Refined long time asymptotics for the Fisher-KPP equation

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 goal of this paper is to understand the convergence rate. We show that a level set {u(t, x) = s} of the solution will move as 2t − (3/2) log t + x∞ + c̄s/ √ t, with a universal constant c̄s that depends on s but not on the initial condition. To this end, we construct a universal approximate solution, which solves the Fisher-KPP equation with an almost O(t−1) precision. Then we prove the almost O(t−1) convergence rate of the solution to the approximate solution.