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 graphic curve $$\Gamma _t$$ V denotes its normal velocity and H curvature. If contacts two boundaries $$\partial _\pm \Omega $$ of with constant slopes, 1993, Altschular Wu (Math Ann 295:761–765, 1993) proved that converges to grim reaper contacting same prescribed slopes. In this paper we case where slopes equaling $$\pm 1$$ times height. When moves infinity, global gradient estimate is impossible due unbounded boundary first special symmetric derive uniform interior estimates by using zero number argument, then use these present general non-symmetric curves, which lead convergence $$C^{2,1}_{loc} ((-1,1)\times {R})$$ topology span $$(-1,1)$$ .