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

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 L1, 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 u0 is nonnegative and belongs to X. Smoothing estimates are also proved to hold both for L1 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.