---
abstract: |
    We prove scattering of solutions just below the energy norm of the $3D$ Klein-Gordon equation

    $$\left\{
    \begin{array}{ccl}
    \partial_{tt} u  - \Delta u + u & = & -|u|^{p-1} u \\
    u(0,x)& = & u_{0}(x)    \\
    \partial_{t} u(0,x) & = & u_{1}(x)
    \end{array}
    \right. \nonumber$$

    with $5 > p >3$: this result extends those obtained in the energy class [@brenscat; @nakanash1; @nakanash2] and those obtained below the energy norm under the additional assumption of spherical symmetry [@triroyradnlkg]. It is well-known that the notion of scattering is closely related to that of decay. The main difficulty is that, unlike the radial case, there is no useful decay estimate available. The idea is to generate some decay estimates, by means of concentration [@bourgjams] and a low-high frequency decomposition [@bourgbook; @almckstt]. On low frequencies we use the physical properties of the solution and we modify the arguments in [@nakanash1; @nakanash2]; on high frequencies we use the local smoothing properties of the solutions: this, combined with an almost conservation law, allows to prove these decay estimates: this is the starting point to establish scattering.
address:
- Korea Advanced Institute Of Science And Technology 
- 'Institute For Advanced Study, Princeton'
author:
- Soonsik Kwon
- Tristan Roy
title: Generation of decay estimates and application to scattering of rough solutions of $3D$ NLKG
---

Introduction
============

In this paper we consider the $p$- defocusing Klein-Gordon equation on $\mathbb{R}^{3}$

$$\begin{array}{ll}
\partial_{tt} u  - \Delta u + u & =  -|u|^{p-1}u \\
\end{array}
\label{Eqn:NlkgWdat}$$

with data $u(0)=u_{0}$, $\partial_{t}u(0)=u_{1}$ lying in $H^{s}$, $H^{s-1}$ respectively. Here $H^{s}$ is the standard inhomogeneous Sobolev space i.e $H^{s}$ is the completion of the Schwartz space $\mathcal{S}(\mathbb{R}^{3})$ with respect to the norm

$$\begin{array}{ll}
\| f \|_{H^{s}} & :=  \| \langle D \rangle^{s} f \|_{L^{2}(\mathbb{R}^{3})}
\end{array}$$

where $\langle D \rangle$ is the operator defined by

$$\begin{array}{ll}
\widehat{\langle D \rangle^{s} f}(\xi) & := ( 1 + |\xi|)^{s} \hat{f}(\xi)
\end{array}$$

and $\hat{f}$ denotes the Fourier transform $$\begin{array}{ll}
\hat{f}(\xi) & := \int_{\mathbb{R}^{3}} f(x) e^{-i x \cdot \xi} \, dx
\end{array}$$ We are interested in the strong solutions of the $p$- defocusing Klein-Gordon equation on some interval $[0,T]$ i.e maps $u$, $\partial_{t} u$ that lie in $C \left([0, \, T], \, H^{s}(\mathbb{R}^{3}) \right)$, $C \left( [0, \,T], \, H^{s-1} ( \mathbb{R}^{3}) \right)$ respectively and that satisfy

$$\begin{array}{ll}
u(t) & = \cos{(t \langle D \rangle)} u_{0} + \frac{\sin(t \langle D \rangle)}{\langle D \rangle} u_{1} - \int_{0}^{t} \frac{\sin \left(
(t-t^{'}) \langle D \rangle \right)}{\langle D \rangle} |u|^{p-1}(t^{'}) u(t^{'}) \, dt^{'}
\end{array}
\label{Eqn:StrongSol}$$

The $p$- defocusing Klein-Gordon equation is closely related to the $p$- defocusing wave equation i.e

$$\begin{array}{ll}
\partial_{tt} v - \triangle v = & - |v|^{p-1} v
\end{array}
\label{Eqn:DefocPWaveEq}$$

with data $v(0)=v_{0}$, $\partial_{t} v(0)=v_{1}$. (\[Eqn:DefocPWaveEq\]) enjoys the following scaling property

$$\begin{array}{ll}
v(t,x) & \rightarrow \frac{1}{\lambda^{\frac{2}{p-1}}} u \left( \frac{t}{\lambda}, \frac{x}{\lambda} \right) \\
v_{0}(x) & \rightarrow \frac{1}{\lambda^{\frac{2}{p-1}}} u_{0}  \left( \frac{x}{\lambda} \right) \\
v_{1}(x) & \rightarrow \frac{1}{\lambda^{\frac{2}{p-1}+1}} u_{1} \left( \frac{x}{\lambda} \right)
\end{array}
\label{Eqn:WaveScaling}$$

We define the critical exponent $s_{c}:= \frac{3}{2} - \frac{2}{p-1}$. One can check that the $\dot{H}^{s_{c}} \times \dot{H}^{s_{c}-1}$ norm of $(u_{0},u_{1})$ is invariant under the transformation (\[Eqn:WaveScaling\]) [^1]. (\[Eqn:DefocPWaveEq\]) was demonstrated to be locally well-posed in [@linsog] and in $H^{s} \times H^{s-1}$, $s >\frac{3}{2} - \frac{2}{p-1}$, $p>3$ by using an iterative argument. In fact their results extend immediately to (\[Eqn:NlkgWdat\]) [^2].

If $p=5$ then $s_{c}=1$ and this is why we say that that the nonlinearity $|u|^{p-1} u$ is $\dot{H}^{1}$ critical. If $3<p<5$ then $s_{c}<1$ and the regime is $\dot{H}^{1}$ subcritical.

It is well-known that smooth solutions to (\[Eqn:NlkgWdat\]) have a conserved energy

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

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 $H^{s} \times H^{s-1}$, $s_{c} \leq s <1$) and in dimension $3$ has been recently studied in [@triroyradnlkg] for radial data. More precisely it was proved that the asymptotic behaviour is linear for $3<p<5$ and for radial data in $H^{s} \times H^{s-1}$, $ s_{p} < s < 1$ with

$$\begin{array}{l}
s_{p} := \left\{
\begin{array}{l}
1- \frac{(5-p)(p-3)}{2(p-1)(p-2)}, \, 4 \geq  p > 3 \\
1- \frac{(5-p)^{2}}{2(p-1)(6-p)}, \, 5 > p \geq 4
\end{array}
\right. \end{array}$$

In this paper we are interested in proving scattering results for general data below the energy norm and in dimension $3$. The main result of this paper is the following one

Let $3<p<5$. Let $(u_{0},u_{1}) \in H^{s} \times H^{s-1}$ and $A \geq 0$ such that

$$\begin{array}{ll}
\|( u_{0},u_{1}) \|_{H^{s} \times H^{s-1}} & \leq A
\end{array}
\label{Eqn:BoundA}$$

Then there exists $\tilde{s}:=\tilde{s}(A,p) < 1$ such that the solution of the $p$-defocusing Klein-Gordon equation on $\mathbb{R}^{3}$ with data $(u_{0},u_{1})$ exists for all time $T$ in $ H^{s} \times H^{s-1} $, $ 1> s > \tilde{s}$. Moreover it scatters as $T$ goes to infinity, i.e there exists $\left( u_{+,0}, u_{+,1} \right) \in H^{s} \times H^{s-1}$ such that

$$\begin{array}{ll}
\lim \limits_{T \rightarrow \infty} \| \left( u(T) , \partial_{t} u(T) \right) - K(T) (u_{+,0}, u_{-,0} ) \|_{H^{s} \times H^{s-1}} & = 0
\end{array}
\label{Eqn:ScatteringRes}$$

and

$$\begin{array}{ll}
\| I u \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}(\mathbb{R})} & \lesssim_{A} 1
\end{array}
\label{Eqn:DecayEstp}$$

Here

$$K(T):= \left(
\begin{array}{ll}
\cos{(T \langle D \rangle)} & \frac{\sin{(T \langle D \rangle)}}{\langle D \rangle} \\
- \langle D \rangle \, \sin{(T \langle D \rangle)} & \cos{ ( T \langle D \rangle)}
\end{array}
\right)$$

and $ \left( 2(p-1)-, 2(p-1)+ \right) \in \left( a_{p}, \, 2(p-1) \right) \times  \left( 2(p-1), \, b_{p} \right) $ (for $a_{p}$ and $b_{p}$ to be determined) that satisfy $ \frac{1}{2(p-1)-} + \frac{3}{2(p-1)+} = \frac{3}{2} -s_{c} = \frac{2}{p-1} $ [^3]

\[Thm:Scat\]

1.  On can find an an explicit value of $\tilde{s}$ if one reads carefully the proof of Theorem \[Thm:Scat\]. Nevertheless we have decided not to do this in order to make the paper easier to read.

2.  $\tilde{s}$ grows to one as $A$ goes to $\infty$.

3.  The reader can check, after reading the proof of Theorem \[Thm:Scat\] that

    -   $a_{p}$ decreases and $b_{p}$ increases as $p$ grows

    -   $\lim_{p \rightarrow 3^{+}} (a_{p},b_{p})=\left( 2(p-1),2(p-1) \right)$.

We set some notation that appear throughout the paper.\
Let $\nabla$ denote the gradient operator. If $w$ is a smooth function then

$$\begin{array}{ll}
Z_{m,s}(J,w) & := \sup_{ \substack {(q,r)-m \\ wave \, adm} } \| \partial_{t} \langle D \rangle^{-m} I w \|_{L_{t}^{q}(J) L_{x}^{r}} + \|
\langle D \rangle^{1-m} I w \|_{L_{t}^{q}(J) L_{x}^{r}}
\end{array}$$

and

$$\begin{array}{ll}
Z(J,w) & := \sup_{m \in [0,1]} Z_{m,s}(J,w)
\end{array}$$

We write $F(w)$ for the following function

$$\begin{array}{ll}
F(w) & := |w|^{p-1} w
\end{array}$$

If $w$ is a solution of (\[Eqn:NlkgWdat\]) with data $(w_{0},w_{1})$ then $w(t)=w^{lin}(t)+ w^{nlin}(t)$ with

$$\begin{array}{ll}
w^{lin}(t) & := \cos{(t \langle D \rangle)} w_{0} + \frac{ \sin{(t \langle D \rangle)}}{ \langle D \rangle} w_{1}
\end{array}$$

and

$$\begin{array}{ll}
w^{nlin}(t) & := - \int_{0}^{t} \frac{ \sin{(t-s) \langle D \rangle}} {\langle D \rangle} F(w) \, ds
\end{array}$$

If $(C,Q) \in (\mathbb{R}^{+})^{2}$ then let $X_{C,Q}:=Y_{0}$ where $(Y_{i})_{ 0 \leq i \leq m}$ is an element of the class of sequences $(Z_{i})_{0 \leq i \leq m}$ satisfying

$$\left\{
\begin{array}{l}
Z_{i} := e^{C^{C} C^{C Z_{i+1}} A^{C} C^{A ^{C}}}   \\
Z_{m} << 1 \\
m \sim Q
\end{array}
\right.$$

Let $I$ be the following multiplier

$$\begin{array}{ll}
\widehat{If}(\xi) & := m(\xi) \hat{f}(\xi)
\end{array}$$

where $m(\xi): =  \eta \left( \frac{\xi}{N} \right)$, $\eta$ is a smooth, radial, nonincreasing in $|\xi|$ such that

$$\begin{array}{ll}
\eta (\xi) & :=  \left\{
\begin{array}{l}
1, \, |\xi| \leq 1 \\
\left( \frac{1}{|\xi|} \right)^{1-s}, \, |\xi| \geq 2
\end{array}
\right.
\end{array}$$

and $N>>1$ is a dyadic number playing the role of a parameter to be chosen. We shall abuse the notation and write $m (|\xi|)$ for $m(\xi)$, thus for instance $m(N)=1$. If $w$ is a function and $B(x_{0},R):= \{ x \in \mathbb{R}, \, |x-x_{0}| \leq R  \}$ then

$$\begin{array}{ll}
E(Iw) & := \frac{1}{2} \int_{\mathbb{R}^{3}} |\partial_{t} I w (x)|^{2} \, dx + \frac{1}{2} \int_{\mathbb{R}^{3}} |I w(x)|^{2} \, dx \\
& + \frac{1}{2} \int_{\mathbb{R}^{3}} |\nabla I w (x)|^{2} \, dx + \frac{1}{p+1} \int_{\mathbb{R}^{3}} | I w(x)|^{p+1} \, dx
\end{array}
\label{Eqn:DefMolNrj}$$

$$\begin{array}{ll}
E(Iw,B(x_{0},R)) & := \frac{1}{2} \int_{B(x_{0},R)} |\partial_{t} Iw (x)|^{2} \, dx + \frac{1}{2} \int_{B(x_{0},R)} |I w(x)|^{2} \, dx    \\
& + \frac{1}{2} \int_{B(x_{0},R)} |\nabla I w (x)|^{2} \, dx + \frac{1}{p+1} \int_{B(x_{0},R)} |Iw(x)|^{p+1} \, dx
\end{array}$$

and

$$\begin{array}{l}
E_{c}(Iw) := \frac{1}{2} \int_{\mathbb{R}^{3}} |\partial_{t} I w (x)|^{2} \, dx + \frac{1}{2} \int_{\mathbb{R}^{2}} |Iw(x)|^{2} \, dx +
\frac{1}{2} \int_{\mathbb{R}^{3}} |\nabla I w (x)|^{2} \, dx
\end{array}$$

Some estimates that we establish throughout the paper require a Paley-Littlewood decomposition. We set it up now. Let $\phi(\xi)$ be a real, radial, nonincreasing function that is equal to $1$ on the unit ball $\left\{ \xi \in \mathbb{R}^{3}: \, |\xi| \leq 1 \right\}$ and that that is supported on $\left\{ \xi \in \mathbb{R}^{3}: \, |\xi| \leq 2 \right\}$. Let $\psi$ denote the function

$$\begin{array}{ll}
\psi(\xi) & := \phi(\xi) - \phi(2 \xi)
\end{array}$$

If $(M,M_{1},M_{2}) \in 2^{\mathbb{N}}$ are dyadic numbers such that $M_{2}>M_{1}$ we define the Paley-Littlewood operators in the Fourier domain by

-   if $M > 1$

    $$\begin{array}{ll}
    \widehat{P_{M} f}(\xi) & := \psi \left( \frac{\xi}{M} \right) \hat{f}(\xi) \
    \end{array}$$

-   if $M=1$

    $$\begin{array}{ll}
    \widehat{P_{1} f }(\xi) & := \phi \left( \xi \right) \hat{f}(\xi)
    \end{array}$$

and

$$\begin{array}{ll}
\widehat{P_{\leq M} f}(\xi) & := \phi \left( \frac{\xi}{M} \right) \hat{f}(\xi) \\
\widehat{P_{> M} f}(\xi) & := \widehat{f}(\xi) - \widehat{P_{\leq M} f}(\xi) \\
\widehat{P_{ << M } f}(\xi) & : = \widehat{P_{ \leq \frac{M}{128} }f}(\xi) \\
\widehat{P_{ \gtrsim  M } f}(\xi) & : = \widehat{P_{ > \frac{M}{128}}f}(\xi) \\
P_{M_{1} < . \leq M_{2}} f  &  := P_{ \geq M_{2}} f -  P_{< M_{1}} f \\
\end{array}$$

Since $ \phi(\xi) + \sum_{M \in  2^{\mathbb{N}^{*}}} \psi \left( \frac{\xi}{M} \right)=1$ we have

$$\begin{array}{ll}
f & = \sum_{M \in 2^{\mathbb{N}}} P_{M} f
\end{array}$$

Notice also that

$$\begin{array}{ll}
f & = P_{\lesssim M}f + P_{\gtrsim M} f
\end{array}$$

Throughout this paper we constantly use Strichartz-type estimates (see [@ginebvelo; @nakamura; @triroyradnlkg] )

[**“Strichartz estimates for Klein-Gordon equations in $L_{t}^{q} L_{x}^{r}$ spaces” [@triroyradnlkg]**]{} Assume that $w$ satisfies the following Klein-Gordon equation on $\mathbb{R}^{3}$

$$\left\{
\begin{array}{ccl}
\partial_{tt} w  - \Delta w + w & = & Q \\
w(0,x)& = & w_{0}(x)    \\
\partial_{t} w(0,x) & = & w_{1}(x)
\end{array}
\right. \label{Eqn:KlGH}$$

Let $T \geq 0$. Then

$$\begin{array}{l}
\| w \|_{L_{t}^{q} L_{x}^{r} ([0,T])} + \| \partial_{t} \langle D \rangle^{-1} w \|_{L_{t}^{q} L_{x}^{q} ([0,T]) } +  \| w \|_{L_{t}^{\infty}
H^{m} [0,T] } +  \| \partial_{t} w \|_{L_{t}^{\infty} H^{m-1} [0, T]} \\ \\
\lesssim \| w_{0} \|_{H^{m}} + \| w_{1} \|_{H^{m-1}} + \| Q \|_{L_{t}^{\tilde{q}} L_{x}^{\tilde{r}} ([0,T])}
\end{array}
\label{Eqn:StrNlkg}$$

under the following assumptions

-   $(q,r)$ is $m$- wave admissible, i.e $ (q,r) \in (2, \infty] \times [2, \infty) $ and $ \frac{1}{q} + \frac{1}{r} \leq \frac{1}{2} $ and it obeys the following constraint $\frac{1}{q} + \frac{3}{r}  =\frac{3}{2} -m $

-   $(q,r)  \neq \left( 2, \infty \right) $

-   $(\tilde{q},\tilde{r})$ satisfies $ \frac{1}{\tilde{q}} + \frac{1}{q}=1 $, $\frac{1}{\tilde{r}} + \frac{1}{r}=1$ and $\frac{1}{\tilde{q}} + \frac{3}{\tilde{r}} -2    = \frac{1}{q} + \frac{3}{r} $.

\[prop:StrEstLtqLxr\]

Next we recall some propositions

[“**Mollified energy at time $0$ is bounded by $N^{2(1-s)}$**” [@triroyradnlkg]]{} Let $w$ be a solution of (\[Eqn:NlkgWdat\]). Then there exists a constant $C_{E} >0$ such that

$$\begin{array}{ll}
E(Iw_{0}) & \lesssim  N^{2(1-s)} \left( \| w_{0} \|^{2}_{H^{s}} + \| w_{1} \|^{2}_{H^{s-1}} + \| w_{0} \|^{p+1}_{H^{s}} \right)
\end{array}
\label{Eqn:BoundInitMolNrj}$$

In particular, if $w$ is a solution of (\[Eqn:NlkgWdat\]) with data $(w_{0},w_{1})$ satisfying (\[Eqn:BoundA\]) with $A \gtrsim 1$, then there exists a constant $C_{E}$ such that

$$\begin{array}{ll}
E(Iw_{0}) & \leq C_{E} N^{2(1-s)} A^{p+1}
\end{array}$$

\[Prop:EstInitMolNrj\]

[**“Almost Conservation Law ”** [@triroyradnlkg]]{} Assume that $w$ satisfies (\[Eqn:NlkgWdat\]). Let $J=[a,b]$ be a time interval. Let $3 \leq p < 5$ and $s \geq \frac{3p-5}{2p}$. Then

$$\begin{array}{ll}
\left| \sup_{t \in J} E(Iw(t)) - E(Iw(a)) \right| & \lesssim \left| \int_{J} \int_{\mathbb{R}^{3}} | \Re (\overline{\partial_{t} Iw}
(IF(w)-F(Iw)) ) | \, dx
\, dt \right| \\
& \\
& \lesssim \frac{Z^{p+1}(J,w)}{N^{\frac{5-p}{2}-}}
\end{array}
\label{Eqn:Acl}$$

\[Prop:Acl\]

In fact, if one looks carefully at the proof of Proposition \[Prop:Acl\], on notices that

$$\begin{array}{ll}
\left| \sup_{t \in J} E(Iw(t)) - E(Iw(a)) \right| & \lesssim \left| \int_{J} \int_{\mathbb{R}^{3}} | \Re (\overline{\partial_{t} Iw}
(IF(w)-F(Iw)) ) | \, dx
\, dt \right| \\
& \\
& \lesssim \frac{\bar{Z}^{p+1}(J,w)}{N^{\frac{5-p}{2}-}}
\end{array}
\label{Eqn:AclRem}$$

with $\bar{Z}(J,w):= \sup_{m \in \{ 0,1,\frac{p-3}{2},\frac{3p-5}{2p} \}} Z_{m,s}(J,w)$.

\[rem:VarMolNrj\]

[**“Estimate of integrals”** [@triroyradnlkg]]{} Let $J$ be a time interval. Let $w$ be a smooth function. Then for $i=1,2$ we have

$$\begin{array}{ll}
| R_{i}(J,w) | & \lesssim \frac{Z^{p+1}(J,w)}{N^{\frac{5-p}{2}-}}
\end{array}
\label{Eqn:BdRi}$$

with

$$\begin{array}{ll}
R_{1}(J,w) : = \int_{J} \int_{\mathbb{R}^{3}} \frac{\nabla Iw(t,x).x}{|x|} \left( F(Iw) - I F(w) \right) \, dx dt
\end{array}$$

and

$$\begin{array}{ll}
R_{2}(J,w) := \int_{J} \int_{\mathbb{R}^{3}} \frac{Iw(t,x)}{|x|} \left( F(Iw) - I F(w) \right) \, dx dt
\end{array}$$

\[Prop:EstInt\]

[**“Almost Morawetz-Strauss Estimate”** [@triroyradnlkg]]{} Let $w$ be a solution of (\[Eqn:NlkgWdat\]). Let $x_{0} \in \mathbb{R}^{3}$ and let $T \geq 0$. Then

$$\begin{array}{ll}
\int_{0}^{T} \int_{\mathbb{R}^{3}} \frac{|Iw(t,x)|^{p+1}}{|x-x_{0}|} \, dx dt & \lesssim \sup_{t \in [0,  T]} E(Iw(t)) + R_{1}([0, \, T],w) +
R_{2}([0, \, T],w)
\end{array}
\label{Eqn:AlmMorStrEst}$$

\[Prop:AlmMor\]

Now we explain the main ideas of this paper.

