  In this paper, we study the gradient estimate for positive solutions to the
following nonlinear heat equation problem $$ u_t-\Delta u=au\log u+Vu, \ \ u>0
$$ on the compact Riemannian manifold $(M,g)$ of dimension $n$ and with
non-negative Ricci curvature. Here $a\leq 0$ is a constant, $V$ is a smooth
function on $M$ with $-\Delta V\leq A$ for some positive constant $A$. This
heat equation is a basic evolution equation and it can be considered as the
negative gradient heat flow to $W$-functional (introduced by G.Perelman), which
is the Log-Sobolev inequalities on the Riemannian manifold and $V$ corresponds
to the scalar curvature.
