The porous medium equation on noncompact manifolds with nonnegative Ricci curvature: A Green function approach

Abstract

We consider the porous medium equation (PME) on complete noncompact manifolds M of nonnegative Ricci curvature. We require nonparabolicity of the manifold and construct a natural space X of functions, strictly larger than $L^1$, in which the Green function on M appears as a weight, such that the PME admits a solution in the weak dual (i.e. potential) sense whenever the initial datum $u_0$ is nonnegative and belongs to X. Smoothing estimates are also proved to hold both for $L^1$ data, where they take into account the volume growth of Riemannian balls giving rise to bounds which are shown to be sharp in a suitable sense, and for data belonging to X as well.