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 problemon the compact Riemannian manifold (M, g) of dimension n and with non-negative Ricci curvature. Here is a constant, V is a smooth function on M with 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.