Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations (Fields Institute Monographs) [1st ed. 2018] 3319984063, 9783319984063

Table of contents :
Preface
Contents
1 Preliminaries
1.1 Functional Spaces and Their Properties
1.1.1 Lp Spaces
1.1.2 Sobolev Spaces
1.2 Linear Elliptic Boundary Value Problems
1.3 Nemytskii Operator
1.4 Maximum Principles and Their Applications
1.4.1 Classical Maximum Principles
1.5 Uniform Estimates and Boundedness of the Solutions of Semilinear Elliptic Equations
1.6 The Sweeping Principle and the Moving Plane Method in a Bounded Domain
1.7 The Sliding and the Moving Plane Method in General Domains
1.8 Variational Solutions of Elliptic Equations
1.9 Elliptic Regularity for the Neumann Problem for the Laplace Operator on an Infinite Edge
2 Trajectory Dynamical Systems and Their Attractors
2.1 Kolmogorov epsilon-Entropy and Its Asymptotics in FunctionalSpaces
2.2 Global Attractors and Finite-Dimensional Reduction
2.3 Classification of Positive Solutions of Semilinear Elliptic Equations in a Rectangle: Two Dimensional Case
2.3.1 Sketch of the Proof of Theorem 2.4
2.4 Existence of Solutions of Nonlinear Elliptic Systems
2.5 Regularity of Solutions
2.6 Boundedness of Solutions as
2.7 Basic Definitions: Trajectory Attractor
2.8 Trajectory Attractor of Nonlinear Elliptic System
2.9 Dependence of the Trajectory Attractor on the UnderlyingDomain
2.10 Regularity of Attractor
2.11 Trajectory Attractor of an Elliptic Equation with a Nonlinearity That Depends on x
2.12 Examples of Trajectory Attractors
2.13 The Trajectory Dynamical Approach for the Nonlinear Elliptic Systems in Non-smooth Domains
2.13.1 Existence of Solutions
2.13.2 Trajectory Attractor for the Nonlinear Elliptic System
2.13.3 Stabilization of Solutions in the Potential Case
2.13.4 Regular and Singular Part of the Trajectory Attractor
2.14 The Dynamics of Fast Nonautonomous Travelling Waves and Homogenization
3 Symmetry and Attractors: The Case N

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Fields Institute Monographs 36 The Fields Institute for Research in Mathematical Sciences

Messoud Efendiev

Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations

Fields Institute Monographs VOLUME 36 The Fields Institute for Research in Mathematical Sciences Fields Institute Editorial Board: James G. Arthur, University of Toronto Kenneth R. Davidson, University of Waterloo Ian Hambleton, Director Huaxiong Huang, Deputy Director of the Institute Lisa Jeffrey, University of Toronto Barbara Lee Keyfitz, Ohio State University Thomas S. Salisbury, York University Juris Steprans, York University Noriko Yui, Queen’s University

The Fields Institute is a centre for research in the mathematical sciences, located in Toronto, Canada. The Institutes mission is to advance global mathematical activity in the areas of research, education and innovation. The Fields Institute is supported by the Ontario Ministry of Training, Colleges and Universities, the Natural Sciences and Engineering Research Council of Canada, and seven Principal Sponsoring Universities in Ontario (Carleton, McMaster, Ottawa, Queen’s, Toronto, Waterloo, Western and York), as well as by a growing list of Affiliate Universities in Canada, the U.S. and Europe, and several commercial and industrial partners.

More information about this series at http://www.springer.com/series/10502

Messoud Efendiev

Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations

123

Messoud Efendiev Institute of Computational Biology Helmholtz Center Munich Neuherberg, Bayern, Germany

ISSN 1069-5273 ISSN 2194-3079 (electronic) Fields Institute Monographs ISBN 978-3-319-98406-3 ISBN 978-3-319-98407-0 (eBook) https://doi.org/10.1007/978-3-319-98407-0 Library of Congress Control Number: 2018955495 Mathematics Subject Classification (2010): 35J61, 35Q92, 35J60, 35J70, 35J92, 37C45, 37L30 © Springer Nature Switzerland AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover illustration: Drawing of J.C. Fields by Keith Yeomans This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book mainly deals with a systematic study of a dynamical system approach to investigate the symmetrization and stabilization properties (as |x| tends to infinity) of nonnegative solutions of nonlinear elliptic (both degenerate and nondegenerate) problems in asymptotically symmetric unbounded domains. To this end, we use a trajectory dynamical systems approach and the concept of its attractor. Recall that a dynamical system (DS) is a system which evolves with respect to time. To be more precise, a DS (S(t), ) is determined by a phase space  which consists of all possible values of the parameters describing the state of the system and an evolution map S(t) :  Ñ  which allows us to find the state of the system at time t ą 0 if the initial state at t = 0 is known. Very often, in biology, ecology, mechanics, and physics, more generally in the modeling of life science problems, the evolution of the system is governed by systems of differential equations. If the system is described by ordinary differential equations (ODEs) u1 (t) = F (t, u(t))

(1)

for some nonlinear function F : R+ ˆ RN Ñ RN , we have a so-called finitedimensional DS. In that case, the phase space  is some (invariant) subset of RN and the evolution operator S(t) is defined by S(t)y0 := y(t),

(2)

where y(t) solves (1). We also recall that, in the case where Eq. (1) is autonomous (i.e., does not depend explicitly on time), the evolution operators S(t) generate a semigroup on the phase space , i.e. S(t1 + t2 ) = S(t1 )S(t2 ),

t1 , t2 P R+ ,

(3)

The qualitative study of DS in finite dimensions goes back to the beginning of the twentieth century with the pioneering works of Poincaré on the N -body problem and

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important contributions of Lyapunov on stability and of Birkhoff on minimal sets and the ergodic theorem. One of the most surprising and significant facts discovered at the very beginning of the theory is that even relatively simple equations can generate very complicated chaotic behaviors. Moreover, these types of systems are extremely sensitive to initial conditions (the trajectories with close but different initial data diverge exponentially). Thus, in spite of the deterministic nature of the system (we recall that it is generated by a system of ODEs, for which we usually have a unique solvability theorem), its temporal evolution is unpredictable on time scales larger than some critical time T0 (which clearly depends on the error of approximation and on the rate of divergence of close trajectories) and can show typical stochastic behaviors. This fact was used by Lorenz to justify the so-called butterfly effect, a metaphor for the imprecision of weather forecasting. The theory of DS in finite dimensions has been extensively developed during the twentieth century, due to the efforts of many mathematicians (such as Anosov, Arnold, LaSalle, Sinai, Smale), and, nowadays, very much is known about the nontrivial dynamics of such systems, at least in low dimensions. In particular, it is known that very often, the trajectories of a chaotic system are localized, up to a transient process, in some subset of the phase space having a very complicated fractal geometric structure (e.g., locally homeomorphic to the Cartesian product of Rm and some Cantor set) which thus accumulates the nontrivial dynamics of the system (this is the so-called strange attractor). The chaotic dynamics on such sets are usually described by symbolic dynamics generated by Bernoulli shifts on the space of sequences. We also note that, nowadays, a mathematician has a large number of tools available for the extensive study of concrete chaotic DS in finite dimensions. In particular, we mention here different types of bifurcation theories (including the KAM theory and the homoclinic bifurcation theory with related Shilnikov chaos), the theory of hyperbolic sets, stochastic description of deterministic processes, Lyapunov exponents and entropy theory, dynamical analysis of time series, etc. In other words, in the mid-1970s of the twentieth century, we already had available highly developed finite-dimensional dynamical systems theory, and it has become vitally important to extend this theory to the evolution processes that are usually governed by partial differential equations (PDEs). In this case, the corresponding phase space  is some infinite-dimensional function space (e.g., F := L2 () or F := L8 () for some domain  Ă RN , or Hölder space). Such DS are usually called infinite-dimensional. These classes of equations in the abstract setting can be written as u1 (t) = F (t, u(t))

(4)

in an infinite-dimensional Banach space . We emphasize that for an infinitedimensional DS generated by PDEs, the situation becomes much more complicated. Indeed, a first important difficulty which arises here is related to the fact that the analytic structure of a PDE is essentially more complicated than that of an ODE and, in particular, in general we do not have a unique solvability theorem as in the case of ODEs, so that even finding the proper phase space and the rigorous

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construction of the associated DS can be a highly nontrivial problem. In order to indicate the level of difficulties arising here, it suffices to recall that, for the threedimensional Navier-Stokes system (which is one of the most important equations of mathematical physics), the required associated DS has not yet been constructed. Nevertheless, there exists a large number of equations for which the problem of the global existence and uniqueness of a solution has been solved. Thus, the question of extending the highly developed finite-dimensional DS theory to infinite dimensions arises naturally. Note that at present there are numerous monographs on infinite-dimensional dynamical systems generated by nondegenerate parabolic and hyperbolic equations (both autonomous and nonautonomous) in bounded domains and its application in various areas of mathematical physics (see, e.g., [1, 35, 67, 92, 98] and the references therein). In those books, the long-time dynamics of solutions is described in terms of a global attractor (uniform attractor in the nonautonomous case). It is important to note that the PDE equations in the books mentioned above possess regular structure (nondegenerate diffusion and nondegenerate chemotaxis) and belonging to the so-called dissipative partial differential equations (sometimes partially dissipative, when the underlying domain is unbounded or with hyperbolic equations in a bounded domain). We emphasize once more that the phase spaces  in those books mentioned above are appropriate infinite-dimensional function spaces. Nevertheless, it was observed in experiments that, up to a transient process, the trajectories of the DS considered are localized inside “very thin” invariant subsets of the phase space having a complicated geometric structure which, thus, accumulates all the nontrivial dynamics of the system. It was conjectured that these invariant sets are, in some proper sense, finite-dimensional and that the dynamics restricted to these sets can be effectively described by a finite number of parameters. Thus (when this conjecture is true), in spite of the infinite-dimensional initial phase space, the effective dynamics (reduced to this invariant set) is finite-dimensional and can be studied by using the algorithms and concepts of the classical finitedimensional DS theory. Indeed, the above finite-dimensional reduction principle of dissipative (or partially-dissipative) PDEs in bounded domains has been given a solid mathematical grounding (based on the concept of the so-called global and exponential attractors) over the last four decades, starting from the pioneering papers of Ladyzhenskaya (see [77]) and continued successfully in [1, 35, 67, 92, 98] (see also references therein), mainly for evolution PDEs with more or less regular structure (e.g., uniformly parabolic, nondegenerate parabolic and nondegenerate hyperbolic) leading to finite dimensionality of their attractors. Here, as mentioned above, boundedness of the underlying domains plays a decisive role in obtaining finite fractal dimensional attractors of the associated semigroups. In [48] (see also references therein), we systematically studied infinite-dimensional dynamical systems and their attractors generated by a quite large class of nondegenerate parabolic type PDEs (both for second order and for fourth order) in unbounded domains (both in weighted and in uniformly local phase spaces) and proved that the fractal dimension of their attractors is infinite, so that the reduction principle fails in general. In spite of this new feature, that is, the infinite dimensionality of attractors, we managed to find asymptotics of their Kolmogorov’s ε-entropy which has logarithmic type asymptotics.

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It is worth mentioning that, in contrast to nondegenerate evolution equations, very little was known about the long-time dynamics of degenerate parabolic equations such as porous-medium equations, parabolic p-Laplacian, doubly nonlinear equations, and degenerate diffusion with chemotaxis and ODE-PDE coupling. In the books [46, 47] that were published very recently, we discussed these classes of degenerate parabolic equations in bounded domains in connections with the modeling of life science problems and observed some very interesting new features from the dynamical systems viewpoint, related to the attractors of such equations which cannot be observed in nondegenerate cases, namely: (a) Infinite dimensionality of the attractor (b) Polynomial asymptotics of its epsilon-Kolmogorov entropy (c) Differences in the asymptotics of the epsilon-Kolmogorov entropy depending on the choice of the underlying phase spaces Furthermore, for the evolution equations that are described by parabolic or hyperbolic type equations, the well-posedness, as already mentioned above, has been proved for a quite large class of nonlinearities. However, the nonlinearities involving the abovementioned PDEs that provides nonuniqueness of solutions usually possess some “pathological” nature (see, for instance, [52]) and in turn lead to multivalued semigroups from the dynamical systems viewpoint. I would like especially to emphasize that, in contrast to such equations where non-unicity appears for “pathological” class of nonlinearities, elliptic boundary value problems in unbounded domains, interpreted as evolution equations, naturally lead to nonuniqueness of solutions even for simple polynomial nonlinearities which as a consequence leads to multivalued semigroups, where the role of time (i.e., t ě 0) is played by one of the unbounded directions of the underlying domain, say t := x1 . Indeed, it was a temptation to apply such a strong dynamical systems tool in order to study asymptotic of solutions of elliptic boundary value problem in unbounded domains. In other words, to consider state of equilibrium of bio-physico-chemicalmechanical systems from the point of view of dynamical systems. In order to apply DS approach to elliptic equations of the form given in Chaps. 3–7 of this book, one has to take the following difficulties into account: firstly, the Cauchy initial value problem for such equations (prescribing both the value of solutions u(0) and its normal derivative u1 (0) on the initial surface t = 0) is not wellposed. More precisely, in general, its unique solution either does not exist or blows up in a finite time. Secondly, the boundary value problem for elliptic equations may have a solution, but in general it is not unique. Indeed, Hadamard was the first to notice that the initial value problem for elliptic equations is illposed (see [66, Bk.I, Ch. II, §18]). In [8] this difficulty was overcome, where the attractor of an elliptic equation was constructed by means of the theory of semigroup of multivalued mappings. Unfortunately, the theory of semigroups of multivalued mappings does not provide theorems on the finite dimensionality of the attractors. Indeed such theorems are an important part of the theory of semigroups of single-valued mappings in infinite-dimensional spaces, which we called above a reduction principle. Since we are mainly interested in the finite fractal dimension

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of the attractors of semigroups associated with elliptic problems (so-called finitedimensional reduction discussed above for parabolic/hyperbolic type equations), such multivalued semigroup approach is not suitable for our purposes. Therefore, the usage of infinite-dimensional dynamical systems methods mentioned above for elliptic problems in unbounded domains as well as finite-dimensional reduction of their dynamics requires new ideas and tools. Fortunately, these difficulties have been overcome in several interesting particular cases. We first mention the pioneering work by Kirchgassner [72] on small solutions of elliptic equations in infinite cylinders. His idea was to construct invariant manifolds, where the elliptic initial value problem is well-posed and a flow, or at least a semiflow, is defined (see also [62]). This idea was extended to large solutions, later, in the “parabolic,” convection dominated limiting case of large wave speeds γ P R (see [31] and [87]). Without such a restriction, the case of elliptic equations in a strip was treated (see [9] and [82]). Note that in [31] a similar problem to [8] was considered in the scalar case, when the elliptic equation contains a small parameter in front of second derivative, while the coefficient in front of first derivative is different from zero. In [8], the elliptic BVP was studied with Cauchy initial conditions u(0) = u0 and u1 (0) = u1 . In this case, the semigroup corresponding to this elliptic equation consists of a single-valued mappings, but the set on which it is defined is not described in an explicit form. We strongly believe that in the dynamical setting the most natural boundary value problem is to prescribe the initial value u(0) at the t = 0 and impose the condition of boundedness of u(t) as t goes to infinity. We will consider throughout of this book this type of elliptic boundary value problem. In this book, we overcome the abovementioned difficulties and some other ones (such as finite-dimensional reduction of dynamics) for an elliptic equation in unbounded domains by usage of the trajectory dynamical systems approach (the details are in Chap. 3) as well as symmetry and monotonicity properties of solutions of an elliptic equation (both nondegenerate and degenerate) in unbounded domains. In what follows, we use the following convention. Convention 1 The attractor of a trajectory dynamical system associated to an elliptic equation is called an elliptic attractor. We specify below some of the topics covered by this book that we are mainly interested in: 1. To construct single-valued dynamical systems for nonlinear elliptic equations in unbounded domains 2. To find appropriate functional phase space (possibly not weighted space) in unbounded domains and determine a topology in the phase space guaranteeing existence of a global attractor 3. To find assumptions on nonlinearities under which all solutions converge as t := x1 goes to infinity to the same limiting one-dimensional profile, irrespectively of the “initial” value u0

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4. To provide assumptions on the data, that is, the geometry of underlying domains, assumptions on nonlinearity, etc., for equations under consideration that guarantee finite-dimensional attractors 5. To find new effects imposed by the dynamical systems approach to nonlinear elliptic equations and to their elliptic attractors which was not observed in the parabolic or hyperbolic PDEs previously considered from the dynamical systems viewpoint 6. To understand whether asymptotic symmetry of the domain implies symmetry of elliptic attractors 7. To understand how smoothness or nonsmoothness of underlying domains is inherited by elliptic attractors 8. To provide assumptions guaranteeing that omega-limit sets of a solution of an elliptic equation is a singleton This book consists of seven chapters. Chapter 1 consists of nine subsections. In Sect. 1.1, we introduce some functional spaces and study their properties that we will use in subsequent chapters. Sections 1.2 and 1.3 are devoted to linear elliptic boundary value problems and the properties of Nemytskii operators in various functional spaces, respectively. Sections 1.4 deals with the classical maximum principles as well as their versions for narrow and small domains, which are used in Sects. 1.6, 1.7, and 1.8. In Sect. 1.5, using maximum principles, we obtain explicit and uniform bounds that ensure the boundedness and the asymptotic dissipation of the solutions of semilinear elliptic equations in bounded (or unbounded) domains. These estimates are crucial to construct the attractive basin of trajectories in the dynamical system approach to elliptic equations. We will use these estimates in the following chapters. Sections 1.6 and 1.7 are devoted to the sweeping principle, the moving plane method as well as sliding methods both in bounded and in unbounded domains and their role in the study symmetry and monotonicity properties of nonnegative solutions of nonlinear elliptic equations in unbounded domains. We use this symmetry and monotonicity properties of nonnegative solutions of nonlinear elliptic equations in unbounded domains for the complete characterization of the elliptic attractor. Sections 1.8 and 1.9 are devoted to variational solutions and elliptic regularity for the Neumann problem for the Laplace operator on an infinite edge. We will use the results of these subsections to study the effects of nonsmoothness of the domain to nonsmoothness of elliptic attractor in Chap. 2. I would like especially to emphasize that, in some cases (of course, for the convenience of the reader), I present some well-known results both in this chapter as well as in the following chapters on the one hand referring to original sources and on the other hand giving proofs with preservation of the original notations of the authors which hopefully will make the book readable and self-consistent. Chapter 2 deals with the general theory of trajectory dynamical system and its attractor and its application for the study the asymptotics of solutions of semilinear elliptic boundary value problems in unbounded domains. Chapter 2 consists of 14 subsections. In Sects. 2.1 and 2.2, we give definition of the fractal

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dimension and Kolmogorov’s ε-entropy as well as their properties. Moreover, the definition of a global attractor for the semigroup as well as finite fractal dimensional reduction of its dynamics are also given in these sections. In Sect. 2.3, we give the complete characterization of asymptotic behavior of all bounded nonnegative solutions for semilinear elliptic boundary value problems in a twodimensional rectangle. Sections 2.4–2.12 are devoted to the trajectory dynamical systems approach to semilinear elliptic system in general unbounded domains, the existence of at least one solution, the regularity of solutions, basic definitions of trajectory attractors, and its regularity as well as to some examples of trajectory attractors and its generalizations. In Sect. 2.13, we study how nonsmoothness of underlying unbounded domain transfers to nonsmoothness of trajectory attractors. Section 2.14 is devoted to the relation of nonautonomous parabolic equations in cylindrical domain with its elliptic counterpart via a traveling wave ansatz, and we study the latter from the dynamical systems viewpoint. In Chaps. 3 and 4, we consider semilinear elliptic boundary value problems in the quarter-space and study asymptotic behavior of its nonnegative solutions when the underlying dimension of the quarter-space is less than or equal to three and four, respectively. Here the dimension of the underlying domain plays a very essential role, because the technique for each dimension providing asymptotic behavior of solutions in terms of the trajectory attractor depends heavily on the dimension of the quarter-space. Indeed we show that under very natural dissipativity assumptions, as well as sign condition on the nonlinearity (any polynomial nonlinearity satisfies this dissipativity condition), the trajectory attractor (that by definition captures all the asymptotic behavior of nonnegative solutions in the quarter-space) consists of all nonnegative solutions of the same semilinear elliptic equation, however, in the half-space. Using the symmetry and monotonicity results for nonnegative solutions in such domains (the technique in this chapter for the proving of symmetry and monotonicity results requires dimension restrictions and is based on the moving plane method considered in Sect. 1.7), we can describe the asymptotic behavior of solution and prove that the elliptic attractor is one-dimensional and any solution of the elliptic BVP in the quarter-space converges to a unique solution of a secondorder ODE, indicating remarkable one-dimensional reduction of dynamics on the attractor. We emphasize that indeed such trajectory DS approach provides an elegant proof of a stabilization and symmetrization results that in general are really difficult to obtain using pure elliptic techniques, due to the unboundedness of the underlying domains in any directions and the possible appearance of oscillations during the limiting procedure. These chapters consists of four subsections, which deals with formulation of the problem (Sects. 3.1 and 4.1), a priori estimates and solvability results (Sects. 3.2 and 4.2), the existence of attractor for associated trajectory dynamical systems (Sects. 3.3 and 4.3), and symmetry and stabilization results (Sects. 3.4 and 4.4), respectively. In Chap. 5, we consider the same elliptic boundary value problem as was studied in Chaps. 3 and 4. However, in this chapter, the dimension of the underlying domain is less than or equal to five. We aim at extending the results from dimensions three and four to the dimension five. Unfortunately we cannot apply the techniques from

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the previous chapters. We show how to avoid technical difficulties arising in this case of dimension five and obtain new type of asymptotics in contrast to previous chapters. Indeed, under the assumptions that all equilibria are stable, we prove that: (a) ANY nonnegative bounded solution of a semilinear elliptic BVP in the quarterspace converges to a uniquely defined solution of a second-order ODE. (b) ANY nonnegative bounded solution in the half-space converges to a uniquely defined constant solution. In Chap. 6, we present completely new Liouville-type theorem for two classes of nonlinearities which give us the possibility to consider the same semilinear elliptic boundary value problem in any dimension. It is worth noting that these new Liouville-type results for any dimension are of independent interest for the solutions in half-spaces with homogeneous Dirichlet boundary conditions or in the whole space. In contrast to the results of the Chap. 5, one of the main points is that the results of Chap. 6 hold in any dimensions without any assumption on a solution other than its boundedness. In Chap. 5, the main results were obtained in lower dimensions of the underlying domains and under the additional assumption that the solutions are stable, which among others requires C 1 -differentiability of the nonlinearities. However, the assumptions on the nonlinearity f (s) imposed in the Chap. 5 are incompatible with the assumptions for the nonlinearity in Chap. 6 which are crucial to prove new Liouville-type results. Using these new Liouville-type results, we completely characterize the limiting behavior of solutions of semilinear elliptic boundary value in the quarter-space as well as in the half-space based on the trajectory dynamical systems approach. It is worth noting that all of these results required completely new ideas, tools, and techniques. Moreover in this Chap. 6, in contrast to the previous Chaps. 3–5, we compare two different approaches, namely, PDEs and dynamical systems approach discussing their individual advantages, as regards describing asymptotic profiles of solutions both in the quarter- and half-spaces in any dimension. To prove there exists a onedimensional elliptic attractor, we use, among others, the sliding method which was discussed in Sect. 1.7. Chapter 6 consists of four subsections. Section 6.1 is devoted to the formulation of the problem as well as the introduction of two classes of nonlinearities: sign-changing and sign-preserving. In Sect. 6.2, we deal with the PDE approach to an elliptic boundary value problem both in the quarter- and halfspaces and study some properties of solutions based on, among others, sliding methods that are relevant for determining the asymptotic of solutions. Section 6.3 deals with new type of Liouville results for bounded nonnegative solutions which on the one hand is of independent interest (it holds in any dimension without any assumption on a solution other than its boundedness) and on the other hand plays a decisive role in the study of uniqueness and one-dimensional symmetry of the limiting profiles of solutions as x1 goes to infinity. In Sect. 6.4, we apply the trajectory DS approach to study symmetrization and stabilization properties (as x1 goes to infinity) of nonnegative solutions for the equations in Sect. 6.1 and compare this DS approach with the PDE approach developed in Sect. 6.2.

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Chapter 7 deals with the asymptotic behavior of solutions of quasilinear elliptic problems over the quarter-space, and with similar problems over the half-space, and consists of four subsections. In Sect. 7.1, we formulate the same type of elliptic boundary value problem for the general p-Laplacian as was done in Chap. 6 for the standard Laplacian and discuss the difficulties that arise in this case. We introduce two classes of nonlinearities: sign-changing and sign-preserving nonlinearities satisfying new assumptions that take into account the p-Laplacian nature of the equations. Since there is no strong comparison principle in general for quasilinear elliptic problem, we have to adopt rather different approaches in many key steps previously done in Chaps. 3–6. In Sect. 7.2, we present some basic results which will be needed in our investigation of the half- and quarter-space problems for pLaplacian. A crucial ingredient here is a simple weak sweeping principle which is a consequence of the weak comparison principle for the p-Laplacian. We show that in many situations, it is possible to use the weak sweeping principle to replace the moving plane or sliding method which are based on the strong comparison principle for the Laplacian case which were frequently used in Chaps. 3–6. Using the weak sweeping principles, we prove new Liouville-type results for p-Laplacian equation in the unbounded domains mentioned above. Section 7.3 deals with the asymptotic behavior of solutions of p-Laplacian equations using these new Liouville-type results for p-Laplacian equations. Section 7.4 is devoted to corresponding results in the case of p-Laplacian with those of the semilinear case, existence of solution, as well as exact multiplicity results. I would like to thank many friends and colleagues who gave me suggestions, advice, and support. In particular, I wish to thank H. Berestycki, X. Cabre, N. Dancer, Y. Du, A. Farina, F. Hamel, H. Kielhöfer, K. Kirchgässner (deceased), H. Matano, A. Mielke, L. Nirenberg, M. Otani, J. Scheurle, C.A. Stuart, J.R.L. Webb, W.L. Wendland, J. Wu, E. Valdinochi, M.I. Vishik (deceased), S. Zelik, and A. Zhigun. Furthermore, I am greatly indebted to my colleagues at the Institute of Computational Biology in the Helmholtz Center Munich and Technical University of Munich, Alexander von Humboldt Foundation, as well as the Fields Monographs book series for their efficient handling of publication. This book was finished when I visited as a Dean’s Distinguished Visiting Professor of the University of Toronto and in the Fields Institute. I would like to express my sincere gratitude for these institutions in Toronto for providing an excellent and unique scientific atmosphere. In particular, my thanks go to my colleagues, friends, and staffs in the Fields Institute, namely, Ian Hambleton, Huaxiong Huang, Esther Berzunza, Tanya Nebesna, Miriam Schoeman, Bryan Eelhart, and Tyler Wilson. Last but not least, I wish to thank my family for permanently encouraging me during the writing of this book. Neuherberg, Bayern, Germany

Messoud Efendiev

Contents

1

2

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Functional Spaces and Their Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Lp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Linear Elliptic Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Nemytskii Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Maximum Principles and Their Applications . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Classical Maximum Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Uniform Estimates and Boundedness of the Solutions of Semilinear Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 The Sweeping Principle and the Moving Plane Method in a Bounded Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 The Sliding and the Moving Plane Method in General Domains . . . 1.8 Variational Solutions of Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Elliptic Regularity for the Neumann Problem for the Laplace Operator on an Infinite Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 3 8 18 23 31 33 36 43 49 55 61

Trajectory Dynamical Systems and Their Attractors . . . . . . . . . . . . . . . . . . . . 71 2.1 Kolmogorov ε-Entropy and Its Asymptotics in Functional Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.2 Global Attractors and Finite-Dimensional Reduction . . . . . . . . . . . . . . . 73 2.3 Classification of Positive Solutions of Semilinear Elliptic Equations in a Rectangle: Two Dimensional Case . . . . . . . . . . . . . . . . . . . 77 2.3.1 Sketch of the Proof of Theorem 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.4 Existence of Solutions of Nonlinear Elliptic Systems . . . . . . . . . . . . . . . 83 2.5 Regularity of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.6 Boundedness of Solutions as |x| Ñ 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.7 Basic Definitions: Trajectory Attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2.8 Trajectory Attractor of Nonlinear Elliptic System . . . . . . . . . . . . . . . . . . . 106 2.9 Dependence of the Trajectory Attractor on the Underlying Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 xv

xvi

Contents

2.10 2.11 2.12 2.13

2.14

Regularity of Attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectory Attractor of an Elliptic Equation with a Nonlinearity That Depends on x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of Trajectory Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Trajectory Dynamical Approach for the Nonlinear Elliptic Systems in Non-smooth Domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13.1 Existence of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13.2 Trajectory Attractor for the Nonlinear Elliptic System . . . . 2.13.3 Stabilization of Solutions in the Potential Case . . . . . . . . . . . . 2.13.4 Regular and Singular Part of the Trajectory Attractor. . . . . . The Dynamics of Fast Nonautonomous Travelling Waves and Homogenization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

116 121 123 127 134 137 143 149 151

3

Symmetry and Attractors: The Case N ď 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 A Priori Estimates and Solvability Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Symmetry and Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 163 165 168 171

4

Symmetry and Attractors: The Case N ď 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 A Priori Estimates and Solvability Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Symmetry and Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177 177 179 181 183

5

Symmetry and Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Statement of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Dynamical System Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Proof of Theorem 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Symmetry of the Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Completion of the Proof of Theorem 5.2 . . . . . . . . . . . . . . . . . . . 5.5 Proof of Theorem 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Positivity of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Completion of the Proof of Theorem 5.3 . . . . . . . . . . . . . . . . . . .

187 187 187 190 193 194 194 196 197 197 198

6

Symmetry and Attractors: Arbitrary Dimension . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The PDE Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Problem (6.1) in the Quarter-Space  = (0, +8) ˆ RN ´2 ˆ (0, +8) . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Problem (6.11) in the Half-Space 1 = (0, +8) ˆ RN ´1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

199 199 203 204 211

Contents

6.3

6.4 7

xvii

Classification Results in the Whole Space RN or in the Half-Space RN ´1 ˆ (0, +8) with Dirichlet Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 The Dynamical Systems’ Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

The Case of p-Laplacian Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Some Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 The Weak Sweeping Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Classification of One-Dimensional Solutions . . . . . . . . . . . . . . 7.2.3 A Liouville Type Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Half- and Quarter-Space Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Asymptotic Convergence in Half-Spaces . . . . . . . . . . . . . . . . . . . 7.3.2 One-Dimensional Symmetry in Half-Spaces . . . . . . . . . . . . . . . 7.3.3 Asymptotic Convergence in Quarter-Spaces . . . . . . . . . . . . . . . 7.4 Comparison with Standard Laplacian (p = 2) . . . . . . . . . . . . . . . . . . . . . . . 7.5 Existence Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

225 225 227 227 228 234 236 236 244 246 249 250

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

Chapter 1

Preliminaries

In this chapter, we present notation and generally known facts (mostly without proofs) that we use to state and derive the results of the subsequent chapters. For the sake of convenience, we introduce the following conventions: • N, N0 , Z, R, R+ and R+ 0 are sets of natural, non-negative integer, integer, real, positive real and non-negative real numbers respectively; • x + := max tx, 0u returns the positive part of a number x P R;

sign(x) :=

$ ’ ’ &1

for x ą 0,

0 ’ ’ %´1

for x = 0, for x ă 0

returns its sign; • The integer and fractional parts of a number x P R are the numbers [x] := max tq P Z| x ě qu and txu := x ´ [x] respectively; • By | ¨ | we denote: – for a number x its absolute value |x| = max tM, ´Mu; – for a vector x = (x1 , . . . , xd ) P Rd its Euclidean norm  |x| :=

d ÿ

 12 |xi |2

;

i=1

– for a multiindex α = (α1 , . . . , αd ) P Nd0 for d P N its absolute value |α| = řd j =1 αj ; – for a Lebesgue measurable set its Lebesgue measure; • By measure we always understand the Lebesgue measure; © Springer Nature Switzerland AG 2018 M. Efendiev, Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations, Fields Institute Monographs 36, https://doi.org/10.1007/978-3-319-98407-0_1

1

2

1 Preliminaries

• In a topological space X, we denote by clX (A) the closure of a set A in X. In Rd we use the notation A instead. BA denotes the topological boundary of A. • In a linear space X, we define x + A := tx + a| a P Au for x P X, A Ă X.

1.1 Functional Spaces and Their Properties This section is devoted to the classical Lebesgue and Sobolev spaces and to some of their modifications. We refer to [2, 101] for a detailed analysis of the classical spaces. Most of the presented spaces are normable (e.g., Lp () and W s,p ()), some p s,p are metriziable, but not normable (e.g., Lloc () and Wloc ()), and some (e.g., 8 Lw´˚ ()) are not even metriziable, though locally convex. Let us, therefore, before looking at concrete examples, briefly recall several facts originating from the general framework in locally convex and normed spaces. We refer to [85] (or some other standard textbook) for these as well as for the other facts from functional analysis that we use in this work. Let X and Y be two locally convex spaces. The (continuous) dual space of X is denoted by X1 , its weak and weak-˚ topologies by σ (X, X1 ) and σ (X1 , X) respectively. X – Y denotes the topological equivalence of X and Y , which means that they are homeomorphic. Let F be an operator, not necessarily linear, between X and Y . F is said to be compact if it maps bounded subsets of X onto relatively compact subsets of Y . It is said to be closed if its graph (F ) := t(x, F (x))| x P Xu is closed in X ˆ Y . We will often consider embeddings of a ‘smaller’ locally convex space into a ‘larger’ one: Definition 1.1 (Embedding) Let X and Y be two locally convex spaces. An injective linear operator ι : X Ñ Y is called an embedding. If such an operator ι exists between X and Y , then X is said to be embedded in Y . Further, we say that (1) X is continuously embedded in Y (and write X ãÑ Y ) if ι is a continuous operator; (2) X is compactly embedded in Y (and write: X ãÑãÑ Y ) if ι is a compact operator; d

(3) X is densely and continuously embedded in Y (and write: X ãÑ Y ) if ι is a continuous operator and ι(X) is dense in Y ; d

(4) X is densely and compactly embedded in Y (and write: X ãÑãÑ Y ) if ι is a compact operator and ι(X) is dense in Y . An important property of the dense and continuous embeddings concerns duality: Theorem 1.1 (Embedding of Dual Spaces) Let X and Y be two locally convex spaces. Then

1.1 Functional Spaces and Their Properties

3

X ãÑ Y ñ Y 1 ãÑ X1 . d

d

(1.1)

It is well known that every compact linear operator between Banach spaces is continuous. It is even weak-to-norm continuous, which means that it is continuous between (X, σ (X, X1 )) and Y . A similar property holds for the weak-˚-to-norm continuity: Theorem 1.2 (Weak-˚-to-Norm Continuity of a Compact Linear Operator) Let X be a normed space, Y a Banach space and F a compact linear operator between X1 and Y . Then F is a continuous operator between (X1 , σ (X1 , X)) and Y .

1.1.1 Lp Spaces We assume in this subsection that  is a nonempty measurable subset of Rd , d P N. Let us denote by L0 () the (linear) space of all equivalence classes of measurable functions on . Each such class consists of functions that are equal almost everywhere in . As usual, we identify the functions from one equivalence class and write u instead of [u]. For p P [1, 8] the function || ¨ ||Lp () : L0 () Ñ [0, 8], $ş 1 p &  |u(x)| dx ||u||Lp () := %ess sup|u(x)| x P

for p P [1, 8), for p = 8

(1.2)

is called the Lp norm. It is a well defined norm on the space ! ) Lp () := u P L0 ()| ||u||Lp () ă 8 . Each Lp space equipped with the Lp norm is a Banach space. The space L2 () is a Hilbert space with the scalar product (u, v)L2 () :=

ż

u(x)v(x) dx for u, v P L2 ().



We write || ¨ ||p instead of || ¨ ||Lp () and even || ¨ || instead of || ¨ ||2 and (¨, ¨) instead of (¨, ¨)L2 () to shorten the notation. Some of the most important results about Lp spaces are: Theorem 1.3 (Hölder Inequality) Let p, q P [1, 8] be such that p1 + q1 = 1. Then for all u P Lp (), v P Lq (), we have uv P L1 (), and the following inequality holds

4

1 Preliminaries

||uv||1 ď ||u||p ||v||q . Theorem 1.4 (Interpolation Inequality for Lp ()) Let 1 ď p1 ă p ă p2 ď 8. Then for all u P Lp1 ()XLp2 () we have u P Lp (), and the following inequality holds θ ||u||θp2 , ||u||p ď ||u||1p´ 1

where θ :=

1 p1 1 p1

´ ´

1 p 1 p2

.

Theorem 1.5 (Dual Representation for Lp ()) Let p P [1, 8). Put p1 :=

#

p p ´1

for p P (1, 8),

8

for p = 1.

Then it holds that 1

(Lp ())1 – Lp (). In particular, consider the linear operator 1

ιp : Lp () Ñ (Lp ())1 , ιp (u)(v) :=

ż

u(x)v(x) dx for all v P Lp ().

 1

ιp is an isometric isomorphism between Lp () and (Lp ())1 . Up to this point, we have considered the L8 () space, as well as other Lp spaces, equipped with the topology produced by the corresponding norm. For some of our applications (see below), the original topology appears too restrictive. We are forced to pass to weaker topologies where it is easier to prove compactness. We start with the following 8 8 Definition 1.2 (L8 w´˚ () Space) We define Lw´˚ () to be the set of all L ()functions equipped with the topology

) ! ι1´1 (O)| O P σ ((L1 ())1 , L1 ()) , where ι1 is the isometric isomorphism defined in Theorem 1.5. Some properties of L8 w´˚ () are collected in Theorem 1.6 and Remark 1.1 below.

1.1 Functional Spaces and Their Properties

5

Theorem 1.6 (Properties of L8 w´˚ ()) (1) The space L8 w´˚ () is a locally convex space; (2) A subset of L8 () is bounded in L8 w´˚ () if and only if it is norm bounded; (3) The topology of L8 (), if restricted to an L8 () ball, is completely w´˚ metrizable; (4) L8 () balls are compact in L8 w´˚ (). Sketch of the proof Observe that ι1 is not only an isometric isomorphism between (L1 ())1 and L8 (), it is, due to Definition 1.2, also a linear homeomorphism between ((L1 ())1 , σ ((L1 ())1 , L1 ())) and L8 w´˚ (). But every homeomorphism preserves metrizability and compactness properties, and every linear homeomorphism preserves locally convex structure and boundedness of subsets. Therefore, the properties (i)–(iv) are consequences of the corresponding properties of the space ((L1 ())1 , σ ((L1 ())1 , L1 ())), which is the dual space of the infinitely dimensional separable Banach space L1 (), equipped with the weak-˚ topology, and the fact that compact metric spaces are complete. l Remark 1.1 (Further Properties of L8 w´˚ ()) (1) The weak-˚ topology is the topology of pointwise convergence, so that the topology of L8 w´˚ () can be also obtained by means of the following convergence notion: A sequence tvn unPN converges in L8 w´˚ () to a v if and only if ż ż u(x)vn (x)dx Ñ u(x)v(x) dx for all u P L1 (); 

nÑ8



(2) A metric for the restriction of the L8 w´˚ () topology to a ball of radius R centered at 0 can be defined in the following way. Let tun unPN be a dense subset of L1 () and let tBn unPN be a sequence of positive real numbers, such that ÿ

Bn ||un ||L1 () ă 8.

nPN

Then the function defined by (8 )

d˚ (v1 , v2 ) :=

ÿ nPN

ˇż ˇ ˇ ˇ ˇ Bn ˇ (v1 (x) ´ v2 (x))un (x) dx ˇˇ

(1.3)



for all v1 , v2 P L8 (), ||v1 ||8 , ||v2 ||8 ď R, is an example of a metric which produces the relative topology; (3) The property 4. from Theorem 1.6 is equivalent to L8 () ãÑãÑ L8 w´˚ (). This is due to the definition of compact embedding. (4) For more information on compactness and metrizability in the weak-˚ topology see [85].

6

1 Preliminaries

Thus, the L8 () balls are metrizable compact subsets of L8 w´˚ (). The following p norm topology for p P [1, 8) intersections of the L8 () topology with the L w´˚ are of independent interest. The definition is as follows: Definition 1.3 (The Space Hp ()) Let p P [1, 8]. We define Hp () to be the set of all L8 ()-functions equipped, for p = 8, with the topology of L8 w´˚ () and, for p P [1, 8), with the intersection of the topologies of the spaces L8 w´˚ () and Lp (). Next theorem contains several properties of the spaces Hp (). Theorem 1.7 (Properties of Hp ()) Let p P [1, 8). Then: (1) The space Hp () is a locally convex space; (2) A subset of L8 () is bounded in Hp () if and only if it is norm bounded; (3) The topology of Hp (), if restricted to an L8 () ball, is completely metrizable; (4) The topologies of Hp () and L2 (), if restricted to an L8 () ball, coincide. Sketch of the proof Let p P [1, 8).  ( (1) We observe that the set lu | u P L1 () , where lu (u1 ) := |u1 (u)| for u1 P (L1 ())1 , is an example of a system of seminorms on (L1 ())1 that generates σ ((L1 ())1 , L1 ()) topology on (L1 ())1 . Hence the locally  ( convex structure 1 () , where ω (v) := () is given by the family ω | u P L of the space L8 u u w´˚ ˇ ˇş ˇ u(x)v(x) dx ˇ for all u P L1 () and v P L8 (). Consequently, the family  (  ωu + || ¨ ||p | u P L1 () is an example of a system of norms that generates the locally convex structure of Hp (); (2) A set is bounded in Hp () if and only if it is bounded in each of the seminorms that defines it locally convex structure. In the particular case of Hp () it follows with the proof of the property (1) that a set is bounded in Hp () if and only if it is bounded in each of the norms ωu and in the Lp norm, which is equivalent p to the boundedness in both L8 w´˚ () and L (). The statement now follows with the property (2) from Theorem 1.6 and the fact that, for  bounded, the L8 norm is stronger then any other Lp norm; (3) It is a consequence of (4); (4) Observe first that, due to the Hölder inequality and Theorems 1.4, we have 1 ||1||

p p´1

1´ 1

1

||v||1 ď ||v||p ď ||v||8 p ||v||1p for all v P L8 ().

This shows that the topologies of Lp () and L1 (), if restricted to an L8 () ball, coincide. To show the property (4) it then suffices to check that the 2 restriction of the L8 w´˚ () topology is weaker than, for example, the L () topology.

1.1 Functional Spaces and Their Properties

7

Since the space L2 () is dense in L1 (), we may assume that tun unPN Ă (8 ) L2 () in the definition of the metric d˚ from (1.3) and choose the sequence tBn unPN to be such that ÿ

Bn ||un ||L2 () ă 8

nPN

holds. Consequently, we obtain with the Hölder inequality that (8 )

d˚ (v1 , v2 ) =

ÿ

nPN

ď

ÿ

ˇż ˇ ˇ ˇ ˇ Bn ˇ (v1 (x) ´ v2 (x))un (x) dx ˇˇ 

Bn ||un ||L2 () ||v1 ´ v2 ||2 .

nPN

This shows that, if restricted to an L8 () ball, the L8 w´˚ () topology is weaker then the topology of L2 (). l L8 ()

Hp (),

Thus, the balls are metrizable subsets of and, for p P [1, 8), a subset of an L8 () ball is compact if an only if it is compact in Lp (). Sometimes, especially in case when  is unbounded, it is useful (see [35, 48]) to consider the local version of an Lp space, the space ! ) p Lloc () := u P L0 ()| u P Lp (K) for all compact sets K Ă  . This space is not normable, though metrizable. Define the function p

|| ¨ ||Lp () : Lloc () Ñ [0, 8], b

||u||Lp () := sup ||u||Lp XBx b

0 (1)

x0 PRd

,

where Bx0 (1) is a unit ball in Rd centered at x0 . || ¨ ||Lp () is a norm on a subspace b

p

of Lloc (), namely on the space ! ) p p Lb () := u P Lloc ()| ||u||Lp () ă 8 . b

Note that L1loc () is the largest of the presented spaces of the Lp type.

8

1 Preliminaries

1.1.2 Sobolev Spaces From now on we assume  to be a nonempty domain (i.e. a nonempty open connected set) in Rd . We denote by D() the (locally convex) space of all test functions over . As a set, D() coincides with C08 (), the set of all infinitely differentiable functions with compact support in . The dual space of D(), 1 the space ş D (), is the space of 1distributions over . Distributions of the form v Ñ  u(x)v(x) dx for u P Lloc () are called regular. In case of a regular distribution, we identify the distribution with the L1loc function that produces it. For u P D 1 () and v P D() we denote by (u, v) the value of u on v. In case when u P L2 () we recover the scalar product in L2 (). For a multiindex α = (α1 , . .. , αd ) we define the differential operator of the order |α|: D (α) := Bxα11 , . . . , Bxαdd , where Bxk is the partial distributional derivative along the variable xk and Bxαkk = (Bxk )αk . Recall that any distribution is infinitely differentiable in the distributional sense. For s P N0 and p P [1, 8] the function

||u||W k,p ()

$ ř ’ &

1 › (α) ›p p ›D u› p |α |ďk L () := › (α) › ’ % max ›D u› |α |ďk

8

for p P [1, 8), for p = 8

is a well defined norm on the space ! ) W k,p () := u P Lp ()| D (α) u P Lp () for |α| ď k , Equipped with the || ¨ ||W k,p () norm, the space W k,p () is the classical Sobolev space of order k. All W k,p () spaces are Banach spaces. The space H k () := W k,2 () is a Hilbert space with the scalar product (u, v)H k () :=

 ÿ  D (α) u, D (α) v for u, v P H k (). |α |ďk

With k = 0 we recover the definitions of the corresponding Lp spaces. One of those subspaces of W k,p () that play an important role in partial differential equations k,p is the space W0 () for  bounded. It consists of functions that ‘vanish on the boundary’ in the sense of trace (see [2]). One of the equivalent ways to define these spaces is: k,p

W0 () := clW k,p () (D())

1.1 Functional Spaces and Their Properties

9

for k P N and p P [1, 8]. For  bounded the seminorm || ¨ ||W k,p () : W k,p () Ñ [0, 8), 0

⎛

||u||W k,p () 0

⎞1

›p ÿ ›› › := ⎝ ›D (α) u› p

p

⎠

L ()

|α |=k

k,p

is an equivalent norm on W0 (). This is a consequence of the Poincaré inequality: Theorem 1.8 (Poincaré Inequality) Let p P [1, 8] and let  be a smooth bounded domain in Rd . Then there exists a positive constant P (, p) that depends only on  and p and such that ||u||p ď P (, p)||Du||p 1,p

holds for all u P W0 (). The norm || ¨ ||W k,p () is called the energy norm. On the space H0k () := W0k,2 () 0 the bilinear form defined via  ÿ  (u, v)H k () := D (α) u, D (α) v for u, v P H0k () 0

|α |=k k,p

is a scalar product. The space W0 () is a closed subspace of W k,p (), thus it is a Banach space, while the space H0k () is a Hilbert space. It is often useful to consider a class of ‘in-between’ spaces, that is, to extend the notion of classical Sobolev spaces of non-negative integer order k to the case s P R+ 0 zN0 . One of the possible contractions uses the Slobodeckij seminorm

[u]θ,p :=

$ 1 ş ş |u(x)´u(y)|p p ’ & dx dy   |x ´y |pθ+d ’ % ess sup

x,y P,x ‰y

|u(x)´u(y)| |x ´y |θ

for p P [1, 8), for p = 8

defined for θ P (0, 1) and p P [1, 8]. For s P R+ 0 zN0 and p P [1, 8] the function || ¨ ||W s,p () : W [s],p () Ñ [0, 8], $ 1 p ’ ’ ř “ (α) ‰p p ’ & ||u|| [s],p + u D ts u,p () W ||u||W s,p () := |α |=s ’ ‰ “ ’ ’ %||u|| [s],8 + max D (α) u W

()

|α |=s

ts u,8

for p P [1, 8), for p = 8

10

1 Preliminaries

is a well defined norm on the space " ” ı W s,p () := u P W [s],p ()| D (α) u

ts u,p

* ă 8 for |α| = [s] .

The spaces W s,p () for s P R+ 0 zN0 are called Sobolev-Slobodeckij spaces. These spaces are Banach spaces. The space H s () := W s,2 () is a Hilbert space with the scalar product (u, v)H s () :=(u, v)H [s] ()    ÿ ż ż D (α) u(x) ´ D (α) u(y) D (α) v(x) ´ D (α) v(y) + dxdy. |x ´ y|2θ+d   |α |=s

If the domain  is suitably regular then, indeed, W s2 ,p is a subset of W s1 ,p for all 0 ď s1 ă s2 ă 8. Just as in case of integer order Sobolev spaces, we can define for s P R+ 0 zN0 and p P [1, 8] the space s,p

W0 () := clW s,p () (D()) . Observe that for all s P R+ and p P [1, 8] the space D() is densely and s,p continuously embedded in the space W0 () by means of the identity operator. This is because the convergence in D() is stronger than the convergence in s,p s,p W0 () and because D() is dense in W0 () (by definition). With (1.1) it 1 d  s,p follows that W0 () ãÑ D 1 (). Now, for s P R+ and p P (1, 8] set W

´s,p

 1 s,p1 () := W0 () , p1 :=

#

p p ´1

for p P (1, 8),

1

for p = 8,

H ´s () := W ´s,2 (). This is the way to define the Sobolev spaces of negative order. For p P (1, 8) it also holds 1  ´s,p s,p1 () – W0 (). W This is a consequence of Theorem 1.9 (Reflexivity of W s,p ()) Let s P R and p P (1, 8). The space W s,p () is reflexive.

1.1 Functional Spaces and Their Properties

11

For all s P R and p P [1, 8] the number γ = s ´ pd is called the Sobolev number (corresponding to the pair s, p). The numbers s, p and γ can be used to compare a Sobolev space with another Sobolev space or with a Hölder space. This is the subject of Theorem 1.10 (Sobolev Embedding Theorem) Let  be smooth and bounded. Let ´8 ă s1 ă s2 ă 8, 1 ď p2 ď p1 ď 8 and let γ1 and γ2 be the Sobolev numbers corresponding to the pairs s1 , p1 and s2 , p2 respectively. Then: (Part I) γ2 ą γ1 ñ W s2 ,p2 () ãÑãÑ W s1 ,p1 (), γ2 = γ1 and p1 ă 8 ñ W s2 ,p2 () ãÑ W s1 ,p1 (), the embedding being the identity operator. In both cases the Sobolev inequality ||u||W s1 ,p1 () ď C0 (s1 , s2 , p1 , p2 )||u||W s2 ,p2 () for all u P W s2 ,p2 ()

(1.4)

holds. The embedding constant C0 (s1 , s2 , p1 , p2 ) depends only on s1 , s2 , p1 , p2 and the domain . (Part II) γ2 ą γ1 and p1 = 8, s1 ą 0 ñ W s2 ,p2 () ãÑãÑ C [s1 ],ts1 u (), the embedding being the identity operator and the Sobolev inequality ||u||

C [s1 ],ts1 u ()

ď C1 (s1 , s2 , p2 )||u||W s2 ,p2 ()

holds. The embedding constant C1 (s1 , s2 , p2 ) depends only on s1 , s2 , p2 and . Remark 1.2 In part II of the Sobolev embedding theorem, the Sobolev spaces are compared with the spaces of continuously differentiable functions C k () and the Hölder spaces C k,θ (). They are continuous versions of the Sobolev spaces W k,8 () and the Sobolev-Slobodeckij spaces W k+θ,8 (), respectively:  ( C 0 () := C() := u :  Ñ R| u continuous on  , ! ) C k () := u P C()|D (α) u P C() for |α| ď k , k P N, # + ˇ ˇ (α) ˇD u(x) ´ D (α) u(y)ˇ sup ă8 , C k,θ () := u P C k ()| |x ´ y|θ x,y P,x ‰y || ¨ ||C k,θ () := || ¨ ||W k+θ,8 () , k P N0 , θ P [0, 1). As in case of Lp spaces, we have an interpolation inequality for a space ‘inbetween’:

12

1 Preliminaries

Theorem 1.11 (Interpolation Inequality for W s,p ()) Let  be smooth and bounded. Let s1 , s, s2 P (0, 8) and p1 , p, p2 P [1, 8] be such that s2 ą s ě s1 , γ 2 ą γ ą γ1 , γ ´ γ1 P θ := γ2 ´ γ1



s ´ s1 ,1 , s2 ´ s1

where γ1 , γ and γ2 are the Sobolev numbers corresponding to the pairs s1 , p1 , s, p and s2 , p2 respectively. Then the following interpolation inequality holds for all u P W s2 ,p2 (): ||u||W s,p () ď I (s1 , s, s2 , p1 , p, p2 )||u||1W´s1θ,p1 () ||u||θW s2 ,p2 () .

(1.5)

The constant I (s1 , s, s2 , p1 , p, p2 ) depends only on s1 , s, s2 , p1 , p, p2 and the domain . The following useful nonlinear version of the Sobolev inequality (1.4) is a consequence of Lemma 1.2 from [46]. Lemma 1.1 Let s P (0, 1), p P [1, 8) and q P (1, 8). Then there exists a constant N(q) that depends only on q and such that it holds ||u||

W

s q ,sq ()

› ›1 › ›q ď N (q) ›|u|q ´1 u›

W s,p ()

for all u P W s,p ().

We conclude this subsection with the definition of a local Sobolev space: ! ) k,p p p Wloc () := u P Lloc ()| D (α) u P Lloc () for |α| ď k . Remark 1.3 Below, we give definitions of several Fréchet spaces which will be useful in the study of nonlinear elliptic systems in unbounded domains that we study in Chap. 2. Let V Ă RN be some open set in RN . We denote by  ( W l,p (V) = w P D 1 (V)| D α w P Lp (V) , H l,p (V) = W l,p (RN )|V . It is well-known that if V is a bounded domain with a ‘smooth’ boundary, then the spaces H l,p (V) and W l,p (V) coincide (see [26]). l,p

Remark 1.4 Let V be a bounded subset in RN . By W0 (V) we denote completion ” ı˚ of C08 (V) in the metric of W l,p (V). W01,2 (V) Ă D 1 (V) we denote by W ´1,2 (V).

1.1 Functional Spaces and Their Properties

13

1,2 We define the spaces Wloc (V) for an arbitrary unbounded domain V as a Frechét subspace of D 1 (V) endowed by the seminorms ,  W 1,2 BR x XV

||v, BR x0 X V|| := ||v||

R Ă R+ , x0 P R.

0

´1,2 (V) for 1 ď p ď 8. We set Analogously, we define Lloc (V), Wloc p

” ı 1,2 θ (V) := Wloc (V) X Lrloc (V) , where V is an open set in RN , r is the same as in the assumption on f (see Sect. 2.4). Seminorms in θ (V) we define in the following way: " ||v, BR X V|| := max ||v|| + x0

W 1,2



 , ||v||   Lr BR BR x XV x XV 0

* ,

0

where R Ă R+ , x0 P RN . By θ0 (V) we denote completion of C08 (V) in topology of θ (V). Let V be an arbitrary open set in Rn . For another open set W in Rn , we define by θ0 (V, W) the completion of C08 (V)|VXW in the space θ (V X W). It is obvious that θ0 (V X W) Ă θ0 (V, W) Ă θ (V X W). We denote by θ (V, W) a completion of the set θ (V)|VXW in the space θ (V X W). Usually, in application to elliptic systems in an unbounded domain  Ă Rn , we will take V :=  and W := B(x0 , R). Definition 1.4 Let V be an open set in Rn . Then ´1,2 (V) := [Wloc (V) + Lloc (V)], q

1 1 + = 1, q r

where r is the same as in the definition of θ (V). The space (V) consists of all g P D 1 () that have a representation ´1,2 (V)]k , g2 P [Lloc (V)]k . g = g1 + g2 , g1 P [Wloc q

(1.6)

The system of seminorms in (V) are given by "› ˇ › ˇ › › ˇ Rˇ ˇg, V X Bx0 ˇ := inf ›g1 , V X BR x0 › +

´1,2

› › › › + ›g2 , V X BR x0 ›

0,q

* ,

´1,2 (V)]k , g2 P [Lloc (V)]k , R P R+ , x0 P Rn . g = g1 + g2 , g1 P [Wloc q

14

1 Preliminaries

Analogously, we define (V, W) as the completion of (V)|VXW in the space (V X W). ν0 (BV) := θ (V){θ0 (V) and the system of seminorms in ν0 (BV) are defined in the following way: ||u0 ||ν0 (BVXBRx ) := inf t||w||+ , w P θ (V), w|BV = u0 u , 0

where R P R+ , x0 P RN . The following Propositions will be used in Chap. 2: Proposition 1.1 Let V be a bounded domain in Rn . Then for any 0 ď δ ď r ´ 1 it holds that θ0 (V) ĂĂ Lr ´δ (V).

(1.7)

Proof Indeed, any function u P θ0 (V) can be extended (by zero) to the function ˜ with u˜ P θ (V) with preserving norm, so that it suffices to prove (1.7) for a domain V ε,r ´ δ (V) ˜ ˜ ˜ sufficiently smooth boundary (V ĂĂ V). Due to the embedding θ (V) Ă W for sufficiently small ε ą 0 (l1 = 1, l2 = 0, p1 = 2, p2 = r) u P W l1 ,p1 X W l2 ,p2 ñ l = θ l1 + (1 ´ θ )l2 ,

θ 1 1´θ = + . p p1 p2

Then, u P W l,p and (Gagliardo-Nirenberg) ||u||W l,p ď C||u||θW l1 ,p1 ||u||1W´l2θ,p2 , θ P [0, 1]. Using this fact, we can state that W ε,r ´δ ĂĂ Lr ´δ for ε ăă 1, δ ăă 1. \ [ Proposition 1.2 Let V be a bounded domain in maps W 1,2 (V) to W ´1,2 (V). Moreover, it holds:

Rn .

Then, the Laplace operator

||u, V||´1,2 ď ||u, V||1,2 for all u P W 1,2 (V). Proof From the definition of W 1,2 (V) it follows that |(u, ϕ)| = |(u, ϕ)| = |(∇u, ∇ϕ)| ď ||u, V||1,2 ||ϕ, V||1,2

(1.8)

1.1 Functional Spaces and Their Properties

15

for all ϕ P D(V). Hence, ||u, V||´1,2 = sup

"

* |(u, ϕ)| |ϕ P D(V) ď ||u, V||1,2 . ||ϕ, V||1,2 \ [

This proves Proposition 1.2.

Proposition 1.3 Let V Ă Rn be a bounded domain. Then, for any u P θ0 (V) and g P (V) holds: |(u, g)| ď ||u, V||+ |g, V|+ , where by (¨, ¨) we denote the scalar product in [L2 (V)]k (naturally continuously extended to D 1 (V)). Proof Let g = g1 + g2 , where g1 P [W ´1,2 (V)]k and g2 P [Lq (V)]k . Then |(u, g)| ď |(u, g1 )| + |(u, g2 )| ď||u, V||1,2 ||g1 , V||´1,2 + ||u, V||0,r |||g2 , V||0,q   ď||u, V||+ ||g1 , V||´1,2 + ||g2 , V||0,q . Taking in the last inequality infimum with respect to g = g1 + g2 , we obtain the assertion of Proposition 1.3. [ \ Remark 1.5 The following holds (see [7]): [D1 X D2 ]˚ = D˚1 + D˚1 , where Di Ă D 1 (), i = 1, 2 are some Banach spaces such that D1 X D2 is dense both in D1 and D2 . Thus, in the case of a bounded domain  Ă Rn , the spaces θ0 () and () are naturally conjugate. Below we give several definitions, which we will be useful in the sequel. Definition 1.5 Let X be a linear topological space [85, 105]. A set B Ă X is called bounded if there exists a neighbourhood of the origin (zero) E in X and N = N (E) such that 1 B Ă E for n ě N. n Corollary 1.1 Let X be a locally convex space [85, 105]. Then, a subset B Ă X is bounded if and only if, for every seminorm || ¨ ||p from the definition of X topology, it holds: ||B||p ď C = C(|| ¨ ||p ).

16

1 Preliminaries

Definition 1.6 Let X be a linear topological space and X˚ be its dual space. A system of seminorms defining the (strong) topology in X˚ is given [86] by ||T ||B = sup|T x|, x PB

where B is a bounded set in X, T P X˚ . The weak topology in X is given by a system of seminorms ||x||T = |T x|, T P X˚ , x P X. We denote by Xw the space X endowed with the weak topology. The weak-˚ topology in X˚ is given by the system of seminorms ||T ||x = |T x|, x P X, T P X˚ .

(1.9)

The following Lemma holds: Lemma 1.2 ([86, 105]) Let X be a Fréchet space (F -space). Then X is reflexive if and only if any bounded subset in X is precompact in Xw . Lemma 1.3 (Eberlein) Let X be an F -space. Then, a set B Ă X is precompact in Xw if and only if B is sequentially compact in Xw , which means that any sequence txn u Ă B has a converging subsequence in Xw . Lemma 1.4 ([85]) Let X be a separable locally convex space. Then, any bounded subset in X˚ is metrisable in the weak-˚ topology. Lemma 1.5 ([86]) Let X be a locally convex space and let B Ă X. It holds: 1. B is bounded in Xw if and only if B is bounded in X. 2. Let B be convex. Then, B is closed in Xw if and only if B is closed in X. Lemma 1.6 ([86]) Let X be a locally convex space and let X0 be a closed subspace of X. Then the weak topology in X0 coincides with the topology induced by Xw . Let us now study the structure of the dual spaces introduced above. Theorem 1.12 Let X be one θ (), θ0 (), (). Then, for every  of the spaces ˚ T Ă X˚ there exists an l P X(, BR such that x0 ) A E T x = l, xXBRx , for all x P X.

(1.10)

0

˚  defines by (1.10) a continuous linear functional T on X. Each l P X(, BR x0 )   R ˚ Here, ă ¨, ¨ ą denotes the canonical paring of X(, BR x0 ) and X(, Bx0 ) . The embedding given by (1.10) is continuous.

1.1 Functional Spaces and Their Properties

17

Proof Since T is continuous, the exists, due to [105] and the definition of continuous seminorms in X, a ball BR x0 Ă R such that › › › › › › › › ď C |T x| ď C ›x,  X BR R ›x| › XB x › x0 X

0

X(,BR x )

.

(1.11)

0

The functional T is well-defined and uniformly continuous in the subspace X()|XBRx of X(, BR is dense in X(, BR x0 ); X()|XBR x0 ). Therefore, T x 0

0

can be uniquely extended to a continuous linear functional l : X(, BR x0 ) Ñ R. A ˚ is obvious. This proves Theorem 1.12. continuous embedding of X(, BR ) Ă X x0 \ [ Corollary 1.2 Let X be the same as in Theorem 1.12. Then, a sequence xn P X converges weakly to some x P X if and only if xn |XBRx á x|XBRx 0

(1.12)

0

R n in the space X(, BR x0 ) for each Bx0 Ă R .

Remark 1.6 Let all assumptions of Corollary 1.2 hold. It is then sufficient to k n check (1.12) on BR x0 Ă R , where Rk Ñ 8 for k Ñ 8. Theorem 1.13 Let X be as in Theorem 1.12. Then, both X and X˚ are reflexive and separable. Proof We prove the statement for the space θ (). For the other spaces the proof can be carried out in the same manner. It is clear that, in order to prove separability of θ (), it suffices to show the separability of ” ık ” ık 1,2 R r R θ ( X BR x0 ) = Wloc ( X Bx0 ) X Lloc ( X Bx0 ) n 1,2 and Lr can be found in [26]. for each BR x0 Ă R . A proof of separability for W R Each space θ (XBx0 ) is isometric isomorph to a closed subspace of the space M = ‰k “ r ‰ “ 1,2 R k W ( X BR x0 ) ˆ L ( X Bx0 ) which consists of pairs tz, zu, consequently, it is separable. This proves the separability of θ (). Next, we prove reflexivity of θ (). According to Lemmas 1.2 and 1.3, it suffices to show that any bounded subset of [θ ()]w is sequentially precompact. Thus, with Corollary 1.2, Remark 1.6 and the Canter diagonal procedure, it is sufficient to show R that θ (, BR x0 ) is reflexive for each Bx0 . Let us prove this. Indeed, the space θ ( X R Bx0 ) is reflexive as a closed subspace of the reflexive B-space M [86], and the space R θ (, BR x0 ) is reflexive as a closed subspace of the reflexive B-space θ ( X Bx0 ). Consequently, θ () is reflexive.

18

1 Preliminaries

˚ The space [θ (, BR x0 )] is separable as the dual space to a reflexive B-space. ˚ Hence, due to Theorem 1.12, [θ (, BR x0 )] is separable as well. This proves Theorem 1.13. \ [

Corollary 1.3 Let X be the same as in Theorem 1.13. Then, any bounded subset B Ă X is a metrisable precompact in the space Xw . In particular, any precompact in Xw is metrisable.

1.2 Linear Elliptic Boundary Value Problems Notation Let  be a bounded region in Rn . For α = (α1 , ¨ ¨ ¨ , αn ) an n-tuple of

n n ź ÿ B αi nonnegative integers, recall that D α = , |α| = αi and let ξ α = Bx i i=1 i=1 n ź (ξi )αi if ξ P C1n . i=1

Every linear differential operator L of order 2m (m P N) has the form Lu =

ÿ

aα (x) ¨ D α u.

(1.13)

|α |ď2m

All coefficients aα (x) are assumed to be real. The partial differential operator Lu =

ÿ

aα (x) ¨ D α u

|α |ď2m

is called elliptic of order 2m if its principal symbol, p0 (x, ξ ) =

ÿ

aα (x) ¨ ξ α

|α |=2m

has the property that p0 (x, ξ ) ‰ 0 for all x P , ξ P Rn zt0u. The differential operator L defined by (1.13) is called uniformly elliptic in , if there is some c ą 0, such that ÿ (´1)m aα (x)ξ α ě C|ξ |2m for every x P , ξ P Rn zt0u. (1.14) |α |=2m

Throughout we assume that B is a smooth (n ´ 1)-manifold. Suppose now that L is elliptic and of order 2m.

1.2 Linear Elliptic Boundary Value Problems

19

Let tmi , 1 ď i ď mu be distinct integers with 0 ď mi ď 2m ´ 1, and suppose that for 1 ď i ď m we prescribe a differential operator Bi of order mi on B, by Bi u(x) =

ÿ

bα,i (x)D α u(x),

i = 1, ¨ ¨ ¨ , m.

(1.15)

|α |ďmi

The family of boundary operators B = tB1 , ¨ ¨ ¨ , Bm u is said to satisfy the Shapiro-Lopatinski covering condition with respect to L provided that the following algebraic condition is satisfied. For each x P B, N P Rn zt0u normal to B at x and ξ P Rn zt0u with xξ, N y = 0, consider the (m + 1) polynomials of a single complex variable τ Ý Þ Ñ p0ÿ (x, ξ + τ N ), bα,i (x) ¨ (ξ + τ N )α ” p0,i (x, ξ, τ ), τ ÞÝÑ

1 ď i ď m.

