Kirchhoff equations from quasi-analytic to spectral-gap data

TL;DR

This paper extends Nishihara's global existence results for Kirchhoff equations by removing convexity constraints and connecting to spectral-gap data, broadening the class of initial conditions for which solutions exist.

Contribution

It generalizes previous results by eliminating convexity assumptions and links the theory to recent spectral-gap data findings.

Findings

Removed convexity constraint from initial data class.

Replaced integrability condition with a standard quasi-analytic condition.

Established connections with spectral-gap data results.

Abstract

In a celebrated paper (Tokyo J. Math. 1984) K. Nishihara proved global existence for Kirchhoff equations in a special class of initial data which lies in between analytic functions and Gevrey spaces. This class was defined in terms of Fourier components with weights satisfying suitable convexity and integrability conditions. In this paper we extend this result by removing the convexity constraint, and by replacing Nishihara's integrability condition with the simpler integrability condition which appears in the usual characterization of quasi-analytic functions. After the convexity assumptions have been removed, the resulting theory reveals unexpected connections with some recent global existence results for spectral-gap data.