\documentclass[reqno]{amsart} 

\AtBeginDocument{{\noindent\small 
{\em Electronic Journal of Differential Equations},
Vol. 2001(2001), No. 54, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2001 Southwest Texas State University.} 
\vspace{1cm}}

\begin{document} 

\title[\hfilneg EJDE--2001/54\hfil Asymptotic behaviour for Schr\"{o}dinger equations]
{Asymptotic behaviour for Schr\"{o}dinger equations with a quadratic
 nonlinearity in one-space dimension} 

\author[Nakao Hayashi \& Pavel I. Naumkin\hfil EJDE--2001/54\hfilneg]
{Nakao Hayashi \& Pavel I. Naumkin}

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

\address{Pavel I. Naumkin \hfill\break
Instituto de F\'{\i }sica y Matem\'{a}ticas\\
Universidad Michoacana, AP 2-82\\
Morelia, CP 58040, Michoac\'{a}n, Mexico}
\email{pavelni@zeus.ccu.umich.mx}

\date{}
\thanks{Submitted May 22, 2001. Published July 25, 2001.}
\subjclass[2000]{35Q55, 74G10, 74G25}
\keywords{Schr\"{o}dinger equation, large time behaviour, quadratic nonlinearity }


\begin{abstract}
 We consider the Cauchy problem for the Schr\"{o}dinger
 equation with a quadratic nonlinearity in one space dimension
 $$
 iu_{t}+\frac{1}{2}u_{xx}=t^{-\alpha}| u_x| ^2,\quad u(0,x) = u_0(x),
 $$
 where $\alpha \in (0,1)$. From the heuristic point of view,
 solutions to this problem should have a quasilinear character 
 when $\alpha \in (1/2,1)$. We show in this paper that the 
 solutions do not have a quasilinear character for all $\alpha \in (0,1)$ 
 due to the special structure of the nonlinear term. 
 We also prove that for $\alpha \in [1/2,1)$ if the initial data 
 $u_0\in H^{3,0}\cap H^{2,2}$ are small, then
 the solution has a slow time decay such as $t^{-\alpha /2}$.
 For $\alpha \in (0,1/2)$, if we assume that the initial data $u_0$ are
 analytic and small, then the same time decay occurs.
\end{abstract}

\maketitle

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}{Remark}[section]
\numberwithin{equation}{section}

\section{Introduction}

In this paper we consider the Schr\"{o}dinger equation, with a quadratic
derivative term,
\begin{equation}
\begin{gathered}
\mathcal{L}u=t^{-\alpha }| u_x| ^2, \quad t, x\in \mathbb{R}\\
u(0,x)=u_0(x) ,\quad x\in \mathbb{R},
\end{gathered} \label{1.1}
\end{equation}
where $\mathcal{L}=i\partial_{t}+\frac{1}{2}\partial_x^2$, and 
$\alpha \in (0,1)$. 
The Cauchy problem for Schr\"{o}dinger equations
with a cubic derivative term was studied in \cite{hny1}. 
There the authors considered
\begin{equation}
\begin{gathered}
\mathcal{L}u=t^{1-\delta }F( u,u_x) ,\quad t,x\in \mathbb{R} \\
u(0,x)=\epsilon u_0(x) ,\quad x\in \mathbb{R},
\end{gathered} \label{1.1a}
\end{equation}
where $0<\delta <1$, $\epsilon $ is a sufficiently small
constant, and the nonlinear interaction term $F$ consists of cubic
nonlinearities.
$$
F( u,u_x) =\lambda_1| u| ^2u+i\lambda
_2| u| ^2u_x+i\lambda_3u^2\bar{u}_x+\lambda_4| u_x| ^2u
+\lambda_5\bar{u}u_x^2+i\lambda_6| u_x| ^2u_x,
$$
where the coefficients $\lambda_1,\lambda_6\in \mathbb{R}$, $\lambda
_2,\lambda_3,\lambda_4,\lambda_5\in \mathbb{C}$, $\lambda
_2-\lambda_3\in \mathbb{R}$, $\lambda_4-\lambda_5\in \mathbb{R}$.
In \cite{hny1}, the authors found a time decay estimate for the solutions 
of this  problem,
\begin{equation}
\| u(t)\|_{\infty }\leq C| t| ^{-1/2}.  \label{1.1b}
\end{equation}
The same result is also true for the case $\delta >1$. From the heuristic
point of view problem (\ref{1.1}) corresponds to problem (\ref{1.1a}),
when $\delta =\alpha +\frac{1}{2}$. Therefore it is natural to make a
conjecture that the solutions of (\ref{1.1}) also have the decay property 
(\ref{1.1b}). However, as we will show in the present paper, due to the
special oscillating structure of the nonlinear term, for $\alpha
\in (0,1) $ the asymptotic behavior of solutions to (\ref{1.1}) do not obey
the estimate (\ref{1.1b}). Our result
stated below depends on the structure of nonlinearity which appears in
the identity
$$( \mathcal{FU}(-t)|u_x|^2) (t,\xi )
=(2\pi )^{1/2}\int e^{-it\xi \eta }(\overline{\mathcal{FU}%
(-t)u_x( t,\eta ) })({\mathcal{FU}}(-t)u_x)( t,\xi +\eta
) d\eta .  \label{1.1c}
$$
In the cases of $u_x^2$ and $\bar{u}_x^2$ we have
\begin{eqnarray*}
\lefteqn{( \mathcal{FU}(-t)u_x^2) (t,\xi ) } \\
&=&(2\pi )^{1/2}e^{\frac{i}{4}t\xi ^2}\int e^{-ity^2}(\mathcal{FU%
}(-t)u_x)( t,\frac{\xi }{2}-y) (\mathcal{FU}(-t)u_x)( t,\frac{\xi }{2}+y) dy
\end{eqnarray*}
and
\begin{eqnarray*}
\lefteqn{( \mathcal{FU}(-t)\bar{u}^2) (t,\xi ) }\\
&=&(2\pi )^{1/2}e^{\frac{3i}{4}t\xi ^2}\int e^{ity^2}(\overline{%
\mathcal{FU}(-t)u_x})( t,\frac{\xi }{2}-y) (\overline{\mathcal{%
FU}(-t)u_x})( t,\frac{\xi }{2}+y) dy,
\end{eqnarray*}
where $\mathcal{U}( t) $ is the linear Schr\"{o}dinger evolution
group
\begin{equation*}
\mathcal{U}( t) \phi =\frac{1}{\sqrt{2\pi it}}\int e^{\frac{i}{2t}%
( x-y) ^2}\phi ( y) dy=\mathcal{F}^{-1}e^{-\frac{it}{%
2}\xi ^2}\mathcal{F}\phi ,
\end{equation*}
$\mathcal{F}\phi \equiv \hat{\phi}=\frac{1}{\sqrt{2\pi }}\int e^{-ix\xi
}\phi (x) dx$ denotes the Fourier transform of the function $%
\phi $. The oscillating function $e^{\pm ity^2}$ yields an additional time
decay term through integration by parts. However, the oscillating function
$e^{\pm it\xi y}$ does not give an additional time decay uniformly with
respect to $\xi $. This is the main reason why we do not have estimate (\ref
{1.1b}) for solutions of (\ref{1.1}). In \cite{hn4} we proved (\ref{1.1b})
for solutions of the Cauchy problem
\begin{equation*}
\mathcal{L}u=\lambda ( \overline{u}_x) ^2+\mu u_x^2,\quad
\text{with }\lambda ,\mu \in \mathbb{C}.
\end{equation*}
However, the nonlinearity $| u_x| ^2$ was out of our
scope. In the present paper we intend to fill up this gap studying the case
of quadratic nonlinearity $t^{-\alpha }| u_x| ^2$. The
methods developed for the nonlinear Schr\"{o}dinger equations with quadratic
nonlinearities $u_x^2$, $| u_x| ^2$ and $\overline{u}%
_x^2$ can be applied also to the study of the large time asymptotic
behavior for other quadratic nonlinear equations, such as Benjamin-Ono and
Korteweg-de Vries equations (in paper \cite{hn12}, mBO equation was reduced
to the cubic nonlinear Schr\"{o}dinger equation). In paper \cite{c}, Cohn
used the method of normal forms of Shatah \cite{s} to study the nonlinear
Schr\"{o}dinger equations with quadratic nonlinearity $\overline{u}_x^2$
and showed that the solution exists on $[0,T)$ with $T$ bounded from below by
$C\varepsilon ^{-6}$, where $\varepsilon $ is the size of the data in some
Sobolev norm. In paper \cite{o} the nonlinearity $u_x^2$ was studied by
the Hopf-Cole transformation. The $L^2$-estimate of solutions
involving the operator $\mathcal{J}=x+it\partial_x$ plays a crucial role
in the large time asymptotic behavior of solutions. However the nonlinearity
$\mathcal{N}( u) $ under consideration does not posses a
self-conjugate structure $e^{i\omega }\mathcal{N}( u) =\mathcal{N}%
( e^{i\omega }u) $ for all $\omega \in \mathbb{R}$, therefore we
can not use the operator $\mathcal{J}=x+it\partial_x$ directly in (\ref
{1.1}). To overcome these obstacles we use the method developed in
\cite{hn5} and apply systematically the operator $\mathcal{I}%
=x\partial_x+2t\partial_{t}$.

We now state our strategy for the proof. If
we put $v=u_x$. Then the problem is written as
\begin{equation*}
\mathcal{L}v=t^{-\alpha }\partial_x| v| ^2,\quad t,x\in \mathbb{R}.
\end{equation*}
By the identity
\begin{equation*}
\partial_x\mathcal{J}| v| ^2=\partial_x( \overline{v}%
\mathcal{J}v+it\overline{v}_xv) =\overline{v}\mathcal{J}\partial
_xv+2\overline{v}_x\mathcal{J}v-v\overline{\mathcal{J}\partial_xv}
\end{equation*}
we have
\begin{equation*}
\mathcal{LJ}v=t^{-\alpha }\mathcal{J}\partial_x| v|
^2=t^{-\alpha }( -| v| ^2+\overline{v}\mathcal{J}%
\partial_xv+2\overline{v}_x\mathcal{J}v-v\overline{\mathcal{J}\partial
_xv}) .
\end{equation*}
Therefore, the operator $\mathcal{J}$ acts on this problem also. Thus
global existence in time of small solutions to the problem can be proved for
$\alpha \in (1/2,1) $ and the derivative $u_x$ should
have the same asymptotic behaviour as the solutions to the corresponding
linear problem (along with time-decay estimate (\ref{1.1b})). Combining this
fact and the identity (\ref{1.1c}) we prove the time decay of solutions.
Roughly speaking, we show there exists a constant $c$ and a positive constant
$\gamma $ such that
\begin{equation*}
| u( t,\sqrt{t}) -ct^{-\alpha/2}| \leq Ct^{-(\alpha/2)-\gamma }.
\end{equation*}
In the case of $\alpha \in (0,1/2) $ we use the fact
that
\begin{equation*}
\partial_x| u| ^2=\frac{1}{it}( \overline{u}\mathcal{J}%
u-u\overline{\mathcal{J}u})
\end{equation*}
which implies that usual derivative yields an additional time decay, in
particular, the fractional derivative $| \partial_x| ^{\beta }$
gives us an additional time decay like $t^{-\beta }$ (see Lemma \ref{L2.4}
below). However we have the derivative loss on the nonlinear term which
requires us to use some analytic function space.