In [@nakanash1; @nakanash2], the author found for all dimension $n$ an upper bound of the $L_{t}^{q} L_{x}^{r}$ norm of the solution of (\[Eqn:NlkgWdat\]) with data in the energy class (i.e $(u_{0},u_{1}) \in H^{1} \times L^{2}$) by a tower-exponential type bound of the energy and for some $(q:=q(n),r:=r(n))$ only depending on the dimension $n$: this decay estimate was the preliminary step to prove scattering. A natural question is\
*Is it possible to prove decay estimates of this form (or a modified form) in the subcritical case and for rough data (i.e for data $
(u_{0},u_{1}) \in H^{s}
\times H^{s-1}, \, s_{c} < s < 1 )$ ?*\
\
Because, if this is possible, then it might help us to prove scattering of solutions of (\[Eqn:NlkgWdat\]) with rough data, by analogy with the scattering theory for data in the energy space. This paper gives a positive answer to this question, at least for $s$ close enough to one.\
Of course, one cannot use the energy conservation law as because it can be infinite on $H^{s} \times H^{s-1}$, $s<1$. Instead we work with a mollified energy (defined in (\[Eqn:DefMolNrj\])) that is finite in these rough spaces (see Proposition \[Prop:EstInitMolNrj\]) and that is expected to be almost conserved [^4]: this is the $I$-method, introduced first in [@almckstt] to study global existence of rough solutions of semilinear Schrödinger equations and inspired by the *Fourier truncation method*, designed in [@bourgbook]. Since we basically introduced this multiplier $I$ to make the energy finite, we aim at proving decay estimates involving not only $u$ but also $I$. It was proved in [@triroyradnlkg] that, under the additional assumption of radial symmetry, we can control pretty easily $\| Iu \|_{L_{t}^{p+2} L_{x}^{p+2} (\mathbb{R})}$ by combining the “Almost Morawetz-Strauss estimate” (\[Eqn:AlmMorStrEst\]) with a pointwise decay estimate around the origin (a radial Sobolev inequality). Unfortunately, there is no useful pointwise estimate to our knowledge in the general case and we shall prove that, by means of concentration [@bourgjams], that a large number of norms are bounded. Several difficulties appear: we mention them now.

First we need to identify the norms we would like to control: we realize, after some tries, that the right candidates are $\| Iu
\|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (\mathbb{R})}$, with\
$\left( 2(p-1)-,2(p-1)+ \right)$ defined in Theorem \[Thm:Scat\].

Then we have to control the errors generated by this multiplier $I$, not only in order to establish upper bounds (see for example (\[Eqn:proc2\]) ) but also lower bounds (see (\[Eqn:IncreaseRate2\]) ). These bounds should not depend on time $T$, if not it would kill the scattering: this can be done by letting the estimates only depend on the parameter $N$, taking advantage of the local smoothing properties of the solutions. If we neglect, in a first approximation, the variation of the mollified energy in order to estimate the norm $\| I u
\|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}}$, we expect to find a tower-type exponential bound of this norm, by analogy with the bound of the $L_{t}^{q} L_{x}^{r}$ norm of the solutions of (\[Eqn:NlkgWdat\]) with data $(u_{0},u_{1}) \in H^{1} \times L^{2}$, say formally (by using (\[Eqn:BoundInitMolNrj\]) )

$$\begin{array}{l}
\| I u \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}} \lesssim \left( ( N^{1-s} A)^{\alpha} \right)^{ ( N^{1-s} A)^{\alpha}}
\end{array}$$

for some $\alpha >0$. Then combining this bound with the polynomial decay ( $\frac{1}{N^{\frac{5-p}{2}-}} $) of the almost conservation law we expect to find after iteration of the local theory (see Proposition \[prop:LocalBd\]) an error of the form

$$\begin{array}{l}
\frac{ \left( ( N^{1-s} A)^{\alpha} \right)^{ ( N^{1-s} A)^{\alpha}} } {N^{\frac{5-p}{2}-}}
\end{array}$$

and it seems at first sight that there do not exist $s_{0}:=s_{0}(p) < 1$ and $N:=N(A,p,s)$ such that we can make the error smaller than the upper bound $N^{2(1-s)} A^{p+1}$ (see Proposition \[Prop:EstInitMolNrj\]) of the mollified energy at initial time for $s \geq s_{0}$, since the denominator grows polynomially whereas the numerator grows exponentially. Notice, however, that

$$\begin{array}{l}
\lim_{s \rightarrow 1^{-}} \frac{ \left( ( N^{1-s} A)^{\alpha} \right)^{ ( N^{1-s} A)^{\alpha}} } { N^{2(1-s)} A^{p+1} N^{\frac{5-p}{2}-}}  =
\frac{ (A^{\alpha})^{A^{\alpha}}}{A ^{p+1} N^{\frac{5-p}{2}-}}
\end{array}$$

and therefore, choosing $N:= C  \left( (A^{\alpha})^{A^{\alpha}} \right)^{\frac{2}{5-p}+}$ (with $C:=C(p)>>1$), we see that there exists $s_{1}:=s_{1}(A,p) <1$ such that the error is small compare to $N^{2(1-s)} A^{p+1}$ for $ s_{1} \leq s < 1$: this is why the range of $s$ for which there is scattering is expected not 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 that, we have to prove that the variation of the mollified energy is small enough in order to construct a solution of which the mollified energy is smaller than the initial one by a nontrivial amount ( $(N^{s-1} \eta_{1} A^{-1})^{\alpha}$, see for example (\[Eqn:SepMolNrj\]) ), and this is why we introduce the parameter $\beta$: see (\[Eqn:DefBeta\]), (\[Eqn:MakeDecr\]) and (\[IneqChainNrj\]). The remainder of the argument is based upon a modification of arguments in [@nakanash1; @nakanash2], modulo errors.

$\textbf{Acknowledgements}:$ The second author would like to thank Kenji Nakanishi for providing him with explanations regarding [@nakanash1; @nakanash2] while he visited Courant Institute in March 2010. This work was done while the first author worked at Princeton University.

Proof of Theorem \[Thm:Scat\]
=============================

Let $\epsilon_{0}>0$ be a *fixed* constant (say $\epsilon_{0}=\frac{1}{1000}$).

In this section we prove Theorem \[Thm:Scat\], assuming that the following propositions are true [^5]. As it was pointed out in the introduction, we modify significantly an inductive argument ( see [@bourgjams; @nakanash1; @nakanash2] ). Unlike the energy space, we have to make a balance between the decay of the mollified energy at a given time and the growth of it, at each inductive step.

[**“ Separation of the localized mollified energy ”**]{} Let $A \gtrsim 1$. Let $J \subset [0,\infty)$. Let $w_{l}$ be a solution of (\[Eqn:NlkgWdat\]). Let $(C,Q,M) \in \mathbb{R}^{+} \times
\mathbb{R}^{+} \times[100, \infty)$. There exist $\eta_{1}:=\eta_{1}(p)>0$, $\alpha:=\alpha(p)>0$, $\bar{\alpha}:=\bar{\alpha}(p)$ and $C_{1}:=C_{1}(p)>> 1$ such that if

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

$$\begin{array}{l}
\| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}(J)} \leq 2 X_{C,Q}
\end{array}
\label{Eqn:InducStr}$$

and

$$\begin{array}{ll}
\|  I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J)} & \geq e^{ (C_{1} M e^{ C_{1} (N^{1-s} A)^{\alpha}} ) ^{ C_{1} (N^{1-s} A )^{\alpha}}}
\end{array}$$

then there exists $T_{l} \in J$ and $v_{l}$, solution of the free Klein-Gordon equation, that satisfy

$$\begin{array}{l}
E(Iw_{l}- Iv_{l},T_{l}) \leq  C_{E} N^{2(1-s)} A^{p+1} - (l+1)(1- \epsilon_{0})(N^{s-1}  \eta_{1} A^{-1})^{\alpha}
\end{array}
\label{Eqn:SepMolNrj}$$

$$\begin{array}{l}
E_{c}(v_{l},T_{l}) \lesssim (N^{s-1} \eta_{1}  A^{-1})^{\alpha}
\end{array}$$

and

$$\begin{array}{l}
\| I v_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}(T_{l},b)} \lesssim \frac{(N^{1-s} \eta_{1} A)^{\alpha}}{M^{\bar{\alpha}}}
\end{array}
\label{Eqn:FreeDecay}$$

assuming that $(N,s)$ satisfies the following conditions

$$\begin{array}{ll}
N^{1-s} A^{\frac{p+1}{2}} \frac{( A^{-1} \eta_{1} N^{s-1})^{(p-1) \alpha} }{ N^{\frac{5-p}{2}-}} & = o \left( N^{s-1} \eta_{1} A^{-1}
\right)^{\alpha}
\end{array}
\label{Eqn:Cond1}$$

$$\begin{array}{ll}
\frac{( N^{1-s} A^{\frac{p+1}{2}} )^{p-1} }{N^{\frac{5-p}{2}-}} & << 1
\end{array}
\label{Eqn:Cond2}$$

and

$$\begin{array}{ll}
\frac{(X_{C,Q})^{2(p-1)-} (N^{1-s} A^{\frac{p+1}{2}})^{p} }{N^{\frac{5-p}{2}-}} & << 1
\end{array}
\label{Eqn:Cond3}$$

Here

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

\[Prop:SepLocMol\]

[**“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 Proposition \[Prop:SepLocMol\]) and that there exists a constant $L(l+1)$ such that

$$\begin{array}{ll}
\| I w_{l+1} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}(J)} &  \leq L(l+1) \\
& \leq X_{C,Q}
\end{array}
\label{Eqn:BoundIw}$$

Then there exist $ C_{2}:= C_{2}(p)>> 1 $ and $\bar{C}:=\bar{C}(p) >> 1 $ such that if

$$\begin{array}{ll}
k & \leq \frac{1}{C_{2}^{C_{2}L(l+1) + 1}}
\end{array}
\label{Eqn:Boundk}$$

and

$$\begin{array}{ll}
\| I v_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([T_{l},b])} \leq k
\end{array}
\label{Eqn:BoundIv}$$

then

$$\begin{array}{l}
\| I  w_{l}  \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([T_{l},b])} \lesssim L(l+1)
\end{array}$$

provided that $(N,s)$ satisfies (\[Eqn:Cond2\]) and

$$\begin{array}{l}
\frac{\bar{C}^{\bar{C} X_{C,Q}} (N^{1-s} A^{\frac{p+1}{2}})^{p} X_{C,Q} } {N^{ \min{ \left( \left( s_{c} -\frac{1}{2} \right)-, (1-s_{c})-
\right) }}} << \eta_{0}
\end{array}
\label{Eqn:Cond4}$$

(with $\eta_{0}$ defined in Proposition \[prop:LocalBd\])

Here, again, $\beta$ is defined in (\[Eqn:DefBeta\]).

\[prop:PerturbArg\]

[**“ Initial mollified energy small implies control of $\| Iw \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (\mathbb{R})}$ and control of $\sup_{t \in \mathbb{R}} E(Iw(t))$ ”**]{} Assume that $w$ satisfies (\[Eqn:NlkgWdat\]) and that there exists $(i, t_{i}) \in
\mathbb{N} \times \mathbb{R}$ such that

$$\begin{array}{ll}
E(Iw (t_{i})) & \leq C_{E} N^{2(1-s)} A^{p+1} - i (1- \epsilon_{0}) (N^{s-1} \eta_{1} A^{-1})^{\alpha} \\
& \leq \frac{1}{10^{8}}
\end{array}
\label{Eqn:BoundInitSmall}$$

then

$$\begin{array}{l}
\| I w \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (\mathbb{R})}  \lesssim 1
\end{array}$$

and

$$\begin{array}{l}
\sup_{t \in \mathbb{R}} E(Iw(t)) \lesssim  \left( C_{E} N^{2(1-s)} A^{p+1} - i (1- \epsilon_{0}) (N^{s-1} \eta_{1} A^{-1})^{\alpha} \right) (1+
\beta)
\end{array}$$

provided that $(N,s)$ satisfies

$$\begin{array}{l}
\frac{ 2 (C_{E} N^{2(1-s)} A^{\frac{p+1}{2}})^{\frac{p-1}{2}} }{\beta N^{\frac{5-p}{2}-}} << 1
\end{array}
\label{Eqn:Cond6}$$

\[prop:EstSmallNrj\]

Let $(C,Q) \in (\mathbb{R}^{+})^{2}$. Let $C_{0} >> 1$ and $N:= \max{( N_{1},N_{2},N_{3},N_{4},N_{5})}$ with

$$\begin{array}{l}
N_{1}  := C_{0} (A^{-1} \eta_{1})^{\frac{2(p-2) \alpha}{5-p}+} A^{\frac{p+2}{5-p}+}
\end{array}$$

$$\begin{array}{l}
N_{2} := C_{0} A^{\frac{p^{2}-1}{5-p}+}
\end{array}$$

$$\begin{array}{l}
N_{3} := C_{0} (X_{C,Q})^{\frac{4(p-1)}{5-p}+} A^{\frac{p(p+1)}{5-p}+}
\end{array}$$

$$\begin{array}{l}
N_{4} := C_{0} ( \bar{C}^{\bar{C} X_{C,Q}} A^{\frac{p(p+1)}{2}} X_{C,Q})^{ \max{ \left( \frac{1}{s_{c} -\frac{1}{2}}+, \frac{1}{1-s_{c}}+
\right)}}
\end{array}$$

and

$$\begin{array}{l}
N_{5} := C_{0} \left( (2 X_{C,Q})^{2(p-1)-} 2 (C_{E} A^{p+1})^{\frac{p+1}{2}} \epsilon_{0}^{-1} (\eta_{1} A^{-1})^{- \alpha}
\right)^{\frac{2}{5-p}+}
\end{array}$$

Let $\tilde{s}:=\tilde{s}(A,C,Q,p) < 1 $ be such that $N^{1-s} \sim 1$ for $s \geq \tilde{s}(A,C,Q,p)$ [^6]. Notice that, for this range of $s$ and for this value of $N$, (\[Eqn:Cond1\]), (\[Eqn:Cond2\]), (\[Eqn:Cond3\]), (\[Eqn:Cond4\]) and (\[Eqn:Cond6\]) are satisfied ( if $C_{0}$ is large enough). Observe also that the following condition is satisfied

$$\begin{array}{ll}
\frac{( 2 X_{C,Q} )^{2(p-1)-} 2 \left( C_{E} N^{2(1-s)} A^{p+1} \right)^{\frac{p-1}{2}}  }   { \beta N^{\frac{5-p}{2}-}} & << 1
\end{array}
\label{Eqn:NControlVar}$$

since $N \geq N_{5}$. Next we define the following statement of induction for $(N,s)$ prescribed above:

$\mathcal{P}_{C,Q}(l)$ is the following: For $i \geq l$, let $\mathcal{C}_{i}$ be the following set

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

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)) {\lesssim_{A}}1 $ and $\| I u \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (\mathbb{R})} {\lesssim_{A}}1$, by time reversal symmetry. Indeed, notice first that it is enough by time reversal symmetry to establish these bounds restricted to $[t_{l}, \infty)$; then it is enough to apply $\mathcal{P}_{C,Q}(0)$ to $u$

-   $\mathcal{P}_{C,Q}(\sim A^{p+1 + \alpha})$ is true: it is enough to apply Proposition \[prop:EstSmallNrj\]

-   we want to prove that $\mathcal{P}_{C,Q}(l+1) \Longrightarrow \mathcal{P}_{C,Q}(l)$. Let $T> t_{l}$. We shall first prove that the bounds (\[Eqn:ControlIwStrich\]) and (\[Eqn:ControlEIw\]) hold if we are restricted to $[t_{l},T]$ [^8]. Then it is enough to let $T$ go to infinity in order to fully prove (\[Eqn:ControlIwStrich\]) and (\[Eqn:ControlEIw\]).\
    We introduce

    $$F_{T}:= \left\{
    T^{'} \in [t_{l}, \,T]:
    \begin{array}{ll}
    \sup_{t \in [t_{l}, \, T^{'}] } E(Iw_{l}(t))  \leq  \\
    (C_{E} N^{2(1-s)} A^{p+1} - l(1 - \epsilon_{0}) (N^{s-1} \eta_{1} A^{-1})^{\alpha})\left( 1+ \frac{\beta}{2} \right) \\
    \| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([t_{l},T^{'}])} \leq X_{C,Q}
    \end{array}
    \right\}
    $$

    We want to show that $F_{T}=[t_{l}, \, T]$ . Indeed

    -   $F_{T} \neq \emptyset $ since $ t_{l} \in F_{T}$, by (\[Eqn:DefCi\]).

    -   $F_{T}$ is closed by continuity

    -   $F_{T}$ is an open set of $[t_{l},T]$. Let $\widetilde{T'} \in F_{T}$. By continuity there exists $ \delta > 0$ such that for all $T^{'} \in
        (\widetilde{T^{'}} - \delta, \widetilde{T^{'}} + \delta) \cap [t_{l}, T]$ we have

        $$\begin{array}{l}
        \sup_{t \in [t_{l}, T^{'}]} E(Iw_{l}(t))  \leq \left( C_{E} N^{2(1-s)} A^{p+1} - l (1- \epsilon_{0}) (N^{s-1} \eta_{1} A^{-1})^{\alpha} \right)
        \left( 1+ \beta \right)
        \end{array}
        \label{Eqn:InducNrjThm}$$

        and

        $$\begin{array}{ll}
        \| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([t_{l},T^{'}])} & \leq 2 X_{C,Q}
        \end{array}
        \label{Eqn:BoundIwlXCQ}$$

        We divide $[t_{l},T^{'}]$ into subintervals $(J_{j})_{1 \leq j \leq m}$ such that $ \| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J_{j})} =
        \eta_{0} $ and $\| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J_{m}) } \leq \eta_{0}$. By Proposition \[prop:LocalBd\] and (\[Eqn:InducNrjThm\]) we have

        $$\begin{array}{ll}
        Z(w_{l},J_{j}) & \lesssim \left[ \left( C_{E} N^{2(1-s)} A^{p+1} - l(1- \epsilon_{0}) (N^{s-1} \eta_{1} A^{-1})^{\alpha} \right) (1+ \beta)
        \right]^{\frac{1}{2}}
        \end{array}$$

        Now, by applying Proposition \[Prop:Acl\], (\[Eqn:NControlVar\]) and (\[Eqn:BoundIwlXCQ\]) we see that in fact

        $$\begin{array}{ll}
        \sup_{t \in [0, T^{'}]} E(Iw_{l}(t))  \leq \left( C_{E} N^{2(1-s)} A^{p+1} - l (1- \epsilon_{0}) (N^{s-1} \eta_{1} A^{-1})^{\alpha} \right)
        \left( 1+ \frac{\beta}{2} \right)
        \end{array}$$

        Let $M$ be such that $ \frac{(N^{1-s} A)^{\alpha}}{M^{\bar{\alpha}}}:= \frac{1}{C_{2}^{C_{2} L(l+1) + 1}}$ ( with $\alpha$, $\bar{\alpha}$ and $C_{2}$ defined in Proposition \[prop:PerturbArg\]). If $ \| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([t_{l},T^{'}]) } \geq 2B $ then this implies that there exists $T_{0} \in [t_{l},T^{'}]$ such that $ \| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([t_{l},T_{0}])} \geq B$ and $\|
        I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([T_{0},T^{'}])} \geq B $. There are two options:

        -   $$\begin{array}{ll}
            B & \geq e^{(C_{1} M e^{C_{1} (N^{1-s} A )^{\alpha}})^{C_{1} (N^{1-s} A)^{\alpha}}}
            \end{array}
            \label{Eqn:BoundB1}$$

            with $C_{1}$ defined in Proposition \[Prop:SepLocMol\]. Then, in this case, applying Proposition \[Prop:SepLocMol\] to $J=[t_{l},T_{0}]$, we see that there exists $T_{l} \in J$ such that

            $$\begin{array}{ll}
            E(Iw_{l}-Iv_{l}, T_{l}) & \leq \left( C_{E} N^{2(1-s)} A^{p+1} - l(1-\epsilon_{0}) (N^{s-1} \eta_{1} A^{-1})^{\alpha} \right)(1+ \beta) - (N^{s-1} A^{-1})^{\alpha} \\
            & \leq C_{E} N^{2(1-s)} A^{p+1} - (l+1)(1- \epsilon_{0})(N^{s-1} \eta_{1} A^{-1})^{\alpha}
            \end{array}
            \label{Eqn:MakeDecr}$$

            by our choice of $\beta$ (see (\[Eqn:DefBeta\])) and

            $$\begin{array}{ll}
            \| I v_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ( T_{l},T^{'} ) } & \leq \frac{1}{C_{2}^{C_{2} L(l+1) +1 }}
            \end{array}$$

            and letting $w_{l+1}$ be the solution of (\[Eqn:NlkgWdat\]) with data $w_{l+1}(T_{l}):=(w_{l}-v_{l})(T_{l})$, we see that, by using $\mathcal{P}_{C,Q}(l+1)$ and Proposition \[prop:PerturbArg\]

            $$\begin{array}{ll}
            B & \leq \| I w_{l} \|_{ L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ( [T_{0},T^{'}] ) } \\
            & \leq \| I w_{l} \|_{ L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ( [T_{l},T^{'}] ) } \\ \
            & \lesssim L(l+1)  \\
            \end{array}$$

            Therefore, in this case,

            $$\begin{array}{ll}
            \| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([t_{l},T^{'}])} & \lesssim L(l+1)
            \end{array}$$

        -   There exists a constant $\tilde{C}$ large enough such that

            $$\begin{array}{l}
            B \leq e^{(C_{1} M e^{C_{1} (N^{1-s} A )^{\alpha}})^{C_{1} (N^{1-s} A)^{\alpha}}} \\
            \leq e^{ \left( \tilde{C}^{\tilde{C}} \tilde{C}^{\tilde{C} \, L(l+1)} A^{\tilde{C}} e^{\tilde{C} \, A^{\tilde{C}}} \right) ^{ \tilde{C} \,
            A^{\tilde{C}} }}
            \end{array}$$

            and, in this case

            $$\begin{array}{ll}
            \| I w_{l} \|_{ L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([t_{l},T^{'}])} & \lesssim e^{ \left( \tilde{C}^{\tilde{C}} \tilde{C}^{\tilde{C} L(l+1)}
            A^{\tilde{C}} e^{\tilde{C} A^{\tilde{C}}} \right) ^{ \tilde{C} A^{\tilde{C}} }}
            \end{array}
            \label{Eqn:BoundIwl}$$

        Therefore (\[Eqn:BoundIwl\]) holds in both cases. Now we iterate, and it is even simpler since we can use (\[Eqn:ContBoundLi\]) and (\[Eqn:ControlEIw\]): we do not even need to introduce the set $F_{T}$. For example, let us sketch the proof to estimate $L(l+1)$ from $L(l+2)$. Let $w^{'}_{l+1} \in \mathcal{C}_{l+1}$. Assume that $\| I w^{'}_{l+1} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (\mathbb{R}^{+})} \geq 2B$. Then this implies that there exists $T_{0} \in \mathbb{R}^{+}$ such that $\| I w^{'}_{l+1} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([0,T_{0}])}
        \geq B $ and\
        $ \| I w^{'}_{l+1} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}([T_{0}, \infty))} \geq B$. Now, by (\[Eqn:ContBoundLi\]) and (\[Eqn:ControlEIw\]) we see that we can apply Proposition \[Prop:SepLocMol\] to $J=[0,T_{0}]$ and we find, after letting $M$ be such that $\frac{(N^{1-s}A )^{\alpha}}{M^{\alpha}} := \frac{1}{ C_{2}^{C_{2} L(l+2) +1} }$ and after slightly modifying the steps from (\[Eqn:BoundB1\]) to (\[Eqn:BoundIwl\]), that

        $$\begin{array}{ll}
        \| I w^{'}_{l+1} \|_{ L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (\mathbb{R}^{+})} & \lesssim e^{ \left( \tilde{C}^{\tilde{C}} \tilde{C}^{\tilde{C} L(l+2)}
        A^{\tilde{C}} e^{\tilde{C} A^{\tilde{C}}} \right) ^{ \tilde{C} A^{\tilde{C}} }}
        \end{array}$$

        and therefore

        $$\begin{array}{ll}
        L(l+1) & \lesssim e^{ \left( \tilde{C}^{\tilde{C}} \tilde{C}^{\tilde{C} \, L(l+2)} A^{\tilde{C}} e^{\tilde{C} \, A^{\tilde{C}}} \right) ^{
        \tilde{C} A^{\tilde{C}} }}
        \end{array}$$

        iterating $ \lesssim A^{p+1 + \alpha}$ times, we see that, letting $C:=\tilde{C}$ and $Q:= \tilde{Q}$ (with $\tilde{Q} \sim A^{p+1 + \alpha}$), we have

        $$\begin{array}{ll}
        \| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([0,T^{'}]) } & \leq X_{C,Q}
        \end{array}$$

        , by using Proposition \[prop:EstSmallNrj\].

