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 ∂ t u = Δ u α , 1 < α < 1 + 1 2 n + 1 on M × ( − ∞ , 0 ] . We also obtain a theorem of Liouville type for positive solutions of the porous medium equation.