Time-Fractional Differential Equations: A Theoretical Introduction [1st ed.] 9789811590658, 9789811590665

Table of contents :
Front Matter ....Pages i-x
Basics on Fractional Differentiation and Integration (Adam Kubica, Katarzyna Ryszewska, Masahiro Yamamoto)....Pages 1-8
Definition of Fractional Derivatives in Sobolev Spaces and Properties (Adam Kubica, Katarzyna Ryszewska, Masahiro Yamamoto)....Pages 9-45
Fractional Ordinary Differential Equations (Adam Kubica, Katarzyna Ryszewska, Masahiro Yamamoto)....Pages 47-71
Initial Boundary Value Problems for Time-Fractional Diffusion Equations (Adam Kubica, Katarzyna Ryszewska, Masahiro Yamamoto)....Pages 73-108
Decay Rate as t →∞ (Adam Kubica, Katarzyna Ryszewska, Masahiro Yamamoto)....Pages 109-119
Concluding Remarks on Future Works (Adam Kubica, Katarzyna Ryszewska, Masahiro Yamamoto)....Pages 121-122
Back Matter ....Pages 123-134

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SPRINGER BRIEFS IN MATHEMATICS

Adam Kubica Katarzyna Ryszewska Masahiro Yamamoto

Time-Fractional Differential Equations A Theoretical Introduction

123

SpringerBriefs in Mathematics Series Editors Nicola Bellomo, Torino, Italy Michele Benzi, Pisa, Italy Palle Jorgensen, Iowa, USA Tatsien Li, Shanghai, China Roderick Melnik, Waterloo, Canada Otmar Scherzer, Linz, Austria Benjamin Steinberg, New York, USA Lothar Reichel, Kent, USA Yuri Tschinkel, New York, USA George Yin, Detroit, USA Ping Zhang, Kalamazoo, USA

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Adam Kubica • Katarzyna Ryszewska • Masahiro Yamamoto

Time-Fractional Differential Equations A Theoretical Introduction

Adam Kubica Warsaw University of Technology Warszawa, Poland

Katarzyna Ryszewska Warsaw University of Technology Warszawa, Poland

Masahiro Yamamoto Graduate School of Mathematical Sciences The University of Tokyo Tokyo, Japan

ISSN 2191-8198 ISSN 2191-8201 (electronic) SpringerBriefs in Mathematics ISBN 978-981-15-9065-8 ISBN 978-981-15-9066-5 (eBook) https://doi.org/10.1007/978-981-15-9066-5 Mathematics Subject Classification: 35R11, 26A33 © The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are solely and exclusively licensed 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, expressed 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. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Recently, fractional differential equations have attracted great attention and many studies have been performed. However, there are not many works that cover the theory of partial differential equations, so that unnecessarily lengthy arguments and also arguments that lack rigor are sometimes presented, and it may be difficult to gain unified views of the related fields. This concise book provides rigorous treatments for time-fractional derivatives in Sobolev spaces and solutions to initial boundary value problems for timefractional partial differential equations and establishes the foundation of the theory for fractional differential equations. The results here should be fundamental also for discussing other important topics such as nonlinear dynamical systems, optimal control, and inverse problems for fractional differential equations. Although our approach can work for more general fractional differential equations, focusing on a differential equation with a single time-fractional derivative, we describe the theory. The parts of the book have been presented as graduate courses at Sapienza University of Rome and The University of Tokyo. The authors thank Professor Bangti Jin (University College London) for valuable comments. The first and second authors are supported partly by the National Science Centre, Poland through 2017/26/M/ST1/00700 Grant, and the third author is partly supported by Grant-in-Aid for Scientific Research (S) 15H05740 of Japan Society for the Promotion of Science and by the National Natural Science Foundation of China (nos. 11771270 and 91730303) and the “RUDN University Program 5–100.” Warszawa, Poland Warszawa, Poland Tokyo, Japan November 2020

Adam Kubica Katarzyna Ryszewska Masahiro Yamamoto

v

Remarks for Readers

• In this book, we minimize the references. In Chap. 2, we use some basic knowledge of functional analysis and operator theory. The readers can consult suitable textbooks on functional analysis, and here we rely on Brezis [5], Kato [13], Pazy [23], and Tanabe [27]. In particular, for the function space, we refer to Adams [2]. Needless to say, there are other reputable source books. • The readers who wish to use this book for quickly understanding a theoretical essence of fractional differential equations are encouraged to skip the proofs of Lemmata 2.2–2.4 and Theorem 2.1. They can skip also Sect. 2.6 of Chap. 2 and Sects. 5.2 and 5.3 of Chap. 5. • We use the following numbering: for example, Theorem 2.1 means the first theorem in Chap. 2.

vii

Contents

1

Basics on Fractional Differentiation and Integration .. . . . . . . . . . . . . . . . . . .

1

2 Definition of Fractional Derivatives in Sobolev Spaces and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Motivations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Preliminaries: Operational Structure of J α . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Definition of Generalized ∂tα in Hα (0, T ) . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Some Functions in Hα (0, T ) . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Definition of ∂tα in H α (0, T ) . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6 Adjoint of the Fractional Derivative in Hα (0, T ). .. . . . . . . . . . . . . . . . . . . . 2.7 Laplace Transform of ∂tα . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

9 9 11 21 26 33 38 43

3 Fractional Ordinary Differential Equations . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Fundamental Inequalities: Coercivity .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Well-Posedness for Single Linear Fractional Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 Alternative Formulation of Initial Value Problem .. . . . . . . . . . . . . . . . . . . . 3.5 Systems of Linear Fractional Ordinary Differential Equations . . . . . . . 3.6 Linear Fractional Ordinary Differential Equations with Multi-Term Fractional Derivatives . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4 Initial Boundary Value Problems for Time-Fractional Diffusion Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Some Results from Theorem 4.1 and Proposition 4.2 .. . . . . . . . . . . . . . . . 4.2.1 Interpolated Regularity of Solutions .. . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.2 The Method by the Laplace Transform for Fractional Partial Differential Equations . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

47 47 48 55 59 62 69 73 73 78 78 78 80 88

ix

x

Contents

4.5 Proofs of Propositions 4.1 and 4.2 . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 96 4.5.1 Proof of Proposition 4.1 . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 96 4.5.2 Proof of Proposition 4.2 . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 98 4.6 Case Where the Coefficients of A(t) Are Independent of Time . . . . . . 101 5 Decay Rate as t → ∞.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Key Lemmata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Completion of Proof of Theorem 5.1 . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

109 109 111 115

6 Concluding Remarks on Future Works . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 121 A Proofs of Two Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 123 Notation . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 129 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 131 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 133

Chapter 1

Basics on Fractional Differentiation and Integration

Derivatives of orders of natural numbers have been widely known and commonly used since the origin of the calculus in the seventeenth century. On the other hand, derivatives whose orders are not necessarily natural numbers seem not to be wellknown but have been considered since Leibniz (e.g., Ross [25]). Moreover, we can refer to a historical paper [1] by N.H. Abel. He discussed the following problem: Given a function f (t), find a function v(t) such that 1 √ π



t

(t − s)−1/2 v(s)ds = f (t),

0 ≤ t ≤ T.

(1.1)

0

This is an integral equation and appears to be related to the following mechanical problem: we consider that a mass goes along a curve on the vertical surface only by gravity starting from a point O to the end point P, and assume that the travel time is given by a function f (t), where t is the distance in the vertical direction from O. Then, determine a shape of such a curve realizing the traveling time f (t). Abel estabilished that for given f ∈ C 1 [0, T ], the solution v to (1.1) exists uniquely and is given by 1 d v(t) = √ π dt



t

(t − s)−1/2 f (s)ds,

0 ≤ t ≤ T,

β > 0.

(1.2)

0

Here and henceforth by (·) we denote the gamma function: 

∞

(β) =

e−t t β−1 dt,

β > 0.

0

We note that (n + 1) = n! := n(n − 1) · · · 3 · 2 · 1 for n ∈ N. We can regard (1.2) as the half-order derivative of f as follows. We note t 1 n that (n+1) 0 (t − s) v(s)ds is a function which is obtained by (n + 1)-times © The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 A. Kubica et al., Time-Fractional Differential Equations, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-15-9066-5_1

1

2

1 Basics on Fractional Differentiation and Integration

integrating v(t). Thus we can interpret

t

√1 d π dt

0 (t

− s)−1/2 f (s)ds as a function

which is differentiated 12 = − 12 + 1-times. That is, setting n = − 12 formally in t √ 1 n (n+1) 0 (t − s) v(s)ds and noting that π = (1/2), we see that the right-hand side of (1.2) means the half-times derivative of f . Thus we can interpret that (1.2) means that the half-order derivative of f is v when f is the half-times integration of v, and establishes the inversion of the half-order integral in terms of the half-order derivative. As for other applications of fractional derivatives, we refer to Gorenflo and Vessella [8]. Although it is later seen that the rigorous treatments are necessary, in this chapter we intuitively discuss fractional calculus as an introduction. Throughout the book we assume 0 < α < 1. In this chapter, as functions we mainly consider u ∈ L1 (0, T ) satisfying L1 (0, T ), if not specified. We set Dtα u(t)

1 d := (1 − α) dt

dtα u(t)

1 := (1 − α)



t

du dt

∈

(t − s)−α u(s)ds

(1.3)

du (s)ds. ds

(1.4)

0

and 

t

(t − s)−α

0

We call Dtα and dtα the Riemann–Liouville derivative and the Caputo derivative respectively. The notations of the derivatives are different from e.g., Kilbas et al. [15], Podlubny [24], and in Chap. 2, the fractional derivative ∂tα in fractional Sobolev spaces is defined and plays an essential role rather than the classical Riemann–Liouville derivative and the Caputo derivative. Throughout this book, we interpret equalities in variables t, x, etc., as that they hold for almost all t, x in domains under consideration, and we do not specify if there is no fear of confusion. In this book, we mainly consider the case 0 < α < 1, but for general α > 0, ∈ N, choosing m ∈ N satisfying m − 1 < α < m, for u ∈ C m [0, T ], we can define dtα and Dtα by dtα u(t)

1 = (m − α)



t

(t − s)m−α−1

0

d mu (s)ds ds m

and Dtα u(t) =

dm 1 (m − α) dt m



t 0

(t − s)m−α−1 u(s)ds.

1 Basics on Fractional Differentiation and Integration

3

Moreover, for α > 0, we define the Riemann–Liouville fractional integral operator by J α f (t) =

1 (α)



t

(t − s)α−1 f (s)ds,

(1.5)

0

which we can regard as an operation of α-times integration. β

Example 1.1 Let 0 < α < 1 and β > 0. Then t β , dtdt = βt β−1 ∈ L1 (0, T ) and we have dtα t β = Dtα t β =

(β + 1) β−α t . (1 − α + β)

(1.6)

In fact, this follows from the classical formula: 

b

(t − a)p−1 (b − t)q−1 dt =

a

(p)(q) (b − a)p+q−1 , (p + q)

p, q > 0, a < b. (1.7)

We can prove also: Lemma 1.1 Let u ∈ C 1 [0, T ]. Then for 0 < α < 1, we have dtα u(t) +

u(0) −α t = Dtα u(t), (1 − α)

0 ≤ t ≤ T.

(1.8)

Proof Since u ∈ C 1 [0, T ], integration by parts yields 

t 0

(t − s)

−α



u(s)(t − s)1−α u(s)ds = 1−α =

s=0 s=t

u(0) 1−α 1 t + 1−α 1−α



1 + 1−α t



t 0

(t − s)1−α

0

(t − s)1−α

du (s)ds ds

du (s)ds. ds

By differentiating both sides, the proof is complete by 1 − α > 0.



The first-order derivative of a constant function vanishes identically. Also the Caputo derivative with order α of a constant function is zero: ∂tα 1 = 0. However, the Riemann–Liouville derivative is not zero. More precisely, Dtα 1 =

t −α . (1 − α)

The fundamental theorem of calculus reads  t du (s)ds = u(t) − u(0), 0 ds

0 < t < T,

4

1 Basics on Fractional Differentiation and Integration

which indicates that the integration is a converse operation to the differentiation. We can have a similar inversion for dtα and Dtα , whose special case is (1.2). More precisely, as an important formula, we can prove J α dtα u(t) = u(t),

0 0 such that |y(s)| ≤ Cδ |t1 − s|, δ ≤ s ≤ T . Therefore integrating by parts, we have α (t1 − δ)−α y(δ) − I2 = − (1 − α) (1 − α)



t1

(t1 − s)−α−1 y(s)ds.

δ

By t1 − δ > 0 and y(s) ≥ 0, 0 ≤ s ≤ T , we see that I2 ≤ 0. Hence by (1.16), we obtain ∂tα y(t1 ) ≤ ε. Here since ε > 0 is arbitrary, we let ε → 0 and the proof of Lemma 1.2 is complete.  Lemma 1.2 does not hold for α > 1. It follows from Lemma 1.2 that if dtα u(t) > 0, 0 ≤ t ≤ T , then u(t) ≥ u(0) for all t ∈ [0, T ]. In fact, assume that there exists t0 ∈ [0, T ] such that u(t0 ) = min0≤t ≤T u(t) < u(0). Then t0 = 0. That is, 0 < t0 ≤ T . Lemma 1.2 yields that dtα u(t0 ) ≤ 0, which contradicts the assumption dtα u(t) > 0. Thus the proof is completed. This is much weaker than the usual property for du dt , and we do not know whether u(t0 ) ≥ u(t1 ) if dtα u(t) ≤ 0, t0 < t < t1 with 0 < t0 < t1 < T . Throughout this book, we set  

1,1 1 W (0, T ) := u ∈ L1 (0, T ); du dt ∈ L (0, T ) , (1.17) 1,1 (0, T ) := {u ∈ W 1,1 (0, T ); u(0) = 0} 0W with the norm

   du  

u W 1,1 (0,T ) = u L1 (0,T ) +  .  dt  1 L (0,T )

We note that u ∈ W 1,1 (0, T ) if and only if there exists an absolutely continuous function  u on [0, T ] such that  u(t) = u(t) for almost all t ∈ (0, T ). Here and henceforth we identify  u with u for u ∈ W 1,1 (0, T ). We close this chapter with the following lemma which generalizes Lemma 1.1 to u ∈ W 1,1 (0, T ), Lemma 1.3 (i) For u ∈ W 1,1 (0, T ), we have J 1−α u ∈ W 1,1 (0, T ) and u(0) −α d 1−α du J t , u(t) = J 1−α (t) + dt dt (1 − α)

0 < t < T.

1 Basics on Fractional Differentiation and Integration

7

(ii) Let un , u ∈ W 1,1 (0, T ) and un −→ u in W 1,1 (0, T ). Then dtα un −→ dtα u in L1 (0, T ). (iii) dtα u = Dtα u for u ∈ W 1,1 (0, T ) satisfying u(0) = 0. (iv) Let α, β > 0. Then J α (J β u) = J α+β u,

u ∈ L2 (0, T ).

By the Sobolev embedding (e.g., formula (1.20) on p. 193 in Kato [13], Theorem 5.4 in Adams [2]), we see that W 1,1 (0, T ) ⊂ C[0, T ] and so u(0) makes 1,1 sense for  t u ∈ W (0, T ). We can interpret Lemma 1.3 (i) so that the exchange of d dt and 0 · · · ds is possible in 

t

(t − s)−α u(s)ds.

0

Proof (i) and (iii) By u ∈ W 1,1 (0, T ), we have 

s

u(s) = 0

du (ξ )dξ + u(0), dξ

0 < s < T.

Hence   s  t du 1 (ξ )dξ + u(0) ds (t − s)−α (1 − α) 0 0 dξ    t  s  t 1 1 du = (t − s)−α (t − s)−α ds u(0) (ξ )dξ ds + (1 − α) 0 (1 − α) 0 0 dξ   t  t 1 t 1−α 1 du (ξ )dξ + u(0). = (t − s)−α ds (1 − α) 0 dξ (1 − α) 1 − α ξ (J 1−α u)(t) =

Here we have exchanged the orders of integrals. Therefore (J 1−α u)(t) = +

1 (1 − α)(1 − α)

1 t 1−α u(0), (1 − α)(1 − α)



t

(t − ξ )1−α

0

du (ξ )dξ dξ

0 < t < T.

Taking the differentiation in t, noting that 1 − α > 0, we see that J 1−α u ∈ W 1,1 (0, T ) and u(0) −α d 1−α du J t . u(t) = J 1−α (t) + dt dt (1 − α)

(1.18)

8

1 Basics on Fractional Differentiation and Integration

Here we use Lemma A.1 in the Appendix, which is called the Young inequality for the convolution:   1 du 1−α du −α (t) = t ∗ (t) J dt (1 − α) dt and        1−α du    du   1 −α  J     t  ≤ < ∞.     dt L1 (0,T ) (1 − α) L1 (0,T ) dt L1 (0,T ) Moreover, (1.18) directly implies conclusion (iii).



Proof of (ii) We have    t du dun 1 (s) − (s) ds (t − s)−α (1 − α) 0 ds ds  d ∗ (un − u) (t). dt

dtα un (t) − dtα u(t) =  =

1 t −α (1 − α)

Again the Young inequality Lemma A.1 for the convolution yields

dtα un ≤

− dtα u L1 (0,T )

  d  1 −α 

t L1 (0,T )  (un − u) ≤  1 (1 − α) dt L (0,T )

T 1−α 1

un − u W 1,1 (0,T ) . (1 − α) 1 − α

Thus the proof of Lemma 1.3 (ii) is complete.



Proof of (iv) Exchanging the orders of the integrals and applying (1.7), we verify   s  t 1 1 (t − s)α−1 (s − ξ )β−1 u(ξ )dξ ds (α) (β) 0 0   t  t (t − s)α−1 (s − ξ )β−1 ds u(ξ )dξ

J α J β u(t) = = =

1 (α)(β) 1 (α)(β)



0

ξ t

0

(α)(β) (t − s)α+β−1 u(ξ )dξ = J α+β u(t). (α + β) 

Chapter 2

Definition of Fractional Derivatives in Sobolev Spaces and Properties

2.1 Motivations We consider a very simple equation dtα u(t) = f (t),

u(0) = a

0 < t < T,

(2.1)

and find u(t) satisfying (2.1) with given f ∈ L2 (0, T ) and a ∈ R. Needless to say, for α = 1, we have 

t

u(t) =

f (s)ds + a,

0 < t < T.

0

If α = 1 and u satisfies (2.1) with given f ∈ L1 (0, T ), then u ∈ C[0, T ], so that u(0) can be calculated by substituting t = 0 into u(t). In other words, for α = 1, for any f ∈ L2 (0, T ) and a ∈ R, there exists a unique u ∈ H 1 (0, T ) satisfying (2.1). The situation is different from the case of 0 < α < 1. In most of the existing works, for f ∈ L2 (0, T ) or f ∈ L1 (0, T ), detailed regularity of the solution to dtα u = f is not clarified and so the initial condition can only be justified if the solutions are proved to possess a certain regularity at t = 0. The class f ∈ L2 (0, T ) is not sufficient for such regularity. For example, Theorem 3.3 (p. 126) in Podlubny [24] assumes that f ∈ C[0, T ]. For fractional differential equations, the Laplace transform: 

∞

(Lu)(p) :=

e−pt u(t)dt,

p > p0 :

some constant

0

© The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 A. Kubica et al., Time-Fractional Differential Equations, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-15-9066-5_2

9

10

2 Definition of Fractional Derivatives in Sobolev Spaces and Properties

is convenient, but the justification of the initial value is indispensable in order to use the Laplace transform. Because by some repetition of calculations, the formula L(dtα u)(p) = pα (Lu)(p) − pα−1 u(0) is known (e.g., p. 106 in [24]) and we have to justify the sense of u(0), which requires a certain smoothness of u at t = 0. Such regularity at t = 0 is not well established for f ∈ L2 (0, T ). Moreover, for such f, we have to justify the solution formula for (2.1) (e.g., Kilbas et al. [15, p. 141]). We illustrate with one simple example. Let 0 < α < 12 and 1

f (t) = t δ− 2 ,

0 < t < T,

where δ > 0 is a constant. We consider 1

dtα u(t) = t δ− 2 ,

u(0) = a.

(2.2)

Since δ − 12 > −1, we can formally apply the solution formula (e.g., [15, p. 141]) and obtain  t 1 1 1 (t − s)α−1 s δ− 2 ds = a + C0 t α+δ− 2 , (2.3) u(t) = a + (α) 0 where we set   δ + 12 . C0 =   α + δ + 12 Moreover, u(t) given by (2.3) cannot satisfy (2.2) if 0 < α < 12 and δ > 0 is small such that α + δ − 12 < 0. Indeed, limt ↓0 u(t) = ∞, and so the initial condition 1

does not follow common sense. Furthermore we formally calculate dtα t α+δ− 2 : 1

dtα t α+δ− 2 =

1 (1 − α)

 0

t

(t − s)−α

α + δ − 12 d α+δ− 1 2 )ds = (s ds (1 − α)



t

(t − s)−α s α+δ− 2 ds. 3

0

However, since α +δ − 32 < −1, the integral does not exist. This means that formula (2.3) does not hold for f ∈ L2 (0, T ) in general, although (2.3) is convergent for e.g., f ∈ L∞ (0, T ) as a solution formula to (2.1). We can verify that (2.3) gives 2 α a solution to (2.1) if du dt ∈ L (0, T ), because we can calculate dt u pointwise. In terms of an extended Caputo derivative, we will further discuss this later (see, e.g., (2.36)). In some references (e.g., Podlubny [24, pp. 78–79]), it is emphasized that the Caputo derivative dtα can admit the initial value problem (2.1), unlike the Riemann– 1 Liouville derivative Dtα . However, this simple example f (t) = t δ− 2 shows that the

2.2 Preliminaries: Operational Structure of J α

11

pointwise Caputo derivative dtα cannot make the initial value problem (2.1) wellposed for general f ∈ L2 (0, T ). Moreover, for guaranteeing the uniqueness of u satisfying dtα u = f ∈ L2 (0, T ), we certainly need some extra conditions. In fact, since dtα 1 = 0 by the definition, if u satisfies dtα u = f in (0, T ), then u + C1 satisfies the same equation with arbitrary constant C1 . These discussions suggest the necessity for reconsidering the pointwise Caputo derivative dtα and redefining the Caputo derivative within the framework of L2 (0, T ). The function space L2 (0, T ) is reasonable and convenient as data space. Hence it is natural to formulate the initial value problem and define dtα for f ∈ L2 (0, T ) in order to establish a more unified theory for fractional differential equations. Thus we construct the theory where the fractional derivatives should be included in L2 (0, T ). This is our main motivation in this book, and we construct a seemingly different fractional derivative ∂tα , but we will prove that it is essentially same as the closure operator of the Caputo derivative in 0 C 1 [0, T ] (see Sect. 2.3). First in Sects. 2.2 and 2.3, we define the generalized fractional derivative ∂tα in Sobolev spaces Hα (0, T ). Then in Chaps. 3 and 4, in terms of such ∂tα , we formulate an initial value problem and prove the well-posedness. Our formulation is similar to Zacher [30] (see also Kubica and Yamamoto [17]), but more essentially relies on the property of the generalized fractional derivative ∂tα defined later in some Sobolev space. These properties are feasible for the applications such as the clarification of the Sobolev regularity of solutions to initial boundary value problems.

2.2 Preliminaries: Operational Structure of J α By the direct observation of the definition (1.4) of the pointwise Caputo derivative, α we need the first derivative du ds (s) in order to define the derivative dt u of order α < 1. In order to define such an adequate fractional derivative, which is denoted by ∂tα , we should fulfill the following: 1. ∂tα should be well-defined in a subspace of the Sobolev space of order α; 2. the norm equivalence between ∂tα u L2 (0,T ) and some conventional norm of u such as the norm in a Sobolev space. For them, • We will interpret J α as the fractional power of the operator defined by  t J u(t) = u(s)ds 0

with the domain D(J ) = L2 (0, T ). • We define ∂tα as the inverse to J α . These issues are done respectively in this section and Sect. 2.3.

12

2 Definition of Fractional Derivatives in Sobolev Spaces and Properties

The arguments in this section and a part of Sect. 2.3 are based on Gorenflo and Yamamoto [9], Gorenflo et al. [11]. By L2 (0, T ) and H α (0, T ) we mean the usual L2 -space and the fractional Sobolev space on the interval (0, T ) (see e.g., [2, Chapter VII]), respectively, and we define the norm in H α (0, T ) by

u

H α (0,T )

  2 := u L2 (0,T ) +

T 0



T 0

|u(t) − u(s)|2 dtds |t − s|1+2α

 12 .

The L2 -norm and the scalar product in L2 are denoted by · L2 (0,T ) and (·, ·) = (·, ·)L2 (0,T ) , respectively. By ∼ we denote a norm equivalence. Since J α is injective in L2 (0, T ), by J −α we denote the algebraic inverse to J α . We set 0H

α

(0, T ) = {u ∈ H α (0, T ); u(0) = 0} if

1 < α ≤ 1. 2

We further define the Banach spaces ⎧ 1 α α < 1, ⎪ ⎨ 0H (0, T1 ), 2 <   T |v(t )|2 2 v ∈ H (0, T ); 0 Hα (0, T ) := dt < ∞ , t ⎪ ⎩ α 1 H (0, T ), 0 < α < 2

α = 12 ,

with the following norm:

v Hα (0,T )

⎧ ⎪ ⎨

 = ⎪ ⎩ v 2

H

1 2 (0,T )

+

T 0

v H α (0,T ) , 0 < α < 1, α = 12 , 1 2 |v(t )|2 , α = 12 . t dt

(2.4)

For later arguments, we need some convenient and not too narrow subspace of Hα (0, T ). In particular, for H 1 (0, T ), such a subspace is more delicate. 2 We recall 0W

1,1

(0, T ) = {u ∈ W 1,1 (0, T ); u(0) = 0}

and introduce the following space:

Wα (0, T ) := u ∈ W 1,1 (0, T ); there exists a constant Cu > 0 such that    du   (t) ≤ Cu t α−1 almost all t,  dt 

 u(0) = 0 .

