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We consider a degenerate hyperbolic equation of Kirchhoff type with a smallparameter epsilon in front of the second-order time-derivative.  In a recent paper, under a suitable assumption on initial data, we proveddecay-error estimates for the difference between solutions of the hyperbolicproblem and the corresponding solutions of the limit parabolic problem. Theseestimates show in the same time that the difference tends to zero both asepsilon -> 0, and as t -> +infinity. In particular, in that case the differencedecays faster than the two terms separately.  In this paper we consider the complementary assumption on initial data, andwe show that now the optimal decay-error estimates involve a decay rate whichis slower than the decay rate of the two terms.  In both cases, the improvement or deterioration of decay rates depends on thesmallest frequency represented in the Fourier components of initial data.

Optimal decay-error estimates for the hyperbolic-parabolic singular perturbation of a degenerate nonlinear equation