        Therefore $\mathcal{P}_{C,Q}(l)$ holds, for these values of $C$ and $Q$.

**Global existence**\
$$\begin{array}{ll}
\sup_{t \in [-T,T]} E(Iu(t)) & \lesssim_{A} 1
\end{array}
\label{Eqn:BdNrjNrjIn}$$ and

$$\begin{array}{ll}
\| I u \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([-T,T])} {\lesssim_{A}}1
\end{array}$$

Now by Plancherel and (\[Eqn:BdNrjNrjIn\])

$$\begin{array}{ll}
\| ( u(T),\partial_{t} u(T)) \|^{2}_{H^{s} \times H^{s-1}} & \lesssim E(Iu(T)) \\
& {\lesssim_{A}}1
\end{array}
\label{Eqn:BounduTMolNrj}$$

This proves global well-posedness of (\[Eqn:NlkgWdat\]) with data $(u_{0},u_{1}) \in H^{s} \times H^{s-1}$ such that $\| (u_{0},u_{1})
\|_{H^{s} \times H^{s-1}} \leq A$ and $1 > s > \tilde{s}:=\tilde{s}(A,p)$. Moreover by letting $T$ go to infinity, we have in fact

$$\begin{array}{ll}
\sup_{t \in \mathbb{R}} E(Iu(t))  & {\lesssim_{A}}1
\end{array}
\label{Eqn:BdRNrj}$$

and

$$\begin{array}{ll}
\| I u \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (\mathbb{R})} {\lesssim_{A}}1
\end{array}
\label{Eqn:BdLgEst}$$

This implies that $\mathcal{P}_{C,Q}(l)$ is true.

**Global estimates**\
Now we claim that

$$\begin{array}{ll}
Z_{m,s}(\mathbb{R},u) & {\lesssim_{A}}1
\end{array}
\label{Eqn:GlobEst}$$

for all $0 \leq m \leq s$. Let $ \mathcal{P}:= (J_{l}=[a_{l}, \, b_{l}])_{1 \leq l \leq L }$ be a partition of $[0, \infty)$ such that $\| Iu
\|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}(J_{l})} = \eta_{0}$, $l < L$ and $\| I u \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}(J_{L})} \leq \eta_{0}$.

$$\begin{array}{ll}
\| I u \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}(J_{l}) } & \leq  \eta_{0}
\end{array}
\label{Eqn:LocLongThm}$$

with $\eta_{0}$ defined in Proposition \[prop:LocalBd\]. Notice that this implies that $L {\lesssim_{A}}1$ by (\[Eqn:BdLgEst\]). Moreover by slightly modifying the proof of Proposition \[prop:LocalBd\] we see that

-   if $m=0$ then $Z_{m,s}(J_{l},u) {\lesssim_{A}}1$: it follows from (\[Eqn:BdRNrj\])

-   $m=s_{c}$: by slightly modifying (\[Eqn:Comput\])

    $$\begin{array}{ll}
    Z_{m,s}(J_{l},u) & \lesssim N^{1-s} A^{\frac{p+1}{2}} + Z_{s_{c},s} (J_{l},u) \left( \eta_{0}^{p-1} + \frac{Z^{p-1}_{s_{c},s}(J_{l},u)
    }{N^{\frac{5-p}{2}-}} \right) \\
    & \lesssim N^{1-s} A^{\frac{p+1}{2}} \\
    & {\lesssim_{A}}1
    \end{array}$$

    by a continuity argument and our choice of $N$.

-   $ 0<  m < s_{c}$: it follows by interpolating between $m=s_{c}$ and $m=0$

-   $m=1$: again, by slightly modifying the proof of Proposition \[prop:LocalBd\] we have

    $$\begin{array}{ll}
    Z_{1,s}(J_{l},u) & \lesssim N^{1-s} A^{\frac{p+1}{2}} + \eta^{p-1}_{0} Z_{1,s}(J_{l},u) + \frac{Z_{1,s}(J_{l},u) N^{1-s}
    A^{\frac{p+1}{2}}}{N^{\frac{5-p}{2}-}} \\ & + \frac{ (N^{1-s} A^{\frac{p+1}{2}})^{p-1} }  {N^{\frac{5-p}{2}-}} Z_{1,s}(J_{l},u) +
    \frac{Z^{p}_{\frac{2}{p}-,s} (J_{l},u) }{N^{\frac{5-p}{2}-}} \\
    & \lesssim N^{1-s} A^{\frac{p+1}{2}} \\
    & {\lesssim_{A}}1
    \end{array}$$

    by a continuity argument and our choice of $N$

-   if $s_{c} < m < 1$ then $Z_{m,s}(J_{l},u) \lesssim N^{1-s} A^{\frac{p+1}{2}} {\lesssim_{A}}1$: if follows by interpolating between $m=s_{c}$ and $m=1$.

Therefore, iterating over $l$, we get (\[Eqn:GlobEst\]).

**Scattering**\
Let

$$\begin{array}{ll}
v(t) & := \left(
\begin{array}{l}
u(t) \\
\partial_{t} u(t)
\end{array}
\right)
\end{array}$$

$$\begin{array}{ll}
v_{0} : = \left(
\begin{array}{l}
u_{0} \\
u_{1}
\end{array}
\right)
\end{array}$$

$$\begin{array}{ll}
\mathbf{K}(t) & := \left(
\begin{array}{cc}
\cos{(t \langle D \rangle)} & \frac{\sin{(t \langle D \rangle)}}{\langle D \rangle} \\ & \\
-\langle D \rangle \sin {(t \langle D \rangle)} & \cos{(t \langle D \rangle)}
\end{array}
\right)
\end{array}$$

and

$$\begin{array}{ll}
\mathbf{u_{nl}}(t) & = \left(
\begin{array}{l}
\int_{0}^{t} \frac{\sin{\left((t-t^{'}) \langle D \rangle \right)}}{\langle D \rangle} \left( |u|^{p-1}(t^{'}) u(t^{'}) \right) \, d t^{'} \\ \\
\int_{0}^{t} \cos{ \left( (t-t^{'}) \langle D \rangle \right)} \left( |u|^{p-1}(t^{'}) u(t^{'}) \right) \, d t^{'}
\end{array}
\right)
\end{array}$$

Then we get from (\[Eqn:StrongSol\])

$$\begin{array}{ll}
\mathbf{v}(t) & = \mathbf{K}(t) \mathbf{v_{0}} - \mathbf{u_{nl}}(t)
\end{array}$$

Recall that the solution $u$ scatters in $H^{s} \times H^{s-1}$ if there exists

$$\begin{array}{ll}
\mathbf{v_{+,0}} & :=  \left(
\begin{array}{l}
u_{0,+} \\
u_{1,+}
\end{array}
\right)
\end{array}
\label{Eqn:ScatBegin}$$

such that

$$\left\|   \mathbf{v}(t)
 -\mathbf{K}(t) \mathbf{v_{+,0}} \right\|_{H^{s} \times H^{s-1}}$$

has a limit as $t \rightarrow \infty$ and the limit is equal to $0$. In other words since $K$ is bounded on $H^{s} \times H^{s-1}$ it suffices to prove that the quantity

$$\left\| \mathbf{K}^{-1}(t) \mathbf{v}(t) -  \mathbf{v_{+,0}}   \right\|_{H^{s} \times H^{s-1}}$$

has a limit as $t \rightarrow \infty$ and the limit is equal to $0$. A computation shows that

$$\begin{array}{ll}
\mathbf{K}^{-1}(t) & = \left(
\begin{array}{cc}
\cos{(t \langle D \rangle )} & -\frac{\sin{(t \langle D \rangle)}}{\langle D \rangle} \\ & \\
\langle D \rangle \sin {(t \langle D \rangle)} & \cos{(t \langle D \rangle)}
\end{array}
\right)
\end{array}$$

But

$$\begin{array}{ll}
\mathbf{K}^{-1}(t) \mathbf{v}(t) &  =  \mathbf{v_{0}} - \mathbf{K}^{-1}(t) \mathbf{u^{nlin}}(t)
\end{array}
\label{Eqn:ScatEnd}$$

By Proposition \[prop:StrEstLtqLxr\]

$$\begin{array}{ll}
\| \mathbf{K}^{-1}(t_{1}) \mathbf{u^{nlin}}(t_{1}) - \mathbf{K}^{-1}(t_{2}) \mathbf{u^{nlin}}(t_{2}) \|_{H^{s} \times H^{s-1}} & \lesssim \|
\mathbf{u^{nlin}}(t_{1}) - \mathbf{u^{nlin}}(t_{2})  \|_{H^{s} \times
H^{s-1}} \\
& \lesssim \| |u|^{p-1} u \|_{L_{t}^{\frac{2}{1+s}} L_{x}^{\frac{2}{2-s}} ([t_{1}, \, t_{2}]) } \\
& \lesssim \| \langle D \rangle^{1-s} I \left(  |u|^{p-1} u  \right) \|_{L_{t}^{\frac{2}{1+s}} L_{x}^{\frac{2}{2-s}} ([t_{1}, \, t_{2}]) }
\end{array}
\label{Eqn:CauchyScat}$$

But, plugging $\langle D \rangle^{1-s}$ into (\[Eqn:StrNlkg\]), we get

$$\begin{array}{l}
 \| \langle D \rangle^{1-s} I \left(  |u|^{p-1} u  \right) \|_{L_{t}^{\frac{2}{1+s}} L_{x}^{\frac{2}{2-s}} ([t_{1}, \, t_{2}]) } \lesssim
 \| \langle D \rangle^{1-s} Iu  \|_{L_{t}^{\frac{2}{s}+} L_{x}^{\frac{2}{1-s}-}([t_{1}, \, t_{2}])}
 \| u \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([t_{1},t_{2}])} \\
 \lesssim \| \langle D \rangle^{1-s} I u \|_{L_{t}^{\frac{2}{s}+} L_{x}^{\frac{2}{1-s}-}([t_{1}, \, t_{2}])}
\left( \| P_{<<N} u \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([t_{1},t_{2}])} + \| P_{\gtrsim N} u \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}
([t_{1},t_{2}])} \right) \\
\lesssim Z_{s,s}([t_{1},t_{2}],u) \left( \| I u \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([t_{1},t_{2}]) } +
 \frac{ \| \langle D \rangle^{1- s_{c}} I u  \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([t_{1},t_{2}])}}{N^{ \frac{5-p}{2} -}}
 \right) \\
\lesssim Z_{s,s}([t_{1},t_{2}],u) \left( \| I u \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([t_{1},t_{2}]) } + \frac{Z^{p-1}_{s_{c},s}
([t_{1},t_{2}],u)} {N^{\frac{5-p}{2} -}} \right)
\end{array}
\label{Eqn:Donemins}$$

Therefore in view of (\[Eqn:BdLgEst\]), (\[Eqn:GlobEst\]), (\[Eqn:CauchyScat\]) and (\[Eqn:Donemins\]), we see that the Cauchy criterion is satisfied by $\mathbf{K}^{-1}(t) v(t)$ and we conclude that $\mathbf{K}^{-1}(t) v(t)$ has a limit in $H^{s} \times H^{s-1}$ as $t$ goes to infinity. Moreover

$$\begin{array}{ll}
\lim \limits_{t \rightarrow \infty} \left\|  \mathbf{v}(t) -  \mathbf{K}(t)\mathbf{v_{+,0}}   \right\|_{H^{s} \times H^{s-1}} & = 0
\end{array}$$

with $\mathbf{v_{+,0}}$ defined in (\[Eqn:ScatBegin\]),

$$\begin{array}{ll}
u_{+,0} & := u_{0} + \int_{0}^{\infty} \frac{\sin{(t^{'} \langle D \rangle)}}{\langle D \rangle} \left( |u|^{p-1}(t^{'}) u(t^{'}) \right) \, d
t^{'}
\end{array}$$

and

$$\begin{array}{ll}
u_{+,1} & := u_{1} - \int_{0}^{\infty} \cos{(t^{'} \langle D \rangle)} \left( |u|^{p-1}(t^{'}) u(t^{'}) \right) \, d t^{'}
\end{array}$$

Proof of Proposition \[prop:EstSmallNrj\] {#Sec:EstSmallNrj}
=========================================

In this section we prove Proposition \[prop:EstSmallNrj\].

The proof is made of three steps:

-   Control of $\| I w \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (\mathbb{R})}$: by looking at the proof of Proposition \[prop:LocalBd\], we realize that

    $$\begin{array}{ll}
    Z_{s_{c},s}(\mathbb{R},w) & \lesssim \frac{1}{10^{8}} + \| \langle D \rangle^{1-s_{c}} I w  \|_{L_{t}^{\frac{2}{s_{c}} +}
    L_{x}^{\frac{2}{1-s_{c}}} (\mathbb{R})} \\
    & \left( \| Iw \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (\mathbb{R})} + \frac{ \|
    \langle D \rangle^{1-s_{c}} I w  \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (\mathbb{R}) } } {N^{\frac{5-p}{2}-}} \right) \\
    & \lesssim \frac{1}{10^{8}} + \left( Z^{p}_{s_{c},s}(w,\mathbb{R}) + \frac{Z^{p}_{s_{c},s} (w,\mathbb{R})}{N^{\frac{5-p}{2}-}}  \right)
    \end{array}$$

    where we used the Sobolev embedding

    $$\begin{array}{ll}
    \| I w \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (\mathbb{R})} & \lesssim  \| \langle D \rangle^{1- s_{c}} I w \|_{L_{t}^{2(p-1)-}
    L_{x}^{\frac{6(p-1)}{2p-3}+ }(\mathbb{R})}
    \end{array}
    \label{Eqn:SobEmbedp}$$

    and, by a continuity argument, we have $Z_{s_{c},s}(\mathbb{R},w) \lesssim 1$ hence

    $$\begin{array}{ll}
    \| I w \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}(\mathbb{R})} & \lesssim 1
    \end{array}
    \label{Eqn:EndIu}$$

-   Control of $\sup_{t \in \mathbb{R}} E(Iw(t))$.

    We define the set

    $$\begin{array}{ll}
    F & : =\left\{ T \in \mathbb{R}, \, \sup_{t \in [-T,T]} E(Iw(t)) \leq \left( C_{E} N^{2(1-s)} A^{p+1} - i(1- \epsilon_{0}) (N^{s-1} \eta_{1}
    A^{-1})^{\alpha} \right) \left( 1+ \frac{\beta}{2} \right) \right\}
    \end{array}$$

    -   $F \neq \emptyset$ since $T_{l} \in F$

    -   $F$ is closed by continuity

    -   $F$ is open: given $T \in F$, there exists by continuity $\delta > 0$ such that for $T^{'} \in (T_{l}, T+ \delta)$

        $$\begin{array}{ll}
        \sup_{t \in [-T^{'},T^{'}]} E(Iw(t)) \leq \left( C_{E} N^{2(1-s)} A^{p+1} - i(1- \epsilon_{0}) (N^{s-1} \eta_{1} A^{-1})^{\alpha} \right) \left(
        1+ \beta \right)
        \end{array}
        \label{Eqn:InducNrjSmall}$$

        Using (\[Eqn:InducNrjSmall\]) and (\[Eqn:EndIu\]), we see that, by looking at the proof of Proposition \[prop:LocalBd\], that $Z_{m,s}(w,[-T^{'}, T^{'}]) << 1$ for all $m \in [0,1]$, since $(N,s)$ satisfies (\[Eqn:Cond2\]). Combining these bound with Proposition \[Prop:Acl\] and (\[Eqn:BoundInitSmall\]), we see that in fact

        $$\begin{array}{l}
        \sup_{t \in [-T^{'},T^{'}]} E(Iw(t)) \left( C_{E} N^{2(1-s)} A^{p+1} - i(1- \epsilon_{0}) (N^{s-1} \eta_{1} A^{-1})^{\alpha} \right) \left( 1+
        \frac{\beta}{2} \right)
        \end{array}$$

        since $(N,s)$ also satisfies (\[Eqn:Cond6\]).

Proof of Proposition \[prop:LocalBd\]
=====================================

In this section we prove Proposition \[prop:LocalBd\]. Let

$$\begin{array}{l}
Y:= \left( \left( C_{E} N^{1-s} A^{p+1}  - l(1- \epsilon_{0}) (N^{s-1} A^{-1})^{\alpha} \right) (1+ \beta) \right)^{\frac{1}{2}}
\end{array}$$

Plugging $\langle D \rangle ^{1-m}I$ into (\[Eqn:StrNlkg\]) we have

$$\begin{array}{ll}
Z_{m,s}(J,w_{l}) & \lesssim  Y  + \| \langle D \rangle^{1-m} I ( |w_{l}|^{p-1} w_{l} ) \|_{L_{t}^{\frac{2}{1+m}} L_{x}^{\frac{2}{2-m}}(J)}
\end{array}
\label{Eqn:PlugNrjStr}$$

There are five cases:

-   $m=0$: it follows from (\[Eqn:InducNrj\])

-   $ m = s_{c}$: we have

    $$\begin{array}{l}
    Z_{s_{c},s}(J,w_{l})  \lesssim Y  +  \| \langle D \rangle^{1-s_{c}} I w_{l} \|_{L_{t}^{\frac{2}{s_{c}}+} L_{x}^{\frac{2}{1-s_{c}}-}(J)}
    \| |w_{l}|^{p-1} \|_{L_{t}^{2-} L_{x}^{2+}(J)} \\
    \lesssim Y + Z_{s_{c},s}(J,w_{l}) \| w_{l} \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}(J)} \\
    \lesssim Y + Z_{s_{c},s}(J,w_{l}) \left(  \| P_{<<N} w_{l} \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J)} + \| P_{\gtrsim N} w_{l}
    \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J)} \right) \\
    \lesssim Y + Z_{s_{c},s}(J,w_{l}) \left( \| I w_{l} \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J)} + \frac{ \| \langle D
    \rangle^{1- s_{c}} I w_{l} \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J)}}{N^{\frac{5-p}{2}-}}  \right) \\
    \lesssim Y +  Z_{s_{c},s}(J,w_{l}) \left( \eta^{p-1}_{0} + \frac{Z^{p-1}_{s_{c},s} (J,w_{l})}{N^{\frac{5-p}{2}-}} \right)
    \end{array}
    \label{Eqn:Comput}$$

    Therefore, by a continuity argument, we conclude that

    $$\begin{array}{l}
    Z_{s_{c},s}(J,w_{l}) \lesssim Y
    \end{array}
    \label{Eqn:BoundZsc}$$

-   $0 <  m  < s_{c}$: It follows by interpolating $m=s_{c}$ and $m=0$

-   $m={1}$. We have

    $$\begin{array}{l}
    Z_{m,s}(J,w_{l}) \lesssim Y + \|  I (|w_{l}|^{p-1} w_{l}) \|_{L_{t}^{1} L_{x}^{2} (J)} \\ \\
    \lesssim Y + \| |w_{l}|^{p-1} w_{l} \|_{L_{t}^{1} L_{x}^{2}(J)} \\ \\
    \lesssim Y + \| |P_{<<N} w_{l}|^{p-1}  P_{<< N} w_{l}  \|_{L_{t}^{1} L_{x}^{2} (J)} +  \| |P_{<<N} w_{l}|^{p-1}
    P_{\gtrsim N} w_{l} \|_{L_{t}^{1} L_{x}^{2}(J)} \\ \\
    + \| |P_{\gtrsim N} w_{l}|^{p-1} P_{<<N} w_{l} \|_{L_{t}^{1} L_{x}^{2} (J)} +  \| |P_{\gtrsim N} w_{l}|^{p-1} P_{\gtrsim N} w_{l} \|_{L_{t}^{1}
    L_{x}^{2}(J)} \\ \\
    \lesssim Y + B_{1} + B_{2} + B_{3} + B_{4}
    \end{array}
    \label{Eqn:0}$$

    But

    $$\begin{array}{l}
    B_{1} \lesssim  \| I w_{l} \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J)} \| I w_{l} \|_{L_{t}^{2+} L_{x}^{\infty-}(J)} \\ \\
    \lesssim \eta^{p-1}_{0} Z_{1,s}(w_{l},J)
    \end{array}
    \label{Eqn:1}$$

    As for $B_{2}$, $B_{3}$ and $B_{4}$, we have (see also [@triroyradnlkg] for a similar argument)

    $$\begin{array}{l}
    B_{2} \lesssim   \| I w_{l} \|^{p-1}_{L_{t}^{p-1} L_{x}^{\frac{6(p-1)}{p-3}} (J) } \| P_{\gtrsim N} w_{l} \|_{L_{t}^{\infty}
    L_{x}^{\frac{6}{6-p}}(J)} \\ \\
    \lesssim Z^{p-1}_{1,s}(J,w_{l}) \frac{\| \langle D \rangle I w_{l} \|_{L_{t}^{\infty} L_{x}^{2}(J)} }{N^{\frac{5-p}{2}-}} \\ \\
    \lesssim \frac{Z^{p-1}_{1,s} (J,w_{l})}{N^{\frac{5-p}{2}-}} Y
    \end{array}
    \label{Eqn:2}$$

    $$\begin{array}{l}
    B_{3} \lesssim  \frac{ \| \langle D \rangle^{1-s_{c}} I w_{l} \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}(J)}}{N^{\frac{5-p}{2}}} \| Iw_{l}
    \|_{L_{t}^{2+} L_{x}^{\infty -}(J)} \\ \\
    \lesssim \frac{Z^{p-1}_{s_{c},s}(J,w_{l})}{N^{\frac{5-p}{2}-}} Z_{1,s}(J,w_{l}) \\ \\
    \lesssim \frac{Y^{p-1} }{N^{\frac{5-p}{2}-}} Z_{1,s}(J,w_{l})
    \end{array}
    \label{Eqn:3}$$

    and

    $$\begin{array}{l}
    B_{4} \lesssim  \| P_{\gtrsim N} w_{l} \|^{p}_{L_{t}^{p} L_{x}^{\frac{2p}{p-2}}(J)} \\
    \lesssim  \frac{ \| \langle D \rangle^{1- \left( \frac{2}{p} \right)} I w_{l} \|^{p}_{L_{t}^{p} L_{x}^{\frac{2p}{p-2}}(J)}
    }{N^{\frac{5-p}{2}-}} \\
    \lesssim \frac{ Z^{p}_{\frac{2}{p},s} (J,w_{l}) }{N^{\frac{5-p}{2}-}} \\
    \lesssim \frac{Y^{p}}{N^{\frac{5-p}{2}-}}
    \end{array}
    \label{Eqn:4}$$

    Therefore, by Proposition \[Prop:EstInitMolNrj\], (\[Eqn:Comput\]), (\[Eqn:0\]), (\[Eqn:1\]), (\[Eqn:2\]), (\[Eqn:3\]), (\[Eqn:4\]) and a continuity argument we also get

    $$\begin{array}{l}
    Z_{1,s}(w_{l},J) \lesssim Y
    \end{array}$$

-   $s_{c}<  m < 1$: it follows by interpolating between $m=s_{c}$ and $m=1$.

