Pattern formation in a flux limited reaction–diffusion equation of porous media type

Abstract

A non-linear PDE featuring flux limitation effects together with those of the porous media equation (non-linear Fokker–Planck) is presented in this paper. We analyze the balance of such diverse effects through the study of the existence and qualitative behavior of some admissible patterns, namely traveling wave solutions, to this singular reaction–diffusion equation. We show the existence and qualitative behavior of different types of traveling waves: classical profiles for wave speeds high enough, and discontinuous waves that are reminiscent of hyperbolic shock waves when the wave speed lowers below a certain threshold. Some of these solutions are of particular relevance as they provide models by which the whole solution (and not just the bulk of it, as it is the case with classical traveling waves) spreads through the medium with finite speed. © 2016, Springer-Verlag Berlin Heidelberg.