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

Abstract

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-Emery 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 λ, α 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.