Classical solutions for partial functional differential equations with nonautonomous past
type: Item
instance of: scholarly article
title: Classical solutions for partial functional differential equations with nonautonomous past
zbMATH Open document ID: 1059.47047
DOI: 10.1007/BF02638385
published in: Archiv der Mathematik
publication date: 2003-03-27T00:00:00Z
review text: In [Discrete Contin. Dyn. Syst. 8, 953--966 (2002; Zbl 1013.35080)], \textit{S. Brendle} and \textit{R. Nagel} studied the partial functional differential equation with nonautonomous past \[ \begin{cases} \frac{\partial}{\partial t} u(t,s)= \frac{\partial}{\partial s} u(t,s)+A(s)\,u(t,s),&\quad s\leq 0,\;\;t\geq 0,\\ \frac{\partial}{\partial t} u(t,0)=Bu(t,0)+\Phi u(t,\cdot),&\quad t\geq 0, \end{cases} \] where \(A(s)\) are unbounded operators on a Banach space \(X\), \(B\) is the generator of a strongly continuous semigroup \((S(t))_{t\geq 0}\) on \(X\) and \(\Phi\), the delay operator, is a linear operator from a space of \(X\)-valued functions on \({\mathbb R}_{-}\) into \(X\). They obtained the mild solutions of the above equation by constructing an appropriate semigroup. The author of the present paper proves that the same semigroup yields the classical solution of the above equation under appropriate assumptions.
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 34K06
Mathematics Subject Classification ID: 35R10
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 1889278
zbMATH Keywords: partial functional differential equation
zbMATH Keywords: backward evolution semigroup
zbMATH Keywords: classical solution
author: Genni Fragnelli
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1889278