Global Solutions for Dissipative Kirchhoff Strings with m(r) = rp (p < 1)

Abstract

We investigate the evolution problemu″+δu′+m|A1/2u|2HAu=0,u0=u0,u′0=u1,where H is a Hilbert space, A is a self-adjoint non-negative operator on H with domain D(A), δ>0 is a parameter, and m(r)=rp with p<1. We prove that this problem has a unique global solution for positive times, provided that the initial data (u0,u1)∈D(Aαi/2)×D(A(αi−1)/2) satisfy a suitable smallness assumption and the non-degeneracy condition m(|A1/2u0|2H)>0 (where p≥2−i and αi=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.