The Bramson correction for Fisher--KPP equations with nonlocal diffusion

TL;DR

This paper proves the logarithmic Bramson correction for the position of solutions to the Fisher--KPP equation with nonlocal diffusion, revealing typical and exotic behaviors of the wave front.

Contribution

It establishes the Bramson correction for nonlocal diffusion Fisher--KPP equations, extending known results to more general diffusion types.

Findings

Solutions with step-like initial data follow a specific logarithmic correction in front position.

Certain singular diffusions show more complex, exotic front behaviors.

Explicit constants for wave speed and decay rate are identified.

Abstract

We establish the logarithmic Bramson correction to the position of solutions to the Fisher--KPP equation with nonlocal diffusion. Solutions with step-like initial data typically resemble a front at position $c_{*} t - \frac{3}{2 \lambda_{*}} \log t + \mathcal{O}(1)$ for explicit constants $c_{*}$ and $\lambda_{*}$. However, certain singular diffusions exhibit more exotic behavior.