Spectral gap global solutions for degenerate Kirchhoff equations

Abstract

We consider the second order Cauchy problem $$u^{\prime \prime }+m(|A^{1/2}u{|}^2)Au=0,u(0)=u_0,u^{\prime }(0)=u_1,$$ 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 $({\overline{u}}_0,{\overline{u}}_1)$, $({\hat{u}}_0,{\hat{u}}_1)$ for which the solution is global, and such that $u_0={\overline{u}}_0+{\hat{u}}_0$, $u_1={\overline{u}}_1+{\hat{u}}_1$. 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 $m$.

16 pages