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

The purpose of this paper is to study gradient estimate of Hamilton - Souplet

• Zhang type for the general heat equation $$u_t={\mathrm{\Delta }}_{V}u+au\log u+bu$$ on noncompact Riemannian manifolds. As its application, we show a Harnak inequality for the heat solution and a Liouville type theorem for a nonlinear elliptic equation. Our results are an extention and improvement of the work of Souplet - Zhang (\cite{SZ}), Ruan (\cite{Ruan}), Yi Li (\cite{Yili}) and Huang-Ma (\cite{HM}).