Hamilton’s gradient estimates and Liouville theorems for porous medium equations
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 $$u_{t}=\Delta\bigl(u^{p}\bigr),\quad 1< p< 1+\frac{1}{\sqrt{n-1}}, $$ posed on $(M^{n}, g)$ with the Ricci curvature bounded from below. Moreover, we also obtain a Liouville type theorem. In …