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 ca
ses 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-simil
ar solution of the Hele-Shaw ow near the free boundary. We also show that t
he free boundary approaches a sphere as t --> infinity, and give the precis
e asymptotic growth rate for the radius.