Front propagation in Fisher-KPP equations with fractional diffusion

TL;DR

This paper investigates the Fisher-KPP equation with fractional diffusion, demonstrating that invasion occurs exponentially fast in time, contrasting with the constant speed in the classical case, and provides rigorous mathematical validation of existing heuristics.

Contribution

It establishes the exponential invasion speed in Fisher-KPP equations with fractional Laplacian, extending understanding beyond classical diffusion models.

Findings

Invasion occurs exponentially fast in time.

Contrasts with classical Laplacian case where invasion is at constant speed.

Provides rigorous mathematical justification for heuristics.

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.