Perturbation of semi-linear evolution equations under weak assumptions at initial time
type: Item
instance of: scholarly article
title: Perturbation of semi-linear evolution equations under weak assumptions at initial time
zbMATH Open document ID: 1088.34054
author: Daniel Daners
published in: Journal of Differential Equations
publication date: 2005-05-12T00:00:00Z
review text: This paper builds some perturbation results for initial value problems of the abstract semilinear evolution equation \[ \dot{u}(t)+A(t)\,u(t)=f(t,\,u(t)),\qquad t\in (s,\,T],\quad u(s)=u_0,\qquad\qquad \] associated with the perturbated equation of the form \[ \dot{u}(t)+A_n(t)\,u(t)=f_n(t,\,u(t)),\qquad t\in (s,\,T],\quad u(s)=u_{0n}, \] in a Banach space \(E\), where \(A(t),\,t\in [0,\,T]\) and \(A_n(t),\,t\in [0,\,T]\), are two unbounded closed operator families in \(E\) generating two evolution systems \(U(t,\,s),\,0\leq s\leq t\leq T\) and \(U_n(t,\,s),\,0\leq s\leq t\leq T\). Under rather general conditions, the author shows that the equation and its perturbated equation have unique solutions denoted by \(u\) and \(u_n\), respectively. Then, the author establishes the perturbation theorems under very weak assumptions at initial time, for instance, he does not assume that \(u_{0n}\to u_0\) in \(E\), not even weakly. As applications of the perturbation theorems, the author proves weak continuity properties and deals with rather general domain perturbation problems for semilinear parabolic and hyperbolic boundary value problems.
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35L20
zbMATH DE Number: 2167769
zbMATH Keywords: semilinear evolution equations
zbMATH Keywords: perturbation
zbMATH Keywords: weak continuity properties
zbMATH Keywords: parabolic boundary value problems
zbMATH Keywords: domain variation
MaRDI profile type: Publication
full work available at URL: j.jde.2004.08.004
OpenAlex ID: W1973935623
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DOI: 10.1016/J.JDE.2004.08.004
description: scientific article; zbMATH DE number 2167769