Qualitative behaviour for flux-saturated mechanisms: travelling waves, waiting time and smoothing effects

This paper is devoted to the analysis of qualitative properties of flux-saturated type operators in dimension one. Specifically, we study regularity properties and smoothing effects, discontinuous interfaces, the existence of travelling wave profiles, sub-and supersolutions and waiting time features. The aim of the paper is to better understand these phenomena through two prototypic operators: the relativistic heat equation and the porous media flux-limited equation. As an important consequence of our results we deduce that solutions to the one-dimensional relativistic heat equation become smooth inside their support in the long time run.