  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
$(-1,1)$.
