  We consider a family of Kirchhoff equations with a small parameter epsilon in
front of the second-order time-derivative, and a dissipation term with a
coefficient which tends to 0 as t -> +infinity.
  It is well-known that, when the decay of the coefficient is slow enough,
solutions behave as solutions of the corresponding parabolic equation, and in
particular they decay to 0 as t -> +infinity.
  In this paper we consider the nondegenerate and coercive case, and we prove
optimal decay estimates for the hyperbolic problem, and optimal decay-error
estimates for the difference between solutions of the hyperbolic and the
parabolic problem. These estimates show a quite surprising fact: in the
coercive case the analogy between parabolic equations and dissipative
hyperbolic equations is weaker than in the noncoercive case.
  This is actually a result for the corresponding linear equations with
time-dependent coefficients. The nonlinear term comes into play only in the
last step of the proof.
