Convergence to a Single Wave in the Fisher-KPP Equation

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-→ +o∞ 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.