To state our results we need some notation. We denote the inverse Fourier
transformation by $\mathcal{F}^{-1}\phi $ $=\check{\phi}$ $=\frac{1}{\sqrt{%
2\pi }}\int e^{ix\xi }\phi ( \xi ) d\xi $. We essentially use the
estimates of the operators $\mathcal{J}=x+it\partial_x=\mathcal{U}(
t) x\mathcal{U}( -t) =itM( t) \partial_x%
\overline{M}( t) $ and $\mathcal{I}=x\partial_x+2t\partial
_{t}$, $M=e^{(ix^2)/(2t)}$. Note that the relation $\mathcal{J}%
\partial_x=\mathcal{I}+2it\mathcal{L}$ is valid, where $\mathcal{L}%
=i\partial_{t}+\frac{1}{2}\partial_x^2$ and $\mathcal{U}(
t) =M( t) \mathcal{D}( t) \mathcal{F}M(
t) $, $\mathcal{D}( t) $ is the dilation operator defined
by $(\mathcal{D}( t) \psi )(x) =(1/\sqrt{it})\psi
( x/t) $. Then since $\mathcal{D}^{-1}( t) =i\mathcal{D%
}( 1/t) $ we have $\mathcal{U}( -t) =\overline{M}%
\mathcal{F}^{-1}\mathcal{D}^{-1}( t) \overline{M}=i\overline{M}%
\mathcal{F}^{-1}\mathcal{D}( 1/t) \overline{M}$.

We denote the usual Lebesgue space $L^p=\{ \phi \in \mathbf{S%
}^{\prime };\| \phi \|_{p}<\infty \} $, where the norm
$\| \phi \|_{p}=( \int_\mathbb{R}| \phi (x) | ^{p}dx) ^{1/p}$ if
$1\le p<\infty $ and $\| \phi \|_{\infty }=$
ess.$\sup \left\{ | \phi (x) |;x\in \mathbb{R}\right\} $ if $p=\infty $.
For simplicity we write $\| \cdot \| =\| \cdot \|_2$.
Weighted Sobolev space is
\begin{equation*}
H_{p}^{m,k}=\big\{ \phi \in \mathbf{S}^{\prime }:\| \phi
\|_{m,k,p}\equiv \big\| \langle x\rangle ^{k}\langle
i\partial_x\rangle ^{m}\phi \big\|_{p}<\infty \big\} ,
\end{equation*}
$m,k\in \mathbb{R}$, $1\le p\le \infty $, $\langle x\rangle =%
\sqrt{1+x^2}$. The fractional derivative $|\partial_x|^{\alpha }$, $%
\alpha \in (0,1)$ is equal to
\begin{equation*}
|\partial_x|^{\alpha }\phi =\mathcal{F}^{-1}|\xi |^{\alpha }\mathcal{F}%
\phi =C\int_\mathbb{R}(\phi (x+z)-\phi (x) )\frac{dz}{%
|z|^{1+\alpha }}.
\end{equation*}
We denote also for simplicity $H^{m,k}=H_2^{m,k}$ and
the norm $\| \phi \|_{m,k}=\| \phi \|_{m,k,2}$.
Different positive constants are denoted by the same letter $C$.
 Denote $\Phi (x) =\int e^{-\frac{i}{2}( \xi -x) ^2}| \xi | ^{\alpha -1}d\xi
$.

Now we state the main results of this paper.

\begin{theorem} \label{Th1.1}
Let $\alpha \in [ 1/2,1) $. We assume that
the initial data $u_0\in H^{3,0}\cap $ $H^{2,2}$ and the
norm $\| u_0\|_{3,0}+\| u_0\|_{2,2}$ is
sufficiently small. Then there exists a unique global solution $u$ of the
Cauchy problem (\ref{1.1}) such that $u\in C( \mathbb{R};H^{3,0}) $.
Moreover there exist unique constant $B$ and
functions $P,Q$ such that $| \xi | ^{1-\alpha }P( \xi
) \in \mathbf{L}^{\infty }( \mathbb{R}) $, $| \xi
| ^{1-\alpha }Q( \xi ) \in \mathbf{L}^{\infty }(
\mathbb{R}) $ and the following asymptotic statement is valid
\begin{equation}
u( t,x) =Be^{\frac{ix^2}{2t}}t^{-\frac{\alpha }{2}}\Phi (
\frac{x}{\sqrt{t}}) +O( t^{-\frac{\alpha }{2}-\gamma }(
\langle \frac{x}{\sqrt{t}}\rangle ^{\alpha -1}+\langle \frac{%
x}{\sqrt{t}}\rangle ^{-\alpha }) )  \label{1.2}
\end{equation}
for all $t\geq 1$, uniformly in $| x| \leq t^{1-\rho }$, and
\begin{equation}
u( t,x) =t^{-\alpha }P( \frac{x}{t}) +e^{\frac{ix^2}{%
2t}}\frac{1}{\sqrt{t}}Q( \frac{x}{t}) +O( t^{-\alpha -\gamma
}+t^{-\frac{1}{2}-\gamma }\langle \frac{x}{t}\rangle ^{-\alpha
})  \label{1.3}
\end{equation}
for all $t\geq 1$, uniformly in $| x| \geq t^{1-\rho }$, where $%
\rho ,\gamma >0$ are small.
\end{theorem}

In the case $\alpha \in (0,1/2) $ we have to assume that
the initial data are analytic. Denote
\begin{equation*}
\mathbf{A}_0=\big\{ \phi \in L^2:\| \phi \|_{%
\mathbf{A}_0}\equiv \sum_{n=0}^{\infty }\frac{1}{n!}\| |
\partial_x| ^{\frac{1}{2}-\alpha }( x\partial_x)
^{n}\phi \|_{1,0}<\infty \big\} .
\end{equation*}

\begin{theorem}
\label{T1.2} Let $\alpha \in (0,1/2) $. We assume that
the initial data $u_0\in \mathbf{A}$ and the norm $\| u_0\|_{%
\mathbf{A}_0}$ is sufficiently small. Then there exists a unique global
solution $u$ of the Cauchy problem (\ref{1.1}) such that $u\in \mathbf{C}%
( \mathbb{R};H^{1,0}) $. Moreover there exist unique
constant $B$ and functions $P,Q$ such that asymptotics (\ref{1.2}) and (\ref
{1.3}) are valid.
\end{theorem}

\begin{remark} \rm
In the region $| x| =t^{1-\rho }$ asymptotics (\ref{1.2})
coincides with (\ref{1.3}).
\end{remark}

In Section 2 we prove some preliminary estimates. In Section 3 we prove
Theorem \ref{Th1.1}. Section 4 is devoted to the proof of Theorem \ref{T1.2}.

\section{Preliminaries}

First we prove some time decay estimates.

\begin{lemma}
\label{L2.1} We have the estimate
\begin{equation*}
\| u_x\|_{\infty }\leq Ct^{-1/2}\| \mathcal{FU}%
( -t) u_x\|_{\infty }+Ct^{-\frac{1+\beta -\gamma }{2}%
}( \| u_x\| +\| | \partial_x| ^{\frac{1%
}{2}-\beta }\mathcal{J}\partial_xu\| ) ,
\end{equation*}
for all $t>0$, where $\beta \in ( 0,\frac{1}{2}] $, $\gamma \in
( 0,\beta ) $. \end{lemma}

\begin{proof}
Denote $w=\mathcal{U}( -t) u_x$. Then since $\mathcal{U}(
t) =M\mathcal{DF}M$, where $M=e^{\frac{ix^2}{2t}}$, $\mathcal{D}\phi
=\frac{1}{\sqrt{it}}\phi ( \frac{x}{t}) $ is the dilation
operator, $\mathcal{J}=x+it\partial_x=\mathcal{U}( t) x%
\mathcal{U}( -t) $, we get
\begin{equation*}
u_x=\mathcal{U}( t) w=M\mathcal{DF}w+M\mathcal{DF}(
M-1) w
\end{equation*}
and by virtue of the H\"{o}lder inequality and Sobolev embedding theorem $%
\| \phi \|_{p}\leq C\| | \partial_x| ^{\frac{1%
}{2}-\frac{1}{p}}\phi \| $ if $2\leq p<\infty $, we have
\begin{eqnarray*}
\lefteqn{\| M\mathcal{DF}( M-1) w\|_{\infty }} \\
&\leq &Ct^{-1/2}\| \mathcal{F}( M-1) w\|
_{\infty }\leq Ct^{-1/2}\| ( M-1) w\|_1\leq
Ct^{-\frac{1+\beta -\gamma }{2}}\| | x| ^{\beta -\gamma
}w\|_1 \\
&\leq &Ct^{-\frac{1+\beta -\gamma }{2}}( \| w\| +\|
xw\|_{\frac{1}{\beta }}) \leq Ct^{-\frac{1+\beta -\gamma }{2}%
}( \| w\| +\| | \partial_x| ^{\frac{1}{2}%
-\beta }xw\| ) \\
&\leq &Ct^{-\frac{1+\beta -\gamma }{2}}( \| u_x\| +\|
| \partial_x| ^{\frac{1}{2}-\beta }x\mathcal{U}(
-t) u_x\| ) \\
&\leq &Ct^{-\frac{1+\beta -\gamma }{2}}( \| u_x\| +\|
| \partial_x| ^{\frac{1}{2}-\beta }\mathcal{J}\partial
_xu\| ) ,
\end{eqnarray*}
therefore the result of the lemma follows. Lemma \ref{L2.1} is proved.
\end{proof}

Denote
\begin{equation*}
\| \phi \|_{\mathbf{Y}}=\sup_{t>0}t^{\alpha }\langle
t\rangle ^{1-2\gamma }\| \partial_{t}\phi \|_{0,1,\infty
}+\sup_{t>0}t^{-\gamma }\| | \xi | ^{\frac{1}{2}-\beta
}\partial_{\xi }\phi \| +\sup_{t>0}\| \phi \|_{0,1,\infty
},
\end{equation*}
where $\beta \in ( 0,\frac{1}{2}] $, $\gamma >0$ is small. In the
next lemma we obtain the asymptotic representation as $\xi \rightarrow 0$
for the integral
\begin{equation*}
I=\int_0^{t}\tau ^{-\alpha }d\tau \int e^{-i\tau \xi \eta }\phi_1(
\tau ,\xi +\eta ) \phi_2( \tau ,\eta ) d\eta
\end{equation*}
which corresponds to the identity (\ref{1.1c}).

\begin{lemma}
\label{L2.2} If $\phi_{l}\in \mathbf{Y,}$ $l=1,2$, then we have
\begin{eqnarray*}
I &=&\Gamma ( 1-\alpha ) | \xi | ^{\alpha -1}(
\sin ( \frac{\pi \alpha }{2}) \int \phi_1( t,\eta )
\phi_2( t,\eta ) | \xi | ^{\alpha -1}d\eta
\\
&& +i\mathop{\rm sign}\xi \cos ( \frac{\pi \alpha }{2}) \int \phi
_1( t,\eta ) \phi_2( t,\eta ) | \eta |
^{\alpha -1}\mathop{\rm sign}\eta \,d\eta ) \\
&&+O( t^{-\gamma }| \xi | ^{\alpha -1}\| \phi
_1\|_{\mathbf{Y}}\| \phi_2\|_{\mathbf{Y}}) .
\end{eqnarray*}
for all $| \xi | \leq t^{-\mu }$, $t\geq 1$, where $\mu =\frac{%
3\gamma }{\alpha ^2}$, $\gamma >0$ is small.
\end{lemma}

