆−α 6 D1 U 0 (s − s)∆−α j i

j=1 α1 −1

6 D1 (1 − ξ )

U 0 (s −

i X j=1

1 s)∆−α i

ξ (j−1)α1

Section 6.3 Sharp conditions for the effective localization of boundary regimes

= D3 (1 − ξ α1 )−1 U 0 (s − s)(hi (s) + Ei (s))

α1 η

6 D3 (1 − ξ α1 )−1 U 0 (s − s)(hti (s) + E(ti , s)) ∀ i < i∞ ,

315

α1 η

(6.3.54)

∀ s : s < s < s + δ0 .

Our next aim is to establish an estimate of type (6.3.54) not only for t = ti , i = 1, 2, . . . , but also for an arbitrary point t < T. To this end, we return to the mapping Γs from (6.3.32) and the sequence {ti } specified by this mapping. In view of the continuity and strict monotonicity of this sequence, Γs maps the segment [ti−1 , ti ] onto [ti , ti+1 ] continuously, monotonically, and bijectively for any i > 1 in the case (b) and for any i : 1 6 i 6 i0 − 1 in the case (a). Moreover, in the case (a), [ti0 −1 , t0 ] is mapped onto [ti0 , T ] continuously and bijectively. Let t be an arbitrary point in the interval [t1 , T ). For the sake of definiteness, we assume that t := tk ∈ (tk , tk+1 ] for some k ∈ N. Thus, clearly, it is possible to reconstruct (in a unique way) a sequence {ti }, i = 0, 1, 2, . . . , k − 1, such that ti+1 = Γ(ti ), ∀ i 6 k − 1, ti ∈ (ti , ti+1 ], t0 ∈ (0, t1 ]. Hence, we get the following set of new shifts ∆j : −η

∆j

= (tj − tj−1 )−η   ξη = E(tj , s) − E(tj−1 , s) + sup h(t, s) . b1 T η−α1 t∈(tj−1 ,tj ]

(6.3.55)

As above, we can show that ∆i+1 < ξ∆i

∀ i 6 k − 1.

By using these new shifts, we now define the corresponding functions E j (s) and hj (s) and then construct Aj (s) and H j (s), as well as the corresponding system (6.3.36), (6.3.37) for the last couple of functions. The only difference observed in this case is that the role of initial function is played by Z h0 (s) := h(t0 , s) = |u(t0 , x)|q+1 dx =⇒ H 0 (s) = h0 (s)∆1 ξ −1 . Ω(s)

We now find the upper bound for this new function. For the position of the point t0 , we have two possibilities: (a)

t0 6 2−1 T ;

(b)

t0 > 2−1 T .

In the first case, we have h(t0 , s) 6 b1 2α1 T −α1 due to estimate (6.3.29). In the case (b), by virtue of definition (6.3.33), we get −η

η−α1 −η h(t0 , s) 6 sup h(t, s) 6 b1 T η−α1 ξ −η t−η ξ t0 1 6 b1 T t∈(0,t1 )

6 b1 (2ξ)−η T −α1 .

316

Chapter 6 Energy methods for the investigation of localized regimes

Thus, we arrive at the estimate H 0 (s) = h0 (s)ξ −1 ∆1 6 ξ −1 T h0 (s) 6 ξ −1 b1 T 1−α1 max(2α1 , (2ξ)−η ).

(6.3.56)

Further, by analogy with estimate (6.3.51), for the functions U j (s) defined by the sequence {∆j }, we obtain the following universal estimate: U j (s) 6 D1 U 0 (s − s) ∀ s : s < s < s + δ 0 ,

∀ j 6 k,

(6.3.57)

where the number δ 0 is defined by the equality (1)

U 0 (δ 0 ) = (D2 + b

)D1−1 ,

(6.3.58)

and the constants D1 and D2 and the function U 0 (s) are the same as in (1) depends on the new initial estimate (6.3.51). In this case, the constant b function H 0 (s), i.e., (1)

b

e5 H 0 (s), = (1 − θ)−1 2C

H 0 (s) is defined in (6.3.56).

Estimate (6.3.57) now implies, in a standard way, inequality (6.3.54) for t = tk . Since any point t < T can chosen as tk , we find Z Z tZ q+1 |u(τ, x)| + |Dx u(t, x)|p+1 dx dt ht (s) + E(t, s) = sup 0 0.

(6.4.1)

This case is especially difficult for analysis because the effectively localized limiting boundary regime does not generate a self-similar solution even for the simplest equation. Note that this class of problems contains, in particular, the standard heat-conduction equation for p = q = 1. Theorem 6.4.1. In problem (6.1.26)–(6.1.29), let conditions (6.1.30),(6.1.31), and (6.4.1) be satisfied. Assume that the boundary function f singularly peaking at time t = T is such that the function F (t) specified by (6.1.34) satisfies the inequality −1

F (t) 6 exp(ω(T − t)−p ) := F0 (t) ∀ t < T,

ω > 0.

(6.4.2)

Then there exists a constant c > 0 that depends only on the known parameters of problem (6.1.26)–(6.1.29) (and does not depend on ω ) such that the inclusion p

Ωs (u) ⊂ Ω ∩ {|x| < 1 + c ω p+1 } is true for the set of intensification of any solution u(t, x). Hence, the boundary regime (6.4.2) is effectively localized if p

c ω p+1 < sup{|x| : x ∈ ∂1 Ω} − 1. Corollary 6.4.1. Under the conditions of Theorem 6.4.1, assume that the estimate − p1

F (t) 6 exp [ω(t)(T − t)

] ∀ t < T,

ω(t) → 0 as t → T,

(6.4.3)

which is stronger than (6.4.2), is satisfied. Then Ωs (u) ⊂ ∂0 Ω for any solution u, i.e., regime (6.4.3) is a localized LS -regime. Remark 6.4.1. For p = q = 1, the condition of effective localization (6.4.2) is more exact and corresponds to the condition of effective localization for the standard heat-conduction equation obtained in [141] [see also (6.1.18) and (6.1.19) and the comments to those relations]. Proof of Theorem 6.4.1. The proof is based on the analysis of properties of the infinite system of differential inequalities (6.2.21), (6.2.22). In view of (6.4.1), we get µ1 = µ2 = 0.

318

Chapter 6 Energy methods for the investigation of localized regimes

1 < ν2 = p1 , we can easily derive the following inequalities from Since ν1 = p+1 system (6.2.21), (6.2.22): 1

hj (s) + Ej (s) 6 C5 hj−1 (s) + C 6 ∆jp+1 (−Ej0 (s)) ∀ j > 1,

(6.4.4)

1

hj (s) 6 (1+γ)hj−1 (s) + C 8 ∆jp+1 (−Ej0 (s)) ∀ γ > 0, ∀ s > 1, where 1

C 6 = C6 + C7 T ν2 −ν1 = C6 + C7 T p(p+1) ,

1

C 8 = C8 + C9 T p(p+1) .

For any L > 0 and p > 0, we denote σ = σ(L, p) :=

∞ X

(i + L)−(p+1) .

(6.4.5)

i=1

We now fix the constants γ > 0, L0 > 0, and ν > 0 such that the relation   L0 (1 + γ)(L0 + 1) exp − 1, that obeys the estimate − p1

ht (s0 ) + E(t, s0 ) 6 D1 exp(ω(T − t)

) ∀ t < T.

Also let the parameter ω satisfy the following technical restriction:  1 T p , ω>ω e := (1 + ν) pσ0

(6.4.7)

(6.4.8)

where σ0 := σ(L0 , p), σ is given by (6.4.5), and ν > 0 is taken from (6.4.6). Then there exist constants c1 > 0 and c2 > 0 depending only on n, p, γ, L0 , and ν such that the following estimate is true :   ω −1 (T − t) p ∀ t < T, ht (s1 ) + E(t, s1 ) 6 c1 D1 exp (6.4.9) 1+ν p

where s1 = s0 + c2 ω p+1 . Proof. We define shifts {∆i } in the system of differential inequalities (6.4.4) as follows: ∆i = pω p (i + L1 )−(p+1) ,

ti = ti−1 + ∆i ,

i = 1, 2, . . . ,

t0 = 0,

(6.4.10)

where L1 > 0 is determined from the following necessary property of the sequence {∆i }: ∞ ∞ X X T = ∆i = p ω p (i + L1 )−(p+1) := p ω p σ(L1 , p), i=1

i=1

Section 6.4

319

Effective localization of singular boundary regimes

i.e., L1 must satisfy the equation σ(L1 , p) = T p−1 ω −p .

(6.4.11)

We now show that this equation is solvable. It follows from definition (6.4.5) that the function σ(L, p) is monotonically decreasing in the argument L and σ(L, p) → 0 as L → ∞. In order that Eq. (6.4.11) be solvable, it suffices to guarantee the validity of the inequality T p−1 ω −p < σ(L0 , p) = σ0 . This inequality is obviously true by virtue of condition (6.4.8). Therefore, the solution L1 = L1 (T, ω) of Eq. (6.4.11) exists and, hence, definition (6.4.10) is correct. In view of the obvious inequalities Z j Z j+1 ds 1 ds > > ∀ j ∈ N, p+1 p+1 p+1 s j s j−1 j it follows from definition (6.4.10) that ω p (j + L1 + 1)−p 6 T − tj =

∞ X

∆i 6 ω p (j + L1 )−p .

(6.4.12)

i=j+1

Therefore, the relations  −1  exp ω(T − tj ) p 6 e · exp(j + L1 )

(6.4.13)

are true. We now introduce the following weighted energy functions: Ai (s) = βi Ei (s), Hi (s) = βi hi (s),

i = 1, 2, . . . ; (6.4.14)

H0 (s) = β0 h0 (s), where h 1 βi = λi+L = exp − 0

(i + L1 )L0 i , (1 + ν)(1 + L0 )

λ0 := exp[−L0 (1 + ν)−1 (1 + L0 )−1 ], ν is taken from (6.4.8). In terms of these functions, system (6.4.4) can be rewritten in the form 1

Hj (s)+Aj (s) 6 λ0 C5 Hj−1 (s) + C 6 ∆jp+1 (−A0j (s)) ∀ j > 1, ∀ s > 1, 1

Hj (s) 6 (1 + γ)λ0 Hj−1 (s) + C 8 ∆jp+1 (−A0j (s)) ∀ j > 1.

(6.4.15)

320

Chapter 6 Energy methods for the investigation of localized regimes

Iterating these inequalities, we obtain Hj (s) + Aj (s) 6 C5 (1 + γ)−1 λj1 H0 (s) 1 p+1

+ C 9 ∆j

j X

λj−i 1

∆  i

∆j

i=1

λ1 := (1 + γ)λ0 ,

1 p+1

(−A0i (s)),

(6.4.16)

C 9 = max{C 6 , (1 + γ)−1 C5 C 8 }.

We impose the following restriction on the choice of the free parameter γ : λ = (1 + γ)λ0 < 1

⇐⇒

γ < λ−1 0 − 1.

By analogy with (6.3.12), we now introduce the family of functions Uj (s) =

j X

λj−i 1

∆ 

i=1

i

∆j

1 p+1



 Ai (s) + Hi (s) ,

j = 1, 2, . . . .

(6.4.17)

The following relations [similar to (6.3.13)] are obvious: Uj (s) − Aj (s) − Hj (s) = θj Uj−1 (s),

j = 1, 2, . . . ,

U0 (s) = 0,

(6.4.18)

where, in view of (6.4.6) and (6.4.10), we have     ∆  1 j +L1 1+L0 j−1 p+1 = (1+γ)λ0 θj = λ 1 6 (1+γ)λ0 = θ0 . (6.4.19) ∆j j −1+L1 L0 Therefore, by virtue of (6.4.18) and (6.4.1), we obtain 1

Uj (s) 6 C5 (1 + γ)−1 λj1 H0 (s) + θ0 Uj−1 (s) + C 9 ∆jp+1 (−Uj0 (s)).

(6.4.20)

By using (6.4.7), (6.4.13), and (6.4.6), we estimate Uj (s0 ) as follows: j ∆  1 X
i p+1 Uj (s0 ) = λj−i βi hi (s0 ) + Ej (s0 ) 1 ∆j i=1 j 1 X j−i  ∆i  p+1   exp (1 − ln λ−1 λ1 6 eD1 0 )(i + L1 ) ∆j i=1

6 eD1

j X

  θ0j−i exp (1 − ln λ−1 0 )(i + L1 )

i=1

  6 eD1 (1 − θ0 )−1 exp (1 − ln λ−1 0 )(j + L1 ) .

(6.4.21)

Inequalities (6.4.20) and (6.4.21) now yield the following system: 1 b0 0 6 θ0 U j−1 (s) + C 9 ∆jp+1 (−U j (s)) ∀ s > 1, 1 − θ0 (6.4.22) 1  p  1 −1 p+1 p+1 − p+1 j = 1, 2, . . . , exp (1 − ln λ0 )p ω ∆j

U j (s) := Uj (s) − U j (s0 ) 6

D1 1 − θ0

where b0 := D1 C5 (1 + γ)−1 λ1 β0 exp(ωT

− p1

).

Section 6.4

321

Effective localization of singular boundary regimes

According to Lemma 9.2.7, the following uniform estimate is true for the solutions of system (6.4.22): " p 1 #  D1 e (1−θ0 ) (1−ln λ0−1 )C 9 p p+1 ω p+1 U j (s) 6 M j (s) = exp + s0 − s 1 1−θ0 (1−θ0 ) C 9 ∆jp+1 p

1

p+1 . ∀ j > 1, ∀ s : s0 6 s 6 s1 := s0 + (1−θ0 )−1 C 9 p p+1 (1−ln λ−1 (6.4.23) 0 )ω Hence, −1  e C5 λ1 β0 exp(ωT p )  + := D1 C10 . (6.4.24) Uj (s1 ) 6 D1 1 − θ0 1+γ

In view of (6.4.14) and (6.4.17), this implies the following estimate for the initial energy functions: i h (j + L ) ∀ j > 1. (6.4.25) Ej (s1 ) + hj (s1 ) 6 D1 C10 exp ln λ−1 1 0 Further, we sum estimates (6.4.25) over j from 1 to i and obtain i h (i + L ) ∀ i > 1, hti (s1 ) + E(ti , s1 ) 6 D1 C11 exp ln λ−1 1 0 where C11 = C10 (1 − λ0 )−1 . We continue the last estimate with regard for definition (6.4.12). This yields h i − p1 ω(T − t ) hti (s1 ) + E(ti , s1 ) 6 D1 C11 exp ln λ−1 ∀ i > 1. i 0 Finally, by using inequality (6.4.12) and relation (6.4.6) for the introduced constants once again, we arrive at the estimate   (L1 + 1) − p1 −1 ln λ0 ω(T − t) ht (s1 ) + E(t, s1 ) 6 D1 C11 exp L1   −1 ∀ t < T, (6.4.26) 6 D1 C11 exp ω1 (T − t) p ω1 =

ω , 1+ν

1

s1 = s0 +

p C 9 p p L0 (1 + ν)−1 ω p+1 . (1 − θ0 )(1 + L0 )

Thus, we have deduced estimate (6.4.9) with 1

c1 = C11

C 9 p p L0 . and c2 = (1 − θ0 )(1 + L0 )(1 + ν)



Lemma 6.4.2. Under the conditions of Lemma 6.4.1, there exist constants c3 > 0, c4 > 0, and α1 > 0 depending only on n, p, d0 , and d1 and such that the estimate ht (s) ˜ + E(t, s) ˜ 6 c3 (T − t)−α1 is true for any solution u(t, x).

∀ t < T,

p

where s˜ = 1 + c4 ω p+1 ,

(6.4.27)

322

Chapter 6 Energy methods for the investigation of localized regimes

Proof. Note that, by virtue of condition (6.4.2), estimate (6.2.9) implies the validity of inequality (6.4.7) for s0 = 1 with D1 = D1 := C2 + C1 h0 (1) exp(−ωT

− p1

).

(6.4.28)

We now fix the constants γ > 0, L0 > 0, and ν > 0 guaranteeing the validity of relation (6.4.6). Assume that the lower bound (6.4.8) takes place for the constant ω from condition (6.4.2). In this case, the conditions of Lemma 6.4.1 are satisfied. This lemma yields the following estimate for the solution: i h −1 ht (s1 ) + E(t, s1 ) 6 c1 D1 exp ω1 (T − t) p (6.4.29) ∀ t < T, ω1 = (1 + ν)−1 ω,

p

s1 = 1 + c2 ω p+1 .

If ω1 satisfies condition (6.4.8), i.e., ω1 > ω˜

⇐⇒

1

ω > (1 + ν)2 (T p−1 σ0−1 ) p ,

then we can apply Lemma 6.4.1 once again with the original inequality (6.4.29) instead of (6.4.7). As a result, we arrive at the estimate i h −1 ∀ t < T, ht (s2 ) + E(t, s2 ) 6 c21 D1 exp ω2 (T − t) p ω2 = (1 + ν)−2 ω,

p

p

p − p+1

s2 = s1 + c2 ω p+1 = 1 + c2 ω p+1 (1 + (1 + ν)

).

It is clear that this iterative procedure can be repeated as long as ω ωk = > ω, ˜ (1 + ν)k i.e., i h 1 k := ln(ω(T −1 p σ0 ) p )·(ln(1+ν))−1 times (here, [a] is the integral part of a). As a result, we obtain   ω − p1 ∀ t < T, ht (sk ) + E(t, sk ) 6 ck1 D1 exp (T − t) (1 + ν)k where k X p − ip sk = 1 + c2 (1 + ν) p+1 ω p+1 .

(6.4.30)

i=1

Note that Lemma 6.4.1 cannot be directly used at the point s = sk because condition (6.4.8) is not satisfied at this point and the following strict inverse inequality is true:  p  1 p ω T k+1 ⇐⇒ T > p ω sk . 1 (p σ0 ) p

(6.4.32) (6.4.33)

(k)

We consider estimate (6.4.33) as final for all s > sk , 0 6 t 6 t0 , and continue the iterative process of estimation of the energy functions of the solu(k) tion u(t, x) in the domain Qk (sk ) = {|x| > sk , t0 < t < T }. By definition (6.4.31), we get 1  Tk p (k) , where Tk = T − t0 . ωk = (1 + ν) pσ0 If we consider the solution u(t, x) only in the domain Qk (sk ), then this solution satisfies condition (6.4.8) and, hence, Lemma 6.4.1 is applicable in this domain. By virtue of this lemma, we find ht (sk+1 ) + E(t, sk+1 ) 6 ck+1 1 D 1 exp p



ωk −1 (T − t) p 1+ν



(k)

∀ t : t0 < t < T,

(6.4.34)

where sk+1 = sk + c2 ωkp+1 . Further, we consider the following initial point:  1 p ω (k+1) (k) t0 := T − p σ0 > t0 (1 + ν)k+2 and write the following relations implied by estimate (6.4.34): h i (k+1) − p1 D exp ω (T − t) ∀ t : t0 6 t 6 T, ht (sk+1 ) + E(t, sk+1 ) 6 ck+1 1 k+1 1   1+ν (k+1) ∀ s > sk+1 , 0 < t 6 t0 ht (s) + E(t, s) 6 ck+1 . 1 1 D exp (p σ0 ) p By analogy with the derivation of (6.4.34) from (6.4.32), the first of these estimates yields the inequality i h − p1 ∀ t > tk+1 ht (sk+2 ) + E(t, sk+2 ) 6 ck+2 D exp ω (T − t) 1 k+2 1 0 , p

where sk+2 = sk+1 + c2 ω p+1 . We use this estimate in the next step of the process of estimation. After the ith step, we arrive at the following relations: ht (sk+i ) + E(t, sk+i ) 6 Rk+i (t), sk+i = 1 + c2 ω

p p+1

k+i X

1

j=0

(1 + ν) p+1

jp

,

324

Chapter 6 Energy methods for the investigation of localized regimes

  ω (k+i−1) − p1    ∀ t : t0 < t < T, ck+i  1 D 1 exp (1+ν)k+i (T − t)     Rk+i (t) = (k+i−j−1) (k+i−j) D1 exp (p σ1+ν ck+i−j ∀ t : t0 < t 6 t0 ,  1/p 1  0)    j = 1, . . . , i, (6.4.35) (k+s)

1/p

where t0 := T − p σ0 ωk+s and ωk+s = ω(1 + ν)−(k+s) . We now define s˜ as follows: p

s˜ := lim sj = 1 + c4 ω j→∞

p p+1

,

c4 = c2

(1 + ν) p+1 p

.

(6.4.36)

(1 + ν) p+1 − 1

˜ + E(t, s). ˜ Since, For any point t < T, we establish the upper bound for ht (s) (j) for the sequence of points constructed above, we have t0 → T as j → ∞ for any chosen point t < T, there exists a number j ∈ N such that (j−1) (j)  t ∈ Γj := t0 , t0 . (6.4.37) In view of the fact that the functions ht (s) and E(t, s) are monotonically nonincreasing in s for any t < T, we conclude that estimate (6.4.35) implies the following relation: ht (s) ˜ + Et (s) ˜ 6 ht (sj+1 ) + E(t, sj+1 ) (j−1)

6 Rj+1 (t) = A1 cj1 ∀ t ∈ t0   1+ν A1 := D1 exp 1 . (p σ0 ) p

(j)  , t0 ,

(6.4.38)

It is easy to see that (j−1)
−α1

A1 cj1 = c3 T − t0

,

α1 =

p ln c1 , ln(1 + ν)

c3 = A1 (p σ0 ω)

α1 p

.

(6.4.39)

This relation enables us to derive the following finial estimate from (6.4.38): ht (s) ˜ + E(t, s) ˜ 6 c3 (T − t)−α1

∀ t < T,

where c3 = A1 (p σ0 ω)

α1 p



 ln c1  (1 + ν)  ln(1+ν) , p σ ω = D1 exp 0 (p σ0 )1/p

Thus, the required estimate (6.4.27) is proved.

α1 =

p ln c1 . ln(1 + ν) 

The remaining part of the proof of Theorem 6.4.1 is similar to the final part of the proof of Theorem 6.3.1. Let s0 : s0 > s˜ be an arbitrary point at which

Section 6.4

325

Effective localization of singular boundary regimes

the inequality of structure (6.3.31) holds with s0 instead of s. ¯ Without loss of generality, we can assume that the constant α1 in estimate (6.4.27) satisfies the inequality α1 > (p + 1)−1 . We fix the numbers 0 < ξ < 1, η > 0, and γ > 0 such that 1 α − 1 (1 + γ)ξ 1 p+1 := θ0 < 1; (6.4.40) < α1 < η. p+1 ˜ s0 ], we can now define a sequence {ti } = {ti }(s) by the For any point s ∈ [s, formula i
−η ξη h s) − E(t , s) + sup s) , (6.4.41) E(t , ti − ti−1 := h(t, i−1 i c3 T η−α1 ti−1 s by using relations (6.2.20). Further, we introduce new functions Aj (s) and Hj (s) according to relations (6.3.5), where α1 is determined from (6.4.27) [or from (6.4.39)] and the shifts {∆i } are specified by (6.4.41). It is easy to see that these new functions satisfy the system  ∆ α 1 1 j Hj (s) + Aj (s) 6 C5 Hj−1 (s) + C 6 ∆jp+1 (−A0j (s)) ∀ j > 1, ∆j−1 (6.4.42) 1

Hj (s) 6 λj Hj−1 (s) + C 8 ∆jp+1 (−A0j (s)) ∀ s > s, where ∆0 = ξ −1 ∆1 ,

H0 (s) = ∆α0 1 h0 (s),

 ∆ α 1 j λj = (1 + γ) . ∆j−1

The following relation is obvious: 1

1

λj λj−1 . . . λi+1 ∆ip+1 = (1 + γ)j−i ∆jp+1

 ∆ α 1 − j

1 p+1

. ∆i By using this relation, we iterate system (6.4.42) and obtain  ∆ α 1 j H0 (s) Hj (s) + Aj (s) 6 C5 (1 + γ)j−1 ∆0  X j  α 1 − 1 1 p+1 p+1 j−i ∆j 0 (−Aj (s)) , + C 9 ∆j (1 + γ) ∆i i=1 n C5 C 8 o C 9 = max C 6 , . (1 + γ)

(6.4.43)

By analogy with (6.3.41), we introduce new energy functions Uj (s),  α1 − 1 j X p+1
j−i ∆j Ai (s) + Hi (s) , j = 1, 2, . . . . Uj (s) := (1 + γ) ∆i i=1

326

Chapter 6 Energy methods for the investigation of localized regimes

Clearly, the following analogs of inequalities (6.3.42) are true: Uj (s) − Aj (s) − Hj (s) = θ(j) Uj−1 (s), where

j = 1, 2, . . . ,

U0 (s) = 0,

 ∆ α 1 − 1 p+1 j α − 1 6 (1 + γ)ξ 1 p+1 = θ0 < 1. θ(j) := (1 + γ) ∆j−1

In view of the last inequality and relation (6.4.43), we arrive at the following analog of relation (6.4.20): Uj (s) 6 C5 T α1 λ

j−1

1

h0 (s) + θ0 Uj−1 (s) + C 9 ∆jp+1 (−Uj0 (s)) ∀ s > s,

(6.4.44)

where λ = (1 + γ)ξ α1 < 1. According to (6.4.41), we estimate the initial value Uj (s) as follows: Uj (s) =

j X

(1 + γ)j−i

 ∆ α 1 −

i=1

=

j

∆i

−(η−α1 ) c3 T η−α1 ξ −η ∆j

6 c3 (1 −

1 p+1

j X


∆αi 1 Ei (s) + hi (s)

(1 + γ)j−i

i=1 −1 η−α1 −η −(η−α1 ) ξ ∆j . θ0 ) T

 ∆ η− j

1 p+1

∆i (6.4.45)

By Lemma 9.2.9, inequalities (6.4.44) and (6.4.45) yield the following universal estimate: Uj (s) 6 C5 T α1 h0 (1) + D4 U 0 (s − s) ∀ j > 1,

∀ s > s,

(6.4.46)

where −(η−α1 )(p+1)

U 0 (s) = s

 (η−α1 )(p+1) c3 T η−α1 eC 9 (η − α1 )(p + 1) D4 = . (1 − θ0 )ξ η (1 − θ0 )

,

Therefore, Uj (s) 6 2D4 U 0 (s − s) ∀ j > 1,

∀ s : s < s 6 s + δ0 ,

(6.4.47)

where δ0 is given by the equality U 0 (δ0 ) = C5 D4−1 T α1 h0 (1). By virtue of (6.4.41), we get the following relation from (6.4.47): 1 hj (s) + Ej (s) 6 2D4 U 0 (s − s)∆−α j


α1 = D5 U 0 (s − s) hj (s) + Ej (s) η −

where D5 = 2D4 ξ α1 c3

α1 η

T

−

(η−α1 )α1 η

.

∀ s ∈ [s, s + δ0 ],

(6.4.48)

Section 6.5

327

Effective localization of singular boundary regimes

Further, we fix an arbitrary number i ∈ N and sum inequalities (6.4.48) over j from 1 to i. As a result, by analogy with (6.3.54), we arrive at the inequality i X 1 1 hti (s) + E(ti , s) 6 2D4 U 0 (s−s) ∆−α 6 2D4 (1 − ξ α1 )−1 ∆−α U 0 (s−s) i j j=1 α1 −1


α1 U 0 (s−s) hi (s) + Ei (s) η ,

∀ s ∈ [s, s+δ]. (6.4.49) Reasoning as in the proof of Theorem 6.3.1 in deriving relation (6.3.59) from inequality (6.3.54), we obtain the following inequality from (6.4.49):
α1 ¯ + E(t, s) ¯ η ∀ t < T. ht (s) + E(t, s) 6 D5 (1 − ξ α1 )−1 U 0 (s − s) ht (s) (6.4.50) = D5 (1−ξ )

By Lemma 9.3.2, we get the following final estimate from inequality (6.4.50): ht (s) + E(t, s) 6 C(s − s) ˜ −η(p+1)

p

∀ s > s˜ = 1 + C4 ω p+1 ,

∀ t < T.

Thus, Theorem 6.4.1 is proved.

6.5

Effective localization of singular boundary regimes for the equations of nonstationary fast-diffusion type

In the present section, we consider the last remaining possibility for the relationship between the parameters p and q in Eq. (6.1.26). Namely, we study the case of “fast” nonstationary diffusion 0 < p < q.

(6.5.1)

It is shown that, for the analyzed equations, any singular boundary regime is localized and, moreover, it is a localized LS -regime on the boundary of the domain. Theorem 6.5.1. Let u(t, x) be an arbitrary energy solution of problem (6.1.26)–(6.1.29), (6.5.1). Then, independently of the power of singularity of the boundary regime, i.e., independently of the function F (t), the inclusion Ωs (u) ⊂ ∂0 Ω holds for the set of intensification of the solution u(t, x). This means that any boundary regime is a localized LS -regime. Moreover, the following universal estimate takes place: Z Z τZ q+1 h(τ, s) + E(τ, s) := |u(τ, x)| dx + |Dx u(t, x)|p+1 dx dt 0

Ω(s)

Ω(s)

− pq+2p+1 q−p

6 D1 h0 (1) + D2 (s − 1)

∀ s > 1,

∀ τ < T, (6.5.2)

where the constants D1 and D2 depend only on p, q, n, and T.

328

Chapter 6 Energy methods for the investigation of localized regimes

Remark 6.5.1. Estimate (6.5.2) is sharp in the entire class of blow-up boundary regimes. In order to prove this assertion, we consider the following Cauchy–Dirichlet problem of model type: vt = (v m )xx

∀ (t, x) ∈ (0, T ) × R1+ ,

v(t, 0) = ∞,

∀ t > 0,

v(0, x) = 0

∀ x ∈ R1+ .