(2.5)

Here Cu > 0 depends on a choice of u. We note that t β ∈ Wα (0, T ) for any β ≥ α.

2.2 Preliminaries: Operational Structure of J α

13

Henceforth by C > 0, C1 > 0, etc., we denote generic constants which are independent of functions under consideration but dependent on α, T , while Cu means that it depends on a function or a quantity u under consideration. Then we can prove: Lemma 2.1 Let 0 < α < 1. Then Wα (0, T ) ⊂ Hα (0, T ). Proof First we prove 

T



0

T

(s γ1 + t γ1 )|t − s|γ2 dsdt < ∞ if γ1 , γ2 > −1.

(2.6)

0

 Proof of (2.6) The conclusion (2.6) is trivial for γ1 ≥ or γ2 ≥ 0. Therefore we can assume that −1 < γ1 , γ2 < 0. We note 



T 0

T



T

(s γ1 + t γ1 )|t − s|γ2 dsdt = 2

0



0

t

 (s γ1 + t γ1 )|t − s|γ2 ds dt.

(2.7)

0

Indeed, 

T



0



T

T

(s γ1 + t γ1 )|t − s|γ2 dsdt =

0



0

0





T



t

+

 (s γ1 + t γ1 )|t − s|γ2 ds dt.

t

On the other hand, 



T

T

(s 

0



T

= 0

t t

γ1



T

+ t )|t − s| ds dt = γ1

γ2

(s 0

 (t γ1 + s γ1 )|s − t|γ2 ds dt.



s

γ1

+ t )|t − s| dt ds γ1

γ2

0

0

In the last equality, we exchange s and t. Then 

T





T

(s 0

 =2 0

=2

0

T



t

γ1



T

+ t )|t − s| ds dt = 2 γ1

γ2

  s γ1 (t − s)γ2 ds dt + 2

0

(γ1 + 1)(γ2 + 1) (γ1 + γ2 + 2)

 0

T

  0

γ1

+ t )|t − s| ds dt γ1

γ2

0

t

t γ1

t γ1 +γ2 +1 dt + 2



t

(s 0

0 T



 (t − s)γ2 ds dt

0 T

t γ1 +γ2 +1 dt < ∞ 1 + γ2

by γ1 > −1 and γ2 > −1. Thus the proof of (2.6) is completed.



14

2 Definition of Fractional Derivatives in Sobolev Spaces and Properties

Now we complete the proof of Lemma 2.1. Indeed, u(t) = Wα (0, T ). Hence   |u(t) − u(s)| =  ≤Cu min |ξ |

α−1

ξ ∈(s,t )

Let

1 2

t s

t

du 0 ds (s)

for u ∈

  t       du   du    (ξ )dξ  ≤   (ξ ) dξ  dξ s dξ

(t − s) ≤ Cu (t α−1 + s α−1 )|t − s|.

< α < 1. Then |u(t) − u(s)|2 ≤ C(t 2α−2 + s 2α−2 )|t − s|1−2α . |t − s|1+2α

Since 2α − 2 > −1 by (2.6) implies

1 2

< α < 1 and 1 − 2α > −1 by 0 < α < 1, the inequality



T 0



T 0

|u(t) − u(s)|2 dsdt < ∞. |t − s|1+2α

Next let 0 < α ≤ 12 . Similarly to (2.7), we can verify  0

T

 0

T

|u(t) − u(s)|2 dsdt = 2 |t − s|1+2α

 0

T



t 0

 |u(t) − u(s)|2 ds dt. |t − s|1+2α

Since    du   (ξ ) ≤ Cu ξ α−1 ≤ Cu s α−1  dξ  for ξ ∈ (s, t), we choose δ > 0 such that 0 < δ < value theorem we can choose η ∈ (s, t) such that

α 1−α

≤ 1. Then, by the mean

|u(t) − u(s)|2 = |u(t) − u(s)|1−δ |u(t) − u(s)|1+δ  1+δ 

1−δ  du    ≤ 2 u L∞ (0,T ) ≤ C(ξ α−1 (t − s))1+δ  dt (η) |t − s| ≤Cs (α−1)(1+δ)(t − s)1+δ for 0 < s < t < T . Consequently, since α + αδ − δ > 0 and 1 + δ − 2α > 0 by α 0 < α ≤ 12 and 0 < δ < 1−α , the inequality (2.6) yields 

T 0

 0

t

|u(t) − u(s)|2 dsdt ≤ C |t − s|1+2α

 0

T

 0

t

 s (α+αδ−δ)−1 (t − s)(1+δ−2α)−1 ds dt < ∞.

2.2 Preliminaries: Operational Structure of J α

15

Thus we proved u ∈ H α (0, T ). Therefore u ∈ Hα (0, T ) = H α (0, T ) for 0 < α < 12 . Next for 12 < α < 1, by u(0) = 0, we readily see that u ∈ Hα (0, T ) = 1 α 0 H (0, T ). Therefore Wα (0, T ) ⊂ Hα (0, T ) if 0 < α < 1 and α = 2 . Moreover, for α = 12 , we have   |u(t)| = 

t 0

  t  1 1 du (s)ds  ≤ Cu s − 2 ds = 2Cu t 2 , ds 0

T 2 and so 0 |u(tt )| dt < ∞, which implies that H 1 (0, T ). Thus the proof of 2 Lemma 2.1 is completed.  In fact, the space Wα (0, T ) is a convenient subspace of Hα (0, T ). Remark 2.1 For H 1 (0, T ), Lions and Magenes [19] use a different notation 2

1 2

(0, T ) (Remark 11.5 (p. 68) in [19] vol. I). However, we use H 1 (0, T ) as well 2 as Hα (0, T ), 0 < α < 1 throughout this book.

0 H0

Henceforth we set 0C

1

[0, T ] = {ϕ ∈ C 1 [0, T ]; ϕ(0) = 0}

X denotes the closure of Z by the norm of X. and Z Lemma 2.2 0C

1 [0, T ]

Hα (0,T )

= Hα (0, T ).

Proof For 12 < α ≤ 1, we see that Hα (0, T ) = 0 H α (0, T ) := {u ∈ H α (0, T ); u(0) = 0}, so that the mollifier (e.g., Adams [2]) yields the conclusion. Let 0 ≤ α ≤ 12 . By Lions and Magenes [19], we have Hα (0, T ) = [0 H 1 (0, T ), L2 (0, T )]1−α which is the interpolation space. Applying Proposition 6.1 (p. 28) in [19], for 12 < γ < 1, we see that [0 H 1 (0, T ), L2 (0, T )]1−γ is dense in [0 H 1 (0, T ), L2 (0, T )]1−α by 1 − γ ≤ 1 − α. Therefore Hγ (0, T ) for

1 2

Hα (0,T )

= Hα (0, T )

< γ < 1. As is already proved, we see that 0C

H (0,T ) 1 [0, T ] γ

= Hγ (0, T ).

16

2 Definition of Fractional Derivatives in Sobolev Spaces and Properties

Both density properties yield 0C

1 [0, T ]

Hα (0,T )

= Hα (0, T ) 

for 0 ≤ α ≤ 12 . Thus the proof of the lemma is completed.

For 0 < α < 12 , the proof of the lemma is more direct by Theorem 11.1 (p. 55) in [19]. Lemma 2.2 is useful, because, thanks to the lemma, in order to prove estimates in Hα (0, T ), it usually suffices to prove them for 0 C 1 [0, T ]. Remark 2.2 By Theorem 11.1 (p. 55) in [19], we see that 0C

1 [0, T ]

H α (0,T )

= H α (0, T ),

We should distinguish Hα (0, T ) from H α (0, T ). For that 0 C 1 [0, T ] is dense in Hα (0, T ). To sum up, 0

H α (0,T ) C 1 [0, T ]

=

0≤α≤ 1 2

1 . 2

(2.8)

< α ≤ 1, the mollifier yields

H α (0, T ), 0 < α ≤ 12 , Hα (0, T ), 12 < α < 1.

(2.9)

By Theorem 2.1 in Gorenflo et al. [11], we know the following theorem. Theorem 2.1 Let 0 < α < 1. (i) J α : L2 (0, T ) −→ Hα (0, T ) is injective and surjective. (ii) There exists a constant C > 0 such that C −1 J α u Hα (0,T ) ≤ u L2 (0,T ) ≤ C J α u Hα (0,T )

(2.10)

for all u ∈ L2 (0, T ). Moreover, we can prove J α L2 (0, T ) ⊂ H1 (0, T ),

α ≥ 1.

However, in this book we limit the range of α to 0 < α ≤ 1 and we omit further characterization of J α L2 (0, T ). In terms of Theorem 2.1, (J α )−1 exists and J −α = (J α )−1 . Then: Theorem 2.2 There exists a constant C > 0 such that C −1 J −α v L2 (0,T ) ≤ v Hα (0,T ) ≤ C J −α v L2 (0,T ) for all v ∈ Hα (0, T ).

(2.11)

2.2 Preliminaries: Operational Structure of J α

17

By Theorem 2.1, we see: Corollary 2.1 Let 0 < α < 1. Then J −α J α u = u,

u ∈ L2 (0, T )

J α J −α u = u,

u ∈ Hα (0, T ).

and

The first equality is directly seen by the definition, while the second equality is verified as follows. For u ∈ Hα (0, T ), Theorem 2.1 (i) yields the existence of w ∈ L2 (0, T ) satisfying u = J α w. Therefore J α J −α u = J α J −α (J α w) = J α w. Hence J α J −α u = u for u ∈ Hα (0, T ). Henceforth we write for example (2.10)

J −α v L2 (0,T ) ∼ v Hα (0,T ) , when there is no fear of confusion. 1  t Now we prove Theorem 2.1. By 2J = J we 2denote the integral (Jy)(t) = 0 y(s)ds for 0 ≤ t ≤ T and by I : L (0, T ) → L (0, T ) the identity mapping. In this section, we consider the space L2 (0, T ) over C with the scalar product (u, v) = T (u, v)L2 (0,T ) = 0 u(t)v(t)dt for complex-valued u, v, and Re η and Im η denote the real and the imaginary parts of a complex number η, respectively. Let η¯ denote the complex conjugate of η ∈ C. Henceforth if there is no fear of confusion, then by (·, ·) we mean (·, ·)L2 (0,T ) , and R(A) and D(A) denote the range and the domain of an operator A: AD(A) = R(A) respectively. Lemma 2.3 For any u ∈ L2 (0, T ), the inequality Re (J u, u) ≥ 0 holds true and R(I + J ) = L2 (0, T ). Proof With the notations Re J u(t) = ϕ(t) and Im J u(t) = ψ(t), the following chain of equalities and a final estimate can be easily obtained: 

T

Re (J u, u) = Re 0



T

= 0



t





T

u(s)ds u(t)dt = Re

0

J u(t) 0

d J u(t)dt dt

  dψ 1 dϕ 1 2 + ψ(t) ϕ(t) dt = (ϕ(t)2 + ψ(t)2 )|tt =T =0 = |J u(T )| ≥ 0. dt dt 2 2

Therefore Re (J u, u) ≥ 0 for u ∈ L2 (0, T ). Next for λ = 0, we have the representation (λI + J )−1 u(t) = λ−1 u(t) − λ−2



t 0

e−(t −s)/λu(s)ds,

0≤t ≤T

(2.12)

18

2 Definition of Fractional Derivatives in Sobolev Spaces and Properties

for u ∈ L2 (0, T ). We will prove (2.12). We set v(t) = (λI + J )−1 u(t), that is, 

t

λv(t) +

v(s)ds = u(t),

0 < t < T.

(2.13)

0

Let u ∈ C 1 [0, T ]. Then v(0) =

1 u(0). λ

(2.14)

Differentiating (2.13) with respect to t, we have dv 1 1 du (t) + v(t) = (t). dt λ λ dt With (2.14), we obtain v(t) =



1 −1t e λ λ

t

1

eλs

0

 du (s)ds + u(0) . dt

Using integration by parts, we have v(t) =

1 1 u(t) − 2 λ λ



t

e−

t−s λ

u(s)ds,

0 < t < T.

(2.15)

0

Next let u ∈ L2 (0, T ). Then the right-hand side of (2.15) defines a function v ∈  t  s

 t  t 2 L (0, T ). Exchanging the orders of the integrals: 0 0 dξ ds = 0 ξ ds dξ , we calculate   t  t  s  s−ξ 1 t 1 v(s)ds = u(s)ds − 2 e− λ u(ξ )dξ ds λ 0 λ 0 0 0   t  t  t ξ s 1 1 = u(s)ds − 2 e− λ ds e λ u(ξ )dξ λ 0 λ 0 ξ  t  t  ξ s s=t 1 1 e− λ u(s)ds + e λ u(ξ )dξ = s=ξ λ 0 λ 0  t  t  ξ 1 −t 1 t 1 λ λ u(s)ds + e e u(ξ )dξ − u(ξ )dξ = λ 0 λ λ 0 0  1 t t s e λ u(s)ds. = e− λ λ 0 This and (2.15) yield λv(t) + which is (2.12).

t 0

v(s)ds = u(t), 0 < t < T for each u ∈ L2 (0, T ),

2.2 Preliminaries: Operational Structure of J α

19

Setting λ = 1, by (2.12) the operator (I + J )−1 u is defined for all u ∈ which implies that R(I + J ) = L2 (0, T ). The proof of Lemma 2.3 is completed.  L2 (0, T ),

In terms of Lemma 2.3, for 0 < α < 1, we can define the fractional power of J , which we denote by J (α) (e.g., Tanabe [27, Chapter 2]). More precisely, we have the formula  sin πα ∞ α−1 J (α)u = λ (λI + J )−1 J u dλ, u ∈ D(J ) = L2 (0, T ) (2.16) π 0 (see also Chapter 2, §3 in [27]). Next we will prove that the fractional power J (α) of the integral operator J coincides with the Riemann–Liouville fractional integral operator J α on L2 (0, T ): Lemma 2.4 (J (α)u)(t) = (J α u)(t), 0 ≤ t ≤ T , u ∈ L2 (0, T ), 0 < α < 1. Proof By (2.12), we have (λI + J )

−1

−1



t

J u(t) = λ

e−(t −s)/λu(s)ds,

u ∈ L2 (0, T )

0 t −s λ ,

and by change of variables η = sin πα π sin πα = π = =

sin πα π sin πα π

 

∞

λα−1 (λI + J )−1 J u(t) dλ

0



∞

0

t

α−2

λ 

0

e 0



t



0



t

x γ e−ax dx =



 u(s)ds dλ

 λα−2 e−(t −s)/λdλ ds

∞

 η−α e−η dη (t − s)α−1 ds

0

u(s) 0

−(t −s)/λ

∞

u(s)

(1 − α) sin πα = π Here we use also  ∞

we obtain

0 t

u(s)(t − s)α−1 ds.

0

(γ + 1) , a γ +1

a > 0, γ > −1.

Now the known formula (1 − α)(α) = lemma.

π sin πα

(2.17)

implies the statement of the

20

2 Definition of Fractional Derivatives in Sobolev Spaces and Properties

Next we consider the differential operator ⎧ ⎪ d 2 u(t) ⎪ ⎨ (Su)(t) = − , 0 < t < T, dt 2

 du ⎪ ⎪ D(S) = u ∈ H 2 (0, T ); u(0) = (T ) = 0 . ⎩ dt

(2.18)

Here we note that the boundary conditions u(0) = du dt (T ) = 0 should be interpreted as the traces of u in the Sobolev space H 2 (0, T ) (see e.g., [2, 19]). It is possible to α define the fractional power S 2 of the differential operator S for 0 ≤ α ≤ 1 in terms of the eigenvalues and the eigenfunctions of the eigenvalue problem for the operator S. More precisely, let 0 < λ1 < λ2 < · · · be the eigenvalues and ψk , k ∈ N the well known corresponding normed eigenfunctions of S. It is easy to derive the explicit √ √ (2k−1)2 π 2 2 formulas for λk , ψk , k ∈ N, namely, λk = and ψk (t) = √ sin λk t. In 4T 2 T

particular, we note that ψk (0) = 0 and ψk ∈ H 2 (0, T ). It is known that {ψk }k∈N is α an orthonormal basis of L2 (0, T ). Then the fractional power S 2 , 0 ≤ α ≤ 1 of the differential operator S is defined by the relations ⎧ α ∞ α2 α ⎪ 2u = S ψk )L2 (0,T ) ψk , u ∈ D(S 2 ), ⎪ k=1 λk (u, ⎨ α α 2 D(S 2 ) = {u ∈ L2 ; ∞ k=1 λk |(u, ψk )L2 (0,T ) | < ∞}, ⎪ 1

 ⎪ ∞ ⎩ u

α = λα |(u, ψk ) 2 |2 2 . D(S 2 )

(2.19)

L (0,T )

k=1 k

α

According to [9], the domain D(S 2 ) can be described as follows: α

D(S 2 ) = Hα (0, T ).

(2.20)

The relation (2.20) holds not only algebraically but also topologically: α

S 2 v L2 (0,T ) ∼ v Hα (0,T ) ,

α

0 ≤ α ≤ 1, v ∈ D(S 2 ).

(2.21)

α

In particular, the inclusion D(S 2 ) ⊂ H α (0, T ) holds true. Now we are ready to prove Theorem 2.1.



Proof of Theorem 2.1 We first state the Heinz–Kato inequality (e.g., Theorem 2.3.4 in Tanabe [27]): let X be a Hilbert space and linear operators A, B in X satisfy D(A) = D(B), Re (Au, u) ≥ 0, Re (Bu, u) ≥ 0 for u ∈ D(A) and R(I + A) = R(I + B) = X. We assume that there exists a constant C > 0 such that

Bu ≤ C Au ,

u ∈ D(A).

Then, for 0 ≤ α ≤ 1, we see that D(Aα ) = D(B α ) and

B α u ≤ eπ

√ α(1−α)

C α Aα u ,

u ∈ D(Aα ).

2.3 Definition of Generalized ∂tα in Hα (0, T )

21

The proof of the theorem is done by using the norm equivalence between α and Hα (0, T ), which is justified by the Heinz–Kato inequality.

·

2 D(S )

First of all, it can be directly verified that D(J −1 ) = J (L2 (0, T )) = 0 H 1 (0, T ), dw(t) , and J −1 v L2 (0,T ) = v H 1 (0,T ) for v ∈ 0 H 1 (0, T ). (J −1 w)(t) = dt Therefore by (2.21) we obtain the norm equivalence

J −1 v L2 (0,T ) ∼ S 2 v L2 (0,T ) , 1

v ∈ 0 H 1 (0, T ) = D(J −1 ) = D(S 2 ). 1

Direct calculations show that 

T 0

d(ϕ + iψ) (ϕ − iψ)dt ≥ 0, dt

1

and so both J −1 and S 2 satisfy the conditions for the Heinz–Kato inequality. Hence the Heinz–Kato inequality yields α

J −α v L2 (0,T ) ∼ S 2 v L2 (0,T ) ,

α

v ∈ D(S 2 ),

α

D(J −α ) = D(S 2 ).

(2.22)

α

By (2.20) and (2.22), we have Hα (0, T ) = D(S 2 ) = D(J −α ) = R(J α ), so that Theorem 2.1 (i) follows. By (2.21) and (2.22), the norm equivalence

J −α v L2 (0,T ) ∼ v Hα (0,T ) holds true for v ∈ D(J −α ) = R(J α ). Next, setting v = J α u ∈ D(J −α ) with any u ∈ L2 (0, T ), by (2.21), we obtain the following norm equivalence: α

u L2 (0,T ) ∼ S 2 (J α u) L2 (0,T ) ∼ J α u H α (0,T ) ,

u ∈ L2 (0, T ). 

Therefore Theorem 2.1 (ii) follows.

2.3 Definition of Generalized ∂tα in Hα (0, T ) Next we show a generalization of Theorem 2.1 (ii). Theorem 2.3 Let 0 ≤ α ≤ 1 and 0 ≤ β ≤ 1. (i) Let 0 < α + β ≤ 1. Then J α : Hβ (0, T ) −→ Hα+β (0, T ) is surjective and

J α u Hα+β (0,T ) ∼ u Hβ (0,T ) ,

u ∈ Hβ (0, T ).

(2.23)

u ∈ Hβ−α (0, T ).

(2.24)

(ii) Let 0 ≤ β − α ≤ 1. Then J α−β u = J −β J α u,

22

2 Definition of Fractional Derivatives in Sobolev Spaces and Properties

Remark 2.3 Since J α is defined by the fractional power of the operator J and J −α by J −α = (J α )−1 for 0 ≤ α ≤ 1, the general theory (e.g., Tanabe [27]) implies J α (J β u) = J α+β u,

u ∈ L2 (0, T ),

provided that α, β, α + β ∈ [−1, 1] and we understand D(J

−γ

)=

L2 (0, T ), Hγ (0, T ),

γ ≤ 0, γ ≥ 0.

Proof (i) We note that Hα (0, T ) = {J α ϕ; ϕ ∈ L2 (0, T )},

0 ≤ α ≤ 1.

First let u ∈ J α (Hβ (0, T )), that is, u = J α ϕ with some ϕ ∈ Hβ (0, T ). By Theorem 2.1 (i), there exists ψ ∈ L2 (0, T ) such that ϕ = J β ψ. Then, noting that J α is defined as fractional power of J , we see that u = J α ϕ = J α (J β ψ) = J α+β ψ, that is, u ∈ Hα+β (0, T ) by Theorem 2.1 (i). Hence J α (Hβ (0, T )) ⊂ Hα+β (0, T ). Next let u ∈ Hα+β (0, T ). Then Theorem 2.1 yields the existence of ϕ ∈ L2 (0, T ) such that u = J α+β ϕ. Since J α+β = J α J β , we obtain u = J α (J β ϕ) ∈ J α (Hβ (0, T )). Hence J α : Hβ (0, T ) −→ Hα+β (0, T ) is surjective. The norm equivalence (2.23) is verified as follows:

J α u H α+β (0,T ) ∼ J −α−β J α u L2 (0,T ) = J −β u L2 (0,T ) ∼ u Hβ (0,T ) , by u Hγ (0,T ) ∼ J −γ u L2 (0,T ) and Theorem 2.1 (ii). (ii) Let u ∈ Hβ−α (0, T ). Replacing β by β − α in Theorem 2.3 (i) and applying it, we see that J α u ∈ Hβ (0, T ) = D(J −β ), and so u ∈ D(J −β J α ). Conversely, let u ∈ D(J −β J α ). Then J α u ∈ Hβ (0, T ). Again applying Theorem 2.3 (i), where we replace β by β − α, we have u ∈ Hβ−α (0, T ), which implies Hβ−α (0, T ) = D(J −β J α ). Since D(J α−β ) = Hβ−α (0, T ) by the definition of J α−β = (J β−α )−1 , we obtain D(J −β J α ) = D(J α−β ). For u ∈ Hβ−α (0, T ), we set v := J α−β u ∈ L2 (0, T ). Then u = J β−α v ∈ Hβ−α (0, T ). Therefore J −β J α u = J −β J α (J β−α v) = J −β J β v. Here we used J α J β−α v = J β v by Theorem 2.3 (i). Hence J −β J α u = v = J α−β u. Thus (2.24) is proved. Thus the proof of Theorem 2.3 is completed.  Now we define the fractional derivative ∂tα in Hα (0, T ).

2.3 Definition of Generalized ∂tα in Hα (0, T )

23

Definition 2.1 For 0 ≤ α ≤ 1, we set ∂tα u := J −α u,

u ∈ Hα (0, T )

with the domain D(∂tα ) = Hα (0, T ). We recall that, D(∂tα ) denotes the domain of the operator ∂tα . Remark 2.4 In this book, we mainly consider ∂tα for the case of 0 < α < 1. For α > 1, ∈ N, on the basis of ∂tα with 0 < α < 1, we can define as follows: Let α = m + γ with m ∈ N and 0 < γ < 1. Then  γ

∂tα u = ∂t

d mu dt m



with

D(∂tα )

d m−1 u d mu = u ∈ H (0, T ); u(0) = · · · = m−1 (0) = 0, m ∈ Hγ (0, T ) dt dt



m

(cf. (3.39)), and we can argue the isomorphism and fractional differential equations in the same way as the later parts of this book, but we omit the details and postpone them to forthcoming publications. By Theorem 2.1, we note that Hα (0, T ) = J α L2 (0, T ). Therefore ∂tα in Hα (0, T ) is well-defined and ∂tα u ∈ L2 (0, T ) for u ∈ Hα (0, T ). Moreover, ∂tα : Hα (0, T ) −→ L2 (0, T ) is surjective. Indeed, let v ∈ L2 (0, T ) be arbitrarily given. By Theorem 2.1, we have ϕ := J α v ∈ Hα (0, T ) and so ∂tα ϕ = v by the definition, which means that ∂tα : Hα (0, T ) −→ L2 (0, T ) is surjective. On the other hand, we can prove J −1 u =

du , dt

u ∈ H1 (0, T ).

