Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations: decay-error estimates

We consider degenerate Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative. It is well known that these equations admit global solutions when epsilon is small enough, and that these solutions decay as t -> +infinity with the same rate of solutions of the limit problem (of parabolic type). In this paper we prove decay-error estimates for the difference between a solution of the hyperbolic problem and the solution of the corresponding parabolic problem. These estimates show in the same time that the difference tends to zero both as epsilon -> 0, and as t -> +infinity. Concerning the decay rates, it turns out that the difference decays faster than the two terms separately (as t -> +infinity). Proofs involve a nonlinear step where we separate Fourier components with respect to the lowest frequency, followed by a linear step where we exploit weighted versions of classical energies.