On a curvature flow in a band domain with unbounded boundary slopes

Lixia Yuan; Wei Zhao

doi:10.3934/dcds.2021115

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2022, Volume 42, Issue 1: 261-283. Doi: 10.3934/dcds.2021115

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On a curvature flow in a band domain with unbounded boundary slopes

Lixia Yuan^1, and
Wei Zhao^2,

1.
Mathematics and Science College, Shanghai Normal University, Shanghai, 200234, China
2.
School of Mathematics, East China University of Science and Technology, Shanghai, 200237, China

^* Corresponding author: Wei Zhao
^* Corresponding author: Wei Zhao

Received: March 2021

Revised: June 2021

Early access: August 2021

Published: January 2022

The first author is supported by by NNSFC (No. 12001375). The second author is supported NNSFC (No. 11761058) and NSFS (No. 19ZR1411700, No. 21ZR1418300)

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Abstract

This paper is devoted to an anisotropic curvature flow of the form $ V = A(\mathbf{n})H + B(\mathbf{n}) $ in a band domain $ \Omega : = [-1,1]\times {\mathbb{R}} $, where $ \mathbf{n} $, $ V $ and $ H $ denote respectively the unit normal vector, normal velocity and curvature of a graphic curve $ \Gamma_t $. We require that the curve $ \Gamma_t $ contacts $ \partial \Omega $ with slopes equaling to the heights of the contact points (which corresponds to a kind of Robin boundary conditions). In spite of the unboundedness of the boundary slopes, we are able to obtain the uniform interior gradient estimates for the solutions by using the zero number argument. Furthermore, when $ t\to \infty $, we show that $ \Gamma_t $ converges to a traveling wave with cup-shaped profile and infinite boundary slopes in the $ C^{2,1}_{\rm{loc}} ((-1,1)\times {\mathbb{R}}) $-topology.

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Mathematics Subject Classification: Primary: 35K93, 35C07; Secondary: 53E10.

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