Gradient Estimates for a Nonlinear Heat Equation on Compact Riemannian Manifold

In this paper, we study gradient estimates for the nonlinear heat equation ut−△u = au log u, on compact Riemannian manifold with or without boundary. We get a Hamilton type gradient estimate for the positive smooth solution to the equation on close manifold, and obtain a Li-Yau type gradient estimate for the positive smooth solution to the equation on compact manifold with nonconvex boundary.