Article

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

08/2011;
Source: arXiv

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.

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### Keywords

classical energies

decay-error estimates

degenerate Kirchhoff equations

difference decays

epsilon -> 0

equations

estimates

global solutions

hyperbolic problem

linear step

lowest frequency

nonlinear step

parabolic type

second-order time-derivative

small parameter epsilon

solutions

solutions decay

t -> +infinity

two terms

weighted versions