\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 61, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/61\hfil Blow-up solutions]
{Blow-up solutions for $N$ coupled Schr\"{o}dinger equations}

\author[J. Chen, B. Guo\hfil EJDE-2007/61\hfilneg]
{Jianqing Chen, Boling Guo}  % in alphabetical order

\address{Jianqing Chen \newline
Department of Mathematics, Fujian Normal University\\
  Fuzhou 350007,  China \newline
 Institute of Applied Physics and
Computational Mathematics \\
PO Box 8009, Beijing 100088, China}
\email{jqchen@fjnu.edu.cn}

 \address{Boling Guo \newline
 Institute of Applied Physics and
Computational Mathematics \\
PO Box 8009, Beijing 100088, China} 
\email{gbl@mail.iapcm.ac.cn}

\thanks{Submitted May 29, 2006. Published April 22, 2007.}
\thanks{J. Chen was supported by grant 10501006 from
 the Youth Foundation of NSFC, \hfill\break\indent
 by the China
 Post-Doc Science Foundation, and by the Program NCETFJ}
\subjclass[2000]{35Q55, 35B35}
\keywords{Blow-up solutions;  coupled Schr\"{o}dinger equations}

\begin{abstract}
 It is proved that blow-up solutions to $N$
 coupled Schr\"{o}dinger equations
 $$
 i\varphi_{jt} + \varphi_{jxx} + \mu_j|\varphi_j|^{p-2}\varphi_j
 +\sum_{k\neq j,\;k=1}^N\beta_{kj}|\varphi_k|^{p_k}|\varphi_j|^{p_j-2}
 \varphi_j=0
 $$
 exist only under the condition that the initial data have  strictly
 negative energy.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}


\section{Introduction}

In this paper, we consider the existence of
blow-up solutions of the  $N$ coupled Schr\"{o}dinger
equations
\begin{equation} \label{SE-j}
\begin{gathered}
i\varphi_{jt} + \varphi_{jxx} + \mu_j|\varphi_j|^{p-2}\varphi_j
+\sum_{k\neq j,\; k=1}^N
\beta_{kj}|\varphi_k|^{p_k}|\varphi_j|^{p_j-2}\varphi_j=0,\\
\varphi_j(x,t)\big|_{t=0}=\psi_j(x),\quad x\in \mathbb{R},
\end{gathered}
\end{equation}
where $i=\sqrt{-1}$,
$\varphi_j=\varphi_j(x,t):\mathbb{R}\times\mathbb{R}_+\to\mathbb{C}$,
$j, k\in\{1, \dots,N\}$ and $\mu_j$, $\beta_{kj}\in \mathbb{R}$.
System of this kind
appears in several branches of physics, such as in the study of
interactions of waves with different polarizations \cite{bz} or in
the description of nonlinear modulations of two monochromatic waves
\cite{npf}.


When $p=4$, $p_j=2$, and $p_k=2$, the solution $\varphi_j$ of
\eqref{SE-j} denotes the $j$th component of the beam in Kerr-like photo
refractive media \cite{aa}. The constants $\beta_{kj}$ is the
interaction between the $k$th and the $j$th component of the beam.
As $\beta_{kj}>0$, the interaction is attractive while the
interaction is repulsive if $\beta_{kj}<0$. Moreover, the system
\eqref{SE-j} is integrable and there are various analytical and
numerical results on solitary wave solutions of the general $N$
coupled Schr\"{o}dinger equations \cite{cho, hio}.

When $2<p<6$, $2\leq p_k+p_j<6$ and $N=2$, the existence and
stability of standing wave, which is a trivial global solution, of
\eqref{SE-j} have been studied by Cipolatti et al \cite{cz}. Also when
$2<p<6$ and $2\leq p_k+p_j<6$, for any $j, k\in\{1, \dots, N\}$,
we know from \cite{caz} that for any
$$
\overrightarrow{\varphi}(x,0)=(\varphi_1(x,0), \dots,
\varphi_N(x,0))=\overrightarrow{\psi}=(\psi_1(x), \dots,
\psi_N(x))\in (H^1(\mathbb{R}))^N,
$$
Equation \eqref{SE-j} admits a unique global
solution $\overrightarrow{\varphi}\in C(\mathbb{R}_+,
(H^1(\mathbb{R}))^N)$.

The main purpose here is to prove the existence of blow-up solutions
of \eqref{SE-j} only under the condition of the initial data with
strictly negative energy. The main result is the following theorem.

\begin{theorem}\label{th11}
Let $p=6$, $p_k + p_j=6$ and $\mu_j\geq 0$, $\beta_{kj}>0$ with
$\theta_{kj}:={\beta_{kj}\over p_j}={\beta_{jk}\over p_k}:=\theta_{jk}$,
$p_k, p_j\geq 2$. If $E(\overrightarrow{\psi})<0$(for the definition
of $E$, see
Proposition \ref{pr21}), then the solution of \eqref{SE-j} with initial
data $\overrightarrow{\psi}$ must blow up in finite time.
\end{theorem}

 We emphasize that when $N=1$, i.e. no coupling terms, the blow up
problem has been studied extensively, see e.g.
\cite{ot, gla,  caz}. But as far as we know, there is no blow-up result to the
 $N$ coupled Schr\"{o}dinger equations. The main contribution here is
  to overcome the additional difficulties created by the
 coupling terms and then prove Theorem \ref{th11}.

