Spectral gap global solutions for degenerate Kirchhoff equations

TL;DR

This paper proves the existence of global solutions for a class of degenerate Kirchhoff equations with spectral gap initial data, extending previous results to less regular nonlinearities and degenerate cases.

Contribution

It introduces a global existence result for spectral gap initial data in degenerate Kirchhoff equations, broadening the scope beyond smooth nonlinearities and strictly hyperbolic cases.

Findings

Existence of global solutions for spectral gap initial data.

Extension of previous results to degenerate and less regular cases.

Solutions can be decomposed into parts with specific spectral properties.

Abstract

We consider the second order Cauchy problem $$u''+m(|A^{1/2}u|^2)Au=0, u(0)=u_{0}, u'(0)=u_{1},$$ where $m:[0,+\infty)\to[0,+\infty)$ is a continuous function, and $A$ is a self-adjoint nonnegative operator with dense domain on a Hilbert space. It is well known that this problem admits local-in-time solutions provided that $u_{0}$ and $u_{1}$ are regular enough, depending on the continuity modulus of $m$, and on the strict/weak hyperbolicity of the equation. We prove that for such initial data $(u_{0},u_{1})$ there exist two pairs of initial data $(\overline{u}_{0},\overline{u}_{1})$, $(\widehat{u}_{0},\widehat{u}_{1})$ for which the solution is global, and such that $u_{0}=\overline{u}_{0}+\widehat{u}_{0}$, $u_{1}=\overline{u}_{1}+\widehat{u}_{1}$. This is a byproduct of a global existence result for initial data with a suitable spectral gap, which extends previous results obtained in the strictly hyperbolic case with a smooth nonlinearity $m$.