Gradient estimates and Liouville theorems for nonlinear parabolic equations on noncompact Riemannian manifolds

被引：34

作者：

Zhu, Xiaobao ^{[1
]}

机构：

[1] Chinese Acad Sci, Inst Math, Acad Math & Syst Sci, Beijing 100190, Peoples R China

来源：

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS | 2011年 / 74卷 / 15期

关键词：

Gradient estimates; Nonlinear parabolic equations; Liouville theorems;

D O I：

10.1016/j.na.2011.05.008

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

Let M be a complete noncompact Riemannian manifold. In this paper, we derive a local gradient estimate for positive solutions of the nonlinear parabolic equation. (Delta - partial derivative/t) u(x, t) + lambda(x, t)u(alpha) (x, t) = 0 on M x (-infinity, 0]. We also obtain a theorem of Liouville type for positive solutions of this nonlinear equation. This paper extends the result of Souplet and Zhang [P. Souplet, Qi S. Zhang, Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds, Bull. Lond. Math. Soc. 38 (2006) 1045-1053]. (C) 2011 Elsevier Ltd. All rights reserved.

引用

页码：5141 / 5146

页数：6

共 7 条

[1] DIFFERENTIAL EQUATIONS ON RIEMANNIAN MANIFOLDS AND THEIR GEOMETRIC APPLICATIONS [J].

CHENG, SY ;

YAU, ST .

COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 1975, 28 (03) :333-354

[2]

Hamilton R. S., 1993, COMMUN ANAL GEOM, V1, P113

[3]

LI JY, 1991, J FUNCT ANAL, V100, P233

[4] ON THE PARABOLIC KERNEL OF THE SCHRODINGER OPERATOR [J].

LI, P ;

YAU, ST .

ACTA MATHEMATICA, 1986, 156 (3-4) :153-201

[5] Gradient estimate for the degenerate parabolic equation ut=ΔF(u)+H(u) on manifolds [J].

Ma, Li ;

Zhao, Lin ;

Song, Xianfa .

JOURNAL OF DIFFERENTIAL EQUATIONS, 2008, 244 (05) :1157-1177

[6] Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds [J].

Souplet, Philippe ;

Zhang, Qi S. .

BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, 2006, 38 :1045-1053

[7]

XU X, ARXIV08053676

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