\epsilon_{0} (N^{s-1} \eta_{1} A^{-1})^{\alpha} }{C_{E} N^{2(1-s)} A^{p+1}}
\end{array}
\label{Eqn:DefBeta}$$

\[Prop:SepLocMol\]

[**“Local Boundedness”**]{} Let $A \gtrsim 1$ and let $J \subset [0, \infty)$ be an interval. Let $w_{l}$ be a solution of (\[Eqn:NlkgWdat\]). Assume that there exist $(C,Q, \alpha) \in (\mathbb{R}^{+})^{3}$ such that (\[Eqn:InducNrj\]) and (\[Eqn:InducStr\]) hold.

Then there exist $\eta_{0}>0$ such that if $\eta \leq \eta_{0}$ and

$$\begin{array}{ll}
\| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J)} & \leq \eta
\end{array}$$

then

$$\begin{array}{l}
Z(J,w_{l}) \lesssim \left( \left( C_{E} N^{2(1-s)} A^{p+1} - l(1- \epsilon_{0}) (N^{s-1} \eta_{1} A^{-1})^{\alpha} \right) (1 + \beta)
\right)^{\frac{1}{2}}
\end{array}
\label{Eqn:BoundZ}$$

with $\beta$ is defined in (\[Eqn:DefBeta\]), provided that $(N,s)$ satisfies (\[Eqn:Cond2\]).

\[prop:LocalBd\]

[**“Perturbation argument”**]{}

Let $A \gtrsim 1$ and $J:=[a,b] \subset [0,\infty)$ be an interval. Let $T_{l} \in J$. Let $w_{l}$ be a solution of (\[Eqn:NlkgWdat\]). Assume that there exist $(\alpha,C,Q) \in (\mathbb{R}^{+})^{3}$ such that (\[Eqn:InducNrj\]) and (\[Eqn:InducStr\]) hold. Let $v_{l}$ be solution of the linear Klein-Gordon equation such that

$$\begin{array}{ll}
E_{c}(Iv_{l}) & \leq (N^{s-1} A^{-1})^{\alpha}
\end{array}
\label{Eqn:BoundEc}$$

Let $w_{l+1}$ be a solution of (\[Eqn:NlkgWdat\]) with the same initial data as $w_{l}-v_{l}$ at $t=T_{l}$. Assume that

$$\begin{array}{l}
\sup_{t \in J} E(Iw_{l+1}(t)) \leq \left( C_{E} N^{2(1-s)} A^{p+1} - (1+ l)(1- \epsilon_{0})(N^{s-1} \eta_{1} A^{-1})^{\alpha} \right)(1+ \beta)
\end{array}
\label{Eqn:InducNrjw}$$

(with $\eta_{1}$ defined in 