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

<p style='text-indent:20px;'>This paper is devoted to an anisotropic curvature flow of the form <inline-formula><tex-math id="M1">\begin{document}$ V = A(\mathbf{n})H + B(\mathbf{n}) $\end{document}</tex-math></inline-formula> in a band domain id="M2">\begin{document}$ \Omega : [-1,1]\times {\mathbb{R}} $\end{document}</tex-math></inline-formula>, where id="M3">\begin{document}$ \mathbf{n} id="M4">\begin{document}$ and id="M5">\begin{document}$ H denote respectively unit normal vector, velocity graphic curve id="M6">\begin{document}$ \Gamma_t $\end{document}</tex-math></inline-formula>. We require that id="M7">\begin{document}$ contacts id="M8">\begin{document}$ \partial with slopes equaling heights contact points (which corresponds kind Robin boundary conditions). In spite unboundedness slopes, we are able obtain <i>uniform interior gradient estimates</i> for solutions by using zero number argument. Furthermore, when id="M9">\begin{document}$ t\to \infty show id="M10">\begin{document}$ converges traveling wave cup-shaped profile <i>infinite</i> id="M11">\begin{document}$ C^{2,1}_{\rm{loc}} ((-1,1)\times {\mathbb{R}}) $\end{document}</tex-math></inline-formula>-topology.</p>