 This paper is organized as follows. In Section 2,
 we give some preliminaries and derive a variant of
 virial identity which generalizes some previous works for
the single equation. Section 3 is devoted to the
 proof of Theorem \ref{th11}.

\subsection*{Notation} As above and henceforth, the integral
$\int_\mathbb{R}\dots dx$ is simply denoted by $\int\dots$. For any
$t$, the function $x\mapsto\varphi_j(x,t)$ is simply denoted by
$\varphi_j(t)$. $\overline{f}$ denotes the complex conjugate of $f$.
$f_x$ and $f_t$ denote the derivative of $f$ with respect to $x$ and
$t$, respectively. By $f^{(m)}$ we denote the $m$th order
derivatives of $f$. $\|\cdot\|_{L^q}$ denotes the norm in
$L^q(\mathbb{R})$ or $(L^q(\mathbb{R}))^N$ which will be understood
from the context. $\hbox{Re}$
denotes the real part and $\mathop{\rm Im}$ the imaginary part.


\section{Preliminaries}

Throughout this paper, we always assume that the conditions of
Theorem \ref{th11} hold. The following proposition is useful in what
follows.

\begin{proposition}\label{pr21}
For any $\overrightarrow{\psi}=(\psi_1(x), \dots, \psi_N(x))\in
(H^1(\mathbb{R}))^N$, there is $T>0$ and a unique solution
$\overrightarrow{\varphi}\in C([0,T), (H^1(\mathbb{R}))^N)$
satisfying \eqref{SE-j}. Moreover, there holds the following
conservation laws:
\begin{gather}\label{eq21}
\int|\varphi_j(t)|^2\equiv\int|\psi_j|^2,\\
\label{eq22}
E(\overrightarrow{\varphi}(t))=\sum_{j=1}^N\int\Big(|\varphi_{jx}|^2-{2\over
p}\mu_j|\varphi_j|^p\Big)-2\sum_{k<j}\theta_{kj}\int|\varphi_k|^{p_k}|\varphi_j|^{p_j}\equiv
E(\overrightarrow{\psi}).
\end{gather}
\end{proposition}

\begin{proof} The existence of the local solution
$\overrightarrow{\varphi}$ follows from \cite{caz}. We only sketch
the proof on the conservative laws. Firstly, multiplying \eqref{SE-j} by
$\overline{\varphi}_j$, integrating over $\mathbb{R}$ and taking
imaginary part, we obtain (\ref{eq21}). Secondly, it is deduced from
multiplying \eqref{SE-j} by $\overline{\varphi}_{jt}$, integrating over
$\mathbb{R}$ and taking real part that
\begin{equation}\label{eq23}
\int\Big(-{1\over 2}|\varphi_{jx}|^2+{\mu_j\over
p}|\varphi_j|^p\Big)_t+\sum_{k\neq j}{\beta_{kj}\over
p_j}\int|\varphi_k|^{p_k}(|\varphi_j|^{p_j})_t=0.
\end{equation}
Similarly, for \eqref{SE-j} with $k$ instead of $j$, we have
\begin{equation}\label{eq24}
\int\Big(-{1\over 2}|\varphi_{kx}|^2+{\mu_k\over
p}|\varphi_k|^p\Big)_t+\sum_{j\neq k}{\beta_{jk}\over
p_k}\int|\varphi_j|^{p_j}(|\varphi_k|^{p_k})_t=0.
\end{equation}
 From (\ref{eq23}) and (\ref{eq24}) it follows that
\begin{equation}\label{eq25}
\sum_{j=1}^N\int\Big(-{1\over 2}|\varphi_{jx}|^2+{\mu_j\over
p}|\varphi_j|^p\Big)_t+\sum_{k<j}\theta_{kj}\int\Big(|\varphi_k|^{p_k}|\varphi_j|^{p_j}\Big)_t=0.
\end{equation}
Then (\ref{eq22}) holds.
\end{proof}

Next we derive a variant of virial identity.

\begin{lemma}\label{le22}
Let $\varphi_j$ be a local smooth solution of \eqref{SE-j} with
$\varphi_j(x,0)=\psi_j(x)$. For real function $\phi\in
W^{3,\infty}(\mathbb{R})$, define $\Phi(x)=\int_0^x\phi(y)dy$. Then
\begin{equation}\label{eq26}
\begin{aligned}
&\sum_{j=1}^N\mathop{\rm Im}\int\phi
\psi_j\overline{\psi}_{jx}- \sum_{j=1}^N\mathop{\rm Im}\int\phi
\varphi_j(t)\overline{\varphi}_{jx}(t)\\
&=\int_0^t\Big\{2\sum_{j=1}^N \int|\varphi_{jx}|^2\phi_x
 -{1\over 2}\sum_{j=1}^N\int|\varphi_j|^2\phi^{(3)}
+{{2-p}\over p}\sum_{j=1}^N\mu_j\int|\varphi_j|^p\phi_x\\
&\quad -(p-2)\sum_{k<j}\theta_{kj}
\int|\varphi_k|^{p_k}|\varphi_j|^{p_j}\phi_x\Big\}d\tau,
\end{aligned}
\end{equation}
and
\begin{equation}\label{eq27}
\int\Phi|\varphi_j|^2=\int\Phi|\psi_j|^2
-2\int_0^t\int\mathop{\rm Im}\phi\varphi_j\overline{\varphi}_{jx}\,dx\,d\tau.
\end{equation}
\end{lemma}

