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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 162, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/162\hfil Nonexistence of asymptotically free solutions]
{Nonexistence of asymptotically free solutions to nonlinear Schr\"odinger
systems}

\author[N. Hayashi, C. Li, P. I. Naumkin \hfil EJDE-2012/162\hfilneg]
{Nakao Hayashi, Chunhua Li, Pavel I. Naumkin}  % in alphabetical order

\address{Nakao Hayashi \newline
Department of Mathematics\\
Graduate School of Science, Osaka University, 
Osaka, Toyonaka, 560-0043, Japan}
\email{nhayashi@math.sci.osaka-u.ac.jp}

\address{Chunhua Li \newline
Department of Mathematics, Graduate School of Science, Yanbian University,
Yanji City, Jilin Province, 133002, China}
\email{sxlch@ybu.edu.cn}

\address{Pavel I. Naumkin \newline
Centro de Ciencias Matem\'aticas\\
UNAM Campus Morelia, AP 61-3 (Xangari)\\
Morelia CP 58089, Michoac\'an, Mexico}
\email{pavelni@matmor.unam.mx}

\thanks{Submitted November 16, 2011. Published September 21, 2012.}
\subjclass[2000]{35Q55}
\keywords{Dispersive nonlinear waves; asymptotically free solutions}

\begin{abstract}
 We consider the nonlinear Schr\"odinger systems
 \begin{gather*}
 -i\partial _tu_1+\frac{1}{2}\Delta u_1=F( u_1,u_2), \\
 i\partial _tu_2+\frac{1}{2}\Delta u_2=F( u_1,u_2)
 \end{gather*}
 in $n$ space dimensions, where $F$ is a  $p$-th order local or nonlocal
 nonlinearity smooth up to order $p$, with $1<p\leq 1+\frac{2}{n}$ for
 $n\geq 2$ and $1<p\leq 2$ for $n=1$. These systems are related to higher
 order nonlinear dispersive wave equations. We prove the non existence of
 asymptotically free solutions in the critical and sub-critical cases.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction} \label{S1}

We study the nonexistence of asymptotically free solutions for the nonlinear
dispersive wave equations
\begin{equation}
\begin{gathered}
( \partial _t^2+\frac{1}{4}\Delta ^2) u=\lambda |
\partial _tu| ^{p-1}\partial _tu,\quad ( t,x) \in {
\mathbb{R}\times\mathbb{R}}^n, \\
u(0,x)=u_{0}( x) ,\partial _tu(0,x)=u_1( x) ,\quad
x\in {\mathbb{R}}^n
\end{gathered}  \label{1.1}
\end{equation}
and
\begin{equation}
\begin{gathered}
( \partial _t^2+\frac{1}{4}\Delta ^2) v=\mu \nabla (
| \nabla v| ^{p-1}\nabla v) ,\quad ( t,x) \in \mathbb{R}\times \mathbb{R}^n, \\
v(0,x)=v_{0}( x) ,\partial _tv(0,x)=v_1( x) ,\quad x\in \mathbb{R}^n,
\end{gathered}  \label{1.1-2}
\end{equation}
where $\lambda ,\mu \in\mathbb{C}$, $1<p\leq 1+\frac{2}{n}$ for $n\geq 2$,
and $1<p\leq 2$ for $n=1$. When we consider the large time asymptotic
behavior of solutions for the above equations, it is known that the critical
power of the nonlinearity $p$ is $1+\frac{2}{n}$, so that $1<p<1+\frac{2}{n}$
is called the sub-critical one.

Related to \eqref{1.1} and \eqref{1.1-2}, the equations
\begin{equation}
\begin{gathered}
( \partial _t^2+( -\Delta ) ^{m}) u=\lambda (
-\Delta ) ^{\frac{m-1}{4}+\frac{m}{2}}( u^{p}) ,\quad
( t,x) \in \mathbb{R}\times\mathbb{R}^n, \\
u(0,x)=u_{0}( x) ,\quad \partial _tu(0,x)=u_1(
x) ,\quad  x\in {\mathbb{R}}^n
\end{gathered}
 \label{P1}
\end{equation}
and
\begin{equation}
\begin{gathered}
( \partial _t^2+( -\Delta ) ^{m}) u=\lambda (
-\Delta ) ^{\frac{m-1}{4}}P( \partial _tu,( -\Delta
) ^{\frac{m}{2}}u) ,\quad ( t,x) \in \mathbb{R}\times\mathbb{R}^n, \\
u(0,x)=u_{0}( x) ,\quad \partial _tu(0,x)=u_1(
x) ,\quad x\in {\mathbb{R}}^n,
\end{gathered}  \label{P2}
\end{equation}
were studied in \cite{Nakamura10}, with $m\geq 1$, and $P$  a homogeneous
polynomial of order $p$ in two variables. If $p$ is an integer and satisfies
\[
p>1+\frac{2}{n-1},
\]
with $n\geq 2$, it was shown in \cite{Nakamura10} that \eqref{P1} and
\eqref{P2} have a unique global solution for small regular data. The
number $1+\frac{2}{n-1}$ is the well-known critical exponent for the nonlinear wave
equation. However, taking into account the time decay rates of solutions to
the linear problem for \eqref{P1} or \eqref{P2} with $m\neq 1$, the critical
exponent $1+\frac{2}{n-1}$ should be replaced by $1+\frac{2}{n}$. Indeed, a
closely related problem to \eqref{P2} written as
\begin{equation}
\begin{gathered}
( \partial _t^2+\frac{1}{m^2}( -\partial _x^2)
^{m}) u=\lambda | \partial _tu| ^{p-1}\partial _tu,
\quad ( t,x) \in \mathbb{R}\times\mathbb{R}, \\
u(0,x)=u_{0}( x) ,\partial _tu(0,x)=u_1( x) ,\quad x\in {\mathbb{R}},
\end{gathered} \label{P3}
\end{equation}
with $0<m\leq 2$, $m\neq 1$, $\lambda \in\mathbb{C}$, and $p>3$ for
$0<m<1$, $p>2+m$ for $1<m\leq 2$ was studied in \cite{HNKobayashi11}, and the
existence of asymptotically free solutions was shown. Thus $p=3$ is a
critical exponent for the problem \eqref{P3} from the point of view of the
scattering problem. Other problems related  to \eqref{1.1-2} in one
dimension were studied in \cite{Yoshikawa05} (see also the literature
cited therein for the case of the initial-boundary value problems).
Therefore, we call $p=1+\frac{2}{n}$ the critical exponent and
$1<p<1+\frac{2}{n}$ sub-critical exponents for our problems \eqref{1.1}
and \eqref{1.1-2}.