Proof of Proposition \[Prop:SepLocMol\]
=======================================

The proof of Proposition \[Prop:SepLocMol\] relies upon several conditions and three lemmas that we show in the next subsections.

[**“ Concentration of $\| Iw_{l}  \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} }$ implies concentration of potential term of the mollified energy ”**]{} Assume that $\| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J) } = \eta$ with $\eta \leq \eta_{0} << 1$ [^9]. Then there exist $\alpha:=\alpha(p)
>0$, a subinterval $K \subset J$, $R>0$, $c:=c(p)>0$ , $C:=C(p)>0$ and $x_{0} \in  \mathbb{R}^{3}$ such that

-   $$\begin{array}{ll}
    R & := C (\eta^{-1} A N^{1-s})^{\alpha}
    \end{array}
    \label{Eqn:DefR}$$

-   $$\begin{array}{ll}
    \int_{|x-x_{0}| \leq R} |Iw_{l}(t,x)|^{p+1} \, dx & \gtrsim (\eta A^{-1} N^{s-1})^{\alpha}
    \end{array}
    \label{Eqn:ConcNrjMol}$$

    for all $t \in K$

-   $$\begin{array}{ll}
    |K| & = c (\eta A^{-1} N^{s-1})^{\alpha}
    \end{array}
    \label{Eqn:SizeK}$$

\[lem:ConcNrj\]

[**“Logarithmic growth of size of intervals on which there is concentration”**]{}

Let $ (J_{j}=[a_{j},a_{j+1}])_{ 1 \leq j \leq \bar{j}} $ be a partition of $J$ such that $\| Iw_{l}  \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}
(J_{j})} = \eta$ with $\eta << 1$, $j< \bar{j}$ and $\| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J_{\bar{j}})} \leq \eta$. Then there exist $t_{j} \in J_{j}$ and $\alpha:=\alpha(p)
> 0$ such that

$$\begin{array}{ll}
\sum_{j=1}^{\bar{j}} \frac{1}{t_{j} + 1} & \lesssim (N^{1-s} A \eta^{-1})^{\alpha}
\end{array}
\label{Eqn:RepartJj}$$

\[lem:UseMorEst\]

[**“If $\| I w_{l} \|_{L_{t}^{2(p-1)} L_{x}^{2(p-1)}(J)}$ large enough then large subinterval s.t Strichartz norms smaller compare to concentration of mollified energy”**]{}

Let $M \geq 100$. If $K \subset J$ is a subinterval, let

$$\begin{array}{l}
\bar{Z}(K,w_{l}) := \| \partial_{t} \langle D \rangle^{-\frac{1}{2}} I w_{l} \|_{L_{t}^{4} L_{x}^{4} (K)} +  \| I w_{l} \|_{L_{t}^{2(p-1)-}
L_{x}^{2(p-1)+} (K)} \\
+ \| \langle D \rangle^{1- \frac{2}{p}} I w_{l} \|_{L_{t}^{p} L_{x}^{\frac{2p}{p-2}} (K) }   + \| \langle D
\rangle^{1-\frac{3p-5}{2p}} I w_{l} \|_{L_{t}^{p} L_{x}^{2p} (K)} \\
+ \| \langle D \rangle^{1-\frac{1}{2}} I w_{l}  \|_{L_{t}^{4+} L_{x}^{4-} (K) } + \| \langle D \rangle^{1- \frac{p-3}{2} } I w_{l}
\|_{L_{t}^{\frac{4}{p-3}} L_{x}^{\frac{4}{5-p} }(K)} \\
+ \| \langle D \rangle^{1-1} I w_{l} \|_{L_{t}^{\frac{4(p-1)}{7-p}} L_{x}^{\frac{4(p-1)}{p-3}} (K) }
\end{array}
\label{Eqn:DefBarZ}$$

Then there exist $\alpha:=\alpha(p) > 0$, $\eta_{1}:= \eta_{1}(p) << 1$, $C_{1}:=C_{1}(p) >> 1$, $R^{'} \in (1,\infty)$ , $[S_{l},T_{l}] \subset
J$, $\bar{x} \in \mathbb{R}^{3}$ and $R^{'}>1$ such that for all $\epsilon < \frac{1}{100}$, if

$$\begin{array}{l}
\| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J)} \geq e^{  \left( C_{1} M  e^{C_{1} (N^{1-s}  A \epsilon^{-1})^{\alpha} } \right)^{C_{1}
(N^{1-s } A )^{\alpha}}}
\end{array}
\label{Eqn:UpperBdIu}$$

then

$$\begin{array}{l}
\bar{Z}([S_{l},T_{l}],w_{l}) \leq 3 (\eta_{1} A^{-1} N^{s-1})^{\alpha} \leq E( Iw_{l}(S_{l}),B(\bar{x},R^{'}))
\end{array}
\label{Eqn:BoundZpr}$$

$$\begin{array}{l}
|T_{l}-S_{l}| \geq M R^{'}
\end{array}
\label{Eqn:LowerBoundST}$$

and

$$\begin{array}{l}
\left\| \frac{Iw_{l}(S_{l})}{\langle x-\bar{x} \rangle} \right\|_{L^{2}} \leq \epsilon
\end{array}
\label{Eqn:UpBdWeight}$$

\[lem:StrVsConcNrj\]

Assuming that Lemma \[lem:ConcNrj\], Lemma \[lem:UseMorEst\] and Lemma \[lem:StrVsConcNrj\] are true, let us prove Proposition \[Prop:SepLocMol\]. We apply Lemma \[lem:StrVsConcNrj\] with [^10]

$$\begin{array}{ll}
\epsilon & := \frac{1}{1000} (N^{s-1} \eta_{1} A^{-1})^{\alpha}  N^{s-1} A^{-\frac{p+1}{2}}
\end{array}$$

The proof is made of several steps:

-   Construction of the free Klein-Gordon equation $v_{l}$

    Let $P(y)$: $$\begin{array}{ll}
    E \left( Iw_{l}(S_{l}), B (y,1) \right) & \leq \frac{1}{100} (N^{s-1} \eta_{1} A^{-1})^{\alpha}
    \end{array}$$ By (\[Eqn:InducNrj\]) there exists $\bar{\bar{x}} \in \mathbb{R}^{3}$ such that, even if it means increasing the value of $\alpha$, there exists $\bar{\bar{x}} \in \mathbb{R}^{3}$ such that $|\bar{x} - \bar{\bar{x}}| \lesssim (N^{1-s} A)^{\alpha}$ and such that $P(\bar{\bar{x}})$ is true. By this observation, notice (see [@nakanash1]) that there exists a constant $C >>1$ and $\Gamma \in [\frac{1}{2}, R^{'} + C
    (N^{1-s} A)^{\alpha} ]$ by continuity such that

    $$\begin{array}{ll}
    E \left( I w_{l}(S_{l}), B(\bar{\bar{x}}, \Gamma)  \right) & = 3 (N^{s-1} \eta_{1} A^{-1})^{\alpha}
    \end{array}$$

    , since

    $$\begin{array}{ll}
    E \left( I w_{l}(S), B(\bar{\bar{x}},0.5)   \right) & \leq E \left( Iw_{l}(S), B (\bar{\bar{x}},1) \right) \\
    & \leq \frac{1}{100} (N^{s-1} \eta_{1} A^{-1})^{\alpha}
    \end{array}$$

    and

    $$\begin{array}{ll}
    E \left( I w_{l}(S), B (\bar{\bar{x}}, R^{'} + C (N^{1-s} A)^{\alpha} )  \right) &   \geq E \left( Iw_{l}(S), B(\bar{x},R^{'})  \right) \\
    & \geq 3  (N^{s-1} \eta_{1} A^{-1})^{\alpha}
    \end{array}$$

    , even if it means increasing the value of $\alpha$.

    Let $v_{l}$ be the solution of the free Klein-Gordon equation with data

    $$\left\{
    \begin{array}{ll}
    v_{l}(S_{l}) & := I^{-1} \left( \chi \left( \frac{x - \bar{\bar{x}}}{\Gamma} \right) Iw_{l}(S_{l}) \right) \\
    \partial_{t} v_{l}(S_{l}) & := I^{-1} \left( \chi \left( \frac{x - \bar{\bar{x}}}{\Gamma} \right)   \partial_{t} I w_{l}(S_{l}) \right)
    \end{array}
    \right.$$

    where $\chi$ is a smooth function such that $\chi(x)=1$ if $|x| \leq 1$ and $\chi(x)=0$ if $|x| \geq 2$

    A computation shows that

    $$\begin{array}{ll}
    E_{c} \left( Iv_{l} , S_{l}   \right) & \leq E \left( Iw_{l}(S_{l}), B(\bar{\bar{x}}, \Gamma ) \right) + O \left( \frac{1}{\Gamma^{2}} \int_{
    \Gamma \leq |x-
    \bar{\bar{x}}| \leq 2 \Gamma} |Iw_{l}(S_{l})|^{2} \, dx \right) + \\
    & O \left( \frac{1}{\Gamma} \int_{ \Gamma \leq |x - \bar{\bar{x}}| \leq 2 \Gamma}
    |Iw_{l}(S_{l})| |\nabla Iw_{l} (S_{l})| \, dx  \right) \\
    & \leq (N^{s-1} \eta_{1} A^{-1})^{\alpha} + O \left( N^{1-s} A^{\frac{p+1}{2}} \left\| \frac{I w_{l}(S_{l})}{\langle x - \bar{\bar{x}} \rangle}
    \right\|_{L^{2}}
    \right) + O \left( \left\| \frac{I w_{l}(S_{l})}{\langle x - \bar{\bar{x}} \rangle} \right\|^{2}_{L^{2}} \right) \\
    & \lesssim (N^{s-1} \eta_{1} A^{-1})^{\alpha}
    \end{array}
    \label{Eqn:IneqNrjS}$$

    and

    $$\begin{array}{l}
    E \left( Iw_{l} - Iv_{l}, S_{l} \right)   \leq \frac{1}{2} \int_{\mathbb{R}^{3}}  \left(  1 - \chi^{2} \left( \frac{x- \bar{\bar{x}}}{\Gamma}
    \right) \right) |\nabla I w_{l}(S_{l})|^{2} \, dx + \frac{1}{p+1} \int_{\mathbb{R}^{3}} \left( 1 - \chi^{p+1} \left( \frac{x-
    \bar{\bar{x}}}{\Gamma} \right)
    \right) |Iw_{l}(S_{l})|^{p+1} \, dx \\
     + \frac{1}{2} \int_{\mathbb{R}^{3}}  \left(  1 - \chi^{2} \left( \frac{x- \bar{\bar{x}}}{\Gamma}  \right) \right) |\partial_{t} Iw_{l}(S_{l})|^{2}
    \,
    dx + \frac{1}{2} \int_{\mathbb{R}^{3}} \left(  1 - \chi^{2} \left( \frac{x- \bar{\bar{x}}}{\Gamma}  \right) \right) |Iw_{l}(S_{l})|^{2} \, dx \\
     + O \left( \frac{1}{\Gamma} \int_{ \Gamma \leq |x-\bar{\bar{x}}| \leq 2 \Gamma}   | Iw_{l}(S_{l})| | \nabla I u(S_{l})| \, dx   + \frac{1}{\Gamma^{2}}
    \int_{ \Gamma \leq |x - \bar{\bar{x}}| \leq 2 \Gamma} |Iw_{l}(S_{l})|^{2} \, dx \right) \\
     \leq  \left( C_{E} N^{2(1-s)} A^{p+1} -l(1-\epsilon_{0}) (N^{s-1} \eta_{1} A^{-1})^{\alpha} \right) (1 + \beta)  - 3 (N^{s-1} \eta_{1} A^{-1})^{\alpha} + O
    \left( N^{1-s} A^{\frac{p+1}{2}} \left\| \frac{I w_{l}(S_{l})}{\langle x - \bar{\bar{x}}
    \rangle} \right\|_{L^{2}} \right)  \\
     + O \left( \left\| \frac{I w_{l}(S_{l})}{\langle x - \bar{\bar{x}} \rangle} \right\|^{2}_{L^{2}} \right) \\
     \leq  C_{E} N^{2(1-s)} A^{p+1} - (1+l)(1- \epsilon_{0})(N^{s-1} A^{-1})^{\alpha} - 2(N^{s-1} \eta_{1} A^{-1})^{\alpha}
    \end{array}
    \label{IneqChainNrj}$$

    , by our choice of $\beta$ (see (\[Eqn:DefBeta\])).

-   Proof of the decay (\[Eqn:FreeDecay\])

    By Sobolev, interpolation ( since $p<5$ and $s >s_{c}$ ), (\[Eqn:LowerBoundST\]) and the following dispersive estimate ( see [@ginebvelo], Lemma 2.1 )

    $$\begin{array}{ll}
    \|  e^{it \langle D \rangle \phi } \|_{B^{-\frac{5}{4}}_{\infty,2}} & \lesssim \frac{1}{|t|^{\frac{3}{2}}} \| \phi \|_{B^{\frac{5}{4}}_{1,2} }
    \end{array}$$

    we have, by letting $\gamma:= 1 - \frac{4}{(p-1)(3-2s)}$,

    $$\begin{array}{l}
    \| Iv_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (T_{l},b)}  \lesssim \| \langle D \rangle^{1-s} I v_{l}
    \|^{\frac{4}{(p-1)(3-2s)}}_{L_{t}^{\frac{8}{3-2s}-} L_{x}^{\frac{8}{3-2s}+} (T_{l},b)} \| \langle  D \rangle^{- \left( \frac{3}{2}++ \right) } I
    v_{l} \|^{1- \frac{4}{(p-1)(3-2s)}}_{L_{t}^{\infty}
    L_{x}^{\infty} (T_{l},b)} \\
    \lesssim (N^{s-1} \eta_{1} A^{-1})^{\alpha} \| I v_{l} \|^{\gamma}_{L_{t}^{\infty} B_{\infty,2}^{- \left( \frac{3}{2} + \right) } (T_{l},b) } \\
    \lesssim (N^{s-1} \eta_{1} A^{-1})^{\alpha} \frac{1}{|T_{l}-S_{l}|^{\frac{3 \gamma}{2}}} \left( \| I v_{l}(S_{l}) \|^{\gamma}_{B^{1-}_{1,2}} +
    \|
    \partial_{t} I v_{l}(S_{l}) \|^{\gamma}_{B^{0-}_{1,2}}
    \right) \\
     \lesssim (N^{s-1} \eta_{1} A^{-1})^{ \alpha}  \frac{1}{|T_{l}-S_{l}|^{\frac{3 \gamma}{2}}} \left( \| \langle \nabla \rangle I v_{l}(S_{l}) \|^{\gamma}_{L^{1}} + \|
    \partial_{t} I v_{l}(S_{l}) \|^{\gamma}_{L^{1}} \right) \\
     \lesssim (N^{s-1} \eta_{1} A^{-1})^{\alpha} \frac{1}{|T_{l}-S_{l}|^{\frac{3 \gamma}{2}}} \left( \Gamma^{\frac{\gamma}{2}} \| I w_{l}(S_{l}) \|_{L^{2}} +
    \Gamma^{\frac{3 \gamma}{2}} \| \nabla I w_{l}(S_{l}) \|_{L^{2}} + \Gamma^{\frac{3 \gamma}{2}} \| \partial_{t} I w_{l} (S_{l}) \|_{L^{2}}
    \right) \\
     \lesssim (N^{1-s}  A )^{\alpha} \frac{1}{M^{\frac{3 \gamma}{2}}}
    \end{array}$$

    , even if it means increasing the value of $\alpha$. Therefore (\[Eqn:FreeDecay\]) follows by letting $\bar{\alpha}:= \frac{3 \gamma}{2}$.

-   Proof of the separation of the localized mollified energy (\[Eqn:SepMolNrj\]),

    Notice first, that, in view of (\[Eqn:StrNlkg\]) and (\[Eqn:IneqNrjS\]), we have

    $$\begin{array}{ll}
    Z(v_{l}, \mathbb{R}) & \lesssim (N^{s-1} \eta_{1} A^{-1})^{\alpha}
    \end{array}
    \label{Eqn:Zv}$$

    and therefore, in view of (\[Eqn:BoundZpr\]), we have

    $$\begin{array}{ll}
    \bar{Z}(\bar{w}_{l}, [S_{l},T_{l}]) & \lesssim (N^{s-1} \eta_{1} A^{-1})^{\alpha}
    \end{array}
    \label{Eqn:Zbarw}$$

    with $\bar{w}_{l}:=w_{l}-v_{l}$. We have

    $$\begin{array}{ll}
    \partial_{t} E(I \bar{w}_{l},t) & = \int_{\mathbb{R}^{3}} \Re \left( \overline{\partial_{t} I \bar{w}_{l}}
    \left( \partial_{tt} I \bar{w}_{l}  - \triangle I \bar{w}_{l} + I \bar{w}_{l} + F(I \bar{w}_{l})  \right) \right) \\
    & = \int_{\mathbb{R}^{3}} \Re \left( \overline{\partial_{t} I \bar{w}_{l}} ( F(I \bar{w}_{l}) - I F(w_{l}) ) \right) \\
    \end{array}$$

    Therefore

    $$\begin{array}{ll}
    E(I\bar{w}_{l},T_{l}) - E(I \bar{w}_{l},S_{l}) & = X_{1,1} + X_{1,2} + X_{2}
    \end{array}$$

    with

    $$\begin{array}{ll}
    X_{1,1} & : = \int_{S_{l}}^{T_{l}} \int_{\mathbb{R}^{3}} \Re{ \left( \overline{\partial_{t} I \bar{w}_{l}}  \left( F(w_{l}) -IF(w_{l})  \right)
    \right)} \,dx \, dt
    \end{array}$$

    $$\begin{array}{ll}
    X_{1,2} & := \int_{S_{l}}^{T_{l}} \int_{\mathbb{R}^{3}} \Re{ \left( \overline{\partial_{t} I \bar{w}_{l}}  \left( F(Iw_{l}) -F(w_{l})  \right)
    \right)} \, dx \, dt
    \end{array}$$

    and

    $$\begin{array}{ll}
    X_{2} & := \int_{S_{l}}^{T_{l}} \int_{\mathbb{R}^{3}} \Re{ \left( \overline{\partial_{t} I \bar{w}_{l}}  \left( F (I\bar{w}_{l}) - F(Iw_{l})
    \right) \right)} \, dx \, dt
    \end{array}$$

It is enough to estimate $X_{1,1}$, $X_{1,2}$ and $X_{2}$. By the fundamental theorem of calculus we have $|F(w_{l}) - F(Iw_{l})| \lesssim \max{
( |Iw_{l}|^{p-1}, |w_{l}|^{p-1})} | I w_{l} -w_{l}|$ and therefore

$$\begin{array}{l}
X_{1,1}  \lesssim \| \partial_{t} I \bar{w}_{l} \|_{L_{t}^{\infty} L_{x}^{2}([S_{l},T_{l}])} \left(
\begin{array}{l}
\| P_{<<N} w_{l} \|^{p-1}_{L_{t}^{\frac{4(p-1)}{7-p}} L_{x}^{\frac{4(p-1)}{p-3} } ([S_{l},T_{l}]) } \| P_{\gtrsim N} w_{l}
\|_{L_{t}^{\frac{4}{p-3}}
L_{x}^{\frac{4}{5-p}} ([S_{l},T_{l}]) } \\
+ \| P_{\gtrsim N} w_{l} \|^{p-1}_{L_{t}^{p} L_{x}^{2p} ([S_{l},T_{l}])} \| P_{\gtrsim N} w_{l} \|_{L_{t}^{p} L_{x}^{p} ([S_{l},T_{l}])}
\end{array}
\right) \\
\lesssim \frac{1}{N^{\frac{5-p}{2}-}} \| \partial_{t} I \bar{w}_{l} \|_{L_{t}^{\infty} L_{x}^{2}([S_{l},T_{l}])} \left(
\begin{array}{l}
\| \langle D \rangle^{1- 1} I w_{l} \|^{p-1}_{L_{t}^{\frac{4(p-1)}{7-p} } L_{x}^{\frac{4(p-1)}{p-3} } ([S_{l},T_{l}]) } \\
\| \langle D \rangle^{1-  \frac{p-3}{2}}  I w_{l}  \|_{L_{t}^{\frac{4}{p-3}} L_{x}^{\frac{4}{5-p}} ([S_{l},T_{l}]) } \\
+ \| \langle  D \rangle^{1- \frac{3p-5}{2p}} I w_{l}  \|^{p}_{L_{t}^{p} L_{x}^{2p} ([S_{l},T_{l}])}
\end{array}
\right) \\
\lesssim N^{1-s} A^{\frac{p+1}{2}} \frac{(A^{-1} \eta_{1} N^{s-1})^{p \alpha}}{N^{\frac{5-p}{2}-}} \\ \\
= o \left( (N^{s-1} \eta_{1} A^{-1})^{\alpha}  \right)
\end{array}$$

since $(N,s)$ satisfies (\[Eqn:Cond1\]).

