Gradient Estimates for the Porous Medium Equations on Riemannian Manifolds

被引：0

作者：

Guangyue Huang

Zhijie Huang

Haizhong Li

机构：

[1] Tsinghua University,Department of Mathematical Sciences

来源：

Journal of Geometric Analysis | 2013年 / 23卷

关键词：

Porous medium equation; Li–Yau type estimate; Harnack inequality; 35B45; 35K55;

D O I：

暂无

中图分类号：

学科分类号：

摘要：

In this paper we study gradient estimates for the positive solutions of the porous medium equation: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_t=\Delta u^m$$\end{document} 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 J. Math. Pures Appl. 91:1–19, 2009). Moreover, our results recover the ones of Davies (in Cambridge Tracts Math vol. 92, 1989), Hamilton (in Comm. Anal. Geom. 1:113–125, 1993) and Li and Xu (in Adv. Math. 226:4456–4491, 2011).

引用

页码：1851 / 1875

页数：24

共 20 条

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