The Bramson correction in the Fisher-KPP equation: from delay to advance

TL;DR

This paper analyzes the asymptotic position of solutions to the Fisher-KPP equation with specific initial decay, revealing detailed logarithmic delays and advances in wavefront location, including critical cases.

Contribution

It introduces a novel approach to precisely quantify logarithmic delays and advances in Fisher-KPP wavefronts for a new class of initial data.

Findings

Logarithmic delays and advances in wavefront positions are characterized.

Critical case with $r=3/2$ involves an additional $ ext{O}( ext{ln} ext{ln} t)$ term.

Construction of new sub- and super-solutions based on formal computations.

Abstract

We consider the solution to the scalar Fisher-KPP equation with front-like initial data, focusing on the location of its level sets at large times, particularly their deviation from points moving at the known spreading speed. We consider an intermediate case for the tail of the initial data, where the decay rate approaches, up to a polynomial term, that of the traveling wave with minimal speed. This approach enables us to capture deviations of the form $-r \ln t$ with $r \< \frac{3}{2}$, which corresponds to a logarithmic delay when $0 \< r \< \frac{3}{2}$ and a logarithmic advance when $r \< 0$. The critical case $r=\frac 32$ is also studied, revealing an extra $\mathcal O(\ln \ln t)$ term. Our arguments involve the construction of new sub- and super-solutions based on preliminary formal computations on the equation with a moving Dirichlet condition. Finally, convergence to the traveling wave with minimal speed is addressed.