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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 169, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/169\hfil Persistence of solutions]
{Persistence of solutions to nonlinear evolution equations in
weighted Sobolev spaces}

\author[X. Carvajal, P.  Gamboa \hfil EJDE-2010/169\hfilneg]
{Xavier Carvajal Paredes, Pedro Gamboa Romero}  % in alphabetical order

\address{Xavier Carvajal \newline
IM UFRJ, Av. Athos da Silveira Ramos, P.O. Box 68530.
CEP 21945-970.  RJ. Brazil}
\email{carvajal@im.ufrj.br, Phone 55-21-25627520 }

\address{Pedro Gamboa Romero \newline
IM UFRJ, Av. Athos da Silveira Ramos, P.O. Box 68530.
CEP:21945-970.RJ. Brazil}
\email{pgamboa@im.ufrj.br,  Phone 55-21-25627520}

\thanks{Submitted October 18, 2010. Published November 24, 2010.}
\subjclass[2000]{35A07, 35Q53}
\keywords{Schr\"{o}dinger equation;
Korteweg-de Vries equation; \hfill\break\indent
global well-posed; persistence property; weighted Sobolev spaces}

\begin{abstract}
 In this article, we prove that the initial value problem associated with
 the Korteweg-de Vries equation is well-posed in weighted
 Sobolev spaces $\mathcal{X}^{s,\theta}$,
 for  $s \geq 2\theta \ge 2$ and the initial value problem
 associated with the nonlinear Schr\"odinger equation is
 well-posed in weighted Sobolev spaces $\mathcal{X}^{s,\theta}$,
 for  $s \geq \theta \geq 1$. Persistence property has been
 proved by approximation of the solutions and using
 a priori estimates.
\end{abstract}

\maketitle
\numberwithin{equation}{section}

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

\section{Introduction}\label{IN}

In this paper we consider the initial value problem (IVP)
for the Korteweg-de Vries (KdV) equation
\begin{equation}\label{IVP}
\begin{gathered}
\partial_t u + u_{xxx} + a(u)u_x=0, \quad (t,x) \in \mathbb{R}\times \mathbb{R},\\
u(x,0)  =  u_0(x),
\end{gathered}
\end{equation}
where $u$ a real-valued function and $a \in C^\infty(\mathbb{R}, \mathbb{R})$
is a real function.

And the initial value problem for the nonlinear Schr\"odinger (NLS)
equation
\begin{equation}\label{NIVP}
\begin{gathered}
\partial_t u = i(\Delta u - F(u))=0, \quad (t,x) \in \mathbb{R}\times \mathbb{R}^n,\\
u(x,0)  =  u_0(x),
\end{gathered}
\end{equation}
where $u$ a complex-valued function and $F$ satisfies:
\begin{itemize}
\item[(F1)] $F \in C^{[s]+1}(\mathbb{C},\mathbb{C})$ with $F(0)=0$.

\item[(F2)] If $s \le n/2$ and if $F(\eta)$ is a polynomial in
 $\eta$ and $\bar{\eta}$, then
$\deg (F)=k \le \chi(s):=1+4/(n-2\sigma)$,
$-\infty\le \sigma \le n/2$. If $s \le n/2$ and if $F(\eta)$
is not a polynomial, then
\begin{equation}\label{eqa}
|D^i F(\eta)| \le c |\eta|^{k-i}, \quad i=0,1,\dots,[s]+1, \quad
\text{as }  |\eta| \to \infty,
\end{equation}
where $[s]+1\le k \le \chi(s)$.
\end{itemize}

The above conditions on $a$ and $F$ guarantee the
well-posedness for  \eqref{IVP} and \eqref{NIVP}
in the usual Sobolev spaces $H^s$, $s \ge 2$ and $H^s$,
$s \ge 1$ respectively, see \cite{K1, KT3}.
We are mainly concerned with the question of the  persistence
property in weighted Sobolev spaces. The aim of this work is to
use Lemmas proved in \cite[Lemmas 3 and 4 ]{NP}
 and to apply this result to show persistence property of
 \eqref{IVP} in
$\mathcal{X}^{s,\theta}$ (see definition in \eqref{SSWW})
for $s \geq 2\theta \ge 2$ and persistence property of  \eqref{NIVP} in
$\mathcal{X}^{s,\theta}$ for $s \geq \theta \ge 1$.
The notation we took are from \cite{CN1}.

In what follows we introduce the notion of well-posedness that
we are going to use throughout this work. We say that
 \eqref{IVP} is locally well-posed in a Banach space $X$,
if the following hold.

\begin{enumerate}
\item There exist $T>0$ and a unique solution $u$ in the
time interval $[-T,T]$ (unique existence).

\item The solution varies continuously depending upon
the initial data (continuous dependence); that is,
continuity of the application
$$
u_{0} \to  u \quad \text{from $X$ to $\mathcal{C}([-T,T];X)$}.
$$
In particular if $u_0^n \to u_0$ when $n \to \infty$, then
\begin{equation}\label{mnx1}
\sup_{t \in[-T,T]}\|u_n(t)-u(t)\|_{H^s}\to 0,
\end{equation}
where $u_n(t)$ is solution of \eqref{IVP} with initial data $u_0^n$.

\item The solution describes a continuous curve in $X$ in the time
interval $[-T,T]$ whenever initial data belongs to $X$
(persistence).

\end{enumerate}

Moreover, we say that \eqref{IVP} is globally well-posed in $X$
if the same properties hold for all time $T>0$. If some of the
hypotheses in the definition of local well-posedness fail, we say
that the IVP is well-posed.

