\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{amssymb}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 96, pp. 1--24.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2016/96\hfil Nonlinear Schr\"odinger equations]
{Defocusing fourth-order coupled nonlinear Schr\"odinger equations}
\author[R. Ghanmi, T. Saanouni \hfil EJDE-2016/96\hfilneg]
{Radhia Ghanmi, Tarek Saanouni}
\address{Radhia Ghanmi \newline
University Tunis El Manar,
Faculty of Sciences of Tunis,
LR03ES04 partial differential equations and applications,
2092 Tunis, Tunisia}
\email{ghanmiradhia@gmail.com}
\address{Tarek Saanouni \newline
University Tunis El Manar,
Faculty of Sciences of Tunis,
LR03ES04 partial differential equations and applications,
2092 Tunis, Tunisia}
\email{Tarek.saanouni@ipeiem.rnu.tn}
\thanks{Submitted February 29, 2016. Published April 12, 2016.}
\subjclass[2010]{35Q55}
\keywords{Nonlinear fourth-order Schr\"odinger system; global well-posedness;
\hfill\break\indent scattering}
\begin{abstract}
We study the initial value problem for some defocusing coupled nonlinear
fourth-order Schr\"odinger equations. We show global well-posedness and
scattering in the energy space.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
%\newtheorem{corollary}[theorem]{Corollary}
%\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\cL}{\mathcal L}
%\newcommand{\dsum}{\sum}
%\newcommand{\dsup}{\sup}
%\newcommand{\dlim}{\lim}
%\newcommand{\dinf}{\inf}
\section{Introduction}
This manuscript is concerned with the initial value problem for
some defocusing fourth-order Schr\"odinger system with power-type nonlinearities
\begin{equation}
\begin{gathered}
i\dot u_j +\Delta^2 u_j+ \Big(\sum_{k=1}^{m}a_{jk}|u_k|^p\Big)|u_j|^{p-2}u_j=0 ;\\
u_j(0,\cdot)= \psi_{j},
\end{gathered} \label{S}
\end{equation}
where $u_j: \R \times \R^N \to \C$ for $j\in[1,m]$ and $a_{jk} =a_{kj}$
are positive real numbers.
Fourth-order Schr\"odinger equations have been introduced by Karpman
\cite{Karpman} and Karpman-Shagalov \cite{Karpman 1} to take into account
the role of small fourth-order dispersion terms in the propagation of intense
laser beams in a bulk medium with Kerr nonlinearity.
The $m$-component classical coupled nonlinear Schr\"odinger system with
power-type nonlinearity
\begin{equation} \label{CNLSp}
i\dot u_j +\Delta u_j= \pm \sum_{k=1}^{m}a_{jk}|u_k|^p|u_j|^{p-2}u_j ,
\end{equation}
arises in many physical problems. This models physical systems in which
the field has more than one component. For example, in optical fibers
and waveguides, the propagating electric field has two components that
are transverse to the direction of propagation. Readers are referred
to various works \cite{Hasegawa, Zakharov} for the derivation and
applications of this system. For mathematical point of view,
well-posedness issues of $(CNLS)_p$ were investigated by many authors.
Indeed, global existence of solutions and scattering hold \cite{ntds,ct,st,T2,T}.
System \eqref{S} is a mixture of the two previous problems.
A solution $\mathbf{u}:= (u_1,\dots ,u_m)$ to \eqref{S} formally
satisfies respectively conservation of the mass and the energy
\begin{gather*}
M(u_j):= \int_{\R^N}|u_j(t,x)|^2\,dx = M(\psi_{j});\\
\begin{aligned}
E(\mathbf{u}(t))&:= \frac{1}{2} \sum_{j=1}^{m}\int_{\R^N}|\Delta u_j|^2\,dx
+ \frac{1}{2p} \sum_{j,k=1}^{m}a_{jk} \int_{\R^N} |u_j(t,x)|^p |u_k(t,x)|^p\,dx\\
&\;= E(\mathbf{u}(0)).
\end{aligned}
\end{gather*}
To use the conservation laws, it is natural to study \eqref{S} in $H^2$,
called energy space.
Problem \eqref{S} is a natural extension of the classical one component
fourth-order Schr\"odinger equation which was first studied in \cite{Fibich},
where various properties of the equation in the subcritical regime were described.
Related references \cite{Artzi} gave sharp dispersive estimates for the biharmonic
Schr\"odinger operator which lead to the Strichartz estimates.
The model case given by a pure power nonlinearity is of particular interest.
Indeed, the question of well-posedness in the energy space $H^2$ was
widely investigated. We denote for $p>1$ the fourth-order Schr\"odinger problem
\begin{equation} \label{NLSp}
i\dot u+\Delta^2u\pm u|u|^{p-1}=0,\quad
u:{\mathbb R}\times{\mathbb R}^N\to{\mathbb C}.
\end{equation}
This equation satisfies a scaling invariance. Indeed, if $u$ is a solution
to \eqref{NLSp} with data $u_0$, then
$ u_\lambda:=\lambda^{\frac4{p-1}}u(\lambda^4\, \cdot\,,\lambda\, \cdot)$
is a solution to \eqref{NLSp} with data $\lambda^{\frac4{p-1}}u_0(\lambda\,\cdot)$.
For $s_c:=\frac N2-\frac4{p-1}$, the space $\dot H^{s_c}$ whose norm is
invariant under the dilatation $u\mapsto u_{\lambda}$ is relevant in this theory.
When $s_c=2$ which is the energy critical case, the critical power is
$p_c:=\frac{N+4}{N-4}$, $N\geq 5$. Pausader \cite{Pausader} established
global well-posedness in the defocusing subcritical case, namely $1< p < p_c$.
Moreover, he established global well-posedness and scattering for radial data
in the defocusing critical case, namely $p=p_c$. The same result without radial
condition was obtained by Miao, Xu and Zhao \cite{Miao 1}, for $N\geq 9$.
See also \cite{Miao 11}, for similar results in the more general case $s_c\geq1$.
The focusing case was treated by the last authors in \cite{Miao}.
They obtained results similar to one proved by Kenig and Merle \cite{Merle,km}
in the classical Schr\"odinger case. See \cite{ts} in the case of exponential
nonlinearity.
In this note, which seems to be one of the first papers studying a system
of nonlinear coupled fourth-order Schr\"odinger equations, we combine in
some meaning the two problems \eqref{NLSp} and $(CNLS)_p$.
Thus, we have to overcome two difficulties. The first one is the presence
of a bilaplacian in Schr\"odinger operator and the second one is the coupled
nonlinearities. It is the purpose of this manuscript to obtaining global
well-posedness in the energy space and scattering of \eqref{S} via
Morawetz estimate.
The rest of the paper is organized as follows. The next section contains
the main results and some technical tools needed in the sequel.
The third and fourth sections are devoted to prove well-posedness
of \eqref{S}. In section five, scattering is established. In appendix,
we give a proof of Morawetz estimate and a blow-up criterion.
We close this section with some notations. Define the product space
$$
H:={H^2({\R^N})\times\dots \times H^2({\R^N})}=[H^2({\R^N})]^m,
$$
where $H^2(\R^N)$ is the usual Sobolev space endowed with the complete norm
$$
\|u\|_{H^2(\R^N)} := \Big(\|u\|_{L^2(\R^N)}^2
+ \|\Delta u\|_{L^2(\R^N)}^2\Big)^{1/2}.
$$
Let us denote the real number
$$
p^*:=\begin{cases}
\frac{N}{N-4} &\text{if } N>4;\\
\infty & \text{if } 1\leq N\leq 4.
\end{cases}
$$
We mention that $C$ will denote a constant which may vary
from line to line and if $A$ and $B$ are non negative real numbers,
$A\lesssim B$ means that $A\leq CB$. For $1\leq r\leq\infty$ and
$(s,T)\in [1,\infty)\times (0,\infty)$, we denote the Lebesgue space
$L^r:=L^r({\mathbb R}^N)$ with the usual norm
$\|\cdot \|_r:=\|\cdot\|_{L^r}$, $\|\cdot\|:=\|\cdot\|_2$ and
$$
\|u\|_{L_T^s(L^r)}:=\Big(\int_{0}^{T}\|u(t)\|_r^s\,dt\Big)^{1/s},\quad
\|u\|_{L^s(L^r)}:=\Big(\int_{0}^{+\infty}\|u(t)\|_r^s\,dt\Big)^{1/s}.
$$
For simplicity, we denote the usual Sobolev space
$W^{s,p}:=W^{s,p}({\mathbb R}^N)$ and $H^s:=W^{s,2}$.
If $X$ is an abstract space $C_T(X):=C([0,T],X)$ stands for the set
of continuous functions valued in $X$ and $X_{rd}$ is the set of radial
elements in $X$, moreover for an eventual solution to \eqref{S},
we denote $T^*>0$ its lifespan.
\section{Background and main results}
In what follows, we give the main results and collect some estimates needed
in the sequel.
\subsection{Main results}
First, local well-posedness of the fourth-order Schr\"odinger problem \eqref{S}
is claimed.
\begin{theorem}\label{existence}
Let $1\leq N\leq8$, $2 \leq p\leq p^*$ and $ \Psi \in H$.
Then, there exist $T^*>0$ and a unique maximal solution to \eqref{S},
$ \mathbf{u} \in C ([0, T^*), H)$. Moreover,
\begin{enumerate}
\item $\mathbf{u}\in \big(L_{\rm loc}^{\frac{8p}{N(p-1)}}([0, T^*],
W^{2,2p})\big)^{(m)}$;
\item $\mathbf{u}$ satisfies conservation of the energy and the mass;
\item $T^*=\infty$ in the subcritical case $(2\leq p
0$ such that if
$\Psi:=(\psi_1,\dots ,\psi_m) \in H$ satisfies
$\xi(\Psi):= \sum_{j=1}^m\int_{\R^N}|\Delta \psi_j|^2\,dx\leq \epsilon_0$,
system \eqref{S} possesses a unique global solution $\mathbf{u}\in C(\R, H)$,
which scatters.
