Hamilton's gradient estimates and Liouville theorems for porous medium equations.

Let $(M^{n}, g)$ be an n-dimensional Riemannian manifold. In this paper, we derive a local gradient estimate for positive solutions of the porous medium equation posed on $(M^{n}, g)$ with the Ricci curvature bounded from below. Moreover, we also obtain a Liouville type theorem. In particular, the results obtained in this paper generalize those in (Zhu in J. Math. Anal. Appl. 402:201-206, ). [ABSTRACT FROM AUTHOR]