Parameter dependent quasi-linear parabolic equations
type: Item
instance of: scholarly article
title: Parameter dependent quasi-linear parabolic equations
zbMATH Open document ID: 1068.35047
author: Cláudia Gentile
author: M. R. T. Primo
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2005-02-22T00:00:00Z
review text: The second-order nonlinear partial differential equation: \[ \begin{gathered} {d\over dt} u^\varepsilon(t)- \text{div}(a^\varepsilon|\nabla u^\varepsilon(t)|^{p- 2}\nabla u^\varepsilon(t))= Bu^\varepsilon(t),\quad t> 0,\\ u^\varepsilon(0)= u_0\in W^{1,p}_0(\Omega)\end{gathered}\tag{1} \] is considered. There \(\Omega\subset\mathbb{R}^n\) is a bounded domain with smooth boundary, \(p> \max\{2,{n\over 2}\}\), \(a_\varepsilon\in L^\infty(\Omega)\), \(a^\varepsilon(x)\geq \delta> 0\) almost everywhere in \(\Omega\) for every \(\varepsilon\in[0,\varepsilon_0]\) and \(a^\varepsilon\) converges to \(a\) in \(L^\infty(\Omega)\) as \(\varepsilon\) goes to \(0\). Following the paper of M. Otani some conditions are assumed on the perturbation \(B\). They imply global existence of solutions of the problem (1) in \(W^{1,p}_0(\Omega)\) and the existence of global attractors \(A_\varepsilon\) in \(W^{1,p}_0(\Omega)\) for every \(\varepsilon\in [0,\varepsilon_0]\). In this paper the upper semicontinuity of the global attractors \(A_\varepsilon\) at \(\varepsilon= 0\) on the \(W^{1,p}_0(\Omega)\) topology is shown.
reviewed by: A. Cichocka
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B41
zbMATH DE Number: 2136557
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: global attractors
zbMATH Keywords: convergence of solutions
zbMATH Keywords: upper semicontinuity
MaRDI profile type: Publication
full work available at URL: j.na.2004.07.040
OpenAlex ID: W2070081439
cites work: Q5665783
cites work: Comparison results for nonlinear parabolic equations with monotone principal part
cites work: Asymptotic behaviour of nonlinear parabolic equations with monotone principal part.
cites work: Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems
cites work: Four inequalities for monotone gradient vector fields
cites work: Infinite-dimensional dynamical systems in mechanics and physics
DOI: 10.1016/J.NA.2004.07.040
description: scientific article; zbMATH DE number 2136557