SHARP GRADIENT ESTIMATES FOR A HEAT EQUATION IN RIEMANNIAN MANIFOLDS

被引：25

作者：

Ha Tuan Dung ^{[1
,2
]}

Nguyen Thac Dung ^{[3
,4
]}

机构：

[1] Hanoi Pedag Univ, Fac Math, 2 Xuan Hoa, Xuan Hoa, Vinh Phuc, Vietnam

[2] Natl Tsing Hua Univ, Dept Math, Hsinchu, Taiwan

[3] Hanoi Univ Sci VNU, Fac Math Mech Informat, Hanoi, Vietnam

[4] Thang Long Univ, Thang Long Inst Math & Appl Sci, Hanoi, Vietnam

来源：

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY | 2019年 / 147卷 / 12期

关键词：

Ancient solution; heat equation; Liouville theorem; sharp gradient estimate; sublinear growth; THEOREM;

D O I：

10.1090/proc/14645

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

In this paper, we prove sharp gradient estimates for a positive solution to the heat equation u(t) = Delta u + au log u in complete noncompact Riemannian manifolds. As its application, we show that if u is a positive solution of the equation u(t) = Delta u and log u is of sublinear growth in both spatial and time directions, then u must be constant. This gradient estimate is sharp since it is well known that u(x, t) = e(x+t) satisfying ut = Delta u.

引用

页码：5329 / 5338

页数：10

共 13 条

[1] A Liouville-Type Theorem for Smooth Metric Measure Spaces [J].

Brighton, Kevin .

JOURNAL OF GEOMETRIC ANALYSIS, 2013, 23 (02) :562-570

[2] AN EXTENSION OF HOPF,E. MAXIMUM PRINCIPLE WITH AN APPLICATION TO RIEMANNIAN GEOMETRY [J].

CALABI, E .

DUKE MATHEMATICAL JOURNAL, 1958, 25 (01) :45-56

[3]

Hamilton R. S, 1993, Commun. Anal. Geom., V1, P113, DOI 10.4310/CAG.1993.v1.n1.a6

[4] Gradient estimates and Liouville type theorems for a nonlinear elliptic equation [J].

Huang, Guangyue ;

Ma, Bingqing .

ARCHIV DER MATHEMATIK, 2015, 105 (05) :491-499

[5] GRADIENT ESTIMATE FOR A NONLINEAR HEAT EQUATION ON RIEMANNIAN MANIFOLDS [J].

Jiang, Xinrong .

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 2016, 144 (08) :3635-3642

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

LI, P ;

YAU, ST .

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

[7] Li-Yau-Hamilton estimates and Bakry-Emery-Ricci curvature [J].

Li, Yi .

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2015, 113 :1-32

[8]

Lin F. H., COMM PURE APPL MATH

[9] Gradient estimates for some f-heat equations driven by Lichnerowicz's equation on complete smooth metric measure spaces [J].

Nguyen Thac Dung ;

Nguyen Ngoc Khanh ;

Quoc Anh Ngo .

MANUSCRIPTA MATHEMATICA, 2018, 155 (3-4) :471-501

[10] 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

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