Hyperbolic--parabolic singular perturbation for nondegenerate Kirchhoff equations with critical weak dissipation

TL;DR

This paper investigates the long-term behavior of solutions to a Kirchhoff-type equation with weak dissipation, establishing conditions under which hyperbolic and parabolic solutions converge, especially in the critical dissipation case.

Contribution

It extends previous results by analyzing the critical case p=1, showing convergence properties for the hyperbolic-parabolic singular perturbation problem.

Findings

Unique global solutions for p<1 and p=1 cases.

Convergence of hyperbolic and parabolic solutions as t→∞ and ε→0.

Different behavior for p>1 where previous results do not hold.

Abstract

We consider the hyperbolic-parabolic singular perturbation problem for a nondegenerate quasilinear equation of Kirchhoff type with weak dissipation. This means that the dissipative term is multiplied by a coefficient b(t) which tends to 0 as t tends to +infinity. The case where b(t) behaves like (1+t)^{-p} with p<1 has recently been considered. The result is that the hyperbolic problem has a unique global solution, and the difference between solutions of the hyperbolic problem and the corresponding solutions of the parabolic problem converges to zero both as t tends to +infinity and as epsilon goes to 0. In this paper we show that these results cannot be true for p>1, but they remain true in the critical case p=1.