Gradient estimates for some evolution equations on complete smooth metric measure spaces

In this paper, we consider the following general evolution equation $$ u_t=\Delta_fu+au\log^\alpha u+bu $$ on a smooth metric measure space $(M^n,g,e^{-f}dv)$. We give a local gradient estimate of Souplet—Zhang type for positive smooth solutions of this equation provided that the Bakry—Émery curvature is bounded from below. When $f$ is constant, we investigate the general evolution equation on compact Riemannian manifolds with nonconvex boundary satisfying an interior rolling $R$-ball condition. We show a gradient estimate of Hamilton type on such manifolds.