Stability of stationary reaction diffusion-degenerate Nagumo fronts I: spectral analysis

TL;DR

This paper proves the spectral stability of monotone stationary fronts in reaction-diffusion equations with density-dependent, degenerate diffusion, using advanced spectral analysis techniques to handle the challenges posed by degeneracy.

Contribution

It introduces a novel spectral analysis approach for degenerate diffusion Nagumo fronts, establishing their stability and spectral properties.

Findings

Spectrum is real with a spectral gap.

Zero eigenvalue is simple and isolated.

Linearization generates an exponentially decaying semigroup.

Abstract

This paper establishes the spectral stability of monotone, stationary front solutions for reaction-diffusion equations where the reaction function is of Nagumo (or bistable) type and with diffusion coefficients which are density dependent and degenerate at zero (one of the equilibrium points of the reaction). These stationary profiles connect the non-degenerate equilibrium point with the degenerate state at zero, they are monotone, and arrive to the degenerate state at a finite point. They are neither sharp nor smooth. The degeneracy of the diffusion precludes the application of standard techniques to locate the essential spectrum of the linearized operator around the wave in the energy space $L^2$. This difficulty is overcome with a suitable partition of the spectrum, the analysis of singular sequences, a generalized convergence of operators technique and refined energy estimates. It is shown that the $L^2$-spectrum of the linearized operator around the front is real and with a spectral gap, that is, a positive distance between the imaginary axis and the rest of the spectrum, with the exception of the origin. Moreover, the origin is a simple isolated eigenvalue, associated to the derivative of the profile as eigenfunction (the translation eigenvalue). Finally, it is shown that the linearization generates an analytic semigroup that decays exponentially outside a one-dimensional eigenspace associated to the zero eigenvalue.