(1.16)

|α |=mi

Let τ1+ , ¨ ¨ ¨ , τm+ be the m complex zeros of p0 (x, ξ + τ N ) which have positive imaginary part. Then tp0,i (τ )um i=1 are assumed to be linearly independent modulo m ź (τ ´ τi+ ) = M + (x, ξ, N , τ ), i.e., after division by M + (x, ξ, N, τ ) all the varii=1

ous remainders are linearly independent. In other words, let 1 (x, ξ, N , τ ) = p0,i

mÿ ´1

bi,k (x, ξ, N ) ¨ τ k ,

i = 1, ¨ ¨ ¨ , m

k=0

be the remainders after division by M + (x, ξ, N , τ ). Then the condition of the Shapiro-Lopatinski implies that D(x, ξ, N ) = det }bik (x, ξ, N )} ‰ 0

(1.17)

for all x P B, and for all N P Rn zt0u normal to B at x and ξ P Rn zt0u with xξ, N y = 0. Definition 1.7 We say that (L, B1 , ¨ ¨ ¨ , Bm ) defines an elliptic boundary value problem of order (2m, m1 , ¨ ¨ ¨ , mm ) if L given by (1.13), is uniformly elliptic and of order 2m, each Bi given by (1.15) has order mi , 0 ď mi ď 2m ´ 1, the mi ’s are distinct, B is non characteristic to Bi at each point and tBi um i=1 satisfy the Shapiro-Lopatinski condition with respect to L (see [79]). We have the following lemma (see [48, 63, 79]). Lemma 1.7 Let (L, B1 , ¨ ¨ ¨ , Bm ) define an elliptic boundary value problem of order (2m, m1 , ¨ ¨ ¨ , mm ). Then

20

1 Preliminaries

˜ l , ¨ ¨ ¨ , Bm ˝  ˜ l , L ˝ Bu ) (L ˝ l , B1 ˝  B N defines an elliptic boundary value problem of order (2k + 2l, m1 + 2l, ¨ ¨ ¨ , mm + ˜ is the Laplace-Beltrami operator, l P N . 2l, 2m + 1) where  Proof The principal symbol of L ˝ l is |ξ |2l ¨ p0 (x, ξ ), so it is clear that L ˝ l is uniformly elliptic. Let x P B and ξ, N P Rn zt0u, with xξ, N y = 0 and N normal to B at x. It ˜ l and L ˝ B at ξ + τ N are is obvious that the principal symbol operators Bi ˝  B N ψl (ξ ) ¨ p0i (x, ξ + τ N ) and τp0 (x, ξ + τ N ) respectively, where ψl (ξ ) ‰ 0. If τ1+ , ¨ ¨ ¨ , τm+ are the m roots of p0 (x, ξ + τ N ) = 0 having positive imaginary |N | part, then τ + , ¨ ¨ ¨ , τm+ , i¨ constitute the m+1 roots of |ξ +τ N |2 ¨p0 (x, ξ +τ N ) = |ξ |

1

0 with positive imaginary part. We must show that if λ1 , ¨ ¨ ¨ , λm+1 P C2 ang h(τ ) is a polynomial with ψl (ξ )

řm

¨ p0i(x, ξ + τ  N ) + λm+1 τp0 (x, ξ + τ N ) = ś i |N | + h(τ ) ¨ τ ´ |ξ | ¨ m i=1 (τ ´ τi )

i=1 λi

(1.18)

then λi = 0, 1 ď i ď m + 1 and h(τ ) ” 0. Due to the assumption that (B1 , ¨ ¨ ¨ , Bm ) satisfy the covering condition it is not difficult to see that λ1 = ¨ ¨ ¨ = λm = 0. But then the right-hand side of (1.18) has more roots with positive imaginary part than does the left-hand side, so that λm+1 = 0 and h(τ ) ” 0. With appropriate smoothness conditions on the coefficients (see Lemma 1.8 below), elliptic boundary value problems induce linear Fredholm operators in Sobolev spaces. Here the spaces W 2m+k ´mi ´1{p,p (B) with the fractional differentiation order 2m + k ´ mi ´ p1 play a decisive role. Before giving a precise definition we wish to point out a priori the most important property of these spaces, i.e. the surjective boundary operator ¯ Ñ C 8 (B) T : C 8 () ¯ its classical boundary value T u on B, which assigns to each function u P C 8 () can be extended uniquely to a continuous linear surjective operator T : W 2m+k,p () Ñ W 2m+k ´mi ´1{p,p (B). Here k ě 0 and m ě 1 are integers, and 1 ă p ă 8 (we are mainly interested in the case p = 2, W 2m,2 () = H 2m ()). Then T u is described naturally as the generalized boundary value of u P W 2m+k,p (). These functions u have generalized derivatives D α u up to order 2m+k on . The functions D α u with |α| ď mi have generalized boundary values which all lie in W 2m+k ´mi ´1{p,p (B), since mi ă 2m. Consequently, Bi u P W 2m+k ´mi ´1{p,p (B) also. The differential

1.2 Linear Elliptic Boundary Value Problems

21

operators L and the boundary operator Bi are thus to be understood in the space of generalized derivatives on  and as generalized boundary values respectively. Definition of the Space W m´1{p,p (B). Let  be an open subset of Rn with sufficiently smooth boundary and tUi uli=1 ¯ with diffeomorphisms ϕi : Ui Ñ Rn , ϕi P C m (Ui ), such be an open covering of  that ϕi (Ui ) = V1 = ty P Rn | |y| ă 1u if Ui Ă , and ¯ = V + = ty P Rn | |y| ă 1, yn ě 0u, ϕi (Ui X ) 1 ϕi (Ui X B) = V˜1 = ty P Rn | |y| ă 1, yn = 0u if Ui X B ‰ H. Let χi (x) be a partition of unity subordinated to tUi uli=1 and let λi (y) := χi (ϕi´1 (y)). For each u(x) P C m (B), 0 ă δ ă 1, p ą 1 we define the norm: # }u}1m´δ,p,B =

ÿ

»

i PI 1

ÿ

ÿ

– 1

ż

1

|αż|ďmż´1

|α | =m´1

V˜1

V˜1

V˜1

|Dyα (λi (y) ¨ ui (y))|p dy 1 +

|Dyα (λi (y) ¨ ui (y)) ´ Dzα (λi (z) ¨ ui (z)) |p

dy 1 dz1 ¨ 1 |y ´ z1 |n+p´1´δp

j* p1 , (1.19)

where ui (y) = u(ϕi´1 (y)), y 1 = (y1 , ¨ ¨ ¨ , yn´1 ), I 1 Ă t1, ¨ ¨ ¨ , lu such that: Ui X ř B ‰ H and 1 implies that the sum is taken over those α for which αn = 0, α = (α1 , ¨ ¨ ¨ , αn ). m´ 1 ,p By definition, the norm in W p (B), p ą 1 is defined as the norm . For more details see [84, 96, 101]. } ¨ }1 1 m´ p ,p,B 

Let us return to the discussion of elliptic boundary value problems. We first recall some results regarding linear Fredholm operators. Let X and Y be real Banach spaces. By L(X, Y ) we denote the Banach space of bounded linear operators from X to Y . An operator T in L(X, Y ) is called Fredholm if the Ker T = tx P X|T x = 0u has finite dimension and the image of T , R(T ) is of finite codimension in Y , that is codim R(T ) = dim Y {R(T ) ă 8. For a Fredholm operator T : X Ñ Y , the numerical Fredholm index of T , ind(T ) is defined by ind(T ) = dim KerT ´ codim(R(T )). Lemma 1.8 Let  Ă Rn be open and bounded with B smooth. Suppose that s ą n{2, aα P H s () if |α| ď 2m, while bα,i P H s+2m´mi (B) and i = 1, ¨ ¨ ¨ m. Then the following three assertions are equivalent:

22

1 Preliminaries

(i) The operator A = (L, B1 , ¨ ¨ ¨ , Bm ) A : H s+2m () ÝÑ H s () ˆ

m ź

H s+2m´mi (B)

(1.20)

i=1

is an elliptic boundary value problem of order (2m, m1 , ¨ ¨ ¨ , mm ) (ii) The operator A = (L, B1 , ¨ ¨ ¨ Bm ) is Fredholm (iii) There is some c ą 0, such that if u P H s+2m (), then « }u}2m+s ď c }Lu}s +

m ÿ

ff }Bi (x, D)u}2m+s ´mi ´ 1 + }u}s . 2

i=1

(1.21)

Proof If each aα P C s () and each bα,i P C 2k+s ´mi (), then a priori estimate (1.21) is contained in [3]. It is not difficult to see that (1.21) also holds under the present smoothness conditions. Thus, in fact a priori estimate (1.21) and equivalence (i) and (iii) follows from [3]. Equivalence (i) and (iii) to (ii) can be proved analogously to [4]. Remark 1.7 Of course, the Fredholm index of (L, B1 , ¨ ¨ ¨ , Bm ) need not be equal  i ´1 to 0. If L is uniformly elliptic and Bi u(x) = B u(x) for 1 ď i ď m, then the BN index A = (L, B1 , ¨ ¨ ¨ , Bm ) : H 2m+s () Ñ H s () ˆ

m ź

H 2m+s ´mi ´ 2 (B) 1

i=1

is 0. (see [78]). Remark 1.8 (C γ -theory). The a priori estimates (1.21) remain valid if we choose the following B- spaces for 0 ă γ ă 1 : ¯ Y = C s,γ (), ¯ Z = C(), ¯ Yj = C 2m+s ´mi ,γ X = C 2m+s,γ (), i.e. }u}X ď constant(}Lu}Y +

m ÿ

}Bj u}Yj + }u}Z ).

(1.22)

j =1

Remark 1.9 The important fact is that the index of corresponding operators is the same in both theories.

1.3 Nemytskii Operator

23

Remark 1.10 As shown in [3, 4] the terms }u}s and }u}Z in (1.21), (1.22) disappear if dim KerA = t0u, where Au = (Lu, B1 u, ¨ ¨ ¨ , Bm u).

1.3 Nemytskii Operator The investigation of nonlinear equations in the following chapters relies on proper¯ and Lp (), H l (). ties of mappings of the form u ÞÑ f (u) in the spaces C α () Definition 1.8 Let  Ă Rn be a domain. We say that a function  ˆ Rm Q (x, u) ÞÝÑ f (x, u) P R satisfies the Carathéodory conditions if u ÞÝÑ f (x, u) is continuous for almost every x P  and x ÞÝÑ f (x, u) is measurable for every u P . Given any f satisfying the Carathéodory conditions and a function u :  Ñ Rm , we can define another function by composition F (u)(x) := f (x, u(x)).

(1.23)

The composed operator F is called a Nemytskii operator. In this section we state ¯ Lp (), H l () with nonlinear some important results on the composition of C α (), functions (some of them without proof [74, 106]). Proposition 1.4 Let  Ă Rn be a bounded domain and  ˆ Rm Q (x, u) ÞÝÑ f (x, u) P R satisfy the Carathéodory conditions. In addition, let |f (x, u)| ď f0 (x) + c(1 + |u|)r ,

(1.24)

where f0 P Lp0 (), p0 ě 1, and rp0 ď p1 . Then the Nemytskii operator a priori estimates F defined by (1.23) is bounded from Lp1 () into Lp0 (), and }F (u)}0,p0 ď C1 (1 + }u}rp1 ).

(1.25)

24

1 Preliminaries

Proof By (1.24) and (1.2) }F (u)}o,p0 ď }f0 (x)}o,p0 + C}1}o,p0 + C}|u|r }o,p0 ď C1 + C

ż

|u|rp0 dx

1 p0

(1.26) (1.27)



= C 1 + }u}r0,p0 r .

(1.28)

Since  is bounded, then by Hölder’s inequality }v}o,q ď C()}v}o,p when 1 ď q ď p, v P Lp (), 1

where C() = (mes()) q p = p1 imply (1.25).

´ p1

(1.29)

. Inequalities (1.26) and (1.29) with q = rp0 and \ [

It is well-known that the notions of continuity and boundedness of a nonlinear operator are independent of one another [74]. It turns out that the following is valid. Theorem 1.14 Let  Ă Rn be a bounded domain and let  ˆ Rm Q (x, u) ÞÝÑ f (x, u) P R satisfy the Carathéodory conditions. In addition, let p P (1, 8) and g P Lq () (where p1 + q1 = 1) be given, and let f satisfy |f (x, u)| ď C|u|p´1 + g(x). Then the Nemytskii operator F defined by (1.23) is a bounded and continuous map from Lp () to Lq (). For a more detailed treatment, the reader could consult [74, 106]. Theorem 1.15 Let  be a bounded domain in Rn with smooth boundary and let  ˆ R Q (x, u) ÞÑ f (x, u) P R satisfy the Carathéodory conditions. Then f induces 1) a continuous mapping from H s () into H s () if f P C s , 2) a continuously differentiable mapping from H s () into H s () if f P C s+1 , where in both cases s ą n{2. Proof First we consider the simplest case, that is f = f (u) is independent of x. ¯ Hence we have By the Sobolev embedding theorem, we have H s () Ă C(). ¯ we can obtain ¯ for every u P H s (). Moreover, if u is in C (s) (), f (u) P C() the derivatives of f (u) by the chain rule, and in the general case, we can use

1.3 Nemytskii Operator

25

approximation by smooth functions. Note that all derivatives of f (u) have the form of a product involving a derivative of f and derivatives of u. The first factor ¯ while any l-th derivative of u lies in H s ´l (), which imbeds into is in C(), 2n { (n ´ 2(s ´l)) () if s ´ l ă n . L 2 We can use this fact and Hölder’s inequality to show that all derivatives of f (u) up to order s are in L2 (); moreover, it is clear from this argument that f is actually continuous from H s () into H s (). A proof of the differentiability in this special case is that f = f (u) is based on the relation f (u) ´ f (v) =

ż1 0

fu1 (v + θ (u ´ v))(u ´ v)dθ

and the same arguments as before. Let us now consider the general case, that is f = f (x, u). Let |α| ď s. We must show that u ÞÝÑ D α F (u)

(1.30)

defines a continuous map of H s () into L2 (). It is not difficult to see that (1.29) is the finite sum of operators of the form u(x) ÞÝÑ g(x, u(x)) ¨ D γ u(x),

(1.31)

where |γ | = γ1 + ¨ ¨ ¨ + γn ď s, while g is a partial derivative of f order at most s. It is obvious that D γ is continuous from H s () into L2 () for |γ | ď s. On the ¯ implies that other hand, the continuous embedding of H s () in C() u(x) ÞÝÑ g(x, u(x)) ¯ Thus is continuous from H s () into C(). u(x) ÞÝÑ g(x, u(x)) ¨ D γ u(x) defines a continuous map of H s () into L2 () and hence so does u ÞÝÑ D α F (u). For p P N, let p˜ be the number of multi-indices α with |α| ď p. Corollary 1.4 An analogous result is valid for a continuity of the operator F (u)(x) = f (x, u(x), ¨ ¨ ¨ , D p u(x)) : H s+p () Ñ H s (), where p, s P N with s ą

n 2

and f :  ˆ Rp˜ Ñ R is C s .

Corollary 1.5 Let p, s P N with s ą

n 2

and

f :  ˆ Rp˜ Ñ R be C s+1 .

26

1 Preliminaries

Then the operator F : H s+p () Ñ H s () defined by F (u)(x) = f (x, u(x), ¨ ¨ ¨ , D p u(x)) is Fréchet differentiable from H s+p () into H s (). We have the following continuity and C 1 -differentiability results for a nonlinear differential operator of the form Au(x) = f (x, u(x), ..D 2p u(x)) in the Hölder spaces. They are based on Theorems 1.16 and 1.17. Let p P N and p˜ denote as before the number of multi-indices with |α| ď p. Let  be a bounded domain in Rn . ¯ ˆ Rp˜ Theorem 1.16 Let the function f (x, y) = f (x, y1 , ¨ ¨ ¨ , yp˜ ) be defined on  which satisfies the following conditions: 1) f (x, 0) = 0 ˇ 2 ˇ ˇ ˇ 2) For any R ą 0, sup ˇ ByBi Bfyj ˇ ď C(R), sup }f }C 1,α () ¯ ď C(R), where C(R) |y |ďR

|y |ďR

is constant depending on R. ¯ 0 ă α ă 1, }ui }C α () Let u1 (x), ¨ ¨ ¨ , up˜ (x) P C α (), ˜ ¯ ď R, i = 1, . . . , p. Then }f (x, u1 (x), ¨ ¨ ¨ , up˜ (x))}C α () ¯ ď C1 (R) ¨

p˜ ÿ

}ui }C α () ¯ .

i=1

Proof Obviously, f (x, y, . . . , yp˜ ) =

ż1

d f (x, ty1 , . . . , typ˜ )dt 0 dt ż1 p˜ ÿ Bf (x, ty1 , . . . , typ˜ ) = yj dt Byj 0 j =1 =

p˜ ÿ

ϕj (x, y1 , . . . , yp˜ ) ¨ yj ,

j =1

where ϕj (x, y1 , . . . , yp˜ ) =

ż1 0

Bf (x, ty1 , . . . , typ˜ ) dt. Byj

Hence f (x, u1 (x), ¨ ¨ ¨ , up˜ (x)) =

p˜ ÿ j =1

ϕj (x, u1 (x), ¨ ¨ ¨ , up˜ (x)) ¨ uj (x).

(1.32)

1.3 Nemytskii Operator

27

¯ 0 ă α ă 1 is a Banach algebra, we have Since C α (), }f (x, u1 (x), ¨ ¨ ¨ up˜ (x))}C α ď

p˜ ÿ

}ϕj (x, u1 (x), ¨ ¨ ¨ up˜ (x))}C α ¨ }uj }C α .

j =1

Hence we have to prove that sup }ϕj (x, u1 (x), ¨ ¨ ¨ , up˜ (x))}C α ď C1 (R).

|y |ďR

Indeed |ϕj (x + ξ, u1 (x + ξ ), ¨ ¨ ¨ , up˜ (x + ξ )) ´ ϕj (x, u1 (x), ¨ ¨ ¨ , up˜ (x))|

(1.33)

ď|ϕj (x + ξ, u1 (x + ξ ), ¨ ¨ ¨ , up˜ (x + ξ )) ´ ϕj (x, u1 (x + ξ ), ¨ ¨ ¨ , up˜ (x + ξ ))| (1.34) + |ϕj (x, u1 (x + ξ ), ¨ ¨ ¨ , up˜ (x + ξ )) ´ ϕj (x, u1 (x), ¨ ¨ ¨ , up˜ (x))|.

(1.35)

The first term on the right-hand side of (1.35) is bounded by C(R)|ξ |α . The second term is bounded by Bϕj ||ϕj (x, u1 (x + ξ ), ¨ ¨ ¨ , up˜ (x + ξ )) ´ ϕj (x, u1 (x), s, up˜ (x))| |y |ďR Byk sup |

ď C(R)R|ξ |α .

(1.36)

The estimates (1.35) and (1.36) yield (1.32). ¯ ˆ R p˜ Theorem 1.17 Let the function f (x, y) = f (x, y1 , ¨ ¨ ¨ , yp˜ ) be defined on  satisfy the following conditions: 1) f (x, 0) = 0, grady f (x, 0) = 0

B f 2) For any R ą 0, sup }f (x, y)}C 2,α () ¯ ď C(R) and sup | B yi B yj B yk | ď C(R), 3

|y |ďR

|y |ďR

where C(R) is constant depending on R. Let as before, u1 (x), ¨ ¨ ¨ , up˜ (x) P ¯ with }ui }C α () C α () ˜ ¯ ď R, i = 1, ¨ ¨ ¨ , p. Then the following estimate holds. }f (x, u1 (x), ¨ ¨ ¨ , up˜ (x))}C α () ¯ ď C2 (R) ¨

p˜ ÿ i=1

}ui }2C α .

(1.37)

28

1 Preliminaries

Proof Obviously we have p˜ ÿ

f (x, y1 , ¨ ¨ ¨ , up˜ ) =

gij (x, y1 , ¨ ¨ ¨ , yp˜ ) ¨ yi ¨ yj ,

i,j =1

so we can write f (x, u1 (x), ¨ ¨ ¨ , up˜ (x)) =

p˜ ÿ

gij (x, u1 (x), . . . , up˜ (x)) ¨ ui (x) ¨ uj (x)

i,j =1

and we have }f (x, u1 (x), ¨ ¨ ¨ , up˜ (x)}C α () ¯ ď

p˜ ÿ

}gij (x, u1 (x), ¨ ¨ ¨ , up˜ (x)}C α () ¯ ¨ }ui }C α ¨ }uj }C α .

(1.38)

i,j =1

Due to Theorem 1.16, we obtain }gij (x, u1 (x), ¨ ¨ ¨ , up˜ (x)}C α () ¯ ď C0 (R).

(1.39)

Hence the estimates (1.38) and (1.39) yield (1.37) }f (x, u1 (x), ¨ ¨ ¨ , up˜ (x)}C α () ¯ ď C2 (R) ¨

p˜ ÿ

}ui }2C α .

i=1

We apply Theorems 1.16 and 1.17 to the operator Au(x) = f (x, u(x), ¨ ¨ ¨ , D 2p u(x)), where the function f (x, y1 , ¨ ¨ ¨ , yp˜ ) satisfy conditions of Theorems 1.16 and 1.17, respectively. Hence we have }Au}C 2p,α ď C(R) ¨ }u}C α . Moreover, as it follows from Theorem 1.17, A P C 1 , A1 (0) = 0 and }A1 (u + h) ´ A1 (u)}L(C 2p,α ,C α ) ď C ¨ }h}C 2p,α () ¯ . Remark 1.11 As shown in the proofs of Theorems 1.16 and 1.17, continuity and differentiability of the operator Au(x) = f (x, u(x), ¨ ¨ ¨ , D 2p u(x)) between ¯ and C α () ¯ remains valid under slightly weaker conditions on a given C 2p,α () function f (x, y1 , ¨ ¨ ¨ , yp˜ ). We leave these as exercises for the reader.

1.3 Nemytskii Operator

29

Below we present the properties of the Nemytskii operators in the spaces H s (S 1 ) or C p,α (S 1 ), where S 1 is the unit circle. We recall some of the properties which will be used often in the sequel. The norm in C α (M) is given by |f (x) ´ f (y)| , |x ´ y|α x ‰y

}f }C α (M) = }f }C + sup

M = S1.

As before, by C k,α (M) we denote the space of Hölder continuous functions, which have derivatives up to order k, with D k f P C α (M). Let F be a superposition operator defined by F (u)(x) = f (x, u(x)),

x P M.

The following theorems are not hard to prove (although not obvious). Theorem 1.18 Let k P R+. Then the superposition operator F : E1 Ñ E2 defined by F (u)(x) = f (x, u(x)) acts as a bounded operator in each of the following cases (see also [48]). 1) 2)

f P C(S 1 ˆ R, R), E1 = C(S 1 ), E2 = C(S 1 ) f P C 1 (S 1 ˆ R, R), E1 = C α (S 1 ), E2 = C α (S 1 ), 0 ă α ă 1.

Theorem 1.19 Let k P R+, 0 ă α ă 1. Then the superposition operator F : E1 Ñ E2 defined by F (u)(x) = f (x, u(x)) is m times continuously differentiable if one of the following cases 1) 2)

D 0,j f P C k (S 1 ˆ R, R), E1 = C k (S 1 ), E2 = C k (S 1 ) D 0,j f P C k+1 (S 1 ˆ R, R), E1 = C k,α (S 1 ), E2 = C k,α (S 1 ),

The j -th derivative of F is given by D 0,j F (x, u(x))h1 (x) . . . hj (x) = D j F (f )(h1 , . . . hj )(x). Analogous results are valid in Sobolev spaces: Theorem 1.20 Let X = Y = H s (S 1 )(s ě 1) be the Sobolev space of real functions x(τ ) on the circumference of a circle, where 0 ď τ ă 2π ; f (τ, x) is a smooth real function, x P R, 0 ď τ ă 2π . Then the operator F : H s (S 1 ) Ñ H s (S 1 ) defined by F x(τ ) = f (τ, x(τ )) is continuous. Proof It is not difficult to see, that

d dτ

k f (τ, x(τ )) =

sup

p+q ďk r1 +...+rq =k ´p rj ě0

Cp,q,r1 ...rq

B p+q f (τ, x(τ )) (r1 ) x (τ ) . . . x (rq ) (τ ), Bτ p . . . Bx q

30

1 Preliminaries

where Cp,q,r1 ...rq are some constants. If x(τ ) P H s , then it follows that the ! l ) x(τ ) ds derivatives d dτ are continuous. Therefore in dτ s f (τ, x(τ )) all l |0 ď l ď s ´ 1 s

x(τ ) terms without ones are continuous. The last term is equal to d dτ ˆ Q(τ ) where s Q(τ ) is a continuous function, hence also square integrable. As a consequence of these arguments we obtain continuity.

Remark 1.12 An analogous result holds for vector functions, and also in the multidimensional case, for functions on arbitrary smooth compact manifold with boundary. The following Lemma on the smoothness relations between u and f (u) plays a decisive role in many applications (see [47]). Lemma 1.9 Let the function f P C 2 (R, R) satisfies C1 |u|p´1 ď f 1 (u) ď C1 |u|p´1 , p ą 1, with C1 and C2 some positive constants. Then, for every s P (0, 1) and 1 ă q ď 8, we have 1{p

}u}W s{p,pq () ď Cp }f (u)}W s,q () where the constant Cp is independent of u. Proof Indeed, let f ´1 be the inverse function to f . Then, due to conditions on f , the function G(v) := sgn(v)|f ´1 (v)|p is nondegenerate and satisfies C2 ď G1 (v) ď C1 , for some positive constants C1 and C2 . Therefore, we have |f ´1 (v1 ) ´ f ´1 (v2 )|p ď Cp |G(v1 ) ´ G(v2 )| ď Cp1 |v1 ´ v2 |, for all v1 , v2 P R. Finally, according to the definition of the fractional Sobolev spaces (see e.g. [96, 101]), }f

´1

pq (v)}W s{p,qp ()

:=}f

´1

pq (v)}Lpq ()

ďC}v}Lq () + Cp1 q

+

ż ż  

ż ż

 

|f ´1 (v(x)) ´ f ´1 (v(y)|pq dx dy |x ´ y|n+sq

|v(x) ´ v(y)|q dx dy |x ´ y|n+sq

q =Cp2 }v}W s,q () ,

where we have implicitly used that f ´1 (v) „ sgn(v)|v|1{p . Lemma 1.9 is proved. \ [

1.4 Maximum Principles and Their Applications

31

1.4 Maximum Principles and Their Applications The maximum principle is one of the most useful and best known tools employed in the study of partial differential equations. Indeed, the maximum principle enables us to obtain information about solutions of differential equations and inequalities without any explicit knowledge of the solutions themselves, and thus can be a valuable tool in scientific research. The intention of Sects. 1.4–1.7 is, on one hand, to survey some extension and applications of the classical maximum principles for elliptic operators and, on the other hand, to prove some new explicit and uniform bounds that ensure the boundedness and asymptotic dissipation of the solutions of semilinear elliptic equations in bounded (unbounded) domains based on the maximum principles. These type of explicit and uniform bounds for the solutions are crucial to the construction of the attractive basin of the trajectories in the dynamical systems approach developed in the subsequent chapters. In Sects. 1.4, 1.6, and 1.7 we mainly follow [97]. We consider first the one-dimensional case. If u2 ě 0 on (a, b) and u is continuous on [a, b], then (a) u attains its largest value either at a or at b. (b) If u attains its largest value at c P (a, b), then u is constant on [a, b]. (c) If a non-constant u attains its maximum at b, then u1´ (b) ą 0, if at a, then u1+ (a) ă 0. (d) u is convex on [a, b]. The one-side derivatives in (c) are assumed to exist. If, for example, u1+ (a) fails to exist, then, instead of u1+ (a) ă 0, one has D + u(a) = lim sup ă 0. x Óa

It is common to refer to all or any of the statements (a)–(d) as maximum principle. We shall call (a), (b), (c), (d) the weak maximum principle, the strong maximum principle, the boundary point lemma and the convexity theorem, respectively. Each of the statements (a)–(d) can be easily verified directly. Let us check (a). Assume, for a direct proof, that there is a point c P (a, b) such that u(c) ą u(a) and u(c) ą u(b). Choose an ε ą 0 sufficiently small so that u(c) ą u(a) + ε(c ´ a)2 and u(c) ą u(b) + ε(b ´ a)2 and define a function v by v(x) = u(x) + ε(x ´ a)2 . By the Weierstrass theorem, v attains its maximum in [a, b], and, since v(c) ą v(a) v(c) ą v(b), v attains its maximum at some ξ P (a, b). Therefore, v 1 (ξ ) = 0 and v 2 (ξ ) ď 0. This, however, is a contradiction to v 2 (ξ ) = u2 (ξ ) + 2ε ą 0, and (a) is proved. There is another way to prove (a)–(d). First, (d) is a well-known result from calculus. Hence, its suffices to prove the implication sequence (d) ñ (c) ñ (b) ñ (a). Let us check (d) ñ (c). Without loss of generality, we may assume that

32

1 Preliminaries

u1´ (b) = 0. Then, by convexity, the graph of u lies above the tangent at b, which means that u(x) ą u(b) and u must be constant. It is interesting to note that, in some sense, (a) ñ (d), namely: if for every γ the function x Ñ u(x) ´ γ x satisfies the weak maximum principle on every interval [α, β] Ă [a, b], then u is convex on [a, b]. As we will see below, the maximum principles also hold for a wide class of general second order elliptic partial differential equations. Indeed, let  be an open connected set in Rn with boundary B =  X (Rn z) Let L be the second order differential operator: n ÿ

L=

i,j =1

aij (x)Dij +

n ÿ

bi (x)Di + c(x)

i=1

B 8 with aij P L8 loc () and bi , c P L (). Here we have used Di = B xi and Dij = B B B xi B xj . Without loss of generality one assumes aij = aj i .

Definition 1.9 We will fix the following notions. • The operator L is called elliptic on  if for every x P  there is λ(x) ą 0 such that n ÿ

aij (x)ξi ξj ě λ(x)|ξ |2 for all ξ P Rn .

i,j =1

• The operator L is called strictly elliptic on  if there is λ ą 0 such that n ÿ

aij (x)ξi ξj ě λ|ξ |2 for all ξ P Rn and x P .

i,j =1

• The operator L is called uniformly elliptic on  if there are , λ ą 0 such that λ|ξ |2 ď

n ÿ

aij (x)ξi ξj ď |ξ |2 for all ξ P Rn and x P .

i,j =1

Remark 1.13 These definitions are not uniform throughout the literature. However, if the aij are bounded on  then strictly elliptic implies uniformly elliptic and most references then agree (see Sect. 1.2). Remark 1.14 The assumption aij P L8 loc () is too weak to expect even solutions of Lu = f P C 8 () to satisfy u P C 2 () and for that reason one usually assumes aij to be more regular. The maximum principle however does not need aij to be continuous. Some notations that we will use are as follows. For r ą 0 and y P Rn we will write an open ball by

1.4 Maximum Principles and Their Applications

33

Br (y) = tx P Rn : |x ´ y| ă ru. For a function u we will use u+ , u´ which are defined by u+ (x) = max(0, u(x)), u´ (x) = max(0, ´u(x)). It is obvious that u = u+ ´ u´ and |u| = u+ + u´ .

1.4.1 Classical Maximum Principles Lemma 1.10 Suppose that L is elliptic and that c ď 0. If u P C 2 () and Lu ą 0 in , then u cannot attain a nonnegative maximum in . A proof can be done with a contradiction argument. We leave it to the reader. Theorem 1.21 (Weak Maximum Principle) Suppose that  is bounded and that L is strictly elliptic with c ď 0. If u P C 2 () X C() and Lu ě 0 in , then a nonnegative maximum is attained at the boundary. Proof Suppose that  Ă t|x1 | ă du. Consider w(x) = u(x) + εeαx1 with ε ą 0. Then Lw =Lu + ε(α 2 a11 (x) + αb1 (x) + c(x))eαx1 ěε(α 2 λ + α}b1 }8 + }c}8 )eαx1 One chooses α large enough to find Lw ą 0. By the previous lemma w cannot have a nonnegative maximum in . Hence sup u ď sup w ď sup w + = sup w + ď sup u+ + εeαd 





B

if  Ă t|x| ă du. The result follows for ε Ñ 0.

B

\ [

The proof of this maximum principle uses local arguments. If we skip the assumption that  is bounded we obtain: Corollary 1.6 Suppose that L is strictly elliptic with c ď 0. If u P C 2 () X C() and Lu ě 0 in , then u cannot attain a strict1 nonnegative maximum in . Theorem 1.22 (Strong Maximum Principle) Suppose that L is strictly elliptic and that c ď 0. If u P C 2 () X C() and Lu ě 0 in , then either u ” sup u or u does not attain a nonnegative maximum in .

34

1 Preliminaries

Proof Let m = sup u. and set  = tx P ; u(x) = mu. We are done if  P t, Hu. Arguing by contradiction we assume that  and z are non-empty. The argument proceeds in three steps. First one fixes an appropriate open ball and in the next step an auxiliary function is defined that is positive on and only on this ball. For the sum of u and this auxiliary function one obtains a contradiction on a second ball by the weak maximum principle. For more details we refer to [97]. [ \ Clearly, the weak maximum principle is a consequence of the strong one. Corollary 1.7 (Positivity Preserving Property) Let  be bounded and suppose that L is strictly elliptic with c ď 0. If u P C 2 () X C() satisfies "

´Lu ě 0 uě0

in , on B,

(1.40)

then either u(x) ą 0 for x P  or u ” 0. Remark 1.15 Let f P C() and ϕ P C(B). A function w P C 2 () X C() satisfying "

´Lw ě f wěϕ

in , on B,

(1.41)

´Lu = f u=ϕ

in , on B.

(1.42)

is called a supersolution for "

A much more useful concept of supersolutions assumes that w P C() and replaces ´Lw ě f by ż 

[(´Lϕ)w ´ ϕf ] dx ě 0 for all ϕ P C08 () with ϕ ě 0.

If the maximum principle holds then one finds that a supersolution for (1.42) lies above a solution for (1.42); w ě u. In particular, since solutions are also supersolutions, if there are two solutions u1 and u2 then both u1 ě u2 and u2 ě u1 hold true. In other words, (1.42) has at most one solution in C 2 () X C(). Assuming more for B one obtains an even stronger conclusion, that is, E. Hopf’s result in 1952. Theorem 1.23 (Hopf’s Boundary Point Lemma) Suppose that  satisfies the interior sphere condition, that is, there is a ball B Ă  with x0 P BB at x0 P B. Let L be strictly elliptic with c ď 0: If u P C 2 () X C() satisfies Lu ě 0 and max u(x) = u(x0 ). Then either u ” u(x0 ) on  or

1.4 Maximum Principles and Their Applications

lim inf t Ó0

35

u(x0 ) ´ u(x0 + tν) ą 0 (possibly + 8) t

for every direction ν pointing into an interior sphere. 0) If u P C 1 ( Y tx0 u), then either Bu(x B ν ă 0. There are two directions in order to weaken the restriction on c. Skipping the sign condition for c but adding one for u one obtains the next result. Theorem 1.24 (Maximum Principle for Nonpositive Functions) Let  be bounded. Suppose that L is strictly elliptic (no sign assumption on c). If u P C 2 () X C() satisfies Lu ě 0 and in  and u ď 0 on , then either u(x) ă 0 for all x P , or u ” 0. Moreover, if  satisfies an interior sphere condition at x0 P B and u P C 1 ( Y tx0 u) with u ă u(x0 ) = 0 in , then B u(x0 ) B ν ă 0 for every direction ν pointing into an interior sphere. Proof Writing c(x) = c+ (x) ´ c´ (x) with c+ , c´ ě 0 one finds that L ´ c+ satisfies the condition of the S.M.P. and moreover from u ď 0 it follows that (L ´ c+ )u ě ´c+ u ě 0. The conclusion for the derivative follows from Theorem 1.23.

\ [

Theorem 1.25 (Maximum Principle When a Positive Supersolution Exists) Let  be bounded. Suppose that L is strictly elliptic (no sign assumption on c) and that there exists w P C 2 () with w ą 0 and ´Lw ě 0 on . If u P C 2 () X C() satisfies Lu ě 0 in , then either there exists a constant t P R such that u ” tw, or u{w does not attain a nonnegative maximum in . Remark 1.16 One may rephrase this for supersolutions as follows. If there exists one function w P C 2 () with ´Lw ě 0 and w ą 0 on  then all functions v P C 2 () X C() such that "

´Lv ě 0 vě0

in , on B,

(1.43)

satisfy either v ” 0 or v ą 0 in . Apply the theorem to ´v. Remark 1.17 If  Ă 1 and if L is defined on 1 and happens to have a positive eigenfunction ϕ with eigenvalue λ1 for the Dirichlet problem on 1 : $ & ´Lϕ = λ1 ϕ ϕ=0 % ϕą0

in 1 , on B1 , in 1 ,

(1.44)

then this w = ϕ may serve in the theorem above. It shows that a maximum principle holds on  for c ă λ1 .

36

1 Preliminaries

Proof Set v = u{w. Then with b˜i = bi + Lu = L(vw) =

ÿ n

řn

aij Dij v +

i,j =1

2aij j =1 w

Dj w and c˜ =

Lw w

one has

Lw v w. b˜i Di v + w i=1 n ÿ

(1.45)

˜ ď 0 with L˜ as in (1.45) and since this L˜ does satisfy Then Lu ď 0 implies Lv the conditions of the Strong Maximum Principle, in particular the sign condition for c˜ = Lw w , one finds that either v is constant or that v does not attain a nonnegative maximum in . \ [ Theorem 1.26 (Maximum Principle for Narrow Domains) Suppose that L is strictly elliptic (no sign assumption on c). Then there is d ą 0 such that if S Ă tx P ; |x| ă du and u P C 2 (S) X C(S) satisfies Lu ě 0 in S, then there exists w P C 2 (S) with w ą 0 and ´Lw ě 0 on S. Proof For w(x) = cos(αx1 ) and |x1 | ď

π 4α

we have w(x) ą 0 and

´Lw = (α 2 a11 ´ c) cos(αx1 ) + αb1 sin(αx1 ) ě (α 2 λ ´ α}b1 }8 ´ }c}8 )

1? 2. 2

The claim follows by taking α large enough and defining d =

π 4α .

\ [

1.5 Uniform Estimates and Boundedness of the Solutions of Semilinear Elliptic Equations Here we obtain explicit and uniform bounds that ensure the boundedness and the asymptotic dissipation of the solutions of semilinear elliptic equations in bounded (unbounded domains). These estimates are crucial to construct the attractive basin of the trajectories in the dynamical system approach (see Chaps. 3–5). First, we give a general boundary maximum principle for non-negative subsolutions: Lemma 1.11 Let  be an open subset of RN and v P C 2 () X C 0 () be a solution of $ &v ě Av p in , vě0 in , % v = v¯ on B,

1.5 Uniform Estimates and Boundedness of the Solutions of Semilinear. . .

37

with A ą 0, p ą 1 and v¯ P L8 (B). Then, for any x P , v(x) ď }v} ¯ L8 (B) . Proof Fix any S ą }v} ¯ L8 (B) .

(1.46)

Let C ą 0 (to be conveniently chosen in what follows), α := 2{(p ´ 1) and R := (S ´1 A´1{(p´1) C)1{α . For any x P B(0, R), we define w(x) :=

A´1{(p´1) CR α . (R 2 ´ |x|2 )α

Of course, S = A´1{(p´1) CR ´α = w(0) ď w(x) for any x P B(0, R),

(1.47)

and lim w(x) = +8.

|x |ÑR ´

(1.48)

Moreover, by a direct computation one sees that there exists Co (N, p) ą 0 such that, if C ě Co (N, p), ´ w + Aw p ě 0 in B(0, R).

(1.49)

v(x0 ) ď S for any x0 P .

(1.50)

We claim that

To prove this, we define ϑx0 (x) := w(x ´ x0 ) and we compare v with ϑx0 , by distinguishing two cases: either B(x0 , R) Ă  or not. If B(x0 , R) Ă , we have that v P C 0 (B(x0 , R)) Ă L8 (B(x0 , R)), and so, by (1.48), there exists R¯ P (0, R) such that ¯ v(x) ď ϑx0 (x) for any x P BB(x0 , R).

(1.51)

On the other hand, if we set c(x) := Ap

ż1 0

  p ´1 tv(x) + (1 ´ t)ϑx0 (x) dt

and z(x) := v(x) ´ ϑx0 (x),

(1.52)

38

1 Preliminaries

¯ and we have that z P C 2 (B(x0 , R)) p

z = (v ´ ϑx0 ) ě A(v p ´ ϑx0 ) = cz.

(1.53)

¯ thanks to (1.51). Hence, by the maxiumum principle, z ď Also, z ď 0 on BB(x0 , R), ¯ 0 in B(x0 , R). In particular, v(x0 ) = z(x0 ) + ϑx0 (x0 ) ď 0 + w(0) = S.

(1.54)

This proves (1.50) when B(x0 , R) Ă . On the other hand, in B(x0 , R) Ć , we set K := B(x0 , R) X . We have that v P C 0 (K) Ă L8 (K), and so, by (1.48), there exists R˜ P (0, R) such that   ˜ X . v(x) ď ϑx0 (x) for any x P BB(x0 , R)

(1.55)

Moreover, if x P B(x0 , R) X (B), we have that v(x) = v(x) ¯ ď }v} ¯ L8 (B) ă S ď w(x ´ x0 ) = ϑx0 (x), thanks to (1.46) and (1.47). This observation and (1.55) give that z ď 0 on BK, where z is defined in (1.52). Also, z is C 2 in the interior of K and continuous up to the boundary, and (1.53) holds true in the interior of K. Accordingly, the maximum principle yields that z ď 0 in K, and therefore, by arguing as in (1.54), we conclude that (1.50) holds true also when B(x0 , R) Ć . Since, by (1.46), S can be taken arbitrarily close to }v} ¯ L8 (B) , the desired result follows from (1.50). \ [ Corollary 1.8 Let r ą 2, c1 ě 0, c2 ą 0. Let  be an open subset of Rn . Let f : R Ñ R be a measurable function such that f (s)s ě ´c1 + c2 s r for any s ě 0.

(1.56)

Let u P C 2 () X C 0 () be a solution of $ &u ě f (u) in , uě0 in , % u = u¯ on B, with u¯ P L8 (B). Then u P L8 () and, for any x P , # 0 ď u(x) ď max

c1 c2

1{r

+ , }u} ¯ L8 (B) .

(1.57)

1.5 Uniform Estimates and Boundedness of the Solutions of Semilinear. . .

39

Proof Let p := r{2 ą 1 and v := u2 . Then v = 2uu + 2|∇u|2 ě 2uf (u) + 0

(1.58)

ě ´2c1 + 2c2 ur = ´2c1 + 2c2 v p in . Now we fix  P (0, 2c2 ), to be taken arbitrarily small in the sequel. Let  := tx P  s.t. 2c2 v p (x) ą 2c1 + v p (x)u. We claim that, for any x P , # 0 ď u(x) ď max

2c1 2c2 ´ 

1{r

+ , }u} ¯ L8 (B) .

(1.59)

To establish it, we notice that if x P z , then u(x) =

a

v(x) ď

2c1 2c2 ´ 

1{(2p)

=

2c1 2c2 ´ 

1{r

(1.60) .

If  = , then (1.59) follows from (1.60), so we may suppose that  is a nonempty open subset of RN , with v P C 2 ( ) X C 0 ( ). Also, by virtue of (1.58), v ě ´2c1 + 2c2 v p ě v p in  .

(1.61)

Now, we claim that # if x P B , then 0 ď v(x) ď max

2c1 2c2 ´ 

+

1{p ,

}u} ¯ 2L8 (B)

.

(1.62)

Indeed, if x P (B ) X , we have that 2c2 v p (x) = 2c1 + v p (x), which implies (1.62). If, on the other hand, x P (B )z, then x P B and so v(x) = u(x) ¯ 2 , which gives (1.62) in this case. Exploiting (1.61) and Lemma 1.11, we conclude that v in  is controlled by }v}L8 (B ) , and so, by (1.62), # v(x) ď max

2c1 2c2 ´ 

+

1{p ,

}u} ¯ 2L8 (B)

40

1 Preliminaries

for any x P  . This and (1.60) prove (1.59), which, in turn, proves (1.57) by taking  as close to 0 as we wish. [ \ We remark that the condition r ą 2 in Corollary 1.8 cannot be removed. As x1 a countarexample, take  := RN + , f (r) = r and u(x) = u(x1 , . . . , xN ) := e . Then u = 1 on B and (1.56) holds true with r = 2 (instead of r ą 2), but u is unbounded. The forthcoming Theorems 1.27 and 1.28 give some a priori bounds on subsolutions: namely assuming that the solution has some bound, then the bound may be made explicit and universal. N 0 Theorem 1.27 Let N ě 2, and u P C 2 (RN ++ ) X C (R++ ) be a solution of

$ u ěˇ f (u) in RN ’ ++ , ’ & ˇ uˇ = u0 , ˇx1 =0 ’ ’ ˇ % uˇ = 0. xN =0

Suppose that u0 P L8 (RN ++ X tx1 = 0u) and that f satisfies f (s) ď 0 for any s P [0, μ] and f (s) ě 0 for any s P [μ, +8), for a suitable μ ą 0. Then, if u is bounded from above, we have that u(x) ď max tμ, }u0 }L8 (RN Xtx =0u) u 1 ++

for any x P RN ++ .

(1.63)

Proof Let M := max tμ, }u0 }L8 (RN Xtx =0u) u and v := u ´ M. 1 ++ We see that v solves $ v ě f (u) in RN ’ ++ , ’ & ˇˇ vˇ = u0 ´ M ď 0, xˇ1 =0 ’ ’ % v ˇˇ = ´M ď 0. xN =0

Also, v is bounded from above if so is u. We claim that tx P RN ++ s.t. v(x) ą 0u = ∅.

(1.64)

1.5 Uniform Estimates and Boundedness of the Solutions of Semilinear. . .

The proof of (1.64) is by contradiction: if not, let o of the set in (1.64): then v ě f (u) ě 0 in o and

41

beˇ a connected component ˇ vˇ ď 0. The maximum B o

principle then gives that v ď 0 in o , while o Ď tv ą 0u. This is a contradiction and so (1.64) is established. In turn, (1.64) implies (1.63). \ [ N 0 Theorem 1.28 Let N ě 2, and u P C 2 (RN ++ ) X C (R++ ) be a solution of

$ u ěˇ f (u) in RN ’ ++ , ’ & ˇ uˇ = u0 , ˇx1 =0 ’ ’ ˇ % uˇ = 0. xN =0

Suppose that u0 P L8 (RN ++ X tx1 = 0u) and that f satisfies f (s)s ě ´c1 + c2 s r for any s ě 0 for suitable c1 ě 0, c2 ą 0 and r ě 2, and that u ě 0 in RN ++ . Then, if u P L8 (RN ), we have that ++ u(x) ď

c1 c2

1{r

+ }u0 }L8 (RN Xtx =0u) e´γ x1 1 ++

for any x P RN ++ ,

(1.65)

where

γ :=

g f f c2 }u0 }r ´2 e L8 (RN ++ Xtx1 =0u) 2

.

Proof We define w := u2 ´ }u0 }2 8

L (RN ++ Xtx1 =0u)

e´2γ x1 ´



c1 c2

2{r .

By a direct computation, one sees that w = 2uu + 2|∇u|2 ´ 4}u0 }2 8

L (RN ++ Xtx1 =0u)

ě 2uf (u) ´ 4}u0 }2 8

L (RN ++ Xtx1 =0u)

ě 2c2 ur ´ 2c1 ´ 4}u0 }2 8

γ 2 e´2γ x1

γ 2 e´2γ x1

L (RN ++ Xtx1 =0u)

γ 2 e´2γ x1

(1.66)

42

1 Preliminaries

in RN ++ . Moreover

c1 2{r = ´ ď0 x1 =0 c2 2{r ˇ c1 ˇ and w ˇ = ´}u0 }2 8 N e´2γ x1 ´ ď 0. L (R++ Xtx1 =0u) xN =0 c2

ˇ ˇ wˇ

u20

´ }u0 }2 8 N L (R++ Xtx1 =0u)

(1.67)

We claim that tx P RN ++ s.t. w(x) ą 0u = ∅.

(1.68)

The proof is by contradiction. Indeed, if tw ą 0u ­= ∅, for any point x P tw ą 0u we have that u2 (x) ą }u0 }2 8

L (RN ++ Xtx1 =0u)

e´2γ x1 +



c1 c2

2{r

and so  u (x) ą r

}u0 }2 8 N e´2γ x1 L (R++ Xtx1 =0u)

+

c1 c2

2{r r {2 .

(1.69)

Since r{2 ě 1, this gives that ur (x) ą }u0 }r 8

L (RN ++ Xtx1 =0u)

e´rγ x1 +

c1 . c2

(1.70)

That is, for any x P tw ą 0u, 2c2 ur (x) ´ 2c1 ą 2c2 }u0 }r 8

L (RN ++ Xtx1 =0u)

= 4}u0 }2 8

L (RN ++ Xtx1 =0u)

e´rγ x1

γ 2 e´rγ x1 .

Thus, recalling (1.66), we infer that w ě 0 in tw ą 0u. Also, recalling (1.67), we see that w ď 0 on Btw ą 0u. Hence, since w is bounded, the maximum principle implies that w ď 0 on tw ą 0u. This is a contradiction and so (1.68) is established. The estimate in (1.65) then easily follows from (1.68). \ [ We remark that the condition r ě 2 was used, in the proof of Theorem 1.28, to pass from (1.69) to (1.70). On the other hand, the case r P (0, 2) may also be treated by using instead the formula (a r {2 + br {2 ) ď 21´(r {2) (a + b)r {2

for any r P (0, 2) and a, b ě 0.

1.6 The Sweeping Principle and the Moving Plane Method in a Bounded Domain

43

Also, it is interesting to notice that Theorem 1.28 provides universal bounds for any solution, without any sign condition on f at the origin. The following Theorem 1.29 will be used in the study of the symmetry and monotonicity properties of positive solutions of a semilinear elliptic equations. Theorem 1.29 (Maximum Principle for Small Domains) Suppose that  is bounded and that L is strictly elliptic (without sign condition for c). Then there exists a constant δ, with δ = δ(n, diam(), λ, }b}Ln () , }c+ }8 ), such that the following holds. If || ă δ and u P C 2 () X C() satisfies Lu ě 0 in  and u ď 0 on B, then u ď 0 in . Proof The operator L ´ c+ satisfies the condition of Theorem 5.5 from [97] and from ´Lu ď 0 it follows that (L ´ c+ )u ě ´c+ u ě ´c+ u+ and hence, since supB u+ = 0, sup u ď 

ď

C diam  + + }c u }Ln () λ 1 C diam  + }c }8 || n sup u+ . λ 

 + If δ = ( C diam }c }8 )´n one finds sup u ď 0. Here we define λ

› ›

2n´2 ›› |b| ›› + 1 , D˚ (x) = (det(aij (x)))1{n , C = exp σn nn › D˚ ›Ln ( + X+ )

which depends on }b}Ln () .

(1.71) \ [

1.6 The Sweeping Principle and the Moving Plane Method in a Bounded Domain Using comparison principles for connected families of sub- and supersolutions one obtains a very powerful tool in deriving a priori estimates. One such result is the moving plane method used by Gidas, Ni and Nirenberg [65] to prove symmetry of positive solutions to " ´u = f (u) in , (1.72) u=0 on B, on domains  satisfying some symmetry conditions. Another one is McNabb’s sweeping principle [80] used to its full extend by Serrin. First let us recall the notion of supersolution for (1.72).

44

1 Preliminaries

Definition 1.10 A function v P C() X C 2 () is called a supersolution for "

´u = f (u) u=h

in , on B,

(1.73)

if v ě h on B and ´v ě f (v). Remark 1.18 A much more useful concept of supersolution assumes v P C() and replaces ´v ě f (v) by ż

((´φ)v ´ φf (v)) dx ě 0 

for all φ P C08 () with φ ě 0. The argument combining the strong maximum principle and continuous perturbations goes as follows. One needs: • a continuous family of supersolutions vt , say t P [0, 1], possibly on a subdomain or on an appropriate family of subdomains t ; • a strong maximum principle on each of the subdomains. Roughly spoken the conclusion is that if v0 ą u on t , then either vt ą u for all t P [0, 1] or there is a t1 P [0, 1] such that vt1 ” u. For a precise statement we have to refine the notion of positivity. For a fixed domain such a result is known as a ‘sweeping principle’. A first reference to this result is a paper of McNabb from 1961, see [80]. In the following version we assume that B = 1 Y 2 , with 1 , 2 closed and disjoint, possibly empty. We let e P C() X C 1 ( Y 2 ) be such that e ą 0 on  Y 1 and BBn e ă 0 on 2 , where n denotes the outward normal. Moreover, we define Ce () =tw P C(); |w| ă ce for some c ą 0u, › › ›w› }w}e =›› ›› . e 8 Theorem 1.30 (Sweeping Principle) Let  be bounded with B P C 2 and B = 1 Y 2 as above, f P C 1 (R) and let u P C() X C 1 ( Y 2 ) X C 2 () be a solution of (1.73). Suppose tvt ; t P [0, 1]u is a family of supersolutions in C() X C 1 ( Y 2 ) X C 2 () for (1.73) such that: 1. t ÞÑ (vt ´ v0 ) P Ce () is continuous (with respect to the } ¨ }e -norm); 2. vt = u on 2 and vt ą u on 1 ; 3. v0 ě u in ;

1.6 The Sweeping Principle and the Moving Plane Method in a Bounded Domain

45

Then either vt ” u for some t P [0, 1], or there exists c ą 0 such that vt ě u + ce on  for all t P [0, 1]. Proof Set I = tt P [0, 1]; vt ě u in u and assume that vt ı u for all t P [0, 1]. By the way, notice that vt ” u for some t P [0, 1] can only occur when 1 is empty. The set I is nonempty by assumption 3 and closed by assumption 1. We will show that I is open. Indeed, if t is such that vt ě u in  then ´(vt ´ u) = f (vt ) ´ f (u) and setting g as (see also [97]) # g(x) := one obtains "

f (x,u2 (x))´f (x,u1 (x)) u2 (x)´u1 (x) Bf (x, u2 (x)) Bu

if u2 (x) ‰ u1 (x), if u2 (x) = u1 (x),

´(vt ´ u) + g ´ (vt ´ u) ě g + (vt ´ u) ě 0 vt ´ u ě 0

in , on B,

and hence by the strong maximum principle vt ´ u ą 0 in  or vt ” u. Moreover, for the first case Hopf’s boundary point Lemma and the assumption that u, vt P C 1 ( Y 2 ) imply that ´ BBn (vt ´ u) ą 0 on 2 . Hence there is ct ą 0 such that vt ´u ě ct e on . By assumption 1 it follows that t lies in the interior of I . Hence I is open, implying I = [0, 1]. Moreover, since I is compact a uniform c ą 0 exists. \ [ Example 1.1 Suppose that f P C 1 is such that f (u) ą 0 for u ă 1 and f (u) ă 0 for u ą 1. Using the above version of the sweeping principle one may show that for λ " 1 every positive solution uλ of "

´u = λf (u) u=0

in , on B,

(1.74)

is near 1 in the interior of . Indeed, let ϕ1 , μ1 denote the first eigenfunction/eigenvalue of ´ϕ = μϕ in B1 and ϕ = 0 on BB1 , where B1 is the unit ball in Rn . This first eigenfunction is radially symmetric and we assume it to be normalized such that ϕ(0) = 1. Now let ε ą 0 and take δε ą 0 such that δε u ď f (u) for u P [0, 1 ´ ε]. ?

Set vt (x) = tϕ1 ( ?λδμε (x ´ x ˚ )) for any x ˚ P  with distance to the boundary ?

μ d(x ˚ , B) ą ?λδ =: r0 . On Br0 (x ˚ ) one finds for t P [0, 1 ´ ε] that ´vt = ε

λδε vt ď λf (vt ). Since vt = 0 ă uλ on BBr0 (x ˚ ) and v0 = 0 ă uλ on Br0 (x ˚ ) it is an appropriate family of subsolutions. From v1´ε ă uλ in Br0 (x ˚ ) one concludes that 1 ´ ε = v1´ε (x ˚ ) ă uλ (x ˚ ). Since the strong maximum principle implies that max uλ ă 1 we may summarize:

46

1 Preliminaries

? μ 1 ´ ε ă uλ (x) ă 1 for x P  with d(x , B) ą ? . λδε ˚

More intricate results not only consider a family of supersolutions but also simultaneously modify the domain. One such result is the result by Gidas, Ni and Nirenberg ([65] or [64]). The idea was used earlier by Serrin in [93]. Before stating the result let us first fix some notations. We will be moving planes in the x1 -direction but of course also any other direction will do. Some sets that will be used are: the moving plane: the subdomain: the reflected point: the reflected subdomain: the starting value for λ : the maximum value for λ :

Tλ = tx P Rn ; x1 = λu, λ = tx P ; x1 ă λu, xλ = (2λ ´ x1 , x2 , . . . , xn ), λ1 = txλ ; x P λ u, λ0 = inftx1 ; x P u, λ˚ = suptλ; μ1 Ă  for all μ ă λu.

Theorem 1.31 (Moving Plane Argument) Assume that f is Lipschitz, that  is bounded and that λ0 , λ˚ and λ are as above. If u P C() X C 2 () satisfies (1.72) and u ą 0 in , then u(x) ă u(xλ ) for all λ ă λ˚ and x P λ , Bu B x1 (x)

ą0

for all x P λ .

Remark 1.19 We will use the moving plane arguments, sweeping principle and sliding techniques (see subsequent sections) in Chaps. 3–7 to study nonlinear elliptic boundary value problems in unbounded domains from the dynamical systems viewpoint.

1.6 The Sweeping Principle and the Moving Plane Method in a Bounded Domain

47

Proof First remark that if u ” 0 on some open set in , then f (0) = 0. If f (0) = 0 then u satisfies ´u = c(x)u(x) with c(x) = f (u(x)){u(x) a bounded function. By using the Maximum Principle for nonpositive functions it follows that u ” 0 in . Hence u ” 0 on any open set in . We will consider wλ (x) = u(xλ ) ´ u(x) for x P λ . Defining gλ via # gλ (x) =

f (u(xλ ))´f (u(x)) u(xλ )´u(x)

if u(xλ ) ‰ u(x),

0

if u(xλ ) = u(x),

we find ´wλ (x) = ´ (u(xλ )) + u(x) = ´(u)(xλ ) + u(x) =f (u(xλ )) ´ f (u(x)) = gλ (x)wλ (x) in λ .

(1.75)

Moreover, for λ ă λ˚ we have u(xλ ) ě 0 = u(x) on B X Bλ and u(xλ ) = u(x) on  X Bλ . Hence wλ (x) ě 0 on Bλ .

(1.76)

Also note that since u is bounded and f is Lipschitz the function gλ is uniformly bounded on  and hence that we may use the maximum principles for linear equations with a uniform bound for }c+ }8 . The two basic ingredients in the proof are the maximum principle for small domains (Theorem 1.29) and again the strong maximum principle for nonpositive functions (Theorem 1.24). Set λ˚ = suptλ P [λ0 , λ˚ ]; wμ (x) ě 0 for all x P μ and μ P [0, λ]u. We will suppose that λ˚ ă λ˚ and arrive at a contradiction. By the maximum principle for small domains (or even the one for narrow domains) one finds from (1.75) and (1.76) that there is λ1 ą λ0 such that wλ (x) ě 0 on  λ for all λ P (λ0 , λ1 ]. Hence λ˚ ą 0. By the strong maximum principle either wλ ą 0 in λ for all λ P (λ0 , λ1 ] or wλ ” 0 on  λ for some λ P (λ0 , λ1 ]. We have found that λ˚ ě λ1 . Next we will show that wλ ” 0 on  λ does not occur for λ P (λ0 , λ1 ). If wλ ” 0 on  λ for some λ P (λ0 , λ˚ ) then u(x) = 0 for x P B(λ1 Y  λ ) and moreover, since part of this boundary  X B(λ1 Y  λ ) lies inside  it contradicts u ą 0 in . For λ = λ˚ we have that wλ˚ (x) ě 0 on  λ and since λ1 ˚ Y  λ˚ ‰  we just found that wλ˚ (x) ı 0 on  λ . By the strong maximum principle for signed functions (Theorem 1.24) it follows that wλ˚ (x) ą 0 in λ˚ and even that B B x1 wλ˚ (x) ą 0 for every x P Bλ˚ X . In order to find a strict bound for wλ away from 0 we restrict ourselves to a compact set as follows. Let δ be as in Theorem 1.29 and let δ be an open neighborhood of B X Bλ˚ such that | X δ | ă 12 δ. By the previous estimate we have for some c ą 0 that wλ˚ (x) ą c(λ˚ ´ x1 ) for x P λ˚ zδ .

48

1 Preliminaries

By continuity we may increase λ somewhat without loosing the positivity. Indeed, by the fact that wλ˚ P C 1 (λ˚ ) there exists ε0 ą 0 such that for all ε P [0, ε0 ] one finds wλ˚ +ε (x) ą 0 for x P Aε , where Aε = tx P ; (x1 + ε, x2 , . . . , xn ) P  λ˚ zδ u Ă λ˚ +ε is a shifted λ˚ .

Since λ ÞÑ |λ | is continuous and Aε Ă λ+ε we may take ε1 P (0, ε0 ) such that | λ˚ +ε z(δ X Aε )| ă

1 δ for ε P [0, ε1 ]. 2

Finally we consider the maximum principle for small domains on the remaining subset of λ+ε defined by Rε = (λ˚ +ε X δ ) Y (λ˚ +ε zAε ) with ε P (0, ε1 ). Since wλ˚ +ε ą 0 on BRε and ´wλ˚ +ε (x) = gλ˚ +ε (x)wλ˚ +ε (x) in Rε we find that wλ˚ +ε ě 0 in Rε and hence wλ ě 0 on  λ for all λ P [λ0 , λ˚ + ε1 ], a contradiction. The conclusion that BBx1 u(x) ą 0 for x P λ˚ follows from the fact that wλ (x) ą 0 on λ for all λ P (λ0 , λ˚ ) and by Hopf’s boundary point Lemma BBx1 u(x) = 1 B \ [ 2 B x1 wλ (x) ą 0 on Tλ X .

Corollary 1.9 Let BR (0) Ă Rn . If f is Lipschitz and if u P C 2 (B) X C(B) satisfies $ & ´u = f (u) u=0 % uą0

in B, on BB, in B,

then u is radially symmetric and B|Bx | u(x) ă 0 for 0 ă |x| ă R.

(1.77)

1.7 The Sliding and the Moving Plane Method in General Domains

49

Proof From the previous theorem we find that x1 BBx1 u(x) ă 0 for x P BR (0) with

x1 ‰ 0. Hence BBx1 u(x) ă 0 for x1 = 0. Since  is radially invariant this holds for every direction and we find BBτ u(x) = 0 in BR (0) for any tangential direction. In other words, u is radially symmetric. Since B|Bx | u(x) = BBτ u(r, 0, . . . , 0) for 0 ă r = |x| ă R the second claim follows from BBxu1 (x) ă 0 for x P BR (0) with x1 ą 0. \ [

1.7 The Sliding and the Moving Plane Method in General Domains In the sliding method introduced in [18] one compares translations of the function rather then reflections. As we will see below, the approach based on the maximal principle for ‘narrow domains’ yields not only a simpler but a more general result. To illustrate the main idea, we now state and prove a monotonicity result for equation # u + f (u) = 0, u=0

in  := tx : |x| ă Ru Ă RN , on B.

(1.78)

but with other boundary conditions, in an arbitrary domain and with general regularity assumptions on the solution. Here we follow [18]. Theorem 1.32 Let  be an arbitrary bounded domain of Rn which is convex in the 2,n x1 -direction. Let u P Wloc () X C() be a solution of u + f (u) = 0 in ,

(1.79)

u = ϕ on B.

(1.80)

The function f is supposed to be Lipschitz continuous. Here we assume that for any three points x 1 = (x11 , y), x = (x1 , y), x 2 = (x12 , y) lying on a segment parallel to the x1 -axis, x11 ă x1 ă x12 , with x 1 , x 2 P B, the following hold ϕ(x 1 ) ă u(x) ă ϕ(x 2 ) if x P 

(1.81)

ϕ(x 1 ) ď ϕ(x) ď ϕ(x 2 ) if x P B.

(1.82)

and

Then, u is monotone with respect to x1 in : u(x1 + τ, y) ą u(x1 , y) for (x1 , y), (x1 + τ, y) P  and τ ą 0.

50

1 Preliminaries

Furthermore, if f is differentiable, then ux1 ą 0 in . Finally, u is the unique 2,n solution of (1.79), (1.80), in Wloc () X C() satisfying (1.81). Condition (1.82) requires monotonicity of ϕ on any segment parallel to the x1 -axis lying on B. It is obviously a necessary condition for the result to hold. Proof Theorem 1.32 is proved by using the sliding technique. For τ ě 0, we let uτ (x1 , y) = u(x1 + τ, y). The function uτ is defined on the set τ =  ´ τ e1 obtained from  by sliding it to the left a distance τ parallel to the x1 -axis. The main part of the proof consists in showing that uτ ą u in τ X  for any τ ą 0.

(1.83)

Indeed, (1.83) means precisely that u is monotone increasing in the x1 direction. Set w τ (x) = uτ (x)´u(x) i.e. w τ (x1 , y) = u(x1 +τ, y)´u(x1 , y); w τ is defined in D τ =  X τ . As before, since uτ satisfies the same Eq. (1.79) in τ as does u in , we see that wτ satisfies an equation # w τ + cτ (x)w τ = 0 wτ ě 0

in D τ , on BD τ

(1.84)

where cτ is the same L8 function satisfying |cτ (x)| ď b, @x P D τ , @τ. The inequality on the boundary BD τ Ă B X Bτ follows the assumptions (1.81)– (1.82). Let τ0 = sup tτ ą 0; D τ ‰ Hu . For 0 ă τ0 ´ τ small, |D τ | is small, that is D τ is a ‘narrow domain’. Therefore, from (1.84) it follows that for 0 ă τ0 ´ τ small, w τ ą 0 in D τ . Next, let us start sliding τ back to the right, that is we decrease τ from τ0 to a critical position τ P [0, τ0 ): let (τ, τ0 ) be a maximal interval, with τ ě 0, such that 1 1 for all τ in τ ă τ 1 ď τ0 , w τ ě 0 in D τ . We want to prove that τ = 0. We argue by contradiction, assuming τ ą 0. By continuity, we have wτ ě 0 in D τ . Furthermore, we know by (1.81) that for any x P  X BD τ , w τ ą 0. It follows easily that w τ ı 0 in every component of the open set D τ . By the strong Maximal Principle it follows from (1.84) that w τ ą 0 in D τ . Indeed, if x = (x1 , y) is any interior point of D τ , then the half line tx1 + t, y; t ě 0u hits BD τ at a point x¯ P Bτ X B. This point x¯ is on the boundary of that component of D τ to which x belongs, and u(x) ¯ ą 0. Now choose δ ą 0 and carve out of D τ a closed set K Ă D τ such that |D τ zK| ă δ{2. We know that wτ ą 0 in K. Hence, for small ε ą 0, w τ ´ε is also positive on K. Moreover, for ε ą 0 small, |D τ ´ε zK| ă δ. Since B(D τ ´ε zK) Ă BD τ ´ε Y K, we see that w τ ´ε ě 0 on B(D τ ´ε zK). Thus, w τ ´ε satisfies # w τ ´ε + cτ ´ε (x)w τ ´ε = 0 w τ ´ε ě 0

in D τ ´ε zK, on B(D τ ´ε zK).

(1.85)

1.7 The Sliding and the Moving Plane Method in General Domains

51

It then follows from Theorem 1.29 (see also [20]) that w τ ´ε ě 0 in D τ ´ε zK and hence in all of D τ ´ε . We have reached a contradiction and thus proved that u is monotone: uτ ą u in D τ , @τ ą 0. (Again this follows from the Eq. (1.84) since we know w τ ě 0 and w τ ı 0 which implies wτ ą 0.) If, further more, f is differentiable, ux1 satisfies a linear equation in , by differentiation (1.79): ux1 + f 1 (u)ux1 = 0 in . Since we already know that ux1 ě 0, ux1 ı 0, we infer from this equation that ux1 ą 0 in . To prove the last assertion of the theorem suppose v is another solution. We argue exactly as before but instead of wτ = uτ ´ u we now take w τ = v τ ´ u. The same proof shows that v τ ě u @τ ě 0. Hence v ě u, and by symmetry we have v = u. l Next, we discuss an extension of the moving plane method to the case of an unbounded domain. Here we follow [14]. Let  = tx P Rn |xn ą ϕ(x1 , . . . , xn´1 )u, where ϕ : Rn´1 Ñ R is a coercive Lipschitz graph, that is, they assume that lim ϕ(x) = +8.

|x |Ñ8 x PRn´1

(1.86)

Theorem 1.33 If  is bounded by a coercive continuous graph (in the above understanding), then, for any locally Lipschitz function f and given any solution u of (1.86) -be it bounded or not-, it satisfies Bu ą 0 in . Bxn Notice that for this result the graph ϕ needs not be globally Lipschitz. Note that, Esteban and Lions get this result for a smooth graph ϕ (see [53] and references therein). But using the improved version of the moving plane method in [17], it was possible to extend this result to the case of merely continuous graph ϕ. This result rests on the method of reflection in moving planes. For λ P R, Tλ = txn = λu, as before, is the moving hyperplane, λ0 = inftλ P R; Tλ X  ‰ Hu, (this number λ0 is finite because of (1.86)) λ = tx P λ ; xn ă λu is the lower cap of  cut out by Tλ .

52

1 Preliminaries

For x P λ , x = (x1 , . . . , xn ), let x λ = (x1 , . . . , xn´1 , 2λ ´ xn ) denote its reflection in , v λ (x) =u(x λ ), w λ (x) =u(x λ ) ´ u(x). These two functions are defined in λ . Notice that since  is defined as an epigraph, the cap λ is always induced in  and is not empty for all λ ą λ0 . The main consequence of the coercivity assumption (1.86) is that λ is always bounded, for all values of λ. Actually, (1.86) is equivalent to the boundedness of λ for all λ. This is the reason why the moving planes method for bounded domains applies as such to epigraphs when (1.86) is satisfied. Indeed, as usual, essentially, one wants to prove that wλ ą 0 in λ for all λ ą λ0 . For this shows that u is nondecreasing with respect to xn , hence that BBxun ě 0 in . Since BBxun satisfies some linear equation, viz. ( BBxun ) + f 1 (u) BBxun = 0, one infers from the strong maximum principle that BBxun ą 0 in . To prove that w λ ą 0 in λ , one uses the Eq. (1.86): v λ satisfies the same equation, v λ + f (v λ ) = 0 in λ . Hence, the function w λ satisfies some linear equation w λ + cλ (x)w λ = 0 in λ ,

(1.87)

where cλ (x) =

f (v λ (x)) ´ f (u(x)) . v λ (x) ´ u(x)

The function cλ is bounded in L8 (λ ), uniformly with respect to λ in bounded intervals. On the contrary, w λ satisfies w λ ě 0, w λ ı 0 on Bλ .

(1.88)

In fact, w λ = 0 on Tλ whereas w λ ą 0 on the remaining part of the boundary, that is BzTλ = tx P B, xn ă λu. We now use the method of [17]. It relies on the following version of the maximum principle in domains of small value (see Theorem 1.29 and [20]). Consider an elliptic operator L = aij (x)

B2 B + bi (x) + c(x) Bxi Bxj Bxi

(1.89)

1.7 The Sliding and the Moving Plane Method in General Domains

53

in a domain . Assume that aij P C(), bi , c P L8 (), that the operator is uniformly elliptic, that is: aij (x)ξi ξj ě C0 |ξ |2

@ξ P Rn , @x P .

Let b ě }bi }L8 () , }c}L8 () . Let us recall that L satisfies the maximum principle in  if for any function w which satisfies Lw ď 0 in  w ě 0 on B one can infer that w ě 0 in . As we discussed in Sect. 1.4, a sufficient condition for L to satisfy the maximum principle is that c(x) ď 0. In general, when c(x) is allowed to be positive, the maximum principle does not hold. However, if the domain is sufficiently small, it still holds, see Theorem 1.29. For a systematic study of conditions under which the maximum principle holds and for some general results related to it we refer also to [20]. Step 1 wλ ą 0 in λ for 0 ă λ ´ λ0 small. Indeed, then, λ has small measure and since }cλ }L8 (λ ) is bounded uniformly on compact sets of λ, the maximum principle applies (by theorem 1.29 above). This could also be seen by using the fact that λ is narrow in the xn direction, if 0 ă λ ´ λ0 is small (see Theorem 1.26 and [20]). Therefore, w λ ą 0 in λ if 0 ă λ ´ λ0 is small. Step 2 Suppose w λ ą 0 in λ for all values of λ in some interval (λ0 , μ). Then, by continuity, wμ ě 0 in μ and since w μ ı 0, we infer from (1.87) that w μ ą 0 in μ . Let K be a compact set included in μ such that the measure of μ zK is small, namely, meas(μ zK) ă

δ 2

for some small enough δ. Then since min w μ (x) = η ą 0, if ε ą 0 is sufficiently x PK

small, by continuity, we see that for all values of λ P [μ, μ + ε], min w λ (x) ě x PK

η ą 0. 2

Hence, w λ ą 0 on K and, furthermore, meas(λ zK) ă δ. By the maximum principle for domains with small value (see Theorem 1.29 and [17, 20]), if δ ą 0 is chosen sufficiently small, then the operator  + cλ (x) satisfies the maximum principle in λ zK. Therefore, since w λ satisfies

54

1 Preliminaries

w λ + cλ (x)w λ = 0 in λ zK on B w λ ě 0, w λ ı 0 on K wλ ą 0 we see that wλ in λ zK, hence in all of λ . We have thus shown that w λ ą 0 for all λ, not only in (λ0 , μ), but also in (λ0 , μ + ε]. We may now conclude. The above two steps show that wλ ą 0 for all λ ą λ0 and this implies that BBxun ą 0 in . As said earlier, this proof does not extend to more general epigraph domains when coercivity condition (1.86) is not satisfied. There are two difficulties. Firstly, λ0 is not necessarily finite and, second, λ is not bounded any more for all values of λ and the problem becomes much more involved. Next, observe that Theorem 1.33 implies, in the case of a half space, a symmetry and monotonicity result. Indeed, consider the case of a half space  = tx P Rn ; xn ą 0u which corresponds to a flat graph ϕ ” 0. Then, for any direction ξ P Rn zt0u transversal to  and entering , that is, such that ξn ą 0, for any system of coordinates y1 , . . . , yn , there is a ηn = ξn and one can read the half space as defined by  = tyn ą ψ(x1 , . . . , xn ) with ψ : Rn´1 Ñ R Lipschitzu. Therefore, if f satisfies the following conditions 1. 2. 3. 4.

there exists a μ ą 0, such that f ą 0 in (0, μ), f (0) = f (μ) = 0; f (s) ě ηs for s P [0, δ] for some η, δ ą 0; f is nonincreasing in [μ ´ δ, μ], δ ą 0; f is Lipschitz continuous on [0, μ];

and  is a half space, for any such direction ξ , Theorem 1.33 implies Bu ą 0 in . Bξ This implies symmetry. Indeed, let x, x 1 P  be two points at equal distance of B, i.e. xn = xn1 . Let en = (0, 0, . . . , 1). Choosing appropriate directions ξ , the previous monotonicity property yields u(x) ău(x 1 + εen ) u(x 1 ) ău(x + εen ) hence, by letting ε Ñ 0, u(x) = u(x 1 ).

