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

TL;DR

This paper derives gradient estimates for a class of evolution equations on complete smooth metric measure spaces, extending known results to more general settings with curvature bounds and specific geometric conditions.

Contribution

It provides new local and global gradient estimates for evolution equations on metric measure spaces, including cases with Bakry-Émery curvature bounds and interior rolling R-ball conditions.

Findings

Established Souplet-Zhang type gradient estimates under curvature bounds.

Derived Hamilton type gradient estimates on compact manifolds with interior R-ball condition.

Extended gradient estimate techniques to more general evolution equations.

Abstract

In this paper, we consider the following general evolution equation $$ u_t=\Delta_fu+au\log^\alpha u+bu $$ 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-\'{E}mery 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.