\begin{proof} Let $\varphi_j$ be a smooth solution of
\eqref{SE-j}. Firstly, multiplying \eqref{SE-j} by
$\phi\overline{\varphi}_{jx}$, integrating over $\mathbb{R}$ and
taking the real part, we obtain
\begin{equation}\label{eq28}
-\mathop{\rm Im}\int\phi\varphi_{jt}\overline{\varphi}_{jx}
+\int\Big({1\over 2}\phi(|\varphi_{jx}|^2)_x+{\mu_j\over
p}\phi(|\varphi_j|^p)_x+\sum_{k\neq
j}\theta_{kj}|\varphi_k|^{p_k}(|\varphi_j|^{p_j})_x\phi\Big)=0.
\end{equation}
 From
\begin{gather*}
-\mathop{\rm Im}\int\phi\varphi_{jt}\overline{\varphi}_{jx}
=-{d\over{dt}}\mathop{\rm Im}\int\phi\varphi_j\overline{\varphi}_{jx}
+\mathop{\rm Im}\int\phi\varphi_j\overline{\varphi}_{jxt},\\
\mathop{\rm Im}\int\phi\varphi_j\overline{\varphi}_{jxt}
=-\mathop{\rm Im}\int\phi_x\overline{\varphi}_{jt}\varphi_j
+\mathop{\rm Im}\int\phi\varphi_{jt}\overline{\varphi}_{jx},
\end{gather*}
we obtain
\begin{equation}\label{eq29}
-\mathop{\rm Im}\int\phi\varphi_{jt}\overline{\varphi}_{jx} =-{1\over 2}
{d\over{dt}}\mathop{\rm Im}\int\phi\varphi_j\overline{\varphi}_{jx}-{1\over
2}\mathop{\rm Im}\int\phi_x\overline{\varphi}_{jt}\varphi_j.
\end{equation}
It is deduced from (\ref{eq28}) and (\ref{eq29}) that
\begin{equation}\label{eq210}
\begin{aligned}
&-{1\over 2} {d\over{dt}}\mathop{\rm Im}\int\phi\varphi_j
\overline{\varphi}_{jx}-{1\over 2}\mathop{\rm Im}
\int\phi_x\overline{\varphi}_{jt}\varphi_j+
\int\Big({1\over 2}\phi(|\varphi_{jx}|^2)_x\\
&+{\mu_j\over
p}\phi(|\varphi_j|^p)_x+\sum_{k\neq
j}\theta_{kj}|\varphi_k|^{p_k}(|\varphi_j|^{p_j})_x\phi\Big)=0.
\end{aligned}
\end{equation}
For \eqref{SE-j} with $k$ instead of $j$, we obtain by a similar argument that
\begin{equation}\label{eq211}
\begin{aligned}
&-{1\over 2}
{d\over{dt}}\mathop{\rm Im}\int\phi\varphi_k\overline{\varphi}_{kx}-{1\over
2}\mathop{\rm Im}\int\phi_x\overline{\varphi}_{kt}\varphi_k+
\int\Big({1\over 2}\phi(|\varphi_{kx}|^2)_x\\
&+{\mu_k\over p}\phi(|\varphi_k|^p)_x+\sum_{j\neq
k}\theta_{jk}|\varphi_j|^{p_j}(|\varphi_k|^{p_k})_x\phi\Big)=0.
\end{aligned}
\end{equation}
Secondly, multiplying the complex conjugate of \eqref{SE-j} by
$\varphi_j\phi_x$, integrating by parts and taking the real part, we
get that
\begin{equation}\label{eq212}
-\mathop{\rm Im}\int\phi_x\overline{\varphi}_{jt}\varphi_j
= \int\Big(-\phi_x|\varphi_{jx}|^2+{1\over 2}|\varphi_j|^2\phi^{(3)}\\
+\mu_j\phi_x|\varphi_j|^p+\sum_{k\neq
j}\beta_{kj}|\varphi_k|^{p_k}|\varphi_j|^{p_j}\phi_x\Big).
\end{equation}
Similarly,
\begin{equation}\label{eq213}
-\mathop{\rm Im}\int\phi_x\overline{\varphi}_{kt}\varphi_k
= \int\Big(-\phi_x|\varphi_{kx}|^2+{1\over 2}|\varphi_k|^2\phi^{(3)}\\
+\mu_k\phi_x|\varphi_k|^p+\sum_{j\neq
k}\beta_{jk}|\varphi_j|^{p_j}|\varphi_k|^{p_k}\phi_x\Big).
\end{equation}
We now obtain from (\ref{eq210})--(\ref{eq213}) that
\begin{equation}\label{eq214}
\begin{aligned}
&-{d\over{dt}}\mathop{\rm Im}\int\phi\varphi_j\overline{\varphi}_{jx}-
2\int\phi_x|\varphi_{jx}|^2+{1\over
2}\int|\varphi_j|^2\phi^{(3)}+{{p-2}\over
p}\mu_j\int\phi_x|\varphi_j|^p\\
&+\sum_{k\neq
j}\beta_{kj}\int|\varphi_k|^{p_k}|\varphi_j|^{p_j}\phi_x
+\sum_{k\neq j}{{2\beta_{kj}}\over
p_j}\int|\varphi_k|^{p_k}(|\varphi_j|^{p_j})_x\phi=0
\end{aligned}
\end{equation}
and
\begin{equation}\label{eq215}
\begin{aligned}
&-{d\over{dt}}\mathop{\rm Im}\int\phi\varphi_k\overline{\varphi}_{kx}-
2\int\phi_x|\varphi_{kx}|^2+{1\over
2}\int|\varphi_k|^2\phi^{(3)}+{{p-2}\over
p}\mu_k\int\phi_x|\varphi_k|^p\\
&+\sum_{j\neq
k}\beta_{jk}\int|\varphi_j|^{p_j}|\varphi_k|^{p_k}\phi_x
+\sum_{j\neq k}{{2\beta_{jk}}\over
p_k}\int|\varphi_j|^{p_j}(|\varphi_k|^{p_k})_x\phi=0.
\end{aligned}
\end{equation}
It follows that
\begin{equation}\label{eq216}
\begin{aligned}
&-{d\over{dt}}\sum_{j=1}^N\mathop{\rm Im}\int\phi\varphi_j\overline{\varphi}_{jx}-
2\sum_{j=1}^N\int\phi_x|\varphi_{jx}|^2+{1\over
2}\sum_{j=1}^N\int|\varphi_j|^2\phi^{(3)}\\
&+{{p-2}\over p}\sum_{j=1}^N\mu_j\int\phi_x|\varphi_j|^p
+(p-2)\sum_{k<j}\theta_{kj}\int|\varphi_k|^{p_k}|\varphi_j|^{p_j}\phi_x=0.
\end{aligned}
\end{equation}
Hence (\ref{eq26}) holds. Finally, multiplying the complex conjugate
of \eqref{SE-j} by $\Phi\varphi_j$, integrating by parts and taking the
imaginary part, we obtain
$$-\hbox{Re}\int\Phi\varphi_j\overline{\varphi}_{jt}
 + \mathop{\rm Im}\int\Phi\varphi_j\overline{\varphi}_{jxx}=0,$$
which implies
\begin{equation}\label{eq217}
{d\over{dt}}\int\Phi|\varphi_j|^2=-
2\mathop{\rm Im}\int\phi\varphi_j\overline{\varphi}_{jx}.
\end{equation}
So (\ref{eq27}) easily follows. The proof is complete.
\end{proof}

