Global Solutions for Dissipative Kirchhoff Strings with mŽr. rpŽp 1.

We investigate the evolution problem u u mŽ A1 2u 2H.Au 0, uŽ0. u0, u Ž0. u1, where H is a Hilbert space, A is a self-adjoint non-negative operator on H with domain DŽA., 0 is a parameter, and mŽr. r p with p 1. We prove that this problem has a unique global solution for positive times, provided that the initial dataŽu0, u1. DŽA i 2. DŽAŽ i 1. 2 . satisfy a suitable smallness assumption and the non-degeneracy condition mŽ A1 2u0 2H. 0Žwhere p 2 i and i 2i 1.. Moreover, we prove for this solution decay with a polynomial rate as t . These results apply to degenerate hyperbolic PDEs with non-local non-linearities.