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This paper is devoted to the analysis of qualitative properties offlux-saturated type operators in dimension one. Specifically, we studyregularity properties and smoothing effects, discontinuous interfaces, theexistence of traveling wave profiles, sub- and super-solutions and waiting timefeatures. The aim of the paper is to better understand these kind of phenomenathroughout two prototypic operators: The relativistic heat equation and theporous media flux-limited equation. As an important consequence of our resultswe deduce that solutions to the one-dimensional relativistic heat equationbecome smooth inside their support on the long time run.

Qualitative behavior for flux-saturated mechanisms: Traveling waves, waiting time and smoothing effects