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(f)u + aulog(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-Emery 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.