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The Influence of Fractional Diffusion in Fisher-KPP Equations

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Abstract

We study the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with power decaying kernel, an important example being the fractional Laplacian. In contrast with the case of the standard Laplacian where the stable state invades the unstable one at constant speed, we prove that with fractional diffusion, generated for instance by a stable Lévy process, the front position is exponential in time. Our results provide a mathematically rigorous justification of numerous heuristics about this model.

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Authors and Affiliations

  1. ICREA and Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada I, Diagonal 647, 08028, Barcelona, Spain

    Xavier Cabré

  2. Institut de Mathématiques, Université de Toulouse et CNRS (UMR 5219), 118 Route de Narbonne, 31062, Toulouse, France

    Jean-Michel Roquejoffre

Authors
  1. Xavier Cabré
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  2. Jean-Michel Roquejoffre
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Correspondence to Xavier Cabré.

Additional information

Communicated by P. Constantin

The first author was supported by grants MICINN MTM2008-06349-C03-01/FEDER, MINECO MTM2011-27739-C04-01, and GENCAT 2009SGR-345.

The second author is supported by the ANR grant PREFERED.

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Cabré, X., Roquejoffre, JM. The Influence of Fractional Diffusion in Fisher-KPP Equations. Commun. Math. Phys. 320, 679–722 (2013). https://doi.org/10.1007/s00220-013-1682-5

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  • DOI: https://doi.org/10.1007/s00220-013-1682-5

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