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Comptes Rendus Mathématique
Volume 347, n° 23-24
pages 1361-1366 (décembre 2009)
Doi : 10.1016/j.crma.2009.10.012
Received : 7 May 2009 ;  accepted : 8 October 2009
Propagation de fronts dans les équations de Fisher–KPP avec diffusion fractionnaire
Front propagation in Fisher–KPP equations with fractional diffusion
 

Xavier Cabré a , Jean-Michel Roquejoffre b
a ICREA et Universitat Politécnica de Catalunya, Dep. de Matemàtica Aplicada I, Av. Diagonal 647, 08028 Barcelone, Espagne 
b Institut de mathématiques, Univ. de Toulouse et CNRS (UMR 5219), 118, route de Narbonne, 31062 Toulouse, France 

Résumé

On s’intéresse dans cette Note à l’équation de Fisher–KPP dans l’espace entier, où le laplacien est remplacé par le générateur d’un semi-groupe de Feller à noyau lentement décroissant, un exemple important étant le laplacien fractionnaire. A la différence de l’équation de Fisher–KPP classique, où l’état stable envahit l’état instable à une vitesse constante en temps, nous montrons que la vitesse d’invasion est exponentielle en temps. Ces résultats apportent une justification mathématiquement rigoureuse à de nombreuses heuristiques sur ce modèle. Pour citer cet article : X. Cabré, J.-M. Roquejoffre, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The full text of this article is available in PDF format.
Abstract

We study in this Note the Fisher–KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with slowly decaying kernel, an important example being the fractional Laplacian. Contrary to what happens in the standard Laplacian case, where the stable state invades the unstable one at constant speed, we prove here that invasion holds at an exponential in time velocity. These results provide a mathematically rigorous justification of numerous heuristics about this model. To cite this article: X. Cabré, J.-M. Roquejoffre, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The full text of this article is available in PDF format.


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