Approximate inertial manifolds for retarded semilinear parabolic equations.
type: Item
instance of: scholarly article
title: Approximate inertial manifolds for retarded semilinear parabolic equations.
zbMATH Open document ID: 1039.35133
DOI: 10.1016/S0022-247X(03)00199-9
author: A. V. Rezunenko
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-07-30T00:00:00Z
review text: The author studies the following abstract version of a semilinear retarded parabolic equation \[ \frac {du}{dt}+Au=B(u_t), \tag{1} \] where \(A\) is a linear positively defined selfadjoint operator in a Hilbert space \(H\) whose inverse is compact, \(u_t(\theta):=u(t+\theta)\) and the nonlinear operator \(B=B_r\) maps the space \(C((-r,0),D(A^\alpha))\), \(0\leq\alpha\leq1/2\) to \(H\) (thus, the number \(r>0\) plays the role of a retardation time). The main result of the paper is that, under the natural assumptions on the nonlinear term \(B\) equation (1) possesses a family of approximate inertial manifolds. We however note that, in contrast to the well-known results on the nonretarded case, in the present paper the error of approximation \(\theta>0\) cannot be taken arbitrarily small and should satisfy the inequality \(\theta\geq \theta_0(r)\) where \(\theta_0(r)>0\) depends on the retardation time \(r\). Nevertheless, the author shows that \(\theta_0(r) \to0\) as the retardation time \(r\) tends to zero and the associated approximate manifolds tend as \(r\to0\) to the approximate inertial manifolds of the corresponding non-retarded problem.
Mathematics Subject Classification ID: 35R10
Mathematics Subject Classification ID: 37L25
zbMATH DE Number: 1957238
zbMATH Keywords: retarded parabolic equations
zbMATH Keywords: approximate inertial manifolds
reviewed by: S. V. Zelik
MaRDI profile type: Publication
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full work available at URL: s0022-247x(03)00199-9
OpenAlex ID: W1988977325
description: scientific article; zbMATH DE number 1957238