  We give asymptotics for the level set equation for mean curvature flow on a
convex domain near the point where it attains a maximum. It is known that
solutions are not necessarily $C^3,$ and we recover this result and construct
non-smooth solutions which are $C^3.$ We also construct solutions having
prescribed behavior near the maximum. We do this by analyzing the asymptotics
for rescaled mean curvature flow converging to a stationary sphere.
