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

We consider a curvature flow $V=H$ in the band domain $Omega :=[-1,1]times R$, 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 cite{AW1} 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 $C^{2,1}_{loc} ((-1,1)times R)$ topology to the {it grim reaper} with span $(-1,1)$.