\begin{proof}
We write $I=\sum_{l=1}^{4}I_{l}$, where
\begin{gather*}
I_1=\int_0^{t^{\nu }/| \xi | }\tau ^{-\alpha }d\tau \int
e^{-i\tau \xi \eta }\phi_1( t,\eta ) \phi_2( t,\eta) d\eta ,
\\
I_2=\int_{t^{\nu }/| \xi | }^{t}\tau ^{-\alpha }d\tau \int
e^{-i\tau \xi \eta }\phi_1( \tau ,\xi +\eta ) \phi_2(\tau ,\eta ) d\eta ,
\\
I_3=\int_0^{t^{\nu }/| \xi | }\tau ^{-\alpha }d\tau \int
e^{-i\tau \xi \eta }( \phi_1( \tau ,\eta ) \phi
_2( \tau ,\eta ) -\phi_1( t,\eta ) \phi_2( t,\eta ) ) d\eta
\\
I_4=\int_0^{t^{\nu }/| \xi | }\tau ^{-\alpha }d\tau \int
e^{-i\tau \xi \eta }( \phi_1( \tau ,\xi +\eta ) -\phi
_1( \tau ,\eta ) ) \phi_2( \tau ,\eta )d\eta ,
\end{gather*}
where $\nu =2\gamma/\alpha$. If $\tau | \xi | \geq 1$, we integrate by parts
with respect to $\eta $ to obtain
\begin{eqnarray*}
\lefteqn{| \int e^{-i\tau \xi \eta }\phi_1( t,x+\eta ) \phi
_2( t,\eta ) d\eta | }\\
&\leq &\langle \tau \xi \rangle ^{-1}| \int e^{-i\tau \xi
\eta }\partial_{\eta }( \phi_1( t,x+\eta ) \phi
_2( t,\eta ) ) d\eta | \\
&\leq &C\langle \tau \xi \rangle ^{-1}t^{\gamma
}\sum_{l=1}^2\| \phi_{3-l}\|_{\infty }\sup_{t>0}t^{-\gamma
}\| | \xi | ^{\frac{1}{2}-\gamma }\partial_{\xi }\phi
_{l}\| \leq C\langle \tau \xi \rangle ^{-1}t^{\gamma
}\| \phi_1\|_{\mathbf{Y}}\| \phi_2\|_{\mathbf{Y%
}},
\end{eqnarray*}
hence changing $\tau | \xi | =z$ we obtain
\begin{eqnarray*}
\lefteqn{| \int_{t^{\nu }/| \xi | }^{\infty }\tau ^{-\alpha }d\tau
\int e^{-i\tau \xi \eta }\phi_1( t,x+\eta ) \phi_2(
t,\eta ) d\eta | }\\
&\leq &Ct^{\gamma }\| \phi_1\|_{\mathbf{Y}}\| \phi
_2\|_{\mathbf{Y}}\int_{t^{\nu }/| \xi | }^{\infty
}\langle \tau \xi \rangle ^{-1}\tau ^{-\alpha }d\tau \leq
Ct^{\gamma }| \xi | ^{\alpha -1}\| \phi_1\|_{%
\mathbf{Y}}\| \phi_2\|_{\mathbf{Y}}\int_{t^{\nu }}^{\infty
}z^{-\alpha -1}dz \\
&\leq &C| \xi | ^{\alpha -1}t^{\gamma -\alpha \nu }\| \phi
_1\|_{\mathbf{Y}}\| \phi_2\|_{\mathbf{Y}}\leq
Ct^{-\gamma }| \xi | ^{\alpha -1}\| \phi_1\|_{%
\mathbf{Y}}\| \phi_2\|_{\mathbf{Y}}.
\end{eqnarray*}
Since
\begin{eqnarray*}
\int_0^{\infty }\tau ^{-\alpha }e^{i\tau \xi \eta }d\tau
&=&\int_0^{\infty }\tau ^{-\alpha }\cos ( \tau \xi \eta ) d\tau
+i\int_0^{\infty }\tau ^{-\alpha }\sin ( \tau \xi \eta ) d\tau
\\
&=&\Gamma ( 1-\alpha ) \sin ( \frac{\pi \alpha }{2})
| \xi \eta | ^{\alpha -1} \\
&&+i\Gamma ( 1-\alpha ) \cos ( \frac{\pi \alpha }{2})
| \xi \eta | ^{\alpha -1}\mathop{\rm sign}( \xi \eta )
\end{eqnarray*}
(see \cite{be}), we find
\begin{eqnarray*}
I_1 &=&\int_0^{\infty }\tau ^{-\alpha }d\tau \int e^{-i\tau \xi \eta
}\phi_1( t,\eta ) \phi_2( t,\eta ) d\eta \\
&&-\int_{t^{\nu }/| \xi | }^{\infty }\tau ^{-\alpha }d\tau \int
e^{-i\tau \xi \eta }\phi_1( t,\eta ) \phi_2( t,\eta
) d\eta \\
&=&\Gamma ( 1-\alpha ) \sin ( \frac{\pi \alpha }{2})
| \xi | ^{\alpha -1}\int \phi_1( t,\eta ) \phi
_2( t,\eta ) | \eta | ^{\alpha -1}d\eta \\
&&+i\Gamma ( 1-\alpha ) \cos ( \frac{\pi \alpha }{2})
| \xi | ^{\alpha -1}\int \mathop{\rm sign}( \xi \eta ) \phi
_1( t,\eta ) \phi_2( t,\eta ) | \eta |
^{\alpha -1}d\eta \\
&&+O( t^{-\gamma }| \xi | ^{\alpha -1}\| \phi
_1\|_{\mathbf{Y}}\| \phi_2\|_{\mathbf{Y}}) .
\end{eqnarray*}
In the same manner we obtain
\begin{eqnarray*}
\lefteqn{| \int_{t^{\nu }/| \xi | }^{t}\tau ^{-\alpha }d\tau \int
e^{-i\tau \xi \eta }\phi_1( \tau ,x+\eta ) \phi_2(
\tau ,\eta ) d\eta | }\\
&\leq &Ct^{\gamma }\| \phi_1\|_{\mathbf{Y}}\| \phi
_2\|_{\mathbf{Y}}\int_{t^{\nu }/| \xi |
}^{t}\langle \tau \xi \rangle ^{-1}\tau ^{-\alpha }d\tau \leq
Ct^{-\gamma }| \xi | ^{\alpha -1}\| \phi_1\|_{%
\mathbf{Y}}\| \phi_2\|_{\mathbf{Y}},
\end{eqnarray*}
hence
\begin{equation*}
| I_2| \leq Ct^{-\gamma }| \xi | ^{\alpha -1}\|
\phi_1\|_{\mathbf{Y}}\| \phi_2\|_{\mathbf{Y}}.
\end{equation*}
To estimate $I_3$ we note that
\begin{equation*}
\| \phi_{l}( t,\xi ) -\phi_{l}( \tau ,\xi )
\|_{0,1,\infty }=\| \int_{\tau }^{t}\partial_{\tau }\phi
_{l}( \tau ,\xi ) d\tau \|_{0,1,\infty }=O( \tau
^{2\gamma -\alpha }\| \phi_{l}\|_{\mathbf{Y}})
\end{equation*}
which implies
\begin{eqnarray*}
| I_3| &=&| \int_0^{t^{\nu }/| \xi | }\tau
^{-\alpha }d\tau \int e^{-i\tau \xi \eta }( \phi_1( \tau ,\eta
) \phi_2( \tau ,\eta ) -\phi_1( t,\eta )
\phi_2( t,\eta ) ) d\eta | \\
&\leq &C\| \phi_1\|_{\mathbf{Y}}\| \phi_2\|_{%
\mathbf{Y}}| \int_0^{t^{\nu }/| \xi | }\tau ^{2\gamma
-2\alpha }d\tau | \leq Ct^{-\gamma }| \xi | ^{\alpha
-1}\| \phi_1\|_{\mathbf{Y}}\| \phi_2\|_{%
\mathbf{Y}}
\end{eqnarray*}
since $\mu \alpha \geq \gamma +\nu $ and $| \xi | \leq t^{-\mu }$%
. Now using the estimate
\begin{eqnarray*}
\| \langle \eta \rangle ^{-1}( \phi ( t,\xi +\eta
) -\phi ( t,\eta ) ) \|_1 &=&\|
\langle \eta \rangle ^{-1}\int_0^{\xi }\partial_{y}\phi (
t,y+\eta ) dy\|_1 \\
&\leq &C| \xi | \| | \xi | ^{\frac{1}{2}-\beta
}\partial_{\xi }\phi \| \leq Ct^{\gamma }| \xi | \|
\phi \|_{\mathbf{Y}}
\end{eqnarray*}
for all $| \xi | \leq 1$, we get
\begin{equation*}
| I_4| \leq C\| \phi_1\|_{\mathbf{Y}}\|
\phi_2\|_{\mathbf{Y}}| \xi | \int_0^{t^{\nu }/|
\xi | }\tau ^{\gamma -\alpha }d\tau \leq Ct^{-\gamma }| \xi
| ^{\alpha -1}\| \phi_1\|_{\mathbf{Y}}\| \phi
_2\|_{\mathbf{Y}}
\end{equation*}
since $\mu ( 1-\gamma ) \geq \gamma +\nu $ and $| \xi
| \leq t^{-\mu }$. Lemma \ref{L2.2} is proved.
\end{proof}

In the next lemma we consider the asymptotic behaviour of the integral
\begin{equation*}
I( t,x) =\int e^{-\frac{it}{2}( \xi -\frac{x}{t})
^2}f( t,\xi ) d\xi
\end{equation*}
as $t\rightarrow \infty $ uniformly with respect to $x\in \mathbb{R}$.
Define $\Phi (x) =\int e^{-\frac{i}{2}( \xi -x)
^2}| \xi | ^{\alpha -1}d\xi $. Note that
\begin{equation*}
\Phi (x) =O( \langle x\rangle ^{-\alpha
}+\langle x\rangle ^{\alpha -1})
\end{equation*}
as $| x| \rightarrow \infty $. Let $\gamma$ be a small positive number and
\begin{gather*}
\beta =\min ( 1/2,\alpha ) -\gamma ,\quad \mu =3\gamma/\alpha ^2,
\\ \rho =\frac{5\gamma }{\alpha ^2( 1-\alpha ) }, \quad
\theta =\frac{6\gamma }{\alpha ^2( 1-\alpha ) ^2},\quad \delta
=\theta +\gamma .
\end{gather*}

\begin{lemma}
\label{L2.3} Let $\partial_{\xi }f( t,\xi ) =O( | \xi
| ^{\alpha -2}) $ and $f( t,\xi ) =t^{1-\alpha }\Psi
( t\xi ) +O( t^{1-\alpha -\delta }) $ for all $|
\xi | \leq t^{\theta -1}$, $\partial_{\xi }f( t,\xi )
=( \alpha -1) B| \xi | ^{\alpha -1}\xi ^{-1}+O(
t^{-\gamma }| \xi | ^{\alpha -2}) $ for all $t^{\theta
-1}\leq | \xi | \leq t^{-\mu }$ and $\| | \xi | ^{%
\frac{1}{2}-\beta }\xi \partial_{\xi }f( t,\xi ) \| \leq
Ct^{\gamma }$, then we have the asymptotic formula
\begin{equation*}
I( t,x) =Bt^{-\frac{\alpha }{2}}\Phi ( xt^{-\frac{1}{2}%
}) +O( t^{-\frac{\alpha }{2}-\gamma }( \langle xt^{-%
\frac{1}{2}}\rangle ^{-\alpha }+\langle xt^{-\frac{1}{2}%
}\rangle ^{\alpha -1}) )
\end{equation*}
for all $t\geq 1$ uniformly in $| x| \leq t^{1-\rho }$ and
\begin{equation*}
I( t,x) =\sqrt{2\pi }t^{-\alpha }e^{-\frac{ix^2}{2t}}\check{\Psi%
}( \frac{x}{t}) +\frac{\sqrt{\pi }}{\sqrt{it}}f( t,\frac{x}{t%
}) +O( t^{-\alpha -\gamma }+t^{-\frac{1}{2}-\gamma }\langle
xt^{-1}\rangle ^{-\alpha })
\end{equation*}
for all $t\geq 1$ uniformly in $| x| \geq t^{1-\rho }$. \end{lemma}