Equation \eqref{1.1} may be transformed into a system of nonlinear
Schr\"odinger equations. In fact, let us define new dependent variables by
\[
u_1=\Big( i\partial _t+\frac{1}{2}\Delta \Big) u, \quad
u_2=\Big(-i\partial _t+\frac{1}{2}\Delta \Big) u.
\]
Now \eqref{1.1} becomes
\begin{equation}
\begin{pmatrix}
( -i\partial _t+\frac{1}{2}\Delta ) u_1 \\
( i\partial _t+\frac{1}{2}\Delta ) u_2
\end{pmatrix}
=\begin{pmatrix}
F( u_1,u_2) \\
F( u_1,u_2)
\end{pmatrix},  \label{1.2}
\end{equation}
where
\[
F( u_1,u_2) =-2^{-p}i\lambda | u_1-u_2|^{p-1}( u_1-u_2),
\]
since $u=-( -\Delta ) ^{-1}( u_1+u_2) $ and
$\partial _tu=\frac{1}{2i}( u_1-u_2) $. We write \eqref{1.2}
in the form
\begin{equation}
\mathbf{L}\mathbf{u}=\mathbf{F}(\mathbf{u}) ,  \label{1.3}
\end{equation}
where
\begin{gather*}
\mathbf{L}=\begin{pmatrix}
\overline{L} & 0 \\
0 & L\end{pmatrix}
=\begin{pmatrix}
-i\partial _t+\frac{1}{2}\Delta & 0 \\
0 & i\partial _t+\frac{1}{2}\Delta \end{pmatrix},
\\
\mathbf{u}=\begin{pmatrix}
u_1 \\
u_2\end{pmatrix}, \quad
\mathbf{a}=\begin{pmatrix}
1 \\
1\end{pmatrix}, \quad
\mathbf{b}=\begin{pmatrix}
1 \\
-1\end{pmatrix},
\\
\mathbf{F}( \mathbf{u}) =-i\mathbf{a}2^{-p}\lambda | (
\mathbf{b}\cdot \mathbf{u}) | ^{p-1}( \mathbf{b}\cdot\mathbf{ u}) .
\end{gather*}
Similarly, if we define
\[
v_1=| \nabla | ^{-1}( -i\partial _t-\frac{1}{2}\Delta) v, \quad
v_2=| \nabla | ^{-1}( i\partial _t-\frac{1}{2} \Delta ) v,
\]
where $| \nabla | =( -\Delta ) ^{1/2}$,
$| \nabla | ^{-1}=( -\Delta ) ^{-1/2}$ and
\[
G( v_1,v_2) =-\mu \frac{\nabla }{| \nabla | }
\Big( | \frac{\nabla }{| \nabla | }(
v_1+v_2) | ^{p-1}\frac{\nabla }{| \nabla | }
( v_1+v_2) \Big) ,
\]
then  \eqref{1.1-2} can be reduced to the following system of
nonlinear Schr\"odinger equations with nonlocal nonlinearities
\begin{equation}
\begin{pmatrix}
( -i\partial _t+\frac{1}{2}\Delta ) v_1 \\
( i\partial _t+\frac{1}{2}\Delta ) v_2
\end{pmatrix}
=\begin{pmatrix}
G( v_1,v_2) \\
G( v_1,v_2)
\end{pmatrix}.  \label{1.5}
\end{equation}
We write this equation in the form
\begin{equation}
\mathbf{Lv}=\mathbf{G}( \mathbf{v})  \label{1.6}
\end{equation}
with
\[
\mathbf{v}=\begin{pmatrix}
v_1 \\
v_2 \end{pmatrix},\quad
\mathbf{G}(\mathbf{v}) =-\mu \mathbf{a}
\frac{\nabla }{| \nabla | }\Big( | \frac{\nabla }{| \nabla
| }( \mathbf{a\cdot v}) | ^{p-1}\frac{\nabla }{|
\nabla | }( \mathbf{a\cdot v}) \Big) .
\]

Multiplying both sides of  \eqref{1.3} by 
$\begin{pmatrix}
-\overline{u_1} \\
\overline{u_2}
\end{pmatrix}$, taking the imaginary parts and integrating in space, we obtain
\begin{equation}
\frac{d}{dt}( \| u_1\| _{\mathbf{L}^2}^2+\|
u_2\| _{\mathbf{L}^2}^2) =2^{1-p}( {\operatorname{Re}}\lambda
) \| u_1-u_2\| _{\mathbf{L}^{p+1}}^{p+1}.  \label{1.4}
\end{equation}
Therefore there exist $C>0$, independent of $t >0$ such that
\begin{equation}
\| u( t) \| _{\mathbf{L}^2}\leq C  \label{1.4a}
\end{equation}
if ${Re}(\lambda )\leq 0$. The Strichartz estimate and \eqref{1.4a} imply
that there exists a unique global solution of \eqref{1.3} for $1<p<1+\frac{4
}{n}$ such that
\[
\mathbf{u}=\begin{pmatrix}
u_1 \\
u_2 \end{pmatrix}
\in \Big( (\mathbf{C}\cap \mathbf{L}^{\infty }) (
\mathbf{R;L}^2) \cap \mathbf{L}_{\rm loc}^{\beta }( \mathbb{R};
\mathbf{L}^{p+1}) \Big) ^2,
\]
where $\beta =\frac{4}{n}\frac{p+1}{p-1}$ (see Appendix \ref{S4}). Note that
the identity \eqref{1.4} can be written as
\[
\frac{d}{dt}( \| \partial _tu\| _{\mathbf{L}^2}^2+
\frac{1}{4}\| \Delta u\| _{\mathbf{L}^2}^2) =2( {
\operatorname{Re}}\lambda ) \| \partial _tu\| _{\mathbf{L}
^{p+1}}^{p+1}.
\]
In the same manner, multiplying both sides of \eqref{1.6} by
$\begin{pmatrix}
-\overline{v_1} \\
\overline{v_2}
\end{pmatrix}$, taking the imaginary parts and integrating in space, we obtain
\begin{align*}
&\frac{d}{dt}( \| v_1\| _{\mathbf{L}^2}^2+\|
v_2\| _{\mathbf{L}^2}^2) \\
&= 2{\operatorname{Re}}\Big( i\mu \int_{\mathbb{R}^n}| \frac{\nabla }{|
\nabla | }( v_1+v_2) | ^{p-1}\Big( \frac{\nabla
}{| \nabla | }( v_1+v_2) \big) \Big(\frac{
\nabla }{| \nabla | }( \overline{v_1}-\overline{v_2}
) \Big) \,dx\Big) ,
\end{align*}
from which we obtain the estimate
\begin{equation}
\frac{d}{dt}\big( \| v_1\| _{\mathbf{L}^2}^2+\|
v_2\| _{\mathbf{L}^2}^2\Big)
\leq 2| \mu | \| \frac{\nabla }{| \nabla | }( v_1+v_2) \| _{
\mathbf{L}^{p+1}}^{p}\| \frac{\nabla }{| \nabla | }(
v_1-v_2) \| _{\mathbf{L}^{p+1}}.  \label{1.7}
\end{equation}
Estimate \eqref{1.7} is not sufficient to ensure the  existence of global
solutions to \eqref{1.6}. We again multiply both sides of \eqref{1.6} by
$\begin{pmatrix}
-\partial _t\overline{v_1} \\
\partial _t\overline{v_2}
\end{pmatrix}$, take the real parts and integrate in space to obtain
\begin{equation}
\frac{d}{dt}\Big( \| \nabla v_1\| _{\mathbf{L}
^2}^2+\| \nabla v_2\| _{\mathbf{L}^2}^2\Big)
 =-\frac{ 4\mu }{p+1}\frac{d}{dt}\| \frac{\nabla }{| \nabla | }(
v_1+v_2) \| _{\mathbf{L}^{p+1}}^{p+1}  \label{1.8}
\end{equation}
for $\mu \in \mathbb{R}$. This identity is equivalent to
\[
\frac{d}{dt}\Big( \| \partial _tv\| _{\mathbf{L}^2}^2+
\frac{1}{4}\| \Delta v\| _{\mathbf{L}^2}^2+\frac{2\mu }{p+1}
\| \nabla v\| _{\mathbf{L}^{p+1}}^{p+1}\Big) =0.
\]
If we assume that $\mu \geq 0$, then \eqref{1.8} yields a-priori estimates
for $\| \nabla v_1\| _{\mathbf{L}^2}^2,\| \nabla
v_2\| _{\mathbf{L}^2}^2$, and $\| \frac{\nabla }{|
\nabla | }( v_1+v_2) \| _{\mathbf{L}^{p+1}}$.
Applying the Sobolev imbedding theorem to \eqref{1.7} we obtain
\begin{align*}
\frac{d}{dt}( \| v_1\| _{\mathbf{L}^2}^2+\|
v_2\| _{\mathbf{L}^2}^2)
&\leq C\| \frac{\nabla }{| \nabla | }( v_1-v_2) \| _{\mathbf{L}^{p+1}}
\\
&\leq  C\| \frac{\nabla }{| \nabla | }(
v_1-v_2) \| _{\mathbf{L}^2}^{1-\frac{n}{2}( \frac{p-1
}{p+1}) }\| \nabla ( v_1-v_2) \| _{\mathbf{L}
^2}^{\frac{n}{2}( \frac{p-1}{p+1}) } \\
&\leq  C\| v_1\| _{\mathbf{L}^2}^{1-\frac{n}{2}( \frac{
p-1}{p+1}) }+C\| v_2\| _{\mathbf{L}^2}^{1-\frac{n}{2}
( \frac{p-1}{p+1}) }.
\end{align*}
Therefore we have the estimate
\begin{equation}
( \| v_1\| _{\mathbf{L}^2}+\| v_2\| _{
\mathbf{L}^2}) ^{1+\frac{n}{2}( \frac{p-1}{p+1}) }\leq
C+Ct.  \label{1.4-2}
\end{equation}
Thus by the method in \cite{TS87}, Equation \eqref{1.6} has a unique global solution
for $\mu \geq 0$, $1<p<1+\frac{4}{n}$ such that
\[
\mathbf{v}=
\begin{pmatrix}
v_1 \\
v_2\end{pmatrix}
\in \Big( (\mathbf{C}\cap \mathbf{L}^{\infty }) (
\mathbf{R;L}^2) \cap \mathbf{L}_{\rm loc}^{\beta }( \mathbb{R};
\mathbf{L}^{p+1}) \Big) ^2,
\]
where $\beta =\frac{4}{n}\frac{p+1}{p-1}$ (see Appendix \ref{S4}). However,
as far as we know, the large time asymptotic behavior of such solutions is
not well established.

