Differential Harnack inequalities for semilinear parabolic equations on Riemannian manifolds I: Bakry-Émery curvature bounded below

Abstract

In this paper, we present a unified method for deriving differential Harnack inequalities for positive solutions of the semilinear parabolic equation∂tu=ΔVu+H(u) on complete Riemannian manifolds with Bakry-Émery curvature bounded below. This method transforms the problem of deriving differential Harnack inequalities into solving a related ODE system. As its application, on the one hand, we obtain new and improved estimates for logarithmic type equation and Yamabe type equation. On the other hand, we establish some sharp estimates for them under non-negative Bakry-Émery curvature condition. As their natural corollary, we also obtain some sharp Harnack inequalities and Liouville type theorems.