\begin{proof}
For $x>0$, we have
\begin{eqnarray*}
f( t,\xi ) &=&f( t,1) +\int_{t^{-\mu }}^{\xi }\partial
_{\eta }f( t,\eta ) d\eta +\int_1^{t^{-\mu }}\partial_{\eta
}f( t,\eta ) d\eta \\
&=&f( t,1) +( \alpha -1) B\int_{t^{-\mu }}^{\xi
}| \eta | ^{\alpha -2}d\eta +O( t^{-\gamma }\int_{t^{-\mu
}}^{\xi }| \eta | ^{\alpha -2}d\eta ) \\
&&+O( \| | \xi | ^{\frac{1}{2}-\beta }\xi \partial
_{\xi }f( t,\xi ) \| ( \int_1^{t^{-\mu }}| \xi
| ^{2\beta -3}d\xi ) ^{1/2}) \\
&=&B| \xi | ^{\alpha -1}+O( 1+t^{-\gamma }| \xi |
^{\alpha -1}+t^{\mu ( 1-\beta ) +\gamma }) \\
&=&B| \xi | ^{\alpha -1}+O( t^{-\gamma }| \xi |
^{\alpha -1})
\end{eqnarray*}
for all $t^{\mu -1}\leq | \xi | \leq 2t^{-\rho }$ since $\mu
( 1-\beta ) +2\gamma \leq \rho ( 1-\alpha ) $. We make
a change of variable of integration $\xi =zt^{-1/2}$, then we have
\begin{equation*}
I( t,x) =t^{-1/2}\int e^{-\frac{i}{2}( z-b)
^2}f( t,zt^{-1/2}) dz,
\end{equation*}
where $b=a\sqrt{t}=x/\sqrt{t}$.
First consider the case $| x| \leq t^{1-\rho }$, i.e. $b\leq t^{%
\frac{1}{2}-\rho }$. We represent
\begin{equation*}
I=Bt^{-\frac{\alpha }{2}}\Phi ( b) +R_1+R_2,
\end{equation*}
where the remainder terms are
\begin{equation*}
R_{j}=t^{-1/2}\int e^{-\frac{i}{2}( z-b) ^2}(
f( t,zt^{-1/2}) -Bt^{\frac{1-\alpha }{2}}| z|
^{\alpha -1}) \varphi_{j}( z) dz,
\end{equation*}
the function $\varphi_1( z) \in \mathbf{C}^{1}( \mathbb{R}
) :\varphi_1( z) =1$ if $z<b/3$ and $\varphi
_1( z) =0$ if $z>2b/3$, $\varphi_2( z)
=1-\varphi_1( z) $. In the remainder term $R_1$ we integrate
by parts via the identity
\begin{equation}
e^{-\frac{i}{2}( z-b) ^2}=\frac{1}{1-iz( z-b) }\frac{%
d}{dz}( ze^{-\frac{i}{2}( z-b) ^2})  \label{2.1}
\end{equation}
to get
\begin{eqnarray}
| R_1| &\leq &Ct^{-\frac{\alpha }{2}-\gamma }\int |
z| ^{\alpha -1}\langle zb\rangle ^{-1}( | \varphi
_1| +| z\varphi_1^{\prime }| ) dz  \notag \\
&&+Ct^{-\frac{\alpha }{2}}\int_{| z| \leq t^{\mu -\frac{1}{2}%
}}| z| ^{\alpha -1}\langle zb\rangle ^{-1}dz  \notag \\
&\leq &Ct^{-\frac{\alpha }{2}-\gamma }\langle b\rangle ^{-\alpha
}\leq Ct^{-\frac{\alpha }{2}-\gamma }\langle a\sqrt{t}\rangle
^{-\alpha }.  \label{2.2}
\end{eqnarray}
In the remainder term $R_2$ we use the identity
\begin{equation}
e^{-\frac{i}{2}( z-b) ^2}=\frac{1}{1-i( z-b) ^2}%
\frac{d}{dz}( ( z-b) e^{-\frac{i}{2}( z-b)
^2})  \label{2.3}
\end{equation}
to find
\begin{eqnarray}
| R_2| &\leq &Ct^{-\frac{\alpha }{2}-\gamma }\int |
z| ^{\alpha -1}\langle z-b\rangle ^{-2}( |
\varphi_2| +| z\varphi_2^{\prime }| ) dz  \notag
\\
&&+Ct^{-\frac{\alpha }{2}}\int_{| z| \leq t^{\mu -\frac{1}{2}%
}}| z| ^{\alpha -1}\langle z-b\rangle ^{-2}dz  \notag
\\
&&+Ct^{-1/2}\int_{| z| >2t^{\frac{1}{2}-\rho
}}\langle z-b\rangle ^{-2}| zt^{-1/2}| |
f^{\prime }( t,zt^{-1/2}) | dz  \notag \\
&=&O( t^{-\frac{\alpha }{2}-\gamma }\langle b\rangle
^{\alpha -1}) =O( t^{-\frac{\alpha }{2}-\gamma }\langle a%
\sqrt{t}\rangle ^{\alpha -1}) ,  \label{2.4}
\end{eqnarray}
since
\begin{eqnarray*}
\lefteqn{\int_{| z| >2t^{\frac{1}{2}-\rho }}\langle
z-b\rangle ^{-2}| zt^{-1/2}| | f^{\prime
}( t,zt^{-1/2}) | dz }\\
&\leq &C\| | \xi | ^{\frac{1}{2}-\beta }\xi f^{\prime
}( t,\xi ) \| ( \int_{| z| >2t^{\frac{1}{2}%
-\rho }}| zt^{-1/2}| ^{2\beta -1}\langle
z-b\rangle ^{-4}dz) ^{1/2} \\
&\leq &Ct^{\frac{1-2\beta }{4}}\langle b\rangle ^{-1}\|
| \xi | ^{\frac{1}{2}-\beta }\xi f^{\prime }( t,\xi )
\| ( \int_{| z| >2t^{\frac{1}{2}-\rho }}z^{2\beta
-3}dz) ^{1/2} \\
&\leq &Ct^{-\frac{1}{4}+\rho ( 1-\beta ) }\langle
b\rangle ^{-1}\| | \xi | ^{\frac{1}{2}-\beta }\xi
f^{\prime }( t,\xi ) \| \leq Ct^{-\gamma }\langle
b\rangle ^{-1}.
\end{eqnarray*}
We consider now the case $| x| >t^{1-\rho }$, i.e. $b>t^{\frac{1}{%
2}-\rho }$. Then we represent $I$ in the form
\begin{equation*}
I=t^{-1/2}\int_{| z| \leq t^{\theta -\frac{1}{2}}}e^{-%
\frac{i}{2}( z-b) ^2}f( t,zt^{-1/2}) dz+%
\sqrt{\frac{\pi }{it}}f( t,a) +R_3+R_4,
\end{equation*}
where the remainder terms are
\begin{gather*}
R_3=t^{-1/2}\int_{| z| >t^{\theta -\frac{1}{2}}}e^{-%
\frac{i}{2}( z-b) ^2}f( t,zt^{-1/2}) \varphi_1( z) dz
\\
R_4=t^{-1/2}\int e^{-\frac{i}{2}( z-b) ^2}(
f( t,zt^{-1/2}) -f( t,a) ) \varphi
_2( z) dz.
\end{gather*}
Consider the integral
\begin{eqnarray*}
\lefteqn{t^{-1/2}\int_{| z| \leq t^{\theta -\frac{1}{2}}}e^{-%
\frac{i}{2}( z-b) ^2}f( t,zt^{-1/2})
dz=\int_{| \xi | \leq t^{\theta -1}}e^{-\frac{i}{2}t( \xi
-a) ^2}f( t,\xi ) d\xi }\\
&=&t^{1-\alpha }\int_{| \xi | \leq t^{\theta -1}}e^{-\frac{i}{2}%
t( \xi -a) ^2}\Psi ( t\xi ) d\xi +O( t^{-\alpha
-\gamma }) \\
&=&t^{-\alpha }e^{-\frac{ix^2}{2t}}\int_{| y| \leq t^{\theta
}}e^{iya}\Psi ( y) dy+O( t^{-\alpha -\gamma }) \\
&=&\sqrt{2\pi }t^{-\alpha }e^{-\frac{ix^2}{2t}}\widehat{\Psi }(
a) +O( t^{-\alpha -\gamma }) . \hspace{5cm}
\end{eqnarray*}
In the remainder term $R_3$ above we integrate by parts via identity
(\ref{2.1}) to get
\begin{eqnarray}
| R_3| &\leq &Ct^{-\frac{\alpha }{2}}\int_{| z| \geq
t^{\theta -\frac{1}{2}}}| z| ^{\alpha -1}\langle
zb\rangle ^{-1}( | \varphi_1| +| z\varphi
_1^{\prime }| ) dz  \notag \\
&&+Ct^{-1/2}\int_{| z| >2t^{\frac{1}{2}-\rho
}}\langle zb\rangle ^{-1}| zt^{-1/2}| |
f^{\prime }( t,zt^{-1/2}) | dz   \label{2.5} \\
&\leq &Ct^{-\alpha +\rho -\theta ( 1-\alpha ) }+Ct^{-\alpha
-\gamma }\leq Ct^{-\alpha -\gamma } \nonumber
\end{eqnarray}
since $\theta ( 1-\alpha ) -\rho \geq \gamma $. In the remainder
term $R_4$ we integrate by parts via (\ref{2.3}) to find
\begin{eqnarray}
| R_4| &\leq &Ct^{-1/2}\int_{b/3}^{\infty }|
f( t,zt^{-1/2}) -f( t,a) | \langle
z-b\rangle ^{-4}dz  \notag \\
&&+t^{-1}\int_{b/3}^{\infty }| f^{\prime }( t,zt^{-\frac{1}{2}%
}) | \langle z-b\rangle ^{-1}dz  \label{2.6} \\
&\leq &C| a| ^{-1}t^{\gamma -\frac{1+\beta }{2}}\leq
C\langle a\rangle ^{-1}t^{-\frac{1}{2}-\gamma },  \nonumber
\end{eqnarray}
since
\begin{eqnarray*}
| f( t,zt^{-1/2}) -f( t,a) |
&=&| \int_{zt^{-1/2}}^{a}\partial_{\xi }f( t,\xi )
d\xi | \leq \int_{zt^{-1/2}}^{a}| \xi | ^{\beta -%
\frac{3}{2}}| \xi | ^{\frac{3}{2}-\beta }| \partial_{\xi
}f( t,\xi ) | d\xi \\
&\leq &C\| | \xi | ^{\frac{1}{2}-\beta }\xi \partial_{\xi
}f( t,\xi ) \| ( \int_{zt^{-1/2}}^{a}|
\xi | ^{2\beta -3}d\xi ) ^{1/2} \\
&\leq &C| a| ^{-1}t^{\gamma -\frac{\beta }{2}}| z-b|
^{\beta }.
\end{eqnarray*}
Collecting estimates (\ref{2.2}), (\ref{2.4})-(\ref{2.6}) we get the
asymptotic statement needed and Lemma \ref{L2.3} is proved.
\end{proof}

In the next lemma we obtain time-decay estimate via additional derivative
for the nonlinear term. We will use this estimate in the proof of Theorem
\ref{T1.2}.

