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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 246, pp. 1--25.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/246\hfil Unique continuation principle]
{Unique continuation principle for high order\\
equations of Korteweg-de Vries type}

\author[P. Isaza \hfil EJDE-2013/246\hfilneg]
{Pedro Isaza}  % in alphabetical order

\address{Pedro Isaza J. \newline
Departamento de Matem\'aticas\\
Universidad Nacional de Colombia\\
A. A. 3840 Medell\'in, Colombia}
\email{pisaza@unal.edu.co}

\begin{abstract}
 In this article we consider the problem of  unique continuation
 for  high-order equations of Korteweg-de Vries type
 which include the kdV hierarchy.  It is proved that if the difference
 $w$ of two solutions  of an equation of this form has certain
 exponential decay for $x>0$ at two different times,
 then $w$ is identically zero.
\end{abstract}

\thanks{Submitted July 21, 2013. Published November 13, 2013.}
\thanks{Supported by DIME, Universidad Nacional de Colombia, 
Medell\'in, grant 15401.}
\subjclass[2000]{35Q53, 37K05}
\keywords{Nonlinear dispersive equations; unique continuation;
\hfill\break\indent  estimates of Carleman type}

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

\section{Introduction}

This article  concerns a unique continuation principle for the
equation
\begin{equation}
\partial _{t}u+ (-1)^{k+1}\partial^ {n}_{x}u+P(u, \partial _{x}u, \dots,
\partial_x^{p} u) =0\,,\quad u=u(x,t),\quad x,t\in\mathbb{R},\label{ec}
\end{equation}
where $n=2k+1$, $k=1,2\dots$, and $P$ is a polynomial in
$u, \partial_xu,\dots,\partial^{p}_xu$, with $p\leq{n-1}$.
In particular, we will focus our attention to the case in which  $P$
has the form
\begin{equation}
\begin{gathered}
P(u, \partial _{x}u, \dots, \partial_x^{n-2} u)
=\sum_{d=2}^{k+1} \sum_{|{ m}|
=2(k+1-d)+1}  a_{d,m}\,\partial_x^{m_1}u\dots\partial_x^{m_d}u
\equiv\sum_{d=2}^{k+1}A_d(z),\\
z=(u,\partial_xu,\dots,\partial_x^{n-2}u),
\end{gathered} \label{poli1}
\end{equation}
where,  for $d\in\mathbb{N}$  and  for integers $m_1,\dots,m_d$,
 ${m}:=(m_1,\dots,m_d)$ is a multi-index with $0\leq m_1\leq\dots\leq m_d$,
$|{ m}|:=m_1+\dots +m_d$, and $a_{d,m}$ is a constant.
We will refer to equation \eqref{ec} with $P$ defined by  \eqref{poli1}
as equation \eqref{ec}-\eqref{poli1}.
We will  also  consider equation \eqref{ec} when the nonlinearity
$P$ has order $p\leq k$.

The type of relation expressed in \eqref{poli1}, between the degree and
the order of each monomial of $P$,  is present in the nonlinearities
of the collection of equations known as the KdV (Korteweg-de Vries) hierarchy.
This set of equations was introduced by Lax \cite{L1}  in the process
to determine the functions $u=u(x,t)$ for which the eigenvalues of the
operator $L:=\frac{d^2}{dx^2}-u(\cdot,t)$ remain constant as $t$ evolves.
This property had been already discovered  by Gardner et al. in \cite{G1}
for the solutions of the Korteweg-de Vries equation
$$
\partial_tu+\partial^3_xu+u\partial_xu=0.
$$
 Lax \cite{L1} showed that this property holds for the solutions
of the equations
\begin{equation}
\partial_tu+[B_k(u),L]=0,\label{kdvH}
\end{equation}
where $[B,L]:=BL-LB$ denotes the commutator of $B$ and $L$, and $B_k(u)$
is the skew-adjoint operator defined by
$$
B_k(u)=b_k\frac{d^{2k+1}}{dx^{2k+1}}
+\sum_{j=0}^{k-1}b_{k,j}(u)\frac{d^{2j+1}}{dx^{2j+1}}
+\frac{d^{2j+1}}{dx^{2j+1}}b_{k,j}(u),
$$
with the coefficients $b_{k,j}(u)$ chosen in such a way that the operator
 $[B_k(u),L]$ has order zero. It was proved in \cite{L2}
that the equations in the KdV hierarchy \eqref{kdvH} can be written
in the form $\partial_tu+\partial_xG_{k+1}(u)=0$, were the functions
$G_k(u)$ are the gradients of the functionals $F_k(u)$ which define
the conservation laws of the KdV equation. The gradients $G_k$
satisfy the following recursion formula due to Lenard
(see \cite{G2} and \cite{P}):
$$
\partial_xG_{k+1}=cJG_k\,,\quad\text{where }
J=\partial_x^3+\frac23u\partial_x+\frac13 \partial_xu\,.
$$
This formula can be applied  to obtain a derivation of the  equations
in the hierarchy. Starting with $G_0(u)=3$, with $k=0$ we get the
transport equation, with $k=1$ the KdV equation, and, with $k=2$,  $k=3$,
and $k=4$,  we respectively find  the equations
\begin{equation} \label{fifthorder}
\begin{gathered}
\partial_tu+\partial_x^5u-10u\partial_x^3u-20\partial_xu\partial^2_xu
+30u^2\partial_xu=0, \\
\begin{aligned}
&\partial_tu+\partial_x^7u+14u\partial_x^5u+42\partial_xu\partial^4_xu
+70\partial_x^2u\partial_x^3u+70u^2\partial^3_xu\\
&+280u\partial_xu\partial^2_xu+70(\partial_xu)^3+140u^3\partial_xu=0,
\end{aligned} \\
\begin{aligned}
&\partial_tu+\partial^9_xu +\sum_{\substack{m_1+m_2=7\\0\leq m_1\leq m_2}}
 a_{2,m}\,\partial^{m_1}_xu\partial_x^{m_2}u\\
&+ \sum_{\substack{m_1+m_2+m_3=5\\0\leq m_1\leq m_2\leq m_3}}
  a_{3,m}\,\partial^{m_1}_xu\partial_x^{m_2}u\partial_x^{m_3}u
+\dots +a_{5,m}u^4\partial_xu=0,
\end{aligned}
\end{gathered}
\end{equation}
for certain constants $a_{d,m}$, with $d=2,\dots, 5$ and  $|m|=2(5-d)+1$.

In spite of  computational difficulties, it is possible to obtain
exact expressions for all the equations in the hierarchy (see \cite{A}).
However, following a simple procedure, and without obtaining the explicit
values for the coefficients, it can be proved  (see \cite{Gr})
that the  equations in the KdV hierarchy \eqref{kdvH} have the
form of \eqref{ec}-\eqref{poli1}.  When $k$ is even we have made the
change of variable $x\mapsto -x$ and thus the linear term $\partial^{n}_xu$
has been transformed into  $(-1)^{k+1}\partial^{n}_xu$ in \eqref{ec}.

The aspects of local and global well-posedness of the initial value
problem (IVP) associated with the general equation \eqref{ec}
 have been considered in  \cite{KPV3} and \cite{KPV4},
where Kenig, Ponce, and Vega proved that the (IVP) is  locally well-posed
in  weighted spaces
$H^s(\mathbb{R})\cap L^2(|x|^m\,dx)$ if $s\geq s_0(k)$,
for some $s_0(k)$ and some integer $m=m(k)$.

For the (IVP) associated to  \eqref{ec}-\eqref{poli1},
in \cite{S}, Saut proved the existence of global solutions
for initial data in Sobolev spaces $H^m(\mathbb{R})$ for $m\geq k$,  integer.
By using a variant of Bourgain
spaces, in \cite{Gr}, Gr\"unrok proved the local well-posedness
for the (IVP) of equation \eqref{ec}-\eqref{poli1} in the context of
the spaces
$$
\widehat {H}_{s}^r(\mathbb{R})
:=\{f\mid \|f\|_{s,r}:=\|(1+\xi^2)^{s/2}\widehat{f}(\xi)\|_{L_{\xi}^{r'}}
<\infty\},
$$
with
$$
r\in(1,{{\frac{2k}{2k-1}}}],  \text{ and }
s>k-{\frac32-\frac1{2k}+\frac{2k-1}{2r'}}.
$$
Here $\widehat{\;}$ denotes the Fourier transform and $1/r+1/r'=1$.
We also refer to the articles \cite{KP,Kw, Li,Pi,Po}, which, among others,
consider the problem of well-posedness for high order equations of
KdV-type and especially for the equations of order five ($k=2$).

Our main goal is to prove  continuation principles  for
  \eqref{ec}-\eqref{poli1} with $n\geq 5$, which include the KdV hierarchy,
and for the equations \eqref{ec} with $n\geq 5$ and $p\leq k$.
Roughly speaking, we will prove that if the difference $w:=u_1-u_2$ of
two sufficiently smooth solutions of  equation \eqref{ec}-\eqref{poli1}
decays as $\exp({-x_+^{4/3^+}})$ at two different times, then $w\equiv 0$.
(Here $x_+:=\frac12({x+|x|})$, and $4/3^+$ means $4/3+\epsilon$ for
arbitrarily small $\epsilon>0$). For \eqref{ec} with $p\leq k$ we have
a similar result if $w$ decays as $\exp({-ax_+^{n/(n-1)}})$ for  $a>0$
sufficiently large at two different times.
This last result is coherent with the decay $\exp({-cx_+^{n/(n-1)}})$
of the fundamental solution of the linear problem associated with
\eqref{ec} (see \cite{SSS}). When the nonlineariy $P$ has higher order
as in \eqref{ec}-\eqref{poli1}, it is then necessary to impose a
stronger decay on $w$.

The aspect of unique continuation  has been studied for a variety
of non-linear dispersive equations, and especially for the KdV
and Schr\"odinger equations.  Saut and Sheurer \cite{SS}
considered a class of nonlinear dispersive equations, which
includes the KdV equation, and proved that if a solution $u$
of one of such equations vanishes in an open set $\Omega$ of
the space-time space, then  $u$ vanishes in all horizontal components
of $\Omega$, that is, in the set $\{(x,t): \exists\, y \text{ with }
(y,t)\in\Omega\}$.

By using methods of complex analysis, Bourgain \cite{B} proved
that if a solution $u$ of the KdV equation is supported in a compact
set $\{(x,t): -B\leq x\leq B,\; t_0\leq t\leq t_1\}$,
then $u$ vanishes identically.

Kenig, Ponce and Vega \cite{KPV1}, considered a solution of the KdV equation
which vanishes only in two half lines $ [B,+\infty)\times\{t_0\}$  and
$[B,+\infty)\times\{t_1\}$, and proved that this solution must be identically
zero. A similar result was proved in \cite{KPV2} for the difference
$w=u_1-u_2$ of two solutions of the KdV equation.
 Escauriaza et al. \cite{EKPV1}  refined this result by only imposing
the condition that  $w(\cdot,t_0)$ and $w(\cdot,t_1)$ decay as
$\exp({-ax_+^{\gamma}})$, for $\gamma=3/2$ and $a>0$ sufficiently large,
together with a additional hypothesis of polynomial decay for
 $u_1$ and $u_2$. This result is obtained by  applying two types of
estimates for the function $w$: Carleman type estimates, which express
 a boundedness of the inverse of the linear operator $\partial_t+\partial_x^3$
in $L^p-L^q$-spaces with exponential weight; and a so-called lower
estimate which bounds the $L^2$-norm of $w$ in a small rectangle at the
origin with the $H^2$-norm of $w$ in a distant
rectangle $[R,R+1]\times[0,1]$.

For the fifth-order equation \eqref{ec} ($k=2$),  Dawson \cite{D}
 proved a result similar to that in \cite{EKPV1} with $\gamma=4/3^+$
for the general case $p\leq n-1=4$, and with $\gamma=5/4$ for the case
$p\leq 2$.

