A geometric approach to bistable front propagation in scalar reaction-diffusion equations with cut-off

Freddy Dumortier, Nikola Popovic, Tasso J. Kaper

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Abstract

'Cut-offs' were 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. In this article, we investigate the effects of a cut-off on fronts propagating into metastable states in a class of bistable scalar equations. We apply the method of geometric desingularization from dynamical systems theory to calculate explicitly the change in front propagation speed that is induced by the cut-off. We prove that the asymptotics of this correction scales with fractional powers of the cut-off parameter, and we identify the source of these exponents, thus explaining the structure of the resulting expansion. In particular, we show geometrically that the speed of bistable fronts increases in the presence of a cut-off, in agreement with results obtained previously via a variational principle. We first discuss the classical Nagumo equation as a prototypical example of bistable front propagation. Then, we present corresponding results for the (equivalent) cut-off Schlogl equation. Finally, we extend our analysis to a general family of reaction-diffusion equations that support bistable fronts, and we show that knowledge of an explicit front solution to the associated problem without cut-off is necessary for the correction induced by the cut-off to be computable in closed form. (c) 2010 Elsevier B.V. All rights reserved.

Original languageEnglish
Pages (from-to)1984-1999
Number of pages16
JournalPhysica D: Nonlinear Phenomena
Volume239
Issue number20-22
DOIs
Publication statusPublished - 15 Oct 2010

Keywords

  • Reaction-diffusion equations
  • Bistable fronts
  • Cut-offs
  • Critical front speeds
  • Geometric desingularization
  • BIFURCATIONS
  • SPEED

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