Indeed, setting v = J −1 u, we have v ∈ L2 (0, T ) by Theorem 2.1 and u = J v, t that is, u(t) = 0 v(s)ds. By v ∈ L2 (0, T ), we see that u ∈ AC[0, T ], that is, u is absolutely continuous on [0, T ] and du dt (t) = v(t) for almost all t ∈ (0, T ), that du −1 is, dt (t) = (J u)(t) for almost all t ∈ (0, T ). Replacing α and β respectively by 1 − α and 1 in Theorem 2.3 (ii), we obtain J −α = J (1−α)−1 = J −1 J 1−α , that is, J −α =

d 1−α ). dt (J

24

2 Definition of Fractional Derivatives in Sobolev Spaces and Properties

Noting that J −α = J −1 J 1−α and summing up, we can state: Theorem 2.4 Let 0 < α < 1. Then ∂tα is an isomorphism between Hα (0, T ) and L2 (0, T ). That is, ∂tα : Hα (0, T ) −→ L2 (0, T ) is injective and surjective, and

∂tα u L2 (0,T ) ∼ u Hα (0,T ) .

(2.25)

d 1−α (J u) = Dtα u, dt

(2.26)

Moreover, ∂tα u = J −α u =

u ∈ Hα (0, T )

and ∂tα u = Dtα u = dtα u

for u ∈ Hα (0, T ) ∩ 0 W 1,1 (0, T ).

(2.27)

Indeed, (2.27) follows from (2.26) and Lemma 1.3. Thus we can calculate ∂tα u by means of Dtα u for u ∈ Hα (0, T ). We are tempted to assert (2.27) for u ∈ Hα (0, T ). However, dtα u does not directly make sense for u ∈ Hα (0, T ) and with suitable extension of dtα , we can transfer (2.27) to Hα (0, T ) (see Theorem 2.5). d Formula w = ∂tα u = dt (J 1−α u) in (2.26) can correspond to the classical inversion for finding w solving J α w = u (e.g., Gorenflo and Vessella [8]) for u ∈ 1,1 (0, T ), but our construction for ∂ α guarantees the formula for u ∈ H (0, T ), 0W α t which is a wider space than the set of all absolutely continuous functions on [0, T ]. Moreover, Proposition 2.1 J α Dtα u(t) = J α dtα u(t) = u(t) for u ∈ 0 W 1,1 (0, T ). This proposition means that J α is the left inverse to Dtα and dtα for u ∈ 1,1 (0, T ). 0W Proof By Lemma 1.3, we have J 1−α

du (t) = dtα u(t) = Dtα u(t), dt

u ∈ 0 W 1,1 (0, T ).

Therefore J α J 1−α

du (t) = J α dtα u(t) = J α Dtα u(t), dt

u ∈ 0 W 1,1 (0, T ).

2.3 Definition of Generalized ∂tα in Hα (0, T )

25

By J α J 1−α = J we have du (t) = J (t) = dt dt

α 1−α du

J J

 0

t

du (s)ds = u(t) ds

as u(0) = 0. Thus the proof of the proposition is completed.



Henceforth we regard ∂tα as an operator with the domain D(∂tα ) = Hα (0, T ) if we do not specify. We can estimate ∂tα u L2 (0,T ) by the Sobolev norm but our theorem establishes also the surjectivity for the convenience for applications to fractional differential equations. We conclude this section with the equivalence of ∂tα with the closed extension of the Caputo derivative operator dtα . As for the closed extension and the closure of an operator, see e.g., Kato [13] (Chapter III, §5). We consider the classical Caputo derivative  t du 1 dtα u(t) = (t − s)−α (s)ds (1 − α) 0 ds with D(dtα ) = 0 C 1 [0, T ]. We consider dtα as an operator from D(dtα ) = 1 2 2 0 C [0, T ] ⊂ L (0, T ) to L (0, T ). α By dt we denote the closure in L2 (0, T ) of dtα with D(dtα ) = 0 C 1 [0, T ], which is the smallest closed extension of dtα . Then we prove: Theorem 2.5 We have D(dtα ) = Hα (0, T ), and dtα = ∂tα = Dtα

on Hα (0, T ).

This theorem means that our definition of ∂tα is consistent with the classical Caputo derivative by considering the closure of the operator. Proof of Theorem 2.5 We first prove that the operator dtα is closable. Indeed, let un ∈ 0 C 1 [0, T ] and un −→ 0 in L2 (0, T ) and dtα un −→ v in L2 (0, T ) with some v ∈ L2 (0, T ). Then by Lemma 1.3 and Theorem 2.4, we have dtα un = ∂tα un , n = 1, 2, 3, . . . Therefore ∂tα un −→ v in L2 (0, T ). Theorem 2.4 yields that un ∈ 0 C 1 [0, T ] ⊂ Hα (0, T ) is a Cauchy sequence in Hα (0, T ). Therefore there exists some u ∈ Hα (0, T ) such that un −→ u in Hα (0, T ). Since un −→ 0 in L2 (0, T ), we see that u = 0, that is, un −→ 0 in Hα (0, T ). Hence again by Theorem 2.4, we obtain ∂tα un = dtα un −→ ∂tα 0 = 0, that is, v = 0. Thus dtα is closable. Now we return to the proof of Theorem 2.5. We recall that u ∈ D(dtα ) if and only if there exist un ∈ 0 C 1 [0, T ] such that un −→ u in L2 (0, T ) and dtα un is a Cauchy sequence in L2 (0, T ). Since dtα un = ∂tα un for un ∈ 0 C 1 [0, T ] by Lemma 1.3 and Theorem 2.4, we see that un is a Cauchy sequence in Hα (0, T ). Therefore there exists  u ∈ Hα (0, T ) such that un −→  u in Hα (0, T ) by Theorem 2.4. Since un −→ u in L2 (0, T ), we obtain  u = u and un −→ u in Hα (0, T ) and dtα un −→ ∂tα u in

26

2 Definition of Fractional Derivatives in Sobolev Spaces and Properties

L2 (0, T ). The definition of dtα yields dtα u = lim dtα un = lim ∂tα un = ∂tα u. n→∞

n→∞

Hence we proved that D(dtα ) ⊂ Hα (0, T ) and dtα u = ∂tα u for u ∈ D(dtα ). H (0,T )

Conversely assume that u ∈ Hα (0, T ). Since 0 C 1 [0, T ] α = Hα (0, T ) by Lemma 2.2, there exist un ∈ 0 C 1 [0, T ] such that un −→ u in Hα (0, T ). Then ∂tα un = dtα un −→ ∂tα u in L2 (0, T ) by Theorem 2.4. That is, un ∈ 0 C 1 [0, T ] is dtα convergent to u and so u ∈ D(dtα ) and dtα u = ∂tα u. Thus the proof of Theorem 2.5 is complete.  Remark 2.5 In Sect. 2.5, we discuss the case where we start the operator dtα with D(dtα ) = C 1 [0, T ], not 0 C 1 [0, T ].

2.4 Some Functions in Hα (0, T ) In this section, we show that the function t γ and some functions defined by the Mittag-Leffler functions are in Hα (0, T ). Some of these inclusions are used in later sections. For α, β > 0, we define 

Eα,β (z) = Eα,1 (z) =

∞

zk k=0 (αk+β) , ∞ zk k=0 (αk+1) ,

(2.28)

z ∈ C.

These functions are called the Mittag-Leffler functions and play important roles in the fractional calculus (e.g., [15, 24]). It is known that Eα,β (z) is an entire function in z with α, β > 0. The MittagLeffler functions have been well studied and here we describe only a few of the important properties used later. Lemma 2.5 (i) Let 0 < α < 2 and β > 0. We assume that πα 2 < μ < min{π, πα}. Then there exists a constant C = C(α, β, μ) > 0 such that |Eα,β (z)| ≤

C , 1 + |z|

μ ≤ | arg z| ≤ π.

(ii) For λ ∈ R and α > 0, m ∈ N, we have dm Eα,1 (−λt α ) = −λt α−m Eα,α−m+1 (−λt α ), dt m

t > 0.

2.4 Some Functions in Hα (0, T )

27

The proof of (i) can be found e.g., in [24]. The proof of (ii) is seen directly  (−λt α )k because Eα,1 (z) is entire in z and we can differentiate Eα,1 (−λt α ) = ∞ k=0 (αk+1) termwise. Proposition 2.2 Let 0 < α < 1 and λ ∈ R. Then Eα,1 (−λt α ) − 1 ∈ Wα (0, T ) ⊂ 0 W 1,1 (0, T ) ∩ Hα (0, T ). Here we recall that Wα (0, T ) is defined by (2.5). Proof The inclusion Wα (0, T ) ⊂ Hα (0, T ) is proved already in Lemma 2.1. Moreover, Lemma 2.5 yields d Eα,1 (−λt α ) = −λt α−1 Eα,α (−λt α ) dt and   d   Eα,1 (−λt α ) ≤ |λ|t α−1 |Eα,α (−λt α )| ≤ C|λ|t α−1 ,  dt  By Eα,1 (0) − 1 = conclusion.

1 (1)

t > 0.

− 1 = 0, the definition of Wα (0, T ) implies the 

Proposition 2.3 Let 0 < α < 1. Then 

t

(t − s)α−1 Eα,α (−λ(t − s)α )f (s)ds ∈ Hα (0, T )

0

and  t     (t − s)α−1 Eα,α (−λ(t − s)α )f (s)ds    0

≤ C f L2 (0,T )

Hα (0,T )

for all f ∈ L2 (0, T ). For f ∈ L2 (0, T ) we cannot expect more regularity than Hα (0, T ). For f ∈ we set

L2 (0, T ),



t

v(t) :=

(t − s)α−1 Eα,α (−λ(t − s)α )f (s)ds.

0

Then ∂tα v is well-defined by v ∈ Hα (0, T ), and in Sect. 3.4 of Chap. 3, we prove that v satisfies ∂tα v(t) = −λv(t) + f (t), (see Proposition 3.2).

0 0

if k ≥ N,

and so R ∈ C[0, ∞) and R(0) = 0. By the termwise differentiation, we obtain ∞ ∞   d (−λ)k (αk + α − 1)s αk+α−2 (−λ)k s α(k−N) R(s) = = s αN+α−2 ds (α(k + 1)) (α(k + 1) − 1) k=N

= s αN+α−2

∞  j =0

k=N

(−λ)N+j s αj = (−λ)N s αN+α−2 Eα,αN+α−1 (−λs α ). (αN + α − 1 + αj )

1 Since αN + α − 2 > −1 by (2.30), we see that dR ds ∈ L (0, T ). Therefore  t  t dR d (t − s)f (s)ds, R(t − s)f (s)ds = dt 0 0 dt

so that the Young inequality for the convolution (Lemma A.1 in the Appendix) yields   t  d   R(t − s)f (s)ds  ≤ C f L2 (0,T ) ,  dt  2 0 L (0,T )

2.4 Some Functions in Hα (0, T )

which means that

t 0

29

R(t − s)f (s)ds ∈ W 1,1 (0, T ). Moreover,

  t   t d   ≤C R(t − s)f (s)ds (t − s)αN+α−2 |f (s)|ds.  dt  0

0

Since αN > 32 − α by (2.30), we have 2αN + 2α − 4 > −1. Therefore the Cauchy– Schwarz inequality and (2.30) yield   t   t  12 d  2αN+2α−4   R(t − s)f (s)ds  ≤ C (t − s) ds f L2 (0,T )  dt 0

=

0

C 1

(2αN + 2α − 3) 2

t

αN+α− 32

f L2 (0,T ) ≤ C1 f L2 (0,T ) .

Again by (2.30), we see that αN + α − 1 > 0, so that at t = 0. Hence we proved 

t

t 0

R(t − s)f (s)ds vanishes

R(t − s)f (s)ds ∈ 0 W 1,1 (0, T ) ∩ Wα (0, T ) ⊂ 0 W 1,1 (0, T ) ∩ Hα (0, T ).

0

Therefore by (2.27) we have  ∂tα

t

 R(t − s)f (s)ds

0

1 = (1 − α)



t

−α

(t − s)

0



d ds



s

 R(s − ξ )f (ξ )dξ ds.

0

Hence   t   s    t   α   −α  d  ∂ R(t − s)f (s)ds  ≤ C (t − s)  R(s − ξ )f (ξ )dξ  ds  t ds 0 0 0  t (t − s)−α ds × C1 f L2 (0,T ) ≤ CT 1−α f L2 (0,T ) . ≤C 0

Therefore   t 2   α  ∂ R(t − s)f (s)ds   t



L2 (0,T )

0

T

≤ 0

C 2 T 2−2α f 2L2 (0,T ) dt,

that is, (2.25) yields   t    R(t − s)f (s)ds    0

≤ C f L2 (0,T ) . Hα (0,T )

(2.31)

30

2 Definition of Fractional Derivatives in Sobolev Spaces and Properties

Moreover, by the definition (1.5) of J α , the Eq. (2.29) implies 

t

(t − s)α−1 Eα,α (−λ(t − s)α )f (s)ds

0

=

N−1 



t

(−λ)k (Jαk+α f )(t) +

R(t − s)f (s)ds.

0

k=0

Since 0 < αk + α ≤ αN for 0 ≤ k ≤ N − 1, Theorem 2.1 yields

Jαk+α f ∈

Hαk+α (0, T ), αk + α < 1, H1 (0, T ), αk + α ≥ 1, k = 0, . . . , N − 1

and so N−1 

(−λ)k Jαk+α f ∈

k=0

N−1 

Hαk+α (0, T ) ∩ H1 (0, T ) = Hα (0, T ).

k=0



Thus with (2.31), the proof of Proposition 2.3 is complete. Moreover, we can prove: Proposition 2.4 Let 0 < α
0 satisfy α + δ −

1 2

1 2

< 0. Then

1

t α+δ− 2 ∈ Hα (0, T ) = H α (0, T ). Proof In view of Theorem 2.1, it suffices to verify 1

t α+δ− 2 = J α w0 (t), where  Γ α + δ + 12 1 t δ− 2 ∈ L2 (0, T ),  w0 (t) := Γ δ + 12 but we prove directly. We set 1 θ = −α − δ + . 2 Then 0 < θ
−1 and γ1 + γ2 + γ3 > −2.



Proof of (2.33) By γ1 , γ2 > −1, we have   T  t  (1 + γ1 )(1 + γ2 ) T γ1 +γ2 +1 γ3 γ1 γ2 s (t − s) ds t γ3 dt = t t dt, (2 + γ1 + γ2 ) 0 0 0 which is convergent by γ1 + γ2 + γ3 + 1 > −1. Thus the proof of (2.33) is complete.  Now we will prove  T t  T  t −θ θ θ 2 |t − s −θ |2 −2θ −2θ |t − s | dsdt = t s dsdt < ∞. 1+2α |t − s|1+2α 0 0 |t − s| 0 0

(2.34)

We choose 0 < μ < 1 such that

 1 1−α < μ < min 1, . 2 − 2θ 1−θ   1 This is possible because 0 < 2−2θ < min 1, 1−α 1−θ by 0 < θ
0 by 0 < α < 2 . θ θ θ By 0 < θ < 1, we can easily see that a + b ≥ (a + b) for a, b ≥ 0 and so |t θ − s θ | ≤ |t − s|θ . Hence

|t θ − s θ |2μ ≤ |t − s|2μθ for 0 ≤ s ≤ t. On the other hand, the mean value theorem and θ < 1 yield |t θ − s θ | ≤ max θ ηθ−1 (t − s) ≤ θ s θ−1 (t − s) s≤η≤t

32

2 Definition of Fractional Derivatives in Sobolev Spaces and Properties

for 0 ≤ s ≤ t. Therefore |t θ − s θ |2(1−μ) ≤ Cs 2(1−μ)(θ−1) |t − s|2(1−μ), and so |t θ − s θ |2 = |t θ − s θ |2μ |t θ − s θ |2−2μ ≤ Cs 2(1−μ)(θ−1)(t − s)2(1−μ)+2μθ . Hence 

T

0

 0

t

|t θ − s θ |2 t −2θ s −2θ dsdt ≤ C |t − s|1+2α

By 0 < α < 12 , 0 < θ
−1,

1 − 2α − 2μ + 2μθ > −1

and {2(1 − μ)(θ − 1) − 2θ } + (−2θ ) + (1 − 2α − 2μ + 2μθ ) > −2, 

so that (2.33) yields (2.34). Thus the proof of Proposition 2.4 is complete.

Remark 2.6 It seems that we cannot directly prove the lemma without introducing the parameter μ ∈ (0, 1). Since Proposition 2.4 proves C0 t

α+δ− 12

∈ Hα (0, T ), where C0 =

  δ+ 12 ,   α+δ+ 21

we

can apply Theorem 2.5 to calculate 1

1

1

∂tα (C0 t α+δ− 2 ) = dtα (C0 t α+δ− 2 ) = Dtα (C0 t α+δ− 2 )  t 1 C0 d = (t − s)−α s α+δ− 2 ds (1 − α) dt 0  1 (1 − α) α + δ + 1 2 d δ+ 1 C0  t 2 = t δ− 2 . = 3 (1 − α) dt  δ+ 2

   Here we used  δ + 32 = δ + 12  δ + 12 . Therefore with the closure of dtα defined in Sect. 2.3 and ∂tα , we can justify 1

1

dtα (C0 t α+δ− 2 ) = t δ− 2 for 0 < α
0 satisfying α + δ −

1 2

(2.36) < 0.

2.5 Definition of ∂tα in H α (0, T )

33

2.5 Definition of ∂tα in H α (0, T ) We have defined ∂tα u for u ∈ Hα (0, T ). Next, it is natural to define the fractional derivative in H α (0, T ) which is wider than Hα (0, T ) for 0 < α < 1, although we work mainly within Hα (0, T ) for discussing fractional differential equations. Since Hα (0, T ) is a set of functions in H α (0, T ) which “vanish” in some sense at t = 0, the following definition is naive but may be reasonable for 0 < α < 1 with α = 12 : we define ∂tα v for v ∈ H α (0, T ) by

∂tα v =

∂tα (v

v ∈ H α (0, T ), 0 < α < 12 , v ∈ H α (0, T ), 12 < α < 1.

∂tα v, − v(0)),

(2.37)

Then ∂tα u is well-defined because Hα (0, T ) = H α (0, T ) for 0 < α < 12 , and H α (0, T ) ⊂ C[0, T ] for 12 < α < 1 which enables us to define v(0). For 0 < α < 1 and α = 12 , we note that this definition gives the same result for v ∈ Hα (0, T ), and so we can extend the domain of ∂tα from Hα (0, T ) to H α (0, T ). On the other hand, the case of α = 1

1 2

1

is delicate, and we do not define ∂t2 u for 1

u ∈ H 2 (0, T )  H 1 (0, T ). For example, we note that 1 ∈ H 2 (0, T ) but 1 ∈ 2

1

1

H 1 (0, T ). We do not define ∂t2 1 in L2 (0, T ), although we can calculate dt2 1 = 0 2

1

and Dt2 1 =

1 − 12 t ( 21 )

pointwise.

By the definition we note that ∂tα (u1 + u2 ) = ∂tα u1 + ∂tα u2 ,

u1 , u2 ∈ H α (0, T )

for 0 < α < 1 and α = 12 . Throughout this book, we always interpret ∂tα u with 0 < α < 1 and α = the sense of (2.37) for u ∈ H α (0, T ). Now we show properties of ∂tα in conventional Sobolev space H α (0, T ).

1 2

in

Proposition 2.5 (i)  ∂tα 1

=

t −α (1−α) ,

0 < α < 12 , 0, 12 < α < 1.

(2.38)

(ii)

∂tα u = for u ∈ C 1 [0, T ].

Dtα u, 0 < α < 12 , dtα u, 12 < α < 1

(2.39)

34

2 Definition of Fractional Derivatives in Sobolev Spaces and Properties

(iii) Let 12 < α < 1 and u ∈ H α (0, T ). Then there exists a sequence ϕk ∈ C 1 [0, T ], k ∈ N such that ϕk −→ u in H α (0, T ) and ∂tα u = limk→∞ ∂tα ϕk in L2 (0, T ). (iv) Let 0 < α < 12 . Then there exists a constant C > 0 such that C −1 u − a H α (0,T ) ≤ ∂tα u L2 (0,T ) + |a| ≤ C( u − a H α (0,T ) + |a|) for all a ∈ R and u ∈ H α (0, T ). (v) Let 12 < α < 1. Then there exists a constant C > 0 such that C −1 u H α (0,T ) ≤ ∂tα u L2 (0,T ) + |u(0)| ≤ C u H α (0,T ) for all u ∈ H α (0, T ). Proof (i) By (2.37), setting u(t) ≡ 1, we have ∂tα 1 = ∂tα (u − u(0)) = ∂tα (1 − 1) = ∂tα 0 = 0 for 12 < α < 1. For 0 < α < 12 , by (2.26) we can directly calculate  t 1 d 1−α d (J 1) = (t − s)−α ds dt dt (1 − α) 0   1 t −α d t 1−α = . = dt (1 − α)(1 − α) (1 − α) ∂tα 1 =

This completes the proof of (i). (ii) Since u ∈ C 1 [0, T ] ⊂ H α (0, T ) = Hα (0, T ) for 0 < α < 12 and u is absolutely continuous, equality (2.26) yields that ∂tα u = Dtα u for 0 < α < 12 . Next we verify (2.39) for 12 < α < 1. By (2.37), we have ∂tα u = ∂tα (u − u(0)). Applying Lemma 1.1 to u − u(0) and noting that u ∈ C 1 [0, T ], we obtain ∂tα (u − u(0)) = dtα (u − u(0)) = =

1 (1 − α)

 0

t

(t − s)−α

1 (1 − α)

 0

t

(t − s)−α

d (u(s) − u(0))ds ds

du (s)ds = dtα u(t). ds

(iii) Since C 1 [0, T ] is dense in H α (0, T ), there exists a sequence ϕk ∈ C 1 [0, T ], k ∈ N such that ϕk −→ u in H α (0, T ). By α > 12 , the Sobolev embedding yields ϕk (0) −→ u(0). Therefore ϕk − ϕk (0) ∈ 0 C 1 [0, T ] and ϕk − ϕk (0) −→ u − u(0) in Hα (0, T ). By the definition (2.37), we see that ∂tα u = ∂tα (u − u(0)) = lim ∂tα (ϕk − ϕk (0)) = lim ∂tα ϕk k→∞

in L2 (0, T ).

k→∞

2.5 Definition of ∂tα in H α (0, T )

35

(iv) For 0 < α < 12 , we note that Hα (0, T ) = H α (0, T ), and so u ∈ H α (0, T ) implies u − a ∈ Hα (0, T ). Therefore, by Theorem 2.2, we have

u − a H α (0,T ) ∼ J −α (u − a) L2 (0,T ) = ∂tα u − ∂tα a L2 (0,T )

(2.40)

if u − a ∈ Hα (0, T ). Hence (2.38) and (2.40) imply

u − a H α (0,T ) ≥ u H α (0,T ) − a H α (0,T ) ≥ C1 ∂tα u L2 (0,T ) − C2 ∂tα a L2 (0,T ) = C1 ∂tα u L2 (0,T ) −

C2

t −α L2 (0,T ) |a| ≥ C1 ∂tα u L2 (0,T ) − C3 |a|, (1 − α)

which is the second inequality in (iv). Next, (2.40) yields

u − a H α (0,T ) ≤ u H α (0,T ) + a H α (0,T ) ≤ C4 ( ∂tα u L2 (0,T ) + ∂tα a L2 (0,T ) ) ≤ C5 ( ∂tα u L2 (0,T ) + |a|), which is the first inequality in (iv). (v) By (2.37) and Theorems 2.1 and 2.2, we have

∂tα u L2 (0,T ) = ∂tα (u−u(0)) L2 (0,T ) ∼ u−u(0) Hα (0,T ) ∼ u−u(0) H α (0,T ) .

Therefore C6 ∂tα u L2 (0,T ) ≥ u H α (0,T ) − u(0) H α (0,T ) and

∂tα u L2 (0,T ) ≤ C6 ( u H α (0,T ) + u(0) H α (0,T ) ). Directly, we see that

u(0) H α (0,T ) = |u(0)| 1 H α (0,T ) ≤ C7 |u(0)|. Moreover, by the Sobolev embedding for 12 < α < 1, we see |u(0)| ≤ C8 u H α (0,T ) . Hence u(0) H α (0,T ) ≤ C7 |u(0)| ≤ C9 u(0) H α (0,T ) . Thus, the proof of (v) and so the proof of Proposition 2.5 is completed.  It seems that the definition of ∂tα in H α (0, T ) with 12 < α < 1 may be artificial, but the definition is related to the closure of the classical Caputo derivative operator. In Sect. 2.3, we introduce the closure in L2 (0, T ) of the Caputo derivative

36

2 Definition of Fractional Derivatives in Sobolev Spaces and Properties

operator dtα with the domain D(dtα ) = 0 C 1 [0, T ] and we establish Theorem 2.5. Our discussion here is similar. We consider the classical Caputo derivative α 0 dt u(t)

1 := (1 − α)



t 0

(t − s)−α

du (s)ds, ds

D(0 dtα ) = C 1 [0, T ].

To avoid confusion, by 0 dtα we denote the classical Caputo fractional derivative with the domain C 1 [0, T ], and we consider 0 dtα as an operator from C 1 [0, T ] ⊂ L2 (0, T ) to L2 (0, T ). First we prove: Lemma 2.6 For

1 2

≤ α < 1, the operator 0 dtα is closable.