We denote the weighted Sobolev space by
\[
\mathbf{H}^{m,s}=\Big\{ f=( f_1,f_2) \in \mathbf{L}
^2\times \mathbf{L}^2;\| f\| _{\mathbf{H}^{m,s}}=
\sum_{j=1}^2\| f_j\| _{\mathbf{H}^{m,s}}<\infty \Big\} ,
\]
where
\[
\| f\| _{\mathbf{H}^{m,s}}=\| ( 1-\Delta ) ^{m/2}( 1+| x| ^2) ^{s/2}f\| _{
\mathbf{L}^2}.
\]
We write $\mathbf{H}^{m}=\mathbf{H}^{m,0}$ for simplicity. As usual, let the
Fourier transform be defined by
\[
\mathcal{F}\phi =\hat{\phi}( \xi ) =\frac{1}{( 2\pi )
^{n/2}}\int_{\mathbb{R}^n}e^{-i(x\cdot \xi )}\phi ( x) dx
\]
and the inverse Fourier transform be given by
\[
\mathcal{F}^{-1}\phi =\frac{1}{( 2\pi ) ^{n/2}}
\int_{\mathbb{R}^n}e^{i(x\cdot \xi )}\phi ( \xi ) d\xi .
\]
Denote by $\mathcal{U}( t) =\mathcal{F}^{-1}e^{-\frac{it}{2}
| \xi | ^2}\mathcal{F}$ the free Schr\"odinger evolution
group.

In what follows, we assume that \eqref{1.3} or \eqref{1.6} has a unique
global solution. To solve the usual scattering problem we need to
find a solution of \eqref{1.3} or \eqref{1.6} in a neighborhood of a free
solution $\mathbf{w}=\begin{pmatrix}
w_1 \\
w_2
\end{pmatrix}$ of the linear equation
\begin{equation}
\mathbf{Lw}=0  \label{1.9}
\end{equation}
with some initial data $\mathbf{w}( 0) =\mathbf{w}_{0}=
\begin{pmatrix}
w_{1,0} \\
w_{2,0}
\end{pmatrix}$.

The purpose of this paper is to show that it is impossible to find a
solution to \eqref{1.3} or \eqref{1.6} in any neighborhood of any free
solution $\mathbf{w.}$ We use the functional used in \cite{BA84} for the
Schr\"odinger equations which is a version of the one used for the
nonlinear Klein-Gordon equation in \cite{GL73} and \cite{MA76}.

Our main results are the following.

\begin{theorem}\label{Theorem 1}
Let ${\operatorname{Im}}\lambda \neq 0$, $\beta =\frac{4}{n}\frac{
p+1}{p-1}$, and let
\[
\mathbf{u}\in \Big( (\mathbf{C}\cap \mathbf{L}^{\infty })
( \mathbf{R;L}^2) \cap \mathbf{L}_{\rm loc}^{\beta }( \mathbf{R
};\mathbf{L}^{p+1}) \Big) ^2
\]
be a solution of \eqref{1.3} with $1<p\leq 1+\frac{2}{n}$ for $n\geq 2$,
$1<p\leq 2$ for $n=1$. Then, there does not exist any free solution
$\mathbf{w}$ of \eqref{1.9} with the initial data $\mathbf{w}_{0}\neq 0$
such that $\mathbf{w}_{0}\in ( \mathbf{H}^{0,1}\cap \mathbf{L}^{1}) ^2$
and
\[
\lim_{t\to \infty }\| \mathbf{u}( t) -\mathbf{w}
( t) \| _{\mathbf{L}^2}=0.
\]
\end{theorem}

For the case of the system \eqref{1.6} we have

\begin{theorem}\label{Theorem 2}
Let ${\operatorname{Re}}\mu \neq 0,\beta =\frac{4}{n}\frac{p+1}{p-1}$ and let
\[
\mathbf{v}\in \Big( (\mathbf{C}\cap \mathbf{L}^{\infty })
( \mathbf{R;L}^2) \cap \mathbf{L}_{\rm loc}^{\beta }( \mathbf{R
};\mathbf{L}^{p+1}) \Big) ^2
\]
be a solution of \eqref{1.6} with $1<p\leq 1+\frac{2}{n}$ for $n\geq 2$,
$1<p\leq 2$ for $n=1$. Then, there does not exist any free solution
$\mathbf{w}$ of \eqref{1.9} with the initial data $\mathbf{w}_{0}\neq 0$
such that $\mathbf{w}_{0}\in ( \mathbf{H}^{0,1}\cap \mathbf{L}^{1}) ^2$
and
\[
\lim_{t\to \infty }\| \mathbf{v}( t) -\mathbf{w}( t) \| _{\mathbf{L}^2}=0.
\]
\end{theorem}

For  \eqref{1.1} and \eqref{1.1-2}, we define free solution to
be a solution $u_{+}( t) $ to the linear dispersive equation
\begin{equation}
\Big( \partial _t^2+\frac{1}{4}\Delta ^2\Big) u_{+}=0  \label{1.10}
\end{equation}
with initial data $u_{+}( 0) =u_{1+}$,
 $\partial _tu_{+}(0) =u_{2+}$. As a consequence of the above results we have the
corollaries for  \eqref{1.1} and \eqref{1.1-2}.

\begin{corollary} \label{Corollary 1}
Let $\operatorname{Im}\lambda \neq 0$, 
$\beta =\frac{4}{n}\frac{p+1}{p-1}$ and let $u$ be a solution
 of \eqref{1.1} with $1<p\leq 1+\frac{2}{n}$ for $n\geq 2$, $1<p\leq 2$
 for $n=1$ such that
\[
\Delta u,\partial _tu\in (\mathbf{C}\cap \mathbf{L}^{\infty })
( \mathbf{R;L}^2) \cap \mathbf{L}_{\rm loc}^{\beta }( \mathbf{R
};\mathbf{L}^{p+1}) .
\]
Then, there does not exist any free solution $u_{+}( t) $ of
\eqref{1.10} with the initial data $( u_{1+},u_{2+}) \neq 0$ such that
$\Delta u_{1+},u_{2+}\in \mathbf{H}^{0,1}\cap \mathbf{L}^{1}$ and
\[
\lim_{t\to \infty }( \| \partial _t( u(
t) -u_{+}( t) ) \| _{\mathbf{L}^2}+\|
\Delta ( u( t) -u_{+}( t) ) \| _{
\mathbf{L}^2}) =0.
\]
\end{corollary}

\begin{corollary}\label{Corollary 2}
Let $\operatorname{Re}\mu \neq 0$, 
$\beta =\frac{4}{n}\frac{p+1}{p-1}$ and let $v$ be a solution of \eqref{1.1-2}
 with $1<p\leq 1+\frac{2}{n}$
for $n\geq 2$, $1<p\leq 2$ for $n=1$ such that
\[
| \nabla | v,\, | \nabla | ^{-1}\partial _tv\in (
\mathbf{C}\cap \mathbf{L}^{\infty }) ( \mathbf{R;L}^2)
\cap \mathbf{L}_{\rm loc}^{\beta }( \mathbb{R};\mathbf{L}^{p+1}) .
\]
Then, there does not exist any free solution $u_{+}( t) $ of
\eqref{1.10} with the initial data $( u_{1+},u_{2+}) \neq 0$ such that
$| \nabla | u_{1+}$, $| \nabla | ^{-1}u_{2+}\in
\mathbf{H}^{0,1}\cap \mathbf{L}^{1}$ and
\[
\lim_{t\to \infty }\Big( \| | \nabla |
^{-1}\partial _t( v( t) -u_{+}( t) )
\| _{\mathbf{L}^2}+\| \nabla ( v( t)
-u_{+}( t) ) \| _{\mathbf{L}^2}\Big) =0.
\]
\end{corollary}

To prove our results using the methods in \cite{GL73} and \cite
{MA76}, we need a-priori lower bounds for the solutions to the linear
problem. Our main point in this paper is to prove the lower bound of time
decay estimates of solutions to the linear problem which is a main tool on
the proof of \cite{GL73} and \cite{MA76}. To get these estimates in the case
of nonlinear Klein-Gordon equations, the finite propagation property of
solutions was used in \cite{GL73} and \cite{MA76}. However, the equations
considered in this paper do not have this property, so instead we study the
large time asymptotic behavior for a linear combination of two types of free
Schr\"odinger evolution groups to get the lower bound of solutions. We
note that the lower bound of time decay estimates of solutions was shown in
\cite{BA84} for a single free Schr\"odinger evolution group.

