Annali di Matematica Pura ed Applicata

, Volume 188, Issue 2, pp 207–233 | Cite as

Nontrivial large-time behaviour in bistable reaction–diffusion equations

  • Jean-Michel Roquejoffre
  • Violaine Roussier-Michon
Article

Abstract

Bistable reaction–diffusion equations are known to admit one-dimensional travelling waves which are globally stable to one-dimensional perturbations—Fife and McLeod [7]. These planar waves are also stable to two-dimensional perturbations—Xin [30], Levermore-Xin [19], Kapitula [16]—provided that these perturbations decay, in the direction transverse to the wave, in an integrable fashion. In this paper, we first prove that this result breaks down when the integrability condition is removed, and we exhibit a large-time dynamics similar to that of the heat equation. We then apply this result to the study of the large-time behaviour of conical-shaped fronts in the plane, and exhibit cases where the dynamics is given by that of two advection–diffusion equations.

Keywords

Reaction–diffusion equations Travelling fronts Nontrivial dynamics 

Mathematics Subject Classification (2000)

35K57 35B40 35B35 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Jean-Michel Roquejoffre
    • 1
  • Violaine Roussier-Michon
    • 2
  1. 1.Institut de Mathématiques (UMR CNRS 5219) and Institut Universitaire de FranceUniversité Paul SabatierToulouse Cedex 4France
  2. 2.Institut de Mathématiques (UMR CNRS 5219)INSA de ToulouseToulouse Cedex 4France

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