Convergence to a single wave in the Fisher-KPP equation

TL;DR

This paper provides a simplified PDE-based proof of the long-term behavior of solutions to the Fisher-KPP equation, showing convergence to a traveling wave in a specific moving frame, originally established by Bramson.

Contribution

It offers a new, simpler PDE proof of the convergence to a traveling wave in the Fisher-KPP equation, replacing probabilistic methods.

Findings

Proves convergence to a traveling wave in the Fisher-KPP equation.

Simplifies the proof of Bramson's result using PDE techniques.

Clarifies the asymptotic behavior of solutions with step-like initial conditions.

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 $2t - ({3}/{2}) \log t +x_\infty$, the solution of the equation converges as $t\to+\infty$ to a translate of the traveling wave corresponding to the minimal speed~$c_*=2$. The constant $x_\infty$ 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.