Gradient estimates for the porous medium equations on Riemannian manifolds

TL;DR

This paper derives local gradient estimates of Li-Yau type for positive solutions of the porous medium equation on Riemannian manifolds with Ricci curvature bounds, leading to improved parabolic Harnack inequalities.

Contribution

It extends gradient estimate techniques to nonlinear porous medium equations on curved spaces, improving and unifying previous results by various researchers.

Findings

Established new local gradient estimates for porous medium equations.

Derived several improved parabolic Harnack inequalities.

Unified previous results and extended them to Riemannian manifolds.

Abstract

In this paper we study gradient estimates for the positive solutions of the porous medium equation: $$u_t=\Delta u^m$$ where $m>1$, which is a nonlinear version of the heat equation. We derive local gradient estimates of the Li-Yau type for positive solutions of porous medium equations on Riemannian manifolds with Ricci curvature bounded from below. As applications, several parabolic Harnack inequalities are obtained. In particular, our results improve the ones of Lu, Ni, V\'{a}zquez and Villani in [10]. Moreover, our results recover the ones of Davies in [4], Hamilton in [5] and Li and Xu in [7].