The global solution of the minimal surface flow and translating surfaces

In this paper, we study evolved surfaces over convex planar domains which are evolving by the minimal surface flow $$u_t=div\left(\frac{Du}{\sqrt{1+\mathrm{\mid }Du{\mathrm{\mid }}^2}}\right)-H(x,Du)\mathrm{.}$$ 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.