Maximal regularity of linear vector-valued parabolic Volterra equations
type: Item
instance of: scholarly article
title: Maximal regularity of linear vector-valued parabolic Volterra equations
zbMATH Open document ID: 0760.45008
published in: Journal of Integral Equations and Applications
publication date: 1992-08-13T00:00:00Z
review text: The author investigates maximal regularity of type \(C^ \alpha\) for a Volterra equation \((*)\;u(t)=f(t)+\int^ t_ 0a(t-s)Au(s)ds\) in a Banach space X, where \(A\) generates a holomorphic semigroup; i.e., conditions are given under which for every \(f\in C^ \alpha_ 0\) there is a unique solution \(u\in C^ \alpha_ 0\) of \((*)\). Applications to viscoelasticity and heat conduction are given. The proofs are based on an interesting result about inversion of Laplace transforms: If \(g\) is holomorphic in the right half plane and \(\|\lambda g(\lambda)\|\leq M\), \(\|\lambda^ 2g'(\lambda)\|\leq M\), then \(g\) is the Laplace transform of a bounded function (if X is reflexive).
reviewed by: Wolfgang Arendt
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 44A10
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 74Hxx
zbMATH DE Number: 40686
zbMATH Keywords: resolvent
zbMATH Keywords: Hölder-continuity
zbMATH Keywords: maximal regularity
zbMATH Keywords: Volterra equation
zbMATH Keywords: Banach space
zbMATH Keywords: holomorphic semigroup
zbMATH Keywords: viscoelasticity
zbMATH Keywords: heat conduction
zbMATH Keywords: Laplace transforms
author: Jan Pruess
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/jiea/1181075601
OpenAlex ID: W2090705478
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DOI: 10.1216/JIEA/1181075601
description: scientific article; zbMATH DE number 40686