1.8 Variational Solutions of Elliptic Equations

55

1.8 Variational Solutions of Elliptic Equations In this section, we formulate auxiliary results concerning the regularity of solutions to linear elliptic equations of the form (2.161), which we will use in Chap. 2. 2 (ω) denotes the space of Let ω Ă Rn be a bounded polyhedral domain. Then HQ all variational solutions v P H 1 (ω) to the problem (1 ´ )v = g, Bn v|Bω = 0,

(1.90)

2 (ω) is a Hilbert space in a natural where the right-hand side g belongs to L2 (ω). HQ manner. For the moment, let G0 be the space of all variational solutions to the problem (1.90), with ω replaced by T1 ´1,T2 +1 = (T1 ´ 1, T2 + 1) ˆ ω. When speaking on T1 ,T2 , we shall not consider T1 ,T2 as a bounded polyhedral domain itself, but as piece of the cylinder R ˆ ω over the bounded polyhedral domain ω. In slight abuse of notation we define: 2 ( Definition 1.11 HQ T1 ,T2 ) is the space of all restrictions of functions from G0 to T1 ,T2 with the norm

}v, T1 ,T2 }2,Q ” inft}u}G0 : u P G0 , u|T1 ,T2 = vu. 3{2

Let HQ (ω) denote the space of traces on the set tt = 0u of functions belonging to 2 ( ) with the norm HQ 0 }u0 , ω}3{2,Q = inft}u, 0 }2,Q : u|t=0 = u0 u. 2 (+ ) denotes the Fréchet space of all distributions u on + Definition 1.12 HQ,loc 2 ( ) for every T ě 0. such that u|T belongs to HQ T 2 2 (+ ) for which HQ,b (+ ) denotes the Banach space of all functions from HQ,loc the following norm is finite:

}u}b = sup }u, T }2,Q . T ě0

3{2

+ 2 2 k k Set V0 = [HQ (ω)]k , + 0 = [HQ,loc (+ )] , F0 = [HQ,b (+ )] , where + := R+ ˆ ω.

Lemma 1.12 Let u be a variational solution to the problem $ & B 2 u + u = g, Bn u| = 0, Bω t % u| = u , u| =u , t=T1

1

t=T2

2

56

1 Preliminaries

2 ( where u1 , u2 P V0 and g P L2 (T1 ,T2 ). Then u P HQ T1 ,T2 ). Moreover, the following estimate holds:

}u, T1 ,T2 }2,Q ď C(}u1 , ω}3{2,Q + }u2 , ω}3{2,Q + }g, T1 ,T2 }0,2 ). 2 ( Proof By definition, there is a function v P HQ T1 ,T2 ) with v|t=T1 = u1 , v|t=T2 = u2 such that

}v, T1 ,T2 }2,Q ď C(}u1 , ω}3{2,Q + }u2 , ω}3{2,Q ). 2 ( We show that w = u ´ v P HQ T1 ,T2 ). The function w satisfies the equation

$ & B 2 w + w = g1 ” g ´ (B 2 v + v) P L2 (T ,T ), t t 1 2 % w| t=T1 = w|t=T2 = 0, Bn w|B ω = 0. Take a cut-off function φ P C08 (R) with φ(t) = 1 for t P (T1 , T2 ) and φ(t) = 0 for t R (T1 ´ ε, T2 + ε), where 0 ă ε ă T2 ´ T1 . It is readily seen that the function $ ´w(2T1 ´ t) for t P (T1 ´ 1, T1 ), ’ ’ & W (t) = φ(t)w(t) ˜ ” φ(t) w(t) for t P (T1 , T2 ), ’ ’ % ´w(2T2 ´ t) for t P (T2 , T2 + 1) belongs to G0 . Indeed, ˜ + φ 2 (t)w(t) ˜ P L2 (T1 ´1,T2 +1 ), Bt2 W + W = φ(t)g˜ 1 (t) + 2φ 1 (t)Bt w(t) ˜ In addition, W satisfies the appropriate boundary where g˜ 1 is defined similarly to w. 2 ( conditions. Hence, according to Definition 1.11, w belongs to HQ \ [ T1 ,T2 ). 2 ( 2 8 Theorem 1.34 The space HQ T1 ,T2 ) X L (T1 ,T2 ) is dense in HQ (T1 ,T2 ).

Proof It suffices to prove that G0 X L8 (T1 ´1,T2 +1 ) is dense in G0 . Take a function u P G0 and a function g P L2 (T1 ´1,T2 +1 ) which satisfy (1.90). Let tgm u Ă L8 (T1 ´1,T2 +1 ) be a sequence such that gm Ñ g in L2 (T1 ´1,T2 +1 ) as m Ñ 8.

(1.91)

Let um P G0 be the variational solution to (1.90) with right-hand side gm . It follows from (1.91) and the definition of the space G0 that um Ñ u in G0 as m Ñ 8.

1.8 Variational Solutions of Elliptic Equations

57

Hence Theorem 1.34 will be proved when we have shown that um L8 (T1 ´1,T2 +1 ). We use the following maximum principle.

P

Lemma 1.13 Let ui P H 1 (T1 ´1,T2 +1 ) for i = 1, 2 be variational solutions to (1.90) with right-hand sides gi P H 1 (T1 ´1,T2 +1 )˚ . Assume, in addition, that xg1 , y ě xg2 , y for all  P H 1 (T1 ´1,T2 +1 ),  ě 0. Then u1 (t, x) ě u2 (t, x) for (t, x) P T1 ´1,T2 +1 a.e.. Proof Consider the function u = u2 ´ u1 . Then xBt u, Bt y + x∇u, Bt y + xu, y ě 0 for all  P H 1 (T1 ´1,T2 +1 ),  ě 0. (1.92) Now introduce u+ (t, x) = maxtu, 0u, u´ (t, x) = maxt´u, 0u. Then u = u+ ´u´ . It is known that u+ , u´ P H 1 (T1 ´1,T2 +1 ) and xu+ , u´ y = 0, xBt u+ , Bt u´ y + x∇u+ , ∇u´ y = 0 (see [104]). Upon replacing  in (1.92) with u´ , we obtain ´xBt u´ , Bt u´ y ´ x∇u´ , ∇u´ y ´ xu´ , u´ y = 0. This implies that xu´ , u´ y = 0 or u´ (t, x) = 0 for (t, x) P T1 ´1,T2 +1 a.e. The lemma is proved. \ [ Corollary 1.10 Let u P H 1 (T1 ´1,T2 +1 ) be a variational solution to (1.90) with right-hand side g P L8 (T1 ´1,T2 +1 ). Then u P L8 (T1 ´1,T2 +1 ). \ [

This completes the proof of Theorem 1.34. Theorem 1.35 The embedding 2 HQ (T1 ,T2 ) Ă Lq (T1 ,T2 )

(1.93)

holds for q ď q0 = 2

n+1 . n´3

2 ( If q ă q0 , then this embedding is compact. Furthermore, if u P HQ T1 ,T2 ), then

u|u|(q0 ´2){2 P H 1 (T1 ,T2 ) and

58

1 Preliminaries q {2

0 }u|u|(q0 ´2){2 , T1 ,T2 }1,2 ď C}u, T1 ,T2 }2,Q .

(1.94)

2 ( 2 ( ˜ P HQ Proof Let u P HQ T1 ,T2 ). By definition, there is a function u T1 ´1,T2 +1 ), u| ˜ T1 ´1,T2 +1 = u, such that

xBt u, ˜ Bt y + x∇ u, ˜ Bt y + xu, ˜ y = xg, ˜ y for all  P H 1 (T1 ´1,T2 +1 ) (1.95) for a certain right-hand side g˜ P L2 (T1 ´1,T2 +1 ) and ˜ T1 ´1,T2 +1 }0,2 . }u, T1 ´1,T2 +1 }2,Q ď C}g, Approximate g˜ P L2 (T1 ´1,T2 +1 ) by a sequence tg˜ m u Ă L8 (T1 ´1,T2 +1 ), g˜ m Ñ g˜ in L2 (T1 ´1,T2 +1 ) as m Ñ 8. Let u˜ m be the solution to the variational problem (1.95) with the right-hand side g˜ replaced with g˜ m . Then u˜ m P L8 (T1 ´1,T2 +1 ) by Corollary 1.10. Hence the function u˜ m |u˜ m |l ´2 belongs to H 1 (T1 ´1,T2 +1 ), where l0 ´ 2 = 4{(n ´ 3) = (q0 ´ 2){2. Replacing u˜ by u˜ m and  by u˜ m |u˜ m |l0 ´2 in (1.95) and arguing as in the proof of estimate (2.180) we obtain the inequality }u˜ m |u˜ m |(l0 ´2){2 , T1 ´1,T2 +1 }21,2 ď C(1 + |xg˜ m , u˜ m |u˜ m |l0 ´2 y|).

(1.96)

By Sobolev’s embedding theorem, H 1 (T1 ´1,T2 +1 ) Ă L2n{(n´2) (T1 ´1,T2 +1 ). It follows that 0 = }u˜ m |u˜ m |(l0 ´2){2 , T1 ´1,T2 +1 }20,2n{(n´2) }u˜ m , T1 ´1,T2 +1 }l0,q 0

= C}u˜ m |u˜ m |(l0 ´2){2 , T1 ´1,T2 +1 }21,2 . By Hölder’s inequality, we further obtain 0 ´1 |xg˜ m , u˜ m |u˜ m |l0 ´2 y| ď }g, T1 ´1,T2 +1 }0,2 }u˜ m , T1 ´1,T2 +1 }l0,q 0 0 0 ď μ}u˜ m , T1 ´1,T2 +1 }l0,q + Cμ }gm , T1 ´1,T2 +1 }l0,2 , 0

where μ ą 0 is arbitrary. Applying these estimates to (1.96) for a sufficiently small μ ą 0, we get 0 0 + }u˜ m |u˜ m |(l0 ´2){2 , T1 ´1,T2 +1 }21,2 ď C}gm , T1 ´1,T2 +1 }l0,2 . }u˜ m , T1 ´1,T2 +1 }l0,q 0

We know that g˜ m Ñ g˜ in L2 (T1 ´1,T2 +1 ), hence the sequence tu˜ m u is bounded in the space Lq0 (T1 ´1,T2 +1 ). Without loss of generality, we may assume that u˜ m Ñ u˜ weakly in Lq0 (T1 ´1,T2 +1 ). Thus u˜ P Lq0 (T1 ´1,T2 +1 ) and

1.8 Variational Solutions of Elliptic Equations

59

}u, T1 ,T2 }0,q0 ď }u, ˜ T1 ´1,T2 +1 }0,q0 ď C}g, ˜ T1 ´1,T2 +1 }0,2 ď C1 }u, T1 ,T2 }2,Q . The statement u|u|l0 ´2 P H 1 (T1 ,T2 ) is shown analogously. We finally prove the compactness of embedding (1.93) for q ă q0 . By the interpolation inequality between H1 and Lq0 , 2 (T1 ,T2 ) Ă H ε,q0 (T1 ,T2 ) HQ

for some ε ą 0. The assertion follows, since the embedding H ε,q0 Ă Lq is compact. The proof of Theorem 1.35 is complete. \ [ Corollary 1.11 The embedding 2 HQ (T1 ,T2 ) Ă C([T1 , T2 ], Lp0 (ω))

holds. Here p0 = 2l0 = 2 + 4{(n ´ 3) is the maximum value of p in (2.162). In fact, the estimate (1.94) together with Sobolev’s embedding theorem imply that 2 ( u|u|(l0 ´2){2 P C([T1 , T2 ], L2 (ω)) if u P HQ T1 ,T2 ). Moreover, we infer from 2 1 2 the embedding HQ Ă H that u P C([T1 , T2 ], L (ω)). Now, employing standard arguments, we finally open that u P C([T1 , T2 ], Lp0 (ω)). 2 ( 2 2 2 Theorem 1.36 Let u P HQ T1 ,T2 ). Then Bt u P L (T1 ,T2 ), Bt ∇u P L (T1 ,T2 ). Moreover, the following estimate holds:

}Bt2 u, T1 ,T2 }0,2 + }Bt ∇u, T1 ,T2 }0,2 ď C}u, T1 ,T2 }2,Q .

(1.97)

˜ T1 ,T2 = u and Proof By definition, there exists a function u˜ P G0 that satisfies u| $ & B 2 u˜ + u˜ ´ u˜ = g(x), Bn u| ˜ Bω = 0, t % u| ˜ = 0, u| ˜ =0 t=T1 ´1

(1.98)

t=T2 +1

for some function g P L2 (T1 ´1,T2 +1 ). }g, T1 ,T2 }0,2 ď C}u, T1 ,T2 }2,Q . Below we only give the formal arguments for deriving estimate (1.97). A rigorous proof can be supplied by exploiting the Galerkin approximation method. Multiplying Eq. (1.98) by Bt2 u˜ and integrating over T1 ´1,T2 +1 , we obtain after an integration by parts ˜ 2 , 1y + x|Bt ∇ u| ˜ 2 , 1y + x|Bt u| ˜ 2 , 1y = xg, Bt2 uy. ˜ x|Bt2 u|

(1.99)

Applying Hölder’s inequality xg, Bt2 uy ˜ ď

1 1 x|g|2 , 1y + x|Bt2 u| ˜ 2 , 1y 2 2

to the right-hand side in (1.99), we then find (1.97). The proof is finished.

\ [

60

1 Preliminaries

From Theorem 1.36 we conclude: Corollary 1.12 We have 2 HQ (T1 ,T2 ) Ă H 1 ((T1 , T2 ), H 1 (ω)) X H 2 ((T1 , T2 ), L2 (ω)).

In particular, the functions t ÞÑ }u(t)}1,2 and t ÞÑ }Bt u(t)}0,2 are defined and 2 ( continuous for every u P HQ T1 ,T2 ). Corollary 1.13 Furthermore, 2 2 HQ (T1 ,T2 ) = H 2 ((T1 , T2 ), L2 (ω)) X L2 ((T1 , T2 ), HQ (ω)).

(1.100)

Remark 1.20 For smooth domains ω Ă Rn , all previous results of this section are consequences of L2 -elliptic regularity for the Laplace operator (e.g., [101]), especially the fact 2 (T1 ,T2 ) = tu P H 2 (T1 ,T2 ) : Bn u|Bω = 0u HQ

(1.101)

and Sobolev’s embedding theorem. For polyhedral domains ω, however, (1.101) is in general not fulfilled. The following deep result may be found in [40]. Theorem 1.37 Let ω Ă Rn be a bounded polyhedral domain. Then there exists an ε satisfying 0 ă ε ď 1{2 such that 2 (ω) Ď H 3{2+ε (ω). HQ

(1.102)

Corollary 1.14 Let ω Ă Rn be a bounded polyhedral domain. Then 2 HQ (T1 ,T2 ) Ď H 3{2+ε (T1 ,T2 ),

(1.103)

where ε is the same as in Theorem 1.37. (1.103) is actually a consequence of (1.100), (1.102). 2 ( Corollary 1.15 Let u P HQ T1 ,T2 ). Then

Bn u|Bω P H ε ((T1 , T2 ) ˆ Bω). This follows from (1.103) and Sobolev’s embedding theorem. In particular, using 2 ( Green’s formula (see [78]) we obtain that Bn u|Bω = 0 for u P HQ T1 ,T2 ). Thus + solutions u to the problem (2.161) which belong to 0 satisfy the homogeneous Neumann boundary condition in the proper sense.

1.9 Elliptic Regularity for the Neumann Problem for the Laplace Operator on. . .

61

1.9 Elliptic Regularity for the Neumann Problem for the Laplace Operator on an Infinite Edge In this section, we discuss elliptic regularity for the Neumann problem for the Laplace operator on an infinite cone  Ă R2 and the infinite wedge R ˆ  Ă R3 . We will use the result of this section in Chap. 2 (see Sect. 2.13) while studying the trajectory attractor for an elliptic system in the half cylinder + = R+ ˆω, where ω is a bounded polyhedral in Rn . In this section we mainly follow [91], preserving the notations of the authors for the convenience of the reader. It is worth to note that the results of this section are of independent interest. Let  Ă R2 be an open cone with angle α. Throughout we shall suppose that  = t(r, θ ); 0 ă θ ă αu. Here (r, θ ) denote polar coordinates in R2 . We further suppose that α ą π (see Remark 1.22 (a)). Since the model cone  arises from flatting out the boundary of ω near a fixed conical point of Bω, we shall consider operators Id ´y ´ M(y, řBy ) on , where γ y = (y1 , y2 ) are Euclidian coordinates in R2 and M(y, By ) = |γ |ď2 bγ (y)By is a second-order partial differential operator subject to the following conditions: For γ P N2 , |γ | = 2, bγ P C 8 () and (a1) }bγ }L8 () ď δ, bγ (0) = 0; (a2) }∇y bγ }L8 (Bδ{K X) ď K, where B = ty P R2 ; |y| ď u. For γ P N2 , |γ | ď 1, bγ = bγ 1 + bγ 2 , where bγ 1 , bγ 2 P C 8 () and (b1) suppbγ 1 Ď Bδ X , }bγ 1 }L8 () ď K; (b2) }bγ 2 }L8 () ď δ for some constant K ą 1 and a certain δ = δ(K) sufficiently small, 0 ă δ ă 1. 2 () denote the space of all variational solutions v to the problem Let HQ (Id ´y ´ M(y, By ))v = g, Bn v|B = 0

(1.104)

with right-hand side g P L2 () (see Sect. 1.8). 2 () is actually independent Remark 1.21 It is a well-known fact that the space HQ of the choice of the operator M(y, By ) satisfying (a), (b) provided that δ ą 0 is small enough (see [83, 89]).

Moreover, from Theorem 1.37 and its corresponding version for a model cone it follows that a solution to (1.104) belongs to H 3{2+ε () for a certain ε ą 0. For the special case M(y, By ) ” 0, it is readily seen that 2 () = HN2 () ‘ spantSu, S(y) = ψ(r)r π {α cos(π θ {α), HQ

(1.105)

where HN2 () = tv P H 2 (); Bn v|B = 0u (see [40, 83]). Here ψ P C08 () is some fixed cut-off function, depending only on the radial coordinate r, such that

62

1 Preliminaries

ψ(r) = 1 in a neighbourhood of 0 and ψ is supported sufficiently close to 0. Notice that S P H 1+π {α ´ε () for any ε ą 0, but S R H 1+π {α (). Lemma 1.14 For δ ą 0 sufficiently small (depending on K), the differential operator 2 Id ´y ´ M(y, By ) : HQ () Ñ L2 ()

(1.106)

2 () is the space given in (1.105). Moreover, we induces an isomorphism, where HQ have the estimate

}v}H 2 () ď C}(Id ´y ´ M(y, By ))v}L2 ()

(1.107)

Q

2 (), where the constant C ą 0 only depends on δ, K. for v P HQ 2 () onto L2 (). Proof It is known that Id ´ is an isomorphism from HQ 2 () into L2 (), and it can be shown Furthermore it is seen that M(y, By ) maps HQ that

}M(y, By )}H 2 ()ÑL2 () ď C(K)δ 1{2

(1.108)

Q

with some constant C(K) ą 0. To prove (1.108) it suffices to observe that › › ÿ › γ › › bγ By S ›› › |γ |=2

L2 ()

ď CK 1´π {α δ π {α

and › ÿ › › γ › › › b B v γ1 y › › |γ |ď1

L2 ()

ď

ÿ

}bγ 1 }L4 () }v}H 1,4 () ď CKδ 1{2 }v}H 2 () . Q

|γ |ď1

In fact, for |γ | = 2, we have bγ (y) = r b˜γ (y) with b˜γ (y) = such that }b˜γ }L8 (Bδ{K X) ď K and ż

γ

B1 X

|bγ (y)By S(y)|2 dy ď C

ż δ {K ż α 0

+C

ş1

0 (y{r)¨∇y bγ (sy) ds

|b˜γ (y)|2 r 2π {α ´1 dθ dr

0

ż1 żα δ {K

|bγ (y)|2 r 2π {α ´3 dθ dr

0

  ď C K 2 (δ{K)2π {α + δ 2 (δ{K)2(π {α ´1) = CK 2(1´π {α) δ 2π {α .

1.9 Elliptic Regularity for the Neumann Problem for the Laplace Operator on. . .

63

2 () Ă H 1,4 () follows from the explicit description of H 2 (), Moreover, HQ Q H 1,4 () and Sobolev’s embedding theorem. Now choose δ ą 0 dependent on K so small that

}M(y, By )}H 2 ()ÑL2 () ă }(Id ´)´1 }L2 ()ÑH 2 () , Q

Q

2 () Ñ L2 (). Then the where (Id ´)´1 stands for the inverse to Id ´ : HQ differential expression Id ´ ´ M(y, By ) in (1.106) induces an isomorphism. The estimate (1.107) immediately follows. \ [

Remark 1.22 2 () = H 2 () when α ă π . In (a) The same argumentation yields that HQ N subsequent discussion we always assume that α ą π . 2 () can uniquely be represented in the (b) From (1.105) it follows that each v P HQ form

v = v0 + dS,

(1.109)

where v0 P HN2 (), d P C. It is important to observe that the coefficient d in (1.109) is independent of the particular cut-off function ψ, i.e., choosing another cut-off function possessing the same properties as ψ we obtain d as before. 2 (R ˆ ) of variational solutions v to Next we discuss the space HQ

(Id ´Bt2 ´ y ´ M(y, By ))v = g, Bn v|RˆB = 0 with right-hand side g P L2 (R ˆ ), where M(y, By ) is a second-order partial differential operator as above, but satisfying the additional conditions supp bγ 1 Ď Bδ 2 {K X , bγ 2 (y) ” 0 for |γ | ď 1.

(1.110)

2 (Rˆ) is independent of the operator M(y, B ) Again it turns out that the space HQ y provided that δ ą 0 is small enough. We need the following result in the cases s = 2, s = 0. For a proof, see [40, 89].

Lemma 1.15 Let  Ă R2 be an open cone, s P R. Then an equivalent norm on H s (R ˆ ) is given by }u}H s (Rˆ) =

"ż 8 ´8

p(τ )}2H s () dτ xτ y2s }κ(τ )´1 u

p(τ ) = Ft Ñτ u(τ ), κ(τ ) = κxτ y , xτ y = (1 + |τ |2 )1{2 and where u

*1{2 ,

64

1 Preliminaries

κλ u(y) = λu(λy), λ ą 0, y P , for u P H s (). Notice that tκλ uλą0 is a strongly continuous group on H s (). It consists of isometries when s = 0. Lemma 1.16 Let  Ă R2 be an open cone as above. Then we have 2 (R ˆ ) HQ

p )u; d P H 2 (R)u, = HN2 (R ˆ ) ‘ tFτ´Ñ1 t txτ yψ(rxτ y)(rxτ y)π {α cos(π θ {α)d(τ (1.111) where HN2 (R ˆ ) = tv P H 2 (R ˆ ); Bn v|RˆB = 0u. Proof Let v be a solution to (1.109) with right-hand side g P L2 (R ˆ ). Upon applying the Fourier transformation Ft Ñτ and afterwards the group action κ(τ )´1 we obtain the equation $ & (Id ´ ´ Mτ (y, By ))κ(τ )´1 vp(τ ) = xτ y´2 κ(τ )´1 gp(τ ) in , % B u(κ(τ )´1 vp(τ ))| = 0 n B

(1.112)

with parameter τ P R, where Mτ (y, By ) = xτ y´2 M(xτ y´1 y, xτ yBy ). Now it is ř ´2+|γ | b (xτ y´1 y)B γ satisfies seen that the operator Mτ (y, By ) = γ y 1ď|γ |ď2 xτ y requirements (a), (b) with the same δ ą 0, K ą 1 as before. Indeed, for |γ | ď 1, we put bγ 1,τ (y) = xτ y´1 bγ (xτ y´1 y), bγ 2,τ (y) = 0 if xτ y ď K{δ and bγ 1,τ (y) = 0, bγ 2,τ (y) = xτ y´1 bγ (xτ y´1 y) if xτ y ą K{δ. Hence we conclude from Eq. (1.112) together with (1.104), (1.109) that p )S(y), S(y) = ψ(r)r π {α cos(π θ {α). κ(τ )´1 vp(τ ) = κ(τ )´1 vp0 (τ ) + d(τ (1.113) Moreover, from (1.107) we derive the estimate p )|2 ď Cxτ y´4 }κ(τ )´1 gp(τ )}2 2 . }κ(τ )´1 vp0 (τ )}2H 2 () + |d(τ L () Therefore, we get ż8

xτ y4 }κ(τ )´1 vp0 (τ )}2H 2 () dτ +

´8 ż8

ďC

´8

ż8 ´8

p )|2 dτ xτ y4 |d(τ

}κ(τ )´1 gp(τ )}2L2 () dτ = C}g}2L2 (Rˆ)

1.9 Elliptic Regularity for the Neumann Problem for the Laplace Operator on. . .

65

showing that v0 P H 2 (R ˆ ), d P H 2 (R) by Lemma 1.15. From (1.113) we finally get p )(κ(τ )S)(y)u v = v0 + Fτ´Ñ1 t td(τ which gives us the decomposition (1.111), observing that the sum on the right-hand 2 (R ˆ ). \ [ side of (1.111) is direct and is obviously contained in HQ Remark 1.23 The proof of Lemma 1.16 shows that }u}1H 2 (Rˆ) Q

=

"ż 8 ´8

xτ y }κ(τ ) 4

´1 p

u(τ )}2H 2 () Q

*1{2 dτ

2 (R ˆ ). Since H 2 () is a cone Sobolev space of is an equivalent norm on HQ Q functions possessing asymptotics of a certain discrete asymptotic type near y = 0, 2 (R ˆ ) is in fact a wedge Sobolev space (see [88–90]). HQ

After these preparations, we are now in position to discuss elliptic regularity and asymptotics for the cylinder R ˆ ω and the half-cylinder + = R+ ˆ ω. Let ω be a bounded, polyhedral domain in R2 . The boundary Bω is in particular smooth except for a finite number of conical points. Only the conical points with an obtuse angle deserve further interest, for H 2 -regularity holds up to conical points with an acute angle (see Remark 1.22 (a)). Let tb1 , . . . , bκ u denote the set of these conical points. Let αj be the angle at bj , αj ą π . For every j , 1 ď j ď κ, we choose an open cone j Ă R2 , open subsets Uj , Vj in R2 with Uj Q bj , Vj Q 0, and a diffeomorphism χj : Uj Ñ Vj such that χj (bj ) = 0 and χj (ωXUj ) =  j XVj . We assume that j = t(r, θ ); 0 ă θ ă αj u. Furthermore, we suppose that the diffeomorphism χj are chosen in such a manner that χj1 (x)t χj1 (x) is a positive scalar multiple of the identity for x P BωXUj . Moreover, χj1 (bj ) P SO(2; R). It can be shown that such a choice is possible, if Uj is sufficiently small. Then the homogeneous Neumann boundary condition is preserved under the diffeomorphisms χj . Notice that the assumption implies that (χj )˚  =  + Mj (y, By ) close to y = 0, where Mj (y, By ) is a second-order differential operator (without zero-order terms). By shrinking Uj , if necessary, we may suppose that Mj (y, By ) satisfies, for  = j , K = 1 and δ ą 0 sufficiently small, the assumptions (a), (b) previous to Lemma 1.14 as well as condition (1.110). Let U0 Ă R2 be an open set not meeting tb1 , . . . , bκ u such that tU0 u Y tUj uκj =1 forms an open covering of ω. Let tφ0 uYtφj uκj =1 be a subordinate partition of unity, ř φ0 + κj =1 φj = 1 on ω, φj = 1 in a neighbourhood of bj for all j , 1 ď j ď κ. Eventually we assume that, for 1 ď j ď κ, ψj = (χj )˚ φj only depends on the radial variable r, i.e., ψj = ψj (r).

66

1 Preliminaries

Remark 1.24 Before we proceed we give an intrinsic interpretation of (1.107): There is a short split exact sequence 2 (ω) ÝÝÝÝÝÑ !κj =1 C ÝÝÝÝÝÑ 0 0 ÝÝÝÝÝÑ HN2 (ω) ÝÝÝÝÝÑ HQ

(1.114)

2 (ω) its sequence (d , . . . , d ) with the surjection assigning to each function u P HQ 1 κ of singular coefficients. Thereby, dj is explained as the coefficient appearing in the front of S for v = (χj )˚ (φj u),  = j .

To see that (1.114) is correctly defined observe that the coefficient dj is not only independent of the choice of the cut-off function ψj (see Remark 1.22 (b)), but also independent of the choice of the diffeomorphism χj meeting all of the assumptions above. A splitting of (1.114) is obtained via (1.105) after having fixed the diffeomorphisms χj and the cut-off functions ψj . More precisely, we may write u = u0 +

κ ÿ

dj (χj )˚ (ψj (r)r π {αj cos(π θ {αj ))

j =1 2 (ω), where u P H 2 (ω), d P C are uniquely determined. The for u P HQ 0 j N coefficients dj can be calculated using the formula

  dj = lim βj´2 r ´π {αj ((χj )˚ (φj u)(r, θ ) ´ u(bj )), cos(π θ {αj )) L2 (0,α ) , r Ñ 0+

j

(1.115) where (¨, ¨)L2 (0,αj ) denotes the scalar product in L2 (0, αj ), u(bj ) is the value of u şα at bj , and βj = t 0 j | cos(π θ {αj )|2 dθ u1{2 . The value u(bj ) = (χj )˚ (φj u)(0) is well-defined by Theorem 1.37. 2 (R ˆ ω) is given by Notice that an equivalent norm on HQ }u}H 2 (Rˆω) Q

*1{2 " κ ÿ 2 2 = }φ0 u}H 2 (Rˆω) + }(χj )˚ (φj u)}H 2 (Rˆ ) . j =1

Q

j

(1.116)

2 (R ˆ ω) if and only if φ u P H 2 (R ˆ ω) for This follows from the fact that u P HQ l Q 2 all l, 0 ď l ď κ, and obviously φ0 u P HQ (R ˆ ω) if and only if φ0 u P H 2 (R ˆ ω), 2 (R ˆ ω) if and only if (χ ) (φ u) P H 2 (R ˆ  ). while, for 1 ď j ď κ, φj u P HQ j ˚ j j Q From Lemma 1.16 and (1.116) we conclude that

2 HQ (R ˆ ω) = HN2 (R ˆ ω) ‘

"ÿ κ

(χj )˚ (Fτ´Ñ1 t txτ yψj (rxτ y)(rxτ y)π {αj

j =1

* 2 p ˆ cos(π θ {αj )dj (τ )u); dj P H (R), 1 ď j ď κ . (1.117)

1.9 Elliptic Regularity for the Neumann Problem for the Laplace Operator on. . .

67

Analogously to (1.114) we have the following lemma. Lemma 1.17 For ω Ă R2 being a bounded, polyhedral domain as above, there is a short split exact sequence 2 0 ÝÝÝÝÑ HN2 (Rˆω) ÝÝÝÝÑ HQ (Rˆω) ÝÝ1ÝÝÝκÑ !κj =1 H 1´π {αj (R) ÝÝÝÝÑ 0, (1.118) (τ ,...,τ )

where the operators τj are given by   τj u(t) = lim βj´2 r ´π {αj ((χj )˚ (φj u)(t, r, θ ) ´ u(t, bj )), cos(π θ {αj )) L2 (0,α ) . r Ñ 0+

j

(1.119) Moreover, a splitting of (1.118) is given by the mapping (d11 , . . . , dκ1 ) ÞÝÑ

κ ÿ

(χj )˚ (Fτ´Ñ1 t tψj (rxτ y)dpj 1 (τ )ur π {αj cos(π θ {αj )).

j =1

(1.120) Proof According to (1.112) and the short exact sequence (1.114), the functions dj P 2 (R ˆ ω) as H 2 (R) appearing in the representation of u P HQ u = u0 +

κ ÿ

(χj )˚ (Fτ´Ñ1 t txτ yψj (rxτ y)(rxτ y)π {αj cos(π θ {αj )dpj (τ )u)

j =1

= u0 +

κ ÿ

(χj )˚ (Fτ´Ñ1 t tψj (rxτ y)dpj 1 (τ )uπ {αj cos(π θ {αj )),

j =1

where u0 P HN2 (R ˆ ω), are uniquely determined, independently of the choice of the diffeomorphisms χj and the cut-off function ψj . Likewise, the same is then true for the functions dj 1 = xDy1+π {αj dj P H 1´π {αj (R). Therefore, the surjection in (1.118) is well-defined. Moreover, it becomes clear that (1.118) is exact and a splitting of it is provided by (1.120). Thus it remains to show (1.119). From (1.115), applied to  = j , v = (χj )˚ (φj u), and Eq. (1.112), in which d = dj , we conclude that dpj (τ )

  = lim βj´2 r ´π {αj (xτ y´1 vp(τ, rxτ y´1 , θ ) ´ xτ y´1 vp(τ, 0)), cos(π θ {αj ) L2 (0,α

j)

r Ñ 0+

  = lim βj´2 (rxτ y)´π {αj xτ y´1 (p v (τ, r, θ ) ´ vp(τ, 0)), cos(π θ {αj ) L2 (0,α ) , r Ñ 0+

j

68

1 Preliminaries

the latter line upon replacing r with rxτ y, i.e., dpj 1 (τ ) =xτ y1+π {αj dpj (τ )   v (τ, r, θ ) ´ vp(τ, 0)), cos(π θ {αj ) L2 (0,α ) , = lim βj´2 r ´π {αj (p r Ñ 0+

j

  dj 1 (t) = lim βj´2 r ´π {αj ((χj )˚ (φj u)(t, r, θ ) ´ u(t, bj )), cos(π θ {αj ) L2 (0,α ) . r Ñ 0+

This proves Lemma 1.17 completely.

j

\ [

From (1.119) we obtain in particular that the trace operation on an edge is local. 2 (R ˆ ω), we have supp(τ u) Ď supp(u) X (R ˆ tb u). Corollary 1.16 For u P HQ j j

Remark 1.25 (a) To interpret the functions dj 1 P H 1+π {αj (R), 1 ď j ď κ, as coefficients in the 2 (Rˆω) close to the edge Rˆtb u, we observe asymptotic expansion of u P HQ j ´1 p that Fτ Ñt tψj (rτ )dj 1 (τ )u = dj 1 (t) when r = 0. (b) It can be shown that βj´2 (r ´π {αj ((χj )˚ (φj u)(t, r, θ ) ´ u(t, bj )), cos(π θ {αj ))L2 (0,αj ) P H 1 (R) 2 (R ˆ ω), and convergence in (1.119) takes place in H 1´π {αj (R). for u P HQ

The final goal in this section is to conclude the form of asymptotics when going over 2 (R ˆ ω) to its factor space H 2 (R ˆ ω). This is achieved by constructing from HQ + Q 2 (R ˆ ω) a suitable splitting of (1.118) in terms of a continuous projection !2 in HQ by means of a reformulation of the asymptotic information. Theorem 1.38 Let ω Ă R2 be a bounded, polyhedral domain as above. Then there 2 (Rˆω) obeying the following properties: exists a continuous projection !2 in HQ (a) (b) (c) (d) (e)

ker !2 = HN2 (R ˆ ω); Ts !2 = !2 Ts for all s P R; suppu Ď R´ implies supp!2 u Ď R´ ; 2 (R ˆ ω), H 2 (R ˆ ω))-continuous; !2 is (HQ,b Q,b 2 2 (R ˆ ω), HQ,loc (R ˆ ω))-continuous. !2 is (HQ,loc

In the proof of Theorem 1.38, we shall make use of the following result. Lemma 1.18 Let  Ă R2 be an open cone. Further let ψ P S(R), ψ1 P S(R), d1 P H 1´π {α (R). Then ψ1 (r)Fτ´Ñ1 t t(ψ(rxτ y) ´ ψ(rτ ))dp1 (τ )ur π {α cos(π θ {α) P HN2 (R ˆ ).

1.9 Elliptic Regularity for the Neumann Problem for the Laplace Operator on. . .

69

Proof Let u(t, r) = ψ1 (r)Fτ´Ñ1 t t(ψ(rxτ y) ´ ψ(rτ ))dp1 (τ )ur π {α cos(π θ {α). Then we have }u}H 2 (Rˆ) N

= =

"ż 8

´8

"ż 8

ďC

*1{2 xτ y2 }κ(τ )´1 (ψ1 (r)(ψ(rxτ y)´ψ(rτ ))dp1 (τ )r π {α cos(π θ {α))}2H 2 () dτ N

*1{2 2 ´1 π {α 2 p xτ y |d(τ )| }ψ1 (rxτ y )(ψ(r)´ψ(rτ {xτ y))r cos(π θ {α)}H 2 () dτ 2

´8

N

"ż 8 ´8

p )|2 dτ xτ y2 |d(τ

*1{2 (1.121)

,

where d = xDy´1´π {α d1 P H 2 (R). Thereby, }ψ1 (rxτ y´1 )(ψ(r) ´ ψ(rτ {xτ y))r π {α cos(π θ {α)}H 2 () ď C N

for a certain constant C ą 0 independent of τ is seen from the fact that ψ2 (r) ÞÑ ψ2 (r)r π {α cos(π θ {α) constitutes a bounded map from tψ2 P S(R+ ); ψ2 (0) = 0u into HN2 (), while tψ1 (rxτ y´1 )(ψ(r) ´ ψ(rτ {xτ y)); τ P Ru for ψ P S(R), ψ1 P S(R+ ) is bounded in tψ2 P S(R+ ); ψ2 (0) = 0u. Hence the right-hand side in (1.121) is finite proving that u P HN2 (R ˆ ). \ [ Proof of Theorem 1.38 Lemma 1.18 allows to replace Fτ´Ñ1 t txτ yψj (rxτ y)(rxτ y)π {αj cos(π θ {αj )dpj (τ )u in (1.117) by ψj 1 Fτ´Ñ1 t tψj (rτ )dˆj 1 (τ )ur π {αj cos(π θ {αj ), i.e., we have 2 (R ˆ ω) = HN2 (R ˆ ω)‘ HQ "ÿ * κ (χj )˚ (ψj 1 (r)Fτ´Ñ1 t tψj (rτ )dpj 1 (τ )ur π {αj cos(π θ {αj )); dj 1 P H 1´π {αj (R) , j =1

where, for each j , 1 ď j ď κ, ψj P S(R), ψj 1 P C08 (R+ ), ψj (0) = ψj 1 (0) = 1, and ψj 1 is supported in Vj when considered as a function on j . If especially the ψj are chosen in a way such that supp F ´1 ψj Ď R´ holds for all j , then

70

1 Preliminaries

!2 u =

κ ÿ

(χj )˚ (ψj 1 (r)Fτ´Ñ1 t tψj (rτ )(τj u)p(τ )ur π {αj cos(π θ {αj ))

(1.122)

j =1

2 (R ˆ ω) is a projection in H 2 (R ˆ ω) meeting all the requirements for u P HQ Q (a)–(e). That !2 is a projection follows from the fact that τj !2 u = τj u holds for 2 (Rˆω), (a), (c) are immediate, (b) is the locality of the trace operator τ (see u P HQ j Corollary 1.16) and the translation invariance of the pseudo-differential operator d1 ÞÑ Fτ´Ñ1 t (ψj (rτ )dp1 (τ )), where r ą 0 is regarded as a parameter, and (d), (e) come from the observation that ψj 1 (r)Fτ´Ñ1 t tψj (rτ )(τj u)p(τ )r π {αj cos(π θ {αj )u 2 (R ˆ  ) and H 2 belongs to HQ,b j Q,loc (R ˆ j ), respectively, for u belonging to 2 2 (R ˆ j ), as an easy calculation reveals. \ [ HQ,b (R ˆ j ) and HQ,loc

The following consequences of Theorem 1.38 supply the projection !+ 2 in 2 HQ,b (R ˆ ω) onto its closed subspace comprising the asymptotic information as well as the short exact sequences used in Chap. 2. Theorem 1.39 Let ω Ă R2 be a bounded, polyhedral domain as above. Then 2 there exists a continuous projection !+ 2 in HQ,b (R+ ˆ ω) obeying the following properties: 2 (a) ker !+ 2 = HN,b (R+ ˆ ω); + + (b) Ts !2 = !2 Ts for all s ě 0. 2 2 Moreover, !+ 2 is (HQ,loc (R+ ˆ ω), HQ,loc (R+ ˆ ω))-continuous.

Proof The theorem follows from Theorem 1.38 (a)–(e) by continuous extension of 2 (Rˆω) and its subsequent factorization to H 2 (R ˆω). the projection !2 to HQ,b + Q,b \ [ Notice that a projection !+ 2 satisfying the requirements of Theorem 1.39 is !+ 2u=

κ ÿ

(χj )˚ (ψj 1 (r)Fτ´Ñ1 t tψj (τ r)((τj+ u)ext )p(τ )ur π {αk cos(π θ {αj )),

j =1

(1.123) 2 (R ˆω), where ψ, ψ are as in (1.122). Here (τ + u) u P HQ,b + 1 ext means an arbitrary j 1´π {αj

extension of τj+ u P Hb

1´π {αj

(R+ ) to a function in Hb

(R).

Corollary 1.17 The short exact sequence (1.118) extends by continuity and factors subsequently to a short split exact sequence (τ ,...,τ )

1´π {αj

2 2 0 ÝÑ HN,b (R+ ˆ ω) ÝÑ HQ,b (R+ ˆ ω) ÝÝ1ÝÝÝκÑ !κj =1 Hb

(R+ ) ÝÑ 0,

where (τ1 , . . . , τκ ) is the vector of trace operators. A splitting is obtained 1´π {αj

from (1.123) by replacing τj+ u by d1j P Hb

(R+ ).

Chapter 2

Trajectory Dynamical Systems and Their Attractors

2.1 Kolmogorov ε-Entropy and Its Asymptotics in Functional Spaces We start with the definition of Kolmogorov ε-entropy, via which we define fractal dimension of the compact set in the metric space. We will use these two concepts in the sequel. Definition 2.1 Let K be a (pre)compact set in a metric space M. Then, due to Hausdorff’s criteria, it can be covered by a finite number of ε-balls in M. Let Nε (K, M) be the minimal number of ε-balls that cover K. Then, we can call Kolmogorov’s ε-entropy of K the logarithm of this number; Hε (K, M) := log2 Nε (K, M). We now give several examples of typical asymptotics for the ε-entropy. Example 2.1 We assume that K = [0, 1]n and M = Rn (more generally, K is an n-dimensional compact Lipschitz manifold of the metric space M). Then Hε (K, M) = (n + o(1)) log2

1 as ε Ñ 0. ε

This example justifies the definition of the fractal dimension. Definition 2.2 The fractal dimension dimF (K, M) is defined as dimF (K, M) := lim sup ε Ñ0

Hε (K, M) . log2 1{ε

© Springer Nature Switzerland AG 2018 M. Efendiev, Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations, Fields Institute Monographs 36, https://doi.org/10.1007/978-3-319-98407-0_2

71

72

2 Trajectory Dynamical Systems and Their Attractors

Hence, for a compact n-dimensional Lipschitz manifold K in a metric space M, dimF (K, M) = n. The following example shows that, for sets that are not manifolds, the fractal dimension may be a non-integer. Example 2.2 Let K be a standard ternary Cantor set in M = [0, 1]. Then 2 dimF (K, M) = ln ln 3 ă 1. Proof Let K be the Cantor set obtained from the segment [0,1] by the sequential removal of the centre thirds. First we remove  all the points between 1/3 and 2/3. Then the centre thirds 1{9, 2{9 and 7{9, 8{9 of the remaining seg  ments [0, 1{3] and r2{3, 1s are deleted. After that the centre parts 1{27, 2{27 ,       7{27, 8{27 , 19{27, 20{27 and 25{27, 26{27 of the four remaining segments r0, 1{9s, r2{9, 1{3s, r2{3, 7{9s and r8{9, 1s, respectively, are deleted. If we continue this process to infinity, it will lead to the standard Cantor Ş8 set K. Next we calculate its fractal dimension. We emphasize that K = M=0 θm , where θ0 = [0, 1], θ1 = [0, 1{3] Y [2{3, 1], θ2 = [0, 1{9] Y [2{9, 1{3] Y [2{3, 7{9] Y [8{9, 1] and so on. Each of the sets θm can be considered as a union of 2m segments of length 3´m . In particular, the cardinality of the covering of the set K with segments of length 3´m is equal to 2m . Consequently ln 2m ln 2 dimF (K, [0, 1]) = limmÑ8 ln(3 m ) = ln 3 It is not difficult to show that (1) (2) (3) (4)

if K1 Ď K2 , then dimF (K1 , M) ď dimF (K2 , M) dimF (K1 Y K2 , M) ď max tdimF (K1 , M); dimF (K2 , M)u dimF (K1 ˆ K2 , M ˆ M) ď dimF (K1 , M) + dimF (K2 , M) let g be a Lipschitzian mapping of one metric space M1 into another M2 . Then dimF (g(K), M2 ) ď dimF (K, M1 ). The next example gives the typical behavior of the entropy in classes of functions with finite smoothness.

Example 2.3 Let V be a smooth bounded domain of Rn and let K be the unit ball in the Sobolev space W l1 ,p1 (V ) and M be another Sobolev space W l2 ,p2 (V ) such that the embedding W l1 ,p1 Ă W l2 ,p2 is compact, i.e. l1 ą l2 ě 0,

1 1 l1 l2 ´ ą ´ . n p1 n p2

Then, the entropy Hε (K, M) has the following asymptotics (see [101]): C1

n{(l1 ´l2 ) n{(l1 ´l2 ) 1 1 ď Hε (K, M) ď C2 . ε ε

Finally, the last example shows the typical behavior of the entropy in classes of analytic functions. Example 2.4 Let V1 Ă V2 be two bounded domains of C3n . We assume that K is the set of all analytic functions φ in V2 such that }φ}C(V2 ) ď 1 and that M = C(V1 ). Then

2.2 Global Attractors and Finite-Dimensional Reduction

73

  n+1 n+1 C1 log2 1{ε ď Hε (K|V1 , M) ď C2 log2 1{ε , (see [73]).

2.2 Global Attractors and Finite-Dimensional Reduction It is well-known that, one of the main concepts of the modern theory of DS in infinite dimensions is that of the global attractor. We give below its definition for an abstract semigroup S(t) acting on a metric space , although, without loss of generality, the reader may think that (S(t), ) is just a DS associated with one of the PDEs described in the introduction. To this end, we first recall that a subset K of the phase space  is an attracting set of the semigroup S(t) if it attracts the images of all the bounded subsets of , i.e., for every bounded set B and every ε ą 0, there exists a time T (depending in general on B and ε) such that the image S(t)B belongs to the ε-neighborhood of K if t ě T . This property can be rewritten in the equivalent form lim distH (S(t)B, K) = 0,

t Ñ8

where distH (X, Y ) := supx PX infy PY d(x, y) is the nonsymmetric Hausdorff distance between subsets of . We now give the definition of a global attractor, following Babin-Vishik (see [1, 35, 48, 98]). Definition 2.3 A set A Ă  is a global attractor for the semigroup S(t) if 1) A is compact in ; 2) A is strictly invariant: S(t)A = A, for all t ě 0; 3) A is an attracting set for the semigroup S(t). Thus, the second and third properties guarantee that a global attractor, if it exists, is unique and that the DS reduced to the attractor contains all the nontrivial dynamics of the initial system. Furthermore, the first property indicates that the reduced phase space A is indeed “thinner” than the initial phase space  (we recall that, in infinite dimensions, a compact set cannot contain, e.g., balls and should thus be nowhere dense). In most applications, one can use the following attractor’s existence theorem. Theorem 2.1 Let a DS (S(t), ) possess a compact attracting set and the operators S(t) :  Ñ  be continuous for every fixed t. Then, this system possesses the global attractor A which is generated by all the trajectories of S(t) which are defined for all t P R and are globally bounded.

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2 Trajectory Dynamical Systems and Their Attractors

The strategy for applying this theorem to concrete equations of mathematical physics is the following. In a first step, one verifies a so-called dissipative estimate which has usually the form }S(t)u0 } ď Q(}u0 } )e´αt + C˚ , u0 P ,

(2.1)

where }¨} is a norm in the function space  and the positive constants α and C˚ and the monotonic function Q are independent of t and u0 P  (usually, this estimate follows from energy estimates and is sometimes even used in order to “define” a dissipative system). This estimate obviously gives the existence of an attracting set for S(t) (e.g., the ball of radius 2C˚ in ), which is however noncompact in . In order to overcome this problem, one usually derives, in a second step, a smoothing property for the solutions, which can be formulated as follows: }S(1)u0 }1 ď Q1 (}u0 } ), u0 P ,

(2.2)

where 1 is another function space which is compactly embedded into . In applications,  is usually the space L2 () of square integrable functions, 1 is the Sobolev space H 1 () of the functions u such that u and ∇x u belong to L2 () and estimate (2.2) is a classical smoothing property for solutions of parabolic equations (for parabolic equations in unbounded domains and for hyperbolic equations, a slightly more complicated asymptotic smoothing property should be used instead of (2.2), see the Sect. 3.2 of the monograph [48] and the references therein. Since the continuity of the operators S(t) usually poses no difficulty (if the uniqueness is proven), then the above scheme gives indeed the existence of the global attractor for most of the PDE of mathematical physics in bounded domains. Remark 2.1 As was shown in [1] the assumption that S(t) :  Ñ  be continuous for every fixed t can be replaced by the closedness of the graph t(u0 , S(t)u0 ), u0 P u. Remark 2.2 Although the global attractor has usually a very complicated geometric structure, there exists one exceptional class of DS for which the global attractor has a relatively simple structure which is completely understood, namely the DS having a global Lyapunov function. We recall that a continuous function L :  Ñ R is a global Lyapunov function if 1) L is non-increasing along the trajectories, i.e. L(S(t)u0 ) ď L(u0 ), for all t ě 0; 2) L is strictly decreasing along all non-equilibrium solutions, i.e. L(S(t)u0 ) = L(u0 ) for some t ą 0 and u0 implies that u0 is an equilibrium of S(t). It is well known that, if a DS possesses a global Lyapunov function, then, at least under the generic assumption that the set R of equilibria is finite, every trajectory u(t) stabilizes to one of these equilibria as t Ñ +8. Moreover, every complete bounded trajectory u(t), t P R, belonging to the attractor is a heteroclinic orbit joining two equilibria. Thus, the global attractor A can be described as follows [1, 48, 98]:

2.2 Global Attractors and Finite-Dimensional Reduction

A=

ď

75

M+ (u0 ),

u0 PR

where M+ (u0 ) is the so-called unstable set of the equilibrium u0 (which is generated by all heteroclinic orbits of the DS which start from the given equilibrium u0 P A). It is also known that, if the equilibrium u0 is hyperbolic (generic assumption [1]), then the set M+ (u0 ) is a κ-dimensional submanifold of , where κ is the instability index of u0 . Thus, under the generic hyperbolicity assumption on the equilibria, the attractor A of a DS having a global Lyapunov function is a finite union of smooth finite-dimensional submanifolds of the phase space . These attractors are called regular (following Babin-Vishik (see [1]). It is also worth emphasizing that, in contrast to general global attractors, regular attractors are robust under perturbations. Moreover, in some cases, it is also possible to verify the so-called transversality conditions (for the intersection of stable and unstable manifolds of the equilibria) and, thus, verify that the DS considered is a Morse-Smale system. In particular, this means that the dynamics restricted to the regular attractor A are also preserved (up to homeomorphisms) under perturbations. In the sequel we will apply Theorem 2.1 or Remark 2.1 (whenever it will be necessary) to a class of PDEs arising in mathematical physics. We especially emphasize that one of the challenging questions in the theory of attractors is, in which sense are the dynamics on the global attractor finite-dimensional? As already mentioned, the global attractor is usually not a manifold, but has a rather complicated geometric structure. So, it is natural to use the definitions of dimensions adopted for the study of fractal sets here. We restrict ourselves to the so-called fractal (or box-counting, entropy) dimension, although other dimensions (e.g., Hausdorff, Lyapunov, etc.) are also used in the attractors’ theory. Here the so-called Mané theorem (which can be considered as a generalization of the classical Whitney embedding theorem for fractal sets) plays an important role in the finite-dimensional reduction theory (see [98]). Theorem 2.2 Let  be a Banach space and A be a compact set such that df (A) ă N for some N P N. Then, for “almost all” 2N + 1-dimensional planes L in , the corresponding projector !L :  Ñ L restricted to the set A is a Hölder continuous homeomorphism. Thus, if the finite fractal dimensionality of the attractor is established, then, fixing a hyperplane L satisfying the assumptions of the Mané theorem and projecting the attractor A and the DS S(t) restricted to A onto this hyperplane (A¯ := !L A and ´1 ¯ which is ¯ ¯ S(t) := !L ˝ S(t) ˝ !L ), we obtain indeed a reduced DS (S(t), A) 2N +1 defined on a finite-dimensional set A¯ Ă L „ R . Moreover, this DS will be Hölder continuous with respect to the initial data. Remark 2.3 Note that, good estimates on the dimension of the attractors in terms of the physical parameters are crucial for the finite-dimensional reduction described above and (consequently) there exists a highly developed machinery for obtaining

76

2 Trajectory Dynamical Systems and Their Attractors

such estimates. The best known upper estimates are usually obtained by the socalled volume contraction method which is based on the study of the evolution of infinitesimal k-dimensional volumes in the neighborhood of the attractor (and, if the DS considered contracts the k-dimensional volumes, then the fractal dimension of the attractor is less than k, (see [1, 98]) Remark 2.4 Lower bounds on the dimension are usually based on the observation that the global attractor always contains the unstable manifolds of the (hyperbolic) equilibria. Thus, the instability index of a properly constructed equilibrium gives a lower bound on the dimension of the attractor, (see [1, 48, 98]). The following Theorem 2.3 plays the decisive role in the study of the dimension of attractor, which in turn does not require differentiability of the associated semigroup in contrast to (see [1, 35, 98]). We especially emphasize for a quite large class of degenerate parabolic system arising in the modelling of life science problem (see [46]) the associate semigroup is not differentiable. We denote by S := S(1). Theorem 2.3 Let H1 and H be Banach spaces, H1 be compactly embedded in H and let K ĂĂ H . Assume that there exists a map S : K Ñ K, such that S(K) = K and the following ‘smoothing’ property is valid }S (k1 ) ´ S (k2 ) }H1 ď C}k1 ´ k2 }H

(2.3)

for every k1 , k2 P K. Then the fractal dimension of K in H is finite and can be estimated in the following way: dF (K, H ) ď H1{4C (B (1, 0, H1 ) , H )

(2.4)

where C is the same as in (2.3) and B(1, 0, H1 ) denotes the unit ball in the space H1 . ε Proof Let tB(ε, ki , H )uN i=1 , ki P K, be some ε-covering of the set K (here and below we denote by B(ε, k, V ) the ε-ball in the space V , centered in k). Then ε according to (2.3), the system tB(Cε, L(ki ), H1 )uN i=1 of Cε-balls in H1 covers the set S(K) and consequently (since S(K) = K) the same system covers the set K.

Cover now every H1 -ball with radius Cε by a finite number of 4ε -balls in H . By definition, the minimal number of such balls equals to Nε{4 (B(Cε, S(ki ), H1 ), H ) = Nε{4 (B(Cε, 0, H1 ), H ) = N1{4C (B(1, 0, H1 ), H ) ” N . Note, that the centers of ε{4-covering thus obtained not necessarily belongs to K but we evidently can construct the ε{2-covering with centers in K and with the same number of balls. Thus, having the initial ε-covering of K in H with the number of balls Nε we have constructed the ε{2-covering with the number of balls Nε{2 = N Nε . Consequently, the ε-entropy of the set K possesses the following estimate.

2.3 Classification of Positive Solutions of Semilinear Elliptic Equations in a. . .

77

In fact the assertion of the theorem is a corollary of this recurrent estimate. Indeed, since K ĂĂ H then there exists ε0 such that K Ă B(ε0 , k0 , H ) and consequently Hε0 (K, N ) = 0.

(2.5)

Iterating the estimate (2.5) n-times we obtain that Hε0 {2n (k, H ) ď n log2 N . Fix now an arbitrary ε ą 0 and choose n = n(ε) in such a way that ε0 ε0 . ďεď n 2n 2 ´1 Then  ε0  Hε (K) ď Hε{2n (K) ď n log2 N ď log2 1 + log2 N . ε

(2.6)

Thus (2.4) is an immediate consequence of (2.6). Theorem 2.3 is proved.

2.3 Classification of Positive Solutions of Semilinear Elliptic Equations in a Rectangle: Two Dimensional Case As we have seen in Chap. 1, positive solutions of semilinear second order elliptic problems have symmetry and monotonicity properties which reflects the symmetry of the operator and of the domain, see also e.g., [65] and [18] for the case of bounded domains and [13, 19, 21, 25] and [23, 30, 65] for the case of unbounded domains. In particular, the symmetry and monotonicity results for the case of a half space have been considered in [13, 21] and the analogous results (including the existence and uniqueness of a nontrivial positive solution) for the case of whole space have been obtained in [23, 30, 65, 75], see also the references therein. The goal of the present section is to give a description of all bounded nonnegative solutions of the following elliptic boundary value problem in a two dimensional rectangle + := t(x, y) P R2 , x ě 0, y ě 0u: # x,y u = f (u), (x, y) P + , u|B+ = 0, u(x, y) ě 0,

(2.7)

where we assume that u P Cb () and the nonlinearity f is smooth enough (f P C 1 (R)) and f (0) = 0.

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2 Trajectory Dynamical Systems and Their Attractors

It is known (see [13]) that, under the above assumptions, every solution u(x, y) of (2.7) (if it exists) should be monotonic with respect to x and y and, consequently, there exist the following limits lim u(x, y) = ψu (y),

x Ñ8

lim u(x, y) = φu (x).

y Ñ8

(2.8)

Moreover functions ψu and φu bounded solutions of one dimensional analogue of problem (2.7) " 2 = f ("), "(0) = 0, "(z) ě 0, z ě 0.

(2.9)

We recall, that every solution of (2.9) stabilizes as z Ñ 8 to some c ě 0 such that f (c) = 0 and, for fixed c there exists not greater than one solution "(z) = "c (z) of this problem. Consequently, the functions ψu and φu in (2.8) should coincide: ψu (z) = φu (z) = "c (z), where the constant c = cu ą 0 satisfies f (c) = 0. Thus, we can rewrite (2.8) in the following form: lim

(x,y)Ñ8

|u(x, y) ´ "c (x, y)| = 0, where "c (x, y) := mint"c (x), "c (y)u. (2.10)

The aim of this notes is to verify the existence and uniqueness of a solution u(x, y) satisfying (2.10). We establish this fact under the following nondegeneracy assumption that f 1 (c) ‰ 0

(2.11)

(in a fact, the existence of a solution "c (z) of Eqs. (2.9) and (2.11) imply that f 1 (c) ą 0, see [15]). Thus, the main result of the chapter is the following theorem. Theorem 2.4 Let the nonlinearity f satisfy the above assumptions, "c be a solution of (2.9) such that f 1 (c) ą 0. Then, there exists a unique solution u(x, y) of (2.7) which satisfies (2.10). The following corollary shows that, generically, Eq. (2.7) has only finite number of different positive solutions. Corollary 2.1 Let the above assumptions hold and let, in addition, inequality (2.11) hold, for every solution c ą 0 of equation f (c) = 0. Then, problem (2.7) has the finite number of different positive bounded solutions.

2.3.1 Sketch of the Proof of Theorem 2.4 For the proof, we need the following lemma.

2.3 Classification of Positive Solutions of Semilinear Elliptic Equations in a. . .

79

Lemma 2.1 Let the assumptions of Theorem 2.4 hold and let "cM (x, y)

:=

# c,

(x, y) P [0, M]2 , (x, y) P + z[0, M]2 ,

"c (x, y),

(2.12)

where M is sufficiently large positive number. Then, the spectrum of the operator x,y ´ f 1 ("cM (x, y)) in + (with the Dirichlet boundary conditions) is strictly negative: σ (x,y ´ f 1 ("cM ), L2 (+ )) ď ´K ă 0.

(2.13)

Indeed, estimate (2.13) can be easily deduced from the standard fact that σ (Bz2 ´ f 1 ("c (z)), L2 (R+ )) ď ´K ď 0

(2.14)

(which is the corollary of the Perron-Frobenius theorem), the minimax principle and the special form of the function "c (x, y). for some positive K and, consequently, ż8

1

1

|φ (z)| + f ("c (z))|φ(z)| dz ě K 2

2

0

ż8 0

|φ(z)|2 dz, φ P C08 (R+ ). (2.15)

We claim that spectrum of the two dimensional operator x,y ´ f 1 ("c (x, y)) in + is also strictly negative: σ (x,y ´ f 1 ("c (x, y))) ď ´K ď 0

(2.16)

Indeed, in order to prove this, it is sufficient to verify that ż +

|(φx1 |2

+ |φy1 |2

1

+ f ("c (x, y))|φ| dx dy ě K 2

ż +

|φ|2 dx dy, φ P C08 (+ ) (2.17)

But the last formula is an immediate corollary of (2.15) (after passing to new variables (x 1 , y 1 ) rotated on π {4 with respect to the initial variables (x, y)). Thus, formula (2.16) is verified. The following two corollaries of Lemma 2.1 are of fundamental significance for what follows. Corollary 2.2 Let the assumptions of Theorem 2.4 hold and let u(x, y) be a positive bounded solution of (2.7) which satisfies (2.10). Then: σess (x,y ´ f 1 (u(x, y)), L2 ()) ď ´K ă 0.

(2.18)

80

2 Trajectory Dynamical Systems and Their Attractors

Indeed, due to (2.10) and (2.12) the operator x,y ´ f 1 (u(x, y)) is a compact perturbation of x,y ´ f 1 ("cM ). Corollary 2.3 Let the assumptions of Corollary 1.1 hold. Then, the rate of decaying in (2.10) is exponential, i.e. there exist positive constants ε ě 0 and C depending on u such that |u(x, y) ´ "c (x, y)| ď Ce´ε(x+y) , (x, y) P + .

(2.19)

Indeed, estimate (2.19) is more or less standard corollary of (2.13), convergence (2.10) and the maximum principle, so we left its rigorous proof to the reader. We are now ready to verify the existence of a solution u(x, y). To this end, we consider the following sequence of auxiliary problems in the domains N := t(x, y) P + , y ď Nu: # x,y uN = f (uN ), u(x, y) ě 0, u(0, y) = u(x, 0) = 0, u(x, N ) = "c (x).

(2.20)

Obviously, for every N P N, this problem has at least one solution uN (x, y) satisfying 0 ď uN (x, y) ď c

(2.21)

(which can be obtained using u´ = 0 and u+ = c as sub and super solutions respectively for problem (2.20), see e.g. [104]). Moreover, this solution is also monotonic with respect to x and y and tends exponentially as x Ñ 8 to "c (y) (analogously to Corollary 1.2). We also note that, due to the elliptic regularity theorem, estimate (2.21) implies that }uN }C 2 (+ ) ď C b

(2.22)

where the constant C is independent of N . Thus, without loss of generality, we may assume that the sequence uN tends in 2 ( ) to a some solution u(x, y) of problem (2.7) as N Ñ 8. As we have Cloc + explained in the introduction, this implies that there exists 0 ď c1 ď c (may be c1 = 0) such that f (c1 ) = 0 and lim

(x,y)Ñ8

|u(x, y) ´ "c1 (x, y)| = 0.

(2.23)

We need to prove that, necessarily, c1 = c. We prove this fact using the special integral identity. In order to derive it, we multiply Eq. (2.20) by Bx uN . Then, we have Bx (|Bx uN |2 ´ |By uN |2 ´ 2F (uN )) = ´2By (Bx uN ¨ By uN )

(2.24)

2.3 Classification of Positive Solutions of Semilinear Elliptic Equations in a. . .

81

where F (u) is a potential of f (u). Integrating this formula over N and using the boundary conditions and the fact that |"c1 (0)|2 = ´2F (c) ě 0, we derive that żN 0

(|"c1 (0)|2

´ |Bx uN (0, y)| ) dy = 2

żN 0

´2

2[F (c) ´ F ("c (y))] + |"c1 (y)|2 dy ż8 0

"c1 (x) ¨ By uN (x, N ) dx.

(2.25)

Since "c1 (x) ě 0 and By uN (x, N ) ě 0, then żN 0

(|"c1 (0)|2 ´ |Bx uN (0, y)|2 ) dy ď C"c

(2.26)

where the constant C"c is independent of N . Moreover, obviously, the function Bx uN (0, y) is strictly increasing with respect to y and Bx uN (0, N ) = "c1 (0). Consequently, (2.26) implies that żN 0

|"c1 (0)2 ´ Bx uN (0, y)2 | dy ď C"c .

(2.27)

We now note that Bx u(0, y) is monotone increasing function (since u(x, y) is monotone with respect to y and u(0, y) = 0) and Bx u(0, y) ă Bx u(0, 8) = "c1 1 (0), @y P R+ .

(2.28)

2 ( )) then Since "c1 1 (0) ă "c1 (0) if c1 ă c, see [15] and uN Ñ u in Cloc + estimates (2.27) and (2.28) imply that the limit function u(x, y) satisfies (2.23) with c = c1 . Thus, the existence of a solution is verified. Let us now verify the uniqueness of the constructed solution u(x, y). To this end, we need the following lemma which is of independent interest also.

Lemma 2.2 Let u(x, y) be an arbitrary solution of (2.7) which satisfies (2.10). Then the spectrum of the linearization of (2.7) on u(t, x) is strictly negative, i.e. σ (x,y ´ f 1 (u)) ď ´Cu ,

(2.29)

for some positive constant Cu , depending on the solution u. Proof Indeed, assume that (2.29) is wrong. Then, according to (2.18), there exists a nonnegative eigenvalue λ0 ě 0 of this operator and the corresponding eigenvector v P L2 (+ ). Moreover, it can be deduced in a standard way, using condition (2.13) and the exponential convergence (2.19) that |v(x, y)| ď Cv e´ε(x+y) , (x, y) P + ,

(2.30)

82

2 Trajectory Dynamical Systems and Their Attractors

for some positive constant Cv , depending on v. We may also assume, without loss of generality, then the eigenvalue λ0 ě 0 is maximal. Then, thanks to the PerronFrobenius theory, function v(x, y) is strictly positive inside of + . We note that the function v1 (x, y) := Bx u(x, y) is also strictly positive and satisfies the equation x,y v1 ´ f 1 (u(x, y))v1 = 0.

(2.31)

Multiplying this equation by the eigenvector v(x, y) and integrating over + , integrating by parts and using the boundary conditions, we derive that ż8

v1 (0, y)Bx v(0, y) dy + λ0

0

ż

v ¨ v1 dx dy = 0.

(2.32)

+

We now recall that v1 (x, y) := Bx u(x, y) ě 0, v(x, y) ě 0 and Bx v(0, y) ą 0 (due to the strict maximum principle). Consequently, (2.32) implies that v1 (0, y) := Bx u(0, y) ” 0.

(2.33)

Since, u(0, y) ” 0 due to the boundary conditions, then (2.33) implies that u(x, y) ” 0 (due to the uniqueness theorem for elliptic equations). This contradiction proves estimate (2.29) and Lemma 2.2. \ [ Now we are ready to verify the uniqueness. Indeed, let u1 (x, y) and u2 (x, y) be two solutions of problem (2.7) which satisfy (2.10). Then, without loss of generality, we may assume that u2 (x, y) ě u1 (x, y).

(2.34)

Indeed, if (2.34) is not satisfied, then, using the sub and supersolution method (parabolic equation method, see e.g., [104]), we may construct the third solution u3 (x, y) such that c ě u3 (x, y) ě maxtu1 (x, y), u2 (x, y)u

(2.35)

which is not coincide with u1 and u2 and for which (2.34) is satisfied. Let us now consider the parabolic boundary value problem in + Bt U = x,y U ´ f (U ), U |B+ = 0, U |t=0 = U0

(2.36)

with the phase space W0 := tU0 P L8 (+ ), u1 (x, y) ď U0 (x, y) ď u2 (x, y)u. Then, this problem generates a semiflow on the phase space W0 :

(2.37)

2.4 Existence of Solutions of Nonlinear Elliptic Systems

St : W0 Ñ W0 , St U0 := U (t)

83

(2.38)

which (according to the general theory, see [1, 98] and [52] and Sect. 2.2) possesses a global attractor A0 Ă W0 . Moreover, due to (2.19) and (2.37), we have the following Lyapunov function on W0 : L(U0 ) :=

ż +

|∇(U0 ´ u1 )|2 + 2Fu1 (U0 ´ u1 , x, y) dx dy

(2.39)

şz where Fu1 (z, x, y) := 0 f (u1 (x, y) + z) ´ f (u1 (x, y)) dz. Thus, the attractor A0 should consist of heteroclinic orbits to the appropriate equilibria, belonging to W0 (see [1]), but as proved in Lemma 2.2, all of these equilibria are exponentially stable which is possible only in the case u1 ” u2 . Therefore, the uniqueness is also proved and Theorem 2.4 is proved. \ [

2.4 Existence of Solutions of Nonlinear Elliptic Systems In this section and in subsequent Sects. 2.5–2.13, we study, mainly following [103, 108], the existence of at least one solution of the following nonlinear elliptic system in an unbounded domain  Ă Rn : " au ´ γ ¨ Du ´ f (x, u) = g(x), x P , (2.40) u|B = u0 , where u = (u1 , .., uk ), f = (f 1 , . . . , f k ) and g = (g 1 , . . . , g k ),  is the Laplacian with respect to x = (x 1 , . . . , x n ), a P L(Rk , Rk ) is the k ˆ k matrix with constant coefficients satisfying a + a ˚ ą 0, and γ ¨ Du is a differential operator of first order with constant coefficients, that is, γ ¨ Du =

n ÿ

γ i Bxi u, γ i P L(Rk , Rk ).

(2.41)

i=1

Systems of the form (2.40) were studied in [76] under some (more restrictive) conditions both on f, g, u0 and the geometry of  Ă Rn . In what follows, we make the following assumptions on the data in (2.40). Condition 2.1 (Assumptions on f ) 1. f P C( ˆ R, R); r 2. f (x, u) ¨ u ě ´C  1 + C2 |u| , C1 , C2 ą 0 are some constants; 3. |f (x, u)| ď C 1 + |u|r ´1 , r ą 2, C ą 0 are some constants. Here ξ ¨ η denotes the inner product in Rk .

84

2 Trajectory Dynamical Systems and Their Attractors

Condition 2.2 (Assumptions on ) There exists a direction l P Rn such that 1. TŤ h  Ă ;  2. T´h  = RN , where Th x = x + hl. hě 0

Condition 2.3 (Assumptions on u0 ) u0 P ν0 (B). Condition 2.4 (Assumption on g) g = (g 1 , . . . , g k ) belongs to the space (): ´1,2 () X Lloc ()], () = [Wloc q

where

1 r

+

1 q

= 1, and r is the same as in the Assumptions on f (Condition 2.1).

For simplicity of presentation, we start with f (x, u) ” f (u), g ” 0, γ ” 0 and a ” I d. As we will see from the proof the existence of solution to the problem (2.40), the general case can be studied in the similar manner. We will also give necessary hints for the proof of the general case. Therefore, we are dealing with the existence of at least one solution of the following semilinear elliptic system: "

u = f (u), x P , u|B = u0 .

(2.42)

We especially emphasize that no assumptions on B are made. Since we do not assume a Lipschitz property for f , uniqueness cannot be guaranteed in general. Theorem 2.5 Let f , , u0 satisfy the assumptions mentioned above. Then the semilinear elliptic boundary value problem (2.42) possesses at least one solution in θ (). Proof We will prove Theorem 2.5 in two stages: Step 1 First we prove that (2.42) is solvable in any bounded domain. To prove the existence of solution in this case, we reduce (2.42) to the case with homogenous boundary condition. Indeed, let v P θ () such that v|B = u0 and ||v||θ() ď 2||u0 ||ν0 (B) . Existence of such v P θ () follows from the definition of ν0 (B) (see Sect. 1.1 for the definitions of θ () and ν0 (B)). We set w = u ´ v. Then, w satisfies " w ´ f (w + v) = v, w|B = 0. Let tej (x)| j P Nu, ej (x) P C08 () - orthogonal basis in L2 () which is complete in θ0 (). Let PN be the orthoprojector: PN [L2 ()] = VN := span te1 , . . . , eN u. We consider the following finite-dimensional problem in VN : PN (wN ´ f (wN + v) ´ v) = 0.

