Asymptotic convergence of the Stefan problem to Hele-Shaw

We discuss the asymptotic behaviour of weak solutions to the Hele-Shaw and one-phase Stefan problems in exterior domains. We prove that, if the space dimension is greater than one, the asymptotic behaviour is given in both cases by the solution of the Dirichlet exterior problem for the Laplacian in the interior of the positivity set and by a singular, radial and self-similar solution of the Hele-Shaw ow near the free boundary. We also show that the free boundary approaches a sphere as t --> infinity, and give the precise asymptotic growth rate for the radius.