Asymptotic limits for mildly degenerate Kirchhoff equations

TL;DR

This paper investigates the long-term behavior of solutions to a mildly degenerate Kirchhoff equation with dissipation, providing new estimates, a renormalization approach, and explicit limit calculations independent of initial conditions.

Contribution

It introduces a novel renormalization technique and explicit limit computation for solutions of the degenerate Kirchhoff equation, advancing understanding of its asymptotic properties.

Findings

New estimate on second-time derivative of solutions

Renormalization yields a non-zero limit as time approaches infinity

Limit norm is explicitly calculated and independent of initial data

Abstract

We study the asymptotic behavior of the solutions of the mildly degenerate Kirchhoff equation with a dissipative term. We obtain a new estimate on second-in-time derivative of the solution. Moreover we renormalize the solution in such a way that the renormalization as a no zero limit as t goes to infinity. Finally we calculate explicitly the norm of this limit and we prove that it does not depend on the initial data.