In this talk I will present some recent results concerning the
asymptotic self-similar patterns of degenerate diffusion in an infinite
porous medium with vanishing at infinity variable density.

The asymptotic pattern turns out to strongly depend on the decay rate of
the density. For "slowly" decaying densities, the picture is similar to
the homogeneous case (Barenblatt-type solutions), whereas for densities,
decaying fast enough, a completely different behavior, typical of problems
in bounded domains, arises.

For intermediate decay rates, both descriptions are correct, providing an
example of matched asymptotics.