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

TL;DR

This paper investigates decay-error estimates for a degenerate hyperbolic equation with a small parameter, showing how initial data assumptions influence the decay rates of solution differences between hyperbolic and parabolic limits.

Contribution

It extends previous work by analyzing the case with different initial data assumptions, revealing how initial conditions affect decay-error rates in hyperbolic-parabolic singular perturbations.

Findings

Optimal decay-error estimates depend on initial data frequency content.

Different initial data assumptions lead to faster or slower decay rates.

The decay rate of solution differences is influenced by the smallest Fourier frequency.

Abstract

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