A Porous Medium Equation Involving the Infinity-Laplacian. Viscosity Solutions and Asymptotic Behavior

Abstract

We study a nonlinear porous medium type equation involving the infinity Laplacian operator. We first consider the problem posed on a bounded domain and prove existence of maximal nonnegative viscosity solutions. Uniqueness is obtained for strictly positive solutions with Lipschitz in time data. We also describe the asymptotic behavior for the Dirichlet problem in the class of maximal solutions. We then discuss the Cauchy problem posed in the whole space. As in the standard porous medium equation (PME), solutions which start with compact support exhibit a free boundary propagating with finite speed, but such propagation takes place only in the direction of the spatial gradient. The description of the asymptotic behavior of the Cauchy Problem shows that the asymptotic profile and the rates of convergence and propagation exactly agree for large times with a one-dimensional PME.