Analysis of a class of degenerate parabolic equations with saturation mechanisms

Abstract

We analyze a family of degenerate parabolic equations with linear growth Lagrangian having the form $u_t=÷(\phi (u)\psi (\nabla u/u))$. Here $|\psi |\le 1$ and saturates at infinity. We present a simple and natural set of assumptions on the functions $\psi ,\phi$, 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 $\phi (u)=u$ we get more detailed information on the spreading rate of compactly supported solutions and some interesting connections with optimal mass transportation theory.

41 pages. Some previous gaps have been solved. arXiv admin note: text overlap with arXiv:1211.6313 by other authors

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