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