0 < m < 1, (6.5.3)

This problem has an explicit solution, namely, the so-called solution of the “razor-blade” structure (see [129]):  x − 2 1−m . (6.5.4) V (t, x) = √ t Introducing a new unknown function u := v m , we rewrite Eq. (6.5.3) in the form 1
u m t − uxx = 0, U (t, x) := V (t, x)m . 1 This equation corresponds to the structure equation (6.1.26) with q = m and p = 1. The energy function h(t, s) corresponding to solution (6.5.4) has the form Z Z q+1 h(t, s) = U (t, x) dx = V (t, x)m(q+1) dx Ω(s) Ω(s) Z Z 2(1+m) m+1 1+3m m+1 1+m s− 1−m dx = t 1−m s− 1−m . (6.5.5) = V dx = t 1−m x>s

Ω(s)

It is easy to see that, for the analyzed equation, we have 1 + 3m pq + 2q + 1 = . q−p 1−m Hence, the asymptotics of the solution U (t, x) in a neighborhood of the intensification set exactly corresponds to the maximum growth rate admitted by the general estimate (6.5.2). To prove Theorem 6.5.1, we first deduce the main system of differential inequalities that controls the behavior of the energy functions corresponding to the solution u(t, x). Lemma 6.5.1. Let u(t, x) be an arbitrary energy solution of problem (6.1.26)– (6.1.34), (6.5.1). Then the corresponding energy functions Ej (s) and hj (s) in (6.2.20) satisfy the following system of differential inequalities: 1

q−p 1− (q+1)(p+1)

hi (s) 6 (1 + γ)hi−1 (s) + C1 ∆ip+1 (−Ei0 (s)) q+1

q−p 1− q(p+1)

+ C2 ∆i(p+1)q (−Ei0 (s)) C1 = C1 (γ) → ∞,

,

∀ s > 1, ∀ γ > 0, ∀ i ∈ N;

C2 = C2 (γ) → ∞ as γ → 0,

(6.5.6)

Section 6.5

329

Effective localization of singular boundary regimes 1

q−p 1− (q+1)(p+1)

hi (s) + Ei (s) 6 2(1 + c0 )hi−1 (s) + C3 ∆ip+1 (−Ei0 (s)) q+1

q−p 1− q(p+1)

+ C4 ∆i(p+1)q (−Ei0 (s))

∀ i ∈ N,

∀ s > 1,

(6.5.7)

where c0 is given by (6.2.24) and the constants C1 , C2 , C3 , and C4 are independent of s. Proof. We first prove the intermediate relation (6.2.18) from Lemma 6.2.2 valid for any p > 0 and q > 0. However, in this case, the second factor in the second term on the right-hand side of this relation is estimated by a different method. Namely, by virtue of Hölder’s inequality, we get Z  p+1 Z r+1 r−p p+1 r+1 , (6.5.8) |u(t, x)| dσ 6 (meas ∂0 Ω(s)) r+1 |u(t.x)| dσ ∂0 Ω(s)

∂0 Ω(s)

where r = p(q + 1)(p + 1)−1 . In view of the inequality in assertion (1) of Proposition 9.1.1, for the function v := |u|r u, we obtain   Z Z Z n X r+1 r r+1 |u(t, x)| dσ 6 k0 (r+1) |u| |Dxi u| dx + |u(t, x)| dx . ∂0 Ω(s)

Ω(s) i=1

By using

r(p+1) p

Z

Ω(s)

= q + 1 and Hölder’s inequality, we get r+1

|u(t, x)|

 Z

p+1

|Dx u|

dσ 6 k 0

∂0 Ω(s)



1 p+1

Z

q+1

|u(t, x)|

dx



p p+1

dx

Ω(s)

Ω(s)

Z |u(t, x)|

+

r+1

 dx ,

(6.5.9)

Ω(s)

where k 0 depends only on k0 , n, p, and q. Integrating inequality (6.5.9) with respect to t, we find Z eti Z |u(t, x)|r+1 dσ dt ti−1 ∂0 Ω(s)

Z

q+1

|u(t, x)|

6 k 0 sup t∈(ti−1 ,e ti ]

Z

e ti

p p+1 dx

Ω(s)

Z

× ti−1 Ω(s)

1 p p+1 e p+1 ∆ |Dx u(t, x)|p+1 dx dt i


q−r ei + k 0 meas Ω(s) q+1 ∆



Z sup

|u(t, x)|

q+1

 r+1 q+1 , dx

(6.5.10)

t∈(ti−1 ,e ti ] Ω(s)

where {ti } is the sequence specified in (6.2.20), t˜i is an arbitrary point from the interval (ti−1 , ti ], and ∆˜ i := t˜i − ti−1 . By virtue of (6.5.8) and (6.5.10),

330

Chapter 6 Energy methods for the investigation of localized regimes

estimate (6.2.18) yields the relation e h(t˜i , s) + c−1 0 Ei (s) Z Z −1 q+1 ˜ := |u(ti , x)| dx + c0

t˜i

ti−1

Ω(s)

Z

|Dx u(t, x)|p+1 dx dt

Ω(s)

r−p p
r−p q+1 e 0 (s)) p+1 meas ∂0 Ω(s) (r+1)(p+1) · ∆˜ (r+1)(p+1) 6 h(ti−1 , s) + d1 (−E i i q h p p 1 ei (s) p+1 ∆˜ p+1 × k 0 sup h(t, s) p+1 · E i

t∈(ti−1 ,e ti ] q−r

+ k 0 (meas Ω(s)) q+1 ∆˜ i ·

r+1

h(t, s) q+1

sup t∈(ti−1 ,t˜i ]

c0 = (i)

(i)

i

1 r+1

,

(6.5.11)

q . (q + 1)d0 (i)

For any i ∈ N and s > 1, there exists δ0 := δ0 (s) : 0 < δ0 < 1/2 such (i) that, for any δ : 0 < δ < δ0 , one can find t˜i = t˜i (δ, s) satisfying the inequality Z (1 − δ)hi (s) 6 |u(t˜i , x)|q+1 dx. (6.5.12) Ω(s)

Moreover, relation (6.5.11) implies the inequality e (1 − δ)hi (s) + c−1 0 Ei (s) r p 1
p ei0 (s) p+1 ∆˜ (r+1)(p+1) E ei (s) (p+1)(r+1) hi (s) (p+1)(r+1) 6 hi−1 (s) + c1 − E i 1
p 1 ei0 (s) p+1 ∆˜ p+1 hi (s) q+1 , + c2 − E (6.5.13) i

where  r−p 1  (q + 1)d1 r+1 (r+1)(p+1) max(meas ∂0 Ω(s)) c1 = , k0 s>1 q

q−r

c2 = c1 (meas Ω) (q+1)(r+1) .

To estimate the terms on the right-hand side of (6.5.13), we apply Young’s inequality with ε and obtain e (1 − δ)hi (s) + c−1 0 Ei (s) 1
p(r+1) ei0 (s) (p+1)r ei (s) + εhi (s) + c3 (ε)∆˜ p+1 − E 6 hi−1 (s) + εE i (q+1)
p(q+1) (6.5.14) + εhi (s) + c4 (ε)∆˜ iq(p+1) − Ei0 (s) (p+1)q ,

where c3 (ε) → ∞ and c4 (ε) → ∞ as ε → 0. It follows from (6.5.14) that e (1 − δ − 2ε)hi (s) + (c−1 0 − ε)Ei (s) q+1 1
p(q+1)
p(r+1) ei0 (s) (p+1)q . ei0 (s) (p+1)r + c4 ∆˜ q(p+1) · − E 6 hi−1 (s) + c3 ∆˜ ip+1 − E i

(6.5.15)

Section 6.5

331

Effective localization of singular boundary regimes

e 0 (s) 6 −E 0 (s) and both δ > 0 and ε > 0 can be made arbitrarily Since −E i i small, we get the required relation (6.5.6) from (6.5.15). We now return to relation (6.5.11) and set t˜i := ti . As a result, by analogy with (6.5.13), we find r

p

p

1

Ei (s) 6 c0 hi−1 (s) + c0 c1 (−Ei0 (s)) p+1 ∆i(r+1)(p+1) Ei (s) (p+1)(r+1) hi (s) (p+1)(r+1) 1

p

1

+ c0 c2 (−Ei0 (s)) p+1 ∆ip+1 hi (s) q+1 ,

(6.5.16)

where c1 and c2 are the same constants as in (6.5.13). We now add inequalities (6.5.13) and (6.5.16). This yields r

p

(1 − δ)hi (s) + Ei (s) 6 (1 + c0 )hi−1 (s) + (1 + c0 )c1 (−Ei0 (s)) p+1 ∆i(r+1)(p+1) p

1

× Ei (s) (p+1)(r+1) · hi (s) (p+1)(r+1) 1

p

1

+ (1 + c0 )c2 (−Ei0 (s)) p+1 ∆ip+1 hi (s) q+1 .

(6.5.17)

Here, we used the following obvious relations: ∆˜ i 6 ∆i

ei (s) 6 Ei (s) (−E e 0 (s)) 6 (−E 0 (s)). and E i i

Further, by analogy with (6.5.14), we estimate the terms on the right-hand side of (6.5.17) by using Young’s inequality with ε. As a result, we arrive at the inequality (1 − δ − 2(1 + c0 )ε)hi (s) + (1 − (1 + c0 )ε)Ei (s) 1

p(r+1)

6 (1 + c0 )hi−1 (s) + (1 + c0 )c3 ∆ip+1 (−Ei0 (s)) (p+1)r q+1

p(q+1)

+ (1 + c0 )c4 ∆iq(p+1) (−Ei0 (s)) q(p+1) .

(6.5.18)

We fix δ > 0 and ε > 0 such that the relations δ + 2(1 + c0 )ε 6 2−1 ,

ε(1 + c0 ) < 2−1

are true. By using inequality (6.5.18), we obtain the required relation (6.5.7) with C3 = 2(1 + c0 )c3 and C4 = 2(1 + c0 )c4 .  Proof of Theorem 6.5.1. To prove the theorem, we analyze relation (6.5.7). We rewrite the corresponding inequality for the interval (0, τ ] with arbitrary τ < T in the form Z  Z τZ q+1 Jτ (s) := sup |u(t, x)| dx + |Dx u(t, x)|p+1 dx dt t∈(0,τ ]

0

Ω(s)

Ω(s)

1− q−p 1 d (q+1)(p+1) 6 2(1 + c0 )h0 (1) + C3 T p+1 − E(τ, s) ds 1− q−p q+1  d q(p+1) ∀ s > 1, ∀ τ < T, + C4 T q(p+1) − E(τ, s) ds 

(6.5.19)

332

Chapter 6 Energy methods for the investigation of localized regimes

or, in view of the inequality d d E(τ, s) > Jτ (s), ds ds we get J τ (s) := Jτ (s) − 2(1 + c0 )h0 (1)  q+1  1 6 C3 T p+1 + C4 T q(p+1)   d 1− q−p 1− q−p  d (q+1)(p+1) q(p+1) . × max − J τ (s) , − J τ (s) ds ds

(6.5.20)

We now consider the behavior of the function J τ (s) solely in the domain of large values of this function, which is nonincreasing in s, i.e., for the values n q+1
o 1 s ∈ S := J τ (s) > C5 := C3 T p+1 + C4 T q(p+1) . (6.5.21) It is easy to see that d Iτ (s) > 1 ∀ s ∈ S. ds Hence, relation (6.5.20) implies the inequality −

 1− q−p d (q+1)(p+1) J τ (s) 6 C5 − J τ (s) ds

∀ s ∈ S.

(6.5.22)

Integrating this differential inequality, we immediately arrive at the following estimate: i−µ−1 h ∀ s > 1 : s ∈ S. (6.5.23) J τ (s) 6 Jτ (1)−µ + µC5−ν (s − 1) where µ=

q−p > 0, pq + 2p + 1

(q + 1)(p + 1) > 0. pq + 2p + 1

ν=

In view of (6.5.23), we get, in particular, the following universal estimate valid for all boundary regimes: Jτ (s) 6 2(1 + c0 )h0 (1) + µ

1 −µ

ν

1 −µ

C5µ (s − 1)

∀ s ∈ S,

∀ τ < T.

This inequality implies, inter alia, the required estimate (6.5.2).

(6.5.24) 

Chapter 7

Method of functional inequalities in peaking regimes for parabolic equations of higher orders 7.1

Boundary peaking regimes for quasilinear parabolic equations of higher orders

We now consider the conditions of localization (effective localization) of singular boundary regimes for general divergent parabolic equations
(B(u))t + A(u) := |u|q−1 u t X + (−1)m Dxα aα (t, x, u, Dx1 u, . . . , Dxm u) = 0, (7.1.1) |α|=m

where m > 1, q > 0, and aα (t, x, ξ) are continuous functions satisfying the following coercivity (parabolicity) conditions and the conditions of growth :  X p+1 X aα (t, x, ξ)ξα > d0 |ξβ | , p > 0, d0 > 0, (7.1.2) |α|=m

aα (t, x, ξ) 6 d1

|β|=m

 X

p |ξβ |

∀ (t, x, ξ) ∈ [0, T ] × Ω × RN (m) .

(7.1.3)

|β|=m

Here, N (m) is the number of different multiindices α := (α1 , α2 , . . . , αn ) of length |α| := α1 + α2 + . . . + αn that does not exceed m. We use the standard notation ∂ |α| , Dxm u = {Dxα u, |α| = m}. Dxα = α1 ∂x1 · · · ∂xαnn The energy methods presented in Chapter 6 cannot be directly applied to equations (7.1.1) for m > 1 because, in this case, it seems to be impossible to deduce a system of differential inequalities of the form (6.2.21), (6.2.22) for the corresponding energy functions. It turns out that a natural approach to the construction of an analog of the method of local energy estimates in the qualitative theory of elliptic and parabolic equations of higher orders is based not on the ordinary differential equations and inequalities but on specific functional inequalities (see [114] and [115]). The adaptation of the method of functional inequalities to the investigation of the asymptotic properties of solutions in the vicinity of the time of singular peaking of the boundary data based on the introduction of an infinite family of subdomains (strips) in the vicinity of peaking time was proposed in [117] and [43].

334

Chapter 7 Method of functional inequalities in peaking regimes

Let Ω be an arbitrary bounded domain in Rn , n > 1. Assume that its boundary ∂Ω = ∂0 Ω ∪ ∂1 Ω satisfy relation (6.1.25). Throughout Chapter 7, in order to simplify the presentation of the results of investigation of the localization of singular boundary regimes in bounded domains Ω, we impose the following restriction on the upper bound of the parameter q : (q + 1)−1 > (p + 1)−1 − mn−1 ⇐⇒

q + 1 < n(p + 1)(n − m(p + 1))−1

(7.1.4)

In the domain (0, T )×Ω, we consider solutions u(t, x) of Eq. (7.1.1) satisfying the following inhomogeneous initial and boundary conditions: u(0, x) = u0 (x) ∈ Lq+1 (Ω) in Ω,

¯ ku0 kq+1 Lq+1 (Ω) := h0 ,

Dxα (u − f ) = 0 on (0, T ) × ∂0 Ω ∀ α : |α| 6 m − 1,

(7.1.5) (7.1.6)

and the following homogeneous Dirichlet condition on (0, T ) × ∂1 Ω: Dxα u = 0 on (0, T ) × ∂1 Ω ∀ α : |α| 6 m − 1.

(7.1.7)

The function f in the Dirichlet condition (7.1.6) is the trace of a function f¯(t, x) on (0, T ) × ∂0 Ω such that

m (Ω, ∂1 Ω) , (7.1.8) f¯(t, ·) ∈ C loc [0, T ); Lq+1 (Ω) ∩ Lp+1, loc [0, T ); Wp+1
f¯t (t, ·) ∈ L1, loc [0, T ); Lq+1 (Ω) , (7.1.9) and moreover, Z F (t) := sup

|f¯(τ, x)|q+1 dx +

Z tZ

0 0, tj → T as j → ∞, and consider three families of energy functions corresponding to the three functions in (7.2.1): Z tj Z Ej (s) = |Dxm u(t, x)|p+1 dx dt ∀ s > 1, j = 0, 1, 2, . . . ; t−1 := 0; tj−1

Z

Ω(s) tj Z

Jj (s) = tj−1

Ω(s)

hj (s) =

|u(t, x)|p+1 dx dt ∀ s > 1, j = 0, 1, 2, . . . , Z sup |u(t, x)|q+1 dx, j = 0, 1, . . . .

(7.2.3)

t∈[tj−1 ,tj ] Ω(s)

We now deduce two systems of functional inequalities for these families of functions. The first of these systems connects the functions Ej (s) and hj (s), j = 1, 2, . . . , whereas the second system relates the functions Jj (s) and hj (s). Each of these systems has its domain of applicability in the analysis of the properties of localization of singular boundary regimes. The procedure of construction of these functional systems is based on the following a priori integral

Section 7.2

337

Energy functions of the solutions and the main system

estimates for the solutions: Lemma 7.2.2. ([117]) Let u(t, x) be an arbitrary energy solution of problem (7.1.1)–(7.1.6). Then, for any δ > 0, s > 1, 0 6 a < b < T and any ε > 0, the following relation is true: Z Z bZ |Dxm u(t, x)|p+1 dx dt |u(b, x)|q+1 dx + k1 a

Ω(s+δ)

Z

Ω(s+δ)

|u(a, x)|q+1 dx + c(ε)δ −m(p+1)

6 (1+ε)

Z bZ a

Ω(s)

Ω(s, δ) := Ω(s) \ Ω(s + δ),

|u(t, x)|p+1 dx dt,

Ω(s,δ)

k1 = d0 (q + 1)q −1 ,

(7.2.4)

where the constant c(ε) → ∞ as ε → 0 and, moreover, c(ε) depends neither on the solution u, nor on the parameters s and δ. Proof. We now introduce a C m -smooth cutoff function η0 (h) as follows: η0 (h) = 0 for h 6 0, η0 (h) = 1 for h > 1,

0 6 η0 (h) 6 1 ∀ h ∈ R1 .

This functions satisfies an additional condition η0q ∈ C m (R1 ),

where q > 0 is specified in (9.1.1).

(7.2.5)

We now fix arbitrary numbers s > 1 and δ > 0 and define the following cutoff function:   |x| − s ∀ x ∈ Ω : |∇ix η| 6 cδ −i ∀ i 6 m. (7.2.6) η(x) = η0 δ Further, we fix a number s1 : 1 < s1 < s and define an additional cutoff function   |x| − s1 η1 (x) := η0 ∀ x ∈ Ω. s − s1 It is clear that supp η1 ∈ {|x| > s1 } and η1 (x) = 1 for |x| > s. Hence, η1 (x)q+1 η(x) = η(x) ∀ x ⊂ Ω. In view of (7.2.5), we find








 |u|q−1 u t , ηu = η1q |u|q−1 u t , η1 ηu = |uη1 |q−1 uη1 t , ηη1 u .

(7.2.7)

Since
m v(t, x) := η1 · u(t, x) ∈ Lp+1, loc [0, T ); Wp+1 (Ω, ∂0 Ω) , by using (7.2.7) and the formula of integration by parts (see [Prop. 3.2 in 16]),

338

Chapter 7 Method of functional inequalities in peaking regimes

we arrive at the equality Z Z Z 
q+1 b

q+1 q−1 |v(b, x)| η dx − |v(a, x)|q+1 η dx |u| u t , ηu dt = q a ZΩ ZΩ q+1 |u(b, x)| η dx − |u(a, x)|q+1 dx (7.2.8) = Ω

Ω

valid for arbitrary numbers 0 6 a < b < T. Hence, substituting u(t, x)η(x) in the integral identity (7.1.11) as a test function and using (7.2.8), the structural conditions (7.1.2) and (7.1.3), and inequality (7.2.6), we obtain Z Z bZ q |u(b, x)|q+1 η dx + d0 |Dxm u(t, x)|p+1 η dx dt q+1 Ω(s) a Ω(s)  Z bZ  p Z p+1 q q+1 m p+1 6 |u(a, x)| η dx + c |Dx u| dx dt q + 1 Ω(s) a Ω(s)  m−1  1 Z Z b X p+1 −(m−i)(p+1) i p+1 . (7.2.9) × δ |Dx u| dx dt a

i=0

Ω(s,δ)

We now estimate the second term on the right-hand side by using assertion (3) of Proposition 9.1.2. This yields Z −(m−i)(p+1) |Dxi u(t, x)|p+1 dx δ Ω(s,δ)  Z −(m−i)(p+1) −i(p+1) 6 cδ δ |u|p+1 dx Ω(s,δ)

Z

|Dxm u|p+1 dx

+

 i Z m |u|

Ω(s,δ)

= cδ −m(p+1)

p+1

 m−i  m

dx

Ω(s,δ)

Z

|u|p+1 dx

Ω(s,δ)

Z +c

|Dxm u|p+1 dx

i Z m −m(p+1) δ

Ω(s,δ)

p+1

|u|

 m−i m dx .

Ω(s,δ)

Integrating this inequality with respect to t, substituting it in (7.2.9), and using Young’s inequality with ε, we get Z Z bZ q+1 |u(b, x)| η dx + k1 |Dxm u(t, x)|p+1 η dx dt a

Ω(s)

Z

Ω(s) q+1

|u(a, x)|

6

Z bZ η dx + ε a

Ω(s)

+ c(ε)δ

−m(p+1)

Z bZ a

Ω(s,δ)

|Dxm u|p+1 dx dt

Ω(s,δ)

|u|p+1 dx dt ∀ ε > 0. (7.2.10)

Section 7.2

339

Energy functions of the solutions and the main system

We consider (7.2.10) as a relation for the following functions: Z Z Ab (s) := |u(b, x)|q+1 dx, Aa (s) = |u(a, x)|q+1 dx, Ω(s)

Ω(s)

Z bZ

|u(t, x)|p+1 dx dt,

B(s) = a

Ω(s)

Z bZ

|Dxm u(t, x)|p+1 dx dt.

H(s) = k1 a

Ω(s)

In this notation, inequality (7.2.10) can be rewritten in the form Ab (s + δ) + H(s + δ)
6 εH(s) + c(ε)δ −m(p+1) B(s) − B(s + δ) + Aa (s).

(7.2.11)

In view of relation (7.2.11) and Lemma 9.3.1, we obtain the required estimate (7.2.4).  Lemma 7.2.3. ([43]) In the notation and under the conditions of Lemma 7.2.2, the following relation is true for any solution u of problem (7.1.1)–(7.1.7): Z Z bZ |u(b, x)|q+1 dx + k1 |Dxm u(t, x)|p+1 dx dt a

Ω(s+δ)

Z

q+1

|u(a, x)|

6

Ω(s+δ)

dx + c3 δ

−κ

 Z bZ a

Ω(s)

Z bZ

q+1

|u|

× a

+ c4

m−1 X

δ

p p+1

Ω(s,δ)

 p+1 q+1

dx

 1 p+1 dt

Ω(s,δ)

−(m−i)

 Z bZ a

i=0

|Dxm u|p+1 q+1

×

 p+θi p+1

Ω(s,δ)

Z bZ

|u| a



|Dxm u|p+1 dx dt

 p+1 q+1

dx

 1−θi p+1 dt ,

(7.2.12)

Ω(s,δ)

where κ =m+

n(p − q) , (p + 1)(q + 1)

θi =

i(p + 1)(q + 1) + n(p − q) . m(p + 1)(q + 1) + n(p − q)

Proof. By using the interpolation inequalities from Proposition 9.1.2, we estimate the integrals on the right-hand side of relation (7.2.9) in a somewhat different way, as compared with Lemma 7.2.2. Namely, Z θi Z (1−θi )(p+1) Z q+1 i p+1 m p+1 q+1 |Dx u(t, x)| dx 6 c |Dx u| dx |u| dx Ω(s,δ)

Ω(s,δ)

+ cδ

i(p+1)(q+1)+n(p−q) − q+1

Ω(s,δ)

Z

q+1

|u(x, t)| Ω(s,δ)

 p+1 q+1

dx

,

340

Chapter 7 Method of functional inequalities in peaking regimes

where θi are specified in (7.2.12). Integrating the last inequality with respect to t and substituting the estimate obtained as a result in (7.2.9), after simple calculations, we arrive at relation (7.2.12).  Lemma 7.2.4 ([117]). The energy functions hj (s) and Jj (s) defined in (7.2.3) satisfy the following system of functional inequalities: ∆Jj (s) δ m(p+1)

hj (s + δ) 6 Rj (s, δ) := (1 + ε)hj−1 (s) + c(ε) 0 Jj (s + δ) 6 c∆1−θ Rj (s, δ)1+ν j

∀ j ∈ N;

∀ s > 1,

where ∆Jj (s) := Jj (s) − Jj (s + δ),

ν=

∀ ε > 0,

(7.2.13)

∀ δ > 0,

(7.2.14)

(p − q)(1 − θ0 ) , q+1

and θ0 is specified in (7.2.12). Proof. Estimate (7.2.13) directly follows from inequality (7.2.4) for a = tj−1 and b 6 tj . Since the solution u(t, x) satisfies the homogeneous boundary condition (7.1.7), assertion (2) of Proposition 9.1.2 yields the following interpolation inequality: Z |u(t, x)|p+1 dx Ω(s+δ)

Z 6c

|Dxm u(t, x)|p+1 dx

θ0  Z

q+1

|u|

Ω(s+δ)

 (1−θ0 )(p+1) q+1

dx

Ω(s+δ)

(7.2.15) ∀ s > 0, δ > 0. Integrating this inequality with respect to t and using estimate (7.2.4), we obtain  Z tj Z θ0 m p+1 0 Jj (s + δ) 6 c |Dx u| dx dt ∆1−θ j tj−1

Ω(s+δ)

 ×

Z

q+1

|u(t, x)|

sup t∈(tj−1 ,tj ] θ0 +

0 6 c∆1−θ Rj (s, δ) j

 (p+1)(1−θ0 ) q+1

dx

Ω(s+δ) (p+1)(1−θ0 ) q+1

0 = c∆1−θ Rj (s, δ)1+ν , j

which corresponds to the required inequality (7.2.14).



Lemma 7.2.5. The energy functions hj (s) and Ej (s) given in (7.2.1) satisfy the following system of functional inequalities: hj (s + δ) 6 (1 + ε)hj−1 (s) + c(ε)∆Ej (s)Γ(Pj (s, δ)) ∀ ε > 0, ∀ j ∈ N, Ej (s + δ) 6 c1 hj−1 (s) + c2 ∆Ej (s)Γ(Pj (s, δ)),

∀ s > 1, δ > 0,

(7.2.16) (7.2.17)

Section 7.2

341

Energy functions of the solutions and the main system

where ∆Ej (s) := Ej (s) − Ej (s + δ);

1−θm−1

1

Γ(h) := h q(p+1) + h (p+1)(q+θm−1 ) ,

−κ(p+1)(q+1) Pj (s, δ) := Ej (s)p−q ∆q+1 , j δ

and κ and θm−1 are specified in (7.2.12). Proof. By using Young’s inequality with ε, we get the following inequalities from relation (7.2.12): p

1

1

hj (s + δ) 6 hj−1 (s) + c3 δ −κ (∆Ej (s)) p+1 hj (s) q+1 ∆jp+1 + c4

m−1 X

p+θi

δ −(m−i) (∆Ej (s)) p+1 hj (s)

1−θi q+1

1−θi

∆jp+1

i=0

h 1 i q+1 p q 6 hj−1 (s) + εhj (s) + c(ε) δ −κ (∆Ej (s)) p+1 ∆jp+1 + c(ε)

m−1 Xh

p+θi

1−θi

δ −(m−i) (∆Ej (s)) p+1 ∆jp+1

i q+1

q+θi

∀ ε > 0. (7.2.18)

i=0

The relations p(q + 1) p−q =1+ , (p + 1)q q(p + 1)

κ(q + 1) m(p + 1)(q + 1) + n(p − q) = , q q(p + 1)

(p − q)(1 − θi ) (p + θi )(q + 1) =1+ , (p + 1)(q + θi ) (q + θi )(p + 1) (m − i)(q + 1) 1 − θi = [m(p + 1)(q + 1) + n(p − q)] q + θi (p + 1)(q + θi ) can be easily verified. Thus, the last inequality immediately implies that hj (s + δ) 6 hj−1 (s) + εhj (s) + c(ε)∆Ej (s)Gj (s, δ), where 1

Gj (s, δ) := Pj (s, δ) q(p+1) +

m−1 X

(7.2.19)

1−θi

Pj (s, δ) (p+1)(q+θi ) .

i=0

By virtue of Lemma 9.3.1 and inequality (7.2.19), we arrive at the following relation: hj (s + δ) 6 (1 + ε)hj−1 (s) + c(ε)∆Ej (s)Gj (s, δ). Since 0 < θi 6 θm−1 for all i 6 m − 1, this inequality yields estimate (7.2.16). To prove (7.2.17), we consider (7.2.12) once again. By analogy with (7.2.18) and (7.2.19), for a = tj−1 and b = tj , this implies the following inequality: Ej (s + δ) 6 k1−1 hj−1 (s) + εhj (s) + c(ε)∆Ej (s)Gj (s, δ).

342

Chapter 7 Method of functional inequalities in peaking regimes

We now add the last inequality to (7.2.19) and obtain hj (s + δ) + Ej (s + δ) 6 (1 + k1−1 )hj−1 (s) + εhj (s) + c(ε)∆Ej (s)Gj (s, δ). In view of Lemma 9.3.1, this inequality leads to estimate (7.2.17) with c1 = (1 + k1−1 )(1 − ε)−1

and c2 = c(ε),

where ε > 0 is an arbitrary number satisfying the inequality −

ε < ε0 := 2

7.3

κ(q+1) q

.



Localized singular boundary regimes: the case of slow diffusion p > q

In the present section, we give an improved proof of the following general result concerning a sufficient condition for the effective localization of singular boundary regimes: Theorem 7.3.1 (see Theorem 1 in [117] and Theorem 1.2 in [45]). Assume that the boundary function f in condition (7.1.5) satisfies conditions (7.1.8) and (7.1.9) and that the corresponding function F (t) in (7.1.10) satisfies the estimate q+1 , ω = const > 0. F (t) 6 ω(T − t)−α0 ∀ t < T, α0 = (7.3.1) p−q Then there exists a constant c < ∞ that depends only on the known parameters n, p, q, d0 , and d1 of problem (7.1.1)–(7.1.9), (7.3.1) and does not depend on ω from relation (7.3.1) such that the inclusion p−q

Ωs (u) ⊂ Ω ∩ {|x| 6 1 + c ω m(p+1)(q+1)+n(p−q) }

(7.3.2)

holds for the set of intensification Ωs (u) of any energy solution u of the analyzed problem with singular peaking. In particular, if Ω ⊂ BR1 := {x : |x| < R1 }, then the boundary regime (7.3.1) is effectively localized whenever ω < ω0 , where ω0 is given by the equality p−q

1 + c ω0m(p+1)(q+1)+n(p−q) = R1 . Proof. We split the proof into several steps. First, we transform the functional system (7.2.13), (7.2.14) obtained in Lemma 7.2.4. We fix an arbitrary number j ∈ N and introduce j nonnegative free parameters δi , i 6 j . Since the function Ji (s) is monotone and the parameters s and δ in system (7.2.13), (7.2.14) can be chosen arbitrarily, we get the following j pairs of inequalities:

Section 7.3

343

Localized singular boundary regimes: the case of slow diffusion

      i−1 i X X Ji (s) 1+ν 0 s + + ε)h δ + c(ε) , Ji s + δk 6 c∆1−θ (1 i−1 k i m(p+1) δi k=1 k=1 i = 1, 2, . . . , j,     i i−1 X X (7.3.3) Ji (s) hi s + δk 6 (1 + ε)hi−1 s + δk + c(ε) m(p+1) , δi k=1 k=1 i = 1, 2, . . . , j. We now perform the iterativeP estimation. In the first inequality in (7.3.3), we estimate the term hj−1 (s + j−1 k=1 δk ) on the right-hand side for i = j by using the second inequality in (7.3.3) with i = j − 1. Then we estimate the P δ ) on the right-hand side of the obtained inequality by term hj−2 (s + j−2 k=1 k using the second inequality in (7.3.3) with i = j − 2, etc. After j steps of this estimation procedure, we arrive at the inequality   j X Jj s + δk k=1

6

0 c∆1−θ j

 c(ε)

j X

−m(p+1) δi (1+ε)j−i Ji (s)

1+ν + (1+ε) h0 (s) . j

(7.3.4)

i=1

We now define one more family of energy functions 1

Ii (s) := Ji (s) 1+ν ,

i = 1, , 2, . . . .

(7.3.5)

As a result of simple calculations, we obtain the following inequalities from relation (7.3.4):   j X Ij s + δk k=1

6 c(ε)

j X ∆

1−θ0 1+ν

i

i=1

(1+ε)j−i Ii (s)1+ν  ∆j  m(p+1) ∆i δ

1−θ0 1+ν

1−θ0

+ c(1+ε)j ∆j1+ν h0 (s)

i

(7.3.6) ∀ ε > 0, ∀ s > 1, ∀ δk > 0, j = 1, 2, . . . . In our analysis of boundary singularities, the main application of the functional system (7.2.13), (7.2.14) and system (7.3.6) obtained as a result of its transformation is the proof of the following assertion: q+1 Lemma 7.3.1. Assume that there exist constants s0 > 1 and γ : p−q >γ >0 such that the estimate Z Z tZ h(t, s0 ) + E(t, s0 ) = |u(t, x)|q+1 dx + |Dxm u|p+1 dx dt Ω(s0 )

0

Ω(s0 )

q+1 − p−q +γ

6 CFγ (t) := C(T − t)

∀ t ∈ (0, T ),

C = const < ∞, holds for some solution u(t, x) of problem (7.1.5)–(7.1.10).