In this article we consider  equations \eqref{ec} and \eqref{ec}-\eqref{poli1}
with arbitrary order $n$   and prove the continuation principles stated
in Theorem \ref{thmI} and \ref{thmII} below. For that, we follow the method traced
in \cite{EKPV1}.  The  greatest difficulty in this process is to manage
the huge amount of terms arising in  the computations  of the operators
involved in the lower estimate. We consider that the main contribution
of our work is the presentation of  a clear and organized procedure to
obtain the lower estimate (see Lemma \ref{lema3} and Theorem \ref{estinferior}).

We now state our main results.

\begin{theorem}\label{thmI} %\label{principal}
For  odd $n\geq 5$ ,  $k=(n-1)/2$,  and $\alpha>\frac{n+1}3$, suppose
that $u_1,\,u_2$ are in $C([0,1];H^{n+1}(\mathbb{R})\cap L^2((1+x_+)^{2\alpha}\,dx))$
are two solutions of  the equation
\begin{equation}
\partial _{t}u+ (-1)^{k+1}\partial^ {n}_{x}u+P(u, \partial _{x}u,
\dots, \partial_x^{n-2} u) =0\label{Lax}
\end{equation}
with $P$ as in \eqref{poli1},
and let $w:=u_1-u_2$.  If
\begin{equation}
w(0), w(1)\in L^2(e^{2x_+^{4/3+\epsilon}}dx)\label{w}
\end{equation}
for some $\epsilon>0$, then $w\equiv0$.
\end{theorem}

 The proof of this theorem  can be adapted to obtain a similar
continuation principle for equation \eqref{ec} when  $p\leq k$.
In this case we require a weaker decay for $w(0)$ and $w(1)$ and
consider some minor modifications in  the  polynomial decay hypothesis
for $u_1$ and $u_2$.    For the sake of simplicity we state this
result without making special emphasis in the optimal value of  $\alpha$.

\begin{theorem} \label{thmII}
For  odd $n\geq 5$, $k=(n-1)/2$,  and $\alpha_0>0$ sufficiently large,
suppose that $u_1,\,u_2\in C([0,1];H^{n+1}(\mathbb{R})
\cap L^2((1+|x|)^{2\alpha_0}\,dx))$ are two solutions of  the equation
\begin{equation}
\partial _{t}u+ (-1)^{k+1}\partial^ {n}_{x}u
+P(u, \partial _{x}u, \dots, \partial_x^p u) =0\label{Lax1}
\end{equation}
with $p\leq k$. Define $w:=u_1-u_2$.  Then,  there is $a>0$, which
depends only on $n$, such that if
\begin{equation}
w(0), w(1)\in L^2(e^{2ax_+^{n/{(n-1)}}}dx),\label{decayhyp2}
\end{equation}
then $w\equiv0$.
\end{theorem}

The article is organized as follows:
In section \ref{exponen} we prove that the exponential decay
for $w$ in the  semi-axis $x>0$  is preserved  in time.
In section \ref{sectioncar} we establish the Carleman type estimates
and in section \ref{sectionlow} we prove the lower estimates.
Finally we give the proofs of Theorems \ref{thmI}  and \ref{thmII}
in section \ref{sectionmain}.

 Throughout the paper the letters $C$ and $c$ will denote diverse positive
constants which may change from line to line, and whose dependence on certain
parameters is clearly established in all cases.
Sometimes, for a parameter $a$, we will use the notations $C_a$, $C(a)$,
 and $c_a$ to make emphasis in the fact that the constants  depend upon $a$.
We frequently write $f(\cdot_s)$ to denote a function $s\mapsto f(s)$.
For a set $A$, $\chi_A$ will denote the characteristic function of $A$.
The symbols $\widehat{\;}$
and $\check{\;}$ will denote the Fourier and the inverse Fourier transform,
 respectively.
The notations $\widehat{\;}^{\,\,{}_{x}}$ and  $\check{\;}^{\,{}_{\xi}}$
will emphasize the facts that the Fourier transform and its inverse are taken
with respect to  specific variables $x$ and $\xi$, respectively.
For $1\leq p,q<\infty$, $A,B\subseteq\mathbb{R}$, $D=A\times B$, and $f=f(x,t)$
we will denote
 $$
\|f\|^p_{L^p_xL^q_t(D)}:=\int_A\Bigl(\int_B|f(x,t)|^q\,dt\Bigr)^{p/q}\,dx\,.
$$
 We will  use  similar definitions when  $p=\infty$ or $q=\infty$ and
also for $\|f\|_{L^q_tL^p_x(D)}$.


\section{Exponential decay}\label{exponen}

In this section we prove that if the difference $w$ of two solutions
of \eqref{Lax} decays exponentially at $t=0$, then this decay is preserved
at all positive times. This property will be crucial for the application
of the Carleman estimates in the proofs of Theorem \ref{thmI} and \ref{thmII}.

\begin{theorem}\label{decaimiento}
For odd $n\geq 5$, $k=(n-1)/2$, and $\alpha>(n+1)/4$,
let $u_1,u_2$ are in $C([0,1];H^{n+1}(\mathbb{R})\cap L^2((1+x_+)^{2\alpha}\,dx)))$
be  two solutions of \eqref{ec}-\eqref{poli1}, and define $w:=u_1-u_2$.
Let $\beta>0$ and suppose that $w(0)\in L^2(e^{\beta x} \, dx)$. Then
\begin{equation}
\sup_{t\in[0,1]}\|w(t)\|_{L^2(e^{\beta x} \, dx)}<\infty.\end{equation}
\end{theorem}

\begin{proof}
Let us denote $z_i=(u_i,\partial_xu_i,\dots\partial_x^{n-2}u_i)$, $i=1,2$.
Then $w$ is a solution of the differential equation
\begin{equation}
\partial _{t}w+ (-1)^{k+1}\partial^ {n}_{x}w+ P(z_1)-P(z_2)=0.\label{apr}
\end{equation}

We will first prove that the theorem is valid provided we can construct
a  sequence $\{\phi_N \}_{N \in\mathbb{N}}$ of  nondecreasing functions
in $ C^{\infty}(\mathbb{R})$ satisfying  for all $x\in\mathbb{R}$ the conditions
\begin{gather}
\varphi_N (x)\to e^{\beta x}\quad \text{as }\, N\to\infty\quad \text{and}
\quad 0\leq \varphi_N (x)\leq C\,e^{\beta x},\label{condfi3}
\\
 \varphi_N (x)\leq C_N (1+x_+)^{(k+2)/4}\,,\label{condfi2}
\\
|\varphi_N ^{(j)}(x)|\leq C_j\varphi'_N (x)\quad\text{for }
j=2,3\dots,n=2k+1 \text{ and } \varphi_N'(x)\leq  C\varphi_N(x),\label{condfi}
\\
\varphi_N (x)\leq C(1+x_+)\varphi'_N (x)\,,\label{condfi1}
\end{gather}
where the constants $C$ and $C_j$ are independent of $N$.

We multiply \eqref{apr} by $\varphi _N u$ and, for $t$ fixed, integrate in $\mathbb{R}$.
Thus, by applying integration by parts we obtain
\begin{equation}
\begin{aligned}
\frac12\frac{d}{dt}\int\varphi_N  w^2
&=-{\frac{2k+1}2}\int\varphi_N '(\partial_x^kw)^2
 +c_{k-1}\int\varphi_N ^{(3)}(\partial_x^{k-1}w)^2+\dots\\
&\quad +c_1\int \varphi_N ^{(2k-1)}(\partial_x w)^2
  +\frac 12\int\varphi_N ^{(2k+1)}w^2\\
&\quad -\int( P(z_1)-P(z_2))\,\varphi_N  w\,.
\end{aligned} \label{apriori}
\end{equation}
The integration by parts is justified as follows:
since there is a constant $C>0$ such that
$\|(1+x_+)^{\alpha}u_i(t)\|_{L^2}\leq C$, and
$\|u_i(t)\|_{H^{n+1}(\mathbb{R})}\leq C$ for all $t\in[0,1]$ and
$i=1,2$, by using integration by parts and  truncation functions,
it can be proved that the following interpolation property holds:
\begin{equation}
\|(1+x_+)^{\alpha(1-\frac j{(n+1)})}\partial_x^j u_i(t)\|_{L^2(\mathbb{R}}\leq C\,,
\quad \text{for all }t\in[0,1]; i=1,2;\quad j=0,\dots,n+1. \label{inter}
\end{equation}
Since $\alpha>(n+1)/4$, it follows that, for $0\leq j\leq k$,
$(1+x_+)^{(k+2)/4}\partial ^j_xw(t)\in L^2(\mathbb{R}$, and thus,
from \eqref{condfi2} and \eqref{condfi}  $\varphi^{(l)}\partial_x^jw(t)\in L^2(\mathbb{R}$
for all positive integers $l$. This implies that all the terms which appear
in the procedure to obtain  \eqref{apriori} are in a right setting
for the application of  the integration by parts.

Let us estimate the last term  on the rand-hand side of \eqref{apriori}.
From \eqref{poli1} we have that
\begin{equation}
P(z)=\sum_{d=2}^{k+1}A_d(z)\quad\text{where}\quad
A_d(z)= \sum_{\substack{|m|=n-2(d-1)\\0\leq m_1\leq\dots\leq m_d}}
 a_{d,m}\,\partial_x^{m_1}u\dots\partial_x^{m_d}u\,,\label{P}
\end{equation}
and thus
\begin{equation}
|\int\bigl(P(z_1)-P(z_2)\bigr)\varphi_N \,w\,dx|
=|\sum_{d=2}^{k+1}\int\bigl(A_d(z_1)-A_d(z_2)\bigr)\varphi_N \,w\,dx|
\equiv|\sum_{d=2}^{k+1}I_d|\,.\label{suma1}
\end{equation}
It is easily seen that
\begin{equation}
\begin{aligned}
A_d(z_1)-A_d(z_2)
&=\sum_{\substack{|m|=n-2(d-1)\\0\leq m_1\leq\dots\leq m_d}}
   a_{d,m}\Bigl(\partial_x^{m_1}w\partial_x^{m_2}u_1\dots\partial_x^{m_d}u_1\\
&\quad +\partial_x^{m_1}u_2\partial_x^{m_2}w\dots
 \partial_x^{m_d}u_1
+\partial_x^{m_1}u_2\partial_x^{m_2}u_2\dots\partial_x^{m_d}w\Bigr).
\end{aligned}\label{Ad}
\end{equation}
We estimate $I_2$, which,   having the derivatives of the highest order,
is the most critical term in \eqref{suma1}. From \eqref{Ad},
\begin{equation}
\begin{aligned}
I_2&=\int\bigl(A_2(z_1)-A_2(z_2)\bigr)\varphi_N \,w\,dx\\
&=\sum_{\substack{m_1+m_2=n-2\\ 0\leq m_1\leq m_2}}a_{2,m}
 \int(\partial_x^{m_1}w\partial_x^{m_2}u_1
 +\partial_x^{m_1}u_2\partial_x^{m_2}w)\varphi_N \,w\,dx.
\end{aligned}\label{Idos}
\end{equation}
We estimate only the second term on the right-hand side of \eqref{Idos},
the other term being similar. We apply integration by parts to reduce the
order of $\partial_x^{m_2}w$ and obtain that
\begin{equation}
\int(\partial_x^{m_1}u_2)\,(\partial_x^{m_2}w)\,\varphi_N \,w
=\sum_{\substack{r_1+r_2+2r_3=m_1+m_2=n-2\\r_1\geq m_1}}
 c_{r_1,r_2}\int(\partial^{r_1}_xu_2)\,\varphi_N ^{(r_2)}
(\partial_x^{r_3}w)^2\label{I2}
\end{equation}
To analyze the terms in this sum we consider the cases $r_2=0$ and $r_2\geq 1$:

(i) If $r_2=0$ and $r_3=0$, then  $r_1=n-2$ and  we bound the corresponding
integral in \eqref{I2}  by
$$
C\|\partial_x^{n-2}u_2(t)\|_{L^{\infty}}\int\varphi_N \,w^2
\leq C\int\varphi_N \,w^2,
$$
where $C$ is independent of $t\in[0,1]$ by the Sobolev embedding
of $H^1(\mathbb{R})$ in $L^\infty(\mathbb{R})$.

