Harnack Estimates for Nonlinear Heat Equations with Potentials in Geometric Flows

Abstract

Let $M$ be a closed Riemannian manifold with a family of Riemannian metrics $g_{ij}(t)$ evolving by geometric flow $\partial_{t}g_{ij} = -2{S}_{ij}$, where $S_{ij}(t)$ is a family of smooth symmetric two-tensors on $M$. In this paper we derive differential Harnack estimates for positive solutions to the nonlinear heat equation with potential: \begin{eqnarray*} \frac{\partial f}{\partial t} = {\Delta}f + \gamma (t) f\log f +aSf, \end{eqnarray*} where $\gamma (t)$ is a continuous function on $t$, $a$ is a constant and $S=g^{ij}S_{ij}$ is the trace of $S_{ij}$. Our Harnack estimates include many known results as special cases, and moreover lead to new Harnack inequalities for a variety geometric flows.