(2.43)

2.4 Existence of Solutions of Nonlinear Elliptic Systems

85

Proposition 2.1 The problem (2.43) possesses at least one solution wN (x) for each N P N. Proof We define a map  : VN Ñ VN , (ξ ) := ´PN (ξ ´ f (ξ + v) + v), ξ P VN . It is not difficult to see that  P C(VN , VN ). We estimate ((ξ ), ξ ) from below to show that ((ξ ), ξ ) ě 0 for all ξ P BBR 0 for sufficiently large R. Indeed,   ´ (ξ, ξ ) ě C ||ξ ||2W 1,2 () + ||ξ ||2L2 () .

(2.44)

Due to the last condition on f , we obtain (f (ξ + v), ξ ) = (f (ξ + v), ξ + v) ´ (f (ξ + v), v)     ě ´C1 + C2 |ξ + v|r , 1 ´ C 1 + |ξ + v|r ´1 |v|, 1     ě ´C1 + 2C3 (|ξ |r ´ |v|r ), 1 ´ C + C3 |ξ |r + C4 |v|r , 1   (2.45) ěC0 ||ξ ||rLr ´ C 1 + ||u0 ||rν0 (B) ; |(v, ξ )| ď||v||θ() ||ξ || ()   ďCε 1 + ||v||2θ() + ε||ξ ||2W 1,2 () + ε||ξ ||2Lr () .

(2.46)

Taking into account (2.44)–(2.46), we obtain   ((ξ ), ξ ) ěC ||ξ ||rLr () + ||ξ ||2W 1,2 () + ||ξ ||2L2 ()   ´ C1 1 + ||u0 ||rν0 (B) , C ą 0.

(2.47)

From (2.47) is follows that, for sufficiently large R ą 0, in  ( BR 0 = ξ P VN | ||ξ ||L2 () ă R , we have ((ξ ), ξ ) |BBR ą 0. 0

(2.48)

Hence, from the classical theorem on Brouwer’s degree (see [47, 106]) it follows that the equation (ξ ) = 0 has a solution ξ = wN for each N in VN .

86

2 Trajectory Dynamical Systems and Their Attractors

Corollary 2.4 Let wN be a solution. Then   ||wN ||θ() ď C 1 + ||u0 ||rν0 (B)

(2.49)

uniformly in N. Indeed, it follows from (2.49). To this end, we have to use (2.47) with taking into account that (wN ) = 0. Now we are in position to pass to the limit in Galerkin approximation. From (2.49) it follows that wN á w in θ (). Our goal is to prove that w is a desirable solution. Accordingly to the definition of weak solution, we have to show that ´(∇w, ∇ϕ) ´ (f (v + w), ϕ) = ´(v, ϕ)

(2.50)

for each ϕ P C08 (). Let ϕ P VN0 for some N0 P N. Then PN0 ϕ = ϕ. Multiplying (2.42) by ϕ, for sufficiently large N ąą 1, we have ´(∇wN , ∇ϕ) ´ (f (v + wN ), ϕ) = ´(v, ϕ). By definition of θ () it follows that ´(∇wN , ∇ϕ) Ñ ´(∇w, ∇ϕ). Next, we show that (f (v + wN ), ϕ) Ñ (f (v + w), ϕ) as N Ñ 8. According to the assumptions on f , we obtain that the map G(ξ ) := f (ξ + v) is a continuous map (see Sect. 1.3) G : Lr ´1 () Ñ L1 (). Since θ () is compactly embedded in Lr ´δ () for sufficiently small δ ą 0, we have that wN Ñ w in Lr ´1 (), and, consequently, f (v + wN ) Ñ f (v + w) in L1 (). Thus, (2.50) holds for all ϕ = PN ϕ for some N Ă N. Since tej (x)u is a complete system in θ (), we obtain that (2.50) holds for any ϕ P C08 (). Thus, we proved that (2.42) has a solution belonging to θ (), in the case  is a bounded domain. \ [

2.4 Existence of Solutions of Nonlinear Elliptic Systems

87

Next, we prove uniform (in  Ă Rn ) a priori estimate of the form (2.51), and, based on this a priori estimate, we will prove existence of at least one solution u P θ () for any unbounded domain Rn . Lemma 2.3 Let  be any open set in RN , u0 P ν0 (B) and u is a solution of (2.42). Then   ||u||θ(XBRx ) ď Cε 1 + ||u0 ||r

  ν0 B XBR+ε x0

0

,

(2.51)

where ε ą 0, Cε -depend on ε, R but independent of  Ă RN and x0 P RN . Proof Let us fix R, ε and x0 P RN . Due to the definition of ν0 (B) there exist v P θ () such that   θ XBR+ε x0

||v||

  ν0 B XBR+ε x0

ď 2||u0 ||

(2.52)

with v|B = u0 . Hence, w = u ´ v P θ0 () and satisfies w ´ f (v + w) = ´v.

(2.53)

 2r ´ |x ´ x0 | r´2 , |x ´ x0 | ă R + 2ε , |x ´ x0 | ě R + 2ε .

(2.54)

We define ψ(x): # R+ ψ(x) = 0,

ε 2

It is easy to compute and show that 1

1

|∇ψ| ď Cψ 2 + r (x). To show (2.52), we multiply (2.53) by ψw and integrate: (w, ψw) ´ (f (v + w), ψw) = (´v, ψw).

(2.55)

  , we obtain Since ψw P θ0  X BR+ε x0 ´(w, ψw) = (∇w, ∇(ψw)) = (ψ∇w, ∇w) + (∇w, w∇ψ)  1  1 ě 2C(ψ|∇w|2 , 1) ´ C1 ψ 2 |∇w|, |∇ψ|ψ ´ 2 |w|  2  ě C(ψ|∇w|2 , 1) ´ C˜ ψ r |w|2 , 1 .

(2.56)

88

2 Trajectory Dynamical Systems and Their Attractors 1

1

To obtain (2.56), we used |∇ψ| ď Cψ 2 + r (x) and Cauchy-Schwartz inequality. Due to the assumptions on f we obtain (ψf (v + w), w) ě C1 (ψ|w|r , 1) ´ C2 ||u0 ||r

 . ν0 B XBR+ε x0

(2.57)

Moreover,   ||ψw||   θ XBR+ε θ XBR+ε x0 x0

|(´v, ψw)| ď || ´ v||

  2  ď ε(ψ|∇w|2 , 1) + ε(ψ|w|r , 1) + Cε 1 + ψ r |w|2 , 1  + || ´ v||2 

θ XBR+ε x0



+ ||u0 ||r

  ν0 B XBR+ε x0

(2.58)

Using (2.56)–(2.58), from (2.55) we obtain (ε ăă 1).  2  (ψ|∇w|2 , 1) + (ψ|w|r , 1) ´ C ψ r |w|2 , 1  ďC1

 1 + || ´ v||2  θ XBR+ε x0



 + ||u0 ||r  ν0 B XBR+ε x0

(2.59)

According to the Hölder inequality ˇ 2 ˇ 1 ˇr  ˇ ˇˇ ˇ 1 ˇ2 ˇˇ ˇ ˇ ˇ ˇ ˇ ˇ r ˇ ψ |w|2 , 1 ˇ = ˇˇ ˇψ r w ˇ , 1 ˇˇ ď ε ˇψ r w ˇ , 1 + Cε = ε(ψ|w|r , 1) + Cε . (2.60) Inserting (2.60) into (2.59) and taking ε ăă 1 we have:  (ψ|∇w| , 1) + (ψ|w| , 1) ď C0 2

r



 1 + ||u0 ||r  ν0 B XBR+ε x0

.

(2.61)

Since ψ(x) ą C ą 0 when x P BR x0 , (2.61) leads to  (|∇w| , 1) + (|w| , 1) ď C0 2

r

which is the assertion of Lemma 2.3.



 1 + ||u0 ||r  ν0 B XBR+ε x0

\ [

Step 2 Now we are ready to prove Theorem 2.5 for any unbounded domain. Indeed, let  be unbounded. We consider

2.4 Existence of Solutions of Nonlinear Elliptic Systems

=

89

! ) k , k :=  X Bk0 , Bk0 := x P RN | |x| ď k .

8 ď k=1

Let ϕk (x) P C08 (RN ) such that 0 ď ϕk (x) ď 1, ϕk (x) = 1 for x P B0k ´1 and ϕk (x) = 0 for x P RN zBk0 . Let uk0 := ϕk v|Bk , where v P θ (), v|B = u0 . It is not difficult to see that uk0 |BXBk´1 = u0 |BXBk´1 , uk0 |Bk zB = 0. 0

0

According to Step 1, @k P N there exists uk solution of (2.42) for k , with uk |Bk = uk0 and ||uk ||θ0 (M ) ď CM uniformly in k ě M +1. Since θ0 (M ) is reflexive, using the Cantor diagonalization procedure, one can extract a subsequence from tuk u (for simplicity we denote it also by tuk u), such that uk á u in θ (M ) @M P N, k ě M + 1. Let us show that u is the desirable solution of (2.42). Indeed, let ϕ P C08 () and supp ϕ P L . Then, for k ą L, we have ´ (∇uk , ∇ϕ) = (f (uk ), ϕ) . Passing to the limit as k Ñ 8 in the last equality (we justify it below), we obtain ´ (∇u, ∇ϕ) = (f (u), ϕ) . Hence u is a solution of (2.42). This proves Theorem 2.5 in the case f = f (u), g ” 0, γ ” 0 and a ” I d. Remark 2.5 In general case, that is, in the presence of f = f (x, u), γ ¨ Du, g = g(x), we proceed in the Step 1 as follows: We consider a map  : VN Ñ VN : (ξ ) := ´PN (aξ + γ ¨ Du ´ f (x, ξ + v) ´ g1 (x)), ξ P VN ,

(2.62)

where g1 (x) := g(x) ´ av ´ γ ¨ Dv. We aim to prove that for sufficiently large R ą 0, ((ξ ), ξ ) ě 0 for all ξ P BB0R . To this end, we have to estimate the terms: (´aξ, ξ ), (f (x, ξ + v), ξ ), (γ ¨ Dξ, ξ ) and (g1 (x), ξ ). Indeed, (´aξ, ξ ) = (a∇ξ, ∇ξ ) =

  1 ((a + a˚ )∇ξ, ∇ξ ) ě C ||ξ ||2W 1,2 () + ||ξ ||2L2 () . 2 (2.63)

90

2 Trajectory Dynamical Systems and Their Attractors

Here (a∇ξ, ∇ξ ) =

k ÿ   aBxi u ¨ Bxi w . i=1

The term (f (x, ξ + v), ξ ) can be estimated as follows (f (x, ξ + v), ξ ) = (f (x, ξ + v), ξ + v) ´ (f (x, ξ + v), v) ě (´C1 + C2 |ξ + v|r , 1) ´ (C(1 + |ξ + v|r ´1 |v|), 1) ě (´C1 + 2C3 (|ξ |r ´ |v|r ), 1) ´ (C + C3 |ξ |r + C4 |v|r , 1)   ě C0 ||ξ ||rLr ´ C 1 + ||u0 ||rν0 (B) .

(2.64)

As for (γ ¨ Dξ, ξ ), it can be estimated in the following way: |(γ ¨ Dξ, ξ )| ď μ||ξ ||2W 1,2 () + μ||ξ ||rLr () + Cμ , Cμ = Cμ (||),

(2.65)

where μ is an arbitrary positive real number. To obtain the above inequality, we use the Hölder inequality with exponents 2, r and r 2r ´2 . We estimate   |(g1 (x), ξ )| ď||ξ, ||+ |g1 , |+ ď Cμ 1 + |g1 , |2+   + μ ||ξ ||2W 1,2 () + ||ξ ||rLr ()

(2.66)

(for the definitions of || ¨ ||+ and | ¨ |+ see Sect. 1.1) Inserting estimates (2.63)–(2.66) into the expression for ((ξ ), ξ ): ((ξ ), ξ ) = ´(aξ, ξ ) ´ (γ ¨ Dξ, ξ ) + (f (x, ξ + v), ξ ) + (g1 (x), ξ ), we obtain that     ((ξ ), ξ ) ě C ||ξ ||2W 1,2 () + ||ξ ||rLr () ´ C1 1 + ||u0 ||rν0 (B) + |g1 , |2+ , (2.67) where C ą 0 and  is a bounded domain. Hence, from (2.66) it follows that, for sufficiently large R ą 0, ((ξ ), ξ ) ě 0 in BR 0,  ( where BR 0 = ξ P VN | ||ξ ||L2 () ď R .

2.5 Regularity of Solutions

91

Remark 2.6 Therefore, for bounded , the presence of f = f (x, u), a P L(Rk , Rk ), γ ¨ Du and g = g(x) do not bring additional difficulties. The case when  is unbounded, repeats word by word the proof of Theorem 2.5 in the special case. Hence, the analog of the estimate (2.51) in this case has the following form: 



||u||θ(XBRx ) ď Cε 1 + ||g||



 XBR+ε x0

0



 + ||u0 ||r  ν0 B XBR+ε x0

,

(2.68)

where  is an arbitrary positive number, and the constant Cε depends upon  and R, as well as the constants appearing in the assumptions on f , but is independent of  Ă Rn and x0 P Rn .

2.5 Regularity of Solutions In this section, we prove several theorems on the additional regularity of solutions for (2.40) depending on the additional regularity of g(x). First, assume that g P [Lq ()]k , where

1 1 + = 1 and r is the same as in the assumptions onf. q r (2.69)

Theorem 2.6 Let u(x) be a solution of (2.40) and the condition (2.69) be satisfied “  ‰k and BxR0 ĂĂ . Then u P H 2,q BxR0 and › › › › ›u, BxR0 ›

›2r › › R+ε › , ď C 1 + ›g, Bx0 ›

2,q

0,q

(2.70)

where BxR+ε ĂĂ . 0 Proof Let ϕ(x) P C08 (), such that ϕ(x) = 1 if |x ´ x0 | ď R and ϕ(x) = 0 if |x ´ x0 | ě R + ε. Multiplying (2.40) by ϕ and performing some elementary calculations, we obtain $ &(ϕu) = 2∇ϕ ¨ ∇u + uϕ + a ´1 (´ϕγ Du + ϕf (x, u) + ϕg) =: g2 (x), %ϕu|B R+ε = 0. x0

Here ∇ϕ ¨ ∇u = › › › › ›g2 , BxR+ε › 0

0,q

řk

i=1 Bi ϕBi u.

Analogously to (2.51),

› › ›r › › › › R+ε › g, B ďC 1 + ›u, BxR+ε + › › x0 › 0 +

0,q

› ›2r › › ďC1 1 + ›g, BxR+ε . › 0 0,q

(2.71)

92

2 Trajectory Dynamical Systems and Their Attractors

From the Lq regularity of the Laplacian it follows that › › › › ›u, BxR0 ›

2,q

› › › › ď C ›ϕu, BxR+ε › 0

2,q

› › › › ď C1 ›g2 , BxR+ε › 0

0,q

(2.72)

.

Then, the estimate (2.70) is a consequence of (2.71) and (2.72). Theorem 2.6 is thus proved. \ [ Let us assume the matrix is self-adjoint, so that a = a ˚ ą 0, and g P [Lp ()]k for some p ą N2 . Moreover, assume also that there exists a v P θ () such that   . Let us mention again that by H l,p () v|B = u0 , v|XBxR+ε P H 2,p  X BxR+ε 0 0

we denote the space W l,p (Rn )| . Then, we have a stronger result than the one stated in Theorem 2.6: Theorem 2.7 Let u P θ () be a solution of (2.40) and all above mentioned “ ‰k assumptions fulfilled. Then, u P L8 ( X BxR0 ) and it holds › › › › ›u,  X BxR0 ›

0,8

› › › › ď Q ›g,  X BxR+ε › 0

0,p

› › › › + Q ›v,  X BxR+ε › 0

H 2,p

 , (2.73)

where Q : R+ Ñ R+ is some monotonic function depending on f and the constants R and ε. Proof For simplicity, we consider the case v ” 0. The general case is standard (see, for example, the proof of Lemma 2.3). Hence, in what follows, we assume that u P θ0 (). To prove the assertion of Theorem 2.7, we need several lemmas, which we prove below. Lemma 2.4 There exists a ϕ P C 2 (Rn ), ϕ ě 0, such that $ ’ ’ &ϕ(x) = 1 if |x ´ x0 | ă R; ϕ(x) = 0 if |x ´ x0 | ě R + ε, 1

1

|∇ϕ(x)| ď Cϕ 2 + r (x), x P Rn , ’ ’ %|ϕ(x)| ď Cϕ 2r (x), x P Rn .

(2.74)

Proof Let h˚ (x) P C08 () such that 0 ď h˚ (x) ď 1 and h˚ (x) = 1 if x P BxR0 and h˚ (x) = 0 if |x ´ x0 | ą R + 2ε . Let ψ(t) =

# tα

t ě 0,

0

t ă 0,

where α is a sufficiently large number. Then, it is not difficult to see that the function ϕ(x) = h˚ (x) + (1 ´ h˚ (x))ψ(R + ε|x ´ x0 |) satisfies all conditions of Lemma 2.4.

\ [

2.5 Regularity of Solutions

93

Lemma 2.5 Let u P θ0 () be a solution of (2.40) and w = ϕau ¨ u. Then, the function p= w belongs to H01,l (Rn ), l =

2r r+2

# w

x P ,

0

xR

and satisfies $ &w pu (x), p=h %w| p B R+ε = 0,

(2.75)

x0

where by vp we denote the extension of a given function v by the zero when x R , and hu (x) := 2ϕa∇u ¨ ∇u + 2∇ϕ ¨ ∇(au ¨ u) + ϕau ¨ u ´ 2γ Du ¨ u + 2ϕf (x, u) ¨ u + 2ϕg ¨ u

(2.76)

p P θ (Rn ). Using the Hölder inequality with the Proof Since u P θ0 (), hence u r 2 exponents l and l , we have ż ›l › › › p BxR+ε ď C › ›∇ w, 0 0,l

BxR+ε 0

› › › › p, BxR+ε |p u|l |∇p u|l dx ď C1 ›u › 0

0,r

› › › › p, BxR+ε ›∇ u › 0

l

0,2

.

 (   p P H01,k BxR+ε Consequently, w for k = min 2r , l = l. In the sequel, we denote, 0 as usual, by ă ¨, ¨ ą and ă ¨, ¨ ą the scalar product in L2 (Rn ) and L2 (), respectively. Let  P C08 (Rn ). Then, p ϕ ą=ă w, p  ą= ´ ă ∇w, ∇ ą ă w, = ´ ă ∇φau ¨ u, ∇ ą ´ ă φ∇(au ¨ u), ∇ ą =ă φau ¨ u,  ą + ă ∇φ ¨ ∇(au ¨ u),  ą ´ ă φ∇(au ¨ u), ∇ ą (2.77) To obtain (2.77), we several times used the Green’s formula ă ∇U1 , U2 ą + ă U1 , ∇U2 ą = 0, 1,l

in which U1 P W01,l ( X BxR+ε ), U2 P W0 ˚ ( X BxR+ε ) and 1l + l1˚ = 1 (see [78]) 0 0 Let us transform ă φ∇(au ¨ u), ∇ ą in the following way: ă φ∇(au ¨ u), ∇ ą =2 ă a∇u, ∇(ϕu) ą ´2 ă ϕa∇u ¨ ∇u,  ą ´ ă ∇ϕ ¨ ∇(au ¨ u),  ą .

(2.78)

94

2 Trajectory Dynamical Systems and Their Attractors

Since u is a solution of (2.40) and ϕu P θ0 ( X BxR+ε ), we have 0 ´ ă a∇u, ∇(ϕu)+ ă ϕγ Du ¨ u,  ą ´ ă ϕf (x, u) ¨ u,  ą =ă ϕg ¨ u,  ą .

(2.79)

Inserting instead of ă φ∇(au ¨ u), ∇ ą in (2.77) its expressions obtained via (2.78) and (2.79), we obtain pu ,  ą . p  ą=ă hu ,  ą =ă h ă w, \ [

This proves Lemma 2.5.

    pu P H ´1,l B R+ε X L1 B R+ε and satisfies Lemma 2.6 The functions h x0 x0   1 pu (x) ě ´C1 1 + |p p 2 ” h(x) for almost all x P Rn . g (x)|w h

(2.80)

    pu P H ´1,l B R+ε follows pu P L1 B R+ε follows from (2.76) and h Proof h x0 x0 from (2.75) and the assertion of Lemma 2.5. Next, we prove the estimate (2.80). Similar to estimates (2.56)–(2.61), using (2.74), we obtain  hu (x) ěC(ϕ|∇u|2 + ϕ|u|r ) ´ C1 ϕ|∇u||u| + |∇ϕ||∇u||u| + |ϕ||u|2  +1 + ϕ|g||u|   1 ěC0 (ϕ|∇u|2 + ϕ|u|r ) ´ C2 1 + |g|(ϕau ¨ u) 2   1 ě ´ C2 1 + |g|w 2 . \ [     Lemma 2.7 Let hi P H ´1,l BxR+ε , i = 1, 2, l ą 1 and wi P H01,l BxR+ε are the 0 0 solutions of This proves Lemma 2.6.

$ &wi = hi , %wi |BB R+ε = 0, x0

respectively. Let it also hold   ă h1 ,  ąěă h2 ,  ą for each  P C08 BxR+ε . 0

(2.81)

2.5 Regularity of Solutions

95

Then, for almost all x P BxR+ε 0 w1 (x) ď w2 (x). Proof For 0 ă δ ăă 1, we define the ‘average’ operator Sδ :     8 R+ε Ñ C B , Sδ : D 1 BxR+ε x0 0 ż ˇ  ˇ ˇ ˇ (Sδ h)(x) := ϕδ (|x ´ y|)h(Tδ y) dy ” (det Tδ )´1 h(z), ϕδ ˇx ´ Tδ´1 zˇ , Rn

where Tδ x ” x0 +

(2.82) R+ε R+ε+2δ (x

supp ϕ Ă [´1, 1] and

ş

Rn

(Sδ˚ )(z) = (det Tδ )´1

´ x0 ) and ϕδ (|z|) :=

1 δn ϕ



|z| δ



, where ϕ P C 8 (R),

ϕ(|z|) dz = 1. It is not difficult to show that

ż Rn

ˇ ˇ ˇ ˇ ϕδ ˇx ´ Tδ´1 zˇ (x) dx, ă Sδ h,  ą=ă h, Sδ˚  ą

and     8 R+ε Sδ˚ : C 8 BxR+ε Ñ C B , x0 0  1,p  and for each  P H0 BxR+ε and 1 ď p ď 8 Sδ˚  Ñ  as δ Ñ 0 in 0  R+ε  1,p H Bx0 . To prove the assertion of Lemma 2.7, we consider v = w1 ´ w2 , where w1 and w2 are due to the Lemma 2.7. Then, v = h1 ´ h2 , v|BBxR+ε = 0. 0

  . Indeed, for all Let hδ = Sδ (h1 ´ h2 ). Then, hδ á h as δ Ñ 0 in H ´1,l BxR+ε 0    P H 1,l˚ BxR+ε 0 | ă Sδ h,  ą ´ ă h,  ą |=| ă h, Sδ˚  ´  ą |ď||h||´1,l ||Sδ˚  ´ ||1,l ˚ Ñ 0. δ Ñ0

  Moreover, from (2.81) and (2.82), it follows that hδ P C 8 BxR+ε , hδ ě 0 and 0 $ &vδ = hδ , %vδ |BB R+ε = 0. x0

96

2 Trajectory Dynamical Systems and Their Attractors

Due to the maximum principle (see Chap. 1), vδ (x) ď 0 for all δ ą 0 and x P BxR+ε . 0  R+ε  1,l  R+ε  ´ 1,l ˚ and H Bx0 , we have Since x is an isomorphism between H0 Bx0 1,l  R+ε  R+ε \ [ vδ á v in H0 Bx0 . Hence, v ď 0 a.e. in Bx0 . This proves Lemma 2.7. p defined in Lemma 2.5 belongs to Lemma 2.8   Assume that the function w Lm BxR+ε for some m ě 1. Then, 0   p P Lk(m) BxR+ε w , 0 where k(m) is defined via # k(m) =

2pnm pn+2(n´2p)m

if pn + 2(n ´ 2p)m ą 0

8

if pn + 2(n ´ 2p)m ă 0

(2.83)

and the following estimate holds: › › › › p BxR+ε ›w, › 0

0,k(m)

" › › › › ď C 1 + ›gp, BxR+ε › 0

0,p

› ›1 * › ›2 p BxR+ε ›w, › 0

(2.84)

0,m

Proof First, consider the following axillary problem: $ &w p p 1 = h(x), %w p 1 |BB R+ε = 0, x0

where h(x) is defined by (2.80). Then, with Lemmas 2.6 and 2.7, we obtain that p p 1 (x) for all most all x P . w(x) ďw

(2.85)

From the Hölder inequality, it follows that › › › › ›h, BxR+ε › 0

0,s

› › › › ď C 1 + ›g, BxR+ε › 0

0,p

› ›1 › R+ε › 2 p Bx 0 › , s= ›w, 0,m

2pm . p + 2m

p1 P Then, the Lp regularity of solutions for the Laplace equation leads to w 2,s ĂĂ Lq for 1 = 1 ´ s , and, as a consequence of the embedding W W 2,s BxR+ε q 2 n 0 p ě 0, it follows that q = k(m). Thus, from (2.85) and w › › › › p BxR+ε ›w, › 0

0,q

› › › › p 1 , BxR+ε ď ›w › 0

0,q

› › ›p R+ε › ďC2 ›h 1 , Bx 0 ›

› › › › p 1 , BxR+ε ď C1 ›w › 0

0,s

This proves Lemma 2.8.

2,s

› › › › ď C3 1 + ›gp, BxR+ε › 0

0,p

› ›1 › R+ε › 2 p Bx 0 › . ›w, 0,m

\ [

2.5 Regularity of Solutions

97

Now we are ready to finish the proof of Theorem 2.7. Analogously to (2.51), we obtain that › › ›4 › › › R+ε › R+ε › p Bx0 › r ď C 1 + ›gp, Bx0 › . ›w, 0, 2

0,p

Thus, all assumptions of Lemma 2.8 are fulfilled with m = 2r ą 1. Let us consider a sequence m0 = 2r , ml+1 = k(ml ). Next, we show that ml = 8 for sufficiently large l. Indeed, from (2.83) and the condition n ´ 2p ă 0, it follows that ml+1 ě 2ml or ml ě 2l ´1 r. Substituting this inequality in (2.83), we obtain that m8 for pn l ą L = log2 r(2p ´n) . In order to obtain the estimate (2.73), we have to iterate the estimate (2.84) [L] + 1 times. The Theorem 2.7 is thus complete. \ [ Theorem 2.8 Let u P θ () be a solution of (2.40) and all assumptions of “  ‰k Theorem 2.7 fulfilled. Then u P H 2,p BxR0 , and it holds › › › › ›u, BxR0 ›

2,p

› › › › ď Q ›g, BxR+ε › 0

(2.86)

,

0,p

where Q : R+ Ñ R+ is some monotonic function depending on f and the constants R and ε. Proof Due to the estimate (2.73) and the conditions on f , we have › › › › ›f (x, u), BxR0 ›

0,8

› › › › ď Q1 ›g, BxR+ε › 0

0,p

(2.87)

.

To finish the proof, we need the following Lemma 2.9. Lemma 2.9 Let u P H 1,m (BxR0 ) for some m ą 1. Then, u P H 1,q(m) (BxR0 ), where $ ! ) &min mn , p n´m q(m) = %p

if n ą m,

(2.88)

if n ď m,

and the following estimate holds › › › › ›u, BxR0 ›

1,q

› › › › ď C ›g, BxR+ε › 0

0,p

› › › › + ›u, BxR+ε › 0

1,m

› › › › + ›f (x, u), BxR+ε › 0

0,p

.

Proof Let ϕ(x) P C08 (Rn ) such that ϕ ” 0 if x R BxR+ε and ϕ(x) = 1 if x P BxR0 . 0 We rewrite(2.40) in the following form $ &(ϕu) = h(x), %u|BB R+ε = 0, x0

(2.89)

98

2 Trajectory Dynamical Systems and Their Attractors

where h(x) := 2

k ÿ

Bi ϕBi u + ϕu + a ´1 ϕ tf (x, u) + g ´ γ Duu .

i=1

From Lm -regularity of solutions of (2.89) and the embedding theorem, we have › › › › ›u, BxR0 ›

1,q

› › › › ď C0 ›u, BxR0 ›

› › › › ďC2 ›g, BxR+ε › 0

0,p

2,m

› › › › ď C ›ϕu, BxR+ε › 0

2,m

› › › › + ›u, BxR+ε › 0

1,m

› › › › ď C1 ›h, BxR+ε › 0

› › › › + ›f (x, u), BxR+ε › 0

0,m

(2.90)

0,p

\ [

This proves Lemma 2.9.

Now we are ready to finish the proof of Theorem 2.8. Let m0 = 2 and ml+1 = q(ml ). Then, it is not difficult to see that ml = p for sufficiently large l. Indeed, it follows from (2.88) that " ml+1 ě min p, ml Hence, ml = p if l ą L =

ln p2 n ln n´2

› › › › ›u, BxR0 ›

1,2

n n ´ ml

*

" ě min p, ml

* n . n´2

. Then, analogously to (2.68), › › › › ď C ›g, BxR+ε › 0

0,p

+1 .

ε instead of ε), we Iterating the estimate (2.90) [L] + 1 times (with m0 = 2 and [L]+1 obtain › › › ›2 › › › › › R› R+ε › R+ε › . ›u, Bx0 › ď C ›g, Bx0 › + ›f (x, u), Bx0 › 1,p

0,p

0,p

Applying analogously to (2.89) the theorem of Lq -regularity, we obtain a formula similar to (2.90) › › › › ›u, BxR0 ›

2,p

› ›2 › › ď C ›g, BxR+ε › 0

0,p

› › › › + ›f (x, u), BxR+ε › 0

0,p

.

(2.91)

Hence, the assertion of Theorem 2.8 follows from (2.91) and (2.87). This proves Theorem 2.8. \ [ Remark 2.7 If the boundary B X BxR+ε is a smooth manifold and the boundary 0 value u0 satisfies (2.87), then, in the same manner as above, one can obtain estimates analogously to (2.86) for the point x0 P B.

2.6 Boundedness of Solutions as |x | Ñ 8

99

2.6 Boundedness of Solutions as |x | Ñ ∞ In this section, we use the estimates obtained in the previous section to show the boundedness of solutions for the problem (2.40) in various functional spaces. To this end, we need several definitions. ř ř Definition 2.4 We denote by b () the Banach space of functions g P (), for which |g; b|+ = sup |g, Bx10 X |+ ă 8. x0 PRn

Analogously, we define νb (B) with |u0 ; b|0 = sup |u0 , Bx10 X B|0 ă 8 x0 PRn

and θb () with › › › › |u; b|+ := ||u, ; b||+ = sup ›u, Bx10 X › ă 8. +

x0 PRn

l,p

Definition 2.5 Let 1 ď p ă 8, l = 0, 1, 2. We denote by Wb () the Banach l,p space of functions g P Wloc () such that › › › › ||g; b||l,p = sup ›g, Bx10 X › x0 PRn

p

l,p

ă 8.

0,p

We denote by Lb () := Wb (). ř Theorem 2.9 Let g(x) P b () and ε ą 0. Then: 1. Any solution u of (2.40) belongs to θb (ε ), where ε = tx P | dist(x, B) ą εu , and the following estimate holds:   }u, ε ; b} ď Cε 1 + |g, ; b|2+ . 2. Let u0 P νb (B). Then, any solution u of (2.40) belongs to θb (), and the following estimate holds:   }u, ; b}+ ď C 1 + |g; b|2+ + |u0 ; b|r0 .

100

2 Trajectory Dynamical Systems and Their Attractors

A proof of this Theorem is an immediate consequence of (2.68). Theorem 2.10 Let a = a ˚ and g P Lb () for p ą n2 . Then, any solution of (2.40) 2,p belongs to Wb (ε ) for each ε ą 0, and the following estimate holds: p

  }u, ε ; b}2,p ď Qε |g, ; b|0,p , where Qε : R+ Ñ R+ is some monotonic function. A proof of this Theorem is an immediate consequence of (2.86). Example 2.5 Let  = Rn and all assumptions of Theorem 2.9 hold. Then, any 2,p solution of (2.40) belongs to Wb () and, consequently, to Cb (). In particular, this means that there does not exist a solution of (2.40) for which lim sup|u(x)| = 8. x Ñ8

Example 2.6 Let  = R+ ˆ Rn´1 , x = (t, x 1 ) and all assumptions of Theorem 2.9 hold. Then, due to Theorem 2.10 and the Sobolev embedding theorem, any solution of (2.40) admits sup |u(t, x 1 )| = C

x 1 PRn´1



1 , ||g; b||0,p t

ă 8 for all t ą 0.

Note that we do not impose any restriction on u0 (x 1 ) as x 1 Ñ 8. 2´ p1 ,p

Assume in addition that u0 P Wloc

(Rn´1 ) and

› › › › ›u0 , Rn´1 , b›

2´ p1 ,p

ă 8.

2,p

Then (see [101]), there exists a v P Wb () such that v|B = u0 . Hence, according 2,p to Theorem 2.7 and Remark 2.7, any solution of (2.40) belongs to Wb () and, consequently, belongs to Cb (). Definition 2.6 By a local solution of (2.40) we call a function u defined in BxR0 and belonging to θ (BxR0 ), which, in turn, satisfies (2.40). We especially emphasize that the proofs of Theorems 2.7 and 2.8 by no means use that u is defined outside of BxR+ε and, therefore, the assertions of Theorems 2.7 0 and 2.8 remain valid for local solutions of (2.40) as well. Thus, we have the following result: Theorem 2.11 Let all assumptions of previous theorem hold. Then, for all ε ą 0 such that BxR+ε ĂĂ , there exists a K = K(ε, R, }g; b}0,p ) such that no local 0 solution u of (2.40) defined in BxR0 exists with

2.7 Basic Definitions: Trajectory Attractor

101

sup |u(x)| ą K

(2.92)

x PBxR

0

p in BxR+ε . that can be extended to a local solution u 0  R+ε  p|BxR = u. Then, due to p P θ Bx 0 and u Proof Assume the contrary. Let u 0

Theorem 2.8 and the Sobolev embedding theorem (W 2,p (BxR0 ) Ă C(BxR0 ) if p ą n2 ). › › › › sup |u(x)| ď Q ›g, BxR+ε › 0

0,p

x PBxR

.

(2.93)

0

› ›  › Estimate (2.93) contradicts the condition (2.92) if K ą Q ›g, BxR+ε . 0 0,p Theorem 2.11 is proved. \ [

2.7 Basic Definitions: Trajectory Attractor In order to construct a trajectory attractor for (2.40), we consider together with (2.40) also # au + γ Du ´ f (u) = σ (x), σ (¨) P .

(2.94)

Here,  Ă D 1 () is an invariant set with respect to Ts : Ts  Ă , s ě 0. Moreover, we assume that  is a compact subset of some functional space  + Ă D 1 (). In this sequel, we usually consider  endowed with its weak topology that is  w (). Remark 2.8 In application, we take  = H+ (g), that is, the hull of the right-hand side of (2.40). A set  we call the space of symbol (2.94). Definition 2.7 A set of solutions of (2.40) which belongs to θ () with the righthand side σ P  and u0 P ν0 (B), we denote by Kσ+ . It is not difficult to see that the set tKσ+ , σ P u enjoys the so-called translation compatibility, that is, Ts Kσ+ Ă KT+s σ Ă θ ().

102

2 Trajectory Dynamical Systems and Their Attractors

+ Thus, the semigroup tTs , s ě 0u acts on the trajectory phase space K of (2.94): + K =

ď σ P

+ + Kσ+ Ă θ (), Ts K Ă K .

+ We endowed the (nonlinear) set K by the induced weak topology: + Ă θ+ ” θ w (). K + is called an attractive set for the semigroup Definition 2.8 A set B Ă K + + tTs , s ě 0u in K , if for every neighbourhood O(B) of B in K there exists a T = T (O(B)) such that + ) Ă O(B) for all s ě T . Ts (K

(2.95)

Remark 2.9 Note that the definition of attracting set as given in Definition 2.8, is not the traditional one (compare with the definitions given in Sect. 2.2). Usually, the condition (2.95) is only required to be fulfilled for bounded subsets (in an + appropriate topology) of K . However, as we will later, due to the estimates for the solutions of (2.40), in some concrete situation, one can use the Definition 2.8, + that is, the attractivity property (2.95) is satisfied for all bounded subsets in K with the same constant T = T (O(B)). + Definition 2.9 A set A Ă K is called a trajectory attractor for (2.94) if

1. A is compact in θ+ ; 2. A is strictly invariant: Ts A = A for all s ě 0; + 3. A is an attracting set for the semigroup tTs , s ě 0u in K . To formulate the main Theorem on the existence of the trajectory attractor for (2.94), we need several Definitions. Definition 2.10 We denote by ω() the ω-limit set of the semigroup tTs , s ě 0u in  (Ts  Ă ) if ω() =

č s ě0

«

ď

hě s

ff Th 

, +

where by [. . .]+ we denote the closure in the topology + . Remark 2.10 Since  is compact, ω() ‰ H (see [1]). Definition 2.11 A family of setsŤtKσ+ , σ P u is closed in the topology (θ+ , ) (sequentially closed), if its graph σ P Kσ+ ˆ σ is sequentially closed in θ + ˆ . Remark 2.11 Note that, in the case of  compact, this property is equivalent to + sequential compactness of K in θ+ [35].

2.7 Basic Definitions: Trajectory Attractor

103

Theorem 2.12 ([35]) Assume that the semigroup tTs , s ě 0u and tKσ+ , σ P u defined above satisfy the following conditions: 1. The sets Kσ+ , σ P  are closed; + 2. The semigroup tTs , s ě 0u possesses in K a compact attracting set. Then (2.94) possesses a trajectory attractor A : A = Aω() =

č s ě0

«

ď

hě 0

+ Th K 

ff , θ+

where Aω() is a trajectory attractor of (2.94) with the space of symbols ω(), and, as usual, [. . .] is the closure in θ w (). The conditions 1) and 2) we check later on. Next, we formulate a theorem which describes the structure of the trajectory attractor for the semigroup tTs , s ě 0u. To this end, we need several Definitions and auxiliary lemmas. Definition 2.12 1. We denote by b+ (l)  () a Fréchet space ( b+ (l)  () Ă ) with the following seminorms |g; x0 , b+ (l)|+ = sup|g,  X Ts B1x0 |+ = C(x0 ) ă 8; x0 P Rn . s ě0

2. By b(l)  () we denote a Fréchet space ( b(l)  () Ă ) with the following seminorms |g; x0 , b(l)|+ = sup|g,  X Ts B1x0 |+ = C(x0 ) ă 8; x0 P Rn . s PR

Analogously are defined the spaces θb+ (l)  () and θb(l)  () and the corresponding  + and ||u; x0 , b(l)||  +. seminorms ||u; x0 , b+ (l)|| Lemma 2.10 A set  Ă () satisfying Ts  Ă  is relatively compact in + () if and only if it is bounded in b+ (l)  (). Analogous statement is valid for the space θ. Proof According to Corollary 1.3, a set  is relatively compact in + if and only if  is bounded in (). We will show that from the boundedness in () and () and Ts  Ă  it follows that  is bounded in the space b+ (l)  (). Indeed, 1 n let Bx0 Ă R . Then, it is not difficult to see that (compare with Lemma 2.14) there exists an s0 = s0 (x0 ), such that Ts0 B1x0 Ă . Then, due to Definition 2.12 and Ts  Ă , we have

104

2 Trajectory Dynamical Systems and Their Attractors

ˇ ˇ ˇ ˇ ˇ  ˇˇ = sup ˇˇ,  X Ts B1x ˇˇ ˇ; x0 , b+ (l) 0 +

s ě0

+

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ď sup ˇ,  X Ts B1x0 ˇ + sup ˇ, Ts Ts0 B1x0 ˇ s ăs0

s ě0

ˇ ˇ ˇ ˇ ˇ ˇ ď ˇ,  X Bsx00 ˇ+ + sup ˇTs , Ts0 B1x0 ˇ s ě0

ˇ ˇ ˇ ˇ ď C(x0 ) + ˇ, Ts0 B1x0 ˇ ă 8.

+

+

+

\ [

This proves Lemma 2.10.

Definition 2.13 The function p ξ P C(R, ) is called a complete symbol for a semigroup tTs , s ě 0u, Ts :  Ñ , if Ts p ξ (t) = p ξ (t + s) for s ě 0, t P R. By Z() we denote the set of all complete symbols of a semigroup tTs , s ě 0u acting in . Lemma 2.11 ([35]) It is valid ! ) ω() = p ξ (0)| p ξ P Z() , that is, for any symbol σ P ω() there exists at least one complete symbol ξ(t) P Z() such that p ξ (0) = σ . Lemma 2.12 For any complete symbol p ξ P Z() there exists a unique function n ) such that ξ P b+ (l) (R  p   , s P R, x P . ξ (s)(x) = ξ(x + s l)|

(2.96)

n Remark 2.12 Since ξ P b(l)  (R ) is, in general, not a regular generalised function, a more precise formulation of (2.96) is the following:

ăp ξ (s), ϕ ą=ă ξ, Ts ϕ ą, ϕ P D(), s P R.

(2.97)

Proof Let p ξ P Z() and ϕ P D(Rn ). Then it follows from the assumption on  and s ě 0 (see Lemma 2.14) such that Ts supp ϕ Ă  and, consequently, Ts ϕ P D(). Consider the generalised function ξ P D 1 (Rn ) which is defined by ă ξ, ϕ ą:=ă p ξ (´s), Ts ϕ ą .

(2.98)

Let us check that (2.98) is well-defined. Let s2 ą s1 ě 0 and Ts1 ϕ, Ts2 ϕ P D(). Then,

2.7 Basic Definitions: Trajectory Attractor

105

ăp ξ (´s2 ), Ts2 ϕ ą= ă p ξ (´s2 ), Ts2 ´s1 Ts1 ϕ ą = ă Ts2 ´s1 p ξ (´s2 ), Ts1 ϕ ą =ăp ξ (´s1 ), Ts1 ϕ ą . Next, we check (2.97). Let s ě 0. Then, for each ϕ P D() it holds: ăp ξ (s), ϕ ą=ă Ts p ξ (0), ϕ ą=ă p ξ (0), Ts ϕ ą=ă ξ, Ts ϕ ą, ăp ξ (´s), ϕ ą=ă p ξ (´s), Ts T´s ϕ ą=ă ξ, T´s ϕ ą . n Let us prove that ξ P b(l)  (R ). Indeed, from Lemma 2.14 it follows that there exists an s0 = s0 (x0 ) P R+ such that Ts0 B1x0 Ă , where B1x0 Ă Rn . From Definition 2.12, formula (2.98) and the compactness of , it follows that

 + =sup|η, Ts B1x |+ = sup|ξ, Ts ´s0 Ts0 B1x |+ |ξ, Rn ; x0 , b(l)| 0 0 s PR

s PR

=sup|p ξ (s ´ s0 ), Ts B1x0 |+ ď |, Ts0 B1x0 | ă 8. s PR

(2.99) \ [

This proves Lemma 2.12.

In the sequel, we will identify the complete symbol p ξ and the corresponding to p ξ n function ξ P b(l)  (R ). n Corollary 2.5 A set Z() is bounded in b(l)  (R ), strictly invariant with respect to the group tTs , s P Ru, that is,

Ts Z() = Z() and compact in the space w (Rn ). n Indeed, boundedness in b(l)  (R ) follows from (2.99), and the relative compactness w n in (R ) is a consequence of Corollary 1.3. The strict invariance and closedness of Z() follows from the definition of the complete symbol, Lemma 2.11 and the analogous properties of ω(). + p(s), u p P C(R, K Definition 2.14 The function u ) is called a complete trajectory of a semigroup tTs , s ě 0u corresponding to a complete symbol p ξ , if the following conditions are satisfied: $ &Ts u p(t) = u p(t + s), s ě 0, t P R, + %u p(s) P Kp , s P R. ξ (s)

106

2 Trajectory Dynamical Systems and Their Attractors

We denote the set of all complete trajectories corresponding to a complete symbol p ξ by Kξ . p P Kξ there exists a unique u P Lemma 2.13 For any complete trajectory u n ) which satisfies θb(l) (R  au + γ Du ´ f (u) = ξ(x), x P Rn and  . p(t)(x) = u(x + t l)| u The proof of Lemma 2.13 is analogous to the proof of Lemma 2.12. Theorem 2.13 ([35]) The trajectory attractor A has the following structure: A = Aω() =

ď

tp u(0)| u P Kξ u .

ξ PZ ()

2.8 Trajectory Attractor of Nonlinear Elliptic System In this section we will apply Theorem 2.12 to a family of equations #

au + γ Du ´ f (u) = σ (x), σ (¨) P 

(2.100)

which will prove existence of a trajectory attractor for system (2.100). Theorem 2.14 A family tKσ+ , σ P u corresponding to (2.100) is a closed set. Proof Let un P Kσ+n , un Ñ u in + and σn Ñ σ in + . We have to show that u P Kσ+ . In other words, we have to show that u is a solution of (2.100) with the right-hand side σ (x). According to definition of solution of (2.100) we have ´xa∇un , ∇φy + xγ Dun , φy ´ xf (un ), φy = xσn , φy

(2.101)

for any φ P D(). Passing to the limit in (2.101) and using arguments similar to which we used in the proof of Theorem 2.5, we have ´xa∇u, ∇φy + xγ Dun , φy ´ xf (un ), φy = xσ, φy for any φ P D(). Theorem 2.14 is proved. + Corollary 2.6 The set K is a sequentially closed subspace of + .

(2.102) \ [

2.8 Trajectory Attractor of Nonlinear Elliptic System

107

+ Theorem 2.15 A semigroup tTs , s ě 0u corresponding to (2.100) possesses in K a compact, attracting (absorbing) set Babs .

Proof The proof of this theorem is based on the following lemma. n Lemma 2.14 For any BR x0 Ă R there holds

lim d(Ts BR x0 , B) = 8.

s Ñ+8

(2.103)

In particular, there exists s0 ą 0 such that Ts B R x0 Ă  for any s ě s0 .

(2.104)

n Proof of Lemma 2.14 Let us start with proving (2.104). Indeed, since BR x0 Ă R = Ť R s ą0 T´s , then for any x P Bx0 there exists neighborhood Ux and sx ě 0, such that Ux Ă T´sx . Let tUxi , i = 1, . . . , N u be a subcovering of tUx , x P BR x0 u. R Hence, according to Condition 2.2, Bx0 Ă T´s , where s = s(BR x0 ) = maxtsxi , i = 1, . . . , N u and consequently R Ts B R x0 Ă ; s = s(Bx0 ).

(2.105)

Since Ts1  Ă Ts2  for s2 ě s1 , the function s ÞÝÑ d(Ts BR x0 , ) is monotone (nondecreasing), therefore the limit (2.103) exists. The next goal is to prove that this limit is 8. Assume, on the contrary, that there is L ă 8 such that d(Ts BR x0 , ) ă L, @s ě 0. Then Ts BR+L X B ‰ H, s P R+ , which contradicts condition (2.105), with R x0 replaced by R + L. This proves Lemma 2.14. n Proof of Theorem 2.15 According to Lemma 2.14, for any ball BR x0 Ă R there exists S = S(BR x0 ), such that

Ts B2R x0 Ă  for any s ě S. Since  is compact in + , it follows from Lemma 2.10 that |; x0 , R, b+ (l)|+ ” sup |,  X Ts BR x0 |+ = C(x0 , R) ă 8. s ě0

Hence, according to estimate (2.51) with ε = R, and s ě S we have

108

2 Trajectory Dynamical Systems and Their Attractors + + R }Ts K ,  X BR x0 }+ = }K , Ts ( X Bx0 )}+ + ď }K , Ts B R x0 }+ 2 ď CR (1 + |, Ts B2R x0 |+ )

ď CR (1 + |; x0 , 2R, b+ |2+ ) ” M(x0 , R). Note that M(x0 , R) ă 8. Let β

0

(2.106)

Ă b+ (l)  () be the set defined by

n }β 0 ,  X BR x0 }+ ď M(x0 , R); x0 P R , R Ă R+ .

(2.107)

By Lemma 2.10, the set β 0 is compact in + . Obviously, β 0 is a closed convex subset of () and consequently (see Lemma 1.4) is closed in the topology of the space + . Thus, according to Corollary 1.3, the set β 0 (endowed with the induced topology of + ) is a compact metric space. We set + . Babs := β 0 X K

Our next goal is to prove that Babs is a desirable compact attracting set. Indeed, + since β 0 is a compact metric space and the set K is sequentially closed, it follows + that Babs is a compact set in K . Next we will prove the attractivity property of Babs . To this end, let us consider Q as a neighborhood of Babs in + . From compactness of Babs it follows that Q contains a neighborhood QN of the form Babs Ă QN =

N ď

tui + Ei u; ui P Babs ,

(2.108)

i=1

where the set Ei belong to the basis of a neighborhood of zero in the space + . Due to the definition of weak topology in () and Theorem 1.1, any Ei can be represented Ei = Ei (Ri , εi ; Li1 , . . . , Lini ) = tu P + : |xLij , u|

R

XBx0i

i ˚ y| ă εi ; j = 1, . . . , ni , Lij P ((, BR 0 )) u.

(2.109) Let R = maxtRi , i = 1, . . . , N u. Then Q(R) := tu P + ; uXBR Ă Babs |XBR u Ă QN Ă Q. 0 0

2.8 Trajectory Attractor of Nonlinear Elliptic System

109

Hence, it suffices to show that + |XBR Ă Babs |XBR Ts K  0

0

for sufficiently large s ě 0. The last statement is an immediate consequence of (2.106) and the definition of Babs . Theorem 2.15 is proved. l Thus, all conditions of Theorem 2.12 are fulfilled and we have Theorem 2.16 A family of equations (2.100) possesses in + a trajectory attractor n A , which consists of restrictions to  of all solutions u P b(l)  (R ) of the following family of equations #

au(x) + γ Du(x) ´ f (u(x)) = ξ(x), x P Rn ξ P Z().

(2.110)

n Remark 2.13 Due to Lemma 2.12, Z() Ă b(l)  (R ). Therefore according to n estimate (2.51) with  = R , it follows that, any solution u P (Rn ) of (2.110) n belongs to the space b(l)  (R ).

Next we will describe the character of convergence to the trajectory attractor A . In the sequel we denote A by Atr . Corollary 2.7 It follows from Theorem 2.15, that for any ball BR x0 Ă  the set + Atr |BRx is an attracting set for a family of sets tTs K |BRx ; s ě R+ u as s Ñ 8 0

0

tr , say in the topology of w (BR x0 ). It means that, for any neighborhoods of A |BR x

O(Atr |BRx ) in the topology of w (BR x0 ), there exists S = S(O), such that

0

0

+ Ts K  |BRx Ă O(Atr |BRx ) for s ě S. 0

0

Since the embeddings #

1´ε,2 (BR ) (BR x0 ) Ť H x0 2´ε (BR ) (BR x0 ) Ť L x0

(2.111)

are compact for any ε ą 0, it follows from Theorem 2.16 that, for any ball BR x0 Ă  the following condition holds. $ + & lim dist H 1´ε,2 (BR ) tTs K |BRx , Atr |BRx u = 0 x s Ñ+8

0

0

0

+ % lim dist L2´ε (BR ) tTs K |BRx , Atr |BRx u = 0, x s Ñ+8

0

0

0

(2.112)

110

2 Trajectory Dynamical Systems and Their Attractors

where dist ... tX, Y u = sup inf }x ´ y}... . x PX y PY

In what follows we present some useful consequences from Theorem 2.16 and also give some useful application of this theorem. Moreover, we will also present the dependence of Atr on the direction l and domain . We start with the following lemma. Lemma 2.15 Let  be a compact subset of + , which is invariant with respect to a semigroup tTs , s ě 0u and let g P . Then a hull of g, that is H+ (g) = [Ts g, s ě 0] + Ť +

(2.113)

is a compact set in + . A proof of this lemma is an immediate consequence of the inclusion H+ (g) Ă . Definition 2.15 A function g P + is called translation compact (in the direction  in + , if the condition (2.113) is satisfied. l) Analogously one can define translation compactness in the other spaces, those are invariant with respect to the semigroup tTs , s ě 0u. Obviously, a set H+ (g) is a translation invariant, that is Ts H+ (g) Ă H+ (g), s ě 0.

(2.114)

Consequently, the set H+ (g) can be chosen as a candidate of space of symbols  for the family of equations (2.100), if g is a translation compact function. Remark 2.14 It is clear that, in the case of translation compactness of g, the set H+ (g) is a minimal set containing g, which can be chosen as a space of symbols for the following family of equations (2.100). Remark 2.15 Since the hull H+ (g) of a translation compact function g is a compact metric space in + , it follows that ω(g) ” ω(H+ (g)) and H+ (g) can be represented in the following way [35]: #

H+ (g) = tTs g; s ě 0u Y ω(g) ω(g) =: tξ P + |Dsn ě 0, sn Ñ 8 as n Ñ 8, ξ = lim Tsn gu,

(2.115)

nÑ8

where by lim is denoted a limit in the topology of space + . Definition 2.16 By Atr g we denote a trajectory attractor of the equation #

au + γ Du ´ f (u) = g(x), u|B = u0

(2.116)

2.8 Trajectory Attractor of Nonlinear Elliptic System

111

with the right-hand side g (we assume that g is a translation compact in + ) and by definition we put Atr g to equal to the trajectory attractor A for the family of Eq. (2.100). Let us recall that, due to the Lemma 2.10, a function g P () is a translation compact in + if and only if, when g P b(l)  (). Thus we have Theorem 2.17 Let g P b(l)  (). Then the equation #

au + γ Du ´ f (u) = g(x), u|B = u0

+ possesses a trajectory attractor Atr g in the space  .

Lemma 2.16 Let g be a translation compact in + . Then any solution u of the Eq. (2.116) is a translation compact in + . Indeed, from estimate (2.51) analogously (2.106), it follows that H+ (u) is a bounded subset of b+ (l)  (), and consequently, according to Lemma 2.10 is a compact in + . For convenience of the reader, below we present some examples of translation compact function g in the Eq. (2.116), for which due to Theorem 2.17 the Eq. (2.116) possesses a trajectory attractor. Example 2.7 Let g = g(x) be a bounded function in Rn , that is, g P L8 (Rn ). Then for any direction l and for any domain  satisfying the condition on  (see Sect. 2.1), a function g is a translation compact in + = w (). Example 2.8 Let h : R Ñ R be a function belonging to L8 (R), and let Vl = Rn´1 be the orthogonal complement of the line  s P Ru ts l, q

in Rn . Assume that a function g0 P Lloc (Vl ), where the exponent q is the same as in the definition of the space . Then it is not difficult to check that the function   ¨ g0 x ´ x ¨ l l g(x) = h(x ¨ l) l ¨ l

(2.117)

n belongs to b(l)  (R ), and consequently, for any domain satisfying the condition on , the restriction of g| is a translation compact in + .

In particular, let x = (t, x 1 ), and let l coincide with the first coordinate vector, that is, l = (1, 0, . . . , 0). Then 1 2 g(t, x 1 ) = sin(t 2 )e|x |

is a translation compact in the direction l in + .

(2.118)

112

2 Trajectory Dynamical Systems and Their Attractors

Remark 2.16 According to Lemma 2.10, a translation compact function in direction l is bounded in this direction. As was shown in the previous example, in the other directions the function need not be bounded.

2.9 Dependence of the Trajectory Attractor on the Underlying Domain In this section we will study the dependence of the constructed attractor for nonlinear elliptic systems (2.100) on the right-hand side as well as on the domain  Ă Rn . Firstly we consider dependence of the trajectory attractor on the domain  Ă Rn . Definition 2.17 Let 1 and 2 be domains satisfying the condition on domain  with the same direction l and let gi P (i ), i = 1, 2 and gi (x) be a translation compact function in the spaces w (). Consider the trajectory attractors Agi of Eq. (2.100) in the domain i with the right-hand side gi (x), i = 1, 2. We say that Ag1 = Ag2 if the set of corresponding complete trajectories coincide, that is ď

pg = A 1

ξ PZ(H+ (g1 ))

p g Ă   (Rn ). Kξ = A 2 b(l)

(2.119)

(In the sequel, we identify the trajectory attractor A with the corresponding sets of p  .) complete trajectories A Theorem 2.18 Let the assumptions of the previous Definition 2.17 hold for two domains i , i = 1, 2 and g1 |1 X2 = g2 |1 X2 .

(2.120)

Then the trajectory attractors Ag1 and Ag2 coincide, that is, Ag1 = Ag2 .

(2.121)

Proof The proof is based on showing that  Z(g1 ) = Z H+



l(g1 )

= Z(g2 ).

Let  = 1 X 2 . Then, it follows from (2.120) that H+

| l(g1 ) 1 X2

= H+

| . l(g1 ) 1 X2

2.9 Dependence of the Trajectory Attractor on the Underlying Domain

113

Hence, ˜ 2 )|1 X2 . ω(g ˜ 1 )|1 X2 = ω(g

(2.122)

Now, let ξ P Z(g1 ). Then, from definition of Z(g1 ) it follows that there exists σn P ω(g ˜ 1 ), such that T´n ξ |1 = σn , n P N. From (2.122) it follows that there ˜ 2 ), such that σn |1 X2 = σ˜ n |1 X2 . Hence, due to Lemma 2.11, exists σ˜ n P ω(g there exists ξn P Z(g2 ) such that ξn |1 X2 = σ˜ n |1 X2 . Thus, we have T´n ξ |1 X2 = ξn |1 X2 , ξn P Z(g2 ), or, equivalently, for ξn P Z(g2 ), we have ξ |T´n (1 X2 ) = Tn ξn |T´n (1 X2 ) . According to Lemma 2.14,

Ť s ě0

T´s (1 X 2 ) = RN , it follows that Tn ξn Ñ ξ in

(RN ), which in turn, due to Corollary 2.5, implies ξ P Z(g2 ). Hence Z(g1 ) Ă Z(g2 ). The inclusion Z(g2 ) Ă Z(g1 ) is obtained in the same way. \ [ Next assume that, a domain  satisfies the condition on  for two different directions l1 and l2 . Let g1 and g2 be translation compact functions in the direction l1 and l2 respectively. Then according to Theorem 2.17, there exists trajectory attractors Alg11 in the directions l1 for the Eq. (2.100) with the right-hand side g1 (x) and Alg22 in the directions l2 for the Eq. (2.100) with the right-hand side g2 (x). Theorem 2.19 The trajectory attractors Alg11 and Alg22 coincide if and only if ωl1 (g1 ) = ωl2 (g2 )

(2.123)

where ωli (gi ) ” ω(Hl+i (gi )), i = 1, 2. Proof Assume that Alg11 = Alg22 . Then formula (2.123) follows from Theorem 2.16 and Lemma 2.11. Now assume that, the formula (2.123) holds. Then, according to Theorem 2.16 and Remark 2.12, it suffices to prove that, Zl1 (g1 ) = Zl2 (g2 ). Let ξ P Zl1 (g1 ). Then @n P N Tnl1 ξ | = σn P ωl1 (g1 ) = ωl2 (g2 ).

(2.124)

Furthermore, using Lemma 2.11 and the arguments similar to the previous theorem, we obtain Tnl1 ξ ÝÑ ξ as n Ñ 8 in (Rn ); ξn P Zl2 (g2 ).

(2.125)

114

2 Trajectory Dynamical Systems and Their Attractors

Since ωli are strictly invariant with respect to semigroup tTsli , s ě 0u, then from equality (2.123) and definition of complete symbol it follows that, the set Zl2 (g2 ) is invariant with respect to semigroup tTsl1 , s ě 0u. Thus, T´l1n ξn P Zl2 (g2 ), which in \ turn, according to Corollary 2.5, implies ξ P Zl2 (g2 ). Theorem 2.19 is proved. [ Corollary 2.8 Let g = g(x) be such that: Hl+1 (g) = Hl+2 (g) ” H+ (g).

(2.126)

Then Alg1 = Alg2 . Proof Note that, according to (2.126) and (2.114) and the definition of ω-limit set we have j č„ď j č„ď l l2 l2 + l2 l1 + 1 Tp ωl1 (g) = Tp Th H (g) Ă Tp Th H (g) s ě0

=

č„ď s ě0

hě s

hě s

Thl1 Tpl2 H+ (g)

j +

s ě0

Ă

hě s

č„ď

s ě0

hě s

Thl1 H+ (g)

j

= ωl1 (g)

and analogously Tsl1 ωl2 (g) Ă ωl1 (g) for s ě 0. Next we aim to show ωl1 (g) = ωl2 (g). Let ξ P ωl1 (g). Then due to the representation (2.115) and Condition (2.126), at least one of the following three condition hold: $ l1 ’ & 1. Ds0 ě 0, such that Ts0 g P ωl2 (g) 2. Tsl1 g = g for all s ě 0 ’ % 3. Ds0 ą 0, p0 ą 0, such that Tsl01 g = Tsl02 g.

(2.127)

In the first case, Tsl1 g P ωl2 (g) for s ě s0 and consequently we have ξ P ωl2 (g). In the second case, ωl1 (g) = ωl2 (g) = tgu. In the third case, due to (2.115) we represent ξ as ξ = lim Tsln1 g. nÑ8

Without loss of generality one can assume that, sn ě ns0 . Then l2 Tsln1 g = Tnp (Tsln1´ns0 g). 0

(2.128)

Since Tsln1´ns0 g P Hl+2 (g) and np0 Ñ 8 as n Ñ 8 we obtain that, ξ P ωl2 (g). Hence ωl1 (g) Ă ωl2 (g). The reverse inclusion can be proved analogously. Consequently, according to Theorem 2.19, we have Alg1 = Alg2 . Corollary 2.8 is proved. \ [

2.9 Dependence of the Trajectory Attractor on the Underlying Domain

115

We now present some examples that illustrate the assumptions on g mentioned above. Example 2.9 Let g P (RN ) such that › › › › ||g|| = sup ›g,  X Bx10 ›

(RN )

x0 PRN

ă8

and there exists h P D 1 (R) such that g(x) = h(|x|). Then g is translation-compact  As a consequence of in any direction l P RN and H+ () does not depend on l. l l Corollary 2.8, we obtain that Ag also does not depend on l P RN . Another interesting example is the case when g P b (RN ) is periodic with respect to ZN , i.e. g(x1 + k1 l1 , . . . , xN + kN lN ) ” g(x1 , . . . , xN ), ki P Z. Here

! ) li , i = 1, . . . , N is a basis in RN which generates ZN . Fundamental

domains of ZN we denote by Z0 , i.e. # Z0 =

x P R |x = N

N ÿ

+

xi li , 0 ď xi ă 1 .

i=1 N N    For arbitrary ! given )direction l P R , we denote by O(l), O(l) P T , a closure of N N N   s P R under the projection ! : R Ñ T = R {ZN and by Z(l) the line s l,  Z0 . Then, it is not difficult to see that  = !´1 O(l)| the set Z(l)

! )  . H+ (g) = g(x + s), s P Z(l) l



In this case, H+ (g) is strictly invariant under Tsl , hence ωl(g) = H+ (g). l

Consequently, due to Theorem 2.19, the attractors if O(l1 ) = O(l2 ). Now let us present a sufficient condition for

 Agl1

and

 Agl2

l

coincide if and only

O(l1 ) = O(l2 ). Definition 2.18 A direction l is called ) irrational with respect to ZN if all coordi! nates of l in the basis li , i = 1, . . . , N are irrational numbers.  = TN . It is known that for irrational with respect to ZN directions l it holds O(l) l1 l2 Hence, for such two directions l1 and l2 , we have Ag = Ag .

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2 Trajectory Dynamical Systems and Their Attractors

Remark 2.17 For a direction l which is not irrational with respect to ZN , it is not known whether these attractors coincide or not. In what follows, we assume that the domain  satisfies Condition 2.2 and the direction l P Rn is fixed. In addition we assume that the right-hand side g is represented by g(x) = g0 (x) + g1 (x),

(2.129)

where g1 is a translation compact function in the direction l in + , and the function g0 P + satisfies Ts g0 ÝÑ 0 as s ÝÑ 8 in + .

(2.130)

Then it is not difficult to show that g is also translation compact in the direction l and ω(g) = ω(g1 ).

(2.131)

As a consequence we obtain Theorem 2.20 Let the assumptions (2.129) and (2.130) hold. Then Ag0 +g1 = Ag1 .

(2.132)

Theorem 2.20 is an immediate consequence of (2.131) and Definition 2.12. Example 2.10 Let x = (t, x 1 ) and let l coincide with the first coordinate in Rn , that is, l = (1, 0, . . . , 0). Consider 1 2 g0 (x) = sin(t 2 )e|x | .

Since Ts sin(t 2 ) á 0 in C(R+ ) as s Ñ 8, it is not difficult to check that g0 (x) satisfies condition (2.130). Let g1 P b+ (l)  (). Then according to Theorem 2.20, we have Ag0 +g1 = Ag1 .

(2.133)

2.10 Regularity of Attractor In this section we study regularity properties of the trajectory attractor constructed above. It will be done under additional assumption, both for the matrix a and for the right-hand side g in Eq. (2.40). Moreover, we will show attraction to the trajectory 2,p attractor in the stronger topology of Wloc (), p ą n2 .

2.10 Regularity of Attractor

117

We state the following three conditions on the involved data: Condition 2.5 (1) a = a ˚ ; n p (2) g P Lloc () with p ą ; 2 p (3) Function g is a translation compact in the direction l either in Lloc () or p p Lloc,w (), that is in Lloc endowed with the weak topology. p

p

Next we formulate criteria in Lloc () and Lloc,w (). Lemma 2.17 The following statements are equivalent: p

p

p

1) A function g P Lloc () is a translation compact in Lloc () (Lloc,w ()) 2) For any ball BR x0 Ă , the restriction of g to the semicylinder R !+ x0 ,R =  X tTs Bx0 ; s ě 0u + is a translation compact in Lloc (!+ x0 ,R ) (Lloc,w (!x0 ,R )). p

p

Proof Implication 1) ñ 2) is obvious. Next we show 2) ñ 1). Indeed, due to p the Eberlein theorem (Lemma 1.3; see also [86]), both in the case Lloc () and p p Lloc,w (), it suffices to show that, H+ (g) is a sequentially compact set in Lloc () p (Lloc,w ()). The sequential compactness of H+ (g) is an immediate consequence of condition (2) and the Cantor diagonal procedure. Lemma 2.17 is proved. \ [ 2,p

2,p

Definition 2.19 Let W1 = Wloc () and W2 = Wloc,w (). Analogously we define m,p m,p a Fréchet space Wb+ (l) () as a subspace of Wloc (). We recall that, the seminorms m,p in Wloc () are defined by: n }u; x0 , R, b+ (l)}m,p = sup }u,  X Ts BR x0 }m,p ă 8; x0 P R , R Ă R+ . s ě0

m,p

m,p

A space Wb(l) () Ă Wloc () is defined by the seminorms n }u; x0 , R, b(l)}m,p = sup }u,  X Ts BR x0 }m,p ă 8; x0 P R , R Ă R+ . s PR

0,p

0,p

We denote by Lb+ (l) () and Lb(l) the spaces Wb+ (l) () and Wb(l) () respectively 2,p

2,p

and by Wb+ (l) and Wb(l) the spaces Wb+ (l) () and Wb(l) () respectively. Below we p p formulate criteria of translation compactness on the spaces Lloc () and Lloc,w (). p p We denote by L1 := Lloc () and L2 := Lloc,w (). Lemma 2.18 p

1) A function g is translation compact in Lloc,w () if and only if when g P Lb+ (l)

118

2 Trajectory Dynamical Systems and Their Attractors α,p

2) Let a function g P Wb+ (l) (ε ) for some α ą 0 and ε ě 0. Then the function g is a translation compact in L1 . Proof Part 1) can be proved analogously to Lemma 2.10. Next we prove part 2. Since the hull H+ (g) is metrizable in L1 , in order to finish proof of the Lemma it suffices to prove that, from any sequence tsn , n P Nu, one can select a subsequence tsk = snk ; k P Nu such that Tsk g ÝÑ ξ in the space L1 . Without loss of generality we can assume that sn Ñ 8 as n Ñ 8 (otherwise, there exists a subsequence sk Ñ s0 , and from continuity of the semigroup tTs , s ě 0u n it follows that Tsk g Ñ Ts0 g in L1 ). Let BR x0 be any ball in R . According to + the definition of topology in L1 and by Cantor diagonal procedure, it suffices to show relative compactness of the sequence Tsn g|XBRx in Lp ( X BR x0 ). From 0

R Lemma 2.14, it follows that, there exists S = S(BR x0 ), such that Ts Bx0 Ă ε for s ě S. Thus, due to the assumption of Lemma 2.18, for sn ě S we have R + }Tsn g,  X BR x0 }H α,p ď C}g, Tsn Bx0 }2,p ď C}g, ε ; TS x0 , R, b (l)}α,p

ď C(x0 , R). Hence, the sequence tTsn g|XBRx , n P N, sn ě Su is bounded in H α,p ( X BR x0 ). 0

p R Since H α,p ( X BR x0 ) is compactly embedded in L ( X Bx0 ) for each α ą 0, this implies that the sequence tTsn g|XBRx , n P N, sn ě Su is relatively compact in 0

Lp ( X BR x0 ). This proves Lemma 2.18. Theorem 2.21 Let Conditions (1), (2), and (3) from Condition 2.5 be satisfied, that p is, a = a ˚ , g P Lloc () for p ą n{2 and g is a translation compact in L1 = p p g of the Eq. (2.116) is a compact set in Lloc,w (). Then the trajectory attractor A n Wi (R ), i = 1, 2, and consequently, Ag is compact in Wi (), i = 1, 2. p + (g) of the function g in the space Li . Since g is a Proof Let us consider the hull H translation compact in Li , hence (analogously to the case of translation compactness p + (g) and ω-limit set of g, that is ω p (g) are compact in Li (). in + ) the hull H p Moreover, a set of complete symbols Z(g) is a bounded subset of the space Lb(l) (Rn ) and is compact in Li (Rn ), However, it is not difficult to show that p + (g) = H+ (g); ω p p (g) = ω(g); Z(g) H = Z(g).