(7.3.7)

344

Chapter 7 Method of functional inequalities in peaking regimes

Then there exists a continuous nonincreasing function K(s) such that K(s) < ∞ ∀ s > 0 and the estimate Z tZ |u(τ, x)|p+1 dxdτ 6 K(s − s0 ) ∀ s > s0 , ∀ t < T. (7.3.8) J(t, s) = 0

Ω(s)

holds uniformly in t ∈ (0, T ). Proof. First, we use estimate (7.3.7) to establish the upper bound for Jj (s0 ). To this end, for s + δ = s0 , we integrate the interpolation inequality (7.2.15) with respect to t. As a result, we obtain Jj (s0 ) 6 cEj (s0 )θ0 hj (s0 )

(1−θ0 )(p+1) q+1

0 ∆1−θ , j

j = 0, 1, 2, . . . .

Estimating the factors on the right-hand side by using inequality (7.3.7), we arrive at the inequality 0 Jj (s0 ) 6 cC 1+ν Fγ (tj )1+ν ∆1−θ , j

j = 0, 1, 2, . . . .

(7.3.9)

Note that this inequality is true for any choice of the sequence {ti } and, hence, it is also true for {∆i }. We now introduce a positive number µ = µ(γ), where γ is the same as in (7.3.7). This number satisfies only the following restriction: q+1 q+1 , 6γ< 1 g 1+ν .

(7.3.11)

It is clear that the absence of these points means that Lemma 7.3.1 is proved. Since the function J(T, s) ¯ is monotonic in the second argument, we conclude that inequality (7.3.11) holds for all s¯ from a certain interval (s0 , s0 ), where s0 > s0 . By analogy with (6.3.32), for every s, ¯ we define a continuous positive increasing and strictly monotone function Φs¯ : [0, t∞ ] → [Φs¯ (0), T ] by the equality (Φs¯ (t) − t)1−θ0 (J(Φs¯ (t, s) ¯ − J(t, s)) ¯ ν+µ = g 1+ν ,

(7.3.12)

where the value Φs¯ (0) is obviously determined by the equality Φs¯ (0)1−θ0 J(Φs¯ (0), s) ¯ ν+µ = g 1+ν

(7.3.13)

Section 7.3

Localized singular boundary regimes: the case of slow diffusion

345

and the quantity t∞ : Φs¯ (t∞ ) := T satisfies the relations (a) t∞ < T

if

(b) t∞ = T

if

lim J(t, s) ¯ 6 c < ∞;

t→T

(7.3.14)

lim J(t, s) ¯ = ∞.

t→T

It is also clear that t < Φs¯ (t) < T

∀ t ∈ [0, t∞ ).

(7.3.15)

We now define a sequence {ti } = {ti (s)} ¯ by the recurrence relation ti := Φ(ti−1 ),

i = 1, 2 . . . ,

t0 > 0,

(7.3.16)

similar to (6.3.33). Here, t0 ∈ [0, Φs¯ (0)) is chosen to guarantee the validity of the equality lim ti = T.

(7.3.17)

i→∞

In the case (b) of (7.3.14), sequence (7.3.16) is infinite and strictly increasing; moreover, it satisfies (7.3.17) for any t0 > 0. In the case (a) of (7.3.14), as follows from the continuity and monotonicity of the function Φs¯ (t), there exists a unique value t0 ∈ [0, Φs¯ (0)) such that, for some finite number i0 , the numbers specified by (7.3.16) satisfy the relation ti0 −1 = t∞ =⇒ ti0 = Φs¯ (ti0 −1 ) = T.

(7.3.18)

The indicated value t0 is taken as the initial value in definition (7.3.16) in the case (a). In the case (b), we set t0 = 0. Hence, relation (7.3.16) uniquely determines the sequence {ti }. Thus, the sequence of shifts {∆i } := {∆i (s)} ¯ is also uniquely determined. In terms of the last sequence, equalities (7.3.12) and (7.3.16) yield the following relation: ∆i (s) ¯ 1−θ0 Ji (s) ¯ ν+µ = g 1+ν ,

∀ i > 1,

(7.3.19)

quite important for our subsequent presentation. Since Ji (s) ¯ 6 Ji (s0 ) ∀ i ∈ N, it follows from definitions (7.3.12) and (7.3.16) and estimate (7.3.9) that (1−θ0 )(1+ν+µ)

cν+µ C (1+ν)(ν+µ) ∆i

Fγ (ti )(1+ν)(ν+µ) > g 1+ν

∀ i > 1, ∆i = ∆i (s), ¯

or, by the definition of Fγ (t), 1+ν

∆i > C1 g (1−θ0 )(1+ν+µ) (T − ti )B , where −

ν+µ

−

(1+ν)(ν+µ)

C1 = c (1−θ0 )(1+ν+µ) C (1−θ0 )(1+ν+µ) ,   (1 + ν)(ν + µ) q+1 −γ B= . p−q (1 − θ0 )(1 + ν + µ)

(7.3.20)

346

Chapter 7 Method of functional inequalities in peaking regimes

By using the equality 1 − θ0 =

m(p + 1)(q + 1) , m(p + 1)(q + 1) + n(p − q)

one can easily show that B =1+

µ(q + 1) − γ(1 + ν)(ν + µ)(p − q) . ν(q + 1)(1 + ν + µ)

This means that conditions (7.3.10) are sufficient for the validity of the inequality B = B(γ, µ) 6 1.

(7.3.21)

Thus, under condition (7.3.10), inequality (7.3.20) yields the relation 1+ν

∆i > C1 g (1−θ0 )(1+ν+µ) T −(1−B) (T − ti ),

i = 1, 2, . . . ,

and, hence, ∆i+1 6 C2 ∆i ,

C2 = T 1−B C1−1 g

i = 1, 2, . . . ,

1+ν 0 )(1+ν+µ)

− (1−θ

.

(7.3.22)

Our first condition imposed on the choice of the parameter g is C2 < 1 ⇐⇒ g > (T 1−B C1−1 )α ,

α=

(1 − θ0 )(1 + ν + µ) . 1+ν

(7.3.23)

For the functions Ii (s) given by (7.3.5), equalities (7.3.12) and (7.3.16) can be rewritten in the form 1−θ

0 − (1+ν)(ν+µ)

1

Ii (s) ¯ = g ν+µ ∆i

,

i = 1, 2, . . . .

(7.3.24)

We now estimate this quantity by using (7.3.22). This yields 1

(1−θ0 )(j−i)

1−θ

0 − (1+ν)(ν+µ)

Ii (s) ¯ 6 g ν+µ C2(1+ν)(ν+µ) ∆j

(1−θ0 )(j−i)

= C2(1+ν)(ν+µ) Ij (s) ¯ ∀ i 6 j.

(7.3.25)

We now consider our main functional relation (7.3.6). We fix a constant b such that 0 < b < 1 and determine the shifts {δk } as follows: δk := δbj−k ,

k = 1, 2, . . . , j,

0 < b < 1,

δ > 0.

(7.3.26)

In view of relations (7.3.19) and (7.3.22)–(7.3.25), after simple calculations, we obtain 1−θ0  j−i j 1+ν δ  + ε)C gc(ε) X (1 1−µ 2 Ij s¯ + 6 m(p+1) Ii (s) ¯ 1−b δ bm(p+1) i=1



1−θ0

(j−1)(1−θ0 ) 1+ν

+ c∆11+ν (1 + ε)j C2

h0 (s) ¯

Section 7.3

Localized singular boundary regimes: the case of slow diffusion

6

gc(ε) δ

I (s) ¯ 1−µ m(p+1) j

j  X (1 + ε)C i=1

1−θ0 1+ν

+ c∆1

1−θ0 ν+µ

2 bm(p+1)

347

j−i

1−θ0

(1 + ε)((1 + ε)C21+ν )j−1 h0 (s). ¯

(7.3.27)

Finally, we choose the values of the free parameters g, b, ε, and µ guaranteeing the validity of the inequality 1−θ0

(1 + ε)C21+ν b−m(p+1) 6 2−1 .

(7.3.28)

Substituting C2 specified in (7.3.22), and (7.3.20) in this relation, we conclude that condition (7.3.28) is equivalent to the inequality g > (1 + ε)1+ν+µ T

(1−B)(1−θ0 )(1+µ+ν) 1+ν

ν+µ

c 1+ν C ν+µ 21+ν+µ b−m(p+1)(1+ν+µ) , (7.3.29)

where C, c, and B are given in (7.3.7), (7.2.15), and (7.3.20), respectively. It is clear that, for any µ specified in (7.3.10) and any ε > 0 and 0 < b < 1, inequalities (7.3.23) and (7.3.29) establish a finite lower bound for any admissible value of the parameter g = g(ε, µ, b). For this choice of ε, µ, b, and g = g(ε, µ, b), inequality (7.3.27) yields the following relation:  δ  2gc(ε) ¯ 1−µ Ij s¯ + 6 m(p+1) Ij (s) 1−b δ  bm(p+1) j−1 1−θ0 h0 (s), ¯ j = 1, 2, . . . . (7.3.30) + c ∆11+ν (1 + ε) 2 In view of (7.3.25) and (7.3.30), we conclude that  δ  2gc(ε) (1−θ0 )(1−µ)(j−i) ¯ 1−µ Ii s¯ + 6 m(p+1) C2 (1+ν)(ν+µ) Ij (s) 1−b δ  bm(p+1) i−1 1−θ0 + c∆11+ν (1 + ε) h0 (s) ¯ ∀ i 6 j. (7.3.31) 2 We now raise these inequalities to the (1 + ν)th power and find their sum over i from 1 to j. This yields     δ δ ¯ − J t0 , + J tj , s¯ + 1−b 1−b 6 B1 δ −m(p+1)(1+ν) Jj (s) ¯ 1−µ + B2 h0 (s) ¯ 1+ν 6 B1 δ −m(p+1)(1+ν) J(tj , s) ¯ 1−µ + B2 h0 (s) ¯ 1+ν ∀ j > 1, ∀ δ > 0, ∀ s¯ > s0 , where 1+ν

B1 = (1 + ν)(2gc(ε))

 −1 (1−θ0 )(1−µ) ν+µ · 1 − C2 ,

0 (1 + ε)1+ν (1 − 2−1 bm(p+1) )−1 . B2 = (1 + ν)c1+ν ∆1−θ 1

(7.3.32)

348

Chapter 7 Method of functional inequalities in peaking regimes

In the case (b) of (7.3.14), i.e., for t0 = 0, we obtain   δ ¯ 6 h¯ 0 , J t0 , s¯ + = 0 , h0 (s) 1−b where h¯ 0 is given in (7.1.5). By Lemmas 9.3.2 and 9.3.3, relation (7.3.32) yields the required estimate (7.3.8) with   m(p+1)(1+ν) − 1+ν ¯ µ , K(s) = K1 (s) := max 2B2 h0 , B3 s (7.3.33) B3 := 2

m(p+1)(1+ν) (1−µ)µ2

1

−

(2B1 ) µ · (1 − b)

m(p+1)(1+ν) µ

.

In the case (a) of (7.3.14), i.e., for t0 > 0, in view of the arbitrariness of the choice of values of the arguments in inequality (7.3.32), we obtain     δ δ δ = J tj , s¯ + J tj , s¯ + + 1−b 2(1 − b) 2(1 − b) 1−µ  −m(p+1)(1+ν)  δ δ J tj , s¯ + 6 B1 2 2(1 − b)   1+ν  δ δ + J t0 , s¯ + + B2 h0 s¯ + (7.3.34) (1 − b) 2(1 − b) ∀ δ > 0, ∀ s¯ > s0 , ∀ j > 1 . We now estimate the second and third terms on the right-hand side. By using inequality (7.2.4) for b = t0 and a = 0, we obtain    −m(p+1) δ δ ¯ h0 s¯ + 6 (1 + ε)h0 + c(ε)J(t0 , s) ¯ . (7.3.35) 2(1 − b) 2(1 − b) Since J(t0 , s) is monotone in s, we get   δ 6 J(t0 , s) ¯ . J t0 , s¯ + 1−b

(7.3.36)

It remains to establish the upper bound for J(t0 , s). ¯ In view of (7.3.16), (7.3.9), and the definition (7.3.7) of the function Fγ (t), we find q+1 −( p−q −γ)(1+ν)

J(t0 , s) ¯ 6 J(Φs¯ (0), s) ¯ 6 c C 1+ν Φs¯ (0)1−θ0 (T − Φs¯ (0))

.

(7.3.37)

On the other hand, according to definition (7.3.13), we obtain 1+ν

−

J(Φs¯ (0), s) ¯ = g ν+µ Φs¯ (0)

1−θ0 ν+µ

.

(7.3.38)

Comparing (7.3.37) and (7.3.38), we conclude that 1+ν

q+1 −γ)(1+ν) ( p−q

g ν+µ (T − Φs¯ (0))

6 c C 1+ν Φs¯ (0)

(1−θ0 )(ν+µ+1) ν+µ

.

Section 7.3

Localized singular boundary regimes: the case of slow diffusion

349

This yields the following a priori universal lower bound for Φs¯ (0): ¯ = const > 0. Φs¯ (0) > Φ

(7.3.39)

¯ depends on known parameters of the problem and does Here, the constant Φ not depend on the solution u and on the point s. ¯ Returning to (7.3.38) and (7.3.37), we conclude that 1+ν

−

¯ J(t0 , s) ¯ 6 B3 = g ν+µ Φ

1−θ0 ν+µ

.

(7.3.40)

We now estimate the terms on the right-hand side of (7.3.34) by using inequalities (7.3.35), (7.3.36), and (7.3.40). This yields    −m(p+1)(1+ν) δ δ J tj , s+ ¯ [J(tj , s) ¯ 1−µ + B4 ] + B5 6 B1 1−b 2  −m(p+1)(1+ν) 1 δ 1−µ 1−µ 6 B1 (1−µ)−1 [J(tj , s)+B ¯ ] + B5 , 4 2 (7.3.41) where B4 = (1 + ν)B2 B1−1 c(ε)1+ν B31+ν (1 − b)−m(p+1)(1+ν) , B5 = B3 + (1 + ν)B2 (1 + ε)1+ν h¯ 1+ν 0 . By virtue of Lemma 9.3.3 and inequality (7.3.41), we arrive at estimate (7.3.8) with    1  1 m(p+1)(1+ν) − 1−µ µ , B6 s K(s) = K2 (s) := max 2 B5 + B4 + B41−µ , B6 = 2

m(p+1)(1+ν) µ2 (1−µ)



2B1 1−µ

1  µ

2 1−b

(7.3.42)

 m(p+1)(1+ν) µ

.



Remark 7.3.1. Analyzing the structure of the functions K1 (s) from (7.3.33) and the functions K2 (s) from (7.3.42), we can easily deduce the following universal estimate for the functions K(s) in (7.3.8):   m(p+1)(1+ν) − 1+ν ¯ µ ∀ s > 0, K(s) 6 max c1 + c2 h0 , c3 s where h¯ 0 =

Z

uq+1 dx and the constants c1 , c2 , and c3 depend only on given 0

Ω

parameters of problem (7.1.1)–(7.1.10), (7.3.7) and are independent of the solution u and the initial function u0 . Lemma 7.3.1 is an efficient tool for the analysis of the asymptotic properties of solutions in the case of appearance of the zone of “flat” singularity, i.e., for the boundary regime (7.3.6). In the zones where this flatness is absent, the energy

350

Chapter 7 Method of functional inequalities in peaking regimes

functions of the investigated singular solution are controlled by other tools, which are described in what follows. Assume that, for some s0 > 1, the energy functions (7.2.1) of the solution u(t, x) satisfy the following estimate h(t, s0 ) + E(t, s0 ) 6 F (t) ∀ t < T,

(7.3.43)

where F (t) is a continuous positive nondecreasing function that does not satisfy estimate (7.3.7), i.e., q+1

F (t)(T − t) p−q > c > 0 ∀ t < T.

(7.3.44)

The function F (t) with property (7.3.44) is used to define a sequence {tj } as follows: F (tj ) := r−1 F (tj−1 ) ∀ j ∈ N,

t0 = 0,

∆j = tj − tj−1 ,

(7.3.45)

where 1 > r > 0 is a free parameter determined in what follows. We also define a sequence {δ¯i } as the minimal monotonically nondecreasing majorant of the sequence {δi }: p−q

1

δ¯i > δi := (F (ti ) q+1 ∆i ) κ(p+1) , i = 1, 2, . . . , κ = m +

n(p − q) . (7.3.46) (p + 1)(q + 1)

Lemma 7.3.2. Assume that estimate (7.3.43) is true for a solution u(t, x) of problem (7.1.5)–(7.1.10) with some s0 > 1 and some function F (t) satisfying (7.3.44). Let hj (t, x) and Ej (t, s) be energy functions of the solution u determined by the sequence {tj } specified in (7.3.45). Then, for any b ∈ (0, 1), there exists a value of the parameter r = r(b) ∈ (0, 1) from the definition (7.3.45) of shifts {∆i } such that the following estimates are true: Ej (s0 + c0 δ¯j ) 6 rjb F (tj ) ∀ j ∈ N,

∀ t < T,

hj (s0 + c0 δ¯j ) 6 rjb F (tj ) ∀ j ∈ N,

∀ t < T,

(7.3.47)

where c0 > 0 is independent of j. Proof. By virtue of condition (7.3.43), we find hj (s0 ) + Ej (s0 ) 6 F (tj ) ∀ j > 1. This inequality yields the following estimates for the functions Pj (s, δ) given by (7.2.16), (7.2.17): Pj (s0 , δ) 6 δ −κ(p+1)(q+1) F (tj )p−q ∆q+1 j

∀ j > 1, ∀ δ > 0.

(7.3.48)

(1)

We define the first series of shifts {δj } by the formula (1)

δj

=ξ

1 − κ(p+1)(q+1)

δ¯j

∀ j > 1,

δ¯j is specified in (7.3.46),

(7.3.49)

Section 7.3

Localized singular boundary regimes: the case of slow diffusion

351

where ξ > 0 is a small parameter whose value is fixed in what follows. By virtue of (7.3.48), we get (1)

∀ j > 1.

Pj (s0 , δj ) 6 ξ

(7.3.50)

Hence, for the function Γ(Pj (s, δ)), we obtain the following estimate: 1−θm−1

1

(1)

Γ(Pj (s0 , δj )) 6 ζ(ξ) := ξ q(p+1) + ξ (p+1)(q+θm−1 ) .

(7.3.51)

Therefore, inequality (7.2.17) implies that (1)

Ej (s0 + δj ) 6 c1 F (tj−1 ) + c2 ζF (tj ) 6 (c1 r + c2 ζ)F (tj ) ∀ j > 1.

(7.3.52)

Similarly, it follows from inequality (7.2.16) that (1)

hj (s0 + δj ) 6 (1+ε)F (tj−1 ) + c(ε)ζF (tj ) 6 ((1+ε)r + c(ε)ζ)F (tj ) ∀ j > 1.

(7.3.53)

We now choose the values of free parameters. First, we fix an arbitrary number b such that 0 < b < 1. Then we choose r = r(b) and ε = ε(b) > 0 for which r1−b < max{(1 + ε), c−1 1 }

(1 + ε)r < rb ,

⇐⇒

c1 r < r b .

(7.3.54)

Further, we fix sufficiently small ξ = ξ(b) > 0 such that the following inequalities are satisfied: c1 r + c2 ζ(ξ) 6 rb ,

(1 + ε)r + c(ε)ζ(ξ) 6 rb ,

(7.3.55)

where c1 , c2 , and c(ε) are the constants specified in (7.2.16) and (7.2.17). For the indicated choice of the free parameters in (7.3.52) and (7.3.53), we arrive at the inequalities (1)

Ej (s0 + δj ) 6 rb F (tj ),

(1)

hj (s0 + δj ) 6 rb F (tj ) ∀ j ∈ N.

(7.3.56)

For j > 2, we can perform the second cycle of similar calculations. Namely, (2) we introduce the second series of shifts {δj }, (2)

δj

b(p−q)

(1)

:= r κ(p+1)(q+1) δj

∀ j > 1,

(7.3.57)

(1)

where δj is specified in (7.3.49). By virtue of (7.3.56), the functions Pj (s, δ) in (7.2.16) obey the following estimate: (1)

(2)

Pj (s0 + δj , δj ) 6 ξ

=⇒

(1)

(2)

Γ(Pj (s0 + δj , δj )) 6 ζ = ζ(ξ).

(7.3.58)

352

Chapter 7 Method of functional inequalities in peaking regimes (1)

Applying estimate (7.2.17) for s = s0 + δj (1)

(2)

(2)

and δ = δj , we get

(1)

(1)

Ej (s0 + δj + δj ) 6 c1 hj−1 (s0 + δj ) + c2 ζEj (s0 + δj ) 6 [by the monotonicity of δ¯i and relations (7.3.56)] 6 c1 rb F (tj−1 ) + c2 ζrb F (tj ) 6 (7.3.45) 6 (c1 r + c2 ζ)rb F (tj ) 6 (7.3.55) 6 r2b F (tj ) ∀ j > 2.

(7.3.59)

Similarly, in view of (7.3.55), inequality (7.2.16) implies that (1)

(2)

hj (s0 + δj + δj ) 6 r2b F (tj ) ∀ j > 2.

(7.3.60)

This completes the second cycle of estimation. For j > 3, we perform the third cycle of iterative estimations. For this cycle, we introduce the third series of shifts (3)

δj

b(p−q)

(2)

:= r κ(p+1)(q+1) δj .

After j cycles of this type, we get     j j X X (i) (i) max{hj s0 + δj , Ej s0 + δj } 6 rjb F (tj ) ∀ j ∈ N. i=1

i=1

This yields the required estimates (7.3.47) with c0 = ξ

1 − κ(p+1)(q+1)

b(p−q)

(1 − r κ(p+1)(q+1) )−1 ,

(7.3.61) 

The proof is thus completed.

Proof of Theorem 7.3.1. According to Lemma 7.2.1, we get the following starting estimate for the energy functions corresponding to the analyzed solution u : q+1 − p−q

h(t, 1) + E(t, 1) 6 C1 h¯ 0 + C2 ω(T − t) q+1 − p−q

= ω(T − t)

q+1

(C2 + C1 h¯ 0 ω −1 (T − t) p−q ) ∀ t ∈ [0, T ). (7.3.62)

For any ω > 0, by tω , 0 < tω < T, we denote a number such that q+1

ω −1 (T − t) p−q 6 1 ∀ t ∈ [tω , T )

⇐⇒

p−q

tω = T − ω q+1 .

(7.3.63)

Section 7.3

Localized singular boundary regimes: the case of slow diffusion

353

We now consider the solution u(t, x) of the analyzed mixed problem as a solution defined only in the domain (tω , T ) × Ω. We first introduce the energy functions of the solution in this domain and then perform the change of the time variable. Hence, after necessary transformations, we arrive at the solution of problem (7.1.1)–(7.1.10) satisfying, by virtue of (7.3.62) and (7.3.63), the following global energy estimate: q+1 − p−q

∀ t ∈ [0, T ),

h(t, 1) + E(t, 1) 6 CF0 (t) := Cω(T − t)

(7.3.64)

C = C2 + h¯ 0 C1 . We now check the applicability of Lemma 7.3.2 for s0 = 1 and F (t) = CF0 (t). We fix r : 0 < r < 1 and define a sequence {ti } by the formula ∆i := ti − ti−1 = ∆1 r

(i−1)(p−q) q+1

p−q

∀ i > 1,

∆1 = T (1 − r q+1 ).

(7.3.65)

It is clear that T − tj =

∞ X i=j+1

∞ X

∆i =

∆1 r

(i−1)(p−q) q+1

= ∆1 r

j(p−q) q+1



p−q

1 − r q+1

−1

i=j+1

= Tr

j(p−q) q+1

=⇒

F0 (tj ) = ωT

q+1 − p−q −j

r

.

(7.3.66)

Therefore, F0 (tj ) = r−1 F0 (tj−1 ) ∀ j > 1, which corresponds to definition (7.3.45) from Lemma 7.3.2. According to definition (7.3.46), the sequence of shifts δ¯i can be defined as follows: 1  κ(p+1)  p−q p−q δ¯i := C q+1 F0 (ti ) q+1 ∆i = δ¯ 1  p−q p−q  p−q  p−q  κ(p+1) − ¯ = δ(ω) = ω q+1 r q+1 1 − r q+1 C q+1

∀ i ∈ N.

(7.3.67)

Relations (7.3.65)–(7.3.67) imply that all assertions of Lemma 7.3.2 are true and, hence, we get the following estimates for the energy functions: − ¯ Ej (1 + c0 δ(ω)) 6 Crjb F0 (tj ) = CωT

b(q+1) p−q

−

(T − tj )

(q+1)(1−b) p−q

, (7.3.68)

¯ hj (1 + c0 δ(ω)) 6 Crjb F0 (tj ) = CωT

b(q+1) − p−q

(q+1)(1−b) − p−q

(T − tj )

∀ j ∈ N, ¯ where δ(ω) and c0 are specified in (7.3.67) and (7.3.61), respectively. We sum the inequalities in (7.3.68) and take into account (7.3.66). This yields ¯ 6 CωT − E(ti , 1 + c0 δ)

b(q+1) p−q

i X j=1

−

(T − tj )

(q+1)(1−b) p−q

354

Chapter 7 Method of functional inequalities in peaking regimes

= CωT

q+1 − p−q

i X

r−j(1−b) 6 CωT

q+1 − p−q −i(1−b)

r

(1 − r1−b )−1

j=1

= Cω b T

b(q+1) − p−q

(1 − r1−b )−1 F0 (ti )1−b

∀ i ∈ N.

(7.3.69)

By using (7.3.66), we obtain ¯ 6 Cω b c¯1 F0 (t)1−b E(t, 1 + c0 δ) where c¯1 = T

−

b(q+1) p−q

∀ t < T,

(7.3.70)

(1 − r1−b )−1 r−(1−b) .

Similarly, inequality (7.3.68) immediately implies the following estimate for ¯ h(t, 1 + c0 δ(ω)): ¯ 6 Cω b c¯2 F0 (t)1−b h(t, 1 + c0 δ)

∀ t < T,

c¯2 = T

−

b(q+1) p−q

r−(1−b) .

(7.3.71)

By virtue of inequalities (7.3.70) and (7.3.71), we conclude that condition (7.3.7) of Lemma 7.3.1 is satisfied for s0 = 1 + c0 δ¯ and γ = b (q+1) p−q . Hence, by this lemma, J(t, 1 + c0 δ¯ + s) < K(s) < ∞

∀s > 0,

∀ t < T,

(7.3.72)

where K(s) is given in (7.3.33). In terms of the energy functions (7.2.1), ¯ we rewrite inequality (7.2.4) for b = t, a = 0, s = 1 + c0 δ(ω) + τ2 , and δ = τ2 in the form ¯ ¯ h(t, 1 + c0 δ(ω) + τ ) + k1 E(t, 1 + c0 δ(ω) + τ)   −m(p+1)    ¯ τ ¯ τ + c(ε) τ J t, 1+c0 δ+ ∀ τ > 0. 6 (1+ε)h 0, 1+c0 δ+ 2 2 2 According to the obtained estimate (7.3.72), we get h(t, 1 + c0 δ¯ + τ ) + k1 E(t, 1 + c0 δ¯ + τ )    −m(p+1)  τ  ¯ τ + c(ε) τ 6 (1+ε)h 0, 1+c0 δ+ K ∀τ > 0, ∀ε > 0. 2 2 2 (7.3.73) ¯ In view of this inequality and the dependence δ¯ = δ(ω) described by relation by (7.3.67), we complete the proof of Theorem 7.3.1.  We now consider boundary peaking regimes for the following one-dimensional parabolic equation of order 2m, m > 1: ut + ∆m,p (u) = 0 (t, x) ∈ (0, T ) × R1+ ,

p > 1,

∆m,p (u) := (−1)m Dxm (|Dxm u|p−1 Dxm u).

(7.3.74)

Section 7.3

355

Localized singular boundary regimes: the case of slow diffusion

It is easy to see that Eq. (7.3.74) is invariant under the group of transformations u → λ−l u,

x → x/λβ ,

t → t/λ ∀ λ > 0,

where l is an arbitrary real number and β = (1 − l(p − 1))(m(p + 1))−1 .

(7.3.75)

Since Eq. (7.3.74) is invariant, it admits a self-similar solution of the form u(t, x) = (T − t)−l v(y),

y=

x , (T − t)β

(7.3.76)

where v(y) is a solution of the ordinary differential equation of order 2m Bm,p (v) := ∆m,p (v) + βyv 0 (y) + lv(y) = 0 ∀ y > 0,

v(∞) = 0.

(7.3.77)

Consider the following Dirichlet problem for Eq. (7.3.77): Dyi v(0) = ai = const ∈ R1 ,

Dyi v(∞) = 0,

i = 0, 1, 2 . . . , m.

(7.3.78)

Theorem 7.3.2 (see Theorem 2.1 in [45]). Under the assumption that β − 2l < 0,

(7.3.79)

m (R1 ) ∩ L (R1 ). problem (7.3.77), (7.3.78) possesses a unique solution v ∈ Wp+1 2 + +

Theorem 7.3.3. Under the additional assumption that β 6 0, the solution v in Theorem 7.3.2 has a compact support for any {ai } ∈ Rm and, moreover, ¯ 1 , . . . , am , p, m) such that there exists a constant K¯ = K(a ∀{ai } ∈ Rm .

sup{y ∈ supp v} 6 K¯

(7.3.80)

Prior to proving these theorems, we establish some corollaries of these assertions. It is easy to see that the solution u(t, x) given by (7.3.76) satisfies the Dirichlet conditions Dxi u(t, 0) = ai (T − t)−γi

i = 0, 1, . . . , m − 1,

(7.3.81)

where γi = l+iβ. Note that, parallel with the solution v(y) of problem (7.3.77), (7.3.78), we also have the following solution of Eq. (7.3.77): m(p+1)  y  ∀y > 0 (7.3.82) vω (y) := ω p−1 v ω for any ω > 0. Hence, for every β 6 0, the function uω (t, x) := (T − t)−l vω (y),

y=

x , (T − t)β

(7.3.83)

356

Chapter 7 Method of functional inequalities in peaking regimes

is a solution of Eq. (7.3.74) with compact support with respect to x for all t ∈ (0, T ) provided that the boundary conditions Dxi uω (t, 0) = fi (t) := ai ω

m(p+1) −i p−1

(T − t)−γi ,

i = 0, 1, . . . , m − 1,

(7.3.84)

are satisfied. We now restrict ourselves to the case where β = 0 ⇐⇒ l = (p−1)−1 . As follows from Theorem 7.3.1 and the form of the self-similar solution (7.3.76), this case corresponds to the limiting localized S -regime. Under this restriction, the boundary condition (7.3.84) takes the form Dxi uω (t, 0) = fi (t) = ai ω

m(p+1) −i p−1

1 − p−1

(T − t)

i = 0, 1, . . . , m − 1.

,

(7.3.85)

Hence, as an extension f¯(t, x) satisfying the condition Dxi f¯(t, 0) = fi (t), we can set x m(p+1) − 1 ∀ x > 0. f¯(t, x) := uω (t, x) = ω p−1 (T − t) p−1 v ω With the help of this function, we define F¯ (t) by using relation (7.1.10) (for q = 1 ¯ and n = 1). Since supp f¯(t, ·) ⊂ [0, Kω], where K¯ is given by (7.3.80), after simple calculations, we arrive at the estimate F¯ (t) 6 K¯ 1 ω

2m(p+1)+(p−1) p−1

2 − p−1

(T − t)

∀ t < T,

(7.3.86)

where K¯ 1 is a constant that depends only on known parameters of the problem and does not depend on ω. By virtue of estimate (7.3.86), Theorem 7.3.1 guarantees the validity of the estimate Ωs (u) ⊂ [0, cω]

(7.3.87)

for the set of intensification Ωs (u) of any solution u(t, x) of Eq. (7.3.74) satisfying the boundary conditions (7.3.85) and the initial condition (7.1.5) with an arbitrary function u0 ∈ L2 (R1+ ). By virtue of (7.3.80)–(7.3.82), we conclude ¯ that Ωs (uω ) = [0, Kω], which proves that estimates (7.3.87) and, hence, (7.3.2) are sharp with respect to the parameter ω > 0. Proof of Theorem 7.3.2. The standard theorems on solvability of the operator equations with monotone coercive operators are not applicable to Eq. (7.3.77) because the operator v → Bm,p (v) contains the term βyv 0 (y) whose coefficient is unbounded in the domain Ω = R+ 1 . Therefore, we construct the solution v as the limit, as j → ∞, of the solutions {vj (y)} of Eq. (7.3.77) on bounded intervals (0, j) with the following boundary conditions: Di vj (0) = ai ,

Di vj (j) = 0,

i = 0, 1, . . . , m − 1;

j = 1, 2.

