# American Institute of Mathematical Sciences

January  2022, 42(1): 261-283. doi: 10.3934/dcds.2021115

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

 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

Received  March 2021 Revised  June 2021 Published  January 2022 Early access  August 2021

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

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.

Citation: Lixia Yuan, Wei Zhao. On a curvature flow in a band domain with unbounded boundary slopes. Discrete & Continuous Dynamical Systems, 2022, 42 (1) : 261-283. doi: 10.3934/dcds.2021115
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