(2.134)

Here H+ (g) = , ω(g), Z(g)- are hull, ω-limit set and set of complete symbols of g respectively; g is a translation compact function in + . Let ξ P ω(g). Then, according to (2.115)

2.10 Regularity of Attractor

119

ξ = + ´ lim Tsn g

(2.135)

nÑ8

for some sequence sn Ñ 8 as n Ñ 8. Since g is a translation compact in Li , i = 1, 2, then a set of limit points tTsn g, n P Nu (as well as any of its subsequences) is nonempty. At the same time, from (2.135) it follows that this limit set consists of a single point tξ u, that is, p (g). ξ = L+ i ´ lim Tsn g P ω

(2.136)

nÑ8

p (g). The inverse inclusion is obvious. Thus ω p (g) = ω(g). Hence we have ω(g) Ă ω All other equalities of (2.134) can be proved analogously. Thus, due to the estimate (2.86) we obtain that any trajectory u(x), x P Rn of the p g of all complete family of equations (2.100) belongs to Wb(l) (Rn ), and the set A trajectories is bounded in this space, which in turn implies that (see Corollary 1.3) p g is relatively compact in W2 (Rn ). The closedness of A p g in W2 (Rn ) can be proved A analogously to Theorem 2.12. Thus, in the case of the mean topology (i = 2), Theorem 2.21 is proved. Let us prove assertion of Theorem 2.21 in the case of the strong topology (i = 1), p g in W1 (Rn ). To this end, we consider arbitrary un P that is, prove compactness of A p g with corresponding complete symbols ξn P Z(g). Then as was proved above, A 2,p without loss of generality we can assume that un á u (weakly) in Wloc,w (Rn ) and p n ξn á ξ (weakly) in Lloc,w (R ), and u P Kξ . To finish the proof of Theorem 2.21 in 2,p

this case, it suffices to prove that, un á u (strongly) in Wloc (Rn ). In other words, for any ball BR x0 we have un |BRx ÝÑ u|BRx in W 2,p (BR x0 ). 0

(2.137)

0

n Note that, firstly ξn Ñ ξ in L+ 1 (R ). Indeed, since g is a translation compact in the strong topology, it follows that Z(g) is compact in L1 (Rn ). Using arguments analogous to those in the proof of equality (2.134), we obtain that ξn Ñ ξ . Let us prove (2.137). Let ϕ(x) P C08 (Rn ), such that, ϕ(x) = 1 if x P BR x0 and ϕ(x) = 0 if x R BR+ε and rewrite Eq. (2.100) in the following way: x0

$ & x (ϕun ) = 2∇x ϕ ¨ ∇x un + ϕa ´1 [´γ Dun + f (un ) + ξn ] ” ξ˜n (x), % ϕun |BR+ε = 0. x 0

(2.138) p Note that, ξ˜n Ñ ξ˜ in Lp (BR+ε x0 ). Indeed, as was proved above, ϕξn Ñ ϕξ in L . 2,p R+ε 1,p R+ε 2,p Since W (Bx0 ) is compactly embedded in W (Bx0 ) and un á u in W , 2,p (BR+ε ) we obtain ∇x un Ñ ∇x u in Lp (BR+ε x0 ). Since p ą n{2 it follows that W x0

120

2 Trajectory Dynamical Systems and Their Attractors

is compactly embedded in C(BR+ε x0 ) and, taking into account the continuity of f , p R+ε we obtain that fn Ñ f in L (Bx0 ). Thus ξ˜n (x) Ñ ξ˜ (x) in Lp (BR+ε x0 ). From Lp -regularity of the Laplacian we obtain that, ϕun ÝÑ ϕu in W 2,p (BR+ε x0 ), which in turn implies the validity of (2.137). This proves Theorem 2.21.

\ [

Our next task is to study convergence of trajectories of the equations (2.100) to the trajectory attractor Ag in the space Wi (), i = 1, 2. Since we do not impose any + conditions (smoothness) on B, we cannot state that (in general), Ts K is a subset of Wi for s ě 0. Consequently, the analog of Definition 2.8 for attraction property to the attractor in the topology of space Wi , i = 1, 2 must be specified. Note that, due to Lemma 2.14 and Theorem 2.8, for any ball BR x0 Ť  we have + |BRx Ă Wi (BR Ts K  x0 ) for s ě 0 0

and consequently the following definition is correct: + Definition 2.20 We say that the attractor Ag attracts the family of sets tTs K ,sě R 0u in the topology of Wi , if, for any Bx0 Ť , the restriction Ag |BRx attracts the 0

+ |BRx , s ě 0u in Wi (BR family of sets tTs K x0 ). In other words, in any neighborhoods 0

of Ag , that is, for O(Ag |BRx ) in Wi (BR x0 ), there exists S = S(O), such that for any 0 s ě S the following equations holds. + Ts K  |BRx Ă O(Ag |BRx ), for s ě S. 0

0

Analogously, we will say that + Tsn un ÝÑ u as sn ÝÑ 8, un P K

in Wi , if for any BR x0 Ť  Tsn un |BRx ÝÑ u|BRx in Wi (BR x0 ). 0

(2.139)

0

Theorem 2.22 Let Conditions (1), (2), and (3) from Condition 2.5 hold. Then the trajectory attractor Ag of equation ( = H+ (g)) #

au + γ Du ´ f (u) = g(x), u|B = u0

+ attracts a family of semitrajectories tTs K , s ě 0u in the topology of Wi .

2.11 Trajectory Attractor of an Elliptic Equation with a Nonlinearity That. . .

121

Proof A proof of Theorem 2.22 is based on the following Lemma 2.19 (see below). + Lemma 2.19 Let un P K , n P N and sn Ñ 8. Then from the sequence tTsn un , s P Nu one can select a subsequence which converges in the topology Wi to some u P Ag .

Proof From Theorem 2.8 it follows that, for any ball BR x0 Ť , the sequence Tsn un |BRx is bounded in the space W 2,p (BR ) and consequently relative compact x0 0

in W2 (BR x0 ). Using Cantor’s diagonal procedure one can extract a subsequence from Tsn un (which we continue to denote as Tsn un ) converging in W2 () to some u P W2 (). By definition of trajectory attractor, it follows that u P Ag . Thus, in the case of the weak topology (i = 2), Lemma 2.19 is proved. Next consider the case of the strong topology (i = 1). In this case, in the same manner as in the proof of Theorem 2.22, one can show that, from Tsn un á u in W2 it follows that Tsn un Ñ u in W1 . Lemma 2.19 is proved. Proof of Theorem 2.21 Assume the contrary. Then there exists a ball BR x0 Ť , + such that, Ag |BRx does not attract a family Ts K |BRx in the space Wi (BR x0 ). This 0 0 in turn (see [67]) implies that, there exists a sequence tTsn un |BRx , n P Nu, un P 0

+ and sn Ñ 8, such that, a set of its limit points in the topology of Wi do not K intersect Ag |BRx . However, this contradicts the assertion of Lemma 2.19. This proves 0 Theorem 2.21.

Corollary 2.9 Let Conditions (1), (2), and (3) from Condition 2.5 hold. Then for any BR x0 Ť  holds + lim dist C(BRx ) tTs K |BRx , Ag |BRx u = 0.

s Ñ+8

0

0

(2.140)

0

2.11 Trajectory Attractor of an Elliptic Equation with a Nonlinearity That Depends on x In this section, we briefly describe the construction of the trajectory attractor of (2.40) for the case when the nonlinearity f = f (x, u) depends on x P . Due to the estimates (2.51) and (2.86), in this general case (f = f (x, u)), all estimates obtained for the case f = f (u) can be easily extended. The only difference here consists in the fact that we have to change the definition of symbols (see Sect. 2.7) in an appropriate manner. Indeed, in order to define the corresponding trajectory dynamical systems for (2.40), we have to take the symbol σ as a pair σ = (f (¨, ¨), g(¨)). Otherwise, a family of tKσ+ | σ P u would not be translation compatible, and, consequently, we would not be in position to define a semigroup tTs , s ě 0u on K . In order to define a semigroup appropriate for this case and to prove the existence of the trajectory attractor, we need several definitions.

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2 Trajectory Dynamical Systems and Their Attractors

Definition 2.21 Let C = Rk ˆ  and denote ¯ C loc (Rk , Rk )) = C loc (C, Rk ). M = C loc (,

(2.141)

A system of seminorms in M is defined by ! ) ¯ X BR ||f ||K,BRx = sup |f (x, u)| u P K, x P  x0 , 0

for arbitrary K Ă Rk compact, x0 P Rk , R P R+ . It is obvious that a function f (x, u) which satisfies the Assumptions on f (see Condition 2.1), belongs to M. Consider the hull of M: ”  ı   H(f ) = H+ (f ) = f Tsl x, u , s ě 0

M

l

.

(2.142)

Definition 2.22 A function f P M is called translation compact in M (in direction  if its hull (2.142) is compact in M. l) Criteria for translation compactness in the spaces (2.142) are formulated in [32, 33]. In what follows, we assume that f is translation compact in M. Remark 2.18 Let f P M and satisfy    f Tsl x, u ” f (x, u), s ě 0. Then H+ (f ) = tf u and, consequently, f is translation compact in M. In order to l construct the trajectory attractor for the Eq. (2.40) instead of the family of equations # au + γ Du ´ f (u) = σ (x), σ (¨) P , we consider # au + γ Du ´ h(t, u) = ξ(x), σ ” (h(¨, ¨), ξ(¨)) P  = H+ (f ) ˆ H+ (g),

(2.143)

where the nonlinearity t(f, g)u on the right-hand side of (2.40) respectively. Analogously to the Sect. 2.7, we will call the set  the space of symbols and its elements th, ξ u we call the symbols of equations (2.143). Remark 2.19 Let f be translation compact in the space M and satisfy Assumptions on f (see Condition 2.1). Then, it is not difficult to see that any function h P H+ (f ) will satisfy Condition 2.1 with the same constants C, C1 and C2 as were for f . Thus, all estimates (2.68) and (2.86) are fulfilled uniformly for the family of

2.12 Examples of Trajectory Attractors

123

equations (2.143). Therefore, taking into the account Remark 2.19, all Definitions and Theorems of Sects. 2.4–2.7 extend to the case when f = f (x, u). For example, the analogs of Theorems 2.17, 2.21, 2.22 are the following: Theorem 2.23 Let f be translation compact in M and satisfy Assumptions on f and g be translation compact in  + . Then, the Eq. (2.40) possesses a trajectory attractor Atf,g u = A ,  = H+ (f ) ˆ H+ (g). Theorem 2.24 Let f be translation compact in M and all assumptions of Theo+ rems 2.21 and 2.22 satisfied. Then, Atf,g u is compact in W+ i () and attracts H in the topology W+ i () (in the Definition 2.20).

2.12 Examples of Trajectory Attractors In this section we present several examples of equations of the form (2.40) having trajectory attractors. First we start with those examples for which Ag is “trivial”. Theorem 2.25 Assume that g ” 0, γ = γ ˚ and the nonlinearity f satisfies Assumptions on f (see Condition 2.1) and in addition can be represented in the form f (u) = αu + F (u), F (u) ¨ u ě 0, @u P Rn ,

(2.144)

where α ą 0. Then for any domain  satisfying Condition 2.2, the trajectory attractor is A0 = t0u. Proof Note that, from (2.144), it follows that f (0) = 0, and thus u ” 0 is a solution of the Eq. (2.40). Our goal is to prove that u ” 0 is the only solution of the Eq. (2.40), belonging to (Rn ). Indeed, let u P (Rn ) be a solution of (2.40). Since g ” 0, Theorem 2.9 follows that, u P b (Rn ). Multiplying Eq. (2.40) in Rk (scalar product) by the function ue´ε|x | , 0 ă ε ! 1 and integrating with respect to x P Rn we obtain xa∇u, ∇(ue´ε|x | )y + αxe´ε|x | u, uy ´ xγ Du, ue´ε|x | y ď 0.

(2.145)

Since a + a ˚ ą 0, we have xa∇x u, ∇x (ue´ε|x | )y ě Cx|∇x u|2 e´ε|x | , 1y ´ C1 εx|∇x u| ¨ |u|e´ε|x | , 1y ě C2 x|∇x u|2 e´ε|x | , 1y ´ C2 ε2 x|u|2 e´ε|x | , 1y.

(2.146)

Integrating the third term in (2.145) by parts and taking into account self-adjointness of γj P L(Rk , Rk ) as well as |∇x e´ε|x | | ď εe´ε|x | , we obtain

124

2 Trajectory Dynamical Systems and Their Attractors

|xγ Dx u, ue´ε|x | y| ď

k 1ÿ |xBxi (γi u ¨ u)e´ε|x | , 1y| d i=1

ď Cεx|∇x u| ¨ |u|e´ε|x | , 1y ď C2 x|∇x u|2 e´ε|x | , 1y ´ C3 εx|u|2 e´ε|x | , 1y. Inserting these estimates into (2.145) for sufficiently small ε ! 1 we have x|u|2 e´ε|x | , 1y ď 0.

(2.147)

Consequently u ” 0. Thus, due to Theorem 2.16, we obtain that A0 = t0u. This proves Theorem 2.25. Remark 2.20 Note that, Condition (2.144) does not imply monotonicity of the Eq. (2.40). For example, all Conditions of Theorem 2.25 are fulfilled by the function F (u) = u|u|2 (2 + sin |u|)2 . Remark 2.21 The assumption γ = γ ˚ is essential for the validity of the assertion of Theorem 2.25. Indeed, let us consider y 2 (t) + γ y 1 (t) ´ y(t) = y(|y|2 ´ 1)2 ,

(2.148)

where y = (y1 , y2 ) and

0 ´1 γ =2 . 1 0

(2.149)

In this case, the Eq. (2.148) admits the nontrivial solution y(t) = (sin t, cos t). The next examples show that, in contrast to the attractors of evolution equations, an elliptic attractor (that is, a trajectory attractor corresponding to a nonlinear elliptic equation in an unbounded domain) is not necessarily connected. We now present a couple of examples that show that, in contrast to the attractors of evolution  equations, Alg is not necessarily connected. Theorem 2.26 Assume that N = 1 or N = 2. Moreover, a = a ˚ ą 0, g = 0 and f satisfies f (u) ¨ u ě 0 in addition. Then, any solution u P θ (RN ) of au = f (u) in RN

(2.150)

is a constant, i.e., u ” u0 P RN with f (u0 ) = 0. Proof Let u P θ (RN )(RN ) be a solution of (2.150). As is known, such solution is bounded in RN . Let us consider y(x) := au(x) ¨ u(x). It is obvious that

2.12 Examples of Trajectory Attractors

125

x y = 2a∇u ¨ ∇u + 2f (u) ¨ u ě 0.

(2.151)

Thus, y(x) is a subharmonic function in RN and for N = 1 and N = 2 it is constant, hence y ” const. Indeed, for N = 1 it is obvious, since y is a convex function, in the case N = 2 it is also clear that y(x) = const. Then, it follows with (2.151) that f (u) = 0. \ [ For the convenience of the reader, we present below a proof of an analog of the Liouville theorem for subharmonic functions in R2 . Lemma 2.20 Any bounded subharmonic function w P L1b (R2 ) is a constant. Proof Without loss of generality we will present a proof for w P C 8 (R2 )XCb (R2 ). The general case can be reduced to w P C 8 (R2 ) X Cb (R2 ) by means of averaged operator (see (2.82)). Thus, let x w = ϕ(x) ě 0.

(2.152)

If ϕ(x) ” 0, then w(x) is a harmonic function and the assertion of Lemma 2.20 follows from Harnack’s inequality [21]. Assume now ϕ(x) ı 0. Then without loss 2 of generality one can assume that, ϕ(0) ‰ 0. Let BR 0 Ă R be an arbitrary ball in R2 with radius R about the origin. Let w(x) = w1 (x) + w2 (x), x P BR 0,

(2.153)

where #

#

x w1 (x) = 0 w1 |BBR = wBBR 0

and

0

x w2 (x) = ϕ(x) w2 |BBR = 0.

(2.154)

0

From the maximum principle (see Chap. 1) it follows that w1 (x) ď C uniformly in ρ R, R P R+ . Let us estimate w2 (x). Since ϕ(x) ı 0, then there exists B0 , ρ ă R, ρ such that, ϕ(x) ě ε for all x P B0 . From the maximum principle it follows that w2 (x) ď w˚ (x) for x P BR 0 , where w2 (x) is a solution of the following problem: #

x w˚ (x) = εχBρ (x), x P BR 0

w˚ |BBR = 0.

0

(2.155)

0

ρ

Here χBρ (x) is the characteristic function of the ball B0 with a radius ρ about the 0 origin. A solution of the Eq. (2.155) can be written explicitly, namely, w˚ (x) =

„ j  2 maxtρ, |x|u ρ ln ´ (|x|2 ´ ρ 2 )´ , x P BR (0). 2 R

126

2 Trajectory Dynamical Systems and Their Attractors ?

R e

ε 2 Thus w(x) ď C + w˚ (x) for all x P BR 0 . In particular, w(0) ď C ´ 2 ρ ln ρ . Letting R Ñ 8, we obtain that w(0) = ´8, which is a contradiction to boundedness of w(x). This proves Lemma 2.20. \ [

Remark 2.22 Simple examples show that, in the case of dimension n ě 3 the assertion of Theorem 2.26 cannot be true (in general). Moreover, γ = 0 is also essential. Indeed, let us consider the following equation (k = 1, n = 1): y 2 (t) + 2y 1 (t) = f (y),

(2.156)

where $ 3 yă0 &y f (y) = sin2 y(1 ´ sin y cos y) 0 ď y ď π % y ą π. (y ´ π )3

(2.157)

One can easily check that this equation admits the family of solutions of the form y(t) =

π + arctan(t + t0 ). 2

(2.158)

Example 2.11 Let us consider in an arbitrary domain  Ă R2 that satisfies the conditions on the domain  (see Sect. 2.4). Then, due to the Theorem 2.26, the trajectory attractor of the equation "

u = u(u ´ 1)2 ¨ ¨ ¨ (u ´ M)2 , u|B = u0 .

consists only of the corresponding equlibria, 

Al0 = tu ” 0, 1, . . . , Mu . Example 2.12 Let us consider in RN the following ‘autonomous’ equation (i.e., its coefficients do not depend explicitly on x): u ´ u = ´2uθ (u) + u|u|2 (1 ´ θ (u)).

(2.159)

Here, θ (u) is a C 8 cut-off function, which is equal to 1 in |u| ď 1 and 0 in |u| ą 2. It is obvious that (2.159) satisfies all assumptions of Theorem 2.17 and possesses a trajectory attractor A. We show that dimF A = 8. Indeed, for |u| ď 1, the Eq. (2.159) takes the form u + u = 0.

(2.160)

2.13 The Trajectory Dynamical Approach for the Nonlinear Elliptic Systems. . .

127

Let W0 be a space of solutions of (2.160) belonging to Cb (RN ) - the space of continuous functions bounded as |x| Ñ 8. Obviously, W0 is a closed subspace of Cb (R2 ), and its dimension is infinite since W0 contains infinitely many linearly independent functions wα (x) = sin(αx1 ) ¨ . . . ¨ sin(αxN ),

N ÿ

αi2 = 1.

i=1

Hence, A contains a unit ball of W0 and thus dimF A = 8.

2.13 The Trajectory Dynamical Approach for the Nonlinear Elliptic Systems in Non-smooth Domains In this section our goal is to study how nonsmoothness of the unbounded domain (see below) is inherited by the corresponding trajectory attractor for elliptic systems. Indeed, in the half cylinder + = R+ ˆω, where ω is a bounded polyhedral domain in Rn , we consider the following elliptic system: $ & a(B 2 u + u) + γ Bt u ´ f (u) = g(t), t

% u|

t=0

= u0 , Bn u|Bω = 0.

(2.161)

Here (t, x) are the variables in + , u = u(t, x) = (u1 , ¨ ¨ ¨ , uk ), g = (g 1 , ¨ ¨ ¨ , g k ), and f (u) are vector-valued functions,  is the Laplacian with respect to the variable x = (x 1 , ¨ ¨ ¨ , x n ) and γ and a are constant k ˆ k matrices with a = a ˚ ą 0. Notice that domains ω with non-degenerate edges and corners on the boundary are admitted. To be more rigorous, the domain ω is said to be polyhedral if any of its boundary points b is either regular or there are polyhedron P Ă Rn , a non-regular boundary point b1 of P , open subsets U , V of Rn with b P U , b1 P V , and a C 8 -diffeomorphism χ : U Ñ V such that χ (b) = b1 and χ (ω X U ) = P X V . On the nonlinear term f (u) we impose the following conditions: $ k k ’ ’ & (1) f P C(R , R ); (2) f (u) ¨ u ě ´C1 + C2 |u|p , 2 ă p ă 2 + 4{(n ´ 3); ’ ’ % (3) |f (u)| ď C(1 + |u|p´1 ).

(2.162)

Here and below u ¨ v denotes the inner product in Rk . We suppose that the right-hand side g belongs to the space [L2loc (+ )]k and has a finite norm

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2 Trajectory Dynamical Systems and Their Attractors

|g|b = sup }g, T }0,2 ă 8, T ě0

(2.163)

where T = (T , T + 1) ˆ ω. Further, we write H 1 (U ) instead of H (1,2) (U ) and }¨, U }t,p instead of } ¨ }H l,p (U ) . A solution u(t, x) to problem (2.161) is defined as a function that belongs to the space     2 2 (T )]k = [H 2 (T , T + 1), L2 (ω) X L2 (T , T + 1), HQ (ω) ]k [HQ for each T ě 0 and satisfies Eq. (2.161) in the sense of distributions. As in 2 (ω) the domain of the Laplace operator ´ in L2 (ω) Sect. 1.8, we denote by HQ with the homogeneous Neumann boundary condition. As we noted in Sect. 1.8, for 2 (ω) ‰ tu P H 2 (ω), B u| polyhedral domains in general HQ n B ω = 0u, in contrast to the case of smooth Bω (see [40]). 2 ( )]k , the Note that the third assumption of (2.162) implies that, for u P [HQ T 2 k function f (u) belongs to [L (T )] . Hence Eq. (2.161) can be considered as an equality in the space [L2loc (+ )]k . The initial data u0 are assumed to belong to the trace space V0 on tt = 0u of 2 functions in [HQ,loc (+ )]k . For domains ω with smooth boundary, the problem (2.161) has been investigated under different assumptions on the nonlinear part f and the right-hand side g in [8, 29, 31, 103]. The main objective of this section is to study the behaviour of the solutions to Eq. (2.161) as t Ñ +8 for polyhedral cross-sections ω. The following estimate is of fundamental significance in that connection: p ´1

}u, T }2,Q ď C(1 + χ (1 ´ T )}u0 }V0

˜ T }0,2 ). + }g, 

(2.164)

˜ T = (maxt0, T ´ 1u, T + 2) ˆ ω, χ (z) is the Heaviside function, i.e., Here  χ (z) = 1 for z ě 0 and χ (z) = 0 for z ă 0, and the constant C is independent of u0 . This estimate makes it possible to apply the methods of the theory of attractors (see [1, 32–34, 36]) to the problem (2.161). Furthermore, estimate (2.164) implies that every solution u(t, x) to the problem (2.161) is bounded as t Ñ 8, i.e., }u}b = sup }u, T }2,Q ă 8. T ě0

(2.165)

The subspace of functions u P [D1 (+ )]k which have finite norm (2.165) is denoted by F0+ . Since the conditions that we impose on the nonlinear function f (see (2.162)) guarantee in general only the existence (but not the uniqueness) of a solution to the problem (2.161), in order to describe the behaviour of solutions as t Ñ +8 we

2.13 The Trajectory Dynamical Approach for the Nonlinear Elliptic Systems. . .

129

construct a trajectory attractor for the dynamical system generated by the semigroup tTs , s ě 0u of positive shifts of the solutions to (2.161) along the t-axis (see Sect. 2.7 and [32, 34, 47, 103]). Here, since g explicitly depends on t, it is natural to study the family of equations of the form (2.161) generated by all positive shifts of this equation with respect to t and their limits in a suitable topology (see Sect. 2.10). The trajectory attractor A attracts the set K + of all trajectories of the above family as t Ñ +8. Recall that the attracting property is usually required only for those subsets of the phase space K + which are, in a certain sense, bounded. But in our case estimate (2.164) allows us to verify the following improved version (see Sect. 2.7 and also [108]): for any neighbourhood O(A) of the attractor A in the space K + , there exists a number T = T (O) such that Ts K + Ă O(A) for all s ě T . Moreover, like an ordinary attractor, the trajectory attractor A is strictly invariant with respect to the semigroup tTs , s ě 0u and is generated by all the trajectories of this semigroup that are defined and bounded for t P R. We also study in this Section the problem of stabilization of the solutions to Eq. (2.161) as t Ñ +8 in the case that the nonlinear part f has a potential (f = ∇F, F : Rk Ñ R). Especially, it is shown that in the autonomous case (g(t, x) ” g(x)) every solution to the problem (2.161) in the whole cylinder  = R ˆ ω is a heteroclinic orbit connecting two stationary solutions (see below). Furthermore, there is a non-canonical splitting of K + into a regular part and + the space Ksing which contains the edge asymptotics of solutions belonging to K + . + becomes invariant with respect to This splitting is chosen in a way such that Ksing the semigruop tTs , s ě 0u of positive shifts along the t-axis: + + Ñ Ksing s ě 0. Ts : Ksing + possesses an We then show that the semigroup tTs , s ě 0u restricted to Ksing attractor Asing which in turn is interpreted as the singular part of the trajectory attractor A of Eq. (2.161). We start to obtain a priori estimates for solutions to the problem (2.161). In the sequel these estimates will be used to prove existence of solutions and to construct the trajectory attractor.

Theorem 2.27 Let u be a solution to (2.161). Then p ˜ T }20,2 ), }u, T }21,2 ď C(1 + χ (1 ´ T )}u0 }V0 + }g, 

where C is independent of u.

(2.166)

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2 Trajectory Dynamical Systems and Their Attractors

2 ( )]k (see Sect. 1.8) Proof By definition of V0 , there exists a function v P [HQ,b + such that supp v Ă 0 , v|t=0 = u0 , and

}v, 0 }2,Q ď C}u0 }V0 ,

(2.167)

where the constant C is independent of u0 . Let us rewrite Eq. (2.161) for the function w = u ´ v, $ & a(Bt w + w) + γ Bt w ´ f (w + v) = g(t) ´ a(Bt v + v) ´ γ Bt v ” h(t), % w| = 0. t=0

(2.168) From the choice of v it follows that }h, T }0,2 ď C(}g, T }0,2 + χ (1 ´ T )}u0 }V0 ).

(2.169)

Let φ(t) = φT (t) be the following cut-off function: φ(t) =

"

(1 ´ |t ´ T ´ 1{2|)2p{(p´2) , for t P (T ´ 1{2, T + 3{2), 0, for t R (T ´ 1{2, T + 3{2).

It is readily seen that φ 1 P L8 (R). Moreover, the following estimate is valid: |φ 1 (t)| ď Cφ(t)1{2+1{p , for t P R.

(2.170)

Multiplying Eq. (2.168) by φw in Rk and integrating over + gives us xaBt2 w, φwy + xaw, φwy + xγ Bt w, φwy ´ xf (v + w), φwy = xh, φwy. (2.171) From the positivity of a and (2.170) it follows that ´xaBt2 w, φwy ě C1 xφ|Bt w|2 , 1y ´ x|φ 1 ||Bt w|, |w|y 1 ě C1 xφ|Bt w|2 , 1y ´ C1 xφ|Bt w|2 , 1y ´ Cxφ 2{p |w|2 , 1y 2 ě C2 xφ|Bt w|2 , 1y ´ Cxφ 2{p |w|2 , 1y.

(2.172)

Applying Hölder’s inequality to the third term in (2.171) we obtain the estimate |xγ Bt w, φwy|ďμxφ|Bt w|2 , 1y + Cμ xφ|w|2 , 1yďμxφ|Bt w|2 , 1y + Cμ xφ 2{p |w|2 , 1y for any μ ą 0 with some constant C ą 0.

2.13 The Trajectory Dynamical Approach for the Nonlinear Elliptic Systems. . .

131

In view of assumption (2.162) on the nonlinear term f (u), we further have xf (w + v), φwy =xf (w + v) ¨ (w + v), φwy ´ xf (v + w), vφy ě ´ C + C1 xφ|w + v|p , 1y ´ Cx1 + |w + v|p´1 , φ|v|y ě ´ C2 (1 + xφ|v|p , 1y) + C3 xφ|w|p , 1y p

ě ´ C4 (1 + χ (1 ´ T )}u0 }V0 ) + C3 xφ|w|p , 1y.

(2.173)

    2  q Here we have employed (2.167) and the embedding HQ T1 ,T2 Ă L T1 ,T2 . Using the positivity of a again, after integration by parts we find ´xaw, φwy ě Cxφ|∇w|2 , 1y. Finally, from (2.169) and Hölder’s inequality we conclude xh, φwy ďxφ|h|2 , 1y + xφ|w|2 , 1y ďC(xφ|g|2 , 1y + χ (1 ´ T )}u0 }2V0 ) + C1 xφ 2{p |w|2 , 1y.

(2.174)

By inserting all the estimates (2.172)–(2.174) into (2.171), a short calculation yields xφ|Bt w|2 , 1y + xφ|∇w|2 , 1y + xφ|w|p , 1y ´ Cxφ 2{p |w|2 , 1y ďC1 (1 + xφ|g|2 , 1y + χ (1 ´ T )}u0 }V0 ).

(2.175)

We estimate the last term of the left-hand side in (2.175) using Hölder’s inequality, xφ 2{p |w|2 , 1y = x|φ 1{p w|2 , 1y ď C(xφ|w|p , 1y2{p ď μxφ|w|p , 1y + Cμ , which holds for any μ ą 0. Choosing μ ą 0 sufficiently small, this estimate inserted into (2.175) yields xφ|Bt w|2 , 1y + xφ|∇w|2 , 1y + xφ|w|p , 1yďC2 (1 + xφ|g|2 , 1y + χ (1 ´ T )}u0 }V0 ). (2.176) Recall that φ(t) ą C0 ą 0 for t P (T , T + 1). Hence from (2.176) we infer that ˜ T }20,2 ). }w, T }21,2 ď C(1 + χ (1 ´ T )}u0 }V0 + }g,  p

Theorem 2.27 is proved.

\ [

Remark 2.23 In a similar manner it follows from (2.176) that p p ˜ T }20,2 ). }u, T }0,p ď C(1 + χ (1 ´ T )}u0 }V0 + }g, 

(2.177)

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2 Trajectory Dynamical Systems and Their Attractors

Theorem 2.28 Let u be a solution to (2.161). Then for each T ě 0 we have 2(p´1)

2(p´1)

}u, T }0,2(p´1) ď C(1 + χ (1 ´ T )}u0 }V0

˜ T }20,2 + }u,  ˜ T } ). + }g,  0,p (2.178) p

The exponent p is defined in (2.162). Proof We fix some T ě 0 and take another cut-off function ϕ(t) P C08 (R) such that ϕ(t) = 1 for t P (T , T + 1], ϕ(t) = 0 for t R (T ´ 1, T + 2), and 0 ď ϕ(t) ď 1. p ´2 Multiplying Eq. (2.168) by ϕw|w|a , where |w|a ” (aw ¨w)1{2 , and afterwards integrating over + , we obtain the following equality: p ´2

xa(Bt2 w + w), φw|w|a

p ´2

y = ´ xϕγ Bt w, w|w|a p ´2

+ xϕh, w|w|a

p ´2

y + xϕf (w + v) ¨ w, |w|a

y.

y

(2.179)

2 ( ˜ T )]k , Bt2 w + w P [L2 ( ˜ T )]k . It is not difficult to see By definition of [HQ p ´2 ˜ T )]k . Hence, all the that the functions w|w|a and f (w + v) also belong to [L2 ( integrals in (2.179) are correctly defined. Moreover, by virtue of Theorem 1.35, we p ´2 ˜ T ). Thus on the left-hand side of (2.179) we can integrate by have w|w|a P H 1 ( parts and get p ´2

xaBt2 w, φw|w|a

y p ´2

= ´ xaBt w, Bt (φw|w|a =´

)y

1 1 p p ´2 p ´4 xφ , Bt (|w|a )y ´ xφ|Bt w|2a , |w|a y ´ (p ´ 2)xφ(aBt w, w)2 , |w|a y p

1 4(p ´ 2) p{2 p{2 p p ´2 = xφ 2 , |w|a y ´ xφ|Bt w|2a , |w|a y ´ xφBt (|w|a ), Bt (|w|a )y 2 p p ˜ T }p ´ C2 xφBt (|w|ap{2 ), Bt (|w|ap{2 )y. ďC1 }w,  0,p Analogously, p ´2

xaw, φw|w|a

p{2

p{2

y ď ´C2 xφ∇(|w|a ), ∇(|w|a )y.

Hence we obtain p ´2

´xa(Bt2 w + w), φw|w|a

p{2

p{2

˜ T } + C2 (xφBt (|w|a ), Bt (|w|a )y y ě ´ C1 }w,  0,p p

p{2

p{2

+ xφ∇(|w|a ), ∇(|w|a )y).

(2.180)

2.13 The Trajectory Dynamical Approach for the Nonlinear Elliptic Systems. . .

133

It follows from Hölder’s inequality that p ´2

|xγ Bt w, φw|w|a

p{2

p{2

y| ď μxφBt (|w|a ), Bt (|w|a )y + Cμ xφ|w|p , 1y

and p ´2

|xh, φw|w|a

y| ď μxφ|w|2(p´1) , 1y + Cμ xφh, 1y ˜ T }20,2 + χ (1 ´ T )}u0 }2V ). ď μxφ|w|2(p´1) , 1y + Cμ (}g,  0 (2.181)

Here μ ą 0 is an arbitrary number. Arguing as for (2.173) above we obtain p ´2

xf (w + v), φw|w|a

y ě ´ C1 (1 + xφ|v|2(p´1) , 1y) + C2 xφ|w|2(p´1) , 1y 2(p´1)

ě ´ C3 (1 + χ (1 ´ T )}u0 }V0

) + C2 xφ|w|2(p´1) , 1y. (2.182)

Now replacing all terms in equality (2.179) by their corresponding bounds in (2.180) to (2.182) and taking μ ą 0 sufficiently small, after a short calculation we get 2(p´1)

xφ|w|2(p´1) , 1y ďC(1 + χ (1 ´ T )}u0 }V0

˜ T }20,2 + }w,  ˜ T }p ). + }g,  0,p \ [

This proves Theorem 2.28. Corollary 2.10 Let u be a solution to (2.161). Then for each T ě 0 we have ˜ T }0,2 ď C(1 + χ (1 ´ T )}u0 }p´1 + }g,  ˜ T }0,2 ). }f (u),  V0

(2.183)

This follows from the estimates (2.177), (2.178). Theorem 2.29 (The Main Estimate) Let u be a solution to the problem (2.161). Then the following estimate holds. ˜ T }2,Q ď C(1 + χ (1 ´ T )}u0 }p´1 + }g,  ˜ T }0,2 ). }u,  V0

(2.184)

Proof Rewrite Eq. (2.168) in the following form: $ & B 2 (ϕw) + (ϕw) = hw (t), t

% ϕw|

t=maxtT ´1,0u = 0, ϕw|t=T +2 = 0, Bn (ϕw)|B ω = 0.

(2.185)

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2 Trajectory Dynamical Systems and Their Attractors

Here ϕ is a cut-off function as in the proof of Theorem 2.28 and hw (t) = ϕ 2 w + 2ϕ 1 Bt w ´ a ´1 (ϕh(t) + ϕf (u) ´ γ Bt w). By (2.166) and (2.183) we have the estimate p ´1

˜ T }0,2 ď C(1 + χ (2 ´ T )}u0 } }hw ,  V0

p T }0,2 ), + }g, 

p T = (maxt0, T ´ 2u, T + 3) ˆ ω. By the L2 -regularity theorem (see where  Sect. 1.8) we obtain ˜ T }2,Q ď C1 }ϕw,  ˜ T }2,Q }w, + X  ˜ T }0,2 ď C}hw ,  p ´1

ď C2 (1 + χ (2 ´ T )}u0 }V0

p T }0,2 ). + }g,  \ [

This completes the proof of Theorem 2.29.

Remark 2.24 Let the condition (2.163) be satisfied. Then each solution u to (2.161) 2 2 ( )]k . More that is in [HQ,loc (+ )]k belongs automatically to the space [HQ,b + precisely, we have the estimate p ´1

}u}b ” sup }u, T }2,Q ď C(1 + }u0 }V0 T ě0

+ }g}b ).

(2.186)

In fact, (2.186) is a consequence of (2.184).

2.13.1 Existence of Solutions In this section we shall prove solvability for the problem (2.161). We first solve the following auxilliary problem in a finite cylinder: $ & a(B 2 u + u) + γ Bt u ´ f (u) = g(t), t

% u|

t=0

= u0 , u|t=M = u1 , Bn u|Bω = 0.

(2.187)

2 ( k Here u0 , u1 P V0 and u P [HQ 0,M )] . Then we shall obtain the solution u to the main problem (2.161) as the limit as M Ñ 8 of solutions uM to the corresponding auxiliary problems (2.187).

Theorem 2.30 Let u be the solution to the problem (2.187). Then the following estimate holds uniformly with respect to M Ñ 8:

2.13 The Trajectory Dynamical Approach for the Nonlinear Elliptic Systems. . . p ´1

}u, T }2,Q ď C(1 + χ (1 ´ T )}u0 }V0

135 p ´1

+ χ (T ´ M + 1)}u1 }V0

˜ T X 0,M }0,2 ). + }g, 

(2.188)

The proof of (2.188) is analogous to that of (2.184) given above in the case of the semibounded cylinder: Theorem 2.31 For every u0 , u1 P V0 , the problem (2.187) has at least one solution. Proof Introduce the space 2 WM = tw P [HQ (0,M )]k : w|t=0 = w|t=M = 0u

and reformulate problem (2.187) with respect to the new function w = u ´ v, where 2 ( k w P WM , v P [HQ 0,M )] : $ & B 2 w + w = a ´1 (´γ Bt w + f (v + w) + g(t)), t % w| = 0, w| = 0, B w| = 0. t=0

t=M

n

(2.189)

Bω

Here g1 = ´a(Bt2 v + v) ´ γ Bt v + g. Let A denote the inverse to the Laplace operator with respect to the variables (t, x) P 0,M and the boundary conditions w|t=0 = 0, w|t=M = 0, Bn w|Bω = 0. Then from the results of Sect. 1.8 we get A : [L2 (0,M )]k Ñ WM . Applying the operator A to both sides of Eq. (2.189) we obtain w + F (w) = h ” ´A(Bt2 + v), where F (w) = ´Aa ´1 (´γ Bt w + f (v + w) + g ´ γ Bt v). Now we use the Leray-Schauder principle in the following form (see [49, 69]): Leray-Schauder Principle Let D be a bounded open set in a Banach space W and let F : D Ñ W be a compact and continuous operator. Further let the point h P D be such that w + sF (w) ‰ h, for all w P BD, s P [0, 1].

(2.190)

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2 Trajectory Dynamical Systems and Their Attractors

Then the equation w + F (w) = h has at least one solution in D. Let BR be an open ball in WM of sufficiently large radius and suppose that ws + sF (ws ) = h, for some ws P BD, s P [0, 1].

(2.191)

Equation (2.191) can be rewritten in the form $ & a(B 2 us + us ) + sγ Bt us ´ sf (us ) = sg(t), t %u | s t=0 = 0, us |t=M = 0, Bn us |B ω = 0,

(2.192)

where us = ws + v. Now (2.192) is of the form (2.187). Moreover, it is not difficult to see that the estimate (2.188) holds uniformly with respect to s P [0, 1]. Hence }ws }WM ď K for all solutions to (2.192) uniformly with respect to s P [0, 1]. Therefore, condition (2.190) is fulfilled if the radius of BR is chosen larger than K. We prove compactness for the operator F . It is sufficient to prove compactness for the nonlinear part Aa ´1 f (w + v). To do this decompose the nonlinear part as a composition of three continuous operators A ˝ F2 ˝ F1 , with one of them being compact: F1 : WM Ñ [L2(p´1) (0,M )]k is the embedding which is compact because 2(p ´ 1) ă q0 (see Theorem 1.35) and F2 w = a ´1 f (v + w). The operator F2 is continuous from [L2(p´1) (0,M )]k to [L2 (0,M )]k in view of condition (2.162) and Krasnoselski’s theorem (see Theorem 1.14 and [74]). Hence the operator F is compact and according to the Leray-Schauder principle the problem (2.187) has at least one solution. \ [ 2 ( )]k . Theorem 2.32 The problem (2.161) has at least one solution u P [HQ,b +

Proof Consider a sequence uM , M = 1, 2, . . . , of solutions to the auxiliary problems (2.187) with u1 |t=M = 0. It follows from Theorem 2.30 that, for every fixed N , }uM , 0,N }2,Q ď C(u0 , N, g) holds uniformly with respect to M ě N . Using Cantor’s diagonalization procedure we extract a subsequence from uM , again denoted by uM , obeying the following property: 2 (0,N )]k uM |0,N Ñ u|0,N weakly in [HQ

2.13 The Trajectory Dynamical Approach for the Nonlinear Elliptic Systems. . .

137

2 ( )]k . We finally show that u is a solution to (2.161). It is for a certain u P [HQ,b + sufficient to prove that, for every  P [C08 (+ )]k , the following equality holds:

´xaBt u, Bt y ´ xa∇u, ∇y + xγ Bt u, y ´ xf (u), y = xg, y.

(2.193)

From the definition of uM we conclude that ´xaBt uM , Bt y ´ xa∇uM , ∇y + xγ Bt uM , y ´ xf (uM ), y = xg, y (2.194) when M is sufficiently large. Taking the limit M Ñ 8 in (2.194) we obtain (2.193). In fact, the only non-trivial part in its proof is to show that xf (uM ), y = xf (u), y holds. Suppose that supp  Ă 0,N . By Theorem 1.35, the embedding 2 (0,N )]k Ă [L2(p´1) (0,N )]k [HQ

is compact. Hence uM Ñ u in [L2(p´1) (0,N )]k and, by condition (2.162), f (uM ) Ñ f (u) in [L2 (0,N )]k . Theorem 2.32 is proved. \ [

2.13.2 Trajectory Attractor for the Nonlinear Elliptic System Now we are going to construct the trajectory attractor for the problem (2.161). For the convenience of the reader we recall and adopt below the main concepts and definitions as well as theorems from Sect. 2.7 to (2.161). See also [33, 34] for more details. Definition 2.23 The right-hand side g of (2.161) is said to be translation-compact in = [L2loc (R+ , L2 (ω))]k if its hull H+ (g) = [Ts g, s ě 0] + , (Ts g)(t) = g(t + s) is compact in + . Here [¨] + means the closure in the space + . The right-hand side g of (2.161) is said to be weakly translation-compact in the space + if its weak hull

138

2 Trajectory Dynamical Systems and Their Attractors + Hw (g) = [Ts g, s ě 0] +w

+ + is compact in + w . Here w denotes the space w equipped with the weak topology.

Remark 2.25 If the function g is translation-compact for the strong topology, then it is weakly translation-compact and + H+ (g) = Hw (g)

(see [103]). Remark 2.26 A functions g that is almost-periodic in t with values in L2 (ω) in the sense of Bochner-Amerio, in particular, a periodic or a quasi-periodic function, is evidently translation-compact in the space + (for its strong topology). Hence translation-compactness is a generalization of the concept of almost-periodicity. Remark 2.27 It follows from the definition of the hull that + + Ts H+ (g) Ď H+ (g), Ts Hw (g) Ď Hw (g) for s ě 0,

(2.195)

+ (g), respectively. i.e., the semigroup tTs , s ě 0u of shifts acts on H+ (g) and Hw

Next we formulate necessary and sufficient conditions for translationcompactness and weak translation-compactness in the space + . Theorem 2.33 ([32]) (1) A function g is weakly translation-compact in + if and only if it is bounded with respect to t Ñ 8, i.e., }g}b ă 8. (2) A function g is translation-compact in + if and only if the following conditions hold: şt+s (a) for any fixed t ą 0 the set t s g(z)dz, s P R+ u is precompact in the space [L2 (ω)]k ; (b) there exists a function β(s), s ě 0, β(s) Ñ 0 as s Ñ +0, such that ż t+1 t

}g(z) ´ g(z + l)}2L2 (ω) dz ď β(|l|) for all t P R+ with t + l P R+ . (2.196)

Remark 2.28 Condition (2.196) is fulfilled, e.g., if }Ts g, (0, 1) ˆ ω}δ,2 ď C, s ě 0, for a suitable δ ą 0.

2.13 The Trajectory Dynamical Approach for the Nonlinear Elliptic Systems. . .

139

To construct the trajectory attractor for the problem (2.161), we consider the family of problems of the form (2.161) obtained from all positive shifts of the initial problem (2.161) together with all limits in the appropriate topology: $ & a(B 2 u + u) + γ Bt u ´ f (u) = σ (t), σ P , t % u| = u , B u| = 0. t=0

0

n

(2.197)

Bω

Here we take  = H+ (g), if g is translation-compact for the strong topology, and + (g) otherwise.  = Hw Definition 2.24 For each σ P , Kσ denotes the space of all solutions to (2.197) with an arbitrary u0 P V0 . Further define K+ as the union of all Kσ+ : K+ =

ď σ P

Kσ+ .

It follows from (2.195) that the semigroup tTs , s ě 0u of non-negative shifts along the t-axis ((Ts v)(t) ” v(t + s)) acts on the space K+ , i.e., Ts K+ Ď K+ for s ě 0. The set K+ is endowed with the relative topology induced from the embedding + K+ Ă + 0 if  = H (g) (in case of the strong topology) and induced from + w + the embedding K Ă (+ 0 ) if  = Hw (g) (in case of the weak topology), + respectively. (For the definition of 0 , see Sect. 1.8.) Definition 2.25 The (global) attractor of the semigroup tTs , s ě 0u acting on the topological space Kσ+ is called the trajectory attractor of the family (2.197). That means that a set A Ă K+ is the trajectory attractor of the family (2.197) if the following conditions hold: (1) A is compact in K+ ; (2) A is strongly invariant with respect to tTs , s ě 0u, i.e., Ts A Ď A for all s ě 0; (3) A is attracting for tTs , s ě 0u, i.e., for any neighbourhood O = O(A ) in the topology of K+ there is an sO ą 0 such that + Ă O for all s ě sO . Ts K 

(2.198)

Remark 2.29 The attracting property is usually required only for (in some sense) bounded subsets of K+ . In view of estimate (2.185), however, the set T1 K+ is already bounded both in F0+ and + 0 . Hence the attracting property (2.198) is

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2 Trajectory Dynamical Systems and Their Attractors

automatically implied for all subsets of K+ , with the same constant sO (see also Remark 2.9). Theorem 2.34 ([32]) Let the following conditions be satisfied: (1) There exists a compact attracting set P Ă K+ for the semigroup tTs , s ě 0u; (2) the set K+ is closed in the space + 0 in case of the strong topology and sequenw in case of the weak topology, respectively. tially closed in the space (+ ) 0 Then the family (2.198) possesses a trajectory attractor A = A in K+ . + (g) Definition 2.26 The trajectory attractor Aw of the family (2.198) with  = Hw (the case of the weak topology) is called the weak trajectory attractor of the initial problem (2.161). Analogously the trajectory attractor A = As of the family (2.198) with  = + H (g) (the case of the strong topology) is called the (strong) trajectory attractor of the initial problem (2.161).

Theorem 2.35 (1) Let condition (2.163) hold. Then the problem (2.161) possesses a weak trajectory attractor Aw . (2) Let the right-hand side g of the problem (2.161) be translation-compact in + (endowed with the strong topology). Then the problem (2.161) possesses a strong trajectory attractor A = As . To this end, we check the conditions of Theorem 2.34. w Lemma 2.21 The set K+ is sequentially closed in the space ( + 0) . w Proof Let um P Kσ+m , um Ñ u in ( + 0 ) . Without loss of generality we may suppose that σm Ñ σ weakly in + , since  is compact in + w . We have to prove that u P Kσ+ . By definition, the functions um (t) are bounded solutions to the following problems:

$ & a(B 2 um + um ) + γ Bt um ´ f (um ) = σm (t), t

%u | 0 0 m t=0 = um , um P V0 .

(2.199)

As in the proof of Theorem 2.32, taking the limit m Ñ 8 in (2.199) we obtain that \ [ u P Kσ+ . Hence the second condition of Theorem 2.34 holds. Let us check the first condition. From the estimate (2.184) we infer that the set P = BR X K+ , where BR is a sufficiently large ball in the space F0+ , is an absorbing set for the semigroup tTs , s ě 0u. First we consider the case of the weak topology. The set

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141

w BR is a compact and metrizable subset of (+ 0 ) . In fact, the ball BR is bounded in + + 0 and 0 is a reflexive and separable Fréchet space, hence BR is semi-compact and metrizable for the weak topology. Due to convexity, BR is a metrizable compact. w From Lemma 2.21 we finally conclude that the set P is compact in (+ 0) . Now let us suppose that the right-hand side g of the problem (2.161) is translation-compact for the strong topology.

Lemma 2.22 Let the previous conditions hold. Then the set Ts P is compact in + 0 for all s ą 0. Proof Without loss of generality we suppose that s = 1. Let tum u, um P BR X Kσ+m , be an arbitrary sequence. We can suppose that σm Ñ σ in + , since the hull  is compact in + (for the strong topology). Further we can suppose that tum u is weakly convergent to some u P Kσ+ , since P is weakly compact. 2 ( )]k for an To prove the lemma it suffices to show that um Ñ u in [HQ T arbitrary T ě 1. The functions um satisfy Eq. (2.199). Multiplying (2.199) by a cut-off function φ P C08 (R) such that φ(t) = 1 for t P (T , T + 1) and φ(t) = 0 for t ‰ (T ´ 1, T + 2), we obtain $ 2 ´1 2 ’ ’ & Bt (φum ) + (φum ) = a (φσm + φf (um ) ´ γ φBt um ) + 2Bt φBt um +(Bt φ)um ” h(t), ’ ’ % φum |t=T ´1 = 0, um |t=T +2 = 0, Bn (φum )|Bω = 0. Then, arguing as in the proof of Theorem 2.32, we can show that f (um ) Ñ f (u) ˜ T )]k and and Bt um Ñ Bt u in + . So hm Ñ h in [L2 ( 2 ˜ (T ) φum Ñ φu in HQ

(see Sect. 1.8). Hence, 2 (T )]k um Ñ u in [HQ

\ [

The proof is finished.

Therefore, all conditions of Theorem 2.34 hold: so the proof of Theorem 2.35 is complete. Corollary 2.11 Let the right-hand side g of the problem (2.161) be translationcompact in + . Then dist2,Q (!T1 ,T2 Ts K+ , !T1 ,T2 A) Ñ 0 as s Ñ +8, where dist2,Q (M, N ) = sup inf }x ´ y, wT1 ,T2 }2,Q . x PM y PN

Here !T1 ,T2 denotes the restriction in t to the interval (T1 , T2 ).

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2 Trajectory Dynamical Systems and Their Attractors

This corollary is immediate from the definition of trajectory attractor. Corollary 2.12 Let the right-hand side g of the problem (2.161) be weakly translation-compact in + . Then dist3{2+ε,2 (!T1 ,T2 Ts K+ , !T1 ,T2 A) Ñ 0 as s Ñ +8, and dist0,q (!T1 ,T2 Ts K+ , !T1 ,T2 A) Ñ 0 as s Ñ +8, where ε ą 0 is sufficiently small and q ă 2(n + 1){(n ´ 3). 2 (w k This corollary follows from the compactness of the embeddings [HQ T1 ,T2 )] Ă

2 (w k q k [H 3{2+ε,2 (wT1 ,T2 )]k and [HQ T1 ,T2 )] Ă [L (wT1 ,T2 )] proved in Sect. 1.8. Next we investigate the structure of the trajectory attractor A. To this end for the convenience of the reader we recall several concepts from Sect. 2.7 again. Let ω() be the ω-limit set (the attractor) of the semigroup tTs , s ě 0u acting on the compact space . It is non-empty and can be represented in the form

ω() =

č„ď t ą0

j Ts 

s ąt



(see [1]). Here [¨] denotes the closure in the space . Definition 2.27 A function ξ(t), t P R, is called a complete symbol of (2.197) if !+ ξs (¨) P ω() for all s P R. Here ξs (t) = ξ(t + s). The operator !+ is the restriction to the half-axis R+ . The set of all complete symbols of (2.197) is denoted by Z(). Lemma 2.23 ([32]) For each σ P ω(), there exists a complete symbol ξ P Z() such that !+ ξ = σ . Definition 2.28 For ξ P Z(), Kξ is the set of all bounded solutions to Eq. (2.197) on the whole axis t P R, where σ (t) is replaced by ξ(t). Theorem 2.36 ([32]) The attractor A has the following structure: A = !+

ď

Kξ

ξ PZ()

Corollary 2.13 ([103]) Let the right-hand side g be translation-compact in + . Then the weak trajectory attractor of the problem (2.161) coincides with the strong trajectory attractor: As = Aw .

2.13 The Trajectory Dynamical Approach for the Nonlinear Elliptic Systems. . .

143

2.13.3 Stabilization of Solutions in the Potential Case In this section we shall study the long-term behaviour of solutions when the righthand side g(t) of (2.161) has the form g(t, x) = g+ (x) + g1 (t, x),

(2.200)

where g+ P L2 (ω) is independent of t and g1 satisfies the condition Ts g1 Ñ 0 as s Ñ +8

(2.201)

in + and ( + )w , respectively. It is not difficult to see that in the first case the function g is strongly translation-compact in + , while in the second case it is weakly translation-compact in + . Theorem 2.37 Suppose that the condition (2.201) holds. Then the problem (2.161) with right-hand side (2.200) possesses a strong and weak trajectory attractor A = Ag , respectively. It coincides with the attractor of the limit autonomous equation a(Bt2 u + u) ´ γ Bt u ´ f (u) = g+ , i.e., A = Ag .

(2.202)

Proof The existence of the trajectory attractor follows immediately from Theorem 2.35. Thus we show (2.202). From condition (2.201) we get that Z(g) ” Z() = ω() = g+ . Here  is the respectively strong and weak hull of the right-hand side g in the space + . Hence formula (2.202) holds in view of Theorem 2.36. Theorem 2.37 is proved. \ [ Now we assume that the nonlinear term f (u) on the left-hand side of Eq. (2.161) is gradient-like, i.e., f (u) = ´∇F (u), F P C(Rk , R).

(2.203)

2 ( )]k , we introduce the function F (t) by For u P [HQ,b + u

Fu (t) =

1 1 (aBt u(t), Bt u(t)) ´ (a∇u(t), ∇u(t)) + (F (u(t)), 1) ´ (g+ , u(t)), 2 2 (2.204)

where (¨, ¨) is the L2 -scalar product in the cross-section.

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2 Trajectory Dynamical Systems and Their Attractors

Theorem 2.38 2 ( )]k , the function F is well-defined and belongs to the (1) For every u P [HQ,b + u

space Hb1,1 (R+ ). (2) If u is a solution to the problem (2.161), then dFu (t) = ´(γ Bt u(t), Bt u(t)) + (g1 (t), Bt u(t)). dt

(2.205)

2 ( )]k . Then, according to Corollary 1.12, the first, the Proof Let u P [HQ,b + second, and the fourth term on the right-hand side of (2.204) are well-defined. It remains to consider the third term. From (2.162) and (2.203) we infer that

|F (u)| ď C(1 + |u|p ). Then Corollary 1.11 yields that (F (u(t)), 1) P Cb (R+ ). Hence Fu (t) is well-defined. When calculating the derivative, using standard methods of distribution theory, we obtain that Fu P Hb1,1 (R+ ), dFu (t) = (a(Bt2 u + u) ´ f (u) ´ g+ , Bt u). dt

(2.206)

Hence the first part of the proof is finished. Now suppose that u is a solution to the problem (2.161). Then (2.203) follows immediately from (2.206). The second part of the proof is also finished. \ [ Theorem 2.39 Suppose that the conditions (2.201) and (2.203) hold. Further suppose that the matrix γ on the left-hand side of (2.161) is sign-definite, i.e., either γ + γ ˚ ą 0 or γ + γ ˚ ă 0, and the function g1 (t) = g1 (t, x) satisfies at least one of the following conditions: $ ş8 ’ ’ & (i) 0 }g1 (t)}0,2 dt ă 8; ş8 (ii) Bt g1 P L1loc (R+ , L2 (ω)) and 0 }Bt g1 (t)}0,2 dt ă 8; ’ ’ % (iii) ř8 }G ,  } ă 8 for some G such that B G = g . 1 N 0,2 1 t 1 1 N =0 (2.207)

2.13 The Trajectory Dynamical Approach for the Nonlinear Elliptic Systems. . .

145

Then every solution u to the problem (2.161) possesses the finite dissipative integral ż8 }Bt u(t)}20,2 dt ă 8. (2.208) 0

Proof We integrate (2.204) over t P [0, T ] and obtain żT żT (γ Bt u, Bt u) dt = Fu (0) ´ Fu (T ) + (g1 , Bt u) dt. 0

0

Now it follows from the sign-definiteness of the matrix γ that ˇż T ˇ żT ˇ ˇ 2 ˇ }Bt u(t)}0,2 dt = C|Fu (T ) ´ Fu (0)| + C ˇ (g1 , Bt u) dt ˇˇ. 0

(2.209)

0

Theorem 2.38 implies that function Fu (T ) is bounded as T Ñ 8. Hence it suffices to show the boundedness of the integral on the right-hand side of (2.209). Suppose that condition (i) of (2.207) holds. Then ˇż T ˇ żT ˇ ˇ ˇ ˇď (g , B u) dt }g1 (t)}0,2 }Bt u(t)}0,2 dt 1 t ˇ ˇ 0

0

ď sup }Bt u(t)}0,2 t P[0,T ]

ď}u}b

ż8

żT

}g1 (t)}0,2 dt

0

}g1 (t)}0,2 dt.

(2.210)

0

şT Thus, | 0 (g1 , Bt u) dt| is bounded as T Ñ 8. Now suppose that condition (ii) of (2.207) holds. Then we obtain by integration by parts ˇż T ˇ ˇż T ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ (g1 , Bt u) dt ˇ ď |(g1 (T ), u(T ))| + |(g1 (0), u(0))| + ˇ (Bt g1 (t), u(t)) dt ˇˇ. ˇ 0

0

(2.211) The integral on the right-hand side of (2.211) is estimated in the same manner as the integral in (2.210). To estimate the first two terms on the right-hand side it suffices to prove that under above assumptions g1 P Cb (R+ , L2 (ω)). Let [N, N + 1] Ă R+ be an arbitrary interval and let [t, T ] be in that interval. Then }g1 (T )}0,2 ď }g1 (t)}0,2 + }g1 (T ) ´ g1 (t)}0,2 żT ď }g1 (t)}0,2 + }Bt g1 (t)}0,2 dt t

ď }g1 (t)}0,2 +

ż8 t

}Bt g1 (t)}0,2 dt.

(2.212)

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2 Trajectory Dynamical Systems and Their Attractors

Integrating (2.212) over t P [N, N + 1], we get }g1 (T )}0,2 ď C}g1 , N }0,2 +

ż8 t

}Bt g1 (t)}0,2 dt ď }g1 }b + }Bt g1 }L1 (R+ ,L2 (ω)) .

Since the constant N was arbitrarily chosen, we find g1 P Cb (R+ , L2 (ω)). Now suppose that condition (iii) of (2.207) holds. Again integrating by parts we obtain that ˇż T ˇ ˇ ˇ ˇ (g1 , Bt u) dt ˇˇ ď |(G1 (T ), Bt u(T ))| + |(G1 (0), Bt u(0))| ˇ 0

ˇ ˇż T ˇ ˇ (G1 (t), Bt2 u(t)) dt ˇˇ. + ˇˇ 0

The first two terms on the right-hand side can be estimated as before. The third term is estimated as follows: ˇż T ˇ żT ˇ ˇ 2 ˇ (G1 (t), Bt u(t)) dt ˇˇ ď }G1 (t)}0,2 }Bt2 u(t)}0,2 dt ˇ 0

0

ď

[T ] ÿ

}G1 , N }0,2 }Bt2 u, N }0,2

N =0

ďC}u}b

8 ÿ

}G1 , N }0,2 .

N =0

\ [

Theorem 2.39 is proved.

Theorem 2.40 Suppose that all the assumptions of the previous theorem hold. Further suppose that the limit problem in the cross-section w $ & av+ ´ f (v+ (x)) = g+ (x), % B v | = 0, n + Bω

(2.213)

has only a finite number of solutions 1 l v+ P V+ = tv+ (x), ¨ ¨ ¨ , v+ (x)u.

Then, for every solution u to the problem (2.161). there exists an equilibrium N (x) P V such that v+ + N (Ts u)(t, x) Ñ v+ (x) in + as s Ñ +8.

(2.214)

2.13 The Trajectory Dynamical Approach for the Nonlinear Elliptic Systems. . .

147

Here + denotes the space + 0 if g is strongly translation-compact in and the + w space (0 ) if g is weakly translation-compact in , respectively. Remark 2.30 As is known (see, for instance, [1]), there exists an open dense subset in L2 (ω) such that V+ is finite for every g+ belonging to this set. Proof Let u be a solution of the problem (2.161). Consider the ω-limit set1 ω(u) of u P + under the action of the semigroup tTs , s ě 0u. Recall that u+ P ω(u) if and only if there exists the sequence tsj uj PN , sj Ñ 8, such that Tsj u Ñ u+ in + .

(2.215)

By Theorem 2.37, tTs , s ě 0u possesses an attractor A in K+ Ă + , hence ω(u) is a nonempty, compact, and connected subset of + (see [1]). Let u+ be in ω(u). Let tsj uj PN be a sequence as in 2.215. Then, for every T ą 0, 2 (T ) as sj Ñ 8. Tsj u Ñ u+ weakly in HQ

In particular, }Tsj Bt u ´ Bt u+ , T }0,2 Ñ 0 as sj Ñ 8. Now the finiteness of the dissipative integral (2.208) implies that }Tsj Bt u, T }0,2 = }Bt u, Tsj T }0,2 Ñ 0 as sj Ñ 8. Therefore, }Bt u+ , T }0,2 = 0 and u+ (t, x) ” u+ (x). From condition (2.201) and Lemma 2.21, however, we conclude that u+ (x) is a solution to the limit problem (2.213). Thus ω(u) Ă V+ .

(2.216)

Since ω(u) is connected and V+ is discrete, we eventually get N u for some N P t1, ¨ ¨ ¨ , lu. ω(u) = tv+

Finally (2.214) is a consequence of the attracting property for tTs , s ě 0u. Theorem 2.40 is proved. \ [

1 Actually

we use here the letter ω in two different senses: for the ω-limit set and for a bounded polyhedral domain in Rn ; + := R ˆ ω,  = R ˆ ω. We hope that it will not lead to a misunderstanding for the readers.

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2 Trajectory Dynamical Systems and Their Attractors

Corollary 2.14 Both in the case of strong translation-compactness and of weak translation-compactness of g, (2.216) implies that $ & limt Ñ+8 }u(t, ¨) ´ v N (¨)}0,p = 0, + 0 % lim }B u(t, ¨)} = 0, t Ñ+8

t

ε,2

where the exponent p0 is defined in Corollary 1.11 and ε ă 1{2. This is obtained similarly to the proof of Corollary 2.12. Corollary 2.15 Suppose that the function g+ satisfies the conditions of Theorem 2.40. Then, any solution u(t), t P R, to Eq. (2.202) in the full cylinder  = R ˆ ω, which is not an equilibrium itself, is a heteroclinic orbit, i.e., there exist two different equilibria wu+ and wu´ belonging to V+ such that Ts u Ñ wu+ as s Ñ +8, Ts u Ñ wu´ as s Ñ ´8.

(2.217)

In fact, in view of estimate (2.184), see Remark 2.23, any solution u(t) to the problem (2.197) is bounded with respect to both t Ñ 8 and t Ñ ´8. Thus the convergence (2.217) follows from Theorem 2.40. Hence, it remains to prove that wu+ ‰ wu´ . Integrating (2.205) over R, where g1 ” 0, we get Fu (+8) ´ Fu (´8) = Fw+ ´ Fw´ = ´

ż R

(γ Bt u, Bt u) dt ‰ 0.

Thus w+ ‰ w ´ . We now give examples for the perturbation term g1 (t, x) satisfying the conditions of Theorem 2.40. Example 2.13 Let g1 (t, x) = ϕ(t)g0 (x),

(2.218)

where g0 P L2 (ω) and ϕ(t) =

| sin(t 2 )| . 1 + t2

Then condition (i) of (2.207) is fulfilled. (2.201) holds for the strong topology. Example 2.14 Let g1 (t, x) be as in (2.218), where ϕ(t) =

t . 1 + t2

Then condition (ii) of (2.207) is fulfilled. (2.201) holds for the strong topology.

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149

Example 2.15 Let g1 (t, x) be as in (2.218), where ϕ(t) = sin(t 3 ). Then condition (iii) of (2.207) is fulfilled. (2.201) holds for the strong topology.

2.13.4 Regular and Singular Part of the Trajectory Attractor In this final section we show that the trajectory attractor A of the problem (2.161) decomposes into a regular part Areg and a singular part Asing . For brevity we suppose that the right-hand side g of the problem (2.161) is strongly translation-compact in + . The case of weak translation-compactness is treated analogously. Let K + = K+ be the union of all solutions to the family (2.197) (see Definition 2.24). Let !+ 2 be the same as in Theorem 1.39. The regular and the singular part of the union K + are introduced by + + + + + + + = !+ Kreg 1 K , Ksing = !2 K , where !1 = Id ´ !2 .

Notice that + 2 Ă [HN,loc (+ )]k , Kreg

(2.219)

+ induced by the embedding K + Ă + coincides with the and the topology on Kreg reg 0 topology induced by the embedding (2.219). From Theorem 1.39 it follows that the semigroup tTs , s ě 0u of positive shifts + and in K + , i.e., acts both in Kreg sing + + + + Ď Kreg and Ts Ksing Ď Ksing for all s ě 0. Ts Kreg

Definition 2.29 The attractor Areg of the semigroup tTs , s ě 0u acting in the topo+ is called the regular trajectory attractor for the problem (2.161) logical space Kreg (see Definition 2.24). Analogously, the attractor Asing of the semigroup tTs , s ě 0u acting in the topo+ logical space Ksing is called the singular trajectory attractor for the problem (2.161) Theorem 2.41 Under the above assumptions, the problem (2.161) possesses a regular trajectory attractor Areg as well as a singular trajectory attractor Asing . Moreover, + Areg = !+ 1 A, Asing = !2 A,

where A is the trajectory attractor for the problem (2.161).

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2 Trajectory Dynamical Systems and Their Attractors

+ Proof We only verify that Asing = !+ 2 A. Areg = !1 A is completely analogous. First we are concerned with the attracting property. Let O = O(!+ 2 A) be a + + ´1 A in K . By Theorem 1.39, (! ) O is a neighbourhood neighbourhood of !+ 2 sing 2 of A in K + . Hence from the attracting property for A we obtain that there exists a sO ą 0 such that

Ts K + Ă !2´1 O for s ě sO .

(2.220)

Applying !+ 2 to both sides of (2.220) and using assertion (b) of Theorem 1.39, we find + + ´1 Ă !+ Ts Ksing 2 (!2 ) O Ď O for s ě sO .

This is the attracting property for !+ 2 A. Secondly, by definition, Ts A = A for all s ě 0. Applying !+ to both sides and using (b) of Theorem 1.39 again we get 2 + T s !+ 2 A = !2 A for s ě 0.

This is the strict invariance of !+ 2 A under the action of tTs , s ě 0u. Finally, the + compactness of !+ A in K is immediate from the compactness of the attractor A 2 sing + and the continuity of !2 . Thus !+ 2 A is the singular trajectory attractor for the problem (2.161). Theorem 2.41 is proved. \ [ Corollary 2.16 Let τj+ , 1 ď j ď κ, be the trace operators supplied by Corollary 1.17. Then the semigroup tTs , s ě 0u of positive shifts acts in the spaces 1´π {αj

τj+ K + Ă [Hloc

(R+ )]k and possesses the attractors Aj = τj+ A there.

This is immediate from the topological isomorphism 2 (τ1+ , ¨ ¨ ¨ , τκ+ ) : !+ 2 HQ,loc (R+ ˆ ω) Ñ

κ à j =1

1´π {αj

Hloc

(R+ )

derived in Sect. 1.9. Note that τj+ A has an invariant meaning, while !+ 2 A depends + on the choice of the projection !2 . Finally we are concerned with the question of stabilization of asymptotics in the case when stabilization of solutions takes place. For that we impose all assumptions of Theorem 2.39, in particular, that f (u) = ´∇F (u) is gradient-like (see (2.203)) and the limit equation av+ ´ f (v+ ) = g+ , Bn v+ |Bω = 0 N in [H 2 (w)]k , N = 1, . . . , l. has only a finite number of solutions v+ = v+ Q

2.14 The Dynamics of Fast Nonautonomous Travelling Waves and. . .

151

N , i.e., Let tdjN uκj =1 be the sequence of singular coefficients to v+ N v+ = v0N +

κ ÿ

djN (χj )˚ (ψj (r)r π {αj cos(π θ {αj ))

j =1

N N 2 k k where v0N = !+ 1 v+ P [HN (w)] and dj P C (see Remark 1.24).

Theorem 2.42 Let the assumptions of Theorem 2.40 be fulfilled. Then, for each N such that solution u to the problem (2.161), there exists an equilibrium v+ N Ts u Ñ v+ in + 0 as s Ñ 8.

(2.221)

Moreover, + N + + N N T s !+ 1 u Ñ !1 v+ = v0 , Ts !2 u Ñ !2 v+

and 1´π {αj

Ts τj+ u Ñ djN in Hloc

(R+ ) as s Ñ 8.

Proof (2.221) is a consequence of Theorem 2.40. The other assertions are immedi+ + ate from the continuity of the operators !+ 1 , !2 and τj in the appropriate spaces. \ [

2.14 The Dynamics of Fast Nonautonomous Travelling Waves and Homogenization We study the elliptic boundary value problem # a(Bt2 + x u) ´ γε Bt u ´ f (u) = g(t, x), u|B = 0; u|t=0 = u0 (x)

(2.222)

in a semicylinder (t, x) P + := R+ ˆ ω, where ω is a smooth bounded domain in Rn , u = (u1 , . . . , uk ) is an unknown vector function, f , g and u0 are given vector functions, and a and γ are given constant k ˆ k matrices such that a + a ˚ ą 0 and γ = γ ˚ ą 0, and ε is a small positive number. In this section we follow [107], thereby preserving the notation (for a similar homogenization setting for parabolic systems see [48]). The problems of (2.222) type arise in the study of travelling wave solutions of nonautonomous evolution equations in a cylindrical domain  := R ˆ ω. Indeed, consider the second order nonautonomous parabolic equation in (t, x) P .

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2 Trajectory Dynamical Systems and Their Attractors

  γ Bη v = a(Bt2 + x u) ´ f (v) ´ g t ´ η, x (2.223) ε   with the fast travelling wave external force g t ´ γε η, x (where γε ąą 1 is the wave speed, and the variable η plays the role of time). Then, the problem of finding the travelling wave solution (modulated by the external travelling wave g)   γ v(η, t, x) := v t ´ η, x ε leads to the elliptic boundary value problem (2.222) in a cylinder . Applying the dynamical approach to studying the problem in the full cylinder, we obtain the axillary problem of the type (2.222) which is of independent interest. We assume that the nonlinear term f (u) in (2.222) satisfies the following assumptions: $ ’ ’ &f (u) ¨ v ě ´C,

f 1 (v) ě ´K, ’ ’ %|f (v)| ď C(1 + |v|q ) for all v P Rk

(2.224)

for some appropriate constants C and K and with the growth exponent q ă qmax = n+2 n´2 . Moreover, we assume that the external force g(t, x) is almost periodic with respect to t with values in L2 (ω), that is, g P Cb (R, L2 (ω)). Recall that (see Sect. 2.8 and [48]) it means that the hull H(g) := [Th g, h P R]Cb (R,L2 (ω)) , (Th g)(t) := g(t + h) is compact in Cb (R, L2 (ω)). Here, we denote by [. . .]H the closure in the space H . The solution of (2.222) is a function which belongs to W 2,2 (T ) for every T ě 0 (T := [T , T + 1] ˆ ω) and has the finite norm ||u||W 2,2 (R b

+)

:= sup ||u, T ||2,2 ă 8, T ě0

and, therefore, we restrict ourselves to consider only the solutions of the problem (2.222) that are bounded with respect to t Ñ 8. It is natural to assume that the 3 initial data u0 belongs to the space V0 := W 2 ,2 (ω) X tu0 |Bω = 0u, which is in fact the space of traces of the functions from Wb2,2 (+ ) X tu|Bω = 0u at t = 0 (see, e.g., [101]). The equations of type (2.222) have been studied, under various assumptions on a, γ , f, g, and ε, in [31]. It is known that (see [31]) under the assumptions (2.224) for each fixed ε ă ε0 ăă 1 the problem (2.222) possesses a unique (bounded with respect to t Ñ 8) solution u(t, x), which satisfies the estimate     ||u, T ||2,2 ď Qε ||u0 ||V0 e´αT + Qε ||g||L2 b

(2.225)

2.14 The Dynamics of Fast Nonautonomous Travelling Waves and. . .

153

with a certain monotonic function Qε depending only on f, a, γ and ε (and independent of u0 and g) and a positive α. Consequently, the problem (2.222) defines a process Ugε (t, τ ) in the phase space V0 by the formula Ugε (t, τ )uτ (x) = u(t, x), where u(t, x), t ě τ is a solution of the problem (2.222) with u(τ, x) = uτ (x). It is well-known (see [35, 48]) that the process can be extended to a semigroup Stε acting in the extended phase space V0 ˆ H(g) in the following way:   Stε (u0 , ξ ) = Uξε (t, 0)u0 , Tt ξ , t ě 0, ξ P H(g), u0 P V0 .