\section{A-priori estimates of solutions to the linear problem
from below} \label{S2}

In this section we prove the estimates 
$\| \mathbf{a\cdot w}\| _{\mathbf{L}^{p+1}}\geq Ct^{-\frac{n}{2}\frac{p-1}{p+1}}$ and
 $\| \mathbf{b\cdot w}\| _{\mathbf{L}^{p+1}}\geq Ct^{-\frac{n}{2}\frac{p-1}{p+1}}$
for the solution $\mathbf{w}( t) =\begin{pmatrix}
w_1( t) \\
w_2( t)
\end{pmatrix}$ of the linear problem
\[
\mathbf{Lw}=0
\]
with the initial data
\[
\mathbf{w}( 0) =\mathbf{w}_{0}
=\begin{pmatrix}
w_{1,0} \\
w_{2,0}
\end{pmatrix}.
\]
Note that
\[
\mathbf{w}( t) =
\begin{pmatrix}
\mathcal{U}( -t) w_{1,0} \\
\mathcal{U}( t) w_{2,0}
\end{pmatrix}.
\]

Since $\mathbf{a\cdot w}=w_1+w_2$ and $\mathbf{b\cdot w}=w_1-w_2$ do
not have the finite propagation property which was used in papers \cite{GL73}
and \cite{MA76}, we need to find the large time asymptotic behavior for
$\mathcal{U}( -t) \phi \pm \mathcal{U}( t) \psi $.
Denote $M( t) =e^{\frac{i}{2t}| x| ^2}$ and 
$D(t) \phi =\frac{1}{( it) ^{n/2}}\phi ( \frac{x}{t}) $. 
We only consider the case $t\geq 0$, since the case $t \leq 0$
can be treated in the same way.

\begin{lemma} \label{Lemma 1} 
Let $0\leq \gamma \leq 1$. For any $\phi \in \mathbf{H} ^{2\gamma }$
\[
\| \mathcal{U}( t) \phi -M( t) D( t)
\mathcal{F}\phi \| _{\mathbf{L}^2}\leq Ct^{-\gamma }\| \widehat{
\phi }\| _{\mathbf{H}^{2\gamma }}
\]
for $t>0$.
\end{lemma}

\begin{proof}
By the identity $\mathcal{U}( t) =M( t) D(t) \mathcal{F}M( t) $ we find
\[
\mathcal{U}( t) \phi =M( t) D( t) \mathcal{F}M( t) \phi
=M( t) D( t) \mathcal{F}\phi +M( t) D( t) \mathcal{F}( M( t)-1) \phi .
\]
The $\mathbf{L}^2$-norm of the last term in the right-hand side of the
above identity is estimated by
\begin{align*}
\| M( t) D( t) \mathcal{F}( M( t)
-1) \phi \| _{\mathbf{L}^2}
&= \| ( M( t) -1) \phi \| _{\mathbf{L}^2} \\
&\leq Ct^{-\gamma }\| \phi \| _{\mathbf{H}^{0,2\gamma
}}=Ct^{-\gamma }\| \widehat{\phi }\| _{\mathbf{H}^{2\gamma }}.
\end{align*}
This proves the lemma.
\end{proof}

In the next lemma we find a lower bound of the norm 
$\| \mathcal{U} ( -t) \phi \pm \mathcal{U}( t) \psi \| _{\mathbf{
L}^{p+1}}$.

\begin{lemma} \label{Lemma 2} For 
$\phi ,\psi \in \mathbf{H}^{0,1}\cap \mathbf{L}^{1}$,
\begin{align*}
&\| \mathcal{U}( -t) \phi \pm \mathcal{U}( t)
\psi \| _{\mathbf{L}^r} \\
&\geq \frac{1}{2}( 2kt) ^{\frac{n}{2}( \frac{2}{r}-1)
}\Big( \| \widehat{\phi }\| _{\mathbf{L}^2( | \xi
| \leq k) }+\| \widehat{\psi }\| _{\mathbf{L}
^2( | \xi | \leq k) }\Big)  \\
&\quad -C( k) ( kt) ^{\frac{n}{2}( \frac{2}{r}
-1) }t^{-\alpha/2}\Big( \| \widehat{\phi }\| _{
\mathbf{H}^{1}}+\| \widehat{\psi }\| _{\mathbf{H}^{1}}+\|
\widehat{\phi }\| _{\mathbf{L}^{\infty }}+\| \widehat{\psi }
\| _{\mathbf{L}^{\infty }}\Big)
\end{align*}
for all $t>0$ and $k>0$, where $2\leq r\leq \infty ,\alpha <\frac{1}{2}$ for
$n=1$ and $\alpha =\frac{1}{2}$ for $n\geq 2$, and $C( k) $ is a
positive constant depending on $k$.
\end{lemma}

