Existence families, functional calculi and evolution equations
type: Item
instance of: scholarly article
title: Existence families, functional calculi and evolution equations
zbMATH Open document ID: 0811.47034
published in: Lecture Notes in Mathematics
publication date: 1994-04-24T00:00:00Z
review text: The subject of the book under review is the theory of regularized semigroups. The notion consists in a generalization of \(C_ 0\)- semigroups and its aim is to solve linear Cauchy problems. More precisely, let \(C\) be a bounded injective operator on a Banach space \(X\). A \(C\)-regularized semigroup is a strongly continuous mapping \(W: [0,\infty)\to {\mathcal L} (X)\) such that \[ W(0)=C, \quad CW(t+s)= W(t) W(s) \qquad (t,s\geq 0). \] Thus, for \(C\) the identity we find the usual notion of a \(C_ 0\)-semigroup. Similar to this classical case one can define the infinitesimal generator \(A\) of a \(C\)-semigroup \(W\). It is a closed linear operator which commutes with \(C\). Moreover, for all \(x\) in the image of \(C\), the Cauchy problem \[ \begin{cases} u'(t)= Au(t) \qquad (t\geq 0)\\ u(0)= x\end{cases} \tag{CP} \] has a unique mild solution (given by \(u(t)= W(t)x\)). Regularized semigroups were introduced by Da Prato (1966). The interest in the subject arose again recently and related concepts were introduced. These lecture notes are based on a series of articles of the author from 1987 to 1994. Here is a sample of subjects treated in the book: Diverse notions of existence and uniqueness, Hille-Yosida theorems, holomorphic regularized semigroups, integrated semigroups, functional calculus, reversibility of solutions, boundary of a holomorphic semigroup, systems, non-autonomous equations. Applications concern the Schrödinger equation, correct systems in the sense of Petrovsky, equipartition of energy. The aim of the book is not to present a development which parallels the theory of \(C_ 0\)-semigroups. The intention of the author is rather to describe a variety of new phenomena which can occur for regularized semigroups. He takes care to explain the diverse notions and to illustrate them by examples and counterexamples.
reviewed by: Wolfgang Arendt
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47-02
Mathematics Subject Classification ID: 47A60
zbMATH DE Number: 556114
zbMATH Keywords: ill-posed problems
zbMATH Keywords: regularized semigroups
zbMATH Keywords: \(C_ 0\)-semigroups
zbMATH Keywords: linear Cauchy problems
zbMATH Keywords: infinitesimal generator
zbMATH Keywords: existence and uniqueness
zbMATH Keywords: Hille-Yosida theorems
zbMATH Keywords: holomorphic regularized semigroups
zbMATH Keywords: integrated semigroups
zbMATH Keywords: functional calculus
zbMATH Keywords: reversibility of solutions
zbMATH Keywords: boundary of a holomorphic semigroup
zbMATH Keywords: systems
zbMATH Keywords: non-autonomous equations
zbMATH Keywords: Schrödinger equation
zbMATH Keywords: correct systems in the sense of Petrovsky
zbMATH Keywords: equipartition of energy
author: R. deLaubenfels
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 556114