Proof We have to prove that w = 0 if un ∈ C 1 [0, T ] = D(0 dtα ), n = 1, 2, 3, . . ., un −→ 0 in L2 (0, T ) and 0 dtα un converge to w in L2 (0, T ). By un ∈ C 1 [0, T ], n = 1, 2, 3, . . ., we see that un − un (0) ∈ 0 C 1 [0, T ] ⊂ Hα (0, T ). Therefore (2.27) implies α 0 d t un

= 0 dtα (un − un (0)) = dtα (un − un (0)) = ∂tα (un − un (0)).

Hence ∂tα (un − un (0)) −→ w in L2 (0, T ), and so Theorem 2.4 implies un − un (0) is a Cauchy sequence in Hα (0, T ), and there exists v ∈ Hα (0, T ) such that un − un (0) −→ v

in Hα (0, T ).

(2.41)

In particular, un − un (0) −→ v in L2 (0, T ). Since un −→ 0 in L2 (0, T ), we have −un (0) −→ v in L2 (0, T ) and so v(t) is a constant function as limit of the constant functions: v(t) = v0 for 0 < t < T with some constant v0 . By v ∈ Hα (0, T ), we see that v0 ∈ Hα (0, T ). For 12 < α < 1, since v ∈ Hα (0, T ) ⊂ C[0, T ] implies v(0) = 0, we obtain v0 = 0. For α = 12 , since v0 ∈ H 1 (0, T ), we have 2  T |v(t )|2  2 T 1 dt < ∞. Consequently, v = 0. The proof of Lemma 2.6 is dt = v 0 0 0 t 0 t complete.  α 2 Now in this section, let 12 ≤ α < 1. By d t , we denote the closure in L (0, T ) α α 1 of 0 dt . More precisely, u ∈ D(d t ) if there exists a sequence un ∈ C [0, T ], n = 1, 2, 3, . . . such that un −→ u in L2 (0, T ) and 0 dtα un , n = 1, 2, 3, . . ., are α convergent to some w ∈ L2 (0, T ). Then we define d t u = w. Finally, we prove:

Theorem 2.6 (i) α α D(d t ) = H (0, T )

if

1 0 and C0 =

  δ+ 12 .   α+δ+ 12

Proposition 2.4 yields

1

t α+δ− 2 ∈ Hα (0, T ) and Proposition 2.6 implies 1

1

1

1

dtα (C0 t α+δ− 2 ) = ∂tα (C0 t α+δ− 2 ) = Dtα (C0 t α+δ− 2 ) = t δ− 2 . Moreover, we can directly verify the above result by the definition of dtα . For this, it suffices to prove 1

1

(C0 t α+δ− 2 , (dtα )∗ ϕ) = (t δ− 2 , ϕ),

ϕ ∈ C01 (0, T ).

Indeed, using Lemma 2.7, integration by parts and  T  η

= 0 0 · · · dt dη, we obtain 1

(C0 t α+δ− 2 , (dtα )∗ ϕ) = C0



T 0

1

t α+δ− 2



−1 (1 − α)



T t

 T  T

(η − t)−α

0

t

· · · dη dt

 dϕ (η)dη dt dη

42

2 Definition of Fractional Derivatives in Sobolev Spaces and Properties

−C0 = (1 − α) =−

1 δ+



T

0

dϕ (η) dη 1

1 2



η

t

α+δ− 12

−α

(η − t)

 dt dη = −

0

η=T

[ϕ(η)ηδ+ 2 ]η=0 +



T

1



1 δ+

1 2

T

1

ηδ+ 2

0

dϕ (η)dη dη

1

ηδ− 2 ϕ(η)dη = (t δ− 2 , ϕ).

0

Proof of Proposition 2.6 Let u ∈ Hα (0, T ) be arbitrary. By Lemma 2.2, there exist un ∈ 0 C 1 [0, T ], n = 1, 2, 3, . . . , such that un −→ u in Hα (0, T ). By Theorem 2.4, we note that ∂tα un = dtα un , n = 1, 2, 3, . . .. By Theorem 2.2, we have dtα un = ∂tα un −→ ∂tα u in L2 (0, T ). By using un (0) = 0, Lemma 2.7, un ∈ 0 C 1 [0, T ] and Definition 2.3 imply (dtα un , ϕ) = (un , (dtα )∗ ϕ),

ϕ ∈ C01 (0, T ), n = 1, 2, 3, . . .

Letting n → ∞, we obtain (u, (dtα )∗ ϕ) = (∂tα u, ϕ),

ϕ ∈ C01 (0, T ).

Since ∂tα u ∈ L2 (0, T ) for u ∈ Hα (0, T ), Definition 2.3 yields that u ∈ D(dtα ) and dtα u = ∂tα u. Combining (2.26), we complete the proof of Proposition 2.6.  The adjoint equality (2.51) for u ∈ 0 C 1 [0, T ] is generalized as follows. Proposition 2.7 (∂tα u, ϕ) = (u, (dtα )∗ ϕ),

u ∈ Hα (0, T ), ϕ ∈ C01 (0, T ).

(2.53)

Proof For u ∈ Hα (0, T ), by Lemma 2.2, we can choose un ∈ 0 C 1 [0, T ], n ∈ N such that un −→ u in Hα (0, T ). Then, by Lemma 2.7 we have (∂tα un , ϕ) = (un , (dtα )∗ ϕ),

ϕ ∈ C01 (0, T ).

Moreover, (2.11) yields that ∂tα un −→ ∂tα u in L2 (0, T ). Therefore letting n → ∞, we complete the proof of Proposition 2.7. 

2.7 Laplace Transform of ∂tα

43

2.7 Laplace Transform of ∂tα For u belonging to a certain class, we can define the Laplace transform (Lu)(p) by 

∞

(Lu)(p) :=

e−pt u(t)dt

0

for Re p > p0 : some constant. The formulae of the Laplace transforms for fractional derivatives are well-known (e.g., [15, 24]), provided that u satisfies a certain conditions. For example, L(dtα u)(p) = pα (Lu)(p) − pα−1 u(0)

(2.54)

for Re p > p0 : some constant. The formula (2.54) is convenient for solving fractional differential equations. However, formula (2.54) requires some regularity for u in order that u(0) is well-defined. Formula (2.54) does not make sense for all u ∈ H α (0, T ) with 0 < α < 12 . Moreover, such needed regularity should be consistent with the regularity which we can prove for solutions to fractional differential equations. In particular, the regularity for the formula concerning the Laplace transform should be not very strong. Thus on the regularity assumption for the formula like (2.54), we have to make adequate assumptions for u. In this section, we state the formula of the Laplace transform for the fractional derivative ∂tα in Hα (0, T ). We set Vα (0, ∞) := {u; u|(0,T ) ∈ Hα (0, T )

for any T > 0,

(2.55)

there exists a constant C = Cu > 0 such that |u(t)| ≤ CeCt for t ≥ 0}. Here u|(0,T ) denotes the restriction of u to (0, T ). Then we can state: Theorem 2.7 The Laplace transform L(∂tα u)(p) can be defined for u ∈ Vα (0, ∞) by  L(∂tα u)(p) = lim

T →∞ 0

T

e−pt ∂tα u(t)dt,

p > Cu

and L(∂tα u)(p) = pα Lu(p), Here Cu > 0 in some constant depending on u.

p > Cu .

44

2 Definition of Fractional Derivatives in Sobolev Spaces and Properties

Proof First for u ∈ Hα (0, T ), by Theorem 2.3 (i), we can see that J 1−α u ∈ H1 (0, T ) ⊂ H 1 (0, T ),

J 1−α u(0) = 0 in the trace sense

(2.56)

and so Dtα u =

d 1−α J u ∈ L2 (0, T ). dt

d 1−α J u for u ∈ Hα (0, T ). Let T > 0 be arbitrarily Theorem 2.4 yields ∂tα u = dt fixed. Then, in terms of (2.56), we integrate by parts to obtain 

T

0

 t=T = J 1−α u(t)e−pt t=0

=



e−pT (1 − α)

T



T

d 1−α (J u)(t)e−pt dt dt  T +p e−pt J 1−α u(t)dt

e−pt ∂tα u(t)dt =

0

0 −α

(T − s)

0

p u(s)ds + (1 − α)



T

e

−pt



0

t

−α

(t − s)

 u(s)ds dt

0

=: I1 + I2 .

Since |u(t)| ≤ C0 eC0 t for t ≥ 0 with some constant C0 > 0, we estimate |I1 | ≤ Ce = Ce

−pT

−(p−C0 )T



T

(T − s)

e

ds = Ce



T

s

−α −C0 s

e

(1 − α) C01−α

ds ≤ Ce



−pT

0

0

= Ce−(p−C0 )T

−α C0 s

−(p−C0 )T



T

s −α eC0 (T −s) ds

0 ∞

s −α e−C0 s ds

0

.

Hence if p > C0 , then lim I1 = 0.   T  t T →∞ T T As for I2 , by 0 · · · ds dt = · · · dt ds, we see 0 0 s   T  T p e−pt (t − s)−α dt u(s)ds (1 − α) 0 s   T  T −s p −pη −α e η dη e−ps u(s)ds. = (1 − α) 0 0 I2 =

2.7 Laplace Transform of ∂tα

45

For p > C0 , since |u(s)| ≤ C0 eC0 s for s ≥ 0, we have    

T −s

e 0

−pη −α

η

    −ps    dη e u(s) ≤ C0

∞

e

−pη −α

η



dη e−(p−C0 )s

0

for all s > 0 and T > 0, and the Lebesgue convergence theorem yields   ∞  ∞ p −pη −α e η dη e−ps u(s)ds lim I2 = T →∞ (1 − α) 0 0  (1 − α) ∞ −ps p e u(s)ds = pα (Lu)(p) = (1 − α) p1−α 0 for p > C0 . Thus the proof of Theorem 2.7 is complete.



Chapter 3

Fractional Ordinary Differential Equations

3.1 Examples First we consider simple fractional ordinary differential equations: Dtα u(t) = −λu(t) + f (t),

0 < t < T,

(3.1)

dtα u(t) = −λu(t) + f (t),

0 < t < T.

(3.2)

We note that (3.1) and (3.2), etc. are considered pointwise. It is known that as a well-posed problem, we have to attach some initial condition, so that we usually discuss the following initial value problems:

Dtα u(t) = −λu(t) + f (t), limt ↓0 J 1−α u(t) = a,

0 < t < T,

dtα u(t) = −λu(t) + f (t), u(0) = a.

0 < t < T,

(3.3)

and

(3.4)

It is proved that there exists a unique solution to (3.3) and (3.4) respectively, and the solutions are given by the following formulae (Kilbas et al. [15], Podlubny [24]). For (3.3)  u(t) = at

α−1

t

Eα,α (−λt ) + α

(t − s)α−1 Eα,α (−λ(t − s)α )f (s)ds,

0 < t < T.

0

© The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 A. Kubica et al., Time-Fractional Differential Equations, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-15-9066-5_3

(3.5)

47

48

3 Fractional Ordinary Differential Equations

For (3.4) 

t

u(t) = aEα,1 (−λt ) + α

(t − s)α−1 Eα,α (−λ(t − s)α )f (s)ds,

0 < t < T.

0

(3.6) Here we recall that Eα,β (z) with α, β > 0 are defined by (2.28). Here we should expect that u(t) given by (3.5) and (3.6) are verified to pointwisely satisfy (3.3) and (3.4) respectively. Moreover, in general, for f ∈ L2 (0, T ), we do not know whether the system

Dtα u(t) = −λu(t) + f (t), u(0) = a

0 < t < T,

possesses a unique solution. Even for a simple initial value problem (3.4) for a fractional ordinary differential equation, there are very few existing works which systematically establish the uniqueness and the existence of solutions in relevant Sobolev spaces with f ∈ L2 (0, T ), which is a natural choice of a function space for f . The purpose of this chapter is to establish the unique existence of solutions to initial value problems for fractional ordinary differential equations. When we will treat the problem pointwise, for example, not in L2 (0, T ), we have to formulate the initial condition in (3.4) and interpret dtα u for f ∈ L2 (0, T ). The latter is not direct as is seen in Sect. 2.1 of Chap. 2. In this book, on the basis of the fractional derivative ∂tα defined in Chap. 2, we will discuss exclusively in the Sobolev spaces Hα (0, T ) with 0 < α < 1. In Sect. 3.4, we will return to formulation (3.4) and formula (3.6), and discuss some details.

3.2 Fundamental Inequalities: Coercivity In the case of α = 1, it is trivial that 

t 0

1 1 du (s)u(s)ds = |u(t)|2 − |u(0)|2 , ds 2 2

0 0 such that

u − a Hα (0,T ) ≤ C(|a| + f L2 (0,T ) )

(3.23)

u H α (0,T ) ≤ C(|a| + f L2 (0,T ) ).

(3.24)

and

In the pointwise sense, the unique existence of solutions to initial value problems for fractional ordinary differential equations with Dtα and ∂tα has been well studied (e.g., Kilbas et al. [15], Podlubny [24]), but such pointwise formulations meet difficulty in several cases such as f ∈ L∞ (0, T ). Thus we need to formulate the initial condition by (3.20). Proof of Theorem 3.5 By Theorem 2.4, we can rewrite (3.20) as J −α (u − a) = λu + f (t),

u − a ∈ Hα (0, T ),

which is equivalent to u(t) = a + λ(J α u)(t) + (J α f )(t) λ =a+ (α)



t 0

(t − s)

α−1

1 u(s)ds + (α)



t

(t − s)α−1 f (s)ds,

0

0 < t < T.

(3.25)

3.3 Well-Posedness for Single Linear Fractional Ordinary Differential Equations

57

By Theorem 2.1, we see that λJ α : L2 (0, T ) −→ Hα (0, T ) is bounded and so is a compact operator from L2 (0, T ) to itself (e.g., [2]). Now we assume that u(t) = λ(J α u)(t) for 0 < t < T , that is, 

λ u(t) = (α)

t

(t − s)α−1 u(s)ds,

0 < t < T.

(t − s)α−1 |u(s)|ds,

0 < t < T.

0

Then  |u(t)| ≤ C

t

0

Applying a generalized Gronwall inequality, we obtain u = 0 in (0, T ). For completeness, we will prove it as Lemma A.2 in the Appendix.  Consequently, the Fredholm alternative yields that there exists a unique solution u ∈ L2 (0, T ) satisfying (3.25). Moreover, since u − a = J α (λu + f ) in L2 (0, T ). Theorem 2.1 (i) implies that u − a ∈ Hα (0, T ). Thus the proof of the unique existence of u is complete. Moreover, (3.25) yields  |u(t)| ≤ |a| + C

t



t

(t − s)α−1 |f (s)|ds + C

0

(t − s)α−1 |u(s)|ds,

0 < t < T.

0

(3.26) We set  R(t) = |a| + C

t

(t − s)α−1 |f (s)|ds.

0

Applying the generalized Gronwall inequality Lemma A.2, we have  |u(t )| ≤ CR(t ) + C

t

(t − s)α−1 R(s)ds

0

     s  t  t (t − s)α−1 |f (s)|ds + C (t − s)α−1 |a| + (s − ξ )α−1 |f (ξ )|dξ ds ≤ C |a| + 0

0

0

(3.27) for 0 ≤ t ≤ T . We take the norms in L2 (0, T ). The Young inequality Lemma A.1 in the Appendix yields  t     (t − s)α−1 |f (s)|ds    0

L2 (0,T )

≤ t −α L1 (0,T ) f L2 (0,T ) ≤

T 1−α

f L2 (0,T ) . 1−α

58

3 Fractional Ordinary Differential Equations

Moreover,  t  

0

0 t

=

   t  t (s − ξ )α−1 |f (ξ )|dξ ds = (s − ξ )α−1 ds |f (ξ )|dξ

s

0

0

ξ

(t − ξ )α |f (ξ )|dξ, α

and so  t  s     α−1  (s − ξ ) |f (ξ )|dξ ds    0

0

L2 (0,T )

 α t   ≤

f L2 (0,T ) α 1 L (0,T )

again by the Young inequality. By exchanging the orders of the integrals, we can similarly obtain  

t

 (t − s)

0 t

= 0

s

α−1

 |f (ξ )|

 (s − ξ )

α−1

|f (ξ )|dξ ds

0 t



(t − s)

α−1

(s − ξ )

ξ

α−1

(α)2 ds dξ = (2α)



t

(t − ξ )2α−1 |f (ξ )|dξ,

0

and  t  s     α−1  (t − s)α−1 (s − ξ ) |f (ξ )|dξ ds    0

L2 (0,T )

0

 t   (α)2  2α−1  ≤ (t − ξ ) |f (ξ )|dξ    (2α) 0

L2 (0,T )

≤ C f L2 (0,T ) .

Consequently, (3.27) implies u L2 (0,T ) ≤ C(|a| + f L2 (0,T ) ). Hence the first inequality in (3.20) yields

∂tα (u − a) L2 (0,T ) ≤ C(|λ| u L2 (0,T ) + f L2 (0,T ) ) ≤ C(|a| + f L2 (0,T ) ). Thus the proof of (3.23) is completed. We can see (3.24) by u − a H α (0,T ) ≤

u − a Hα (0,T ) and

u − a H α (0,T ) ≥ u H α (0,T ) − a H α (0,T ) = u H α (0,T ) − which can be seen by a H α (0,T ) = a L2 (0,T ) =

√ T |a|,

√ T |a| for constant a.

3.4 Alternative Formulation of Initial Value Problem

59

3.4 Alternative Formulation of Initial Value Problem In this book, we exclusively formulate an initial value problem by (3.20). On the other hand, in this section, we discuss

dtα u = p(t)u + f (t), u(0) = a.

0 < t < T,

(3.28)

We recall (1.17):

 du 1 1 ∈ L (0, T ) W (0, T ) := u ∈ L (0, T ); dt    with u W 1,1 (0,T ) = u L1 (0,T ) +  du dt L1 (0,T ) . 1,1

Here and henceforth we assume p ∈ L∞ (0, T ) and f ∈ L2 (0, T ). In (3.28), if u ∈ W 1,1 (0, T ), then the initial condition u(0) = a can be immediately justified by W 1,1 (0, T ) ⊂ C[0, T ]. Moreover, by u ∈ W 1,1 (0, T ), we can prove that dtα u ∈ L1 (0, T ) exists. In many monographs (e.g., Diethelm [6], Kilbas et al. [15], Podlubny [24]), the initial value problem is formulated by (3.28). Now we consider the formulation:

α ∂t (u − a) = p(t)u + f (t), 0 < t < T , (3.29) u − a ∈ Hα (0, T ), and compare with (3.28). Then we can prove: Proposition 3.1 (i) Let p ∈ L∞ (0, T ) and f ∈ L2 (0, T ). If u ∈ W 1,1 (0, T ) satisfies (3.28), then u satisfies (3.29). (ii) Let p ∈ C 2 [0, T ] and f ∈ W 1,1 (0, T ). Then the unique solution u to (3.29) is in W 1,1 (0, T ) and satisfies (3.28). In (ii) we note that dtα u is defined by (1.4) because u ∈ W 1,1 (0, T ). The proposition means that under assumptions f ∈ W 1,1 (0, T ) and p ∈ C 2 [0, T ], the formulations (3.28) and (3.29) are equivalent. In (3.29), we note that in general we cannot prove that u ∈ W 1,1 (0, T ) if p ∈ L∞ (0, T ) and f ∈ L2 (0, T ). Proof of Proposition 3.1 (i) Let u ∈ W 1,1 (0, T ) satisfy (3.28). Then Lemma 1.3 (i) implies that dtα u = dtα (u − a) = Dtα (u − a) pointwise. Therefore (3.28) yields Dtα (u − a) = p(t)u + f (t),

0 < t < T.

(3.30)

Since u − a ∈ 0 W 1,1 (0, T ), applying J α to (3.30), by Proposition 2.1 and pu + f ∈ L2 (0, T ) we obtain u − a = J α Dtα (u − a) = J α (pu + f ) ∈ R(J α ) = Hα (0, T ).

60

3 Fractional Ordinary Differential Equations

Hence u − a ∈ Hα (0, T ). Therefore (2.26) implies ∂tα (u − a) = Dtα (u − a). Equation (3.30) yields (3.29). Thus we complete the proof of Proposition 3.1 (i).  Proof of Proposition 3.1 (ii) For f ∈ W 1,1 (0, T ) and p ∈ C 2 [0, T ], we can prove that u ∈ W 1,1 (0, T ) and u(0) = a. The proof is done in Theorem 3.6 (iii) later for a more general case, and is postponed. Then Theorem 2.4 implies ∂tα (u − a) = dtα (u − a) for u − a ∈ Hα (0, T ). By u − a ∈ W 1,1 (0, T ), the pointwise dtα defined by (1.4) yields ∂tα (u =

− a) =

1 (1 − α)



t

dtα (u

1 − a)(t) = (1 − α)

(t − s)−α

0



t

(t − s)−α

0

d (u − a)(s)ds ds

du (s)ds = dtα u(t). ds

Therefore (3.29) implies dtα u(t) = p(t)u + f (t) for 0 < t < T . Since u ∈ W 1,1 (0, T ) ⊂ C[0, T ], we have the initial condition u(0) = a in the sense of limt →0 u(t) = a. Thus u satisfies (3.28) and the proof of (ii) is complete.  Now we discuss a simple case (3.4) and clarify in which sense the solution formula (3.6) should be understood. In fact, we prove: Proposition 3.2 Let f ∈ L2 (0, T ). Then u given by (3.6) satisfies

∂tα (u − a) = −λu + f (t), u − a ∈ Hα (0, T ).

0 < t < T,

(3.31)

The existing references [15, 24] give a representation formula (3.6) for the solution to (3.4), but it can be justified pointwise only if f has certain regularity such as f ∈ W 1,1 (0, T ). In other words, for f ∈ L2 (0, T ) it is more consistent for us to interpret (3.6) as solution formula for the initial value problem (3.31), not (3.4). Proof We set  u1 (t) = aEα,1 (−λt ), α

t

u2 (t) = u2 (f )(t) =

(t − s)α−1 Eα,α (−λ(t − s)α )f (s)ds.

0

Since u1 − a ∈ 0 W 1,1 (0, T ) ∩ Hα (0, T ) by Proposition 2.2, Eq. (2.27) yields ∂tα (u1 − a) = dtα (u1 − a) = dtα u1 = −λEα,1 (−λt α ) = −λu1 (t). Next we have to prove

∂tα u2 = −λu2 + f (t), u2 ∈ Hα (0, T )

0 < t < T,

3.4 Alternative Formulation of Initial Value Problem

61

for each f ∈ L2 (0, T ). Proposition 2.3 immediately u2 ∈ Hα (0, T ). We will verify the first equation in the above. First let f ∈ C0∞ (0, T ). Then 

t

u2 (f )(t) =

(t − s)α−1 Eα,α (−λ(t − s)α )f (s)ds

0



t

=

s α−1 Eα,α (−λs α )f (t − s)ds.

0

Hence 

t

|u2 (f )(t)| ≤ C 0

s α−1 ds f C[0,T ] =

C α t f C[0,T ] . α

In particular, u2 (f )(0) = 0. Moreover, using f ∈ C0∞ (0, T ), we can justify: du2 (f ) (t) = dt



t

s α−1 Eα,α (−λs α )

0

df (t − s)ds. dt

Therefore    t  du2 (f )  C  ≤C s α−1 ds f C 1 [0,T ] = t α f C 1 [0,T ] , (t)  dt  α 0 so that u2 (f ) ∈ 0 W 1,1 (0, T ). Consequently, u2 (f ) ∈ Hα (0, T ) ∩ 0 W 1,1 (0, T ) for f ∈ C0∞ (0, T ). Therefore (2.27) yields ∂tα u2 (f )(t)

=

Dtα u2 (f )(t)

1 d = (1 − α) dt



t

(t − s)−α u2 (f )(s)ds,

0 < t < T.

0

Exchanging the orders of the integrals, we calculate d 1 (1 − α) dt



t

(t − s)−α u2 (f )(s)ds

0

   s  t 1 d (t − s)−α (s − ξ )α−1 Eα,α (−λ(s − ξ )α )f (ξ )dξ ds = (1 − α) dt 0 0  t   t d 1 −α α−1 α = (t − s) (s − ξ ) Eα,α (−λ(s − ξ ) )ds f (ξ )dξ. dt 0 (1 − α) ξ On the other hand, using the power series of Eα,α (−ληα ) for η ≥ 0, we can directly verify 1 (1 − α)



t 0

(t − η)−α ηα−1 Eα,α (−ληα )dη = Eα,1 (−λt α ),

t > 0,

62

3 Fractional Ordinary Differential Equations

which means 1 (1 − α) =

1 (1 − α)

 

t

(t − s)−α (s − ξ )α−1 Eα,α (−λ(s − ξ )α )ds

ξ t −ξ

((t − ξ ) − η)−α ηα−1 Eα,α (−ληα )dη = Eα,1 (−λ(t − ξ )α ).

0

Consequently, ∂tα u2 (f )(t)

d = dt



t

Eα,1 (−λ(t − ξ )α )f (ξ )dξ.

