Convergence to the Grim Reaper for a Curvature Flow with Unbounded Boundary Slopes

We consider a curvature flow $V=H$ in the band domain $\Omega :=[-1,1]\times \mathbb{R}$, where, for a graphic curve ${\Gamma }_t$, $V$ denotes its normal velocity and $H$ denotes its curvature. If ${\Gamma }_t$ contacts the two boundaries ${\partial }_{\pm }\Omega$ of $\Omega$ with constant slopes, in 1993, Altschular and Wu proved that ${\Gamma }_t$ converges to a {\it grim reaper} contacting ${\partial }_{\pm }\Omega$ with the same prescribed slopes. In this paper we consider the case where ${\Gamma }_t$ contacts ${\partial }_{\pm }\Omega$ with slopes equaling to $\pm 1$ times of its height. When the curve moves to infinity, the global gradient estimate is impossible due to the unbounded boundary slopes. We first consider a special symmetric curve and derive its uniform interior gradient estimates by using the zero number argument, and then use these estimates to present uniform interior gradient estimates for general non-symmetric curves, which lead to the convergence of the curve in $C_{loc}^{2,1}((-1,1)\times \mathbb{R})$ topology to the {\it grim reaper} with span $(-1,1)$.