Asymptotic profiles for the Kirchhoff equation

Abstract

The first aim of this paper is to find asymptotic profiles for 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 the small amplitude solutions with data belonging to $H^{\sigma (p),p}\times H^{\sigma (p)-1,p}$, where $\sigma (p)=n(\frac{2}{p}-1)+1$, $p\in [1,\frac{2(n-1)}{n+1})$ and $n\ge 4$.