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 (Math Ann 295:761–765, 1993) proved that \(\Gamma _t\) converges to a 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 \(C^{2,1}_{loc} ((-1,1)\times \mathbb {R})\) topology to the grim reaper with span \((-1,1)\).

我们考虑带域\(\Omega :=[-1,1]\times \mathbb {R}\) 中的曲率流\(V=H\ )，其中，对于图形曲线\(\Gamma _t\ ) , V表示法向速度，H表示曲率。如果\（\伽玛_t \）接触的两个边界\（\部分_ \ PM \欧米茄\）的\（\欧米茄\）具有恒定的斜坡，在1993年，Altschular和吴（数学安295：761-765，1993年) 证明\(\Gamma _t\)收敛到一个死神以相同的规定斜率接触\(\partial _\pm \Omega \)。在本文中，我们考虑\(\Gamma _t\)接触\(\partial _\pm \Omega \)的斜率等于其高度的\(\pm 1\)倍。当曲线移动到无穷大时，由于边界斜率无界，全局梯度估计是不可能的。我们首先考虑一个特殊的对称曲线，并通过使用零数参数推导出它的均匀内部梯度估计，然后使用这些估计来给出一般非对称曲线的均匀内部梯度估计，这导致曲线在\( C 1-4 {2,1} _ {LOC}（（-1,1）\倍\ mathbb {R}）\）拓扑到死神与跨度\（（ - 1,1）\） 。