Harnack Estimates for Nonlinear Heat Equations with Potentials in Geometric Flows

TL;DR

This paper establishes differential Harnack estimates for positive solutions to a nonlinear heat equation with potentials on manifolds evolving under geometric flows, unifying and extending previous results.

Contribution

It derives new Harnack inequalities for nonlinear heat equations with potentials on manifolds undergoing geometric flows, generalizing many existing results.

Findings

Includes known Harnack estimates as special cases

Provides new inequalities for various geometric flows

Enhances understanding of heat equations in evolving geometries

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.