Convergence to a single wave in the Fisher-KPP equation

Abstract

The authors study the large time asymptotics of a solution of the Fisher-KPP reaction-diffusion equation, with an initial condition that is a compact perturbation of a step function. A well-known result of Bramson states that, in the reference frame moving as 2t−(3/2)log t+x _∞, the solution of the equation converges as t → +∞ to a translate of the traveling wave corresponding to the minimal speed c _* = 2. The constant x _∞ depends on the initial condition u(0, x). The proof is elaborate, and based on probabilistic arguments. The purpose of this paper is to provide a simple proof based on PDE arguments.

Keywords

Traveling waves / KPP / Front propagation / Asymptotic analysis / Reaction-diffusion

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