Monotonicity and convergence results in order-preserving systems in the presence of symmetry
type: Item
instance of: scholarly article
title: Monotonicity and convergence results in order-preserving systems in the presence of symmetry
zbMATH Open document ID: 0958.37061
author: Toshiko Ogiwara
author: Hiroshi Matano
published in: Discrete and Continuous Dynamical Systems
publication date: 2000-08-16T00:00:00Z
review text: The authors deal with the following question: Suppose that a group \(G\) acts on a space \(X\) and that a mapping \(F:X\to X\) is \(G\)-equivariant, that is \(F\circ g=g\circ F\) for every \(g\in G\). Can we say that solutions of the equation \(F(u)= 0\) are \(G\)-invariant? Here the authors establish a general theorem concerning symmetry or monotonicity properties of stable equilibria. The authors apply their theory to travelling waves and pseudo-travelling waves for a certain class of quasilinear diffusion equations. The authors also establish another useful general theorem, which they call the ``convergence theorem'', which roughly states that stability implies asymptotic stability and present various important applications of this result.
Mathematics Subject Classification ID: 37L20
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 22E70
Mathematics Subject Classification ID: 35K99
Mathematics Subject Classification ID: 37C80
zbMATH DE Number: 1492556
zbMATH Keywords: monotonicity
zbMATH Keywords: order preserving dynamical systems
zbMATH Keywords: pseudo-travelling waves
zbMATH Keywords: stability
MaRDI profile type: Publication
full work available at URL: dcds.1999.5.1
OpenAlex ID: W2002055895
DOI: 10.3934/DCDS.1999.5.1
description: scientific article; zbMATH DE number 1492556