  We consider a mean curvature flow in a cone, that is, a hypersurface in a
cone which moves toward the opening with normal velocity equaling to the mean
curvature, and the contact angle between the hypersurface and the cone boundary
being $\varepsilon$-periodic in its position. First, by constructing a family
of self-similar solutions, we give a priori estimates for the radially
symmetric solutions and prove the global existence. Then we consider the
homogenization limit as $\ve\to 0$, and use {\it the slowest self-similar
solution} to characterize the solution, with error $O(1)\ve^{1/6}$, in some
finite time interval.
