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Archiv der Mathematik

, Volume 105, Issue 5, pp 479–490 | Cite as

Gradient estimates of Hamilton–Souplet–Zhang type for a general heat equation on Riemannian manifolds

  • Nguyen Thac Dung
  • Nguyen Ngoc Khanh
Article

Abstract

The purpose of this paper is to study gradient estimates of Hamilton–Souplet–Zhang type for the following general heat equation
$$u_t=\Delta_V u + au\log u+bu$$
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).

Keywords

Gradient estimates General heat equation Laplacian comparison theorem V-Bochner–Weitzenböck Bakry–Emery Ricci curvature 

Mathematics Subject Classification

Primary 58J35 Secondary 35B53 35K0 

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Mechanics and Informatics (MIM)Hanoi University of Sciences (HUS-VNU)Thanh XuanVietnam

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