Our main focus in this work will be to show the  persistence
property. In \cite{CN1} they proved the persistence property for
an equation mixed Korteweg-de Vries - Nonlinear Schr\"odinger with
a weight of low regularity.  To accomplish this they used an
abstract interpolation lemma (\cite[Lemma 2.2]{CN1}).

The interpolation lemma proved in \cite{CN1} is quite  general and
applies to several equations provided they satisfy certain \emph{a
priori} estimates. These \emph{a priori} estimates are related to
the conserved quantities and are as follows.
\begin{gather}\label{ap-1}
\|u(t)\|_{L^2} \le C \|u_0\|_{L^2}. \\
\label{ap-2}
\|u(t)\|_{\dot{H}^1} \le  C\|u_0\|_{\dot{H}^1}+  A_1(\|u_0\|_{L^2}).\\
\label{ap-3}
\|u(t)\|_{\dot{H}^2} \le C A_2(\|u_0\|_{\dot{H}^2},
 \|u_0\|_{\dot{H}^1}, \|u_0\|_{L^2}).\\
\label{ap-4}
\|u(t)\|_{L^2(d\dot{\mu}_r)} \le C \|u_0\|_{L^2(d\dot{\mu}_r)}+
A_3(\|D_{x}^a u_0\|_{L^2}, \|D_{x}^{a-1} u_0\|_{L^2},\dots,
\|u_0\|_{L^2}),
\end{gather}
where $a=a(r) \ge 1$, $r \in \mathbb{Z}^{+}$, $A_j$ are continuous
functions with $A_1(0)=0$, $A_2(0,0,0)=0$ and $A_3(0,\dots, 0)=0$.

It can be inferred that, if one has local well-posedness  result
for given data in $H^s$ and if the model under consideration
satisfies  \emph{a priori} estimates \eqref{ap-1}-\eqref{ap-4},
then with the help of an abstract interpolation lemma, it is easy
to prove persistence property in weighted Sobolev spaces.

A typical example of  \eqref{IVP} that satisfies the  properties
\eqref{ap-1}--\eqref{ap-4} listed above is the IVP associated to
the generalized Korteweg-de Vries (gKdV) equation ($a(x)=x^k$ in
\eqref{IVP})
\begin{equation}\label{IVP-1}
\begin{gathered}
\partial_t u + \partial_{xxx}u +u^k\partial_xu=0, \quad (t,x) \in \mathbb{R}^2,\;
 k = 1, 2, 3, \dots \\
u(x,0)  =  u_0(x).
\end{gathered}
\end{equation}
Another typical example is the IVP  associated to the Nonlinear
Schr\"odinger (NLS) equation, \eqref{NIVP} when $F(x)=\mu
|x|^{\alpha-1}$.
\begin{equation}\label{nls-1}
\begin{gathered}
i\partial_t u +\Delta u = \mu |u|^{\alpha-1}u, \quad
 \mu = \pm 1, \; \alpha >1,\; x \in \mathbb{R}^n,\; t\in \mathbb{R} \\
u(x,0) = u_0(x),
\end{gathered}
\end{equation}
the local well-posedness has been studied in \cite{HNT}
for given data in the weighted Sobolev spaces. More precisely,
the following result that deals with the persistence property
has been proved in \cite{HNT}.

\begin{theorem}\label{thm1}
Suppose that $u_0\in H^{s}(\mathbb{R}^n)\cap L^2(|x|^{2m}dx)$, $m\in
\mathbb{Z}^+$, with $m\leq \alpha-1$ if $\alpha$ is not an odd integer.
\begin{itemize}
\item[(A)] If $s\geq m$, then there exist
$T=T(\|u_0\|_{s,2})>0$ and a unique solution $u=u(x,t)$ of
\eqref{nls-1} with
\begin{equation}\label{eq-m1}
u\in C([-T, T]; H^s\cap L^2(|x|^{2m}dx))
\cap L^q([-T, T];L_s^p\cap L^p(|x|^{2m}dx)).
\end{equation}

\item[(B)] If $1\leq s <m$, then \eqref{eq-m1} holds with $[s]$
instead of $m$, and
\begin{equation}\label{eq-m2}
\Gamma^{\beta}u=(x_j+2it\partial_{x_j})^{\beta}u\in C([-T, T]; L^2)\cap L^q([-T, T]; L^p),
\end{equation}
for any $\beta \in (\mathbb{Z}^+)^n$ with $|\beta|\leq m$.
\end{itemize}
\end{theorem}

 The power $m$ of the weight in Theorem \ref{thm1} is
assumed to be a positive integer. In the recent work of Nahas and
Ponce \cite{NP}, this restriction in $m$ is relaxed by proving
that the persistence property holds for positive real $m$. To be
more precise, the result in \cite{NP} is the following.

\begin{theorem}\label{thm2}
Suppose that $u_0\in H^s(\mathbb{R}^n)\cap L^2(|x|^{2m}dx)$, $m>0$, with
$m\leq \alpha-1$ if $\alpha$ is not an odd integer.
\begin{itemize}
\item[(A)]
If $s\geq m$, then there exist $T=T(\|u_0\|_{s,2})>0$ and a unique
solution $u=u(x,t)$ of  \eqref{nls-1} with
\begin{equation}\label{eq-m3}
u\in C([-T, T]; H^s\cap L^2(|x|^{2m}dx))\cap L^q([-T, T];
L_s^p\cap L^p(|x|^{2m}dx)).
\end{equation}

