New Differential Harnack Inequalities for Nonlinear Heat Equations

Abstract

This paper deals with constrained trace, matrix and constrained matrix Harnack inequalities for the nonlinear heat equation ω _t = Δω + aω ln ω on closed manifolds. A new interpolated Harnack inequality for ω _t = Δω − ω ln ω+εRω on closed surfaces under ε-Ricci flow is also derived. Finally, the author proves a new differential Harnack inequality for ω _t = Δω − ω ln ω under 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.

Keywords

Harnack inequality / Nonlinear heat equation / Ricci flow