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 ∂tu=Δuα,1<α<1+12n+1 on M×(−∞,0]. We also obtain a theorem of Liouville type for positive solutions of the porous medium equation.