Spectral gap global solutions for degenerate Kirchhoff equations

Abstract

We consider the second order Cauchy problem u ″ + m ( | A 1 / 2 u | 2 ) A u = 0 , u ( 0 ) = u 0 , u ′ ( 0 ) = u 1 , 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 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 ( u ¯ 0 , u ¯ 1 ) , ( u ̂ 0 , u ̂ 1 ) for which the solution is global, and such that u 0 = u ¯ 0 + u ̂ 0 , u 1 = u ¯ 1 + 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 .