only to depend on $p$, but also on the size of the initial data (i.e $A$): see Theorem \[Thm:Scat\].

Lastly we deal with quantities that are not conserved. This implies that we have to modify significantly the induction on energy method designed in [@bourgjams], in order to establish a finite bound of $\| Iu \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}}$. In particular, we design a relation that allows to control not only $\| Iw \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}}$ but also $\sup_{t \in \mathbb{R}} E(Iw)$ (with $w$ solution of (\[Eqn:NlkgWdat\]) ), assuming that we control the mollified energy of $w$ at one time (see the definition of $\mathcal{P}_{C,Q}(l)$, just below (\[Eqn:NControlVar\])) and the goal is to prove that this relation is true for large levels of mollified energy at one time by induction, using the small mollified energy (at one time) theory: see Proposition \[prop:EstSmallNrj\]. Also, one has to make sure that the methods of concentration, combined with the Almost Morawetz-Strauss estimate, allow to make the mollified energy decrease at one time at a non-decreasing rate if the $L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}$ norm of the solutions is large, in order to reach the small energy level and apply the small energy theory (see Proposition \[prop:EstSmallNrj\]). In order to do 