GRADIENT ESTIMATES AND LIOUVILLE THEOREMS FOR LINEAR AND NONLINEAR PARABOLIC EQUATIONS ON RIEMANNIAN MANIFOLDS

doi:10.1016/S0252-9602(16)30017-0

数学物理学报(英文版) ›› 2016, Vol. 36 ›› Issue (2): 514-526.doi: 10.1016/S0252-9602(16)30017-0

GRADIENT ESTIMATES AND LIOUVILLE THEOREMS FOR LINEAR AND NONLINEAR PARABOLIC EQUATIONS ON RIEMANNIAN MANIFOLDS

朱晓宝

Department of Mathematics, School of Information, Renmin University of China, Beijing 100872, China

收稿日期:2014-10-13 修回日期:2015-06-24 出版日期:2016-04-25 发布日期:2016-04-25
作者简介:Xiaobao ZHU,E-mail:zhuxiaobao@ruc.edu.cn
基金资助:
This work is partially supported by the National Science Foundation of China (41275063 and 11401575).

GRADIENT ESTIMATES AND LIOUVILLE THEOREMS FOR LINEAR AND NONLINEAR PARABOLIC EQUATIONS ON RIEMANNIAN MANIFOLDS

Xiaobao ZHU

Department of Mathematics, School of Information, Renmin University of China, Beijing 100872, China

Received:2014-10-13 Revised:2015-06-24 Online:2016-04-25 Published:2016-04-25
Supported by:
This work is partially supported by the National Science Foundation of China (41275063 and 11401575).

摘要/Abstract

摘要：

In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations (Δ-∂/∂t)u(x, t)+q(x, t)u(x, t)=0 and nonlinear parabolic equations (Δ-∂/∂t)u(x, t)+h(x, t)u^p(x, t)=0(p>1) on Riemannian manifolds. As applications, we obtain some theorems of Liouville type for positive ancient solutions of such equations. Our results generalize that of Souplet-Zhang ([1], Bull. London Math. Soc. 38(2006), 1045-1053) and the author ([2], Nonlinear Anal. 74(2011), 5141-5146).

关键词: Gradient estimate, linear parabolic equation, nonlinear parabolic equation, Liouville type theorem

Abstract:

Key words: Gradient estimate, linear parabolic equation, nonlinear parabolic equation, Liouville type theorem

中图分类号:

35K55

朱晓宝. GRADIENT ESTIMATES AND LIOUVILLE THEOREMS FOR LINEAR AND NONLINEAR PARABOLIC EQUATIONS ON RIEMANNIAN MANIFOLDS[J]. 数学物理学报(英文版), 2016, 36(2): 514-526.

Xiaobao ZHU. GRADIENT ESTIMATES AND LIOUVILLE THEOREMS FOR LINEAR AND NONLINEAR PARABOLIC EQUATIONS ON RIEMANNIAN MANIFOLDS[J]. Acta mathematica scientia,Series B, 2016, 36(2): 514-526.

参考文献

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GRADIENT ESTIMATES AND LIOUVILLE THEOREMS FOR LINEAR AND NONLINEAR PARABOLIC EQUATIONS ON RIEMANNIAN MANIFOLDS

GRADIENT ESTIMATES AND LIOUVILLE THEOREMS FOR LINEAR AND NONLINEAR PARABOLIC EQUATIONS ON RIEMANNIAN MANIFOLDS

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