Spectral gap global solutions for degenerate Kirchhoff equations

We consider the second order Cauchy problem for the Kirchhoff equation. It is well known that this problem admits local-in-time solutions provided that the initial data are regular enough, depending on the continuity modulus of the nonlinear term, and on the strict/weak hyperbolicity of the equation. We prove that for such initial data (u0 , u1) there exist two pairs of initial data (v0 , v1) and (w0 , w1)for which the solution is global, and such that u0 = v0+w0 and u1 =v1+w1. 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.