Gradient estimates for some evolution equations on complete smooth metric measure spaces
PUBLICATIONES MATHEMATICAE-DEBRECEN(2020)
Abstract
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.
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Key words
gradient estimates,Bakry-Emery curvature,complete smooth metric measure space,Harnack-type inequalities,Liouville-type theorems
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