\begin{proof}
By H\"older's inequality
\begin{equation}
\begin{split}
&\| \mathcal{U}( -t) \phi \pm \mathcal{U}( t)
\psi \| _{\mathbf{L}^2( | x| \leq kt) }
\\
&\leq \| \mathcal{U}( -t) \phi \pm \mathcal{U}(
t) \psi \| _{\mathbf{L}^r( | x| \leq kt)
}\Big( \int_{| x| \leq kt}dx\Big) ^{\frac{r-2}{2r}}   \\
&= ( 2kt) ^{\frac{n}{2}( 1-\frac{2}{r}) }\|
\mathcal{U}( -t) \phi \pm \mathcal{U}( t) \psi
\| _{\mathbf{L}^r( | x| \leq kt) }   \\
&\leq ( 2kt) ^{\frac{n}{2}( 1-\frac{2}{r}) }\|
\mathcal{U}( -t) \phi \pm \mathcal{U}( t) \psi
\| _{\mathbf{L}^r}.
\end{split}\label{2.1}
\end{equation}
Hence in order to get the desired estimate from below we need to find a
lower bound for the norm $\| \mathcal{U}( -t) \phi \pm
\mathcal{U}( t) \psi \| _{\mathbf{L}^2( |
x| \leq kt) }$. By Lemma \ref{Lemma 1} with $\gamma =\frac{1}{2}$
we find
\begin{align*}
&\| \mathcal{U}( -t) \phi \pm \mathcal{U}( t)
\psi -( M( -t) D( -t) \widehat{\phi }\pm M(
t) D( t) \widehat{\psi }) \| _{\mathbf{L}
^2( | x| \leq kt) } \\
&\leq \| \mathcal{U}( -t) \phi \pm \mathcal{U}(
t) \psi -( M( -t) D( -t) \widehat{\phi }\pm
M( t) D( t) \widehat{\psi }) \| _{\mathbf{L
}^2} \\
&\leq 2| t| ^{-1/2}( \| \widehat{\phi }
\| _{\mathbf{H}^{1}}+\| \widehat{\psi }\| _{\mathbf{H}
^{1}}) .
\end{align*}
Therefore changing the variable of integration by $\xi =\frac{x}{t}$, we obtain
\begin{equation}
\begin{aligned}
&\| \mathcal{U}( -t) \phi \pm \mathcal{U}( t)
\psi \| _{\mathbf{L}^2( | x| \leq kt) }
\\
&\geq \| M( -t) D( -t) \widehat{\phi }\pm
M( t) D( t) \widehat{\psi }\| _{\mathbf{L}
^2( | x| \leq kt) }
-Ct^{-1/2}( \| \widehat{\phi }\| _{\mathbf{H}
^{1}}+\| \widehat{\psi }\| _{\mathbf{H}^{1}})   \\
&\geq \| e^{-\frac{it}{2}| \xi | ^2}( -i) ^n
\widehat{\phi }( -\xi ) \pm e^{\frac{it}{2}| \xi |
^2}\widehat{\psi }( \xi ) \| _{\mathbf{L}^2(
| \xi | \leq k) }
-Ct^{-1/2}( \| \widehat{\phi }\| _{\mathbf{H}
^{1}}+\| \widehat{\psi }\| _{\mathbf{H}^{1}}) .
\end{aligned}  \label{2.2}
\end{equation}
By a direct computation we have
\begin{equation}
\begin{aligned}
&\| e^{-\frac{it}{2}| \xi | ^2}( -i) ^n
\widehat{\phi }( -\xi ) \pm e^{\frac{it}{2}| \xi |
^2}\widehat{\psi }( \xi ) \| _{\mathbf{L}^2(
| \xi | \leq k) }^2   \\
&= \| \widehat{\phi }\| _{\mathbf{L}^2( | \xi |
\leq k) }^2+\| \widehat{\psi }\| _{\mathbf{L}^2(
| \xi | \leq k) }^2\pm 2\operatorname{Re}\int_{| \xi |
\leq k}( -i) ^n\widehat{\phi }( -\xi ) \overline{
\widehat{\psi }( \xi ) }e^{-it| \xi | ^2}d\xi .
\end{aligned}\label{2.3}
\end{equation}
Integration by parts and using the identity
\[
e^{-it| \xi | ^2}=\frac{1}{n-2it| \xi | ^2}\nabla
\cdot \xi e^{-it| \xi | ^2}
\]
yields
\begin{align*}
\int_{| \xi | \leq k}F( \xi ) e^{-it| \xi| ^2}d\xi 
&=\int_{| \xi | \leq k}F( \xi ) \frac{
1}{n-2it| \xi | ^2}\nabla \cdot \xi e^{-it| \xi |
^2}d\xi \\
&= \int_{| \xi | \leq k}\nabla \cdot \Big( \frac{\xi F( \xi
) }{n-2it| \xi | ^2}e^{-it| \xi | ^2}\Big) d\xi\\
&\quad -\int_{| \xi | \leq k}e^{-it| \xi | ^2}\xi \cdot
\nabla \frac{F( \xi ) }{n-2it| \xi | ^2}d\xi ,
\end{align*}
for any $F\in \mathbf{L}^{\infty }$ with $\nabla F\in \mathbf{L}^{1}$.
Therefore,
\[
\big| \int_{| \xi | \leq k}\nabla \cdot \Big( \frac{\xi F(
\xi ) }{n-2it| \xi | ^2}e^{-it| \xi |
^2}\Big) d\xi \big| \leq C( k) t^{-1/2}\|
F\| _{\mathbf{L}^{\infty }}
\]
and by a direct calculation
\[
\xi \cdot \nabla \frac{F( \xi ) }{n-2it| \xi | ^2}=
\frac{4it| \xi | ^2F( \xi ) }{( n-2it| \xi
| ^2) ^2}+\frac{\xi \cdot \nabla F( \xi ) }{
n-2it| \xi | ^2}.
\]
Hence
\begin{align*}
&| \int_{| \xi | \leq k}F( \xi ) e^{-it|
\xi | ^2}d\xi | \\
&\leq \int_{| \xi | \leq k}\frac{ 2| F( \xi ) | +| \xi \cdot \nabla F( \xi
) | }{n+2t| \xi | ^2}d\xi +C( k) t^{-
\frac{1}{2}}\| F\| _{\mathbf{L}^{\infty }} \\
&\leq Ct^{-\alpha }\int_{| \xi | \leq k}( \frac{|
F( \xi ) | }{| \xi | ^{2\alpha }}+| \nabla
F( \xi ) | ) d\xi +C( k) t^{-\frac{1}{2}
}\| F\| _{\mathbf{L}^{\infty }} \\
&\leq Ct^{-\alpha }\| F\| _{\mathbf{L}^{\infty }}\int_{|
\xi | \leq k}| \xi | ^{-2\alpha }d\xi +Ct^{-\alpha }\|
\nabla F\| _{\mathbf{L}^{1}}+C( k) t^{-1/2}\|
F\| _{\mathbf{L}^{\infty }} \\
&\leq Ct^{-\alpha }( C( k) \| F\| _{\mathbf{L}
^{\infty }}+\| \nabla F\| _{\mathbf{L}^{1}}) ,
\end{align*}
where $\alpha <1/2$ for $n=1$ and $\alpha =1/2$ for $n\geq 2$.
Therefore taking $F( \xi ) =\widehat{\phi }( -\xi )
\overline{\widehat{\psi }( \xi ) }$ in the above estimate, we obtain
\begin{equation}
\begin{aligned}
&\big| \int_{| \xi | \leq k}\widehat{\phi }( -\xi )
\overline{\widehat{\psi }( \xi ) }e^{-it| \xi |
^2}d\xi \big|   \\
&\leq Ct^{-\alpha }\Big( ( k^{n-2\alpha }+1) \| \widehat{
\phi }( -\xi ) \overline{\widehat{\psi }( \xi ) }
\| _{\mathbf{L}^{\infty }}+\| \nabla ( \widehat{\phi }(
-\xi ) \overline{\widehat{\psi }( \xi ) }) \| _{
\mathbf{L}^{1}}\Big)   \\
&\leq C( k) t^{-\alpha }\Big( \| \widehat{\phi }\| _{
\mathbf{H}^{1}}+\| \widehat{\phi }\| _{\mathbf{L}^{\infty
}}\Big) \Big( \| \widehat{\psi }\| _{\mathbf{H}^{1}}+\|
\widehat{\psi }\| _{\mathbf{L}^{\infty }}\Big) .
\end{aligned}  \label{2.4}
\end{equation}
We apply \eqref{2.4} to \eqref{2.3} and use \eqref{2.2} to obtain
\begin{align*}
&\| \mathcal{U}( -t) \phi \pm \mathcal{U}( t)
\psi \| _{\mathbf{L}^2( | x| \leq kt) } \\
&\geq \Big( \| \widehat{\phi }\| _{\mathbf{L}^2( |
\xi | \leq k) }^2+\| \widehat{\psi }\| _{\mathbf{L}
^2( | \xi | \leq k) }^2\Big) ^{1/2}
-\Big( 2| \int_{| \xi | \leq k}\widehat{\phi }( -\xi
) \overline{\widehat{\psi }( \xi ) }e^{-it| \xi
| ^2}d\xi | \Big) ^{1/2}\\
&\quad -2| t| ^{-\frac{1}{2}}( \| \widehat{\phi }\| _{\mathbf{H}^{1}}
+\| \widehat{\psi }\| _{\mathbf{H}^{1}}) \\
&\geq \frac{1}{2}\| \widehat{\phi }\| _{\mathbf{L}^2(
| \xi | \leq k) }+\frac{1}{2}\| \widehat{\psi }
\| _{\mathbf{L}^2( | \xi | \leq k) } \\
&\quad -C( k) t^{-\alpha/2}\Big( \| \widehat{\phi }
\| _{\mathbf{H}^{1}}+\| \widehat{\psi }\| _{\mathbf{H}
^{1}}+\| \widehat{\phi }\| _{\mathbf{L}^{\infty }}+\|
\widehat{\psi }\| _{\mathbf{L}^{\infty }}\Big) .
\end{align*}
Finally by \eqref{2.1} we obtain
\begin{align*}
&\| \mathcal{U}( -t) \phi \pm \mathcal{U}( t)
\psi \| _{\mathbf{L}^r}\geq ( 2kt) ^{\frac{n}{2}(
\frac{2}{r}-1) }\| \mathcal{U}( -t) \phi \pm \mathcal{U
}( t) \psi \| _{\mathbf{L}^2( | x| \leq
kt) } \\
&\geq \frac{1}{2}( 2kt) ^{\frac{n}{2}( \frac{2}{r}-1)
}( \| \widehat{\phi }\| _{\mathbf{L}^2( | \xi
| \leq k) }+\| \widehat{\psi }\| _{\mathbf{L}
^2( | \xi | \leq k) }) \\
&\quad -C( k) ( kt) ^{\frac{n}{2}( \frac{2}{r}
-1) }t^{-\alpha/2}( \| \widehat{\phi }\| _{
\mathbf{H}^{1}}+\| \widehat{\psi }\| _{\mathbf{H}^{1}}+\|
\widehat{\phi }\| _{\mathbf{L}^{\infty }}+\| \widehat{\psi }
\| _{\mathbf{L}^{\infty }}),
\end{align*}
which proves Lemma \ref{Lemma 2}.
\end{proof}

