  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 $m$.
