Differential Harnack inequalities for nonlinear heat equations with potentials under the Ricci flow

TL;DR

This paper establishes new differential Harnack inequalities for nonlinear heat equations with potentials evolving under Ricci flow, extending previous results for linear cases and applying to closed surfaces.

Contribution

It introduces novel differential Harnack inequalities for nonlinear heat equations with potentials under Ricci flow, including an interpolated version for the b5-Ricci flow.

Findings

Derived differential Harnack inequalities for nonlinear heat equations with potentials.

Extended previous linear Harnack inequalities to nonlinear cases.

Applied results to the b5-Ricci flow on closed surfaces.

Abstract

We prove several differential Harnack inequalities for positive solutions to nonlinear backward heat equations with different potentials coupled with the Ricci flow. We also derive an interpolated Harnack inequality for the nonlinear heat equation under the $\varepsilon$-Ricci flow on a closed surface. These new Harnack inequalities extend the previous differential Harnack inequalities for linear heat equations with potentials under the Ricci flow.