0

Since d Eα,1 (−λ(t − ξ )α ) = −λ(t − ξ )α−1 Eα,α (−λ(t − ξ )α ) dt by Lemma 2.5 (ii) with m = 1, using f ∈ C0∞ (0, T ), we have  ∂tα u2 (f )(t) =

t

−λ(t − ξ )α−1 Eα,α (−λ(t − ξ )α )f (ξ )dξ + Eα,1 (0)f (t),

0

that is, ∂tα u2 (f ) = −λu2 (f ) + f (t),

0

3 , 2

(k1 − 1)α ≤

Similarly to the proof of (iii), in terms of (3.41), we will improve the regularity of u − a. First by Lemma 3.3, u − a ∈ (Hα (0, T ))N yields Lα (u − a) ∈ (H2α (0, T ))N . Consequently, setting v1 = Lα (u − a) and w1 = G, by (3.41) we see u − a = v1 +w1 with v1 ∈ (H2α (0, T ))N and w1 ∈ (Wα (0, T ))N . Therefore, continuing this argument, in view of Lemmata 3.3 and 3.4, we obtain u − a = vk + wk for k ∈ N,

68

3 Fractional Ordinary Differential Equations

where vk ∈ (H(k+1)α (0, T ))N and wk ∈ (Wα (0, T ))N provided that (k − 1)α ≤ 2. Hence +W , u−a =V

 ∈ (Hk1 α (0, T ))N , V

 ∈ (Wα (0, T ))N . W

 ∈ (Hk1 α (0, T ))N implies By the Sobolev embedding and (3.47), we see that V  dV ∈ (Hk1 α−1 (0, T ))N ⊂ (L∞ (0, T ))N , dt  ∈ (Wα (0, T ))N . Thus u − a ∈ (Wα (0, T ))N , and the proof of which means that V Theorem 3.6 is complete.  Now we prove Lemmata 3.2 and 3.3. Proof of Lemma 3.2 The first inclusion was proved as Lemma 1.3 (i). By P ∈ (C 1 [0, T ])N×N , we see that P (W 1,1 (0, T ))N ) ⊂ (W 1,1 (0, T ))N and so Lα (W 1,1 (0, T ))N = J α (P (W 1,1 (0, T ))N ) ⊂ J α (W 1,1 (0, T ))N . Thus Lα (W 1,1 (0, T ))N ⊂ (W 1,1 (0, T ))N follows and the proof of Lemma 3.2 is complete.  Proof of Lemma 3.3 In terms of (2.4), (3.39) and (3.40), using P ∈ (C 2 [0, T ])N×N , we obtain P (Hβ (0, T ))N ⊂ (Hβ (0, T ))N ,

0 < β ≤ 2.

(3.48)

Indeed, for (3.48) we need the assumption P ∈ (C 2 [0, T ])N×N , not P ∈ (C 1 [0, T ])N×N . In order to prove Lemma 3.3, thanks to (3.48), it suffices to prove J α Hβ (0, T ) ⊂ Hα+β (0, T ) for 0 < β ≤ 2 and 0 < α < 1.

(3.49) 

Proof of (3.49) For 0 < α + β ≤ 1 and 0 < α < 1, estimate (3.49) is already proved in Theorem 2.3 (i). We have to prove more general cases. Let α + β > 1. It is sufficient to prove (3.49) for β = 1 + δ with 0 ≤ δ ≤ 1. Let v ∈ Hβ (0, T ) be arbitrarily given. Then dv dt ∈ Hδ (0, T ). Hence v ∈ Hβ (0, T ) = H1+δ (0, T ) implies 1,1 v ∈ W (0, T ) and v(0) = 0. Moreover, d α dv J v = Jα dt dt

3.6 Linear Fractional Ordinary Differential Equations with Multi-Term. . .

69

by Lemma 1.3 (i) and so d α J v ∈ J α (Hδ (0, T )). dt If α + δ ≤ 1, then Theorem 2.3 (i) yields J α (Hδ (0, T )) ⊂ Hα+δ (0, T ). By the definition of Hα+δ+1 (0, T ), we see J α v ∈ Hα+β (0, T ).

(3.50)

Next we assume 1 < α + δ < 2. We can represent α + δ = 1 + δ1 with some δ1 ∈ (0, 1). By 0 ≤ δ ≤ 1 and Theorem 2.1 (i), we obtain Hδ (0, T ) = J δ L2 (0, T ) and it follows from α + δ = 1 + δ1 and Lemma 1.3 (iv) that J α Hδ (0, T ) = J α (J δ L2 (0, T )) = J α+δ L2 (0, T ) = J δ1 (J 1 L2 (0, T )). 1

d Since dt (J δ1 (J 1 w)) = J δ1 d(Jdt w) = J δ1 w for all w ∈ L2 (0, T ), the definition of H1+δ1 (0, T ) yields J 1+δ1 w ∈ H1+δ1 (0, T ) for each w ∈ L2 (0, T ). Therefore

J α Hδ (0, T ) ⊂ H1+δ1 (0, T ) = Hα+δ (0, T ). Thus (3.50) holds for all (α, β) satisfying 0 < β ≤ 2 and 0 < α < 1, and so the proof of (3.49) is complete. Thus the proof of Lemma 3.3 is finished. 

3.6 Linear Fractional Ordinary Differential Equations with Multi-Term Fractional Derivatives We have discussed the unique existence of the solutions to initial value problems for fractional ordinary differential equations on the basis of ∂tα in the Sobolev spaces Hα (0, T ), and the method is widely applicable for example, to nonlinear equations. Rather than comprehensive discussions, here we are restricted to linear fractional ordinary differential equations with multi-term time fractional derivatives: m 

α

rj (t)∂t j (u − a)(t) = p(t)u(t) + f (t),

0 0, we have

u L2 (0,T ) ≤ C(|a| + f L2 (0,T ) ),

(3.55)

u − a Hα1 (0,T ) ≤ C(|a| + F L2 (0,T ) )

(3.56)

and

u H α1 (0,T ) ≤ C(|a| + f L2 (0,T ) ),

if 0 < α < 1 and α =

1 . 2

(3.57)

Proof We can rewrite (3.51) as J −α1 (u − a)(t) +

m 

rj (t)J −αj (u − a)(t) = p (t)u(t) + f(t),

0 < t < T.

j =2

(3.58) Here and henceforth we set rj (t) =

rj (t) , r1 (t)

j = 2, . . . , m,

p (t) =

p(t) , r1 (t)

f (t) f(t) = , r1 (t)

0 ≤ t ≤ T.

Therefore (3.51) with (3.52) is equivalent to w(t) = −

m 

rj (t)J α1 −αj w(t)+ p J α1 w(t)+ p (t)a+f(t),

0 α2 > · · · > αm > 0, we have (t − s)α1 −αj −1 = (t − s)α1 −α2 +(α2 −αj )−1 ≤ T α2 −αj (t − s)(α1 −α2 )−1 and (t − s)α1 −1 = (t − s)(α1 −α2 )−1 (t − s)α2 ≤ T α2 (t − s)(α1 −α2 )−1 , and so 

t

|v(t)| ≤ C

(t − s)α1 −α2 −1 |v(s)|ds,

0 < t < T.

0

Noting that α1 −α2 > 0, we apply the generalized Gronwall inequality (Lemma A.2) to verify that v = 0 in (0, T ). Thus the proof of the unique existence is complete. The estimates (3.55)–(3.57) are proved similarly to Theorem 3.5. 

Chapter 4

Initial Boundary Value Problems for Time-Fractional Diffusion Equations

4.1 Main Results Let  ⊂ Rn be a bounded domain with boundary ∂ of C 2 -class and let ν = ν(x) = (ν1 , . . . , νn ) be the unit outward normal vector to ∂ at x = (x1 , . . . , xn ). Henceforth let ∂i =

∂ , ∂xi

∂i2 =

∂2 , ∂xi2

i = 1, . . . , n

∇ = (∂1 , . . . , ∂n ),

∂s =

∂ . ∂s

We would like to discuss an initial boundary value problem for a time-fractional partial differential equation, which we can write as follows at the expense of rigor: ⎧ α ⎨ dt u(x, t) + A(t)u(x, t) = F (x, t), u(x, t) = 0, x ∈ ∂, 0 < t < T , ⎩ u(x, 0) = a(x), x ∈ .

x ∈ , 0 < t ≤ T , (4.1)

Let −A(t) be a uniform elliptic differential operator of the second order with (x, t)-dependent coefficients: (−A(t)u)(x, t) =

n  i,j =1

∂i (aij (x, t)∂j u(x, t)) +

n 

bj (x, t)∂j u(x, t) + c(x, t)u(x, t),

j =1

© The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 A. Kubica et al., Time-Fractional Differential Equations, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-15-9066-5_4

73

74

4 Initial Boundary Value Problems for Time-Fractional Diffusion Equations

for x ∈  and t > 0, where aij = aj i , bj , c ∈ C 2 ([0, T ]; C 1 ()), and there exists a constant μ0 > 0 such that n  i,j =1

aij (x, t)ξi ξj ≥ μ0

n 

ξj2 ,

x ∈ , 0 ≤ t < ∞, (ξ1 , . . . , ξn ) ∈ Rn .

j =1

(4.2) Remark 4.1 We can relax the regularity conditions of the coefficients similarly to [17, 30], by approximating them by smooth functions. However, we omit the details for simplicity. The main purpose of this chapter is to formulate the initial boundary value problem for t-dependent A(t) with initial value a and non-homogeneous term F in L2 -spaces. We refer to Kubica and Yamamoto [17], Zacher [30] for such treatments, and here we describe a unified approach within the framework by means of ∂tα in the Sobolev space Hα (0, T ). In particular, for more regular F , for example F ∈ L∞ (0, T ; L2 ()), there are several works on the well-posedness for fractional partial differential equations and we can refer to Gorenflo et al. [11], Li et al. [18], Luchko [21], Sakamoto and Yamamoto [26]. Here we do not intend any complete list of the references. We mainly assume that a and F are in some L2 -spaces. Then in general we cannot prove that u ∈ C([0, T ]; L2 ()). Therefore, similarly to the case of fractional ordinary differential equations, we must be careful in interpreting the initial condition u(·, 0) = a in (4.1), which cannot imply that u(·, t) −→ a in L2 () as t → 0. Henceforth for H01 () ⊂ L2 (), identifying the dual of L2 () of L2 () with itself, we define H −1 () = (H10 ()) . Then H01 () ⊂ L2 () ⊂ H −1 () algebraically and topologically (e.g., Brezis [5]). In terms of the definition ∂tα in Hα (0, T ) defined in Chap. 2, we formulate an initial boundary value problem as follows: ∂tα (u − a)(x, t) + A(t)u(x, t) = F u(·, t) ∈ H01 (),

in H −1 (), 0 < t < T , 0 < t < T,

u − a ∈ Hα (0, T ; H −1 ()).

(4.3) (4.4) (4.5)

For 12 < α < 1, we have Hα (0, T ) ⊂ H α (0, T ) ⊂ C[0, T ] by the Soloblev embedding, and so (4.5) implies that u − a ∈ C([0, T ]; H −1()) and u(x, 0) = a(x) is satisfied in the sense of limt →0 u(·, t) = a in H −1 ().

4.1 Main Results

75

Remark 4.2 Moreover, in view of (2.38), interpreting ∂tα u by (2.37), we note that (4.3) is equivalent to ⎧ α ⎪ ⎪ ∂t u + A(t)u = F + ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

t −α (1−α) a,

∂tα u + A(t)u = F,

in H −1 (), 0 < t < T , if 0 < α < 12 , −1 in H (), 0 < t < T , if 12 < α < 1.

(4.6)

We note that if u ∈ C 1 ((0, T ]; C())∩C([0, T ]; C())∩L2 (0, T ; C 2 ()) satisfies (4.1), then u satisfies also (4.3)–(4.5). Now we are ready to state our main results. Theorem 4.1 We assume regularity aij , bj , c ∈ C 2 ([0, T ]; C 1 ()), 1 ≤ i, j ≤ n. For F ∈ L2 (0, T ; H −1 ()) and a ∈ L2 (), there exists a unique solution u ∈ L2 (0, T ; H01 ()) satisfying u − a ∈ Hα (0, T ; H −1 ()) to (4.3)–(4.5). Moreover, there exists a constant C > 0 such that

u − a Hα (0,T ;H −1 ()) + ∇u L2 (0,T ;L2 ()) ≤ C( a L2 () + F L2 (0,T ;H −1 ()) ). (4.7) In Zacher [30], and Kubica and Yamamoto [17] (Theorem 1.1), with the same regularity conditions on a and F , the unique existence of the solution u to (4.3)– (4.5) is proved with the regularity J 1−α (u − a) ∈ 0 H 1 (0, T ; H −1 ()),

u ∈ L2 (0, T ; H01 ()).

Noting that 0 H 1 (0, T ; H −1 ()) = H1 (0, T ; H −1()) by the definition, we can verify that the class of solutions in [17, 30] is the same as in Theorem 4.1, that is, J 1−α (u − a) ∈ 0 H 1 (0, T ; H −1 ()),

u ∈ L2 (0, T ; H01 ())

if and only if u − a ∈ Hα (0, T ; H −1 ()),

u ∈ L2 (0, T ; H01 ()).

Indeed, setting α := 1 − α and β := α in Theorem 2.3 (i) of Chap. 2, we see that J 1−α : Hα (0, T ) −→ H1 (0, T ) is surjective and isomorphism. Therefore J 1−α (u − a) ∈ H1 (0, T ; H −1()) = 0 H 1 (0, T ; H −1 ()) if and only if u − a ∈ Hα (0, T ; H −1 ()), which means that the solution classes coincide. In particular, for 12 < α < 1, we have

u − a C([0,T ];H −1 ()) ≤ C F L2 (0,T ;H −1 ()).

(4.8)

76

4 Initial Boundary Value Problems for Time-Fractional Diffusion Equations

Indeed, the estimate (4.8) is directly seen by (4.7), because Hα (0, T ) ⊂ H α (0, T ) ⊂ C[0, T ] if 12 < α < 1 by the Sobolev embedding. Remark 4.3 By (2.40), we can rewrite (4.7) as

∂tα u L2 (0,T ;H −1 ()) + u L2 (0,T ;H 1 ()) 0

≤ C( a L2 () + F L2 (0,T ;H −1 ()) ),

α =

1 . 2

(4.9)

Here ∂tα is defined by (2.37). Proof of (4.9) For 0 < α < 12 , by (2.38) we have ∂tα (u − a) = ∂tα u − ∂tα a = ∂tα u −

t −α a. (1 − α)

Therefore, since t −α L2 (0,T ) < ∞ for 0 < α < 12 , inequality (4.7) yields  

∂tα u L2 (0,T ;H −1 ()) ≤ ∂tα (u − a) L2 (0,T ;H −1 ()) +  

  t −α a (1 − α) L2 (0,T ;H −1 ())

≤ C( F L2 (0,T ;H −1 ()) + a L2 (0,T ;H −1 ()) ).

Next, let

1 2

< α < 1. Then (4.9) is seen by the definition (2.37).



Remark 4.4 In Sakamoto and Yamamoto [26], in a special case where A(t) is symmetric (i.e., bj = c = 0, 1 ≤ j ≤ n) and the coefficients of A(t) are independent of t and F = 0, then for (4.1) it is proved that lim u(·, t) − a L2 () = 0

t →0

for each a ∈ L2 (). See also Sect. 4.5. For more regular a and F , we can prove: Proposition 4.1 We assume all the conditions in Theorem 4.1. Moreover, let p > α2 , and let F ∈ Lp (0, T ; H −1()) and a ∈ H01 (). Then there exists a constant C > 0 such that α

u − a L∞ (0,T ;L2 ()) ≤ C(t 2 a H 1 () + t κ F Lp (0,T ;H −1 ()) ). 0

Here κ=

pα − 2 > 0. 2p

(4.10)

4.1 Main Results

77

In particular, lim u(·, t) − a L2 () = 0.

(4.11)

t →0

The proposition means that with more regular a and F , we can prove the continuity of u(·, t) at t = 0 in L2 (), and so the initial condition (4.5) holds in the usual sense. Remark 4.5 For dtα u = div (p(x, t)∇u(x, t)) + F (x, t) in (4.1) with a = 0, Jin, Li and Zhou (Theorem 2.1 in [12]) proves that u ∈ C([0, T ]; L2 ()) ∩ Lp (0, T ; H 2 ()) and dtα u ∈ Lp (0, T ; L2 ()) if F ∈ Lp (0, T ; L2 ()) with p > α1 . In fact, we can interpret that p > α1 is a critical condition for the continuity of u in t. Proposition 4.1 requires a stronger assumption p > α2 for p, but the weaker spatial regularity is needed. The continuity of u at t = 0 should be exploited more but here we omit details. For more regular a and F , we can improve the regularity of u. Theorem 4.2 We assume regularity aij ∈ C 2 ([0, T ]; C 1 ()), bj , c ∈ C 2 ([0, T ]; C 1 ()), 1 ≤ i, j ≤ n. For F ∈ L2 (0, T ; L2 ()) and a ∈ H01 (), there exists a unique solution u ∈ L2 (0, T ; H 2() ∩ H01 ()) satisfying u − a ∈ Hα (0, T ; L2 ()) to (4.3)–(4.5). Moreover, there exists a constant C > 0 such that

u − a Hα (0,T ;L2 ()) + u L2 (0,T ;H 2 ()) ≤ C( F L2 (0,T ;L2 ()) + a H 1 () ). 0

Theorems 4.1 and 4.2 are corresponding results to the classical results for the parabolic equation (i.e., α = 1) for which we refer to Evans [7], Lions and Magenes [19], for example. In the case of F = 0 in Theorem 4.2, we can further prove: Proposition 4.2 We assume all the conditions in Theorem 4.2 and F = 0 and max aij L∞ (×(0,∞)) ,

1≤i,j ≤n

max bj L∞ (×(0,∞)) ,

1≤j ≤n

c L∞ (×(0,∞)) < ∞.

Then there exists a constant C > 0 such that

u(·, t) L2 () ≤ CeCt u(·, 0) H 1 () , 0

for all the solutions u to (4.3)–(4.5).

t ≥0

78

4 Initial Boundary Value Problems for Time-Fractional Diffusion Equations

4.2 Some Results from Theorem 4.1 and Proposition 4.2 4.2.1 Interpolated Regularity of Solutions Our classes of solutions in Theorems 4.1 and 4.2 are flexible for interpolated regularity properties. Although we can choose a general uniform elliptic operator, we introduce the Laplacian with the homogeneous Dirichlet boundary condition: −A0 u(x) =

n 

∂k2 u(x),

D(A0 ) = H 2 () ∩ H01 ().

k=1 1

−1

Then it is known that D(A02 ) = H01 () and H −1 () = (H01 ()) = D(A0 2 ). Moreover, we can verify that for 0 ≤ θ ≤ 1, the interpolation spaces are given by: [Hα (0, T ; L2 ()), L2 (0, T ; H 2() ∩ H01 ())]1−θ = [Hα (0, T ; D(A00 )), L2 (0, T ; D(A0 ))]1−θ 2−2θ = Hαθ (0, T ; D(A1−θ ()) 0 )) ⊂ Hαθ (0, T ; H

(e.g., Yamamoto [29]). Here we consider only the case of a = 0 and F ∈ L2 (0, T ; L2 ()). Therefore for each θ ∈[0, 1], in Theorem 4.2 we have u∈Hαθ (0, T ; H 2−2θ ()). 1

In particular, by D(A02 ) = H01 (), we see that u ∈ H 1 α (0, T ; H01()) by choosing 2

θ = 12 in Theorem 4.2, which is included in Theorem 1.4 in Kubica and Yamamoto [17].

4.2.2 The Method by the Laplace Transform for Fractional Partial Differential Equations Let the elliptic operator A(t) in (4.1) be t-independent: (Av)(x) = −

n  i,j =1

∂i (aij (x)∂j v(x)) −

n 

bj (x)∂j v(x) − c(x)v(x),

x ∈ ,

j =1

where aij , bj , c ∈ C 2 () and (4.2) is satisfied. A convenient way for constructing a solution to (4.3)–(4.5) is by the Laplace transform (see Sect. 2.7 of Chap. 2), which relies on formulae of Laplace transforms of time-fractional derivatives. For the rigorous treatments, we have to specify the class of solutions u admitting such formulae for the Laplace transforms, but it is not often clarified in view of the consistency with the expected regularity of solutions to (4.3)–(4.5).

4.2 Some Results from Theorem 4.1 and Proposition 4.2

79

Within our framework, we can apply Theorem 2.7 of Chap. 2 concerning the Laplace transform of ∂tα in Vα (0, ∞). Here we sketch such treatments. As related arguments, see Kian and Yamamoto [14]. For arbitrary T > 0, let u ∈ L2 (0, T ; H 2() ∩ H01 ()) satisfy u − a ∈ Hα (0, T ; L2 ()) and (4.3)–(4.5) with a ∈ H01 () and F = 0. In particular, ∂tα (u − a) + Au = 0 in  × (0, T ). Taking the scalar products in L2 () of both sides with arbitrarily fixed ϕ ∈ C0∞ () and integrating by parts, we have (∂tα (u − a)(·, t), ϕ)L2 () + (Au(·, t), ϕ)L2 () = 0,

t > 0.

Then ∂tα {((u − a)(·, t), ϕ)L2 () } + (u(·, t), A∗ ϕ)L2 () = 0,

t > 0.

(4.12)

Here ∗

(A v)(x) = −

n 

∂i (aij (x)∂j v) +

i,j =1

n 

∂j (bj (x)v(x)) − c(x)v(x),

x ∈ .

j =1

By u − a ∈ Hα (0, T ; L2 ()), we see that ((u − a)(·, t), ϕ)L2 () ∈ Hα (0, T ). Proposition 4.2 yields u(·, t) L2 () ≤ CeCt a H 1 () for all t ≥ 0. Hence 0

|((u − a)(·, t), ϕ)L2 () |,

|(u(·, t), A∗ ϕ)L2 () | ≤ CeC1 t ,

t ≥ 0.

Here C > 0 depends on a and ϕ. Consequently, ((u(·, t) − a), ϕ)L2() ∈ Vα (0, ∞) and the Laplace transform of (u(·, t), A∗ ϕ)L2 () : ∗

L((u(·, t), A ϕ)L2 () )(p) :=



∞ 0

e−pt (u(·, t), A∗ ϕ)L2 () dt

exists for p > C1 . We take the Laplace transforms of both sides of (4.12) and apply Theorem 2.7, so that pα (L(u − a)(·, p), ϕ)L2 () + ((Lu)(·, p), A∗ ϕ)L2 () = 0,

p > C1 ,

that is, ({pα L(u − a)(·, p) + A(Lu)(·, p)}, ϕ)L2 () = 0,

p > C1

80

4 Initial Boundary Value Problems for Time-Fractional Diffusion Equations

for all ϕ ∈ C0∞ (), which means pα L(u − a)(·, p) + A(Lu)(·, p) = 0,

p > C1

in L2 (). Thus we can justify the method by the Laplace transform for an initial boundary value problem (4.3)–(4.5).

4.3 Proof of Theorem 4.1 The proof is based on what is called the Galerkin approximation (e.g., Evans [7], Lions and Magenes [19]). The proof is composed of: (i) Construction of approximating solutions in a family of finitely dimensional subspaces: For this step, Theorem 3.6 (iii) plays an important role. (ii) Uniform boundedness of the approximating solutions: The coercivity Theorems 3.3 and 3.4 are essential. (iii) Then we can prove the existence of solution as a weak convergent limit of a subsequence of the sequence of the approximating solutions. The uniqueness of the solution follows from the coercivity. First we prove the existence of solutions. Let 0 < λ1 ≤ λ2 ≤ · · · be the eigenvalues of − with the zero Dirichlet boundary condition, which are numbered according to the multiplicities, that is, λm appears -times if the multiplicity of λm is . By ρk ∈ H 2 (), k ∈ N, we denote an eigenfunction of − for λk such that

1, k = , (ρk , ρ )L2 () = δk := 0, k = . Then {ρk }k∈N is a basis in each L2 (), H01 (), (ρk , ρ )L2 () = δk ,

H −1 () ρk , ρ H01 ()

= δk .

(4.13)

Moreover, ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

a=

∞

k=1 (a, ρk )L2 () ρk

in L2 (),

a 2L2 () =

∞

k=1 |(a, ρk )L2 () |

there exists a constant C > 0 such that  −1 () and a= ∞ k=1 H −1 () a, ρk H01 () ρk in H  −1 2 2 C −1 ∞ ⎪ k=1 λk |H −1 () a, ρk H01 () | ≤ a H −1 () ⎪ ⎪  ⎪ −1 2 ⎪ ≤C ∞ a ∈ H −1 (), ⎪ k=1 λk |H −1 () a, ρk H01 () | , ⎪ ⎪   ⎪ 2 2 2 ⎩ C −1 ∞ ≤C ∞ k=1 λk |(a, ρk )| ≤ a 1 k=1 λk |(a, ρk )| , H0 ()

2,

a ∈ H01 (). (4.14)

4.3 Proof of Theorem 4.1

81

We note that H −1 () ϕ, ψH01 ()

= (ϕ, ψ)L2 ()

for ϕ ∈ L2 () and ψ ∈ H01 ()

(e.g., p.136 in [5]). By the density of C0∞ (0, T ; H −1()) in L2 (0, T ; H −1()), for any ε > 0, we can choose F ε ∈ C0∞ (0, T ; H −1 ()) such that

F ε − F L2 (0,T ;H −1 ()) < ε.