\end{theorem}
In the next subsection, we give some standard estimates needed in the paper.
\subsection{Tools}
We start with some properties of the free fourth-order Schr\"odinger kernel.
\begin{proposition}\label{fre}
Denoting the free operator associated to the fourth-order fractional
Schr\"odinger equation
$$
e^{it\Delta^2}u_0:=\mathcal F^{-1}(e^{it|y|^{4}})*u_0,
$$
yield
\begin{enumerate}
\item
$e^{it\Delta^2}u_0$ is the solution to the linear problem associated to \eqref{NLSp};
\item
$e^{it\Delta^2}u_0 \mp i\int_0^te^{i(t-s)\Delta^2}u|u|^{p-1}\,ds$
is the solution to the problem \eqref{NLSp};
\item
$(e^{it\Delta^2})^*=e^{-it\Delta^2}$;
\item
$e^{it\Delta^2}$ is an isometry of $L^2$.
%\item$T_{0,0}=I_d$.
\end{enumerate}
\end{proposition}
Now, we give the so-called Strichartz estimate \cite{Pausader}.
\begin{definition} \label{def2.7} \rm
A pair $(q,r)$ of positive real numbers is said to be admissible if
$$
2\leq q,r\leq \infty,\quad (q,r,N) \neq(2, \infty, 4)\quad \text{and} \quad
\frac{4}{q} = N\Big(\frac{1}{2} - \frac{1}{r}\Big).$$
\end{definition}
\begin{proposition}
Let two admissible pairs $(q,r)$, $(a,b)$. There exists a positive real
number $C:=C_{q,a}$ such that for any $T>0$,
\begin{gather}
\|u\|_{L_T^q(W^{2,r})}\leq C \Big( \|u_0\|_{H^2} + \|i\dot u
+ \Delta^2 u \|_{L_T^{ a^\prime}(W^{2,b^\prime})}\Big);\label{S1}\\
\|\Delta u\|_{L^q_T(L^r)}\leq C \Big(\|\Delta u_0\|_{L^2} + \|i\dot u
+ \Delta^2 u\|_{L^2_T(\dot W^{1,\frac{2N}{N +2}})}\Big)\label{S2}.
\end{gather}
\end{proposition}
The following Morawetz estimate which is essential in proving scattering,
is proved in the appendix, in the spirit of \cite{Miao 1,Miao 2}.
\begin{proposition}\label{prop2''}
Let $ 5\leq N\leq 8$, $2\leq p\leq p^*$ and $\mathbf{u}\in C(I,H)$ be the solution
to \eqref{S}. Then,
\begin{enumerate}
\item if $N>5$,
\begin{equation}\label{mrwtz1}
\sum_{j=1}^m\int_I\int_{\R^N\times\R^N}
\frac{|u_j(t,x)|^2|u_j(t,y)|^2}{|x-y|^5}dx\,dy\,dt\lesssim_u1;
\end{equation}
\item if $N=5$,
\begin{equation}\label{mrwtz2}
\sum_{j=1}^m\int_I\int_{\R^5}|u_j(t,x)|^4dxdt\lesssim_u1.\end{equation}
\end{enumerate}
\end{proposition}
Let us gather some useful Sobolev embeddings \cite{Adams}.
\begin{proposition}\label{injection}
The continuous injections hold
\begin{enumerate}
\item $ W^{s,p}(\R^N)\hookrightarrow L^q(\R^N)$ whenever
$1
0$ and $\frac{1}{p}\leq \frac{1}{q} + \frac {s}{N}$;
\item $W^{s,p_1}(\R^N)\hookrightarrow W^{s - N(\frac{1}{p_1}
- \frac{1}{p_2}),p_2}(\R^N)$ if $1\leq p_1\leq p_2 < \infty$.
\end{enumerate}
\end{proposition}
Now, we give some fractional Gagliardo-Nirenberg inequality \cite{hmon}.
\begin{lemma}\label{gn}
Let $1
0$ and $X\in C([0, T], \R_+)$ such that
$$
X\leq a + b X^{\theta}\quad on \quad [0,T],
$$
where $a, b>0$, $\theta>1$,
$a<(1 - \frac{1}{\theta})\frac{1}{(\theta b)^{\frac{1}{\theta}}}$ and
$X(0)\leq \frac{1}{(\theta b)^{\frac{1}{\theta -1}}}$. Then
$$
X\leq \frac{\theta}{\theta - 1}a\quad \text{on } [0, T].
$$
\end{lemma}
\begin{proof}
The function $f(x):=bx^\theta-x +a$ is decreasing on
$[0,(b\theta)^{\frac1{1-\theta}}]$ and increasing on
$[(b\theta)^\frac1{1-\theta} ,\infty)$. The assumptions imply that
$f((b\theta)^\frac1{1-\theta})< 0$ and $f(\frac\theta{\theta-1}a)\leq0$.
As $f(X(t))\geq 0$, $f(0) > 0$ and $X(0)\leq(b\theta)^\frac1{1-\theta}$,
we conclude the proof by a continuity argument.
\end{proof}
\section{Local well-posedness}
This section is devoted to prove Theorem \ref{existence}.
The proof contains two steps. First we prove existence of a unique local
solution to \eqref{S}, second we establish global existence in the
subcritical case.
\subsection{Local existence and uniqueness}
We use a standard fixed point argument.
\smallskip
(1) \emph{Subcritical case $2\leq p
0$, we denote the space
\begin{align*}
E_{T,R}:=\Big\{&\mathbf{u}\in (C_T(H^2)\cap
L^{\frac{8p}{N(p-1)}}_T(W^{2,2p}))^{(m)}:\\
&\|\mathbf{u}\|_{(L^\infty_T(L^2)\cap L^{\frac{8p}{N(p-1)}}_T(L^{2p})) ^{(m)}}
+\|\Delta\mathbf{u}\|_{(L^\infty_T(L^2)\cap L^{\frac{8p}{N(p-1)}}_T(L^{2p}))^{(m)}}
\leq R\Big\}
\end{align*}
endowed with the distance
$$
d(\mathbf{u},\mathbf{v}):=\sum_{j=1}^m\Big(\|u_j-v_j\|_{L_T^\infty(L^2)}
+\| u_j-v_j\|_{L^{\frac{8p}{N(p-1)}}_T(L^{2p})}\Big).
$$
Define the function
$$
\phi(\mathbf{u})(t) := T(t){\Psi} - i \sum_{k=1}^{m}a_{jk}\int_0^tT(t-s)
\Big(|u_k|^p|u_1|^{p-2}u_1,\dots ,|u_k|^p|u_m|^{p-2}u_m\Big)\,ds,
$$
where $T(t){\Psi} := (e^{it\Delta^2}\psi_{1},\dots ,e^{it\Delta^2}\psi_{m})$.
We prove the existence of some small $T, R >0$ such that $\phi$ is a contraction
of $E_{T,R}$.
\smallskip
\noindent$\bullet$ First step $3C\|\Psi\|_H$ and $T>0$ sufficiently small via the fact that
$2\leq p< p^*$, $\phi$ is a contraction of $ E_{T, R}$.
\smallskip
\noindent$\bullet$ Second step $1\leq N\leq3$.
In this case, we use the Sobolev embedding $ H^2\hookrightarrow L^\infty$.
Applying Strichartz estimate to $\mathbf{u}, \mathbf{v} \in E_{T,R}$ yields
\begin{equation}
\begin{aligned}
d(\phi(\mathbf{u}),\phi(\mathbf{v}))
&\lesssim \sum_{j, k=1}^{m}\||u_k|^p|u_j|^{p-2}u_j
- |v_k|^p |v_j|^{p-2}v_j\|_{L_T^1(L^2)}\\
&\lesssim \sum_{j, k=1}^{m}\||u_k|^{p - 1}|u_j|^{p - 1}
+|u_k|^p|u_j|^{p - 2}\|_{L_T^\infty(L^\infty)}\|\mathbf{u}
- \mathbf{v}\|_{L_T^1(L^2)} \\
&\lesssim T\|\mathbf{u}\|_{L_T^\infty(H^2)}^{2(p -1)}
\|\mathbf{u} - \mathbf{v}\|_{L_T^\infty(L^2)}\\
&\lesssim TR^{2(p-1)}d(\mathbf{u},\mathbf{v}).
\end{aligned} \label{01}
\end{equation}
It remains to estimate
\begin{align*}
(B)
&:=\|\Delta (f_{j,k}(\mathbf{u}))\|_{L_T^{1}(L^2)}\\
&\lesssim \|D(f_{j,k})(\mathbf{u})\Delta\mathbf{u}\|_{L_T^1(L^2)}
+\||\nabla\mathbf{u}|^2D^2(f_{j,k})(\mathbf{u})\|_{L_T^1(L^2)}.
\end{align*}
Thanks H\"older and Sobolev inequalities, we obtain
\begin{align*}
\|D(f_{j,k})(\mathbf{u})\Delta\mathbf{u}\|_{L_T^1(L^2)}
&\lesssim \|\Delta\mathbf{u}\|_{L_T^{1}(L^{2})}\| |u_k|^{p-1}|u_j|^{p -1}
+ {|u_k|^{p}|u_j|^{p-2}}\|_{L_T^{\infty}(L^\infty)}\\
&\lesssim T\|\Delta\mathbf{u}\|_{L_T^{\infty}(L^{2})}
\|\mathbf{u}\|_{L_T^\infty( H^2)}^{2(p-1)}\\
&\lesssim TR^{2p-1}.%\|\Delta\mathbf{u}\|_{L_T^{\infty}(L^{2})}.