\section{Proof of Theorem \ref{th11}}

In this section, we will borrow an idea from \cite{gla, ot} to
prove Theorem \ref{th11}. Firstly we introduce two lemmas from
\cite{ot}.

\begin{lemma}[{\cite[Lemma 2.1]{ot}}] \label{le31}
Let $u\in H^1(\mathbb{R})$ and $\rho$ be a real valued function in
$W^{1,\infty}(\mathbb{R})$. Then for any $r>0$, we have
\begin{equation}\label{eq31}
\|\rho u\|_{L^\infty(|x|>r)}\leq \|u\|^{1/2}_{L^2(|x|>r)}
\Big(2\|\rho^2u_x\|_{L^2(|x|>r)}+\|u(\rho^2)_x\|_{L^2(|x|>r)}\Big)^{1/2}.
\end{equation}
\end{lemma}

\begin{lemma}[{\cite[Lemma 2.3]{ot}}] \label{le32}
Let $v(x)$ be in $L^2$. We define $R(x)$ such that $R(x)=|x|$ for
$|x|<1$ and $R(x)=1$ for $|x|>1$. Put
$v_\varepsilon(x)=\varepsilon^{-1/2}v(x/\varepsilon)$ for
$\varepsilon>0$. Then for any $\delta>0$, there exists an
$\varepsilon_0>0$ such that $\|Rv_\varepsilon\|_{L^2}\leq \delta$
for $0<\varepsilon<\varepsilon_0$.
\end{lemma}

