Hamilton’s gradient estimates and Liouville theorems for porous medium equations

Abstract

Abstract Let ( M n , g ) $(M^{n}, g)$ 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 = Δ ( u p ) , 1 < p < 1 + 1 n − 1 , $$u_{t}=\Delta\bigl(u^{p}\bigr),\quad 1< p< 1+\frac{1}{\sqrt{n-1}}, $$ posed on ( M n , g ) $(M^{n}, g)$ 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, 2013).

Keywords

• porous medium equation
• Hamilton’s gradient estimate
• Liouville type theorem