黎曼流形上多孔介质方程的梯度估计
Gradient estimates for the porous medium equations on Riemannian manifolds
摘要：
本文研究多孔介质方程：$$u_t=Δu^m$$正解的梯度估计，其中$m>1$是热传导方程的非线性形式.我们得到了里奇曲率下有界的黎曼流形上多孔介质方程正解的Li-Yau型局部梯度估计.作为应用，得到了几个抛物型Harnack不等式.特别地，我们的结果改进了Lu，Ni，Vázquez和Villani在[10]中的结果.此外，我们的结果恢复了Davies在[4]，Hamilton在[5]和Li和Xu在[7]中的结果。
英文摘要：
In this paper we study gradient estimates for the positive solutions of the porous medium equation: $$u_t=Δu^m$$ where $m>1$, which is a nonlinear version of the heat equation. We derive local gradient estimates of the Li-Yau type for positive solutions of porous medium equations on Riemannian manifolds with Ricci curvature bounded from below. As applications, several parabolic Harnack inequalities are obtained. In particular, our results improve the ones of Lu, Ni, Vázquez and Villani in [10]. Moreover, our results recover the ones of Davies in [4], Hamilton in [5] and Li and Xu in [7].
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