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

Abstract

In this paper, we consider the following general evolution equation $$u_t={\Delta }_{f}u+au{\log }^{\alpha }u+bu$$ on smooth metric measure spaces $(M^n,g,e^{-f}dv)$. We give a local gradient estimate of Souplet-Zhang type for positive smooth solution of this equation provided that the Bakry-Émery curvature bounded from below. When $f$ is constant, we investigate the gereral evolution on compact Riemannian manifolds with no nconvex boundary satisfying an "\emph{interior rolling $R$-ball}" condition. We show a gradient estimate of Hamilton type on such manifolds.