Gradient estimates and entropy monotonicity formula for doubly nonlinear diffusion equations on Riemannian manifolds

In the first part of this paper, we prove the sharp global Li-Yau type gradient estimates for positive solutions to doubly nonlinear diffusion equation(DNDE) on complete Riemannian manifolds with nonnegative Ricci curvature. As an application, one can obtain a parabolic Harnack inequality. In the second part, we obtain a Perelman-type entropy monotonicity formula for DNDE on compact Riemannian manifolds with nonnegative Ricci curvature. These results generalize some works of Ni (JGA 2004), Lu-Ni-Vazquez-Villani (JMPA 2009) and Kotschwar-Ni (Annales Scientifiques de l'Ecole Normale Superieure 2009). Copyright (c) 2013 John Wiley & Sons, Ltd.