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This paper is concerned with quasilinear parabolicreaction-diffusion-advection systems on extended domains. Frameworks forwell-posedness in Hilbert spaces and spaces of continuous functions arepresented, based on known results using maximal regularity. It is shown thatspectra of travelling waves on the line are meaningfully given by the familiartools for semilinear equations, such as dispersion relations, and basicconnections of spectra to stability and instability are considered. Inparticular, a principle of linearized orbital instability for manifolds ofequilibria is proven. Our goal is to provide easy access for applicants tothese rigorous aspects. As a guiding example the Gray-Scott-Klausmeier modelfor vegetation-water interaction is considered in detail.

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