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

TL;DR

This paper establishes decay-error estimates for solutions of degenerate Kirchhoff equations with a small parameter, showing the difference between hyperbolic and parabolic solutions diminishes over time and as the parameter approaches zero.

Contribution

It provides the first decay-error estimates for hyperbolic-parabolic singular perturbations in degenerate Kirchhoff equations, revealing faster decay of the difference than individual solutions.

Findings

The difference between hyperbolic and parabolic solutions tends to zero as epsilon approaches zero.

The difference decays faster than the individual solutions as time progresses.

Decay-error estimates hold uniformly over time, showing the convergence of solutions.

Abstract

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.