If $r_2=0$ and $r_3\geq 1$, then the maximum value of $r_1$ in \eqref{I2}
is $n-4$. Therefore, using the fact that
 $\varphi_N \leq  C(1+x_+)\varphi'_N $ we bound the corresponding integral
in \eqref{I2} by
 \begin{equation}
C\max_{0\leq r_1\leq n-4}\|(1+x_+)\partial^{r_1}_xu_2(t)\|_{L^{\infty}}
\int\varphi_N '\,(\partial_x^{r_3}w)^2.\label{apriori3}
\end{equation}
From \eqref{inter}  it can be seen that if $\Psi\in C^{\infty}(\mathbb{R})$
is a truncation function with $\Psi\equiv0$ in $(-\infty,1]$, and
$\Psi\equiv 1$ in $[2,+\infty)$, then
$\Psi(\cdot)(1+x_+)^{\alpha(1-\frac {j+1}{n+1})}\partial_x^j u_i(t)\in H^1(\mathbb{R})$,
$i=1,2$,  $j=0,1,\dots,n$, and
 \begin{equation}
\begin{aligned}
&\|\Psi(\cdot)(1+x_+)^{ \alpha(1-\frac {j+1}{n+1})}
\partial_x^j u_i(t)\|_{H^1(\mathbb{R})}\\
&\leq C(\|(1+x_+)^{\alpha}u_i(t)\|_{L^2(\mathbb{R}}\,,
\|u_i(t)\|_{H^{n+1}(\mathbb{R})})\leq C\,\quad\text{for all }t\in[0,1].
\end{aligned} \label{alpfa1}
\end{equation}
 In particular, for $j=0,\dots,n-4$,
$\alpha(1-\frac {j+1}{n+1})> \frac{n+1}4(1-\frac{n-3}{n+1})= 1$,
and thus from the Sobolev embedding of $H^1$ in $L^\infty$ we conclude that
 \begin{equation}
\max_{0\leq j\leq n-4}\|(1+x_+)\partial_x^ju_i(t)\|_{L^\infty(\mathbb{R})}
\leq C,\label{Idos1}
\end{equation}
 with $C$ independent of $t\in[0,1]$. Thus \eqref{apriori3} is bounded
by $C\int\varphi_N '(\partial_x^{r_3}w)^2$.


 (ii) If $r_2\geq1$, then $r_1\leq n-3$. From \eqref{condfi},
$\varphi^{(r_2)}\leq C_{r_2}\,\varphi' \leq C\varphi'$ for $1\leq r_2\leq n-2$.
Thus we bound the corresponding term in \eqref{I2} by
 \begin{equation}
C\max_{0\leq r_1\leq n-3}\|\partial_x^{r_1}u_2(t)\|_{L^{\infty}}
\int\varphi_N '\,(\partial_x^{r_3}w)^2\leq C \int\varphi_N '
(\partial_x^{r_3}w)^2.\label{Idos3}
\end{equation}

 Now, let us determine the maximum value of $r_3$ appearing in \eqref{I2}.
For that, we observe that  $r_1+r_2+2r_3=n-2$ is odd, and thus the maximum
value of $r_3$ occurs when $(r_1,r_2)=(0,1)$ or $(1,0)$, which then gives
$r_3\leq (n-3)/2=k-1$.

 Gathering the estimates of the cases (i) and (ii) above, and taking
into account that $r_3\leq k-1$, we conclude that
 \begin{equation}
|I_2|\leq C\sum_{j=1}^{k-1}\int\varphi_N '(\partial_x^jw)^2\,dx
 +C\int\varphi_N  w^2\,.
\end{equation}
 Proceeding in a similar way, we obtain the same bound for
$|I_3|,\dots |I_{k+1}|$, and thus  for the left hand side of \eqref{suma1}.
 Therefore, returning to \eqref{apriori} and using the fact that,
from condition \eqref{condfi}, $|\phi^{(j)}|\leq C\,\phi'$, $j=1,\dots 2k+1$,
we obtain
\begin{equation}
\frac12\frac{d}{dt}\int\varphi_N  w^2
\leq-{{\frac{2k+1}2}}\int\varphi_N '(\partial_x^kw)^2
+ C\sum_{j=1}^{k-1}\int\varphi_N '(\partial_x^jw)^2\,dx
+C\int\varphi_N  w^2 \label{apriori2}
\end{equation}

To handle the terms in \eqref{apriori2} having derivatives
$\partial_x^jw$ with $j=1,\dots, k-1$, we will show that given
$\varepsilon>0$ there is a constant $C_\varepsilon>0$ such that
for $j=1,\dots,k-1$
\begin{equation}
\int\varphi_N '( \partial_x^{j}w)^2
\leq \varepsilon\int\varphi_N '(\partial^k_xw)^2+C_\varepsilon\int\varphi_N
w^2\,.\label{apriori0}
\end{equation}
In fact, we first prove that
\begin{equation}
\int\varphi_N '(\partial_x^jw)^2
\leq\varepsilon\int\varphi_N '(\partial_x^{j+1}w)^2+C_\varepsilon
\int\varphi_N  w^2\,.\label{apriori1}
\end{equation}
This can be seen by induction: by applying integration by parts,
Young's inequality $|xy|\leq  \frac1{2\varepsilon} x^2+\frac{\varepsilon}{2}y^2$,
and the properties of $\varphi_N$ we can see that \eqref{apriori1} is
valid for $j=1$. If we assume that \eqref{apriori1} is valid for $j-1$, then,
again  integrating by  parts and  using Young's inequality,
\begin{align*}
\int\varphi_N '(\partial_x^jw)^2
&=\frac12\int\varphi_N ^{(3)}(\partial_x^{j-1}w)^2
 -\int\varphi_N '\partial_x^{j-1}w\,\,\partial_x^{j+1}w\\
&\leq C\int\varphi_N '(\partial_x^{j-1}w)^2
 +\frac 1{2\varepsilon}\int \varphi_N '(\partial_x^{j-1}w)^2
 +\frac\varepsilon 2\int\varphi_N '(\partial_x^{j+1}w)^2.
\end{align*}
If we apply the induction hypothesis at level $j-1$, with
$\frac{1/2}{C+1/{(2\varepsilon)}}$ instead of $\varepsilon$, then we have
\begin{equation}
\int\varphi_N '(\partial_x^jw)^2
\leq {\frac12}\int\varphi_N '(\partial_x^jw)^2
+C_\varepsilon\int \varphi_N  w^2
+\frac\varepsilon2\int\varphi_N '(\partial_x^{j+1}w)^2\,,
\end{equation}
which gives \eqref{apriori1}. From a repeated application of  \eqref{apriori1}
we obtain \eqref{apriori0}.

Therefore, taking into account that the first term on the right-hand side
of  \eqref{apriori2}  is negative,  we can apply \eqref{apriori0}
with $\varepsilon$ sufficiently small,  to absorb with this negative
term the integrals containing $(\partial_x^jw)^2$ in \eqref{apriori2}.
Thus we conclude that
\begin{equation}
\frac{d}{dt}\int\varphi_N  w^ 2\leq C_\varepsilon\int\varphi_N  w^2,
\label{tres1}
\end{equation}
which, from Gronwall's inequality and \eqref{condfi3} implies that
\begin{equation*}
\int\varphi_N  w(t)^2\,dx\leq C\int\varphi_N  w(0)^2\,dx
\leq C\int e^{\beta x}w(0)^2dx\,,\quad\text{for all } t\in[0,1] ,
\end{equation*}
where $C$ is independent of $t\in[0,1]$ and $N$.
Since $\varphi_N (x)\to e^{\beta x}$ as $N\to\infty$, the conclusion
of the theorem will follow by applying Fatou's Lemma on the left-hand
side of the former inequality.

 In this way, the proof of theorem \ref{decaimiento} will be complete
if we construct a sequence of functions $\varphi_N $, satisfying the
conditions \eqref{condfi3} to \eqref{condfi1}. For that we proceed as follows:

Let $\tilde\phi \in C^\infty(\mathbb{R})$ be a nonincreasing function such that
$\tilde\phi(x)=1$ for $x\in (-\infty, 0]$, and $\tilde\phi(x)=0$
for $x\in[1,\infty)$. For each $N\in\mathbb{N}$ let
$\phi_{N}(x)\equiv\phi(x) :=\tilde\phi(x-N)$.
Thus $\phi$ is supported in $(-\infty, N+1]$, and $(1-\phi)$ in   $[N,+\infty)$.
We define
 \begin{gather}
\theta_N (x)\equiv\theta(x):=\phi \beta e^{\beta x}+(1-\phi)\beta e^{\beta N},
\label{tetan} \\
\varphi_N (x)\equiv\varphi(x):=\int_{-\infty}^x\theta(x')\,dx'\,.\label{fifi}
\end{gather}
Let us show that $\varphi_N $ satisfies the conditions
\eqref{condfi3}--\eqref{condfi1}.

Taking into account the support of $(1-\phi)$, we see that
$0\leq \theta(x)\leq \phi\beta e^{\beta x}+(1-\phi)\beta e^{\beta x}=
\beta e^{\beta x }$. Thus, by integrating $\varphi'$ we have that
$0\leq\varphi(x)\leq e^{\beta x}$. Besides, from the definition of
$\varphi$ it is clear that $\varphi_N (x)\to e^{\beta x}$ as $N\to\infty$.
Thus $\varphi$ satisfies \eqref{condfi3}.

To prove \eqref{condfi2} it suffices to observe that for $x\leq N$,
 $\varphi(x)\leq e^{\beta N}\leq C_{N}(1+x_+)^{(k+2)/4}$, while for $x>N$,
\begin{equation}
\varphi(x)\leq \int_{-\infty}^{N+1}\beta e^{\beta x'} dx'
+\int_N ^x\beta e^{\beta N} dx'\leq e^{\beta(N+1)}
+ x\beta e^{\beta N}\leq C_{N}(1+x_+)^{(k+2)/4},\label{es}
\end{equation}
since $k\geq 2$. Thus we have \eqref{condfi2}.

We proceed now to prove \eqref{condfi1}.
For $x\leq N$,
$\varphi(x)=e^{\beta x}=\frac1\beta\varphi'(x)\leq C (1+x_+)\varphi'(x)$.
If $x> N$, then, from \eqref{es}, and using the fact that $x\geq 1$,
 we see that
\begin{equation}
\varphi(x)\leq  e^{\beta(N+1)}+ x\beta e^{\beta N}
\leq (\frac1{\beta}+1)e^\beta x \beta  e^{\beta N}\,.\label{fiprima2}
\end{equation}
On the other hand, for $x>N$,
\begin{equation}
x\varphi'(x)=x\theta(x)\geq N\phi\beta e^{\beta N}+x(1-\phi)\beta e^{\beta N}.
\label{fiprima}\end{equation}
Therefore, from \eqref{fiprima2} and \eqref{fiprima}, taking into account
the supports of $\phi$ and $(1-\phi)$ we observe   that  for $x>N+1$,
$ x\varphi'(x)\geq C\varphi(x)$, while for $N<x<N+1$,  we conclude  that
\begin{equation*}
x\varphi'(x)\geq N\phi\beta e^{\beta N}+N(1-\phi)\beta e^{\beta N}
=N\beta e^{\beta N}\geq  { \frac12}(N+1)
\beta e^{\beta N}  \geq { \frac12}x\beta e^{\beta N}
\geq C\varphi(x),\label{thetaphi}
\end{equation*}
from which \eqref{condfi1} follows.

