Evolutionary integral equations and applications.

*(English)*Zbl 0784.45006
Monographs in Mathematics 87. Basel: Birkhäuser Verlag (ISBN 3-7643-2876-2/hbk; 978-3-0348-0498-1/pbk; 978-3-0348-0499-8/ebook). xxvi, 366 p. (1993).

An evolutionary equation describes the evolution of a system with time. Examples of such equations are initial value problems for systems of ordinary differential equations, partial differential equations of parabolic or hyperbolic type, integral or integrodifferential equations of Volterra type, and functional differential equations of retarded or neutral type. This excellent book presents a general approach to linear evolutionary systems, with an emphasis on infinite-dimensional systems with time delays, such as those occurring in linear viscoelasticity with or without thermal effects. It gives a very natural and mature extension of the usual semigroup approach to a more general class of infinite- dimensional evolutionary systems. This is the first appearance in the form of a monograph of this recently developed theory. A substantial part of the results are due to the author, or are even new.

The book is divided into three major chapters: one chapter on equations of scalar type, another on nonscalar equations and a third chapter on equations on the whole real line. These chapters are preceded by an introduction giving an overview of the book, and some preliminary background results.

The first chapter, that is made up of sections 1-5, deals with equations of scalar type. These are equations where the time delay effects are one- dimensional, but the equation is infinite-dimensional in the spatial direction. The basic equation is an integrated version of the ordinary semigroup equation, namely \[ u(t)=f(t)+\int^ t_ 0a(t-s)Au(s)ds,\;t\in R^ +=[0,\infty). \tag{1} \] Here \(u\) and \(f\) belong to a Banach space \(X\), \(A\) is a densely defined, usually unbounded operator in \(X\), and \(a\in L^ 1_{\text{loc}}(\mathbb{R}^ +)\) is a kernel describing the evolution of the equation in the time direction. For example, by taking \(a(t)=1\) and differentiating the equation once one gets an abstract differential equation (generating a semigroup), and by taking \(a(t)=t\) and differentiating the equation twice one gets an abstract second order problem (generating a cosine family). The theory presented in this chapter contains the semigroup theory and the cosine family theory as two special cases. Everything centers around the resolvent \(S\) of the equation (1) appearing in the variation of constants formula \[ u(t)={d\over dt}\int^ t_ 0S(t-s)f(s)ds,\;t\in\mathbb{R}^ +, \tag{2} \] for the solution of (1). This resolvent is the natural extension of the notion of a semigroup (in the case where \(a=1)\) and a cosine family (in the case where \(a(t)=t)\). It is the solution of the resolvent equation \[ S(t)x=x+\int^ t_ 0a(t-s)AS(s)xds=x+A\int^ t_ 0a(t-s)S(s)x ds,\;t\in\mathbb{R}^ +, \tag{3} \] for each \(x\) in the domain of \(A\).

Section 3 describes how the well-posedness of equation (1) is equivalent to the existence of the resolvent \(S\), and extends the standard generation theorems for semigroups and cosine families to this more general setting. Several other basic results are also presented here, including some perturbation results. This section ends with a discussion of the integral resolvent \(R\), i.e., the solution of the integral resolvent equation \[ R(t)x=a(t)x+A\int^ t_ 0a(t-s)R(s)x ds,\;t\in\mathbb{R}^ +. \] This resolvent appears in the integral variation of constants formula for (1), namely \[ u(t)=f(t)+A\int^ t_ 0R(t- s)f(s)ds,\;t\in\mathbb{R}^ +. \] If \(S\) is differentiable, then \(R(t)Ax=\dot S(t)x\) and \(R(t)x=(a*S)(t)x\).

Section 2 deals with analytic resolvents, i.e., resolvents that have an analytic extension to a sector in the complex plane containing the positive real axis. This generalizes the concept of an analytic semigroup. The next section discusses equations of parabolic type. In the semigroup case this notion coincides with the notation of an analytic semigroup, but in the more general context of this book the notion of a parabolic equation has a life of its own. A typical assumption is that \(A\) generates an analytic semigroup, and that the kernel \(a\) has some monotonicity properties. Parabolicity is used to get results on maximal regularity.

Section 4 is titled “Subordination”, and it deals with the question of different kernels that are related to each other through a “subordination principle”. This very important subject has largely been ignored in the theory of evolutionary equations up to now; its main applications have been in other fields of mathematics such as potential theory. Some notions studied here are completely monotone kernels, Bernstein functions, completely positive kernels, and subordinate resolvents. Section 5, which is the final section in the chapter on equations of scalar type, contains a very readable presentation of the mathematical basis of linear viscoelasticity with or without thermal effects.

