Hamilton-type gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

Let (M,g) be a complete noncompact Riemannian manifold. In this note, we derive a local Hamilton-type gradient estimate for positive solution to a simple nonlinear parabolic equation partial derivative tu - Delta u + au log u + qu on M x (0,a), where a is a constant and q is a C (2) function. This result can be compared with the ones of Ma (JFA, 241, 374-382 (2006)) and Yang (PAMS, 136, 4095-4102 (2008)). Also, we obtain Hamilton's gradient estimate for the Schodinger equation. This can be compared with the result of Ruan (JGP, 58, 962-966 (2008)).