We turn to $X_{1,2}$. We use an argument in [@triroyradnlkg]: on low frequencies we use the smoothness of $F$ ($F$ is $C^{1}$) and on high frequencies, we use the regularity of $u$ (in $H^{s}$). We have, by the fundamental theorem of calculus

$$\begin{array}{l}
F (w_{l}) := F \left( P_{<< N} w_{l} + P_{\gtrsim N} w_{l}  \right) \\
= F \left( P_{<<N} w_{l} \right) + \left( \int_{0}^{1} |P_{<< N} w_{l} + y P_{\gtrsim N} w_{l}|^{p-1} \, dy \right) P_{\gtrsim N} w_{l} \\
+ \left( \int_{0}^{1} \frac{ P_{<< N} w_{l} + y P_{\gtrsim N} w_{l} }{ \overline{P_{<<N} w_{l} + y P_{\gtrsim N} u} } |P_{<< N} w_{l} + y
P_{\gtrsim N} w_{l}|^{p-1} \, dy \right) \, \overline{P_{\gtrsim N} w_{l}}
\end{array}
\label{Eqn:Decomp}$$

Therefore

$$\begin{array}{ll}
X_{1,2} & \lesssim \| \partial_{t} I \bar{w}_{l} \|_{L_{t}^{\infty} L_{x}^{2}([S_{l},T_{l}])} (X_{1,2,1} + X_{1,2,2} + X_{1,2,3} ) \\
& \lesssim N^{1-s} A^{\frac{p+1}{2}}  (X_{1,2,1} + X_{1,2,2} + X_{1,2,3} )
\end{array}$$

with

$$\begin{array}{ll}
X_{1,2,1} & :=  \| P_{\gtrsim N} F(P_{<<N} w_{l}) \|_{L_{t}^{1} L_{x}^{2}([S_{l},T_{l}])} \\
X_{1,2,2} & := \| | P_{<< N} w_{l} |^{p-1} P_{\gtrsim N} w_{l} \|_{L_{t}^{1} L_{x}^{2}([S_{l},T_{l}])} \\
X_{1,2,3} & :=  \| P_{\gtrsim N} w_{l} \|^{p}_{L_{t}^{p} L_{x}^{2p} ([S_{l},T_{l}])}
\end{array}$$

We have

$$\begin{array}{ll}
X_{1,2,1} & \lesssim \frac{1}{N} \| \nabla F(P_{<<N} w_{l}) \|_{L_{t}^{1} L_{x}^{2}([S_{l},T_{l}])} \\
& \lesssim \frac{1}{N} \| P_{<<N} w_{l} \|^{p-1}_{L_{t}^{\frac{4(p-1)}{7-p}} L_{x}^{\frac{4(p-1)}{p-3}} ([S_{l},T_{l}])} \| \nabla P_{<<N} w_{l}
\|_{L_{t}^{\frac{4}{p-3}} L_{x}^{\frac{4}{5-p}} ([S_{l},T_{l}])} \\
& \lesssim \frac{1}{N^{\frac{5-p}{2}-}} \| \langle D \rangle^{1- 1} I w_{l} \|^{p-1}_{L_{t}^{\frac{4(p-1)}{7-p} } L_{x}^{\frac{4(p-1)}{p-3}}
([S_{l},T_{l}]) } \| \langle D \rangle^{1- \frac{p-3}{2}} I w_{l} \|_{L_{t}^{\frac{4}{p-3}} L_{x}^{\frac{4}{5-p} }([S_{l},T_{l}])}  \\
& \lesssim \frac{(N^{s-1} \eta_{1} A^{-1})^{\alpha p}}{N^{\frac{5-p}{2}-}}
\end{array}
\label{Eqn:AX123}$$

$$\begin{array}{ll}
X_{1,2,2} & \lesssim \left\| |P_{<<N} w_{l}|^{p-1} P_{\gtrsim N} w_{l} \right\|_{L_{t}^{1} L_{x}^{2} ([S_{l},T_{l}])} \\
& \lesssim \frac{1}{N^{\frac{5-p}{2}-}} \| \langle D \rangle^{1-1} I w_{l} \|^{p-1}_{L_{t}^{\frac{4(p-1)}{7-p}} L_{x}^{\frac{4(p-1)}{p-3}}
([S_{l},T_{l}])}
\| \langle D \rangle^{1- \frac{p-3}{2} } I w_{l}  \|_{L_{t}^{\frac{4}{p-3}} L_{x}^{\frac{4}{5-p}} ([S_{l},T_{l}]) } \\
& \lesssim \frac{(N^{s-1} \eta_{1} A^{-1})^{\alpha p}}{N^{\frac{5-p}{2}-}}
\end{array}
\label{Eqn:BX123}$$

$$\begin{array}{ll}
X_{1,2,3} & \lesssim \| P_{\gtrsim N} w_{l} \|^{p}_{L_{t}^{p} L_{x}^{2p} ([S_{l},T_{l}])} \\
& \lesssim \frac{ \| \langle D \rangle^{1- \frac{3p-5}{2p}} I w_{l} \|^{p}_{L_{t}^{p} L_{x}^{2p}([S_{l},T_{l}])}} {N^{\frac{5-p}{2}-}} \\
& \lesssim \frac{(N^{s-1} \eta_{1} A^{-1})^{\alpha p}}{N^{\frac{5-p}{2}-}} \\
\end{array}
\label{Eqn:CX123}$$

Therefore $X_{1,2} =  o \left( (N^{s-1} \eta_{1} A)^{\alpha} \right) $, since $(N,s)$ satisfies (\[Eqn:Cond1\]).

Now we turn to $X_{2}$. By (\[Eqn:Zbarw\]) and (\[Eqn:BoundZpr\]), we have

$$\begin{array}{ll}
X_{2} & \lesssim \| \partial_{t} \langle D \rangle^{-\frac{1}{2}} I \bar{w}_{l}  \|_{L_{t}^{4} L_{x}^{4} ([S_{l},T_{l}])} \| \langle D
\rangle^{\frac{1}{2}}
(F(I\bar{w}_{l}) -F(Iw_{l}) ) \|_{L_{t}^{\frac{4}{3}} L_{x}^{\frac{4}{3}} ([S_{l},T_{l}])  } \\
& \lesssim \| \partial_{t} \langle D \rangle^{-\frac{1}{2}} I \bar{w}_{l}  \|_{L_{t}^{4} L_{x}^{4} ([S_{l},T_{l}])}  \left( \| I w_{l}
\|^{p-1}_{L_{t}^{2(p-1)-}
L_{x}^{2(p-1)+} ([S_{l},T_{l}])} + \| I \bar{w}_{l} \|^{p-1}_{ L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([S_{l},T_{l}]) } \right) \\
& \left( \| \langle D \rangle^{\frac{1}{2}} I w_{l} \|_{L_{t}^{4+} L_{x}^{4-}([S_{l},T_{l}])} + \| \langle D \rangle^{\frac{1}{2}} I
\bar{w}_{l} \|_{L_{t}^{4+} L_{x}^{4-}([S_{l},T_{l}])}    \right) \\
& =  o \left( (N^{s-1} \eta_{1} A^{-1})^{\alpha} \right)
\end{array}$$

Therefore (\[Eqn:SepMolNrj\]) holds since again $(N,s)$ satisfies (\[Eqn:Cond1\]). Notice also that, in view of (\[Eqn:Zv\]), we have

$$\begin{array}{ll}
E(Iv_{l},T_{l}) & \lesssim (N^{s-1} \eta_{1} A^{-1})^{\alpha}
\end{array}$$

even if it means increasing the value of $\alpha$.

Proof of Lemma \[lem:ConcNrj\]
------------------------------

The proof is made of three steps:

-   Control of auxiliary norms

    $$\begin{array}{ll}
    \bar{Z}(J):= & \| \langle D \rangle^{1- \frac{1}{2}} I w_{l} \|_{L_{t}^{4} L_{x}^{4} (J)} + \| \langle D \rangle^{1- \frac{1}{2}} I w_{l}
    \|_{L_{t}^{2(p-1)}
    L_{x}^{\frac{6(p-1)}{2p-3}} (J) } + \| \langle D \rangle^{1-\frac{1}{2}} I w_{l} \|_{L_{t}^{\infty -} L_{x}^{3 +}(J)} \\
    & + \| \langle D \rangle^{1-\frac{1}{2}} I w_{l}  \|_{L_{t}^{\frac{2(p-1)}{p-4} -} L_{x}^{\frac{6(p-1)}{p+2} +} (J) }
    \end{array}$$

    Let $ Y:= \left( (C_{E} N^{2(1-s)} A^{p+1} -l(1-\epsilon_{0}) (N^{s-1} \eta_{1} A^{-1})^{\alpha}) (1+ \beta) \right)^{\frac{1}{2}} $. We have

    $$\begin{array}{l}
    \bar{Z}(J) \lesssim \sup_{t \in J} E^{\frac{1}{2}}(Iw_{l}(t)) + \| \langle D \rangle^{1-\frac{1}{2}} I (|w_{l}|^{p-1} w_{l}) \|_{L_{t}^{\frac{4}{3}} L_{x}^{\frac{4}{3}} (J) } \\
    \lesssim Y + \| \langle D  \rangle^{1-\frac{1}{2}} I w_{l} \|_{L_{t}^{4} L_{x}^{4}(J)} \left( \| P_{<<N} w_{l} \|^{p-1}_{L_{t}^{2(p-1)-}
    L_{x}^{2(p-1)+} (J)} + \| P_{\gtrsim N} w_{l} \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}(J)} \right) \\
    \lesssim Y + \bar{Z}(J) \left( \| I w_{l} \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J)} + \frac{\| \langle D \rangle^{\frac{1}{2}} I w_{l}
    \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{\frac{6(p-1)}{2p-1}+} (J) }}{N^{\frac{5-p}{2}}} \right) \\
    \lesssim Y + \eta^{p-1} \bar{Z}(J) + \frac{\bar{Z}^{p}(J)}{N^{\frac{5-p}{2}}}
    \end{array}$$

    Applying a continuity argument, we see that

    $$\begin{array}{ll}
    \bar{Z}(J) & \lesssim Y \\
    & \lesssim N^{1-s} A^{\frac{p+1}{2}}
    \end{array}
    \label{Eqn:EstZ}$$

    since $(N,s)$ satisfies (\[Eqn:Cond2\]).

-   Lower bound of the size of $J$

    If $p \geq 4$ then, by (\[Eqn:EstZ\]) we have [^11]

    $$\begin{array}{ll}
    \eta & = \| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J)} \\
    & \lesssim |J|^{\frac{5-p}{2(p-1)} \pm }  \| \langle D \rangle^{1-\frac{1}{2}} I w_{l} \|_{L_{t}^{\frac{2(p-1)}{p-4}-} L_{x}^{\frac{6(p-1)}{p+2}+} (J)} \\
    & \lesssim |J|^{\frac{5-p}{2(p-1)} \pm} A^{\frac{p+1}{2}} N^{1-s}
    \end{array}
    \label{Eqn:LowerBdJ1}$$

    and if $p < 4$

    $$\begin{array}{ll}
    \eta & = \| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J)} \\
    & \lesssim |J|^{\frac{1}{2(p-1)} \pm}  \| \langle D \rangle^{1-\frac{1}{2}} I w_{l}  \|_{L_{t}^{\infty -} L_{x}^{3 +} (J)} \\
    & \lesssim |J|^{\frac{1}{2(p-1)} \pm} A^{\frac{p+1}{2}} N^{1-s}
    \end{array}
    \label{Eqn:LowerBdJ2}$$

    Therefore there exists a constant $\alpha:=\alpha(p)>0$ such that

    $$\begin{array}{l}
    |J| \gtrsim ( A^{-1} N^{s-1} \eta)^{\alpha}
    \end{array}
    \label{Eqn:LowerBoundJ}$$

-   Lower bound of $\| P_{M} w_{l} \|_{L_{t}^{\infty} L_{x}^{\infty}(J)}$ for some $M \in 2^{\mathbb{N}}$ such that $ M \leq f(A,N)$ We have, by (\[Eqn:EstZ\])

    $$\begin{array}{ll}
    \eta = & \| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}(J)} \\
    &  \leq \| \langle D \rangle^{1- \left (\frac{1}{2} + \right)} I w_{l} \|^{\frac{2}{p-1}}_{L_{t}^{4-}
    L_{x}^{4+} (J)} \| \langle D \rangle^{- \left(\frac{1}{p-3} + \right)}  I w_{l} \|^{1- \frac{2}{p-1}}_{L_{t}^{\infty} L_{x}^{\infty} (J)} \\
    & \lesssim A^{\frac{p+1}{p-1}} N^{\frac{2(1-s)}{p-1}} \| \langle D \rangle^{- \left( \frac{1}{p-3} + \right)}  I w_{l} \|^{1-
    \frac{2}{p-1}}_{L_{t}^{\infty} L_{x}^{\infty} (J)}
    \end{array}$$

    Therefore, even if it means increasing the value of $\alpha$, we have

    $$\begin{array}{l}
    \| \langle D \rangle^{- \left( \frac{1}{p-3} + \right)}  I w_{l} \|_{L_{t}^{\infty} L_{x}^{\infty} (J)} \gtrsim   (\eta A^{-1} N^{s-1})^{\alpha}
    \end{array}$$

    and, by the pigeonhole principle, this implies that there exists $M \in 2^{\mathbb{N}}$ such that

    $$\begin{array}{l}
    \| P_{M} I w_{l} \|_{L_{t}^{\infty} L_{x}^{\infty} (J)} \gtrsim \langle M \rangle^{\frac{1}{p-3}+} (\eta A^{-1} N^{s-1})^{\alpha}
    \end{array}
    \label{Eqn:UpBdPM1}$$

    Moreover

    $$\begin{array}{ll}
    \| P_{M} I w_{l} \|_{L_{t}^{\infty} L_{x}^{\infty} (J)} & \lesssim \langle M \rangle^{\frac{1}{2}} \| \langle D \rangle I u \|_{L_{t}^{\infty}
    L_{x}^{2} (J)} \\
    & \lesssim \langle M \rangle^{\frac{1}{2}} A^{\frac{p+1}{2}} N^{1-s}
    \end{array}
    \label{Eqn:UpBdPm2}$$

    Therefore, combining (\[Eqn:UpBdPM1\]) and (\[Eqn:UpBdPm2\]), we see that, even if it means increasing the value of $\alpha$, we have $$\begin{array}{l}
    \langle M \rangle \lesssim  (A N^{1-s} \eta^{-1})^{\alpha}
    \end{array}
    \label{Eqn:UpperBdM}$$

-   Control of $| P_{M}  I w_{l} (t,\bar{x}) |$ for some $\bar{x} \in \mathbb{R}^{3}$ and for all $t \in K$, $K \subset J$ to be defined shortly. By (\[Eqn:UpBdPM1\]), there exists $(\bar{t}, \bar{x})$ such that

    $$\begin{array}{l}
    |P_{M} I w_{l}(\bar{t},\bar{x})| \gtrsim  \langle M \rangle^{\frac{1}{p-3}+}  (\eta  A^{-1} N^{s-1})^{\alpha}
    \end{array}
    \label{Eqn:aux1}$$

    But, by the fundamental theorem of calculus and (\[Eqn:InducNrj\])

    $$\begin{array}{ll}
    | P_{M} I w_{l}(t,\bar{x}) - P_{M} I w_{l}(\bar{t},\bar{x})| & \lesssim \sup_{s \in (\bar{t},t)} \| \partial_{s} I w_{l}(s) \|_{L_{x}^{2}}
    \langle M \rangle^{\frac{3}{2}}
    |t - \bar{t}| \\
    & \lesssim (N^{1-s} A \eta^{-1})^{\alpha}  |t - \bar{t}|
    \end{array}
    \label{Eqn:aux2}$$

    , even if it means increasing the value of $\alpha$. Therefore, in view of (\[Eqn:LowerBoundJ\]), (\[Eqn:aux1\]) and (\[Eqn:aux2\]) there exist $c<<1$ such that for all $t \in [\bar{t}, \bar{t} + \delta] $ (in this case let $K:= [\bar{t}, \bar{t}+ \delta]$) or for all $t \in
    [\bar{t} - \delta, \bar{t}]$ ( in this case let $ K:= [\bar{t}- \delta, \bar{t}]$ ), even if it means increasing the value of $\alpha$, we have

    $$\begin{array}{ll}
    |P_{M} Iw_{l}(t,x_{0})| & \gtrsim  M^{\frac{1}{p-3}+} \eta^{\frac{p-1}{p-3}} A^{-\alpha_{3}} N^{\frac{s-1}{p-3}} \\
     & \gtrsim (\eta A^{-1} N^{s-1})^{\alpha}
    \end{array}
    \label{Eqn:PM}$$

    and $[\bar{t}, \bar{t} + \delta] \subset J$ or $[\bar{t} -\delta, \bar{t}] \subset J$ with

    $$\begin{array}{l}
    \delta  := c (\eta A^{-1}  N^{s-1})^{\alpha}
    \end{array}$$

-   lower bound of potential mollified energy. Let $R>1$ to be fixed shortly. Let $\Psi:= \psi$ if $M > 1$ and $\Psi:= \phi$ is $M=1$. By (\[Eqn:PM\]) we have [^12]

    $$\begin{array}{ll}
    M^{3}(A+B) & \gtrsim (\eta A^{-1} N^{s-1})^{\alpha}
    \end{array}$$

    with

    $$\begin{array}{l}
    A:= \int_{|y| \leq R} |\check{\Psi}(My)| |Iw_{l}(t,x_{0}-y)| \, dy \\
    \end{array}$$

    and

    $$\begin{array}{l}
    B:=  \int_{|y| \geq R} |\check{\Psi}(My)| |Iw_{l}(t,x_{0}-y)| \, dy
    \end{array}$$

    We have

    $$\begin{array}{ll}
    A  & \lesssim \left( \int_{|y| \leq R} |\check{\Psi} (My) |^{\frac{p+1}{p}} \, dy \right)^{\frac{p}{p+1}} \left( \int_{|y| \leq R}
    |Iw_{l}(t,x_{0}-y)|^{p+1}
    \, dy \right)^{\frac{1}{p+1}} \\
    & \lesssim  \left( \int_{|y-x_{0}| \leq R} |Iw_{l}(t,y)|^{p+1} \, dy \right)^{\frac{1}{p+1}} M^{-\frac{3p}{p-1}}
    \end{array}$$

    and

    $$\begin{array}{ll}
    B & \lesssim M^{-\frac{3}{2}} \| \check{\Psi} \|_{L^{2}(|y| \geq M R)} \| I w_{l} (t) \|_{L_{x}^{2}} \\
    & \lesssim  M^{-\frac{3}{2}} \| \check{\Psi} \|_{L^{2}(|y| \geq M R)}  N^{1-s} A^{\frac{p+1}{2}}
    \end{array}$$

    Therefore, by the inequality $ \| \check{\Psi} \|_{L^{2}(|y| \geq  MR)} \lesssim \frac{1}{(MR)^{\frac{3}{2}}}$, we see that there exists $C>0$ such that, even if it means increasing the value of $\alpha$, if

    $$\begin{array}{ll}
    R & := C (\eta^{-1} A  N^{1-s})^{\alpha}
    \end{array}$$

    then

    $$\begin{array}{l}
    M^{\frac{3}{2}} \| \check{\Psi} \|_{L^{2}_{(|y| \geq MR)}} N^{1-s} A^{\frac{p+1}{2}} << (\eta A^{-1} N^{s-1})^{\alpha}
    \end{array}
    \label{Eqn:aux3}$$

    and therefore ( even if it means increasing the value of $\alpha$0 we have

    $$\begin{array}{l}
    \int_{|y-x_{0}| \leq R} |Iw_{l}(t,y)|^{p+1} \, dy \gtrsim (\eta A^{-1} N^{s-1})^{\alpha}
    \end{array}$$

    for all $t \in K$.

Proof of Lemma \[lem:UseMorEst\]
--------------------------------

Recall that, by Lemma \[lem:ConcNrj\], on each $J_{j}$, there exist $x_{j} \in \mathbb{R}^{3}$ and $K_{j}=[t_{j},t_{j+1}] \subset J_{j}$ such that

$$\begin{array}{ll}
\int_{|x- x_{j}| \leq R } |I w_{l}(t,x)|^{p+1} \, dx & \gtrsim (\eta A^{-1} N^{s-1})^{\alpha}
\end{array}
\label{Eqn:LowerPotNrj}$$

$t \in K_{j}$, with $R$ satisfying (\[Eqn:DefR\]) and

$$\begin{array}{ll}
|K_{j}| & = c (\eta A^{-1} N^{s-1})^{\alpha}
\end{array}
\label{Eqn:SizeKj}$$

The proof is made of three steps:

-   We construct (see [@nakanash1]) a set $\mathcal{P}:= \{ j_{1},....,j_{l} \} \subset [1,...,M]$. Initially $j_{1}=1$. Then let $j_{k+1}$ be the minimal $j \in [1,M]$ such that

    $$\begin{array}{ll}
    |x_{j} - x_{j_{k+1}}| & \geq |t_{j}-t_{j_{k+1}}| + 2R
    \end{array}
    \label{Eqn:Boundxjxk}$$

    for $j:=j_{1},...,j_{k}$. Observe that $J=(J_{j})_{1 \leq j \leq M} = \bigcup_{j_{k} \in \mathcal{P}} A_{j_{k}}$ with $A_{j_{k}}:= \left\{ l \in
    [1,..,m], \, l \geq j_{k}, \, and  \, |x_{j_{k}} - x_{l}| < |t_{j_{k}}-t_{l}| + 2R \right\}$

-   Weighted long time estimate and lower bound of mollified energy, restricted to a ball.