(4.15)

T Indeed, we can construct such F ε by the mollifier 0 F (x, s)χε (t − s)ds, where ∞ χε ∈ C0∞ (R), ≥ 0 satisfies supp χε ⊂ (−ε, ε) and −∞ χε (t)dt = 1 for small ε > 0. Let N ∈ N be fixed arbitrarily. We look for uεN (x, t) =

N 

ε pN,k (t)ρk (x),

FNε (x, t) =

k=1

N 

H −1 () F

ε

(·, t), ρk H 1 () ρk (x) 0

k=1

(4.16) satisfying  (∂tα (uεN − aN ), ρ )L2 () + (A(t)uεN , ρ )L2 () = H −1 () FNε , ρ H 1 () , uεN − aN ∈ Hα (0, T ; L2 ()),

1 ≤  ≤ N, 0 ≤ t ≤ T ,

0

(4.17)

where we set aN =

N 

ck ρk ,

ck = (a, ρk )L2 () .

(4.18)

k=1 ε and summing up over  = 1, . . . , N, Multiplying the first equation in (4.17) by pN, we have

(∂tα (uεN − aN ), uεN )L2 () + (A(t)uεN , uεN )L2 () = H −1 () FNε , uεN H 1 () , 0

0 ≤ t ≤ T.

(4.19)

We rewrite (4.17) as 

 ε ε ε ∂tα (pN, − c ) = N k=1 k (t)pN,k (t) + f (t), ε − c ∈ Hα (0, T ), 1 ≤  ≤ N, pN,

0 < t < T,

(4.20)

82

4 Initial Boundary Value Problems for Time-Fractional Diffusion Equations

where ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

fε (t) = H −1 () F ε (·, t), ρ H 1 () , 0  k (t) = ni,j =1 (∂i (aij (·, t)∂j ρk ), ρ (·))L2 ()  + ni=1 (bi (·, t)∂i ρk , ρ )L2 () + (c(·, t)ρk , ρ )L2 () ⎪   ⎪ ⎪ = − ni,j =1  aij (x, t)(∂j ρk )(x)(∂i ρ )(x)dx ⎪ ⎪ ⎩ n + i=1 (bi (·, t)∂i ρk , ρ )L2 () + (c(·, t)ρk , ρ )L2 () . By the regularity of F ε and aij , we know that k ∈ C 2 [0, T ] and fε ∈ W 1,1 (0, T ). Consequently, we can apply Theorem 3.6 (iii) to see that there exists a unique ε ε solution pN, ∈ W 1,1 (0, T ) to (4.20) and pN, (0) = c for  = 1, . . . , N. ε ε ε 1,1 (0) = ck , we see that Next we estimate uN Since pN,k ∈ W (0, T ) and pN,k ε 1,1 pN,k − ck ∈ Hα (0, T ) ∩ 0 W (0, T ) and so (2.27) yields ε ε ε − ck ) = dtα (pN,k − ck ) = dtα pN,k , ∂tα (pN,k

N ∈ N, 1 ≤ k ≤ N.

(4.21)

ε (dtα pN,k )ρk = dtα uεN .

(4.22)

Therefore, ∂tα (uεN − aN ) =

N 

ε ∂tα (pN,k − ck )ρk =

k=1

N  k=1

By (4.19) we have (t − s)α−1 (dtα uεN (·, s), uεN (·, s))L2 () n 

−

(t − s)α−1 (∂i (aij (·, s)∂j uεN (·, s)), uεN (·, s))L2 ()

i,j =1

−

n 

(t − s)α−1 (bj (·, s)∂j uεN (·, s), uεN (·, s))L2 ()

j =1

− (t − s)α−1 (c(·, s)uεN (·, s), uεN (·, s))L2 () = H −1 () FNε (·, s), uεN (·, s)H 1 () (t − s)α−1 . 0

(4.23)

By (4.2) we obtain −

n 

(t − s)α−1 (∂i (aij (·, s)∂j uεN (·, s)), uεN (·, s))L2 ()

i,j =1

=

n  i,j =1

 (t − s)α−1 

aij (x, s)(∂j uεN (x, s))∂i uεN (x, s))dx

≥ C0 (t − s)α−1 ∇uεN (·, s) 2L2 () .

(4.24)

4.3 Proof of Theorem 4.1

83

Moreover, fixing small δ > 0, we can choose a constant Cδ > 0 such that we have        n   n ε ε  − (b (·, s)∂ u (·, s), u (·, s)) |∂j uεN (x, s)||uεN (x, s)|dx 2 () ≤ C j j L N N    j =1   j =1 

 ≤δ 

|∇uεN (x, s)|2 dx + Cδ



|uεN (x, s)|2 dx.

(4.25)

Similarly we see |H −1 () FNε (·, s), uεN (·, s)H 1 () | ≤ δ ∇uεN (·, s) 2L2 () + Cδ FNε (·, s) 2H −1 () . 0

(4.26) On the other hand, by Theorem 3.4 (i) and uεN ∈ W 1,1 (0, T ; L2 ()), we obtain 

t

0

(t − s)α−1 (dtα uεN (·, s), uεN (·, s))L2 () ds ≥

(α) ( uεN (·, t) 2L2 () − aN 2(L2 ())N ). 2

Therefore application of (4.24)–(4.26) in (4.23) yields 

uεN (·, t ) 2L2 () +  ≤ aN 2L2 () + Cδ  + Cδ 0

t

0

t

0 t

(t − s)α−1 ∇uεN (·, s) 2L2 () ds

(t − s)α−1 ∇uεN (·, s) 2L2 () ds 

(t − s)α−1 uεN (·, s) 2L2 () ds + Cδ

0

t

(t − s)α−1 FNε (·, s) H −1 () ds,

0 < t < T.

Fixing δ > 0 sufficiently small and absorbing the second term on the right-had side into the left-hand side, we obtain

uεN (·, t) 2L2 ()



≤ uεN (·, t) 2L2 () +  ≤ C aN 2L2 () 

+C

0

0 t 0

t

+C

t

(t − s)α−1 ∇uεN (·, s) 2L2 () ds

(t − s)α−1 uεN (·, s) 2L2 () ds

(t − s)α−1 FNε (·, s) 2H −1 () ds.

(4.27)

84

4 Initial Boundary Value Problems for Time-Fractional Diffusion Equations

The generalized Gronwall inequality Lemma A.2 in the Appendix, implies

uεN (·, t) 2L2 () ≤ C aN 2L2 ()  +C

t



t

+C 0

 (t − s)α−1

0

(t − s)α−1 FNε (·, s) 2H −1 () ds

s

0

 (s − ξ )α−1 FNε (·, ξ ) 2H −1 () dξ ds,

0 < t < T.

Here  

t

(t − s)

0

0

s

 (s − ξ )

0 t

= =

 α−1



FNε (·, ξ ) 2H −1 ()

(α)2 (2α)



t 0

t

α−1

FNε (·, ξ ) 2H −1 () dξ

ds

 (t − s)α−1 (s − ξ )α−1 ds dξ

ξ

(t − ξ )2α−1 FNε (·, ξ ) 2H −1 () dξ.

Since (t − s)2α−1 ≤ C(t − s)α−1 ,

0 ≤ s ≤ t ≤ T,

we have 

uεN (·, t) 2L2 ()

≤

C aN 2L2 ()

t

+C 0

(t − s)α−1 FNε (·, s) 2H −1 () ds.

T Taking 0 · · · dt and applying the Young inequality on the convolution (Lemma A.1 in the Appendix), we have

uεN (·, t) 2L2 (0,T ;L2 ()) ≤ C( aN 2L2 () + FNε 2L2 (0,T ;H −1 ()) ).

(4.28)

By (4.22), applying Theorem 3.3 (i) we obtain 

T 0

 (∂tα (uεN − aN )(·, t), uεN (·, t))L2 () dt =  ≥ C0 0

T

T 0

(dtα uεN (·, t), uεN (·, t))L2 () dt 

uεN (·, t) 2L2 () dt

− aN 2L2 ()

.

(4.29)

4.3 Proof of Theorem 4.1

85

On the other hand, estimating the second term on the left-hand side and the righthand side of (4.19) by (4.29) and (4.25)–(4.26), we have 



T

0

uεN (·, t) 2L2 () dt



T

≤ Cδ 

0

+ Cδ

T

0

+ 0

T

∇uεN (·, t) 2L2 () dt 

∇uεN (·, t) 2L2 () dt + Cδ

T 0

uεN (·, t) 2L2 () dt

FNε (·, t) 2H −1 () dt + C aN 2L2 () .

Therefore, fixing δ > 0 small and absorbing the first term on the right-hand side into the left-hand side, we reach

∇uεN 2L2 (0,T ;L2 ()) ≤ C( aN 2L2 () + uεN 2L2 (0,T ;L2 ()) + FNε 2L2 (0,T ;H −1 ()) ). Applying (4.28), we obtain

∇uεN 2L2 (0,T ;L2 ()) ≤ C( aN 2L2 () + FNε 2L2 (0,T ;H −1 ()) ).

(4.30)

Here C > 0 is independent of N ∈ N and ε > 0. Now we estimate ∂tα (uεN − aN ) L2 (0,T ;H −1 ()). Since ∂tα (uεN − aN ) =

N 

ε ∂tα (pN,k − ck )ρk ,

k=1

we have H −1 ()

∂tα (uεN − aN ), ρ H 1 () =

N 

0

ε ∂tα (pN,k (t) − ck )H −1 () ρk , ρ H 1 () = 0 0

k=1

for  ≥ N + 1. For any ψ ∈ H01 (), we set ψN = Therefore (4.17) yields α ε H −1 () ∂t (uN

N

=1 H −1 () ψ, ρ H01 () ρ .

− aN ), ψH 1 () 0

= − H −1 () A(t)uεN , ψN H 1 () 0

+ H −1 () FNε , ψN H 1 () , 0

ψ ∈ H01 ().

86

4 Initial Boundary Value Problems for Time-Fractional Diffusion Equations

Hence

∂tα (uεN − aN )(·, t ) H −1() =

sup ψ∈H01 (), ψ H 1 () =1

|H −1 () ∂tα (uεN − aN )(·, t ), ψH 1 () | 0

0

≤

sup ψ∈H01 (), ψ H 1 () =1

|H −1 () A(t )uεN , ψN H 1 () | 0

0

+

sup ψ∈H01 (), ψ H 1 () =1

|H −1 () FNε (·, t ), ψN H 1 () | 0

0

≤C

sup ψ∈H01 (), ψ H 1 () =1

|(∇uεN (·, t ), ∇ψN )L2 () |

0

+

sup ψ∈H01 (), ψ H 1 () =1

FNε (·, t ) H −1 () ∇ψN L2 ()

0

≤ C( uεN (·, t ) L2()

+ FNε (·, t ) H −1() ).

Here we used also ψN H 1 () ≤ ψ H 1 () ≤ 1. 0 0 Since aN L2 () ≤ a L2 () and FNε L2 (0,T ;H −1 ()) ≤ F ε L2 (0,T ;H −1 ()) , combining (4.28) and (4.30), we obtain

∂tα (uεN − aN ) 2L2 (0,T ;H −1 ()) + uεN 2L2 (0,T ;H 1 ()) 0

≤ C( a 2L2 () + F ε 2L2 (0,T ;H −1 ()) ).

(4.31)

The sequences {uεN − aN }N∈N and {uεN }N∈N are bounded in H α (0, T ; H −1 ()) and in L2 (0, T ; H01()) respectively. Therefore we can extract a subsequence N  of N ∈ N and uε ∈ L2 (0, T ; H01 ()) and v ε ∈ Hα (0, T ; H −1 ()) such that uεN  −→ uε weakly in L2 (0, T ; H01 ()) and uεN  − aN  −→ v ε weakly in Hα (0, T ; H −1 ()). Since aN  −→ a strongly in L2 (), we have uεN  −→ a + v ε weakly in Hα (0, T ; H −1 ()). Therefore uεN  −→ uε and uεN  −→ a + v ε in the sense of distribution, that is, in (C0∞ ( × (0, T ))) . Hence uε = a + v ε , that is, v ε = uε − a. Therefore by (4.31) we have

uε − a 2H (0,T ;H −1 ()) + uε 2L2 (0,T ;H 1 ()) α 0  ε 2 ε 2  α

u ≤ lim inf − a +

u

  N −1 1 2 N N L (0,T ;H ()) H (0,T ;H ())  N →∞

≤ C( F ε 2L2 (0,T ;H −1 ())

0

+ a 2L2 () ).

(4.32)

4.3 Proof of Theorem 4.1

87

Henceforth by XN we denote the closed subspace in H01 () spanned by ρ1 , . . . , ρN . For any ψ ∈ XN , it follows from (4.17) that  



T H −1 ()

0 T

= 0

∂tα (uεN − aN ), ψH 1 () dt + 0

ε H −1 () FN (·, t), ψH01 () dt,

T 0

ε H −1 () A(t)uN , ψH01 () dt

N ∈ N.

Consequently, letting N → ∞, we obtain 

T

0

 α ε H −1 () ∂t (u

− a), ψH 1 () dt + 0



T

= 0

H −1 () F

ε

T 0

H −1 () A(t)u, ψH01 () dt

(4.33)

(·, t), ψH 1 () dt. 0

Since ψN ∈ XN and N ∈ N are chosen arbitrarily, it follows that (4.33) holds for each ψ ∈ H01 (), so that ∂tα (uε − a) + A(t)uε = F ε

in H −1 (), 0 < t < T .

By (4.32) we choose a sequence {εn }n∈N satisfying limn→∞ εn = 0 and u ∈ L2 (0, T ; H01 ()) such that ∂tα (uεn −a) −→ ∂tα (u−a) weakly in L2 (0, T ; H −1 ()) and uεn −→ u weakly in L2 (0, T ; H01()) as n → ∞. Since Fεn −→ F in L2 (0, T ; H −1 ()) as n → ∞ by (4.15), we can verify (4.7) and ∂tα (u − a)(·, t) + A(t)u(·, t) = F (·, t)

in H −1 (), 0 < t < T .

Finally, we have to prove the uniqueness of the solution to (4.3)–(4.5). To this end, it is necessary to verify that if u ∈ Hα (0, T ; H −1 ()) ∩ L2 (0, T ; H01()) satisfies ∂tα u −

n 

∂i (aij (x, t)∂j u) =

i,j =1

n 

bj (x, t)∂j u + c(x, t)u

in H −1 (),

(4.34)

j =1

then u = 0 in  × (0, T ). From (4.34) and integration by parts in view of u(·, t) ∈ H01 (), we have ⎛ α H −1 () ∂t u(·, s), u(·, s)H01 ()

= H −1 () 

n  j =1

+⎝

n 

⎞ aij (·, s)∂j u(·, s), ∂i u(·, s)⎠

i,j =1

L2 ()

bj (·, s)∂j u(·, s), u(·, s)H 1 () + H −1 () c(·, s)u, uH 1 () , 0

0

0 < s < T.

88

4 Initial Boundary Value Problems for Time-Fractional Diffusion Equations

Therefore 

t 0



⎛

t

+

α H −1 () ∂t u(·, s), u(·, s)H01 () (t

(t − s)α−1 ⎝

0

n 

− s)α−1 ds ⎞

aij (x, s)(∂j u(x, s))∂i u(x, s)dx ⎠ ds

i,j =1

⎞ ⎛  t   n bj (x, s)(∂j u(x, s))u(x, s)dx ⎠ ds = (t − s)α−1 ⎝ 0



t

+ 0

 (t − s)

 j =1

 2

α−1

c(x, s)u (x, s)dx ds. 

Applying (4.2) and (4.25), we derive  

t 0 t

+ 0

α H −1 () ∂t u(·, s), u(·, s)H01 () (t

− s)α−1 ds

(t − s)α−1 ∇u(·, s) 2L2 () ds 

t

≤ Cδ 0

 (t − s)α−1 ∇u(·, s) 2L2 () ds + Cδ

t 0

(t − s)α−1 u(·, s) 2L2 () ds.

By u ∈ Hα (0, T ; H −1 ()) ∩ L2 (0, T ; H01 ()), applying Theorem 3.4 (ii) to the first term on the left-hand side and choosing δ > 0 sufficiently small, we have  t (α) (α) (t − s)α−1 ∇u(·, s) 2L2 () ds

u(·, t) 2L2 () ≤

u(·, t) 2L2 () + (1 − Cδ) 2 2 0  t ≤ Cδ (t − s)α−1 u(·, s) 2L2 () ds, 0 < t < T . 0

The generalized Gronwall inequality Lemma A.2 in the Appendix implies u = 0 in (0, T ). Thus the proof of Theorem 4.1 is complete.

4.4 Proof of Theorem 4.2 For F ∈ L2 (0, T ; L2 ()) and ε > 0, let F ε ∈ C0∞ (0, T ; L2 ()) satisfy F ε − F L2 (0,T ;L2 ()) < ε. In terms of Theorem 3.6 (iii) and (v), we can argue similarly ε to the proof of Theorem 4.1 to construct an approximating sequence, that is, pN,k ∈

4.4 Proof of Theorem 4.2

89

ε (0) = c := (a, ρ ) (Wα (0, T ))N and pN,k k k L2 () for 1 ≤ k ≤ N. Here we have to estimate

uεN (x, t) :=

N 

ε pN,k (t)ρk (x)

in L2 (0, T ; H 2())

k=1

 and ∂tα (uεN − aN ) in L2 (0, T : L2 ()), where aN = N k=1 ck ρk . ε Using (4.21), multiplying the first equation in (4.20) by dtα pN, , summing over  = 1, . . . , N, we obtain (dtα uεN (·, t), dtα uεN (·, t))L2 () −

n 

(∂i (aij (·, t)∂j uεN (·, t)), dtα uεN (·, t))L2 ()

i,j =1

−

n  (bj (·, t)∂j uεN (·, t), dtα uεN (·, t))L2 () − (c(·, t)uεN (·, t), dtα uεN (·, t))L2 () j =1

= (FNε (·, t), dtα uεN (·, t))L2 () .

(4.35)

The main part of the proof is the estimation of the second term on the left-hand side. By uεN (·, t) ∈ H01 (), integration by parts yields ⎛ −⎝

n 

⎞ ∂i (aij (·, t)∂j uεN (·, t)), dtα uεN (·, t)⎠

i,j =1

⎛ =⎝

⎞

n 

aij (·, t)∂j uεN (·, t), dtα ∂i uεN (·, t)⎠

i,j =1

Now we consider we have n 

L2 ()

. L2 ()

n

ε α ε i,j =1 aij (∂j uN )∂i dt uN (·, t).

Since aij = aj i , 1 ≤ i, j ≤ n,

aij (∂j uεN )∂i dtα uεN (·, t)

i,j =1

=

 i>j

aij ((∂j uεN )∂i dtα uεN + (∂i uεN )∂j dtα uεN ) +

n  i=1

aii (∂i uεN )(∂i dtα uεN ) =: J1 + J2 .

90

4 Initial Boundary Value Problems for Time-Fractional Diffusion Equations

We set g(t) = J1 =



1 −α . (1−α) t

aij ((∂j uεN )∂i dtα uεN + (∂i uεN )∂j dtα uεN )(x, t)

i>j

=



 0

i>j

=−

t

aij (x, t)



Here we write uεN (s) = uεN (x, s), etc. Then

g(t − s)((∂j uεN )(t)∂i ∂s uεN (x, s) + (∂i uεN )(t)∂j ∂s uεN (x, s))ds



t

aij (x, t) 0

i>j

g(t − s)((∂i ∂s uεN (s))(∂j uεN (s) − ∂j uεN (t))

+ (∂j ∂s uεN (s))(∂i uεN (s) − ∂i uεN (t))ds  t  + aij (x, t) g(t − s)((∂i ∂s uεN (s))∂j uεN (s) + (∂j ∂s uεN (s))∂i uεN (s))ds 0

i>j

=: J11 + J12 . We have J11 = −





t

aij (x, t) 0

i>j

g(t − s){∂s (∂i uεN (s) − ∂i uεN (t))(∂j uεN (s) − ∂j uεN (t))

+ ∂s (∂j uεN (s) − ∂j uεN (t))(∂i uεN (s) − ∂i uεN (t))}ds  t  =− aij (x, t) g(t − s)∂s ((∂i uεN (s) − ∂i uεN (t))(∂j uεN (s) − ∂j uεN (t)))ds 0

i>j

=−



aij (x, t)[g(t − s)(∂i uεN (s) − ∂i uεN (t))(∂j uεN (s) − ∂j uεN (t))]s=t s=0

i>j

−



 0

i>j

=



t

aij (x, t)

dg (t − s)(∂i uεN (s) − ∂i uεN (t))(∂j uεN (s) − ∂j uεN (t))ds dξ

aij (x, t)g(t)(∂i uεN (0) − ∂i uεN (t))(∂j uεN (0) − ∂j uεN (t))

i>j



t

+ 0

−

 dg (t − s) aij (x, t)(∂i uεN (s) − ∂i uεN (t))(∂j uεN (s) − ∂j uεN (t))ds. dξ i>j

Here we used lim g(t − s)(∂i uεN (s) − ∂i uεN (t))(∂j uεN (s) − ∂j uεN (t)) = 0, s↑t

which can be justified by ∂i uεN (x, ·), ∂j uεN (x, ·) ∈ Wα (0, T ) following from Theorem 3.6 (v).

4.4 Proof of Theorem 4.2

91

On the other hand, J12 = 

=





t

aij (x, t) 0

i>j

g(t − s)∂s (∂i uεN (x, s)∂j uεN (x, s))ds

aij (x, t)dtα (∂i uεN ∂j uεN )(x, t).

i>j

Consequently, 1 aij g(t)(∂i uεN (0) − ∂i uεN (t))(∂j uεN (0) − ∂j uεN (t)) 2

J1 = + +

i=j



1 2

t

−

0

 dg (t − s) aij (x, t)(∂i uεN (s) − ∂i uεN (t))(∂j uεN (s) − ∂j uεN (t))ds dξ i=j

1 aij dtα (∂i uεN ∂j uεN )(x, t). 2 i=j

Similarly, we can calculate J2 =





0

i

=−



+

i

=−

t 0



t

aii 0

g(t − s)∂i ∂s uεN (s)∂i uεN (t)ds

g(t − s)∂i ∂s uεN (s)(∂i uεN (s) − ∂i uεN (t))ds

g(t − s)∂i ∂s uεN (s)∂i uεN (s)ds

 t 1 aii g(t − s)∂s ((∂i uεN (s) − ∂i uεN (t))(∂i uεN (s) − ∂i uεN (t)))ds 2 0 i

+



aii

i



t

aii

1 aii 2 i



t 0

g(t − s)∂s (|∂i uεN (s)|2 )ds

1 1 = aii g(t)(∂i uεN (t) − ∂i uεN (0))2 + 2 2 i

1 + aii dtα (|∂i uεN (t)|2 ). 2 i



t 0

−

 dg (t − s) aii (∂i uεN (s) − ∂i uεN (t))2 ds dξ i

92

4 Initial Boundary Value Problems for Time-Fractional Diffusion Equations

Thus n 

aij ∂j uεN (x, t)∂i dtα uεN (x, t)

i,j =1

=

n 1  g(t)aij (x, t)(∂i uεN (0) − ∂i uεN (t))(∂j uεN (0) − ∂j uεN (t)) 2 i,j =1

+

+

1 2



t

−

0

n  dg (t − s) aij (x, t)(∂i uεN (s) − ∂i uεN (t))(∂j uεN (s) − ∂j uεN (t))ds dξ i,j =1

n 1  aij dtα ((∂i uεN )∂j uεN )(x, t). 2 i,j =1

By g(t) > 0, − dg dξ (t − s) > 0 for 0 < s < t < T and (4.2), we obtain n 

2

aij (x, t)∂j uεN (x, t)∂i dtα uεN (x, t) ≥

i,j =1



t

=

n 

g(t − s)

n 

g(t − s)

0

 +

t

∂s (aij (x, s)∂i uεN (s)∂j uεN (s))ds

i,j =1 t

−

aij (x, t)dtα ((∂i uεN )∂j uεN )(x, t)

i,j =1

0



n 

(∂s aij (x, s))∂i uεN (s)∂j uεN (s)ds

i,j =1 n 

g(t − s)

0

(aij (x, t) − aij (x, s))∂s ((∂i uεN (s))∂j uεN (s))ds =: S1 + S2 + S3 .

i,j =1

(4.36) Here we have 

t

S1 = 0

⎛



t

=

n 

g(t − s)

g(s)∂t ⎝

0

∂s (aij (x, s)∂i uεN (s)∂j uεN (s))ds

i,j =1 n 

(4.37) ⎞

aij (x, t − s)∂i uεN (t − s)∂j uεN (t − s)⎠ ds

i,j =1



t

= ∂t

g(s) 0

− g(t)

n 

aij (x, t − s)∂i uεN (t − s)∂j uεN (t − s)ds

i,j =1 n 

i,j =1

aij (x, 0)∂i uεN (x, 0)∂j uεN (x, 0).

4.4 Proof of Theorem 4.2

93

Furthermore 

t

S2 = −

n 

g(t − s)

0

(∂s aij (x, s))∂i uεN (s)∂j uεN (s)ds

i,j =1

and  S3 =

t

n 

0 i,j =1

rij (x, t, s)∂s (∂i uεN (s)∂j uεN (s))ds,

where rij (x, t, s) =

(t − s)−α (aij (x, t) − aij (x, s)), (1 − α)

1 ≤ i, j ≤ n.

We directly obtain 

t

|S2 | ≤ C 0

(t − s)−α |∇uεN (·, s)|2 ds.