The second chapter in the book consists of four sections, titled “Hyperbolic Equations of Nonscalar Type”, “Nonscalar Parabolic Equations”, “Parabolic Problems in \(L^ p\)-Spaces”, and “Viscoelasticity and Electrodynamics with Memory”. It deals with nonscalar equations of the type \[ u(t)=f(t)+\int^ t_ 0A(t-s)u(s)ds,\;t\in\mathbb{R}^ +, \tag{4} \] where this time we do not deal with just one unbounded operator \(A\), but rather with a one-parameter family of unbounded operators \(A(t)\). The variation of constants formula (2) remains the same, but the resolvent equation (3) becomes (in a weak sense) \[ S(t)x=x+\int^ t_ 0A(t-s)S(s)x ds=x+\int^ t_ 0S(t-s)A(s)x ds,\;t\in\mathbb{R}^ +. \] Depending exactly on in which sense this equation is satisfied one talks about a pseudo-resolvent or a resolvent. Some of the results deal with problems of variational type, and others extend the earlier results for parabolic equations of scalar type to nonscalar equations. In the \(L^ p\)-section the forcing function \(f\) belongs to some \(L^ p\)-space (or it is the convolution of an \(L^ p\)-function and a scalar function \(a)\), and the solution is sought in the same \(L^ p\)- space. In this connection the kernel is required to be parabolic and have a main part, i.e., it should be a certain perturbation of a parabolic kernel of the type studied in Chapter I. Important roles are played by imaginary powers of operators and by the vector-valued Marcienkiewicz multiplier theorem. The last section in this chapter is a continuation of the discussion on linear viscoelasticity given in Section 5.

The final chapter of the book deals with equations on the line, and consists of the sections “Integrability of Resolvents”, “Limiting Equations”, “Admissibility of Function Spaces”, and “Further Applications and Complements”. In this chapter the equations (1) and (4) are required to hold for all \(t\in\mathbb{R}=(-\infty,\infty)\), and not just for all \(t\in\mathbb{R}^ +\). The questions addressed in this chapter are related to the input-output stability of the system, and to its asymptotic behavior as time tends to infinity. These questions are closely related to the (uniform or strong) integrability on all of \(\mathbb{R}^ +\) of the resolvent \(S\) or the integral resolvent \(R\). The last section contains some applications of the theory to specific problems, such as the Timoshenko beam, and some short discussions on how the theory given here can be applied to some related problems, linear or nonlinear, that have been excluded due to space considerations.

This book is written in a clear and consistent mathematical way. It requires the reader to have a fair knowledge of real analysis, harmonic analysis, and some functional analysis. It is not a book that one reads in a few days. Rather, it should be considered as an investment with lasting value.

The book is divided into three major chapters: one chapter on equations of scalar type, another on nonscalar equations and a third chapter on equations on the whole real line. These chapters are preceded by an introduction giving an overview of the book, and some preliminary background results.

The first chapter, that is made up of sections 1-5, deals with equations of scalar type. These are equations where the time delay effects are one- dimensional, but the equation is infinite-dimensional in the spatial direction. The basic equation is an integrated version of the ordinary semigroup equation, namely \[ u(t)=f(t)+\int^ t_ 0a(t-s)Au(s)ds,\;t\in R^ +=[0,\infty). \tag{1} \] Here \(u\) and \(f\) belong to a Banach space \(X\), \(A\) is a densely defined, usually unbounded operator in \(X\), and \(a\in L^ 1_{\text{loc}}(\mathbb{R}^ +)\) is a kernel describing the evolution of the equation in the time direction. For example, by taking \(a(t)=1\) and differentiating the equation once one gets an abstract differential equation (generating a semigroup), and by taking \(a(t)=t\) and differentiating the equation twice one gets an abstract second order problem (generating a cosine family). The theory presented in this chapter contains the semigroup theory and the cosine family theory as two special cases. Everything centers around the resolvent \(S\) of the equation (1) appearing in the variation of constants formula \[ u(t)={d\over dt}\int^ t_ 0S(t-s)f(s)ds,\;t\in\mathbb{R}^ +, \tag{2} \] for the solution of (1). This resolvent is the natural extension of the notion of a semigroup (in the case where \(a=1)\) and a cosine family (in the case where \(a(t)=t)\). It is the solution of the resolvent equation \[ S(t)x=x+\int^ t_ 0a(t-s)AS(s)xds=x+A\int^ t_ 0a(t-s)S(s)x ds,\;t\in\mathbb{R}^ +, \tag{3} \] for each \(x\) in the domain of \(A\).

