\begin{array}{l}
w_{i} \, \, solution \, \, of \, \, (\ref{Eqn:NlkgWdat}) \\
\exists t_{i} \in \mathbb{R}^{+} \, \, s.t \, \, E \left( I w_{i}(t_{i}) \right) \leq C_{E} N^{2(1-s)} A^{p+1} - i(1- \epsilon_{0}) (N^{s-1}
\eta_{1} A^{-1})^{\alpha}
\end{array}
\right\}
\end{array}
\label{Eqn:DefCi}$$

with $\eta_{1}$ defined in Proposition \[Prop:SepLocMol\]. Then there exists an $L:=L_{N,A} (i)$ [^7] such that

$$\begin{array}{l}
\inf{ \{ \bar{C}, \, \, w_{i} \in \mathcal{C}_{i} \, \, and \, \, \| I w_{i} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (\mathbb{R}^{+})} \leq \bar{C}
\}} = L(i)
\end{array}
\label{Eqn:ControlIwStrich}$$

$$\begin{array}{ll}
L(i) & \leq X_{C,Q}
\end{array}
\label{Eqn:ContBoundLi}$$

and

$$\begin{array}{ll}
\sup_{t \in \mathbb{R}+} E(Iw_{i}(t)) & \leq  (C_{E} N^{2(1-s)} A^{p+1} -i (1 - \epsilon_{0}) (N^{s-1} \eta_{1} A^{-1})^{\alpha} ) \left( 1+
\frac{\beta}{2} \right)
\end{array}
\label{Eqn:ControlEIw}$$

We shall prove that $\mathcal{P}_{C,Q}(l)$ is true for some constants $C:=C(A,p)$ and $Q:=Q(A,p)$ (to be chosen shortly), for all $ \sim
A^{p+1+ \alpha} \geq l \geq 0$: this will be done by induction on $l$. Once we have proved that $\mathcal{P}(l)$ is true for this range of $l$, then this implies that we have $\sup_{t \in \mathbb{R}+} E(Iu(t)) {\lesssim_{A}}1$, $\| I u \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (\mathbb{R}+)}
{\lesssim_{A}}1$: indeed, it is enough to apply $\mathcal{P}_{C,Q}(0)$ to $u$, using (\[Eqn:BoundInitMolNrj\]). In fact, we also have $\sup_{t \in
\mathbb{R}} E(Iu(t)) 