A geometric approach to bistable front propagation in scalar reaction-diffusion equations with cut-off
type: Item
instance of: scholarly article
title: A geometric approach to bistable front propagation in scalar reaction-diffusion equations with cut-off
zbMATH Open document ID: 1206.35144
author: Freddy Dumortier
author: Tasso J. Kaper
author: Nikola Popović
published in: Physica D
publication date: 2010-11-05T00:00:00Z
review text: This paper is built on the results obtained by the same authors during 2007. The cut-offs problems are introduced to model front propagation in reaction-diffusion systems in which the reaction is effectively deactivated at points where the concentration lies below some threshold. This paper is devoted to the investigation of effects of a cut-off on fronts propagating into metastable states in a class of bistable scalar equations. The main problem considered here is the parabolic equation with cut-off of the form \[ \varphi_t=\varphi_{xx}+ f(\varphi) \Theta\bigg(\varphi,\varepsilon,\frac{\varphi}{\varepsilon}\bigg), \] where \((t,x)\in\mathbb R^+\times\mathbb R\), \(\varphi (t,x)\in\mathbb R\), and \(f:\mathbb R\to\mathbb R\) denotes a smooth reaction function which vanishes at the three rest states at \(\varphi^+\), \(\varphi^o\), \(\varphi^-\) in the above stated equation. The cut-off function satisfies \[ \Theta\bigg(\varphi,\varepsilon,\frac{\varphi}{\varepsilon}\bigg)<1, \quad \text{if }\varphi<\varepsilon, \] \[ \Theta\bigg(\varphi,\varepsilon,\frac{\varphi}{\varepsilon}\bigg)\equiv 1, \quad \text{if }\varphi>\varepsilon, \] where \(\Theta \) is bounded at \(\varphi=\varepsilon \) and \(0<\varepsilon\ll1\) denotes cut-off parameter. The method of geometric desingularization from dynamical systems theory is applied here in order to calculate explicitly the change in front propagation speed that is induced by the cut-off. It is shown that the speed of bistable fronts increases in the presence of a cut-off, in agreement with results obtained previously via a variational principle. The classical Nagumo equation is discussed as a prototypical example of bistable front propagation. Corresponding results for Schlögl equation with cut-off \[ \varphi_t=\varphi_{xx}- [2(1-\sigma)\varphi+ (\sigma-3)\varphi^2+ \varphi^3]H(\varphi- \varepsilon), \quad \sigma\in (0,1), \] where \(H\) is the Heaviside cut-off, are considered as well. An interesting conclusion is that the know…
reviewed by: Dimitar A. Kolev
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35K58
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 5810798
zbMATH Keywords: critical front speeds
zbMATH Keywords: geometric desingularization
zbMATH Keywords: Schlögl equation
MaRDI profile type: Publication
full work available at URL: j.physd.2010.07.008
OpenAlex ID: W1983829093
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DOI: 10.1016/J.PHYSD.2010.07.008
description: scientific article; zbMATH DE number 5810798