  For a mean curvature flow of complete graphical hypersurfaces
$M_{t}=\operatorname{graph} u(\cdot,t)$ 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.

Differential Geometry