\begin{lemma}
\label{L2.4} We have the estimate
\begin{eqnarray*}
\lefteqn{\| | \partial_x| ^{\frac{1}{2}-\beta }( u_x%
\overline{v_x}) \|_{1,0} }\\
& \leq &Ct^{\beta -1}\| | \partial_x| ^{\frac{1}{2}-\beta
}u\|_{1,0}\| | \partial_x| ^{\frac{1}{2}-\beta
}v\|_{1,0} \\
& &+Ct^{-1}( t^{\beta }\| \mathcal{FU}( -t)
u_x\|_{\infty }+\| | \partial_x| ^{\frac{1}{2}%
-\beta }u\|_{1,0}) \| | \partial_x| ^{\frac{1%
}{2}-\beta }\mathcal{J}\partial_xv\|_{1,0} \\
&& +Ct^{-1}( t^{\beta }\| \mathcal{FU}( -t)
v_x\|_{\infty }+\| | \partial_x| ^{\frac{1}{2}%
-\beta }v\|_{1,0}) \| | \partial_x| ^{\frac{1%
}{2}-\beta }\mathcal{J}\partial_xu\|_{1,0}
\end{eqnarray*}
for all $t>0$, where $\beta \in ( 0,1/2]$. \end{lemma}

\begin{proof}
Application of the Fourier transformation yields
\begin{equation*}
\mathcal{F}( u_x\overline{v_x}) =\frac{1}{\sqrt{2\pi }}\int
\hat{u}( t,\xi +\eta ) \overline{\hat{v}( t,\eta ) }%
( \xi +\eta ) \eta d\eta ,
\end{equation*}
then changing $i\eta \hat{u}( t,\eta ) =e^{-\frac{it}{2}\eta
^2}\phi ( t,\eta ) $ and $i\eta \hat{v}( t,\eta )
=e^{-\frac{it}{2}\eta ^2}\psi ( t,\eta ) $ we obtain
\begin{equation}
\mathcal{FU}( -t) ( u_x\overline{v_x}) =\frac{1}{%
\sqrt{2\pi }}\int e^{-it\xi \eta }\phi ( t,\xi +\eta ) \overline{%
\psi ( t,\eta ) }d\eta ,  \label{2.7}
\end{equation}
whence integrating by parts with respect to\thinspace $\eta $ we get
\begin{eqnarray*}
\lefteqn{\| | \partial_x| ^{\frac{1}{2}-\beta }( u_x%
\overline{v_x}) \| =C\| | \xi | ^{\frac{1}{2}%
-\beta }\mathcal{FU}( -t) ( u_x\overline{v_x}) \| }\\
&=&C\| | \xi | ^{\frac{1}{2}-\beta }\int e^{-it\xi \eta
}\phi ( t,\xi +\eta ) \overline{\psi ( t,\eta ) }d\eta
\| \\
&\leq &C\| \langle t\xi \rangle ^{-1}| \xi | ^{%
\frac{1}{2}-\beta }\| ( \| \int e^{-it\xi \eta }\phi (
t,\xi +\eta ) \overline{\psi ( t,\eta ) }d\eta \|
_{\infty }\\
&&+\| \int e^{-it\xi \eta }\phi_{\xi }( t,\xi +\eta )
\overline{\psi ( t,\eta ) }d\eta \|_{\infty } \\
&& +\| \int e^{-it\xi \eta }\phi ( t,\xi +\eta )
\overline{\psi_{\eta }( t,\eta ) }d\eta \|_{\infty
}) \\
&\leq &Ct^{\beta -1}\| \phi \| \| \psi \| +Ct^{\beta
-1}\| \phi \|_{\infty }\| | \xi | ^{\frac{1}{2}%
-\beta }\partial_{\xi }\psi \| +Ct^{\beta -1}\| \psi \|
_{\infty }\| | \xi | ^{\frac{1}{2}-\beta }\partial_{\xi
}\phi \| \\
&\leq &Ct^{\beta -1}\| u_x\| \| v_x\| +Ct^{\beta
-1}\| \mathcal{FU}( -t) u_x\|_{\infty }\|
| \partial_x| ^{\frac{1}{2}-\beta }\mathcal{J}\partial
_xv\| \\
&&+Ct^{\beta -1}\| \mathcal{FU}( -t) v_x\|_{\infty
}\| | \partial_x| ^{\frac{1}{2}-\beta }\mathcal{J}%
\partial_xu\|
\end{eqnarray*}
and
\begin{eqnarray*}
\lefteqn{\| | \partial_x| ^{\frac{3}{2}-\beta }( u_x%
\overline{v_x}) \| =C\| | \xi | ^{\frac{3}{2}%
-\beta }\mathcal{FU}( -t) ( u_x\overline{v_x})
\| }\\
&=&C\| | \xi | ^{\frac{3}{2}-\beta }\int e^{-it\xi \eta
}\phi ( t,\xi +\eta ) \overline{\psi ( t,\eta ) }d\eta
\| \\
&\leq &Ct^{-1}( \| | \xi | ^{\frac{1}{2}-\beta }\int
e^{-it\xi \eta }\phi_{\xi }( t,\xi +\eta ) \overline{\psi (
t,\eta ) }d\eta \|  \\
&&+C\| | \xi | ^{\frac{1}{2}-\beta }\int e^{-it\xi
\eta }\phi ( t,\xi +\eta ) \overline{\psi_{\eta }( t,\eta
) }d\eta \| ) \\
&\leq &Ct^{-1}\| \langle \xi \rangle ^{\frac{1}{2}-\beta
}\phi \| \| \partial_{\xi }\psi \|_1+Ct^{-1}\|
\phi \| \| | \xi | ^{\frac{1}{2}-\beta }\partial_{\xi
}\psi \|_1 \\
&&+Ct^{-1}\| \langle \xi \rangle ^{\frac{1}{2}-\beta }\psi
\| \| \partial_{\xi }\phi \|_1+Ct^{-1}\| \psi
\| \| | \xi | ^{\frac{1}{2}-\beta }\partial_{\xi
}\psi \|_1 \\
&\leq &Ct^{-1}\| | \partial_x| ^{\frac{1}{2}-\beta
}u\|_{1,0}\| | \partial_x| ^{\frac{1}{2}-\beta }%
\mathcal{J}\partial_xv\|_{1,0} \\
&&+Ct^{-1}\| | \partial_x| ^{\frac{1}{2}-\beta }v\|
_{1,0}\| | \partial_x| ^{\frac{1}{2}-\beta }\mathcal{J}%
\partial_xu\|_{1,0}.
\end{eqnarray*}
Lemma \ref{L2.4} is proved.
\end{proof}

\section{Proof of Theorem \ref{Th1.1}}

By virtue of the method in \cite{h}, \cite{ho} (see also the proof of
a-priori estimates below in Lemma \ref{L3.2}) we easily obtain the local
existence of solutions in the functional space
\begin{equation*}
\mathbf{X}_{T}=\big\{ \phi \in \mathbf{C}( ( -T,T) ;
L^2( \mathbb{R}) ) :\sup_{t\in ( -T,T)
}\| \phi ( t) \|_{\mathbf{X}}<\infty \big\} ,
\end{equation*}
where the norm in $\mathbf{X}$ is
\begin{eqnarray*}
\| u\|_{\mathbf{X}} &=&\langle t\rangle ^{-\gamma
}\| u\|_{3,0}+\langle t\rangle ^{-\gamma }\|
\mathcal{I}u\|_{1,0}+\langle t\rangle ^{-3\gamma }\|
\mathcal{I}^2u\| \\
&&+t^{\alpha }\langle t\rangle ^{1-2\gamma }\| \partial_{t}%
\mathcal{FU}( -t) u_x( t) \|_{0,1,\infty },
\end{eqnarray*}
with $\mathcal{I}=x\partial_x+2t\partial_{t}$.

\begin{theorem}
\label{T3.1} Let the initial data $u_0\in H^{3,0}\cap H%
^{2,2}$. Then for some time $T>0$ there exists a unique solution $u\in
\mathbf{X}_{T}$ of the Cauchy problem (\ref{1.1}). If we assume in addition
that the norm of the initial data $\| u_0\|_{3,0}+\|
u_0\|_{2,2}=\varepsilon ^2$ is sufficiently small, then there
exists a unique solution $u\in \mathbf{X}_{T}$ of (\ref{1.1}) for some time $%
T>1$, such that the following estimate $\sup_{t\in [0,T]}\| u\|_{%
\mathbf{X}}<\varepsilon $ is valid.
\end{theorem}

In the next lemma we obtain the estimates of global solutions in the norm $%
\mathbf{X}$.

\begin{lemma}
\label{L3.2} Let $\alpha \in [1/2,1) $. We assume that
the initial data $u_0\in H^{3,0}\cap H^{2,2}$ and the
norm $\| u_0\|_{3,0}+\| u_0\|_{2,2}=\varepsilon
^2$ is sufficiently small. Then there exists a unique global solution of
the Cauchy problem (\ref{1.1}) such that $u\in \mathbf{C}( \mathbb{R};%
H^{3,0}) $ and the following estimate is valid
\begin{equation}
\sup_{t>0}\| u\|_{\mathbf{X}}<\varepsilon .  \label{3.1}
\end{equation}
\end{lemma}

\begin{proof}
Applying the result of Theorem \ref{T3.1} and using a standard continuation
argument we can find a maximal time $T>1$ such that the inequality
\begin{equation}
\| u\|_{\mathbf{X}}\leq \varepsilon  \label{3.2}
\end{equation}
is true for all $t\in [0,T]$. If we prove (\ref{3.1}) on the whole time
interval $[0,T]$, then by the contradiction argument we obtain the desired
result of the lemma. In view of the local existence Theorem \ref{T3.1} it is
sufficient to consider the estimates of the solution on the time interval $%
t\ge 1$ only.

As a consequence of (\ref{3.2}) we have
\begin{eqnarray*}
\| \mathcal{FU}( -t) u_x( t) \|
_{0,1,\infty } &\leq &C\varepsilon +\int_0^{t}\| \partial_{\tau }%
\mathcal{FU}( -\tau ) u_x( \tau ) \|
_{0,1,\infty }d\tau \\
&\leq &C\varepsilon +C\varepsilon \int_0^{t}\langle \tau
\rangle ^{\gamma -1}\tau ^{-\alpha }d\tau \leq C\varepsilon .
\end{eqnarray*}
Note that $\mathcal{J}\partial_x=\mathcal{I}+2it\mathcal{L}$, where $%
\mathcal{J}$ $=x+it\partial_x$. Hence
\begin{equation*}
\| \mathcal{J}\partial_xu\|_{1,0}\leq \| \mathcal{I}%
u\|_{1,0}+Ct\| \mathcal{L}u\|_{1,0}\leq \| \mathcal{I%
}u\|_{1,0}+Ct^{1/2}\| u_x\|_{\infty }\|
u\|_{2,0}
\end{equation*}
and
\begin{equation*}
\| \mathcal{J}\partial_x\mathcal{I}u\| \leq \| \mathcal{I}%
^2u\| +Ct\| \mathcal{LI}u\| \leq \| \mathcal{I}%
^2u\| +Ct^{1/2}\| u_x\|_{\infty }(
\| u_x\| +\| \mathcal{I}u_x\| ) .
\end{equation*}
Then by Lemma \ref{L2.1} with $\beta =\frac{1}{2}$, using estimate (\ref{3.3}%
) we find
\begin{eqnarray*}
\| u_x\|_{1,0,\infty } &\leq &Ct^{-1/2}\|
\mathcal{FU}( -t) u_x\|_{0,1,\infty }+Ct^{\frac{\gamma }{%
2}-\frac{3}{4}}( \| u\|_{2,0}+\| \mathcal{J}\partial
_xu\|_{1,0}) \\
&\leq &C\varepsilon t^{-1/2}+C\varepsilon t^{3\gamma -\frac{1}{4}%
}\| u_x\|_{\infty },
\end{eqnarray*}
whence
\begin{equation}
\| u_x\|_{1,0,\infty }\leq C\varepsilon t^{-1/2}.
\label{3.4}
\end{equation}
Therefore by virtue of (\ref{3.2}) we have also the estimates
\begin{equation}
t^{-\gamma }\| \mathcal{J}\partial_xu\|_{1,0}+t^{-3\gamma
}\| \mathcal{J}\partial_x\mathcal{I}u\| \leq C\varepsilon .
\label{3.3}
\end{equation}

