Interpolating between constrained Li-Yau and Chow-Hamilton Harnack inequalities for a nonlinear parabolic equation

TL;DR

This paper introduces a family of Harnack inequalities that interpolate between two known inequalities for a nonlinear parabolic equation on a 2D manifold, extending previous work to a nonlinear setting.

Contribution

It establishes a one-parameter family of inequalities connecting Li-Yau and Chow-Hamilton Harnack inequalities for nonlinear equations under Ricci flow.

Findings

Unified framework for Harnack inequalities in nonlinear setting

Extension of previous linear results to nonlinear equations

Potential applications in geometric analysis and PDEs

Abstract

We establish a one-parameter family of Harnack inequalities connecting the constrained trace Li-Yau differential Harnack inequality for a nonlinear parabolic equation to the constrained trace Chow-Hamilton Harnack inequality for this nonlinear equation with respect to evolving metrics related to Ricci flow on a 2-dimensional closed manifold. This result can be regarded as a nonlinear version of the previous work of Y. Zheng and the author (Arch. Math. 94 (2010), 591-600).