A class of degenerate parabolic equations with saturation mechanisms

In this talk we analize a family of degenerate parabolic equations with linear growth Lagrangian having the form $u_t= \mbox{div}\, (u \psi(\nabla u/u))$. Here $|\psi|\le 1$ and saturates at infinity. This includes as a prototypical case the so-called relativistic heat equation, which features a finite propagation speed property. Under a number of natural assumptions on $\psi$ we are able to analyze in a detailed way the finite propagation speed property for these models, their behavior concerning traveling fronts and some connections that they bear with optimal mass transportation theory, among other distinctive features. These are important issues from the point of view of mathematical modeling.