Regular solutions for time dependent abstract integrodifferential equations with singular kernel
type: Item
instance of: scholarly article
title: Regular solutions for time dependent abstract integrodifferential equations with singular kernel
zbMATH Open document ID: 0638.45007
DOI: 10.1016/0022-247X(88)90382-4
author: B. George
published in: Journal of Mathematical Analysis and Applications
publication date: 1988-01-01T00:00:00Z
review text: The problems of existence, uniqueness and regularity of solutions to the linear integrodifferential equation \[ u'(t)=A(t)u(t)+ \int^{t}_{0} B(t,s)u(s)ds+f(t),\quad 0<t\leq T \] depending on the regularity of the initial value \(u(0)=u_ 0\) are considered. For each \(t\in [0,T]\) the operator \(A(t): F\subset E\to E\) generates an analytic semigroup \(e^{sA(t)}\) in E and \(B(t,s)\) \((0\leq s\leq t\leq T)\) is a linear operator from F to E. A(t) and f(t) are Hölder continuous with exponent \(\alpha\in (0,1)\) and, for each \(y\in F\) \(B(t,s)y\) is \(\alpha\)-Hölder continuous, with respect to t and \(L^ p\), with respect to s, for some \(p>1\). As an example of applications a linear parabolic Volterra equation is given.
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 45J05
zbMATH DE Number: 4040546
zbMATH Keywords: time dependent abstract integrodifferential equations with singular kernel
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: regularity
zbMATH Keywords: analytic semigroup
zbMATH Keywords: linear parabolic Volterra equation
reviewed by: Izabela Foltyńska
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4040546