  We consider Kirchhoff equations with a small parameter epsilon in front of
the second-order time-derivative, and a dissipative term whose coefficient may
tend to 0 as t -> + infinity (weak dissipation).
  In this note we present some recent results concerning existence of global
solutions, and their asymptotic behavior both as t -> + infinity and as epsilon
-> 0. Since the limit equation is of parabolic type, this is usually referred
to as a hyperbolic-parabolic singular perturbation problem.
  We show in particular that the equation exhibits hyperbolic or parabolic
behavior depending on the values of the parameters.
