Hamilton–Souplet–Zhang's gradient estimates and Liouville theorems for a nonlinear parabolic equation

Ma, Bingqing; Zeng, Fanqi

doi:10.1016/j.crma.2018.04.003

Differential geometry

Hamilton–Souplet–Zhang's gradient estimates and Liouville theorems for a nonlinear parabolic equation
[Estimations du gradient de Hamilton–Souplet–Zhang et théorèmes de Liouville pour une équation non linéaire parabolique]

Présenté par : the Editorial Board

Ma, Bingqing ^1,² ; Zeng, Fanqi ³

¹ College of Physics and Materials Science, Henan Normal University, Xinxiang 453007, People's Republic of China
² Department of Mathematics, Henan Normal University, Xinxiang 453007, People's Republic of China
³ School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, People's Republic of China

Comptes Rendus. Mathématique, Tome 356 (2018) no. 5, pp. 550-557.

Résumé
Abstract

Dans la présente Note, nous étudions les estimations du gradient de Hamilton–Souplet–Zhang pour les solutions positives de l'équation non linéaire parabolique

u_{t} = Δ u + λ u^{α}

sur une variété riemannienne non compacte, où λ et α sont deux constantes réelles. Nous en déduisons, comme application, un théorème de type Liouville.

In this paper, we study Hamilton–Souplet–Zhang's gradient estimates for positive solutions to the nonlinear parabolic equation

u_{t} = Δ u + λ u^{α}

on noncompact Riemannian manifolds, where

λ, α

are two real constants. As an application, we obtain a Liouville-type theorem.

Reçu le : 2017-06-23
Accepté le : 2018-04-06
Publié le : 2018-04-11

DOI : 10.1016/j.crma.2018.04.003

Affiliations des auteurs :

Ma, Bingqing ^{1, 2} ; Zeng, Fanqi ³

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Ma, Bingqing; Zeng, Fanqi. Hamilton–Souplet–Zhang's gradient estimates and Liouville theorems for a nonlinear parabolic equation. Comptes Rendus. Mathématique, Tome 356 (2018) no. 5, pp. 550-557. doi : 10.1016/j.crma.2018.04.003. http://www.numdam.org/articles/10.1016/j.crma.2018.04.003/

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Cité par Sources :

^☆ The research of the first author was supported by NSFC (Nos. 11401179, 11671121).