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Abstract
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In this paper, we establish a local elliptic gradient estimate for
positive bounded solutions to a parabolic equation concerning the
-Laplacian
on an
-dimensional
complete Riemannian manifold with the Bakry–Émery Ricci curvature
bounded below, which is
weaker than the
-Bakry–Émery
Ricci curvature
bounded below considered by Chen and Zhao (2018). As applications,
we obtain the local elliptic gradient estimates for the cases that
and
.
Moreover, we prove parabolic Liouville theorems for the solutions satisfying
some growth restriction near infinity and study the problem about
conformal deformation of the scalar curvature. In the end, we also derive
a global Bernstein-type gradient estimate for the above equation with
.
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Keywords
gradient estimate, Liouville theorem, $V$-Laplacian,
Bakry–Émery Ricci curvature, parabolic equation
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Mathematical Subject Classification 2010
Primary: 58J35
Secondary: 35B53, 35K05
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Milestones
Received: 12 October 2018
Revised: 12 December 2019
Accepted: 10 July 2020
Published: 14 January 2021
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