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In this paper, we prove sharp gradient estimates for positive solutions tothe weighted heat equation on smooth metric measure spaces with compactboundary. As an application, we prove Liouville theorems for ancient solutionssatisfying the Dirichlet boundary condition and some sharp growth restrictionnear infinity. Our results can be regarded as a refinement of recent resultsdue to Kunikawa and Sakurai.

Sharp gradient estimates on weighted manifolds with compact boundary