Asymptotic profiles for the Kirchhoff equation

The first aim of this paper is to find asymptotic profiles for the Kirchhoff equation. More precisely, it will be shown that there exists a small amplitude solution which is not asymptotically free. The second aim is to prove the existence of scattering states for small amplitude solutions with data belonging to H-sigma(p),H- p x H sigma((p)-1, p), where sigma(p) = n(2/p - 1) + 1, p is an element of [1, 2(n - 1)/(n + 1)) and n = 4.