Translating surfaces of the non-parametric mean curvature flow in Lorentz manifold $M^2\times \mathbb{R}$

In this paper, for the Lorentz manifold $M^2\times \mathbb{R}$, with $M^2$ a $2$-dimensional complete surface with nonnegative Gaussian curvature, we investigate its space-like graphs over compact strictly convex domains in $M^2$, which are evolving by the non-parametric mean curvature flow with prescribed contact angle boundary condition, and show that solutions converge to ones moving only by translation.