Gradient estimates of porous medium equations under the Ricci flow

Abstract

In this paper we consider an n-dimensional manifold Mn evolving under the Ricci flow and establish gradient estimates for positive solutions of porous medium equations on Mn. As applications, we derive Harnack type inequalities. In particular, our results generalize gradient estimates for positive solutions of the heat equations in Liu (Pacific J Math 243:165–180 [18]).