Local existence and regularity of solutions for some partial functional integrodifferential equations with infinite delay in Banach spaces
type: Item
instance of: scholarly article
title: Local existence and regularity of solutions for some partial functional integrodifferential equations with infinite delay in Banach spaces
zbMATH Open document ID: 1167.45008
author: Khalil Ezzinbi
author: Hamidou Toure
author: Issa Zabsonre
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-04-02T00:00:00Z
review text: The authors study the existence and regularity of solutions for the following partial functional integrodifferential equations with infinite delay \[ \begin{cases} u'(t)=Au(t)+\displaystyle\int_0^t\gamma(t-s)u(s)\,ds+f(u_t)\qquad \text{for}\quad t\geq0\\ u_0=\varphi\in\mathcal{B} \end{cases} \] in Banach spaces: \(A\;:\;D(A)\rightarrow X\) is a closed linear operator on a Banach space \(X\), \(\gamma(t)\) for \(t\geq0\) is a closed linear operator with domain \(D(\gamma)\supset D(A)\), being time-independent. \(\mathcal{B}\) is a linear space of functions \(\psi\;:\;]-\infty,0]\rightarrow X\) satisfying some axioms. The history \(u_t\in\mathcal{B}\), \(t\geq0\), is defined by \(u_t(\theta)=u(t+\theta)\), \(\theta\in]-\infty,0]\). \(f\) is a continuous function. The existence of a resolvent operator for the homogeneous equation (1), with \(f=0\), is admitted in the sense of \textit{R. Grimmer} [Trans. Am. Math. Soc. 273, 333--349 (1982; Zbl 0493.45015)]. The delayed part is assumed to be locally Lipschitz. Local existence of the mild solutions of equation (1), their global continuation and their possible blow up in finite time is proved. Later is proved the existence of the strict solutions and also the existence of the mild solution in nonautonomous case. The results are applied to partial functional integrodifferential equations with diffusion.
reviewed by: Roland Duduchava
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 45J05
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 5539219
zbMATH Keywords: semigroup
zbMATH Keywords: resolvent operator
zbMATH Keywords: mild solution
zbMATH Keywords: strict solutions
zbMATH Keywords: partial functional integrodifferential equations
zbMATH Keywords: infinite delay
zbMATH Keywords: Banach spaces
MaRDI profile type: Publication
full work available at URL: j.na.2008.05.006
OpenAlex ID: W2026747024
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DOI: 10.1016/J.NA.2008.05.006
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description: scientific article; zbMATH DE number 5539219