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

Abstract

We consider degenerate Kirchhoff equations with a small parameter ε in front of the second-order time-derivative. It is well known that these equations admit global solutions when ε is small enough, and that these solutions decay as t→+∞ 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 ε→0+, and as t→+∞. Concerning the decay rates, it turns out that the difference decays faster than the two terms separately (as t→+∞).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.