(2.226)

It is not difficult to prove (see [1, 48]) using the dissipative estimate (2.225) and the compactness of H(g) in Cb (R, L2 (ω)) that the semigroup (2.226) possesses a (global) attractor Aε in V0 ˆ H(g). The projection of Aε := !1 Aε of this attractor to the first component (V0 ) is called (uniform) attractor for (2.222) (see [1, 48]). Note that the attractor Aε is generated by all bounded solutions of the family of equations a(Bt2 u + x u) ´

γ Bt u ´ f (u) = ξ(t, x), ξ P H(g), ε

(2.227)

which is defined in a full cylinder  = R ˆ ω: ď

Aε =

Kξε |t=0 ,

(2.228)

ξ PH(g)

where Kξε is a set of all bounded in Wb2,2 () solutions of (2.227) with the right -hand side ξ , or, which is the same, the set of all travelling wave solutions of the evolution equation (2.223) (with g replaced by ξ ). This justifies the attractor’s approach to study the travelling wave solutions. The main goal of this section is to investigate the behaviour of the attractors Aε as ε Ñ 0. To this end, we make the time rescaling t Ñ εt and write the problem in the following more convenient form: # a(ε2 Bt2 u + x u) ´ γ Bt u ´ f (u) = gε (t, x), u|t=0 = u0 , u|B+ = 0,

(2.229)

  where gε (t, x) = g εt , x . Obviuosly, the attractors of the Eqs. (2.222) and (2.229) coincide. The feature of (2.229)   is that the right-hand side becomes rapidly t-oscilating external force g εt , x . Let ż 2 T g(t, x) dt T Ñ8 T ´T

gp(x) := lim

154

2 Trajectory Dynamical Systems and Their Attractors

and write the limit (ε = 0) equation in the following form: # γ Bt u = ax ´ f (u) ´ gp(x) u|t=0 = u0 , u|B+ = 0,

(2.230)

Equation (2.230) is an autonomous dissipative reastion-diffusion equation and, consequently (see [1, 35, 48, 98]), possesses a (global) attractor A0 in the phase space L2 (ω). The main result of this section is the following Theorem Theorem 2.43 Let the above assumptions hold. Then, the attractors Aε converge to the attractors A0 in the space W 1´δ,2 (ω) for every δ ą 0 in the following sense:   distW 1´δ,2 (ω) Aε , A0 Ñ 0 as ε Ñ 0.

(2.231)

(see [1, 48, 67, 98]). Assume now that the limit attractor A0 is exponential, that is, distL2 (ω) (St B, A0 ) ď C(B)e´νt

(2.232)

for every bounded subset B Ă L2 (ω). Here, St is a semigroup generated by the autonomous equation (2.230), ν ą 0, and the constant C(B) depends on ||B||L2 (ω) . It is known that (2.232) is valid for generic gp(x) at least if the Eq. (2.230) possesses a Lyapunov function, that is, a = a ˚ and f (u) = ∇u F (u). Theorem 2.44 Let the assumptions of the previous theorem hold and let the limit attractor be exponential. Moreover, let the almost-periodic function g(t, x) ´ gp(x) have a bounded primitive in L2 (ω), that is, C(T ) :=

żT 0

(g(t, x) ´ gp(x)) dt, ||C(T )||L2 (ω) ď C(g), for all T ě 0.

(2.233)

Then, the following error estimate holds: distL2 (ω) (Aε , A0 ) ď Cg εχ ,

(2.234)

where 0 ă χ ă 1, and Cg can be calculated explicitly. Remark 2.31 Note that the assumption (2.233) is obviously valid for any periodic function g but may be wrong for more general almost periodic ones. Some sufficient conditions on g satisfying (2.233) will be given in the end of this section. Remark 2.32 Error estimate (2.234) for the differences between the regular attractors for semigroups possessing global Lyapunov functions and depending regularly on the parameter ε is given in [1]. These results have been extended in [48] to regular attractors for a quite large class of autonomous reaction-diffusion equations

2.14 The Dynamics of Fast Nonautonomous Travelling Waves and. . .

with spatially oscillating coefficients homogenizations.

x  ε

155

in the main diffusion term and their

To prove Theorems 2.43 and 2.44, we derive several estimates for the solutions of the following auxiliary problem # a(ε2 Bt2 u + x u) ´ γ Bt u ´ f (u) = gε (t, x),

(2.235)

u|t=0 = u0 , u|B+ = 0,

Lemma 2.24 Let the above assumptions hold. Then, for every ε ă ε0 ăă 1 the problem (2.235) possesses a unique solution which satisfies the following estimate:     ||u, T ||Vε ď Qε ||u0 ||V0 e´αT + Qε ||g||L2 ,

(2.236)

b

where, by definition, › ›2 › › ||u, T ||2Vε := ε4 ›Bt2 u, T › + }Bt u, T }20,2 + }u, T }22,2 , 0,2

||u0 ||2Vε 0

:=

ε||u0 ||2V0

+ ||u0 ||20,2 .

Note that in (2.236) the monotonic function Q and the exponent α are independent of ε ă ε0 . Remark 2.33 The uniform estimates of the form (2.236) are more or less known (see [31]), therefore we omit their proof here. Lemma 2.25 Let the assumptions of Lemma 2.24 hold. In addition, assume that p(t, x) be the solutions of the right-hand side g satisfies (2.233). Let uε (t, x) and u p(0, x) = u0 (x). the problems (2.235) and (2.230), respectively, with uε (0, x) = u Then: 1. 1

p(t, x)||0,2 ď C1 ε 2 eK1 t , ||uε (t, x) ´ u

(2.237)

where the constants C1 and K1 are independent of ε and uniform with respect to bounded in Vε0 sets of u0 (x). 2. If the nonlinear term satisfies the additional regularity   4 |f 1 (u)| ď C 1 + |u| n´2 ,

(2.238)

and primitive the Cs (T ) is bounded not only in L2 (ω), but in W01,2 (ω) as well, 1

then the estimate (2.237) remains true with ε1 instead of ε 2 on the right-hand side, that is,

156

2 Trajectory Dynamical Systems and Their Attractors

p(t, x)||0,2 ď CεeK1 t . ||uε (t, x) ´ u

(2.239)

p(t, x). Then, v(t, x) satisfies Proof Let v(t, x) := uε (t, x) ´ u # u)) ´ hε (t, x), γ Bt v = ax v ´ (f (uε ) ´ f (p v|t=0 = 0, v|B+ = 0,

(2.240)

where hε (t, x) := ε2 Bt uε (t, x) + (gε (t, x) ´ gp(x)). Multiplying (2.240), by v(t, x) and integrating over (t, x) P [0, T ] ˆ ω and using quasimonotony assumption f 1 ě ´K, as well as the positivity of matrices γ and a, we obtain

żT ||v(t, x)||21,2 dt α ||v(T , x)||20,2 + ďK

żT 0

0

||v(t, x)||20,2 dt +

żT (hε (t, x), v(t, x)) dt

(2.241)

0

with an appropriate positive α. Next, we estimate the last integral on the right-hand side of (2.241). To this end, we decompose it in a sum of two integrals I1 (T ) := ε2 I2 (T ) :=

żT

żT 0

0

(Bt2 uε , v) dt,

(g (t, x) ´ gp(x), v) dt.

şT Let Cε (T ) := 0 (gε (t, x) ´ gp(x)) dt. Then, obviously due to (2.233), we have T  Cε (T ) = εC ε , and consequently, ||Cε (T )||0,2 ď Cε,

(2.242)

and the assumptions of Lemma 2.25 (2) imply that ||Cε (T )||1,2 ď Cε. Integrating I1 by parts and using the facts that Bt uε and Bt v are uniformly bounded with respect to ε in L2b (+ ) (according to Lemma 2.24), we obtain I1 = ´ε

2

żT 0

(Bt uε , Bt v) dt + ε2 (Bt uε (T , x), v(T , x)) ď Cμ T ε2 + μ||v(T , x)||20,2 , (2.243)

where μ ą 0 can be chosen to be arbitrary small. Moreover, to prove (2.243), we also used uniform boundedness with respect to ε of the term ε }Bt uε (T , x)}0,2 . Indeed, it follows from the interpolation inequality

2.14 The Dynamics of Fast Nonautonomous Travelling Waves and. . .

ε }Bt uε (T , x)}0,2

› › › › ď C ε2 ›Bt2 uε ›

L2 ([T ,T +1],L2 )

157

1  1 2 2 }Bt uε }L2 ([T ,T +1],L2 ) ď C1 .

Thus, it remains to estimate I2 . Integrating by parts again, we have I2 = ´

żT

(Cε (t), Bt v) dt + (Cε (T ), v(T , x)).

(2.244)

0

The second term on the right-hand side of (2.244) can easily be estimated by Hölder inequality and (2.242), leading to (Cε (T ), v(T , x)) ď Cμ ε2 + μ||v(T , x)||20,2 . Estimating the first term on the right-hand side of (2.244), which we denote by I21 , by Hölder inequality as well as (2.242) and the uniform boundedness of Bt v in L2b (+ ) one can easily derive I21 ď cεT , which in contrast to the previous estimate leads to a linear growth with respect to ε and not ε2 . Inserting all previously obtained estimates into (2.241) and applying Gronwall inequality, we obtain the rough estimate (2.237), 1 with the rate of converging ε 2 , instead of ε1 . Our next task is to derive a sharper estimate for the I21 under the assumption Lemma 2.25 (2). For this purpose, we take the expression for the term Bt v from the Eq. (2.240) and insert it to I21 : I21

=´

żT (Cε (T ), γ

´1

ax v(t, x)) dt +

0

+

żT 0

żT

(Cε (T ), γ ´1 (f (uε ) ´ f (p u)) dt

0

(Cε (t), γ ´1 Cε1 ) dt = J1 + J2 + J3 .

Let us now derive estimate the terms Jj , j = 1, 2, 3. To estimate J1 , we integrate by parts with respect to x and use the estimate ||∇x Cε (t)||0,2 ď Cε together with Hölder inequality, which leads to J1 =

żT 0

(∇x Cε (t), γ

´1

ax v(t, x)) d ď Cμ ε T + μ 2

żT 0

||v(t, x)||21,2 dt (2.245)

Here, we essentially used the fact that Cε in zero on the boundary Bω (without this fact, we would obtain the additional boundary term which would be an obstacle for the desired estimate). To estimate J2 , we use uniform boundedness of uε (t, x) with respect to ε in W 1,2 (ω) (due to the Lemma 2.24), the growth assumption (2.238), ş1 and Sobolev embedding theorem. Indeed, since f (uε ) ´ f (p u) = v 0 f 1 (suε + (1 ´ s)p u) ds, then, applying Hölder inequality with the exponents p1 = p2 = n2n ´2 and p3 = n2 and the embedding W 1,2 Ă Lp1 , we will have

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2 Trajectory Dynamical Systems and Their Attractors

J2 ď C

żT 0

ď C1

  4 1 + ||uε (t, x)||0,p1 + ||p u(t, x)||0,p1 n´2 ||v(t, x)||0,p1 ||Cε (t)||0,p1 dt

żT 0

ď C2 ε T + μ 2

4

(1 + ||uε (t, x)||1,2 + ||p u(t, x)||1,2 ) n´2 ||v(t, x)||1,2 ||Cε (t)||1,2 dt żT 0

||v(t, x)||21,2 dt.

As for the estimate for J3 : J3 =

1 (Cε (T ), γ ´1 Cε (T )) ď Cε2 . 2

(2.246)

Inserting the estimates (2.245)–(2.246) in (2.244), we obtain that I2 (T ) ď Cε2 (1 + T ) + μ

żT 0

||v(t, x)||21,2 dt + μ||v(T , x)||20,2 .

(2.247)

To prove Lemma 2.25, it remains to insert the estimates (2.243) and (2.247) in (2.241), taxing μ small enough and applying the Gronwall inequality, which in turn leads to (2.239). \ [ After these preliminaries, we are in position to start the proof of Theorem 2.43. Proof It follows from the estimate (2.236) that the attractors Aε are uniformly bounded in the norms of Vε0 and, in particular, in the norm of W01,2 (ω): ||Aε ||1,2 ď C, ε ă ε0 .

(2.248)

Thus, in order to prove the upper semi-continuity (2.231), it is sufficient to verify that if uε P Aε and uεn Ñ u˚ as εn Ñ 0 weakly in W01,2 , then u˚ P A0 . It would indicate that we have the upper semicontinuity in the weak topology of W 1,2 and, consequently, due to compact embedding of W 1´δ,2 in W 1,2 , it would also indicate the upper semicontinuity in W 1´δ,2 (ω). Indeed, due to the (2.228), there exists a sequence ξn P H(g) and complete bounded solutions uεn (t, x), t P R of the equations a(εn2 Bt2 uεn

+ x uεn ) ´ γ Bt uεn ´ f (uεn ) = ξn

t εn

.

(2.249)

Our goal now is to pass to the limit as n Ñ 8 in (2.249). For this purpose, we recall that due to Lemma 2.24 uεn (t, x) are uniformly bounded in Vε for each T P R. Hence, passing to a subsequence if necessary, we may assume that for each T P R, p(t, x) and uεn (t, x) á u p(t, x) in W 2,2 (T ), Bt uεn (t, x) á Bt u

(2.250)

2.14 The Dynamics of Fast Nonautonomous Travelling Waves and. . .

159

p(t, x) is bounded with respect to t, that is, Bt u p, x u p P and the limit function u p(0, x) = u˚ (x). It remains to prove that the function ω p (t, x) satisfies L2b () and u the limit equation (2.230). Then, (2.228) implies that u˚ P A0 . Note that the convergence (2.250) together with restriction (2.224) admits to pass to the limit in the left-hand side of (2.249) in a standard way (see [1]). In order to pass to the limit on the right-hand side of (2.249), we use the following Lemma. Lemma 2.26 Let g P Cb (R, L2 (ω)) be almost periodic with a hull H(g). Let also ξn P H(g). Then, for each T P R ξn

t εn

á gp as εn Ñ 0

weakly in L2 (T ). Proof Indeed, since the almost periodic flow is strictly ergodic, it follows from Birkhoff-Khinchin ergodic theorem that the L2 -limit gp(x) = lim

żT

T Ñ8 ´T

ξ(t, x) dt

is uniform with respect to ξ P H(g). The assertion of Lemma 2.26 is a simple corollary of this fact. \ [ Proof We are now ready to prove Theorem 2.44. In fact, the assertion of this Theorem is a simple corollary of Lemma 2.25. Indeed, assume that (2.233) is valid for the initial right-hand side g(t, x). Then, it is not difficult to verify that it is valid uniformly with respect to ξ P H(g) and, consequently, the estimate (2.237) is also valid uniformly with respect to ξ P H(g).  Namely, let uε,ξ (t, x) be a solution p(t, x) be the of the Eq. (2.235) with right hand side ξ εt , x , ξ P H(g) and let u p(0, x) = u0 (x)) of the limit problem (2.230). corresponding solution (uε,ξ (0, x) = u Then, it holds 1

p(t, x)||0,2 ď Cε 2 eK1 t ||uε,ξ (t, x) ´ u

(2.251)

uniformly with respect to ξ P H(g) and bounded in Vε0 sets of initial data u0 (x). Assume now that ϕ P Aε . According to the attractor’s structure theorem, there exists a complete bounded trajectory uε (t, x), t P R of the Eq. (2.235) with the right-hand p(t, x) side ξ P H(g). Let us fix an arbitrary T ą 0 and consider the trajectory u p(0, x) = uε (´T , x). Then (since Aε are uniformly of the limit equation such that u bounded in Vε0 ), (2.251) implies that 1

p(T , x)||0,2 ď Cε 2 eK1 T . ||ϕ ´ u

(2.252)

160

2 Trajectory Dynamical Systems and Their Attractors

On the other hand, since A0 is an exponential attractor, it follows that u(T , x), A0 ) ď C1 e´νT , distL2 (ω) (p

(2.253)

where ν ą 0 is some positive constant. Combining (2.252) and (2.253), we deduce that distL2 (ω) (ϕ, A0 ) ď C1 e´νT + Cε 2 eK1 T . 1

(2.254)

Taking the optimal value for T (solving the equation C1 e´νT = Cε 2 eK1 T ) in the estimate (2.254), we have 1

distL2 (ω) (ϕ, A0 ) ď C2 εχ , χ =

ν . 2(K1 + ν)

Since ϕ P Aε is arbitrary, we obtain distL2 (ω) (Aε , A0 ) ď C2 εχ . \ [

This proves Theorem 2.44.

Let us formulate some sufficient conditions (see Proposition 2.2 below) for almost periodic right-hand sides g satisfying the assumption of Theorem 2.44. For this purpose, we recall (see [48] and references therein) that every almost periodic function belonging to Cb (R, L2 (ω)) possesses a Fourier expansion g(t, x) =

8 ÿ

gαk (x)eiαk t ,

(2.255)

k=´8

where tαk u Ă R is a countable set of Fourier modes for g and the corresponding amplitudes gαk (x) P L2 (ω) satisfy 8 ÿ k=´8

||gαk (x)||20,2 ă 8.

Moreover, gp(x) = g0 (here, we define gα ” 0 if α R tαk u). Proposition 2.2 Let the Fourier amplitudes gαk of an almost periodic function g P Cb (R, L2 (ω)) satisfy ÿ αk

1 ||gαk (x)||0,2 ă 8. |α | ‰0 k

(2.256)

2.14 The Dynamics of Fast Nonautonomous Travelling Waves and. . .

161

Then, the function C(T ) satisfies the inequality (2.233). Analogously, if ÿ αk

1 ||gαk (x)||1,2 ă 8, |α | ‰0 k

(2.257)

then ||C(T )||1,2 ď C. Proof Let us verify (2.233) using the Fourier expansion (2.255). Indeed, subtracting gp(x) = g0 in (2.255) and integrating over t, we derive that C(t) =

ÿ

gαk (x)

αk ‰0

 1  iαk t e ´1 . iαk

(2.258)

Taking the L2 -norm from both sides of (2.258) and using (2.256), we obtain (2.233). The second part of the Proposition 2.2 can be verified analogously. Proposition 2.2 is proved. \ [ Corollary 2.17 Let the assumptions of Lemma 2.24 hold and the function g satisfy (2.256). Then, the estimate (2.237) holds. If, in addition, C(t) belongs to W01,2 (ω) and the assumptions (2.238) and (2.257) hold, then the improved estimate (2.239) is valid. Example 2.16 Let us consider the case of quasiperiodic functions. By definition, g P Cb (R, L2 (ω)) is called a quasiperiodic function if there exists a finite vector m of řmfrequencies β = (β1 , . . . , βm ) P R , mm ą 1 such that αk = (β, l(k)) = i=1 βi (l(k))i for the appropriate l(k) P Z and βi are rationally independent. Then, (2.255) reads as follows ÿ gl (x)ei(β,l)t . g(t, x) = l PZm

Moreover, it is known that for every such g P Cb (R, L2 (ω)) there exists a 2π periodic with respect to every zi , i = 1, .., m function  P Cb (Rm , L2 (ω)) such that ÿ gl (x)ei(z,l)t . (2.259) g(t, x) = (β1 z1 , . . . , βm zm ), (z, x) = l PZm

In fact, (2.259) gives another definition of a quasiperiodic function. The condition (2.256) reads in our case as follows: I :=

ÿ

||gl (x)||0,2 ă 8. |(β, l)| l PZm ,l ‰0

(2.260)

162

2 Trajectory Dynamical Systems and Their Attractors

Recall that, due to the theory of Diophantine approximation for every δ ą 0 and for almost every β P Rm (with respect to the Lebesgue measure) the following estimate is valid: |(β, l)| ě Cβ |l|´m´δ , l ‰ 0.

(2.261)

Assume that β P Rm is chosen such a way that (2.261) holds. Then, the sum I in (2.260) can be estimated by

I ďC

ÿ l PZm

 |l|

m+δ

||gl (x)||0,2 ď C

ÿ

1  2(m+δ ´α)

2

|l|

l ‰0

ÿ

l ‰0

1 2

|l|

2α

||gl (x)||20,2 (2.262)

Note that the first term on the right-hand side of (2.262) is finite if 2(m + δ ´ α) ă 3m , and the second term is a finite for every function g such that ´m, that is, if α ą 2+δ the function  from the representation (2.259) belong to Cbα (Rm , L2 (ω)). Thus, for every β satisfying (2.261) and every periodic  P Cbα (Rm , L2 (ω)) with α ą δ + 3m 2 , the function (2.259) satisfies the assumption (2.256). Remark 2.34 The assumption (2.233) can weakened in the following way: ||C(T )||0,2 ď CT 1´β , β ą 0.

(2.263)

Then ||Cε (T )||0,2 ď εβ CT 1´β , and, arguing as in the proof of Lemma 2.25 and . Note that (2.263) Theorem 2.44, we derive the estimate (2.234) with χ = 2(kβν 1 +ν) looks not very restrictive because this estimate with β = 0: ||C(T )||0,2 ď CT is obviously valid for every almost periodic function g.

Chapter 3

Symmetry and Attractors: The Case N ď 3

3.1 Introduction As it was shown in Chap. 1, positive solutions of semilinear second order elliptic problems have symmetry and monotonicity properties which reflect the symmetry of the operator and of the domain, see e.g. [18, 65] for the case of bounded domains and [13, 19, 21] for the case of unbounded domains (such as  = Rn ,  = R+ ˆ Rn´1 , cylindrical domains, etc.). These results have been extended to the case of positive solutions of second order parabolic problems in bounded symmetric domains in [10, 11]. Moreover, the symmetrization and stabilization properties of such solutions as t Ñ 8 were investigated using a combination of moving planes method with the classical methods of dynamical systems theory (such as ω-limit sets, attractors, etc.). The main goal of this chapter is to apply the dynamical system approach to study the symmetrization and stabilization (as |x| Ñ 8) properties of positive solutions of elliptic problems in asymptotically symmetric unbounded domains. To the best of our knowledge, the use of dynamical systems methods for elliptic problems was initiated in the pioneering paper of K. Kirchgässner [72], where a local center manifold for a semilinear elliptic equation on a strip was constructed. As it was indicated in the Preface, one of the main difficulties which arises in the dynamical study of elliptic equations, is the fact that the corresponding Cauchy problem is not well posed for such equations, and, consequently, the straightforward interpretation of the elliptic equation as an evolution equation leads to semigroups of multivalued maps even in the case of cylindrical domains, see [8] (for global attractors of multivalued flows associated with subdifferentials see [71]). The use of multivalued maps can be overcome using the so-called trajectory dynamical approach [15, 61, 81] (see also an alternative approach in [12, 31, 92]). Recall that, under this approach, which we introduced in Chap. 2, one fixes a signed direction l in Rn , which will play the role of time. The space K + of all bounded solutions of the elliptic problem in the unbounded domain  is then considered as a trajectory © Springer Nature Switzerland AG 2018 M. Efendiev, Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations, Fields Institute Monographs 36, https://doi.org/10.1007/978-3-319-98407-0_3

163

3 Symmetry and Attractors: The Case N ď 3

164

 phase space for the semi-flow Thl of translations along the direction l defined via   h P R+ , u P K + . (Thl u)(x) := u(x + hl), 

In order for the trajectory dynamical system (Thl , K + ) to be well defined, one evidently needs the domain  to be invariant with respect to positive translations along the l directions:    Thl  Ă , Thl x := x + hl.

In this chapter, we apply the trajectory dynamical systems approach, which we developed in Chap. 2, to a more detailed study of the asymptotic behavior of positive solutions of the following model elliptic boundary problem in an unbounded domain x := (x1 , x2 , x3 ) P + := R+ ˆ R+ ˆ Rn : # x u ´ f (u) = 0; u|x1 =0 = u0 ; u|x2 =0 = 0;

(3.1)

It is assumed that the nonlinear term f (u) satisfies the following conditions: $ 1 ’ ’ &1. f P C (R, R),

2. f (v).v ě ´C + α|v|2 , α ą 0, ’ ’ %3. f (0) ď 0

(3.2)

(see Remark 3.2 for some relaxed conditions). As mentioned above, we consider non-negative solutions of problem (3.1): u(x) ě 0, x P + and study their behavior as x1 Ñ +8. Thus, in this case, the x1 -axis will play the role of time (l := (1, 0, 0)). Moreover, we restrict our consideration to bounded (with respect to x Ñ 8) solutions of (3.1). More precisely, a bounded solution 2+β of (3.1) is understood to be a function u P Cb (+ ), for some fixed 0 ă β ă 1, which satisfies (3.1) in the classical sense (in fact, due to interior estimates, this assumption is equivalent to u P Cb (+ ), but we prefer to work with classical solutions). Therefore, the boundary data is assumed to be non-negative u0 (x2 , x3 ) ě 0 and to belong to the space 2+β

u0 P Cb

(0 ), (x2 , x3 ) P 0 := R+ ˆ Rn .

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165

Here and below, we use the notation 2+β

Cb

(V ) := tu0 : }u0 }C 2+β := sup }u0 }C 2+β (B 1 XV ) ă 8u, b

ξ PV

ξ

where Bξr denotes the ball of radius r, centered in ξ .

3.2 A Priori Estimates and Solvability Results We start with proving a dissipative estimate with respect to the a priori estimate for positive solutions of (3.1), which allows us to apply the trajectory dynamical systems approach. 2+β

Theorem 3.1 Let u0 P Cb (0 ) and let the first and second compatibility conditions be valid on B0 (i.e. u0 (0, x3 ) = 0 and Bx22 u0 (0, x3 ) = f (0)). Then, (3.1) has at least one non-negative bounded solution, every such solution u satisfies the estimate }u}C 2+β (Bx1 X+ ) ď Q(}u0 }C 2+β )e´γ x1 + Cf ,

(3.3)

b

x = (x1 , x2 , x3 ) P + , where γ ą 0, Q is an appropriate monotonic function and Cf is independent of u0 . Proof Let us first verify the a priori estimate (3.3). To this end, we consider, as usual, the function w(t, x) = u2 (t, x), which evidently satisfies the equation x w = 2f (u).u + 2∇x u.∇x u ě ´2C + 2αw, w|x1 =0 = u20 , w|x2 =0 = 0. (3.4) Consider also the auxiliary linear problem x w1 = ´2C + 2αw1 , w1 |x1 =0 = w|x1 =0 = u20 , w1 |x2 =0 = 0,

(3.5)

with the same boundary conditions as w. Lemma 3.1 The linear equation (3.5) has a unique bounded solution w1 (x), which satisfies the following estimate: }w1 }C ( Bx1 ) ď C1 }u0 }2Cb (0 ) e´αx1 + C2 .

(3.6)

Proof The proof of Lemma 3.1 is standard and is based on the maximum principle. Indeed, let us decompose w1 (x) = v(x) + v1 (x), where v1 (x) is a solution of the non-homogeneous equation (3.5), with zero boundary conditions and v(x) is a

3 Symmetry and Attractors: The Case N ď 3

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solution of the homogeneous equation, with non-zero boundary conditions. Then, evidently, }v1 }Cb (+ ) ď C2 ,

(3.7)

where C2 depends only on the constant C on the right-hand side of (3.5). In order to obtain the exponential decay of v(x), we introduce the functions ψh (x) := 1{ cosh(ε(x1 ´ h)), where ε ą 0 is sufficiently small and h ě 0. Then, it is not difficult to verify that |ψh1 (x)| ď εψh (x), |ψh2 (x)| ď 3ε2 ψh (x), ψh (0) ď e´εh .

(3.8)

Let vh (x) := ψh (x)v(x). Then this function satisfies the equation x vh + L1h (x)vh + L2h (x)Bx1 vh ´ 2αvh = 0, vh |x1 =0 = ψh (0)u20 ,

(3.9)

and (3.8) implies that |L1h (x)| ď Cε2 and |L2h (x)| ď Cε (where C is independent of h). Consequently, if ε ą 0 is small enough, the classical maximum principle works for Eq. (3.9) and, therefore, }vh }Cb (+ ) ď }vh }Cb (0 ) .

(3.10)

Estimate (3.6) is an immediate corollary of (3.7), (3.10) and the third estimate of (3.8). Lemma 3.1 is proved. l Having estimate (3.6), applying the comparison principle to the solutions w and w1 of (3.4) and (3.5), respectively, and using the evident fact that w = u2 is nonnegative, we derive that }u}2C(B 1 ) ď }w}C(Bx1 ) ď }w1 }C(Bx1 ď C1 }u0 }C 2+β ( ) e´αx1 + C2 . x

b

0

(3.11)

Recall that, due to classical interior estimates for the Laplace equation (see e.g. [3, 76]), we have the following estimate for every small positive δ ą 0: }u}C 2´δ (+ XBx1 ) ď C(}f (u)}L8 (XBx1 ) + }u}L8 (XBx2 ) + χ (2 ´ x1 )}u0 }C 2+β (0 XBx2 ) ) ď Q(}u}L8 (+ XBx2 ) ) + Cχ (2 ´ x1 )}u0 }C 2´δ (0 XBx2 ) , x P + ,

(3.12)

where the monotonic function Q and the constant C depend only on f and α, but are independent of x P  and on the actual solution u, and χ (z) is a classical Heaviside function (which equals zero for z ď 0 and one for z ą 0).

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Recall that we assume the first compatibility condition, u0 |x2 =0 = 0, to be valid. This assumption is necessary in order to obtain C 2´δ -regularity in (3.12), in the case where x is close to the edge B0 ). Inserting now estimate (3.11) into the right-hand side of (3.12), we derive the analogue of estimate (3.3) for C 2´δ -norm: }u}C 2´δ (Bx1 X+ ) ď Q(}u0 }C 2´δ )e´γ x1 + Cf .

(3.13)

b

In order to derive estimate (3.3), it is sufficient to use the elliptic interior estimate in the form }u}C 2+β (+ XBx1 ) ď C(}f (u)}C 1 (XBx1 ) + }u}C(XBx2 ) + χ (2 ´ x1 )}u0 }C 2+β (0 XBx2 ) ) ď Q(}u}C 1 (+ XBx2 ) ) + Cχ (2 ´ x1 )}u0 }C 2+β (0 XBx2 ) , x P + (here we have implicitly used the second compatibility condition Bx22 u|x2 =0 = f (0), in order to obtain C 2+β -regularity near the edge B0 ). Inserting estimate (3.13) into the last interior estimate, we derive inequality (3.3) for the C 2+β -norm. Let us verify now the existence of a positive solution of problem (3.1). To this end, we consider a sequence of bounded domains N + , N P N, defined via N +1 N + := + X B0

and a sequence of cut-off functions φN (x) ” 1, if x P B0N and φN (x) ” 0, if x R B0N +1 , 0 ď φ ď 1. Consider also the family of auxiliary elliptic problems N N x uN ´ f (uN ) = 0, x P N + , u |B N X0 = u0 φN , u |B N z0 = 0. +

+

(3.14) Note that, according to our construction, uN |BN ě 0 and, according to assump+ N ” 0 is a subsolution and w N ” R is a supersolution of (3.14), if R tions (3.2), w´ + is large enough. Thus (see e.g. [104]), problem (3.14) has at least one non-negative solution R ě uN (x) ě 0. Note that R is, in fact, independent of N . Consequently, applying again the interior regularity theorem, we derive that }uN }C 2+β (B 1 XN ) ď C, x

+

(3.15)

with C = C(f, u0 ) independent of N and x P N + . Having the uniform estimate (3.15), one can easily take the limit as N Ñ 8 in Eq. (3.3) and construct a bounded non-negative solution u(x) of the initial equation (3.1). Theorem 3.1 is proved. l

3 Symmetry and Attractors: The Case N ď 3

168

3.3 The Attractor Now we can apply the trajectory dynamical systems approach to (3.1). To this end, let us consider the union K + of all bounded positive solutions of (3.1), which 2+β correspond to every u0 P Cb . The set K + is nonempty due to Theorem 3.1. Then a semigroup of positive shifts, (Th u)(x1 , x2 , x3 ) := u(x1 + h, x2 , x3 ), acts on the set K + : T h : K + Ñ K + , K + Ă Cb

2+β

(+ ).

(3.16)

This semigroup acting on K + is called the trajectory dynamical system corresponding to (3.1). Our next task is to construct the global attractor for this system. Firstly, 2+β we note that the uniform topology of Cb is too strong for our purposes. That is + why we endow the space K with a local topology, according to the embedding K + Ă Cloc (+ ), 2+β

2+β

where, by definition,  := Cloc (+ ) is a Fréchet space generated by the seminorms } ¨ }C 2+β (Bx1 X+ ) , x0 P + . Recall briefly the definition of the attractor 0 from Chap. 2 adapted in our case. Definition 3.1 The set Atr Ă K + is called the global attractor for the trajectory dynamical system (3.16) (i.e., trajectory attractor for problem (3.1)), if the following conditions are valid: 2+β

1. The set Atr is compact in Cloc (+ ); 2. It is strictly invariant with respect to Th : Th Atr = Atr ; 3. Atr attracts bounded subsets of solutions, when x1 Ñ 8. That means that, for 2+β every bounded (in the uniform topology of Cb ) subset B Ă K + and for every 2+β neighborhood O(Atr ) in the Cloc topology, there exists H = H (B, O) such that Th B Ă O(Atr ) if h ě H. Note that the first assumption of the definition claims that the restriction Atr |1 is compact in C 2+β (1 ), for every bounded 1 Ă + , and the third one is equivalent to the following: For every bounded subdomain 1 Ă + , for every B - bounded subset of K + and for every neighborhood O(Atr |1 ) in the C 2+β (1 )-topology of the restriction of Atr to this domain, there exists H = H (1 , B, O) such that (Th B)|1 Ă O(Atr |1 ) if h ě H.

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169

Theorem 3.2 Let the assumptions of Theorem 3.1 hold. Then Eq. (3.1) possesses a trajectory attractor Atr which has the following structure: Atr = !+ K(),

(3.17)

where (x1 , x2 , x3 ) P  := RˆR+ ˆRn and K() denotes the union of all bounded p(x) P Cb2+β () of non-negative solutions u p(x) ě 0, p ´ f (p x u u) = 0, x P , u|B = 0, u

(3.18)

i.e. the attractor Atr consists of all bounded non-negative solutions u of (3.1) in p in . + , which can be extended to bounded non-negative solution u Proof According to Theorem 2.1, in order to verify that a semigroup Th : K + Ñ K + possesses an attractor, it should be verified that this semigroup is continuous for every fixed h ě 0, and that this semigroup possesses a compact attracting (or absorbing) set in K + . The continuity of the semigroup Th on K + is obvious in our situation. Indeed, the semigroup Th of positive shifts along the x1 axis, is evidently 2+β continuous (for every fixed h) as a semigroup in Cloc (+ ), therefore, its restriction + to K is also continuous. Thus, it remains to construct a compact absorbing set for 2+β Th : K + Ñ K + . Let BR be the R-ball centered in 0 in the space Cb (+ ). Then, estimate (3.3) implies that the set MR := K + X BR will be an absorbing set for the semigroup (3.16) (more precisely, for R = 2Cf , where Cf is defined in (3.3)). But this set is not compact in . That is why we construct a new set, VR := T1 MR Ă MR Ă K + . Evidently, this set is also absorbing. We claim also that this set is precompact in . Indeed, by definition, the set VR consists of all bounded solutions u of Eq. (3.1), p, defined not for x1 ě 0, but for x1 ě which can be extended to bounded solution u ´1, such that }p u}C 2+β ([´1,8]ˆ ) ď R.

(3.19)

0

b

Note that, due to (3.2), f P C 1 ; consequently, we may apply the interior estimate p, not only with the exponent 2+β, but with an arbitrary (see (3.12)) for the solution u one, 2 + β 1 , with β 1 ă 1. In particular, if we fix β 1 ą β, then the interior estimate together with (3.19), yields }u}

2+β 1

Cb

(T1 + )

= }p u}

2+β 1

Cb

(+ )

ď R1

3 Symmetry and Attractors: The Case N ď 3

170

where the constant R1 depends only on R and f . Consequently, we have proved that 2+β 1

VR Ă Cb

(+ ) 2+β 1

and is bounded in it. Now note that the embedding Cb (+ ) Ă  is compact if β 1 ą β and, consequently, VR is indeed precompact in . (This was the main reason to endow the trajectory phase space with the ‘local’ topology of , not with the ‘uni2+β 2+β 1 2+β form’ topology of Cb (+ ). Indeed, the embedding Cb (+ ) Ă Cb (+ ) is evidently non-compact and we cannot construct the compact absorbing set in this topology. Moreover, elementary examples show that problem (3.1) indeed may not possess an attractor in a ‘uniform’ topology—this we leave to the reader.) Thus, the precompact absorbing set VR is already constructed and it remains to find the compact one. The simplest way is to take the compact absorbing set V1R := [VR ] , where [¨] denotes the closure in . Indeed, since VR Ă MR Ă K + and MR is evidently closed in , V1R Ă K + and, consequently, it is a compact absorbing set for the semigroup (3.16). Thus, (due to the attractor existence Theorem 2.1 for abstract semigroups), the semigroup (3.16) possesses an attractor Atr , which can be defined by the formula Atr = Xhě0 r Ys ěh Ts V1R s .

(3.20)

It remains to prove representation (3.17). As we will see below, this follows p(x), x P , be a non-negative bounded solution of from (3.20). Indeed, let u p), h P N, is uniformly problem (3.18). Then in particular the sequence !+ (T´h u bounded in Cb2+α (+ ) and, consequently, according to the attractor’s definition, p) Ñ Atr in , as h Ñ 8. Th !+ (T´h u p) = !+ u p. Thus, !+ u p P Atr and, consequently, On the other hand, Th !+ (T´h u !+ K() Ă Atr . Let us prove the reverse inclusion. Let u P Atr . Then, (3.20) implies that there exist a sequence hn Ñ +8 and a sequence of solutions un P V1R , such that u = ´ lim Thn un . nÑ8

(3.21)

Note that the solution Thn un is defined not only in + , but also in the domain Thn + := (´hn , 8) ˆ 0 , and }un }C 2+β (T b

hn  + )

ď R.

(3.22)

3.4 Symmetry and Stabilization

171

Consequently, arguing as in the proof of the compactness of V1R , we deduce that 2+β the sequence Thn un , n ě n0 , is precompact in Cloc (Thn0 +1 + ), for every n0 P N. Taking a subsequence, if necessary, and using Cantor’s diagonal procedure and the pP fact that hn Ñ 8, we may assume that this sequence converges to some function u 2+β 2+β Cloc (), in the spaces Cloc (Thn0 +1 + ), for every n0 P N. Then, (3.22) implies 2+β

p P Cb (). Moreover, since Thn un are non-negative solutions of (3.1), by that u p is a non-negative solution of Eq. (3.18) letting n Ñ 8, we easily obtain that u p = u. Thus, u P !+ K(). Theorem 3.2 is and formula (3.21) implies that !+ u proved. l

3.4 Symmetry and Stabilization To obtain additional information on the behaviour of solutions of the initial problem (3.1), we use the description of non-negative bounded solutions in the halfspace (see below). Proposition 3.1 Let assumptions (3.2) hold and let n = 1, that is, + = R+ ˆ p(x) of Eq. (3.18) depends only R+ ˆ R. Then any non-negative bounded solution u on the variable x2 , i.e. u(x) = V (x2 ), where V (z) is a bounded solution of the following problem: V 2 (z) ´ f (V (z)) = 0, z ą 0, V (0) = 0, V (z) ě 0.

(3.23)

The proof of this proposition is given in [21] (see also Sect. 1.7), for the case where p(x) is strictly positive inside . The general case can be reduced to the the solution u one above, using the following version of the strong maximum principle (see also Sect. 1.4). Lemma 3.2 [21] Let V Ă Rn be a (connected) domain with sufficiently smooth boundary and let w P C 2 (V ) X C(V ) satisfy the following inequalities: x w(x) ´ l(x)w(x) ď 0, x P V , w(x) ě 0, x P V . Assume also that |l(x)| ď K for x P V . Then either v(x) ” 0, or v(x) ą 0, for every interior point x P V . In order to apply the lemma to Eq. (3.18), we rewrite it in the following form: p ´ l(x)p x u u = f (0) ď 0, l(x) :=

f (p u(x)) ´ f (0) . p(x) u

3 Symmetry and Attractors: The Case N ď 3

172

p(x) is bounded, l(x) is also bounded in . Thus, Since f P C 1 and the solution u p(x) ” 0 (which is evidently symmetric), or u p(x) ą according to Lemma 3.2, either u 0, in the interior of  and, thus, Proposition 3.1 follows from the result of [21] mentioned above. Proposition 3.1 is proved. l Denote by RV the set of all bounded non-negative solutions V (z) of problem (3.23). Then Proposition 3.1 implies that Atr = RV . Let us now study the positive solutions of problem (3.23). It is well known that every non-negative bounded solution of this problem is monotonically increasing, V (z1 ) ě V (z2 ), if z1 ě z2 , and, consequently, there exists the limit z0 = z0 (V ) := lim V (z), f (z0 ) = 0, 0 ď V (z) ď z0 , z ě 0. zÑ+8

(3.24)

Moreover, it follows from Lemma 3.2 that either V (z) ” 0, or V 1 (z) ą 0, for every z ě 0. Multiplying Eq. (3.23) by V 1 and integrating over [0, z], we obtain the explicit expression for the derivative V (z), V 1 (z)2 = ´2F (V (z)) + C,

(3.25)

şV where F (V ) := ´ 0 f (V ) dV . Letting z Ñ +8 in (3.25) and taking into account (3.24), one can easily derive that C = 2F (z0 ). Therefore, we obtain the following equation for V (z), stabilizing to z0 : V 1 (z)2 = 2(F (z0 ) ´ F (V (z)).

(3.26)

Assume now that F (z0 ) ą 0 (in the other case V (z) ” 0). Then, the solution Vz0 (z) of (3.26), which satisfies (3.24), exists if and only if F (z0 ) ´ F (z) ą 0, for every z P (0, z0 ). Moreover, such a solution is unique, because Vz0 satisfies (3.23) with the initial conditions a Vz0 (0) = 0, Vz10 (0) = 2F (z0 ). (3.27) Denote by R+ f := tz0 P R+ : f (z0 ) = 0, F (z0 ) ´ F (z) ą 0 for every z P (0, z0 )u. (3.28) Note that the set (3.28) is totally disconnected in R. Indeed, otherwise it should contain a segment [α, β] P R+ f , β ą α ě 0. Then, f (z0 ) ” 0, for z0 P [α, β] and, consequently, F (z0 ) = F (β), for every z0 P [α, β], which evidently contradicts the fact that β P R+ f . Thus, we obtain the following proposition.

3.4 Symmetry and Stabilization

173

Proposition 3.2 There exists a homeomorphism τ : (RV , Cloc (R+ )) Ñ (R+ f , R) 2+β

Moreover, the set R+ f and, consequently, RV , are totally disconnected. Indeed, (3.27) defines a homeomorphism between R+ f and the set RV (0) := t(0, V 1 (0)) : V P RV u of values at t = 0, for functions from RV . Recall that RV consists of solutions of the second order ODE (3.23) and, consequently, by a classical theorem on continuous dependence of solutions of ODE’s, the set RV is homeomorphic to RV (0) and this homeomorphism is given by the solution operator S : (V (0), V 1 (0)) Ñ V (t) of Eq. (3.23). Proposition 3.2 is proved. l Remark 3.1 Note that, although for generic functions f the sets R+ f „ RV are finite, these sets may be uncountable, for some very special choices of the nonlinearity f . The simplest example of such an f is the following: f (z) = ´ dist(z, K),

(3.29)

where K is a standard Cantor set on [0, 1] and dist(z, K) denotes the distance from z to K. Indeed, it is easy to verify that, for this case, R+ = K and, consequently, RV consists of a continuum of elements. To be rigorous, function (3.29) is only Lipschitz continuous (but not in C 1 ) and does not satisfy the second assumption of (3.2), but by slightly modifying this function, one can construct the function f˜, which will satisfy all our assumptions and R+˜ = R+ f = K. Indeed, for instance, f define $ ’ for z P K, ’ &0 1 ´ f˜(z) = e (z´a)(z´b) for z P (a, b), (3.30) ’ ’ 1 %e´ z´1 for z ą 1, where the numbers a and b are such that 0 ă a ă b ă 1, a, b P K and (a, b) Ă (0, 1)zK. We now state the main result of this section. Theorem 3.3 Let the assumptions of Theorem 3.1 hold. Then, for every nonnegative bounded solution u of problem (3.1), there exists a solution V (x2 ) = Vu (x2 ) P RV of problem (3.23), such that, for every fixed R and x = (x1 , x2 , x3 ), }u ´ Vu }C 2+β (BxR X+ ) Ñ 0, xh := (x1 + h, x2 , x3 ), h

when h Ñ 8.

174

3 Symmetry and Attractors: The Case N ď 3

Proof Indeed, consider the ω-limit set of the solution u P K + , under the action of the semigroup Th of shift in the x1 direction, ω(u) = Xhě0 r Ys ěh Ts us .

(3.31)

Recall that Th possesses the attractor Atr in K + , and, consequently, the set (3.31) is nonempty. Thus, ω(u) Ă Atr . It follows now from Proposition 3.1 that ω(u) Ă RV . Note that on the one hand, the set ω(u) must be connected (see e.g. [67]) and on the other, it is a subset of RV , which is totally disconnected (by Proposition 3.2). Therefore, ω(u) consists of a single point, Vu Ă RV : ω(u) = tVu u. The assertion of the theorem is a simple corollary of this fact and of our definition of the topology in K + . Theorem 3.3 is proved. l Remark 3.2 We discuss assumptions (3.2) imposed on the nonlinear term f (u), in order to obtain the results of Chap. 3. Note first that sign condition (3.2)(3) is, evidently, essential in order to prove the solvability of (3.1) in the class of positive bounded solutions, for every positive bounded initial data u0 (and in fact, it is also essential for Proposition 3.1 and Lemma 3.2, see e.g. [21]). It is easy to see that the assumption f P C 1 is not necessary either for proving the existence of a positive solution of problem (3.1), or for applying the trajectory dynamical system approach to this problem, and can, therefore, be weakened to f P C(R, R). Note, however, that the (local) Lipschitz continuity of the nonlinear term is very important for the symmetry result, formulated in Proposition 3.1 (see [21]) and, consequently, for all results, obtained in Chap. 3. Thus, assumptions (3.2)(1) and (3.2)(3) seem to be close to optimal in order to derive the results of Chap. 3. In contrast, the dissipativity assumption (3.2)(2) is far from optimal and has been imposed in such a form only in order to avoid additional technicalities and to make the trajectory approach for the study of the behavior of positive solutions clearer. In fact, it can be proved, using the standard sub- and super-solutions technique and some monotonicity results for positive solutions of elliptic equations, that under assumptions (3.2)(1) and (3.2)(3), problem (3.1) has at least one positive bounded solution, for every positive bounded initial value u0 , if and only if its one dimensional analogue, V 2 (z) ´ f (V (z)) = 0, z ą 0, V (0) = M,

(3.32)

3.4 Symmetry and Stabilization

175

is solvable in the class of bounded non-negative solutions, for every M ě 0. Recall that (3.32) is a second order ODE of Newtonian type and can be easily analyzed using e.g. the phase portrait technique. In the case where n = 1, that is, + = R+ ˆ R+ ˆ R, using the explicit description of the set of bounded positive solutions of the Eq. (3.1) in , one can easily show that the attractor Atr exists if and only if the set K of all bounded positive solutions of the problem V 2 (z) ´ f (V (z) = 0, z ą 0, V (0) = 0,

(3.33)

is globally bounded in C(R+ ). Combining (3.32) and (3.33), we derive, after straightforward analysis of the corresponding phase portrait, that, under the assumptions (3.2) (1) as well as (3.2) (3), problem şv (3.1) possesses the trajectory attractor Atr if and only if the potential F (v) := ´ 0 f (u) du achieves its global maximum on [0, 8), i.e. if there exists v0 ě 0, such that F (z0 ) = max F (v). v PR+

Hence, all results of Chap. 3 remain valid, in the case when condition (3.2)(2) is replaced by (3.2). Evidently, condition (3.2)(2) is sufficient, but not necessary for (3.2). Remark 3.3 Note that neither our concrete choice of of the domain + = RˆR+ ˆ Rn nor the concrete choice of the ‘time’ direction x1 are essential for the trajectory dynamical system approach. Indeed, let us replace the ‘time’ direction x1 by any  Then the fixed direction l P Rn and (and correspondingly (Th u)(x) := u(x + hl). above construction seems to be applicable if the domain + satisfies the following assumptions: 1. Th + Ă + (it is necessary in order to define the restriction Th to the trajectory phase space K+ ). 2.  = Yhď0 T´h + (it is required in order to obtain representation (3.17)). Further generalizations of the result of Theorem 3.3 to higher dimensions, as well as other classes of equations, will be discussed in the following chapters.

Chapter 4

Symmetry and Attractors: The Case N ď 4

4.1 Introduction In the previous chapter, we showed that nonnegative solutions of elliptic equations in “asymptotically symmetric” domains are “asymptotically symmetric” as well (see Theorem 3.3). However, in order to prove Theorem 3.3, we imposed a restriction on the dimension (less or equal 3) of the underlying domain, which was crucial for our proof. The goal of this chapter is to extend Theorem 3.3 for higher dimensions. As we will see below, this is not straightforward and requires both new ideas and stronger assumptions on the nonlinearity. Moreover, even under these stronger assumptions, we manage to extend the results of the previous Chap. 3 only up to dimension 4 of the underlying domain (see below and [50]). We consider the following elliptic boundary problem in an unbounded domain + := tx = (x1 , x2 , x3 , x4 ) | 0 ď x1 , x4 ă +8, x2 , x3 P R}, # x u ´ f (u) = 0; u|x1 =0 = u0 ; u|x4 =0 = 0.

(4.1)

Here x denotes the four-dimensional Laplacian. It is assumed that the nonlinear term satisfies $ 2 ’ ’ &1. f P C (R, R), (4.2) 2. f (v).v ě ´C + α|v|2 , α ą 0, ’ ’ %3. for some μ ą 0, f (u) ď 0 in [0, μ] and f (u) ě 0 in [μ, 8[ We suppose also that u0 (x2 , x3 , x4 ) ě 0 and will consider only nonnegative solutions of (4.1):

© Springer Nature Switzerland AG 2018 M. Efendiev, Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations, Fields Institute Monographs 36, https://doi.org/10.1007/978-3-319-98407-0_4

177

4 Symmetry and Attractors: The Case N ď 4

178

u(x) ě 0

x P +

It is assumed also that 2+β

u0 P Cb

(0 ), (x2 , x3 , x4 ) P 0 := R ˆ R ˆ R+

for some 0 ă β ă 1. Here and below we write 2+β

Cb

(0 ) := tu0 : }u0 }C 2+β = sup }u0 }C 2+β (B 1 X0 ) ă 8u b

ξ P0

ξ

where Bξr means a ball of radius r centered at ξ . 2+β

A bounded solution of (4.1) is understood to be a function u P Cb (+ ) which satisfies (4.1) in a classical sense. (In fact due to the interior estimates this assumption is equivalent to u P Cb (+ )). The main goal of this chapter is to study the behaviour of a bounded solution of (4.1) when x1 Ñ 8. We are particularly interested in the symmetrization and stabilization properties of a bounded solution of (4.1) as x1 Ñ 8. As we mentioned above, we apply to (4.1) the dynamical systems approach, which was initiated in the seminal work of Kirchgässner [72] where a local center manifold for a semilinear elliptic equation on a strip was constructed. As we already mentioned both in the Preface and in Chap. 3, one of the main difficulties arising in a dynamical study of elliptic equations is that the corresponding Cauchy problem is not well posed and thus, in general (4.1) can only be rigorously defined as a semigroup of multivalued maps (see [8]). The use of multivalued maps can be overcome by using a new approach, the so-called trajectory dynamical one (see Chaps. 2–3 for details). Under this approach one considers the set K + of all bounded solutions of (4.1), endowed with a suitable topology (in our 2+β case it will be Cloc (+ ) for some 0 ă β ă 1), as a (trajectory) phase space for the semigroup defined by the translation (Th u)(x1 , x2 , x3 , x4 ) := u(x1 + h, x2 , x3 , x4 ),

hě0

If an associated attractor exists it is called a trajectory attractor for (4.1). For the case of second order elliptic systems there is another way to avoid the use of multivalued maps. In [31] the problem under consideration was studied with Cauchy initial conditions u|t=0 = u0 , Bt u|t=0 = u1 In this case under certain assumptions a solution will be unique and one can define a semigroup St : (u(0), Bt u(0)) ÞÑ (u(t), Bt u(t))

4.2 A Priori Estimates and Solvability Results

179

But the set on which it is defined is not described in explicit form. As for the application of the trajectory dynamical approach to evolution problems we refer to [12, 12, 32, 52, 92]. In the previous Chap. 3, we considered (4.1) from the viewpoint of dynamical systems, when + = tx = (x1 , x2 , x3 ) P R3+ |0 ď x1 , x2 ă 8, x3 P Ru. Such a restriction on the dimension of + (dim + ď 3) was crucial because in Chap. 3 symmetrization and stabilization to the attractor for solutions of (4.1) among others, were based on the paper [21], where a surprising link with a problem concerning the Schrödinger operator led to the restriction on dimension of the underlying domain + , namely n ď 3. Whether or not the symmetry result in Rn+ = tx = (x1 , . . . , xn )|xn ě 0u holds for n ě 4 under the assumptions (4.2) was open at that time. In this section, using recent development on the De Giorgi conjecture (see [16, 22] and references therein, for symmetry results for parabolic equations we refer to [11, 68]), especially the paper [5] and a personal communication with X. Cabre, we prove the existence of the trajectory attractor for (4.1) in + := tx = (x1 , x2 , x3 , x4 ) | 0 ď x1 , x4 ă +8, x2 , x3 P Ru and analyse the symmetrization and stabilization of solutions (4.1) to the attractor. The chapter is organised as follows. The dissipative a priori estimate with respect to x1 Ñ 8 for the positive solutions of (4.1) is derived in Sect. 4.2, which allows us to apply the trajectory approach to our situation, and, in particular, gives the existence of at least one nonnegative solution. In Sect. 4.3, we construct the trajectory dynamical system (Th , K + ) associated with problem (4.1) and prove existence of a global attractor for this dynamical system. Section 4.4 is devoted to a more comprehensive study of the four dimensional case. Following [5] we establish that the trajectory attractor Atr consists of functions u(x1 , x2 , x3 , x4 ) := V (x4 ) which satisfy the ordinary differential equation V 2 (z) ´ f (V (z)) = 0, z ą 0, V (0) = 0, V (z) ě 0

(4.3)

In this section stabilisation of solutions of (4.1) to Atr is also discussed.

4.2 A Priori Estimates and Solvability Results In this section, we prove that the problem (4.1) possesses at least one non-negative bounded solution u and derive the so-called dissipative estimate for such solutions, which is of fundamental significance in order to apply the dynamical approach to the elliptic equation (4.1). For the convenience of the reader, we present a sketch of the proof. It uses the same techniques and ideas and leads to the same results, as in Theorem 3.1. As was mentioned in Remark 3.3, the dimension of + doesn’t play an important role in the proof of existence of trajectory attractor for Eq. (4.1). The

4 Symmetry and Attractors: The Case N ď 4

180

difference between the assumptions (4.2) and (3.2) on the nonlinearity is related to the symmetry result which is formulated in Sect. 4.4. The main result of this section is the following theorem. 2+β

Theorem 4.1 Let u0 P Cb (0 ) and let the first and second compatibility conditions at B0 be valid (i.e. u0 (x2 , x3 , 0) = 0 and Bx24 u0 (x2 , x3 , 0) = f (0)). Then (4.1) possesses at least one nonnegative bounded solution and every such solution u satisfies the estimate }u}C 2+β (Bx1 X+ ) ď Q(}u0 }C 2+β )e´γ x1 + Cf .

(4.4)

b

Here x = (x1 , x2 , x3 , x4 ) P + , γ ą 0, Q is an appropriate monotonic function, and Cf is independent of u0 . Sketch of the Proof First we verify the a priori estimate (4.4). To this end we consider the function w(t, x) = u2 (t, x) which evidently satisfies the equation x w = 2f (u).u + 2∇x u.∇x u ě ´2C + 2αw, w|x1 =0 = u20 , w|x2 =0 = 0 (4.5) Consider also the auxiliary linear problem with the same boundary conditions as for the function w, x w1 = ´2C + 2αw1 , w1 |x1 =0 = w|x1 =0 = u20 , w1 |x2 =0 = 0.

(4.6)

Then applying the comparison principle to the solutions w and w1 of (4.5) and (4.6) respectively and the evident fact that w = u2 is non-negative, we derive }u}2C(B 1 ) ď }w}C(Bx1 ) ď }w1 }C(Bx1 ) ď C1 }u0 }C 2+β ( ) e´αx1 + C2 x

b

0

(4.7)

The latter inequality in (4.7) is due to the fact that a solution of linear equation (4.6) admits the following estimate }w1 }C(Bx1 ) ď C1 }u0 }2Cb (0 ) e´αx1 + C2 Consequently, the estimate (4.4) can be obtained using a classical elliptic interior estimate (see [72], as well as Lemma 3.1). A proof of the existence of at least one nonnegative solution for problem (4.1) is based on the approximation of + by a sequence of bounded domains and the sub-super solution techniques and follows exactly the same arguments as in the proof of Theorem 3.1. This proves Theorem 4.1. l

4.3 The Attractor

181

4.3 The Attractor In this section, we study the behaviour of the non-negative solutions of problem (4.1) when x1 Ñ 8 applying the dynamical system approach to the elliptic boundary value problem (4.1) for general unbounded domains + . This approach we already applied in the Chaps. 2 and 3 for several class of semilinear elliptic equations in unbounded domains taking into account both different geometries of underlying domains and different class of their solutions. Recall that in such a case we fix some direction in our unbounded domain + and interpret it as the ‘time’ direction. In our case this will be the x1 direction, the x1 variable will then play the role of ‘time’ variable and we (formally) consider (4.1) as an ‘evolutionary’ equation in an unbounded domain 0 . The main difficulty which arises here is the fact that the solution of (4.1) may be not unique and consequently we cannot construct the semigroup corresponding to the ‘evolutionary’ equation (4.1) in the standard way. One of the possible ways to overcome this difficulty, as was mentioned in the beginning of this chapter, is to use the trajectory approach which takes into account the dynamical system for (4.1) in another way. Namely, let us consider the union K + 2+β of all bounded positive solutions of (4.1) which corresponds to every u0 P Cb . Then a semigroup of positive shifts (Th u)(x1 , x2 , x3 , x4 ) := u(x1 + h, x2 , x3 , x4 ) acts on the set K + : T h : K + Ñ K + , K + Ă Cb

2+β

(+ )

(4.8)

This semigroup acting on K + is called the trajectory dynamical system, corresponding to (4.1). Our next task is to construct the attractor for this system. Firstly we note 2+β that the uniform topology of Cb is too strong for our purposes. That is why we endow the space K + with a local topology according to the embedding K + Ă Cloc (+ ) 2+β

2+β

where by definition  := Cloc (+ ) is a Fréchet space generated by the seminorms } ¨ }C 2+β (Bx1 X+ ) , x0 P + . 0 For the convenience of the reader we recall briefly the definition of the attractor adapted to our case. Definition 4.1 The set Atr Ă K + is called the attractor for the trajectory dynamical system (4.8) (= trajectory attractor for the problem (4.1)) if the following conditions are valid.

4 Symmetry and Attractors: The Case N ď 4

182 2+β

1. The set Atr is compact in Cloc (+ ). 2. The set Atr is strictly invariant with respect to Th : Th Atr = Atr 3. Atr attracts bounded subsets of solutions when x1 Ñ 8. This means that for every bounded subset B Ă K + and for every neighbourhood O(Atr ) in the 2+β Cloc topology there exists H = H (B, O) such that Th B Ă O(Atr ) if h ě H Note that the first condition of the definition asserts that the restriction Atr |1 is compact in C 2+β (1 ) for every bounded 1 Ă + and the third one is equivalent to the following: For every bounded subdomain 1 Ă + , for every B – bounded subset of K + and for every neighbourhood O(Atr |1 ) in the C 2+β (1 )-topology of the restriction Atr to this domain, there exists H = H (1 , B, O) such that (Th B)|1 Ă O(Atr |1 ) if h ě H Theorem 4.2 Let the assumptions of Theorem 4.1 hold. Then the Eq. (4.1) possesses the trajectory attractor Atr which has the following structure: Atr = !+ K()

(4.9)

where (x1 , x2 , x3 , x4 ) P  := R ˆ R ˆ R ˆ R+ and the symbol K() means the p(x) P Cb2+β () of union of all bounded nonnegative solutions u p(x) ě 0 p ´ f (p p|B = 0, u x u u) = 0, x P , u

(4.10)

That is, the attractor Atr consists of all bounded nonnegative solutions u of (4.1) in p in . + which can be extended to a bounded nonnegative solution u The proof of this theorem repeats word by word the proof of Theorem 3.2, we omit it. Remark 4.1 Note that neither our concrete choice of the domain + = R+ ˆ R ˆ R ˆ R+ nor the concrete choice of the ‘time’ direction x1 are essential for the use of the trajectory dynamical system approach. Indeed, let us replace the ‘time’ direction  x1 by any fixed direction l P R4 and (and correspondingly (Th u)(x) := u(x + hl). Then the above construction seems to be applicable if the domain + satisfies the following assumptions: 1. Th + Ă + (this is necessary in order to define the restriction Th to the trajectory phase space K + ). 2.  = Yhď0 T´h + (this is required in order to obtain the representation (4.9)).

4.4 Symmetry and Stabilization

183

4.4 Symmetry and Stabilization In this section based on a recent development on a conjecture of E. DeGiorgi, especially the paper [5], we give the main result. Theorem 4.3 Let assumptions (4.2) hold. Then for every nonnegative bounded solution u of the problem (4.1) there is a solution V (x4 ) = Vu (x4 ) of the problem (4.3) such that for every fixed R and x = (x1 , x2 , x3 , x4 ) }u ´ Vu }C 2+β (BxR X+ ) Ñ 0, xh := (x1 + h, x2 , x3 , x4 ) h

when h Ñ 8. The proof of Theorem 4.3 will be carried out in three steps, based on the following Propositions 4.1, 4.2, 4.3. Proposition 4.1 ([21]) Let Q Ă Rn be a (connected) domain with sufficiently smooth boundary and let w P C 2 (Q) X C(Q) satisfy the following inequalities x w(x) ´ l(x)w(x) ď 0, x P Q, w(x) ě 0, x P Q Assume also that |l(x)| ď K for x P Q. Then either v(x) ” 0 or v(x) ą 0 for every interior point x P Q. Proposition 4.2 Let the assumptions (4.2) hold. Then any non-negative bounded p(x) of the Eq. (4.10) is symmetric, that is depends only on the variable x4 , solution u p(x) = V (x4 ) where V (z) is a bounded solution of the following problem: u V 2 (z) ´ f (V (z)) = 0, z ą 0, V (0) = 0, V (z) ě 0

(4.11)

Proof We rewrite the Eq. (4.10) in the following form p ´ l(x)p u = f (0) ď 0, l(x) := x u

f (p u(x)) ´ f (0) p(x) u

p(x) is bounded then l(x) is also bounded in Since f P C 2 and the solution u p(x) ” 0 (which is . Thus, according to Proposition Proposition 4.1 either u p(x) ą 0 in the interior of . Hence it remains to prove evidently symmetric) or u p(x) ą 0 in . We emphasize again that we cannot Proposition 4.2 in the case where u apply the arguments of Chap. 3, which leads to the restrictions for the dimension of the underlying domain + Ă Rn , n ď 3. Here we mainly follow [5]. Following [21], to show the symmetry result in R4+ we have to prove that p) ě 0. To this end, we note that from the assumptions (4.2) it follows that f (sup u p ą 0 (see [21]). We define in R3 Bx4 u

4 Symmetry and Attractors: The Case N ď 4

184

u(x ¯ 1 , x2 , x3 ) :=

p(x1 , x2 , x3 , x4 ), lim u

x4 Ñ+8

(x1 , x2 , x3 ) P R3 .

p = M, it suffices to show that Obviously, this limit exists. Since sup u¯ = sup u f (sup u) ¯ ě 0. For this purpose we proceed as follows. We remark that, Bx4 u satisfies p ´ f 1 (p p = 0 in R4+ u)Bx4 u x Bx4 u

(4.12)

2

8 4 Multiplying (4.12) by ξB (x) p , where ξ P C0 (R+ ) and integrating by parts, we obtain x4 u

ż

ż 2 2ξ(x) ξ (x) p ¨ ∇ξ + f 1 (p p|2 dx u)ξ 2 dx = |∇x Bx4 u ∇x Bx4 u p p)2 Bx4 u (Bx4 u

Using the Cauchy-Schwarz inequality the last equality leads to ż

u)ξ 2 )dx ě 0 for all ξ P C08 (R4 ) (|∇x ξ |2 + f 1 (p

(4.13)

As was shown in [5] from (4.13) one can deduce (here for the first time we need f P C2) ż R3



 |∇x η|2 + f 1 (u)η ¯ 2 dx ě 0

for all η P C08 (R3 )

and as a result (see [5]) we obtain the existence of a strictly positive solution ϕ such that x ϕ ´ f 1 (u)ϕ ¯ = 0.

(4.14)

Note that the function u(x ¯ 1 , x2 , x3 ) defined by (4.4) also satisfies ¯ = 0. x u¯ ´ f (u)

(4.15)

Next, we prove that, u¯ defined by (4.4) satisfies ż

|∇ u| ¯ 2 dx ď CR 2

(4.16)

BR

for every ball BR Ă R3 of radius R and centered at the origin. To this end, we first prove, p ď 0 in R4+ . x u

(4.17)

p) ě 0, then it leads immediately to a symmetry result; that is Indeed, if f (sup u p(x1 , x2 , x3 , x4 ) = V (x4 ) (see [21]). Assume that f (sup u p = p) ă 0. Since x u u

4.4 Symmetry and Stabilization

185

p ă 0. Hence (4.17) holds. Now f (p u), it follows from conditions (4.2) that x u using (4.17) we show that ż

p|2 dx ď CR 3 |∇x u

QR

(4.18)

for every cylinder QR = BR ˆ (b, b + R) Ă R4+ , where C is a constant independent p ě 0, we have u p(´x u p) ď M(´x u p) and hence of R and b. Indeed, since ´x u ż QR

p|2 dx = |∇x u ď ´M

ż QR

ż QR

=

ż B QR

ż

p(´x u p)dx + u

pdx + x u

(p u ´ M)

ż B QR

p u

B QR

p u

Bp u dσ Bν

Bp u dσ Bν

Bp u dσ ď CR 3 , Bν

that is (4.18) holds. By letting b Ñ 8 in (4.18), it is not difficult to see that ż

|∇x u| ¯ 2 dx ď CR 2 ,

BR

Bx u¯

that is (4.16) holds. To show f (sup u) ¯ ě 0 we define σi (x1 , x2 , x3 ) := ϕi , i = 1, 2, 3. Then, due to (4.14), (4.15) and (4.16), σi satisfies the following conditions σi div(ϕ 2 ∇σi ) = 0 in R3 and ż

(ϕσi )2 dx ď CR 2 , @R ą 1.

Then a Liouville-type theorem due to [21] implies that σi is necessarily constant, hence each partial derivative of u¯ is constant, that is Bxi u¯ = ci ϕ, i = 1, 2, 3. In particular, u¯ is either constant, which leads to f 1 (M) ě 0, or u(x ¯ 1 , x2 , x3 ) = u(c ˜ 1 x1 + c2 x2 + c3 x3 ). Then, the bounded function u(θ ˜ ), which is a monotone function of only one variable satisfies the ODE u˜ 2 + f (u(θ ˜ ) = 0. It is then a classical result that f (sup u) ¯ = 0. This proves Proposition 4.2. l Denote by RV the set of all bounded non-negative solutions V (z) of problem (4.11). Then Proposition 4.2 implies that Atr = RV .

4 Symmetry and Attractors: The Case N ď 4

186

Proposition 4.3 The set Atr (trajectory attractor) endowed with the local topology 2+β Cloc (+ ) is totally disconnected. The proof of Proposition 4.3 repeats word by word the proof of Proposition 3.2 and arguments (3.23)–(3.27). We omit it. Now we are in the position to prove Theorem 4.3. Proof Indeed, consider the ω-limit set of the solution u P K + under the action of the semigroup Th of shifts in the x1 direction: ω(u) =

č hě 0

r Ys ěh Ts us

(4.19)

Recall that Th possesses the attractor Atr in K + , consequently the set (4.19) is nonempty and ω(u) Ă Atr It follows now from the Proposition 4.2 that ω(u) Ă RV . Note that the set ω(u) must be connected [1] but also it is a subset of the set RV which is totally disconnected (due to Proposition 4.2). Therefore ω(u) consists of a single point Vu Ă RV : ω(u) = tVu u The assertion of the theorem is a simple corollary of this fact and of our definition l of the topology in K + . Theorem 4.3 is proved. Remark 4.2 We would like to emphasize especially that, in order to prove the main result of Chap. 4, that is, Theorem 4.3, we were forced to impose in (4.2) a strong condition on the nonlinearity f . This assumption on f we used in order to prove the symmetry result (see Proposition 4.2). Note that, in the case when the underlying domain + has dimension less or equal 3, weaker assumptions on f , such as (3.2), are sufficient (see Remark 3.2).

Chapter 5

Symmetry and Attractors

5.1 Introduction In this chapter, symmetry results in the half-space and in RN will be used towards the characterization of the asymptotic profiles of solutions in the quarter-space and in the half-space, respectively. As we have seen in the previous Chaps. 3 and 4, here the dimension of the underlying domain plays an important role and to extend the results on the symmetrization and stabilization of solutions of semilinear elliptic equations for dimensions less or equal 3 to the case of dimensions less or equal 4 requires nontrivial arguments and assumptions on the nonlinearities. The goal of this chapter is to extend the results from Chaps. 3 and 4 to the case of dimensions less or equal 5. As we will see below, to this end we need new arguments and we cannot use the techniques from Chaps. 3 and 4. Similar to the previous chapters, we will apply the trajectory dynamical systems approach in order to study the asymptotic profiles of solutions for this new case of dimension 5 or higher. Moreover, in contrast to the previous chapters, we will also study the case when the asymptotic profile is a constant.

5.1.1 Statement of Results For N P N, N ě 2, we consider the half-space N RN + := tx = (x1 , . . . , xN ) P R s.t. x1 ą 0u

and the quarter-space N RN ++ := tx = (x1 , . . . , xN ) P R s.t. x1 ą 0 and xN ą 0u.