(7.3.88)

Section 7.3

Localized singular boundary regimes: the case of slow diffusion

357

m (R1 ) with supp g ⊂ [0, 1) satisfying First, we introduce a function g ∈ Wp+1 + the boundary conditions (7.3.88). Then we define an operator (j) m m Bm,p,g : Wp+1 (Ωj , ∂Ωj ) → (Wp+1 (Ωj , ∂Ωj ))∗

by the equality (j) m Bm,p,g (w) := Bm,p (g + w) ∀ w ∈ Wp+1 (Ωj , ∂Ωj ), m (Ω , ∂Ω ), where Ωj = (0, j). For any w1 = v1 − g and w2 = v2 − g from Wp+1 j j as a result of integration by parts, we get (j) (j) hBm,p,g (w1 ) − Bm,p,g (w2 ), w1 − w2 i

 Z β = h∆m,p (v1 ) − ∆m,p (v2 ), v1 − v2 i + l − |v1 − v2 |2 dx 2 Ωj Z  Z β |v1 − v2 |2 dx, c0 > 0. > c0 |Dm (v1 − v2 )|p+1 dx + l − 2 Ωj Ωj (7.3.89) (j)

In view of (7.3.79), this implies that the operator Bm,p,g is strictly monotone. m (Ω , ∂Ω ), we have Moreover, for any w = v − g ∈ Wp+1 j j Z h (j) hBm,p,g (w), wi = |Dm v|p+1 − |Dm v|p Dm v · Dm g Ωj

 i β 2 w + (βyg 0 + lg)w dy. + l− 2 Thus, by using Hölder’s inequality and Young’s inequality with ε, we obtain Z Z m p+1 (j) |Dm g|p+1 dy |D v| dy − c1 (ε) hBm,p,g (w), wi > (1 − ε) Ω1 Ωj Z Z  β w2 dx − c1 (δ) (β 2 |g 0 |2 + l2 g 2 ) dy + l− −δ 2 Ωj Ω1 ∀ ε > 0, In view of the fact that Z Z m p+1 |D v| dy > Ωj

Ωj

|Dxm w|p+1 dy

Z −

∀ δ > 0.

g p+1 dy,

Ω1

we conclude that the last inequality yields the following relation: Z Z  β (j) −1 m p+1 −1 hBm,p,g (w), wi > 2 |D w| dy + 2 l− w2 dy − L, (7.3.90) 2 Ωj Ωj m (Ω ) and does where L is a constant that depends only on l, β, and kgkwp+1 1 not depend on w and j.

358

Chapter 7 Method of functional inequalities in peaking regimes

By virtue of the Poincaré inequality, condition (7.3.90) implies, in particular, (j) the coercivity of Bm,p,g : (j) hBm,p,g (w), wi > bj kwkp+1 wm

p+1 (Ωj )

− L,

bj > 0.

(7.3.91)

In view of (7.3.89) and (7.3.91), the standard theorem on equations with monotone operators (see [89, Chap. 2]) guarantees the existence and uniqueness of the solution vj of problem (7.3.77), (7.3.88). By virtue of (7.3.90), this solution satisfies the following a priori estimate: Z Z  β (m) p+1 (vj − g)2 dy 6 2L, f (m) := Dm f, |(vj − g) | dy + l − 2 Ωj Ωj (7.3.92) which is uniform in j ∈ N. Thus, to prove Theorem 7.3.2, it suffices to show that {vj } is a Cauchy m (Ω ) for any j ∈ N, i.e., sequence in the space Wp+1 j0 0 m (Ω ) → 0 kvj − vk kwp+1 as j → ∞ and k → ∞. j0

(7.3.93)

To this end, we fix constants τ > j0 , δ > 0, τ + δ < min(j, k) and introduce a monotonically nonincreasing cutoff function ξ(y) > 0 such that ξ(y) = 1 for y < τ, ξ(y) = 0 for y > τ + δ, and |Dyi ξ| := |ξ (i) | 6 c0 δ −i for all y ∈ [τ, τ + δ] and all i 6 m − 1. It is clear that the function (vj − vk )ξ belongs to both spaces m (Ω , ∂Ω ) and W m (Ω , ∂Ω ). Therefore, we use this function as a test Wp+1 j j k k p+1 function in the integral identities specifying the solutions vj and vk . Thus, subtracting the equalities obtained as a result and integrating by parts, we get  Z   β (m) p−1 (m) (m) p−1 (m) (m) (m) 2 (vj − vk ) ξdy (|vj | vj − |vk | vk )(vj − vk ) + l − 2 R1+ Z τ +δ m X (m) (m) (m) (m) (m−i) (m−i) ) dy = (|vj |p−1 vj − |vk |p−1 vk ) bi,m ξ (i) (vj − vk τ

+

i=1

β 2

Z

τ +δ

yξ 0 (y)(vj − vk )2 dy := R1 + R2 ,

(7.3.94)

τ

where bi,m are the corresponding binomial coefficients. In what follows, by C we denote any constants that depend on m, p, l, and the constant L from estimate (7.3.92) and do not depend on j, k, τ, and δ. Thus, by using the mean-value theorem for the function f (s) = |s|p−1 s, the Hölder and Young inequalities, and estimate (7.3.92), we find Z τ +δ (m) (m) R1 6 C |vj − vk |p+1 dy τ

Z m  X +C δ −i(p+1) i=1

τ

τ +δ

(m−i) |vj

−

(m−i) p+1 vk | dx

 p1 .

(7.3.95)

Section 7.3

359

Localized singular boundary regimes: the case of slow diffusion

By virtue of the Gagliardo–Nirenberg interpolation inequality [41], [101] (see also Proposition 9.1.2) and the Young inequality, we can estimate the terms on the right-hand side of (7.3.95) as follows: Z τ +δ 1  p |Dm−i (vj − vk )|p+1 dy δ −i(p+1) τ

6 Cδ

−ν

Z

τ +δ 2

 p+1 2p

+ Cδ −µi

|vj − vk | dy Z

Z

τ τ +δ

+C

τ +δ

|vj − vk |2 dy

ρi

τ

|Dm (vj − vk )|p+1 dy,

(7.3.96)

τ

where ν =

2(p+1)m−1 p

> 0, µi = i(p + 1)(p − θi )−1 > 0,

ρi = 2−1 (1 − θi )(p + 1)(p − θi )−1 > 0,

and θi =

2(p + 1)(m − i) + p − 1 . 2(p + 1)m + p − 1

We substitute estimate (7.3.96) in (7.3.95) and apply (7.3.95) in order to estimate the right-hand side of equality (7.3.94). In view of the strict monotonicity of the function f (s) = |s|p−1 s, after simple calculations, we get     β β (j,k) (j,k) (j,k) I(τ ) := I (τ ) := I1 (τ ) + l − I2 (τ ) := I1 (τ ) + l − I2 (τ ) 2 2   Z τ Z τ β |vj − vk |2 dy := |Dm (vj − vk )|p+1 dy + l − 2 0 0 6 C(I1 (τ + δ) − I1 (τ )) + C(τ + δ)δ −1 (I2 (τ + δ) − I2 (τ )) m X p+1 +C δ −µi (I2 (τ + δ) − I2 (τ ))ρi + Cδ −ν (I2 (τ + δ) − I2 (τ )) 2p . i=1

(7.3.97)

Setting δ = τ in this inequality and using the uniform estimate (7.3.92), we arrive at the inequality   min{j, k} −γ I(τ ) 6 C(I(2τ ) − I(τ )) + Cτ ∀ τ ∈ 1, , (7.3.98) 2 where γ = min {ν, µi } > 0. Finally, inequality (7.3.98) implies that 16i6m

min{j, k} , 2 Iterating this inequality i times, we obtain I(τ ) 6 λI(2τ ) + λτ −γ

∀τ : 1 < τ
0 and τ¯ > j0 such that I(τ ) 6 λi I(2i τ ) + C1 τ −γ

∀τ : 1 < τ 6

C1 τ¯ −γ 6 2−1 ε

=⇒

τ¯ = τ¯ (j0 , ε).

360

Chapter 7 Method of functional inequalities in peaking regimes

We also fix i = i0 such that 2λi0 L 6 2−1 ε

=⇒

i0 = i0 (ε, L),

where L is the constant from the a priori estimate (7.3.92). In view of (7.3.99), we get the estimate I(τ¯ ) := I (j,k) (τ¯ ) 6 ε ∀ j > j¯ := τ¯ 2i0 ,

¯ ∀ k > j.

(7.3.100)

By the Gagliardo–Nirenberg interpolation inequality, this estimate yields the required property of convergence (7.3.93).  Proof of Theorem 7.3.3. We introduce a cutoff function η(y) = 1 − ξ(y), where ξ(y) is the same as in Theorem 7.3.2. It is clear that supp Di η ⊂ {τ 6 y 6 τ + δ} i = 1, . . . , m − 1.

supp η ⊂ {y > τ },

m (Ω , ∂Ω ) ∀ j ∈ N, we substitute the test function ηv in the Since ηvj ∈ Wp+1 j j j integral identity specifying the solution vj and obtain    Z Z ∞ β τ +δ 0 β m p+1 2 vj η dy − yη (y)vj2 dy |D vj | + l− 2 2 τ τ Z τ +δ m X m p−1 m = |D vj | D vj bi,m η (i) (y)Dm−i vj dy := R. (7.3.101) τ

i=1

By virtue of the Hölder inequality, we find Z

τ +δ m

p+1

|D vj |

R6C

dy

p  p+1 m X

τ

δ

−i

Z

1  p+1

τ +δ

|D

m−i

p+1

vj |

dy

.

τ

i=1

(7.3.102) By using the Gagliardo–Nirenberg interpolation inequality, we obtain Z

τ +δ

|D

m−j

p+1

vj |



1 p+1

dy

τ

Z

θi Z  p+1

τ +δ

|D

6C

m−i

vj |

p+1

vj2 dy

dy

i)  (1−θ 2

τ

τ

+ Cδ

τ +δ

p−1 −(m−i)− 2(p+1)

Z

τ +δ

vj2 dy

 12 ,

(7.3.103)

τ

where θi is specified in (7.3.96). We now introduce the energy function associated with the solution vj : Z   β  2 (j) v dy ∀ τ ∈ (0, j). E(τ ) := E (τ ) := |Dm vj |p+1 + l − 2 j y>τ

Section 7.4

361

Localized boundary regimes: the case p = q

If we substitute estimate (7.3.103) in (7.3.102) and apply the Young inequality, then we get p−1

−(m+

)

2(p+1) R 6 C(E(τ ) − E(τ + δ))1+γ δ m X +C δ −i (E(τ ) − E(τ + δ))1+γi ,

(7.3.104)

i=1

where γ=

p−1 , 2(p + 1)

γi =

(1 − θi )(p − 1) i(p − 1) = . 2(p + 1) 2(p + 1)m + p − 1

Since η 0 (y) > 0 and β 6 0, it follows from (7.3.101) and estimate (7.3.104) that −(m+

p−1

2(p+1) E(τ ) 6 C(E(τ ) − E(τ + δ))1+γ δ m X +C δ −j (E(τ ) − E(τ + δ))1+γi

)

i=1

∀ τ : 0 < τ < j,

∀ δ > 0 : τ + δ < j.

(7.3.105)

By using (7.3.92), we arrive at the following estimate uniform in j : E(0) := E (j) (0) < 2L ∀ j ∈ N. By virtue of assertion (3) of Lemma 9.3.2, relation (7.3.105) implies that supp E (j) (·) is bounded uniformly in j, which completes the proof of Theorem 7.3.3. 

7.4

Localized boundary regimes: the case p = q

In the present section, we propose a modified proof of the following general assertion: Theorem 7.4.1 (see Theorem 1.3 in [43] and Theorem 1.2 in [44]). Assume that the boundary function f in the boundary-value problem (7.1.1)–(7.1.7) with p = q satisfies conditions (7.1.8)–(7.1.9) with p = q and that the function F (t) constructed for given f in (7.1.10), satisfies the estimate 1 − m(p+1)−1

F (t) 6 exp(w(T − t)

) := F0 (t) ∀ t < T,

(7.4.1)

where w = const > 0. Then there exists a constant K that depends only on the initial parameters of the problem under consideration and does not depend, in particular, on the parameter w from (7.4.1) such that   m(p+1)−1 ¯ : |x| < 1 + Kw m(p+1) Ωs (u) ⊂ x ∈ Ω (7.4.2) for any solution u of the problem.

362

Chapter 7 Method of functional inequalities in peaking regimes

In Sec. 7.4.1, we present arguments demonstrating certain sharpness of the limit boundary regime specified by Theorem 7.4.1. First, we prove the theorem. As in the case p > q, the proof is based on the consecutive analysis of the process of localization of singularity in the areas of “flat” and “nonflat” growth of energy in the vicinity of the peaking time. Lemma 7.4.1. Assume that there exists s0 > 1 such that the estimate h(t, s0 ) + E(t, s0 ) 6 C(T − t)−A

∀ t < T,

1 < A = const < ∞,

(7.4.3)

holds for a solution u(t, x) in a sense of Definition 7.1.1. Then Ωs (u) ⊂ {x : |x| 6 s0 } for the set of singularity and there exists a nonincreasing continuous function K(s), K(s) < ∞ ∀ s > 0, such that estimate (7.3.8) is true. Proof. We now follow the procedure used in the proof of Lemma 7.3.1. We again start from relation (7.3.6). For p = q, this relation takes the form Z tj Z j X p+1 Jj (s + δk ) :=
|u(t, x)| dxdt j P k=1

tj−1 Ω s+

δk

k=1

6 c(1 + ε)j ∆j h0 (s) + c(ε)

j X ∆i Ji (s)(1 + ε)j−i  ∆j  m(p+1)

i=1

∀ ε > 0,

∀ s > 1,

∆i

δi

∀ δk > 0,

(7.4.4)

j = 1, 2 . . . ,

By virtue of (7.4.3) and Lemma 7.2.1, we arrive at the starting estimate Jj (s0 ) 6 c C(T − tj )−A ∆j

∀ j ∈ N.

(7.4.5)

We now fix an arbitrary value s¯ > s0 and study the behavior of the function J(t, s) ¯ as t → T. We define a sequence {ti } by the following recurrence relation similar to (7.3.16): (ti − ti−1 )(J(ti , s) ¯ − J(ti−1 , s)) ¯ µ = g, i = 1, 2, . . . ,

t0 > 0,

(7.4.6)

where g > 0 is a number determined in what follows and µ is an arbitrary number from the interval 0 < µ < min{1, (A − 1)−1 }. As above, we distinguish the following two cases: (a) J(t, s) ¯ 6 c0 = const < ∞ ∀ t < T. (b) J(t, s) ¯ → ∞ as t → ∞.

(7.4.7)

Section 7.4

363

Localized boundary regimes: the case p = q

In the case (b), we set t0 = 0. In the case (a), by analogy with Sec. 7.3, we define t0 > 0 as a unique number such that 0 6 t0 6 t¯, where t¯J(t¯, s)µ := g and ti0 = T for some number i0 < ∞. Since the functions Ji (s) are monotone, relations (7.4.5) and (7.4.6) imply that (cC)µ ∆µ+1 (T − ti )−Aµ > g i In view of the fact that T − ti > ∆i+1 and

Aµ 1+µ

∀ i ∈ N. 6 1, the last inequality yields

1

∆i > g

1 1+µ

µ − 1+µ

(c C)

(T − ti )

Aµ 1+µ

g 1+µ ∆i+1

>

µ

(cC) 1+µ T =⇒

∆i+1 6 C1 ∆i

∀ i ∈ N,

Aµ 1− 1+µ

C1 = (cC)

µ 1+µ

T

(7.4.8) Aµ 1− 1+µ

g

1 − 1+µ

.

Our first restriction on the choice of the number g is C1 = const < 1

g > (cC)µ T 1−µ(A−1) .

⇐⇒

(7.4.9)

We now rewrite relation (7.4.6) in the form 1

1 −µ

Ji (s) ¯ = g µ ∆i

∀ i ∈ N.

(7.4.10)

Combining relations (7.4.8) and (7.4.10), we obtain (j−i) µ

Ji (s) ¯ 6 C1

Jj (s) ¯ ∀ i 6 j.

(7.4.11)

We fix an arbitrary number j ∈ N and define shifts δk > 0, k = 1, 2, . . . , j, by using relation (7.3.26). Substituting the introduced quantities δk and ∆k in (7.4.4), after simple calculations with the use of relations (7.4.6)–(7.4.11), we find   δ Jj s¯ + 1−b  j  c(ε) X 1+ε j−i ¯ 6 m(p+1) ∆i Ji (s) ¯ µ Ji (s) ¯ 1−µ + c(1+ε)j ∆j h0 (s) m(p+1) δ b i=1 1−µ j−i j  µ X g c(ε) (1+ε)C 1−µ 1 6 m(p+1) Jj (s) ¯ (7.4.12) ¯ + c(1+ε)j C1j−1 ∆1 h0 (s). m(p+1) δ b i=1

We now fix free parameters ε > 0 and 0 < b < 1 and impose the second restriction on the choice of g as follows: 1−µ

1−µ

1−µ

−

A(1−µ)

µ+1 (1 + ε)C1 µ (1 + ε)(cC) 1+µ T µ := 6 2−1 1−µ bm(p+1) bm(p+1) g (µ+1)µ  µ(1+µ)  1−µ 1+ε (cC)µ T 1−µ(A−1) . =⇒ g > m(p+1) 2b

(7.4.13)

364

Chapter 7 Method of functional inequalities in peaking regimes

In view of inequality (7.4.12), we obtain   2g c(ε) δ 6 m(p+1) Jj (s) ¯ 1−µ + c(1 + ε)(C1 (1 + ε))j−1 ∆1 h0 (s) Jj s¯ + ¯ . (7.4.14) 1−b δ Since inequality (7.4.14) is true for every j ∈ N, we get   δ 2g c(ε) 1−µ (j−i) Jj (s) ¯ 1−µ Ji s¯ + 6 m(p+1) C1 µ 1−b δ + c(1 + ε)(C1 (1 + ε))i−1 ∆1 h0 (s) ¯ ∀i 6 j

(7.4.15)

by virtue of (7.4.11). We now sum inequalities (7.4.15) over i from 1 to j and obtain the following analog of inequality (7.3.32): J(tj , s¯ +

δ ) 1−b

6 B3 δ −m(p+1) Jj (s) ¯ 1−µ + B4 h0 (s) ¯ + J(t0 , s¯ +

δ ) ∀ j > 1, 1−b

(7.4.16)

where 1−µ

B3 = 2g c(ε)(1 − C1 µ )−1

and B4 = c(1 + ε)∆1 (1 − C1 (1 + ε))−1 .

The final part of the proof of Lemma 7.4.1 is reduced to the analysis of the functional inequality (7.4.16) and is similar to the final part of the proof of Lemma 7.3.1 presented after relation (7.3.32).  The analysis of the localization of singularity in the zones of “nonflat” increase in the total energy of solution is performed on the basis of system (7.2.16), (7.2.17). In the investigated case p = q, this system takes the form hj (s + δ) 6 (1 + ε)hj−1 (s) + c(ε)∆Ej (s) G(∆j , δ) ∀ ε > 0, ∀ j ∈ N, (7.4.17) Ej (s + δ) 6 c1 hj−1 (s) + c2 ∆Ej (s) G(∆j , δ) ∀ s > 1, ∀ δ > 0,

(7.4.18)

where G(∆j , δ) := (

1 ∆j )p m(p+1) δ

+(

1 ∆j ) mp+m−1 . m(p+1) δ

Lemma 7.4.2. Assume that the following estimate holds for the solution u(t, x) (in a sense of Definition 7.1.1) for some s0 > 1: 1

h(t, s0 ) + E(t, s0 ) 6 C F0 (t) = C exp(w(T − t)− l ) ∀ t < T,

7.4.19

where C = const > 0, w = const > 0, and l = m(p + 1) − 1. Then, for any b, 0 < b < 12 , there exist constants K1 = K1 (b) < ∞ and K2 = K2 (b) independent of constants C and w and such that the following estimates are true: l

E(t, s0 + K1 w l+1 ) 6 C K2 F0 (0)b F0 (t)1−b h(t, s0 + K1 w

l l+1

l

1−b

) 6 C K2 F0 (0) F0 (t)

∀ t : 0 < t < T, (7.4.20) ∀ t < T.

Section 7.4

365

Localized boundary regimes: the case p = q

Proof. By analogy with (7.3.45), we define a sequence {ti } by the following recurrence relation: i h 1 (7.4.21) F0 (tj ) = exp w(T − tj )− l := r−1 F0 (tj−1 ) ∀ j > 1, t0 = 0, where r : 0 < r < 1 is a free parameter whose value is fixed in what follows. First, we establish the explicit upper and lower bounds for the intervals ∆j = tj − tj−1 specified in (7.4.21). We rewrite (7.4.21) in the form 1

g0 (tj ) − g0 (tj−1 ) = k := ln r−1 ,

g0 (t) := w(T − t)− l .

(7.4.22)

Since the function g0 (t) is convex, (7.4.22) yields the following relation: kg00 (tj )−1 6 ∆j 6 kg00 (tj−1 )−1

∀ j ∈ N.

(7.4.23)

The relation g00 (t) = wl−1 (T − t)−

l+1 l

= l−1 w−l g0 (t)l+1

is obvious. It follows from (7.4.23) that klwl g0 (tj )−(l+1) 6 ∆j 6 klwl g0 (tj−1 )−(l+1) . We also note that definition (7.4.22) implies that   g0 (0) g0 (tj ) = kj + g0 (0) = k j + k

∀ j ∈ N.

By virtue of (7.4.25) and the first inequality in (7.4.24), we get l+1  klwl klwl klwl kj ∆j > > = l+1 l+1 kj + g0 (0) g0 (tj ) (kj) (k + g0 (0))l+1 j l+1

(7.4.24)

(7.4.25)

∀ j > 1. (7.4.26)

On the other hand, by using the second inequality in (7.4.24) and relation (7.4.25), we obtain  l+1 kj 2l+1 lwl klwl ∀ j > 2. (7.4.27) 6 ∆j 6 (kj)l+1 kj − k + g0 (0) k l j l+1 For ∆1 , relation (7.4.22) immediately implies that l    w . ∆1 = T 1 − 1 w + kT l

(7.4.28)

We now fix a number j and define a shift δj as follows: δj := ω

1 − m(p+1)

1

∆jm(p+1) ,

0 < ω < 1.

(7.4.29)

A sufficiently small value of the parameter ω is specified in what follows.

366

Chapter 7 Method of functional inequalities in peaking regimes

We rewrite system (7.4.17), (7.4.18) in the form hi (s + δj ) 6 (1 + ε)hi−1 (s) + c(ε)∆Ei (s)G(∆i , δj ) ∀ ε > 0,

∀ i 6 j, ∀ s > s0 ,

(7.4.30)

Ei (s + δj ) 6 c1 hi−1 (s) + c2 ∆Ei (s)G(∆i , δj ) ∀ ε > 0, ∀ i 6 j, ∀ s > s0 . (7.4.31) We now establish the upper bound for G(∆i , δj ). By using estimates (7.4.26) and (7.4.27), we find −m(p+1)

∆i δj

−m(p+1)

= ∆j δj

−1 · ∆i ∆−1 j = ω∆i ∆j

6 ω2l+1 l(k + g0 (0))l+1 γ −l i−(l+1) j l+1

for all i, 2 6 i 6 j.

Let [a] be the integral part of a. This enables us to write   j −m(p+1) ∆ i δj 6 k1 ω ∀ j > 2, ∀ i : + 1 6 i 6 j, 2

(7.4.32)

k1 = 4l+1 l(k + g0 (0))l+1 k −l . In view of (7.4.32), we obtain   j ∀ j > 2, ∀ i : + 1 6 i 6 j. G(∆i , δj ) 6 ω1 (ω) := (k1 ω) + (k1 ω) 2 (7.4.33) Inequalities (7.4.33), (7.4.19), and (7.4.31) now imply that 1 mp+m−1

1 p

Ei (s0 + δj ) 6 c1 CF0 (ti−1 ) + c2 ω1 (ω)CF0 (ti ) 6 (7.4.21)   j + 1 6 i 6 j. 6 (c1 r + c2 ω1 (ω))CF0 (ti ) ∀ j > 2, ∀ i : 2 (7.4.34) Similarly, we derive the following relation from (7.4.30): hi (s0 + δj ) 6 (1 + ε)CF0 (ti−1 ) + c(ε)ω1 (ω)CF0 (ti )   j + 1 6 i 6 j. 6 ((1 + ε)r + c(ε)ω1 (ω))CF0 (ti ) ∀ j > 2, ∀ i : 2 (7.4.35) By analogy with (7.3.54) and (7.3.55), we choose the values of free parameters as follows: For any b : 0 < b < 1/2, we find sufficiently small r = r(b) > 0 and ε = ε(b) > 0 such that the relations (1 + ε)r < r2b ,

c1 r < r2b

⇐⇒

1

1−2b r < [max{(1 + ε)−1 , c−1 1 }]

(7.4.36)

are satisfied. Then we fix sufficiently small ω > 0 such that the inequalities (1 + ε)r + c(ε)ω1 (ω) 6 r2b ,

c1 r + c1 ω1 (ω) 6 r2b ,

ω1 (ω) is specified in (7.4.33),

(7.4.37)

Section 7.4

Localized boundary regimes: the case p = q

367

are true. Inequalities (7.4.34) and (7.4.35) can be transformed as follows:   j 2b (7.4.38) Ei (s0 + δj ) 6 r CF0 (ti ) ∀ j > 2, ∀ i : + 1 6 i 6 j, 2   j hi (s0 + δj ) 6 r2b CF0 (ti ) ∀ j > 2, ∀ i : (7.4.39) + 1 6 i 6 j. 2 We now return to system (7.4.30), (7.4.31). For s = s0 + δj , by virtue of (7.4.33), this system yields the following relations: hi (s0 + 2δj ) 6 (1 + ε)hi−1 (s0 + δj ) + c(ε)Ei (s0 + δj )ω1 (w), Ei (s0 + 2δj ) 6 c1 hi−1 (s0 + δj ) + c2 Ei (s0 + δj )ω1 (w). Thus, in view of (7.4.38)–(7.4.39), we obtain Ei (s0 + 2δj ) 6 Cr2b (c1 F0 (ti−1 ) + c2 ω1 (ω)F0 (ti )) 6 [by virtue of (7.4.21)] 6 Cr2b F0 (ti )(c1 r + c2 ω1 (ω)) 6 [by virtue of (7.4.37)]   j 4b 6 Cr F0 (ti ) ∀ j > 2, ∀ i : + 2 6 i 6 j; 2   j + 2 6 i 6 j. hi (s0 + 2δj ) 6 Cr4b F0 (ti ) ∀ j > 2, ∀ i : 2 With regard for the last inequalities, for the next iteration, we get Ei (s0 + 3δj ) 6 c1 hi−1 (s0 + 2δj ) + c2 Ei (s0 + 2δj )ω1 (ω) 6 Cr2b (c1 F0 (ti−1 ) + c2 ω1 (ω)F0 (ti )) 6 Cr6b F0 (ti ),   j 6b + 3 6 i 6 j, hi (s0 + 3δj ) 6 Cr F0 (ti ) ∀ j > 2, ∀ i : 2

(7.4.40)

Since [ 2j ]+[ j+1 2 ] = j ∀ j ∈ N, we perform the outlined procedure of iterative estimation [ j+1 2 ] times and obtain    j+1  j+1 Ej (s0 + δj ) 6 Cr2b 2 F0 (tj ) ∀ j > 2, 2   (7.4.41)  j+1  j+1 δj ) 6 Cr2b 2 F0 (tj ) ∀ j > 2. hj (s0 + 2 It follows from (7.4.21) that r2b[

j+1 ] 2

F0 (tj ) = r2b[

j+1 ]−j 2

F0 (0) 6 r−j(1−b) F0 (0).

(7.4.42)

Moreover, by virtue of (7.4.27), we arrive at the following estimate for the shifts δj in definition (7.4.29): 1

l

1

l

δj 6 2l l+1 k − l+1 ω − l+1 w l+1 j −1

∀ j > 2,

l = m(p + 1) − 1.

368

Chapter 7 Method of functional inequalities in peaking regimes

Therefore,   l 1 j+1 δj 6 k2 ω − l+1 w l+1 2

∀ j > 2,

1

l

k2 = l l+1 k − l+1 .

(7.4.43)

By using inequalities (7.4.41), (7.4.42), and (7.4.43), we find Ej (s1 ) 6 Cr−j(1−b) F0 (0),

l

1

K1 = k2 ω − l+1 ,

s1 = s0 + K1 w l+1 ,

hj (s1 ) 6 CF0 (0)r−j(1−b) ,

∀j > 2.

(7.4.44)

In view of (7.4.19) and (7.4.21), we conclude that E1 (s1 ) 6 E1 (s0 ) 6 CF0 (t1 ) = Cr−1 F0 (0) .

(7.4.45)

We now sum the first inequalities in (7.4.44) from j = 2 to j = i and add inequality (7.4.45). This yields E(ti , s1 ) 6 CBi F0 (0)r−(1−b)i

∀ i > 1,

(7.4.46)

where r−(1−b) (1 − r(1−b)(i−1) ) r−(1−b) − 1 (1−b) −1 −b ) := B ∀ i > 1. 6 r + (1 − r

Bi = r−1+(1−b)i +

In view of (7.4.21), we get the following relation from estimate (7.4.45): E(ti , s1 ) 6 CF0 (0)b BF0 (ti )1−b

∀ i > 1.

Since the function E(t, s1 ) is monotone in t, we conclude that E(t, s1 ) 6 CK2 F0 (0)b F0 (t)1−b

K2 = Br−(1−b) .

∀ t > 0,

(7.4.47)

In view of the definition (7.2.3) of the functions hj (s), the following relation immediately follows from the second estimate in (7.4.44): h(t, s1 ) 6 CK2 F0 (0)b F0 (t)1−b

∀ t > 0.

(7.4.48)

Hence, we have established the required estimates (7.4.20) with K2 from (7.4.47) and K1 from (7.4.44).  We now directly proceed to the proof of Theorem 7.4.1. To do this, we iteratively apply Lemma 7.4.2. The application of this lemma with initial estimates (7.4.47) and (7.4.48) instead of (7.4.19) gives the following relations for the second iteration: E(t, s2 ) 6 CK2 F0 (0)b K2 F1 (0)b F1 (t)1−b ∀ t : 0 < t < T, (7.4.49) h(t, s2 ) 6 CK2 F0 (0)b K2 F1 (0)b F1 (t)1−b ∀ t : 0 < t < T, where 1

F1 (t) = F0 (t)1−b = exp[w1 (T − t)− l ], l l+1

s 2 = s 1 + K 1 w1

w1 = (1 − b)w, l

l

= s1 + K1 (1 − b) l+1 w l+1 .