Let us estimate norms $\| u\|_{3,0}$, $\| \mathcal{I}%
u\|_{1,0}$ and $\| \mathcal{I}^2u\| \mathbf{.}$
Differentiating three times equation (\ref{1.1}) we get for $h_0=(
1+\partial_x^{3}) u$%
\begin{equation*}
\mathcal{L}h_0=t^{-\alpha }( \overline{u}_x\partial
_xh_0+u_x\partial_x\overline{h_0}) +R_0
\end{equation*}
where
\begin{equation*}
\mathcal{L}=i\partial_{t}+\frac{1}{2}\partial_x^2,R_0=t^{-\alpha
}( -| u_x| ^2+3u_{xx}\overline{u}_{xxx}+3u_{xxx}%
\overline{u}_{xx}) .
\end{equation*}
Via (\ref{3.2}), (\ref{3.4}) we have the estimate
\begin{equation*}
\| R_0\| \leq Ct^{-\alpha }\| u_x\|_{1,0,\infty
}\| h_0\| \leq C\varepsilon ^2t^{\gamma -1}.
\end{equation*}
Applying the operator $\mathcal{I}$ to both sides of equation (\ref{1.1})
and using the commutator relations $\mathcal{LI}=( \mathcal{I}+2)
\mathcal{L}$ and $[ \mathcal{I},t^{-\alpha }] =-2\alpha
t^{-\alpha }$, we find
\begin{equation}
\mathcal{L}h_{k}=t^{-\alpha }( \overline{u}_x\partial
_xh_{k}+u_x\partial_x\overline{h_{k}}) +R_{k},  \label{3.5}
\end{equation}
where $k=1,2$, $h_1=( 1+\partial_x) \mathcal{I}u$, $h_2=\mathcal{I}^2u$,
\begin{equation*}
R_1=t^{-\alpha }( \overline{u}_{xx}\mathcal{I}u_x+u_{xx}\overline{%
\mathcal{I}u}_x+2( 1-\alpha ) ( 1+\partial_x)
| u_x| ^2),
\end{equation*}
and
\begin{equation*}
R_2=2t^{-\alpha }( | \mathcal{I}u_x| ^2+(
2-\alpha ) \mathcal{I}| u_x| ^2+2( 1-\alpha
) ^2| u_x| ^2) .
\end{equation*}
By (\ref{3.2}) and (\ref{3.3}) we have
\begin{equation*}
\| \mathcal{I}u_x\overline{\mathcal{I}u_x}\| \leq Ct^{-\frac{1%
}{2}}\| \mathcal{I}u_x\| ^{\frac{3}{2}}\| \mathcal{JI}%
u_x\| ^{1/2}\leq C\varepsilon ^2t^{3\gamma -\frac{1}{2}},
\end{equation*}
then by virtue of (\ref{3.2}), (\ref{3.4}) we estimate the remainder terms
\begin{equation*}
\| R_1\| \leq Ct^{-1/2}\| u_x\|
_{1,0,\infty }( \| u\|_{1,0}+\| \mathcal{I}u\|
_{1,0}) \leq C\varepsilon ^2t^{\gamma -1}
\end{equation*}
and
\begin{equation*}
\| R_2\| \leq Ct^{-1/2}\| u_x\|_{\infty
}( \| u\|_{1,0}+\| \mathcal{I}u\|_{1,0})
+Ct^{-1/2}\| \mathcal{I}u_x\overline{\mathcal{I}u_x}%
\| \leq C\varepsilon ^2t^{3\gamma -1}.
\end{equation*}
To cancel the higher-order derivative $t^{-\alpha }\bar{u}_x\partial_xh_{k}$,
 we multiply (\ref{3.5}) by $E\equiv
e^{-t^{-\alpha }\bar{u}}$. The other higher-order derivative $t^{-\alpha
}u_x\partial_x\overline{h_{k}}$ will be eliminated via integration by
parts. Since $E( \mathcal{L}-t^{-\alpha }\overline{u}_x\partial
_x) =( \mathcal{L}-g) E$, where $g=-t^{-\alpha }\overline{%
u}_{xx}+\frac{1}{2}t^{-2\alpha }( \overline{u}_x)
^2-t^{-2\alpha }| u_x| ^2$,  from equation (\ref{3.5}) we obtain
\begin{equation}
\mathcal{L}Eh_{k}=t^{-\alpha }u_xE\partial_x\overline{h_{k}}%
+ER_{k}+gEh_{k}.  \label{3.6}
\end{equation}
Note that $\| E\|_{1,0,\infty }\leq C$ and $\| g\|
_{\infty }\leq C\varepsilon t^{-1}$ by virtue of (\ref{3.2}), (\ref{3.4}).
Applying the energy method to (\ref{3.6}) we obtain
\begin{equation*}
\frac{d}{dt}\| Eh_{k}\| ^2\leq Ct^{-\alpha }| \int
u_xE\partial_x( \overline{h_{k}}) ^2dx| +C(
\| ER_{k}\| +\| gEh_{k}\| ) \|
Eh_{k}\| ,
\end{equation*}
whence integration by parts yields
\begin{equation}
\frac{d}{dt}\| Eh_{k}\| \leq C\varepsilon t^{-1}\|
Eh_{k}\| +C\| R_{k}\| ,  \label{3.7}
\end{equation}
where $k=0,1,2$. Integrating (\ref{3.7}) with respect to time $t\in [
1,T] $ we obtain the estimate
\begin{equation}
\langle t\rangle ^{-\gamma }\| u\|_{3,0}+\langle
t\rangle ^{-\gamma }\| \mathcal{I}u\|_{1,0}+\langle
t\rangle ^{-3\gamma }\| \mathcal{I}^2u\| <\frac{%
\varepsilon }{2}.  \label{3.8}
\end{equation}
for all $t\in [ 0,T] $. We now estimate $\| \partial_{t}%
\mathcal{FU}( -t) u_x( t) \|_{0,1,\infty }$.
We apply the Fourier transformation to equation (\ref{1.1}), then changing
the dependent variable $\mathcal{F}u_x=e^{-\frac{it}{2}\xi ^2}w$, in
view of (\ref{2.7}) we obtain
\begin{equation}
iw_{t}( t,\xi ) =-\frac{i\xi t^{-\alpha }}{\sqrt{2\pi }}\int
e^{-it\xi \eta }w( t,\xi +\eta ) \overline{w( t,\eta )
}d\eta ,  \label{3.9}
\end{equation}
where $w( t,\xi ) =\mathcal{FU}( -t) u_x$. When $t\in (0,1)$ we get
\begin{equation*}
\| \xi \int e^{-it\xi \eta }w( t,\xi +\eta ) \overline{%
w( t,\eta ) }d\eta \|_{0,1,\infty }\leq C\| w\|
_{0,2}^2\leq C\| u\|_{3,0}^2\leq C\varepsilon ^2
\end{equation*}
and if $t\geq 1$, we integrate by parts with respect to $\eta $,
\begin{eqnarray*}
\lefteqn{\| \xi \int e^{-it\xi \eta }w( t,\xi +\eta ) \overline{%
w( t,\eta ) }d\eta \|_{0,1,\infty } }\\
&\leq &C\langle t\rangle ^{-1}\| \int e^{-it\xi \eta
}\partial_{\eta }( w( t,\xi +\eta ) \overline{w(
t,\eta ) }) d\eta \|_{0,1,\infty } \\
&\leq &C\langle t\rangle ^{-1}\| \partial_{\eta }w\|
_{0,1}\| w\|_{0,1} \\
&\leq &C\langle t\rangle ^{-1}\| \mathcal{J}\partial
_xu\|_{1,0}\| u\|_{2,0}\leq C\varepsilon
^2\langle t\rangle ^{2\gamma -1};
\end{eqnarray*}
therefore,
\begin{equation}
t^{\alpha }\langle t\rangle ^{1-2\gamma }\| \partial_{t}%
\mathcal{FU}( -t) u_x( t) \|_{0,1,\infty }<%
\frac{\varepsilon }{2}.  \label{3.10}
\end{equation}
By (\ref{3.8}) and (\ref{3.10}) we see that estimate (\ref{3.1}) is true for
all $t\in [ 0,T] $. The contradiction obtained proves (\ref{3.1})
for all $t>0$. \end{proof}

To complete the proof of Theorem \ref{Th1.1} we evaluate the large time
asymptotic estimate of the solution $u$. Note that by Lemma \ref{L2.1}
derivative $u_x$ has a quasi linear asymptotic formula
\begin{equation*}
u_x=M\mathcal{D}w+O( t^{\gamma -\frac{3}{4}}\| \mathcal{J}%
\partial_xu\| ) =M\mathcal{D}w+O( t^{2\gamma -\frac{3}{4}%
}) ,
\end{equation*}
where $M=e^{\frac{ix^2}{2t}}$, $\mathcal{D}\phi =\frac{1}{\sqrt{it}}\phi
( \frac{x}{t}) $. For the solution $u( t,x) $ we have
\begin{equation*}
u( t,x) =\mathcal{F}^{-1}e^{-\frac{it}{2}\xi ^2}v=\frac{M}{%
\sqrt{2\pi }}\int e^{-\frac{it}{2}( \xi -\frac{x}{t})
^2}v( t,\xi ) d\xi ,
\end{equation*}
where $v=\mathcal{FU}( -t) u$. In the same way as in the proof of
(\ref{3.10}) we have the estimate
\begin{equation*}
\| \partial_{t}\mathcal{FU}( -t) \mathcal{I}u_x(
t) \|_{0,1,\infty }\leq Ct^{5\gamma -\alpha -1}.
\end{equation*}
To apply Lemma \ref{2.3} we need to prove the representation
\begin{equation*}
\partial_{\xi }v( t,\xi ) =( \alpha -1) B| \xi
| ^{\alpha -1}\xi ^{-1}+O( t^{-\gamma }| \xi |
^{\alpha -2})
\end{equation*}
for all $t^{\theta -1}\leq | \xi | \leq t^{-\mu }$, $\partial
_{\xi }v( t,\xi ) =O( | \xi | ^{\alpha -2})
$ and $v( t,\xi ) =t^{1-\alpha }\Psi ( t\xi ) +O(
t^{1-\alpha -\delta }) $ for all $| \xi | \leq t^{\theta
-1}$, with $\delta \geq \theta +\gamma $. From (\ref{3.3}) we get $\|
\xi \partial_{\xi }v\| \leq \| \partial_x\mathcal{J}u\|
\leq Ct^{\gamma }$. We have
\begin{equation*}
\partial_{\xi }v( t,\xi ) =\xi ^{-1}( 2t\partial_{t}-v-%
\widehat{\mathcal{I}}v) ,
\end{equation*}
where $\widehat{\mathcal{I}}=\mathcal{FIF}^{-1}=-\partial_{\xi }\xi
+2t\partial_{t}$.