We are now in a position to prove the theorem. Observe that $p=6$,
$p_j+p_k=6$ for $j, k\in\{1, \dots, N\}$ and the solution
$\varphi_j(x,t)$ of \eqref{SE-j} has the following scaling invariance.
More precisely, if we put
\begin{equation}\label{eq32}
\varphi_{\varepsilon j}(x,t)=\varepsilon^{-1/2}\varphi_j(x/\varepsilon,
t/\varepsilon^2),\quad \varphi_{\varepsilon
k}(x,t)=\varepsilon^{-1/2}\varphi_k(x/\varepsilon,
t/\varepsilon^2)
\end{equation}
for $\varepsilon>0$, then $\varphi_{\varepsilon j}$ and
$\varphi_{\varepsilon k}$ also satisfy \eqref{SE-j} and
\eqref{SE-j} with $k$ instead of $j$  and with
initial data $\varphi_{\varepsilon j}(x,0)=\psi_{\varepsilon
j}=\varepsilon^{-1/2}\psi_j(x/\varepsilon)$ and
$\varphi_{\varepsilon k}(x,0)=\psi_{\varepsilon k}
 =\varepsilon^{-1/2}\psi_k(x/\varepsilon)$, respectively. The proof
is divided into two steps. In the first step, we show that if
$-E(\overrightarrow{\psi})$ is large and
$\|\overrightarrow{\psi}\|_{L^2(|x|>1)}$ is small (but
$\|\overrightarrow{\psi}\|_{L^2(|x|<1)}$ may be large), then
$\|\overrightarrow{\varphi}(t)\|_{L^2(|x|>1)}$ is small for all
$t>0$.

In the second step, for any initial data $\overrightarrow{\psi}$
with negative energy, we use the scaling transform (\ref{eq32}) to
choose $\varepsilon>0$ so small that
$-E(\overrightarrow{\psi}_\varepsilon)$
($\overrightarrow{\psi}_\varepsilon=(\psi_{\varepsilon 1}, \dots,
\psi_{\varepsilon N}))$ is sufficiently large and
$\|\overrightarrow{\psi}_\varepsilon\|_{L^2(|x|>1)}$ is small
enough. Then the proof of the second step is reduced to the first
step and we complete the proof.

Let $\phi: [0,\infty)\to \mathbb{R}_+$ be a function with bounded
third order derivatives and be such that
$$
\phi(s)=\begin{cases}
 s, &  0\leq |s| < 1, \\
s-(s-1)^3, &  1< s < 1+{\sqrt{3}\over 3}, \\
s-(s+1)^3, &  -(1+{\sqrt{3}\over 3})< s <-1, \\
\hbox{smooth}, \phi'<0, &  1+{\sqrt{3}\over 3}\leq |s| < 2,\\
0, & \quad 2\leq |s|.
\end{cases}
$$
  Putting $\Phi(x)=\int_0^x\phi(y)dy$ and
$E_0=E(\overrightarrow{\psi})$, we have the following proposition.

\begin{proposition}\label{pr33}
Let $\varphi_j(t)$ be a solution of \eqref{SE-j} in
$C([0,T), H^1(\mathbb{R}))$ with $\varphi_j(0)=\psi_j$. Put $a_0=3/(16M)$. If
$\varphi_j(t)$ satisfies
\begin{equation}\label{eq33}
\sum_{j=1}^N\|\varphi_j(t)\|^4_{L^2(|x|>1)}\leq 2a_0,\quad 0\leq t < T,
\end{equation}
then we have
\begin{equation}\label{eq34}
\begin{aligned}
&-\sum_{j=1}^N\mathop{\rm Im}\int\phi\varphi_j(t)\overline{\varphi}_{jx}(t)
+\sum_{j=1}^N\mathop{\rm Im}\int\phi
\psi_j\overline{\psi}_{jx}\\
&\leq\Big(2E_0+4M(1+M)^2\sum_{j=1}^N\|\psi_j\|^6_{L^2}
+{M\over 2}\sum_{j=1}^N\|\psi_j\|^2_{L^2}\Big)t,
\end{aligned}
\end{equation}
where
$M=\|\phi_{xx}\|_{L^\infty}+\|\phi^{(3)}\|_{L^\infty}+\sum_{k,j=1}^N\beta_{kj}
+\sum_{j=1}^N\mu_j$.
\end{proposition}

