Spectral stability of traveling fronts for reaction diffusion-degenerate Fisher-KPP equations

TL;DR

This paper proves spectral stability of traveling fronts in reaction-diffusion equations with degenerate diffusion, using regularization and energy estimates to analyze the spectrum in weighted spaces.

Contribution

It introduces a parabolic regularization method to analyze spectral stability of monotone traveling fronts with degenerate diffusion in Fisher-KPP type equations.

Findings

All fronts with speed above a threshold are spectrally stable in weighted spaces.

The point spectrum stability is established through energy estimates.

A new technique locates the compression spectrum of the linearized operator.

Abstract

This paper establishes the spectral stability in exponentially weighted spaces of smooth traveling monotone fronts for reaction diffusion equations of Fisher-KPP type with nonlinear degenerate diffusion coefficient. It is assumed that the former is degenerate, that is, it vanishes at zero, which is one of the equilibrium points of the reaction. A parabolic regularization technique is introduced in order to locate a subset of the compression spectrum of the linearized operator around the wave, whereas the point spectrum is proved to be stable with the use of energy estimates. Detailed asymptotic decay estimates of solutions to spectral equations are required in order to close the energy estimates. It is shown that all fronts traveling with speed above a threshold value are spectrally stable in an appropriately chosen exponentially weighted $L^2$-space.