Hamilton's gradient estimates and Liouville theorems for porous medium equations on noncompact Riemannian manifolds

Let M be a complete noncompact Riemannian manifold of dimension n. In this paper, we derive a local gradient estimate for positive solutions of porous medium equation partial derivative(t)u = Delta u(alpha), 1 < alpha < 1 + 1/root 2n + 1 on M x (-infinity, 0]. We also obtain a theorem of Liouville type for positive solutions of the porous medium equation. (C) 2013 Elsevier Inc. All rights reserved.