Finally, we verify \eqref{condfi}. We observe that that for $j=1,2\dots$,
and fixed $\beta>0$,
 \begin{align*}
 |\varphi^{(j+1)}|
&=|\theta^{(j)}|=|\phi \beta^{1+j}e^{\beta x}
 +\sum_{l=1}^j c_{j,l}\phi^{(l)}\beta^{j-l}\beta e^{\beta x}
 - \phi^{(j)}\beta e^{\beta N}|\\
&\leq \beta^j\phi\beta e^{\beta x}+C_j(1+\beta)^{j-1}(\beta e^{\beta(N+1)}
 +\beta e^{\beta N})\chi_{[N,N+1]}\\
&\leq \beta^j\theta+C_j\beta e^{\beta N}\chi_{[N,N+1]}\\
&= \beta^{j}\theta+C_j(\beta\phi e^{\beta N}+(1-\phi)
 \beta e^{\beta N})\chi_{[N,N+1]}\\
&\leq \beta^j\theta +C_j\theta =C_{j}\varphi',
 \end{align*}
 where $C_j$ depends on $\beta$ and $j$ but is independent of $N$.
Thus, the first inequality in \eqref{condfi} is proved.
For the  inequality $\phi' \leq C\phi$   in \eqref{condfi} we proceed
by integrating the inequality $\phi''\leq C\phi'$ already established.
This completes the proof of Theorem \ref{decaimiento}.
 \end{proof}

\begin{remark}\label{remdec} {\rm
For the case of equation \eqref{ec}, with $p\leq k$, we can establish
a result similar to Theorem \ref{decaimiento},  by making minor modifications
and some simplifications in the former proof. In the simple case $p\leq 1$,
for example for the equation
$$
\partial_tu+(-1)^{k+1}\partial_x^nu=-u\partial_xu,
$$
it is possible to follow the procedure of the proof of Theorem \ref{decaimiento}
to establish, without the hypothesis of polynomial decay, that the exponential
decay at $t=0$ is preserved for $t\in[0,1]$.
This can be done by taking $\varphi_N(x):=\int_{-\infty}^x\theta_N(x')\,dx'$
as in \eqref{fifi}, with
$\theta_N(x)\equiv\theta(x):=\phi \beta e^{\beta x}
+(1-\phi)\beta e^{-\beta(x-2N) }$, instead of the functions $\theta_N$
defined in \eqref{tetan}. This functions $\varphi_N$ are bounded
and satisfy \eqref{condfi3} and \eqref{condfi} which is enough for this case.
}\end{remark}

\section{Estimates of Carleman type}\label{sectioncar}

In this section we obtain boundedness properties of the linear operator
$(\partial_t+(-1)^{k+1}\partial_x^n)^{-1}$, and its spatial derivatives
up to order $n-1$, in spaces of the type $L^p-L^q$ with exponential
weight $e^{\lambda x}$. We  keep our exposition simple since we only use
 values of  $p$ and $q$ in the set $\{1,2,+\infty\}$.

Let $D:=\mathbb{R}\times[0,1]$ and, for $R\in \mathbb{R}$,
let $D_R:=\{(x,t)\mid x\geq R\, ,\,t\in[0,1]\}$. We will denote
\begin{gather*}
\|\cdot\|_{L^p_xL^q_T}:=\|\cdot\|_{L^p_xL^q_t(D)},\quad
\|\cdot\|_{L^p_{x\geq R}L^q_T}:=\|\cdot\|_{L^p_xL^q_t(D_R)}, \\
\|\cdot\|_{L^q_TL^p_x}:=\|\cdot\|_{L^q_tL^p_x(D)},\quad
\|\cdot\|_{L^q_TL^p_{x\geq R}}:=\|\cdot\|_{L^q_tL^p_x(D_R)}.
\end{gather*}

\begin{theorem}\label{car}
For $n=2k+1$ with $k\in\mathbb{N}$, let $v$ be a function
in $C([0,1];H^{n}(\mathbb{R}))\cap C^1([0,1];L^2(\mathbb{R})$
such that $\operatorname{supp}v(t)\subset[-M,M]$ for all $t\in[0,1]$,
for some $M>0$. Then, for $\lambda>2$ we have
\begin{gather}
\| e^{\lambda x} v\|_{L_T^\infty L_x^2} \leq C\| e^{\lambda x}(|v(0)|+|v(1)|)\|_L^2(\mathbb{R}
+C\| e^{\lambda x}(\partial_t+(-1)^{k+1}\partial_x^{n})v\|_{L^1_T L^2_x}.
\label{car1}
\\
\begin{aligned}
\sum_{j=1}^{n-1}\| e^{\lambda x}\partial_x^jv\|_{L^\infty_x L^2_T}
&\leq C\lambda^{n-1}\|( |J^{n} ( e^{\lambda x} v(1))| +|J^{n}( e^{\lambda x} v(0))|\|_L^2(\mathbb{R}\\
&\quad +\| e^{\lambda x}(\partial_t+(-1)^{k+1}\partial_x^{n})v\|_{L^1_x L^2_T},
\end{aligned}\label{car2}
\end{gather}
where $C$ is independent of $\lambda>2$ and $M$, and
 $(Jf)\widehat{\;}:= (1+|\xi|^2)^{1/2}\widehat f$.
\end{theorem}

Reasoning formally, suppose that $ e^{\lambda x}(\partial_t+(-1)^{k+1}\partial^n_x)g=h$,
and denote $f= e^{\lambda x} g$ and
$T_0=[ e^{\lambda x} (\partial_t+(-1)^{k+1}\partial^n_x)e^{-\lambda x}]^{-1}$.
Then,  $f=T_0h$. Since $ e^{\lambda x}\partial_xe^{-\lambda x}f=(\partial_x-\lambda)f$,
we have that $ e^{\lambda x}\partial_x^ne^{-\lambda x}f=(\partial_x-\lambda)^nf$,
and thus the multipier operator representing $T_0$ via the Fourier
transform is given by
\begin{equation}
(T_0h)\widehat{\;}\,(\xi,\tau)
=\frac{\widehat h}{i\tau+(-1)^{k+1}(i\xi-\lambda)^n}
\equiv m_0\widehat h.
\end{equation}
We will write $m_0$ as
\begin{equation}
m_0=\frac{-i}{\tau-(\xi+i\lambda)^n}.\label{tres}
\end{equation}
Since for a positive integer $j$,
$ e^{\lambda x}\partial^j_xg=(\partial_x-\lambda)^jf$, we have that
\begin{equation}
\begin{aligned}
 e^{\lambda x}\partial^j_xg &=(\partial_x-\lambda)^jT_0h\equiv T_jh
=[(i\xi-\lambda)^jm_0\widehat h]\,\check{\;}\\
&=\Bigl[\frac{-i^{j+1}(\xi+i\lambda)^j\,}{\tau-(\xi+i\lambda)^n}
\widehat h\Bigr]\,{}^{{\check{\;}}}\,\equiv[m_j\widehat h]\,\check{\;}.
\end{aligned} \label{cuatro}
\end{equation}


\begin{lemma}\label{lema1}
Let $h\in L^1(\mathbb{R}^2)$. Then there is $C>0$ independent of $h$ and $\lambda>0$
such that
\begin{equation}
\|T_0h\|_{L_t^\infty L_x^2}\leq C\|h\|_{L^1_t L^2_x}.
\label{cinco}
\end{equation}
\end{lemma}

\begin{proof}
Let $a(\xi)$ and $b(\xi)$ be the real and imaginary parts of
$-(\xi+i\lambda)^n$, respectively. Then
\[
m_0=\frac{-i}{\tau+a(\xi)+ib(\xi)}.
\]
We recall that for $a\in \mathbb{R}$ and $b\neq 0$
\begin{equation}
\begin{aligned}
&\bigl(\frac{1}{\tau+a+ib}\bigr)\,\check{\;}^{{}_\tau}\,(t)\\
&=c e^{-iat}[e^{-bt}\chi_{[0,\infty]}(t)\chi_{(0,\infty)}(b)
+e^{bt}\chi_{[-\infty,0)}(t)\chi_{(-\infty, 0)}(b)]
=:G_{a,b}(t).
\end{aligned}\label{siete}
\end{equation}
 Since $|G_{a,b}|\leq c$, by taking inverse Fourier transform
in the variable $\tau$ and using convolutions, it follows that for $t\in\mathbb{R}$
\[
|[T_0h(t)]\widehat{\;}\,(\xi)|
=|\int_{-\infty}^\infty G_{a(\xi),b(\xi)} (t-s)\widehat{h(s)}(\xi)\,ds|
\leq C\int_{-\infty}^{\infty}|\widehat{h(s)}(\xi)|\,ds,
\]
for those values of $\xi$ such that $b(\xi)\neq 0$ (a finite set).
In this way, applying Plancherel's identity and Minkowski's integral
inequality we obtain \eqref{cinco}.
\end{proof}

\begin{lemma}\label{lema2}
Let $h\in L^1(\mathbb{R}^2)$. Then there is $C>0$, independent of $h$ and $\lambda>2$,
such that
\begin{equation}
\|T_jh\|_{L^\infty_x L^2_t}\leq C\|h\|_{L^1_x L^2_t}\quad\text{for }
 j=1,\dots,n-1.\label{seis}
\end{equation}
\end{lemma}

\begin{proof}
From \eqref{tres} and \eqref{cuatro}, it follows that
\begin{equation}
(T_jh)\widehat{\;}=(i\xi-\lambda)^jm_0\widehat h
=\frac{-i^{j+1}(\xi+i\lambda)^j\widehat h}{\tau-(\xi+i\lambda)^n}
=m_j\widehat h.\label{ocho}
\end{equation}
Let us denote $\theta:=(\xi+i\lambda)/\tau^{1/n}$. Then
\begin{equation}
m_j=\frac{C}{\tau^{1-j/n}}\frac{\theta^j}{1-\theta^n}
=\frac{C}{\tau^{1-j/n}}\sum_{l=1}^n\frac{c_l}{\theta-r_l},
\end{equation}
where $r_l:=a_l+ib_l$, $l=1,\dots,n$, are the $n^{\text{th}}$-roots of $ 1$,
and the $c_l$ can be computed by L'Hopital's rule to obtain that
\[
c_l=\lim_{\theta\to r_l}\frac{(\theta-r_l)\theta^j}{1-\theta^n}
=-\frac{1}{nr_l^{n-j-1}}.
\]
Therefore,
\[
m_j=\frac{C}{\tau^{1-(j+1)/n}}\sum_{l=1}^n\frac{c_l}{\xi+i\lambda-\tau^{1/n}r_l}
=\frac{C}{\tau^{1-(j+1)/n}}\sum_{l=1}^n\frac{c_l}{\xi-\tau^{1/n}a_l
+i(\lambda-\tau^{1/n}b_l)}.
\]
Taking the inverse Fourier transform with respect to the variable $\xi$,
 observing that for fixed $\lambda$, $\lambda-\tau^{1/m}b_l\neq 0$ for all $l$,
except for a finite number of values of $\tau$, and applying \eqref{siete}
(with $\xi$ and $x$ instead of $\tau$ and $t$, respectively) we obtain a
collection of bounded functions $G_1,\dots,G_l$ of $x$ and $\tau$ such that
\begin{equation*}
[m_j(\cdot_{\xi},\tau)]\,\check{\;}^{{}_{\,\xi}}\,(x)
=\frac{C}{\tau^{1-(j+1)/n}}\sum_{l=1}^n c_l\,G_l(x,\tau).
\end{equation*}
If $|\tau|>1$, then it is clear that
\begin{equation}
	|[m_j(\cdot,\tau)]\,\check{\:}^{{}_{\,\xi}}\,(x)|\leq C,\label{inve}
\end{equation}
with $C$ independent of $\lambda$, $x$ and $|\tau|>1$.
 We can use \eqref{ocho} to prove that this function is bounded also
for $|\tau|\leq 1$. To do this we will consider only the case $j=n-1=2k$,
 the other cases being similar. Let us observe from \eqref{cuatro} that
 \begin{align*}
\bigl |m_{2k}-\frac{i^{n}}{\xi+i\lambda}\bigr |
&=\bigl|\frac{(\xi+i\lambda)^{n-1}}{\tau-(\xi+i\lambda)^n}
+\frac1{\xi+i\lambda}\bigr|\\
&=\bigl|\frac\tau{(\tau-(\xi+i\lambda)^n)
(\xi+i\lambda)}\bigr|\leq \frac2{|\xi+i\lambda|^{n+1}}
\end{align*}
which belongs to $L_\xi^1$,
since $\lambda>2$ and $|\tau|\leq 1$. From the Fourier inversion
formula it can be seen that $|[|\xi+i\lambda|^{-n-1}]\check{\;}(x)|\leq C$,
 with $C$ independent of $\lambda>2$. Thus, by taking inverse
Fourier transform with respect to the variable $\xi$  and taking into
account that from \eqref{siete}
$[(\xi+i\lambda)^{-1}]\,\check{\;}^{{}_{\,\xi}}\,(x)$
is a bounded function of $x$, with bound independent of $\lambda$,
we see, together with the estimate already obtained for $|\tau|\leq 1$,
that \eqref{inve} is  valid for all $x$ and all but a finite number
of values of $\tau$.

