Attractors of reaction diffusion systems on infinite lattices
type: Item
instance of: scholarly article
title: Attractors of reaction diffusion systems on infinite lattices
zbMATH Open document ID: 1041.37040
DOI: 10.1023/B:JODY.0000009745.41889.30
author: Wolf-Jürgen Beyn
author: Sergei Yu. Pilyugin
published in: Journal of Dynamics and Differential Equations
publication date: 2004-03-15T00:00:00Z
review text: The authors show that for a dynamical system that is generated by an implicit discretization of the system \(u_t= Du_{xx}+ f(u)\), where \(u(t,x)\in \mathbb{R}^k\), \(D\subset \mathbb{R}^{k\times k}\) and \(x\in\mathbb{R}\), that is \[ {1\over h} (u^{n+1}_m- u^n_m)= {D\over m^2} (u^{n+1}_{m+1}- 2u^{n+1}_{m+1}+ u^{n+1}_{m-1})+ f(u^{n+1}_m), \] \(m\in\mathbb{R}\), \(n\in\mathbb{N}\) there exists global attractor \(A_{gl}\). It is shown that the size of the absorbing set containing \(A_{gl}\), is uniformly bounded for small time and space steps. The authors also establish upper semicontinuity of \(A_{gl}\) for the scalar case. Finally, the authors present an example in which the global attractor is infinite dimensional. It is based on the fact, the unstable manifold of the origin and the latter one may be infinite dimensional in case of a bimodal nonlinearity.
Mathematics Subject Classification ID: 37L60
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 37L30
zbMATH DE Number: 2057860
zbMATH Keywords: infinite lattices
zbMATH Keywords: global attractors
zbMATH Keywords: Sobolev space
zbMATH Keywords: infinite-dimensional attractor
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: b:jody.0000009745.41889.30
OpenAlex ID: W1972073758
description: scientific article; zbMATH DE number 2057860