Gradient Estimates and Entropy Formulae for Weighted p-Heat Equations on Smooth Metric Measure Spaces

Let (M, g, e−f d v) be a smooth metric measure space. In this paper, we consider two nonlinear weighted p-heat equations. Firstly, we derive a Li-Yau type gradient estimates for the positive solutions to the following nonlinear weighted p-heat equation ∂u∂t = efdiv(e−f|▿u|p−2▽u) on M × [0,∞), where 1 < P < ∞ and f is a smooth function on M under the assumption that the m-dimensional nonnegative Bakry-Émery Ricci curvature. Secondly, we show an entropy monotonicity formula with nonnegative m-dimensional Bakry-Émery Ricci curvature which is a generalization to the results of Kotschwar and Ni [9], Li [7].