\end{align*}
Using the Sobolev injection $ H^2\hookrightarrow W^{1,4}$, we obtain
\begin{align*}
\||\nabla\mathbf{u}|^2(f_{j,k})_{ii}(\mathbf{u})\|_{L_T^1(L^{2})}
&\lesssim \||\nabla\mathbf{u}|^2(|u_k|^{p-2}|u_j|^{p-1}
+ |u_k|^p|u_j|^{p-3})\|_{L_T^1(L^{2})}\\
&\lesssim T\|\nabla \mathbf{u}\|_{L_T^\infty(L^{4})}^2
\|\mathbf{u}\|_{L_T^\infty( H^2)}^{2p-3}\\
&\lesssim TR^{2p-1}.
\end{align*}
This implies
\begin{align*}
&\|\phi(\mathbf{u})\|_{(L_T^{\frac{8p}{N(p-1)}}(L^{2p})
\cap L_T^\infty(L^2))^{(m)}}
+\|\Delta (\phi(\mathbf{u}))\|_{(L_T^{\frac{8p}{N(p-1)}}(L^{2p})
\cap L_T^\infty(L^2))^{(m)}}\\
&\leq C\Big(\|\Psi\|_H+ TR^{2p-1}\Big).
\end{align*}
Choosing $R>C\|\Psi\|_H$ and $T>0$ sufficiently small, $\phi$ is a contraction
of $ E_{T, R}$.
Finally, thanks to a classical fixed point Theorem, We deduce the existence
of a fixed point $\mathbf{u}\in B_T(R)$, which is a solution to \eqref{S}.
Moreover, uniqueness follows thanks to \eqref{00} and \eqref{01}.
\smallskip
(2) \emph{Critical case $40$ and a solution $\mathbf{v} = (v_1,\dots ,v_m)$ to
\eqref{P1} on $C\big([s, s+\tau], H)$. Using the conservation of energy we
see that $\tau$ does not depend on $s$. Thus, if we let $s$ be close
to $T^*$ such that $T^*< s + \tau$, this fact contradicts the maximality of $T^*$.
\section{Scattering}
This section we establish Theorem \ref{sctr} about the scattering
of \eqref{S}. For any time slab $I$, take the Strichartz space
$$
S(I):=C(I, H^2)\cap{L^{\frac{8p}{N(p -1)}}(I, W^{2, 2p})}
$$
endowed the norm
$$
\|u\|_{S(I)}:= \|u\|_{L^\infty(I, H^2)}
+ \|u\|_{L^{\frac{8p}{N(p -1)}}(I, W^{2, 2p})}.
$$
The first intermediate result is as follows.
\begin{lemma} \label{lem4.1}
For any time slab $I$, we have
$$
\| \mathbf{u}(t) - e^{it\Delta^2}\Psi\|_{(S(I))^{(m)}}
\lesssim\|\mathbf{u}\|_{\big(L^\infty(I, L^{2p})
\big)^{(m)}}^{\frac{2pN(p-1)-8p}{N(p-1)}}
\|\mathbf{u}\|_{\big(L^{\frac{8p}{N(p-1)}}
(I, W^{2,2p})\big)^{(m)}}^{\frac{8p - N(p-1)}{N(p-1)}},
$$
where $e^{it\Delta^2}(\Psi_1,\dots ,\Psi_m):=(e^{it\Delta^2}\Psi_1,
\dots ,e^{it\Delta^2}\Psi_m)$.
\end{lemma}
\begin{proof}
Using Strichartz estimate, we have
$$
\| \mathbf{u}(t) - e^{it\Delta^2}\Psi\|_{(S(I))^{(m)}}
\lesssim \sum_{j,k=1}^m \|f_{j,k}(\mathbf{u})\|_{L^{\frac{8p}{p(8 - N) + N}}
(I, W^{2,\frac{2p}{2p -1}})}.
$$
Thanks to H\"older inequality, we obtain
\[
\|f_{j,k}(\mathbf{u})\|_{L^\frac{2p}{2p -1}_x}
\lesssim \big\||u_k|^p|u_j|^{p - 1}\big\|_{L^\frac{2p}{2p -1}_x}
\lesssim\|u_k\|_{L^{2p}_x}^p\|u_j\|_{L^{2p}_x}^{p -1}.
\]
Letting $\theta:= \frac{8p - N(p - 1)}{2N(p - 1) }$, we obtain the inequality
$$
\frac 12 \leq \theta\leq p - \frac 12.
$$
The left part of the inequality follows from $p\leq p^*$.
Denoting $X:= p-1$, the right part of the claim is equivalent to
$$
T(X):= NX^2+ (N-4)X -4\geq 0.
$$
$T$ has two roots $X_1=-1<00$, there exist $T_\epsilon>0$ and $ n_\epsilon\in \N$
such that
\begin{equation}\label{chi}
\|\chi(\mathbf{u}_n - \mathbf{u})\|_{(L_{T_\epsilon}^\infty
(L^2))^{(m)}}<\epsilon \quad \text{for all } n>n_\epsilon.
\end{equation}
Indeed, denoting the functions $\mathbf{v}_n:= \chi \mathbf{u}_n$ and
$\mathbf{v}=(v_1,\dots,v_m) :=\chi \mathbf{u}$, we compute
$v_j^n(0) = \chi \varphi_j^n$ and
\begin{align*}
i\dot v_j^n + \Delta^2 v_j^n
&= \Delta^2\chi u_j^n + 2 \nabla \Delta\chi \nabla u_j^n
+ \Delta\chi\Delta u_j^n + 2 \nabla \chi \nabla\Delta u_j^n \\
&\quad + 2\big(\nabla\Delta\chi\nabla u_j^n + \nabla\chi\nabla\Delta u_j^n
+ 2\sum_{i=1}^N \nabla\partial_i \chi\nabla\partial_i u_j^n\big)\\
&\quad + \chi\big(\sum_{k=1}^m|u_k^n|^p|u_j^n|^{p - 2}u_j^n\big).
\end{align*}
Similarly, $v_j(0)=\chi\phi_j$ and
\begin{align*}
i\dot v_j + \Delta^2 v_j
&= \Delta^2\chi u_j + 2 \nabla \Delta\chi \nabla u_j
+ \Delta\chi\Delta u_j + 2 \nabla \chi \nabla\Delta u_j \\
&\quad + 2\big(\nabla\Delta\chi\nabla u_j + \nabla\chi\nabla\Delta u_j
+ 2\sum_{i=1}^N \nabla\partial_i \chi\nabla\partial_i u_j\big)\\
&\quad + \chi\big(\sum_{k=1}^m|u_k|^p|u_j|^{p - 2}u_j\big).
\end{align*}
Denoting $\mathbf{w}_n=(w_1^n,\dots,w_m^n):= \mathbf{v}_n - \mathbf{v}$ and
${\bf z}_n=(z_1^n,\dots,z_m^n):= \mathbf{u}_n - \mathbf{u}$, we have
\begin{align*}
i\dot w_j^n + \Delta^2 w_j^n
&= \Delta^2\chi z_j^n+ 4 \nabla \Delta\chi \nabla z_j^n
+ \Delta\chi\Delta z_j^n + 4 \nabla \chi \nabla\Delta z_j^n \\
&\quad + 4\sum_{i=1}^N \nabla\partial_i \chi\nabla\partial_i z_j^n
+ \chi\big(\sum_{k=1}^m|u_k^n|^p|u_j^n|^{p - 2}u_j^n
- \sum_{k=1}^m|u_k|^p|u_j|^{p - 2}u_j\big).
\end{align*}
Thanks to Strichartz estimate, we obtain
\begin{align*}
&\|\mathbf{w}_n\|_{\big(L_T^\infty(L^2)
\cap L^{\frac{8p}{N(p-1)}}_T(L^{2p})\big)^{(m)}}\\
&\lesssim \|\chi(\varphi_n - \varphi)\|_{(L^2)^{(m)}}
+ \|\Delta^2\chi {\bf z}_n\|_{(L^1_T(L^2))^{(m)}}
+ \|\nabla \Delta\chi \nabla {\bf z}_n\|_{(L^1_T(L^2))^{(m)}}\\
&\quad + \|\nabla \chi \nabla\Delta {\bf z}_n\|_{(L^1(L^2))^{(m)}}
+\| \nabla\partial_i \chi\nabla\partial_i {\bf z}_n\|_{(L^1(L^2))^{(m)}} \\
&\quad +\sum_{j,k=1}^m\big\|\chi\big(|u_k^n|^p|u_j^n|^{p- 2}u_j^n
- |u_k|^p|u_j|^{p - 2}u_j\big)\big\|_{L^{\frac{8p}
{p(8-N) + N}}_T(L^{\frac{2p}{2p-1}})}.
