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

Let （M,g, e^-fdv） be a smooth metric measure space. In this paper, we con- sider 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 equationand f is a smooth function on M under the assumptionthat the m-dimensional nonnegative Bakry-Emery Ricci curvature. Secondly, we show an entropy monotonicity formula with nonnegative m-dimensional Bakry-Emery Ricci curva- ture which is a generalization to the results of Kotschwar and Ni [9], Li [7].     

光滑度量测度空间上的加权扩散方程的梯度估计

本文主要考虑了一类加权非线性扩散方程正解的梯度估计.在m-维Bakry-(E)mery Ricci曲率下有界的假设下,得到加权多孔介质方程(γ＞1)正解的Li-Yau型梯度估计,此外对于加权快速扩散方程(0＜γ＜1),证明了Hamilton型椭圆梯度估计,结论分别推广了Lu,Ni,Vázquez and Villani在文[1]和Zhu在文[2]中的结果.