Spectral stability of monotone traveling fronts for reaction diffusion-degenerate Nagumo equations

TL;DR

This paper proves the spectral stability of monotone traveling fronts in reaction-diffusion Nagumo equations with density-dependent, degenerate diffusivity, using advanced spectral analysis techniques to handle degeneracy and non-hyperbolic points.

Contribution

It establishes spectral stability for various types of monotone traveling fronts in degenerate Nagumo reaction-diffusion equations, overcoming challenges posed by degeneracy.

Findings

Spectral stability of stationary diffusion-degenerate fronts.

Spectral stability of traveling diffusion-degenerate fronts.

Spectral stability of non-degenerate fronts.

Abstract

This paper establishes the spectral stability of monotone traveling front solutions for reaction-diffusion equations where the reaction function is of Nagumo (or bistable) type and with diffusivities which are density dependent and degenerate at zero (one of the equilibrium points of the reaction). Spectral stability is understood as the property that the spectrum of the linearized operator around the wave, acting on an exponentially weighted space, is contained in the complex half plane with non-positive real part. Three different types of monotone waves are studied: (i) stationary diffusion-degenerate fronts, connecting the two stable equilibria of the reaction; (ii) traveling diffusion-degenerate fronts connecting zero with the unstable equilibrium; and, (iii) non-degenerate fronts. In the first two cases, the degeneracy is responsible of the loss of hyperbolicity of the asymptotic coefficient matrices of the spectral problem at one of the end points, precluding the application of standard techniques to locate the essential spectrum. This difficulty is overcome with a suitable partition of the spectrum, a generalized convergence of operators technique, the analysis of singular (or Weyl) sequences and the use of energy estimates. The monotonicity of the fronts, as well as detailed descriptions of the decay structure of eigenfunctions on a case by case basis, are key ingredients to show that all traveling fronts under consideration are spectrally stable in a suitably chosen exponentially weighted $L^2$ energy space.