Gradient estimates for a simple nonlinear heat equation on manifolds

In this paper, we study the gradient estimate for positive solutions to the following nonlinear heat equation problem $$u_t-\mathrm{\Delta }u=au\log u+Vu,u>0$$ on the compact Riemannian manifold $(M,g)$ of dimension $n$ and with non-negative Ricci curvature. Here $a\le 0$ is a constant, $V$ is a smooth function on $M$ with $-\mathrm{\Delta }V\le A$ 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.