Abstract Cauchy problems: three approaches

246 Pages·2001·1.159 MB·English

by  Irina V. Melnikova, Alexei Filinkov

CHAPMAN & HALL/CRC
Monographs and Surveys in
Pure and Applied Mathematics 120
ABSTRACT
CAUCHY
PROBLEMS:
Three Approaches
IRINA V. MELNIKOVA
ALEXEI FILINKOV
CHAPMAN & HALL/CRC
Boca Raton London New York Washington, D.C.
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Library of Congress Cataloging-in-Publication Data
Filinkov, A.I. (Aleksei Ivanovich)
Abstract Cauchy problems : three approaches / Alexei Filinkov, Irina
V. Melnikova
p. cm. (Chapman & Hall/CRC monographs and surveys in pure and
applied mathematics ; 120)
Includes bibliographical references and index.
ISBN 1-58488-250-6 (alk. paper)
1. Cauchy problem. I. Melnikova, I.V. (Irina Valerianovna) II.
Title. III. Series.
[DNLM: 1. Hepatitis B virus. QW 710 G289h]
QA377 .F48 2001
515′ .35—dc21 2001017069
CIP
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To our teacher
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Contents
Preface
Intro duction
0 Illustration and Motivation
0.1 Heat equation
0.2 The reversed Cauchy problem for the
Heat equation
0.3 Wave equation
1 Semigroup Metho ds
1.1 C0-semigroups
1.1.1 Definitions and main properties
1.1.2 The Cauchy problem
1.1.3 Examples
1.2 Integrated semigroups
1.2.1 Exponentially bounded integrated semigroups
1.2.2 (n,ω)-well-posedness of the Cauchy problem
1.2.3 Local integrated semigroups
1.2.4 Examples
1.3 κ-convoluted semigroups
1.3.1 Generators of κ-convoluted semigroups
1.3.2 Θ-convoluted Cauchy problem
1.4 C-regularized semigroups
1.4.1 Generators of C-regularized semigroups
1.4.2 C-well-posedness of the Cauchy problem
1.4.3 Local C-regularized semigroups
1.4.4 Integrated semigroups and
C-regularized semigroups
1.4.5 Examples
1.5 Degenerate semigroups
1.5.1 Generators of degenerate semigroups
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1.5.2 Degenerate 1-time integrated semigroups
1.5.3 Maximal correctness class
1.5.4 (n,ω)-well-posedness of a degenerate
Cauchy problem
1.5.5 Examples
1.6 The Cauchy problem for inclusions
1.6.1 Multivalued linear operators
1.6.2 Uniform well-posedness
1.6.3 (n,ω)-well-posedness
1.7 Second order equations
1.7.1 M,N-functions method
1.7.2 Integrated semigroups method
1.7.3 The degenerate Cauchy problem
2 Abstract Distribution Metho ds
2.1 The Cauchy problem
2.1.1 Abstract (vector-valued) distributions
2.1.2 Well-posedness in the space of distributions
2.1.3 Well-posedness in the space of
exponential distributions
2.2 The degenerate Cauchy problem
2.2.1 A-associated vectors and degenerate
distribution semigroups
2.2.2 Well-posedness in the sense of distributions
2.2.3 Well-posedness in the space of
exponential distributions
2.3 Ultradistributions and new distributions
2.3.1 Abstract ultradistributions
2.3.2 The Cauchy problem in spaces of
abstract ultradistributions
2.3.3 The Cauchy problem in spaces of
new distributions
3 Regularization Metho ds
3.1 The ill-posed Cauchy problem
3.1.1 Quasi-reversibility method
3.1.2 Auxiliary bounded conditions (ABC) method
3.1.3 Carasso’s method
3.2 Regularization and regularized
semigroups
3.2.1 Comparison of the ABC and
the quasi-reversibility methods
3.2.2 ‘Differential’ and variational
methods of regularization
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3.2.3 Regularizing operators and local C-regularized semi-
groups
3.2.4 Regularization of ‘slightly’ ill-posed problems
Bibliographic Remark
Bibliography
Glossary of Notation

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Preface
Many mathematical models in physics, engineering, finance, biology, etc.,
involve studying the Cauchy problem
u(cid:1)(t)=Au(t)+F(t), t∈[0,T), T ≤∞, u(0)=x,
where A is a linear operator on a Banach space X and F is an X-
valuedfunctionthatrepresentsthedeterministicorstochasticinfluenceofa
medium. Thefirstandmainstepinsolvingsuchproblemsconsistsofstudy-
ing the homogeneous Cauchy problem
u(cid:1)(t)=Au(t), t∈[0,T), T ≤∞ , u(0)=x. (CP)
The problem (CP) has been comprehensively studied in the case when the
operator A generates a C0-semigroup. A C0-semigroup generated by a
bounded operator A is nothing but an exponential operator-function
(cid:1)∞ Aktk
eAt = .
