  We study the large-time behavior of solutions of a one-phase Stefan-type
problem with anisotropic diffusion in periodic media on an exterior domain in a
dimension $n \geq 3$. By a rescaling transformation that matches the expansion
of the free boundary, we deduce the homogenization of the free boundary
velocity together with the homogenization of the anisotropic operator.
Moreover, we obtain the convergence of the rescaled solution to the solution of
the homogenized Hele-Shaw-type problem with a point source and the convergence
of the rescaled free boundary to a self-similar profile with respect to the
Hausdorff distance.