Hence, we can apply basic properties of convolution and Plancherel's identity
to conclude that
 \[
\|T_{2k}h\|_{L^\infty_x L^2_t}\leq C\|h\|_{L^1_x L^2_t},
\]
which concludes the proof of Lemma \ref{lema2}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{car}]
We extend $v$ to all $t\in\mathbb{R}$ with value zero in $\mathbb{R}-[0,1]$,
and call this extension again $v$. For $\varepsilon>0$, we consider a
function $\eta:=\eta_\varepsilon\in C^\infty(\mathbb{R}_t)$ such that
$\eta_\varepsilon=0$ in $\mathbb{R}-[0,1]$, $\eta_\varepsilon=1$ in $[\varepsilon,1-\varepsilon]$,
 $\eta'\geq0$ in $[0,\varepsilon]$,   $\eta'\leq0$ in $[1-\varepsilon,1]$,
and $\eta_{\epsilon'}\leq \eta_\varepsilon$ if $\varepsilon'<\varepsilon$.
Define $g:=\eta_\varepsilon(\cdot_t)v$. Then
\begin{equation*}
 e^{\lambda x}(\partial_t+(-1)^{k+1}\partial_x^n)g
= e^{\lambda x}\eta_\varepsilon'v+\eta_\varepsilon e^{\lambda x}(\partial_t+(-1)^{k+1}\partial_x^n)v
\equiv h_{1,\varepsilon}+h_{2,\varepsilon}\equiv h_1+h_2.\end{equation*}
Then,  from \eqref{cuatro},
\begin{equation} e^{\lambda x}\partial^j_xg=T_jh_1+T_jh_2\quad
\text{ for }j=0,\dots,n-1.\label{docea}
\end{equation}
From Lemma \ref{lema2},
\begin{equation}
\|T_jh_2\|_{L^\infty_x L^2_t}\leq C\|h_2\|_{L^1_x L^2_t}
=C\|\eta_\varepsilon e^{\lambda x}(\partial_t+(-1)^{k+1}\partial_x^n)v\|_{L^1_x L^2_t}.\label{doce}
\end{equation}
For $T_jh_1$,  we see from \eqref{cuatro} that
$(T_jh_1)\widehat{\;}=C(\xi+i\lambda)^j(T_0h_1)\widehat{\;}$,
and apply  the Sobolev embedding from $H^1(\mathbb{R})$ to $L^\infty(\mathbb{R})$,
 Plancherel's identity, and Lemma \ref{lema1} to conclude that
\begin{equation}
\begin{aligned}
\|T_jh_1\|_{L^\infty_x L^2_T}
&\leq C\|JT_jh_1\|_{L^2_T L^2_x}\\
&\leq C\|(1+|\xi|)(|\xi|^j+ \lambda^j)(T_0h_1)
 \widehat{\;}{\,}^x\|{{}_{L^2_T L^2_\xi}} \\
&\leq C\|(1+\lambda)^j(1+|\xi|)^{j+1}(T_0h_1)\widehat{\;}{\,}^x\|{{}_{L^2_T
 L^2_\xi}} \\
&\leq C(1+\lambda)^j\|T_0J^{j+1}h_1\|{{}_{L^\infty_t L^2_x}} \\
&\leq C\lambda^j\|J^{j+1}h_1\|_{L^1_t L^2_x} \\
&=C\lambda^j\|\eta_\varepsilon' J^{j+1}( e^{\lambda x} v)\|_{L^1_t L^2_x}\\
&\leq C\lambda^{n-1}\|\eta_\varepsilon' J^{n}( e^{\lambda x} v)\|_{L^1_t L^2_x}.\label{trece}
\end{aligned}
\end{equation}
Hence, from \eqref{docea}, \eqref{doce}, and \eqref{trece} we have that
\begin{equation*}
\|\eta_\varepsilon  e^{\lambda x}\partial^j_x v  \|_{L^\infty_x L^2_T}
 \leq C\lambda^{n-1}\|\eta_\varepsilon' J^{n}( e^{\lambda x} v)\|_{L^1_t L^2_x}
 + C\|\eta_\varepsilon e^{\lambda x}(\partial_t+(-1)^{k+1}\partial_x^n)v\|_{L^1_x L^2_t}.
\end{equation*}
We now make $\varepsilon\to 0$ in this inequality and apply Fatou's Lemma on the
left-hand side and the monotone convergence theorem in the second term
of the right-hand side.  For the first term on the right-hand side we
use the fact that $|\eta'_{\varepsilon}|$ acts as an  approximation of the
identity  on each one of the time intervals $(0,\varepsilon)$ and $(1-\varepsilon,1)$.
Thus,  we obtain  \eqref{car2} after adding up in $j$.

The proof of \eqref{car1} is similar but we use Lemma \ref{lema1}
instead of Lemma \ref{lema2}. This completes the proof.
\end{proof}

\section{Lower estimates}\label{sectionlow}

In this section we prove that the $L^2$-norm of $w$ in a small
rectangle $Q=[0,1]\times[r,1-r]$, with $r\in(0,1)$ can be bounded
by a multiple of the $H^{n-1}$ norm of $w$ in a distant rectangle
$[R,R+1]\times[0,1]$.

\begin{lemma}\label{lema3}
Let $\phi\in C^\infty([0,1])$ be a function with $\phi(0)=\phi(1)=0$,
and for $R>1$ and $ \alpha>0$ define
 $\psi_\alpha(x,t):=\psi(x,t):=\alpha(\frac xR+\phi(t))^2$, $x\in\mathbb{R}$,
$t\in[0,1]$. For $n=2k+1$, suppose that
$g\in C([0,1];H^n(\mathbb{R}))\cap C^1([0,1];L^2(\mathbb{R}))$ is such that $g(0)=g(1)=0$
and  the support of $g$ is contained in the set
\begin{equation*}
A_{1,5}:=\{(x,t): t\in[0,1],\;1\leq \frac xR+\phi(t)\leq 5\}.
\end{equation*}
Then, there are  constants $C=C(n)>0$  and
$\overline C=\overline C(n,\|\phi'\|_{L^\infty},\|\phi''\|_{L^\infty})>1$
such that
\begin{equation}
\sum_{j=0}^{n-1}\frac{\alpha^{n-j-\frac12}}{R^{n-j}}\|e^{\psi_\alpha}
\partial_x^jg\|\leq C\|e^{\psi_\alpha}(\partial_t+(-1)^{k+1}\partial_x^n)g\|\quad
\text{for all }\alpha>\overline CR^{n/(n-1)},\label{quince}
\end{equation}
where $\|\cdot\|:=\|\cdot\|_{L^2(D)}$.
\end{lemma}

\begin{proof}
Let $f:=e^\psi g$ and observe that
$e^\psi\partial_x^j g=e^\psi\partial_x^je^{-\psi}f=(\partial_x-\psi_x)^jf$, and
$e^\psi\partial_t g=(\partial_t-\psi_t)f$. Therefore, if we denote
$$
T:= e^\psi(\partial_t+(-1)^{k+1}\partial_x^n)e^{-\psi}
=(\partial_t-\psi_t)+(-1)^{k+1}(\partial_x-\psi_x)^n,
$$
then, to prove the inequality in \eqref{quince} we must prove that
\begin{equation}
\sum_{j=0}^{n-1}\frac{\alpha^{n-j-\frac12}}{R^{n-j}}\|
(\partial_x-\psi_x)^jf\|\leq C\|Tf\|.\label{dxmb}
\end{equation}
Let $B:=-\psi_x=-\frac{2\alpha} R\varphi$, where
$\varphi(x,t):=\frac xR+\phi(t)$. We will study the operator
$(\partial_x-\psi_x)^n=(\partial_x+B)^n$ in the following manner.

Each one of the terms in the expansion of this operator is of the form
$T_1\dots T_n $, where each $T_i$ is either $\partial_x$ or $B$.
For $m$ and $l$ with $m+l=n$ we will denote by $[m,l]$ the sum of the
terms $T_1\dots T_n$ in this  expansion with $T_i=\partial_x$ for $m$
values of $i$, and $T_i=B$ for $l$ values of $i$. The number of terms of
 $[m,l]$ in the binomial expansion of  $(\partial_x+B)^n$ is then given by
$\binom nl=\binom nm:=n!/m!\,l!$.
Applying integration by parts in $D$, for the class of functions satisfying
the hypotheses given for $g$, we observe that $[m,l]$ is a symmetric
operator if $m$ is even and is an antisymmetric operator if $m$ is odd.

In this way, we write
\begin{equation}
(-1)^{k+1}T=(-1)^{k+1}(\partial_t-\psi_t)
+\sum_{\substack{0\leq m\leq n\\l+m=n}}[m,l]:=S+A,
\end{equation}
where
\begin{equation}\begin{gathered}
S=[n-1,1]+[n-3,3]+\dots+[2,n-2]+[0,n]+(-1)^k\psi_t \equiv S_1+(-1)^k\psi_t,\\
A=[n,0]+[n-2,2]+\dots+[3,n-3]+[1,n-1]-(-1)^k\partial_t
\equiv A_1-(-1)^k\partial_t
\end{gathered}\label{SAdos}.
\end{equation}
Let us denote by $\langle\cdot,\cdot\rangle$ the inner product in the (real)
space $L^2(D)$. Then
\begin{equation}
\|Tf\|^2= \langle (S+A)f,(S+A)f\rangle
=\|Sf\|^2+\|Af\|^2+2\langle Sf,Af\rangle
\geq 2\langle Sf,Af \rangle.\label{Te}
\end{equation}
Now,
\begin{equation}
\langle Sf,  Af \rangle
=\langle S_1f,  A_1f \rangle- (-1)^k\langle S_1f   ,
\partial_tf \rangle+(-1)^k\langle \psi_t f  ,
A_1f \rangle-\langle \psi_tf  , \partial_tf  \rangle.\label{suno}
\end{equation}
We will now estimate each one of the four terms on the right hand side
of \eqref{suno}.

To estimate $\langle S_1f,A_1f\rangle$ we observe that this product
is a sum of terms of the form $\langle [m,n-m]f,[r,n-r]f\rangle$,
with $m$ even and $r$ odd, say $m=2k_1$, $r=2k_2+1$, $k_1, k_2\in\{0,\dots,k\}$.
Using the fact that $B_{xx}=\psi_{xxx}=0$, we can apply integration by parts
to obtain
\begin{equation}
\langle [m,n-m]f,[r,n-r]f\rangle
=\sum_{j=0}^{k_1+k_2}\int_DP_{k_1,k_2,j}(B,B_x)(\partial_x^jf)^2
 \end{equation}
where  $P_{k_1,k_2,j}(B,B_x)=c_{k_1,k_2,j}B^{(n-m)+(n-r)-\nu}B_x^\nu$,
with $\nu+2j=m+r$.
Since $B=-\psi_x=-\frac{2\alpha}R\varphi$,
$B_x=-\psi_{xx} =-\frac{2\alpha}{R^2}$, we have that
\begin{equation}
\begin{aligned}
P_{k_1,k_2,j}(B,B_x)
&=-c_{k_1,k_2,j}(2\alpha)^{2n-m-r}
\frac{\varphi^{2n-m-r-\nu}}{R^{2n-m-r+\nu}}  \\
&=-(2\alpha)^{2n-2k_1-2k_2-1}c_{k_1,k_2,j}
\frac{\varphi^{2n-4k_1-4k_2+2j-2}}{R^{2n-2j}} \\
&\equiv \alpha^{2n-2k_1-2k_2-1}Q_{k_1,k_2,j}
\end{aligned} \label{p}
\end{equation}
 Therefore,
\begin{align}
 \langle S_1f,A_1f\rangle
&=\sum_{k_1,k_2=0}^k\langle[2k_1,n-2k_1]f,[2k_2+1,n-2k_2-1]f\rangle \nonumber \\
 &=\sum_{k_1,k_2=0}^k\alpha^{2n-2k_1-2k_2-1}\sum_{j=0}^{k_1+k_2}
 \int Q_{k_1,k_2,j}(\partial_x^jf)^2 \nonumber \\
 &=\sum_{j=0}^{2k}\sum_{\substack{k_1+k_2\geq j\\0\leq k_1,k_2\leq k}}
 \alpha^{2n-2k_1-2k_2-1}\int Q_{k_1,k_2,j}(\partial_x^jf)^2. \label{mayorj}
\end{align}
 For each $j$ in the former expression we will separate the term having
the highest power of $\alpha$. Thus we write
 \begin{equation}
\begin{aligned}
\langle S_1f,A_1f\rangle
&=\kern-2pt\sum_{j=0}^{2k}\kern-1pt\alpha^{2n-2j-1}
\sum_{\substack{k_1+k_2=j\\0\leq k_1,k_2\leq k}}
\int Q_{k_1,k_2,j}(\partial_x^jf)^2\\
&\quad +\sum_{j=0}^{2k}\sum_{\substack{k_1+k_2> j\\0\leq k_1,k_2\leq k}}
 \alpha^{2n-2k_1-2k_2-1}\int Q_{k_1,k_2,j}(\partial_x^jf)^2\\
&\equiv\sum_{j=0}^{2k} I_j+\sum_{j=0}^{2k} II_j.
\end{aligned}\label{inf3}
\end{equation}
We will concentrate upon the terms $I_j$ and will refer to the terms
 $II_j$ as lower order terms (l.o.t.).


