Remarks on differential Harnack inequalities

In this paper, we give a generalization of (global and local) differential Harnack inequalities for heat equations obtained by Li and Xu [J.F. Li, X.J. Xu, Differential Harnack inequalities on Riemannian manifolds I: linear heat equation, Adv: Math. 226 (5) (2011) 4456-4491] and Baudoin and Garofalo [F. Baudoin, N. Garofalo, Perelman's entropy and doubling property on Riemannian manifolds, J. Geom. Anal. 21 (2011) 1119-1131]. From this we can derive new Harnack inequalities and new lower bounds for the associated heat kernel. Also we provide some new entropy formulas with monotonicity. (C) 2013 Elsevier Inc. All rights reserved.