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

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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).

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Authors and Affiliations

  1. Department of Mathematics, Mechanics and Informatics (MIM), Hanoi University of Sciences (HUS-VNU), No. 334, Nguyen Trai Road, Thanh Xuan, Hanoi, Vietnam

    Nguyen Thac Dung & Nguyen Ngoc Khanh

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  1. Nguyen Thac Dung
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  2. Nguyen Ngoc Khanh
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Correspondence to Nguyen Thac Dung.

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Dung, N.T., Khanh, N.N. Gradient estimates of Hamilton–Souplet–Zhang type for a general heat equation on Riemannian manifolds. Arch. Math. 105, 479–490 (2015). https://doi.org/10.1007/s00013-015-0828-4

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  • DOI: https://doi.org/10.1007/s00013-015-0828-4

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