For each $j$ we will now compute the term $I_j$. If  $m\leq n$ and $l=n-m$,
then $[m,l]$ is a sum of operators of the form $T=T_1T_2\dots T_n$  where
$T_i=\partial_x$ for $m$ indices $i$, and $T_i=B$ for the remaining $l$
indices $i$.  We apply the product rule for derivatives to expand $T$
and consider the terms  in its expansion  containing the derivatives of
highest order: $\partial^m_x$ and $\partial^{m-1}_x$.
This leads to (see the illustration below)
\begin{equation}
\begin{aligned}
T&=T_1\dots T_n=B^l\partial_x^m+[B^{r_1}(\partial_xB^{l-r_1})
 +B^{r_2}(\partial_xB^{l-r_2})+\dots\\
&\quad +B^{r_m}(\partial_xB^{l-r_m})]\partial_x^{m-1}+\text{l.d.t.},
\end{aligned}\label{T}
\end{equation}
where $r_1,r_2,\dots r_n$ depend upon the position of the $m$ operators
$\partial_x$ in the expression of $T$, and the notation l.d.t.
stands for ``lower derivative terms".

To illustrate this, let us take for example the case with $n=9$, $m=3$, $l=6$,
and consider the operator
$T=B\partial_xB\partial_xBB\partial_xBB$.
 Then,
\[
T=B^6\partial_x^3+B^4(\partial_xB^2)\partial_x^2
+B^2(\partial_xB^4)\partial_x^2+B(\partial_xB^5)\partial_x^2+\text{l.d.t.}
\]
But, the operator  $T^{(*)}:=T_n\dots T_2 T_1$,  is also present in the
expansion of $[m,l]$, and
\begin{equation}
\begin{aligned}
T^{(*)}&=T_n\dots T_1\\
&=B^l\partial_x^m+[B^{l-r_1}(\partial_xB^{r_1})+B^{l-r_2}(\partial_xB^{r_2})
 +\dots\\
&\quad   +B^{l-r_m}(\partial_xB^{r_m})]\partial_x^{m-1}+\text{l.d.t.}
\end{aligned}\label{T*}
\end{equation}
Therefore, if $T\neq T^*$ we have from \eqref{T} and \eqref{T*}  that
 \begin{equation}
T+T^{(*)}=2B^l\partial_x^m+m(\partial_xB^l)\partial_x^{m-1}
+\text{l.d.t.}\label{TT}
\end{equation}
 It can be seen that if $T=T^{(*)}$, then  the same expression is valid.
 Since there are ${\binom nm}$ terms in the expansion of $[m,l]$
we conclude from \eqref{TT} that
\begin{equation}
 [m,l]={\binom nm}[B^l\partial^m_x+\frac {ml}2B^{l-1}B_x\partial_x^{m-1}]
+\text{l.d.t.}\label{ml}
\end{equation}
In this way, for $m=2k_1$, $r=2k_2+1$, $j=k_1+k_2=\frac12(m+r-1)$, $l=n-m$,
$s=n-r$, we apply integration by parts to observe that
\begin{align*}
&\langle[m,l]f,[r,s]f \rangle\\
&={{{\binom nm}{\binom nr}} }
 \Bigl(\int B^{l+s}\partial_x^mf\,\partial_x^rf
 +\frac{sr}2 \int B^{l+s-1}B_x\,\partial_x^mf\,\partial_x^{r-1}f
 +\frac{ml}2 \\
&\quad\times \int B^{l+s-1}B_x\,\partial_x^{m-1}f\partial_x^{r}f\Bigr)
 +\text{l.o.t.}\\
&=\frac12{{{\binom nm}{\binom nr}}} (-1)^{\frac12(m+1-r)}
 \Bigl((m-r)(l+s)+{rs}-{ml}\Bigr)\\
&\quad\times \int B^{l+s-1}  B_x(\partial_x^{\frac12(m+r-1)}f)^2+\text{l.o.t.}\\
&=\frac12{{{\binom nm}{\binom nr}}}(-1)^{k_1-k_2}n(m-r)\int B^{l+s-1}
 B_x(\partial_x^{j}f)^2+\text{l.o.t.}\\
&=(2\alpha)^{2n-2j-1}{{\frac12}}{{{\binom nm}{\binom nr}}}(-1)^{j}n(m-r)
\int( \frac\varphi R)^{2n-2j-2}\,\frac{-1}{R^2}
(\partial_x^{j}f)^2+\text{l.o.t.}
 \end{align*}
According to the definition of $I_j$ in \eqref{inf3}, and from the
first equality in \eqref{mayorj},  to obtain $I_j$ we add the high order
terms terms in the former expression with $k_1+k_2=j$  and
$0\leq k_1,k_2\leq k$. In this way, we obtain that
 \begin{equation}
I_j= {A^n_j\,{\frac {n}2}}\,\frac{(2\alpha)^{2n-2j-1}}{R^{2n-2j}}
\int \varphi^{2n-2j-2}(\partial_x^{j}f)^2,\label{high}
 \end{equation}
Where
\begin{equation}
A^n_j:=(-1)^{j}\sum_{\substack{k_1+k_2=j\\ 0\leq k_1,k_2\leq k}}
 (2k_2+1-2k_1){\binom n{2k_1}}{\binom n{2k_2+1}}.\label{Asubnj2}
\end{equation}

 We will prove in Lemma \ref{lemaaj} below that
  \begin{equation}
A_j^n=n{\binom{n-1}j}\geq n,\quad \text{for all integers } n\geq 3
\text{ odd and all }j\leq n-1,\label{comb1}
\end{equation}
and in particular all the coefficients $A_j^n$ are positive.

Regarding the lower terms $II_j$, we see from \eqref{inf3} and \eqref{p} that
 \begin{equation}
II_j=-\sum_{\substack{k_1+k_2> j\\0\leq k_1,k_2\leq k}}c_{k_1,k_2,j}
\frac{(2\alpha)^{2n-2(k_1+k_2)-1}}{R^{2n-2j}}
\int \varphi^{2n-4k_1-4k_2+2j-2}(\partial^j_xf)^2,\label{low}
 \end{equation}
Thus, from \eqref{inf3}, using \eqref{high} and \eqref{low} we have
\begin{equation}
\begin{aligned}
&\langle S_1f,A_1f\rangle\\
&=\sum_{j=0}^{2k}{A^n_j\,{\frac {n}2}}\,\frac{(2\alpha)^{2n-2j-1}}{R^{2n-2j}}
\int \varphi^{2n-2j-2}(\partial_x^{j}f)^2 \\
&\quad -\sum_{j=0}^{2k}\sum_{\substack{k_1+k_2> j\\0\leq k_1,k_2\leq k}}
c_{k_1,k_2,j} \frac{(2\alpha)^{2n-2(k_1+k_2)-1}}{R^{2n-2j}}
\int \varphi^{2n-4k_1-4k_2+2j-2}(\partial^j_xf)^2.
\end{aligned} \label{IjmasIIj}
\end{equation}
 We now turn our attention to the product $\langle \psi_tf,A_1f\rangle$
in \eqref{suno}.  For $r=2k_1+1$, $k_1=0,\dots,k$ and $s=n-r$,
taking into account that $\psi_{txx=0}$, and using integration by parts
we see that
 \begin{equation}
 \langle [r,s]f,\psi_tf]\rangle=\sum_{j=0}^{k_1}
\int P_{k_1,j}(B,B_x,\psi_t,\psi_{tx})(\partial^j_xf)^2,\label{ers}
 \end{equation}
where
 \begin{equation}
P_{k_1,j}(B,B_x,\psi_t,\psi_{tx})
=c'_{k_1,j}B^{s-\nu}B_x^{\nu}\psi_t+c''_{k_1,j}B^{s-(\nu-1)}
B_x^{\nu-1}\psi_{tx}
\end{equation}
with $\nu+2j=r$.
 Since $\psi_t=2\alpha(\frac xR+\phi)\phi'=2\alpha\varphi\phi'$ and
$\psi_{tx}=2\alpha\frac{\phi'}R$, we have that
 \begin{align*}
P_{k_1,j}(B,B_x,\psi_t,B_t)
&=(2\alpha)^{s+1}\Bigl(c'_{k_1,j}\frac{\varphi^{s-\nu}}{R^{s-\nu}}
 \frac1{R^{2\nu}}\,\varphi\phi'+  c''_{k_1,j}
 \frac{\varphi^{s-\nu+1}}{R^{s-\nu+1}}\frac1{R^{2\nu-2}}\frac{\phi'}R\Bigr)\\
&\equiv c_{k_1,j}\,\alpha^{s+1}\frac{\varphi^{s-\nu+1}}{R^{s+\nu}}\phi'
= \alpha^{n-2k_1}\,c_{k_1,j}\frac{\varphi^{n-4k_1+2j-1}}{R^{n-2j}}\phi'\\
&\equiv \alpha^{n-2k_1}Q_{k_1,j}.
 \end{align*}
In this way, from \eqref{ers} and proceeding as we did to obtain \eqref{mayorj},
\begin{equation}
\begin{aligned}
&\langle \psi_tf,A_1f\rangle \\
&=\sum_{k_1=0}^k \langle [2k_1+1, n-(2k_1+1)]f,\psi_tf\rangle
 \\
&=\sum_{k_1=0}^k \alpha^{n-2k_1}\sum_{j=0}^{k_1}
 \int Q_{k_1,j}(\partial^j_xf)^2=\sum_{j=0}^{k}
 \sum_{k_1=j}^k\alpha^{n-2k_1}\int Q_{k_1,j}(\partial^j_xf)^2 \\
&=\sum_{j=0}^{k}\alpha^{n-2j}\int Q_{j,j}(\partial^j_xf)^2
 +\sum_{j=0}^k\sum_{k_1=j+1}^k\alpha^{n-2k_1}\int Q_{k_1,j}(\partial^j_xf)^2
 \\
&=\sum_{j=0}^{k}c_{j,j}\frac{\alpha^{n-2j}}{R^{n-2j}}
 \int \varphi^{n-2j-1}\phi'(\partial^j_xf)^2\\
&\quad +\sum_{j=0}^k\sum_{k_1=j+1}^kc_{k_1,j}\frac{\alpha^{n-2k_1}}{R^{n-2j}}
 \int \varphi^{n-4k_1+2j-1}\phi'(\partial^j_xf)^2.
\end{aligned}\label{psisubt}
\end{equation}
 To estimate the term $\langle S_1f,\partial_t f\rangle $ in \eqref{suno}
we notice that, since $S_1$ is the sum of operators of the form $T_1,\dots T_n$
as described above,  from the product rule for differentiation we see that
$\partial_t[(S_1f)]=S_{1t}f+ S_1\partial_t f$, where $S_{1t}$ is  a sum of
compositions of the form $Q_1Q_2\dots Q_n$ where one of the $Q_i's$ is $B_t$
and the others are either $B$ or $\partial_x$. In this way,
by applying  integration by parts we conclude that
$$
\langle S_1f   ,  \partial_tf \rangle=-\langle S_{1t}f
+S_1\partial_tf,f\rangle=-\langle S_{1t}f,f\rangle
-\langle \partial_tf, S_1f\rangle.
$$
Therefore, $\langle S_1f,\partial_tf \rangle=-\frac12\langle S_{1t}f,f\rangle$.
Proceeding as we did to obtain \eqref{ers} to \eqref{psisubt} we see that
$\langle S_{1t}f,f\rangle$ has the same form as the right hand side
of \eqref{psisubt}.