Section 7.4

369

Localized boundary regimes: the case p = q

The obtained estimates (7.4.49) are used instead of (7.4.19) for the next application of Lemma 7.4.2. After the j th iteration of the lemma, we arrive at the formulas j−1 Y j max{E(t, sj ), h(t, sj )} 6 CK2 Fi (0)b Fj (t) ∀ t : 0 < t < T , (7.4.50) i=0 1

Fj (t) := exp[(1 − b)j w(T − t)− l ]. We now optimize estimate (7.4.50) by choosing the number of iterations minimizing the right-hand side of the estimate. We choose this optimal number j0 = j0 (t) to guarantee the validity of the following relation: 1  T −t l ln T − ln(T − t) < (1 − b)j0 =⇒ j0 6 (1 − b)j0 +1 6 . T l ln(1 − b)−1 In this case, we have (1 − b)j0 6

(T − t)1/l T 1/l (1 − b)

and estimate (7.4.50) implies that E(t, s∞ ) + h(t, s∞ ) 6 2C exp[ln K2 · j0 +

w

1

T 1/l b

] exp[(1 − b)j0 w(T − t)− l ]

6 A1 (T − t)−A ∀ t : 0 < t < T,   ln T w A1 = exp ln K2 · + , l ln(1 − b)−1 T 1/l (1 − b)b A = l−1 (ln(1 − b)−1 )−1 , l

(7.4.51) l

l

where s∞ := lim sj = 1 + K3 w l+1 , K3 = k2 ω − l+1 (1 − (1 − b) l+1 )−1 . j→∞

The singular regime specified by estimate (7.4.51) is already not exponential. Actually, it is governed by a power law, i.e., is “flat” in a sense of Lemma 7.4.1. Applying this lemma, we complete the proof of Lemma 7.4.2.  We now briefly dwell upon the problem of sharpness of the limiting S -regime studied in Theorem 7.4.1. Consider the following Cauchy–Dirichlet problem for a one-dimensional linear parabolic equation of order 2m, m > 1 (see [43]): ut + (−1)m Dx2m u = 0 (t, x) ∈ (0, T ) × R1+ ,

(7.4.52)

u(t, 0) = f (t) → ∞ as t → T,

(7.4.53)

u(t, 0) = · · · = Dx u(t, 0) = · · · =

Dxm−1

∀ t ∈ (0, T ),

u(0, x) = 0. The solution of this linear problem has the following explicit form: Z t u(t, x) = 2(−1)m f (τ ) · Dx2m−1 ϕ(t − τ, x) dτ, 0

(7.4.54) (7.4.55)

(7.4.56)

370

Chapter 7 Method of functional inequalities in peaking regimes

where ϕ(t, x) is the fundamental self-similar solution of Eq. (7.4.52), 1 x (7.4.57) ϕ(t, x) = t− 2m Φ(y), and y = 1/2m . t Here, Φ(y) is a unique symmetric solution (i.e., Φ(y) = Φ(−y)) of the ordinary linear differential equation (−1)m+1 Φ(2m) (y) + (2m)−1 (Φ(y)y)0 = 0 in R1

(7.4.58)

with the following normalization: Z ∞ Φ(y) dy = 1. −∞

It is clear that the asymptotic properties of the solution u specified in (7.4.56) are determined by the asymptotic properties of the kernel Φ(y) from (7.4.57) as y → ±∞. First, by using simple properties of the Fourier transform Z∞ 1 F : Φ(y) → G(z) := √ Φ(y) exp(−izy) dy, 2π −∞

and Eq. (7.4.58), we conclude that G(z) satisfies the following ordinary differential equation: z 2m G(z) +

1 zG(z)0 = 0. 2m

(7.4.59) √ The normalization condition for Φ(y) yields the equality G(0) = 1/ 2π. By using this equality, we solve Eq. (7.4.59) and obtain 1 G(z) = √ exp(−z 2m ). 2π Hence, as a result of the inverse Fourier transformation, in view of the symmetry properties of the function G(z) : G(z) = G(−z), we arrive at the following representation for Φ(y): Z 1 ∞ Φ(y) = (7.4.60) exp(−z 2m ) cos(yz) dz. π 0 The direct analysis of the asymptotic properties of the function Φ(y) given by (7.4.60) is quite complicated. Therefore, we construct the main part of the asymptotic expansion of the function Φ(y) as |y| → ∞. To this end, we ˜ use the methods of standard asymptotic analysis and seek a solution Φ(y) of Eq. (7.4.58) in the form ˜ Φ(y) = exp(−a|y|α )(1 + g(y)),

g(y) → 0 as |y| → ∞.

(7.4.61)

It is clear that 0 ˜ (y Φ(y)) = (−aα)|y|α exp(−a|y|α )(1 + g1 (g)),

g˜1 (y) → 0 as |y| → ∞;

Section 7.4

371

Localized boundary regimes: the case p = q

˜ 2m (y) = (−aα)2m |y|2m(α−1) exp(−a|y|α )(1 + g˜2 (y)), Φ g˜2 (y) → 0 as |y| → ∞. Hence, in view of Eq. (7.4.58), we get α = 2m(α − 1)

⇐⇒

α=

2m ∈ (1, 2) (2m − 1)

(7.4.62)

and (−1)m+1 (−aα)2m +

1 (−aα) = 0 2m

(−1)m (aα)2m−1 + 1/2m = 0.

⇐⇒

(7.4.63) Substituting (7.4.62) in (7.4.63), we find a2m−1 = (−1)m−1 (2m)−1 ·



2m − 1 2m

2m−1 .

This algebraic equation of degree (2m − 1) has (2m − 1) roots,      2kπ 2kπ  for m = 2j + 1, j ∈ N, cos 2m−1 + i sin 2m−1 ak = l(m) ×      cos (2k−1)π + i sin (2k−1)π for m = 2j, j ∈ N, 2m−1 2m−1 2m

where l(m) = (2m − 1)(2m)− 2m−1 , k = 1, 2, . . . , 2m − 1. It is easy to see that, for every m ∈ N, there are roots with Re ak > 0 and there are no roots with Re ak = 0. Hence, by k = k(m) ∈ [1, 2m − 1], we denote the number such that 0 < Re ak¯ = min{Re ak > 0}. It is clear that ak¯ specifies the leading term of the asymptotic expansion of any solution of Eq. (7.4.58) admitting normalization. Thus, there exist constants A1 = A1 (m) and A2 = A2 (m) such that the solution Φ(y) given by (7.4.60) admits the following representation: Φ(y) = exp(−am,1 |y|α )[A1 cos(am,2 |y|α ) + A2 sin(am,2 |y|α )]·(1+g(y)), (7.4.64) g(y) → 0 as |y| → ∞, am,1 := Re ak¯ := Re ak(m) > 0, am,2 := Im ak¯ . ¯ Differentiating this expansion (2m − 1) times and using the relation (2m − 1)(α − 1) = 1, we arrive at the asymptotic equality Φ(2m−1) (y) = y exp(−am,1 |y|α )[B1 cos(am,2 |y|α ) + B2 sin(am,2 |y|α )](1 + g(y)), ¯

(7.4.65)

where g(y) ¯ → 0 as |y| → ∞; B1 = B1 (m), B2 = B2 (m) : B12 + B22 := B 2 6= 0.

372

Chapter 7 Method of functional inequalities in peaking regimes

We now consider the solution u(t, x) of problem (7.4.52)–(7.4.55) with boundary regime 1 f (t) = exp((T − t)−γ ), γ = . (7.4.66) 2m − 1 By virtue of representation (7.4.56) and equality (7.4.65), the leading term u(t, ˜ x) of the asymptotic expansion of this solution as t → T has the form u(t, ˜ x) = B3 xIx (t) + B4 xJx (t), B32 + B42 6= 0, Z t Ix (t) := exp[(T − τ )−γ − am,1 xα (t − τ )−γ ]

(7.4.67)

0

× cos(am,2 xα (t − τ )−γ )(t − τ )−γ (t − τ )− Z Jx (t) :=

t

2m+1 2m

dτ ;

2m+1 2m

dτ.

exp[(T − τ )−γ − am,1 xα (t − τ )−γ ]

0

× sin(am,2 xα (t − τ )−γ )(t − τ )−γ (t − τ )−

We now consider the problem of convergence (divergence) as t → T for the oscillating integrals specifying the functions Ix (t) and Jx (t). In the integral Ix (t), we change the variable of integration, α

α

v = (T − t) 2m · (t − τ )− 2m

(T − τ ) = (T − t)(1 + v −

=⇒

2m α

).

As a result of standard calculations, we obtain 1 2m Ix (t) = (T − t) 2m α " !# Z ∞ α v α × exp (T − t)− 2m 2m α − am,1 x v v0 (1 + v α ) 2m α

α

1

× cos(am,2 xα (T − t)− 2m v)v − 2m dv,

α

v0 = (T − t) 2m t− 2m .

We now perform the change parameters in Ix (t):  k 1 α ; k := am,1 xα =⇒ x = x(k) := am,1 α

s := (T − t)− 2m

t = t(s) := T − s−
− α 2m v0 (t(s)) := v¯ 0 (s) = T s α − 1 2m . =⇒

2m α

, (7.4.68)

As a result, we obtain a new energy function given by the formula Z 1 2m − 1 ∞ ¯ s α Ik (s) := Ix(k) (t(s)) = exp[sψk (v)] cos(βkvs)v − 2m dv, (7.4.69) α v¯ 0 (s) where β = am,2 a−1 m,1 ,

ψk (v) =

v 1+v α

2m α




α 2m

− kv,

the parameter s varies within the interval (T − 2m , ∞), and v¯ 0 (s) → 0 as s → ∞.

Section 7.4

373

Localized boundary regimes: the case p = q

First, we consider the case where ⇐⇒

k>1

−1

α . x > am,1

(7.4.70)

It is clear that ψk (v) < c0 = c0 (k) < 0 ∀ v > v¯ 0 . Moreover, it is obvious that the relation
α 1 2m 1 v − 2m 6 v¯ 0 (s)− 2m = T s α − 1 4m2 holds in the entire interval of integration in (7.4.69). These inequalities imply the uniform boundedness of the integral in (7.4.69) as s → ∞. Therefore, under condition (7.4.70), the function Ix (t) is also uniformly bounded as t → T. Similarly, we can prove that Jx (t) is uniformly bounded. It follows from (7.4.67) that the set of singularity Ωs (u) of the solution u of problem (7.4.52)–(7.4.55), (7.4.66) satisfies the following inclusion: n 1 o −α Ωs (u) ⊂ x : x 6 am,1 . Hence, our main problem is to prove that n 1 o −α 6= ∅. Ωs (u) ∩ x : 0 < x < am,1

(7.4.71)

We fix a point x such that −1

α 0 < x < am,1

⇐⇒

k = k(x) < 1 [k is specified in (7.4.68)].

(7.4.72)

It is easy to see that ψk (v) > 0 ∀ v ∈ (0, v1 ),

v1 = k −

2m α

−1




α 2m

.

(7.4.73)

Furthermore, ψk (v1 ) = 0 and ψk (v) < 0 ∀ v > v1 . In the interval (0, v1 ), the function ψk reaches its maximum value at a point vmax such that
α
α+2m 2m 2m (7.4.74) vmax = k − α+2m − 1 2m , ψk (vmax ) = k k − α+2m − 1 2m . We now rewrite the integral in (7.4.69) in the form Z ∞ 2m − 1 ¯ α s Ek (s), Ek (s) := Ik (s) = exp[sψk,s (v)] cos(βkvs) dv, α v¯ 0 (s)

(7.4.75)

ψk,s (v) := ψk (v) − s−1 (2m − α−1 ) ln v. It is clear that, for all v > v¯ 0 (s),
2m α −s−1 ln v 6 −s−1 ln v¯ 0 = s−1 ln T s α − 1 → 0 as s → ∞. (7.4.76) 2m Therefore, the function ψk,s (v) reaches its maximum value in the interval (v¯ 0 (s), ∞) at a single point (s)

(s)

vmax = vmax (k) = vmax + ω1 (s), where vmax = vmax (k) is defined in (7.4.74) and ω1 (s) → 0 as s → ∞.

(7.4.77)

374

Chapter 7 Method of functional inequalities in peaking regimes

It is also obvious that (s)

ψk,s (vmax ) = ψk (vmax ) + ω2 (s),

(7.4.78)

where ψk (vmax ) is specified in (7.4.74) and ω2 (s) → 0 as s → ∞. Moreover, we have (s)

00 ψk,s (vmax ) = ψk00 (vmax ) + ω3 (s),

(7.4.79)

where ψk00 (vmax ) = −

2m
α+4m α + 2m 2m−α − 2m α vmaxα 1 + vmax α

and ω3 (s) → 0 as s → ∞.

We now expand the function ψk,s (v) in the Taylor series in the vicinity of (s) the point v = vmax and represent the function Ek (s) defined by (7.4.75) in the form Z ∞ (s) (s) (s) (s) 00 Ek (s) = exp[s(ψk,s (vmax ) + 2−1 ψk,s (vmax )(1 + µs (v − vmax ))(v − vmax )2 ] v¯ 0 (s) (s)

(s)

× cos(βk(v − vmax )s + βkvmax s) dv,

(7.4.80)

where µs (τ ) is a family of continuous functions with the following property: |µs (τ )| < c|τ | ∀ τ ∈ R1 ,

∀ s > s0 ,

c = const < ∞.

(7.4.81)

(s)

Since the continuous function g(s) := svmax → ∞ as s → ∞, there exists an unbounded set S of points s such that (s)

βksvmax = 2nπ,

where n(s) ∈ N.

(7.4.82)

In the integral from (7.4.80), we now perform the change of integration variable: (s) v → v + vmax . In view of relation (7.4.82), this yields (s)

Ek (s) = Ek (s), (τ )

(τ )

Ek (s) := exp(sψk,τ (vmax )) Z ∞ (τ ) 00 × exp[2−1 sψk,τ (vmax )(1 + µτ (v))v 2 ] cos(βkvs) dv (τ )

v¯ 0 (s)−vmax

∀ s ∈ S,

∀ τ ∈ S.

(7.4.83)

With regard for properties (7.4.77), (7.4.78), and (7.4.79), we get (∞)

Ek

(s) = exp(sψk (vmax )) Z ∞ × exp[2−1 sψk00 (vmax )(1 + µ∞ (v))v 2 ] cos(βkvs) dv, v¯ 0 (s)−vmax

(7.4.84) where the function µ∞ satisfies estimate (7.4.81).

Section 7.4

375

Localized boundary regimes: the case p = q (∞)

We associate the asymptotic properties of the function Ek (s) with the behavior of the following model function as s → ∞: Z ∞ (∞) ¯ exp(2−1 ψk00 (vmax )sv 2 ) cos(βkvs) dv ∀ s > 0. Ek (s) := exp(sψk (vmax )) −∞

(7.4.85) (∞) Lemma 7.4.3. The function E¯ k (s) admits the following explicit representation: (∞) E¯ k (s) = As−1/2 exp(sR(k)),

A = const > 0.

¯ Moreover, there exists a value k¯ = k(m), 0 < k¯ < 1, such that R(k) > 0 for ¯ ¯ ¯ 0 < k < k, R(k) = 0, and R(k) < 0 for k > k. Proof. In the integral on the right-hand side of (7.4.85), we introduce a new integration variable z −2−1 ψk00 (vmax )sv 2 = z 2 =⇒ v = 00 −1 (−2 ψk (vmax )s)1/2 and rewrite this integral in the form Z ∞ 1 1 1 (as)− 2 exp(−z 2 ) cos(bs1/2 z) dz := (as)− 2 f (s 2 ),

(7.4.86)

−∞

where a = −2−1 ψk00 (vmax ) and b = βka−1/2 . We now show that the function f (s) in (7.4.86) satisfies an ordinary differential equation. Indeed, it is easy to see that Z ∞ Z ∞ d 0 2 −1 f (s) = −b exp(−z ) sin(bsz)z dz = 2 b sin(bsz) (exp(−z 2 )) dz. dz −∞ −∞ Integrating by parts, we obtain f 0 (s) + 2−1 b2 sf (s) = 0.

(7.4.87)

Solving this equation and using (7.4.86), we obtain Z ∞  b2  f (s) = d exp − s2 , d = f (0) = exp(−z 2 ) dz = π 1/2 . 4 −∞

(7.4.88)

We now return to definition (7.4.85) and use relations (7.4.86) and (7.4.88). This yields 1

21/2 π 1/2 s− 2 exp(sR(k)), (−ψk00 (vmax ))1/2 (7.4.89) β 2 k2 R(k) := ψk (vmax ) − . 2(−ψk00 (vmax ))

1 (∞) E¯ k (s) = (as)− 2 exp(sψk (vmax ))f (s1/2 ) =

376

Chapter 7 Method of functional inequalities in peaking regimes

By using the values of ψk (vmax ) from (7.4.74) and ψk00 (vmax ) from (7.4.79), as a result of simple calculations we get 2m

2m

R(k) =

2(α + 2m)(1 − k α+2m )2 − αβ 2 k α+2m 2m

2(α + 2m)(1 − k α+2m )

2m−α 2m

.

(7.4.90)

Since α + 2m = 2mα, we conclude that 1

R(k) =

1

4m(1 − k α )2 − β 2 k α 1

2m−2

.

4m(1 − k α ) 2m−1 This implies that R(k) < 0 for k > k¯ and

i2 h 8m 1/2 β  1+ −1 < 1. R(k) > 0 for k < k¯ := 1 − 4m β Lemma 7.4.3 is thus proved.

(7.4.91) 

The following general conditional assertion on the sharpness of the limiting localized S -regime (7.4.66) in problem (7.4.52–(7.4.55) and, hence, of the general S -regime specified in Theorem 7.4.1 follows from Lemma 7.4.3: Proposition 7.4.1. Suppose that the following hypothesis is true: (s) (∞) (H) The functions Ek (s) and E¯ k (s) defined by (7.4.83) and (7.4.85) have the same leading term in the asymptotic expansion for all k ∈ (0, 1) as s → ∞.

Let k¯ ∈ (0, 1) be the number specified in (7.4.91). Then the set of intensification Ωs (u) of the solution u of problem (7.4.52)–(7.4.55) contains the set ¯ −1 )1/α }. {x : 0 ≤ x < (ka m,1

Chapter 8

Nonlocalized regimes with singular peaking

8.1

Propagation of blow-up waves

In the present chapter, we study nonlocalized boundary regimes with peaking. For these regimes, the set of singularity Ωs (u) of any solution u coincides with the entire domain Ω of the problem. Therefore, in order to describe the singularity of these regimes, it is necessary to introduce additional characteristics of the behavior of the corresponding solutions in a vicinity of the peaking time. Good understanding of these characteristics can be obtained by analyzing the family of explicit solutions (7.3.76) of the model equation (7.3.74). For any l > (p − 1)−1 , the boundary regime (7.3.81) generating solution (7.3.76), is obviously a nonlocalized boundary peaking regime. In this case, as follows from Theorems 7.3.2 and 7.3.3, u(t, x) = 0 ∀(t, x) : x > a(T − t)−β

∀ t < T,

(8.1.1)

and, for any ε > 0, u(t, (a − ε)(T − t)−β ) → ∞ as t → T,

(8.1.2)

where a = const < ∞ is the boundary of the support of the solution v(y) of elliptic problem (7.3.77), (7.3.78) specified in Theorem 7.3.2. Hence, in this case, we observe a wavelike unbounded expansion of the support of the solution u of problem (7.3.74), (7.3.81) and a similar propagation of the area of unbounded growth of the solution. It is clear that the rate of propagation of the support of solution (8.1.1) is finite only in the case where the initial function u0 is finite [for the solution u(t, x) from (7.3.76), we have u0 (x) = T −l v(xT −β )]. In the general case where u0 is not finite, it is quite natural to assume that (8.1.1) should be replaced by the following property: There exists a constant L < ∞ that depends only on u0 such that u(t, x) 6 L ∀(t, x) : x > a(T − t)−β

∀ t < T.

(8.1.3)

In view of properties (8.1.2) and (8.1.3), we introduce the following characteristic of the nonlocalized boundary regime: Definition 8.1.1. A continuous function χL (t) such that χL (t) → ∞ as t → T is called the effective length of propagation of the blow-up wave for a continuous

378

Chapter 8 Nonlocalized regimes with singular peaking

solution u of a problem with nonlocalized boundary regime if χL (t) = sup{x : u(t, x) > L} ∀t < T,

L = const < ∞.

A fairly complete classification of nonlocalized boundary regimes in terms of the function of effective length of propagation of the blow-up waves was given for various types of model quasilinear parabolic equations of the second order in [109, Chapters 3 and 6]. For the energy solutions of divergent quasilinear parabolic equations (7.1.1) of general type and any order, an adequate definition of the function of effective length of propagation of the blow-up waves was introduced in [44]. Moreover, the order-exact upper bounds for the propagation of this wave were established for some singular boundary regimes in [44] and [45]. In the present chapter, we study the propagation of blow-up waves for arbitrary nonlocalized boundary regimes (with arbitrary strong peaking) and establish the order-exact upper bounds for the corresponding function χL (t). Clearly, the propagation of the blow-up wave as t → T for the nonlocalized boundary regime can be studied only in the case where the original domain Ω is unbounded. For this reason, unlike the previous chapter, we now consider the solutions u(t, x) of problem (7.1.1)–(7.1.9) in an unbounded domain Ω with boundary ∂Ω = ∂0 Ω ∪ ∂1 Ω. Remark 8.1.1. For any unbounded domain Ω ⊂ Rn , the existence of a generalized energy solution u ∈ Lp+1 (0, T, W (Ω)) of the homogeneous problem (7.1.5)–(7.1.7) (with f = 0) for the inhomogeneous equation (7.1.1) [i.e., for Eq. (P) from Remark 7.1.1], where X W (Ω) = {v : kvkW (Ω) := kvkLq+1 (Ω) + kDα vkLp+1 (Ω) < ∞}, |α|=m

follows from the results presented in [15] provided that condition (7.1.4) is satisfied and the operator A is potential, i.e., under the condition that there exists a function F (t, x, ξ) such that aα (t, x, ξ) =

8.2

∂F (t, x, ξ) ∂ξα

∀ α : |α| = m.

Estimates for the blow-up wave in the equation of slow-diffusion type

In an unbounded domain Ω ⊂ Rn , we consider the initial boundary-value problem (7.1.1)–(7.1.7) for p > q > 0 with a singularly peaking boundary regime characterized by the function F (t) specified in Definition 7.1.10. We consider

Section 8.2

Estimates for the blow-up wave in the equation

379

the boundary regimes with arbitrary strong peaking, i.e., assume that the inequality q+1 ln(F (t)) := g(T −t) > ln(T −t)−1 := gs (T −t) ∀ t : 0 6 t0 6 t < T, p−q (8.2.1) is satisfied. This inequality is opposite to inequality (7.3.1) in Theorem 7.3.1. Without loss of generality, we can assume that t0 = 0. For the generalized energy solutions under consideration, our definition of the function χL (t) of propagation of the blow-up wave is transformed into the following definition, which is based solely on the energy functions associated with the solution: Definition 8.2.1. For any constant L  1, a continuous function χL (t) specified by the equality Z Z tZ |u(τ, x)|q+1 dx + |Dxm u(τ, x)|p+1 dτ dx = L sup 0χL (t)} (8.2.2) is called the effective depth of propagation of the blow-up wave associated with a solution u(t, x) of problem (7.1.1)–(7.1.10). Definition 8.2.2. We say that a monotonically decreasing continuous function g : (0, T ] → [g(T ), ∞) is regularly decreasing if there exist constants k0 > 0 and v0 > g(T ) such that the inequality   p−q exp w [g −1 (w−k) − g −1 (w)] q+1   p−q (8.2.3) > (1−δ) exp v [g −1 (v−k) − g −1 (v)] ∀ w > v > v0 +k, q+1 holds for every k > k0 and some δ = δ(k) : 0 6 δ < 1. In this section, for the function χL (t) of propagation of the blow-up wave, we establish general exact-order upper estimates in terms of a regularly decreasing (in a sense of Definition 8.2.2) majorant g0 (τ ) of the function g(τ ) defined by (8.2.1). In view of (8.2.1), the estimate q+1 ln τ −1 ∀ τ ∈ (0, T ), g0 (τ ) > (8.2.4) p−q is true for any majorant g0 (τ ) and, hence, the inverse function g0−1 (v) satisfies the estimate   (p − q) g0−1 (v) > exp − v ∀ v > g0 (T ). (8.2.5) (q + 1) For any k > 0, we define a function   p−q v [g0−1 (v − k) − g0−1 (v)] ∀ v > g0 (T ) + k Gk (v) := exp q+1

(8.2.6)

380

Chapter 8 Nonlocalized regimes with singular peaking

and the minimal monotonically decreasing majorant of the function Gk (v), which is defined as follows: G¯ k (v) :=

(0)

¯ ∀ v > v0 := g0 (T ) + k. max Gk (v)

(8.2.7)

(0) ¯ v0 6v (1 − δ)Gk (v) ∀ w, v : w > v > v0 + k, p − q  v ∀ v > v0 + k. 0 < c1 6 Gk (v) < c2 exp q+1

(8.2.8) (8.2.9)

It is clear that every function satisfying estimates (8.2.8) and (8.2.9) belongs to one of the two following classes: (A) there exist a constant β, 0 < β < as i → ∞, such that

p−q q+1 ,

Gk (vi ) > exp(βvi ) ∀ i ∈ N,

and a sequence {vi }, vi → ∞

0 0, there exists a finite number vβ such that Gk (v) 6 exp(βv) ∀ v > vβ .

(8.2.11)

It is easy to see that the class (A) includes, in particular, all functions generated by strongly singular boundary regimes g0 (τ ), i.e., by the regimes satisfying the condition g0 (τ )(ln τ −1 )−1 → ∞ as τ → 0. The class (B) contains the functions Gk (v) generated by weakly singular regimes, i.e., by the functions g0 (τ ) such that g0 (τ ) (p − q)g0 (τ ) → 1 as τ → 0. := gs (τ ) (q + 1) ln τ −1 In the class (A), we also introduce the following subclass of functions: ¯ there exist constants β, 0 < β < (A)

p−q q+1 ,

and v0 > 0 such that

Gk (v) > exp(βv) ∀ v > v0 .

(8.2.12)

We now deduce estimates for the propagation of the blow-up waves.We present the results of these calculations in the form of two main lemmas (Lemma 8.2.2 and Lemma 8.2.1) for the regimes from the classes (A) and (B) respectively.

Section 8.2

381

Estimates for the blow-up wave in the equation

Thus, Lemma 7.2.1 yields the following energy estimate, which is global for the considered solution u: h(t, 1) + E(t, 1) 6 C0 F0 (t) := C0 exp(g0 (T − t)) ∀ t > 0, C0 = const < ∞. (8.2.13) According to relation (7.3.45), we define a sequence {ti } by the following recurrence relation: F0 (tj ) := r−1 F0 (tj−1 ) ∀ j > 1, (0)

(8.2.14)

j > 1.

(8.2.15)

0 6 t0 < F0−1 (r−1 F0 (0)) = T − g0−1 (g0 (T ) + ln r−1 ) := t1 . This yields ∆j = tj − tj−1 = g0−1 (g0 (T − tj ) − ln r−1 ) − (T − tj ),

Denote vj := g0 (T − tj ). This enables us to rewrite (8.2.15) in the form ∆j = g0−1 (vj − k) − g0−1 (vj ),

k = ln r−1 .

(8.2.16)

Hence, we can represent the shifts δ¯i from (7.3.46) in the following form: 1

δ¯j = (C00 G¯ k (vj )) κ(p+1) ,

p−q

k = ln r−1 ,

j > 1,

C¯ 00 = C0q+1 ,

(8.2.17)

where G¯ k (v) is the function from definition (8.2.7). Since the function G¯ k (v) is monotone, the sequence {δ¯i } is monotonically nondecreasing and, therefore, Lemma 7.3.2 can be used in the analyzed case. By virtue of this lemma, the energy functions of the considered solution satisfy estimates (7.3.47). In view of (8.2.13) and (8.2.17), these estimates take the form 1

Ej (1 + c0 (C00 G¯ k (g0 (T − tj ))) κ(p+1) ) 6 C0 rjb F0 (tj ) = C0 r−(1−b)j F0 (t0 ) = C0 F0 (t0 )b F0 (tj )1−b = C0 exp[bg0 (T − t0 )] exp((1 − b)g0 (T − tj )) j = 1, 2, . . . , 1

hj (1 + c0 (C00 G¯ k (g0 (T − tj ))) κ(p+1) ) 6 C0 exp[bg0 (T − t0 )] exp((1 − b)g0 (T − tj )),

(8.2.18)

where k = ln r−1 . Since the function G¯ k (v) is monotone, we can find the sum of estimates (8.2.18) for Ej over j from j = 1 to j = i and obtain the following inequality: 1

E(ti ,1 + c0 (C00 G¯ k (g0 (T − ti ))) κ(p+1) ) 1

6 c1 C0 exp((1 − b)g0 (T − ti )) + E(t0 , 1 + c0 (C00 G¯ k (g0 (T − ti ))) κ(p+1) ) (0)

(0)

1

6 c1 C0 exp[(1 − b)g0 (T − ti )] + E(t1 , 1 + c0 (C00 G¯ k (g0 (T − t1 ))) κ(p+1) )

382

Chapter 8 Nonlocalized regimes with singular peaking (0)

6 c1 C0 exp[(1 − b)g0 (T − ti )] + C0 exp[bg0 (T )] exp[(1 − b)g0 (T − t1 )] 6 (1 + c1 )C0 exp[(1 − b)g0 (T − ti )] ∀ i > 1,

(8.2.19)

where c1 = (1 − r1−b )−1 exp[bg0 (T − t0 )]. By using estimate (8.2.18) and the property of monotonicity of the function g0 (T − t), we get 1

h(ti , 1 + c0 (C00 G¯ k (g0 (T − ti ))) κ(p+1) ) 1

= max{ sup h(t, 1 + c0 C¯ 0 G¯ k (g0 (T − ti ))) κ(p+1) , 06t6t0 1

max hj (1 + c0 (C00 G¯ k (g0 (T − ti ))) κ(p+1) )},

16j6i

(0)

6 C0 max{exp[bg0 (T )] exp[(1 − b)g0 (T − t1 )], exp[bg0 (T − t0 )] exp[(1 − b)g0 (T − ti )]} 6 C0 exp[(1 − b)g0 (T − ti )] exp[bg0 (T − t0 )] 6 c2 C0 exp((1 − b)g0 (T − ti )) ∀ i > 1,

(8.2.20)

b

c2 = F0 (t0 ) = exp[bg0 (T − t0 )]. (0)

(0)

As a result of variation of t0 ∈ [0, t1 ], where t1 is specified in (8.2.14), we conclude that inequalities (8.2.19) and (8.2.20) imply the following estimates: 1

E(t, 1 + c0 (C00 G¯ k (g0 (T − t))) κ(p+1) ) (0)

6 c3 C0 exp[(1 − b)g0 (T − t)] ∀ t ∈ [t1 , T ), 1

h(t, 1 + c0 (C00 G¯ k (g0 (T − t))) κ(p+1) ) (0)

6 c4 C0 exp((1 − b)g0 (T − t)) ∀ t ∈ [t1 , T ),

(8.2.21)

where c3 = 2(1 − r1−b )−1 r−b exp(bg0 (T )) and c4 = 2−1 (1 − r1−b )c3 < c3 . In addition, by using (8.2.13), we obtain h(t, 1) + E(t, 1) 6 C0 exp(bg0 (T −t)) exp((1−b)g0 (T −t)) (0)

6 C0 r−b exp(bg0 (T )) exp((1−b)g0 (T −t)) ∀ t ∈ [0, t1 ]. (8.2.22) We now fix an arbitrary number t¯ < T and write the following corollaries of estimates (8.2.21) and (8.2.22): max{h(t, 1+s0 (t¯)), E(t, 1 + s0 (t¯))} (1)

(1)

6 min{C0 F0 (t), C1 F0 (t)} 6 C1 F0 (t) ∀ t ∈ (0, t¯ ],

(8.2.23)

Section 8.2

383

Estimates for the blow-up wave in the equation

where (1)

F0 (t) = exp(g1 (T − t)),

g1 (T − t) = (1 − b)g0 (T − t),

C1 = 2C0 exp(bg0 (T )) r−b (1 − r1−b )−1 , and

 1 c (C 0 G¯ (g (T − t))) κ(p+1) 0 0 k 0 s0 (t) := 0

(0)

∀ t ∈ [t1 , T ), (0)

(8.2.24)

∀ t ∈ (0, t1 ].

