Instability of solutions in a degenerate reaction diffusion equation

TL;DR

This paper analyzes the spectral stability of solutions in degenerate reaction-diffusion systems, providing analytical and numerical methods to determine stability conditions for travelling and stationary waves.

Contribution

It offers a comprehensive spectral analysis framework, including explicit Evans functions and numerical approaches, for stability in degenerate reaction-diffusion equations.

Findings

Stable travelling fronts can occur.

Travelling pulses are typically unstable.

Analytical and numerical spectral descriptions are provided.

Abstract

We study the spectral stability of travelling and stationary front and pulse solutions in a class of degenerate reaction-diffusion systems. We characterise the essential spectrum of the linearised operator in full generality and identify conditions under which it lies entirely in the left-half plane. For a number of special cases we obtain analytical results, including explicit Evans functions, and complete spectral descriptions for certain stationary waves. In regimes where analytical methods are not available, we compute the point spectrum numerically using a Riccati-Evans function approach. Our results show that stable travelling fronts can occur, while travelling pulses are typically unstable.