\item[(B)] If $1\leq s <m$, then \eqref{eq-m3} holds with $[s]$
instead of $m$, and
\begin{equation}\label{eq-m4}
\Gamma^b\Gamma^{\beta}u\in C([-T, T]; L^2)\cap L^q([-T, T]; L^p),
\end{equation}
where $\Gamma^b = e^{i|x|^2/4t}2^bt^bD^b(e^{i|x|^2/4t}.)$
with $|\beta|=[m]$ and $b=m-[m]$.
\end{itemize}
\end{theorem}


Kato \cite{KT1} studied the IVP associated to the  gKdV
equation for given data in the weighted Sobolev spaces
and proved the following result.

\begin{theorem}[Kato]\label{kato}
Let $r\in \mathbb{Z}^+$, then the IVP for  \eqref{IVP-1}
is locally well-posed in weighted Sobolev
spaces $\mathcal{X}^{2r,r}$, and globally well-posed in
$\mathcal{X}^{2r,r}$ if the initial data satisfies
$\|u_0\|_{L^2}< \gamma$.
\end{theorem}

In this work we are interested in removing the requirement
that the power of the weight in Theorem~\ref{kato} is integer,
by proving the similar result for the non integer values of
$r \ge 1$ and also we present a proof simples for the persistence
in weighted Sobolev spaces for the generalized non-linear
Schr\"odinger equation \eqref{nls-1} for the non integer
values of $r \ge 1$. In \cite{NP} they cover all possible
values of the parameters $s, \theta$ in the spaces
$\mathcal{X}^{s,\theta}$. The main results of this paper
are the following.

\begin{theorem}\label{mainthm}
Problems \eqref{IVP} and \eqref{NIVP} are local well-posed
in weighted Sobolev spaces $\mathcal{X}^{s,\theta}$, for
$s \ge 2\theta \ge 2$ and $\mathcal{X}^{s,\theta}$,
for $s \ge \theta \ge 1$ respectively.
\end{theorem}

Without loss of generality in the proof of Theorem \ref{mainthm},
we will restrict our attention to  \eqref{IVP-1} and \eqref{nls-1}.
As an application of Theorem~\ref{mainthm} we have the following
result.

\begin{theorem}\label{mainthm1}
Problem  \eqref{IVP} is globally well-posed in
$\mathcal{X}^{s,\theta}$, for $s \ge 2 \theta \ge 2$,
if the initial data satisfies $\|u_0\|_{L^2}< \gamma$.
\end{theorem}

Is not difficult to see that a similar proof as in \cite{CN1}
proves local well-posedness for  \eqref{IVP}, in weighted Sobolev
spaces $\mathcal{X}^{s,\theta}$, $s\ge 2$ and $\theta \in [0,1]$.

For other results about persistence, for the problems \eqref{IVP-1} and \eqref{nls-1} see the work of Nahas and Ponce in \cite{NP0}, see also Nahas, \cite{JN} for persistence of the modified Korteweg-de Vries equation (k=2 in \eqref{nls-1}).
%And for other results about the IVP for \eqref{nls-1} see \cite{[O-T], [SS], T1, Ts, Cz}
\subsection*{Notation and Background:}
 We follow the  notation
introduced in earlier paper \cite{CN1}. For the sake of clarity we
recall them here. We use $dx$ to denote the Lebesgue measure on
$\mathbb{R}$ and, for $\theta \geq 0$, we use
\begin{gather*}
d\mu_\theta(x):= (1 + |x|^2)^ \theta \,dx,\\
d\dot{\mu}_\theta(x):= |x|^{2 \theta} \,dx
\end{gather*}
to denote the Lebesgue-Stieltjes measures on $\mathbb{R}$. Hence, given a
set $X$, a measurable function $f \in L^2(X;d\mu_\theta)$ means
that
$$
\|f\|_{L^2(X ; d\mu_\theta)}^2= \int_X |f(x)|^2 \,d\mu_\theta(x)<
\infty.
$$
When $X= \mathbb{R}$, we write: $L^2(d\mu_\theta) \equiv L^2(\mathbb{R};d\mu_\theta)$,
and for simplicity
$$
L^2 \equiv L^2(d\mu_0), \quad L^2(d\mu) \equiv L^2(d\mu_1).
$$
Analogously, for the measure $d\dot{\mu}_\theta$. We will use the
Lebesgue space-time $L_{x}^{p}\mathcal{L}_{\tau}^{q}$ endowed with
the norm
$$
\|f\|_{L_{x}^{p}\mathcal{L}_{\tau}^{q}}
= \big\| \|f\|_{\mathcal{L}_{\tau}^{q}} \big\|_{L_{x}^{p}}
= \Big( \int_{\mathbb{R}} \Big( \int _{0}^{\tau} |f(x,t)|^{q} dt \Big)^{p/q} dx
\Big)^{1/p} \quad (1 \leq p,q < \infty).
$$
When the integration in the time variable is on the whole real line,
we use the notation $\|f\|_{L_{x}^{p}L_t^{q}}$.
The notation $\|u\|_{L^p}$ is used when there is no doubt about
the variable of integration. Similar notations when $p$ or $q$ are
$\infty$.