© Springer Nature Switzerland AG 2018 M. Efendiev, Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations, Fields Institute Monographs 36, https://doi.org/10.1007/978-3-319-98407-0_5

187

188

5 Symmetry and Attractors

The purpose of this chapter is to develop asymptotic results, as x1 Ñ +8, of N solutions of elliptic PDEs on either RN + or R++ . These results extend some previous work of [15, 50] by making use of some symmetry results and classifications of [45, 60]. Following are the main results of this chapter. First, we show that the solutions of elliptic equations on the quarter-space are asymptotic to one-dimensional solutions, up to dimension 5, according to our next result: Theorem 5.1 Let f P C 1 (R). Let us assume that there exist c1 ě 0, c2 ą 0 and rě2 such that f (s)s ě ´c1 + c2 s r , for any s ě 0,

(5.1)

and that there exists μ ą 0 such that f (s) ď 0 for any s P [0, μ] and f (s) ě 0 for any s P [μ, +8). N 0 Let N ď 5 and u P C 2 (RN ++ ) X C (R++ ) be a non-negative solution of

$ u = f (u) in RN ’ ++ , ’ ’ ’ ’ & u|tx1 =0u = u0 , ’ ’ ’ ’ ’ % u|txN =0u = 0.

(5.2)

Then, there exists uniquely defined w : R Ñ R such that, as x1 Ñ +8, u converges 2 (RN ) to w(x ). In particular, we have that either w vanishes identically, or in Cloc N ++ it is a bounded solution of $ ’ w 2 (t) = f (w(t)) for any t P R, ’ ’ ’ ’ & w(0) = 0, ’ ’ ’ ’ ’ % w 1 (t) ą 0 for any t P R. Remark 5.1 Nonlinearities satisfying (5.1) have been extensively studied in the dynamical system framework, and they are sometimes called “dissipative” in the literature. The next result shows that the stable solutions on the half-spaces are asymptotic to constants, up to dimension 4: Theorem 5.2 Let f P C 1 (R). Let us assume that there exist c1 ě 0, c2 ą 0 and rě2 such that

5.1 Introduction

189

f (s)s ě (´c1 + c2 s r )+ , for any s ě 0,

(5.3)

the set E := ts ě 0 s.t. f (s) = 0u is non-empty.

(5.4)

and that

N 0 Let N ď 4 and u P C 2 (RN + ) X C (R+ ) be a non-negative stable solution of

$ N ’ &u = f (u) in R+ , ’ % u| tx1 =0u = u0 .

(5.5)

2 (RN ) to a constant c P E. Then, as x1 Ñ +8, u converges in Cloc +

Remark 5.2 In the statement of Theorem 5.2 we used the standard notation v + (x) := maxtv(x), 0u,

(5.6)

and the classical language of the calculus of variation, according to which a solution u of (5.5) is stable if ż |∇ψ(x)|2 + f 1 (u(x))ψ 2 (x) dx ě 0 (5.7) RN +

for any ψ P Cc8 (RN + ). Such stability condition is classical and widely studied (see, e.g., [44], Sect. 7 in [59], and the references therein). We observe that solutions minimizing the associated energy functional are always stable. Moreover, Theorem 5.2 remains valid (with the same proof) if the stability assumption is weakened to the so-called “stability outside a compact set” K Ă RN + , namely if ) whose support does not intersect one requires (5.7) to hold for any ψ P Cc8 (RN + K (these kind of assumptions play a role in the study of solutions with finite Morse index, which are always stable outside a compact set, see [55, 57]). Remark 5.3 From (5.6), we have that condition (5.3) implies (5.1). Condition (5.4) is also necessary to make sense of Theorem 5.2, because if E is empty there cannot be any c to which u is asymptotic, since such c should satisfy 0 = c = f (c). In fact, if (5.3) holds but (5.4) does not hold, there does not exist any non-negative solution of u = f (u) in RN + (see Remark 5.9). A particular case comprised by Theorem 5.2 is when f has the special form f (r) = r

m ź

(r ´ pj )2 ,

j =1

(5.8)

190

5 Symmetry and Attractors

with 0 ď p1 ď ¨ ¨ ¨ ď pm . In this case, our Theorem 5.2 provides a positive answer, at least for stable solutions and in dimension up to 4. In fact, in this case, Theorem 5.2 gets even stronger, since it is not necessary to assume that u is nonnegative: more precisely, for the nonlinearity in (5.8) we have: N 0 Theorem 5.3 Let N ď 4, 0 ď p1 ď ¨ ¨ ¨ ď pm , and u P C 2 (RN + ) X C (R+ ) be a stable solution of

$ m ź ’ ’ u = u (u ´ pj )2 in RN ’ +, & ’ ’ ’ %

j =1

u|tx1 =0u = u0 .

2 (RN ) to a constant c P E. Then, as x1 Ñ +8, u converges in Cloc +

Remark 5.4 We think that our Theorems 5.1 and 5.2 leaves open some very intriguing questions. For instance, we wonder whether the results remain true in higher dimension or not. In particular, related to the proof of Theorem 5.2, we wonder if there exists positive and bounded, non-constant, or non-stable, solutions of u = f (u) in the whole of RN , with f ě 0, or, in particular, with f as in (5.8).

5.2 The Dynamical System Approach The goal of this Section is to apply the dynamical systems approach in order to study the symmetrization and stabilization (as |x| Ñ 8) properties of nonnegative solutions we consider non-negative solutions of problem (5.2) (in the same manner one can handle (5.5)). u(x) ě 0, x P RN ++ and study their behavior when x1 Ñ +8. Thus, in this case, the x1 -axis will play the role of time (l := (1, 0, . . . , 0) P RN ). Moreover, we restrict our consideration to bounded (with respect to x Ñ 8) solutions of (5.2). More precisely, a bounded 2+β solution of (5.2) is understood to be a function u P Cb (RN ++ ), for some fixed 0 ă β ă 1, which satisfies (5.2) in a classical sense (in fact, due to interior estimates, this assumption is equivalent to u P Cb (RN ++ ), but we prefer to work with classical solutions). Therefore, the boundary data is assumed to be non-negative u0 (x2 . . . xN ) ě 0 and to belong to the space 2+β

u0 P Cb

(0 ), (x2 . . . xN ) P 0 := RN ++ X tx1 = 0u .

5.2 The Dynamical System Approach

191

Here and below, we use the notation 2+β

Cb

(V ) := tu0 : }u0 }C 2+β := sup }u0 }C 2+β (B 1 XV ) ă 8u, ξ PV

b

ξ

where Bξr denotes the ball of radius r, centered in ξ . The dissipative estimate for nonnegative solutions of (5.2), which allows us to apply the trajectory dynamical systems approach, can be obtained in the same manner as in the previous Sections. We omit these details. 2+β

Theorem 5.4 Let u0 P Cb

(0 ) and let the first and second compatibility

conditions be valid on B0 (i.e. u0 (x2 . . . xN ´1 , 0) = 0 and BB2u0 (x2 . . . xN ´1 , 0) xN = f (0)). Then (5.2) has at least one non-negative bounded solution, every such solution u satisfies the estimate 2

}u}C 2+β (B 1 XRN x

++ )

ď Q(}u0 }C 2+β )e´γ x1 + Cf ,

(5.9)

b

x P RN ++ , where γ ą 0, Q is an appropriate monotonic function and Cf is independent of u0 . Remark 5.5 It is interesting enough to note that, if r ą 2, then one can show that without sub- and supersolution technique (without sign condition on f at the origin) that any weak solution of (5.2) in an appropriately defined space is necessarily bounded (see Sect. 2.6). Now we can apply the trajectory dynamical systems approach to (5.2). To this end, as in the previous chapters, we consider the union K + of all bounded positive 2+β solutions of (5.2), which correspond to every u0 P Cb . Then a semigroup of positive shifts, (Th u)(x1 , x 1 , xN ) := u(x1 + h, x 1 , xN ), x 1 = (x2 , . . . , xN ´1 ) P RN ´2 ,

(5.10)

acts on the set K + : T h : K + Ñ K + , K + Ă Cb

2+β

(RN ++ ).

This semigroup acting on K + is called the trajectory dynamical system corresponding to (5.2). Our next task is to construct the global attractor for this system. To this end, we endow the space K + with a local topology, according to the embedding K + Ă Cloc (RN ++ ), 2+β

2+β

where, by definition,  := Cloc (RN ++ ) is a Fréchet space generated by the N seminorms } ¨ }C 2+β (B 1 XRN ) , x0 P R++ . x0

++

192

5 Symmetry and Attractors

For convenience of the reader, we recall briefly the definition of the attractor adapted in our case. Definition 5.1 The set Atr Ă K + is called the global attractor for the trajectory dynamical system (5.10) (= trajectory attractor for problem (5.2)), if the following conditions are valid. 2+β

1. The set Atr is compact in Cloc (RN ++ ). 2. It is strictly invariant with respect to Th : Th Atr = Atr 3. Atr attracts bounded subsets of solutions, when x1 Ñ 8. That means that, for 2+β every bounded (in the uniform topology of Cb ) subset B Ă K + and for every 2+β neighborhood O(Atr ) in the Cloc topology, there exists H = H (B, O) such that Th B Ă O(Atr ) if h ě H. Note that the first assumption of the definition claims that the restriction Atr |1 is compact in C 2+β (1 ), for every bounded 1 Ă RN ++ , and the third one is equivalent to the following: + For every bounded subdomain 1 Ă RN ++ , for every B – bounded subset of K 2+β and for every neighborhood O(Atr |1 ) in the C (1 )-topology of the restriction of Atr to this domain, there exists H = H (1 , B, O) such that (Th B)|1 Ă O(Atr |1 ) if h ě H. Theorem 5.5 Let the assumptions of Theorem 5.4 hold. Then Eq. (5.2) possesses a trajectory attractor Atr which has the following structure: Atr = !RN K(RN + ), ++

(5.11)

N where (x1 , x 1 , xN ) P RN + and K(R+ ) denotes the union of all bounded non-negative 2+β N p(x) P Cb (R+ ) of solutions u

p(x) ě 0, p ´ f (p u) = 0, x P RN x u + , u|xN =0 = 0, u

(5.12)

i.e. the attractor Atr consists of all bounded non-negative solutions u of (5.2) in p in RN RN ++ , which can be extended to bounded non-negative solution u +. The proof of this theorem is identical to Theorem 3.2. We omit the details. Remark 5.6 Note that neither our concrete choice of RN ++ as the domain, on which we study our problem, nor the concrete choice of the ‘time’ direction x1 are not essential for the of the trajectory dynamical system approach. Indeed, let us replace the ‘time’ direction x1 by any fixed direction l P RN and (and correspondingly  Then the above construction is applicable if the domain + (Th u)(x) := u(x + hl). satisfies the following assumptions:

5.3 Proof of Theorem 5.1

193

1. Th + Ă + (it is necessary in order to define the restriction Th to the trajectory phase space K+ ). 2. RN + = Yhď0 T´h + (it is required in order to obtain representation (5.11)).

5.3 Proof of Theorem 5.1 To obtain additional information on the behaviour of solutions of the initial problem (5.2), we use the description of non-negative bounded solutions in the halfspace (see below). Proposition 5.1 Let assumptions of Theorem 5.1 hold and let N ď 5. Then any p(x) of Eq. (5.12) depends only on the variable xN , non-negative bounded solution u i.e. u(x) = V (xN ), where V (z) is a bounded solution of the following problem: V 2 (z) ´ f (V (z)) = 0, z ą 0, V (0) = 0, V (z) ě 0.

(5.13)

The proof of this proposition is given in [56], for the case where the solution p(x) is strictly positive inside . The general case can be reduced to the one above, u using the following version of the strong maximum principle. Lemma 5.1 Let V Ă Rn be a (connected) domain with sufficiently smooth boundary and let w P C 2 (V ) X C(V ) satisfy the following inequalities: x w(x) ´ l(x)w(x) ď 0, x P V , w(x) ě 0, x P V . Assume also that |l(x)| ď K for x P V . Then either v(x) ” 0, or v(x) ą 0, for every interior point x P V . In order to apply the lemma to Eq. (5.12), we rewrite it in the following form: p ´ l(x)p x u u = f (0) ď 0, l(x) :=

f (p u(x)) ´ f (0) . p(x) u

p(x) is bounded, l(x) is also bounded in RN Since f P C 1 and the solution u + . Thus, p(x) ” 0 (which is evidently symmetric), or u p(x) ą according to Lemma 5.1, either u 0, in the interior of RN and, thus, Proposition 5.1 follows from the result of [56] + mentioned above. Proposition 5.1 is proved. Denote by RV the set of all bounded non-negative solutions V (z) of problem (5.13). Then Proposition 5.1 implies that Atr = RV . In the same manner as in the previous sections, one can obtain: Proposition 5.2 There exists a homeomorphism

194

5 Symmetry and Attractors

τ : (RV , Cloc (R+ )) Ñ (R+ f , R) 2+β

Moreover, the set R+ f and, consequently, RV , are totally disconnected. We now state the main result of this section. Theorem 5.6 Let the assumptions of Theorem 5.1 hold. Then, for every nonnegative bounded solution u of problem (5.2), there exists a solution V (xN ) = Vu (xN ) P RV of problem (5.13), such that, for every fixed R and x = (x1 , x 1 , xN ), }u ´ Vu }C 2+β (B R XRN xh

++ )

Ñ 0, xh := (x1 + h, x 1 , xN ),

when h Ñ 8. Proof The proof of this theorem repeats word by word the proof of Theorem 3.3. Still, since it is short, we give it here for the convenience of the reader. Indeed, consider the ω-limit set of the solution u P K + , under the action of the semigroup Th of shift in the x1 direction, ω(u) = Xhě0 r Ys ěh Ts us .

(5.14)

Recall that Th possesses the attractor Atr in K + , and, consequently, the set (5.14) is nonempty. Thus, ω(u) Ă Atr . It follows now from Proposition 5.1 that ω(u) Ă RV . Note that on the one hand, the set ω(u) must be connected (see e.g. [1]) and on the other, it is a subset of RV , which is totally disconnected (by Proposition 5.2). Therefore, ω(u) consists of a single point, Vu Ă RV : ω(u) = tVu u. The assertion of the theorem is a simple corollary of this fact and of our definition of the topology in K + . Theorem 5.6 is proved, which in turn implies the assertion of Theorem 5.1. \ [

5.4 Proof of Theorem 5.2 5.4.1 Symmetry of the Profiles We recall a result of [45]:

5.4 Proof of Theorem 5.2

195

Theorem 5.7 Let N ď 4 and f P C 1 (R). Suppose that u P C 2 (RN ) X L8 (RN ) is a stable solution of u = f (u) in the whole of RN . Assume that f (u(x)) ě 0 for almost any x P RN .

(5.15)

Then u is constant. Proof This is Theorem 1.1 of [45]. In fact, there it was assumed that f (r) ď 0 for any r P R,

(5.16)

but of course one can replace u with ´u and so change (5.16) into f (r) ě 0 for any r P R.

(5.17)

In fact, the same proof works by requiring (5.17) in the domain of u, that is (5.15). \ [ Remark 5.7 It is worth noticing that, when N ď 2 the stability condition in Theorem 5.7 may be dropped. Indeed, if N = 1, it just follows by an argument on convex functions, and if N = 2, one defines w := u2 and checks by a computation that w is bounded and its Laplacian has a sign (hence is constant by the Liouville theorem in R2 , see Theorem 2.26 and Lemma 2.20). Then, we have: Theorem 5.8 Let f P C 1 (R). Let us assume that f (s) ě 0 for any s ě 0,

(5.18)

the set E := ts ě 0 s.t. f (s) = 0u is non-empty.

(5.19)

and that

Let N ď 4 and u be a bounded and non-negative stable solution u P C 2 (RN +) X C 0 (RN + ) of $ N ’ &u = f (u) in R+ , ’ % u| tx1 =0u = u0 . 2 (RN ) to a constant c P E. Then, as x1 Ñ +8, u converges in Cloc ++

Proof Let Ks+ be the set of all non-negative stable solutions v P C 2 (RN +) X

N 8 N + C 0 (RN + ) X L (R+ ) of v = f (v) in R+ . Notice that E Ď Ks , so, by (5.19), + + we have that Ks is non-empty. For any v P Ks and any h ě 0, we define

Th v(x1 , x2 , . . . , xN ) := v(x1 + h, x2 , . . . , xN ).

196

5 Symmetry and Attractors

Since the above PDE and the stability condition (5.7) are translation invariant, we 2+β have that Th (Ks+ ) Ď Ks+ . Also, Th is continuous in the Cloc (RN ++ ) topology, due to elliptic estimates. Therefore, by Theorem 5.5, there exists a global stable attractor As , that is, as x1 Ñ +8, any v P K+ converges to an element of As . On the other hand, if w P As , it is a bounded and non-negative stable solution of w = f (w) in the whole of RN . Since we know from Theorem 5.7 that w has to be constant, we are done. \ [

5.4.2 Completion of the Proof of Theorem 5.2 Notice that condition (5.3) implies (5.18) and (5.1) (recall Remark 5.3). Hence, thanks to Theorem 5.8, we have that all the solutions of (5.2) are bounded, and so Theorem 5.2 is a consequence of Theorem 5.8. \ [ Remark 5.8 Notice that the assumption that u is non-negative has been used only at the end of Theorem 5.8 to say that w is non-negative (so to fulfill (5.15) and be able to use Theorem 5.7). If one knows that such a w is non-negative for other reasons, the assumption that u is non-negative in Theorems 5.2 and 5.8 can be dropped. This observation will be important in the forthcoming proof of Theorem 5.3. Remark 5.9 If (5.3) holds but (5.4) does not hold, we have that: there does not exist any non-negative solution of u = f (u) in RN +,

(5.20)

hence condition (5.4) is natural and cannot be avoided. To prove this, we argue as follows. If (5.4) is violated, there exists c3 ą 0 such that f (s) ą c3 for any  1{r s P [0, A], where A := 2(1 + c1 ){c2 . Let "

c3 c2 , c4 := min r (1 + A) 2



A 1+A

r * .

We claim that, for any s ě 0, f (s) ě c4 (1 + s)r

(5.21)

5.5 Proof of Theorem 5.3

197

To prove (5.21), we distinguish two cases. If s P [0, A], we have that c4 (1 + s)r ď c4 (1 + A)r ď c3 ď f (s) which proves (5.21) in this case. On the other hand, if s ě A, we have that 1 ď s{A, hence

r s 1+A r r r + s = c4 c4 (1 + s) ď c4 s A A

1+A r r c1 c2 = ´c1 + c1 + c4 s ď ´c1 + r s r + s r A A 2 c1 c2 c2 = ´c1 + s r + s r ď ´c1 + c2 s r ď f (s), 2(1 + c1 ) 2 which completes the proof of (5.21). Now, if u were a non-negative solution of u = f (u) in RN + , by Theorem 5.8, we know that, as x1 Ñ +8, u converges to some v which is a non-negative solution of v = f (v) in the whole of RN . Let now w := 1 + v. By (5.21), we have w + = w = v = f (v) ě c4 (1 + v)r = c4 (w + )r . Accordingly, by Lemma 2 of [28], we have that 0 ě w + = w = 1 + v, that is v ď ´1. Since we knew v to be non-negative, this contradiction proves (5.20).

5.5 Proof of Theorem 5.3 5.5.1 Positivity of Solutions Global solutions of the PDE with the special nonlinearity in (5.8) are always nonnegative, as pointed out by the following result: Lemma 5.2 Let N P N. Let w be a solution of w = w

m ź

(w ´ pj )2 in the whole

j =1

of RN . Then 0 ď w ď pm . Proof The proof is inspired by some arguments in [54]. Let v := ´w. We have v = v

m ź

(v + pj )2

j =1

198

5 Symmetry and Attractors

and so, by Kato’s Inequality (see Lemma A.1 in [28]), m ź

v + ě v

(v + pj )2 χtv ě0u

j =1

ěv

2m+1

χtv ě0u = (v + )2m+1

in the sense of distribution. Therefore, by Lemma 2 of [28], v + ď 0 almost everywhere, that is w ě 0, as desired. Now, let qj := pm ´ pj and w := u ´ pm . Notice that qj ě 0 and that w = ψ(w + pm )(w + pm )

m ź

(w + qj )2 .

j =1

Accordingly, by Kato’s Inequality, w + ě ψ(w + pm )(w + pm )

m ź j =1

(w + qj )χtwě0u

ě w 2m+1 χtwě0u = (w + )2m+1 . Once again, by Lemma 2 of [28], we have that w + ď 0 almost everywhere, that is u ď pm , as desired. \ [

5.5.2 Completion of the Proof of Theorem 5.3 We proceed as in the proof of Theorem 5.2. The only difference here is that the function w involved in the proof of Theorem 5.8 is non-negative, thanks to Lemma 5.2 (recall Remark 5.8). \ [

Chapter 6

Symmetry and Attractors: Arbitrary Dimension

6.1 Introduction Let  be the domain of RN (N ě 2) defined by  =(0, +8) ˆ RN ´2 ˆ (0, +8) =tx = (x1 , x 1 , xN ) P RN | x1 ą 0, x 1 = (x2 , . . . , xN ´1 ) P RN ´2 , xN ą 0u. This chapter is devoted to the study of the large space behavior, that is as x1 Ñ +8, of the nonnegative bounded classical solutions u of the equation $ in , & u + f (u) = 0 1 for all x1 ą 0 and x 1 P RN ´2 , u(x1 , x , 0) = 0 % 1 1 u(0, x , xN ) = u0 (x , xN ) for all x 1 P RN ´2 and xN ą 0,

(6.1)

where the function u0 : RN ´2 ˆ (0, +8) Ñ R+ = [0, +8) is given, continuous and bounded. The solutions u are understood to be bounded, of   class C 2 () and to be continuous on  z t0uˆRN ´2 ˆt0u . From standard elliptic 2,β estimates, they are then automatically of class Cb ([, +8) ˆ RN ´2 ˆ R+ ) for all  ą 0 and β P [0, 1). Here and below, for any closed set F Ă RN and β P [0, 1), we write ) ! 2,β (6.2) Cb (F ) := u : F Ñ R | }u}C 2+β (F ) = sup }u}C 2,β B(x,1)XF  ă 8 , b

x PF

© Springer Nature Switzerland AG 2018 M. Efendiev, Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations, Fields Institute Monographs 36, https://doi.org/10.1007/978-3-319-98407-0_6

199

200

6 Symmetry and Attractors: Arbitrary Dimension

where B(x, 1) means the open Euclidean ball of radius 1 centered at x. Problems sets in the half-space 1 = (0, +8) ˆ RN ´1 will also be considered in this chapter, see (6.11) below. The value of u at x1 = 0 is then given and the goal is to describe, as it was in the previous chapters, the limiting profiles of u as x1 Ñ +8. If the equation were parabolic in the variable x1 , we would then be reduced to characterize the ω-limit set of the initial condition u0 . However, problem (6.1) is an elliptic equation in all variables, including x1 , and the “Cauchy” problem (6.1) with the “initial value” u0 at x1 = 0 is ill-posed. There might indeed be several solutions u with the same value u0 at x1 = 0. Nevertheless, under some assumptions on the nonlinearity f , we will see that the behavior as x1 Ñ +8 of any solution u of (6.1) or of similar problems in the half-space 1 = (0, +8) ˆ RN ´1 is well-defined and unique (roughly speaking, no oscillation occur). In some cases, we will prove that all solutions u converge as x1 Ñ +8 to the same limiting one-dimensional profile, irrespectively of u0 . To do so, we will use two different approaches. The first one is a pure PDE approach based on comparisons with suitable sub-solutions and on Liouville type results. This chapter indeed contains new Liouville type results of independent interest for the solutions of some elliptic equations in half-spaces RN ´1 ˆ (0, +8) with homogeneous Dirichlet boundary conditions, or in the whole space RN (see Sect. 6.2 for more details). The second approach is a dynamical systems’ approach which says that x1 can all the same be viewed as a time variable for a suitably defined dynamical system whose global attractor can be proved to exist and can be characterized. Let us now describe more precisely the types of assumptions we make on the functions f , which are always assumed to be locally Lipschitz-continuous from R+ to R. The first class of functions we consider corresponds to functions f such that $ ’ D μ ą 0, f ą 0 on (0, μ), f ď 0 on [μ, +8), ’ & D 0 ă μ1 ă μ, f is nonincreasing on [μ1 , μ], ” ı ” ı f (s) ’ ’ % either f (0) ą 0 or f (0) = 0 and lim inf ą0 . s s Ñ 0+

(6.3)

Under assumption (6.3) on f , it is immediate to see that there exists a unique solution V P C 2 (R+ ) of the one-dimensional equation " 2 V (ξ ) + f (V (ξ )) = 0 for all ξ ě 0, V (0) = 0 ă V (ξ ) ă μ = V (+8) for all ξ ą 0.

(6.4)

Furthermore, V 1 (ξ ) ą 0 for all ξ ě 0. Under assumption (6.3), the behavior of the nontrivial solutions u of (6.1) as x1 Ñ +8 is uniquely determined, as the following theorem shows.

6.1 Introduction

201

Theorem 6.1 Let N be any integer such that N ě 2 and assume that f satisfies (6.3). Let u be any nonnegative and bounded solution of (6.1), where u0 : RN ´2 ˆ (0, +8) Ñ R+ is any continuous and bounded function such that u0 ı 0 in RN ´2 ˆ (0, +8). Then  lim

R Ñ+8

inf

(R,+8)ˆRN´2 ˆ(R,+8)

 u ěμ

(6.5)

and u(x1 + h, x 1 , xN ) Ñ V (xN ) as h Ñ +8 in Cb ([A, +8) ˆ RN ´2 ˆ [0, B]) (6.6) for all A P R, B ą 0 and β P [0, 1), where V P C 2 (R+ ) is the unique solution of (6.4). 2,β

Notice that property (6.5) means that the non-trivial nonnegative solutions u of (6.1) are separated from 0, irrespectively of u0 , far away from the boundary B. If u0 ď μ, then since f ď 0 on [μ, +8) and u = μ on B+ , where + =  X tu ą μu, it follows that from the maximum principle applied in + (see [13], since RN z+ contains the closure of an infinite open connected cone), that actually u ď μ in + whence + = H and u ď μ in . In this case, it also follows from Theorem 6.1 and standard elliptic estimates that the convergence (6.6) 2,β holds not only locally in xN , but in Cb ([A, +8) ˆ RN ´2 ˆ R+ ) for all A P R and β P [0, 1). However, without the assumption u0 ď μ, it is not clear that this last convergence property holds globally with respect to xN in general. The second class of functions f we consider corresponds to the following assumption: $ & f ě 0 on R+ , % @ z P E, lim inf s Ñz +

f (s) ą 0, s´z

(6.7)

where E = tz P R+ ; f (z) = 0u

(6.8)

denotes the set of zeroes of f . A typical example of such a function f is f (s) = | sin s| for all s ě 0, with E = π N. More generally speaking, under the assumption (6.7), it follows immediately that the set E is at most countable. Furthermore, it is easy to check that, for each z P Ezt0u, there exists a unique solution Vz P C 2 (R+ ) of the one-dimensional equation " 2 Vz (ξ ) + f (Vz (ξ )) = 0 for all ξ ě 0, Vz (0) = 0 ă Vz (ξ ) ă z = Vz (+8) for all ξ ą 0. Furthermore, Vz1 (ξ ) ą 0 for all ξ ě 0.

(6.9)

202

6 Symmetry and Attractors: Arbitrary Dimension

The following theorem states any solution of (6.1) is asymptotically onedimensional as x1 Ñ +8. Theorem 6.2 Let N be any integer such that N ě 2 and assume that f satisfies (6.7). Let u be any nonnegative and bounded solution of (6.1), where u0 : RN ´2 ˆ (0, +8) Ñ R+ is any continuous and bounded function such that u0 ı 0 in RN ´2 ˆ (0, +8). Then there exists R ą 0 such that inf

(R,+8)ˆRN´2 ˆ(R,+8)

uą0

and there exists z P Ezt0u such that u(x1 + h, x 1 , xN ) Ñ Vz (xN ) as h Ñ +8 in Cb ([A, +8) ˆ RN ´2 ˆ R+ ) (6.10) for all A P R and β P [0, 1), where Vz P C 2 (R+ ) is the unique solution of (6.9) with the limit Vz (+8) = z. 2,β

This result shows that any non-trivial bounded solution u of (6.1) converges to a single one-dimensional profile as x1 Ñ +8. More precisely, given u, the real number z defined by (6.10) is unique and, in the proof of Theorem 6.2, the explicit expression of z will be provided. Observe that the asymptotic profile may now depend on the solution u (unlike in Theorem 6.1) but Theorem 6.2 says that the oscillations in the x1 variable are excluded at infinity, for any solution u. The last two results are concerned with the analysis of the asymptotic behavior, as x1 Ñ +8, of the nonnegative bounded classical solutions u of "

u + f (u) = 0 in 1 = (0, +8) ˆ RN ´1 , u(0, x2 , . . . , xN ) = u0 (x2 , . . . , xN ) for all (x2 , . . . , xN ) P RN ´1 ,

(6.11)

in the half-space 1 , where the function u0 : RN ´1 Ñ R+ is given, continuous and bounded. The solutions u of (6.11) are understood to be bounded, of class C 2 (1 ) 2,β and to be continuous on 1 . They are then automatically of class Cb ([, +8) ˆ RN ´1 ) for all  ą 0 and β P [0, 1). Firstly, under the same assumptions (6.7) as in the previous theorem, the behavior as x1 Ñ +8 of any non-trivial solution u of (6.11) is well-defined: Theorem 6.3 Let N be any integer such that N ě 2 and assume that f satisfies (6.7). Let u be any nonnegative and bounded solution of (6.11), where u0 : RN ´1 Ñ R+ is any continuous and bounded function such that u0 ı 0 in RN ´1 . Then there exists z P Ezt0u such that u(x1 + h, x2 , . . . , xN ) Ñ z as h Ñ +8 in Cb ([A, +8) ˆ RN ´1 ) 2,β

for all A P R and β P [0, 1).

(6.12)

6.2 The PDE Approach

203

Notice that the conclusion implies in particular that u is separated from 0 far away from the boundary t0u ˆ RN ´1 of 1 . Furthermore, as in Theorem 6.2, the real number z in (6.12) is uniquely determined by u and its explicit value will be given during the proof. In Theorem 6.3, if instead of (6.7) the function f now satisfies assumption (6.3), then u may not converge in general to a constant as x1 Ñ +8. Furthermore, even if u does converge to a constant as x1 Ñ +8, that constant may not be equal to the real number μ given in (6.3). For instance, if there exists ρ P (μ, +8) such that f (ρ) = 0, then the constant function u = ρ solves (6.1) with u0 = ρ. Therefore, under assumption (6.3), the asymptotic profile of a solution u of problem (6.11) in the half-space 1 depends on u and is even not clearly well-defined in general. The situation is thus very different from Theorem 6.1 about the existence and uniqueness of the asymptotic behavior of the solutions of problem (6.1) in the quarter-space . However, under (6.3) and an additional appropriate assumption on f , the following result holds: Theorem 6.4 Let N be any integer such that N ě 2 and assume that, in addition to (6.3), f is such that lim inf s Ñz ´

f (s) ą 0 for all z ą μ such that f (z) = 0. s´z

(6.13)

Let u be any nonnegative and bounded solution of (6.11), where u0 : RN ´1 Ñ R+ is any continuous and bounded function such that u0 ı 0 in RN ´1 . Then there exists z ě μ such that f (z) = 0 and u(x1 + h, x2 , . . . , xN ) Ñ z as h Ñ +8 in Cb ([A, +8) ˆ RN ´1 ) 2,β

for all A P R and β P [0, 1). Remark 6.1 It is worth noticing that all above results hold in any dimension N ě 2.

6.2 The PDE Approach In this section, we use a pure PDE approach to prove the main results announced in Sect. 6.1. In Sect. 6.2.1, we deal with the case of problem (6.1) set in the quarterspace  = (0, +8) ˆ RN ´2 ˆ (0, +8), while Sect. 6.2.2 is concerned with problem (6.11) set in the half-space 1 = (0, +8) ˆ RN ´1 .

204

6 Symmetry and Attractors: Arbitrary Dimension

6.2.1 Problem (6.1) in the Quarter-Space  = (0, +∞) ˆ RN ´2 ˆ (0, +∞) Let us first begin with the Proof of Theorem 6.1 The proof is divided into three main steps: we first prove that u is bounded from below away from 0 when x1 and xN are large, uniformly with respect to x 1 P RN ´2 . Then, we pass to the limit as x1 Ñ +8 and use a classification result, which finally leads to the uniqueness and one-dimensional symmetry of the limiting profiles of u as x1 Ñ +8. Step 1 It consists in the following result, which we state as a lemma since it will be used several times in the chapter. Notice that only the third line of (6.3) is needed. Lemma 6.1 Let N be any integer such that N ě 2 and assume that either f (0) ą 0, or f (0) = 0 and lim infs Ñ0+ f (s){s ą 0. Let u be any nonnegative and bounded solution of (6.1), where u0 : RN ´2 ˆ(0, +8) Ñ R+ is any continuous and bounded function such that u0 ı 0 in RN ´2 ˆ (0, +8). Then there exist R ą 0 and  ą 0 such that u ě  in [R, +8) ˆ RN ´2 ˆ [R, +8).

(6.14)

Proof First of all, observe that, since f (0) ě 0, the strong maximum principle implies that the function u is either positive in , or identically equal to 0 in . But since u is continuous up to t0u ˆ RN ´2 ˆ (0, +8) and since u0 ı 0, it follows that u ą 0 in . In the sequel, for any x P RN and R ą 0, denote B(x, R) the open Euclidean ball of centre x and radius R. For each R ą 0, let λR be the principal eigenvalue of the Laplace operator in B(0, R) with Dirichlet boundary condition on BB(0, R), and let ϕR be the normalized principal eigenfunction, that is $ ϕR + λR ϕR = 0 in B(0, R), ’ ’ & in B(0, R), ϕR ą 0 ’ }ϕR }L8 (B(0,R)) = ϕR (0) = 1, ’ % on BB(0, R). ϕR = 0

(6.15)

Notice that λR Ñ 0 as R Ñ +8. If f (0) = 0, we can then choose R ą 0 large enough so that λR ă lim inf s Ñ 0+

f (s) . s

If f (0) ą 0, we simply choose R = 1. Then, fix a point x0 P  in such a way that B(x0 , R) Ă . Since u is continuous and positive on B(x0 , R), there holds

6.2 The PDE Approach

205

minB(x0 ,R) u ą 0. Therefore, it follows from the choice of R that there exists  ą 0 small enough, such that the function u(x) =  ϕR (x ´ x0 ) is a subsolution in B(x0 , R), that is u + f (u) ě 0 and u ă u in B(x0 , R).

(6.16)

Next, let xr0 be any point in  such that B(r x0 , R) Ă , x0,1 , xr01 , xr0,N ) with xr0,1 ě R and xr0,N ě R. For all t P [0, 1], call that is xr0 = (r x0 ´ x0 ) yt = x0 + t (r and observe that B(yt , R) Ă  for all t P [0, 1). Define ut (x) = u(x ´ yt + x0 ) =  ϕR (x ´ yt ) for all t P [0, 1] and x P B(yt , R). By continuity and from (6.16), there holds ut ă u in B(yt , R) for t P [0, t0 ], where t0 ą 0 is small enough. On the other hand, for all t P [0, 1], the function ut is a subsolution of the equation satisfied by u, that is ut + f (ut ) ě 0 in B(yt , R). We shall now use a sliding method (see Chap. 1 and [13, 18]) to conclude that ut ă u in B(yt , R) for all t P [0, 1). Indeed, if this were not true, there would then exist a real number t ˚ P (0, 1) such that the inequality ut ˚ ď u holds in B(yt ˚ , R) with equality at some point x ˚ P B(yt ˚ , R). Since B(yt ˚ , R) Ă , u ą 0 in  and ut ˚ = 0 on BB(yt ˚ , R), one has x ˚ P B(yt ˚ , R). But since ut ˚ is a subsolution of the equation satisfied by u, the strong maximum principle yields ut ˚ = u in B(yt ˚ , R) and also on the boundary by continuity, which is impossible. One has then reached a contradiction. Hence, ut ă u in B(yt , R) for all t P [0, 1). By continuity, one also gets that u1 ď u in B(y1 , R). Therefore, for any xr0 = (r x0,1 , xr01 , xr0,N ) with xr0,1 ě R and xr0,N ě R, there holds x0 ) =  ϕR (0) = . u(r x0 ) ě u1 (r In other words, (6.14) holds and the proof of Lemma 6.1 is complete.

l

206

6 Symmetry and Attractors: Arbitrary Dimension

Step 2 Let (x1,n )nPN be any sequence of positive numbers such that x1,n Ñ +8 as n Ñ +8, and let (xn1 )nPN be any sequence in RN ´2 . From standard elliptic estimates, there exists a subsequence such that the functions un (x) = u(x1 + x1,n , x 1 + xn1 , xN ) 2,β

converge in Cloc (RN + ), for all β P [0, 1), to a bounded classical solution u8 of "

u8 + f (u8 ) = 0 in RN +, u8 = 0 on BRN +,

N ´1 ˆ [0, +8). Furthermore, u ě 0 and u ı 0 in RN from Step where RN 8 8 + =R + 1, since

u8 ě  ą 0 in RN ´1 ˆ [R, +8) from Lemma 6.1. Thus, u8 ą 0 in RN ´1 ˆ (0, +8) from the strong maximum principle. It follows from Theorems 1.1 and 1.2 of Berestycki, Caffarelli and Nirenberg [13]1 (see also [6, 37]) that u8 is unique and has one-dimensional symmetry. By uniqueness of the problem (6.4), one gets that u8 (x) = V (xN ) for 1 all x P RN + , and the limit does not depend on the sequences (x1,n )nPN or (xn )nPN . Property (6.6) of Theorem 6.1 then follows from the uniqueness of the limit. Step 3 Let us now prove formula (6.5). One already knows from Lemma 6.1 that   m := lim inf u ě  ą 0. R Ñ+8

(R,+8)ˆRN´2 ˆ(R,+8)

Let (xn )nPN = (x1,n , xn1 , xN,n )nPN be a sequence in  such that (x1,n , xN,n )nPN Ñ (+8, +8) and u(xn ) Ñ m as n Ñ +8. Up to extraction of a subsequence, the functions vn (x) = u(x + xn ) 2 (RN ) to a classical bounded solution v of converge in Cloc 8

v8 + f (v8 ) = 0 in RN such that v8 ě m in RN and v8 (0) = m ą 0. Thus, f (m) ď 0, whence m ě μ due to (6.3). The proof of Theorem 6.1 is thereby complete. l

1 In

[13], the function f was assumed to be globally Lipschitz-continuous. Here, f is just assumed to be locally Lipschitz-continuous. However, since u is bounded, it is always possible to find a Lipschitz-continuous function f˜ : R+ Ñ R satisfying (6.3) and such that f˜ and f coincide on the range of u.

6.2 The PDE Approach

207

The proof of Theorem 6.2 is based on two Liouville type results for the bounded nonnegative solutions u of the elliptic equation u + f (u) = 0 in the whole space RN or in the half-space RN ´1 ˆ R+ with Dirichlet boundary condition on RN ´1 ˆ t0u. Theorem 6.5 Let N be any integer such that N ě 1 and assume that the function f satisfies (6.7). Let u be a bounded nonnegative solution of u + f (u) = 0 in RN .

(6.17)

Then u is constant. The following result is concerned with the one-dimensional symmetry of nonnegative bounded solutions in a half-space with Dirichlet boundary conditions. Theorem 6.6 Let N be any integer such that N ě 1 and assume that the function f satisfies (6.7). Let u be a bounded nonnegative solution of "

N ´1 ˆ R , u + f (u) = 0 in RN + + =R N ´1 ˆ t0u. u = 0 on BRN + =R

(6.18)

Then u is a function of xN only. Furthermore, either u = 0 in RN ´1 ˆ R+ or there exists z ą 0 such that f (z) = 0 and u(x) = Vz (xN ) for all x P RN ´1 ˆ R+ , where the function Vz satisfies equation (6.9). These results are of independent interest and will be proved in Sect. 6.3. Notice that one of the main points is that they hold in any dimensions N ě 1 without any other assumption on u than its boundedness. In low dimensions N ď 4, and under the additional assumption that u is stable, the conclusion of Theorem 6.5 holds for any nonnegative function f of class C 1 (R+ ), see Dupaigne and Farina [43]. Consequently, because of the monotonicity result in the direction xN due to Berestycki, Caffarelli and Nirenberg [13] and Dancer [39] (since f (0) ě 0), it follows that the conclusion of Theorem 6.6 holds for any nonnegative function f of class C 1 (R+ ), provided that N ď 5, see Farina and Valdinoci [60]. However, observe that the nonnegativity and the C 1 character of f are incompatible with (6.7) for any positive zero z of f . Furthermore, assumption (6.7) is crucially used in the proof of Theorem 6.5 and 6.6. It is actually not true that these theorems stay valid in general when f is just assumed to be nonnegative and locally Lipschitzcontinuous. For instance, non-constant solutions of (6.17), which are even stable, exist for power-like nonlinearities f in high dimensions (see [57] and the references therein).

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6 Symmetry and Attractors: Arbitrary Dimension

With these results in hand, let us turn to the Proof of Theorem 6.2 Observe that, from (6.7), either f (0) ą 0, or f (0) = 0 and lim infs Ñ0+ f (s){s ą 0. Therefore, Lemma 6.1 gives the existence of R ą 0 and  ą 0 such that (6.14) holds. Set M = lim

AÑ+8



 u .

sup

(6.19)

[A,+8)ˆRN´2 ˆ[0,+8)

Our goal is to prove that the conclusion of Theorem 6.2 holds with z = M. Since u is bounded and satisfies (6.14), M is such that  ď M ă +8. Furthermore, there exists a sequence (xn )nPN = (x1,n , xn1 , xN,n )nPN of points in  such that x1,n Ñ +8 and u(xn ) Ñ M as n Ñ +8. Assume first, up to extraction of a subsequence, that the sequence (xN,n )nPN converges to a nonnegative real number xN,8 as n Ñ +8. From standard elliptic estimates, the functions un (x) = u(x1 + x1,n , x 1 + xn1 , xN ) converge in Cloc (RN ´1 ˆ R+ ) for all β P [0, 1), up to extraction of another subsequence, to a bounded nonnegative solution u8 of the problem (6.18) in the half-space RN ´1 ˆ [0, +8), such that 2,β

u8 (0, 0, xN,8 ) = M =

sup RN´1 ˆ[0,+8)

u8 ą 0.

It follows from Theorem 6.6 that u8 (x) = Vz (xN ) is a one-dimensional increasing solution of (6.9), whence z = M. But since VM is (strictly) increasing, it cannot reach its maximum M at the finite point xN,8 . This case is then impossible. Therefore, one can assume without loss of generality that xN,n Ñ +8 as n Ñ +8. From standard elliptic estimates, the functions un (x) = u(x + xn ) 2,β

converge in Cloc (RN ) for all β P [0, 1), up to extraction of another subsequence, to a bounded nonegative solution u8 of the problem (6.17) in RN , such that u8 (0) = M = sup u8 ą 0. RN

Theorem 6.5 implies that u8 = M in RN , whence f (M) = 0. Furthermore, since the limit M is unique, the convergence of the functions un to the constant M holds for the whole sequence. Now, in order to complete the proof of Theorem 6.2, we shall make use of the following lemma of independent interest:

6.2 The PDE Approach

209

Lemma 6.2 Let g : R+ Ñ R be a locally Lipschitz-continuous nonnegative function. Then, for each z ą 0 such that g(z) = 0 and for each  P (0, z], there 1 exist R 1 = Rg,z, ą 0 and a classical solution v of $ ’ ’ ’ ’ &

v + g(v) 0 ď v v ’ ’ ’ ’ v v(0) = max % 1

= ă = ě

0 in B(0, R 1 ), z in B(0, R 1 ), 0 on BB(0, R 1 ), z ´ .

(6.20)

B(0,R )

The proof of this lemma is postponed at the end of this section. Let us now finish 1 the proof of Theorem 6.2. Fix an arbitrary  in (0, M]. Let R 1 () = Rf,M, be as in Lemma 6.2 and let v be a solution of (6.20) with g = f and z = M. Since the functions un (x) = u(x + xn ) converge locally uniformly in RN to the constant M and since max

B(0,R 1 ())

v ă M,

there exists n0 P N large enough so that B(xn0 , R 1 ()) Ă  and v(x ´ xn0 ) ă u(x) for all x P B(xn0 , R 1 ()).

(6.21)

But since v solves the same elliptic equation as u, the same sliding method as in Theorem 6.1 implies that u ě v(0) ě M ´  in [R 1 (), +8) ˆ RN ´2 ˆ [R 1 (), +8).

(6.22)

Lastly, choose any sequence (r x1,n )nPN converging to +8, and any sequence (r xn1 )nPN in RN ´2 . Up to extraction of a subsequence, the functions rn (x) = u(x1 + xr1,n , x 1 + xrn1 , xN ) u converge in Cloc (RN ´1 ˆ R+ ) for all β P [0, 1) to a bounded nonnegative solution r8 of problem (6.18) in the half-space RN ´1 ˆ R+ . The function u r8 satisfies u r8 ď M in RN ´1 ˆ R+ by definition of M, while u 2,β

 lim

AÑ+8

inf

RN´1 ˆ[A,+8)

 r8 ě M u

because  ą 0 in (6.22) can be arbitrarily small. Theorem 6.6 and the above estimates imply that r8 (x) = VM (xN ) for all x P RN ´1 ˆ R+ . u

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6 Symmetry and Attractors: Arbitrary Dimension

Since this limit does not depend on any subsequence, and due to (6.19), (6.22) and standard elliptic estimates, it follows in particular that u(x1 + h, x 1 , xN ) Ñ VM (xN ) in Cb ([A, +8) ˆ RN ´2 ˆ R+ ) as h Ñ +8, 2,β

for all A P R and β P [0, 1). The proof of Theorem 6.2 is thereby complete.

l

Remark 6.2 Instead of Theorem 6.5, if f is just assumed to be nonnegative and locally Lipschitz-continuous on R+ and if u8 is a solution of (6.17) which reaches its maximum and is such that f (maxRN u8 ) = 0 (as in some assumptions of [14, 21]), then u8 is constant, from the strong maximum principle. Therefore, it follows from similar arguments as in the above proof that if, instead of (6.7) in Theorem 6.2, the function f is just assumed to be nonnegative, locally Lipschitzcontinuous and positive almost everywhere on R+ and if u is a bounded nonnegative solution of (6.1) such that f (M) = 0, where M is defined by (6.19), then either M = 0 and u(x1 + h, x 1 , xN ) Ñ 0 in C 2 ([A, +8) ˆ RN ´2 ˆ R+ ) as h Ñ +8 for all A P R, or the conclusion (6.10) holds with z = M. Proof of Lemma 6.2 The proof uses classical variational arguments, which we sketch here for the sake of completeness (see also e.g. [24] for applications of this method). Let g and z be as in Lemma 6.2 and let gr be the function defined in R by $ & g(0) if s ă 0, g(s) ˜ = g(s) if 0 ď s ď z, % 0 if s ą z. The function gr is nonnegative, bounded and Lipschitz-continuous on R. Set G(s) =

żz s

gr(τ ) dτ ě 0

for all s P R. The function G is nonnegative and Lipschitz-continuous on R. Let r be any positive real number. Define Ir (v) =

1 2

ż B(0,r)

|∇v|2 +

ż G(v) B(0,r)

for all v P H01 (B(0, r)). The functional Ir is well-defined in H01 (B(0, r)) and it is coercive, from Poincaré’s inequality and the nonnegativity of G. From Rellich’s and Lebesgue’s theorems, the functional Ir has a minimum vr in H01 (B(0, r)). The function vr is a weak and hence, from the elliptic regularity theory, a classical C 2 (B(0, r)) solution of the equation "

vr + gr(vr ) = 0 in B(0, r), . vr = 0 on BB(0, r).

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211

Since gr ě 0 on (´8, 0], it follows from the strong maximum principle that vr ě 0 in B(0, r). Furthermore, either vr = 0 in B(0, r), or vr ą 0 in B(0, r). Similarly, since gr = 0 on [z, +8), one gets that vr ă z in B(0, r). Consequently, gr(vr ) = g(vr ) in B(0, r). It also follows from the method of moving planes and Gidas, Ni and Nirenberg [65] that vr is radially symmetric and decreasing with respect to |x| (provided that vr ı 0 in B(0, r)). In all cases, there holds 0 ď vr (0) = max vr ă z. B(0,r)

In order to complete the proof of Lemma 6.2, it is sufficient to prove that, given  in (0, z], there exists r ą 0 such that vr (0) ě z ´ . Let  P (0, z] and assume that maxB(0,r) vr = vr (0) ă z ´  for all r ą 0. Observe that the function G is nonincreasing in R, and actually decreasing and positive on the interval [0, z), from (6.7). Therefore, Ir (vr ) ě αN r N G(z ´ )

(6.23)

for all r ą 0, where αN ą 0 denotes the Lebesgue measure of the unit Euclidean ball in RN . For r ą 1, let wr be the test function defined in B(0, r) by wr (x) =

"

z if |x| ă r ´ 1, z (r ´ |x|) if r ´ 1 ď |x| ď r.

This function wr belongs to H01 (B(0, r)) and |∇wr |2 and G(wr ) are supported on the shell B(0, r)zB(0, r ´ 1). Thus, there exists a constant C independent of r such that Ir (wr ) ď C (r n ´ (r ´ 1)n )

(6.24)

for all r ą 1. But since Ir (vr ) ď Ir (wr ), by definition of vr , and since G(z´) ą 0, inequalities (6.23) and (6.24) lead to a contradiction as r Ñ +8. Therefore, there exists a radius R 1 ą 0 such that vR 1 (0) ě z ´ , and vR 1 solves (6.20). The proof of Lemma 6.2 is now complete. l

6.2.2 Problem (6.11) in the Half-Space 1 = (0, +∞) ˆ RN ´1 Let us now turn to problem (6.11) set in the half-space 1 = (0, +8) ˆ RN ´1 . This section is devoted to the proof of Theorems 6.3 and 6.4. A useful ingredient in the proof of these two theorems is the following result, which is the analogue of

212

6 Symmetry and Attractors: Arbitrary Dimension

Lemma 6.1 in the case of half-spaces. Its proof is very similar to that of Lemma 6.1 and it is left to the reader. Lemma 6.3 Let N be any integer such that N ě 2 and assume that either f (0) ą 0, or f (0) = 0 and lim infs Ñ0+ f (s){s ą 0. Let u be any nonnegative and bounded solution of (6.11), where u0 : RN ´1 Ñ R+ is any continuous and bounded function such that u0 ı 0 in RN ´1 . Then there exist R ą 0 and  ą 0 such that u ě  in [R, +8) ˆ RN ´1 . Proof of Theorem 6.3 Assume here that f satisfies (6.7). First of all, it follows from Lemma 6.3 that there exists R ą 0 such that inf[R,+8)ˆRN´1 u ą 0. Call M = lim

AÑ+8

 sup [A,+8)ˆRN´1

 u ą 0.

We shall now prove that the conclusion of Theorem 6.3 holds with z = M. Choose a sequence (xn )nPN = (x1,n , . . . , xN,n )nPN in 1 such that x1,n Ñ +8 and u(xn ) Ñ M as n Ñ +8. As in the proof of Theorem 6.2, it follows from Theorem 6.5 that the functions un (x) = u(x + xn ) 2,β

converge in Cloc (RN ) for all β P [0, 1) to the constant M, whence f (M) = 0. 1 Then, for any  P (0, M], let R 1 () = Rf,M, be as in Lemma 6.2 and let v be a solution of (6.20) with g = f and z = M. As in the proof of Theorem 6.2, there exists n0 P N large enough so that B(xn0 , R 1 ()) Ă 1 and (6.21) holds. The sliding method yields u ě v(0) ě M ´  in [R 1 (), +8) ˆ RN ´1 . Since  ą 0 is arbitrarily small, the definition of M implies that, for all A P R, u(x1 + h, x2 , . . . , xN ) Ñ M as h Ñ +8 uniformly with respect to (x1 , x2 , . . . , xN ) P [A, +8) ˆ RN ´1 . The convergence 2,β also holds in Cb ([A, +8) ˆ RN ´1 ) for all β P [0, 1) from standard elliptic estimates. The proof of Theorem 6.3 is thereby complete. l In the case when f satisfies (6.3) and (6.13), the conclusion is similar to that of Theorem 6.3, as the following proof of Theorem 6.4 will show. As a matter of fact, it is also based on a Liouville type result for the bounded nonnegative solutions of (6.17), which is the counterpart of Theorem 6.5 under assumptions (6.3) and (6.13).

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213

Theorem 6.7 Let N be any integer such that N ě 1 and assume that the function f satisfies (6.3) and (6.13). Then any bounded nonnegative solution u of (6.17) is constant. The proof is postponed in Sect. 6.3 and we now complete the Proof of Theorem 6.4 First of all, it follows from Lemma 6.3 that there exist  ą 0 and R ą 0 such that u ě  in [R, +8) ˆ RN ´1 . Call now m = lim



AÑ+8

inf

[A,+8)ˆRN´1

 u .

One has m P [, +8). Let (xn )nPN = (x1,n , . . . , xN,n )nPN be a sequence in 1 such that x1,n Ñ +8 and u(xn ) Ñ m as n Ñ +8. Up to extraction of a subsequence, the functions un (x) = u(x + xn ) 2,β

converge in Cloc (RN ) for all β P [0, 1) to a classical bounded solution u8 of (6.17) such that u8 ě m ą 0 in RN and u8 (0) = m. Theorem 6.7 then implies that u8 is constant in RN , whence it is identically equal to m and f (m) = 0. Call now M 1 = sup u 1

and let g : [0, +8) Ñ R be the function defined by g(s) =

"

´f (M 1 + 1 ´ s) if 0 ď s ď M 1 + 1 ´ m, 0 if s ą M 1 + 1 ´ m.

The function g is Lipschitz-continuous and nonnegative. The real number z = M1 + 1 ´ m is positive and fulfills g(z) = f (m) = 0. Choose any  in (0, z]. From Lemma 6.2, there exist R 1 ą 0 and a classical solution v of (6.20) in B(0, R 1 ), that is $ ’ ’ ’ ’ &

v + g(v) = 0 ď vă v= ’ ’ ’ ’ v ě v(0) = max % 1 B(0,R )

0 in B(0, R 1 ), z in B(0, R 1 ), 0 on BB(0, R 1 ), z ´ .

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6 Symmetry and Attractors: Arbitrary Dimension

The function V = M 1 + 1 ´ v then satisfies $ ’ V + f (V ) = 0 in B(0, R 1 ), ’ ’ ’ & m ă V ď M 1 + 1 in B(0, R 1 ), V = M 1 + 1 on BB(0, R 1 ), ’ ’ ’ ’ V (0) = min V ď m + . % B(0,R 1 )

2,β

Since un (x) = u(x +xn ) Ñ u8 (x) = m as n Ñ +8 in Cloc (RN ) for all β P [0, 1), it follows that there exists n0 P N large enough so that B(xn0 , R 1 ) Ă 1 and V (x ´ xn0 ) ą u(x) for all x P B(xn0 , R 1 ). Since V = sup1 u + 1 ą sup1 u on BB(0, R 1 ), it follows from the elliptic maximum principle and the sliding method that u(x) ď V (0) ď m +  for all x P [R 1 , +8) ˆ RN ´1 . Owing to the definition of m, one concludes that, for all A P R, u(x1 + h, x2 , . . . , xN ) Ñ m as h Ñ +8 uniformly with respect to (x1 , x2 , . . . , xN ) P [A, +8)ˆRN ´1 and the convergence 2,β holds in Cb ([A, +8) ˆ RN ´1 ) for all β P [0, 1) from standard elliptic estimates. The proof of Theorem 6.4 is thereby complete. l

6.3 Classification Results in the Whole Space RN or in the Half-Space RN ´1 ˆ (0, +∞) with Dirichlet Boundary Conditions This section is devoted to the proof of the Liouville type results for the bounded nonnegative solutions u of problems (6.17) or (6.18). Theorems 6.6 and 6.7 are actually corollaries of Theorem 6.5. We then begin with the proof of the latter. Proof of Theorem 6.5 Let u be a bounded nonnegative solution of (6.17) under assumption (6.7). Denote m = inf u ě 0. RN

Since f (m) ě 0, the constant m is a subsolution for (6.17). It follows from the strong elliptic maximum principle that either u = m in RN , or u ą m in RN .

6.3 Classification Results in the Whole Space RN or in the Half-Space. . .

215

Let us prove that the second case, that is u ą m, is impossible. That will give the desired conclusion. Assume that u ą m in RN and let us get a contradiction. Let us first check that f (m) = 0.

(6.25)

This could be done by considering a sequence along which u converges to its minimum; after changing the origin, the limiting function would be identically equal to m from the strong maximum principle, which would yield (6.25). Let us choose an alternate elementary parabolic argument. Assume that f (m) ą 0 and let ξ : [0, T ) Ñ R be the maximal solution of " 1 ξ (t) = f (ξ(t)) for all t P [0, T ), ξ(0) = m. The maximal existence time T satisfies 0 ă T ď +8 (and T = +8 if f is globally Lipschitz-continuous). Since ξ(0) ď u in RN , it follows from the parabolic maximum principle for the equation vt = v + f (v), satisfied by both ξ and u in [0, T ) ˆ RN , that ξ(t) ď u(x) for all x P RN and t P [0, T ). But ξ 1 (0) = f (ξ(0)) = f (m) ą 0. Hence, there exists τ P (0, T ) such that ξ(τ ) ą m, whence u(x) ě ξ(τ ) ą m for all x P RN , which contradicts the definition of m. Therefore, (6.25) holds. Now, as in the proof of Lemma 6.1, because of property (6.7) at z = m, there exist R ą 0 and  ą 0 such that (m +  ϕR ) + f (m +  ϕR ) ě 0 in B(0, R) and m+ ϕR ă u in B(0, R), where ϕR solving (6.15) is the principal eigenfunction of the Dirichlet-Laplace operator in B(0, R). Since m +  ϕR = m on BB(0, R) and u ą m in RN , the same sliding method as in the proof of Lemma 6.1 implies that m +  ϕR (x ´ y) ď u(x) for all x P B(y, R) and for all y P RN . Therefore, u ě m +  ϕR (0) ą m in RN , which contradicts the definition of m. As a conclusion, the assumption u ą m is impossible and, as already emphasized, the proof of Theorem 6.5 is thereby complete. l The proof of Theorem 6.6 also uses the sliding method and Theorem 6.5, combined with limiting arguments as xN Ñ +8 and comparison with non-small subsolutions.

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6 Symmetry and Attractors: Arbitrary Dimension

Proof of Theorem 6.6 Let u be a bounded nonnegative solution of (6.18) under assumption (6.7). Since f (0) ě 0, it follows from the strong maximum principle that either u = 0 in RN ´1 ˆ R+ , or u ą 0 in RN ´1 ˆ (0, +8). Let us then consider the second case. Since f (0) ě 0, it follows from Corollary 1.3 of Berestycki, Caffarelli and Nirenberg [21] that u is increasing in xN . Denote M=

sup

u.

RN´1 ˆ[0,+8)

There exists a sequence (xn )nPN = (x1,n , . . . , xN,n )nPN in RN ´1 ˆ R+ such that xN,n Ñ +8 and u(xn ) Ñ M as n Ñ +8. From standard elliptic estimates, the functions un (x) = u(x + xn ), 2,β

which satisfy the same equation as u, converge in Cloc (RN ) for all β P [0, 1), up to extraction of a subsequence, to a solution u8 of (6.17) such that u8 (0) = M. Theorem 6.5 implies that u8 = M in RN , whence f (M) = 0. It follows then from Berestycki, Caffarelli and Nirenberg [14] (see also Theorem 1.4 in [21]) that u depends on xN only. In other words, the function u(x) is equal to VM (xN ), which completes the proof of Theorem 6.6. l Let us complete this section with the Proof of Theorem 6.7 Assume that the function f satisfies (6.3) and (6.13) and let u be a bounded nonnegative solution of (6.17). As already underlined, it first follows from the strong maximum principle that either u ” 0 in RN , or u ą 0 in RN . Let us then consider the second case. Applying the sliding method and using the same notations as in the proof of Lemma 6.1, one gets the existence of R ą 0 and  ą 0 such that u(x) ě ϕR (x ´ y) for all x P B(y, R) and for all y P RN , whence m = inf u ą 0. RN

Let (xn )nPN be a sequence in RN such that u(xn ) Ñ m as n Ñ +8. Up to extraction of a subsequence, the functions un (x) = u(x + xn ) 2,β

converge in Cloc (RN ) for all β P [0, 1) to a classical bounded solution u8 of (6.17) in RN such that u8 ě m in RN and u8 (0) = m. Hence, f (m) ď 0, whence m ě μ from (6.3) and since m ą 0.

6.4 The Dynamical Systems’ Approach

217

Call now M = sup u. RN

If M = m, then u is constant, which is the desired result. Assume now that M ą m. The function v =M ´u is a nonnegative bounded solution of v + g(v) = 0 in RN , where the function g : [0, +8) Ñ R is defined by g(s) =

"

´f (M ´ s) if 0 ď s ď M ´ m, ´f (m) if s ą M ´ m.

Because of (6.3) and (6.13), the function g is Lipschitz-continuous and fulfills property (6.7). Theorem 6.5 applied to g and v implies that the function v is actually constant. Hence, u is also constant, which actually shows that the assumption M ą m is impossible. As a conclusion, M = m and u is then constant. l

6.4 The Dynamical Systems’ Approach The goal of this section is to apply the dynamical systems’ (shortly DS) approach to study the symmetrization and stabilization (as x1 Ñ +8) properties of the nonnegative solutions (6.1) in  = (0, +8) ˆ RN ´2 ˆ (0, +8) and (6.11) in 1 = (0, +8) ˆ RN ´1 . To this end we apply as aforementioned the DS approach which we developed in Chaps. 2–5. One of the main difficulties which arises in the dynamical study of (6.1) in  or (6.11) in 1 is the fact that the corresponding Cauchy problem is not well posed for (6.1) in  and for (6.11) in 1 , and consequently the straightforward interpretation of (6.1) and (6.11) as an evolution equation leads to semigroups of multivalued maps even in the case of cylindrical domains, see [8]. The usage of multivalued maps can be overcome using the so-called trajectory dynamical

218

6 Symmetry and Attractors: Arbitrary Dimension

approach (see Chaps. 2–5 and [12, 35, 92] and the references therein). Under this approach, one fixes a signed direction l in RN , which will play role of time. Then the space K + of all bounded nonnegative classical solutions of (6.1) in  or (6.11) in 1 (in the sense described in Sect. 6.1) is considered as a trajectory phase space  for the semi-flow (Thl )hPR+ of translations along the direction l defined via 

   h P R+ , u P K + . Thl u (x) = u(x + hl),

As already mentioned in a previous chapter, the trajectory dynamical system  l +  Th , K to be well defined, one needs the domains  and 1 to be invariant with respect to positive translations along the l directions, that is       Thl () Ă  resp. Thl (1 ) Ă 1 , Thl (x) := x + hl, for all h ě 0. In our case, the x1 -axis will play the role of time, that is l = (1, 0, . . . 0, 0). For the sake of simplicity of the notation, we then set 

Thl = Th . To apply the DS approach for our purposes, we apply the following Lemma 6.4 (see below), which also has an independent interest. For that purpose, let us introduce a few more notations. For any locally Lipschitz-continuous function f from R+ to R, such that f (0) ě 0, let Zf be defined by Zf = tz0 P R+ | f (z0 ) = 0 and F (z) ă F (z0 ) for all z P [0, z0 )u,

(6.26)

where F (z) =

żz f (σ )dσ. 0

The set Zf is then a subset of the set E of zeroes of f , defined in (6.8). Lastly, by Rf we denote the set of all bounded, nonnegative solutions V P C 2 (R+ ) of " 2 V (ξ ) + f (V (ξ )) = 0 for all ξ ě 0, V (0) = 0, V ě 0, V is bounded.

(6.27)

Lemma 6.4 Let f be a locally Lipschitz-continuous function from R+ to R, such that f (0) ě 0. Then the set Rf is homeomorphic to Zf and as a consequence is totally disconnected. The proof of Lemma 6.4 is identical to the proof of Proposition 3.2. We omit it.

6.4 The Dynamical Systems’ Approach

219

Below we state the main results of this Sect. 6.4, that is Theorems 6.8–6.11, which are obtained by the dynamical systems’ approach. To this end we define a class of functions K + to which the solutions of (6.1) as well as (6.11) belong to. Namely, a bounded nonnegative solution of (6.1) (resp. (6.11)) is understood to be a solution u of class C 2 () (resp. C 2 (1 )) and continuous on  z t0uˆRN ´2 ˆt0u (resp. on 1 ). The set K + is endowed with the local topology according to the embedding of K + in 2,β 2,β Cloc (ztx1 = 0u) (resp. Cloc (1 ztx1 = 0u)) for all β P [0, 1). Actually, as already 2,β emphasized in Sect. 6.1, all solutions u P K + are automatically in Cb ([, +8) ˆ 2,β RN ´2 ˆ[0, +8)) (resp. Cb ([, +8)ˆRN ´1 )) for all β P [0, 1) and for all  ą 0, 2,β where we refer to (6.2) for the definition of the sets Cb (F ). The first two theorems are concerned with the case of functions f fulfilling the condition (6.3). Theorem 6.8 Let N be any integer such that N ě 2 and let f be a locally Lipschitz-continuous  function  from R+ to R, satisfying (6.3). Then the trajectory dynamical system Th , K + associated to (6.1) possesses a global attractor Atr in 2,β 2,β K + which is bounded in Cb () and then compact in Cloc () for all β P [0, 1). Moreover Atr has the following structure   r Atr = ! K +    + r r = RN where  + , K  is the set of all bounded nonnegative solutions of (6.18) in r = RN  + , and ! denotes the restriction to . Hence, Atr Ă tx ÞÑ 0, x ÞÑ V (xN )u and Atr = tx ÞÑ V (xN )u if f (0) ą 0, where V is the unique solution of (6.4). Lastly, for any bounded nonnegative solution u of (6.1) in , the functions Th u 2,β converge as h Ñ +8 in Cloc () for all β P [0, 1) either to 0 or to x ÞÑ V (xN ), and they do converge to the function x ÞÑ V (xN ) if f (0) ą 0. The next theorem deals with the analysis of the asymptotic behavior as x1 Ñ +8 of the nonnegative bounded classical solutions u P K + of Eq. (6.11) in the halfspace 1 = (0, +8) ˆ RN ´1 . For any M ě 0, we define + = K + X t0 ď u ď Mu. KM

Theorem 6.9 Let N be any integer such that N ě 2 and assume that, in addition to (6.3), the given locally Lipschitz-continuous function f fromR+ to R + satisfies (6.13). Then, for every M ě μ, the trajectory dynamical system Th , KM 2,β associated to (6.11) possesses a global attractor Atr , which is bounded in Cb (1 ) 2,β and then compact in Cloc (1 ) for all β P [0, 1), and satisfies

220

6 Symmetry and Attractors: Arbitrary Dimension +  Ă1   Atr = !1 KM

(6.28)

Ă1  = K +  Ă1  X t0 ď u ď Mu, K +  Ă1  is the set of Ă1 = RN , K +  where  M Ă1 = RN , and ! 1 denotes the all bounded nonnegative solutions of (6.17) in   restriction to 1 . Hence, Atr = tz P [0, M] | f (z) = 0u = E X [0, M]. Lastly, for any bounded nonnegative solution u of (6.11) in 1 such that 0 ď u ď M, 2,β the functions Th u converge as h Ñ +8 in Cloc (1 ) for all β P [0, 1) to some z P E X [0, M] which is uniquely defined by u. The next two theorems, which are concerned with the case of functions f fulfilling the condition (6.7), are based on the new Liouville type Theorems 6.5 and 6.6 which were already stated in Sect. 6.2. Theorem 6.10 Let N be any integer such that N ě 2, let f be any locally Lipschitz-continuous function from R+ to R satisfying (6.7) and assume that, for problem (6.1) in the quarter-space , the set K + is not empty. Then, for every  + associated sufficiently large M ě 0, the trajectory dynamical system Th , KM 2,β to (6.1) possesses a global attractor Atr , which is bounded in Cb () and then 2,β compact in Cloc () for all β P [0, 1) and has the following structure +r  Atr = ! KM

(6.29)

  +r r X t0 ď u ď Mu. Hence, where KM  = K+  Atr = tx ÞÑ Vz (xN ) | z P [0, M], f (z) = 0u. Lastly, for any bounded nonnegative solution u of (6.1) in  such that 0 ď u ď M, 2,β the functions Th u converge as h Ñ +8 in Cloc () for all β P [0, 1) to some function x ÞÑ Vz (xN ), where z P E X [0, M] is uniquely defined by u and E denotes the set of zeroes of the function f . Analogously to Theorem 6.9 we have the following Theorem 6.11 in the case of the half-space 1 = (0, +8) ˆ RN ´1 . Theorem 6.11 Let N be any integer such that N ě 2, let f be any locally Lipschitz-continuous function from R+ to R satisfying (6.7), and assume that, for problem (6.11) in the half-space 1 , the set K + is not empty. Then, for every  + sufficiently large M ě 0, the trajectory dynamical system Th , KM associated 2,β to (6.11) possesses a global attractor Atr , which is bounded in Cb (1 ) and then 2,β compact in Cloc (1 ) for all β P [0, 1), and satisfies (6.28). Lastly, for any bounded nonnegative solution u of (6.11) in 1 such that 0 ď u ď M, the functions Th u

6.4 The Dynamical Systems’ Approach

221

2,β

converge as h Ñ +8 in Cloc (1 ) for all β P [0, 1) to some z P E X [0, M] which is uniquely defined by u. In what follows we prove Theorem 6.8. A proof of Theorems 6.9–6.11 can be done in the same manner as in Theorem 6.8 with some minor modifications (see Remark 6.4). Proof of Theorem 6.8 Let K + be the set of all bounded nonnegative solutions of (6.1) in . Due to the assumptions (6.3), the set K + is not empty (the function x ÞÑ V (xN ), where V is the unique solution of (6.4), belongs to K + ) and due to the translation invariance of (6.1), it follows that Th : K + Ñ K + is well defined for all h ě 0, where (Thu)(x1 , x1 , xN ) := u(x1 + h, x 1 , xN ). To show that Th , K + possesses a global attractor, it suffices to show (see [8] and the references therein) that • for any fixed h ą 0, Th is a continuous map in K + (we recall that K + is endowed 2,β with local topology according to the embedding of K + in Cloc (ztx1 = 0u)) for all β P [0, 1); 2,β • the semi-flow (Th )hě0 possesses a compact attracting (absorbing) set in Cloc (), 2,β which is even bounded in Cb (), for all β P [0, 1). 2,β

Note that, the continuity of Th in Cloc (ztx1 = 0u) is obvious, because the shift operator is continuous in this topology, as well as its restriction to K + . As for the existence of compact attracting (absorbing) set for the semi-flow (Th )hě0 , it follows from the fact that the set of all bounded nonnegative solutions of (6.18) in r = RN Yhě0 T´h () =  + under the assumption (6.3) on f is uniformly bounded. Indeed, as already recalled in Sect. 6.2.1 and according to a result of [13], under the assumption (6.3), any bounded solution of (6.18) in RN + which is positive in RN ´1 ˆ (0, +8) has one-dimensional symmetry, that is u(x1 , x 1 , xN ) = V (xN ) where 0 ď V ă μ is the unique solution of (6.4). On the other hand, since f (0) ě 0, any bounded nonnegative solution of (6.18) is either positive in RN ´1 ˆ(0, +8), + r or identically 0 in RN + , and it cannot be 0 if f (0) ą 0. Thus, the set K  of all N 8 r = R+ is bounded in L (), r namely bounded nonnegative solutions of (6.18) in  sup

r) uPK + (

}u}8 ď μ. 2,β

Then the existence of a compact absorbing set for (Th )hě0 in Cloc (), which is even 2,β bounded in Cb (), for all β P [0, 1) is a consequence of the uniform boundedness r = RN of all solutions in  + and of standard elliptic estimates. Hence the   of +(6.18) semigroup Th , K possesses a global attractor Atr in K + which is bounded in 2,β 2,β Cb () and compact in Cloc () for all β P [0, 1).

