Vol. 257, No. 1, 2012

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

Jia-Yong Wu

Vol. 257 (2012), No. 1, 199–218
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
Authors
Jia-Yong Wu
Department of Mathematics
Shanghai Maritime University
Haigang Avenue 1550
Shanghai 201306
China