  We consider the hyperbolic-parabolic singular perturbation problem for a
nondegenerate quasilinear equation of Kirchhoff type with weak dissipation.
This means that the dissipative term is multiplied by a coefficient b(t) which
tends to 0 as t tends to +infinity.
  The case where b(t) behaves like (1+t)^{-p} with p<1 has recently been
considered. The result is that the hyperbolic problem has a unique global
solution, and the difference between solutions of the hyperbolic problem and
the corresponding solutions of the parabolic problem converges to zero both as
t tends to +infinity and as epsilon goes to 0.
  In this paper we show that these results cannot be true for p>1, but they
remain true in the critical case p=1.