\end{align*}
Thanks to the Rellich Theorem, up to subsequence extraction, we have
$$
\epsilon:=\|\chi(\varphi_n - \varphi)\|_{L^2_x}\to0\quad\text{as } n\to\infty.
$$
Moreover, by the conservation laws via properties of $\chi$,
\begin{align*}
\mathcal{I}_1
&:=\|\Delta^2\chi {\bf z}_n\|_{(L^1_T(L^2))^{(m)}}
+ \|\nabla \Delta\chi \nabla {\bf z}_n\|_{(L^1_T(L^2))^{(m)}}
+ \|\nabla \chi \nabla\Delta {\bf z}_n\|_{(L^1_T(L^2))^{(m)}}\\
&\quad + \| \nabla\partial_i \chi\nabla\partial_i {\bf z}_n\|_{(L^1_T(L^2))^{(m)}}\\
&\lesssim \| {\bf z}_n\|_{(L^1_T(L^2))^{(m)}}
+ \| \nabla {\bf z}_n\|_{(L^1_T(L^2))^{(m)}}
\lesssim CT,
\end{align*}
where
$$
C:= \|\mathbf{u}\|_{(L^\infty(\R,H^2))^{(m)}}
+ \|\mathbf{u}_n\|_{(L^\infty(\R,H^2))^{(m)}} .
$$
Arguing as previously, we have
\begin{align*}
\mathcal{I}_2
&:=\|\chi(|u_k^n|^p|u_j^n|^{p-2}u_j^n - |u_k|^p|u_j|^{p-2}u_j)\|_{L^{\frac{8p}
{p(8 - N) + N}}_T(L^{\frac{2p}{2p -1}})}\\
&\lesssim \|\chi(|u_k^n|^{p -1}|u_j^n|^{p-1} - |u_k|^p|u_j|^{p-2})|\mathbf{u}_n
- \mathbf{u}|\|_{L^{\frac{8p}{p(8 - N) + N}}_T(L^{\frac{2p}{2p -1}})}\\
&\lesssim \|\chi(\mathbf{u}_n - \mathbf{u})\|_{L^{\frac{8p}{p(8 - N)
+ N}}_T((L^{2p})^{(m)})}\Big( \|u_k^n\|_{L^\infty_T(L^{2p})}
^{p-1}\|u_j^n\|_{L^\infty_T(L^{2p})} ^{p-1} \\
&\quad + \|u_k\|_{L^\infty_T(L^{2p})} ^{p}\|u_j\|_{L^\infty_T(L^{2p})} ^{p-2}\Big)\\
&\lesssim T^{\frac{8p - 2N(p-1)}{8p}}\|\mathbf{w}_n \|_{L^{\frac{8p}{N(p -1)}}_T
((L^{2p})^{(m)})}\Big( \|u_k^n\|_{L^\infty_T(H^2)} ^{2(p-1)}
\|u_j^n\|_{L^\infty_T(H^2)} ^{2(p-1)} \\
&\quad + \|u_k\|_{L^\infty_T(H^2)} ^{2p}\|u_j\|_{L^\infty_T(H^2)} ^{2(p-2)}\Big)\\
&\lesssim T^{\frac{4p -N(p-1)}{4p}}\|\mathbf{w}_n
\|_{L^{\frac{8p}{N(p -1)}}_T((L^{2p})^{(m)})}.
\end{align*}
As a consequence
\begin{align*}
\|\mathbf{w}_n\|_{\big(L_T^\infty(L^2)
\cap L^{\frac{8p}{N(p-1)}}_T(L^{2p})\big)^{(m)}}
&\lesssim \epsilon + CT + T^{\frac{4p -N(p-1)}{4p}}
\|\mathbf{w}_n \|_{L^{\frac{8p}{N(p -1)}}((L^{2p})^{(m)})}\\
&\lesssim \frac{\epsilon + T}{1 - T^{\frac{4p - N(p-1)}{4p}}}.
\end{align*}
The claim is proved.
\smallskip
By an interpolation argument it is sufficient to prove the decay for
$r:= 2 +\frac{4}{N}$. We recall the following Gagliardo-Nirenberg inequality
\begin{equation}\label{GN}
\|u_j(t)\|_{2 + \frac{4}{N}}^{2 + \frac{4}{N}}\leq C \|u_j(t)\|_{H^2}^2
\Big(\sup_x \|u_j(t)\|_{L^2(Q_1(x))}\Big)^{4/N},
\end{equation}
where $Q_a(x)$ denotes the square centered at $x$ whose edge has length $a$.
We proceed by contradiction. Assume that there exist a sequence $(t_n)$ of
positive real numbers and $\epsilon >0$ such that $\lim_{n\to \infty}t_n =\infty$ and
\begin{equation} \label{IN}
\|u_j(t_n)\|_{L^{2 + \frac{4}{N}}}>\epsilon\quad \text{for all } n\in \N.
\end{equation}
By \eqref{GN} and \eqref{IN}, there exist a sequence $(x_n)$ in $\R^N$
and a positive real number denoted also by $\epsilon>0$ such that
\begin{equation}\label{IN1}
\|u_j(t_n)\|_{L^2(Q_1(x_n))}\geq\epsilon,\quad \text{for all } n\in \N.
\end{equation}
Let $\phi_j^n(x):=u_j(t_n,x +x_n)$. Using the conservation laws, we obtain
$$
\sup_n\|\phi_j^n\|_{H^2}<\infty.
$$
Then, up to a subsequence extraction, there exists $\phi_j\in H^2$
such that $\phi_j^n$ convergence weakly to $\phi_j$ in $H^2$.
By Rellich Theorem, we have
$$
\lim_{n\to \infty}\|\phi_j^n - \phi_j\|_{L^2(Q_1(0))}=0.
$$
Moreover, thanks to \eqref{IN1} we have, $\|\phi_j^n\|_{L^2(Q_1(0))}\geq \epsilon$.
So, we obtain
$$
\|\phi_j\|_{L^2(Q_1(0))}\geq \epsilon.
$$
We denote by $\bar{u}_j\in C(\R, H^2)$ the solution of \eqref{S} with data
$\phi_j$ and ${u}_j^n\in C(\R, H^2)$ the solution of \eqref{S} with data
$\phi_j^n$. Take a cut-off function $\chi \in C_0^\infty(\R^N)$ which satisfies
$0\leq \chi\leq1$, $\chi=1$ on $Q_1(0)$ and
$\operatorname{supp}(\chi)\subset Q_2(0)$. Using a continuity argument,
there exists $T>0$ such that
$$
\inf_{t\in[0, T]}\|\chi \bar{u}_j(t) \|_{L^2(\R^N)}\geq \frac{\epsilon}{2}.
$$
Now, taking account \eqref{chi}, there is a positive time denoted also
$T$ and $n_\epsilon\in \N$ such that
$$
\|\chi(u_j^n - \bar{u}_j)\|_{L_T^\infty(L^2)}
\leq \frac{\epsilon}{4}\quad \text{for all } n\geq n_\epsilon.
$$
Hence, for all $t\in [0, T]$ and $n\geq n_\epsilon$,
$$
\|\chi u_j^n(t)\|_{L^2}\geq \|\chi \bar{u}_j(t)\|_{L^2} - \|\chi(u_j^n
- \bar{u}_j)(t)\|_{L^2}\geq \frac{\epsilon}{4}.
$$
Using a uniqueness argument, it follows that $u^n_j(t,x)=u_j(t+t_n,x+x_n)$.
Moreover, by the properties of $\chi$ and the last inequality,
for all $t\in[0, T]$ and $n\geq n_\epsilon$,
$$
\|u_j(t+t_n)\|_{L^2(Q_2(x_n))}\geq \frac{\epsilon}{4}.
$$
This implies that
$$
\|u_j(t)\|_{L^2(Q_2(x_n))}\geq \frac{\epsilon}{4},\quad \text{for all }
t\in [t_n, t_n + T]\text{ and all } n\geq n_\epsilon.
$$
Moreover, as $\lim_{n\to \infty}t_n=\infty$,
we can suppose that $t_{n +1}- t_n>T$ for $n\geq n_\epsilon$.
Therefore, thanks to Morawetz estimates \eqref{mrwtz1}, we obtain for $N>5$,
the contradiction
\begin{align*}
1 &\gtrsim \int_0^\infty\int_{\R^N\times\R^N}
\frac{|u_j(t,x)|^2|u_j(t,y)|^2}{|x - y|^5}\,dx\,dy\,dt\\
&\gtrsim \sum_n\int_{t_n}^{t_{n} +T}\int_{Q_2(x_n)\times Q_2(x_n)}
|u_j(t,x)|^2|u_j(t,y)|^2\,dx\,dy\,dt\\
&\gtrsim \sum_nT\big(\frac{\epsilon}{4}\big)^4 = \infty.
\end{align*}
Using \eqref{mrwtz2}, for $N=5$, we write
\begin{align*}
1
&\gtrsim \int_0^{\infty}\|u_j(t)\|_{L^4(\R^5)}^4dt\\
&\gtrsim \sum_n\int_{t_n}^{t_n+T}\|u_j(t)\|_{L^4(Q_2(x_n))}^4dt\\
&\gtrsim \sum_n\int_{t_n}^{t_n+T}\|u_j(t)\|_{L^2(Q_2(x_n))}^4dt\\
&\gtrsim \sum_n(\frac\varepsilon4)^4T=\infty.
\end{align*}
This completes the proof of Proposition \ref{prop1}.
\end{proof}
Finally, we are ready to prove scattering. By the two previous lemmas,
via the fact that $2\leq p0$ such that for any initial
data $\Psi \in H$ and any interval $I=[0, T]$, if
$$
\sum_{j=1}^{m}\|e^{it\Delta^2}\psi_{j}\|_{W(I)}< \delta,
$$
then there exits a unique solution $\mathbf{u}\in C(I, H)$ of \eqref{S}
which satisfies
$\mathbf{u}\in \big(M(I)\cap L^{\frac{2(N+4)}{N}}(I\times \R^N)\big)^{(m)}$.
Moreover,
\begin{gather*}
\sum_{j=1}^{m}\|u_j\|_{W(I)}\leq 2\delta;\\
\sum_{j=1}^{m}\|u_j\|_{M(I)} + \sum_{j=1}^{m}\|u_j\|_{L^\infty(I, H^2)}\leq C(\|\Psi\|_{H^2}+\delta^{\frac{N+4}{N-4}}).