\section{Proof of Theorem \ref{Theorem 1}} \label{S3}

By the contradiction, suppose that there
exists a free solution $\mathbf{w}$ of \eqref{1.9} defined by the initial
data such that $\mathbf{w}_{0}\neq 0$: $\mathbf{w}_{0}\in ( \mathbf{H}
^{0,1}\cap \mathbf{L}^{1}) ^2$ satisfying
\begin{equation}
\lim_{t\to \infty }\| \mathbf{u}( t) -\mathbf{w}
( t) \| _{\mathbf{L}^2}=0.  \label{3.0}
\end{equation}
Define the functional
\[
\mathbf{H}_{u}( t) =\operatorname{Re}\int_{\mathbb{R}^n}i\mathbf{w}
\cdot \overline{\mathbf{u}}dx=\operatorname{Re}\sum_{j=1}^2\int_{\mathbb{R}
^n}iw_j\overline{u}_jdx
\]
as in \cite{BA84}. By \eqref{1.9} and \eqref{1.2} we have
\begin{align*}
\frac{d}{dt}\mathbf{H}_{u}( t)
& =\operatorname{Re}\int_{\mathbb{R}^n}\Big(
\begin{pmatrix}
i\partial _t & 0 \\
0 & i\partial _t
\end{pmatrix}
\mathbf{w}\cdot \overline{\mathbf{u}}+\mathbf{w\cdot }\overline{
\begin{pmatrix}
-i\partial _t & 0 \\
0 & -i\partial _t
\end{pmatrix}
 \mathbf{u}}\Big) dx \\
&= \operatorname{Re}\int_{\mathbb{R}^n}\Big(
\begin{pmatrix}
\frac{1}{2}\Delta  & 0 \\
0 & -\frac{1}{2}\Delta
\end{pmatrix}
\mathbf{w}\cdot \overline{\mathbf{u}}+\mathbf{w\cdot }\overline{
\begin{pmatrix}
-i\partial _t & 0 \\
0 & -i\partial _t
\end{pmatrix}
\mathbf{u}}\Big) dx \\
&= \operatorname{Re}\int_{\mathbb{R}^n}\mathbf{w\cdot }\overline{
\begin{pmatrix}
-i\partial _t+\frac{1}{2}\Delta  & 0 \\
0 & -( i\partial _t+\frac{1}{2}\Delta )
\end{pmatrix}
\mathbf{u}}dx \\
&= \operatorname{Re}\int_{\mathbb{R}^n}\mathbf{w\cdot }\overline{
\begin{pmatrix}
-2^{-p}i\lambda | u_1-u_2| ^{p-1}( u_1-u_2)
\\
2^{-p}i\lambda | u_1-u_2| ^{p-1}( u_1-u_2)
\end{pmatrix}}dx.
\end{align*}
Letting
\[
W=w_1-w_2, \quad U=u_1-u_2,
\]
from the above identity we have
\begin{equation}
\begin{split}
&\frac{d}{dt}\mathbf{H}_{u}( t)\\
&=2^{-p}\operatorname{Re}\Big( i
\overline{\lambda }\int_{\mathbb{R}^n}| U| ^{p-1}\overline{U}
W\, dx\Big)    \\
&= 2^{-p}\operatorname{Re}\Big( i\overline{\lambda }\int_{\mathbb{R}^n}(
| U| ^{p-1}\overline{U}W-| W| ^{p+1}) dx\Big)
+2^{-p}( \operatorname{Im}\lambda ) \int_{\mathbb{R}^n}|
W| ^{p+1}dx.
\end{split} \label{3.1}
\end{equation}
Due to the inequality
\begin{align*}
\big| | a| ^{p-1}a-| b| ^{p-1}b\big|
&\leq C( | a| ^{p-1}+| b| ^{p-1}) |a-b|  \\
&\leq C( | a-b| ^{p-1}+| b| ^{p-1}) |a-b| ,
\end{align*}
where $a,b\in\mathbb{C}$ and the H\"older inequality we obtain
\begin{equation}
\begin{aligned}
&\Big| 2^{-p}\operatorname{Re}\Big( i\overline{\lambda }\int_{\mathbb{R}
^n}( | U| ^{p-1}\overline{U}W-| W|
^{p+1}) dx\Big) \Big|    \\
&= \Big| 2^{-p}\operatorname{Re}\Big( i\overline{\lambda }\int_{\mathbb{R}
^n}( | U| ^{p-1}\overline{U}-| W| ^{p-1}
\overline{W}) W\,dx\Big) \Big|    \\
&\leq C\int_{\mathbb{R}^n}( | U-W| ^{p}| W|
+| U-W| | W| ^{p}) dx   \\
&\leq C\| U-W\| _{\mathbf{L}^2}^{p}\| W\| _{\mathbf{
L}^{\frac{2}{2-p}}}+C\| U-W\| _{\mathbf{L}^2}\| W\|
_{\mathbf{L}^{2p}}^{p}   \\
&\leq C( \delta ) | t| ^{\frac{n}{2}( 1-p)
}\| U-W\| _{\mathbf{L}^2}( 1+\| U-W\| _{\mathbf{
L}^2}^{p-1}) ,
\end{aligned}\label{3.2}
\end{equation}
since $1<p\leq 1+\frac{2}{n}\leq 2$ for $n\geq 2$, $1\leq p\leq 2$ for $n=1$,
and $\| W\| _{\mathbf{L}^r}\leq C| t| ^{\frac{n}{2}
( \frac{2}{r}-1) }\| \mathbf{w}_{0}\| _{\mathbf{L}^{
\frac{r}{r-1}}}$ for $r\geq 2$, where $C( \delta ) $ is a
constant depends on $\delta \mathbf{=}\| \mathbf{w}_{0}\| _{
\mathbf{L}^{1}}+\| \mathbf{w}_{0}\| _{\mathbf{H}^{0,1}}$.

