  We consider the evolution of hypersurfaces in $\mathbb{R}^{n+1}$ with normal
velocity given by a positive power of the mean curvature. The hypersurfaces
under consideration are assumed to be strictly mean convex (positive mean
curvature), complete, and given as the graph of a function. Long-time existence
of the $H^{\alpha}$-flow is established by means of approximation by bounded
problems.
