Mildly degenerate Kirchhoff equations with weak dissipation: global existence and time decay

TL;DR

This paper investigates a degenerate Kirchhoff equation with weak dissipation, proving global existence of solutions and their decay over time, aligning with the behavior of the associated parabolic limit problem.

Contribution

It establishes the global existence and decay rates of solutions for a degenerate Kirchhoff equation with weak dissipation, a novel analysis in this context.

Findings

Unique global solutions exist for suitable parameters.

Solutions decay to zero at the same rate as the parabolic limit.

Decay behavior matches that of the associated parabolic problem.

Abstract

We consider the hyperbolic-parabolic singular perturbation problem for a degenerate quasilinear Kirchhoff equation with weak dissipation. This means that the coefficient of the dissipative term tends to zero when t tends to +infinity. We prove that the hyperbolic problem has a unique global solution for suitable values of the parameters. We also prove that the solution decays to zero, as t tends to +infinity, with the same rate of the solution of the limit problem of parabolic type.