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