Spectral stability of the critical front in the extended Fisher-KPP equation

TL;DR

This paper analyzes the spectral stability of the critical front in the extended Fisher-KPP equation, confirming marginal spectral stability and nonlinear stability with a sharp decay rate, using advanced functional analytic techniques.

Contribution

It refines previous stability results by precisely characterizing the spectral properties and decay rates of the critical front in the extended Fisher-KPP equation.

Findings

The critical front is marginally spectrally stable with no unstable eigenvalues.

The critical front exhibits nonlinear stability with a decay rate of t^{-3/2}.

The analysis involves regularizing singular perturbations and tracking eigenvalues near the essential spectrum.

Abstract

We revisit the existence and stability of the critical front in the extended Fisher-KPP equation, refining earlier results of Rottsch\"afer and Wayne [28] which establish stability of fronts without identifying a precise decay rate. We verify that the front is marginally spectrally stable: while the essential spectrum touches the imaginary axis at the origin, there are no unstable eigenvalues and no eigenvalue (or resonance) embedded in the essential spectrum at the origin. Together with the recent work of Avery and Scheel [3], this implies nonlinear stability of the critical front with sharp $t^{-3/2}$ decay rate, as previously obtained in the classical Fisher-KPP equation. The main challenges are to regularize the singular perturbation in the extended Fisher-KPP equation and to track eigenvalues near the essential spectrum, and we overcome these difficulties with functional analytic methods.