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6 Symmetry and Attractors: Arbitrary Dimension

To prove the convergence part of Theorem 6.8, as we will see below, it is r Assuming for a moment that this sufficient to show that Atr = ! K + (). representation is true, we obtain from the previous considerations that Atr Ă tx ÞÑ 0, x ÞÑ V (xN )u,

(6.30)

and Atr is then equal to the singleton tx ÞÑ V (xN )u if f (0) ą 0. Hence, for any 2,β bounded nonnegative solution u of (6.1), since tTh u, h ě 1u is bounded in Cb () 2,β and compact in Cloc () for all β P [0, 1), the ω-limit set ω(u) of u is not empty and it is an invariant and connected subset of Atr . Since Atr is totally disconnected,2 it follows that either ω(u) = tx ÞÑ 0u or ω(u) = tx ÞÑ V (xN )u, the latter being necessarily true if f (0) ą 0. r To complete the proof of Theorem 6.8, it remains to show that Atr = ! K + (). p of (6.18) p P Atr for any bounded nonnegative solution u First we prove that ! u r Indeed, for such a u r = RN p P K + (). p, the family (! (T´h u p))hě0 in  + , that is u 2,β 2,β is uniformly bounded in Cb () and compact in Cloc () for all β P [0, 1) and, according to definition of the attractor, there holds 2,β p) ÝÑ Atr in Cloc Th ! (T´h u () as h Ñ +8,

p) = ! u p. Hence ! u p P Atr .  for all β P [0, 1). On the other hand, Th ! (T´h u Next we prove the reverse inclusion. To this end, let us recall that Th , K + 2,β possesses an absorbing set which is bounded in Cb () and then compact in 2,β Cloc () for all β P [0, 1), say B˚ Ă K + , and, as a consequence, Atr = ω(B˚ ) =

č hě 0

«

ď

s ěh

ff Ts B ˚ ,

2,β

where [ ] means the closure in Cloc () (see [8, 35] and the references therein). Let now u P Atr . The property Atr = ω(B˚ ) implies that there exist an increasing sequence (hk )k PN Ñ +8 and a sequence of solutions (uk )k PN in B˚ , such that u = lim Thk uk k Ñ+8

(6.31)

2,β

in Cloc () for all β P [0, 1). Note that the solution Thk uk is defined not only in , but also in the domain (´hk , +8) ˆ RN ´2 ˆ R+ , and that

2 This

property is obvious here due to (6.30). See Remark 6.4 for a comment about the other situations, corresponding to Theorems 6.9, 6.10 and 6.11.

6.4 The Dynamical Systems’ Approach

223

sup }Thk uk }C 2,β [´h +,8)ˆRN´2 ˆR ă +8 +) k b ( k PN

(6.32)

for all  ą 0 and β P [0, 1), from standard elliptic estimates. Consequently, for every k0 P N and β P [0, 1), the sequence (Thk uk )k ąk0 is precompact in  2,β  Cloc [´hk0 , 8) ˆ RN ´2 ˆ R+ . Taking a subsequence, if necessary, and using Cantor’s diagonal procedure and the fact hk Ñ 8, we can say that this sequence  2,β 2,β  N ´2 ˆ R p P Cloc converges to u (RN + for every + ) in the spaces Cloc [´hk0 , 8) ˆ R p P Cb2,β (RN k0 P N and for every β P [0, 1). Then (6.32) implies that u + ) for every β P [0, 1). Lastly, the functions Thk uk are nonnegative solutions of (6.1) in p is a (´hk , +8) ˆ RN ´2 ˆ R+ and by letting k Ñ +8, we easily obtain that u r = RN bounded nonnegative solution of (6.18) in  . Finally, formula (6.31) implies + that p = u. ! u r and the representation formula Atr = ! K + () r is proved. Thus u P ! K + ()  The proof of Theorem 6.8 is thereby complete. l Remark 6.3 As far as Theorems 6.1 and 6.2 on the one hand, and Theorems 6.8 and 6.10 on the other hand, are concerned, we especially emphasize that the DS approach simplified in a very elegant way most of the computations regarding the asymptotic behavior of the solutions of (6.1) as x1 Ñ +8. However, the DS approach and Theorem 6.8 (resp. Theorem 6.10) do not provide as in PDE approach the fact that only the limiting profile V (xN ) (resp. Vz (xN ) for some z P Ezt0u) is selected, even if f (0) = 0, as soon as u0 ı 0 on t0u ˆ RN ´2 ˆ (0, +8). In the PDE proof of Theorems 6.1 and 6.2, it is indeed shown that the condition u0 ı 0 implies that u is separated from 0 for large enough x1 and xN . This property is not shown in the DS proof of Theorems 6.8 and 6.10. Similar comments also hold for Theorems 6.3, 6.4, 6.9 and 6.11, where the PDE proof provides the convergence to a non-zero zero of f , what the DS proof does not. Lastly, in some of the results obtained through the PDE approach, the convergence of the solutions as x1 Ñ +8 is proved to be uniform with respect to the variables (x 1 , xN ), while the DS approach only provides local convergence, due to the necessity of using the local topology to get the existence of a global attractor. Remark 6.4 In Theorem 6.10 (resp. Theorem 6.11) under assumption (6.7), the + phase space KM , which is invariant under the semigroup Th , is not empty for any sufficiently large M, because K + is assumed to be not empty (one can take M as any nonnegative real number such that M ě }U }8 , where U is any fixed element in K + ). Then, in the same manner as in the proof of Theorem 6.8, using + r + both the representation formula Atr = ! KM () (resp. Atr = !1 KM (RN )), the Liouville theorems of Sect. 6.2 and Lemma 6.4, one obtains the desired conclusions. In particular, for Theorem 6.10 (resp. Theorem 6.11) about problem (6.1) in 

224

6 Symmetry and Attractors: Arbitrary Dimension

(resp. (6.11) in 1 ), the total disconnectedness of Atr follows from the representation formula (6.29) (resp. (6.28)), from Theorem 6.6 (resp. Theorem 6.5) and from Lemma 6.4 with, here, Zf = E (resp. condition (6.7) again). Note that for Theorem 6.9 in the case of assumptions (6.3) and (6.13), the total disconnectedness of Atr follows from Theorem 6.7 and assumption (6.13) again. Remark 6.5 Note that neither the concrete choice of the domain  (or 1 ) nor the concrete choice of the “time” direction x1 are essential for the use of the trajectory dynamical system’ approach. Indeed, let us replace the “time” direction x1 by any  fixed direction l P RN and correspondingly Thl u = u(¨ + hl ) for h P R+ and u P + K . Then the above construction seems to be applicable if the domain  satisfies the following assumptions: • Th  Ă  (this is necessary in order to define the restriction of Th to the trajectory + phase space K + or KM ). N • Yhě0 T´h  = R+ , or RN (this is required in order to obtain representation + N formulas of the type Atr = ! K + (RN + ) or Atr = ! K (R ), with possibly + + KM instead of K ).

Chapter 7

The Case of p-Laplacian Operator

7.1 Introduction We are interested in quasilinear elliptic problems over a half-space of the form "

p u + f (u) = 0 in R+ ˆ RN ´1 , u(0, x2 , . . . , xN ) = u0 (x2 , . . . , xN ),

(7.1)

and similar problems over a quarter-space $ & p u + f (u) = 0 in R+ ˆ RN ´2 ˆ R+ , u(0, x2 , . . . , xN ) = u0 (x2 , . . . , xN ), % u(x1 , x2 , . . . , xN ´1 , 0) = 0.

(7.2)

Here u0 ě 0 may be regarded as a given bounded continuous function, R+ = [0, +8), and p u = div(|∇u|p´2 ∇u) is the usual p-Laplacian operator, and we always assume that p ą 1. If u ı 0 is a bounded nonnegative solution to either (7.1) or (7.2), we want to understand the behavior of u(x1 , x2 , . . . , xN ) as x1 Ñ 8. For some general classes of nonlinearities f , we show that, in the half-space case, limx1 Ñ8 u(x1 , x2 , . . . , xN ) always exists and is a positive zero of f ; and in the quarter-space case, lim u(x1 , x2 , . . . , xN ) = V (xN ),

x1 Ñ8

where V is a solution of the one-dimensional problem p V + f (V ) = 0 in R+ , V (0) = 0, V (t) ą 0 for t ą 0, V (+8) = z,

© Springer Nature Switzerland AG 2018 M. Efendiev, Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations, Fields Institute Monographs 36, https://doi.org/10.1007/978-3-319-98407-0_7

(7.3)

225

226

7 The Case of p-Laplacian Operator

where z is a positive zero of f . These features of (7.1) and (7.2) were previously investigated and shown in Chap. 6 (see also [51]) for the special case p = 2. The nonlinearities f considered in Chap. 6 are mainly of two types. The first type consists of functions which are nonnegative (see (6.7) in Chap. 6 for details), and the second type are sign-changing functions satisfying f (s) ą 0 in (0, a) and f (s) ď 0 in (a, +8) for some constant a ą 0 (see (6.2) in Chap. 6). All the functions considered in Chap. 6 are locally Lipschitz continuous. Among other techniques, the arguments in Chap. 6 rely on various forms of the comparison principles, and in particular, the strong comparison principle plays a key role. These rely crucially on the assumption that p = 2. In this chapter, we show that most of the results of Chap. 6 continue to hold for the corresponding general p-Laplacian problems. Since there is no strong comparison principle in general for the p-Laplacian problem when p ­= 2, we have to take rather different approaches in many key steps. A crucial ingredient here is a simple weak sweeping principle, which is a consequence of the weak comparison principle for the p-Laplacian, and is a variant of Serrin’s famous sweeping principle for the Laplacian equations. We will show that in many situations, it is possible to use the weak sweeping principle to replace the moving plane or sliding method, which are based on the strong comparison principle for the Laplacian case and frequently used in Chap. 6. The weak sweeping principle was used, for example, in [38, 41] and [42] for related problems. The techniques in these papers are further developed here to treat (7.1) and (7.2). As in Chap. 6, to obtain a good understanding of the asymptotic behavior of the solutions to (7.1) and (7.2), we need a thorough classification of the solutions of (7.3), and also some Liouville type results over the entire RN . These preparations will be done in Sect. 7.2. The results in this section are mostly of interests on their own. The Liouville theorem here (see Theorem 7.5) improves the corresponding result of Chap. 6 (see Remark 7.3 for details) when p = 2. For convenience of the reader, we also present a version of the weak sweeping principle in this section in a form that is easy to use. In Sect. 7.3, we make use of the results in Sect. 7.2 to study (7.1) and (7.2). The main results for the half-space problem are Theorems 7.6 and 7.7, which imply, respectively, Theorems 6.3 and 6.4 of Chap. 6 in the special case p = 2, except that we require the extra condition that the zeros of f are isolated, but on the other hand, our conditions on f near its zeros are less restrictive then those in Chap. 6 due to our better Liouville theorem. Our main results for the quarter-space problem are Theorems 7.10 and 7.11. Our Theorem 7.11 extends Theorem 6.1 of Chap. 6. In proving these results, apart from the techniques developed for treating the halfspace problems, we also need two one-dimensional symmetry results for half-space problems of the form p u + f (u) = 0 in RN ´1 ˆ R+ , u = 0 on txN = 0u.

(7.4)

For sign-changing nonlinearities considered in Theorem 7.11, the one-dimensional symmetry of positive solutions of (7.4) follows from a modification of the arguments used in [42] (see Theorem 7.9 below). However, for nonnegative nonlinearities considered in Theorem 7.10, we are able to prove such symmetry only under

7.2 Some Basic Results

227

the extra assumption that BxN u ě 0 (see Theorem 7.8 and Remark 7.4). As a consequence our Theorem 7.10 also requires the solution u to satisfy BxN u ě 0. Such a monotonicity condition for u is not needed in Chap. 6 for the case p = 2. We believe that here this monotonicity condition for u is also unnecessary. If the half-space tx P RN : x1 ą 0u in (7.1) is replaced by an unbounded Lipschitz domain of the form tx P RN : x1 ą φ(x2 , . . . , xN )u as in [13], then our techniques can be extended to obtain analogous results, except for those on one-dimensional symmetry in Sect. 7.2. Similarly, our results on (7.2) can be extended when the quarter-space tx P RN : x1 ą 0, xN ą 0u is replaced by tx P RN : x1 ą φ(x2 , . . . , xN ), xN ą 0u, with φ a Lipschitz map. We note that for this case, the limiting problem as x1 Ñ 8 is again a half space problem over tx P RN : xN ą 0u.

7.2 Some Basic Results In this section, we present some basic results which will be needed in our investigation of the half- and quarter-space problems. Most of the results here are also of independent interests.

7.2.1 The Weak Sweeping Principle Several key steps in our arguments are based on a weak sweeping principle for p-Laplacian equations, which is a variant of Serrin’s sweeping principle for the Laplacian case (see [94]). Such a weak sweeping principle was used before in [38, 41] and [42]. Here we state it in a form that is convenient to use; its proof is almost identical to that of Lemma 2.7 in [38]. Proposition 7.1 Suppose that D is a bounded smooth domain in RN , h(x, s) is measurable in x P D, continuous in s, and for each finite interval J , there exists a continuous increasing function L(s) such that h(x, s) + L(s) is nondecreasing in s for s P J and x P D. Let ut and vt , t P [t1 , t2 ], be functions in W 1,p (D) X C(D) and satisfy in the weak sense, ´p ut ě h(x, ut ) + 1 (t), ´p vt ď h(x, vt ) ´ 2 (t) in D, @t P [t1 , t2 ], ut ě vt +  on BD, @t P [t1 , t2 ], where 1 (t) + 2 (t) ě  ą 0.

228

7 The Case of p-Laplacian Operator

Moreover, suppose that ut0 ě vt0 in D for some t0 P [t1 , t2 ] and t Ñ ut , t Ñ vt are continuous from the finite closed interval [t1 , t2 ] to C(D). Then ut ě vt on D, @t P [t1 , t2 ].

7.2.2 Classification of One-Dimensional Solutions In this sub-section, we classify all the solutions of the one dimensional p-Laplacian problem of the following form: p V + f (V ) = 0 in R+ , V (0) = 0, V ě, ı 0, }V }8 ă +8.

(7.5)

We always assume that "

f : R+ Ñ R is continuous, f (0) ě 0, and it is locally Lipschitz continuous except possibly at its zeros.

(7.6)

By a solution of (7.5) we mean a function V P C 1 (R+ ) satisfying (7.5) in the weak sense; that is V (0) = 0 and

ż8 0

1 p ´2

|V |

1 1

V φ dx =

ż8 0

f (V )φdx for all φ P C01 (0, 8).

Theorem 7.1 Suppose that f satisfies (7.6), and that whenever z P Zf := tz P R+ : f (z) = 0u, we have lim inf s Œz

f (s) f (s) ą ´8, lim sup ă +8. (s ´ z)p´1 (z ´ s)p´1 s Õz

(7.7)

Then each solution V of (7.5) has the following properties: (i) V 1 (t) ą 0 @t P R+ (and hence V P C 2 (0, 8)), (ii) z0 := V (+8) P Zf+ := Zf zt0u,

1 1 p (iii) F (z0 ) = p´ p V (0) ą 0, F (z) ă F (z0 ) @z P [0, z0 ), şz where F (z) = 0 f (s)ds.

Proof Let V P C 1 (R+ ) be a solution to (7.5). Due to f (0) ě 0 and (7.7), and }V }8 ă 8, we have ´p V = f (V ) ě cV p´1 for some constant c ą ´8. Hence we can use the strong maximum principle (see Theorem 5 in [102]) to conclude that V (t) ą 0 in (0, +8) and V 1 (0) ą 0. Thus V 1 (t) ą 0 for all small positive t. It follows that either V 1 (t) ą 0 for all t ą 0, or there is a first t0 ą 0 such that V 1 (t) ą 0 in (0, t0 ) and V 1 (t0 ) = 0. If the latter happens, we define

7.2 Some Basic Results

229

V˜ (t) :=

"

V (t), t P [0, t0 ], V (2t0 ´ t), t P (t0 , 2t0 ].

Then it is easily checked that V˜ P C 1 satisfies the following in the weak sense: p V˜ + f (V˜ ) = 0 in [0, 2t0 ], V˜ (0) = V (0), V˜ 1 (0) = V 1 (0). We will show in a moment that V˜ ” V on [0, 2t0 ], which implies V 1 (2t0 ) = ´V 1 (0) ă 0 and hence V must change sign as t increases across 2t0 , a contradiction to the assumption that V is nonnegative in R+ . This proves (i) provided we can show V˜ ” V on [0, 2t0 ]. We now set to show V˜ ” V . We first observe that g(V (t0 )) ­= 0, for otherwise with z0 := V (t0 ) we obtain from (7.7) that f (V (t)) ď c(z0 ´ V (t))p´1 for some constant c and all t ă t0 and close to t0 . Hence W := z0 ´ V satisfies ´p W = ´f (V ) ě ´cW p´1 and W ą 0 for all such t. We can now apply Theorem 5 in [102] again to conclude that W 1 (t0 ) ă 0, i.e., V 1 (t0 ) ą 0, a contradiction to V 1 (t0 ) = 0. We further notice that actually f (V (t0 )) ą 0 must hold. Indeed, if f (V (t0 )) ă 0, then from the equation we obtain V 2 (t) ą 0 for all t ă t0 and close to t0 (note that V is C 2 in (0, t0 ) since V 1 (t) ą 0 in this interval). It follows that V 1 (t0 ) ą V 1 (t) ą 0 for t ă t0 and close to t0 . This contradiction shows that f (V (t0 )) ą 0. We show that V 1 (t) ď 0 for all t ą t0 and close to t0 . Otherwise we can find tn decreasing to t0 as n Ñ 8 such that V 1 (tn ) ą 0. Fix a large n so that f (V (t)) ą 0 for t P [t0 , tn ]. By continuity we can find t n P [t0 , tn ) satisfying V 1 (t) ą 0 in (t n , tn ], V 1 (t n ) = 0. As before from standard elliptic regularity we know that V is C 2 in (t n , tn ], and it satisfies in this interval (p ´ 1)|V 1 (t)|p´2 V 2 (t) = ´f (V (t)) ă 0. Thus V 2 (t) ă 0 for t P (t n , tn ], which implies V 1 (t n ) ą V 1 (tn ) ą 0, a contradiction. This proves what we wanted. We claim that the above conclusion can be strengthened to V 1 (t) ă 0 for all t ą t0 and close to t0 . If V 1 (t) ” 0 in a small right neighborhood of t0 , then from the equation we deduce f (V (t)) ” 0 in this interval, a contradiction to f (V (t0 )) ą 0. Therefore if the above claim is false then we can find a sequence tn decreasing to t0 as n Ñ 8 such that V 1 (tn ) = 0, and another sequence sn decreasing to t0 such that V 1 (sn ) ă 0. Fix n large so that f (V (t)) ą 0 for t P [t0 , tn ]. Then choose sm P (0, tn ). We can now find s m P (sm , tn ] such that V 1 (t) ă 0 in [sm , s m ), V 1 (s m ) = 0.

230

7 The Case of p-Laplacian Operator

Similar to above, we deduce V 2 (t) ă 0 for t P [sm , s m ) and hence V 1 (s m ) ă V 1 (sm ) ă 0. This contradiction proves our claim. We are now ready to show V˜ ” V . Since V is C 2 in (0, t0 ), we have

p´1 1 p p´1 1 |V (t)| + F (V (t)) ” constant = |V (0)|p for t P [0, t0 ]. p p (7.8) Taking t = t0 we deduce F (z0 ) =

p´1 1 |V (0)|p ą 0. p

Thus for t P (0, t0 ] we have F (V (t)) ă F (z0 ) which implies F (z) ă F (z0 ) for z P (0, z0 ). Moreover, it follows from (7.8) that for t P (0, t0 ),

V 1 (t) [F (z0 ) ´ F (V (t))]1{p

= C0 :=

p p´1

1{p .

Integrating over (0, t) we obtain ż V (t)

[F (z0 ) ´ F (z)]´1{p dz = C0 t, t P (0, t0 ).

(7.9)

0

Since F (z0 ) ´ F (z) = f (z0 )(z0 ´ z) + o(|z0 ´ z|) near z = z0 , and f (z0 ) ą 0, we find that M0 :=

ż z0

[F (z0 ) ´ F (z)]´1{p dz ă 8.

0

This implies that t0 = M0 {C0 and V (t) for t P [0, t0 ] is uniquely determined by (7.9). Let (t0 , T ) be the largest interval in which V 1 (t) ă 0. Then (7.8) holds in [0, T ] and we deduce ´V 1 (t) [F (z0 ) ´ F (V (t))]1{p

= C0 , t P (t0 , T ).

Integrating over (t0 , t) we obtain ż z0

[F (z0 ) ´ F (z)]´1{p dz = C0 (t ´ t0 ).

V (t)

It follows from the definition of M0 that ż V (t) 0

[F (z0 ) ´ F (z)]´1{p dz = M0 ´

ż z0 V (t)

[F (z0 ) ´ F (z)]´1{p dz = C0 (2t0 ´ t),

7.2 Some Basic Results

231

or equivalently, ż V (2t0 ´t)

[F (z0 ) ´ F (z)]´1{p dz = C0 t, t P (2t0 ´ T , t0 ).

0

Comparing this with (7.9) we immediately obtain T ě 2t0 and V (2t0 ´ t) ” V (t) in [0, t0 ]. This completes the proof of (i). Next we prove (ii) and (iii). Since now we know that V is increasing and bounded, z0 := limt Ñ+8 V (t) is well-defined. It follows from elementary consideration and (7.5) that f (z0 ) = 0. Letting t Ñ +8 in (7.8) we deduce F (z0 ) =

p´1 1 p V (0) . p

Hence p´1 1 p |V (t)| = F (z0 ) ´ F (V (t)). p For any z P (0, z0 ), there is a unique t0 ą 0 such that V (t0 ) = z, and thus F (z0 ) ´ F (z) =

p´1 1 |V (t0 )|p ą 0. p l

Remark 7.1 Let us note that functions satisfying (7.7) need not be locally Lipschitz at z P Zf . On the other hand, locally Lipschitz continuous functions satisfy (7.7) automatically when p P (1, 2], but when p ą 2, locally Lipschitz continuous functions may or may not satisfy (7.7). Theorem 7.2 Let f be as in Theorem 7.1, and define Zf˚ := tz0 P Zf+ : F (z) ă F (z0 ) @z P [0, z0 )u. Then for every z0 P Zf˚ , (7.5) has a unique solution satisfying V (+8) = z0 . Proof Let z0 P Zf˚ . Then from (7.7) we easily deduce that F (z0 ) ´ F (z) ď C(z0 ´ z)p for some constant C and all z ă z0 close to z0 . This implies that ż z0 0

[F (z0 ) ´ F (z)]´1{p dz = +8.

232

7 The Case of p-Laplacian Operator

It follows that the formula ż V (t)

[F (z0 ) ´ F (z)]´1{p dz = C0 t, C0 =



0

p p´1

1{p ,

uniquely defines a function V (t), t P R+ ; moreover V (t) is increasing, V (+8) = z0 , and it satisfies 1

p V + f (V ) = 0 in (0, 8), V (0) = 0, V (0) =



p F (z0 ) p´1

1{p .

If V˜ is any solution of (7.5) satisfying V˜ (+8) = z0 , then by Theorem 7.1 we have 1 ˜1 p V˜ 1 (t) ą 0 for all t ą 0, and F (z0 ) = p´ p V (0) . Thus making use of (7.8) we see that V˜ (t) is determined by the same integral formula used for V (t) above. It follows that V˜ ” V . This proves the uniqueness. l From the above two theorems, we immediately obtain Corollary 7.1 Under the assumptions on f in Theorem 7.1, the set Zf˚ and the solutions of (7.5) are in one-to-one correspondence. Next we focus on two types of nonlinearities f , namely (F1 ) and (F2 ) defined below. (F1 ): We say that f is of type (F1 ), if it satisfies (7.6), is nonnegative on R+ , and not identically zero in any open interval of R+ . (F2 ): We say that f is of type (F2 ) if it satisfies (7.6), f (u) ą 0 in (0, a), f (u) ď 0 in (a, +8), and is not identically zero in any open interval of R+ . Note that in the following two theorems, condition (7.7) is not needed. Theorem 7.3 Suppose that f is of type (F1 ), and V is a solution of (7.5). Then V (t) ą 0 in (0, +8), it is nondecreasing, and has the properties in (ii) and (iii) of Theorem 7.1. Proof Since f (V ) ě 0, we can apply the strong maximum principle to conclude that V (t) ą 0 in (0, +8) and V 1 (0) ą 0. Let [0, t0 ) be the largest interval such that V 1 (t) ą 0 for t P [0, t0 ). If t0 = 8, then V is C 2 in [0, 8) and the conclusions follow easily as before. So suppose now t0 ă 8. Then clearly V 1 (t0 ) = 0. We observe that if V 1 (t1 ) = 0 for some t1 ą t0 then V 1 (t) ” 0 on [t0 , t1 ]. Indeed, if this is not the case, say V 1 (s) ­= 0 for some s P (t0 , t1 ), then we can find a largest interval (s0 , s1 ) Ă (t0 , t1 ) such that V 1 (t) ­= 0 in (s0 , s1 ) and V 1 (s0 ) = V 1 (s1 ) = 0. It follows that V (s0 ) ­= V (s1 ). Since V is C 2 on (s0 , s1 ), we can apply (7.8) to deduce F (V (s0 )) = F (V (s1 )), which is a contradiction to V (s0 ) ­= V (s1 ). Thus we have either V 1 (t) ” 0 for t ě t0 or there exists t1 ě t0 such that V 1 (t) = 0 in [t0 , t1 ] and V 1 (t) ­= 0 for t ą t1 . In the former case, the conclusions of the theorem follow readily. We show next that the latter case cannot happen. Otherwise,

7.2 Some Basic Results

233

V (8) = limt Ñ8 V (t) exists and V (8) ­= V (t1 ). Since V is C 2 over (t1 , 8), we can apply (7.8) to deduce F (V (8)) = F (V (t1 )), a contradiction. l Theorem 7.4 Suppose that f is of type (F2 ), and V is a solution of (7.5). Then V (t) ą 0 in (0, +8), it is nondecreasing, and V (+8) = a. Such V exists and is unique. Proof Let V be a solution of (7.5). Since f (s) ą 0 in (0, a), by the strong maximum principle we find that V (t) ą 0 in (0, +8) and V 1 (0) ą 0. Hence V 1 (t) ą 0 for all small positive t. We have two possibilities: (i) V 1 (t) ą 0 for all t ą 0, (ii) there is a first t0 ą 0 such that V 1 (t) ą 0 in [0, t0 ) and V 1 (t0 ) = 0. In case (i), we denote z0 := V (+8) and obtain, as before, f (z0 ) = 0, F (z0 ) =

p´1 1 p V (0) , F (V (t)) ă F (z0 ) @t ě 0. p

Thus z0 ě a. But if z0 ą a, then there is a unique t0 ą 0 such that V (t0 ) = a and V (t) ą a for t ą t0 . Thus we obtain from (7.8) şz that F (a) = F (V (t0 )) ă F (z0 ), which is impossible since F (z0 ) ´ F (a) = a0 f (s)ds ă 0 by our assumption on f . So in case (i), we have V (+8) = a. Let us now consider case (ii). If z1 := V (t0 ) ą a, then we can find a unique t1 P (0, t0 ) such that V (t1 ) ş= a and so, z by (7.8) we obtain F (a) = F (V (t1 )) ă F (z1 ), which is impossible as a1 f (s)ds ă 0. So we must have V (t0 ) ď a. If V (t0 ) ă a, then there is a maximal interval I = (t0 , t1 ) in (t0 , 8) such that V (t) ă a in I . We must have V 1 (t) ­= 0 in I , since if V 1 (s0 ) = 0 for some s0 P I then we can argue as in the proof of Theorem 7.3 above to deduce V 1 (t) ” 0 in (t0 , s0 ), which implies f (V (t0 )) = 0, a contradiction to the assumption that f (u) ą 0 in (0, a). If V 1 (t) ą 0 in I then for any fixed t P I , a ą V (t) ą V (t0 ) which implies F (V (t)) ą F (V (t0 )). On the other hand, since 1 1 p V is C 2 in I , we can apply (7.8) to deduce F (V (t0 )) = F (V (t)) + p´ p |V (t)| ą 1 F (V (t)). This contradiction shows that we must have V (t) ă 0 in I (note that we already know V 1 (t) ­= 0 in I ). Thus I = (t0 , 8) and V (8) exists and satisfies V (8) ă V (t0 ) ă a. We may now apply (7.8) to obtain F (V (8)) = F (V (t0 )), şV (t0 ) which is impossible since V (8 ) f (t)dt ą 0. This proves that V (t0 ) ă a cannot occur. Thus we necessarily have V (t0 ) = a. We show next that V (t) ď a for t ě t0 . Otherwise there exists t1 ą a such that V (t1 ) ą a and V 1 (t1 ) ą 0. Let [t1 , t2 ) be the largest interval in [t1 , 8) such that V 1 (t) ą 0 in [t1 , t2 ). Then either t2 = 8 and V 1 (8) = limt Ñ8 V 1 (t) = 0, or t2 ă 8 and V 1 (t2 ) = 0. In either case we have a ă V (t1 ) ă V (t2 ) and V 1 (t2 ) = 0. Applying (7.8) over [t1 , t2 ) we deduce şV (t ) F (V (t1 )) = F (V (t2 )), which is impossible since V (t12) f (t)dt ă 0. This proves that V (t) ď a for t ě t0 . We are now ready to prove that V (t) ” a for t ě t0 . If this is not true, then there exists t1 ą a such that V (t1 ) ă a and V 1 (t1 ) ă 0. Similar to the argument in the last paragraph, we let [t1 , t2 ) be the largest interval in [t1 , 8) such that V 1 (t) ă 0 in [t1 , t2 ). Then either t2 = 8 and V 1 (8) = limt Ñ8 V 1 (t) = 0, or t2 ă 8 and V 1 (t2 ) = 0. In either case we have a ą V (t1 ) ą V (t2 ) and V 1 (t2 ) = 0. Applying (7.8) over [t1 , t2 ) we deduce F (V (t1 )) = F (V (t2 )), which is impossible

234

7 The Case of p-Laplacian Operator

şV (t ) since V (t21) f (t)dt ą 0. This proves that V (t) ” a for t ě t0 . The existence and uniqueness of the solution of (7.5) with the above properties follows from (7.8), which shows that V (t) is uniquely determined by the formula ż V (t)

[F (a) ´ F (z)]´1{p dz = C0 t for t ą 0 such that V (t) ă a.

0

Note that this formula indicates that case (i) happens if +8, and case (ii) happens when this integral is finite.

şa

0 [F (a)

´ F (z)]´1{p dz = l

7.2.3 A Liouville Type Result Let us denote σN,p := (p ´ 1) NN ´p when N ą p, and for N ď p, we assume that σN,p stands for an arbitrary number in [1, +8). As in [41], we say that f is quasi-monotone in [0, 8) if for any bounded interval [s1 , s2 ] Ă [0, 8), there exists a continuous increasing function L(s) such that f (s) + L(s) is non-decreasing in [s1 , s2 ]. Theorem 7.5 Suppose that f is of type (F1 ), is quasi-monotone, and for each z P Zf , lim inf s Œz

f (s) ą 0. (s ´ z)σN,p

(7.10)

Let u be a bounded nonnegative solution of p u + f (u) = 0 in RN (N ě 1). Then u must be a constant. Proof Let u be given as in the theorem. Denote m := inf u ě 0. RN

Since ´p (u ´ m) = f (u) ě 0 and u ´ m ě 0 in RN , we can apply the strong maximum principle to conclude that either u ” m, or u ą m in RN . We show that the second alternative cannot happen. Otherwise, we can find a sequence xn P RN such that u(xn ) Œ m as n Ñ 8. Without loss of generality we may assume that u(xn ) = min|x |=|xn | u(x). We claim that f (m) = 0. Consider the sequence of solutions un (x) := u(x + xn ). Since }un }8 = }u}8 ă +8, by standard elliptic regularity results [99, 100] we know that un is bounded in C 1,μ (K) for any compact subset of RN . Hence we may use a standard diagonal process to extract 1 (RN ), and v satisfies a subsequence, still denoted by un , such that un Ñ v in Cloc

7.2 Some Basic Results

235

v ě m, v(0) = m and p v + f (v) = 0 in RN . Thus ´p (v ´ m) ě 0 in RN and v(0) ´ m = 0. By the strong maximum principle we conclude that v ” m and hence f (m) = 0. We also claim that |xn | Ñ +8 as n Ñ +8. Otherwise by passing to a subsequence we may assume that xn Ñ x0 and then the function v obtained in the last paragraph is nothing but u(x + x0 ), and thus we have u ” m, a contradiction to our assumption that u ą m. By (7.10) with z = m, either f (s) ą 0 for all s ą m, or there exists m1 ą m such that f (s) ą 0 in (m, m1 ) and f (m1 ) = 0. In either case, we choose R := |xn | with large enough n such that a := u(xn ) P (m, m1 ); taking m1 = +8 when f (s) ą 0 for all s ą m. We next choose tnk u so that Rk := |xnk | is an increasing sequence converging to +8, with R1 ą R. Now we consider the boundary value problem p w + f (w) = 0 in R ă |x| ă Rk , w|t|x |=R u = a, w|t|x |=Rk u = m.

(7.11)

For each k ě 1, w = u is an upper solution of (7.11), and w ” m is a lower solution. Hence the standard upper and lower solution argument as in the proof of [42, Theorem 3.7] implies that (7.11) has a minimal solution wk satisfying m ď wk ď u in tR ă |x| ă Rk u. We may now proceed as on page 1894 of [38] to conclude that wk ď wk+1 ď u when R ă |x| ă Rk , k = 1, 2, . . . , and w(x) := limk Ñ8 wk (x) is well-defined for |x| ą R, and it satisfies m ď w ď u, p w + f (w) = 0 for |x| ą R, w = a when |x| = R. Moreover, as each wk is radially symmetric (as a minimal solution) so is w. We may then write w = w(r). Denoting φ(r) = r N ´1 |w 1 (r)|p´2 w 1 (r) we obtain φ 1 (r) = ´r N ´1 f (w) ď 0 for r ě R. Thus φ(r) is nonincreasing and there are two possibilities: (i) φ(r) is negative for all large r, or (ii) φ(r) ě 0 in [R, +8). If case (ii) occurs, then w1 (r) ě 0 for r ě R, and hence for all large n so that |xn | ą R, we have u(xn ) ě w(|xn |) ě a ą m, a contradiction to our initial choice of xn . If case (i) occurs, say φ(r) ă 0 for r ě R0 ą R, then w 1 (r) ă 0 for such r and hence m ď w(r) ď m0 ă m1 for all r ě R 0 , where R 0 := |xn0 | with n0 chosen so large that m0 := w(R 0 ) ď u(xn0 ) ă m1 . Due to (7.10) applied to z = m, and the fact that f (s) ą 0 in (m, m0 ], we can find c ą 0 and σ P (p ´ 1, (p ´ 1) (N ´Np)+ ) such that f (s) ě c(s ´ m)σ in [m, m0 ].

236

7 The Case of p-Laplacian Operator

It follows that ´p (w ´ m) ě c(w ´ m)σ , w ´ m ě 0 in t|x| ą R 0 u. Denote w(x) ˜ = w(|x|) ´ m and set v(x) := w(c ˜ ´p x). Clearly v satisfies ´p v ě v σ , v ě 0 in t|x| ě cp R 0 u. By [27] and [95, Theorem II], (see [38, Proposition 2.3]), we necessarily have v ” 0, which clearly is a contradiction. Thus we have proved that u ” m. l Remark 7.2 Condition (7.10) is sharp. For each ξ ą σN,2 = N {(N ´ 2), there are examples with f (s) = cs ξ for small positive s, such that the problem u+f (u) = 0 has a non-constant positive solution in RN which decays to 0 at infinity; see [38, p. 1892] for more details. Remark 7.3 For the case p = 2 Theorem 7.5 was previously proved in Chap. 6, see Theorem 6.5. There, however, a more restrictive condition on f was used instead of (7.10), namely lim inf s Œz

f (s) ą 0. s´z

7.3 Half- and Quarter-Space Problems 7.3.1 Asymptotic Convergence in Half-Spaces We start by introducing some notations. For any closed set D Ă RN , the space Cb1 (D) consists of functions u : D Ñ R such that }u}C 1 (D) := sup }u}C 1 (B1 (x)XD) ă +8. b

x PD

Theorem 7.6 Suppose that f is of type (F1 ), is quasi-monotone, its zeros are isolated and for each z P Zf+ , (7.10) holds. Moreover, assume that lim inf s Œ0

f (s) ą 0. s p ´1

(7.12)

Let u be any nontrivial nonnegative bounded solution of (7.1). Then there exists z P Zf+ such that lim u(x1 + h, x2 , . . . , xN ) = z in Cb1 ([A, +8) ˆ RN ´1 )

hÑ8

for every A P R.

7.3 Half- and Quarter-Space Problems

237

Proof By the assumptions on f , f (s) is positive on (0, δ0 ] for some δ0 ą 0. Therefore we can use the strong maximum principle, much as before, to conclude that u ą 0 in (0, +8) ˆ RN ´1 . Moreover, using (7.12) we can find some η ą 0 such that f (s) ě ηs p´1 @s P [0, δ0 ].

(7.13)

We divide the proof below into several steps. Step 1 m := limAÑ+8 inf[A,+8)ˆRN´1 u ě δ0 . It is well-known that for any bounded domain D of RN , "ż * ż 1,p p p λ1 (D) := inf |Du| dx : u P W0 (D), |u| dx ą 0 D

D

is achieved by some positive function φ which satisfies ´p φ = λ1 (D)φ p´1 in D, φ|BD = 0. Moreover, λ1 (D) ą 0, and such φ is unique if we require }φ}8 = 1. Take D = B1 (0), then it is well-known from the rearrangement theory (see [K]) that φ is radially symmetric and φ(0) = }φ}8 = 1. We now fix λ ą 0 large enough such that λ´p λ1 (B1 (0)) ă η{2. For arbitrary x0 P [λ+1, +8)ˆRN ´1 , since u ą 0 in the closed ball Bλ (x0 ) which is contained in [1, +8) ˆ RN ´1 , there exists δ P (0, δ0 ) such that u(x) ě δ, @x P Bλ (x0 ). We now let t1 = δ, t2 = δ0 , and for t P [t1 , t2 ], we define vt (x) = tφ(λ´1 (x ´ x0 )), x P Bλ (x0 ). Clearly, 0 ď vt (x) ď δ0 , @x P Bλ (x0 ), @t P [t1 , t2 ].

238

7 The Case of p-Laplacian Operator

Let δ1 P (0, 1) be so small that t2 φ(x) ă δ{2 whenever 1 ě |x| ě 1 ´ δ1 . Then vt (x) ď δ{2, @x P BBλ(1´δ1 ) (x0 ), @t P [t1 , t2 ]. Moreover, by the definition of vt and (7.13), we obtain, for t P [t1 , t2 ], p ´1

´p vt = λ´p λ1 (B1 (0))vt p ´1

ď ηvt

p ´1

ď (1{2)ηvt

´ ζ ď f (vt ) ´ ζ, @x P Bλ(1´δ1 ) (x0 ),

where ζ =

min

p ´1

(1{2)ηvt1

x PBλ(1´δ1 ) (x0 )

ą 0.

Therefore, in view of vt1 ď δ ď u in Bλ(1´δ1 ) (x0 ), we can apply Proposition 7.1 with D = Bλ(1´δ1 ) (x0 ),  = mintδ{2, ζ u to conclude that u ě vt , @x P Bλ(1´δ1 ) (x0 ), @t P [t1 , t2 ]. In particular, u(x0 ) ě vt2 (x0 ) = δ0 . Since x0 P [λ + 1, +8) ˆ RN ´1 is arbitrary, this implies that lim

inf

AÑ+8 [A,+8)ˆRN´1

u ě δ0 .

Step 2 For every A P R, u(x1 + h, x2 , . . . , xN ) Ñ m as h Ñ +8 uniformly with respect to (x1 , x2 , . . . , xN ) P [A, +8) ˆ RN ´1 . By Step 1 above, M := lim

sup

AÑ+8 [A,+8)ˆRN´1

u ě m ą 0.

We show next that f (M) = 0. This follows from a standard consideration as n ) such that x n Ñ +8 described below. Choose a sequence x n = (x1n , x2n , . . . , xN 1 n and u(x ) Ñ M as n Ñ +8. Then by passing to a subsequence, un (x) = u(x +xn ) 1 (RN ) to a function U which satisfies converges in Cloc p U + f (U ) = 0, }u}8 ě U ě 0 in RN , U (0) = M. We can now apply Theorem 7.5 to conclude that U ” M, which infers f (M) = 0. Next we set to show that

7.3 Half- and Quarter-Space Problems

239

u(x1 + h, x2 , . . . , xN ) Ñ M as h Ñ +8 uniformly with respect to (x1 , x2 , . . . , xN ) P [A, +8) ˆ RN ´1 for every A P R. Clearly this implies M = m. We will prove the above conclusion by applying the weak sweeping principle. The construction of the required lower solution is rather involved; for clarity we divide the remaining arguments into two sub-steps. Sub-step 2.1 Construction of a lower solution. To construct a suitable lower solution which can be used to bound u from below via the weak sweeping principle, we first modify f suitably and then use the modified f to form a Dirichlet problem over a ball. The required lower solution will be a positive solution of this auxiliary Dirichlet problem. The modification of f is needed in order to create a gap from f , so that the condition 1 (t) + 2 (t) ě  in Proposition 7.1 is met; see (7.14) below. Since the zeros of f are isolated, we can find 0 ă M0 ă M such that f (s) ą 0 in [M0 , M). Define F1 (s) =

żM f (t)dt. s

Clearly F1 (s) ą 0 in [0, M). For any small  ą 0, we consider g(s) = g (s) := f (s) ´ s σ in [0, M], where σ = maxt1, σN,p u with σN,p as given in (7.10) in the case f (0) = 0, and σ = 1 when f (0) ą 0. There exists M P (M0 , M) such that g(M ) = 0 and g(s) ą 0 in [M0 , M ). Set G(s) = G (s) :=

ż M g(t)dt. s

Clearly G(s) ą 0 in [M0 , M ), and M Ñ M as  Ñ 0. Since G (s) Ñ F1 (s) uniformly in [0, M] as  Ñ 0, and F1 (s) ě F1 (M0 ) ą 0 in [0, M0 ], we thus find that there exists 0 ą 0 sufficiently small such that for each  P (0, 0 ], $ M ´  ą M 0 , ’ ’ & G (s) ą 0 in [0, M ), ’ G (s) ě G (M ´ ) for s P [0, M ´ ), ’ %  G (s) is decreasing in [M0 , M ). Let us also notice that due to (7.10), we always have f (s) ą g (s) ą 0 for small positive s, say s P (0, s0 ), and s0 can be chosen independent of  P (0, 0 ]. Set $ & g(0) for s ă 0, g(s) ˜ = g(s) for s P [0, M ], % 0 for s ą M ,

240

7 The Case of p-Laplacian Operator

and ˜ G(s) =

ż M

g(t)dt. ˜

s

˜ Clearly G(s) ě 0 for all s P R. Motivated by Lemma 6.2 in Chap. 6, we consider the functional ż ż 1 p ˜ Ir (v) = |∇v| + G(v) p Br (0) Br (0) p

for all v P H0 (Br (0)). It is well-known that a critical point of Ir corresponds to a weak solution of ˜ = 0 in Br (0), v|BBr (0) = 0. p v + g(v) Since g˜ ě 0 in (´8, 0] and g˜ = 0 for s ě M , by the weak maximum principle, any such solution satisfies 0 ď v ď M . Consequently for any such solution we have g(v) ˜ = g(v). Moreover, by elliptic regularity for p-Laplacian equations [99, 100] we know that such a solution also belongs to C 1,α (B r (0)). Let us observe that the functional Ir is well-defined and is coercive. Thus by standard argument we know that it has a minimizer vr , which is a critical point of Ir and thus, as discussed above, is a nonnegative solution to p vr + g(vr ) = 0 in Br (0), vr |BBr (0) = 0. Since vr is a minimizer, by well-known rearrangement theory (see, e.g. [70]) it must be radially symmetric and decreasing away from the center of the domain. Thus 0 ď vr (|x|) ď vr (0) ď M in Br (0). We claim that there exists r ą 0 such that vr (0) ě M ´ . Otherwise vr ď M ´  for all r ą 0. Hence, recalling G(s) ě G(M ´ ) for s P [0, M ´ ], we obtain ż ż Ir (vr ) ě G(vr ) ě G(M ´ ) = αN r N G(M ´ ), @r ą 0, Br (0)

Br (0)

where αN stands for the volume of B1 (0). On the other hand, for r ą 1 define wr (x) =

"

M for |x| ă r ´ 1, M (r ´ |x|) for r ´ 1 ď |x| ď r.

1,p

This function belongs to W0 (Br (0)), and |∇wr |p and G(wr ) are supported on the annulus tr ´ 1 ď |x| ď ru. Thus there exists a constant C independent of r such that Ir (wr ) ď C[r N ´ (r ´ 1)N ] @r ą 1.

7.3 Half- and Quarter-Space Problems

241

Since vr is the minimizer of Ir , we have Ir (vr ) ď Ir (wr ). Thus αN G(M ´ )r N ď C[r N ´ (r ´ 1)N ] @r ą 1. Since G(M ´ ) ą 0, the above inequality does not hold for large r. This contradiction shows that vr (0) ě M ´  for all large r. We will use vr as the lower solution in the weak sweeping principle. Sub-step 2.2 Completion of the proof by the weak sweeping principle. Fix  P (0, 0 ] and then choose r ą 0 such that vr (0) ě M ´ . Let x ˚ be an arbitrary point in [r +1, +8)ˆRN ´1 . Let x n be the sequence used earlier with the properties 1 (RN ). x1n Ñ +8 and u(x n ) Ñ M. We recall that un (x) := u(x ´ x n ) Ñ M in Cloc 0 ) with x 0 ą r + 1 such that u(x) ě Thus we can find a point x 0 = (x10 , x20 , . . . , xN 1 0 t 0 M in Br (x ). We now define x = tx + (1 ´ t)x ˚ and ut (x) = u(x + x t ). Clearly x1t ě r + 1 and thus Br (x t ) Ă [1, +8) ˆ RN ´1 for all t P [0, 1]. Since u ą 0 on the compact set Yt P[0,1] Br (x t ), we can find δ ą 0 such that u ě δ on this set. Let r1 P (0, r) be chosen so that vr P (0, δ{2) on t|x| = r1 u. Denote D := Br1 (0). Then we have, for t P [0, 1], vr + δ{2 ď ut on BD and ´p vr = g(vr ) ď f (vr ) ´ ζ, ´p ut = f (ut ) in D, where ζ :=

inf [f (vr (x)) ´ g(vr (x))] ą 0.

x PBr1 (0)

(7.14)

Moreover, u0 ě M ě vr on D. Thus we can apply Proposition 7.1 to conclude that ut ě vr in D for all t P [0, 1]. In particular, u(x ˚ ) ě vr (0) ě M ´ . Since  P (0, 0 ] and x ˚ P [r + 1, +8) ˆ RN ´1 are arbitrary, and M ´  Ñ M as  Ñ 0, this implies that lim

inf

AÑ+8 [A,+8)ˆRN´1

u ě M.

In view of the definition of M, the above inequality implies the required conclusion in Step 2. Finally we note that the uniform convergence proved in Step 2 implies convergence in Cb1 ([A, +8) ˆ RN ´1 ) for any A P R by standard elliptic estimates. l Theorem 7.7 Suppose that f is a type (F2 ) function which is quasimonotone, (7.12) holds and for each z P Zf+ ztau,

242

7 The Case of p-Laplacian Operator

lim sup s Õz

f (s) ă 0. (z ´ s)σN,P

(7.15)

In addition, we assume that the zeros of f are isolated. Let u be any nontrivial nonnegative bounded solution of (7.1). Then there exists z P Zf+ such that lim u(x1 + h, x2 , . . . , xN ) = z in Cb1 ([A, +8) ˆ RN ´1 )

hÑ8

for every A P R. Proof Since f (s) ą 0 in (0, a), we can apply the strong maximum principle in the set t0 ď u ă au to conclude that u is either identically 0 or it is positive in (0, +8) ˆ RN ´1 . Since u is nontrivial, we must have u ą 0. We divide the proof below into three steps. Step 1 m := limAÑ+8 inf[A,+8)ˆRN´1 u ě a. For any  P (0, a), by our assumptions on f , there exists η ą 0 such that f (s) ě ηs p´1 @s P [0, a ´ ]. Hence we can repeat Step 1 in the proof of Theorem 7.6, with δ0 = a´, to conclude that m = lim

inf

AÑ+8 [A,+8)ˆRN´1

u ě a ´ .

Letting  Ñ 0 we obtain m ě a. 1 (RN ) along some sequence x n with Step 2 m P Zf+ and u(x + x n ) Ñ m in Cloc n Ñ +8. Let x n = (x n , . . . , x n ) be a sequence in RN such that x n Ñ +8 xN N 1 1 and u(x n ) Ñ m as n Ñ +8. Up to extraction of a subsequence, the functions 1 (RN ) to a bounded solution U of  u+f (u) = un (x) = u(x +x n ) converge in Cloc p N 0 in R with the properties that m ď U and U (0) = m. Clearly we also have U ď }u}8 ă M := }u}8 +1. It follows that v := M ´U is a bounded nonnegative solution of

p v + f˜(v) = 0 in RN , where f˜(s) =

"

´f (M ´ s) for 0 ď s ď M ´ m, (s ´ M + m)p´1 ´ f (m) for s ą M ´ m.

By our assumptions on f , we see that f˜ satisfies all the conditions of Theorem 7.5. Hence we must have v ” constant. It follows that U ” m and hence f (m) = 0.

7.3 Half- and Quarter-Space Problems

243

Step 3 For every A P R, lim u(x1 + h, x2 , . . . , xn ) = m uniformly in [A, +8) ˆ RN ´1 .

hÑ+8

Let f˜ be defined as above and denote M˜ := M ´ m. As in Step 2 of the proof of Theorem 7.6, for any  ą 0 we can find r ą 0 and a radially symmetric function v˜r satisfying 0 ď v˜r ď v˜r (0) ď M˜  , v˜r (0) ě M˜  ´ , and ´p v˜r = g(v˜r ) in Br (0), v˜r = 0 on BBr (0), with g(s) = f˜(s) ´ s σ ă f˜(s) in (0, M˜  ], where M˜  ă M˜ and converges to M˜ as  Ñ 0. Now clearly vr := M ´ v˜r satisfies ´p vr = ´g(M ´ vr ) in Br (0), vr = M on BBr (0). Since ´g(M ´ s) ą f (s) for s P (0, M˜  ], and u ď M ´ 1 in R+ ˆ RN ´1 , and u ă M ´M˜  in some Br (x 0 ) (by the conclusion in Step 2 and the fact that M ´M˜  ą m), we can use the weak sweeping principle to show that u ď vr (0) ď M ´ M˜  in [r + 1, +8) ˆ RN ´1 . It follows that lim

sup

AÑ+8 [A,+8)ˆRN´1

u ď M ´ M˜  .

Letting  Ñ 0, we deduce lim

sup

AÑ+8 [A,+8)ˆRN´1

u ď M ´ M˜ = m.

In view of the definition of m, we obtain lim u(x1 + h, x2 , . . . , xN ) = m

hÑ+8

uniformly in [A, +8) ˆ RN ´1 for any A P R. As in Theorem 7.6, the convergence in the Cb1 norm is a consequence of the above uniform convergence and standard elliptic estimates. l

244

7 The Case of p-Laplacian Operator

7.3.2 One-Dimensional Symmetry in Half-Spaces For treating the quarter-space problems, we will need some one-dimensional symmetry results for the half-space problem (7.4). We want to stress that due to the lack of a strong comparison principle for general p-Laplacian equations, such partial symmetry results are not readily available as in the Laplacian case treated in Chap. 6. In Theorem 1.8 of the recent paper [58], a one-dimensional symmetry result has been established for the case N = 2, 3, with 1 + NN+2 ă p ď 2, and with f (u) locally Lipschitz continuous, and satisfying f (u) ą 0 for u ă a, f (u) ă 0 for u ą a. Theorem 7.8 Suppose that f is of type (F1 ) which is quasi-monotone, and moreover (7.12) holds, the zeros of f are isolated, and at each of its zero f satisfies (7.7). Let u be a nonnegative bounded solution of (7.4) satisfying BxN u ě 0. Then either u ” 0 or u(x) ” V (xN ), where V is the positive solution of the one dimensional problem (7.5) satisfying V (8) = }u}8 . Proof Due to (7.12), we can apply the strong maximum principle to conclude that either u ” 0 or u ą 0 for xN ą 0. So we only need to consider the case u ą 0. By Theorem 7.6, lim u(x 1 , xN ) = m

xN Ñ8

(7.16)

uniformly in x 1 P RN ´1 for some positive constant m P Zf+ . Since BxN u ě 0, we necessarily have u(x) ď m for all x P  := RN ´1 ˆ R+ . Thus, since f (m) = 0, u ” m is an upper solution of (7.4) satisfying u ď u. By standard upper and lower solution argument we know that (7.4) has a solution in the order interval [u, u]. We show next that there is a maximal solution u˚ in this order interval in the sense that any solution v in [u, u] satisfies v ď u˚ . For this purpose, we choose a sequence of increasing numbers Rn Ñ 8 and define Bn+ = tx P RN : |x| ă Rn , xN ą 0u. Set φn (x) = maxt0, m(|x| ´ Rn + 1)u. We now consider the auxiliary problem ´ p u = f (u) in Bn+ , u = φn on BBn+ .

(7.17)

Clearly u|Bn+ is a lower solution to (7.17) and m is an upper solution to (7.17). Hence by standard upper and lower solution argument (7.17) has a maximal solution un in the order interval [u, m]. One then easily sees that un ě un+1 on Bn+ and u˚ := limnÑ8 un is a solution of (7.4). Clearly u˚ P [u, u]. If v is any solution of (7.4) in [u, u], then it is evident that v|Bn+ is a lower solution of (7.17) and it follows that un ě v (since the standard upper and lower solution argument implies the existence of a solution in [v, m] which is less than or equal to un ). It follows

7.3 Half- and Quarter-Space Problems

245

that v ď u˚ . Thus u˚ is the maximal solution in [u, u]. The maximality implies that u˚ is a function of xN only. Indeed, if this is not true, then there exist two points 0 ) and ((x 1 )2 , x 0 ) in RN ´1 ˆ R such that ((x 1 )1 , xN + N 0 0 ) ă u˚ ((x 1 )2 , xN ). u˚ ((x 1 )1 , xN

Define u˜ ˚ (x) = u˚ (x 1 + (x 1 )2 ´ (x 1 )1 , xN ). Then u˜ ˚ is a solution of (7.4) and v(x) := maxtu˚ (x), u˜ ˚ (x)u ě, ı u˚ (x). Moreover, v is a lower solution to (7.4) satisfying v ď u = m. Hence (7.4) has a solution v ˚ in the order interval [v, u]. It follows that v ˚ is a solution in [u, u] which satisfies v ˚ ě, ı u˚ , a contradiction to the maximality of u˚ . Thus u˚ is a positive solution of (7.5). Since u ď u˚ ď m, we necessarily have u˚ (8) = m. Next we construct a suitable lower solution which is below u. Since the zeros of f are isolated, there exists a small δ ą 0 such that f (u) ą 0 in [m ´ δ, m). We now choose m P (m ´ δ, m) and then modify f (u) over [0, m] to obtain a new function f (u) ě 0 so that f (u) = f (u) in [0, m ´ δ], 0 ă f (u) ď f (u) in (m ´ δ, m ), f (m ) = 0. Since u˚ is a positive solution of (7.5), by Theorem 7.1 we see that F (z) ă F (m) for z P [0, m). In view of the above construction of f , it follows that F (z) ă F (m ) for z P [0, m ). Thus we can apply Theorem 7.2 to see that the following 1-d problem ´p v = f (v) in R+ , v(0) = 0, v(8) = m has a unique positive solution v . For any given R ą 0, define V,R (t) =

"

0, t P [0, R], v (t ´ R), t P [R, 8).

It is easily checked that V,R is a lower solution of (7.4). Using (7.16) and m ă m, we can find a large R = R so that u(x) ą m for xN ě R . Hence in view of v (xN ) ď m we find that V,R (xN ) ď u(x) for all xN ě 0. Thus we may define u = V,R , and u is a lower solution of (7.4) that satisfies u ď u. By an analogous argument to the one used above for u˚ , we can show that in the order interval [u , u] the problem (7.4) has a minimal solution u˚ , and moreover the minimality of u˚ implies that u˚ is a function of xN only. Thus u˚ is a positive solution of (7.5), and u˚ (8) is a zero of f (u) contained in [m , m]. But m is the only zero of f in this range. Hence u˚ (8) = m. By Theorem 7.2, (7.5) has a unique solution with limit m at infinity. Thus we necessarily have u˚ = u˚ . Since u˚ ď u ď u˚ , this implies l that u is a function of xN only. This completes the proof.

246

7 The Case of p-Laplacian Operator

Remark 7.4 From the proof we easily see that the condition BxN u ě 0 in Theorem 7.5 can be replaced by }u}8 ď M := limxN Ñ8

sup x 1 PRN´1

u(x 1 , xN ),

which is a consequence of BxN u ě 0. We conjecture that these extra conditions are unnecessary. The following result is an extension of Theorem 1.1 in [42], where instead of (F2 ), it was assumed that f (s) ą 0 in (0, a) and f (s) ă 0 in (a, +8). See also Theorem 1.8 of [58]. Theorem 7.9 Suppose that f is of type (F2 ), is quasi-monotone and (7.12) holds. Let u be a nonnegative bounded solution of (7.4). Then either u ” 0 or u(x) ” V (xN ), where V is the unique one dimensional solution determined by Theorem 7.4. Proof Since f (s) ą 0 in (0, a), we can apply the strong maximum principle in the set t0 ď u ă au to conclude that either u ” 0 or u ą 0 in RN ´1 ˆ (0, +8). So it suffices to consider the case that u ą 0. Let M := }u}8 P (0, +8). Since (F2 ) holds, either there exists b ě M such that f (b) = 0, or f (s) ă 0 for all s ě M. In the latter case, we can find ap ě a such that f (p a ) = 0 and f (s) ă 0 for s ą ap. Then one can use the weak sweeping principle as in the proof of Proposition 2.2 of [41] to deduce that u ď ap and hence M ď ap. Thus we can always find a constant b ě M such that f (b) = 0. Clearly we must have b ě a. Now we can repeat the arguments used to prove Lemma 3.2 of [42] to conclude that (7.4) has a maximal positive solution u among all bounded positive solutions. We can also repeat the argument in the proof of Lemma 3.1 of [42] to show that (7.4) has a minimal positive solution u among all bounded positive solutions. Thus we have in particular that u ď u ď u. As in [42], being the maximal and minimal solution of (7.4), u and u must be functions of xN only, and thus are positive solutions of (7.5). By Theorem 7.4, u = u = V . Thus u ” V . l Remark 7.5 It is interesting to compare Theorem 7.9 above with [13, Theorem 1.1]. Here we do not require f to be non-increasing in a small left neighborhood of a, but we require that f is not identically zero in any open interval.

7.3.3 Asymptotic Convergence in Quarter-Spaces Theorem 7.10 Suppose that f is of type (F1 ) satisfying all the conditions in Theorem 7.8. Let u be any nontrivial nonnegative bounded solution of (7.2) with BxN u ě 0. Then

7.3 Half- and Quarter-Space Problems

247

lim u(x1 + h, x2 , . . . , xN ) = V (xN ) in Cb1 ([A, +8) ˆ RN ´2 ˆ R+ )

hÑ8

(7.18)

for every A P R, where V is a solution of the one dimensional problem (7.5). Proof Since u is nontrivial, we can apply the strong maximum principle to conclude that u ą 0. Let m = lim

inf

AÑ+8 [A,+8)ˆRN´2 ˆ[A,+8)

u.

Then a simple modification of the arguments in Steps 1 and 2 of the proof of Theorem 7.6 shows that lim u(x1 + h, x2 , . . . , xN ´1 , xN + h) = m P Zf+

hÑ+8

(7.19)

uniformly in [A, +8) ˆ RN ´2 ˆ [A, +8) for every A P R. Now let x1n be any sequence of positive numbers converging to +8 as n Ñ +8, and y n be any sequence in RN ´2 . Define un (x1 , y, xN ) = u(x1 + x1n , y + y n , xN ). By standard elliptic estimates we can find a subsequence of un that converges to some v in 1 (RN ´2 ˆ R ), and v is a bounded nonnegative solution of Cloc + p v + f (v) = 0 in RN ´1 ˆ R+ , v = 0 on R N ´1 ˆ t0u. In view of (7.19), we have limxN Ñ8 v(x 1 , xN ) = m uniformly for x 1 P RN ´1 . Moreover, due to BxN u ě 0 we have BxN v ě 0. Hence }v}8 = m. We may now apply Theorem 7.8 to conclude that v ” V where V is the unique solution of (7.5) satisfying V (8) = m. The uniqueness of V implies that the entire sequence un converges to V . Moreover, in view of (7.19) and V (8) = m, we find that lim u(x1 , x2 , . . . , xN ) = V (xN )

x1 Ñ+8

uniformly for (x2 , . . . , xN ) P RN ´2 ˆ R+ . From this and standard elliptic estimates we see that (7.18) holds. l Remark 7.6 The condition BxN u ě 0 in Theorem 7.10 is required only in order to be able to apply Theorem 7.8. Thus if we can drop the corresponding condition in Theorem 7.8, then we can remove this condition here. Theorem 7.11 Suppose that f is of type (F2 ), is quasi-monotone, and (7.12) holds. Let u be any nontrivial nonnegative bounded solution of (7.2). Then lim

inf

R Ñ8 [R,8)ˆRN´2 ˆ[R,8)

uěa

248

7 The Case of p-Laplacian Operator

and lim u(x1 + h, x2 , . . . , xN ) = V (xN ) in Cb1 ([A, +8) ˆ RN ´2 ˆ [0, B])

hÑ8

(7.20)

for every A P R and B ą 0, where V is the unique one dimensional solution determined by Theorem 7.4. If we assume further that the zeros of f are isolated and (7.15) holds for each z P Zf+ ztau, then (7.20) can be strengthened to lim u(x1 + h, x2 , . . . , xN ) = V (xN ) in Cb1 ([A, +8) ˆ RN ´2 ˆ R+ )

hÑ8

(7.21)

for every A P R, which implies that lim

inf

R Ñ8 [R,8)ˆRN´2 ˆ[R,8)

u = a.

Proof Since f (s) ą 0 in (0, a) and (7.12) holds, we can apply the strong maximum principle in the set t0 ď u ă au to conclude that u is either identically 0 or it is positive in (0, +8)ˆRN ´2 ˆ(0, +8). Thus u ą 0 in (0, +8)ˆRN ´2 ˆ(0, +8). By (7.12), for any  P (0, a) we can find η ą 0 such that (7.13) holds with δ0 = a ´ . Then by using the weak sweeping principle in the same way as in Step 1 of the proof of Theorem 7.6, we can show that m = lim

inf

AÑ+8 [A,+8)ˆRN´2 ˆ[A,+8)

u ě a ´ .

Letting  Ñ 0 we deduce m = lim

inf

AÑ+8 [A,+8)ˆRN´2 ˆ[A,+8)

u ě a.

(7.22)

Now let x1n be any sequence of positive numbers converging to +8 as n Ñ +8, and y n be any sequence in RN ´2 . Define un (x1 , y, xN ) = u(x1 + x1n , y + y n , xN ). By standard elliptic estimates we can find a subsequence of un that converges to 1 (RN ´2 ˆ R ), and v is a bounded nonnegative solution of some v in Cloc + p v + f (v) = 0 in RN ´1 ˆ R+ , v = 0 on R N ´1 ˆ t0u. By (7.22) we know that v ı 0. Hence we can apply Theorem 7.9 to conclude that v(x) ” V (xN ), where V is the unique solution determined by Theorem 7.4. The uniqueness of v implies that the limit does not depend on the sequences tx1n u and ty n u. This proves (7.20). Under the extra assumptions, the arguments used in Steps 2 and 3 of the proof of Theorem 7.7 can be easily modified to show that (7.19) holds. Thus we necessarily have m = a and

7.4 Comparison with Standard Laplacian (p = 2)

249

lim u(x1 , x2 , . . . , xN ) = V (xN )

x1 Ñ+8

uniformly for (x2 , . . . , xN ) P RN ´2 ˆ R+ . From this and standard elliptic estimates we see that (7.21) holds. l

7.4 Comparison with Standard Laplacian (p = 2) Consider the quasilinear elliptic problem p u + f (u) = 0 in Q, u = 0 on BQ,

(7.23)

where Q = (0, 8) ˆ (0, 8) ˆ RN ´2 is a quarter-space in RN (N ě 2), p u = div(|∇u|p´2 ∇u) is the usual p-Laplacian operator with p ą 1, and f ě 0 is a continuous function over R+ := [0, 8). If further f is locally Lipschitz continuous in R+ , and lim

s Œz

f (s) ą 0 whenever f (z) = 0, s´z

(7.24)

then it follows from Theorem 6.2 that, in the special case p = 2, for every bounded positive solution of (7.23), one has lim u(x1 , x2 , . . . , xN ) = Vz (x2 ),

(7.25)

lim u(x1 , x2 , . . . , xN ) = Vz˜ (x1 ),

(7.26)

x1 Ñ8

and x2 Ñ8

where z, z˜ P (0, 8) are zeros of f , and whenever z0 is a positive zero of f , Vz0 denotes the unique positive solution of the one-dimensional problem p V + f (V ) = 0 in R+ , V (0) = 0, V (t) ą 0 for t ą 0, V (8) = z0 .

(7.27)

This result was extended to the general case p ą 1 in Theorem 7.8 (see previous section) under suitable conditions on f and the extra assumption that Bxi u ě 0 in Q for i = 1, 2. (In Chap. 6, it was actually assumed that u = 0 for x2 = 0 and u ě, ı 0 for x1 = 0. However, this latter condition is only used to guarantee that the solution is positive in Q. The conclusions there remain valid for positive solutions satisfying u = 0 on BQ.) We assume that tz ą 0 : f (z) = 0u = tz1 , . . . , zk u,

(7.28)

250

7 The Case of p-Laplacian Operator

and prove that for each zi , i = 1, . . . , k, (7.23) has a bounded positive solution satisfying lim u(x1 , x2 , . . . , xN ) = Vzi (x2 ), lim u(x1 , x2 , . . . , xN ) = Vzi (x1 ).

x1 Ñ8

x2 Ñ8

(7.29) Moreover, if p = 2, we prove that (7.23) has no other bounded positive solutions. Therefore, in (7.25) and (7.26), we necessarily have z = z˜ . If further f is nonincreasing in (zi ´ , zi ), then (7.23) has a unique bounded positive solution satisfying (7.29). When p = 2, the existence of a positive solution of (7.23) satisfying (7.29) was established in Sect. 2.3, where the special case k = N = 2 was considered.

7.5 Existence Result In this section, we construct a bounded positive solution of (7.23) satisfying (7.29). We need to be more precise about the conditions imposed on f . We assume that (7.28) holds, "

f : R+ Ñ R is continuous, nonnegative and locally Lipschitz continuous except possibly at tz1 , . . . , zk u,

(7.30)

and for i = 1, . . . , k, lim inf s Œzi

f (s) f (s) ą 0, lim sup ă +8, σ p ´1 N,p (s ´ zi ) s Õzi (zi ´ s)

(7.31)

where σN,p = (p ´ 1)

N if N ą p, N ´p

and σN,p stands for an arbitrary number in [1, 8) if N ď p. Moreover, we assume lim inf s Œ0

f (s) ą 0 if f (0) = 0. s p ´1

(7.32)

Let us note that since f is nonnegative, we automatically have lim inf s Œzi

f (s) ě 0. (s ´ zi )p´1

Moreover, in the special case p = 2, the second condition in (7.31) is automatically satisfied if f is locally Lipschitz continuous, and the first condition in (7.31) is

7.5 Existence Result

251

less restrictive than (7.24). By Theorems 7.1 and 7.2 from Sect. 7.2, we have the following result. Proposition 7.2 Let f satisfy (7.28), (7.30), (7.31) and (7.32). Then for every zi , i = 1, . . . , k, (7.27) has a unique solution with z0 replaced by zi . Moreover, the unique solution is a strictly increasing function. To construct a positive solution of (7.23) with the desired properties, a key step is the following result. Lemma 7.1 With f as in Proposition 7.2, for each zi and any given small δ ą 0, 1,p there exists R = Rδ ą 0 and a function v P W0 (B), with B = BR := tx P RN : |x| ă Ru, satisfying (i) (ii) (iii) (iv)

p v + f (v) ě 0 in B, v = 0 on BB, 0 ă v ă zi in B, v(x1 , x2 , . . . , xN ) ď mintVzi (x1 + R + 1), Vzi (x2 + R + 1)u in B, supB v ě zi ´ δ.

Proof We define g(s) = f (s) ´ s σ for small  ą 0 as in sub-step 2.1 of the proof of Theorem 7.6. The argument there (with M = zi ) shows that for R = R large enough, there exists a positive solution to p v + g(v) = 0 in BR , v = 0 on BBR , which satisfies v(0) = supBR v P (zi ´ δ, zi ) provided that  ą 0 is small enough. Thus v has properties (i), (ii) and (iv). It remains to prove (iii). We may use the week sweeping principle as in sub-step 2.2 of the proof of Theorem 7.6, with u = Vzi (x1 ) or Vzi (x2 ). It results that v(x1 , x2 , . . . , xN ) ď Vzi (x1 + R + 1) and v(x1 , x2 , . . . , xN ) ď Vzi (x2 + R + 1). \ [ Theorem 7.12 Let f satisfy (7.28), (7.30), (7.31) and (7.32). Then for each zi , (7.23) has a bounded positive solution u satisfying (7.29). Proof Let δ ą 0 be small enough such that f (s) ą 0 in [zi ´δ, zi ). Then let R = Rδ and v be given by Lemma (7.1). Fix x0 P RN such that the ball BR+1 (x0 ) := tx P RN : |x ´ x0 | ă R + 1u is contained in Q. Then define vx0 (x) =

"

v(x ´ x0 ) if x P BR (x0 ), 0 otherwise

Since f (0) ě 0, it is clear that vx0 is a subsolution of (7.23). Define u=

sup

BR+1 (x0 )ĂQ

vx0 .

252

7 The Case of p-Laplacian Operator

Then u is again a subsolution of (7.23), and it satisfies u(x) ě zi ´ δ when x1 ě R + 1, x2 ě R + 1.

(7.33)

Define u = mintVzi (x1 ), Vzi (x2 )u. Then u is a supersolution to (7.23), and by Lemma 7.1, we have u ě u in Q. Therefore we can apply the standard sub- and supersolution argument to conclude that (7.23) has a positive solution u satisfying u ď u ď u in Q. As in the proof of Theorem 7.10, we find that lim u(x1 + h, x2 + h, . . . , xN ) = m

hÑ8

and m is a positive zero of f . By (7.33) and the choice of δ, we necessarily have m = zi . On the other hand, we have u ď u ă zi in Q. Therefore we are able to use Remark 7.6 in the proof of Theorem 7.10 to conclude that lim u(x1 , x2 , . . . , xN ) = Vzi (x2 )

x1 Ñ8

uniformly for (x2 , . . . , xN ) P R+ ˆ RN ´2 . We similarly have lim u(x1 , x2 , . . . , xN ) = Vzi (x1 )

x2 Ñ8

uniformly for (x1 , x3 , . . . , xN ) P R+ ˆ RN ´2 . Thus (7.29) holds.

\ [

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