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arXiv · 2009.01506

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

Montie Avery·Louis Garénaux
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Abstract

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

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Montie Avery, Louis Garénaux. 2021-09-24. Spectral stability of the critical front in the extended Fisher-KPP equation. https://arxiv.org/abs/2009.01506

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