Hamilton-Souplet-Zhang's gradient estimates for two weighted nonlinear parabolic equations

In this paper, we consider gradient estimates for positive solutions to the following weighted nonlinear parabolic equations on a complete smooth metric measure space with only Bakry-Aemery Ricci tensor bounded below: One is u(t) = Delta(f)u + au log u + bu with a, b two real constants, and another is u(t) = Delta(f)u + lambda u(alpha) with lambda, alpha two real constants. We obtain local Hamilton-Souplet-Zhang type gradient estimates for the above two nonlinear parabolic equations. In particular, our estimates do not depend on any assumption on f.