GRADIENT ESTIMATES AND ENTROPY FORMULAE FOR WEIGHTED p-HEAT EQUATIONS ON SMOOTH METRIC MEASURE SPACES

摘要

Let(M,g,e fdv) 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 аuаt = 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-′Emery Ricci curvature.Secondly,we show an entropy monotonicity formula with nonnegative m-dimensional Bakry-′Emery Ricci curvature which is a generalization to the results of Kotschwar and Ni [9],Li [7].