\end{gather*}
Furthermore, the solution depends continuously on the initial data in the
sense that there exists $\delta_0$ depending on $\delta$, such that for
any $\delta_1\in (0,\delta_0)$, if
$\sum_{j=1}^{m}\|\psi_{j} - \varphi_{j}\|_{H^2}\leq \delta_1$ and
$\mathbf{v}$ is the local solution of \eqref{S} with initial data
$\varphi:=(\varphi_{1},\dots ,\varphi_{m})$, then $\mathbf{v}$
is defined on $I$ and for any admissible couple $(q,r)$,
$$
\|\mathbf{u} - \mathbf{v}\|_{(L^q(I, L^r))^{(m)}}\leq C\delta_1.
$$
\end{proposition}
\begin{proof}
The proposition follows from a contraction mapping argument.
For $\mathbf{u}\in( W(I))^{(m)}$, we let $\phi(\mathbf{u})$ given by
\begin{align*}
\phi(\mathbf{u})(t)
&:= T(t){\Psi} -i \sum_{k=1}^{m}a_{jk}
\int_0^tT(t-s)\Big(|u_k|^{\frac{N}{N-4}}|u_1|^{\frac{8-N}{N-4}}u_1(s),\\
&\quad \dots, |u_k|^{\frac{N}{N-4}}|u_m|^{\frac{8-N}{N-4}}u_m\Big)\,ds.
\end{align*}
Define the set
$$
X_{M,\delta} := \Big\{ \mathbf{u}\in (M(I))^{(m)}:
\sum_{j=1}^{m}\|u_j\|_{W(I)}\leq 2\delta, \,
\sum_{j=1}^{m}\|u_j\|_{L^{\frac{2(N+4)}{N}}(I,L^{\frac{2(N+4)}{N}})}\leq 2M\Big\}
$$
where $M := C \|\Psi\|_{(L^2)^{(m)}}$ and $\delta>0$ is sufficiently small.
Using Strichartz estimate, we obtain
\[
\|\phi(\mathbf{u}) - \phi(\mathbf{v})\|_{\big({L^{\frac{2(N+4)}{N}}
(I,L^{\frac{2(N+4)}{N}})}\big)^{(m)}}
\lesssim \sum_{j,k=1}^{m}\big\|f_{j,k}(\mathbf{u})
- f_{j,k}(\mathbf{v})\big\|_{L^{\frac{2(N+4)}{N+8}}(I,L^{\frac{2(N+4)}{N+8}})}.
\]
Using H\"older inequality and denoting the quantity
\[
(\mathcal{J}):= \big\|f_{j,k}(\mathbf{u})
- f_{j,k}(\mathbf{v})\big\|_{L^{\frac{2(N+4)}{N+8}}(I,L^{\frac{2(N+4)}{N+8}})},
\]
we obtain
\begin{align*}
(\mathcal{J})
&\lesssim \big\|\Big(|u_k|^{\frac{4}{N-4}}|u_j|^{\frac{4}{N-4}}
+ |u_k|^{\frac{N}{N-4}}|u_j|^{\frac{8-N}{N-4}}\Big)
|\mathbf{u} - \mathbf{v}|\big\|_{L_T^{\frac{2(N+4)}{N+8}}
(L^{\frac{2(N+4)}{N+8}})}\\
&\lesssim \|\mathbf{u} - \mathbf{v}\|_{L_T^{\frac{2(N+4)}{N}}
(L^{\frac{2(N+4)}{N}})}\Big(\|u_k\|_{L_T^{\frac{2(N+4)}{N- 4}}
(L^{\frac{2(N+4)}{N - 4}})}^{\frac{4}{N-4}} \|u_j\|_{L_T^{\frac{2(N+4)}{N - 4}}
(L^{\frac{2(N+4)}{N- 4}})}^{\frac{4}{N-4}}\\
&\quad + \|u_k\| _{L_T^{\frac{2(N+4)}{N - 4}}(L^{\frac{2(N+4)}{N - 4}})}
^{\frac{N}{N-4}}\|u_j\|_{L_T^{\frac{2(N+4)}{N - 4}}(L^{\frac{2(N+4)}{N - 4}})
}^{\frac{8-N}{N-4}}\Big).
\end{align*}
By Proposition \ref{injection}, we have the Sobolev embedding
$$
\| u\|_{L^{\frac{2(N+4)}{N-4}}(I, L^{\frac{2(N + 4)}{N - 4}})}
\lesssim \|\nabla u\|_{L^{\frac{2(N + 4)}{N-4}}(I, L^{\frac{2N(N + 4)}
{N^2 -2N + 8}})},
$$
hence
\begin{align*}
(\mathcal{J})
&\lesssim \|\mathbf{u} - \mathbf{v}\|_{L_T^{\frac{2(N+4)}{N}}
(L^{\frac{2(N+4)}{N}})}\Big(\|u_k\|_{W(I)}^{\frac{4}{N-4}}
\|u_j\|_{W(I)}^{\frac{4}{N-4}}+\|u_k\| _{W(I)} ^{\frac{N}{N-4}}
\|u_j\|_{W(I)}^{\frac{8-N}{N-4}}\Big)\\
&\lesssim \delta^{\frac{8}{N-4}}\|\mathbf{u} - \mathbf{v}\|_{L_T
^{\frac{2(N+4)}{N}}(L^{\frac{2(N+4)}{N}})}.
\end{align*}
Then
$$
\|\phi(\mathbf{u}) - \phi(\mathbf{v})\|_{\big({L^{\frac{2(N+4)}{N}}
(I,L^{\frac{2(N+4)}{N}})}\big)^{(m)}}
\lesssim \delta^{\frac{8}{N-4}} \|\mathbf{u} - \mathbf{v}
\|_{\big({L^{\frac{2(N+4)}{N}}(I,L^{\frac{2(N+4)}{N}})}\big)^{(m)}}.
$$
Moreover, taking in the previous inequality ${\bf v=0}$, we obtain for
small $\delta>0$,
\begin{align*}
\|\phi(\mathbf{u})\|_{\big({L^{\frac{2(N+4)}{N}}(I,L^{\frac{2(N+4)}{N}})}\big)^{(m)}}
&\leq C\|\Psi\|_{(L^2)^m}+ 2\delta^{\frac{8}{N-4}} M\\
&\leq (1+ 2\delta^{\frac{8}{N-4}}) M\\
&\leq 2M.
\end{align*}
With a classical Picard argument, there exists
$\mathbf{u}\in (L^{\frac{2(N+4)}{N}}(I,L^{\frac{2(N+4)}{N}}))^m$
a solution to \eqref{S} satisfying
$$
\|\mathbf{u}\|_{\big(L^{\frac{2(N+4)}{N}}(I,L^{\frac{2(N+4)}{N}})\big)^{(m)}}\leq 2M.
$$
Taking account of Strichartz estimate we obtain
\[
\| \mathbf{u}\|_{(M(I))^{(m)}}
\lesssim \|\Delta \Psi\|_{({L^2})^{(m)}} +\sum_{j,k=1}^{m}
\| \nabla f_{j,k}(\mathbf{u})\|_{L_T^2(L^{\frac{2N}{N +2}})}.
\]
Let $(\mathcal{J}_1):= \| \nabla f_{j,k}(\mathbf{u})\|_{L_T^2(L^{\frac{2N}{N +2}})} $.
Using H\"older inequality and Sobolev embedding with, yields
\begin{align*}
(\mathcal{J}_1)
&\lesssim \big\| |\nabla \mathbf{u}|
\Big( |u_k|^{\frac{4}{N-4}}|u_j|^{\frac{4}{N-4}}
+ |u_k|^{\frac{N}{N-4}}|u_j|^{\frac{8-N}{N-4}}\Big)
\big\|_{L_T^2(L^{\frac{2N}{N +2}})}\\
&\lesssim \|\nabla\mathbf{u}\|_{L_T^{\frac{2(N+4)}{N - 4}}
(L^{\frac{2N(N+4)}{N^2 - 2N +8}})}\Big(\|u_k\|_{L_T^{\frac{2(N+4)}{N- 4}}
(L^{\frac{2(N+4)}{N - 4}})}^{\frac{4}{N-4}} \|u_j\|_{L_T^{\frac{2(N+4)}{N - 4}}
(L^{\frac{2(N+4)}{N- 4}})}^{\frac{4}{N-4}}\\
&\quad + \|u_k\| _{L_T
^{\frac{2(N+4)}{N - 4}}(L^{\frac{2(N+4)}{N - 4}})}
^{\frac{N}{N-4}}\|u_j\|_{L_T^{\frac{2(N+4)}{N - 4}}
(L^{\frac{2(N+4)}{N - 4}})}^{\frac{8-N}{N-4}}\Big)\\
&\lesssim \|\mathbf{u}\|_{(W(I))^{(m)}}\Big(\|u_k\|_{W(I)}
^{\frac{4}{N-4}} \|u_j\|_{W(I)}^{\frac{4}{N-4}}+\|u_k\| _{W(I)}
^{\frac{N}{N-4}}\|u_j\|_{W(I)}^{\frac{8-N}{N-4}}\Big).
\end{align*}
Then
\begin{align*}
&\|\mathbf{u}\|_{(M(I))^{(m)}}\\
&\lesssim \|\Psi\|_H
+\sum_{j,k=1}^m\|\mathbf{u}\|_{(W(I))^{(m)}}
\Big(\|u_k\|_{W(I)}^{\frac{4}{N-4}} \|u_j\|_{W(I)}^{\frac{4}{N-4}}+\|u_k\| _{W(I)}
^{\frac{N}{N-4}}\|u_j\|_{W(I)}^{\frac{8-N}{N-4}}\Big) \\
&\lesssim \|\Psi\|_H + \delta^{\frac{N + 4}{N - 4}}.