Similarly to (\ref{3.5}) we get
\begin{equation*}
\mathcal{LI}u=t^{-\alpha }( \overline{u}_x\mathcal{I}u_x+u_x%
\overline{\mathcal{I}u_x}) +2( 1-\alpha ) t^{-\alpha
}| u_x| ^2,
\end{equation*}
hence
\begin{eqnarray*}
\widehat{\mathcal{I}}v( t,\xi ) &=&\widehat{\mathcal{I}}v(
0,\xi ) +\frac{i}{\sqrt{2\pi }}\int_0^{t}\tau ^{-\alpha }d\tau \int
e^{-i\tau \xi \eta }\widehat{\mathcal{I}}w( \tau ,\xi +\eta )
\overline{w( \tau ,\eta ) }d\eta \\
&&+\frac{i}{\sqrt{2\pi }}\int_0^{t}\tau ^{-\alpha }d\tau \int e^{-i\tau
\xi \eta }w( \tau ,\xi +\eta ) \overline{\widehat{\mathcal{I}}%
w( \tau ,\eta ) }d\eta \\
&&+\frac{2i( 1-\alpha ) }{\sqrt{2\pi }}\int_0^{t}\tau ^{-\alpha
}d\tau \int e^{-i\tau \xi \eta }w( \tau ,\xi +\eta ) \overline{%
w( \tau ,\eta ) }d\eta .
\end{eqnarray*}
Since $t\partial_{t}v=O( t^{1-\alpha }\langle t\xi \rangle
^{-1}) =O( t^{-\gamma }| \xi | ^{\alpha -1}) $
for $t^{\theta -1}\leq | \xi | $ and $t\partial_{t}v=O(
| \xi | ^{\alpha -1}) $ for $| \xi | \leq
t^{\theta -1}$, applying Lemma \ref{L2.2} we get
\begin{eqnarray*}
v_{\xi }( t,\xi ) &=&-\xi ^{-1}( \widehat{\mathcal{I}}%
v+v) +O( t^{-\gamma }| \xi | ^{\alpha -2}) \\
&=&-\frac{i}{\sqrt{2\pi }}\int_0^{t}\tau ^{-\alpha }d\tau \int e^{-i\tau
\xi \eta }\widehat{\mathcal{I}}w( \tau ,\xi +\eta ) \overline{%
w( \tau ,\eta ) }d\eta \\
&&-\frac{i}{\sqrt{2\pi }}\int_0^{t}\tau ^{-\alpha }d\tau \int e^{-i\tau
\xi \eta }w( \tau ,\xi +\eta ) \overline{\widehat{\mathcal{I}}%
w( \tau ,\eta ) }d\eta \\
&&+\frac{2i\alpha }{\sqrt{2\pi }}\int_0^{t}\tau ^{-\alpha }d\tau \int
e^{-i\tau \xi \eta }w( \tau ,\xi +\eta ) \overline{w( \tau
,\eta ) }d\eta +O( t^{-\gamma }| \xi | ^{\alpha
-2}) \\
&=&G( t) | \xi | ^{\alpha -1}\xi ^{-1}+O(
t^{-\gamma }| \xi | ^{\alpha -2})
\end{eqnarray*}
for all $t^{\theta -1}\leq | \xi | \leq t^{-\mu }$, and $\partial
_{\xi }v( t,\xi ) =O( | \xi | ^{\alpha -2})
$ for all $| \xi | \leq t^{\theta -1}$, where
\begin{eqnarray*}
G( t) &=&\frac{2i}{\sqrt{2\pi }}\Gamma ( 1-\alpha )
\sin ( \frac{\pi \alpha }{2}) \Re \int \widehat{\mathcal{I}}%
w( t,\eta ) \overline{w( t,\eta ) }| \eta |
^{\alpha -1}d\eta \\
&&+\frac{2i\alpha }{\sqrt{2\pi }}\Gamma ( 1-\alpha ) \sin (
\frac{\pi \alpha }{2}) \int | w( t,\eta ) |
^2| \eta | ^{\alpha -1}d\eta ,
\end{eqnarray*}
since $| w( t,\eta ) | ^2$sign$\eta $ and $\Re
( \widehat{\mathcal{I}}w( t,\eta ) \overline{w( t,\eta
) }) $sign$\eta $ are odd functions. We have by (\ref{3.10})
\begin{equation*}
\| w( t,\eta ) -w( \tau ,\eta ) \|
_{0,1,\infty }\leq \| \int_{\tau }^{t}\partial_{s}w( s,\eta
) ds\|_{0,1,\infty }\leq C\int_{\tau }^{t}s^{2\gamma -\alpha
-1}ds\leq C\tau ^{2\gamma -\alpha }
\end{equation*}
for all $1\leq \tau \leq t$. Therefore there exists a limit $%
W=\lim_{t\rightarrow \infty }w( t) $ in $H_{\infty
}^{0,1}( \mathbb{R}) $ such that
\begin{equation*}
\| w( t,\eta ) -W\|_{0,1,\infty }\leq C\tau ^{2\gamma
-\alpha }.
\end{equation*}
Similarly to (\ref{3.10}) we get by (\ref{3.3})
\begin{equation}
\| \partial_{t}\widehat{\mathcal{I}}w\|_{0,1,\infty }\leq
C\varepsilon t^{5\gamma -\alpha -1}  \label{3.11}
\end{equation}
for all $t\geq 1$. Hence there exists a limit $K=\lim_{t\rightarrow \infty }%
\widehat{\mathcal{I}}w( t) $ in $H_{\infty }^{0,1}(
\mathbb{R}) $ such that
\begin{equation*}
\| \widehat{\mathcal{I}}w( t,\eta ) -K\|_{0,1,\infty
}\leq C\tau ^{5\gamma -\alpha }.
\end{equation*}
Thus
\begin{equation*}
\partial_{\xi }v( t,\xi ) =B_1| \xi | ^{\alpha
-1}\xi ^{-1}+O( t^{-\gamma }| \xi | ^{\alpha -2}) ,
\end{equation*}
where
\begin{eqnarray*}
B_1 &=&\frac{2i}{\sqrt{2\pi }}\Gamma ( 1-\alpha ) \sin (
\frac{\pi \alpha }{2}) \Re \int K( \eta ) \overline{W(
\eta ) }| \eta | ^{\alpha -1}d\eta \\
&&+\frac{2i\alpha }{\sqrt{2\pi }}\Gamma ( 1-\alpha ) \sin (
\frac{\pi \alpha }{2}) \int | W( \eta ) |
^2| \eta | ^{\alpha -1}d\eta .
\end{eqnarray*}
Also we have
\begin{eqnarray*}
v( t,\xi ) &=&\int_0^{t}\tau ^{-\alpha }d\tau \int e^{-i\tau
\xi \eta }w( \tau ,\xi +\eta ) w( \tau ,\eta ) d\eta \\
&=&\int_0^{t}\tau ^{-\alpha }d\tau \int e^{-i\tau \xi \eta }| W(
\eta ) | ^2d\eta +O( t^{-\gamma }\min ( | \xi
| ^{\alpha -1},t^{1-\alpha }) ) \\
&=&t^{1-\alpha }\int_0^{1}| z| ^{-\alpha }dz\int e^{-iz\eta
t\xi }| W( \eta ) | ^2d\eta +O( t^{1-\alpha
-\delta }) \\
&=&t^{1-\alpha }\Psi ( t\xi ) +O( t^{1-\alpha -\delta
})
\end{eqnarray*}
for $| \xi | \leq t^{\theta -1}$, where $\delta =\alpha -2\gamma
, $ $\Psi (x) =\int_0^{1}| z| ^{-\alpha }dz\int
e^{-iz\eta x}| W( \eta ) | ^2d\eta $. Now
application of Lemma \ref{L2.3} yields asymptotics (\ref{1.2}) for the
solution $u( t,x) $. Using Lemma \ref{L3.2} we get the result of
Theorem \ref{Th1.1} with $B=B_1/\sqrt{2\pi }$, $P=\frac{1}{\sqrt{2\pi }}%
\check{\Psi}$, $Q=\frac{1}{\sqrt{2\pi }}V$. Theorem \ref{Th1.1} is proved.

\section{Proof of Theorem \ref{T1.2}}

By  the method in \cite{h1} (see also the proof of a-priori
estimates below in Lemma \ref{L4.2}), we easily obtain the local existence of
solutions in the analytic functional space
\begin{equation*}
\mathbf{A}_{T}=\big\{ \phi \in \mathbf{C}( [ -T,T] ;\mathbf{%
L}^2( \mathbb{R}) ) :\sup_{t\in [ -T,T]
}\| \phi ( t) \|_{\mathbf{A}_{t}}<\infty \big\} ,
\end{equation*}
where the norm $\mathbf{A}_{t}$ is defined as
\begin{eqnarray*}
\| u\|_{\mathbf{A}_{t}} &=&\langle t\rangle ^{-\gamma
}\| | \partial_x| ^{\frac{1}{2}-\beta }u\|
_{3,0}+t^{\alpha }\langle t\rangle ^{1-\gamma }\| \partial
_{t}\mathcal{FU}( -t) u_x( t) \|_{0,1,\infty }
\\
&&+\sum_{n=1}^{\infty }\frac{\langle t\rangle ^{-n\gamma }}{n!}%
\| | \partial_x| ^{\frac{1}{2}-\beta }\mathcal{I}%
^{n}u\|_{1,0}.
\end{eqnarray*}
Denote
\begin{equation*}
\| u\|_{\mathbb{Z}}=\sum_{n=0}^{\infty }\frac{t^{-n\gamma }}{n!}%
\| | \partial_x| ^{\frac{1}{2}-\beta }\mathcal{I}%
^{n}u\|_{1,0}.
\end{equation*}

\begin{theorem}
\label{T4.1} Let $\alpha \in (0,1) $. We assume that the initial
data $u_0\in \mathbf{A}_0$. Then for some time $T>0$ there exists a
unique solution $u\in \mathbf{A}_{T}$ of the Cauchy problem (\ref{1.1}). If
we assume in addition that the norm of the initial data $\|
u_0\|_{\mathbf{A}_0}=\varepsilon ^2$ is sufficiently small,
then there exists a unique solution $u\in \mathbf{A}_{T}$ of (\ref{1.1}) for
some time $T>1$, such that the following estimate $\sup_{t\in [0,T]}\|
u\|_{\mathbf{A}}<\varepsilon $ is valid.
\end{theorem}

In the next lemma we obtain the estimates of global solutions in the norm $%
\mathbf{A}_{t}$.

\begin{lemma}
\label{L4.2} Let the initial data $u_0\in \mathbf{A}_0$ are such that
the norm $\| u_0\|_{\mathbf{A}_0}=\varepsilon ^2$ is
sufficiently small. Then there exists a unique global solution of the Cauchy
problem (\ref{1.1}) such that $u\in \mathbf{A}_{\infty }$. Moreover the
following estimate is valid
\begin{equation}
\| u\|_{\mathbf{A}_{t}}<\varepsilon  \label{4.1}
\end{equation}
for all $t>0$. \end{lemma}

\begin{proof}
As in Lemma \ref{L3.2} we argue by contradiction and find a maximal time $%
T>1 $ such that the estimate
\begin{equation}
\| u\|_{\mathbf{A}_{t}}\leq \varepsilon   \label{4.2}
\end{equation}
is valid for all $t\in [ 0,T] $. Via Theorem \ref{T4.1} it is
sufficient to consider $t\geq 1$. As above in Lemma \ref{L3.2} we can
estimate the norms $\| u_x\|_{1,0,\infty }$, $\| |
\partial_x| ^{\frac{1}{2}-\beta }\mathcal{J}\partial_xu\|
_{1,0}$, $\| | \partial_x| ^{\frac{1}{2}-\beta }\mathcal{J%
}\partial_x\mathcal{I}u\|_{1,0}$ via the norm $\| u\|_{%
\mathbb{Z}}$. Indeed we have the estimate
\begin{eqnarray*}
\| | \partial_x| ^{\frac{1}{2}-\beta }\mathcal{J}\partial
_xu\|_{1,0} &\leq &\| | \partial_x| ^{\frac{1}{2}%
-\beta }\mathcal{I}u\|_{1,0}+2t\| | \partial_x| ^{%
\frac{1}{2}-\beta }\mathcal{L}u\|_{1,0} \\
&\leq &C\| u\|_{\mathbb{Z}}+C\| u\|_{\mathbb{Z}%
}^2\leq C\varepsilon
\end{eqnarray*}
and
\begin{eqnarray*}
\| | \partial_x| ^{\frac{1}{2}-\beta }\mathcal{J}\partial
_x\mathcal{I}u\|_{1,0} &\leq &\| | \partial_x| ^{%
\frac{1}{2}-\beta }\mathcal{I}^2u\|_{1,0}+2t\| | \partial
_x| ^{\frac{1}{2}-\beta }\mathcal{LI}u\|_{1,0} \\
&\leq &C\| u\|_{\mathbb{Z}}+C\| u\|_{\mathbb{Z}%
}^2\leq C\varepsilon .
\end{eqnarray*}