Estimates (8.2.23) complete the first cycle of estimation of the energy functions h(t, s) and E(t, s). Note that the new blow-up regime specified by the established estimate (8.2.23) is less steep than the initial regime F0 (t). In particular, the following inequality may be true: (1) min{C0 F0 (t), C1 F0 (t)} 6 ω0 exp(gs (T − t)) ∀ t ∈ (0, t¯),

(8.2.25)

where gs (τ ) is given by (8.2.1) and 0 < ω0 = const is independent of t¯. In this case, it is possible to apply Theorem 7.3.1. According to this theorem, there exist constants c and R¯ that depend only on known parameters and do not depend on t¯ such that E(t¯, 1 + s0 (t¯) + s) ¯ + h(t¯, 1 + s0 (t¯) + s) ¯ 6 R¯

∀ t¯ < T,

p−q

s¯ = cω0m(p+1)(q+1)+n(p−q) . This inequality is equivalent to the following estimate of propagation of the blow-up wave: ¯ χL (t¯) 6 1 + s0 (t¯) + s¯ L = R.

(8.2.26)

In fact, this is the final estimate for the blow-up waves in case (8.2.25). We now show that estimate (8.2.25) and, hence, (8.2.26) are true for the regimes g0 (τ) in which the corresponding functions Gk (v) belong to the class (B). In this case, inequality (8.2.11) implies that  p − q   ϕ0 (v − k) − ϕ0 (v) 6 exp − −β v q+1 (8.2.27) p−q , ϕ0 (v) := g0−1 (v). ∀ v > vβ , ∀ β : 0 < β < q+1 This immediately yields the inequality ϕ0 (v) =

∞ X

(ϕ0 (v + ik) − ϕ0 (v + (i + 1)k)) 6 L2 exp(−Γv)

i=0

∀ v > vβ ,

p−q − β, Γ := q+1

exp(Γk) L2 = , exp(Γk) − 1

(8.2.28)

and, hence, g0 (T − t) 6 Γ−1 ln(L2 (T − t)−1 ) = L3 + g¯0 (T − t) ∀ t ∈ [tβ , T ),

(8.2.29)

384

Chapter 8 Nonlocalized regimes with singular peaking

where tβ := T − g0−1 (vβ ),

g¯0 (τ ) :=

(q + 1) ln τ −1 , (p − q) − β(q + 1)

L3 = Γ−1 ln L2 = (q + 1)(p − q − β(q + 1))−1 ln L2 . We now fix an arbitrary number ρ : (1 − b) < ρ < 1 and set β = β¯ := ρ−1 (q + 1)−1 (p − q)(ρ − 1 + b).

(8.2.30)

In view of (8.2.29), we can easily show that g1 (τ ) := (1 − b)g0 (τ ) = (1 − b)L3 + ρgs (τ ), where gs (τ ) is given by (8.2.1). Thus, we have (1)

C1 F0 (t) 6 ω0 exp(ρgs (T − t)) ∀ t ∈ (tβ¯ , T ),

ρ < 1,

(8.2.31)

where ω0 = C1 exp((1 − b)L3 ). This inequality, Lemma 7.3.1, and inequality (8.2.23) imply the following estimate (uniform in t¯ < T ): J(t, 1 + s0 (t¯) + δ) 6 K(δ) ∀ δ > 0,

∀ t < t¯ < T,

(8.2.32)

where K(δ) is the function from (7.3.8), (7.3.33) corresponding to the constants C = ω0 from (8.2.31) and γ = (1 − ρ)

(q + 1) . p−q

Further, reasoning as in the case of getting estimate (7.3.73) from inequality (7.3.72) at the end of the proof of Theorem 7.3.1, we obtain the following relation from (8.2.32):  −m(p+1)   δ δ K h(t, 1+s0 (t¯)+δ) + k1 E(t, 1+s0 (t¯)+δ) 6 (1+ε)h¯ 0 + c(ε) 2 2 (8.2.33) ∀ t < t¯ < T, ∀ ε > 0, ∀ δ > 0, where h¯ 0 is given in (7.1.5) and the constant c = c(ε) is independent of t, t¯ < T. We now fix δ = δ0 > 0 and ε = ε0 > 0 and obtain the following inequality from (8.2.33): χL (t¯) 6 1 + s0 (t¯) + δ0 where

∀ t¯ < T,

 −m(p+1)   δ0 δ0 ¯ K . L = (1 + ε)h0 + c(ε0 ) 2 2

(8.2.34)

Section 8.2

385

Estimates for the blow-up wave in the equation

Thus, we have deduced estimate (8.2.26) for the majorant g0 (τ ) from the class (B). This proves the following assertion: Lemma 8.2.1. Assume that a function g(τ ) monotonically nonincreasing on [0, T ) is specified in (7.1.10), (8.2.1) by a blow-up boundary regime f (t, x) of problem (7.1.1)–(7.1.7) and that g0 (τ ) is a majorant of the function g(τ ) from the class (B) regularly decreasing in a sense of Definition 8.2.2 [see (8.2.11)]. Then, for any constants δ0 > 0 and ε0 > 0, the blow-up wave of any solution of the considered problem satisfies estimate (8.2.34), where the function K(s) is defined by (7.3.8) and (7.3.33) and the function s0 (t) is defined by (8.2.24). We now return to estimate (8.2.23). We have already considered the case where the right-hand side of this inequality satisfies (8.2.25). We now consider (0) the case where (8.2.25) is not true, i.e., there exists a nonempty set σ ⊂ (t1 , t¯) such that (1) C1 F0 (t) > ω0 exp(gs (T − t)) ∀ t ∈ σ. In this case, we perform a new cycle of iterative estimations for the functions h(t, s) and E(t, s). As a starting point of this cycle, we use (8.2.23) as the initial estimate instead of (8.2.13). Moreover, instead of sequence (8.2.14), we (1) introduce a new sequence {ti } specified as follows: (1)

(1)

(1)

(1)

F0 (tj ) := r−1 F0 (tj−1 ) ∀ j > 1, (1)

(1)

(0)

t0 ∈ [0, t2 ), (0)

where F0 (t) := exp[g1 (T − t)] and the value t2

(8.2.35)

is given by the formula

(0)

g1 (T − t2 ) := g1 (T ) + k ⇐⇒

(0)

g0 (T − t2 ) := g0 (T ) + (1 − b)−1 k,

k = ln r−1

(8.2.36)

(0) t1 .

by analogy with It follows from (8.2.35) and (8.2.36) that (1)

(1)

(1)

(1)

(1)

∆j := tj −tj−1 = g1−1 (g1 (T −tj ) − k) − (T −tj )   k (1) (1) = g0−1 g0 (T −tj ) − − (T −tj ) ∀ j > 1. 1−b (1)

Denote vj

(8.2.37)

(1)

:= g0 (T − tj ). Thus, we get   k (1) (1) (1) −1 ∆j = g0 vj − − g0−1 (vj ) ∀ j > 1. 1−b

(8.2.38)

Further, in view of relations (8.2.37) and (8.2.38), we conclude that, in the new cycle, the function      (p − q)(1 − b) k (1) v g0−1 v − −g0−1 (v) Gk (v) := exp q+1 1−b (8.2.39) k (1) ∀ v > g0 (T ) + := v0 , 1−b

386

Chapter 8 Nonlocalized regimes with singular peaking

is an analog of the function Gk (v) in the first cycle. Hence, a monotonically (1) nondecreasing sequence of shifts {δ¯j } can be defined as follows: 1

(1) (1) (1) δ¯j := (C10 G¯ k (vj )) κ(p+1) , p−q q+1

C10 = C1

,

(8.2.40)

C1 is specified in (8.2.23),

(1) (1) G¯ k (v) := max Gk (v), ¯

(1)

v0 = g0 (T ) +

(1)

¯ v0 6v6v (1)

k . 1−b

(1)

We now introduce the energy functions Ej (s) and hj (s) associated with (1)

the new sequence {tj }. By virtue of Lemma 7.3.2, inequality (8.2.23) yields the following estimates for these new energy functions: (1)

(1)

(1)

(1)

Ej (1+s0 (t¯)+s1 (tj )) 6 C1 rjb F0 (tj ) (1)
(1)
= C1 exp bg1 (T −t0 ) exp (1−b)g1 (T −tj ) (1)

∀ j : tj 6 t¯,

(8.2.41)
(1) (1) (1) (1)
hj (1+s0 (t¯)+s1 (tj )) 6 C1 exp bg1 (T −t0 ) exp (1−b)g1 (T −tj ) (1) ∀ j : tj 6 t¯,

where s1 (t) =

 1 c0 (C 0 G¯ (1) (g0 (T − t))) κ(p+1) 1 k

∀ t ∈ [t2 , T )

 0

∀ t ∈ [0, t2 ].

(0)

(0)

These estimates are similar to estimates (8.2.18) from the first cycle. In addition, we denote g2 (T − t) := (1 − b)g1 (T − t). This immediately yields the following relation from (8.2.23): max{h(t, 1 + s0 (t¯)), E(t, 1 + s0 (t¯))} 6 C1 exp(bg1 (T − t)) exp((1 − b)g1 (T − t)) (0)

6 C1 r−b exp(bg1 (T )) exp(g2 (T − t)) ∀ t ∈ (0, t2 ). By using this relation and the procedure of getting estimates (8.2.23) from (8.2.18) and (8.2.22), we can prove that inequalities (8.2.41) imply the following relations: max{h(t, 1 + s0 (t¯) + s1 (t¯)), E(t, 1 + s0 (t¯) + s1 (t¯))} (1)

(2)

6 min{C0 F0 (t), C1 F0 (t), C2 F0 (t)} (2) 6 C2 F0 (t) ∀ t ∈ (0, t¯),

(8.2.42)

where C2 = 2C1 exp(bg1 (T )(1 − r1−b )−1 )r−b ,

(2)

F0 (t) = exp(g2 (T − t)).

Section 8.2

Estimates for the blow-up wave in the equation

387

It may happen that the blow-up regime specified by estimate (8.2.42) has a sufficiently gentle slope in a sense that the following estimate is true: (1)

(2)

min{C0 F0 (t), C1 F0 (t), C2 F0 (t)} 6 ω0 exp(gs (T − t)) ∀ t < t¯.

(8.2.43)

This estimate is similar to estimate (8.2.25). Thus, by using Theorem 7.3.1, we arrive at the final estimate ¯ χL (t¯) 6 1 + s0 (t¯) + s1 (t¯) + s, ¯ L = R, (8.2.44) which is similar to estimate (8.2.26). If inequality (8.2.43) is not true, then we use estimate (8.2.42) as the initial estimate for the third cycle of estimation of the energy functions. Finally, after the ith cycle, we get   i−1 i−1 X X max h(t, 1 + sj (t¯)), E(t, 1 + sj (t¯)) j=0

j=0



6 min C0 F0 (t),

(1) C1 F0 (t), . . . ,

∀ t ∈ (0, t¯ ],

(i) (i) Ci F0 (t) 6 Ci F0 (t)

(8.2.45)

(i)

F0 (t) := exp(gi (T − t)),

where gi (T −t) = (1−b)i g0 (T −t), the coefficients Ci are given by the recurrence relations 2Ci−1 exp[b(1 − b)i−1 g0 (T )] 2Ci−1 exp[bgi−1 (T )] = 6 D0 Ci−1 Ci = b 1−b (1 − r )r (1 − r1−b )rb (8.2.46) D0 := 2(1 − r1−b )−1 r−b exp[bg0 (T )],

∀ i > 1,

the functions sj (t) in (8.2.45) are given by the formula  p−q 1 (0) c C κ(p+1)(q+1) G¯ (j) (g (T − t)) κ(p+1) ∀ t ∈ [tj , t¯), 0 j 0 k sj (t) :=  (0) 0 ∀ t ∈ (0, tj ], (0)

where the numbers tj

(8.2.47)

are given by the equality (0)

g0 (T − tj ) := g0 (T ) + k(1 − b)−j ,

(8.2.48)

(j) and the function G¯ k (v) is the minimal monotonically nondecreasing majorant (j) of the function Gk (v) k (j) (j) (j) G¯ k (v) := max Gk (v 0 ), v0 = g0 (T ) + (8.2.49) (j) (1 − b)j v 6v 0 6v 0

given by the equality      p−q k (j) −1 −1 j Gk (v) := exp v− − g0 (v) (1 − b) v g0 q+1 (1 − b)j k (j) (0) ∀ v > g0 (T ) + = v0 = g0 (T − tj+1 ). (1 − b)j

(8.2.50)

388

Chapter 8 Nonlocalized regimes with singular peaking

We now determine the maximal admissible number i0 = i0 (t¯) of the outlined cycles. We estimate this number both from above and from below. To this end, (0) we vary the initial point t0 = t0 (t¯) ∈ [0, t1 ) and obtain the equality (0) ¯ 0 (t)

ti

= t¯

(8.2.51)

At the same time, by virtue of (8.2.48), we get (0)

g0 (T − t¯) = g0 (T − ti0 ) = g0 (T − t0 ) + k(1 − b)−(i0 −1) .

(8.2.52)

Since g0 (T ) 6 g0 (T − t0 ) 6 g0 (T ) + k, this yields following inequalities: g0 (T ) + k(1 − b)−(i0 −1) 6 g0 (T − t¯), g0 (T ) + k + k(1 − b)−(i0 −1) > g0 (T − t¯). These inequalities imply the following estimate for i0 : ln[g0 (T − t¯) − g0 (T )] − ln k ln[g0 (T − t¯) − g0 (T ) − k] − ln k > i0 − 1 > . −1 ln(1 − b) ln(1 − b)−1 (8.2.53) It is clear that the actual number of required iterations may be smaller than i0 (t¯) because the corresponding intermediate blow-up regime is already localized for some i < i0 (t¯), i.e., it is a regime for which Theorem 7.3.1 is applicable [see inequalities (8.2.25) and (8.2.43)]. We denote this actual optimal number of iterations by iop = iop (t¯) and establish the required sharp upper bound for the function iop −1

Sop (t¯) :=

X

iop −1

sj (t¯) = c0

X

j=0

p−q

1

(j)

Cjκ(p+1)(q+1) G¯ k (g0 (T − t¯)) κ(p+1) ,

(8.2.54)

j=0

(0) where G¯ k (v) := G¯ k (v). ¯ > Gk (v), ¯ we By using (8.2.46), definition (8.2.49), and the inequality G¯ k (v) obtain   iop −1 1 X j κ(p+1) Sop (t¯) 6 s0 (t¯) 1 + D1 Hj , (8.2.55) j=1

where s0 (t) is given in (8.2.24) and p−q

D1 = D0κ(p+1)(q+1) ,

(j)

(0)

Hj = Hj (v) ¯ := Gk (v (j) )(Gk (v)) ¯ −1 .

(j) Here, v¯ = g0 (T − t¯) and v (j) is the minimum number such that G¯ k (v) ¯ = (j) (j) (j) (j) −j Gk (v ) in the interval (v0 , v], ¯ v0 = g0 (T ) + k(1 − b) . We denote

(1 − b)−j = J + ξ,

where J = [(1 − b)−j ], 0 6 ξ < 1.

Section 8.2

389

Estimates for the blow-up wave in the equation

We now estimate Hj from above. Since the function g0 is regularly decreasing (see Definition 8.2.2), we obtain ϕ0 (v (j) − (1 − b)−j k) − ϕ0 (v (j) ) = ϕ0 (v (j) − ξk) − ϕ0 (v (j) ) + 6

J X

[ϕ0 (v (j) − (l − 1)k − ξk − k) − ϕ0 (v (j) − (l − 1)k − ξk)]

l=1 ϕ0 (v (j)

− k) − ϕ0 (v (j) )

    J X p−q p−q (l−1)k exp ξk + (1−δ)−1 (ϕ0 (v (j) −k)−ϕ0 (v (j) ))· exp q+1 q+1 l=1   p−q 6 (1−δ)−1 Ak (ϕ0 (v (j) −k)−ϕ0 (v (j) )) · exp (8.2.56) (1−b)−j k , q+1  −1   p−q k −1 Ak := exp + (1 − δ). q+1 By using (8.2.56), we can estimate Hj as follows: (j)

(0)

(0)

(0)

Hj = Hj (v) ¯ = Gk (v (j) ) · Gk (v (j) )−1 · Gk (v (j) ) · Gk (v) ¯ −1  p−q  (j) v − (1−b)j v (j) 6 (1−δ)−1 Ak exp − q+1  (0) (0) −j − (1−b) k Gk (v (j) )Gk (v) ¯ −1 .

(8.2.57)

(j)

As for the location of the points v (j) in the intervals [v0 , v], ¯ there are only two possibilities: either v (j) > 2b−1 (1 − b)−j k,

j = 1, 2, . . . , i0 ,

(8.2.58)

where i0 is given in (8.2.53), or there exists a number j1 < i0 such that the inequality in (8.2.58) is satisfied for all j < j1 and v (j1 ) < 2b−1 (1 − b)−j1 k.

(8.2.59)

In the first case, i.e., in the case where (8.2.58) is true, we continue estimate (8.2.57) and obtain   (p − q)b (j) (0) (0) −1 ¯ −1 Hj 6 (1 − δ) Ak exp − v Gk (v (j) )Gk (v) (q + 1)2   (p − q) b (j) −2 v 6 (1 − δ) Ak exp − (q + 1) 2 0   (p − q) b (1 − b)−j 6 Bk exp − ∀ j > 1, (8.2.60) (q + 1) 2

390

Chapter 8 Nonlocalized regimes with singular peaking

where −2

Bk = (1 − δ)

 (p − q) b g0 (T ) . Ak exp (q + 1) 2 

In view of (8.2.60), we find iop −1

X

1

1

D1j Hjκ(p+1) 6 Bkκ(p+1)

∞ X

−j

D1j µ(1−b)

1

= Bkκ(p+1) D2 ,

(8.2.61)

j=1

j=1

where  µ = exp −

(p − q) b 2κ(q + 1)(p + 1)

 D2 = D2 (D1 , b) < ∞.

< 1,

Hence, in the case where (8.2.58) is true, by using (8.2.61), we get the following estimate from (8.2.55): (1) Sop (t¯) 6 Sop (t¯) := D3 s0 (t¯),

1

D3 = (1 + D2 Bkκ(p+1) ).

(8.2.62)

By virtue of inequalities (8.2.45), (8.2.62), and (8.2.53), we arrive at the following estimate: (1) (1) h(t¯, 1 + Sop (t¯)) + E(t¯, 1 + Sop (t¯))

6

1+ ln k C0 D0 ln(1−b) (g0 (T − t¯ )

ln D

0 − ln(1−b)

− g0 (T ))

6 C10 g0 (T − t¯ )m1 ,

 kg0 (T − t¯ ) exp b(g0 (T − t¯ ) − g0 (T )) (8.2.63) 

where ln D0 , m1 = ln(1 − b)−1

C10 =

1+ ln k C0 D0 ln(1−b)



 g0 (T ) + k exp . b

We now consider the second case of possible location of the points v (j) , i.e., the case where there is a number j1 < i0 such that inequality (8.2.59) is true. In this case, we have  2(p − q)    k (j ) (j ) Gk 1 (v (j1 ) ) 6 exp ϕ0 v0 1 − b(q + 1) (1 − b)j1  2(p − q)  (8.2.64) = T exp := L1 . b(q + 1) Hence, the quantity Hj1 (v) ¯ from (8.2.57) can be estimated as follows: (0)

Hj1 (v) ¯ 6 L1 Gk (v) ¯ −1 .

(8.2.65)

We now assume that the analyzed boundary regime g0 (T − t) belongs to the class (A), i.e., inequality (8.2.10) holds with a positive constant β < p−q q+1 . Assume that v¯ coincides with some vi in the sequence specified by (8.2.10).

Section 8.2

391

Estimates for the blow-up wave in the equation

Thus, in view of (8.2.65), we obtain ¯ 6 L1 exp(−β v) ¯ Hj1 (v) (j )

6 L1 exp(−βv0 1 ) = L1 exp(−βg0 (T )) exp(−βk(1 − b)−j1 ). By virtue of (8.2.65), we find 1

1

−j1

6 M1κ(p+1) D1j1 µ¯ (1−b) D1j1 Hjκ(p+1) 1 where M1 = L1 exp(−βg0 (T )),

µ¯ = exp(

,

(8.2.66)

−βk ) < 1. κ(p + 1)

By using (8.2.66), by analogy with (8.2.61) and (8.2.62), we get (2) ¯ 3 s0 (t¯), Sop (t¯) 6 Sop (t¯) := D

¯ t¯ = ti := T − g0−1 (vi ) = T − g0−1 (v),

(8.2.67)

¯3 = D ¯ 3 (D1 , β). D

This proves the following assertion: Lemma 8.2.2. Assume that g(τ ) > 0, is a function monotonically nonincreasing in the interval [0, T ) (g(τ ) → ∞ as τ → 0) and specified in (7.1.10), (8.2.1) by the blow-up boundary regime f (t, x) of problem (7.1.1)–(7.1.7). Let g0 (τ ) be any regularly decreasing [in a sense of Definition 8.2.2] majorant of the function g(τ ) and let the function Gk determined for given g0 by (8.2.6), belong to class (A) [see (8.2.10)]. Then the energy functions associated with the solution u of the analyzed boundary-value problem satisfy the following inequality: h(ti , 1+S(ti )) + E(ti , 1+S(ti )) 6 C10 F10 (ti ) := C10 exp(g10 (T −ti )) ∀ i ∈ N,

(8.2.68)

g10 (τ ) := m1 ln g0 (τ ),

where ti := T −g0−1 (vi ), {vi } is the sequence from the definition of the class (A) [see (8.2.10)], S(t) := Ds0 (t), ¯ 3 }, D = max{D3 , D

(8.2.69) s0 (t) is given in (8.2.24),

¯ 3 are given in (8.2.62) and (8.2.67), respectively, and the constants D3 and D m1 and C10 are given in (8.2.63). By using Lemmas 8.2.2 and 8.2.1, we can establish, in a certain sense, sharp upper bounds for the propagation of blow-up waves in any nonlocalized blow-up boundary regimes. We now present two examples.

392

Chapter 8 Nonlocalized regimes with singular peaking

Example 1.  g(τ ) 6 g0 (τ ) =

ξ+q+1 p−q



ln τ −1

 In this case, we have g0−1 (v) = exp −

∀ τ > 0, (p−q)v ξ+q+1



  (p − q)v g0−1 (v − k) − g0−1 (v) = ck exp − , ξ+q+1

ξ = const > 0.

(8.2.70)

and  ck = exp

(p − q)k ξ+q+1

 − 1.

Therefore, Gk (v) = G¯ k (v) = ck exp



 (p − q)ξv . (q + 1)(ξ + q + 1)

(8.2.71)

Thus, the function Gk (v) specified by regime (8.2.70) belongs to the class (A) with the following constant: β=

ξ(p − q) p−q . < (q + 1)(ξ + q + 1) q+1

We now apply Lemma 8.2.2. By using (8.2.68), we obtain h(t, 1 + S (0) (t¯)) + E(t, 1 + S (0) (t¯)) 6 C10 F10 (t) ∀ t < t¯ < T,

(8.2.72)

where ξ − m(p+1)(q+1)+n(p−q)

1

S (0) (t¯) = DGk (g0 (T − t¯)) κ(p+1) = K(T − t¯) 1 κ(p+1)

K = Dck

,

,

D and ck are specified in (8.2.69) and (8.2.71), respectively, and m1  (ξ + q + 1) −1 ln(T − t) . F10 (t) = p−q At the point s0 = 1 + S (0) (t¯) for any t¯ < T, estimate (8.2.72) determines a weakly singular boundary regime satisfying condition (7.3.1) of Theorem 7.3.1. By virtue of this theorem, we get the following estimate for the blow-up wave: ξ

− χL (t¯) 6 1 + K(T − t) m(p+1)(q+1)+n(p−q) + δ

∀ t¯ < T,

∀ δ > 0,

(8.2.73)

where L = L(C10 , δ) = const < ∞. Estimate (8.2.73) for m > 1 and its sharpness, in a certain sense, were proved in [45]. The upper bound of the blow-up wave for the model boundary regime considered in the next example was also established in [45].

Section 8.2

393

Estimates for the blow-up wave in the equation

Example 2. g(τ ) 6 g0 (τ ) := λτ −η

∀ τ > 0,

η = const > 0, λ = const > 0.

It is clear that ϕ0 (v) := g0−1 (v) = v

− η1

(8.2.74)

1

λη .

Since this function is convex, we get   p−q > exp −(1−ν) v ϕ0 (v−k) − ϕ0 (v) > = kλ η v q+1 (8.2.75) ∀ ν : 0 < ν < 1, ∀ k > 0, ∀ η > 0, ∀ v > v0 = v0 (k, η, λ). 1+η −1 − η

1 η

−kϕ00 (v)

This implies that condition (8.2.10) with arbitrary β=

ν(p − q) p−q :0 v¯0 := 2k.

It is easy to see that min Gˆ k (v) = Gk (vmin ), v>0

vmin =

(1 + η)(q + 1) . η(p − q)

Moreover, Gˆ k (v) is a monotonically increasing function for v > vmin such that Gˆ k (v) → ∞ as v → ∞. If (0)

v0 := g0 (T ) + k = λT −η + k, then we define v1 as follows: (0) v1 > vmin : Gˆ k (v1 ) := Gˆ k (v0 ).

(8.2.76)

Then Gˆ k (v) > G¯ k (v) ∀ v > v1 , where G¯ k (v) is given in definition (8.2.24). At the same time, by virtue of Lemma 8.2.2, we find h(t, 1 + S (0) (t¯)) + E(t, 1 + S (0) (t¯)) 6 C10 exp(g10 (T − t)) ∀ t : t1 < t < t¯,

∀ t¯ < T,

(8.2.77)

394

Chapter 8 Nonlocalized regimes with singular peaking

where

1

S (0) (t) = DGˆ k (g0 (T − t)) κ(p+1) ,

t1 = T − ϕ0 (v1 ),

D is given in (8.2.69), and the function g10 (τ ) := m1 η ln τ −1 satisfies condition (8.2.70) with ξ = ξ0 > 0 given by the formula m1 λ =

ξ0 + q + 1 p−q

=⇒

ξ0 = m1 λ(p − q) − q − 1.

Hence, for any t¯ < T, estimate (8.2.77) specifies an intermediate blow-up boundary regime considered in Example 1. Therefore, by virtue of (8.2.77), we find ξ0

− χL (t¯) 6 1 + S (0) (t¯) + K(T − t¯) m(p+1)(q+1)+n(p−q) + δ

∀ t¯ ∈ (t1 , T ) ∀ δ > 0,

(8.2.78)

L = L(δ).

In view of the fact that the functions Gk (v) and, hence, Gˆ k (v) belong to the class (A), it is easy to see that ξ0

S (0) (t)(T − t) m(p+1)(q+1)+n(p−q) → ∞ as t → T. Thus, it follows from (8.2.78) that χL (t¯) 6 1 + (1 + ω(t¯))S (0) (t¯) ∀ t¯ ∈ (t1 , T ), where ω(t¯) → 0 as t¯ → T. By using this inequality and the expression for S (0) (t) from (8.2.77), we derive the following estimate for all t < T :   (1+η) 1 λ(p − q) − κ(p+1) −η κ(p+1) (T − t) exp χL1 (t) 6 1 + K1 λ , (8.2.79) (T − t) κ(q + 1) where L1 < ∞ and K1 < ∞ depend only on known parameters of the problem. The procedure of step-by-step estimation of the blow-up wave described in Example 2 and based on Lemma 8.2.2 can be applied to any general strongly blow-up boundary regimes. Theorem 8.2.1. Assume that g(τ ) > 0 is a function monotonically nonincreasing in the interval (0, T ) (so that g(τ ) → ∞ as τ → ∞) and that this function is determined in (8.2.1), (7.1.10) by a blow-up boundary regime f (t, x) of problem (7.1.1)–(7.1.7). Let g0 (τ ) be any regularly decreasing [in a sense of Definition 8.2.2] majorant of g(τ ) and let the function Gk (v) determined by ¯ [see (8.2.12)]. Then there exist cong0 (τ ) in (8.2.6) belong to the subclass (A) stants k, K, and L depending only on known parameters of the problem such that the blow-up wave of any solution u of the analyzed problem satisfies the estimate 1

χL (t¯) 6 1 + KGk (g0 (T − t¯)) κ(p+1)

∀ t¯ < T,

κ from (7.2.12),

(8.2.80)

Section 8.2

395

Estimates for the blow-up wave in the equation

¯ it follows Proof. In view of the fact that Gk belongs to the class (A), estimate (8.2.68) established in Lemma 8.2.2 that h(t, 1 + Ds0 (t¯)) + E(t, 1 + Ds0 (t¯)) 6 C10 F10 (t) ∀ t : 0 < t < t¯ < T, (8.2.81) where s0 (t) is a function from (8.2.24). If the regime F10 (t) from (8.2.81) satisfies conditions (7.3.1) of Theorem 7.3.1, then, by virtue of this theorem, inequality (8.2.81) yields estimate (8.2.80). If F10 (t) does not satisfy relation (7.3.1), then we can use estimate (8.2.81) instead of (8.2.13) as the starting point in the cycle of calculations performed in Lemma 8.2.2 and leading to estimate (8.2.6). It is easy to see that i cycles of these calculations result in the relation h(t¯, 1 + Ds0 (t¯) +

i−1 X

s1j (t¯)) + E(t¯, 1 + Ds0 (t¯) +

j=0

6

(i) C10 D0i F1,0 (t¯)

i−1 X

s1j (t¯))

j=0

:=

where

C10 D0i exp((1 p−q

i

− b) g10 (T − t¯)) ∀ t¯ < T,

j(p−q)

(8.2.82)

1

(j) κ(p+1)(q+1) s1j (t) = c0 C10 D0κ(p+1)(q+1) G¯ k1 (g10 (T − t)) κ(p+1) , (j) and G¯ k1 is the minimal monotonically nonincreasing majorant of the function      p−q k (j) −1 −1 (1 − b)j v Gk1 (v) := exp (v) . g10 v− − g 10 p+1 (1 − b)j

According to (8.2.53), for any t¯ < T, we define a natural number i10 = i10 (t¯) by the inequalities ln(g10 (T − t¯ ) − g10 (T )) − ln k ln(g10 (T − t¯ ) − g10 (T ) − k) − ln k > i − 1 > 10 ln(1−b)−1 ln(1−b)−1 (8.2.83) and set i = i10 (t¯) in (8.2.82). Moreover, by analogy with (8.2.63), we obtain h(t¯, 1+Ds0 (t¯) +

iX 10 −1 j=0

s1j (t¯)) + E(t¯, 1+Ds0 (t¯) +

iX 10 −1

s1j (t¯))

j=0

6 C20 exp(g20 (T − t¯)) ∀ t¯ < T,

(8.2.84)

2 , and where g20 (τ ) = m1 ln g10 (τ ), C20 = C10 D01 = C0 D01   ln k 1+ ln(1−b) g0 (T ) + k exp . D01 := D0 b

The following estimate is obvious:   p−q j ¯ s1j (t¯) 6 C10 exp (j ln D0 + (1 − b) m1 ln g0 (T − t¯)) , κ(p + 1)(q + 1)

396

Chapter 8 Nonlocalized regimes with singular peaking

where

p−q

1

κ(p+1)(q+1) T κ(p+1) . C¯ 10 = c0 C10

Therefore, as a result of simple calculations, we obtain iX 10 −1

s1j (t¯) 6 C¯ 10 exp (m ¯ 1 ln g0 (T − t¯)) (1 + ϕ1 (t¯)),

(8.2.85)

j=0

where m ¯1=

m1 (p−q) κ(p+1)(q+1)

and h i p−q ln D0 ¯ exp − κ(p+1)(q+1) (bm1 ln g0 (T − t¯) − ln(1−b) ln g (T − t )) 10 −1 h i ϕ1 (t¯) = . (p−q) ln D0 exp κ(p+1)(q+1) − 1

(0) It is clear that ϕ1 (t¯) → 0 as t¯ → T. We also recall that the function Gk (v) ¯ belongs to the class (A) and, hence,   p−q β(p − q) s0 (t¯) > c0 C0κ(p+1)(q+1) exp g0 (T − t¯) . (8.2.86) κ(p + 1)(q + 1)

In view of (8.2.85) and (8.2.84), we arrive the following relation: h(t, 1 + S1 (t¯)) + E(t, 1 + S1 (t¯)) 6 C20 F20 (t) ∀ t < t¯ < T,

(8.2.87)

where F20 (t) := exp(g20 (T − t)) = exp(m1 ln(m1 ln g0 (T − t))), S1 (t) := Ds0 (t) + C¯ 10 exp(m ¯ 1 ln g0 (T − t))(1 + ϕ1 (t)) = Ds0 (t) + S¯1 (t). It is possible to rewrite inequality (8.2.87) in the form h(t, 1 + S1 (t¯)) + E(t, 1 + S1 (t¯)) 6 ω(t)gs (T − t),

∀ t < t¯ < T,

(8.2.88)

where   q+1 ln(T − t)−1 ω(t) = C20 exp m1 ln(m1 ln g0 (T − t)) − p−q and gs (τ ) is given by (8.2.1). By Theorem 7.3.1, it follows from (8.2.88) that there exist constants C and K that depend only on the known parameters of the problem and do not depend on t and t¯ such that p−q

h(t, 1 + Ds0 (t¯) + S¯1 (t¯) + Cω(t¯) κ(p+1)(q+1) ) p−q

+ E(t, 1 + Ds0 (t¯) + S¯1 (t¯) + Cω(t¯) κ(p+1)(q+1) ) 6 K

∀ t < t¯ < T.

