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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

How travelling waves attract the solutions of KPP-type equations


Authors: Michaël Bages, Patrick Martinez and Jean-Michel Roquejoffre
Journal: Trans. Amer. Math. Soc. 364 (2012), 5415-5468
MSC (2010): Primary 35K57; Secondary 35B35, 35K55, 42A38
Published electronically: May 18, 2012
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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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05554-1
PII: S 0002-9947(2012)05554-1
Received by editor(s): May 16, 2010
Received by editor(s) in revised form: November 29, 2010
Published electronically: May 18, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.