  In this paper, we study the gradient estimates of Li-Yau-Hamilton type for
positive solutions to both drifting heat equation and the simple nonlinear heat
equation problem $$ u_t-\Delta u=au\log u, \ \ u>0 $$ on the compact Riemannian
manifold $(M,g)$ of dimension $n$ and with non-negative (Bakry-Emery)-Ricci
curvature. Here $a\leq 0$ is a constant. The latter heat equation is a basic
evolution equation which is the negative gradient heat flow to the functional
of Log-Sobolev inequality on the Riemannian manifold. We derive various
versions of gradient estimates which generalize Hamilton's gradient estimate.
An question concerning the Hamilton type gradient estimate for the simple
nonlinear heat equation is addressed.