To conclude the computation of \eqref{suno} we see that
\begin{equation}
\langle \psi_t f,\partial_t f\rangle=-\frac12\int\psi_{tt}f^2
=-\alpha\int ((\phi')^2+\varphi\phi'')f^2.\label{fini}
\end{equation}
We return to \eqref{suno} and compare the terms of the form
$\alpha^N/R^M$ (for diverse integer powers $N$ and $M$) in
\eqref{IjmasIIj},  \eqref{psisubt}, and \eqref{fini}, especially,
we compare the highest order terms
$$\frac{\alpha^{2n-2j-1}}{R^{2n-2j}}\quad
\text{(in \eqref{IjmasIIj})}\quad\text{and} \quad
\frac{\alpha^{n-2j}}{R^{n-2j}}\quad\text{(in \eqref{psisubt})}.
$$
Since by the hypotheses in this lemma,   $1\leq\varphi\leq5$
in the support of $f$,  there is a constant
$\overline C=\overline C(n, \|\phi'\|_{L^\infty}+\|\phi''\|_{L^\infty})>1$
such that if we take $\alpha>\overline CR^{n/(n-1)}$, then the  leading terms
in \eqref{IjmasIIj} with coefficient $\frac12 nA_j^n(2\alpha)^{2n-2j-1}/R^{2n-2j}$
absorb the other terms appearing in \eqref{IjmasIIj},
as well as all terms in \eqref{psisubt}, and \eqref{fini}.
Therefore, from \eqref{suno} and \eqref{Te} we have
\begin{equation}
\|Tf\|^2\geq 2\langle Sf,Af\rangle
\geq C\sum_{j=0}^{n-1} A^n_j \frac{\alpha^{2n-2j-1}}{R^{2n-2j}}
\int(\partial_x^jf)^2,\label{SfAf}
\end{equation}
where $C$ depends only upon $n$.

To prove \eqref{dxmb} we must obtain an expression similar to \eqref{SfAf}
with the integrals $\int(\partial_x^jf)^2$ replaced by
$\int((\partial_x+B)^jf)^2$.
To do that, using integration by parts,  we observe that for $m=1,\dots,n-1$,
\begin{equation}
\begin{aligned}
&\int((\partial_x+B)^mf)^2\\
&=\int (\partial_x^mf)^2+\sum_{j=0}^{m-1}
 \sum_{\substack{r,s\geq 0\\r+2s+2j=2m}}c_{m,s,j}
 \int B^rB_x^s(\partial_x^jf)^2 \\
&=\int (\partial_x^mf)^2+\sum_{j=0}^{m-1}
 \sum_{\substack{r,s\geq0\\r+2s+2j=2m}}c_{m,s,j}
 \frac{(2\alpha)^{r+s}}{R^{r+2s}}\int\varphi^r(\partial_x^jf)^2 \\
&=\int (\partial_x^mf)^2+\sum_{j=0}^{m-1}
 \sum_{s=0}^{m-j}c_{m,s,j}\frac{(2\alpha)^{2m-2j-s}}{R^{2m-2j}}
 \int\varphi^{2m-2s-2j}(\partial_x^jf)^2\,.
\end{aligned}\label{cambio}
\end{equation}
Since $\alpha>\overline CR^{n/(n-1)}>1$, and in the support of $f$,
$1\leq\varphi\leq5$,  from \eqref{cambio} it follows that
\begin{equation}
\int\big((\partial_x+B)^mf\big)^2
\leq \int (\partial_x^mf)^2+C' \sum_{j=0}^{m-1}
\frac{\alpha^{2m-2j}}{R^{2m-2j}}\int(\partial^j_xf)^2,
\label{ecu9}
\end{equation}
where $C'>0$ depends only on $n$.
Now, from  \eqref{SfAf} we see that, if $K_{n-1}$ is a constant
with $0<K_{n-1}\leq A^n_{n-1}=A^n_0=n$, then
\begin{equation*}
\|Tf\|^2\geq CK_{n-1}\frac{\alpha}{R^2} \int(\partial^{n-1}_xf)^2
+ C\sum_{j=0}^{n-2} A^n_j \frac{\alpha^{2n-2j-1}}{R^{2n-2j}}
\int(\partial_x^jf)^2,
\end{equation*}
Thus, applying \eqref{ecu9} with $m=n-1$, we conclude that
\begin{align*}
&\|Tf\|^2\\
&\geq CK_{n-1}\frac{\alpha}{R^2}\Big[ \int((\partial_x+B)^{n-1}f)^2
 - C' \sum_{j=0}^{n-2}\frac{\alpha^{2n-2-2j}}{R^{2n-2-2j}}
\int(\partial^j_xf)^2\Big] \\
&\quad +C\sum_{j=0}^{n-2} A^n_j \frac{\alpha^{2n-2j-1}}{R^{2n-2j}}
 \int(\partial_x^jf)^2\\
&=CK_{n-1}\frac{\alpha}{R^2} \int((\partial_x+B)^{n-1}f)^2
 +C\sum_{j=0}^{n-2} (A^n_j -K_{n-1}C')\frac{\alpha^{2n-2j-1}}{R^{2n-2j}}
 \int(\partial_x^jf)^2.
\end{align*}
Therefore, by choosing
$K_{n-1}=\min \{A_{n-1}^n, \frac 12A_{n-2}^n/C',\dots, \frac 12A_{0}^n/C'\}$,
we obtain that
\begin{equation}
\|Tf\|^2\geq C K_{n-1}\frac{\alpha}{R^2}
\int((\partial_x+B)^{n-1}f)^2+C\sum_{j=0}^{n-2} \frac 12 A^n_j
\frac{\alpha^{2n-2j-1}}{R^{2n-2j}}\int(\partial_x^jf)^2,\label{repla}
\end{equation}
and in this way we obtain an expression similar to \eqref{SfAf} with the
first integral term $\int(\partial^{n-1}_xf)^2$ replaced by
$\int((\partial_x+B)^{n-1})f)^2$. Notice that from \eqref{comb1},
$K_{n-1}>0$. Proceeding in a similar manner, using \eqref{repla}, successively
applying \eqref{ecu9} with $m=n-2,n-3,\dots, 1$, and taking adequate values
of $K_{n-2},\dots, K_1>0$, we can perform the replacement of the remaining
integrals. Since the $\min\{K_1,\dots,K_{n-1}\}>0$,  \eqref{dxmb}
follows and the  proof  is complete.
 \end{proof}

 We now prove that all the coefficients $A_j^n$ defined in \eqref{Asubnj2}
are positive.

  \begin{lemma}\label{lemaaj}
 For integer $k\geq 1$, let $n=2k+1$. Then, for  $j=0,\dots,n-1$,
\begin{equation}
  A^n_j:=(-1)^{j}\sum_{\substack{k_1+k_2=j\\ 0\leq k_1,k_2\leq k}}
 (2k_2+1-2k_1){\binom n{2k_1}}{\binom n{ 2k_2+1}}=n{\binom{n-1}j}.\label{anjanj}
\end{equation}
\end{lemma}

\begin{proof}
 Using the formula $(1+x)^n=\sum_{r=0}^n{\binom nr}x^r$ we see that for
$x,\beta\in\mathbb{R}$, $x\neq 0$,
 \begin{align*}
h_\beta(x)&:=\frac14\Bigl((1+\beta x)^n+(1-\beta x)^n\Bigr)
\Bigl((1+\beta/ x)^n-(1-\beta/ x)^n\Bigr)\\
 &=\sum_{k_1,k_2=0}^k
 {{\binom n{2k_1}}}{{\binom n{2k_2+1}}}\beta^{2k_1+2k_2+1}x^{2k_1-2k_2-1}\\
&=\sum_{j=0}^{2k}\sum_{\substack{k_1+k_2=j\\ 0\leq k_1,k_2\leq k}}
 {{\binom n{2k_1}}{\binom n{2k_2+1}}}\beta^{2j+1}x^{2k_1-2k_2-1}.
 \end{align*}
Therefore,
 \begin{equation}
\begin{aligned}
h'_\beta(1)&=\sum_{j=0}^{2k}\beta^{2j+1}
\sum_{\substack{k_1+k_2=j\\ 0\leq k_1,k_2\leq k}} (2k_1-2k_2-1)
{{\binom n{2k_1}}{\binom n{2k_2+1}}}\\
&=\sum_{j=0}^{2k}(-1)^{j+1}A_j^n\beta^{2j+1}.
\end{aligned}\label{expaj}
\end{equation}
But
\begin{align*}
\frac4{n\beta}h'_\beta(x)
&= \Bigl( (1+\beta x)^{n-1}-(1-\beta x)^{n-1}\Bigr)
 \Bigl( (1+\beta/x)^n-(1-\beta/x)^n\Bigr)\\
&\quad +\Bigl( (1+\beta x)^n+(1-\beta x)^n   \Bigr)
\Bigl((1+\beta/x)^{n-1}+(1-\beta/x)^{n-1}   \Bigr)\Bigl(-\frac 1{x^2}\Bigr).
\end{align*}
Therefore, with $x=1$,
\begin{align*}
\frac4{n\beta}h'_\beta(1)
&= \Bigl( (1+\beta )^{n-1}-(1-\beta )^{n-1}\Bigr)\Bigl( (1+\beta)^n-(1-\beta)^n\Bigr)\\
&\quad -\Bigl( (1+\beta )^n+(1-\beta )^n   \Bigr) \Bigl((1+\beta)^{n-1}+(1-\beta)^{n-1}   \Bigr)\\
&= -2(1+\beta)^{n-1}(1-\beta)^n-2(1+\beta)^n(1-\beta)^{n-1}
=-4(1-\beta^2)^{n-1}.
\end{align*}
In this way,
\begin{equation*}
h'_\beta(1)=\sum_{j=0}^{n-1}(-1)^{j+1}n{\binom{n-1}j}\beta^{2j+1},
\end{equation*}
which, with together with \eqref{expaj}, gives \eqref{anjanj}.
\end{proof}

In the following theorem we apply Lemma \ref{lema3} to the difference $w$
of two solutions of equation \eqref{ec} to  obtain a bound of the
$L^2$-norm of $w$ in a small rectangle with the $H^{n-1}$ norm of $w$
in a distant rectangle $[R-1,R]\times[0,1]$.

\begin{theorem}\label{estinferior}
 Let $\varepsilon>0$. For $r\in(0,1)$, $R>2$ and $u_1,u_2\in C([0,1];H^n(\mathbb{R}))$
solutions of equation \eqref{ec} define $w=u_1-u_2$, $Q=[0,1]\times[r,1-r]$,
and
 \begin{equation}
 A_R(w)=\int_0^1\int_{R}^{R+1 }\sum_{j=0}^{n-1}(\partial_x^jw)^2\,dx\,dt\,.\label{Aw}
 \end{equation}
Suppose that $\|w\|_{L^2(Q)}\geq\delta>0$. Then there exist constants $C>0$,
 $C_*=C_*(n,r)>0$, and $R_0>1$ such that
\begin{equation}
 \|w\|_{L^2(Q)}\leq Ce^{C_* R^\gamma}A_R(w) \quad\text{for all }R>R_0\,,\label{hd}
 \end{equation}
where
 \begin{equation}
 \gamma\equiv\gamma_{n,p}=\begin{cases}
\frac{n}{n-1}&\text{if }\,p=0,\dots,k=\frac{n-1}2,\\[4pt]
 \frac{2(n-p)}{2(n-p)-1}+\epsilon&\text{if }\,p=k+1,\dots,n-1.
\end{cases}\label{inf12}
\end{equation}
\end{theorem}

