Quasi-Linear Parabolic Reaction-Diffusion Systems: A User's Guide to Well-Posedness, Spectra, and Stability of Travelling Waves

Abstract

This paper is concerned with quasi-linear 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 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 practitioners to these rigorous methods. As a guiding example the Gray--Scott--Klausmeier model for vegetation-water interaction is considered in detail.