Spectral gap global solutions for degenerate Kirchhoff equations Original Research Article

We consider the second order Cauchy problem View the MathML sourceu″+m(|A1/2u|2)Au=0,u(0)=u0,u′(0)=u1, Turn MathJax on where m:[0,+∞)→[0,+∞)m:[0,+∞)→[0,+∞) is a continuous function, and AA 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 u0u0 and u1u1 are regular enough, depending on the continuity modulus of mm, and on the strict/weak hyperbolicity of the equation. We prove that for such initial data (u0,u1)(u0,u1) there exist two pairs of initial data View the MathML source(u¯0,u¯1), View the MathML source(û0,û1) for which the solution is global, and such that View the MathML sourceu0=u¯0+û0, View the MathML sourceu1=u¯1+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 mm.