How travelling waves attract the solutions of KPP-type equations
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- by Michaël Bages, Patrick Martinez and Jean-Michel Roquejoffre PDF
- Trans. Amer. Math. Soc. 364 (2012), 5415-5468 Request permission
Abstract:
We consider in this paper a general reaction-diffusion equation of the KPP (Kolmogorov, Petrovskiĭ, Piskunov) type, posed on an infinite cylinder. Such a model will have a family of pulsating waves of constant speed, larger than a critical speed $c_*$. The family of all supercritical waves attracts a large class of initial data, and we try to understand how. We describe in this paper the fate of an initial datum trapped between two supercritical waves of the same velocity: the solution will converge to a whole set of translates of the same wave, and we identify the convergence dynamics as that of an effective drift, around which an effective diffusion process occurs. In several nontrivial particular cases, we are able to describe the dynamics by an effective equation.References
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Additional Information
- Michaël Bages
- Affiliation: Institut de Mathématiques (UMR CNRS 5219), Université Paul Sabatier, 31062 Toulouse Cedex 4, France
- Patrick Martinez
- Affiliation: Institut de Mathématiques (UMR CNRS 5219), Université Paul Sabatier, 31062 Toulouse Cedex 4, France
- Jean-Michel Roquejoffre
- Affiliation: Institut de Mathématiques (UMR CNRS 5219), Université Paul Sabatier, 31062 Toulouse Cedex 4, France
- Received by editor(s): May 16, 2010
- Received by editor(s) in revised form: November 29, 2010
- Published electronically: May 18, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 5415-5468
- MSC (2010): Primary 35K57; Secondary 35B35, 35K55, 42A38
- DOI: https://doi.org/10.1090/S0002-9947-2012-05554-1
- MathSciNet review: 2931334