Almost periodicity of parabolic evolution equations with inhomogeneous boundary values
type: Item
instance of: scholarly article
title: Almost periodicity of parabolic evolution equations with inhomogeneous boundary values
zbMATH Open document ID: 1197.47055
author: Mahmoud Baroun
author: Roland Schnaubelt
author: Lahcen Maniar
published in: Integral Equations and Operator Theory
publication date: 2009-11-23T00:00:00Z
review text: Under consideration is the existence and uniqueness of the almost periodic solution to the inhomogeneous evolution boundary problem in Banach spaces \[ u'(t)=A_m(t) u(t) + g(t), \;B(t) u(t)=h(t), \quad t\in{\mathbb R}, \] where typically \(A_m(t)\) is an elliptic partial differential operator acting in \(L^p(\Omega)\) and \(B(t)\) is a boundary operator mapping \(W^2_p(\partial\Omega)\) into a space like \(W^{1-1/p}_p(\partial\Omega)\). The problem is to show that the solution \(u\) inherits the almost periodicity of the inhomogeneities \(g\) and \(h\). The main results concern the case that the evolution family of the problem has exponential dichotomies on time intervals \((-\infty, -T]\) and \([T, \infty)\). The main theorem gives a Fredholm alternative for mild solution to the problem in a space of continuous functions asymptotically almost periodic on \({\mathbb R}_-\) and \({\mathbb R}_+\). These results are obtained by the technique of evolution equations in extrapolation spaces and periodic equations in the context of exponential dichotomies.
reviewed by: I. V. Melnikova
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 35K90
zbMATH DE Number: 5636884
zbMATH Keywords: asymptotic almost periodicity
zbMATH Keywords: Fredholm operators
zbMATH Keywords: evolution equations
zbMATH Keywords: inter- and extrapolation
zbMATH Keywords: exponential dichotomy
MaRDI profile type: Publication
full work available at URL: s00020-009-1704-z
OpenAlex ID: W1982868240
DOI: 10.1007/S00020-009-1704-Z
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description: scientific article; zbMATH DE number 5636884