On the abstract Cauchy problem for the operator \(\frac{d^ 2}{dt^ 2}- A\)
type: Item
instance of: scholarly article
title: On the abstract Cauchy problem for the operator \(\frac{d^ 2}{dt^ 2}- A\)
zbMATH Open document ID: 0532.47028
DOI: 10.1007/BF01204911
author: Ioana Ciorănescu
published in: Integral Equations and Operator Theory
publication date: 1984-01-01T00:00:00Z
review text: Let X be a Banach space and A a densely defined operator in X with a non- void resolvent set; denote by \(D_\infty =\cap^\infty_{n=0}D(A^ n)\) endowed with the Fréchet topology given by the seminorm system \(\{\| x\|_ n=\sum^{n}_{k=0}\| A^ kx\|,\quad n=0,1,2,...\}.\) It is proved that the Cauchy problem for the operator \(d^ 2/dt^ 2-A\) is well-posed in the sense of distributions iff \(A|_{D_\infty}\) generates a locally-equicontinuous cosine function of class \(C^\infty\) in \(L_ s(D_\infty)\). This is an extension of \textit{T. Ushijima}'s smoothness result of distribution semi- groups on \(D_\infty[\)J. Fac. Sci., Univ. Tokyo, Sect. IA 19, 65-127 (1972; Zbl 0239.47032)].
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47D99
Mathematics Subject Classification ID: 47E05
zbMATH DE Number: 3844447
zbMATH Keywords: operator cosine function
zbMATH Keywords: densely defined operator
zbMATH Keywords: Fréchet topology
zbMATH Keywords: Cauchy problem
zbMATH Keywords: well-posed
zbMATH Keywords: locally-equicontinuous cosine function
zbMATH Keywords: distribution semi-groups
MaRDI profile type: Publication
cites work: Problèmes de Cauchy abstraits et applications à quelques problèmes mixtes
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full work available at URL: https://doi.org/10.1007/bf01204911
OpenAlex ID: W2006508111
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description: scientific article; zbMATH DE number 3844447