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

Jia-Yong Wu

Vol. 257 (2012), No. 1, 199–218

DOI: 10.2140/pjm.2012.257.199

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 𝜀-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.

Keywords

Harnack inequality, interpolated Harnack inequality, nonlinear heat equation, nonlinear backward heat equation, Ricci flow

Mathematical Subject Classification 2010

Primary: 53C44

Milestones

Received: 25 June 2011

Revised: 22 February 2012

Accepted: 22 May 2012

Published: 19 June 2012