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 \$\$u\_t={\backslash}Delta\_V u + au{\backslash}log u+bu\$\$ u t = $Δ$ V u + a u log u + b u 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).