\end{align*}
By Proposition \ref{injection}, $M(I)\hookrightarrow W(I)$ and
\begin{equation}\label{*}
\| \mathbf{u}\|_{(W(I))^{(m)}}\lesssim\| \mathbf{u}\|_{(M(I))^{(m)}}.
\end{equation}
Thanks to Strichartz estimates, it follows that
\begin{align*}
\| \mathbf{u}-e^{it\Delta^2}\Psi\|_{(W(I))^{(m)}}
&\lesssim \|\sum_{k=1}^m\int_0^tT(t-s)(f_{1,k}(\mathbf{u}),\dots,
f_{m,k}(\mathbf{u}))\,ds\|_{(W(I))^{(m)}}\\
&\lesssim \|\sum_{k=1}^m\int_0^tT(t-s)(f_{1,k}(\mathbf{u}),\dots,
f_{m,k}(\mathbf{u}))\,ds\|_{(M(I))^{(m)}}\\
&\lesssim \| \mathbf{u}\|_{(W(I))^{(m)}}^{\frac{N + 4}{N - 4}}.
\end{align*}
So, by Lemma \ref{Bootstrap},
$$
\| \mathbf{u}\|_{(W(I))^{(m)}}\leq 2\delta.
$$
Now, take an admissible couple $(q,r)$ and denote
$ (\mathcal{J}_2):=\|\mathbf{u} - \mathbf{v}\|_{(L^q(I, L^r))^{(m)}}
-\|\Psi - \varphi\|_{(L^2)^{(m)}}$.
By H\"older inequality and Strichartz estimate, we have
\begin{align*}
&(\mathcal{J}_2)\\
&\lesssim \sum_{j,k=1}^{m}\|f_{j,k}(\mathbf{u}) - f_{j,k}
(\mathbf{v})\|_{L^{\frac{2(N+4)}{N+8}}(I,L^{\frac{2(N+4)}{N+8}})}\\
&\lesssim \sum_{j,k=1}^{m}\big\|\Big(|u_k|^{\frac{4}{N-4}}|u_j|
^{\frac{4}{N-4}} + |u_k|^{\frac{N}{N-4}}|u_j|^{\frac{8-N}{N-4}}\Big)
|\mathbf{u} - \mathbf{v}|\big\|_{L^{\frac{2(N+4)}{N+8}}(I,L^{\frac{2(N+4)}{N+8}})}\\
&\lesssim \sum_{j,k=1}^{m}\|\mathbf{u} - \mathbf{v}\|_{L^{\frac{2(N+4)}{N}}
(I,L^{\frac{2(N+4)}{N}})}\Big(\|u_k\|_{L^{\frac{2(N+4)}{N- 4}}(I,
L^{\frac{2(N+4)}{N - 4}})}^{\frac{4}{N-4}} \|u_j\|_{L^{\frac{2(N+4)}{N - 4}}
(I,L^{\frac{2(N+4)}{N- 4}})}^{\frac{4}{N-4}}\\
&\quad + \|u_k\| _{L^{\frac{2(N+4)}{N - 4}}(I,L^{\frac{2(N+4)}{N - 4}})}
^{\frac{N}{N-4}}\|u_j\|_{L^{\frac{2(N+4)}{N - 4}}(I,
L^{\frac{2(N+4)}{N - 4}})}^{\frac{8-N}{N-4}}\Big)\\
&\lesssim \delta^{\frac{8}{N - 4}}\| \mathbf{u}
- \mathbf{v}\|_{\big({L^{\frac{2(N+4)}{N}}(I,L^{\frac{2(N+4)}{N}})}\big)^{(m)}}.
\end{align*}
The proof ends by taking $\delta$ small enough.
\end{proof}
We are ready to prove Theorem \ref{glb}.
\begin{proof}[Proof of Theorem \ref{glb}]
Denote the homogeneous Sobolev space $\mathbf{H}=(\dot{H}^2)^{(m)}$.
Using the previous proposition via \eqref{*}, it suffices to prove
that $\|\mathbf{u}\|_\mathbf{H}$ remains small on the whole interval
of existence of $\mathbf{u}$, which is a consequence of the inequalities
\[
\|\mathbf{u}\|_\mathbf{H}^2\leq E(\mathbf{u}(t))
=E(\Psi) \leq C\big( \xi(\Psi) + \xi(\Psi)^{\frac{N}{N - 4}}\big) .
\]
Now, we prove scattering. Let $\mathbf{v}(t)= e^{-it\Delta^2}\mathbf{u}(t)$.
Taking account of Duhamel formula
$$
\mathbf{v}(t) := {\Psi} + i \sum_{k=1}^{m}\int_0^t e^{-is\Delta^2}
\Big(|u_k|^{\frac{N}{N-4}}|u_1|^{\frac{8-N}{N-4}}u_1(s),\dots,
|u_k|^{\frac{N}{N-4}}|u_m|^{\frac{8-N}{N-4}}u_m\Big)\,ds.
$$
Therefore, for $0 T$.
\end{proposition}
\begin{proof}
Let $\eta>0$ be a small real number and $ M:=\| \mathbf{u}\|_{(Z([0, T]))^{(m)}}$.
The first step is to establish \eqref{G}. In order to do so, we subdivide $[0, T]$
into $n$ slabs $I_j$ such that
$$
n \sim (1 + \frac{M}{\eta})^{\frac{2(N+4)}{N - 4}}\quad\text{and}\quad
\| \mathbf{u}\|_{(Z(I_j))^{(m)}} \leq \eta.
$$
Denote $ (\mathcal{A}):=\| \mathbf{u}\|_{(M([t_j, t]))^{(m)}}$ and
$ I_j = [t_j, t_{j+1}]$. For $t\in I_j$, by Strichartz estimate and
arguing as previously,
\begin{align*}
(\mathcal{A})- \|\mathbf{u}(t_j)\|_\mathbf{H}
&\lesssim \sum_{i,k=1}^m\| \nabla f_{i,k}(\mathbf{u})\|_{\big(L^2([t_j, t],
L^{\frac{2N}{N +2}})\big)^{(m)}}\\
&\lesssim \sum_{j,k=1}^{m}\|\nabla\mathbf{u}\|_{L^{\frac{2(N+4)}{N - 4}}([t_j, t],
L^{\frac{2N(N+4)}{N^2 - 2N +8}})}\\
&\quad\times \Big(\|u_k\|_{L^{\frac{2(N+4)}{N- 4}}([t_j, t],
L^{\frac{2(N+4)}{N - 4}})}^{\frac{4}{N-4}} \|u_j\|_{L^{\frac{2(N+4)}{N - 4}}
([t_j, t],L^{\frac{2(N+4)}{N- 4}})}^{\frac{4}{N-4}}\\
&\quad + \|u_k\| _{L^{\frac{2(N+4)}{N - 4}}([t_j, t],L^{\frac{2(N+4)}{N - 4}})}
^{\frac{N}{N-4}}\|u_j\|_{L^{\frac{2(N+4)}{N - 4}}[t_j, t],
(L^{\frac{2(N+4)}{N - 4}})}^{\frac{8-N}{N-4}}\Big)\\
&\lesssim \|\mathbf{u}\|_{(W([t_j, t]))^{(m)}}\|\mathbf{u}\|_{(Z([t_j,
t]))^{(m)}}^{\frac{8}{N-4}}\\
&\lesssim \|\mathbf{u}\|_{(M([t_j, t]))^{(m)}}\|\mathbf{u}\|_{(Z([t_j, t])
)^{(m)}}^{\frac{8}{N-4}}\lesssim \eta^{\frac{8}{N-4}}\|\mathbf{u}
\|_{(M([t_j, t]))^{(m)}}.
\end{align*}
Take $( \mathcal{B}):=\|\mathbf{u}\|_{\big({L^{\frac{2(N+4)}{N}}([t_j, t],
L^{\frac{2(N+4)}{N}})}\big)^{(m)}}$. Applying Strichartz estimates, we obtain
{\small\begin{align*}
&(\mathcal{B})-C\|\mathbf{u}(t_j)\|_{(L^2)^{(m)}}\\
&\lesssim \sum_{j,k=1}^{m}\||u_k|^{\frac{N}{N - 4}}|u_j|
^{\frac{8 - N}{N - 4}}u_j\|_{L^{\frac{2(N + 4)}{N + 8}}
([t_j, t], L^{\frac{2(N + 4)}{N + 8}})}\\
&\lesssim \sum_{j,k=1}^{m}\|u_k\| _{L^{\frac{2(N + 4)}{N - 4 }}([t_j, t],
L^{\frac{2(N + 4)}{N - 4}})}^{\frac{N}{N - 4}}
\|u_j\| _{L^{\frac{2(N + 4)}{N - 4 }}([t_j, t], L^{\frac{2(N + 4)}{N - 4}})}
^{\frac{8 - N}{N - 4}}\|u_j\|_{L^{\frac{2(N + 4)}{N }}([t_j, t],
L^{\frac{2(N + 4)}{N}})}\\
&\lesssim \|\mathbf{u}\| _{\big(L^{\frac{2(N + 4)}{N - 4 }}([t_j, t],
L^{\frac{2(N + 4)}{N - 4}})\big)^{(m)}}^{\frac{8 }{N - 4}}\|\mathbf{u}
\|_{\big(L^{\frac{2(N + 4)}{N }}([t_j, t], L^{\frac{2(N + 4)}{N}})\big)^{(m)}}\\
&\lesssim \eta^{\frac{8 }{N - 4}}\|\mathbf{u}\|_{\big(L^{\frac{2(N + 4)}{N }}
([t_j, t], L^{\frac{2(N + 4)}{N}})\big)^{(m)}}.