\begin{proof} From the energy conserved identity
\begin{align*}
-\sum_{j=1}^N\int_{|x|<1}|\varphi_{jx}|^2
&=E(\overrightarrow{\varphi}(t))-\sum_{j=1}^N\int_{|x|>1}|\varphi_{jx}|^2\\
&\quad +{1\over 3}\sum_{j=1}^N\mu_j\int|\varphi_j|^6+
2\sum_{k<j}\theta_{kj}\int|\varphi_k|^{p_k}|\varphi_j|^{p_j},
\end{align*}
we obtain by (\ref{eq26}) that
\begin{align*}
&-\sum_{j=1}^N\mathop{\rm Im}\int\phi\varphi_j(t)\overline{\varphi}_{jx}(t)
+\sum_{j=1}^N\mathop{\rm Im}\int\phi
\psi_j\overline{\psi}_{jx}\\
&=\int_0^t\Big\{2E_0
-\sum_{j=1}^N\int_{|x|>1}2\big(1-\phi_x\big)|\varphi_{jx}|^2
+{2\over 3}\sum_{j=1}^N\mu_j\int\big(1-\phi_x\big)|\varphi_j|^6\\
&\quad-{1\over
2}\sum_{j=1}^N\int|\varphi_j|^2\phi^{(3)} +4\sum_{k<j}
\theta_{kj}\int\big(1-\phi_x\big)|\varphi_k|^{p_k}|\varphi_j|^{p_j}\Big\}d\tau.
\end{align*}
By Lemma \ref{le31} with $\rho(x)=(1-\phi_x)^{1/4}$ and H\"{o}lder
inequality, we obtain
\begin{equation}\label{eq35}
\begin{aligned}
&\int_{|x|>1}(1-\phi_x)|\varphi_j|^6\leq
\|\varphi_j\|^2_{L^2(|x|>1)}\|\rho\varphi_j\|^4_{L^\infty(|x|>1)}\\
&\leq\|\varphi_j\|^4_{L^2(|x|>1)}\Big(2\|\rho^2\varphi_{jx}\|_{L^2(|x|>1)}
+\|\varphi_j(\rho^2)_x\|_{L^2(|x|>1)}\Big)^2\\
&\leq
8\|\varphi_j\|^4_{L^2(|x|>1)}\|\rho^2\varphi_{jx}\|^2_{L^2(|x|>1)}
+ 2\|\varphi_j\|^6_{L^2(|x|>1)}\|(\rho^2)_x\|^2_{L^\infty(|x|>1)}.
\end{aligned}
\end{equation}
On the other hand, we have from the definition of $\phi$ and $\rho$
that $|(\rho^2)_x|\leq \sqrt{3}$ for $1<|x|<1+1/\sqrt{3}$. For
$|x|>1+1/\sqrt{3}$, we also have $|(\rho^2)_x|\leq{1\over
2}\|\phi_{xx}\|_{L^\infty}$. It follows that
$|(\rho^2)_x|\leq\sqrt{3}(1+{1\over 2}\|\phi_{xx}\|_{L^\infty})$. So
\begin{equation}\label{eq36}
\begin{aligned}
&\int_{|x|>1}(1-\phi_x)|\varphi_j|^6\\
&\leq 8\|\varphi_j\|^4_{L^2(|x|>1)}\|\rho^2\varphi_{jx}\|^2_{L^2(|x|>1)}
+ 6(1+{1\over
2}\|\phi_{xx}\|_{L^\infty})^2\|\varphi_j\|^6_{L^2(|x|>1)}.
\end{aligned}
\end{equation}
It is deduced from
$$
\int\big(1-\phi_x\big)|\varphi_k|^{p_k}|\varphi_j|^{p_j}\leq {p_k\over 6}
\int\big(1-\phi_x\big)|\varphi_k|^6+{p_j\over
6}\int\big(1-\phi_x\big)|\varphi_j|^6
$$
that
\begin{equation}\label{eq37}
\begin{aligned}
&2E_0
-\sum_{j=1}^N\int_{|x|>1}2\big(1-\phi_x\big)|\varphi_{jx}|^2
+{2\over 3}\sum_{j=1}^N\mu_j\int\big(1-\phi_x\big)|\varphi_j|^6\\
&-{1\over 2}\sum_{j=1}^N\int|\varphi_j|^2\phi^{(3)} +4\sum_{k<j}
\theta_{kj}\int\big(1-\phi_x\big)|\varphi_k|^{p_k}|\varphi_j|^{p_j}\\
&\leq 2E_0
-\sum_{j=1}^N\int_{|x|>1}2\big(1-\phi_x\big)|\varphi_{jx}|^2
+{2\over
3}\sum_{j=1}^N\mu_j\int\big(1-\phi_x\big)|\varphi_j|^6-{1\over
2}\sum_{j=1}^N\int|\varphi_j|^2\phi^{(3)}\\
&\quad +{2\over 3}\sum_{j<k}
\beta_{jk}\int\big(1-\phi_x\big)|\varphi_k|^{p_k}+{2\over
3}\sum_{k<j}
\beta_{kj}\int\big(1-\phi_x\big)|\varphi_j|^{p_j}.
\end{aligned}
\end{equation}
Using (\ref{eq36}) and the choice of $M$, we obtain that
\begin{equation}\label{eq38}
\begin{aligned}
&-\sum_{j=1}^N\mathop{\rm Im}\int\phi\varphi_j(t)\overline{\varphi}_{jx}(t)
+\sum_{j=1}^N\mathop{\rm Im}\int\phi
\psi_j\overline{\psi}_{jx}\\
&=\int_0^t\Big(2E_0
+4M(1+M)^2\sum_{j=1}^N\|\varphi_j\|^6_{L^2(|x|>1)}+{M\over
2}\sum_{j=1}^N\|\varphi_j\|^2_{L^2(|x|>1)}\Big)d\tau\\
&\leq\int_0^t\Big(2E_0
+4M(1+M)^2\sum_{j=1}^N\|\varphi_j\|^6_{L^2}+{M\over
2}\sum_{j=1}^N\|\varphi_j\|^2_{L^2}\Big)d\tau\\
&=\Big(2E_0+4M(1+M)^2\sum_{j=1}^N\|\psi_j\|^6_{L^2}
+{M\over 2}\sum_{j=1}^N\|\psi_j\|^2_{L^2}\Big)t.
\end{aligned}
\end{equation}
The proof is complete. \end{proof}


