  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.