\end{align*}}
If $\eta$ is sufficiently small, the conservation of the mass yields
$$
\|\mathbf{u}\|_{\big({L^{\frac{2(N+4)}{N}}([t_j, t],
L^{\frac{2(N+4)}{N}})}\big)^{(m)}} \leq C\|\Psi\|_{(L^2)^{(m)}}
$$
and
$$
\| \mathbf{u}\|_{(M([t_j, t]))^{(m)}} \leq C \|\mathbf{u}(t_j)\|_\mathbf{H}.
$$
Applying again Strichartz estimates, yields
$$
\| \mathbf{u}\|_{\big( L^\infty([t_j, t], \mathbf{H})\big)^{(m)}}
\leq C \|\mathbf{u}(t_j)\|_\mathbf{H} .
$$
In particular, $\|\mathbf{u}(t_{j +1})\|_\mathbf{H}
\leq C \|\mathbf{u}(t_j)\|_\mathbf{H}$. Finally,
$$
\| \mathbf{u}\|_{\big( L^\infty([t_j, t], \mathbf{H})\big)^{(m)}}
+\| \mathbf{u}\|_{(M([t_j, t]))^{(m)}} \leq2 C^n\|\Psi\|_\mathbf{H}<+\infty.
$$
The first step is done. Choose $t_0\in I_n$, Duhamel's formula gives
\begin{align*}
\mathbf{u}(t)
& = e^{i(t - t_0)\Delta^2}\mathbf{u}(t_0)
- i \sum_{k=1}^{m}a_{jk}\int_{t_0}^te^{i(t - s)\Delta^2}\\
&\quad\times \Big(|u_k|^{\frac{N}{N - 4}}
|u_1|^{\frac{8 - N}{N - 4}}u_1,\dots,|u_k|^{\frac{N}{N - 4}}|u_m|
^{\frac{8 - N}{N - 4}}u_m\Big)\,ds.
\end{align*}
Thanks to Sobolev inequality and Strichartz estimate,
\begin{align*}
\|e^{i(t - t_0)\Delta^2}\mathbf{u}(t_0)\|_{(W([t_0, t]))^{m}}
&\leq \|\mathbf{u}\|_{(W([t_0, t]))^{m}} +C \sum_{j,k=1}^{m}
\big\||u_k|^{\frac{N}{N - 4}}|u_j|^{\frac{8 - N}{N - 4}}u_j \big\|_{N([t_0, t])}\\
&\leq \|\mathbf{u}\|_{(W([t_0, t]))^{m}} +C\|\mathbf{u}
\|_{(W([t_0, t]))^{m}}^{\frac{N + 4}{N - 4}}.
\end{align*}
The dominated convergence theorem ensures that
$ \|\mathbf{u}\|_{(W([t_0, T]))^{m}}$ can be made arbitrarily small
as $t_0\to T$, then
$$
\|e^{i(t - t_0)\Delta^2}\mathbf{u}(t_0)\|_{(W([t_0, T]))^{m}}\leq \delta,
$$
where $\delta$ is as in Proposition \ref{prop5.1}. In particular, we
can find $t_1\in (0, T)$ and $T^{\prime }>T$ such that
$$
\|e^{i(t - t_0)\Delta^2}\mathbf{u}(t_0)\|_{(W([t_1, T']))^{m}}
\leq \delta.
$$
Now, it follows from Proposition \ref{prop5.1} that there exists
$\mathbf{v}\in C([t_1, T'], H)$ such that $\mathbf{v}$ solves \eqref{S}
with $ p = \frac{N}{N -4}$ and $\mathbf{u}(t_1) = \mathbf{v}(t_1)$.
By uniqueness, $\mathbf{u} = \mathbf{v}$ in $[t_1, T)$ and $\mathbf{u}$
can be extended in $[0, T']$.
\end{proof}
\subsection{Morawetz estimate}
In what follows we give a classical proof, inspired by
\cite{Colliander, Miao 2}, of Morawetz estimates. Let
$\mathbf{u}:=(u_1,\dots ,u_m)\in H$ be a solution of
$$
i\dot u_j +\Delta^2 u_j+ \Big(\sum_{k=1}^{m}a_{jk}|u_k|^p\Big)|u_j|^{p-2}u_j=0
$$
in $N_1$-spatial dimensions and $\mathbf{v}:=(v_1,\dots ,v_m)\in H$ be
a solution to
\[
i\dot v_j +\Delta^2 v_j+ \Big(\sum_{k=1}^{m}a_{jk}|v_k|^p\Big)|v_j|^{p-2}v_j =0
\]
in $N_2$-spatial dimensions. Define the tensor product
$\mathbf{w}:= (\mathbf{u}\otimes\mathbf{v})(t,z)$ for $z$ in
$$
\R^{N_1 +N_2}:= \{ (x,y)\quad\text{s. t}\quad x\in \R^{N_1}, y\in \R^{N_2}\}
$$
by the formula
$$
(\mathbf{u}\otimes\mathbf{v})(t,z) = \mathbf{u}(t,x)\mathbf{v}(t,y) .
$$
Denote $F(\mathbf{u}):= \big(\sum_{k=1}^{m}a_{jk}|u_k|^p\big)|u_j|^{p-2}u_j$.
A direct computation shows that
$\mathbf{w}:=(w_1,\dots ,w_n)= \mathbf{u}\otimes\mathbf{v}$ solves the equation
\begin{equation}\label{tensor1}
i\dot w_j +\Delta^2 w_j+ F(\mathbf{u})\otimes v_j
+ F(\mathbf{v})\otimes u_j:=i\dot w_j +\Delta^2 w_j+ h=0
\end{equation}
where $\Delta^2:= \Delta_x^2 + \Delta_y^2$.
Define the Morawetz action corresponding to $\mathbf{w}$ by
\begin{align*}
M_a^{\otimes_2}
&:= 2\sum_{j=1}^m\int_{\R^{N_1}\times \R^{N_2}}\nabla a(z).
\Im(\overline{u_j\otimes v_j(z)}\nabla (u_j\otimes v_j)(z))\,dz\\
&= 2\int_{\R^{N_1}\times \R^{N_2}}\nabla a(z).\Im({\bf \bar{w}}(z)
\nabla (\mathbf{w})(z))\,dz,
\end{align*}
where $\nabla: =(\nabla_x,\nabla_y)$. It follows from equation \eqref{tensor1} that
\begin{align*}
\Im(\dot{\bar{w}}_j\partial_i w_j)
&=\Re (-i\dot{\bar{w}}_j\partial_i w_j) \\
&= - \Re \big((\Delta^2 \bar{w}_j +\sum_{k=1}^{m}a_{jk}
|\bar{u}_k|^p|\bar{u}_j|^{p-2}\bar{u}_j \bar{v}_j
+\sum_{k=1}^{m}a_{jk}|\bar{v}_k|^p|\bar{v}_j|^{p-2}\bar{v}_j
\bar{u}_j)\partial_i w_j\big);
\end{align*}
\begin{align*}
\Im( \bar{w}_j\partial_i\dot w_j)
&=\Re (-i \bar{w}_j\partial_i\dot w_j)\\
&=\Re \big(\partial_i(\Delta^2 w_j
+\sum_{k=1}^{m}a_{jk}|u_k|^p|u_j|^{p-2}u_j v_j
+\sum_{k=1}^{m}a_{jk}|v_k|^p|v_j|^{p-2}v_j u_j) \bar{w}_j\big).
\end{align*}
Moreover, denoting the quantity
$ \big\{ h,w_j\big\}_p:=\Re \big(h\nabla\bar{w}_j - w_j\nabla\bar{h} \big)$,
we compute
\begin{align*}
\big\{ h,w_j\big\}_p^i
& = -\partial_i\Big(\sum_{k=1}^{m}a_{jk}|\bar{u}_k|^p
|\bar{u}_j|^{p-2}\bar{u}_j \bar{v}_j +\sum_{k=1}^{m}a_{jk}
|\bar{v}_k|^p|\bar{v}_j|^{p-2}\bar{v}_j \bar{u}_j\Big) w_j\\
&\quad + \Big(\sum_{k=1}^{m}a_{jk}|u_k|^p|u_j|^{p-2}u_j v_j
+\sum_{k=1}^{m}a_{jk}|v_k|^p|v_j|^{p-2}v_j u_j\Big) \partial_i\bar{w}_j.
\end{align*}
It follows that
\begin{align*}
\dot M_a^{\otimes_2}
&= 2\sum_{j=1}^m\int_{\R^{N_1}\times \R^{N_2}}\partial_i
a\Re \big(\bar{w}_j\partial_i \Delta^2 w_j - \partial_iw_j
\Delta^2\bar{w}_j\big)\,dz \\\
&\quad - 2\sum_{j=1}^m \int_{\R^{N_1}
\times \R^{N_2}}\partial_i a \big\{h,w_j\big\}_p^i\,dz\\
&= -2\sum_{j=1}^m\int_{\R^{N_1}\times \R^{N_2}}\big[\Delta
a\Re(\bar{w}_j \Delta^2 w_j) +2\Re( \partial_i a\partial_i\bar{w}_j
\Delta^2 w_j)\big] \,dz \\
&\quad - \sum_{j=1}^m 2 \int_{\R^{N_1}\times
\R^{N_2}}\partial_i a \big\{h,w_j\big\}_p^i\,dz\\
&:= \mathcal{I}_1+ \mathcal{I}_2 - 2\sum_{j=1}^m \int_{\R^{N_1}
\times \R^{N_2}}\partial_i a \big\{h,w_j\big\}_p^i\,dz.
