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In this paper, we generalize the well-known Kelvin–Nevanlinna–Royden criterion to check the p-hyperbolicity of the class of weighted manifolds for any p ∈ (1,∞). Besides, we introduce a new definition of a weighted cohomology and use it to derive a cohomological criterion for the weighted p-parabolicity. As a result, we acquire a global comparison principle for the weighted p-Laplacian and also receive a Liouville-type result for weighted p-harmonic functions. Moreover, we give a decay estimate for the weighted p-Laplacian on smooth metric measure spaces. Some applications to study the weighted p-parabolicity and f-minimal hypersurfaces are also given. [ABSTRACT FROM AUTHOR]

In this paper, we prove sharp gradient estimates for positive solutions to the weighted heat equation on smooth metric measure spaces with compact boundary. As an application, we prove Liouville theorems for ancient solutions satisfying the Dirichlet boundary condition and some sharp growth restriction near infinity. Our results can be regarded as a refinement of recent results due to Kunikawa and Sakurai. [ABSTRACT FROM AUTHOR]

In this paper, we establish gradient estimates for the positive bounded solutions to a general type of nonlinear parabolic equation concerning the weighted Laplacian ∂ ∂ t − a (x , t) − Δ f u (x , t) = F (u (x , t)) on a smooth metric measure space with the metric evolving under the (k , ∞) -super Perelman–Ricci flow and the Yamabe flow. Applications of our results include Liouville type results and gradient estimates for some important geometric partial differential equations such as the equations involving gradient Ricci solitons and the Einstein-scalar field Lichnerowicz type equations. Our results generalize and improve many previous works. [ABSTRACT FROM AUTHOR]