SHARP GRADIENT ESTIMATES FOR A HEAT EQUATION IN RIEMANNIAN MANIFOLDS

被引：21
|
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.
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页码：5329 / 5338
页数：10
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