k!
k=1
Ingeneral,C0-semigroupsinheritsomemainpropertiesofexponentialfunc-
tions. As it turns out, the generation of a C0-semigroup by the operator
A is closely related to the uniform well-posedness of the Cauchy problem
(CP). However, many operators that are important in applications do not
generate C0-semigroups.
The focus of this book is the Cauchy problem (CP) with operators A,
which do not generate C0-semigroups. We focus on three approaches to
treating such problems:
• semigroup methods
• abstract distribution methods
• regularization methods
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The strategic concept of the first two approaches is the relaxation of
the notion of ‘well-posedness’ so that a Cauchy problem that is not well-
posed in the classical sense becomes well-posed in some other sense. In
the first approach, using semigroups more general than C0-semigroups –
integrated,C-regularized,andκ-convolutedsemigroups–onecanconstruct
solution operators on some subsets of D(A). The corresponding solutions
are stable with respect to norms that are stronger than the norm of X.
Using distribution semigroups in the second approach, one can construct
a family of generalized solution operators on X. The essence of the third
approach is the following: assuming that for some x there exists a solution
u(·) of (CP), for given xδ ((cid:8)x−xδ(cid:8)≤δ) we construct a solution uε(·) of a
well-posed (regularized) problem depending on a regularization parameter
εandchooseε=ε(δ)inawaythatuε →uasδ →0. Souε(·)canbetaken
asanapproximatesolutionandtheoperatorRε,tdefinedbyRε,txδ :=uε(t)
is a regularizing operator.
The motivation for writing this book was twofold: first, to give a self-
containedaccountoftheabove-mentionedthreeapproaches,whichisacces-
sible to nonspecialists. Second, to demonstrate the profound connections
between these seemingly quite different methods. In particular, we demon-
strate that integrated semigroups are primitives of generalized solution op-
erators,andmanyregularizingoperatorscoincidewithCε-regularizedsemi-
groups.
The book’s three chapters are devoted respectively to semigroup meth-
ods, abstract distribution methods, and regularization methods. We dis-
cuss not only the Cauchy problem (CP), but also the following important
generalizations: the degenerate Cauchy problem, the Cauchy problem for
inclusion and the Cauchy problem for the second order equation.
Acknowledgments
This project was supported by the Australian Research Council. The
firstauthorisgratefultotheDepartmentofPureMathematicsofAdelaide
University for their hospitality during her visits to Adelaide. The first
authorwasalsopartiallysupportedbygrantRFBRN99-01-00142(Russian
Federation).
The authors thank Professor Alan Carey, Professor Mike Eastwood,
Dr. John van der Hoek and Dr. Nick Buchdahl for their support and
encouragement.
The authors thank Dr. Maxim Alshansky, Dr. Uljana Anufrieva, Dr.
Alexander Freyberg and Dr. Isna Maizurna for participation in some parts
of this project
The authors also thank Cris Carey for editing the manuscript and Ann
Ross for technical support.