(4.38)

To estimate |S3 |, we integrate by parts and apply    α(t − s)−α−1  (t − s)−α  (aij (x, t) − aij (x, s)) − ∂s aij (x, s) |∂s rij (x, t, s)| =  (1 − α) (1 − α) ≤C|t − s|−α by aij ∈ C 1 ([0, T ]; C()), so that ⎤s=t ⎡  n   |S3 | =  ⎣ rij (x, t, s)∂i uεN (s)∂j uεN (s)⎦ i,j =1

 −

t

n 

0 i,j =1

  ε ε (∂s rij )(x, t, s)∂i uN (s)∂j uN (s)ds 

≤ C|∇uεN (x, 0)|2 + C



t 0

s=0

(t − s)−α |∇uεN (x, s)|2 ds.

94

4 Initial Boundary Value Problems for Time-Fractional Diffusion Equations

Therefore (4.36)–(4.38) imply n 

2

aij (x, t)∂j uεN (x, t)∂i dtα uεN (x, t)

i,j =1



n 

t

≥ ∂t

g(s) 0

− g(t)

aij (x, t − s)∂i uεN (x, t − s)∂j uεN (x, t − s)ds

i,j =1 n 

aij (x, 0)∂i uεN (x, 0)∂j uεN (x, 0)

i,j =1

− C|∇uεN (x, 0)|2



t

−C 0

(t − s)−α |∇uεN (x, s)|2 ds,

x ∈ , 0 < t < T . (4.39)

T  Taking 0  · · · dxdt and applying the Young inequality (Lemma A.1 in the Appendix) to the fourth term on the right-hand side of (4.39), we have 

n 

T

2 0



i,j =1

T

≥

(aij (·, t)∂j uεN (·, t), dtα ∂i uεN (·, t))L2 () dt

g(s) 0

n 

(aij (T − s)∂i uεN (T − s), ∂j uεN (T − s))L2 () ds

i,j =1



T

−C 0

g(t)dt ∇a 2L2 () − C( ∇a L2 () + ∇uεN 2L2 (0,T ;L2 ()) ).

Since 

T

g(s) 0



(aij (·, T − s)∂i uεN (·, T − s), ∂j uεN (·, T − s))L2 () ds

i,j =1 T

=

n 

g(T − s)

0

≥ CT

n 

(aij (·, s)∂i uεN (·, s), ∂j uεN (·, s))L2 () ds

i,j =1 −α

∇uεN 2L2 (0,T ;L2 ())

4.4 Proof of Theorem 4.2

95

by (4.2), we obtain 

T

[the second term on the left-hand side of (4.35)]dt 0

⎛



T

= 0

⎝

n 

(4.40)

⎞ aij (·, t)∂j uεN (·, t), dtα ∂i uεN (·, t)⎠

i,j =1

dt L2 ()

≥ − C( ∇uεN 2L2 (0,T ;L2 ()) + ∇a 2L2 () ). By the Poincaré inequality and uεN (·, t) ∈ H01 (), for small δ > 0, we can estimate 

T

|[the third and the fourth terms on the left-hand side of (4.35)]dt

0



T

≤ 0

(Cδ ∇uεN (·, t) 2L2 () + δ dtα uεN (·, t) 2L2 () )dt

(4.41)

and    

T 0

 

(FNε (·, t), dtα uεN (·, t))L2 () dt 

≤ Cδ FNε 2L2 (0,T ;L2 ()) + Cδ dtα uεN (·, t) 2L2 (0,T ;L2 ()) .

(4.42)

Hence, applying (4.40)–(4.42) in (4.35), we derive

dtα uεN 2L2 (0,T ;L2 ()) ≤ C ∇a 2L2 () + Cδ ∇uεN 2L2 (0,T ;L2 ()) + δ dtα uεN 2L2 (0,T ;L2 ()) + Cδ FNε 2L2 (0,T ;L2 ()) . Fixing δ > 0 sufficiently small and absorbing the third term on the right-hand side into the left-hand side and applying (4.30), which was verified in the proof of Theorem 4.1, we obtain

∂tα (uεN − a) 2L2 (0,T ;L2 ()) = dtα uεN 2L2 (0,T ;L2 ()) ≤ C( ∇aN 2L2 () + FNε 2L2 (0,T ;L2 ()) ),

(4.43)

where the constant C > 0 is independent of N ∈ N and ε > 0. Based on the uniform bound (4.43) in N and ε, we can argue similarly to (4.31)– (4.33). We omit the details, so that we can complete the proof of Theorem 4.2.

96

4 Initial Boundary Value Problems for Time-Fractional Diffusion Equations

We note that the uniqueness of solutions is proved for less regular class, and the uniqueness holds also within the stronger regularity of solutions in Theorem 4.2.

4.5 Proofs of Propositions 4.1 and 4.2 4.5.1 Proof of Proposition 4.1 As is already proved, u ∈ Hα (0, T ; H −1 ()) ∩ L2 (0, T ; H01 ()) satisfies

∂tα (u − a) + A(t)u = F in H −1 (), 0 < t < T , u(·, t) ∈ H01 (), u − a ∈ Hα (0, T ; H −1 ()).

Then we have α H −1 () ∂s (u −

a)(·, s), u − aH 1 () + H −1 () A(s)(u − a)(·, s), u − aH 1 () 0

0

+ H −1 () A(s)a, u − aH 1 () = H −1 () F (·, s), (u − a)(·, s)H 1 () 0

0

for 0 < s < T . Here we note that for any ε > 0 there exists a constant Cε > 0 such that ⎧ |H −1 () A(s)a, u(·, s) − aH 1 () | ≤ C a H 1 () (u − a)(·, s) H 1 () ⎪ ⎪ 0 0 0 ⎪ ⎪ ⎪ ≤ ε (u − a)(·, s) 2H 1 () + Cε a 2H 1 () , ⎪ ⎪ ⎪ 0 0   ⎪ ⎪  ⎨ n j =1 H −1 () bj (·, s)∂j (u(·, s) − a), u(·, s) − aH01 ()  ⎪ + |H −1 () c(·, s)(u(·, s) − a), u(·, s) − aH 1 () | ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ≤ C u(·, s) − a L2 () u(·, s) − a H01 () ⎪ ⎪ ⎪ ⎩ ≤ ε u(·, s) − a 2 1 + Cε u(·, s)0 − a 2 2 . L () H () 0

(4.44) Here we used

bj (·, s)∂j (u(·, s) − a) H −1 () =

sup

ψ H 1 () =1

|H −1 () bj (·, s)∂j (u(·, s) − a), ψH 1 () |

0

=

sup

ψ H 1 () =1

|(u(·, s) − a, ∂j (bj ψ))L2 () |

0

≤

sup

ψ H 1 () =1 0

u(·, s) − a L2 () ∂j (bj ψ) L2 () ≤ C u(·, s) − a L2 () .

0

4.5 Proofs of Propositions 4.1 and 4.2

97

Multiplying (t −s) and integrating in s over (0, t) and applying (4.2) and (α) Theorem 3.4 (ii), we obtain α−1



u(·, t) − a 2L2 () + 

t

≤C 0

t

0

(t − s)α−1 ∇(u(·, s) − a) 2L2 () ds

(t − s)α−1 |H −1 () A(s)a, u(·, s) − aH 1 () |ds 0

    t n     b (·, s)∂ (u(·, s) − a), u(·, s) − a ds +  (t − s)α−1 −1 1 j j H () H0 ()    0 j =1  t    +  H −1 () c(·, s)(u(·, s) − a), u(·, s) − aH 1 () (t − s)α−1 ds  0 0  t (t − s)α−1 F (·, s) H −1 () u(·, s) − a H 1 () ds. +C 0

0

Applying (4.44) and

F (·, s) H −1 () u(·, s) − a H 1 () ≤ ε u(·, s) − a 2H 1 () + Cε F (·, s) 2H −1 () , 0

0

we obtain 

u(·, t) − a 2L2 () + 

t

≤ε 0

t 0

(t − s)α−1 u(·, s) − a 2H 1 () ds

(t − s)α−1 u(·, s) − a 2H 1 () ds + Cε 0



t

+ Cε

(t − s)

α−1

0

0



a 2H 1 () ds 0



t

+ Cε 0

t 0

(t − s)α−1 u(·, s) − a 2L2 () ds

(t − s)α−1 F (·, s) 2H −1 () ds.

Choosing ε > 0 sufficiently small, we absorb the first term on the right-hand side into the left-hand side, so that

u(·, t) − a 2L2 ()



≤ u(·, t) − a 2L2 () + C 

t

≤C 0

t 0

(t − s)α−1 u(·, s) − a 2H 1 () ds

(t − s)α−1 u(·, s) − a 2L2 () ds 

+ Ct α a 2H 1 () + C 0

t 0

(t − s)α−1 F (·, s) 2H −1 () ds.

0

98

4 Initial Boundary Value Problems for Time-Fractional Diffusion Equations

Since F ∈ Lp (0, T ; H −1 ()) with p > α2 , we can choose q > 1 satisfying p 1 and q(α −1) > −1. Indeed, q = p−2 and p > −1. Therefore the Hölder inequality yields



t

 (t − s)

α−1

0

F (·, s) 2H −1 () ds

t

≤

(t − s)

qα−q

imply q(α −1) =

2 α

 q1 

t

ds

0

0

1 p 2

+ q1 =

p p−2 (α −1)

p ( F (·, s) 2H −1 () ) 2 ds

>

 p2

θ

≤ Ct q F 2Lp (0,t;H −1 ()) ,

where θ = q(α − 1) + 1 > 0. Thus θ

u(·, t) − a 2L2 () ≤ C(t α a 2H 1 () + t q F 2Lp (0,T ;H −1 ())) 0



t

+C 0

(t − s)α−1 u(·, s) − a 2L2 () ds,

0 < t < T.

Lemma A.2 in the Appendix implies θ

u(·, t) − a 2L2 () ≤ C((t α + t 2α ) a 2H 1 () + (t q + t

α+ qθ

0

) F 2Lp (0,T ;H −1 ()) ),

which completes the proof of Proposition 4.1.

4.5.2 Proof of Proposition 4.2 The proof is based on an estimate similar to Proposition 4.1. By (4.3), we have ∂sα (u − a)(·, s) + A(s)u(x, s) = 0

in L2 (), 0 < s < t.

Multiplying both sides by (u − a)(x, s)(t − s)α−1 and integrating over  × (0, t), we obtain   t  α (∂s (u − a)(x, s))(u − a)(x, s)dx (t − s)α−1 ds 0

 −



⎛ t  ⎝

0



t

+ 0

⎛  ⎝

n   i,j =1 n   i,j =1

⎞ ∂i (aij (x, s)∂j u(x, s))u(x, s)dx ⎠ (t − s)α−1 ds ⎞ ∂i (aij (x, s)∂j u(x, s))a(x)dx ⎠ (t − s)α−1 ds

4.5 Proofs of Propositions 4.1 and 4.2



t

−

⎞ ⎛   n ⎝ bj (x, s)∂j u(x, s)(u(x, s) − a(x))dx ⎠ (t − s)α−1 ds

0

−

99

 t  0

 j =1

 c(x, s)u(x, s)(u(x, s) − a(x))dx (t − s)α−1 ds = 0.



Since (u − a)(x, ·) ∈ Hα (0, T ) for x ∈  and u(·, s) ∈ H01 () for 0 < s < t, we integrate by parts and use Theorem 3.4 (ii) where we note  α H −1 () ∂s (u(·, s)−a), u(·, s)−aH 1 () = 0



(∂sα (u(x, s)−a(x))(u(x, s)−a(x))dx,

by ∂sα (u − a) ∈ L2 (0, T ; L2 ()), so that  C0 u(·, t) − a 2L2 () + 

t

−

⎛  ⎝

0



0

⎛  ⎝

n 

⎞ aij (x, s)∂j u(x, s)∂i u(x, s)dx ⎠ (t − s)α−1 ds

 i,j =1

⎞

aij (x, s)∂j u(x, s)∂i a(x)dx ⎠ (t − s)α−1 ds

 i,j =1

t

−

n 

t

⎛ ⎞   n ⎝ bj (x, s)∂j u(x, s)u(x, s)dx ⎠ (t − s)α−1 ds

0

 j =1

⎛ ⎞  t   n ⎝ bj (x, s)∂j u(x, s)a(x)dx ⎠ (t − s)α−1 ds + 0

− +

 j =1

 t 

 c(x, s)u(x, s)2 dx (t − s)α−1 ds

0



0



 t 

 c(x, s)u(x, s)a(x)dx (t − s)α−1 ds = 0.

By (4.2) and using |u||a| ≤ 12 (|u|2 + |a|2), we obtain C0 u(·, t) 2L2 () − C1 a 2L2 () + C2 −

 t 0

−

 t 0

C3 ε|∇u(x, s)|(t − s)

α−1 2

C3 ε|∇u(x, s)|(t − s)

α−1 2





 

 t  0

 |∇u(x, t)|2 (t − s)α−1 dx ds



α−1 1 |∇a(x)|(t − s) 2 ε



α−1 1 |u(x, s)|(t − s) 2 ε

dxds  dxds

100

4 Initial Boundary Value Problems for Time-Fractional Diffusion Equations

 α−1 1 2 |a(x)|(t − s) − C3 ε|∇u(x, s)|(t − s) dxds ε 0   t  t 2 α−1 |u(x, s)| (t − s) dxds − C3 |a(x)|2(t − s)α−1 dxds ≤ 0. − C3  t

0

α−1 2



0





Hence, after applying the Cauchy–Schwarz inequality, choosing ε > 0 suffit  ciently small and absorbing 0  C32 ε2 |∇u(x, s)|2 (t − s)α−1 dxds into the term t  C2 0  |∇u(x, s)|2 (t − s)α−1 dxds, we obtain

u(·, t) 2L2 () + C4 ≤C

 t  0

+C

0

|∇u(x, t)|2 (t − s)α−1 dxds 



(|∇a(x)| + |a(x)| )dx (t − s)α−1 dxds 2

2



 t 0

 t

|u(x, s)|2 (t − s)α−1 dxds,

t > 0.



Therefore 

u(·, t) 2L2 ()

t

+ 0

u(·, s) 2H 1 () (t − s)α−1 ds 0



≤ Ct α a 2H 1 () + C 0

t 0

u(·, s) 2L2 () (t − s)α−1 ds,

t > 0,

that is, 

u(·, t) 2L2 () ≤ Ct α a 2H 1 () + C 0

t 0

u(·, s) 2L2 () (t − s)α−1 ds,

t > 0.

Consequently, noting that the constant C > 0 can be chosen uniformly in all t > 0 and applying Lemma A.2 in the Appendix, we can take a constant C0 > 0 such that 

u(·, t) 2L2 ()

≤ C0 t

α

a 2H 1 () 0

+ C0 e

t

C0 t

 (t − s)

s ds a 2H 1 ()

α−1 α

0

  (α)(α + 1) 2α t

a 2H 1 () ≤ C0 t α + C0 eC0 t (2α + 1) 0 for all t ≥ 0. Thus the proof of Proposition 4.2 is complete.

0

4.6 Case Where the Coefficients of A(t) Are Independent of Time

101

4.6 Case Where the Coefficients of A(t) Are Independent of Time In the case where the coefficients of A(t) are independent of t, we can represent the solution to the initial boundary value problem by the Mittag-Lefller functions (e.g., Sakamoto and Yamamoto [26]). In this section, we show that such a represented solution coincides with the solution established by Theorem 4.1. Let −Av(x) =

n 

∂i (aij (x)∂j v(x)) + c(x)v(x),

x ∈ ,

i,j =1

where aij = aj i ∈ C 1 (), 1 ≤ i, j ≤ n, c ∈ L∞ (), c ≤ 0 in , and (4.2) is assumed to hold. We consider the operator A with the domain D(A) = H 2 () ∩ H01 (). Then it is known that there exist eigenvalues of A which can be numbered with their multiplicities 0 < λ1 ≤ λ2 ≤ · · · −→ ∞. Henceforth if there is no fear of confusion, then we write the scalar product (f, g) for f, g ∈ L2 (), in place of (f, g)L2 () . Let ϕk , k ∈ N be an eigenfunction of A for λn such that

 (ϕk , ϕ ) :=

ϕk (x)ϕ(x)dx = 

1, k = , 0, k = .

Moreover, it is known that ϕk , k ∈ N, is an orthonormal basis in L2 (). We can define a fractional power Aγ for γ ∈ R (e.g., Pazy [23]), and for a ∈ D(Aγ ), we have Aγ a =

∞ 

γ

λk (a, ϕk )ϕk

k=1

where the series is convergent in L2 (), and +

A a L2 () = γ

∞  k=1

,1 2

2γ λk (a, ϕk )2

.

102

4 Initial Boundary Value Problems for Time-Fractional Diffusion Equations

For a ∈ L2 () and F ∈ L2 (0, T ; H −1()), we consider v(x, t) =

∞ 

(a, ϕk )Eα,1 (−λk t α )ϕk (x)

k=1

+

∞  

t 0

k=1

 H −1 () F (·, s),

ϕk H 1 () (t − s) 0

α−1

Eα,α (−λk (t − s) )ds ϕk (x)

=: v1 (x, t) + v2 (x, t).

α

(4.45)

We will verify: Theorem 4.3 Let a ∈ L2 () and F ∈ L2 (0, T ; H −1 ()). Then v given by (4.45) satisfies v − a ∈ Hα (0, T ; H −1 ()), v ∈ L2 (0, T ; H01 ()) and (4.3)–(4.5). The theorem means that (4.45) coincides with the solution by Theorem 4.1. We can similarly prove that v defined by (4.45) is the same as u in Theorem 4.2 if a ∈ H01 () and F ∈ L2 (0, T ; L2 ()), but we omit the proof. Proof It is sufficient to prove that v − a ∈ Hα (0, T ; H −1 ()) and v satisfies (4.3). First Step First we note that D(A− 2 ) = H −1 () = (H01 ()) and there exist constants C1 , C2 > 0 such that 1

1

C1 w H −1 () ≤ A− 2 w L2 () ≤ C2 w H −1 () for w ∈ H −1 (), and

A

− 12

w = 2

∞ 

2 λ−1 k (w, ϕk ) .

k=1

Since a =

∞

k=1 (a, ϕk )ϕk

in L2 (), we have

∞  v1 (x, t) − a(x) = (a, ϕk )(Eα,1 (−λk t α ) − 1)ϕk (x).

(4.46)

k=1

Second Step Theorem 2.4 implies

(v1 − a)(x, ·) Hα (0,T ) ∼ ∂tα (v1 − a)(x, ·) L2 (0,T ) for almost all x ∈ . Moreover, since Eα,1 (−λk t α ) − 1 ∈ Hα (0, T ) ∩ 0 W 1,1 (0, T ) by Proposition 2.2, we see that dtα (Eα,1 (−λk t α ) − 1) = ∂tα (Eα,1 (−λk t α ) − 1),

4.6 Case Where the Coefficients of A(t) Are Independent of Time

103

so that ∞  − a)(x, t) = (a, ϕk )∂tα (Eα,1 (−λk t α ) − 1)ϕk (x)

∂tα (v1

k=1

=−

∞ 

λk (a, ϕk )Eα,1 (−λk t α )ϕk (x).

(4.47)

k=1

Therefore Lemma 2.5 (i) implies

∂tα (v1 − a)(·, t) 2H −1 () =

∞  (a, ϕk )2 |Eα,1 (−λk t α )|2 λk k=1

≤Ct −α

∞  k=1

⎛

⎞2 α 2 t λ (a, ϕk )2 ⎝ k α ⎠ ≤ Ct −α a 2L2 () , 1 + λk t 1 2

that is,

∂tα (v1 − a) 2L2 (0,T ;H −1 ()) ≤ C a 2L2 () .

(4.48)

Hence v1 − a ∈ Hα (0, T ; H −1()). Third Step First we note an important property of Eα,1 , which is a consequence of Lemma 2.5 (ii) and what is called the complete monotonicity: d Eα,1 (−ηα ) ≤ 0, dη

Eα,α (−ηα ) ≥ 0,

η ≥ 0,

0 0 and θ > 0 such that

u(·, t) L2 () ≤ Ce−θt a L2 () ,

t >0

for each a ∈ L2 () by the classical energy estimate. For 0 < α < 1, we know that

u(·, t) L2 () ≤

C

a L2 () , tα

t >0

© The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 A. Kubica et al., Time-Fractional Differential Equations, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-15-9066-5_5

(5.3)

109

5 Decay Rate as t → ∞

110

in the case where the coefficients of A(t) are independent of t (e.g., [26]). On the other hand, Vergara and Zacher [28] proved (5.3) for t-dependent A(t). For t-dependent A(t), in a conventional energy estimate, we multiply the equation by u − a and integrate over  and we may obtain only a weak estimate

u(·, t) L2 () ≤

C α

t2

a H 1 () , 0

t > 0.

This is weaker than we can expect in terms of (5.3) for t-independent A(t), because the decay rate is α2 . In this chapter, we modify the arguments in Vergara and Zacher [28] to prove (5.3) within our framework of solutions in the case where the coefficients of A(t) are dependent on time t. More precisely, we prove: Theorem 5.1 In (5.2), we assume aij = aj i ∈ C 2 ([0, ∞); C 1 ()),

1 ≤ i, j ≤ n,

(5.4)

c ∈ L∞ ( × (0, ∞)), and there exists a constant μ0 > 0 such that n 

aij (x, t)ξi ξj ≥ μ0

i,j =1

n 

ξj2 ,

x ∈ , t ≥ 0, ξ1 , ..., ξn ∈ R

(5.5)

j =1

and c(x, t) ≤ 0,

x ∈ , t ≥ 0.

(5.6)

Then there exists a constant C > 0 such that

u(·, t) L2 () ≤ Ct −α a L2 () for each solution u to (5.1). For the proof, we have to return to the construction of the solution to (5.1) by the Galerkin approximation. By means of a conventional energy estimate which can be obtained by multiplying the fractional equation by the solution u to integrate over  by parts, we cannot reach the desired estimate u(·, t) L2 () ≤ Ct −α u(·, 0) L2 () . Thus we need another kind of energy inequality, namely Lemma 5.1 by Vergara and Zacher [28]. The lemma is technical and is shown in Sect. 5.2. Next, repeating the argument by the Galerkin approximation based on the lemma, we can complete the proof of the desired decay estimate.

5.2 Key Lemmata

111

5.2 Key Lemmata The proof is based on another kind of energy inequality: Lemma 5.1 We set G(s) =

1 1 2 s −s+ . 2 2

We assume that u ∈ C( × [0, T ]), u(x, ·) ∈ W 1,1 (0, T )

for x ∈ ,

(5.7)

and t 1−α ∂t u ∈ L∞ ( × (0, T )).

(5.8)

Then for each t ∈ (0, T ] we have ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

dtα u(·, t) 2L2 () +

+

t 2α (1−α) 0 (t

2t −α 2 (1−α) u(·, t) L2 () G



−

s)−α−1 u(·, t) 2L2 () G

u(·,0) L2 ()

u(·,t ) L2 ()



≤2





u(·,s) L2 ()

u(·,t ) L2 ()

 ds

α  u(x, t)dt u(x, t)dx,

if u(·, t) L2 () = 0, if u(·, t) L2 () = 0.

dtα u(·, t) 2L2 () ≤ 0,

(5.9) Furthermore 

u(·, t) L2 () dtα u(·, t) L2 () ≤



u(x, t)dtα u(x, t)dx.

(5.10)

Proof Choosing M > 0 sufficiently large, we have uM := u+M > 0 on ×[0, T ]. We note that uM satisfies (5.7) and (5.8). We have  2 uM (x, t)dtα uM (x, t)dx − dtα uM (·, t) 2L2 () 

=

2 (1 − α)

2 = (1 − α)

 

t

(t − s)−α (∂s uM )(x, s)(uM (x, t) − uM (x, s))dsdx

 0





|uM (x, t)| 

2 0

t

(t − s)

−α

 ∂s

uM (x, s) uM (x, t)



  uM (x, s) 1− ds dx. uM (x, t)

5 Decay Rate as t → ∞

112

Hence, since G (s) = s − 1, we obtain  2 uM (x, t)dtα uM (x, t)dx − dtα uM (·, t) 2L2 () 

2 =− (1 − α)



 |uM (x, t)|

t

2

(t − s)

−α

0



∂ ∂s

    uM (x, s) G ds dx. uM (x, t)

We integrate by parts to have        t−h uM (x, s) uM (x, s) ∂ (t − s)−α G ds = lim G ds h→0+ 0 uM (x, t) ∂s uM (x, t) 0      t−h uM (x, s) uM (x, s) s=t−h (t − s)−α−1 G ds + (t − s)−α G = lim −α h→0+ uM (x, t) uM (x, t) s=0 0        t−h uM (x, s) uM (x, t − h) uM (x, 0) = lim −α (t − s)−α−1 G ds + h−α G − t −α G . h→0+ uM (x, t) uM (x, t) uM (x, t) 0 

t

(t − s)−α

∂ ∂s

We shall show that lim h−α |uM (x, t)|2 G

h→0+



uM (x, t − h) uM (x, t)

 = 0,

t > 0.

