  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 traveling wave profiles, sub- and super-solutions and waiting time
features. The aim of the paper is to better understand these kind of phenomena
throughout 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 on the long time run.
