Front propagation in Fisher-KPP equations with fractional diffusion

Front propagation in Fisher-KPP equations with fractional diffusion. 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. Cabre, J.-M. Raquejoffre, C R. Acad. Sci. Paris, Ser. I 347 (2009). (C) 2009 Academie des sciences. Publie par Elsevier Masson SAS.