Asymptotic stability of the critical Fisher-KPP front using pointwise estimates

TL;DR

This paper presents a simplified proof of the nonlinear asymptotic stability of the Fisher-KPP critical front, demonstrating algebraic decay of perturbations using pointwise semigroup methods and Evans function analysis.

Contribution

It introduces a new proof technique based on pointwise estimates, highlighting the role of the Evans function in stability analysis.

Findings

Perturbations decay at rate t^{-3/2} in weighted L-infinity space

The proof simplifies previous approaches to Fisher-KPP stability

Absence of an embedded zero of the Evans function at the origin is key

Abstract

We propose a simple alternative proof of a famous result of Gallay regarding the nonlinear asymptotic stability of the critical front of the Fisher-KPP equation which shows that perturbations of the critical front decay algebraically with rate $t^{-3/2}$ in a weighted $L^\infty$ space. Our proof is based on pointwise semigroup methods and the key remark that the faster algebraic decay rate $t^{-3/2}$ is a consequence of the lack of an embedded zero of the Evans function at the origin for the linearized problem around the critical front.