| Title:
|
Gradient estimates of Li Yau type for a general heat equation on Riemannian manifolds (English) |
| Author:
|
Khanh, Nguyen Ngoc |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
52 |
| Issue:
|
4 |
| Year:
|
2016 |
| Pages:
|
207-219 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
gradient estimates |
| Keyword:
|
general heat equation |
| Keyword:
|
Laplacian comparison theorem |
| Keyword:
|
$V$-Bochner-Weitzenböck |
| Keyword:
|
Bakry-Emery Ricci curvature |
| MSC:
|
35B53 |
| MSC:
|
58J35 |
| idZBL:
|
Zbl 06674900 |
| idMR:
|
MR3610650 |
| DOI:
|
10.5817/AM2016-4-207 |
| . |
| Date available:
|
2016-12-20T21:47:36Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145929 |
| . |
| Reference:
|
[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. Zbl 1270.58010, MR 2995205, 10.1007/s10455-012-9327-z |
| Reference:
|
[2] Davies, E.B.: Heat kernels and spectral theory.Cambridge University Press, 1989. Zbl 0699.35006, MR 0990239 |
| Reference:
|
[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. Zbl 1329.58023, MR 3413923, 10.1007/s00013-015-0828-4 |
| Reference:
|
[4] Huang, G.Y., Ma, B.Q.: Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds.Arch. Math. (Basel) 94 (2010), 265–275. Zbl 1194.58020, MR 2602453, 10.1007/s00013-009-0091-7 |
| Reference:
|
[5] Li, P., Yau, S.T.: On the parabolic kernel of the Schrödinger operator.Acta Math. 156 (1986), 152–201. Zbl 0611.58045, MR 0834612 |
| Reference:
|
[6] Li, Y.: Li-Yau-Hamilton estimates and Bakry-Emery Ricci curvature.Nonlinear Anal. 113 (2015), 1–32. Zbl 1310.58015, MR 3281843 |
| Reference:
|
[7] Negrin, E.R.: Gradient estimates and a Liouville type theorem for the Schrödinger operator.J. Funct. Anal. 127 (1995), 198–203. Zbl 0842.58078, MR 1308622, 10.1006/jfan.1995.1008 |
| . |