As usual, $H^s \equiv H^s(\mathbb{R}^n)$, $\dot{H}^s \equiv \dot{H}^s(\mathbb{R}^n)$
are the classic Sobolev spaces in $\mathbb{R}^n$, endowed respectively
with the norms
$$
\|f\|_{H^s}:= \|\widehat{f}\|_{L^2(d\mu_s)},
\quad
\|f\|_{\dot{H}^s}:=\|\widehat{f}\|_{L^2(d\dot{\mu}_s)}.
$$

In this work, we study the solutions of \eqref{IVP}
in the Sobolev spaces with weight $\mathcal{X}^{s,\theta}$,
defined as
\begin{equation}\label{SSWW}
\mathcal{X}^{s,\theta}:=H^s \cap L^2(d\mu_\theta),
\end{equation}
with the norm
$$
\| f\|_{\mathcal{X}^{s,\theta}}:=\| f\|_{H^{s}}
+\|f\|_{L^2(d\mu_\theta)}.
$$

\begin{remark}\label{IWSS} \rm
We remark that, $\mathcal{X}^{s,1} \subseteq
\mathcal{X}^{s,\theta}$, for all $s\in \mathbb{R}$ and  $\theta\in [0,1]$.
\end{remark}

Indeed, using H\"older's inequality
$$
\|f\|_{L^2(d\dot{\mu}_\theta)}
\leq \|f\|_{L^2}^{1-\theta}\;\|f\|_{L^2(d\dot{\mu})}^\theta.
$$

\begin{remark}\label{IWRR} \rm
Let $b \in\mathbb{R}$ to denote
$$
D^b f(x)= {((2\pi|\xi|)^b \hat{f}\;)}^\vee(x).
$$
\end{remark}

We follow the notation of the classical $\psi$. d.o's in $\mathcal{S}_{1,0}^m$:
$$
\mathcal{S}_{1,0}^m:=\{a \in\mathcal{C}^\infty(\mathbb{R}^{2n}):
 |\partial_{x}^{\alpha}\partial_{\xi}^{\beta} a(x,\xi)| \leq
C_{\alpha,\beta}(1+|\xi|)^{m-|\beta|}\quad\forall \alpha,\beta
\in (\mathbb{Z}^+)^n\}.
$$

The proof of the following lemmas can be found in \cite{NP}.

\begin{lemma}\label{oper}
If $a\in\mathcal{S}_{1,0}^{0}$ and $\langle x\rangle:=(1+|x|^2)^{1/2}$, then
$$
a(x,D): L^{2}(\mathbb{R}^{n};d\mu_{b})\to L^{2}(\mathbb{R}^{n};d\mu_{b}),
\quad \forall\; b\geq 0.
$$
is the differential, limited operator.
\end{lemma}

\begin{lemma}\label{opera-l}
Let $a, b>0$. Suppose that $ D^a f\in L^2(\mathbb{R}^n)$
and $\langle x\rangle^b f= (1+|x|^2)^{b/2} f\in L^2(\mathbb{R}^n)$.
Then
\begin{equation}
\|\langle x\rangle^{\theta b} D^{(1-\theta)a}
f\|_{L^2}\leq C \|\langle x\rangle^b f\|_{L^2}^{\theta}
\|D^{a} f\|_{L^2}^{1-\theta}.
\end{equation}
\end{lemma}

\section{Statement of the well-posedness result}

 In this  section we prove the well-posedness of the
Cauchy problem \eqref{IVP} in the weighted Sobolev space
$\mathcal{X}^{s,\theta}$, for $\theta \ge 1$ and $s \ge 2\theta$.

\begin{lemma}\label{l1.2}
If  $u_0 \in {L^2(d\dot{\mu}_{\theta})}$, $\theta \in [0,1]$,
$\lambda > 0$ and $u_0^{\lambda}(x)= \mathcal{F}^{-1}(
{\bf\chi}_{\{|\xi|< \lambda \}} \widehat{u_0})(x)$,
then
\begin{equation}\label{2.40}
\|u_0^\lambda\|_{L^2(d\dot{\mu}_{\theta})}
\leq\|u_0\|_{L^2(d\dot{\mu}_{\theta})}.
\end{equation}
\end{lemma}

If $\theta=0$, \eqref{2.40} is a direct consequence of
Plancherel's theorem and definition of $u_0^\lambda$.
If $\theta=1$, using properties of Fourier transform we obtain
$$
|\widehat{x u_0^{\lambda}}(\xi)|=|\partial_{\xi}
\widehat{u_0^{\lambda}}(\xi)|=|{\bf\chi}_{\{|\xi|< \lambda
\}} \partial_{\xi} \widehat{u_0}(\xi)|={\bf\chi}_{\{|\xi|<
\lambda \}}|\widehat{x u_0}(\xi)|.
$$
Thus by Plancherel's equality
\begin{equation*}
\int_{\mathbb{R}}x^2|u_0^{\lambda}(x)|^2 dx = \int_{\mathbb{R}}|\widehat{x
u_0^{\lambda}}(\xi)|^2 d\xi \le \int_{\mathbb{R}}|\widehat{x u_0}(\xi)|^2
d\xi=\int_{\mathbb{R}}|x u_0(x)|^2 dx.
\end{equation*}
When $\theta \in (0,1)$, we obtain \eqref{2.40} by interpolation
between the cases $\theta=0$ and $\theta=1$, see \cite{B-L}.

