Local Aronson-Bénilan gradient estimates and Harnack inequality for the porous medium equation along Ricci flow

Abstract

In this paper, we prove some new local Aronson-Bénilan type gradient estimates for positive solutions of the porous medium equation $u_{t}=Δ u^{m}, m>1$ coupled with Ricci flow, assuming that the Ricci curvature is bounded. As application, the related Harnack inequality is derived. Our results generalize known results. These results may be regarded as the generalizations of the gradient estimates of Lu-Ni-Vázquez-Villani and Huang-Huang-Li to the Ricci flow.