Hamilton type gradient estimates for a general type of nonlinear parabolic equations on Riemannian manifolds

Fanqi Zeng; Fanqi Zeng

doi:10.3934/math.2021610

AIMS Mathematics

2021, Volume 6, Issue 10: 10506-10522. doi: 10.3934/math.2021610

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Research article

Hamilton type gradient estimates for a general type of nonlinear parabolic equations on Riemannian manifolds

Fanqi Zeng ^,

School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, China

Received: 19 February 2021 Accepted: 14 July 2021 Published: 20 July 2021
MSC : 35B09, 35B45, 35R01, 53C44

In this paper, we prove Hamilton type gradient estimates for positive solutions to a general type of nonlinear parabolic equation concerning $ V $-Laplacian:

$ (\Delta_{V}-q(x, t)-\partial_{t})u(x, t) = A(u(x, t)) $

on complete Riemannian manifold (with fixed metric). When $ V = 0 $ and the metric evolves under the geometric flow, we also derive some Hamilton type gradient estimates. Finally, as applications, we obtain some Liouville type theorems of some specific parabolic equations.
- parabolic equation,
- gradient estimate,
- Liouville type theorem,
- geometric flow,
- curvature
Citation: Fanqi Zeng. Hamilton type gradient estimates for a general type of nonlinear parabolic equations on Riemannian manifolds[J]. AIMS Mathematics, 2021, 6(10): 10506-10522. doi: 10.3934/math.2021610

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Abstract

In this paper, we prove Hamilton type gradient estimates for positive solutions to a general type of nonlinear parabolic equation concerning $ V $-Laplacian:

$ (\Delta_{V}-q(x, t)-\partial_{t})u(x, t) = A(u(x, t)) $

on complete Riemannian manifold (with fixed metric). When $ V = 0 $ and the metric evolves under the geometric flow, we also derive some Hamilton type gradient estimates. Finally, as applications, we obtain some Liouville type theorems of some specific parabolic equations.

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