  In this paper, we study evolved surfaces over convex planar domains which are
evolving by the minimal surface flow $$u_{t}=
div\left(\frac{Du}{\sqrt{1+|Du|^2}}\right)-H(x,Du).$$ Here, we specify the
angle of contact of the evolved surface to the boundary cylinder. The
interesting question is to find translating solitons of the form $u(x,t)=\omega
t+w(x)$ where $\omega\in \mathbb R$. Under an angle condition, we can prove the
a priori estimate holds true for the translating solitons (i.e., translator),
which makes the solitons exist. We can prove for suitable condition on $H(x,p)$
that there is the global solution of the minimal surface flow. Then we show,
provided the soliton exists, that the global solutions converge to some
translator.
