Differential Harnack Estimates for Parabolic Equations

TL;DR

This paper establishes differential Harnack inequalities for positive solutions of certain nonlinear parabolic equations on manifolds evolving under Ricci flow, extending previous results on the conjugate heat equation.

Contribution

It introduces new differential Harnack estimates for nonlinear parabolic equations in the Ricci flow setting, generalizing earlier work on conjugate heat equations.

Findings

Proved differential Harnack inequalities for nonlinear parabolic equations.

Extended previous results on conjugate heat equations under Ricci flow.

Provided insights into the behavior of positive solutions in evolving geometries.

Abstract

Let $(M,g(t))$ be a solution to the Ricci flow on a closed Riemannian manifold. In this paper, we prove differential Harnack inequalities for positive solutions of nonlinear parabolic equations of the type $$\ppt f=\Delta f-f \ln f +Rf.$$ We also comment on an earlier result of the first author on positive solutions of the conjugate heat equation under the Ricci flow.