\begin{proof}[Proof of Theorem \ref{th11}]
 We assume the solution
$\varphi_j(t)$ of \eqref{SE-j} exists for all $t\geq 0$ and then
derive a contradiction. The proof is divided into two steps.\\
{\bf Step 1.} In this step, we assume the initial data
$\overrightarrow{\varphi}(0)=\overrightarrow{\psi}$ satisfies
\begin{gather}\label{eq39}
 \eta=-2E_0-4M(1+M)^2\sum_{j=1}^N\|\psi_j\|^6_{L^2}-{M\over
2}\sum_{j=1}^N\|\psi_j\|^2_{L^2}>0, \\
 \label{eq310}
4\Big(\sum_{j=1}^N\int\Phi|\psi_j|^2\Big)^2
\Big({4\over\eta}\sum_{j=1}^N\|\psi_{jx}\|^2_{L^2}+1\Big)^2\leq
a_0,
\end{gather}
where $M$ and $a_0$ are defined as in Proposition \ref{pr33}.

We first prove that if the initial data $\varphi_j(0)=\psi_j$
satisfies (\ref{eq39}) and (\ref{eq310}), then $\varphi_j(t)$
satisfies (\ref{eq33}) for all $t\geq 0$. We prove this by
contradiction. Since $\eta>0$ and $1\leq 2\Phi(x)$ for $|x|>1$, we
have from (\ref{eq310}) that
\begin{equation}\label{eq311}
\sum_{j=1}^N\|\psi_j\|^4_{L^2(|x|>1)}\leq a_0.
\end{equation}
Define $T_0$ as
$$
T_0=\sup\{t>0;\ \sum_{j=1}^N\|\varphi_j(s)\|^4_{L^2(|x|>1)}\leq
2a_0, 0\leq s<t\}.
$$
By (\ref{eq311}) we know that $T_0>0$. If
$T_0=+\infty$, then we are done. Assuming now that $T_0<+\infty$,
the continuity in $L^2$ of $\varphi_j(t)$ implies
\begin{equation}\label{eq312}
\sum_{j=1}^N\|\varphi_j(T_0)\|^4_{L^2(|x|>1)}= 2a_0.
\end{equation}
As $\varphi_j(t)$ satisfies all the assumptions in Proposition
\ref{pr33} on $[0,T_0)$, we get from (\ref{eq27}), (\ref{eq39}) and
Proposition \ref{pr33} that for $0<t<T_0$,
\begin{equation}\label{eq313}
\begin{aligned}
\sum_{j=1}^N\int\Phi|\varphi_j(t)|^2
&\leq\sum_{j=1}^N\int\Phi |\psi_j|^2
-2\int_0^t\mathop{\rm Im}\sum_{j=1}^N\int\phi\varphi_j\overline{\varphi}_{jx}\,dx\,d\tau\\
&\leq\sum_{j=1}^N\int\Phi |\psi_j|^2
-2t\mathop{\rm Im}\sum_{j=1}^N\int\phi\psi_j\overline{\psi}_{jx}-\eta
t^2.
\end{aligned}
\end{equation}
This inequality yields
\begin{align*}
\sum_{j=1}^N\int\Phi|\varphi_j(t)|^2
&\leq-\eta\Big(t+
{1\over\eta}\mathop{\rm Im}\sum_{j=1}^N\int\phi\psi_j
\overline{\psi}_{jx}\Big)^2\\
&\quad +{1\over\eta}\Big(\mathop{\rm Im}\sum_{j=1}^N\int\phi\psi_j
\overline{\psi}_{jx}\Big)^2
+\sum_{j=1}^N\int\Phi|\psi_j|^2.
\end{align*}
Noticing that
\begin{equation}\label{eq314}
\Big(\mathop{\rm Im}\sum_{j=1}^N\int\phi\psi_j
\overline{\psi}_{jx}\Big)^2\leq
2\sum_{j=1}^N\int|\phi\psi_j|^2\int|\psi_{jx}|^2
\end{equation}
and the fact of $\phi^2\leq 2\Phi$, we deduce that
\begin{equation}\label{eq315}
\sum_{j=1}^N\int\Phi|\varphi_j(t)|^2\leq
\Big({4\over\eta}\sum_{j=1}^N\|\psi_{jx}\|^2_{L^2}+1\Big)
\sum_{j=1}^N\int\Phi|\psi_j|^2,
\quad 0\leq t<T_0.
\end{equation}
Since $1\leq 2\Phi(x)$ for $|x|>1$, (\ref{eq310}) and
(\ref{eq315}) imply
\begin{align*}
\Big(\sum_{j=1}^N\|\varphi_j(t)\|^2_{L^2(|x|>1)}\Big)^2&\leq
\Big(2\sum_{j=1}^N\int\Phi|\varphi_j(t)|^2\Big)^2\\
&\leq
4\Big(\sum_{j=1}^N\int\Phi|\psi_j|^2\Big)^2\Big({4\over\eta}
\sum_{j=1}^N\|\psi_{jx}\|^2_{L^2}+1\Big)^2\\
&\leq a_0,\quad 0\leq t<T_0.
\end{align*}
 This and the continuity in
$L^2$ of $\varphi_j(t)$ yield
\begin{equation}\label{eq316}
\sum_{j=1}^N\|\varphi_j(T_0)\|^4_{L^2(|x|>1)}\leq a_0,
\end{equation}
which contradicts to (\ref{eq312}). So if the initial data
$\overrightarrow{\varphi}(0)=\overrightarrow{\psi}$ satisfies
(\ref{eq39}) and (\ref{eq310}), then $\varphi_j(t)$ satisfies
(\ref{eq33}) for all $t\geq 0$.

