The Influence of Fractional Diffusion in Fisher-KPP Equations |
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Authors: | Xavier Cabré Jean-Michel Roquejoffre |
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Institution: | 1. ICREA and Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada I, Diagonal 647, 08028, Barcelona, Spain 2. Institut de Mathématiques, Université de Toulouse et CNRS (UMR 5219), 118 Route de Narbonne, 31062, Toulouse, France
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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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