Local estimates for positive solutions of porous medium equations

TL;DR

This paper develops new gradient estimates for positive solutions of porous medium and fast diffusion equations on manifolds with specific curvature conditions, advancing understanding of their behavior.

Contribution

It introduces novel Aronson-Bénilan type gradient estimates for these equations on curved manifolds, extending previous results to more general geometric settings.

Findings

Derived new gradient estimates for porous medium equations

Extended estimates to manifolds with curvature conditions

Enhanced understanding of solution behavior on curved spaces

Abstract

We derive several new gradient estimates of Aronson-B{\'e}nilan type for positive solutions of porous medium equation and fast diffusion equation on a complete manifold that satisfies the curvature dimension condition