Large-time behavior of one-phase Stefan-type problems with anisotropic diffusion in periodic media

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.