(5.11)

Indeed, since    −α  −α  h |uM (x, t)|2 G uM (x, t − h)  = h |uM (x, t) − uM (x, t − h)|2 ,   uM (x, t) 2 assumption (5.8) yields   t   ∂uM  |uM (x, t − h) − uM (x, t)| =  (x, ξ )dξ  t −h ∂ξ   t  t   α−1  1−α ∂uM α−1   (x, ξ ) ξ dξ ≤ C(t − h)

t 1−α ∂t uM L∞ (×(0,T )) dξ ≤ ξ ∂ξ t −h t −h ≤ h(t − h)α−1 t 1−α ∂t uM L∞ (×(0,T )) , so that (5.11) holds. Thus, letting h → 0+, we obtain  2 

=

uM (x, t)dtα uM (x, t)dx − dtα uM (·, t) 2L2 ()

2α (1 − α)

2t −α + (1 − α)

 

t

(t − s)−α−1 |uM (x, t)|2 G

 0



|uM (x, t)|2 G 





 uM (x, 0) dx. uM (x, t)

 uM (x, s) dsdx uM (x, t)

5.2 Key Lemmata

113

Hence, by setting τ = 0 or = s, the Cauchy–Schwarz inequality yields 

 |uM (x, t)|2 G 

  = 

 uM (x, τ ) dx uM (x, t)

 1 1 2 2 |uM (x, t)| + |uM (x, τ )| − uM (x, t)uM (x, τ ) dx 2 2

1 1 ≥ uM (·, t) 2L2 () + uM (·, τ ) 2L2 () − uM (·, t) L2 () uM (·, τ ) L2 () 2 2 , +

uM (·, τ ) L2 () 2 . = uM (·, t) L2 () G

uM (·, t) L2 () Thus (5.9) holds for uM . If u(·, t) L2 () =  0, then letting M → 0, we obtain (5.9) for u. Next, we assume u(·, t) L2 () = 0. Since G(η) ≥ 0 for η ≥ 0, inequality (5.9) for uM yields  dtα u(·, t)

+

M 2L2 ()

≤2 

 =2 

(u(x, t) + M)dtα (u(x, t) + M)dx

(u(x, t) + M)dtα u(x, t)dx.

The right-hand side tends to 0 as M → 0 by u(·, t) L2 () = 0. Moreover, dtα u(·, t) + M 2L2 () = =

1 (1 − α)

 0

t

1 (1 − α)



t 0

(t − s)−α



d ds

 (u(x, s) + M)2 dx ds

 

   −α 2 (t − s) ∂s u(x, s)dx ds ∂s ( u(·, s) L2 () + 2M

= dtα u(·, t) 2L2 () +

2M (1 − α)

 t 0





(t − s)−α ∂s u(x, s)dxds −→ dtα u(·, t) 2L2 ()

as M → 0 by |∂s u(x, s)| ≤ Cs α−1 by (5.8). Hence dtα u(·, t) 2L2 () ≤ 0, and (5.9) holds for u(·, t) L2 () = 0. Finally, we have to prove (5.10). We can assume that u(·, t) L2 () = 0, because (5.10) is trivial if u(·, t) L2 () = 0. By (5.8), we can directly verify d 1 ds u(·, s) L2 () ∈ L (0, T ). First, dtα u(·, t) 2L2 () − 2 u(·, t) L2 () dtα u(·, t) L2 ()  t 2 d = (t − s)−α ( u(·, s) L2 () − u(·, t) L2 () ) u(·, s) L2 () ds (1 − α) 0 ds , + + ,  t

u(·, s) L2 () d u(·, s) L2 () 2 −α 2 (t − s) u(·, t) L2 () −1 = ds (1 − α) 0

u(·, t) L2 () ds u(·, t) L2 ()

5 Decay Rate as t → ∞

114

2 = (1 − α)



t

(t − s)

0

2α =− (1 − α)



t

−α

d

u(·, t) 2L2 () ds

- + G +

(t − s)

−α−1

u(·, t) 2L2 () G

u(·, s) L2 ()

u(·, t) L2 ()

u(·, s) L2 ()

,. ds ,

ds

u(·, t) L2 () + ,

u(·, s) L2 () s=t 2  + (t − s)−α u(·, t) 2L2 () G (1 − α)

u(·, t) L2 () s=0 + ,  t

u(·, s) L2 () −2α −α−1 2 ≥ (t − s)

u(·, t) L2 () G ds (1 − α) 0

u(·, t) L2 () , +

u(·, 0) L2 () 2t −α 2 . Here we used also (5.8). Hence

u(·, t) L2 () G − (1 − α)

u(·, t) L2 () 0

+ ,  t

u(·, s) L2 () 2α −α−1 2 + (t − s)

u(·, t) L2 () G ds (1 − α) 0

u(·, t) L2 () + ,

u(·, 0) L2 () 2t −α 2

u(·, t) L2 () G + (1 − α)

u(·, t) L2 () dtα u(·, t) 2L2 ()

≥ 2 u(·, t) L2 () dtα u(·, t) L2 () .

Combining with (5.9), we reach (5.10). Thus the proof of Lemma 5.1 is complete. Moreover, we show a generalized extremum principle under weaker assumptions than Lemma 1.2. Lemma 5.2 Let f ∈ W 1,1 (0, T ) attain its maximum over the interval [0, T ] at a point t0 ∈ (0, T ]. If for each κ ∈ (0, T ), there exists β ∈ (0, 1] such that f ∈ 1

W 1, 1−β (κ, T ), then (dtα f )(t0 ) ≥ 0 for every α ∈ (0, β). Proof Firstly, we introduce the function g(t) := f (t0 ) − f (t) for t ∈ [0, T ]. We notice that g(t) ≥ 0 and (dtα g)(t) = −(dtα f )(t) for t ∈ [0, T ]. Then g ∈ W 1,1 (0, T ) and g(t0 ) = 0. Hence, for κ ∈ (0, t0 ), the Hölder inequality yields  |g(t)| ≤ t

t0

     dg     (s) ds ≤  dg  1 |t − t0 |β  ds   dt  1−β L (κ,T )

for t ∈ [κ, t0 ].

(5.12)

Therefore for small h > 0 satisfying κ < t0 − h < t0 , we have    dg   1 hβ . |g(t0 − h)| ≤   ds  1−β L (κ,T )

(5.13)

5.3 Completion of Proof of Theorem 5.1

115

We note that 0 < α < β. For fixed α ∈ (0, β), we obtain (dtα g)(t0 ) =

1 (1 − α)



κ

(t0 − s)−α

0

dg 1 (s)ds + ds (1 − α)



t0 κ

(t0 − s)−α

dg (s)ds. ds

1 We fix ε > 0 arbitrarily. It follows from g ∈ W 1,1 (0, T ) that dg ds ∈ L (0, T ). By the Young inequality, we deduce that there exists sufficiently small κ > 0 such that the first integral is smaller than ε. As for the second one, integration by parts, g ≥ 0 on [0, T ] and g(t0 ) = 0 yield



t0

dg (s)ds = lim h→0+ ds



t0 −h

dg (s)ds ds κ κ  t0 −h −α −α = − (t0 − κ) g(κ) + lim h g(t0 − h) − lim α (t0 − s)−α−1 g(s)ds (t0 − s)−α

(t0 − s)−α

h→0+

−α

≤ lim h h→0+

h→0+

κ

g(t0 − h) = 0.

For the last limit, we used (5.13) by noting 0 < α < β. Thus the proof of Lemma 5.2 is completed. 

5.3 Completion of Proof of Theorem 5.1 We apply the argument in Sect. 4.3 of Chap. 4 for proving the existence of u to (5.1), and we use the same notations. Since the non-homogeneous term F (x, t) is identically zero, we need not introduce the approximation parameter ε > 0 like Sect. 4.3 of Chap. 4. We recall that the function uN (x, t) :=

N 

pN, (t)ρ (x)

=1

satisfies  

(dtα uN )(x, t)ρ (x)dx +

 =

c(x, t)uN (x, t)ρ (x)dx 

N   i,j =1 

aij (x, t)∂j uN (x, t)∂i ρ (x)dx

5 Decay Rate as t → ∞

116

for  = 1, ..., N. In terms of (4.20), we apply Theorem 3.6 (iv) to see pN, − c ∈ Wα (0, T ). We multiply this identity by pN, (t) and sum it up from  = 1 to N:  

uN (x, t)(dtα uN )(x, t)dx +

 =

N  

aij (x, t)(∂j uN )(x, t)(∂i uN )(x, t)dx

i,j =1 

c(x, t)|uN (x, t)|2 dx. 

Using (5.5) and (5.6), we obtain  

uN (x, t)(dtα uN )(x, t)dx + μ0 ∇uN (·, t) 2L2 () ≤ 0.

We define uN,η = uN + ηρN+1 , where η > 0 is a constant. Using the fact that dtα (ηρN+1 ) = 0 and the orthogonality of {ρN }∞ N=1 , we obtain  

uN,η (x, t)dtα uN,η (x, t)dx + μ0 ∇uN (t, ·) 2L2 () ≤ 0.

By Theorem 3.6 (iv) in Chap. 3, we see that the functions uN,η satisfy the assumptions of Lemma 5.1, so that (5.10) and the Poincaré inequality yield

uN,η (·, t) L2 () dtα uN,η (·, t) L2 () + c0 uN (·, t) 2L2 () ≤ 0. Here the constant c0 > 0 is independent of N and η. Since uN,η = uN + ηρN+1 and (uN , ρN+1 ) =

N 

pN, (t)(ρ , ρN+1 ) = 0,

=1

we have

uN,η (·, t) 2L2 () = uN (·, t) 2L2 () + η2 ,

(5.14)

5.3 Completion of Proof of Theorem 5.1

117

that is, c0 uN,η (·, t) 2L2 () = c0 uN (·, t) 2L2 () + c0 η2 . Therefore

uN,η (·, t) L2 () dtα uN,η (·, t) L2 () + c0 uN,η (·, t) 2L2 () = uN,η (·, t) L2 () dtα uN,η (·, t) L2 () + c0 uN (·, t) 2L2 () + c0 η2 ≤ c0 η2 by (5.14). Consequently,

uN,η (·, t) L2 () dtα uN,η (·, t) L2 () + c0 uN,η (·, t) 2L2 () ≤ η2 c0 . We note that uN,η (·, t) 2L2 () ≥ η2 > 0, and we can divide by uN,η (·, t) L2 () to obtain dtα uN,η (·, t) L2 () + c0 uN,η (·, t) L2 () ≤ ηc0 .

(5.15)

With the scalar product (·, ·) in L2 (0, T ), we have (dtα uN,η (·, t) L2 () , ξ ) + (c0 uN,η (·, t) L2 () , ξ ) ≤ (ηc0 , ξ )

(5.16)

for all ξ ∈ C0∞ (0, T ) satisfying ξ ≥ 0. In view of (3.12), by Lemma 2.7 and integration by parts, we know that (2.51) holds for u ∈ W 1,1 (0, T ) and ξ ∈ C0∞ (0, T ): (dtα uN,η (·, t) L2 () , ξ ) = ( uN,η (·, t) L2 () , (dtα )∗ ξ ) −

1 (1 − α)



T

t 0

−α

 ξ(t)dt uN,η (·, 0) L2 () .

By the definition of uN,η , we can readily verify that uN,η (·, t) L2 () −→

uN (·, t) L2 () in L1 (0, T ) and uN,η (·, 0) L2 () −→ uN (·, 0) L2 () as η → 0. Therefore lim (dtα uN,η (·, t) L2 () , ξ )

η→0

= ( uN (·, t) L2 () , (dtα )∗ ξ ) −

1 (1 − α)

 0

T

(5.17)

 t −α ξ(t)dt uN (·, 0) L2 () .

Since uN (·, t) L2 () ∈ W 1,1 (0, T ) by Theorem 3.6 (iv), we apply (2.51) to the right-hand side of (5.17) for uN (·, t) L2 () , and we obtain lim (dtα uN,η (·, t) L2 () , ξ ) = (dtα uN (·, t) L2 () , ξ ).

η→0

5 Decay Rate as t → ∞

118

Therefore, letting η → 0 in (5.16), we reach (dtα uN (·, t) L2 () + c0 uN (·, t) L2 () , ξ ) ≤ 0

ξ ∈ C0∞ (0, T ), ξ ≥ 0. (5.18)

We choose the mollifier χε with ε > 0 defined by (3.19). We note θ (t) := dtα uN (·, t) L2 () + c0 uN (·, t) L2 () ∈ L1 (0, T ). Again using the letter θ , we denote the zero extension of θ to R, and θ ∈ L1 (R). Setting ξ(t) = χε (τ − t) with arbitrarily fixed τ ∈ (0, T ) and ε > 0, we see that ξ ≥ 0and ξ ∈ C0∞ (0, T ) with sufficiently small ε > 0. By (5.18) we obtain ∞ θε (τ ) := −∞ θ (t)χε (τ − t)dt ≤ 0 for small ε > 0. We know that θε −→ θ in L1 (R) as ε → 0 (e.g., [2]), so that we can choose a subsequence {εn }n∈N with limn→∞ εn = 0 such that limn→∞ θεn (τ ) = θ (τ ) for almost all τ ∈ (0, T ). Hence, letting n → ∞ in (5.18), we reach dtα uN (·, t) L2 () + c0 uN (·, t) L2 () ≤ 0

for almost all t ∈ (0, T ).

(5.19)

Using the power series of Eα,1 (−c0 t α ), one can directly verify that v(t) = uN (·, 0) L2 () Eα,1 (−c0 t α ) satisfies dtα v(t) + c0 v(t) = 0,

v(0) = uN (·, 0) L2 () .

Now we will apply Lemma 5.2 to obtain that

uN (·, t) L2 () ≤ v(t)

for every t ∈ [0, T ].

(5.20)

Proof of (5.20) We set f (t) := uN (·, t) L2 () − v(t). Then dtα f (t) + c0 f (t) ≤ 0,

t ∈ [0, T ],

f (0) = 0.

(5.21)

It suffices to verify f (t) ≤ 0 for 0 ≤ t ≤ T . Assume contrarily that there exists t0 ∈ (0, T ] such that f (t0 ) = max f (t) > 0. From Theorem 3.6 (iv), we deduce t ∈[0,T ]

that f ∈ W 1,∞ (κ, T ) for each κ > 0. Thus by Lemma 5.2, we have (dtα f )(t0 ) ≥ 0, which means that dtα f (t0 ) + c0 f (t0 ) > 0. This is a contradiction with (5.21). Thus we have verified (5.20). Next, let ξ ∈ C0∞ (0, T ), ≥ 0 be arbitrary. Then (5.20) yields 

T 0

 ξ(t) uN (·, t) 2L2 () dt ≤

T 0

ξ(t)v 2 (t)dt.

5.3 Completion of Proof of Theorem 5.1

119

There exists a subsequence {uN } (still indexed by N) such that uN  u weakly in L2 (0, T ; H01()), where u is the unique solution to (5.1) given by Theorem 4.1. Thus by the weak lower semi-continuity of the L2 -norm, we have 

T 0

 ξ(t) u(·, t) 2L2 () dt

T

≤

ξ(t)v 2 (t)dt

0

for each ξ ∈ C0∞ (0, T ), ≥ 0. Hence

u(·, t) L2 () ≤ v(t)

for almost all t ∈ [0, T ].

Since T > 0 is arbitrary and the solution u is uniquely defined in [0, ∞) by Theorem 4.1, we reach u(·, t) L2 () ≤ v(t) for almost all t ≥ 0, which means

u(·, t) L2 () ≤ a L2 () Eα,1 (−c0 t α )

for almost all t ≥ 0.

Finally, the asymptotic behavior of the Mittag-Leffler function implies

u(·, t) L2 () ≤ C1 t −α a L2 () , Thus the proof of Theorem 5.1 is complete.

t > 0. 

Chapter 6

Concluding Remarks on Future Works

The references on fractional differential equations are rapidly increasing and it is impossible to refer to them to some substantial extent. Thus in this book with limited pages, we concentrate on establishing one possible theory for the initial boundary value problems for time-fractional partial differential equations. Our theory redefines the time fractional derivative ∂tα keeping the consistency with the classical Riemann–Liouville derivative and the Caputo derivative, and ∂tα should be handled in a convenient way for the applications (e.g., the isomorphism between Hα (0, T ) and L2 (0, T )). We are obliged to give up comprehensive comparisons with other results. Also as for possible future studies, we are restricted to mention a few topics as follows. 1. In this book, we discuss only the case of the order 0 < α < 1 of the timefractional derivative ∂tα . From the physical viewpoint, the case of α > 1, in particular, 1 < α < 2 is important. The corresponding theory for the case α > 1 can be constructed similarly; see Remark 2.4 of Chap. 2. However, we postpone the details to future works. 2 Here we do not discuss nonlinear equations ∂tα (u − a) + A(t)u = F (x, t, u, ∇u). Needless to say, one purpose of the linear theory developed in this book is the applications to the nonlinear theory. 3. Here we argue only the homogeneous Dirichlet boundary condition. For example, the Neumann boundary condition should be considered and initial nonhomogeneous boundary value problems are demanded. The theory for common partial differential equations such as α = 1, 2 has been established satisfactorily (e.g., Lions and Magenes [19]). Such a theory is not only mathematically interesting but also is a basis for e.g., control problems, and widely applicable. As for non-homogeneous boundary value problems for fractional partial differential equations, we refer only to Yamamoto [29] and the references therein. © The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 A. Kubica et al., Time-Fractional Differential Equations, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-15-9066-5_6

121

122

6 Concluding Remarks on Future Works

4. There are various topics on inverse problems where we are required to determine parameters such as coefficients and orders of the derivatives of fractional differential equations by data of solutions which are over-determining in view of the well-posedness of initial boundary value problems. As for surveys, see the three chapters [16a–c] in the handbook edited by Kochubei, Luchko and Machado. The current book has established the foundations for future studies of inverse problems for fractional differential equations, and we can greatly expect the comprehensive development of studies of inverse problems.

Appendix A

Proofs of Two Inequalities

For convenience, we prove two inequalities which have been used in this book. We set  t (ρ ∗ v)(t) = ρ(t − s)v(s)ds, 0 < t < T . 0

Lemma A.1 (The Young Inequality for the Convolution) Let 1 ≤ p ≤ ∞, ρ ∈ L1 (0, T ) and v ∈ Lp (0, T ). Then

ρ ∗ v Lp (0,T ) ≤ ρ L1 (0,T ) v Lp (0,T ) . Proof The proof for p = ∞ is straightforward. Let p = 1. Since 

T 0



t





· · · ds dt =

0

T

0



T

 · · · dt ds

(A.1)

s

we have  t    T  t    ρ(t − s)v(s)ds  dt ≤ |ρ(t − s)||v(s)|ds dt   0 0 0 0    T  T −s |ρ(t − s)|dt |v(s)|ds = |ρ(η)|dη |v(s)|ds 

ρ ∗ v L1 (0,T ) =  = 

T



T



0

≤ 0

T s T 0

T



0

0

|ρ(η)|dη |v(s)|ds = ρ L1 (0,T ) v L1 (0,T ) .

For the second equality from the last, we changed the integral variables η = t − s. Thus the proof in the case of p = 1 is complete.

© The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 A. Kubica et al., Time-Fractional Differential Equations, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-15-9066-5

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124

A Proofs of Two Inequalities

Finally, let 1 < p < ∞. Let q ∈ (1, ∞) satisfy 1 < p < ∞. We have to estimate 

T

0

1 p

+

1 q

= 1, which is possible by

 t p    ρ(t − s)v(s) dt.   0

Then 1

|ρ(t − s)v(s)| = |ρ(t − s)| q (|ρ(t − s)|

1− q1

|v(s)|),

and the Hölder inequality yields 

t



t

|ρ(t − s)||v(s)|ds =

0

1

|ρ(t − s)| q (|ρ(t − s)|

1− q1

|v(s)|)ds

0



t

≤

|ρ(t − s)|ds

 q1 

0

t

|ρ(t − s)|

p q−1 q

|v(s)| ds p

 p1

0



1 q

≤ ρ L1 (0,T )

t

|ρ(t − s)||v(s)| ds p

 p1 .

0

Therefore, using (A.1), we obtain  t p   T  t p   p  ρ(t − s)v(s)ds  dt ≤ ρ q 1 |ρ(t − s)||v(s)| ds dt   L (0,T ) 0 0 0 0   T  T p q = ρ L1 (0,T ) |ρ(t − s)|dt |v(s)|p ds 

T



p q

= ρ L1 (0,T )

0

T 0



s

T −s

 |ρ(η)|dη |v(s)|p ds,

0

where we changed the integral variables η = t − s. Hence p

p

p

p+q

p

ρ ∗ v Lp (0,T ) ≤ ρ Lq 1 (0,T ) ρ L1 (0,T ) v Lp (0,T ) = ρ L1q(0,T ) v Lp (0,T ) , which implies the conclusion by complete.

+

1 p

1 q

= 1. Thus the proof of Lemma A.1 is 

Lemma A.2 (A Generalized Gronwall Inequality) Let C0 > 0 be a constant and 0 < α < 1. Moreover, let r ∈ L1 (0, T ), ≥ 0 in (0, T ). We assume that u ∈ L1 (0, T ) satisfies 

t

0 ≤ u(t) ≤ r(t) + C0 0

(t − s)α−1 u(s)ds,

0 ≤ t ≤ T.

A Proofs of Two Inequalities

125

Then  u(t) ≤ r(t) + C1 e

t

C2 t

(t − s)α−1 r(s)ds,

0 ≤ t ≤ T.

(A.2)

0

Here the constants C1 > 0 and C2 > 0 are dependent on α, C0 , but independent of T > 0. We note that if C0 > 0 is independent of T > 0, then (A.2) holds for t > 0 with C1 , C2 > 0 which are independent of t > 0. Proof We set 

s

(Lr)(s) = C0

(s − ξ )α−1 r(ξ )dξ,

0 0 are independent of s. Hence max |Eα,α ((θ s)α )| ≤ C1 eC2 t

0≤s≤t

for all t ≥ 0. That is, (A.7) yields  ∞  θ αk (t − s)αk−1     ≤ C1 eC2 t (t − s)α−1 ,    (αk)

0 ≤ s ≤ t.

k=1

Letting N → ∞ in (A.6), we obtain 

t

u(t) ≤ r(t) + C1 eC2 t

(t − s)α−1 r(s)ds

0

θ α(N+1) N→∞ ((N + 1)α)



t

+ lim

0

(t − s)(N+1)α−1 u(s)ds,

0 ≤ t ≤ T.

A Proofs of Two Inequalities

127

Let us choose N0 ∈ N such that (N0 + 1)α − 1 > 0. Then for each N ≥ N0 , we have  t   t    (t − s)(N+1)α−1 u(s)ds  ≤ t (N+1)α−1 |u(s)|ds ≤ T (N+1)α−1 u L1 (0,T ) .   0

0

Consequently,    t α N   θ N+1 (N+1)α−1   ≤ θ T α−1 (θ T )

u L1 (0,T ) (t − s) u(s)ds  ((N + 1)α)  ((N + 1)α) 0

for each t ∈ [0, T ] and N ≥ N0 . The Stirling formula yields (θ T α )N = 0. N→∞ ((N + 1)α) lim

Thus the proof of Lemma A.2 is complete.



Notation

· L2 (0,T ) , (·, ·)

L2 -norm, L2 -scalar product in L2 (0, T )

· L2 () , (·, ·)L2 ()

L2 -norm, L2 -scalar product in L2 ()

Dtα , 0 < α < 1

Riemann–Liouville derivative, see (1.3):  t 1 d Dtα u(t) = (t − s)−α u(s)ds (1 − α) dt 0

dtα , 0 < α < 1

Caputo derivative, see (1.4):  t 1 du (t − s)−α dtα u(t) = (s)ds (1 − α) 0 ds

J α, α > 0

Riemann–Liouville fractional integral operator, see (1.5):  t 1 J α f (t) = (t − s)α−1 f (s)ds (α) 0

∂tα , 0 < α < 1

Definition 2.1 in Sect. 2.3 of Chap. 2

W 1,1 (0, T ), 0 W 1,1 (0, T )

Definition (1.17)

Wα (0, T )

Definition (2.5)

0C

1 [0, T ], H (0, T ), H α (0, T ), α α (0, T ), H 1 (0, T ) H 0 0

Section 2.2 of Chap. 2

Eα,β (z)

Mittag-Leffler functions, see (2.28)

D(A)

The domain of an operator A

R(A) ∂i =

The range of an operator A, i.e., R(A) = AD(A) ∂ 2 ∂xi , ∂i

=

∂2 ∂xi2

∇ = (∂1 , . . . , ∂n )

, ∂s =

∂ ∂s

Partial derivatives Gradient

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129

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Index

A

G

Adjoint operator

39

Galerkin approximation 80 Gamma function 1 Generalized extremum principle Generalized Gronwall inequality

C

Caputo derivative 2 Closable 25 Closure 25 Coercivity 48 Complete monotonicity

114 124

H

Heinz–Kato inequality 20 Homogeneous Dirichlet boundary condition 121

103

D I Decay rate

110 Initial boundary value problem 73 Initial condition 55 Initial nonhomogeneous boundary value problems 121 Initial value problem 47 Initial value problem for a linear fractional ordinary differential equation 55 Interpolation space 15, 78 Inverse problems 122

E

Energy estimate 109 Extremum principle 5

F L Fractional ordinary differential equation Fractional power 19 Fractional Sobolev space 12, 65 Fredholm alternative 57

47 Laplace transform

9, 43

© The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 A. Kubica et al., Time-Fractional Differential Equations, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-15-9066-5

133

134

Index

M

R

Mittag-Leffler functions 26 Mollifier 15, 53 Multi-term time fractional derivatives

Riemann–Liouville derivative 2 Riemann–Liouville fractional integral operator 3

N

Nonlinear equations

S

121

P

Poincaré inequality

69

Sobolev embedding

7

Y

95

Young inequality

123