    We chop $J$ into subintervals $(\tilde{J}_{l})_{1 \leq l \leq L }$ such that $\| I w_{l} \|_{L_{t}^{2(p-1) -} L_{x}^{2(p-1) +}(\tilde{J}_{l})} =
    \eta_{0}$, $ \| I w_{l}  \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (\tilde{J}_{L})} \leq \eta_{0}$. By (\[Eqn:InducStr\]) we have

    $$\begin{array}{ll}
    L & \lesssim X^{2(p-1)-}_{C,Q}
    \end{array}
    \label{Eqn:BoundL}$$

    and combining this bound with Proposition \[prop:LocalBd\], Proposition \[Prop:AlmMor\], Proposition \[Prop:EstInt\] and (\[Eqn:BoundL\]) we see that

    $$\begin{array}{ll}
    \int_{J} \int_{\mathbb{R}^{3}} \frac{|Iw_{l}(t,x)|^{p+1}}{|x-x_{j}|} \, dx \, dt & \lesssim N^{2(1-s)} A^{p+1}
    \end{array}
    \label{Eqn:proc2}$$

    since $(N,s)$ satisfies (\[Eqn:Cond3\]). Therefore

    $$\begin{array}{ll}
    \int_{J} \int_{R + |t-t_{j}| >|x-x_{j}|} \frac{|I w_{l}(t,x)|^{p+1}}{1+ R+|t-t_{j}|} \, dx \, dt & \lesssim N^{2(1-s)} A^{p+1}
    \end{array}
    \label{Eqn:StrLongEst}$$

    This proves the weighted long-time estimate. Next, let us prove the following result

    : Let $x_{a} \in \mathbb{R}^{3}$ and let $t_{b} \geq t_{a}$. Then

    $$\begin{array}{ll}
    E \left( Iw_{l}, B(x_{a}, R + t_{b} -t_{a}) \right) & \geq E \left( Iw_{l}, B(x_{a},R)  \right) + o \left( (\eta A^{-1} N^{s-1})^{\alpha}
    \right)
    \end{array}
    \label{Eqn:IncreaseRate2}$$

    Indeed, by integrating the identity

    $$\begin{array}{l}
    \partial_{t} \left( \frac{1}{2} |\partial_{t} I w_{l}|^{2} + \frac{1}{2} |\nabla I w_{l}|^{2} + \frac{|Iw_{l}|^{p+1}}{p+1} + \frac{|Iw_{l}|^{2}}{2} \right)
    - \partial_{x_{i}} \left( \Re ( \overline{\partial_{t} I w_{l}} \partial_{x_{i}} I w_{l} )   \right) \\
    + \Re \left( \overline{\partial_{t} I w_{l}} (I F(w_{l}) - F(Iw_{l})) \right)=0 \\
    \end{array}$$

    inside the truncated cone $M:= \{ (t,x), \, t \in [t_{a}, t_{b}], \,  t-t_{a} + R \geq |x- x_{a}|  \}$ we have

    $$\begin{array}{l}
    E \left( I w_{l}, B( x_{a}, R + t_{b} - t_{a}) \right) - E \left( I w_{l}, B(x_{a}, R) \right) \\
    = \frac{1}{ \sqrt{2}} \int_{\partial M} \frac{|Iw_{l}|^{2}}{2} + \frac{|Iw_{l}|^{p+1}}{p+1} \, d \sigma \\
    + \frac{1}{\sqrt{2}} \int_{\partial M} \left| \frac{x-x_{a}}{|x-x_{a}|} \partial_{t} I w_{l} + \nabla I w_{l}   \right|^{2} \, d \sigma -
    \int_{M} \Re \left( \overline{\partial_{t} I w_{l}} (I F(w_{l}) - F(Iw_{l}))   \right) \, dx dt
    \end{array}
    \label{Eqn:IncreaseRate}$$

    Therefore, by repeating the steps from (\[Eqn:BoundL\]) to (\[Eqn:proc2\]), we get (\[Eqn:IncreaseRate2\]) from (\[Eqn:IncreaseRate\]) and (\[Eqn:Acl\]). Consequently, by (\[Eqn:DefR\]), (\[Eqn:ConcNrjMol\]), (\[Eqn:SizeKj\]), (\[Eqn:LowerPotNrj\]) and (\[Eqn:StrLongEst\]) we have

    $$\begin{array}{ll}
    N^{2(1-s)} A^{p+1}{\operatorname{card}}{\mathcal{P}}  & \gtrsim  \sum_{j_{k} \in \mathcal{P}} \int_{J} \int_{\mathbb{R}^{3}}
    \frac{|Iw_{l}(t,x)|^{p+1}}{|x-x_{j_{k}}|} \, dx \, dt \\
    & \gtrsim (N^{s-1} A^{-1} \eta)^{\alpha} \sum_{j_{k} \in \mathcal{P}} \int_{J} \int_{|x-x_{j_{k}}| \leq |t-t_{j_{k}}| + 100R}
    \frac{|I w_{l}(t,x)|^{p+1}} {1+ |t-t_{j_{k}}| } \, dx \, dt \\
    & \gtrsim (N^{s-1} A^{-1} \eta)^{\alpha} \sum_{j_{k} \in \mathcal{P}} \sum_{l \in A_{j_{k}}} \int_{K_{l}} \frac{1}{1+ t} \, dt \\
    & \gtrsim (N^{s-1} A^{-1} \eta)^{\alpha} \sum_{j=1}^{\bar{j}} \frac{1}{1+t_{j}}
    \end{array}
    \label{Eqn:Finite1}$$

    It suffices to estimate ${\operatorname{card}}{\mathcal{P}}$. Let $j_{k_{max}}:= \max_{j_{k} \in \mathcal{P}} j_{k}$. By applying - $ {\operatorname{card}}{\mathcal{P}}$ - times (\[Eqn:IncreaseRate2\]), by the construction of $\mathcal{P}$ and by (\[Eqn:LowerPotNrj\]) we have

    $$\begin{array}{ll}
    N^{2(1-s)} A^{p+1} & \gtrsim E \left( I w_{l}(t_{j_{k}}), \bigcup_{j_{k} \in \mathcal{P}} B(x_{j_{k}}, R + |t_{j_{k_{max}}} - t_{j_{k}}| ) \right)\\
    & \gtrsim \sum_{j_{k} \in \mathcal{P}} E \left( Iw_{l}(t_{j_{k}}), B(x_{j_{k}},R) \right) +
    o \left( (\eta A^{-1} N^{s-1})^{\alpha} \right) {\operatorname{card}}{(\mathcal{P})} \\
    & \gtrsim {\operatorname{card}}{(\mathcal{P})} (\eta A^{-1} N^{s-1})^{\alpha}
    \end{array}
    \label{Eqn:Finite2}$$

    Therefore, even if it means increasing the value of $\alpha$, we get (\[Eqn:RepartJj\]) from (\[Eqn:Finite1\]) and (\[Eqn:Finite2\]).

 Proof of Lemma \[lem:StrVsConcNrj\]
------------------------------------

The proof is made of several steps:

-   Long-time weighted decay estimates.

    Let $x_{0} \in \mathbb{R}^{3}$.\
    We chop $J$ into subintervals $(\tilde{J}_{l})_{1 \leq l \leq L}$ such that $ \| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (\tilde{J}_{l}) }
    = \eta_{0} $, $1 \leq l < L$ and $ \| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J_{L})} \leq \eta_{0} $. Then, by repeating the steps from (\[Eqn:BoundL\]) to (\[Eqn:proc2\]), we get

    $$\begin{array}{ll}
    \int_{J} \int_{\mathbb{R}^{3}} \frac{|Iw_{l}|^{p+1}}{|x-x_{0}|} \, dx \, dt & \lesssim N^{2(1-s)} A^{p+1}
    \end{array}
    \label{Eqn:LongTimeEstLem}$$

    Therefore (see Lemma $5.3$, [@nakanash1]) we have

    $$\begin{array}{ll}
    \int_{J} \int_{\mathbb{R}^{3}} \left( \frac{|Iw_{l}|^{2}}{\langle x -x_{0} \rangle^{2}} \right)^{\frac{p+1}{2}} \, dx \frac{1}{\langle t
    \rangle} \, dt & \leq X_{1} + X_{2}
    \end{array}$$

    with

    $$\begin{array}{l}
    X_{1} := \int_{J} \int_{|t| > |x-x_{0}|} \left( \frac{|I w_{l}|^{2}}{\langle x -x_{0} \rangle^{2}}  \right)^{\frac{p+1}{2}} dx \frac{1}{\langle
    t \rangle} \, dt
    \end{array}$$

    and

    $$\begin{array}{l}
    X_{2} := \int_{J} \int_{|t| \leq |x-x_{0}|} \left( \frac{|I w_{l}|^{2}}{\langle x -x_{0} \rangle^{2}} \, \right)^{\frac{p+1}{2}} dx
    \frac{1}{\langle t \rangle} \, dt
    \end{array}$$

    But, by Hölder (in space) and (\[Eqn:LongTimeEstLem\]) we have

    $$\begin{array}{ll}
    X_{1}  & \lesssim \int_{J}  \int_{|t| > |x - x_{0}|}  \frac{|I w_{l}|^{p+1}}{\langle t \rangle} \, dx \, dt \\
    & \lesssim N^{2(1-s)} A^{p+1}
    \end{array}$$

    and

    $$\begin{array}{ll}
    X_{2} & \lesssim \int_{J}   \sup_{|x -x_{0}| \geq |t|} \left(  \frac{1}{ \langle  x -x_{0} \rangle^{2}} \right)^{\frac{p+1}{2}}
    \left( \int_{|x-x_{0}| \geq |t|} |Iw_{l}|^{2}  \right)^{\frac{p+1}{2}} \frac{1}{\langle t \rangle} \, dt \\
    & \lesssim (N^{2(1-s)}A ^{p+1} )^{\frac{p+1}{2}}
    \end{array}$$

    Therefore, even it means increasing $\alpha$, we have

    $$\begin{array}{l}
    \int_{J} \int_{\mathbb{R}^{3}} \left( \frac{|I w_{l}|^{2}}{\langle x \rangle^{2}} \, dx \right)^{\frac{p+1}{2}} \frac{1}{\langle t \rangle} \,
    dt \lesssim (N^{1-s} A)^{\alpha}
    \end{array}
    \label{Eqn:LongTimeHardy}$$

-   We chop $J$ into subintervals $(J_{j}=[t_{j}, \, t_{j+1}])_{1 \leq j \leq m}$ such that $\| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J_{j})} = \eta$ with $\eta$ defined in Lemma \[lem:ConcNrj\]. Therefore, using (\[Eqn:InducNrj\]), we get, by Proposition \[prop:LocalBd\]

    $$\begin{array}{ll}
    \bar{Z}(J_{j},w_{l}) & \lesssim N^{1-s} A^{\frac{p+1}{2}}
    \end{array}$$

    with $\bar{Z}$ defined in (\[Eqn:DefBarZ\]). By (\[Eqn:ConcNrjMol\]), there exists $\bar{x}_{j} \in \mathbb{R}^{3}$ and $\bar{t}_{j} \in
    J_{j}$ such that

    $$\begin{array}{ll}
    E \left( Iw_{l}(\bar{t}_{j}), B(\bar{x}_{j},R) \right)  & \gtrsim (N^{s-1} \eta A^{-1})^{\alpha}
    \end{array}$$

    and, by (\[Eqn:IncreaseRate\]), we see that (with $t_{a}:=\bar{t}_{j}$, $t_{b}:=t$, $t \in J_{j}$ ) there exists $\eta_{1}
    > 0 $ such that

    $$\begin{array}{ll}
    E \left( I w_{l}(t), B(\bar{x}_{j}, R + |t -\bar{t}_{j}|)   \right) & \geq 3 (N^{s-1} \eta_{1} A^{-1})^{\alpha}
    \end{array}
    \label{Eqn:LowerBdNrjBall}$$

-   We chop each subinterval $J_{j}$ into subsubintervals $(J_{j,k}= [t_{j,k}, t_{j,k+1}])_{ 1 \leq k < K_{j}}$ such that $\bar{Z}(J_{j,k},w_{l}) \sim  (N^{s-1} \eta_{1} A^{-1})^{\alpha}$ and $\bar{Z}(J_{j,k},w_{l}) \leq 3 (N^{s-1} \eta_{1} A^{-1})^{\alpha}$. Notice that, even if it means increasing the value of $\alpha$, we have

    $$\begin{array}{ll}
    K_{j} & \lesssim (N^{1-s} \eta_{1} A)^{\alpha}
    \end{array}
    \label{Eqn:UpBdK}$$

    Next, given $\epsilon \leq \frac{1}{100}$, we claim that there exists $C_{0}>0$ such that

    $$\begin{array}{ll}
    \log{(C_{0})} & \lesssim N^{2(1-s)} (A \epsilon^{-1})^{p+1}
    \end{array}
    \label{Eqn:BoundLogC0}$$

    and such that for all $1 \leq k \leq K_{j}$ there exists $\tilde{t}_{j,k} \in [t_{j,k}, t_{j,k} + C_{0} \langle t_{j,k} - t_{j} \rangle]$ such that (\[Eqn:UpBdWeight\]) holds: if not, for all $C$ such that $\log{(C)}
    >> N^{2(1-s)} (A \epsilon^{-1})^{p+1} $ , we would have, by (\[Eqn:LongTimeHardy\])

    $$\begin{array}{ll}
    N^{2(1-s)} A^{p+1} & \gtrsim \int_{t_{j,k}}^{t_{j,k} +  C \langle t_{j,k} - t_{j} \rangle } \left\| \frac{Iw_{l}(t)}{\langle x - x_{j} \rangle}
    \right\|^{p+1}_{L^{2} (\mathbb{R}^{3})} \frac{1}{ \langle t \rangle} \, dt \\
    & \gtrsim \log{C} \, \epsilon^{p+1}
    \end{array}$$

    and it would yield a contradiction.

-   Now we claim that there exists $(k_{0},j_{0})$ such that

    $$\begin{array}{ll}
    |t_{j_{0},k_{0}+1} -t_{j_{0},k_{0}}| & \geq M^{'} \left( R + |t_{j_{0},k_{0}}-t_{j_{0}}| \right)
    \end{array}$$

    with $M^{'}$ such that

    $$\begin{array}{l}
    M^{'}  - C_{0} =100 M(C_{0}+1)
    \end{array}
    \label{Eqn:Mprime}$$

    If not, this would imply, by simple induction on $k$, that $|t_{j,k} - t_{j}| \leq (2M^{'})^{k} R$ for all $j$ and therefore, by (\[Eqn:UpBdK\]), there would exist $C_{1} \lesssim 1$ such that we have $|J_{j}| \leq (2M^{'})^{C_{1} (N^{1-s} A)^{\alpha}}$ for all $j$. But, by Lemma \[lem:UseMorEst\], this would imply that

    $$\begin{array}{ll}
    \frac{\log{\left( 1 + m (2M^{'})^{C_{1} (N^{1-s} A)^{\alpha}} \right)}} {(2M^{'})^{C_{1} (N^{1-s} A)^{\alpha}}} & \leq \sum_{j=1}^{m}
    \frac{1}{1+ j(2M^{'})^{C_{1} (N^{1-s} A)^{\alpha}} } \\
    & \lesssim (N^{1-s} \eta^{-1} A)^{\alpha}
    \end{array}$$

    Therefore $ \log{(m)}  \leq C_{1} (2M^{'})^{C_{1} (N^{1-s} A)^{\alpha}} $ (even if it means increasing the value of $C_{1}$) and, combining this inequality with (\[Eqn:BoundLogC0\]) and (\[Eqn:Mprime\]), this yields a contradiction with (\[Eqn:UpperBdIu\]) even if, again, it means increasing the value of $C_{1}$. Consequently,

    $$\begin{array}{ll}
    t_{j_{0},k_{0}+1} -\tilde{t}_{j_{0},k_{0}} & \gtrsim t_{j_{0},k_{0}+1} - t_{j_{0},k_{0}} - C_{0} \langle t_{j_{0},k_{0}}
    - t_{j_{0}} \rangle \\
    & \geq M^{'} \left(  R + | t_{j_{0},k_{0}} -t_{j_{0}}  | \right)   - C_{0} \langle t_{j_{0},k_{0}} - t_{j_{0}} \rangle \\
    & \geq M \left( R + | \tilde{t}_{j_{0},k_{0}} -t_{j_{0}} | \right)
    \end{array}$$

    and, choosing $R^{'}:= R + | \tilde{t}_{j_{0},k_{0}} -t_{j_{0}} | $, $S_{l}:= \tilde{t}_{j_{0},k_{0}}$ and $T_{l}:=t_{j_{0},k_{0}+1}$, we have (\[Eqn:BoundZpr\]), (\[Eqn:LowerBoundST\]) and (\[Eqn:UpBdWeight\]).

Proof of Proposition \[prop:PerturbArg\] {#Sec:ProofPerturb}
========================================

In this section we prove Proposition \[prop:PerturbArg\]. The proof is made of several steps:

-   We divide $J$ into subintervals $(K_{l}=[t_{l},t_{l+1}]_{1 \leq l \leq m})$ such that $ \| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2^(p-1)+} (K_{l})} = \eta_{0}$ and $ \| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2^(p-1)+} (K_{m})} \leq
    \eta_{0}$ (with $\eta_{0}$ defined in Proposition \[prop:LocalBd\] ). Notice that, in view of (\[Eqn:InducNrj\]), we have $Z(K_{l},u)
    \lesssim N^{1-s} A^{\frac{p+1}{2}}$ and, by iteration, in view of (\[Eqn:BounduPerturb\]), we have

    $$\begin{array}{ll}
    Z(w_{l},J) & \lesssim X_{C,Q}^{2(p-1)-} N^{1-s} A^{\frac{p+1}{2}}
    \end{array}
    \label{Eqn:BounduPerturb}$$

-   Again, we divide $J$ into subintervals $(J_{j}:=[t_{j},t_{j+1}])_{1 \leq j \leq k}$ such that\
    $\| I w_{l+1} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}(J_{j})} = \eta_{0}$, $1 \leq j < k$ and $ \| I w_{l+1} \|_{L_{t}^{2(p-1)-}
    L_{x}^{2(p-1)+}(J_{k})} \leq \eta_{0}$. Notice that

    $$\begin{array}{ll}
    k & \lesssim_{A} L^{2(p-1)-}(l+1)
    \end{array}
    \label{Eqn:Boundkprime}$$

    in view of (\[Eqn:InducStr\]). Notice also that, in view of Proposition \[prop:LocalBd\], we have [^13]

    $$\begin{array}{ll}
    Z(w_{l+1},J_{j}) & \lesssim N^{1-s} A^{\frac{p+1}{2}} X_{C,Q}^{2(p-1)-}
    \end{array}
    \label{Eqn:BoundwPerturb}$$

-   Let $\Gamma:= w_{l} - v_{l} - w_{l+1}$. A simple computation shows that

    $$\begin{array}{ll}
    \partial_{tt} I \Gamma - \triangle I \Gamma + I \Gamma & = (I F(w_{l+1}) - F(Iw_{l+1})) + ( F(Iw_{l}) - I F(w_{l})) + ( F(Iw_{l+1}) - F(Iw_{l}))
    \end{array}$$

    Therefore we conclude that

    $$\begin{array}{ll}
    I \Gamma(t) & = I \Gamma_{l}(t_{j}) + X_{1} + X_{2} + X_{3}
    \end{array}$$

    with

    $$\begin{array}{ll}
    I \Gamma_{l}(t_{j},t) & := \cos{ ((t-t_{j})\langle D \rangle)} I \Gamma(t_{j}) + \frac{\sin{ \left((t -t_{j}) \langle D  \rangle \right)
    }}{\langle D \rangle}
    \partial_{t} I \Gamma(t_{j})
    \end{array}$$

    $$\begin{array}{ll}
    X_{1} & := - \int_{t_{j}}^{t} \frac{\sin{(t-s) \langle D \rangle}}{\langle D \rangle} (I F(w_{l+1}) - F(Iw_{l+1})) \, ds
    \end{array}$$

    $$\begin{array}{ll}
    X_{2} & := - \int_{t_{j}}^{t} \frac{\sin{(t-s) \langle D \rangle}} { \langle D \rangle} (I F(w_{l}) - F(Iw_{l})) \, ds
    \end{array}$$

    and

    $$\begin{array}{ll}
    X_{3} & := - \int_{t_{j}}^{t} \frac{\sin{(t-s) \langle D \rangle}} { \langle D \rangle} ( F(Iw_{l}) - F(Iw_{l+1}) ) \, ds
    \end{array}$$

    Notice that

    $$\begin{array}{ll}
    \Gamma_{l}(t_{j+1},t) & = \Gamma_{l}(t_{j},t) + Y_{1} + Y_{2} + Y_{3}
    \end{array}$$

    with

    $$\begin{array}{ll}
    Y_{1} & := - \int_{t_{j}}^{t_{j+1}} \frac{\sin{(t-s) \langle D \rangle}} { \langle D \rangle} (IF(w_{l+1}) -F(Iw_{l+1})) \, ds
    \end{array}$$

    $$\begin{array}{ll}
    Y_{2} & := - \int_{t_{j}}^{t_{j+1}} \frac{\sin{(t-s) \langle D \rangle}} { \langle D \rangle} (I F(w_{l}) - F(Iw_{l})) \, ds
    \end{array}$$

    and

    $$\begin{array}{ll}
    Y_{3} & := - \int_{t_{j}}^{t_{j+1}} \frac{\sin{(t-s) \langle D \rangle}} { \langle D \rangle} (F(Iw_{l}) - F(Iw_{l+1})) \, ds
    \end{array}$$

    In view of the assumptions, it is enough to estimate $ \| I \Gamma \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J)}$. By the Sobolev embedding

    $$\begin{array}{ll}
    \| I \Gamma \|_{ L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J) } & \lesssim  \|  \langle D \rangle^{1- s_{c}} I \Gamma \|_{ L_{t}^{2(p-1)-}
    L_{x}^{\frac{6(p-1)}{2p-3} +} (J) }
    \end{array}
    \label{Eqn:SobEmbedGamma}$$

    it is enough to estimate $\sup_{(q,r) - \left( \frac{1}{2} \right) \, - \, wave \, adm} \| \langle D \rangle^{1- s_{c}} I
    \Gamma \|_{L_{t}^{q} L_{x}^{r}} $. By summation over $j$, it is enough to estimate\
    $Z(j,t):= \sup_{(q,r) -  \frac{1}{2} \, - \, wave \, adm} \| \langle D \rangle^{1- s_{c} } I \Gamma \|_{L_{t}^{q} L_{x}^{r}([t_{j},t])}$ for all $1 \leq j < k$ and for $ t_{j} \leq t \leq t_{j+1}$. Let $ Z_{l}(j):= \sup_{(q,r) - \, \frac{1}{2} \, - \, wave \, adm } \| \langle D
    \rangle^{1- s_{c} } \Gamma_{l}(t_{j}) \|_{L_{t}^{2(p-1)} L_{x}^{\tilde{r}}([t_{j},b])}$. There exists a constant $\bar{C}:=\bar{C}(p) >1$ such that

    $$\begin{array}{ll}
    Z(j,t) & \leq Z_{l}(j) + \bar{C} \sup_{(q,r)- \frac{1}{2}   \, wave \, adm}  \| \langle D \rangle^{1- s_{c}} X_{1}
    \|_{L_{t}^{q} L_{x}^{r} ([t_{j},t])} \\
    & + \bar{C}  \sup_{(q,r)- \frac{1}{2} \, wave \, adm} \| \langle D \rangle^{1- s_{c} } X_{2} \|_{L_{t}^{q} L_{x}^{r} ([t_{j},t]) } \\
    & + \bar{C}  \sup_{(q,r)- \frac{1}{2} \, wave \, adm}  \| \langle D \rangle^{1- s_{c} } X_{3} \|_{ L_{t}^{q} L_{x}^{r} ([t_{j},t]) }
    \end{array}$$

    and

    $$\begin{array}{ll}
    Z_{l}(j+1) & \leq Z_{l}(j) + \bar{C} \sup_{(q,r)- \, \frac{1}{2} \, - wave \, adm } \| \langle D \rangle^{1- s_{c} } Y_{1} \|_{L_{t}^{q}
    L_{x}^{r} (J_{j}) }
    \\
    & + \bar{C} \sup_{(q,r)- \, \frac{1}{2} \, - wave \, adm} \| \langle D \rangle^{1- s_{c} } Y_{2} \|_{L_{t}^{q} L_{x}^{r } (J_{j}) }  \\
    & + \bar{C} \sup_{(q,r)- \, \frac{1}{2} \, - wave \, adm} \| \langle D \rangle^{1- s_{c} } Y_{3} \|_{L_{t}^{q} L_{x}^{r} (J_{j}) }
    \end{array}$$

