  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 behaviour 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 behaviour 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.
