Geodesic
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, Bingqing1, 2 ; Zeng, Fanqi3
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.

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

ut=Δu+λuα
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

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

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.04.003

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 
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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://geodesic.mathdoc.fr/articles/10.1016/j.crma.2018.04.003/

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