-   We are interested in estimating $ \| \langle D \rangle^{1- s_{c}  } X_{2} \|_{L_{t}^{q}
    L_{x}^{r} (J) } $. We write $X_{2}: = X_{2,1} + X_{2,2}$ with

    $$\begin{array}{ll}
    X_{2,1} & : = - \int_{t_{j}}^{t} \frac{\sin{(t-s) \langle D \rangle}}{\langle D \rangle} \left( I F(w_{l}) - F(w_{l}) \right)
    \end{array}$$

    and

    $$\begin{array}{ll}
    X_{2,2} & : = - \int_{t_{j}}^{t} \frac{\sin{(t-s) \langle D \rangle}}{\langle D \rangle} \left( F(w_{l}) - F(Iw_{l}) \right)
    \end{array}$$

    First we deal with $ \| \langle D \rangle^{1- s_{c} } X_{2,2} \|_{L_{t}^{q} L_{x}^{r}(J) } $. By Proposition \[prop:LocalBd\] we have

    $$\begin{array}{l}
    \| \langle D \rangle^{1- s_{c}} X_{2,2} \|_{ L_{t}^{q} L_{x}^{r} ([t_{j},t]) }  \lesssim
    \| \langle D \rangle ^{1- s_{c}} ( F(w_{l})-F(Iw_{l})) \| _{L_{t}^{4+}  L_{x}^{4-}  (J) } \\
    \lesssim \| w_{l} \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J) } \| \langle D \rangle^{1- s_{c}} P_{\gtrsim N} w_{l}
    \|_{L_{t}^{4+} L_{x}^{4-} (J) } \\
     \lesssim  \| \langle D \rangle^{1- s_{c}} I w_{l}  \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J) }
     \frac{ \| \langle D \rangle^{1- \frac{1}{2}}
    I w_{l} \|_{L_{t}^{4+} L_{x}^{4-}(J) }}{N^{s_{c}-\frac{1}{2}}} \\
    \lesssim  \frac { \left( X_{C,Q}^{2(p-1)-} N^{1-s} A^{\frac{p+1}{2}} \right)^{p}}{N^{s_{c} -\frac{1}{2} \, -}}
    \end{array}$$

    Next we deal with $ \| \langle D \rangle^{1- s_{c}} X_{2,1} \|_{L_{t}^{q} L_{x}^{r}(([t_{j},t])) } $. By (\[Eqn:Decomp\]), it is enough to bound $ \| \langle D \rangle^{1- s_{c} } X_{2,1,1} \|_{L_{t}^{q} L_{x}^{r}([t_{j},t]) }$, $ \| \langle D \rangle^{1- s_{c}} X_{2,1,2}
    \|_{L_{t}^{q} L_{x}^{r}([t_{j},t]) } $ and $ \| \langle D \rangle^{1- s_{c}} X_{2,1,3} \|_{L_{t}^{q} L_{x}^{r}([t_{j},t]) }$ with

    $$\begin{array}{ll}
    X_{2,1,1} & := - \int_{t_{j}}^{t} \frac{\sin{(t-s) \langle D \rangle}}{\langle D \rangle} \left( \overbrace{ (I-1) \left( F ( P_{<< N} w_{l} )
    \right) }^{Z_{1}} \right) \, ds
    \end{array}$$

    $$\begin{array}{ll}
    X_{2,1,2} & := - \int_{t_{j}}^{t} \frac{\sin{(t-s) \langle D \rangle}}{\langle D \rangle} \left( \overbrace{   (I-1) \left( \int_{0}^{1} |
    P_{<<N} w_{l}(s) + y P_{\gtrsim N} w_{l}(s)|^{p-1} \,  P_{\gtrsim N} w_{l} (s) \, dy \right) }^{Z_{2}} \right) \, ds
    \end{array}$$

    and

    $$\begin{array}{ll}
    X_{2,1,3} & := - \int_{t_{j}}^{t} \frac{\sin{(t-s) \langle D \rangle}}{\langle D \rangle}  \left( \overbrace { (I -1) \left( \int_{0}^{1} \frac{
    P_{<< N} w_{l} + y P_{\gtrsim N} w_{l} }{ \overline{P_{<<N} w_{l} + y P_{\gtrsim N} w_{l}} } |P_{<< N} w_{l} + y P_{\gtrsim N} w_{l}|^{p-1} \,
    \overline{P_{\gtrsim N} w_{l} (s)} \, dy \right) }^{Z_{3}} \right) \, ds
    \end{array}$$

    Again, we use the smoothness of $F$ ($F$ is $C^{1}$) to deal with $X_{2,1,1}$. We have

    $$\begin{array}{l}
     \| \langle  D  \rangle^{1-s_{c}} X_{2,1,1} \|_{L_{t}^{q} L_{x}^{r} ([t_{j},t]) }  \lesssim
    \| \langle D \rangle^{1- s_{c}} Z_{1} \|_{L_{t}^{\frac{4}{3}} L_{x}^{\frac{4}{3}} ([t_{j},t])} \\
    \lesssim \frac{ \| \nabla F ( P_{<<N} w_{l} ) \|_{ L_{t}^{\frac{4}{3}} L_{x}^{\frac{4}{3}} ([t_{j},t]) } }
    {N^{s_{c}-}} \\
    \lesssim  \frac{1}{N^{s_{c}-}} \| P_{<<N} w_{l} \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([t_{j},t])} \| \nabla P_{<< N} w_{l} \|_{
    L_{t}^{4+} L_{x}^{4-} ([t_{j},t])}  \\
    \lesssim \frac{1}{N^{ \left( s_{c} -\frac{1}{2} \right) \, -}}  \| \langle D \rangle^{1-s_{c}} I w_{l} \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}
    ([t_{j},t])} \| \langle D \rangle^{1-\frac{1}{2}}
    I w_{l} \|_{L_{t}^{4+} L_{x}^{4-} ([t_{j},t]) } \\
    \lesssim \frac { \left( X^{2(p-1)-}_{C,Q} N^{1-s} A^{\frac{p+1}{2}} \right)^{p}}{N^{ \left( s_{c} -\frac{1}{2} \right) \, -}}
    \end{array}$$

    As for $X_{2,1,2}$, we have

    $$\begin{array}{l}
    \| \langle  D  \rangle^{1- s_{c} } X_{2,1,2} \|_{L_{t}^{q} L_{x}^{r} ([t_{j},t]) }  \lesssim \int_{0}^{1} \left\| \langle D \rangle^{1-s_{c}}
    Z_{2}  \right\|_{L_{t}^{\frac{4}{3}} L_{x}^{\frac{4}{3}} ([t_{j},t])} \, dy \\
    \lesssim
     \int_{0}^{1}
    \left[
    \begin{array}{l}
    \| \langle D \rangle^{1-s_{c}} ( |P_{<< N} w_{l} + y P_{\gtrsim N} w_{l}|^{p-1} ) \|_{L_{t}^{?} L_{x}^{?} ([t_{j},t])}
    \| P_{\gtrsim N} w_{l} \|_{L_{t}^{?} L_{x}^{?} ([t_{j},t])} \\
    + \left\| | P_{<<N} w_{l} + y P_{\gtrsim N} w_{l} |^{p-1} \right\|_{L_{t}^{2-} L_{x}^{2+} ([t_{j},t])} \| \langle D  \rangle^{1-s_{c}}
    P_{\gtrsim N} w_{l} \|_{L_{t}^{4+} L_{x}^{4-} ([t_{j},t])}
    \end{array}
    \right] \, dy \\
    \lesssim  \frac{ \| w_{l}  \|_{ L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J) }^{p-1}  \|  \langle D  \rangle^{1- \frac{1}{2}} I w_{l} \|_{ L_{t}^{4+}
    L_{x}^{4-} (J) }}{N^{ \left( s_{c}- \frac{1}{2} \right) \, -  }} + \frac{ \| w_{l}  \|_{ L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J) }^{p-1}  \| \langle
    D  \rangle^{1- \frac{1}{2}} I w_{l} \|_{
    L_{t}^{4+} L_{x}^{4-} (J) }}{N^{ ( 1 -s_{c}) \, -  }}   \\
    \lesssim  \frac { \left( X^{2(p-1)-}_{C,Q}  N^{1-s} A^{\frac{p+1}{2}} \right)^{p}}{N^{ \min{ \left( \left( s_{c} -\frac{1}{2} \right) \, -, \,
    \left( 1- s_{c} \right)- \right) }}} +
    \end{array}$$

    by using the product rule followed by a two-variable Leibnitz rule\
    (see Appendix with $f:= P_{<< N} w_{l}$ and $g:= P_{\gtrsim N} w_{l}$, $F(f,g)=|f + y g|^{p-1}$ and $q=p-2$)

    $X_{2,1,3}$ is treated in a similar fashion: we get

    $$\begin{array}{ll}
    \| \langle  D  \rangle^{1- s_{c} } X_{2,1,3} \|_{L_{t}^{q} L_{x}^{r} ([t_{j},t]) } & \leq  \frac { \left( X^{2(p-1)-}_{C,Q} N^{1-s}
    A^{\frac{p+1}{2}} \right)^{p}}{N^{  \min{ \left( \left( s_{c} -\frac{1}{2} \right) \, -, \, \left( 1- s_{c} \right)- \right)}}}
    \end{array}$$

    Finally we have

    $$\begin{array}{ll}
    \| \langle  D  \rangle^{1- s_{c} } X_{2} \|_{L_{t}^{q} L_{x}^{r}(([t_{j},t])} & \leq \frac { \left( X^{2(p-1)-}_{C,Q} N^{1-s} A^{\frac{p+1}{2}}
    \right)^{p}}{N^{  \min{ \left( \left( s_{c} -\frac{1}{2} \right) \, -, \, \left( 1- s_{c} \right)- \right)}}}
    \end{array}
    \label{Eqn:EstX2}$$

    Similarly

    $$\begin{array}{ll}
    \| \langle D \rangle^{1-s_{c}} Y_{2} \|_{L_{t}^{q} L_{x}^{r} (J_{j})} & \lesssim \frac { \left( X^{2(p-1)-}_{C,Q} N^{1-s} A^{\frac{p+1}{2}}
    \right)^{p}}{N^{  \min{ \left( \left( s_{c} -\frac{1}{2} \right) \, -, \, \left( 1- s_{c} \right)- \right)}}}
    \end{array}
    \label{Eqn:EstY2}$$

-   In order to estimate $  \| \langle  D  \rangle^{1- s_{c}} X_{1} \|_{L_{t}^{q} L_{x}^{r} ([t_{j},t]) } $ and $\| \langle D \rangle^{1-s_{c}} Y_{1} \|_{L_{t}^{q} L_{x}^{r} (J_{j})} $, we perform a similar decomposition to that in the proof of (\[Eqn:EstX2\]), using (\[Eqn:BoundwPerturb\]) instead of (\[Eqn:BounduPerturb\]). We also have

    $$\begin{array}{ll}
    \| \langle  D  \rangle^{1- s_{c}} X_{1} \|_{L_{t}^{q} L_{x}^{r} ([t_{j},t]) } & \lesssim \frac { \left( X^{2(p-1)-}_{C,Q} N^{1-s}
    A^{\frac{p+1}{2}} \right)^{p}}{N^{ \left( s_{c} -\frac{1}{2} \right) \, -}}
    \end{array}
    \label{Eqn:EstX1}$$

    . and

    $$\begin{array}{ll}
    \| \langle D \rangle^{1-s_{c}} Y_{1} \|_{L_{t}^{q} L_{x}^{r} (J_{j})} & \lesssim \frac { \left( X^{2(p-1)-}_{C,Q} N^{1-s} A^{\frac{p+1}{2}}
    \right)^{p}}{N^{ \left( s_{c} -\frac{1}{2} \right) \, -}}
    \end{array}$$

-   We are interested in estimating $  \| \langle  D  \rangle^{ 1 - s_{c}} X_{3} \|_{L_{t}^{q} L_{x}^{r} ([t_{j},t])} $ and $  \| \langle  D  \rangle^{ 1 - s_{c}} Y_{3} \|_{L_{t}^{q} L_{x}^{r} (J_{j})} $. Since $s_{c} > \frac{1}{2}$, there exist $\theta:= \theta(p)$, $m < \frac{1}{2}$, $(q,r)$ $m$- wave admissible such that

    $$\begin{array}{ll}
    \| \langle D  \rangle^{1-s_{c}} I v_{l} \|_{L_{t}^{4+} L_{x}^{4-} ([t_{j},t])} & \lesssim
    \| I v_{l} \|^{\theta}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}}    \| \langle D \rangle^{1-m}  I v_{l}  \|^{1- \theta}_{L_{t}^{q} L_{x}^{r}} \\
    & \lesssim k^{\theta}
    \end{array}$$

    in view of (\[Eqn:BoundEc\]) and (\[Eqn:BoundIv\]). Therefore

    $$\begin{array}{l}
    \| \langle D \rangle^{1- s_{c} } X_{3} \|_{L_{t}^{q} L_{x}^{r} ([t_{j},t])}  \lesssim
     \left(
    \begin{array}{l}
    \| I w_{l} \|^{p-1}_{ L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([t_{j},t]) } + \| I v_{l} \|^{p-1}_{ L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([t_{j},t]) } \\
    + \| I w_{l+1} \|^{p-1}_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([t_{j},t]) }
    \end{array}
    \right) \\
    \left( \| \langle D \rangle^{1 - s_{c} } I v_{l} \|_{ L_{t}^{4+} L_{x}^{4-} ([t_{j},t]) } +
    \| \langle D \rangle^{1 - s_{c}} I \Gamma \|_{ L_{t}^{4+} L_{x}^{4- } ([t_{j},t]) } \right)   \\
    \lesssim  \left( Z^{p-1}(j,t) + k^{p-1} + \eta^{p-1}_{0} \right) \left( k^{\theta} +  \| \langle  D  \rangle^{1 - s_{c} } I \Gamma \|_{
    L_{t}^{4+} L_{x}^{4-} ([t_{j},t])} \right)
    \end{array}$$

    Similarly

    $$\begin{array}{ll}
    \| \langle D \rangle^{1- s_{c}} Y_{3} \|_{L_{t}^{q} L_{x}^{r} ([t_{j},b]) } \lesssim \left( Z^{p-1}(j,t_{j+1}) + k^{p-1} + \eta^{p-1}_{0}
    \right) \left( k^{\theta} + \| \langle D \rangle^{1- s_{c}} I \Gamma \|_{L_{t}^{4+} L_{x}^{4-} (J_{j})}    \right)
    \end{array}
    \label{Eqn:EstY3}$$

    Therefore there exists a positive constant (that we still denote by $\bar{C}$) such that

    $$\begin{array}{ll}
    Z(j+1,t) \leq Z_{l}(j) + \bar{C}  \left( \eta^{p-1}_{0} +  k^{p-1}  + Z^{p-1}(j,t) \right) \left( Z(j,t) + k^{\theta} \right)  + \bar{C} \frac{
    \left( X^{2(p-1)-}_{C,Q} N^{1-s} A^{\frac{p+1}{2}} \right)^{p} } {N^{   \min{ \left( \left( s_{c}-\frac{1}{2} \right) \,-, (1-s_{c}) \,- \right)
    }}}
    \end{array}
    \label{Eqn:Zjplusone}$$

    for $t \in [t_{j},t_{j+1}]$ and

    $$\begin{array}{ll}
    Z_{l}(j+1) & \leq Z_{l}(j) + \bar{C} \left( \eta^{p-1}_{0} + k^{p-1} + Z^{p-1}(j,t_{j+1})  \right) \left( Z(j,t_{j+1}) + k^{\theta} \right) +
    \bar{C} \frac{ \left(  X^{2(p-1)-}_{C,Q}  N^{1-s} A^{\frac{p+1}{2}} \right)^{p} } {N^{  \min{ \left( \left( s_{c}-\frac{1}{2} \right) \,-,
    (1-s_{c}) \,-  \right) }}}
    \end{array}
    \label{Eqn:Zljplusone}$$

    We claim that

    $$\begin{array}{ll}
    Z_{l}(j) & \leq \eta_{0}
    \end{array}
    \label{Eqn:BoundZl}$$

    for all $j$, provided that $C_{2}$ is large enough in (\[Eqn:Boundk\]). Indeed, by induction:

    -   $Z_{l}(1)=0$ since $w_{l+1}(T_{l}):=(w_{l}-v_{l})(T_{l})$

    -   assume that for all $k < j$ we have $Z_{l}(k) \leq \eta_{0}$; then, by (\[Eqn:Zjplusone\]) and a continuity argument we see that,even if it means increasing the value of $\bar{C}$

        $$\begin{array}{ll}
        Z(j,t) & \leq \bar{C} \max{ \left( Z_{l}(j), \, k^{\theta}, \, \frac{  \left( X^{2(p-1)-}_{C,Q}  N^{1-s} A^{\frac{p+1}{2}} \right)^{p} }
        { \min{ \left( \left( s_{c}-\frac{1}{2} \right) \,-, (1-s_{c}) \,-  \right) }} \right)} \\
        & \leq \bar{C} \eta_{0}
        \end{array}
        \label{Eqn:BoundIntermZj}$$

        Plugging (\[Eqn:BoundIntermZj\]) into (\[Eqn:Zljplusone\]) we have (again, even if it means increasing the value of $\bar{C}$)

        $$\begin{array}{ll}
        Z_{l}(j+1) & \leq Z_{l}(j) + \bar{C} \left( 3 \eta^{p-1}_{0} \right) \left( \max{ \left( Z_{l}(j), \, k^{\theta}, \, \frac{ ( X_{C,Q}^{2(p-1)-}
        N^{1-s} A^{\frac{p+1}{2}} )^{p} }{N^{ \min{ \left( \left( s_{c}-\frac{1}{2} \right) \,-, (1-s_{c}) \,-  \right) }} } \right)} + k^{\theta}
        \right) \\
        & + \frac{ \left( X_{C,Q}^{2(p-1)-} N^{1-s} A^{\frac{p+1}{2}} \right)^{p}}{N^{\min{ \left(
        \left( s_{c}-\frac{1}{2} \right) \,-, (1-s_{c}) \,-  \right) }}} \\
        & \leq \bar{C} \max{\left( Z_{l}(j), \, k^{\theta}, \,  \frac{ \left( X^{2(p-1)-}_{C,Q}  N^{1-s} A^{\frac{p+1}{2}} \right)^{p}}{N^{\min{ \left(
        \left( s_{c}-\frac{1}{2} \right) \,-, (1-s_{c}) \,-  \right) }}} \right)}
        \end{array}$$

        and, therefore by simple iteration, in view of (\[Eqn:Boundk\]) and (\[Eqn:Cond4\]), we see that, even if it means increasing $\bar{C}$,

        $$\begin{array}{ll}
        Z_{l}(j+1) & \leq \bar{C}^{\bar{C}  L(l+1)} \max{ \left( k^{\theta}, \, \frac{\left(  X^{2(p-1)-}_{C,Q} N^{1-s} A^{\frac{p+1}{2}} \right)^{p}}
        {N^{ \min{ \left( \left( s_{c}-\frac{1}{2}
        \right) \,-, (1-s_{c}) \,-  \right) }} } \right)} \\
        & \leq \eta_{0}
        \end{array}$$

        provided that $C_{2} >> 1$ is large enough in (\[Eqn:Boundk\]). Therefore (\[Eqn:BoundZl\]) holds, if $k=j$\
        By (\[Eqn:Boundkprime\]), (\[Eqn:BoundZl\]), (\[Eqn:BoundIntermZj\]), (\[Eqn:BoundIv\]), (\[Eqn:BoundIw\]) and (\[Eqn:SobEmbedGamma\]) we see that

        $$\begin{array}{ll}
        \| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([T_{l},b])} & \lesssim L(l+1)
        \end{array}$$

Appendix
========

Proof of Theorem \[Thm:Scat\] {#Sub:Scat}
-----------------------------

In this subsection, we give a short proof of Theorem \[Thm:Scat\] for small data $( A \leq \frac{1}{1000})$. It relies upon well-known arguments: see [@linsog] for example. Assume that $w$ satisfies (\[Eqn:NlkgWdat\]) with data $(w_{0},w_{1})$ such that $\|(w_{0},w_{1})\|_{H^{s} \times H^{s-1}} \leq A \leq \frac{1}{1000}$.

