Gradient Estimates and Harnack Inequality for a Nonlinear Parabolic Equation on Complete Manifolds

Let M be a noncompact complete Riemannian manifold.In this paper,we consider the following nonlinear parabolic equation on M ut(x,t)=△u(x,t)+au(x,t)ln u(x,t)+bu^α(x,t).We prove a Li–Yau type gradient estimate for positive solutions to the above equation;as an application,we also derive the corresponding Harnack inequality.These results generalize the corresponding ones proved by Li(J Funct Anal 100:233–256,1991).