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

doi:10.1016/S0252-9602(13)60055-7

Abstract

Abstract:

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 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 curva-ture which is a generalization to the results of Kotschwar and Ni [9], Li [7].

Key words: gradient estimates, weighted p-heat equation, entropy monotonicity formula, m-Bakry-Émery Ricci curvature

CLC Number:

58J35

WANG Yu-Zhao, YANG Jie, CHEN Wen-Yi. GRADIENT ESTIMATES AND ENTROPY FORMULAE FOR WEIGHTED p-HEAT EQUATIONS ON SMOOTH METRIC MEASURE SPACES[J].Acta mathematica scientia,Series B, 2013, 33(4): 963-974.

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