New differential Harnack inequalities for nonlinear heat equations

TL;DR

This paper establishes new differential Harnack inequalities for nonlinear heat equations on closed manifolds and surfaces, improving existing bounds with exponential correction terms under various geometric flows.

Contribution

It introduces novel constrained and interpolated Harnack inequalities for nonlinear heat equations, extending previous results to broader settings without curvature restrictions.

Findings

Derived constrained trace, matrix, and constrained matrix Harnack inequalities.

Established a new interpolated Harnack inequality under the $oldsymbol{ ext{ε}}$-Ricci flow.

Proved a differential Harnack inequality under Ricci flow without curvature conditions.

Abstract

We prove constrained trace, matrix and constrained matrix Harnack inequalities for the nonlinear heat equation $\omega_t=\Delta\omega+a\omega\ln \omega$ on closed manifolds. We also derive a new interpolated Harnack inequality for the equation $\omega_t=\Delta\omega-\omega\ln\omega+\varepsilon R\omega$ on closed surfaces under the $\varepsilon$-Ricci flow. Finally we prove a new differential Harnack inequality for the equation $\omega_t=\Delta\omega-\omega\ln\omega$ under the Ricci flow without any curvature condition. Among these Harnack inequalities, the correction terms are all time-exponential functions, which are superior to time-polynomial functions.