Differential geometry
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.

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.

Accepted:
Published online:
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, Volume 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/

[1] Cheng, S.-Y.; Yau, S.-T. Differential equations on Riemannian manifolds and their geometric applications, Commun. Pure Appl. Math., Volume 28 (1975), pp. 333-354

[2] Davies, E.B. Heat Kernels and Spectral Theory, Cambridge Tracts in Math., vol. 92, Cambridge University Press, UK, 1989

[3] Hamilton, R. A matrix Harnack estimate for the heat equation, Commun. Anal. Geom., Volume 1 (1993), pp. 113-125

[4] Huang, G.Y.; Huang, Z.J.; Li, H. Gradient estimates for the porous medium equations on Riemannian manifolds, J. Geom. Anal., Volume 23 (2013), pp. 1851-1875

[5] Huang, G.Y.; Huang, Z.J.; Li, H. Gradient estimates and differential Harnack inequalities for a nonlinear parabolic equation on Riemannian manifolds, Ann. Glob. Anal. Geom., Volume 43 (2013), pp. 209-232

[6] Huang, G.Y.; Li, H. Gradient estimates and entropy formulae of porous medium and fast diffusion equations for the Witten Laplacian, Pac. J. Math., Volume 268 (2014), pp. 47-78

[7] Li, P.; Yau, S.-T. On the parabolic kernel of the Schrödinger operator, Acta Math., Volume 156 (1986), pp. 153-201

[8] Li, J. Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds, J. Funct. Anal., Volume 100 (1991), pp. 233-256

[9] Li, J.F.; Xu, X.J. Differential Harnack inequalities on Riemannian manifolds I: linear heat equation, Adv. Math., Volume 226 (2011), pp. 4456-4491

[10] Ma, L. Gradient estimates for a simple elliptic equation on complete non-compact Riemannian manifolds, J. Funct. Anal., Volume 241 (2006), pp. 374-382

[11] Souplet, P.; Zhang, Q.S. Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds, Bull. Lond. Math. Soc., Volume 38 (2006), pp. 1045-1053

[12] Xu, X.J. Gradient estimates for $ut=ΔF(u)$ on manifolds and some Liouville-type theorems, J. Differ. Equ., Volume 252 (2012), pp. 1403-1420

[13] Yang, Y.Y. Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds, Proc. Amer. Math. Soc., Volume 136 (2008), pp. 4095-4102

[14] Yang, Y.Y. Gradient estimates for the equation $Δu+cu−α=0$ on Riemannian manifolds, Acta Math. Sin. Engl. Ser., Volume 26 (2010), pp. 1177-1182

[15] Zhang, J.; Ma, B.Q. Gradient estimates for a nonlinear equation $Δfu+cu−α=0$ on complete noncompact manifolds, Commun. Math., Volume 19 (2011), pp. 73-84

[16] Zhu, X.B. Gradient estimates and Liouville theorems for nonlinear parabolic equations on noncompact Riemannian manifolds, Nonlinear Anal., Volume 74 (2011), pp. 5141-5146

Cited by Sources:

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