Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

Let M be a complete noncompact Riemannian manifold. We consider gradient estimates for the positive solutions to the following non-linear parabolic equation partial derivative u/partial derivative t = Delta(u)(f) + au log u + bu on M x [0, + infinity), where a, b are two real constants, f is a smooth real-valued function on M and Delta(f) = Delta - Delta f Delta. Under the assumption that the N-Bakry-Emery Ricci tensor is bounded from below by a negative constant, we obtain a gradient estimate for positive solutions of the above equation. As an application, we obtain a Harnack inequality and a Gaussian lower bound of the heat kernel of such an equation.