Section 3 describes how the well-posedness of equation (1) is equivalent to the existence of the resolvent \(S\), and extends the standard generation theorems for semigroups and cosine families to this more general setting. Several other basic results are also presented here, including some perturbation results. This section ends with a discussion of the integral resolvent \(R\), i.e., the solution of the integral resolvent equation \[ R(t)x=a(t)x+A\int^ t_ 0a(t-s)R(s)x ds,\;t\in\mathbb{R}^ +. \] This resolvent appears in the integral variation of constants formula for (1), namely \[ u(t)=f(t)+A\int^ t_ 0R(t- s)f(s)ds,\;t\in\mathbb{R}^ +. \] If \(S\) is differentiable, then \(R(t)Ax=\dot S(t)x\) and \(R(t)x=(a*S)(t)x\).

Section 2 deals with analytic resolvents, i.e., resolvents that have an analytic extension to a sector in the complex plane containing the positive real axis. This generalizes the concept of an analytic semigroup. The next section discusses equations of parabolic type. In the semigroup case this notion coincides with the notation of an analytic semigroup, but in the more general context of this book the notion of a parabolic equation has a life of its own. A typical assumption is that \(A\) generates an analytic semigroup, and that the kernel \(a\) has some monotonicity properties. Parabolicity is used to get results on maximal regularity.

Section 4 is titled “Subordination”, and it deals with the question of different kernels that are related to each other through a “subordination principle”. This very important subject has largely been ignored in the theory of evolutionary equations up to now; its main applications have been in other fields of mathematics such as potential theory. Some notions studied here are completely monotone kernels, Bernstein functions, completely positive kernels, and subordinate resolvents. Section 5, which is the final section in the chapter on equations of scalar type, contains a very readable presentation of the mathematical basis of linear viscoelasticity with or without thermal effects.

The second chapter in the book consists of four sections, titled “Hyperbolic Equations of Nonscalar Type”, “Nonscalar Parabolic Equations”, “Parabolic Problems in \(L^ p\)-Spaces”, and “Viscoelasticity and Electrodynamics with Memory”. It deals with nonscalar equations of the type \[ u(t)=f(t)+\int^ t_ 0A(t-s)u(s)ds,\;t\in\mathbb{R}^ +, \tag{4} \] where this time we do not deal with just one unbounded operator \(A\), but rather with a one-parameter family of unbounded operators \(A(t)\). The variation of constants formula (2) remains the same, but the resolvent equation (3) becomes (in a weak sense) \[ S(t)x=x+\int^ t_ 0A(t-s)S(s)x ds=x+\int^ t_ 0S(t-s)A(s)x ds,\;t\in\mathbb{R}^ +. \] Depending exactly on in which sense this equation is satisfied one talks about a pseudo-resolvent or a resolvent. Some of the results deal with problems of variational type, and others extend the earlier results for parabolic equations of scalar type to nonscalar equations. In the \(L^ p\)-section the forcing function \(f\) belongs to some \(L^ p\)-space (or it is the convolution of an \(L^ p\)-function and a scalar function \(a)\), and the solution is sought in the same \(L^ p\)- space. In this connection the kernel is required to be parabolic and have a main part, i.e., it should be a certain perturbation of a parabolic kernel of the type studied in Chapter I. Important roles are played by imaginary powers of operators and by the vector-valued Marcienkiewicz multiplier theorem. The last section in this chapter is a continuation of the discussion on linear viscoelasticity given in Section 5.

The final chapter of the book deals with equations on the line, and consists of the sections “Integrability of Resolvents”, “Limiting Equations”, “Admissibility of Function Spaces”, and “Further Applications and Complements”. In this chapter the equations (1) and (4) are required to hold for all \(t\in\mathbb{R}=(-\infty,\infty)\), and not just for all \(t\in\mathbb{R}^ +\). The questions addressed in this chapter are related to the input-output stability of the system, and to its asymptotic behavior as time tends to infinity. These questions are closely related to the (uniform or strong) integrability on all of \(\mathbb{R}^ +\) of the resolvent \(S\) or the integral resolvent \(R\). The last section contains some applications of the theory to specific problems, such as the Timoshenko beam, and some short discussions on how the theory given here can be applied to some related problems, linear or nonlinear, that have been excluded due to space considerations.

This book is written in a clear and consistent mathematical way. It requires the reader to have a fair knowledge of real analysis, harmonic analysis, and some functional analysis. It is not a book that one reads in a few days. Rather, it should be considered as an investment with lasting value.

Reviewer: O.Staffans (Espoo)

##### MSC:

45N05 | Abstract integral equations, integral equations in abstract spaces |

45-02 | Research exposition (monographs, survey articles) pertaining to integral equations |

45A05 | Linear integral equations |

45M05 | Asymptotics of solutions to integral equations |

45J05 | Integro-ordinary differential equations |

45K05 | Integro-partial differential equations |

45D05 | Volterra integral equations |

76A10 | Viscoelastic fluids |

74Hxx | Dynamical problems in solid mechanics |