Therefore, since all the assumptions in Proposition \ref{pr33} hold
with $T=\infty$, $\varphi_j(t)$ satisfies (\ref{eq33}) with
$T_0=\infty$, which implies that
$\sum_{j=1}^N\int\Phi|\varphi_j(t)|^2$ goes to negative in finite
time. This is a contradiction. Hence if the initial data
$\overrightarrow{\varphi}(0)=\overrightarrow{\psi}$ satisfies
(\ref{eq39}) and (\ref{eq310}), then $\overrightarrow{\varphi}(t)$
must blow up in finite
time.

\noindent{\bf Step 2.} In this step, we prove the theorem for all
the initial data with negative energy. The main idea is to use the
scaling invariance of the \eqref{SE-j}. In the first place, for
$\varepsilon>0$, let $\varphi_{\varepsilon
j}(x,t)=\varepsilon^{-1/2}\varphi_j(x/\varepsilon,t/\varepsilon^2)$.
Put $\varphi_{\varepsilon j}(x,0)=\psi_{\varepsilon
j}(x)=\varepsilon^{-1/2}\psi_j(x/\varepsilon)$. Then
$\varphi_{\varepsilon j}$ is also a solution of \eqref{SE-j} with
initial data $\psi_{\varepsilon j}$ in
$C([0,+\infty),H^1(\mathbb{R}))$. Moreover,
$\varphi_{\varepsilon j}(t)$ satisfies
\begin{gather}\label{eq317}
\|\varphi_{\varepsilon j}(t)\|_{L^2}=\|\psi_{\varepsilon
j}\|_{L^2}=\|\psi_j\|_{L^2}, \quad t\geq 0; \\
\label{eq318}
E(\overrightarrow{\varphi}_\varepsilon
(t))=\varepsilon^{-2}E(\overrightarrow{\psi}),\quad t\geq 0.
\end{gather}
In the second place, we show that there exists an $\varepsilon>0$
such that
\begin{gather}\label{eq319}
\eta_\varepsilon=-2E(\overrightarrow{\psi}_\varepsilon)-4M(1+M)^2\sum_{j=1}^N\|\psi_{\varepsilon
j}\|^6_{L^2}-{M\over 2}\sum_{j=1}^N\|\psi_{\varepsilon
j}\|^2_{L^2}>0; \\
\label{eq320}
4\Big(\sum_{j=1}^N\int\Phi|\psi_{\varepsilon
j}|^2\Big)^2\Big({4\over\eta_\varepsilon}\sum_{j=1}^N\|\psi_{\varepsilon
jx}\|^2_{L^2}+1\Big)^2\leq a_0.
\end{gather}
Using (\ref{eq318}), (\ref{eq319}) follows by choosing
$\varepsilon>0$ such that
\begin{equation}\label{eq321}
\varepsilon^2<-2E_0\Big(4M(1+M)^2\sum_{j=1}^N\|\psi_
j\|^6_{L^2}+{M\over 2}\sum_{j=1}^N\|\psi_ j\|^2_{L^2}\Big)^{-1}.
\end{equation}
Now we have from (\ref{eq317}) and (\ref{eq318}) that for some
$\varepsilon_1>0$ and $0<\varepsilon<\varepsilon_1$,
$$
{4\over\eta}\sum_{j=1}^N\|\psi_{\varepsilon
jx}\|^2_{L^2}\leq C_0(\varepsilon_1),
$$
$C_0(\varepsilon_1)$
denotes positive constant $C_0$ depending on $\varepsilon_1$. On
the other hand, Lemma \ref{le32} implies that there exists an
$\varepsilon_2>0$ with $\varepsilon_2<\varepsilon_1$ and
\begin{equation}\label{eq322}
\sum_{j=1}^N\int\Phi|\psi_{\varepsilon j}|^2\leq
2\sum_{j=1}^N\|R\psi_{\varepsilon j}\|_{L^2}^2\leq {1\over
4}(C_0(\varepsilon_1)+1)^{-1}a_0^{1\over 2}
\end{equation}
for $0<\varepsilon<\varepsilon_2$, where $R$ is defined as in
Lemma \ref{le32}.

Thus if $0<\varepsilon<\varepsilon_2$ and satisfying (\ref{eq321}),
then $\overrightarrow{\varphi}_\varepsilon(0)=\overrightarrow{\psi}$
satisfies (\ref{eq319}) and (\ref{eq320}). Therefore the proof of
the theorem in the general case is reduced to  Step 1 when we
consider $\varphi_{\varepsilon j}(x,t)$ instead of $\varphi_j(x,t)$.
The
proof of Theorem \ref{th11} is complete.
\end{proof}

\subsection*{Acknowledgement}
The authors want to thank the anonymous referee for the helpful comments.


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\end{document}