(8.2.89)

By using (8.2.86), one can easily show that the functions s0 (t), S¯1 (t), and ω(t) have the following properties: p−q

− S¯1 (t¯) · ω(t¯) κ(p+1)(q+1) → ∞ as t¯ → T,

s0 (t¯)S¯1 (t¯)−1 → ∞ as t¯ → T.

Section 8.2

397

Estimates for the blow-up wave in the equation

Therefore, inequality (8.2.89) yields the following estimate for the blow-up waves: χL (t) 6 (D + δ)s0 (t) ∀ t < T,

∀ δ > 0,

L = L(K, δ),

which corresponds to the required estimate (8.2.80). Example 3. g(τ ) 6 g0,l (τ ) := al exp(al−1 exp(. . . a1 exp τ −1 ) . . .),

ai > 0,

l ∈ N. (8.2.90)

We can easily show that −1  (1) −1 −1 −1 −1 −1 := (fl (v))−1 . ϕ0,l (v) := g0,l (v) = ln(a1 ln(a2 ln(a3 . . . ln(al v) . . .) (8.2.91) Setting (j) fl (v)

:=

−1 ln(a−1 j ln(aj+1 . . . ln



v al

 . . . ),

we conclude that  ϕ00,l (v) :=

dϕ0,l (v) (1) = −v −1 (fl (v))−1  dv

l Y

−1 (j) fl (v)

.

(8.2.92)

j=1

Since the function ϕ0,l (v) is convex, we get −kϕ00,l (v) 6 ϕ0,l (v − k) − ϕ0,l (v) 6 −kϕ00,l (v − k). This means that the function  Gk (v) = exp

 p−q v (ϕ0,l (v − k) − ϕ0,l (v)) q+1

¯ By virtue of Theorem 8.2.1, we arrive at the following belongs to the class (A). estimate for the blow-up wave:   (p − q) χL (t) 6 1 + K exp g0,l (T − t) κ(p + 1)(q + 1) Y − 1 l κ(p+1) 1 (j) κ(p+1) fl (g0,l (T − t)) × (T − t) j=1

In particular, if l = 1, then g0,1 (τ ) = a1 exp τ −1 and the estimate takes the following form:   2 (p − q)a1 −1 χL (t) 6 1 + K exp exp(T − t) · (T − t) κ(p+1) . κ(p + 1)(q + 1)

398

Chapter 8 Nonlocalized regimes with singular peaking

We now consider two examples of nonlocalized singular regimes obtained as perturbations of the limiting localized S -regime   q+1 −1 Fs (t) = exp ln(T − t) = exp(gs (T − t)), p−q i.e., consider the class of regimes of the form g(τ ) = gs (τ )(1 + ω(τ )),

ω(τ ) > 0,

(8.2.93)

where ω(τ ) → 0 as τ → 0 and, nevertheless, ϕ(τ ) := gs (τ )ω(τ ) → ∞ monotonically as τ → 0. It is clear that, for regimes (8.2.93), we get g(τ ) 6 gs (τ )(1 + ω0 (τ0 )) ∀ τ < τ0 ,

ω0 (τ0 ) := max ω(τ ). 0 a ln l + ϕ(τ ) − ϕ(lτ ). Thus, for l > 1, in view of the monotonicity of the function ϕ(τ ), we get   kξ , k ξ > a ln l ⇐⇒ l 6 exp a which proves (8.2.96). Finally, we establish the upper bound for (1 + ω1 (τ¯ ))−1 in (8.2.95). It is easy to see that ω1 (τ ) = −τ ϕ0 (τ ) > 0 because the function ϕ(τ ) is monotone. In addition, it follows from the monotonicity of the function ω(τ ) that ω1 (τ ) = ω(τ ) − τ ln τ¯ 1 ω 0 (τ ) 6 ω(τ ) < ω0 (τ0 ) = sup ω(τ ). 0 0, (8.3.1) F (t) > exp w(T − t) m(p+1)−1 i.e, the inequality opposite to (7.4.1). In the analyzed case p = q, the definition of regular monotonicity (Definition 8.2.2) is transformed into the following definition: Definition 8.3.1. We say that a continuous function ϕ(v) monotonically decreasing on (v0 , ∞), v0 > 0, and such that ϕ(v) → 0 as v → ∞ is regularly decreasing if there exist numbers λ ∈ (0, 1), k0 = k0 (λ) > 0, and δ = δ(λ) ∈ [0, 1) such that the following relations are true: 1

1

v2 (ϕ(v2 − k) − ϕ(v2 )) m(p+1) > (1 − δ)v1 (ϕ(v1 − k) − ϕ(v1 )) m(p+1) ∀ v2 , v1 : λ−1 v1 > v2 > v1 > max{k + v0 , λ−1 k}

(8.3.2)

for any k > k0 . Lemma 8.3.1. Assume that, for some s0 > 1, the energy functions associated with a solution u(t, x) of problem (7.1.1)–(7.1.7) satisfy the inequality h(t, s0 ) + E(t, s0 ) 6 CF0 (t) := C exp(g0 (T − t)) ∀ t < T,

(8.3.3)

where the function g0 (τ ) > 0 is such that its inverse function ϕ0 (v) is regularly monotone [in a sense of Definition 8.3.1] for all v > v0 = g0 (T ). Then, for any number b ∈ (0, 2−1 ), there exist positive constants K1 = K1 (b), K2 = K2 (b), and k = k(b) such that the following estimates hold: (0) (0) max{h(t, s0 + K1 U¯ k (g0 (T − t))), E(t, s0 + K1 U¯ k (g0 (T − t)))}

6 CK2 F0 (0)b F0 (t)1−b

(0)

∀ t : t1 < t < T, (8.3.4)

Section 8.3

401

Blow-up waves in quasihomogeneous parabolic equations

where (0)

(0)

t1 = T − ϕ0 (v0 ),

(0)

v0 := g0 (T ) + k,

(0) U¯ k (v) :=

(0)

¯ max Uk (v),

(0)

¯ v0 6 v6v (0)

1

(0)

Uk (v) := (v − v0 )(ϕ0 (v − k) − ϕ0 (v)) m(p+1)

(0)

∀ v > v0 .

Proof. As in the proof of Lemma 7.4.2, we now study the properties of the functional system (7.4.17), (7.4.18). As earlier, we define a sequence {ti } by the recurrence relation F0 (tj ) := r−1 F0 (tj−1 ) ∀ j > 1,

0 6 t0 < T − ϕ0 (g0 (T ) + k),

(8.3.5)

where k = ln r−1 and r > 0 is a constant defined in what follows. This yields the following relation: ∆j = tj − tj−1 = ϕ0 (vj −k) − ϕ0 (vj ) = ϕ0 (v0 +k(j −1)) − ϕ0 (v0 +kj), (8.3.6) where vj := g0 (T − tj ). Let λ ∈ (0, 1) be the number from the definition of regular monotonicity of the function ϕ0 (v). It is clear that vj v0 + kj j = < 6 λ−1 vi v0 + ki i

∀ i : [λj] + 1 6 i 6 j.

By using this inequality and condition (8.3.2), we arrive at the following relation for ϕ0 (v):  m(p+1) vj ϕ0 (vi − k) − ϕ0 (vi ) ∆i = 6 (1 − δ)−1 ∆j ϕ0 (vj − k) − ϕ0 (vj ) vi 6 A1 := (λ(1 − δ))−m(p+1)

∀ j ∈ N, ∀ i : [λj] + 1 6 i 6 j.

(8.3.7)

We now fix a number j ∈ N and define a shift δj by using relation (7.4.29), where ω > 0 is a free parameter specified in what follows. At the same time, we get j pairs of inequalities of the form (7.4.30), (7.4.31). To analyze the obtained system, we estimate the quantity G(∆i , δj ) ∀ i 6 j. By virtue (8.3.7), we find −m(p+1) −m(p+1) ∆i ∆ i δj = ∆ j δj 6 ωA1 ∀ i : [λj] + 1 6 i 6 j. ∆j Hence, the following estimate is true: 1

1

G(∆i , δj ) 6 ω1 (ω) = (ωA1 ) p + (A1 ω) m(p+1)−1 ,

[λj] + 1 6 i 6 j.

(8.3.8)

By using (8.3.3) and definition (8.3.5), we conclude that the analyzed system of functional inequalities implies the following inequalities: Ei (s0 + δj ) 6 c1 CF0 (ti−1 ) + c2 ω1 (ω)CF0 (ti ) 6 (c1 r + c2 ω1 (ω)) CF0 (ti ) ∀ i > [λj] + 1,

402

Chapter 8 Nonlocalized regimes with singular peaking

and hi (s0 + δj ) 6 ((1 + ε)r + c(ε)ω1 (ω)) CF0 (ti ) ∀ i > [λj] + 1. We fix b0 : 0 < b0 < 2−1 and then choose sufficiently small ε = ε(b0 ), ω = ω(b0 ), and r = r(b0 ) such that relations (7.4.36) and (7.4.37) are true. This yields Ei (s0 + δj ) 6 r2b0 CF0 (ti ) ∀ i > [λj] + 1, hi (s0 + δj ) 6 r2b0 CF0 (ti ) ∀ i > [λj] + 1. We use these estimates as initial estimates instead of (8.3.3). As a result, after the second cycle of calculations (analogous to the corresponding cycle in the proof of Lemma 7.4.2), we arrive at the inequalities Ei (s0 + 2δj ) 6 r4b0 CF0 (ti ),

hi (s0 + 2δj ) 6 r4b0 CF0 (ti ) ∀ i : 2 + [λj] 6 i 6 j.

If we continue this procedure, then, after [(1 − λ)j] cycles, we obtain Ej (s0 + (1 − λ)j δj ) 6 Cr−1 r−j(1−2b0 (1−λ)) F0 (t0 ), hj (s0 + (1 − λ)j δj ) 6 Cr−1 F0 (ti )1−2b0 (1−λ) F0 (t0 )2b0 (1−λ) .

(8.3.9)

Clearly, it follows from definition (8.3.5) that i = k −1 (g0 (T −ti ) − g0 (T −t0 )) ∀ i 6 j. (8.3.10) Therefore, by using (8.3.6), we obtain g0 (T −ti ) = ikg0 (T −t0 )

jδj = ω =ω

1 − m(p+1)

=⇒

1

j∆jm(p+1)

1 − m(p+1) −1

k

(g0 (T − tj ) − g0 (T − t0 ))

 1 m(p+1) × ϕ0 (g0 (T − tj ) − k) − ϕ0 (g0 (T − tj )) 

= k −1 ω

1 − m(p+1)

(0)

· Sk (g0 (T − tj )).

(8.3.11)

In view of (8.3.9), this yields max{hj (s0 + (1 − λ)k −1 ω

1 − m(p+1)

(0)

Uk (g0 (T − tj ))),

Ej (s0 + (1 − λ)k −1 ω

1 − m(p+1)

(0)

Uk (g0 (T − tj )))}

6 CF0 (tj )1−b F0 (t0 )b ,

(8.3.12)

where b = 2b0 (1 − λ). We start from (8.3.12) and perform calculations similar to (8.2.19) and (8.2.20). As a result, we obtain (0)

(0)

max{h(ti , s0 + c0 U¯ k (g0 (T − ti ))), E(ti , s0 + c0 U¯ k (g0 (T − ti )))} 6 (1 + c1 )CF0 (t0 )b · F0 (tj )1−b

∀ i > 1,

(8.3.13)

Section 8.3

403

Blow-up waves in quasihomogeneous parabolic equations

where c0 = (1 − λ)k −1 ω

1 − m(p+1)

and c1 = (1 − r1−b )−1 .

These estimates yield the required estimates (8.3.4) for K1 = c0 and K2 = (1 + c1 )r−1−b .  Lemma 8.3.2. Under the conditions of Lemma 8.3.1, there exist constants K3 , C10 , and m1 that depend only on known parameters of the problem [including b and δ specified in (8.3.4) and (8.3.2), respectively ] and do not depend on t¯ < T such that the following energy estimates are satisfied: (0)

(0)

max{h(t¯, s0 + K3 U¯ k (g0 (T − t¯))), E(t¯, s0 + K3 U¯ k (g0 (T − t¯)))} 6 C10 (ln F0 (t¯))m1 := C10 F1,0 (t¯) ∀ t¯ < T, (8.3.14) (0) (0) where the functions Uk (v) and U¯ k (v) are defined in (8.3.4).

Proof. We prove the lemma by using an iterative procedure. According to Lemma 8.3.1, for any t¯ < T, the following estimates are true: max{h(t, s0 + s0 (t¯)), E(t, 1 + s0 (t¯))} (1)

(1)

6 min(CF0 (t), C1 F0 (t)) 6 C1 F0 (t) ∀ t : 0 6 t 6 t¯,

(8.3.15)

where (0)

s0 (t¯) = K1 U¯ k (g0 (T − t¯)), g1 (τ ) = (1 − b)g0 (τ ),

(1)

F0 (t) = exp(g1 (T − t)), C1 = K2 F0 (0)b C.

We also note that ϕ1 (w) :=

g1−1 (w)

=

g0−1



w 1−b



:= g0−1 (v) = ϕ0 (v),

w = v. 1−b

Hence, the following two inequalities are equivalent:   1   m(p+1) k − ϕ0 (v2 ) (α0 ) v2 ϕ0 v2 − 1−b     1 m(p+1) k > (1 − δ)v1 ϕ0 v1 − − ϕ0 (v1 ) 1−b   k λ−1 k −1 ¯ ∀ v1 , v2 : v¯ = g0 (T − t) > λ v1 > v2 > v1 > max +v0 , ; 1−b 1−b   1 m(p+1) (α1 ) w2 ϕ1 (w2 − k) − ϕ1 (w2 )   1 m(p+1) > (1 − δ)w1 ϕ1 (w1 − k) − ϕ1 (w1 ) −1 ∀ w1 , w2 : w¯ := g1 (T − t¯) > λ−1 w 1 := λ (1− b)v1 > w2 > w1 k , w0 := (1 − b)v0 . > max k + w0 , λ

404

Chapter 8 Nonlocalized regimes with singular peaking

Since ϕ0 (v) satisfies condition (8.3.2), inequality (α0 ) holds for all v1 < v2 in the indicated interval and this interval is nonempty if   k k + v0 , v¯ := g0 (T − t¯) > max . (8.3.16) 1−b λ(1 − b) At the same time, by virtue of (α1 ), the function ϕ1 (w) satisfies condition (8.3.2) for the same values of the parameters δ, k, and λ as the function ϕ0 (v) on the corresponding nonempty set of values w1 < w2 . Therefore, by using inequality (8.3.15) as the initial estimate instead of (8.3.3), in view of Lemma 8.3.1, we get the relation max{h(t, s0 + s0 (t¯) + s1 (t¯)), E(t, 1 + s0 (t¯) + s1 (t¯))} (1)

(2)

(2)

6 min(CF0 (t), C1 F0 (t), C2 F0 (t)) 6 C2 F0 (t) ∀ t : 0 6 t 6 t¯, (8.3.17) where (2)

F0 (t) := exp(g2 (T − t)),

g2 (τ ) = (1 − b)g1 (τ ) = (1 − b)2 g0 (τ ),

(2)

(1) s1 (t¯) = K1 U¯ k (g0 (T − t¯)),

C2 = K2 F0 (0)b C1 , (1) U¯ k (v) :=

(1) Uk (v)

max

(1) v0 6v 0 6v

(1)

Uk (v 0 ),

  ϕ0 v −

k := (1 − b)(v − 1−b k (1) . v0 := g0 (T ) + 1−b (1) v0 )



 1 m(p+1) , − ϕ0 (v)

If the following analog of inequality (8.3.16) is true:   k k ¯ v¯ = g0 (T − t) > max , + v0 , (1 − b)2 λ(1 − b)2 then estimate (8.3.17) can be used as the initial inequality for the subsequent iterative application of Lemma 8.3.1. Hence, in the case where   k k v¯ := g0 (T − t¯) > max + v0 , , i ∈ N, (8.3.18) (1 − b)i−1 λ(1 − b)i−1 we can perform i iterations of this kind. As a result, we get the following estimate:      i−1 i−1 X X 0 0 ¯ max h t, s + sj (t) , E t, s + sj (t) j=0

6

(i) Ci F0 (t)

j=0

= Ci exp((1 − b)i g0 (T − t)),

0 < t 6 t¯.

(8.3.19)

Section 8.3

Blow-up waves in quasihomogeneous parabolic equations

405

Here, Cj 6 D0 Cj−1 sj (t) :=

∀ j 6 i, D0 = const = K2 F0 (0)b ,  (0) K1 U¯ (j) (g0 (T − t)) ∀ t ∈ [tj , t¯), k (0)

∀ t < tj ,

0 (0)

where tj

(8.3.20)

is given in (8.2.48), (j) U¯ k (v) := (j)

(j)

max Uk (v 0 ),

(j) v0 6v 0 6v

(j)

Uk (v) := (1 − b)j (v − v0 )(ϕ0 (v −

1 k ) − ϕ0 (v)) m(p+1) , j (1 − b)

(j)

(0)

v0 := g0 (T ) + k(1 − b)−j = g0 (tj ). In view of (8.3.18), for arbitrarily large values v, ¯ the maximal admissible number of iterations iop = iop (t¯) is specified by the inequalities k k > v¯ > . i op λ(1 − b) λ(1 − b)iop −1 Therefore,  i0 =

 ln v¯ − ln λ−1 − ln k . ln(1 − b)−1

In view of (8.3.20), this yields the inequality ln D0
Ciop exp (1 − b)iop g0 (T − t¯) 6 C10 (g0 (T − t¯)) ln(1−b)−1 ,

(8.3.21)

where C10 := C0 exp

  ln D0 (ln λ−1 + ln k) . exp − λ ln(1 − b)−1

k

We now estimate the quantity iop −1

Sop (t¯) :=

X

sj (t¯).

j=0

from above. Conditions (8.3.2) now imply that iop −1    X (0) −1 j ¯ Sop (t) 6 (1 − δ) K1 (1 − b) v¯ − v0 ϕ0 v¯ − j=0

 1  m(p+1) k , − ϕ ( v) ¯ 0 (1 − b)j

406

Chapter 8 Nonlocalized regimes with singular peaking

where v¯ = g0 (T − t¯). This yields iop −1   X (0) ¯ 1+ Hj , Sop (t¯) 6 (1 − δ)−1 K1 Uk (v) j=1

Hj = (1 − b)j

ϕ0 (v¯ −

k ) (1−b)j

− ϕ0 (v) ¯

1 ! m(p+1)

.

ϕ0 (v¯ − k) − ϕ0 (v) ¯

We now estimate Hj . We denote d = [(1 − b)−j ]. By using condition (8.3.2), we get ϕ0 (v¯ − dk) − ϕ0 (v) ¯ =

d X

(ϕ0 (v¯ − (i − 1)k − k) − ϕ0 (v¯ − (i − 1)k))

i=1

6 (1 − δ)−m(p+1) m(p+1) d  X v¯ · (ϕ0 (v¯ − k) − ϕ0 (v)) ¯ × v¯ − (i − 1)k i=1

6

d (1 −

δ)m(p+1) λm(p+1)

(ϕ0 (v¯ − k) − ϕ0 (v)) ¯

6 A1 (1 − b)−j (ϕ0 (v¯ − k) − ϕ0 (v)), ¯ where A1 is given in (8.3.7). The last inequality implies that 1

Hj 6 A1m(p+1) · (1 − b)jγ

∀ j 6 iop ,

γ = (l + 1)−1 ,

l = m(p + 1) − 1.

This yields the estimate 1

A1m(p+1) K1 (0) (0) U (g0 (T − t¯)) := Sop (t¯), Sop (t¯) 6 (1 − δ)(1 − (1 − b)γ ) k

(8.3.22)

whence, in view of (8.3.21), we get (0) (0) max{h(t¯, s0 + Sop (t¯)), E(t¯, s0 + Sop (t¯))} 6 C10 F1,0 (t¯) ∀ t¯ < T,

F1,0 (t) = exp(g10 (T − t)),

(8.3.23)

g10 (τ ) = m1 ln g0 (τ ),

m1 = ln D0 (ln(1 − b)−1 )−1 . This proves the required estimate (8.3.14).



By analogy with the case p > g, it is convenient to split the set of all blow-up boundary regimes into two classes as follows:

Section 8.3

407

Blow-up waves in quasihomogeneous parabolic equations

(A1 ) there exist a constant β > 0 and a sequence {vi } : vi → ∞ as i → ∞ such that −m(p+1)(1−β)

ϕ0 (vi −k) − ϕ0 (vi ) := g0−1 (vi −k) − g0−1 (vi ) > vi

∀ i ∈ N; (8.3.24)

(B1 ) for every β : 0 < β < β0 :=

m(p + 1) − 1 , m(p + 1)

there exists v0 = v0 (β) > g0 (T ) + k such that ϕ0 (v − k) − ϕ0 (v) 6 v −m(p+1)(1−β)

∀ v > v0 .

(8.3.25)

Moreover, in the class (A1 ), we introduce the following subclass of regimes satisfying the uniform estimate (8.3.24): ¯ 1 ) there exist β > 0 and v0 > 0 such that (A ϕ0 (v − k) − ϕ0 (v) > v −m(p+1)(1−β)

∀ v > v0 .

(8.3.26)

In addition, we consider the following class, which is wider than (B1 ): ¯ 1 ) there exist β ∈ (0, β0 ) and v0 > g0 (T ) + k such that estimate (8.3.25) is (B valid. ¯ 1 ) contain arbitrarily strong blow-up Note that the classes (A1 ) and (A regimes g0 (r) and the class (B1 ) contains weakly blow-up nonlocalized regimes obtained as perturbations of the limiting localized regime g0 (τ ) = τ

1 − m(p+1)−1

(ϕ0 (v) := g0−1 (v) = v −m(p+1)+1 ).

By using Lemmas 8.3.1 and 8.3.2, we establish the upper bounds of the blow¯ 1 ) and (B1 ). First, we consider up waves for the regimes from the classes (A ¯ 1 ). We fix the corresponding β ∈ (0, β0 ) and note that the the case ϕ0 (v) ∈ (B ¯ 1 ) implies that definition of the class (B ϕ0 (v) =

∞ X

(ϕ0 (v + lk) − ϕ0 (v + (l + 1)k)) 6

l=0

∞ X

(v + lk)−α0

∀ v > v0 ,

l=1

where α0 = m(p + 1)(1 − β) > 1. Thus, if we continue this inequality, then, as a result of simple calculations, we arrive at the estimate ϕ0 (v) 6 c0 v −α0 +1

∀ v > v0 ,

c0 = c0 (k, β).

This implies that 1 0 −1

−α

g0 (τ ) 6 c0

·τ

1 0 −1

−α

:= c1 τ

1 − m(p+1)(1−β)−1

∀ τ < τ0 := ϕ0 (v0 ).

(8.3.27)

408

Chapter 8 Nonlocalized regimes with singular peaking

Applying Lemma 8.3.2 and using inequality (8.3.27), we obtain (0) (0) max{h(t, s0 + K3 U¯ k (g0 (T − t¯))), E(t¯, s0 + K3 U¯ k (g0 (T − t¯)))} m

1 − m(p+1)(1−β)−1

1 6 C10 cm 1 (T − t)

∀ t < t¯ < T.

(8.3.28)

For any t¯ < T, estimate (8.3.28) determines, for all t < t¯, an intermediate peaking regime satisfying condition (7.4.3) from Lemma 7.4.1. Therefore, by using this lemma and condition (8.3.2), we obtain the following upper bound for the blow-up wave of the analyzed solution: (0) χL (t¯) 6 s0 + (1 − δ)−1 K3 U¯ k (g0 (T − t¯)) + γ,

∀ t¯ < T,

∀ γ > 0,

(8.3.29)

where L = L(γ) also depends on the other known parameters on the right-hand side of estimate (8.3.28) but does not depend on t¯ < T. ¯ 1 ). We again start from estimate We now consider the case where ϕ0 (v) ∈ (A (8.3.14) established in Lemma 8.3.2. This estimate can be rewritten in the form (0) max{h(t, s0 + K3 (1 − δ)−1 Uk (g0 (T − t¯)), (0) E(t, s0 + K3 (1 − δ)−1 Uk (g0 (T − t¯)))} 1

− 6 C10 exp W (t¯)(T − t) m(p+1)−1




∀ t < t¯ < T,

(8.3.30)

where 1

W (t¯) = max w(t),

w(t) = m1 ln g0 (T − t) (T − t) m(p+1)−1 .

0 v0 .

Thus, it is easy to see that W (t¯)

m(p+1)−1 m(p+1)

(0)

(Uk (g0 (T − t¯)))−1 6 c = c(β) ∀ t¯ < T.

If we continue the relation deduced above for χL (t¯) and apply the last inequality, then we get (0)

χL (t¯) 6 s0 + K4 Uk (g0 (T − t¯)) ∀ t¯ < T, where K4 is a constant that depends only on the known parameters.

(8.3.31)

Section 8.3

409

Blow-up waves in quasihomogeneous parabolic equations

The established estimates (8.3.29) and (8.3.31) show that the following general assertion is true: Theorem 8.3.1. Assume that p = q > 0 in problem (7.1.1)–(7.1.7) and that g(τ ) > 0, g(τ ) → ∞ as τ → 0, is a monotonically nonincreasing function defined in (8.3.1), (7.1.10) by the blow-up boundary regime f (t, x) from (7.1.6). Let g0 (τ ) be any majorant of the function g(τ ) such that (a) the function g0−1 (v) := ϕ0 (v) inverse to g0 (τ ) is regularly decreasing in a sense of Definition 8.3.1, ¯ 1 ) [see (8.3.26)] or (B ¯ 1) (b) ϕ0 (v) belongs to at least one of the classes (A [see (8.3.25)]. Then there exist constants K and L that depend only on the known parameters of the problem such that the estimate χL (t) 6 1 + KUk (g0 (T − t)) ∀ t < T, where

1

Uk (v) := v(ϕ0 (v − k) − ϕ0 (v)) m(p+1) , is true for the blow-up wave of any solution u of the analyzed problem. We now consider some examples illustrating the general result formulated in Theorem 8.3.1. Example 6. 1

g(τ ) = wτ − l ,

0 < l < l0 := m(p + 1) − 1.

We now set g0 (τ ) := g(τ ). Hence, ϕ0 (v) = ( wv )−l and, as in Example 2 from Sec. 8.2, we get klwl v −(l+1) = −kϕ00 (v) 6 ϕ0 (v−k)−ϕ0 (v) 6 −kϕ00 (v−k) = klwl (v−k)−(l+1)

∀ v > k.

(8.3.32)

By using these relations, we can easily show that  is satisfied  condition (8.3.2) l+1

for any k0 > 0 and λ ∈ (0, 1) and for δ ∈

1 − (1 − λ) l0 +1 , 1 . By using ¯ 1 ) for any (8.3.32), we can easily prove that ϕ0 (v) belongs to the class (A l0 −l β ∈ (0, l0 +1 ) and some v0 = v0 (k, w, l). Similarly, it is easy to see that ϕ0 (v) ¯ 1 ) for any β ∈ ( l0 −l , l0 ) and some v0 = v0 (k, w, l). belongs to the class (B l0 +1 l0 +1 Hence, by Theorem 8.3.1, we obtain l0

l −l

− l(l0 +1)

χL (t) 6 1 + K 0 w l0 +1 (T − t)

0

∀ t < T,

(8.3.33)

410

Chapter 8 Nonlocalized regimes with singular peaking

where K 0 < ∞ is independent of w and t. For m > 1, this estimate was obtained for the first time in [44]. Moreover, the estimates for the blow-up waves were also obtained in the same work for the model peaking regimes considered in the next two examples. Example 7. g(τ ) = w exp τ −α ,

α > 0,

w = const > 0.

Here, we also set g0 (τ ) = g(τ ). Hence, in this case,  v − 1 α ∀ v > w. ϕ0 (v) = g0−1 (v) = ln w Since the function ϕ0 (v) is convex, we get k αv(ln

v α+1 α w)

= −kϕ00 (v) 6 ϕ0 (v − k) − ϕ0 (v) 6 −kϕ00 (v − k) =

k α(v − k)(ln( v−k w ))

α+1 α

∀ v > k + w. (8.3.34)

By using relations (8.3.34), we can easily show that condition (8.3.2) is satisfied for any k0 > 0 and λ ∈ (0, 1) and for v0 = (1 − λ)−2 w

1

and δ ∈ (1 − 2−1 (1 − λ) l0 +1 , 1).

It is also not difficult to check that  ϕ0 (v) appearing in (8.3.34) belongs to the l0 ¯ class (A1 ) for any β ∈ 0, l0 +1 and for the corresponding v0 = v0 (k, α, β). Therefore, by virtue of Theorem 8.3.1, after simple calculations we obtain   l0 α+1 − l 1+1 l0 −α 0 0 l +1 l +1 0 0 w (T − t) exp χL (t) 6 1 + K α0 ∀ t < T, (T − t) l0 + 1 (8.3.35) where K 0 < ∞ depends only on the known parameters of the problem and does not depend on w, α, and t. In [44], estimate (8.3.35) was established for α = 1. In the next example, we consider a regime with blow-up peaking obtained as a perturbation of the regime presented in Example 6 that does not admit an explicit expression for the inverse function ϕ0 (v). Example 8. 1

g(τ ) = g0 (τ ) = wτ − l (ln τ −1 )α ,

α ∈ R1 .

