Geodesic
Gradient estimates of Li Yau type for a general heat equation on Riemannian manifolds
Archivum mathematicum, Tome 52 (2016) no. 4, pp. 207-219 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In this paper, we consider gradient estimates on complete noncompact Riemannian manifolds $(M,g)$ for the following general heat equation \[ u_t=\Delta _V u+au\log u+bu \] where $a$ is a constant and $b$ is a differentiable function defined on $M\times [0, \infty )$. We suppose that the Bakry-Émery curvature and the $N$-dimensional Bakry-Émery curvature are bounded from below, respectively. Then we obtain the gradient estimate of Li-Yau type for the above general heat equation. Our results generalize the work of Huang-Ma ([4]) and Y. Li ([6]), recently.
In this paper, we consider gradient estimates on complete noncompact Riemannian manifolds $(M,g)$ for the following general heat equation \[ u_t=\Delta _V u+au\log u+bu \] where $a$ is a constant and $b$ is a differentiable function defined on $M\times [0, \infty )$. We suppose that the Bakry-Émery curvature and the $N$-dimensional Bakry-Émery curvature are bounded from below, respectively. Then we obtain the gradient estimate of Li-Yau type for the above general heat equation. Our results generalize the work of Huang-Ma ([4]) and Y. Li ([6]), recently.
DOI : 10.5817/AM2016-4-207
Classification : 35B53, 58J35
Keywords: gradient estimates; general heat equation; Laplacian comparison theorem; $V$-Bochner-Weitzenböck; Bakry-Emery Ricci curvature 
@article{10_5817_AM2016_4_207,
     author = {Khanh, Nguyen Ngoc},
     title = {Gradient estimates of {Li} {Yau} type for a general heat equation on {Riemannian} manifolds},
     journal = {Archivum mathematicum},
     pages = {207--219},
     year = {2016},
     volume = {52},
     number = {4},
     doi = {10.5817/AM2016-4-207},
     mrnumber = {3610650},
     zbl = {06674900},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5817/AM2016-4-207/}
}
TY  - JOUR
AU  - Khanh, Nguyen Ngoc
TI  - Gradient estimates of Li Yau type for a general heat equation on Riemannian manifolds
JO  - Archivum mathematicum
PY  - 2016
SP  - 207
EP  - 219
VL  - 52
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.5817/AM2016-4-207/
DO  - 10.5817/AM2016-4-207
LA  - en
ID  - 10_5817_AM2016_4_207
ER  - 
%0 Journal Article
%A Khanh, Nguyen Ngoc
%T Gradient estimates of Li Yau type for a general heat equation on Riemannian manifolds
%J Archivum mathematicum
%D 2016
%P 207-219
%V 52
%N 4
%U http://geodesic.mathdoc.fr/articles/10.5817/AM2016-4-207/
%R 10.5817/AM2016-4-207
%G en
%F 10_5817_AM2016_4_207
Khanh, Nguyen Ngoc. Gradient estimates of Li Yau type for a general heat equation on Riemannian manifolds. Archivum mathematicum, Tome 52 (2016) no. 4, pp. 207-219. doi: 10.5817/AM2016-4-207

[1] Chen, Q., Jost, J., Qiu, H.B.: Existence and Liouville theorems for $V$-harmonic maps from complete manifolds. Ann. Global Anal. Geom. 42 (2012), 565–584. | DOI | MR | Zbl

[2] Davies, E.B.: Heat kernels and spectral theory. Cambridge University Press, 1989. | MR | Zbl

[3] Dung, N.T., Khanh, N.N.: Gradient estimates of Hamilton - Souplet - Zhang type for a general heat equation on Riemannian manifolds. Arch. Math (Basel) 105 (2015), 479–490. | DOI | MR | Zbl

[4] Huang, G.Y., Ma, B.Q.: Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds. Arch. Math. (Basel) 94 (2010), 265–275. | DOI | MR | Zbl

[5] Li, P., Yau, S.T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156 (1986), 152–201. | MR | Zbl

[6] Li, Y.: Li-Yau-Hamilton estimates and Bakry-Emery Ricci curvature. Nonlinear Anal. 113 (2015), 1–32. | MR | Zbl

[7] Negrin, E.R.: Gradient estimates and a Liouville type theorem for the Schrödinger operator. J. Funct. Anal. 127 (1995), 198–203. | DOI | MR | Zbl

Cité par Sources :