Since $\mathbf{w}_{0}\neq 0$, there exists a $k>0$ such that $\|
\widehat{w_{1,0}}\| _{\mathbf{L}^2( | \xi | \leq
k) }+\| \widehat{w_{2,0}}\| _{\mathbf{L}^2( |
\xi | \leq k) }>0$. We apply Lemma \ref{Lemma 2} with $r=p+1$ to
the difference $w_1( t) -w_2( t) =\mathcal{U}
( -t) w_{1,0}-\mathcal{U}( t) w_{2,0}$ to find
\begin{equation}
\begin{aligned}
&\| w_1( t) -w_2( t) \| _{\mathbf{L}^{p+1}}^{p+1}   \\
&\geq ( \frac{1}{2}) ^{p+1}( 2kt) ^{\frac{n}{2}(
1-p) }\Big( \| \widehat{w_{1,0}}\| _{\mathbf{L}^2(
| \xi | \leq k) }+\| \widehat{w_{2,0}}\| _{
\mathbf{L}^2( | \xi | \leq k) }\Big) ^{p+1}
\\
&\quad-C( k) ( kt) ^{\frac{n}{2}( 1-p)
}t^{-\alpha \frac{p+1}{2}}\Big( \| \mathbf{w}_{0}\| _{\mathbf{H}
^{0,1}}^{p+1}+\| \mathbf{w}_{0}\| _{\mathbf{L}^{1}}^{p+1}\Big)
 \\
&\geq C( k,\gamma ) t^{\frac{n}{2}( 1-p) }-C(
k,\delta ) t^{\frac{n}{2}( 1-p) -\alpha \frac{p+1}{2}}
\end{aligned} \label{3.3a}
\end{equation}
for all $t>0$, where $C( k,\gamma ) $ is a constant depending on
$k$ and $\mathbf{\gamma =}\| \widehat{w_{1,0}}\| _{\mathbf{L}
^2( | \xi | \leq k) }+\| \widehat{w_{2,0}}
\| _{\mathbf{L}^2( | \xi | \leq k) }$ and
$C( k,\delta ) $ is a constant depending on $k$ and $\delta $.
Integrating \eqref{3.1} in time, and using \eqref{3.2} and \eqref{3.3a}, we
obtain
\begin{align*}
| \mathbf{H}_{u}( 2T) -\mathbf{H}_{u}( T)| 
&\geq 2^{-p}| \operatorname{Im}\lambda | | \int_{T}^{2T}\int_{
\mathbb{R}^n}| W( t,x) | ^{p+1}dxdt| \\
&\quad -C\int_{T}^{2T}| t| ^{\frac{n}{2}( 1-p) }\|
U-W\| _{\mathbf{L}^2}^{p-1}( 1+\| U-W\| _{\mathbf{L}
^2}) dt \\
&\geq 2^{-p}| \operatorname{Im}\lambda | \int_{T}^{2T}\|
w_1( t) -w_2( t) \| _{\mathbf{L}
^{p+1}}^{p+1}dt \\
&\quad -C( \delta ) \int_{T}^{2T}| t| ^{\frac{n}{2}(
1-p) }\| \mathbf{u}( t) \mathbf{-w}( t)
\| _{\mathbf{L}^2}^{p-1}dt \\
&\geq 2^{-p}| \operatorname{Im}\lambda | \int_{T}^{2T}( C(
k,\gamma ) t^{\frac{n}{2}( 1-p) }-C( k,\delta )
t^{\frac{n}{2}( 1-p) -\alpha \frac{p+1}{2}}) dt \\
&\quad -C( \delta ) \int_{T}^{2T}t^{n( 1-p)/2}\| \mathbf{u}( t) \mathbf{-w}( t) \| _{
\mathbf{L}^2}^{p-1}dt.
\end{align*}
By  \eqref{3.0}, it follows that for any $\varepsilon $
satisfying $0<\varepsilon \leq 2^{-p-2}| \operatorname{Im}\lambda |
C( k,\gamma ) /C( \delta ) $, there exists a $T_1>0$
such that
\[
\| \mathbf{u}( t) \mathbf{-w}( t) \| _{
\mathbf{L}^2}<\varepsilon ^{\frac{1}{p-1}}
\]
for $t\geq T_1$. Let $T_2>0$ be such that
\[
C( k,\gamma ) t^{\frac{n}{2}( 1-p) }-C( k,\delta
) t^{\frac{n}{2}( 1-p) -\alpha \frac{p+1}{2}}\geq \frac{1}{2
}C( k,\gamma ) t^{\frac{n}{2}( 1-p) }
\]
for $t\geq T_2$. Hence
\begin{equation}
\begin{aligned}
| \mathbf{H}_{u}( 2T) -\mathbf{H}_{u}( T)|
&\geq ( 2^{-p-1}| \operatorname{Im}\lambda | C( k,\gamma
) -C( \delta ) \varepsilon ) \int_{T}^{2T}t^{\frac{n}{
2}( 1-p) }dt   \\
&\geq ( 2^{-p-1}| \operatorname{Im}\lambda | C( k,\gamma
) -C( \delta ) \varepsilon ) \int_{T}^{2T}t^{-1}dt
 \\
&\geq 2^{-p-2}| \operatorname{Im}\lambda | C( k,\gamma ) \log
2>0
\end{aligned}  \label{3.4}
\end{equation}
for $T\geq \max \{ T_1,T_2\} >0$. On the other hand, by the
definition of $\mathbf{H}_{u}( t) $ and \eqref{3.0} we find
\begin{equation}
\begin{aligned}
| \mathbf{H}_{u}( t) |
 &= | \operatorname{Re}\int_{
\mathbb{R}^n}( i\mathbf{w}\cdot ( \overline{\mathbf{u}}-
\overline{\mathbf{w}}) ) dx| \leq C\| \mathbf{w}
{}( t) \| _{\mathbf{L}^2}\| \mathbf{u}(
t) \mathbf{-w}( t) \| _{\mathbf{L}^2}   \\
&\leq C\| \mathbf{w}_{0}\| _{\mathbf{L}^2}\| \mathbf{u}
( t) \mathbf{-w}( t) \| _{\mathbf{L}
^2}\to 0
\end{aligned} \label{3.3}
\end{equation}
for $t\to \infty $. From \eqref{3.3} and \eqref{3.4} we have the
desired contradiction. This completes the proof.


\section{Proof of Theorem \ref{Theorem 2}}

As in the proof of Theorem \ref{Theorem 1}, suppose that there exists a free
solution $\mathbf{w}$ of \eqref{1.9} defined by the initial data such that
$\mathbf{w}_{0}\neq 0$, $\mathbf{w}_{0}\in ( \mathbf{H}^{0,1}\cap
\mathbf{L}^{1}) ^2$ satisfying
\begin{equation}
\lim_{t\to \infty }\| \mathbf{v}( t) -\mathbf{w}
( t) \| _{\mathbf{L}^2}=0.  \label{4.1}
\end{equation}
Define the functional
\[
\mathbf{G}_{v}( t) =\operatorname{Re}\int_{\mathbb{R}^n}( iw_1
\overline{v}_1-iw_2\overline{v}_2) dx
\]
and denote
\[
\Omega =\frac{\nabla }{| \nabla | }( w_1+w_2) ,\quad
V=\frac{\nabla }{| \nabla | }( v_1+v_2) .
\]
Then by \eqref{1.9} and \eqref{1.5} we obtain
\begin{equation}
\begin{split}
\frac{d}{dt}\mathbf{G}_{v}( t)
&=\operatorname{Re}\int_{\mathbb{R}^n}
\mathbf{w\cdot }\overline{\mathbf{Lv}}dx   \\
&= -\operatorname{Re}\Big( \overline{\mu }\int_{\mathbb{R}^n}( \mathbf{
a\cdot w}) \frac{\nabla }{| \nabla | }| V| ^{p-1}
\overline{V}dx\Big)
=\operatorname{Re}\Big( \overline{\mu }\int_{\mathbb{R}
^n}\Omega | V| ^{p-1}\overline{V}dx\Big)   \\
&= \operatorname{Re}\Big( \overline{\mu }\int_{\mathbb{R}^n}\Omega ( |
V| ^{p-1}\overline{V}-| \Omega | ^{p-1}\overline{\Omega }
) dx\Big) +( \operatorname{Re}\overline{\mu }) \int_{\mathbb{R}
^n}| \Omega | ^{p+1}dx.
\end{split} \label{3.5}
\end{equation}
As in \eqref{3.2} we find
\begin{equation}
\begin{split}
&\big| \operatorname{Re}\Big( \overline{\mu }\int_{\mathbb{R}^n}\Omega (
| V| ^{p-1}\overline{V}-| \Omega | ^{p-1}\overline{
\Omega }) dx\Big) \big|   \\
&\leq C\| V-\Omega \| _{\mathbf{L}^2}^{p}\| \Omega
\| _{\mathbf{L}^{\frac{2}{2-p}}}+C\| V-\Omega \| _{\mathbf{L
}^2}^{p-1}\| \Omega \| _{\mathbf{L}^{\frac{4}{3-p}}}^2
\\
&\leq C( \delta ) | t| ^{\frac{n}{2}( 1-p)
}\| V-\Omega \| _{\mathbf{L}^2}^{p-1}( 1+\| V-\Omega
\| _{\mathbf{L}^2}) .
\end{split} \label{3.6}
\end{equation}
Applying Lemma \ref{Lemma 2} to $\Omega =\frac{\nabla }{| \nabla
| }( w_1+w_2) $, we obtain
\begin{equation}
\| \Omega ( t) \| _{\mathbf{L}^{p+1}}^{p+1}\geq
C( k,\gamma ) t^{\frac{n}{2}( 1-p) }-C( k,\delta
) t^{\frac{n}{2}( 1-p) -\alpha \frac{p+1}{2}}  \label{3.6a}
\end{equation}
for all $t>0$, since the norm $\| \frac{\nabla }{| \nabla | }
\cdot \| _{\mathbf{L}^{p+1}}$ is equivalent to $\| \cdot \|
_{\mathbf{L}^{p+1}}$ (see \cite{Stein}). Integrating \eqref{3.5} in time,
and using \eqref{3.6} and \eqref{3.6a}, we obtain
\begin{align*}
&| \mathbf{G}_{v}( 2T) -\mathbf{G}_{v}( T)|\\
&\geq | \operatorname{Re}\mu | \int_{T}^{2T}\| \Omega (
t) \| _{\mathbf{L}^{p+1}}^{p+1}dt
  -C( \mathbf{\delta }) \int_{T}^{2T}| t| ^{\frac{n}{2}
( 1-p) }\| \mathbf{v}( t) \mathbf{-w}(
t) \| _{\mathbf{L}^2}^{p-1}dt \\
&\geq | \operatorname{Re}\mu | \int_{T}^{2T}\Big( C( k,\gamma
) t^{\frac{n}{2}( 1-p) }-C( k,\delta ) t^{\frac{n
}{2}( 1-p) -\alpha \frac{p+1}{2}}\Big) dt \\
&\quad -C( \delta ) \int_{T}^{2T}t^{\frac{n}{2}( 1-p)
}\| \mathbf{v}( t) \mathbf{-w}( t) \| _{
\mathbf{L}^2}^{p-1}dt.
\end{align*}
By   \eqref{4.1}, it follows that for any $\varepsilon $
satisfying $0<\varepsilon \leq 2^{-2}| \operatorname{Re}\mu | C(
k,\gamma ) /C( \delta ) $, there exists a $T_1>0$ such
that
\[
\| \mathbf{u}( t) \mathbf{-w}( t) \| _{\mathbf{L}^2}<\varepsilon ^{\frac{1}{p-1}}
\]
for $t\geq T_1$. Again, let $T_2>0$ such that
\[
C( k,\gamma ) t^{\frac{n}{2}( 1-p) }-C( k,\delta
) t^{\frac{n}{2}( 1-p) -\alpha \frac{p+1}{2}}\geq \frac{1}{2
}C( k,\gamma ) t^{\frac{n}{2}( 1-p) }
\]
for $t\geq T_2$. Therefore,
\begin{equation}
\begin{split}
| \mathbf{G}_{v}( 2T) -\mathbf{G}_{v}( T)|
&\geq ( 2^{-1}| \operatorname{Re}\mu | C( k,\gamma )
-C( \delta ) \varepsilon ) \int_{T}^{2T}t^{\frac{n}{2}
( 1-p) }dt   \\
&\geq  ( 2^{-1}| \operatorname{Re}\mu | C( k,\gamma )
-C( \delta ) \varepsilon ) \int_{T}^{2T}t^{-1}dt   \\
&\geq  2^{-2}| \operatorname{Re}\mu | C( k,\gamma ) \log 2>0
\end{split} \label{3.7a}
\end{equation}
for $T\geq \max \left\{ T_1,T_2\right\} >0$. On the other hand, by the
definition of $\mathbf{G}_{v}( t) $ and \eqref{4.1} we find
\begin{equation}
\begin{split}
| \mathbf{G}_{v}( t) |
& =\big| \operatorname{Re}\int_{
\mathbb{R}^n}( iw_1( \overline{v}_1-\overline{w}_1)
-iw_2( \overline{v}_2-\overline{w}_2) ) dx\big|\\
&\leq C\| \mathbf{w}{}( t) \| _{\mathbf{L}
^2}\| \mathbf{v}( t) -\mathbf{w}( t) \| _{\mathbf{L}^2}\\
&\leq C\| \mathbf{v}( t) \mathbf{-w}(t) \| _{\mathbf{L}^2}\to 0
\end{split}  \label{3.7}
\end{equation}
as $t\to \infty $. Therefore we have the desired contradiction by
\eqref{3.7a} and \eqref{3.7}. This completes the proof.