Lemmas \ref{lem-e} and \ref{kdv-0} tells nothing new;
we present a proof for the sake of completeness

\subsection{A priori estimates for the nonlinear Schr\"odinger equation}

\begin{lemma}\label{lem-a}
If $u \in \mathbb{S}(\mathbb{R}^n)$, $r \ge 1$. Then
\begin{equation*}
\int_{\mathbb{R}^n} \langle x\rangle^{2r-2}|D_{x}u|^2\,dx \le
\Big(\int_{\mathbb{R}^n}\langle x\rangle^{2r}|u|^2\,dx
\Big)^{1-\frac{1}{r}}\Big(\int_{\mathbb{R}^n}|D^{r}u|^2\,dx \Big)^{1/r}.
\end{equation*}
\end{lemma}

\begin{proof}
We apply the Lemma \ref{opera-l}, taking  $a=b=r$ and  $\theta= 1-\dfrac{1}{r}$, then $r \ge 1$
since $0 \le \theta \le1$.
\end{proof}

\begin{lemma}\label{lem-c}
If $u \in \mathbb{S}(\mathbb{R}^n)$.
Then
\[
\int_{\mathbb{R}^n} \langle x\rangle^{2b}|\nabla u(t,x)|^2dx
\leq   b_{r,n}\int_{\mathbb{R}^n} \langle x\rangle^{2b}|D_{x} u(t,x)|^2dx
+ b_{r,n}\int_{\mathbb{R}^n} \langle x\rangle^{2b}|u(t,x)|^2dx.
\]
\end{lemma}

\begin{proof}
Since $\widehat{D_{x} u}(\xi)=|\xi| \widehat{u}(\xi)$, we consider
$a(x,\xi):= \dfrac{\xi_j}{1+|\xi|}$ and using the Lemma \ref{oper},
we can see that the operator $a(x,\xi)$ is bounded and
$a \in S_{1,0}^0$.
\end{proof}

\begin{lemma}\label{lem-e}
If $u$ is a solution of the IVP for the NLS \eqref{nls-1}
with $u_0 \in\mathcal{X}^{s,r}$, $s \ge r \ge 1$. Then
\begin{equation}\label{mm1}
\int_{\mathbb{R}^n}\varphi|u|^2\,dx \le \big\{
C_{r,n}\sup_{t\in[-T,T]}\|u(t)\|_{H^r(\mathbb{R}^n)}^2+\|u(0)\|_{L^2(d\mu_r)}^2
\big\} e^{c_{r,n}T}.
\end{equation}
\end{lemma}

\begin{proof}
Consider $\varphi(x):= (1+|x|^2)^r= \langle x\rangle^{2r}$
to $x \in\mathbb{R}^n$
Multiplying the term $\varphi\overline{u}$ where
$u \in S(\mathbb{R}^n)$ in equation \eqref{nls-1} and after
integrating on $\mathbb{R}^n$, we obtain taking real part
\begin{equation}\label{ab1}
2\Re\big\{\int_{\mathbb{R}^n}u_t\varphi\overline{u}\,dx \big\}
-2\Re\big\{i\int_{\mathbb{R}^n}\Delta u\varphi\overline{u}\,dx\big\}
= -2\mu\Re\big\{i\int_{\mathbb{R}^n}|u|^{\alpha}\varphi\,dx\big\}
\end{equation}
observe that $\partial_t{u.\overline{u}}= 2\Re\{u.\overline{u}_t\}$.
Replacing in \eqref{ab1}, we obtain
\begin{equation}\label{ab2}
\partial_t{\int_{\mathbb{R}^n}\varphi\,dx |u|^2}
= 2\Re\big\{{i\int_{\mathbb{R}^n}\Delta u\varphi\overline{u}\,dx\big\}},
\end{equation}
on the other hand
\begin{equation} \label{ab3}
\begin{aligned}
\int_{\mathbb{R}^n}\varphi\partial_{x_i}^2u \overline{u}\,dx
&=-\int_{\mathbb{R}^n} \partial_{x_i}(\varphi\overline{u})\partial_{x_i}u\,dx \\
&= \int_{\mathbb{R}^n}(\varphi\partial_{x_i}^2 \overline{u}+2\partial_{x_i}\varphi
  \partial_{x_i}\overline{u}+\partial_{x_i}^2\varphi
 \overline{u})u\,dx,
\end{aligned}
\end{equation}
of \eqref{ab3}, we obtain
\begin{equation*}%\label{ab4}
\int_{\mathbb{R}^n}\varphi\Delta u \overline{u}\,dx
=\int_{\mathbb{R}^n}(\varphi\Delta\overline{u}+2\nabla\varphi.\nabla\overline{u}+
\Delta\varphi\overline{u}\;)u\,dx,
\end{equation*}
which leads us to
\begin{equation}\label{ab5}
2i\int_{\mathbb{R}^n}\varphi \Im\{\Delta u\overline{u}\}\,dx
= \int_{\mathbb{R}}\Delta\varphi|u|^2\,dx
+2\int_{\mathbb{R}^n}\nabla\varphi.\nabla\overline{u} u\,dx,
\end{equation}
of \eqref{ab2} and \eqref{ab5}, we obtain
$$
\partial_t\int_{\mathbb{R}^n}\varphi|u|^2\,dx
=  i\int_{\mathbb{R}}\Delta\varphi|u|^2\,dx
+2i\int_{\mathbb{R}^n}\nabla\varphi.\nabla\overline{u} u\,dx,
$$
and taking real part
\begin{equation}\label{ab6}
\partial_t\int_{\mathbb{R}^n}\varphi|u|^2\,dx
= 2\Re\big\{i\int_{\mathbb{R}^n}\nabla\varphi.
\nabla\overline{u} u\,dx\big\}.
\end{equation}
Notice that
\begin{equation}\label{cd1}
|\nabla\varphi| \le 2r\langle x\rangle^{2r-1},
\end{equation} so
\begin{equation}
\begin{aligned}\label{ab7}
\big|\Im\big\{\int_{\mathbb{R}^n}\nabla\varphi.
\nabla\overline{u} u\,dx\big\}\big|
&\le \int_{\mathbb{R}^n}|\nabla\varphi\|\nabla u\|u|\,dx \\
 &\le 2r \int_{\mathbb{R}^n}\langle x\rangle^{r}|u|\langle x\rangle^{r-1} |\nabla u|\,dx \\
 &\le r \int_{\mathbb{R}^n}\varphi|u|^2\,dx
+r\int_{\mathbb{R}^n}\langle x\rangle^{2r-2} |\nabla u|^2\,dx.
\end{aligned}
\end{equation}
Applying Lemma~\ref{lem-c}, \eqref{ab2} and \eqref{ab7}, we have
\begin{equation}\label{cd3}
\partial_t\int_{\mathbb{R}^n}\varphi|u|^2\,dx \le c_{r,n}
\int_{\mathbb{R}^n}\varphi|u|^2\,dx+c_{r,n}
\int_{\mathbb{R}^n} \langle x\rangle^{2r-1}|D_{x}u|^2\,dx,
\end{equation}
and using Lemma~\ref{lem-a}, we obtain
\begin{equation*}
\partial_t\int_{\mathbb{R}^n}\varphi|u|^2\,dx
 \le c_{r,n}\int_{\mathbb{R}^n}\varphi|u|^2\,dx
+c_{r,n}\int_{\mathbb{R}^n}|D_{x}u|^2\,dx+c_{r,n}\int_{\mathbb{R}^n} |D^{r}u|^2\,dx.
\end{equation*}
Thus
\begin{equation}\label{cd5}
\partial_t\int_{\mathbb{R}^n}\varphi|u|^2\,dx
 \le c_{r,n}\int_{\mathbb{R}^n}\varphi|u|^2\,dx
+c_{r,n}\|u\|^2_{H^r(\mathbb{R}^n)},
\end{equation}
applying Gronwall, we obtain the result.
\end{proof}