Irina Melnikova
Alexei Filinkov
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Intro duction
We aim to present semigroup methods, abstract distribution methods, and
regularization methods for the abstract Cauchy problem
u(cid:1)(t)=Au(t), t∈[0,T), T ≤∞, u(0)=x, (CP)
where A is a linear operator on a Banach space X, u(·) is an X-valued
function and x ∈ X. We also demonstrate the connections among these
methods.
In Chapter 0 we use Heat and Wave equations to illustrate some of the
main ideas, notions, and connections from the discussion below.
In Section 1.1 we give a brief account of the theory of C0-semigroups.
Here the reader will find a proof of the following fundamental result of this
theory.
Theorem 1.1.1SupposethatAisacloseddenselydefinedlinearoperator
on X. Then the following statements are equivalent:
(I) the Cauchy problem (CP) is uniformly well-posed on D(A): for any
x ∈ D(A) there exists a unique solution, which is uniformly stable
with respect to the initial data;
(II) the operator A is the generator of a C0-semigroup {U(t), t≥0};
(III) Miyadera-Feller-Phillips-Hille-Yosida (MFPHY) condition holds for
the resolvent of operator A: there exist K >0, ω ∈R so that
(cid:2) (cid:2)
(cid:2)(cid:2)R(k)(λ)(cid:2)(cid:2) ≤ Kk! ,
A (Reλ−ω)k+1
for all λ∈C with Reλ>ω, and all k =0,1,... .
In this case the solution of (CP) has the form
u(·)=U(·)x, x∈D(A),
and sup (cid:8)u(t)(cid:8)≤K(cid:8)x(cid:8) for some constant K =K(T).
t∈[0,T]
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Condition (III) is usually used as the criterion for the uniform well-
posedness of (CP).
Section 1.2 is devoted to n-times integrated semigroups connected with
the well-posedness of (CP) on subsets from D(An+1). Here is the main
result for exponentially bounded semigroups.
Theorem 1.2.4 Let A b e a densely defined linear op erator on X with
nonempty resolvent set. Then the following statements are equivalent:
(I) A is the generator of an exp onentially b oundedn-times integrated semi-
group {V(t), t≥0};
(I I) the Cauchy problem (CP) is (n,ω)-well-p osed: for any x ∈ D(An+1)
there exists a unique solution such that
∃K >0, ω ∈ R : (cid:8)u(t)(cid:8) ≤Keωt(cid:8)x(cid:8)An,
where (cid:8)x(cid:8)An =(cid:8)x(cid:8)+(cid:8)Ax(cid:8)+...+(cid:8)Anx(cid:8).
Statements (I), (II) of this theorem are equivalent to the MFPHY-type
condition (cid:2)(cid:3) (cid:4) (cid:2)
(cid:2)(cid:2) RA(λ) (k ) (cid:2)(cid:2)≤ Kk!
(cid:2) (cid:2)
λn (λ−ω)k +1
for all λ>ω, and k =0,1,... .
In the particular case when A generates a C0-semigroup U, the n-times
integrated semigroup generated by A is the n-th order primitive of U. In
general (and in contrast to C0-semigroups), integrated semigroups may be
not exponentially bounded, may be locally defined, degenerate, and their
generators may be not densely defined. In Section 1.2 we consider non-
degenerate exponentially bounded and local integrated semigroups, their
connections with the Cauchy problem, and discuss several examples. We
also show that an n-times integrated semigroup with a densely defined
generator A can be defined as a family of bounded linear operators V(t)
satisfying
(cid:5)
tn t
V(t)x− x = V(s)Axds
n!
(cid:5)0
t
= AV(s)xds, 0≤t<T, x∈D(A),
0
and (cid:5)
tn t
V(t)x− x=A V(s)xds, x∈X.
n!
0
Replacingfunctiontn/n!byacontinuouslydifferentiablefunctionΘ(t),one
arrives at the notion of κ-convoluted semigroup, where κ=Θ(cid:1).
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