Hamilton-Souplet-Zhang type gradient estimates for porous medium type equations on Riemannian manifolds

TL;DR

This paper derives new Hamilton-Souplet-Zhang type gradient estimates for porous medium equations on Riemannian manifolds, extending previous results and applying them to ancient solutions.

Contribution

It introduces novel gradient estimates for porous medium equations on manifolds, generalizing prior work and including special cases like the heat equation.

Findings

New gradient estimates for porous medium equations on Riemannian manifolds

Extension of Souplet-Zhang results to porous medium equations

Applications to Liouville theorems for ancient solutions

Abstract

In this paper, by employ the cutoff function and the maximum principle, some Hamilton-Souplet-Zhang type gradient estimates for porous medium type equation are deduced. As a special case, an Hamilton-Souplet-Zhang type gradient estimates of the heat equation is derived which is different from the result of Souplet-Zhang. Furthermore, our results generalize those of Zhu. As application, some Livillous theorems for ancient solution are derived.