Spectral gap global solutions for degenerate Kirchhoff equations

We consider the second order Cauchy problem u +m(|Au|)Au = 0, u(0) = u0, u(0) = u1, where m : [0,+∞) → [0,+∞) 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 u0 and u1 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 (u0, u1) there exist two pairs of initial data (u0, u1), (û0, û1) for which the solution is global, and such that u0 = u0 + û0, u1 = u1 + û1. 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. Mathematics Subject Classification 2000 (MSC2000): 35L70, 35L80, 35L90.