The Bramson delay in a Fisher-KPP equation with log-singular non-linearity

TL;DR

This paper investigates the asymptotic behavior of traveling wave solutions in a Fisher-KPP reaction-diffusion equation with a log-singular nonlinearity, revealing a phase transition in the Bramson shift based on the singularity's strength.

Contribution

It characterizes the Bramson delay for equations with log-singular nonlinearities and identifies a phase transition depending on the singularity's nature, introducing new analytical techniques.

Findings

Derived precise decay estimates for traveling wave profiles.

Identified a phase transition in Bramson shift behavior.

Revealed different asymptotic regimes for singular nonlinearities.

Abstract

We consider a class of reaction-diffusion equations of Fisher-KPP type in which the nonlinearity (reaction term) $f$ is merely $C^1$ at $u=0$ due to a logarithmic competition term. We first derive the asymptotic behavior of (minimal speed) traveling wave solutions that is, we obtain precise estimates on the decay to zero of the traveling wave profile at infinity. We then use this to characterize the Bramson shift between the traveling wave solutions and solutions of the Cauchy problem with localized initial data. We find a phase transition depending on how singular $f$ is near $u=0$ with quite different behavior for more singular $f$. This is in contrast to the smooth case, that is, when $f \in C^{1,\delta}$, where these behaviors are completely determined by $f'(0)$. In the singular case, several scales appear and require new techniques to understand.