Regularity and asymptotic behavior of solutions of nonautonomous differential equations
type: Item
instance of: scholarly article
title: Regularity and asymptotic behavior of solutions of nonautonomous differential equations
zbMATH Open document ID: 0834.35021
DOI: 10.1007/BF02219357
published in: Journal of Dynamics and Differential Equations
publication date: 1996-04-01T00:00:00Z
review text: Regularity and asymptotic behavior of solutions and the existence of a nonautonomous approximate inertial manifold is obtained for abstract evolution equations of the form \[ u'+ Au+ N(t, u)=0, \] in which \(A\) is a selfadjoint operator with compact resolvent in a Hilbert space \(H\). The operator \(N(t, u)= G(u)+ F(t, u)\) is nonlinear with \(G\) a monotone gradient that is locally Lipschitz from \(D(A^{1/2})\) into \(H\), and \(F:\mathbb{R}^+\times H\to H\) a Lipschitz perturbation which is Hölder continuous in \(t\). Weak solutions are shown to be uniformly locally Hölder into \(D(A)\) with equicontinuity in families of solutions with \(|u(0) |\leq r\). A priori estimates of \(|Au(t)|\) are also verified, and there is a global attractor whose component elements form an equicontinuous family of solutions.
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35G20
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 800421
zbMATH Keywords: monotone operator
zbMATH Keywords: abstract evolution equations
zbMATH Keywords: Lipschitz perturbation
author: Michael W. Smiley
reviewed by: Boling Guo
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 800421