Nonlinear Volterra integrodifferential equations
type: Item
instance of: scholarly article
title: Nonlinear Volterra integrodifferential equations
zbMATH Open document ID: 0816.45010
author: Melvin L. Heard
author: Samuel M. III. Rankin
published in: Journal of Mathematical Analysis and Applications
publication date: 1995-07-20T00:00:00Z
review text: This is a continuation of the authors' earlier paper [J. Math. Anal. Appl. 139, No. 1, 78-109 (1989; Zbl 0681.45010)]. Consider the semilinear Volterra integro-differential equation \[ u'(t) + Au(t) = \int^ t_ 0 a(t,s)g \bigl( s,u(s) \bigr) ds + f \bigl( t,u(t) \bigr), \quad t > 0, \] with \(u(0) = u_ 0\), where \(-A\) generates an analytic semigroup in a Banach space \(X\), where \(g\) is a polynomial differential operator in divergence form having the same order as \(A\) and the kernel \(a\) is smooth on \(\mathbb{R}^ + \times \mathbb{R}^ +\). Under certain technical assumptions on \(u_ 0\), \(g,f\) the authors prove the existence of a unique local weak solution, i.e. of \(u\) such that \(u \in C ([t_ 0, t_ 0 + T]\); \(X_{{1 \over 2} + \gamma})\), \[ \begin{multlined} u(t) = T(t - t_ 0) u_ 0 + \int^ t_{t_ 0} A^{{1 \over 2}} T(t - s) \left( \int^ s_{t_ 0} a(s, \tau) q \bigl( \tau, u(\tau) \bigr) d \tau \right) ds + \int^ t_{t_ 0} T(t - s)f \bigl( s,u(s) \bigr) ds, \\ t_ 0 \leq t \leq t_ 0 + \delta, \end{multlined} \] where \(\gamma \in (0, {1 \over 2})\) and \(X_ \alpha = D(A^ \alpha)\) equipped with the graph norm. After a review of the theory of Bessel potential spaces the authors apply their result to the case \(A = - \Delta\).
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 45J05
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 710341
zbMATH Keywords: semilinear Volterra integro-differential equation
zbMATH Keywords: analytic semigroup
zbMATH Keywords: Banach space
zbMATH Keywords: local weak solution
zbMATH Keywords: Bessel potential spaces
reviewed by: Stig-Olof Londen
MaRDI profile type: Publication
full work available at URL: jmaa.1994.1446
OpenAlex ID: W2084478697
DOI: 10.1006/JMAA.1994.1446
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description: scientific article; zbMATH DE number 710341