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

Bingqing Ma; Fanqi Zeng

doi:10.1016/j.crma.2018.04.003

Differential geometry

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

Presented by: the Editorial Board

Bingqing Ma ^1,²; Fanqi Zeng 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, Volume 356 (2018) no. 5, pp. 550-557.

Abstract
Résumé

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.

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.

Received: 2017-06-23
Accepted: 2018-04-06
Published online: 2018-04-11

DOI: 10.1016/j.crma.2018.04.003

Author's affiliations:

Bingqing Ma ^{1, 2}; Fanqi Zeng 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

@article{CRMATH_2018__356_5_550_0,
     author = {Bingqing Ma and Fanqi Zeng},
     title = {Hamilton{\textendash}Souplet{\textendash}Zhang's gradient estimates and {Liouville} theorems for a nonlinear parabolic equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {550--557},
     publisher = {Elsevier},
     volume = {356},
     number = {5},
     year = {2018},
     doi = {10.1016/j.crma.2018.04.003},
     language = {en},
}

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Bingqing Ma; Fanqi Zeng. Hamilton–Souplet–Zhang's gradient estimates and Liouville theorems for a nonlinear parabolic equation. Comptes Rendus. Mathématique, Volume 356 (2018) no. 5, pp. 550-557. doi : 10.1016/j.crma.2018.04.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.04.003/

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Cited by Sources:

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

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