Analysis of a class of degenerate parabolic equations with saturation mechanisms

We analyze a family of degenerate parabolic equations with linear growth Lagrangian having the form $u_t=\div (\varphi(u)\psi(\nabla u/u))$. Here $|\psi|\le 1$ and saturates at infinity. We present a simple and natural set of assumptions on the functions $\psi,\varphi$, under which: 1) these equations fall in the framework provided by \cite{ACMEllipticFLDE, ACMMRelat} and hence they are well posed, 2) we can ensure finite propagation speed for these models, 3) a Rankine--Hugoniot analysis on traveling fronts is also performed. On the particular case of $\varphi(u)=u$ we get more detailed information on the spreading rate of compactly supported solutions and some interesting connections with optimal mass transportation theory.