We denote τ = ϕ0 (v) = g0−1 (v), v = g0 (τ ) and rewrite the definition of the

Section 8.3

411

Blow-up waves in quasihomogeneous parabolic equations

function g(τ ) in the form ϕ0 (v) =

 w l v

(ln ϕ0 (v)−1 )αl .

(8.3.36)

Differentiating this equality with respect to v, we get   lwl dϕ0 αlwl (ln ϕ0 (v)−1 )αl−1 = − l+1 (ln ϕ0 (v)−1 )αl . 1+ l dv v ϕ0 (v) v We now derive the expression for v l ϕ0 (v) from (8.3.36) and substitute the resulting expression in the last relation. This yields   αl lwl dϕ0 1+ = − (ln ϕ0 (v)−1 )αl . dv ln ϕ0 (v)−1 v l+1 Since ϕ0 (v) → 0 as v → 0, for arbitrarily small ε > 0, we can find v0 = v0 (ε, α) for which the following inequalities are true: −(1 + ε)lwl −(1 − ε)lwl dϕ0 −1 αl 6 (ln ϕ (v) ) 6 (ln ϕ0 (v)−1 )αl 0 dv v l+1 v l+1

∀ v > v0 . (8.3.37)

By using (8.3.37), we can show that condition (8.3.2) is satisfied for ϕ0 (v) and ¯ 1 ). Hence, Theorem 8.3.1 yields the estimate that ϕ0 (v) belongs to the class (A l0

αl0

l −l

− l(l0 +1)

χL (t) 6 1 + K 0 w l0 +1 (ln(T − t)−1 ) l0 +1 (T − t)

0

∀ t < T.

(8.3.38)

Example 9. Finally, for p = q > 0, we estimate the blow-up wave generated by the supersingular regime g0 (τ ) := g0,l (τ ) (8.2.90) from Example 3. It is easy to see that the corresponding inverse function ϕ0 (v) := ϕ0,l (v) [see (8.2.90)] ¯ 1 ) [see (8.3.26)] for any belongs to the class (A   m(p + 1) − 1 β ∈ 0, . m(p + 1) Thus, by using (8.2.92) and Theorem 8.3.1, we obtain χL (t) 6 1 + Kg0,l (T − t)

m(p+1)−1 m(p+1)

(T − t)

1 m(p+1)

Y l

(j) fl (g0,l (T

− 1 m(p+1) − t))

j=1

∀ t < T, where the constant K < ∞ is also independent of l .

∀ l ∈ N,

Chapter 9

Appendix: Formulations and proofs of the auxiliary results 9.1

Interpolation inequalities

Proposition 9.1.1 (see, e.g., [39]). Let Ω be an arbitrary bounded domain in Rn , n > 1, with C 1 -smooth boundary ∂Ω. Then (1) there exists a constant k0 > 0 that depends only on Ω such that kvkL1 (∂Ω) 6 k0 kvkW11 (Ω)

∀ v ∈ W11 (Ω);

(2) for any numbers q and p : 0 < q 6 p < ∞, there exist constants k1 and k2 depending on p, q, and Ω such that the following inequality is true for any 1 (Ω): function v ∈ Wp+1 ||v||Lp+1 (∂Ω) 6 k1 ||Dv||θLp+1 (Ω) ||v||1−θ Lq+1 (Ω) + k2 ||v||Lq+1 (Ω) ,

(9.1.1)

where θ = [n(p − q) + (q + 1)(p + 1)]−1 (n(p − q) + q + 1). Proposition 9.1.2 (see [41], [101], and Proposition A.1 in [36]). Suppose that numbers 1 6 r 6 ∞, p > 0, q ∈ (0, p), m > 0, m ∈ N, and 0 6 j < m, j ∈ N, are such that 1 1 m−j − < . r N p If Ω ⊂ RN is a bounded domain with piecewise smooth boundary ∂Ω, then there exist positive constants c1 and c2 depending only on Ω, r, m, q, j, and N such that the following inequality is satisfied for any u ∈ Lq (Ω) with Dm u ∈ Lr (Ω): kDj ukLp (Ω) 6 c1 kDm ukθLr (Ω) kuk1−θ Lq (Ω) + c2 kukLq (Ω) , where  θ=

j 1 1 + − N q p



1 m 1 + − q N r

(9.1.2)

−1 .

Moreover, there exists a positive constant c that depends only on r, j, m, q, and N and does not depend on Ω such that (1) if Ω = K(λ) is an N -dimensional cube with side of length λ, then inequal−j−N ( 1q − p1 ) ; ity (9.1.2) is true for c1 = c and c2 = cλ (2) if Ω = RN , then inequality (9.1.2) holds for c2 = 0;

Section 9.2

413

Systems of differential inequalities

(3) if N > 1 and Ω = B(0, r2 ) \ B(0, r1 ), where 0 < r1 < r2 , 2r1 > r2 , and B(0, r) := {|x| < r}, then inequality (9.1.2) is true for c1 = c and −j−N ( 1q − p1 ) . c2 = c(r2 − r1 )

9.2

Systems of differential inequalities

Lemma 9.2.1. Suppose that a family of absolutely continuous nonnegative and monotonically nonincreasing functions {Ui (s)}, i = 1, 2, . . . , satisfies a system of differential inequalities Ui (s) 6 λUi−1 (s) + k(−Ui0 (s))1+γ Ui (0) 6 Ki ,

i = 1, 2, . . . ,

∀ s > 0,

0 < λ < 1,

k = const > 0,

γ > 0,

U0 (s) := 0,

(9.2.1) (9.2.2)

where {Ki } is a nondecreasing sequence of positive numbers. Then the indicated functions Ui (s) satisfy the following estimates : Ui (s) 6 Mi (s) := a1 k

− γ1

h

1

a2 (kKiγ ) 1+γ − s

i 1+γ γ

+

∀ i 6 j,

(9.2.3)

where a1 = (1 − λ)

1 γ



1 − 1+γ

a2 = γ −1 (1 + γ)(1 − λ)

γ 1+γ

1+γ γ

,

,

f (s)+ := max(0, f (s)),
1 and, in particular, supp Ui ∈ [1, bi ] and bi = a2 kKiγ 1+γ . Proof. It is easy to see that the indicated functions Mi (s) are solutions of the following auxiliary Cauchy problem: Mi (s) = λMi (s) + k(−Mi0 (s))1+γ Mi (0) = Ki ,

∀ s > 0,

i = 1, 2, . . . , j.

(9.2.4) (9.2.5)

We prove estimate (9.2.3) by induction. For i = 1, the estimate can be verified by the direct integration of inequality (9.2.1) for the corresponding i. Assume that estimate (9.2.3) holds for all i 6 l − 1, i.e., 0 6 Ui (s) 6 Mi (s) ∀ s > 0,

∀ i 6 l − 1.

It is necessary to prove that Ul (s) 6 Ml (s) ∀ s > 0.

(9.2.6)

414

Chapter 9 Appendix: Formulations and proofs of the auxiliary results

Suppose that this is not true, i.e., there exist an interval (s1 , s2 ), 0 6 s1 , such that Ul (s1 ) = Ml (s1 ) Ul (s) > Ml (s) > 0 ∀ s ∈ (s1 , s2 ).

(9.2.7)

Since the sequence {Ki } is monotone, the assumption of induction implies that the inequalities Ml (s) > Ml−1 (s) > Ul−1 (s) ∀ s > 0 are true. In view of (9.2.7), the last inequalities imply that Ul (s) > Ul−1 (s) ∀ s ∈ (s1 , s2 ). Hence, the function Ul (s) is a solution of the problem Ul (s) 6 λUl (s) + k(−Ul0 (s))1+γ

∀ s ∈ (s1 , s2 ),

Ul (s1 ) = Ml (s1 ). Integrating this inequality, we arrive at the estimate Ul (s) 6 Ml (s) ∀ s ∈ (s1 , s2 ). 

This contradicts assumption (9.2.7).

We now consider the “Cauchy problem” for the corresponding differential inequality:  M (s) 6 max k1 (−M 0 (s))1+γ1 , k2 (−M 0 (s))1+γ2 ∀ s > 0, (9.2.8) M (0) 6 K,

k1 > 0, k2 > 0, γ1 > 0, γ2 > 0, K > 0.

(9.2.9)

Lemma 9.2.2. Let γ2 > γ1 . Then the following estimate is true for any nonnegative nonincreasing absolutely continuous function M (s) satisfying inequalities (9.2.8) and (9.2.9) for almost all s > 0 : 1+γ2 γ −γ1

e = K(k e 1 , k2 , γ1 , γ2 ) := k 2 (1) if K 6 K 1 1 h (1) (1) − γ1 a2

M (s) 6 M (s) := a1 k1 (1)

where a1 =



γ1 1+γ1

 1+γ1 γ1

1+γ1 2 −γ1

−γ

k2

k1 K γ1

1 1+γ1




, then

−s

1 i1+γ γ 1

+

∀ s > 0,

(9.2.10)

(1)

and a2 = γ1−1 (1 + γ1 );

e then (2) if K > K,  2 i1+γ 1 − 1 h γ   a(2) k γ2 a(2) (k2 K γ2 )1+γ2 −s 2 1

M (s) 6 M (s) :=

2

2

∀ s : 0 < s < s, ˜

1 i1+γ  1 − 1 h γ1  a(1) k γ1 a(1) (k1 K e γ1 )1+γ1 + s−s ˜ ∀ s > s, ˜ 1 1 2

+

(9.2.11)

Section 9.2

415

Systems of differential inequalities (2)



 1+γ2

(2)

, a2 = γ2− 1(1+γ2 ), and s˜ is given by the equality i 1+γ2 1 h 1 γ (2) (2) − γ e a1 k2 2 a2 (k2 K γ2 ) 1+γ2 − s˜ 2 = K 1 h γ2 γ2 i (2) 1+γ ⇐⇒ s˜ = a2 k2 2 K 1+γ2 − K˜ 1+γ2 .

where a1 =

γ2 1+γ2

γ2

Remark 9.2.1. In particular, it follows from (9.2.10) and (9.2.11) that M (s) = 0 ∀ s > s0 ,

(9.2.12)

where s0 =

 1 (1)  a2 (k1 K γ1 ) 1+γ1 1  a2(2) (k2 K γ2 ) 1+γ2 +

e for K 6 K, γ2 −γ1 γ1 γ2



γ

k1 2 γ k2 1



1 γ2 −γ1

e for K > K.

Proof. It is easy to see that inequality (9.2.8) can be rewritten in the following equivalent form: (  1   1 ) 1+γ1 1+γ2 M (s) M (s) , M 0 (s) 6 − min k1 k2  1+γ1 1+γ2 1 −  (k −1 M (s)) 1+γ2 for M > M ˜ = k γ2 −γ1 k γ2 −γ1 , 2 1 2 =− ∀ s > 0. 1   −1 ˜, (k1 M (s)) 1+γ1 for M < M (9.2.13) In view of the fact that the function M (s) in (9.2.10), (9.2.11) is a maximum solution of problem (9.2.13), (9.2.9), we arrive at the assertion of Lemma 9.2.2.  Remark 9.2.2. It is easy to see that the function M (s) given by (9.2.10) and (9.2.11) can be represented in the form M (s) = max(M 1 (s), M 2 (s)),

(9.2.14)

where 1 h 1 (2) − γ2 (2) a2 (k2 K γ2 ) 1+γ2

M 2 (s) = a1 k2

−s

2 i 1+γ γ 2

+

,

1 (1) − γ1

M 1 (s) = a1 k1

1+γ1 γ

[s0 − s]+ 1 ,

and s0 is a constant from (9.2.12). We now consider the following sequence of Cauchy problems:   1   1  M j (s) 1+γ1 M j (s) 1+γ2 0 , ∀ s > 0, M j (s) = −fj (M j (s)) := − min kj,1 kj,2 (9.2.15) M j (0) = Kj , j = 1, 2, . . . ,

416

Chapter 9 Appendix: Formulations and proofs of the auxiliary results

where kj,2 = c2 εγj 2 ,

kj,1 = c1 εγj 1 ,

−(1−δ)

Kj = c3 εj

,

0 < δ < 1,

{εj } is an arbitrary monotonically decreasing sequence such that εj → 0 as j → ∞. It is clear that δγ1   1   1 1+γ1 1+γ1 1+γ1 = c1 cγ31 εj kj,1 Kjγ1 → 0 as j → ∞, δγ2   1   1 1+γ2 1+γ2 1+γ2 = c2 cγ32 εj kj,2 Kjγ2 → 0 as j → ∞.   1   1 e j = k 1+γ2 k −(1+γ1 ) γ2 −γ1 = c1+γ2 c−(1+γ1 ) γ2 −γ1 ε−1 . K 1 2 j,1 j,2 j

(9.2.16)

It is easy to see that the inequality ⇐⇒

ej Kj > K

 1  1+γ2 −(1+γ1 ) γ2 −γ1 −1 εj c c3 ε−1+δ > c 1 2 j

is true only for  j 6 j0 = max i : εi >

−1 c3 δ



2 −(1+γ1 ) c1+γ c2 1



1 δ(γ2 −γ1 )

 := C0 .

(9.2.17)

We also note that γ2 kj,1

!

1 γ2 −γ1

 =

γ1 kj,2

cγ12 cγ21



1 γ2 −γ1

= const.

(9.2.18)

By using Lemma 9.2.2 and relations (9.2.16)–(9.2.18), we can represent any solution of problem (9.2.15) in the form  δγ1 1+γ1  1+γ1 γ1 −1   r ε [r ε − s] ∀ s > 0, ∀ j > j0 , 11 12  + j j   1+γ1 δγ2 (9.2.19) M j (s) := r11 ε−1 [r22 + r23 ε 1+γ2 − s] γ1 ∀ j < j0 , ∀ s > s˜j , + j j    1+γ2 δγ2   r ε−1 [r ε 1+γ2 − s] γ2 ∀ j < j0 , ∀ s : 0 < s < s˜j , 31 j 23 j + where r11 =

1 (1) − γ a1 c1 1 ,

(2)

r23 = a2

r12 =

c2 cγ32 (2)




1 1+γ2

1 1+γ2

s˜j = a2 kj,2

(1) a2

1


c1 cγ31 1+γ1 , 1 (2) − γ

r22

γ2 − γ1 = γ1 γ2



cγ12 cγ21



1 γ2 −γ1

r31 = a1 c2 2 , j0 is given in (9.2.17),  γ2  δγ2 γ2  δγ2  1+γ2 1+γ2 1+γ2 1+γ2 e Kj − Kj = r23 εj − C0 , ,

,

Section 9.2

417

Systems of differential inequalities

and C0 is given in (9.2.17). For the solution M j of problem (9.2.15), we also mention the following property obtained from representation (9.2.19): supp M j = [0, sj ],

sj1 < sj2

∀ j1 > j2 ;

sj → 0 as j → ∞.

(9.2.20)

Lemma 9.2.3. For any j1 > j2 , the function M j1 (s) − M j2 (s) is monotone and strictly decreasing on the segment [0, sj1 ] and, hence, there exists a single point sj1 ,j2 ∈ (0, sj1 ) such that M j1 (sj1 ,j2 ) = M j2 (sj1 ,j2 ). Proof. We now consider the functions sj (M ) ∀ j ∈ N inverse to M j (s). By virtue of (9.2.15), these functions satisfy the relations s0j (M ) = −(fj (M ))−1 := −gj (M ),

0 < M < M j (0) = Kj ,

sj (Kj ) = 0 ∀ j ∈ N. Since the sequence of functions gj (M ) is monotone and strictly decreasing as j → ∞, we get s0j1 (M ) − s0j2 (M ) = gj2 (M ) − gj1 (M ) > 0 ∀ M < M j2 (0). 

This proves the assertion of the lemma.

By using the sequence {M j (s)}, we now construct the following monotone sequence: fj (s) = max {M i (s)} ∀ j ∈ N, ∀ s > 0. M (9.2.21) 16i6j

(j)

By Lemma 9.2.3, for every j ∈ N, there exist finitely many lj points sl , (j) (j) (j) l = 1, 2, . . . , lj , such that s1 > s2 > . . . > slj > 0 and fj (s) = M i (s) ∀ s ∈ [s(j) , s(j) ) ∀ l 6 lj , M l l+1 l

(j)

slj +1 = 0.

(9.2.22)

Lemma 9.2.4. The sequence il , l = 1, 2, . . . , lj , specified by (9.2.22) is monotonically increasing, i.e., il+1 > il ∀ l 6 lj . Hence, the following estimate is true for the numbers lj : lj 6 j ∀ j ∈ N. (9.2.23) Proof. We prove the lemma by contradiction. Suppose that il+1 < il for some l < lj and consider the corresponding curves γ l+1 := (s, M il+1 (s)), γ l := (s, M il (s))

(9.2.24)

418

Chapter 9 Appendix: Formulations and proofs of the auxiliary results

in the quadrant (s, M ) ∈ R2 : s > 0, M > 0. By virtue of property (9.2.20) and proposition (9.2.24), we also find M il+1 (sil ) > M il (sil ) = 0. On the other hand, it follows from definition (9.2.22) that (j)

(j)

M il (sl ) > M il+1 (sl ).

(9.2.25)

The last two inequalities imply that the curves γ l+1 and γ l necessarily have (j) a point of intersection on the segment [sl , sil ]. In view of (9.2.22), we obtain (j)

(j)

M il+1 (sl+1 ) > M il (sl+1 ). Together with (9.2.25), this guarantees the existence of a point of intersection (j) (j) of the curves γ l+1 and γ l on the interval [sl+1 , sl ). This means that the curves γ l and γ l+1 have two different intersection points, which contradicts Lemma 9.2.3.  Lemma 9.2.5. Assume that a family of nonnegative absolutely continuous monotonically nonincreasing functions {Mj (s)}, j = 1, 2, . . . , satisfy the following system of differential inequalities:  Mj (s) 6 λMj−1 (s) + (1 − λ) max kj,1 (−Mj0 (s))1+γ1 , kj,2 (−Mj0 (s))1+γ2 (9.2.26) ∀ s > 0, Mj (0) 6 Kj , j = 1, 2, . . . , M0 (s) := 0, where 0 < λ < 1 and kj,1 , kj,2 , and Kj are the constants from (9.2.15). Then the following uniform estimate is true for any j ∈ N : fj (s) ∀ i 6 j, Mi (s) 6 M

∀ s > 0,

(9.2.27)

fj (s) are given in (9.2.21). where the functions M Proof. We prove the lemma by induction. For j = 1, estimate (9.2.27) can be directly verified. Suppose that fj−1 (s) ∀ i 6 j − 1, Mi (s) 6 M

∀ s > 0.

(9.2.28)

It is necessary to prove estimate (9.2.27). If this is not true, then there exists an interval (a, b), a > 0, such that fj (s) ∀ s ∈ (a, b); Mj (s) > M fj (s) ∀ s 6 a, Mj (s) 6 M

(9.2.29) fj (a). Mj (a) = M

Thus, we assume that (j)

(j)

a ∈ [sl+1 , sl ),

l 6 lj ,

(9.2.30)

Section 9.2

419

Systems of differential inequalities

where lj is a constant from (9.2.22). By virtue of (9.2.22), we get fj (s) = M i (s) ∀ s ∈ [a, s(j) ). M l l

(9.2.31)

In view of assumptions (9.2.28) and (9.2.29), for the monotonically nondecreasfi (s), i = 1, 2, . . . , we find ing sequence M (j)

fj−1 (s) 6 M fj (s) < Mj (s) ∀ s ∈ [a, s ). Mj−1 (s) 6 M l

(9.2.32)

By using the differential inequality (9.2.26) and relation (9.2.32), we arrive at the inequality  (j) Mj (s) 6 max kj,1 (−Mj0 (s))1+γ1 , kj,2 (−Mj0 (s))1+γ2 ∀ s ∈ [a, sl ). Since the sequences {kj,1 } and {kj,2 } are monotonically nondecreasing, this inequality implies the following relation:  (j) Mj (s) 6 max kil ,1 (−Mj0 (s))1+γ1 , kil ,2 (−Mj0 (s))1+γ2 ∀ s ∈ [a, sl ). (9.2.33) Moreover, proposition (9.2.29) and property (9.2.31) imply that Mj (a) = M il (a).

(9.2.34)

By virtue of Lemma 9.2.2 and relations (9.2.33) and (9.2.34), we obtain (j)

Mj (s) 6 M il (s) ∀ s ∈ [a, sl ), which contradicts assumption (9.2.29) in view of relation (9.2.31).



Lemma 9.2.6. Under the conditions of Lemma 9.2.5, the following universal a priori estimate holds for the functions Mj (s) satisfying system (9.2.26) : −

Mj (s) 6 max{D1 s

(1+γ1 )(1−δ) δγ1

, D2 } ∀ s > 0,

∀ j ∈ N,

(9.2.35)

where the constants D1 and D2 depend only on the known parameters c1 , c2 , c3 , γ1 , γ2 , and δ of system (9.2.26) and do not depend on j ∈ N. Proof. Consider the following family of functions N τ (s) that depends only on a continuous parameter τ :  h i 1+γ1 δγ1  −1 r τ 1+γ1 − s γ1  r τ ∀ s > 0, ∀ τ 6 C0 ,  11 12   +   h i 1+γ1  δγ2   −1 r + r τ 1+γ2 − s γ1  ∀ τ > C0 , r τ 11 22 23   +  δγ2   δγ2 (9.2.36) N τ (s) := 1+γ2 − C 1+γ2 , ∀ s > r τ 23  0    i 1+γ2 h δγ2   −1 r τ 1+γ2 − s γ2  ∀ τ > C0 , r τ  31 23    δγ2   δγ2   1+γ  ∀ s : 0 < s < r23 τ 1+γ2 − C0 2 ,

420

Chapter 9 Appendix: Formulations and proofs of the auxiliary results

where C0 is a constant from (9.2.17) and the constants r11 , r12 , r22 , r31 , and r23 are taken from (9.2.19). This family is wider than the family of functions {M j (s)} in (9.2.19). It is clear that N εj (s) = M j (s) ∀ j ∈ N. Therefore, max{M i (s)} 6 max{N τ (s)} 6 max{N τ (s)} τ >εj

i6j

τ >0

(1)

6 max{ max N τ (s), max N τ (s)} 6 max{ max N τ (s), C1 }, τ 6C0

C1 :=

τ >C0

(1) N C0 (0)

τ 6C0

1+γ1 γ1

= r11 r12

where

1+γ1 γ

δγ1

(1)

(9.2.37) −(1−δ) C0 ,

N τ (s) = r11 τ −1 [r12 τ 1+γ1 − s]+ 1

∀ s > 0. (1)

We now find the envelope N (s) of the family of curves N τ (s). It is known that, for this purpose, it is necessary to exclude the parameter τ from the system ∂ (1) (1) N (s) = 0, N − N τ (s) = 0. ∂τ τ As a result of standard calculations, we obtain N (s) = C2 s

−

(1+γ1 )(1−δ) δγ1

,

1+γ1 γ

C2 = r11 r12 1 δ

1+γ1 γ1

(1 − δ)

(1+γ1 )(1−δ) δγ1

.

(9.2.38)

Further, we show that (1)

N τ (s) 6 N (s) ∀ τ > 0,

∀ s > 0.

(9.2.39)

(1)

Since N τ (s) > 0 and N (s) > 0, in order to prove (9.2.39), it suffices to show that γ1

(1)

γ1

ϕ(s) := N (s) 1+γ1 − (N τ (s)) 1+γ1 > 0 ∀ τ > 0,

∀ s > 0.

(9.2.40)

As a result of simple calculations, we get δγ1

δ ϕ0 (s) = 0 only for s = sc := r12 (1 − δ)τ 1+γ1

∀ τ > 0.

It is easy to check that ϕ(sc ) = 0. This yields (9.2.40) and, hence, (9.2.39). Therefore, estimate (9.2.37) can be continued as follows: max{M i (s)} 6 max{N (s), C1 } ∀ j ∈ N, i6j

(9.2.41)

whence, by virtue of (9.2.5) and definition (9.2.21), we get the validity of estimate (9.2.35) with D1 = C2 , where C2 is defined in (9.2.38), and D2 = C1 , where C1 is defined in (9.2.37). 

Section 9.2

421

Systems of differential inequalities

Lemma 9.2.7. Assume that a family of nonnegative absolutely continuous monotonically nonincreasing functions {Mi (s)}, i = 1, 2, . . . , satisfies the following system of differential inequalities : Mi (s) 6 λMi−1 (s) + (1 − λ)ki (−Mi0 (s)) ∀ s > 1, Mi (1) 6 D1 exp(D2 ki−1 ),

i = 1, 2, . . . ,

M0 (s) = 0,

(9.2.42)

0 < λ < 1,

where {ki } is a monotonically decreasing sequence such that ki → 0 as i → ∞, D1 < ∞, and D2 < ∞. Then the functions Mi (s) have the following upper bounds : Mi (s) 6 M i (s) = D1 exp[ki−1 (D2 + 1 − s)] (9.2.43) ∀ s : 1 < s < s0 = 1+D2 , ∀ i > 1. Proof. It is easy to see that the function M i (s) is a solution of the problem 0

M i (s) = ki (−M i (s)) ∀ s > 1, M i (1) = D1 exp(D2 ki−1 ). It is clear that the sequence {M i (s)} is strictly monotone and increasing on the interval [1, s0 ), s0 = 1 + D2 . Therefore, we can prove the lemma by induction by analogy with the proof of Lemma 9.2.1.  Lemma 9.2.8. Assume that a family of nonnegative absolutely continuous monotonically nonincreasing functions {Mj (s)} satisfies the following relations : Mj (s) 6 λMj−1 (s) + (1 − λ)kj (−Mj0 (s)) ∀ s > s, Mj (s) 6 exp Kj ,

j = 1, 2, . . . ,

M0 (s) = 0,

0 < λ = const < 1,

(9.2.44)

where the sequences {kj }, {Kj−1 }, and {rj := kj Kj } monotonically tend to zero as j → ∞. Then the functions Mi (s) have the uniform upper bound f(s) := max{M fl (s)} fj (s) := max{M l (s)} 6 M Mi (s) 6 M l∈N

l6j

∀ i 6 j,

∀ j ∈ N,

(9.2.45)

∀ s : s < s 6 s + k1 K1 ,

where M l (s) := exp[kl−1 (Kl kl + s − s)] is a solution of the problem 0

M l (s) = −ki M l (s) ∀ s > s;

M l (s) = exp Kl .

(9.2.46)

Proof. We omit the proof of the lemma because it is, in fact, almost identical to the proof of Lemma 9.2.5. 

422

Chapter 9 Appendix: Formulations and proofs of the auxiliary results

Lemma 9.2.9. Assume that the conditions of Lemma 9.2.8 are satisfied and, in addition, that Ki = f (ki ) ∀ i ∈ N, where f (h) ∀h > 0, is an arbitrary continuously differentiable monotonically decreasing function. Then the following f(s) given in (9.2.45): estimate is true for the function M f(s) 6 N e (s) ∀ s > s, M ¯ e (s) is the envelope of the family of curves {N τ (s)} : N τ (s) = M i (s) if where N τ = ki ∀ i ∈ N and ki is a constant from (9.2.44). Proof. We omit the proof of this lemma because it is similar to the proof of Lemma 9.2.6.  Corollary 9.2.1. Under the conditions of Lemma 9.2.9, if f (s) = r+g ln s−1 , r > 0, g > 0, then
e (s) = exp (r + g)g g (s − s)−g ∀ s > s. N

9.3

Functional inequalities

Lemma 9.3.1 (see [125]). Assume that nonnegative functions A(s), H(s), B(s), and D(s) nonincreasing on (0, ∞) satisfy the following functional inequality: A(s + δ) + H(s + δ) 6 εH(s) + c(ε)

l X

δ −αi (B(s)−B(s+δ))B(s)βi + D(s)

(9.3.1)

i=1

∀ s > 0, ∀ δ > 0, ∀ ε > 0, αi > 0, βi > 0, where c(ε) → ∞ as ε → 0. Then the functions A(s) and H(s) have the following upper bound: A(s + δ) + H(s + δ) 6 (1+ε)D(s) + c1 (ε)(B(s)−B(s+δ))

l X B(s)βi i=1

δ αi

,

(9.3.2)

∀ s > 0, ∀ δ > 0, ∀ ε < ε0 := min 2−αi , i6l

where c1 (ε) → ∞ as ε → 0. Proof. Inequality (9.3.1) yields the following series of inequalities:     δ δ A s + j−1 + H s + j−1 2 2      l  j αi  X δ δ 2 δ βi 6 εH s + j + D s + j + c (ε) B s+ j 2 2 δ 2   i=1   δ δ (9.3.3) × B s + j − B s + j−1 , j = 1, 2, . . . . 2 2

Section 9.3

423

Functional inequalities

(9.3.3) with j = 1 as a starting point. We Further, we consider inequality
δ estimate the term H s + 2 on the right-hand side of this inequality by using
inequality (9.3.3) with j = 2. Then we estimate the term H s+ 4δ on the righthand side of the obtained relation by using (9.3.3) with j = 3, etc. Repeating the procedure k times, we arrive at the following relation: A(s + δ) + H(s + δ)  X    k−1 δ δ r k ε D r + r+1 6ε H s+ k + 2 2 r=0        l X k−1 1+r αi X δ βi δ δ 2 B s + r+1 − B s + r . εr B s + r+1 + c (ε) δ 2 2 2 i=1 r=0

In view of the monotonicity of the functions D(s) and B(s), this yields A(s + δ) + H(s + δ)    k−1  δ X r δ k ε 6ε H s+ k +D s+ k 2 2 r=0     X k−1 l  αi  X
r δ 2 δ βi B s + k − B(s + δ) 2αi ε . + c (ε) B s+ k δ 2 2 r=0 i=1 (9.3.4) Setting ε < ε0 := min 2−αi in (9.3.4) and passing to the limit as k → ∞, we i6l

obtain A(s + δ) + H(s + δ) 6

l
X B(s)βi D(s) + c (ε)K1 B(s) − B(s + δ) , 1−ε δ αi i=1

where K1 = max{2αi (1 − ε2αi )−1 }. i6l



The last relation is equivalent to estimate (9.3.2).

Lemma 9.3.2 (see [126]). Assume that a nonnegative continuous nonincreasing function f : [0, ∞) → R1+ satisfies the relation f (s + δ) 6 a δ −α f (s)θ

∀ s > 0, δ > 0;

where a > 0, α > 0, and θ > 0 are real numbers. Then the function f has the following properties : α

1

α

(1) for θ < 1 : f (s) 6 2 (1−θ)2 a 1−θ s− 1−θ

∀ s > 0; 1

(2) for θ = 1 : f (s) 6 f (0) exp(1 − s/(ae) α ) ∀ s > 0; (3) for θ > 1 : f (s0 ) = 0,

θ

1

s0 = 2 θ−1 (a(f (0))θ−1 ) α .

424

Chapter 9 Appendix: Formulations and proofs of the auxiliary results

Lemma 9.3.3. Assume that a nonnegative continuous nonincreasing func1 satisfies the relation tion f : [0, ∞) → R+ f (s + δ) 6 aδ −α f (s)θ + R

∀ s > 0,

δ > 0,

(9.3.5)

where 0 < θ < 1, a > 0, α > 0, and R = const < ∞. Then the following a priori estimate is true for the function f : n o α 1 α f (s) 6 max 2R, 2 θ(1−θ)2 (2a) 1−θ · s− 1−θ ∀ s > 0.

Proof. It is clear that relation (9.3.5) is equivalent to the inequality f (s) 6 aδ −α f (s − δ)θ + R

∀ s > 0,

∀ δ : 0 < δ < s.

(9.3.6)

Note that the following inequality immediately follows from (9.3.6): f (s) 6 2aδ −α f (s − δ)θ

∀ s : f (s) > 2R,

∀ δ < s.

Since the function f (s) is monotone, we complete the proof of the lemma by using Lemma 9.3.2. 

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