On some flux saturated diffusion equations (Theory of evolution equations and applications to nonlinear problems)

Abstract

In this paper we review some aspects of the theory of flux saturated diffusion equations. After derivation ofthis type ofequations and a summary ofwell-posedness results, the focus will be in some recent results about qualitative properties of solutions, including waiting time phenomena and creation of singularities. Flux-saturated diffusion equations are a class of parabolic equations of the forrn u_{t}=div a(u, nabla u), (0.1) which have a hyperbolic scaling for large values of the modulus of the gradient, in the sense that displaystyle frac{1}{$psi${0}(mathrm{v})}lim{trightarow+infty}mathrm{a}(z, tmathrm{v})cdot mathrm{v}=: $varphi$(z) for all zgeq 0, (0.2) where $psi$_{0} : mathbb{R}^{N}mapsto[0, +infty) is a positively 1-homogeneous convex function, with $psi$_{0}(0)= 0 and $psi$_{0}>0 otherwise and $varphi$ is a locally Lipschitz function with $varphi$(0)=0 and $varphi$(z)>0 if zneq 0. We will mainly consider the following three different equations: The porous medium relativistic heat equation, u_{t}= $nu$ div (displaystyle frac{u^{m}nabla u}{sqrt{u^{2}+$nu$^{2}c^{-2}|nabla u|^{2}), min (1, +infty), (PMRHE) the speed limitedporous medium equation u_{t}= $nu$ div (displaystyle frac{unabla u^{M-1}{sqrt{1+$nu$^{2}c^{-2}|nabla u^{M-1}|^{2}mathrm{I}, mathrm{M}in(1, +infty) (SLPME) and the nonlinear diffusion in transparent media, u_{t}= cdiv (u^{m}displaystyle frac{nabla u}{|nabla u|}), min mathbb{R}. (NDTM) where $nu$>0 is a kinematic viscosity and c>0 represents a characteristic limiting speed. In Section 1 we recall two different derivations ofthese equations: a physical one developed by Ph. Rosenau and a different one which comes from the mass transportation theory, first pointed out by Y. Brennier. In Section 2 we collect some results about Equations (PMRHE) and (SLPME), including existence and uniqueness of entropy solutions, regularity of solutions, propagation of discontinuity fronts and propagation of the support and waiting time phenomena. Finally, in Section 3 we analyze Equation (NDTM). We state the existence and uniqueness of solutions for the Neumann problem and we show some qualitative properties of the solutions with some examples, such as creation ofdiscontinuities in the interior of the support of solutions.