Stability for solutions to nonlinear nonautonomous evolution equations
type: Item
instance of: scholarly article
title: Stability for solutions to nonlinear nonautonomous evolution equations
zbMATH Open document ID: 1307.34097
author: Seyedeh Marzieh Ghavidel
published in: Journal of Evolution Equations
publication date: 2014-09-18T00:00:00Z
review text: Consider the autonomous problem \[ {\dot{u}(t)+Au(t)\ni 0\text{, }t\geq 0 }\quad u(0)=x \] where \(A\) is an accretive operator in a real Banach space \(X\). In [\textit{M. Pierre}, Nonlinear Anal., Theory, Methods Appl. 2, 107--117 (1978; Zbl 0371.47056)], the author derived an upper bound for the expression d\(\left( u(t),F\right) \), where \(F\subseteq X\) is closed. The goal of this article is to extend the ideas of Pierre's article to the nonautonomous problem \[ \dot{u}(t)+A(t)u(t)\ni 0\text{, }0\leq s\leq t\quad u(s)=x \] where \(F\) is replaced by a time-dependent closed set \(K(t)\). One aim is to find a nonautonomous extension of a certain technical assumption made by Pierre. In addition to this assumption, it is assumed that \(A(t)\subseteq X\times X\), \(t\geq 0\), is a family of nonlinear, multivalued, \(\omega \)-accretive operators, and \(t_{n}\uparrow t,x_{n}\in K\left( t_{n}\right) \Longrightarrow \lim\limits_{n\rightarrow \infty }\)d\(\left( x_{n},K(t)\right) =0\). The inequality derived is of the form \[ {\text{d}\left( u(t),K(t)\right) \leq e^{\alpha (t-s)}\left[ \text{d}\left( x,K(s)\right) +\beta (t-s)\right] \text{}} \] for certain nonnegative real constants \(\alpha \) and \(\beta \), where \(u\) is a mild solution to the nonautonomous problem. Finally, the result is applied to a nonautonomous nonlinear partial differential delay equation.
reviewed by: Daniel C. Biles
Mathematics Subject Classification ID: 34G25
Mathematics Subject Classification ID: 35R10
Mathematics Subject Classification ID: 47H06
Mathematics Subject Classification ID: 47J35
zbMATH DE Number: 6345569
zbMATH Keywords: flow invariance
zbMATH Keywords: evolution equation
zbMATH Keywords: accretive operator
zbMATH Keywords: nonautonomous
zbMATH Keywords: Cauchy problem
MaRDI profile type: Publication
full work available at URL: s00028-014-0232-8
OpenAlex ID: W2080930989
cites work: Q3774170
cites work: Nonlinear evolution equations in Banach spaces
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cites work: Q4540535
cites work: Nonlinear evolution operators in Banach spaces
cites work: Q4023811
cites work: Nonlinear evolution operators and semigroups. Applications to partial differential equations
cites work: Invariant closed subsets for nonlinear semigroups
DOI: 10.1007/S00028-014-0232-8
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description: scientific article; zbMATH DE number 6345569