Asymptotic behaviour for the Kirchhoff equation

In this paper we study the asymptotic behaviour of the solution u of the Kirchhoff eqation with small data. More precisely we show that {fx293-01} every k ε Nwhere v is a suitable solution of an appropriate wave equation. Moreover we give some estimates on $$\mathop {\lim }\limits_{t \to \infty } \parallel \nabla u\parallel _2$$ .