Local Aronson-Benolan type gradient estimates for the porous medium type equation under Ricci

TL;DR

This paper derives new local Aronson-Bénilan type gradient estimates for positive solutions of the porous medium equation under Ricci flow, leading to generalized Harnack inequalities and extending previous gradient estimate results.

Contribution

It introduces novel local gradient estimates for the porous medium equation under Ricci flow, generalizing prior results by Lu-Ni-Vázquez-Villani and Huang-Huang-Li.

Findings

Derived new local gradient estimates for porous medium solutions under Ricci flow.

Established related Harnack inequalities.

Extended known gradient estimate results to Ricci flow context.

Abstract

In this paper, we investigate some new local Aronson-B\'enilan type gradient estimates for positive solutions of the porous medium equation $$ u_{t}=\Delta u^{m}, $$ under Ricci flow. As application, the related Harnack inequalities are derived. Our results generalize known results. These results in the paper can be regard as generalizing the gradient estimates of Lu-Ni-V\'{a}zquez-Villani and Huang-Huang-Li to the Ricci flow.