Let $\bar{Z}(T):= \| (w(T),\partial_{t} w(T)) \|_{H^{s} \times H^{s-1}} + \| w \|_{L_{t}^{2(p-1)} L_{x}^{2(p-1)} ([0,T]) } + \| \langle D
\rangle^{s-\frac{1}{2}} w \|_{L_{t}^{2(p-1)} L_{x}^{\frac{6(p-1)}{2p-3}} ([0,T]) } + \| \langle D \rangle^{s- \frac{1}{2}} w \|_{L_{t}^{4}
L_{x}^{4} [0,T]}$. Then, plugging $\langle D \rangle^{s-\frac{1}{2}}$ into (\[Eqn:StrNlkg\]), we get

$$\begin{array}{l}
\| w \|_{L_{t}^{2(p-1)} L_{x}^{2(p-1)} ([0,T]) } \lesssim  \| \langle D \rangle^{s-\frac{1}{2}} w \|_{L_{t}^{2(p-1)}
L_{x}^{\frac{6(p-1)}{2p-3}}([0,T]) }
\end{array}$$

and

$$\begin{array}{l}
\bar{Z}([0,T],w)  \lesssim \| (w_{0},u_{1}) \|_{H^{s} \times H^{s-1}} + \|  \langle D \rangle ^{s-
\frac{1}{2}} ( |w|^{p-1} w ) \|_{L_{t}^{\frac{4}{3}} L_{x}^{\frac{4}{3}} ([0,T]) } \\
 \lesssim \frac{1}{1000} +  \| \langle D \rangle^{s-\frac{1}{2}} w \|_{L_{t}^{4} L_{x}^{4} ([0,T])} \| w \|^{p-1}_{L_{t}^{2(p-1)}
L_{x}^{2(p-1)} ([0,T])} \\
\lesssim \frac{1}{1000} + Z^{p}([0,T],w)
\end{array}
\label{Eqn:ControlZ}$$

we get, by a continuity argument $Z([0,T]) \lesssim 1$. This proves global well-posedness. As for scattering, following the steps from (\[Eqn:ScatBegin\]) to (\[Eqn:ScatBegin\]) we have, for $t_{2}>t_{1}$

$$\begin{array}{ll}
\| \mathbf{K}^{-1} w^{nlin}(t_{1}) - \mathbf{K}^{-1} w^{nlin}(t_{2}) \|_{H^{s} \times H^{s-1}} & \lesssim \| w^{nlin}(t_{1}) - w^{nlin}(t_{2})
\|_{H^{s}
\times H^{s-1}} \\
& \lesssim \| \langle D \rangle^{s-\frac{1}{2}} (|w|^{p-1} w) \| _{L_{t}^{\frac{4}{3}} L_{x}^{\frac{4}{3}} ([t_{1},t_{2}]) } \\
& \lesssim \| \langle  D \rangle^{s- \frac{1}{2}} w \|_{L_{t}^{4} L_{x}^{4} ([t_{1},t_{2}])} \| w \|_{L_{t}^{2(p-1)} L_{x}^{2(p-1)}([t_{1},t_{2}])} \\
\end{array}$$

This proves, in view of (\[Eqn:ControlZ\]), that the Cauchy criterion is satisfied by $\mathbf{K}^{-1} w^{nlin}(t)$ and therefore $\mathbf{K}^{-1} w^{nlin}(t)$ has a limit as $t \rightarrow \infty$ and therefore $w$ scatters.

A two-variable fractional Leibnitz rule
---------------------------------------

In this subsection we prove a two-variable Leibnitz rule

[**“A two-variable Leibnitz rule”**]{} Let $F \in C^{1} \left( \mathbb{C}^{2}, \mathbb{C} \right)$ such that $F(0,0)=0$ and such that for all $\lambda \in [0,1]$ and for all $(z_{1},z_{2},w_{1},w_{2}) \in \mathbb{C}^{4}$ we have

$$\begin{array}{l}
| F^{'} ( \lambda z_{1} + (1- \lambda) z_{2}, \lambda w_{1} + (1-\lambda) w_{2} ) | \lesssim |z_{1}|^{\alpha} + |z_{2}|^{\alpha} +
|w_{1}|^{\alpha} + |w_{2}|^{\alpha}
\end{array}$$

for some $\alpha > 0$. Then

$$\begin{array}{ll}
\| F(f,g) \|_{H^{s,p}} &  \lesssim  \| f \|^{\alpha}_{L^{p_{1}}} \| f \|_{H^{s,p_{2}}} + \| g \|^{\alpha}_{L^{\tilde{p}_{1}}}  \| f
\|_{H^{s,\tilde{p}_{2}}} \\
& + \| f \|^{\alpha}_{L^{r_{1}}} \| g \|_{H^{s,r_{2}}} +  \| g \|^{\alpha}_{L^{\tilde{r}_{1}}} \| g \|_{H^{s,\tilde{r}_{2}}}
\end{array}
\label{Eqn:LeibnForm}$$

assuming that $(p, p_{1}, p_{2}, \tilde{p}_{1}, \tilde{p}_{2}, r_{1},r_{2}, \tilde{r}_{1}, \tilde{r}_{2}) \in (1, \infty)^{9}$ and

$$\begin{array}{l}
\frac{1}{p}  = \frac{\alpha}{p_{1}} + \frac{1}{p_{2}} = \frac{\alpha}{\tilde{p}_{1}} + \frac{1}{\tilde{p}_{2}} \\
\frac{1}{p} = \frac{\alpha}{r_{1}} + \frac{1}{r_{2}} = \frac{\alpha}{\tilde{r}_{1}} + \frac{1}{\tilde{r}_{2}}
\end{array}$$

The proof relies upon a simple modification of the one-variable fractional Leibnitz rules (see [@christweins]). We recall the following inequalities (see for example [@taylor] ): given $q: \mathbb{R}^{3} \rightarrow \mathbb{C}$ a function, we have

$$\begin{array}{l}
\int_{\mathbb{R}^{3}} | P_{N_{2}} q(x) - P_{N_{2}} q(y) | | \check{\psi_{N_{1}}} (x-y) | \, dy \lesssim \min{ \left( \frac{N_{2}}{N_{1}}, 1
\right) } M_{h} ( \tilde{P}_{N_{2}} q)(x)
\end{array}$$

and

$$\begin{array}{ll}
\int_{\mathbb{R}^{3}} | P_{N_{2}} q(x) -P_{N_{2}} q(y) |  | \check{ \psi_{N_{1}}} (x-y) | H(y) \, dy \\
\lesssim \min{ \left( \frac{N_{2}}{N_{1}},1 \right)} \left(  M_{h}( \tilde{P}_{N_{2}} q )(x) M_{h} H(x)  + M_{h} ( |\tilde{P}_{N_{2}} q| H )(x)
\right)
\end{array}
\label{Eqn:EstHy}$$

with $(N_{1},N_{2}) \in 2^{\mathbb{N}^{*}} \in \mathbb{N}$, $H$ a nonnegative function, $\psi_{M}(\xi):= \psi \left( \frac{\xi}{M}\right)$ (if $M \in 2^{\mathbb{N}}$), $\tilde{P}_{M} := P_{ \frac{M}{2} \leq \cdot \leq 2 M}$ (if $M \in 2^{\mathbb{N}^{*}}$), $\tilde{P}_{1}:= P_{\leq 2}$ and $ ( M_{h}(f) )(x):= \sup_{r >0 } \frac{1}{|B(x,r)|} \int_{B(x,r)} |f(y)| \, dy$.\
\
Recall also the Paley-Littlewood inequalities (see [@stein])

$$\begin{array}{l}
\| F(f,g) \|_{H^{s,p}} \lesssim \| P_{1} ( F(f,g)) \|_{L^{p}} + \left\| \left( \sum_{N_{1} \in 2^{\mathbb{N}^{*}}} N_{1}^{2s} |P_{N_{1}} (
F(f,g)) |^{2} \right)^{\frac{1}{2}} \right\|_{L^{p}}
\end{array}$$

and

$$\begin{array}{l}
\left\| \left( \sum_{N_{1} \in 2^{\mathbb{N}}} N_{1}^{2s} |\tilde{P}_{N_{1}} f|^{2} \right)^{\frac{1}{2}} \right\|_{L^{p}} \lesssim \| f
\|_{H^{s,p}}
\end{array}
\label{Eqn:Paleyf}$$

We write

$$\begin{array}{l}
P_{N_{1}} ( F(f,g) )(x) \\  \
= \int_{\mathbb{R}^{3}} F(f(y),g(y)) \check{\psi_{N_{1}}}(x-y) \, dy \\
= \int_{\mathbb{R}^{3}}  \left( F(f(y),g(y)) - F(f(x),g(x)) \right)  \check{\psi_{N_{1}}}(x-y) \, dy \\
= A_{1} + A_{2} + A_{3} + A_{4}
\end{array}$$

with

$$\begin{array}{l}
A_{1} := \int_{0}^{1} \int_{\mathbb{R}^{3}} | \partial_{z} F( \lambda f(y) + (1- \lambda) f(x), \lambda g(y) + (1- \lambda) g(x) ) | |  f(y) -
f(x)) | \,  | \check{\psi_{N_{1}}}(x-y) | \, d\lambda \, dy
\end{array}$$

$$\begin{array}{l}
A_{2} := \int_{0}^{1} \int_{\mathbb{R}^{3}} | \partial_{\bar{z}} F ( \lambda f(y) + (1- \lambda) f(x), \lambda g(y) + (1 - \lambda) g(x) ) | |
f(y) - f(x) | \,  | \check{\psi_{N_{1}}}(x-y) | \, d \lambda \, dy
\end{array}$$

$$\begin{array}{l}
A_{3} := \int_{0}^{1} \int_{\mathbb{R}^{3}} | \partial_{w}  F ( \lambda f(y) + (1- \lambda) f(x), \lambda g(y) + (1 - \lambda) g(x) ) |  |g(y)
-g(x) |  | \check{\psi_{N_{1}}}(x-y) | \, d \lambda \, dy
\end{array}$$

and

$$\begin{array}{l}
A_{4} := \int_{0}^{1} \int_{\mathbb{R}^{3}} | \partial_{\bar{w}} F ( \lambda f(y) + (1- \lambda) f(x), \lambda g(y) + (1 - \lambda) g(x) ) |
|g(y)-g(x) |  | \check{\psi_{N_{1}}}(x-y) | d \lambda \, dy
\end{array}$$

Let us deal for example with $A_{1}$.

$$\begin{array}{ll}
\sum_{N_{1} \in 2^{\mathbb{N}^{*}}} N_{1}^{2s} A_{1}^{2} & \lesssim  \sum_{N_{1} \in 2^{\mathbb{N}^{*}}}
N_{1}^{2s} \left( \int_{\mathbb{R}^{3}} |f(y)|^{\alpha} |f(y)-f(x)|  | \check{\psi_{N_{1}}}(x-y) | \, dy \right)^{2} \\
& + \sum_{N_{1} \in 2^{\mathbb{N}^{*}}} N_{1}^{2s} |f(x)|^{2 \alpha}  \left( \int_{\mathbb{R}^{3}} |f(y)-f(x)|  | \check{\psi_{N_{1}}}(x-y) | \, dy \right)^{2} \\
& + \sum_{N_{1} \in 2^{\mathbb{N}^{*}}} N_{1}^{2s} \left( \int_{\mathbb{R}^{3}} |g(y)|^{\alpha} |f(y)-f(x)|  | \check{\psi_{N_{1}}}(x-y) | \, dy  \right)^{2} \\
& + \sum_{ N_{1} \in 2^{\mathbb{N}^{*}} } N_{1}^{2s} |g(x)|^{2 \alpha}  \left( \int_{\mathbb{R}^{3}} |f(y)-f(x)|  | \check{\psi_{N_{1}}}(x-y) |
\, dy
\right)^{2} \\
& \lesssim A^{2}_{1,1} + A^{2}_{1,2} + A^{2}_{1,3} + A^{2}_{1,4}
\end{array}$$

We have

$$\begin{array}{l}
A^{2}_{1,1}  \lesssim \sum_{N_{1} \in 2^{\mathbb{N}^{*}}} N_{1}^{2s} \left( \sum_{N_{2} \in 2^{\mathbb{N}}} \int_{\mathbb{R}^{3}}
|f(y)|^{\alpha}
|P_{N_{2}} (f)(y)- P_{N_{2}} (f)(x)| | \check{\psi_{N_{1}}}(x-y) | \, dy \right)^{2} \\
\lesssim \left(
\begin{array}{l}
\sum_{N_{1} \in 2^{\mathbb{N}^{*}}} N_{1}^{2s} \left(  \sum_{N_{2} \leq N_{1}} \int_{\mathbb{R}^{3}} |f(y)|^{\alpha} | P_{N_{2}}(f)(y) -
P_{N_{2}}(f)(x) | | \check{\psi_{N_{1}}}(x-y) | \, dy \right)^{2} \\
+ \sum_{N_{1} \in 2^{\mathbb{N}^{*}}} N_{1}^{2s} \left(  \sum_{N_{2} \geq N_{1}} \int_{\mathbb{R}^{3}} |f(y)|^{\alpha} | P_{N_{2}}(f)(y) -
P_{N_{2}}(f)(x) | | \check{\psi_{N_{1}}}(x-y) | \, dy \right)^{2}
\end{array}
\right) \\
\lesssim A^{2}_{1,1,1} + A^{2}_{1,1,2}
\end{array}$$

But, by (\[Eqn:EstHy\])

$$\begin{array}{l}
A^{2}_{1,1,1} \lesssim \left(
\begin{array}{l}
( M_{h} ( |f(x)|^{\alpha}) )^{2} \sum_{N_{1} \in 2^{\mathbb{N}^{*}} } N_{1}^{2s} \left( \sum_{N_{2} \leq N_{1}}
\frac{N_{2}}{N_{1}} M_{h} ( (\tilde{P}_{N_{2}} f)(x) )  \right)^{2} \\
+ \sum_{N_{1} \in 2^{\mathbb{N}^{*}}} N_{1}^{2s} \left( \sum_{N_{2} \leq N_{1}} \frac{N_{2}}{N_{1}} M_{h} (( |\tilde{P}_{N_{2}} f| |f|^{\alpha})
(x) ) \right)^{2}
\end{array}
\right)
\end{array}$$

Now, by Young’s inequality we have (since $s<1$)

$$\begin{array}{ll}
\sum_{N_{1} \in 2^{\mathbb{N}^{*}}} N_{1}^{2s} \left( \sum_{N_{2} \leq N_{1}} \frac{N_{1}}{N_{2}} |a_{N_{2}}|  \right)^{2}
& = \sum_{N_{1} \in 2^{\mathbb{N}^{*}}} \left( \sum_{N_{2} \leq N_{1}} \left( \frac{N_{2}}{N_{1}} \right)^{1-s} N^{s}_{2} |a_{N_{2}}| \right)^{2} \\
& \lesssim \sum_{N_{1} \in 2^{\mathbb{N} }} N_{1}^{2s} |a_{N_{1}}|^{2}
\end{array}$$

which implies that

$$\begin{array}{l}
A^{2}_{1,1,1} \lesssim  \left( M_{h} ( |f(x)|^{\alpha}) \right)^{2} \sum_{N_{1} \in 2^{\mathbb{N} }} N_{1}^{2s}  \left( M_{h} (
(\tilde{P}_{N_{1}} f)(x) ) \right)^{2} + \sum_{N_{1} \in 2^{\mathbb{N} }} N_{1}^{2s} \left( M_{h} ( (  |\tilde{P}_{N_{1}} f| |f|^{\alpha})(x) )
\right)^{2}
\end{array}
\label{Eqn:BoundA2111}$$

and therefore, by Fefferman-Stein maximal inequality [@FefferStein], Hölder’s inequality and (\[Eqn:Paleyf\]) we have

$$\begin{array}{l}
\| A_{1,1,1} \|_{L^{p}} \lesssim  \| f \|^{\alpha}_{L^{p_{1}}} \| f \|_{H^{s,p_{2}}}
\end{array}$$

Also, by (\[Eqn:EstHy\]) we have

$$\begin{array}{l}
A^{2}_{1,1,2} \lesssim \left(
\begin{array}{l}
( M_{h} ( |f(x)|^{\alpha}) )^{2} \sum_{N_{1} \in 2^{\mathbb{N}^{*}} } N_{1}^{2s} \left( \sum_{N_{2} > N_{1}}
 M_{h} ( (\tilde{P}_{N_{2}} f)(x) )  \right)^{2} \\
+ \sum_{N_{1} \in 2^{\mathbb{N}^{*}}} N_{1}^{2s} \left( \sum_{N_{2} > N_{1}} M_{h} (( |\tilde{P}_{N_{2}} f| |f|^{\alpha}) (x) ) \right)^{2}
\end{array}
\right)
\end{array}$$

But, by Young’s inequality (since $s>0$)

$$\begin{array}{l}
\sum_{N_{1} \in 2^{\mathbb{N}^{*}}} N_{1}^{2s} \left( \sum_{N_{2} > N_{1}} |a_{N_{2}}|   \right)^{2} = \sum_{N_{1} \in 2^{\mathbb{N}^{*}}}
\left( \sum_{N_{2} > N_{1}}  \left( \frac{N_{1}}{N_{2}} \right)^{s} N_{2}^{s} |a_{N_{2}}|  \right)^{2} \\
\lesssim \sum_{N_{1} \in 2^{\mathbb{N}}} N_{1}^{2s} |a_{N_{1}}|^{2}
\end{array}$$

and therefore (\[Eqn:BoundA2111\]) also holds if $A_{1,1,1}$ is substituted for $A_{1,1,2}$.\
The other terms ( $A_{1,2}$, $A_{1,3}$, $A_{1,4}$ and then $A_{2}$, $A_{3}$, $A_{4}$) are treated in a similar fashion.\
We also have $ \| P_{1}(F(f,g)) \|_{L^{p}} \lesssim  \| F(f,g) \|_{L^{p}} $. Then writing $F(f,g)= F(f,g)-F(0,0)$ and applying the fundamental theorem of calculus, we see that (\[Eqn:LeibnForm\]) holds if $s=0$.

[99]{}

J. Bourgain, *Global well-posedness of defocusing $3D$ critical NLS in the radial case*, JAMS 12 (1999), 145-171

J. Bourgain, *New global well-posedness results for non-linear Schrödinger equations*, AMS Publications, 1999

P.Brenner, *On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations*, Math. Z. 186 (1984), 383-391

P. Brenner, *On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon Equations*, J. Differential Equations 56 (1985), 310-344

M. Christ and M. Weinstein, *Dispersion of small amplitude solutions of the general Korteweg-de-Vries equation*, J. Func. Analysis 100 (1991), 87-109

J.Colliander, M.Keel, G.Staffilani, H.Takaoka, T.Tao, *Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation*, Math. Res. Letters 9 (2002), pp. 659-682

C. Fefferman and E. Stein, *Some maximal inequalities*, Amer. J. Math 93 (1971), 107-115

J. Ginebre and G. Velo, *Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schrödinger equation*, Ann. Inst. H. Poincare Phys Theor 43 (1985), 399-442

M. Keel and T. Tao, *Endpoint Strichartz estimates*, Amer. J. Math, 120 (1998), 955-980

H. Lindblad, C. D Sogge, *On existence and scattering with minimal regularity for semilinear wave equations*, J. Func.Anal 219 (1995), 227-252

C. Morawetz, *Time decay for the nonlinear Klein-Gordon equation*, Proc. Roy. Soc. A 306 (1968), 291-296

C. Morawetz and W. Strauss, *Decay and scattering of solutions of a nonlinear relativistic wave equation*, Comm. Pure Appl. Math. 25 (1972), pp 1-31

M. Nakamura and T. Ozawa, *The Cauchy Problem for Nonlinear Klein-Gordon Equations in the Sobolev Spaces*, Publ. Res. Inst. Math. Sci., 37 (2001), 255-293

K. Nakanishi, *Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions $1$ and $2$*, Journal of Functional Analysis 169 (1999), 201-225

K. Nakanishi, *Remarks on the energy scattering for nonlinear Klein-Gordon and Schrodinger equations*, Tohoku Math J. 53 (2001), 285-303

E. M. Stein, *Harmonic Analysis*, Princeton University Press, 1993

W. A. Strauss, *Nonlinear wave equations*, CBMS Regional Conference Series in Mathematics, no 73, Amer. Math. Soc. Providence, RI, 1989

T. Roy, *Global well-posedness for the radial defocusing cubic wave equation on $\mathbb{R}^{3}$ and for rough data*, EJDE, 166, 2007, 1-22, arXiv:0708.2288

T. Roy, *Introduction to scattering for radial $3D$ NLKG below energy norm*, J. Differential Equations 248 (2010), no. 4, 893–923., arXiv:0809.3835

M. Taylor, *Tools for PDE, Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials*, Mathematical Surveys and Monographs, 81. American Mathematical Society, Providence, RI, 2000.

[^1]: Here $\dot{H}^{m}$ denotes the standard homogeneous Sobolev space endowed with the norm $\| f \|_{\dot{H}^{m}}: = \| D^{m} f \|_{L^{2}(\mathbb{R}^{3})}$

[^2]: by rewriting for example (\[Eqn:NlkgWdat\]) in the “wave” form $\partial_{tt}u - \triangle u= -|u|^{p-1} u -u$

[^3]: Nevertheless, because our primary goal is to prove scattering and to use the decay estimates (such as (\[Eqn:DecayEstp\])) as tools, we have chosen not to determine $a_{p}$ and $b_{p}$: it is left to the reader. He can check, after reading the proof of Theorem \[Thm:Scat\], that $a_{p}$ decreases and $b_{p}$ increases as $p$ grows; $\lim_{p \rightarrow 3^{+}} a_{p} = 2(p-1) $ and $ \lim_{ p \rightarrow 3^{+} } b_{p}= 2(p-1)$

[^4]: since the multiplier approaches the identity operator as $N$ goes to infinity

[^5]: Recall that $C_{E}$ is defined in Proposition \[Prop:EstInitMolNrj\]

[^6]: $C$ and $Q$ will be chosen shortly so that eventually $\tilde{s}$ will only depend on $A$ and $p$

[^7]: There are two important points. First the upper bound $L(i)$ does not depend on the function $w \in \mathcal{C}_{i}$. Second $L$ depend on the integer $i$: $N$ and $A$ are fixed parameters. This is why $N$ and $A$ are subscripts, in the definition of $L$. In the sequel we forget these subscripts, to make the argument easier to read

[^8]: The reader can check that the bounds (\[Eqn:ControlIwStrich\]) and (\[Eqn:ControlEIw\]) also hold if we are restricted to $[T^{'},t_{l}]$ (for $T^{'}<t_{l}$) by slightly modifying the proof of (\[Eqn:ControlIwStrich\]) and (\[Eqn:ControlEIw\]) on $[t_{l},T]$.

[^9]: Recall that $\eta_{0}$ is defined in Proposition \[prop:LocalBd\]

[^10]: the value of $\epsilon$ is not randomly chosen: see (\[IneqChainNrj\])

[^11]: The sign $\pm$ in (\[Eqn:LowerBdJ1\]) and in (\[Eqn:LowerBdJ2\]) denotes the plus or minus sign. It is not important to know whether it is a plus or minus sign: see the next computations. Therefore we have decided not to determine the sign.

[^12]: Here $\check{f}$ denotes the inverse Fourier transform of a function $f$

[^13]: In fact we have a better bound: $Z(w_{l+1},J_{j}) \lesssim  N^{1-s} A^{\frac{p+1}{2}}$. $X_{1}$, $X_{2}$ will be estimated by using the bounds (\[Eqn:BounduPerturb\]), (\[Eqn:BoundwPerturb\]) respectively. Since we want an upper bound of $X_{1}+ X_{2}$, a sharper estimate for $Z(w_{l+1},J_{j})$ is not needed.
