Differential Harnack inequalities for semilinear parabolic equations on [formula omitted] metric measure spaces

Abstract

We present a unified method to derive differential Harnark inequalities for locally positive weak solutions to semilinear parabolic equations on RCD∗(K,N)metric measure spaces, as introduced by Gigli (2015) and Erbar et al. (2015). As its application, on the one hand, we recover many known differential Harnack inequalities on heat equation, on the other hand, for logarithmic type equation and Yamabe type equation, we get some sharp differential Harnack inequalities on RCD∗(0,N)metric measure spaces. As its corollary, we get some sharp Harnack inequalities and Liouville theorems.