Convergence to a single wave in the Fisher-KPP equation

2016/04/11 by Nolen, James, Roquejoffre, Jean-Michel, Ryzhik, Lenya · 2 citations
#35B40 #35C07 #35K57 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1604.02994

Abstract

We 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.

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