Dimension of attractors and invariant sets in reaction diffusion equations
type: Item
instance of: scholarly article
title: Dimension of attractors and invariant sets in reaction diffusion equations
zbMATH Open document ID: 1288.35103
author: Martino Prizzi
published in: Topological Methods in Nonlinear Analysis
publication date: 2014-01-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/1102.4062
reviewed by: Guy Katriel
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35K58
zbMATH DE Number: 6242607
zbMATH Keywords: Dirichlet boundary condition
zbMATH Keywords: Cwickel-Lieb-Rozenblum inequality
MaRDI profile type: Publication
arXiv ID: 1102.4062
review text: This paper studies reaction diffusion equations of the form NEWLINE\[NEWLINEu_t+\beta(x)u-\Delta u =f(x,u)NEWLINE\]NEWLINE on a possibly unbounded domain \(\Omega\subset\mathbb R^3\) with Dirichlet boundary condition. Under some assumptions it is proved here that any invariant set of the semilflow generated by the equation in \(H_0^1\) has finite Hausdorff dimension. The conditions on the nonlinearity \(f(x,u)\) are general and do not include a dissipativity assumption, so that the invariant set in question is not necessarily an attractor. When the nonlinearity satisfies a dissipativity condition and the invariant set is a global attractor, a more explicit bound on the dimension of this attractor is derived. An important component of the proofs of these results is the application of spectral-theoretic results like the Lieb-Thirring and Cwickel-Lieb-Rozenblum inequality.
description: scientific article; zbMATH DE number 6242607