Abstract
Abstract This paper derives elliptic gradient estimates for positive solutions to a nonlinear parabolic equation defined on a complete weighted Riemannian manifold. Applications of these estimates yield Liouville-type theorem, parabolic Harnack inequalities and bounds on weighted heat kernel on the lower boundedness assumption for Bakry-Émery curvature tensor.
Highlights
We shall consider positive solutions to the nonlinear parabolic equation: Δf − ∂ u(x, t) ∂t +p(x, t)uβ(x, t) q(x, t ) u(x, t) = (1.1)on a weighted Riemannian manifold (MN, g, e−f dv), otherwise known as a smooth metric measure space
The m-weighted Ricci tensor otherwise known as the Bakry-Émery tensor in the literature is defined for some constant m > 0 as 1 ∇f m
Elliptic gradient estimates for Eq (1.1) can be applied to get information that will be useful in solving the Yamabe problem on weighted manifolds [30]
Summary
On a weighted Riemannian manifold (MN, g, e−f dv), otherwise known as a smooth metric measure space. Gradient estimates for both elliptic and parabolic equations have become fundamental tools in geometric analysis In their celebrated work [1], Li and Yau established parabolic gradient estimates on solutions to the linear heat equation on Riemannian manifolds having Ricci curvature bounded from below. They applied their results to get Harnack inequalities and various estimates on the heat kernel. The technique of gradient estimates emanated from the study by Yau [13] (see [14,15]), in which a gradient estimate for harmonic functions was first established using the maximum principle This estimate was applied to obtain a Liouville theorem. It is in order to give background information about this space
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