Basin boundaries and bifurcations near convective instabilities: a case study
type: Item
instance of: scholarly article
title: Basin boundaries and bifurcations near convective instabilities: a case study
zbMATH Open document ID: 1065.35054
author: B. Sandstede
author: Arnd Scheel
published in: Journal of Differential Equations
publication date: 2005-02-16T00:00:00Z
review text: The paper aims to consider the transition between convective and absolute nonlinearities in a nonlinear parabolic (reaction-diffusion) equation. As an example, a model equation is taken: \[ u_t=u_{xx}+u_x+\mu u+\kappa u^3. \] The analysis is focused on the case \(0<\mu<1/4\), when the equation, if considered on a finite interval \(0<x<l\), features convective instability of the trivial solution, \(x=0\). As a measure of proximity to the absolute instability, the distance of the trivial solution to a nearest border of its attraction basin is estimated. It is shown that this distance becomes exponentially small with the increase of the domain's size \(l\) if the nonlinearity in the equation is also unstable, \(\kappa >0\). In the opposite case of the stable nonlinearity, \(\kappa <0\), the distance to the border of the attraction basin is bounded away from zero. Bifurcations accounting for the transition to the absolute instability from its convective counterpart are investigated too.
reviewed by: B. A. Malomed
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 2134395
zbMATH Keywords: absolute instability
zbMATH Keywords: convective instability
zbMATH Keywords: attraction basin
Wikidata QID: Q60143906
MaRDI profile type: Publication
full work available at URL: j.jde.2004.02.016
OpenAlex ID: W2223254556
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DOI: 10.1016/J.JDE.2004.02.016
description: scientific article; zbMATH DE number 2134395