We first estimate $\| \partial_{t}\mathcal{FU}( -t)
u_x( t) \|_{0,1,\infty }$. We apply the Fourier
transformation to equation (\ref{1.1}), then changing the dependent variable
$\mathcal{F}u_x=e^{-\frac{it}{2}\xi ^2}w$, in view of (\ref{2.7}) we
obtain
\begin{equation}
iw_{t}( t,\xi ) =\frac{i\xi t^{-\alpha }}{\sqrt{2\pi }}\int
e^{-it\xi \eta }w( t,\xi +\eta ) \overline{w( t,\eta )
}d\eta ,  \label{4.3}
\end{equation}
where $w( t,\xi ) =\mathcal{FU}( -t) u_x$. For $%
t\geq 1$, we integrate by parts with respect to $\eta $%
\begin{eqnarray*}
\lefteqn{\| \xi \int e^{-it\xi \eta }w( t,\xi +\eta ) \overline{%
w( t,\eta ) }d\eta \|_{0,1,\infty } }\\
&\leq &C\langle t\rangle ^{-1}\| \int e^{-it\xi \eta
}\partial_{\eta }( w( t,\xi +\eta ) \overline{w(
t,\eta ) }) d\eta \|_{0,1,\infty }\leq C\langle
t\rangle ^{-1}\| \partial_{\eta }w\|_{0,1}\|
w\|_{0,1} \\
&\leq &C\langle t\rangle ^{-1}\| \mathcal{J}\partial
_xu\|_{1,0}\| u\|_{2,0}\leq C\varepsilon
^2\langle t\rangle ^{2\gamma -1};
\end{eqnarray*}
therefore,
\begin{equation}
t^{\alpha +1-2\gamma }\| \partial_{t}\mathcal{FU}( -t)
u_x( t) \|_{0,1,\infty }<\varepsilon  \label{4.4}
\end{equation}
for all $t\geq 1$. In the same manner
\begin{equation}
t^{\alpha +1-5\gamma }\| \partial_{t}\mathcal{FU}( -t)
\mathcal{I}u_x( t) \|_{0,1,\infty }<\varepsilon .
\label{4.5}
\end{equation}
We estimate the norm $\| u\|_{\mathbb{Z}}=\sum_{n=0}^{\infty }%
\frac{t^{-n\gamma }}{n!}\| | \partial_x| ^{\frac{1}{2}%
-\beta }\mathcal{I}^{n}u\|_{1,0}$ for all $t\geq 1$. Note that
\begin{eqnarray*}
\lefteqn{\sum_{n=0}^{\infty }\frac{t^{-\gamma n}}{n!}\| ( \mathcal{I}%
+a) ^{n}u\|_{1,0} }\\
&\leq &\sum_{n=0}^{\infty }\frac{t^{-\gamma n}}{n!}\sum_{j=0}^{n}C_{n}^{j}%
| a| ^{n-j}\| \mathcal{I}^{j}u\|
_{1,0}=\sum_{n=0}^{\infty }\sum_{j=0}^{n-j}\frac{| a| ^{n-j}}{%
( n-j) !}\frac{t^{-\gamma n}}{j!}\| \mathcal{I}^{j}u\|
_{1,0} \\
&=&\sum_{j=0}^{\infty }\frac{t^{-\gamma j}}{j!}\| \mathcal{I}%
^{j}u\|_{1,0}\sum_{k=j}^{\infty }\frac{| at| ^{k-j}}{%
( k-j) !}=e^{| a| t^{-\gamma }}\sum_{j=0}^{\infty }%
\frac{t^{-\gamma j}}{j!}\| \mathcal{I}^{j}u\|_{1,0}\leq C\|
u\|_{\mathbb{Z}}
\end{eqnarray*}
We have by Lemma \ref{L2.4} denoting $C_{n}^{m}=\frac{n!}{m!(
n-m!) }$%
\begin{eqnarray*}
\| \mathcal{L}u\|_{\mathbb{Z}} &=&\| t^{-\alpha }|
u_x| ^2\|_{\mathbb{Z}}=\sum_{n=0}^{\infty }\frac{%
t^{-n\gamma }}{n!}\| | \partial_x| ^{\frac{1}{2}-\beta }%
\mathcal{I}^{n}t^{-\alpha }| u_x| ^2\|_{1,0} \\
&=&t^{-\alpha }\sum_{n=0}^{\infty }\frac{t^{-n\gamma }}{n!}\| |
\partial_x| ^{\frac{1}{2}-\beta }( \mathcal{I}-2\alpha )
^{n}| u_x| ^2\|_{1,0} \\
&=&t^{-\alpha }\sum_{n=0}^{\infty }\frac{t^{-n\gamma }}{n!}%
\sum_{j=0}^{n}C_{n}^{j}\| | \partial_x| ^{\frac{1}{2}%
-\beta }( \mathcal{I}-2\alpha ) ^{n-j}u_x( \mathcal{I}%
-2\alpha ) ^{j}\overline{u}_x\|_{1,0} \\
&=&\sum_{n=0}^{\infty }\sum_{j=0}^{n}\frac{t^{-\alpha -3n\gamma }}{(
n-j) !j!}\| | \partial_x| ^{\frac{1}{2}-\beta
}( ( \mathcal{I}-1-2\alpha ) ^{n-j}u)_x(
( \mathcal{I}-1-2\alpha ) ^{j}\overline{u})_x\|
_{1,0} \\
&\leq &C\sum_{n=0}^{\infty }\sum_{j=0}^{n}\frac{t^{-\alpha -3n\gamma +\beta
-1}}{( n-j) !j!}\| | \partial_x| ^{\frac{1}{2}%
-\beta }( \mathcal{I}-1-2\alpha ) ^{n-j}u\|_{1,0} \\
&&( \| | \partial_x| ^{\frac{1}{2}-\beta }(
\mathcal{I}-1-2\alpha ) ^{j}u\|_{1,0}+\| | \partial
_x| ^{\frac{1}{2}-\beta }( \mathcal{I}-1-2\alpha ) ^{j}%
\mathcal{J}\partial_xu\|_{1,0}) \\
&\leq &Ct^{-\alpha +\beta -1}\| u\|_{\mathbb{Z}}( \|
u\|_{\mathbb{Z}}+\| \mathcal{I}u\|_{\mathbb{Z}}+t\|
\mathcal{L}u\|_{\mathbb{Z}}) ,
\end{eqnarray*}
hence we get
\begin{eqnarray*}
\| \mathcal{L}u\|_{\mathbb{Z}} &\leq &Ct^{-\alpha +\beta
-1}\| u\|_{\mathbb{Z}}^2+Ct^{-\alpha +\beta -1}\|
u\|_{\mathbb{Z}}\| \mathcal{I}u\|_{\mathbb{Z}} \\
&\leq &Ct^{-\gamma -1}( \| u\|_{\mathbb{Z}}+\|
\mathcal{I}u\|_{\mathbb{Z}}) .
\end{eqnarray*}
In particular we have
\begin{equation*}
\| \mathcal{J}\partial_xu\|_{\mathbb{Z}}\leq C\|
u\|_{\mathbb{Z}}+C\| \mathcal{I}u\|_{\mathbb{Z}}.
\end{equation*}
Using the commutation relations $\mathcal{I}^{n}\partial_x=\partial
_x( \mathcal{I}+1) ^{n}$, $\mathcal{LI}^{n}=( \mathcal{I}%
+2) ^{n}\mathcal{L}$, $\mathcal{I}^{n}t^{-\alpha }=t^{-\alpha }(
\mathcal{I}-2\alpha ) ^{n}$ applying operator $\mathcal{I}$ to
equation (\ref{1.1}) we get
\begin{equation}
\mathcal{LI}^{n}u=t^{-\alpha }( \mathcal{I}+2( 1-\alpha )
) ^{n}| u_x| ^2.  \label{4.6}
\end{equation}
Via (\ref{4.3}) we obtain
\begin{eqnarray*}
\lefteqn{\sum_{n=0}^{\infty }t^{-\alpha -\gamma n}( n!) ^{-1}\|
| \partial_x| ^{\frac{1}{2}-\beta }( \mathcal{I}%
+2-2\alpha ) ^{n}| u_x| ^2\|_{1,0} }\\
&\leq &C\sum_{n=0}^{\infty }t^{-\alpha -\gamma n}( n!)
^{-1}\sum_{m=0}^{n}C_{n}^{m}( 2-2\alpha ) ^{n-m} \\
&&\times \sum_{j=0}^{m}C_{m}^{j}\| | \partial_x| ^{\frac{1%
}{2}-\beta }( ( \mathcal{I}+1) ^{j}u)_x\overline{%
( ( \mathcal{I}+1) ^{m-j}u)_x}\|_{1,0} \\
&\leq &Ct^{\beta -\alpha -1}\| u\|_{\mathbb{Z}}^2+Ct^{\beta
-\alpha -1}\| u\|_{\mathbb{Z}}\| \mathcal{J}\partial
_xu\|_{\mathbb{Z}} \\
&\leq &C\varepsilon t^{-\gamma -1}( \| u\|_{\mathbb{Z}%
}+\| \mathcal{I}u\|_{\mathbb{Z}})\,.
\end{eqnarray*}
Using Lemma \ref{L2.4} we obtain
\begin{eqnarray*}
\lefteqn{\| | \partial_x| ^{\frac{1}{2}-\beta }( (
\mathcal{I}+1) ^{j}u)_x\overline{( ( \mathcal{I}%
+1) ^{m-j}u)_x}\|_{1,0} }\\
&\leq &Ct^{\beta -1}\| | \partial_x| ^{\frac{1}{2}-\beta
}( \mathcal{I}+1) ^{j}u\|_{1,0}\| | \partial
_x| ^{\frac{1}{2}-\beta }( \mathcal{I}+1) ^{m-j}u\|
_{1,0} \\
&&+Ct^{\beta -1}\| | \partial_x| ^{\frac{1}{2}-\beta
}( \mathcal{I}+1) ^{j}u\|_{1,0}\| | \partial
_x| ^{\frac{1}{2}-\beta }( \mathcal{I}+1) ^{m-j}\mathcal{J%
}\partial_xu\|_{1,0} \\
&&+Ct^{\beta -1}\| | \partial_x| ^{\frac{1}{2}-\beta
}( \mathcal{I}+1) ^{j}\mathcal{J}\partial_xu\|
_{1,0}\| | \partial_x| ^{\frac{1}{2}-\beta }(
\mathcal{I}+1) ^{m-j}u\|_{1,0}.
\end{eqnarray*}
Hence by the energy method, in view of (\ref{4.2}) we find
\begin{equation}
\frac{d}{dt}\| u\|_{\mathbb{Z}}\leq -\gamma t^{-1-\gamma
}\| \mathcal{I}u\|_{\mathbb{Z}}+C\varepsilon t^{-\gamma
-1}\| u\|_{\mathbb{Z}}+C\varepsilon t^{-\gamma -1}\|
\mathcal{I}u\|_{\mathbb{Z}}\leq C\varepsilon t^{\beta -\alpha
-1}\| u\|_{\mathbb{Z}}.  \label{4.7}
\end{equation}
Integration of (\ref{4.7}) with respect to time $t\geq 1$ yields $\|
u\|_{\mathbb{Z}}<\frac{\varepsilon }{2}$ for all $t\geq 1$. The norm $%
\| | \partial_x| ^{\frac{1}{2}-\beta }u\|_{3,0}$
is estimated in the same manner as in the proof of Lemma \ref{L3.2}.
Therefore, Lemma \ref{L4.2} is proved.
\end{proof}

Now we complete the proof of Theorem \ref{T1.2} by applying Lemmas
\ref{L2.2} and \ref{L2.3} as in the previous section.

\noindent\textbf{Acknowledgments.} The second author was
partially supported by a grant from CONACYT.

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\end{document}