\end{align*}
Similar computations as those in \cite{Miao 2}, give
\begin{align*}
\mathcal{I}_1 + \mathcal{I}_2
&= 2 \sum_{j=1}^m \Re \int_{\R^{N_1}\times \R^{N_2}}
\Big\{2\big(\partial_{ik}^x\Delta_x a \partial_i\bar{u}_j\partial_k u_j|v_j|^2
+ \partial_{ik}^y\Delta_y a \partial_i\bar{v}_j\partial_k v_j|u_j|^2 \big) \\
&\quad - \frac{1}{2}(\Delta_x^3 + \Delta_y^3) a |u_jv_j|^2
+ \big(\Delta_x^2 a |\nabla u_j|^2|v_j|^2
+ \Delta_y^2 a |\nabla v_j|^2|u_j|^2\big) \\
&\quad - 4\big(\partial_{ik}^x a \partial_{i_1 i}\bar{u}_j\partial_{i_1k}u_j|v_j|^2
+\partial_{ik}^y a \partial_{i_1 i}\bar{v}_j\partial_{i_1k}v_j|u_j|^2 \big)
\Big\}\,dz.
\end{align*}
Now, we take $a(z):=a(x,y) = |x - y|$ where $(x,y)\in\R^{N}\times \R^{N}$.
Then calculation done in \cite{Miao 2}, yield
$$
\dot M_a^{\otimes_2}\leq2 \sum_{j=1}^m
\Re \int_{\R^{N_1}\times \R^{N_2}}\Big(- \frac{1}{2}(\Delta_x^3
+ \Delta_y^3)a|u_jv_j|^2 - 2\partial_i a\{h,w_j\}_p^i \Big)\,dz.
$$
Hence, we obtain
$$
\sum_{j=1}^m \int_0^T \int_{\R^{N_1}\times \R^{N_2}}
\Big((\Delta_x^3 + \Delta_y^3)a|u_jv_j|^2
+4\partial_i a\{h,w_j\}_p^i \Big)\,dz\,dt \leq\sup_{[0,T]}|M_a^{\otimes_2}|.
$$
Then
\begin{align*}
&\sum_{j=1}^m \int_0^T \int_{\R^{N_1}\times \R^{N_2}}
\Big((\Delta_x^3 + \Delta_y^3)a|u_jv_j|^2 +4(1 - \frac{1}{p})
\Delta_x a\sum_{k=1}^ma_{jk}|u_k|^p|u_j|^p|v_j|^2 \\
&\quad + 4(1 - \frac{1}{p})\Delta_y a\sum_{k=1}^ma_{jk}|v_k|^p|v_j|^p|u_j|^2
\Big)\,dz\,dt \leq\sup_{[0,T]}|M_a^{\otimes_2}|.
\end{align*}
Taking account of the equalities $\Delta_x a = \Delta_ya=(N-1)|x-y|^{-1}$ and
$$
\Delta_x^3 a = \Delta_y^3a
=\begin{cases}
C\delta(x-y),&\text{if } N=5;\\
3(N -1)(N - 3)(N - 5)|x-y|^{-5},&\text{if } N> 5,
\end{cases}
$$
when $N =5$, choosing $u_j = v_j$, we obtain
$$
\sum_{j=1}^m \int_0^T \int_{\R^5} |u_j(t,x)|^4\,dx\,dt
\lesssim \sup_{[0,T]}|M_a^{\otimes_2}|.
$$
If $N>5$, it follows that
$$
\sum_{j=1}^m \int_0^T\int_{\R^{N}\otimes \R^{N}}
\frac{|u_j(t,x)|^2|u(y,t)|^2}{|x - y|^5}\,dx\,dy\,dt
\lesssim \sup_{[0,T]}|M_a^{\otimes_2}|.
$$
This completes the proof.
\begin{thebibliography}{99}
\bibitem{Adams} R. Adams;
\emph{Sobolev spaces}, Academic. New York, (1975).
\bibitem{Artzi} M. Ben-Artzi, H. Koch, J. C. Saut;
\emph{Dispersion estimates for fourth order Schr\"odinger equations},
C. R. Math. Acad. Sci. S\'er. 1. Vol. 330, 87-92, (2000).
\bibitem{ct} B. Cassano and M. Tarulli;
\emph{$H^s$ -scattering for systems of N-defocusing weakly coupled NLS
equations in low space dimensions},
J. Math. Anal. Appl. Vol. 430, 528-548, (2015).
\bibitem{Colliander} J. Colliander, M. Grillakis, N. Tzirakis;
\emph{Tensor products and correlation estimates with applications to nonlinear
Schr\"odinger equations}, Comm. Pure Appl. Math. Vol. 62, no. 1, 920-968, (2009).
\bibitem{Fibich} G. Fibich, B. Ilan, G. Papanicolaou;
\emph{Self-focusing with fourth-order dispersion}, SIAM J. Appl. Math, Vol.
62, no. 4, 1437-1462, (2002).
\bibitem{Guo} B. Guo, B. Wang;
\emph{The global Cauchy problem and scattering of solutions for nonlinear
Schr\"odinger equations in $H^s$}, Diff. Int. Equ, Vol. 15, no. 9,
1073-1083, (2002).
\bibitem{hmon} H. Hajaiej, L. Molinet, T. Ozawa, B. Wan;
\emph{Necessary and sufficient conditions for the fractional
Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and
generalized boson equations}, RIMS Kokyuroku Bessatsu 26, 159-175, (2011).
\bibitem{Hasegawa} A. Hasegawa and F. Tappert;
\emph{Transmission of stationary nonlinear optical pulses in dispersive
dielectric fibers II. Normal dispersion}, Appl. Phys. Lett. Vol. 23, 171-172, (1973).
\bibitem{Karpman} V. I. Karpman;
\emph{Stabilization of soliton instabilities by higher-order dispersion:
fourth-order nonlinear Schr\"odinger equation}, Phys. Rev. E. Vol. 53, no.
2, 1336-1339, (1996).
\bibitem{Karpman 1} V. I. Karpman, A. G. Shagalov;
\emph{Stability of soliton described by nonlinear Schr\"odinger type equations
with higher-order dispersion}, Phys D. Vol. 144, 194-210, (2000).
\bibitem{km} C. E. Kenig, F. Merle;
\emph{Global well-posedness, scattering and blow-up for the energy-critical
focusing non-linear wave equation}, Acta Math. Vol. 201, no. 2, 147-212, (2008).
\bibitem{Merle} C. E. Kenig, F. Merle;
\emph{Global wellposedness, scattering and blow up for the energy critical,
focusing, nonlinear Schr\"odinger equation in the radial case},
Invent. Math. Vol. 166, 645-675, (2006).
\bibitem{Levandosky} S. Levandosky, W. Strauss;
\emph{Time decay for the nonlinear Beam equation}, Meth. Appl. Anal.
Vol. 7, 479-488, (2000).
\bibitem{Lions} P. L. Lions;
\emph{Sym\'etrie et compacit\'e dans les espaces de Sobolev},
J. Funct. Anal. Vol. 49, no. 3, 315-334, (1982).
\bibitem{Miao 11} C. Miao, J. Zheng;
\emph{Scattering theory for the defocusing fourth-order Schr\"odinger equation},
Nonlinearity. Vol. 29, 2, 692-736, (2016).
\bibitem{Miao} C. Miao, G. Xu, L. Zhao;
\emph{Global well-posedness and scattering for the focusing energy-critical
nonlinear Schr\"odinger equations of fourth-order in the radial case},
J. Diff. Equ. Vol. 246, 3715-3749, (2009).
\bibitem{Miao 1} C. Miao, G. Xu, L. Zhao;
\emph{Global well-posedness and scattering for the defocusing energy-critical
nonlinear Schr\"odinger equations of fourth-order in dimensions $d \geq 9$},
J. Diff. Equ. Vol. 251, no. 12, 15, 3381-3402, (2011).
\bibitem{Miao 2} C. Miao, H. Wu, J. Zhang;
\emph{Scattering theory below energy for the cubic fourth-order
Schr\"odinger equation}, Mathematische Nachrichten,
Vol. 288, no. 7, 798-823, (2015).
\bibitem{ntds} N. V. Nguyen, R. Tian, B. Deconinck, N. Sheils;
\emph{Global existence for a coupled system of Schr\"odinger equations with
power-type nonlinearities}, J. Math. Phys. Vol. 54, 011503, (2013).
\bibitem{Pausader} B. Pausader;
\emph{Global well-posedness for energy critical fourth-order Schr\"odinger
equations in the radial case}, Dyn. Part. Diff. Equ. Vol. 4, no. 3, 197-225, (2007).
\bibitem{Pausader 1} B. Pausader;
\emph{The cubic fourth-order Schr\"odinger equation},
J. F. A, Vol. 256, 2473-2517, (2009).
\bibitem{Pausader 2} B. Pausader;
\emph{The focusing energy-critical fourth-order Schr\"odinger equation with
radial data}, Discrete Contin. Dyn.
Syst. Ser. A, Vol. 24, no. 4, 1275-1292, (2009).
\bibitem{ts} T. Saanouni;
\emph{A note on fourth-order nonlinear Schr\"odinger equation},
Ann. Funct. Anal. Vol. 6, no. 1, 249-266, (2015).
\bibitem{T2} T. Saanouni;
\emph{A note on coupled nonlinear Schr\"odinger equations},
Adv. Nonl. Anal., Vol. 3, no. 4, 247-269, (2014).
\bibitem{T} T. Saanouni;
\emph{A note on coupled focusing nonlinear Schr\"odinger equations},
Published online in Applicable Analysis, (2015).
\bibitem{st} Saanouni;
\emph{On defocusing coupled nonlinear Schr\"odinger equations},
arXiv:1505.07059v1 [math.AP], (2015).
\bibitem{Zakharov} V. E. Zakharov;
\emph{Stability of periodic waves of finite amplitude on the surface of
a deep fluid}, Sov. Phys. J. Appl. Mech. Tech. Phys. Vol. 4, 190-194, (1968).
\end{thebibliography}
\end{document}