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

Stabilité spectrale du front critique de l'équation de Fisher-KPP étendue

Montie Avery (1) , Louis Garénaux (2)

1 UMN-MATH - School of Mathematics
2 IMT - Institut de Mathématiques de Toulouse UMR5219

We revisit the existence and stability of the critical front in the extended Fisher-KPP equation, refining earlier results of Rottschäfer and Wayne 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, 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.

Mots clés: Travelling waves; Pulled fronts; Singular perturbation; Embedded eigenvalues

Domaines: Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]; Systèmes dynamiques [math.DS]

Montie Avery, Louis Garénaux. Spectral stability of the critical front in the extended Fisher-KPP equation. 2020. ⟨hal-02928072⟩