Gradient Estimates for a Nonlinear Diffusion Equation on Complete Manifolds

This paper deals with the gradient estimates of the Hamilton type for the positive solutions to the following nonlinear diffusion equation: \begin{align} u_t=\triangle u + \nabla\phi \cdot \nabla u + a(x)u\ln u + b(x)u \nonumber \end{align} on a complete noncompact Riemannian manifold with a Bakry-Emery {\rm Ric}ci curvature bounded below by $-K$ ($K \ge 0$), where $\phi$ is a $C^2$ function, $a(x)$ and $b(x)$ are $C^1$ functions with certain conditions.