\section{Appendix} \label{S4}

In this section we prove the  existence of global solutions to the systems 
\eqref{1.3}) and \eqref{1.6}. We introduce the following space-time norm
\[
\| \phi \| _{\mathbf{L}^{q}( \mathbf{I};\mathbf{L}
^r) }=\| \| \phi ( t,x) \| _{\mathbf{L}
_x^r}\| _{\mathbf{L}_t^{q}( \mathbf{I}) },
\]
where $\mathbf{I}$ is a bounded or unbounded time interval.

To prove the local existence of $\mathbf{L}^2$-solutions, we write 
\eqref{1.3} as a system of integral equations
\begin{equation}
\begin{gathered}
u_1( t) =\overline{\mathcal{U}( t) }\phi
_1+i\int_{0}^{t}\overline{\mathcal{U}( t-\tau ) }F(u_1(\tau
),u_2(\tau ))d\tau , \\
u_2( t) =\mathcal{U}( t) \phi _2-i\int_{0}^{t}
\mathcal{U}( t-\tau ) F(u_1(\tau ),u_2(\tau ))d\tau ,
\end{gathered}  \label{NLSI}
\end{equation}
where $\mathcal{U}( t) $ is the free Schr\"odinger evolution
group. As in \cite{TS87}, we treat the problem in $\mathbf{L}^2$ space by
applying the results for a single nonlinear Schr\"odinger equation with
power nonlinearities.

Define the space
\[
\mathbf{X}( \mathbf{I}) =(\mathbf{C}\cap \mathbf{L}^{\infty
}) ( \mathbf{I;L}^2) \cap \mathbf{L}^{\beta }(
\mathbf{I};\mathbf{L}^{p+1})
\]
with the norm
\[
\| u\| _{\mathbf{X}( \mathbf{I})}
=\sum_{j=1}^2\Big( \| u_j\| _{\mathbf{L}^{\infty
}( \mathbf{I};\mathbf{L}^2) }+\| u_j\| _{
\mathbf{L}^{\beta }( \mathbf{I};\mathbf{L}^{p+1}) }\Big) ,
\]
on a time-interval $\mathbf{I}=[ -T,T] $, where
$\beta =\frac{4}{n }\frac{p+1}{p-1}$, $1<p<1+\frac{4}{n}$.
We now prove the following result.

\begin{theorem}\label{Theorem 4} 
For any $\rho >0$ there exists a $T( \rho ) >0$
such that for any initial data 
$\phi =( \phi _1,\phi _2) \in \mathbf{L}^2$ with the norm 
$\| \phi \| _{\mathbf{L}^2}\leq \rho $, the Cauchy problem for \eqref{1.3} 
has a unique solution $\mathbf{u}=( u_1,u_2) \in \mathbf{X}( \mathbf{I}) $ with
$\mathbf{I}=[ -T( \rho ) ,T( \rho )] $.
\end{theorem}

\begin{proof}
We denote the right-hand sides of \eqref{NLSI} by 
$\Phi _j( \mathbf{u}) $ for $j=1,2$. Applying the Strichartz inequality
\[
\| \int_{0}^{t}\mathcal{U}( t-\tau ) g( \tau )
d\tau \| _{\mathbf{L}_t^r( \mathbf{I};\mathbf{L}_x^{q}) }
\leq C\| g\| _{\mathbf{L}_t^{s}( \mathbf{I};\mathbf{L}_x^{l}) }
\]
for $2\leq r\leq \infty $, $1\leq s\leq 2$,
$\frac{1}{q}=\frac{1}{2}-\frac{2}{nr}$,
$\frac{1}{l}=\frac{1}{2}+\frac{2}{n}( 1-\frac{1}{s}) $
(see \cite{TH}), we estimate $\Phi _j( \mathbf{u}) $ via the
H\"older inequality in space and in time. We choose $r=\beta $, $q=p+1$,
$s=\frac{\beta }{\beta -1}$, $l=\frac{p+1}{p}$ and $\beta =\frac{4}{n}\frac{
p+1}{p-1}$, then
\begin{align*}
\| \Phi _j( \mathbf{u}) \| _{\mathbf{X}(\mathbf{I}) }
&\leq C\| \phi \| _{\mathbf{L}^2}+C\|
\mathbf{F}( \mathbf{u}) \| _{\mathbf{L}^{s}( \mathbf{I}
;\mathbf{L}^{\frac{p+1}{p}}) } \\
&\leq C\| \phi \| _{\mathbf{L}^2}+C\Big( \int_{I}\|
\mathbf{u}\| _{\mathbf{L}^{p+1}}^{ps}dt\Big) ^{1/s} \\
&\leq C\| \phi \| _{\mathbf{L}^2}+CT^{\frac{1}{s}-\frac{p}{
\beta }}\| \mathbf{u}\| _{\mathbf{X}( \mathbf{I})
}^{p}.
\end{align*}
Note that $\beta -ps>0$ since $p<1+\frac{4}{n}$. Similarly, we find the
estimate for the difference
\[
\| \Phi _j( \mathbf{u}) -\Phi _j( \mathbf{u}')
\| _{\mathbf{X}( \mathbf{I}) }\leq CT^{
\frac{1}{s}-\frac{p}{\beta }}( \| \mathbf{u}\| _{\mathbf{X}
( \mathbf{I}) }^{p-1}+\| \mathbf{u}^{\prime }\| _{
\mathbf{X}( \mathbf{I}) }^{p-1}) \| \mathbf{u}-\mathbf{
u}^{\prime }\| _{\mathbf{X}( \mathbf{I}) }.
\]
Therefore the conclusion of the theorem follows from the contraction mapping
principle if we take $T>0$ sufficiently small which depends only on the size
$\rho $ of the initial data.
\end{proof}

The  existence of global solutions for \eqref{1.3} follows from Theorem
 \ref{Theorem 4} and a-priori estimates \eqref{1.4a}. Similarly, a-priori
estimates \eqref{1.4-2} and Theorem \ref{Theorem 4} ensure the
existence of  global solutions to \eqref{1.6}.

\subsection*{Acknowledgements}
The work of P.I.N. is partially supported by CONACYT and PAPIIT. The authors
would like to thank an anonymous  referee for the useful suggestions and comments.

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\end{document}