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

Abstract

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 $$ \partial _t u = \Delta u + au\log u + qu $$ on M × (0,∞), where a is a constant and q is a C2 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)).