Quasilinear parabolic reaction-diffusion systems: user's guide to well-posedness, spectra and stability of travelling waves

TL;DR

This paper provides a comprehensive guide to the well-posedness, spectral analysis, and stability of travelling waves in quasilinear parabolic reaction-diffusion systems, with applications to ecological models.

Contribution

It introduces a framework for analyzing spectra and stability of travelling waves in quasilinear systems, extending known methods to more complex models.

Findings

Spectra of travelling waves can be analyzed using dispersion relations.

A principle of linearized orbital instability is established.

The Gray-Scott-Klausmeier model is used as a detailed example.

Abstract

This paper is concerned with quasilinear parabolic reaction-diffusion-advection systems on extended domains. Frameworks for well-posedness in Hilbert spaces and spaces of continuous functions are presented, based on known results using maximal regularity. It is shown that spectra of travelling waves on the line are meaningfully given by the familiar tools for semilinear equations, such as dispersion relations, and basic connections of spectra to stability and instability are considered. In particular, a principle of linearized orbital instability for manifolds of equilibria is proven. Our goal is to provide easy access for applicants to these rigorous aspects. As a guiding example the Gray-Scott-Klausmeier model for vegetation-water interaction is considered in detail.