\subsection{A priori estimates for the generalized Korteweg-de Vries
equation}

\begin{lemma}\label{kdv-0}
If $u$ is a solution of the IVP for  \eqref{IVP-1} with
$u_0 \in\mathcal{X}^{s,\theta}$, $s \ge 2 \theta \ge 2$. Then
\begin{equation*}
\begin{aligned}
\int_{\mathbb{R}} \varphi|u|^2\,dx &\le C_{\theta,k}\{\sup_{t\in[-T,T]}
\|u\|_{H^1(\mathbb{R})}^2+\sup_{t\in[-T,T]}\|u\|_{H^{2\theta}(\mathbb{R})}^2\}
e^{c_{\theta,k}T}\\
&\quad + \|u(0)\|_{L^2(d\mu_{\theta})}^2\;e^{c_{\theta,k}T}.
\end{aligned}
\end{equation*}
\end{lemma}

\begin{proof}
Let $u \in S(\mathbb{R})$. In  \eqref{IVP-1} consider
$k \in\mathbb{N}$, $s\geq 2\theta$,  $\theta\geq 1$.
Now multiply the equation by the term $\varphi u$ and after
integrating on $\mathbb{R}$, where $\varphi(x):= (1+|x|^2)^{\theta}$.
\begin{equation}\label{kdv-1}
\begin{aligned}
\partial_t{\int_{\mathbb{R}} \varphi|u|^2\,dx}
&=-2\int_{\mathbb{R}} \varphi u u_{xxx}\,dx-2
\int_{\mathbb{R}} \varphi u^{k+1}u_x\,dx\\
&= -\frac12 \int_{\mathbb{R}} \varphi_{xxx}u^2\,dx
+3\int_{\mathbb{R}} \varphi_xu_{xx}u\,dx-\frac{2}{k+2}
\int_{\mathbb{R}} \varphi\partial_xu^{k+2}\,dx
\\
&= \int_{\mathbb{R}} \varphi_{xxx}u^2\,dx-3
\underbrace{\int_{\mathbb{R}} \varphi_x|u_x|^2\,dx}_{I_3}-\frac{2}{k+2}
\underbrace{\int_{\mathbb{R}} \varphi\partial_xu^{k+2}\,dx}_{I_4}.
\end{aligned}
\end{equation}
Is obvious that
\[
\int_{\mathbb{R}} \varphi_{xxx}|u|^2\,dx\leq  C_{\theta}\int_{\mathbb{R}} \varphi|u|^2\,dx.
\]
Applying interpolation
\begin{equation}\label{kdv-2}
\begin{gathered}
|I_3|\leq  C_{\theta}\|u\|_{H^1(\mathbb{R})}^2
+\Big(\int_{\mathbb{R}} |x|^{2\theta}|u|^2\,dx \Big)^{1-\frac{1}{2\theta}}
\Big(\int_{\mathbb{R}} |D_{x}^{2\theta}|^2\,dx \Big)^{1/(2\theta)}\\
|I_4|\leq  C_{\theta}\sup_{t\in [-T,T]}\|u(t)\|_{H^1(\mathbb{R})}^k
\int_{\mathbb{R}} \varphi|u|^2\,dx.
\end{gathered}
\end{equation}
Using Young
\begin{equation}\label{kdv-3}
\begin{aligned}
|I_3|&\leq  C_{\theta,k}\Big(\sup_{t\in [-T,T]}
\|u(t)\|_{H^1(\mathbb{R})^2}+\sup_{t\in [-T,T]}\|u(t)\|_{H^{2\theta}(\mathbb{R})}^2 \Big)\\
&\quad +  C_{\theta,k}\Big(1+\sup_{t\in [-T,T]}\|u(t)\|_{H^1(\mathbb{R})}^k \Big)
 \int_{\mathbb{R}} \varphi|u|^2\,dx.
\end{aligned}
\end{equation}
Applying similar ideas to the case Nonlinear Schr\"odinger (NLS)
equation and using Gronwall, we complete the proof.
\end{proof}



