J. Nonl. Evol. Equ. Appl. 2015 (1), pp. 1-19, published on March 11, 2015:

Gradient estimates for heat-type equations on evolving Manifolds


A. Abolarinwa

Department of Mathematics, University of Sussex, Brighton, BN1 9QH, UK

Received on March 26, 2014, Revised version on Aug 6, 2014
Accepted August 20, 2014

Communicated by Alexander Pankov

Abstract.  In this paper we study gradient estimates for all positive solutions of a class ofheat-type equations defined on a Riemannian manifold whose metric is evolving by the generalized geometric flow. In particular, we apply Laplacian comparison theorem and maximum principle to obtain Li-Yau type gradient estimates (local and global). As a corollary, differential Harnack inequalities are derived. The reason for this on the one hand is to show the advantage of using Ricci flow and on the other hand is to prepare ground for applications to other special cases of geometric flow.
Keywords:Gradient estimates, Harnack inequalities, heat-type equations, geometric flows, Ricci flow.
2010 Mathematics Subject Classification:   35K55, 53C21, 53C44, 58J35.

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