Gradient estimates of Hamilton–Souplet–Zhang type for a general heat equation on Riemannian manifolds
- 210 Downloads
The purpose of this paper is to study gradient estimates of Hamilton–Souplet–Zhang type for the following general heat equation
on noncompact Riemannian manifolds. As its application, we show a Harnack inequality for the positive solution and a Liouville type theorem for a nonlinear elliptic equation. Our results are an extension and improvement of the work of Souplet and Zhang (Bull London Math Soc 38:1045–1053, 2006), Ruan (Bull London Math Soc 39:982–988, 2007), Li (Nonlinear Anal 113:1–32, 2015), Huang and Ma (Gradient estimates and Liouville type theorems for a nonlinear elliptic equation, Preprint, 2015), and Wu (Math Zeits 280:451–468, 2015).
$$u_t=\Delta_V u + au\log u+bu$$
KeywordsGradient estimates General heat equation Laplacian comparison theorem V-Bochner–Weitzenböck Bakry–Emery Ricci curvature
Mathematics Subject ClassificationPrimary 58J35 Secondary 35B53 35K0
Unable to display preview. Download preview PDF.
- 2.E. Calabi, An extension of E.Hopf’s maximum principle with an application to Riemannian geometry, Duke Math. Jour., 25 (157), 45–56Google Scholar
- 4.N. T. Dung, Hamilton type gradient estimate for a nonlinear diffusion equation on smooth metric measure spaces, ManuscriptGoogle Scholar
- 6.G. Y. Huang and B. Q. Ma, Gradient estimates and Liouville type theorems for a nonlinear elliptic equation (2015, preprint). arXiv:1505.01897v1
© Springer Basel 2015