  We study mean convex mean curvature flow $M_s$ of local spacelike graphs in
the flat slicing of de Sitter space. We show that if the initial slice is of
non-negative time and is graphical over a large enough ball, and if $M_s$ is of
bounded mean curvature, then as $s$ goes to infinity, $M_s$ becomes graphical
in expanding balls, over which the gradient function converges to $1$. In
particular, if $p_s$ is the point lying over the center of the domain ball in
$M_s$, then $(M_s,p_s)$ converges smoothly to the flat slicing of de Sitter
space. This has some relation to the mean curvature flow approach to the cosmic
no hair conjecture.
