\nabla  u
(t,x)|^{2} \, dx + \frac{1}{2} \int_{\mathbb{R}^{3}} | u(t,x) |^{2} \, dx \\
& + \frac{1}{p+1} \int_{\mathbb{R}^{3}} |u(t,x)|^{p+1} \, dx
\end{array}
\label{Eqn:DefEnergy}$$

In fact by standard limit arguments the energy conservation law remains true for solutions $(u,\partial_{t}u) \in H^{s} \times H^{s-1}$, $s \geq
1$.

Since the lifespan of the local solution depends only on the $H^{s} \times H^{s-1}$ norm of the initial data $(u_{0},u_{1})$ (see [@linsog]) then it suffices to find an a priori pointwise in time bound in $H^{s} \times H^{s-1}$ of the solution $(u,\partial_{t}u)$ to establish global well-posedness. The energy captures the evolution in time of the $H^{1} \times L^{2}$ norm of the solution. Since it is conserved we have global existence of (\[Eqn:NlkgWdat\]).

The scattering theory (i.e the linear asymptotic behaviour of the solution of (\[Eqn:NlkgWdat\]) ) in the energy space (i.e with data $(u_{0},u_{1}) \in H^{1} \times L^{2}$ ) has attracted much attention from the community. In particular scattering was proved in [@bren; @brenscat; @nakanash1; @nakanash2] for all dimension $n$ and for all exponent $p$ that is $\dot{H}^{1}$ subcritical and $L^{2}$ supercritical, i.e $ 1 + \frac{4}{n} < p < 1 +\frac{4}{n-2} $.

The scattering theory below energy norm (i.e for data in $H^{s} \times 