\subsection{Proof of Theorems \ref{mainthm} and \ref{mainthm1}}

\begin{proof}[Proof of Theorem~\ref{mainthm} (case gKdV)]

The case NLS follows a similar argument.
Let $u_0 \in \mathcal{X}^{s,\theta}$, $s \ge 2 \theta \ge 2$, $u_0 \neq 0$, we know that that there exists an function $u
\in C([-T,T], H^{s})$ such that \eqref{IVP-1} is local
well-posed in $H^{s}$. Is well know that $\mathbf{S}(\mathbb{R})$ is dense
in $\mathcal{X}^{s,\theta}$. Then for $u_0 \in
\mathcal{X}^{s,\theta}$ there exist a sequence $(u_0^{\lambda})$
in $\mathbf{S}(\mathbb{R})$ such that
\begin{equation}\label{converg}
u_0^{\lambda} \to u_0 \quad \textrm{in}\;\mathcal{X}^{s,\theta}.
\end{equation}

By \eqref{mnx1} (continuous dependence) the sequence of solutions
$u^{\lambda}(t)$ associated to IVP \eqref{IVP} with
initial data $u_0^{\lambda}$
\begin{equation}\label{IVPmnx}
\begin{gathered}
\partial_t u^\lambda + u^\lambda_{xxx} + (u^{\lambda})^ku^\lambda_x=0, \quad (t,x) \in \mathbb{R}^2,\\
u^\lambda(x,0)  =  u_0^\lambda(x),
\end{gathered}
\end{equation}
satisfy
\begin{equation}\label{converg1}
\sup_{t \in [-T,T]} \|u^{\lambda}(t)-u(t)\|_{H^s}
\stackrel{\lambda \to \infty}\to  0, \quad s \ge 2\theta \ge 2.
\end{equation}
The solutions $u^{\lambda}$ of  \eqref{IVPmnx}
satisfy the conditions \eqref{ap-1}-\eqref{ap-4} of Section
\ref{IN}. Therefore, Lemma~\ref{kdv-0} gives
\begin{align*}
\int_{\mathbb{R}} \varphi|u^{\lambda}|^2\,dx
&\le C_{\theta,k}\{\sup_{t\in[-T,T]}\|u^{\lambda}\|_{H^1(\mathbb{R})}^2
+\sup_{t\in[-T,T]}\|u^{\lambda}\|_{H^{2\theta}(\mathbb{R})}^2\}e^{c_{\theta,k}T}\\
&\quad + \|u^{\lambda}(0)\|_{L^2(d\mu_{\theta})}^2\;e^{c_{\theta,k}T},
\end{align*}
Taking the limit when $\lambda \to \infty$, \eqref{converg1} implies
\begin{align*}
\int_{\mathbb{R}} \varphi|u|^2\,dx
&\le C_{\theta,k}\{\sup_{t\in[-T,T]}\|u\|_{H^1(\mathbb{R})}^2+\sup_{t\in[-T,T]}\|u\|_{H^{2\theta}(\mathbb{R})}^2\}e^{c_{\theta,k}T}\\
&\quad + \|u(0)\|_{L^2(d\mu_{\theta})}^2\;e^{c_{\theta,k}T}.
\end{align*}
Thus $u(t) \in \mathcal{X}^{s,\theta}$, $t \in [-T,T]$, which proves the persistence. The local well-posedness
theory in $H^{s}$ implies the uniqueness and continuous dependence
upon the initial data in $H^s$, this imply uniqueness in
$\mathcal{X}^{s,\theta}$.

Now we will prove
continuous dependence in the norm
$\|\cdot\|_{L^2(d\dot{\mu}_{\theta})}$. Let $u(t)$ and $v(t)$ be
two solutions in $\mathcal{X}^{s,\theta}$, of
 \eqref{nls-1} with initial dates $u_0$ and $v_0$
respectively, let $u^{\lambda}(t)$, $v^{\lambda}(t)$ be the
solutions associated with \eqref{nls-1} with initial dates
$u_0^{\lambda}$ and $v_0^{\lambda}$ respectively such that
$u_{0}^{\lambda}, v_{0}^{\lambda} \in \mathbf{S}(\mathbb{R})$,
\begin{equation}\label{converpx}
u_{0}^{\lambda} \to u_0, \quad v_{0}^{\lambda} \to v_0 \quad
 \textrm{in } \mathcal{X}^{s,\theta}
\end{equation}
and with $\lambda \gg 1$, we have
\begin{align*}
\|u(t)-v(t)\|_{L^2(d\dot{\mu}_{\theta})}
&\le  \|u(t)-u^{\lambda}(t)\|_{L^2(d\dot{\mu}_{\theta})}
 +\|u^{\lambda}(t)-v^{\lambda}(t)\|_{L^2(d\dot{\mu}_{\theta})}\\
&\quad +\|v^{\lambda}(t)-v(t)\|_{L^2(d\dot{\mu}_{\theta})}.
\end{align*}
The convergence
\begin{equation}\label{converg131}
\sup_{t \in [-T,T]} \|u^{\lambda}(t)-u(t)\|_{H^s}
\to  0, \quad
\sup_{t \in [-T,T]} \|v^{\lambda}(t)-v(t)\|_{H^s}
\to 0,
\end{equation}
as $\lambda \to \infty$,
where $ s \ge 2\theta \ge 2$, implies for $\lambda \gg1$ that
$$
|u(x,t)-u^{\lambda}(x,t)|\le 2 |u(x,t)| \quad \textrm{and} \quad
|v(x,t)-v^{\lambda}(x,t)|\le 2 |v(x,t)|.
$$
The Dominated Convergence Lebesgue's Theorem gives
 \begin{align*}
\|u(t)-u^{\lambda}(t)\|_{L^2(d\dot{\mu}_{\theta})}\to 0 \quad
\textrm{and} \quad\|v^{\lambda}(t)-v(t)\|_{L^2(d\dot{\mu}_{\theta})}
\to 0.
\end{align*}
Let $w^\lambda:=u^{\lambda}-v^{\lambda}$, then $w^\lambda$
satisfies the equation
\begin{align*}
w^\lambda_t &+
w^\lambda_{xxx}+(u^{\lambda})^{k}w^\lambda_{x}
+v^{\lambda}_{x}A(u^{\lambda}, u^{\lambda}){w}^\lambda=0,
\end{align*}
where $A(x,y)=x^{k-1}+x^{k-2}y+\dots +x y^{k-2}+y^{k-1}$.

