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

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