  We prove that the classical hyperbolic Kirchhoff equation admits
global-in-time solutions for some classes of initial data in the energy space.
We also show that there are enough such solutions so that every initial datum
in the energy space is the sum of two initial data for which a global-in-time
solution exists.
  The proof relies on the notion of spectral gap data, namely initial data
whose components vanish for large intervals of frequencies. We do not pass
through the linearized equation, because it is not well-posed at this low level
of regularity.
