Smooth convergence to the enveloping cylinder for mean curvature flow of complete graphical hypersurfaces

For a mean curvature flow of complete graphical hypersurfaces $M_t=\mathrm{graph}u(\cdot ,t)$ defined over domains ${\Omega }_t$, the enveloping cylinder is $\partial {\Omega }_t\times \mathbb{R}$. We prove the smooth convergence of $M_t-h{e}_{n+1}$ to the enveloping cylinder under certain circumstances. Moreover, we give examples demonstrating that there is no uniform curvature bound in terms of the inital curvature and the geometry of ${\Omega }_t$. Furthermore, we provide an example where the hypersurface increasingly oscillates towards infinity in both space and time. It has unbounded curvature at all times and is not smoothly asymptotic to the enveloping cylinder. We also prove a relation between the initial spatial asymptotics at the boundary and the temporal asymptotics of how the surface vanishes to infinity for certain rates in the case ${\Omega }_t$ are balls.