Then, we multiply the above equation by $\varphi \bar{w}^\lambda$,
integrate on $\mathbb{R}$, to obtain by Gronwall's Lemma that
\begin{equation}\label{eqf}
\int_{\mathbb{R}} \varphi |w^\lambda(t,x)|^2\,dx \leq
\big\{ \int_{\mathbb{R}} \varphi |w^\lambda(0,x)|^2\,dx+c_\theta
\sup_{t\in [-T,T]} \|w^\lambda(t)\|_{H^{2\theta}}^2  \big\} e^{k_0T},
\end{equation}
where $k_0$ is a constant to $\lambda \gg 1$.
Observe that the convergence \eqref{converpx} and \eqref{converg131}
imply
$$
\|w^\lambda(0)\|_{L^2(d\mu_\theta)}
=\|u_0^\lambda-v_0^\lambda\|_{L^2(d\mu_\theta)}
\le 2 \|u_0-v_0\|_{L^2(d\mu_\theta)},
$$
and
$$
\|w^\lambda(t)\|_{H^{2\theta}}
=\|u^\lambda(t)-v^\lambda(t)\|_{H^{2\theta}}
\le 2 \sup_{t\in [-T,T]} \|u(t)-v(t)\|_{H^{2\theta}},
$$
if $\lambda \gg1$,
which together with \eqref{eqf} gives the continuous dependence.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{mainthm1}]
Is a direct consequence of the proof of Theorem \ref{mainthm}
and the global theory for the gKDV equation (see \cite{KT1}).
\end{proof}

\section*{Acknowledgements}
The authors thank the anonymous referee for constructive
remarks and also for the suggestion to improve Lemma \ref{lem-e}
and Theorem \ref{mainthm}.

This research was supported by the following grants:
E-26/111.564/2008 ``Analysis, Geometry and Applications'', from FAPERJ, Brazil;
E-26/110.560/2010 ``Nonlinear Partial Diferential Equations'',
from Pronex-FAPERJ, Brazil;
and 303849/2008-8 from the National Council of Technological and 
Scientific Development (CNPq), Brazil.

\begin{thebibliography}{00}

\bibitem{B-L} J. Berg, J. Lofstrom;
\emph{Interpolation Spaces}, Springer, Berlin, 1976.

\bibitem{CN1} X. Carvajal and W. Neves;
 \emph{Persistence of solutions to higher order
nonlinear Schr\"odinger equation},
J. Diff. Equations. \textbf{249}, (2010), 2214-2236.

\bibitem{HNT} N. Hayashi, K. Nakamitsu, M. Tsutsumi;
 \emph{Nonlinear Schr\"odinger equations in weighted Sobolev spaces},
Funkcialaj Ekvacioj, \textbf{31} (1988) 363-381.

\bibitem{K1}  T. Kato;
 \emph{On the Korteweg-de Vries equation}, Manuscripta Math.
\textbf{28} (1979), 89-99.

\bibitem{KT1}  T. Kato;
\emph{On the Cauchy Problem for the
(Generalized) Korteweg-de-Vries Equation}, Studies in Applied
Mathematics, Advances in Math. Supplementary Studies, \textbf{08}
(1983), 93-127.

\bibitem{K-M}  T. Kato and K. Masuda;
 \emph{Nonlinear Evolution Equations and Analycity. I},
Ann. Inst. H. Poincaré Anal. Non Linéaire \textbf{03} (1986), 455-467.

\bibitem{KT3}  T. Kato;
\emph{On nonlinear Schr\"odinger equations, II. $H^s$-solutions
and unconditional well-posedness},
J. Anal. Math. \textbf{67} (1995), 281-305.

\bibitem{JN} J. Nahas;
 \emph{A decay property of solutions to the mKdV equation},
PhD. Thesis, University of California-Santa Barbara, June 2010.

\bibitem{NP0} J. Nahas, G. Ponce;
\emph{On the persistent properties of solutions of nonlinear
dispersive equations in weighted Sobolev spaces},
to appear in RIMS Kokyuroku Bessatsu (RIMS Proceeding).

\bibitem{NP} J. Nahas, G. Ponce;
\emph{On the persistent properties of solutions to semi-linear
Schr\"odinger equation}, Comm. Partial Diff. Eqs.  \textbf{34}
(2009), no. 10-12, 1208-1227.

\end{thebibliography}

\end{document}