 Notice that for the KdV Hierarchy, $p=n-2$,  and thus we have an exponential
$e^{C_* R^{4/3_+}}$ in \eqref{hd}.

\begin{proof}
From \eqref{ec}, by a process similar to that used to obtain \eqref{Ad},
it can be seen that $w$ is a solution of the equation
\begin{equation}
\partial_tw+(-1)^{k+1}\partial^n_xw=-[P(z_1)-P(z_2)]
=-\sum_{j=0}^pF_j\partial_x^jw \,,\label{Fjj}
\end{equation}
where  each $F_j$ is a polynomial in $\partial_x^{j_1}u_1$ and
$\partial_x^{j_2}u_2$ with $j_1,j_2\leq p\leq n-1$.
From the embedding of $H^1(\mathbb{R})$ in $L^\infty(\mathbb{R})$,
the functions $F_j=F_j(x,t)$  are bounded  in  $D=\mathbb{R}\times[0,1]$.
Let $\phi\in C^\infty([0,1])$ be a function such that $\phi\equiv 0$ in
$[0,\frac r2]\cup[1-\frac r2,1]$, $\phi\equiv 4$ in $[r,1-r]$, $\phi$
increasing in $[\frac r2,r]$, and decreasing in $[1-r, 1-\frac r2]$.
Let us choose functions
$\widetilde\mu,\, \widetilde\theta\in C^\infty(\mathbb{R})$, such that
$\widetilde\mu\equiv0$ in $(-\infty,1]$, $\widetilde\mu\equiv1$ in $[2,\infty)$,
$\widetilde\theta\equiv1$ in $(-\infty,0]$,  $\widetilde\theta\equiv0$ in
$[1,\infty)$, and define
$\mu_R(x,t)\equiv\mu(x,t):=\widetilde\mu(\frac xR+\phi(t))$ and
$\theta_R(x)\equiv\theta(x):=\widetilde\theta(x-R)$.
Let $g:=\mu\theta w$. Then, it can be seen that in the support of $\mu\theta$,
$1\leq\frac xR+\phi(t)\leq5$. Besides $g(0)=g(1)=0$. Thus $g$ satisfies
the hypotheses of Lemma \ref{lema3}. With $\psi=\alpha(\frac xR+\phi(t))^2$,
$\alpha>0$,  as in the statement of Lemma \ref{lema3}, we compute
\begin{align*}
&e^\psi(\partial_tg+(-1)^{k+1}\partial^n_xg)\\
&= e^\psi\Bigl(\mu_t\theta w+\mu\theta\partial_t w
 +\mu\theta(-1)^{k+1}\partial^n_xw
 +\sum_{r=1}^n c_{n,r}\partial_x^r(\mu\theta)\partial_x^{n-r}w\Bigr)\\
&=e^\psi\Bigl(-\sum_{j=0}^pF_j\mu\theta\partial^j_xw+\mu_t\theta w
 + \sum_{r=1}^nc_{n,r}\mu\partial_x^r\theta\partial_x^{n-r}w\\
&\quad +\sum_{r=1}^n\sum_{s=1}^rc_{n,r}c_{r,s}\partial_x^s\mu\partial_x^{r-s}
 \theta\partial_x^{n-r}w\Bigr)\\
&=e^\psi\Bigl(-\sum_{j=0}^pF_j\partial_x^j(\mu\theta w)
 +\sum_{j=1}^pF_j\sum_{r=1}^jc_{j,r}\partial_x^r(\mu\theta)\partial_x^{j-r}w \\
&\quad +\mu_t\theta w+\mu \sum_{r=1}^nc_{n,r}\partial_x^r\theta\partial_x^{n-r}w
 +\sum_{r=1}^n\sum_{s=1}^rc_{n,r}c_{r,s}\partial_x^s\mu\partial_x^{r-s}
 \theta\partial_x^{n-r}w\Bigr)\\
&=  e^\psi\Bigl(-\sum_{j=0}^pF_j\partial_x^jg
 +\mu\sum_{j=1}^pF_j\sum_{r=1}^jc_{j,r}\partial_x^r\theta\partial_x^{j-r}w\\
&\quad +\sum_{j=1}^p\sum_{r=1}^j\sum_{s=1}^rF_jc_{j,r}c_{r,s}
 \partial_x^s\mu\partial_x^{r-s}\theta\partial_x^{j-r}w\\
&\quad +\mu_t\theta w+\mu \sum_{r=1}^nc_{n,r}\partial_x^r\theta\partial_x^{n-r}w
 +\sum_{r=1}^n\sum_{s=1}^rc_{n,r}c_{r,s}\partial_x^s
 \mu\partial_x^{r-s}\theta\partial_x^{n-r}w\Bigr).
\end{align*}
To obtain a bound for the $L^2$-norm of the right-hand side of the former
expression we take into account the following facts: The derivatives
$\partial_x^s\mu$ ($s\geq 1$) are supported in
$\subseteq  \{(x,t)\mid 1<\frac xR+\phi(t)<2\}$ and  thus
$ e^\psi\leq e^{4\alpha}$ in this set whose area is of order $R$. Also,
$\partial_x^r\theta$ ($r\geq 1$)  is supported in $[R,R+1]\times[0,1]$, and
$e^\psi\leq e^{25\alpha}$  in this rectangle. Besides, $w$, the functions
$F_j$, and all the derivatives of $\mu$, $\theta$, and $w$,  are bounded
by a constant independent of $R$.   From this considerations, and applying
Lemma \ref{lema3} we obtain
\begin{equation}
\begin{aligned}
&\sum_{j=0}^{n-1}\frac{\alpha^{n-j-1/2}}{R^{n-j}}\|e^\psi\partial_x^jg\| \nonumber\\
&\leq  C\|e^\psi(\partial_t+(-1)^{k+1}\partial_x^n)g\| \nonumber \\
&\leq C\sum_{j=0}^p\|e^\psi\partial_x^jg\|+Ce^{25\alpha}
 \sum_{j=0}^{n-1}\|\partial^j_xw\|_{L^2([R,R+1]\times[0,1])}
+CR^{1/2}e^{4\alpha}.
\end{aligned}\label{inf9}
\end{equation}
Let $\overline C$ be the constant in Lemma \ref{lema3}.
Then $\overline C=\overline C(n,r)$. If we take
$\alpha=(1+\overline C)R^{1+s}$, with $s\geq\frac1{n-1}$, then
$\alpha>\overline CR^{\frac {n}{n-1}}$, and for $j=1,\dots,p$
$$
\frac{\alpha^{n-j-1/2}}{R^{n-j}}
=(1+\overline C)^{n-j-\frac12}R^{-\frac12+s(n-j-\frac12)}
\geq R^{s(n-p-\frac12)-\frac12}
$$
and therefore, after discarding the terms with $j>p$ on the left-hand
side of \eqref{inf9}, and bearing in mind the definition of $A_R(w)$
given in \eqref{Aw}, we have that
\begin{equation}
\sum_{j=0}^{p}R^{s(n-p-\frac12)-\frac12}\|e^\psi\partial_x^jg\|
\leq C\sum_{j=0}^p\|e^\psi\partial_x^jg\|+Ce^{25\alpha}A_R(w)
+CR^{1/2}e^{4\alpha}.\label{inf10}
\end{equation}
We will choose $s\geq \frac1{n-1}$ in such a way that $s(n-p-1/2)-1/2>0$;
 that is, $s>1/(2n-2p-1)$. Then, by making $R$ sufficiently large we can
make the left-hand side of \eqref{inf10} more than twice the first
term on the right-hand side, allowing the absorsion of this last term to
obtain that
\begin{equation}
\sum_{j=0}^{p}R^{s(n-p-\frac12)-\frac12}\|e^\psi\partial_x^jg\|
\leq Ce^{25\alpha}A_R(w)+CR^{1/2}e^{4\alpha}.\label{inf4}
\end{equation}


To choose the appropriate value of $s$ we see that if $p\leq{(n-1)}/2=k$,
then
$$
\frac1{(2n-2p-1)}\leq \frac1n<\frac1{n-1},
$$
and we choose $s=1/(n-1)$. Then with
$\alpha=(1+\overline C)R^{1+s}=(1+\overline C)R^{(n-1)/n}$,
we have \eqref{inf4} for large $R$.

If $(n-1)/2\leq  p\leq n-1$, that is if $ {(n+1)}/2\leq p\leq n-1$,
then, with $\epsilon>0$ and $s=1/(2n-2p-1)+\epsilon$; that is,
with $\alpha=(1+\overline C)R^{\frac{2(n-p)}{2(n-p)-1}+\epsilon}$,
we obtain \eqref{inf4} for large $R$.

 Since $\frac xR+\phi(t)\geq 4$ in $Q:=[0,1]\times[r,1-r]$ and $\mu\equiv1$,
and $\theta\equiv1$ in $Q$, we can replace the left-hand side of \eqref{inf4}
 by a smaller amount to conclude that for $R$ sufficiently large
\begin{equation}
e^{16\alpha}\|w\|_{L^2(Q)}\leq Ce^{25\alpha}A_R(w)+CR^{1/2}e^{4\alpha},
\label{inf11}
\end{equation}
Hence, with $\gamma$ as in \eqref{inf12}, and $\alpha=(1+\overline C)R^\gamma$,
\begin{equation*}
e^{16(1+\overline C)R^\gamma}\|w\|_{L^2(Q)}\leq Ce^{25(1+\overline C)R^\gamma}
A_R(w)+CR^{1/2}e^{4(1+\overline C)R^\gamma},
\end{equation*}
 Since $\|w\|_{L^2(Q)}\geq\delta>0$, by making $R$ sufficiently large we
can absorb the last term on the right-hand side of the former inequality
with the left-hand side to obtain \eqref{hd} with $C_*=9(1+\overline C)$,
which completes the proof of Theorem \ref{estinferior}.
 \end{proof}

 \section{Proofs of Theorems \ref{thmI} and \ref{thmII}}\label{sectionmain}


 For Theorem \ref{thmI} we present a proof  which can be adapted with minor
changes to prove Theorem \ref{thmII}.

\begin{proof}[Proof of Theorem \ref{thmI}]
From hypothesis \eqref{w},  $e^{x_+^{4/3+\epsilon}}w(0) \in L^2(\mathbb{R})$,
then it follows that $\|e^{ax_+^{4/3+\epsilon/2}}w(0)\|_{L^2(\mathbb{R}}\leq C_a<\infty$
for all $a>0$. The same property holds for $w(1)$. Also, by an interpolation
argument similar to that in \eqref{inter},
 \begin{equation}
 \| e^{ax_+^{4/3+\epsilon/2}}\partial^j_xw(i)\|_{L^2(\mathbb{R}}\leq C_a<\infty \quad
\text{for all } a>0, \; j=1,\dots,n,\; i=0,1.\label{aa}
 \end{equation}
 Suppose that $w$ does not vanish identically in $D:=\mathbb{R}\times[0,1]$.
Then,  there is a rectangle $Q:=[x_0,x_0+1]\times[r,1-r]$, for some
$x_0\in\mathbb{R}$ and $r\in(0,1)$, such that $\|w\|_{L^2(Q)}>0$.
 If we consider  translations $\tilde u_i$ of $u_i$, defined by
$\tilde u_i(x,t):=u_i(x+x_0,t)$, $i=1,2$, then, it can be seen that $\tilde u_1$,
$\tilde u_2$, and $\tilde w:= \tilde u_1-\tilde u_2$, satisfy the hypotheses
 of Theorem \ref{thmI}. In this way, making a translation if necessary, we
can suppose without loss of generality that  $Q=[0,1]\times[r,1-r]$.

Let $\eta\in C^\infty(\mathbb{R})$ be a function supported in $(0,1)$ and such that
$\int\eta=1$. For $R>1$ and $N>4R+1$, define
$\phi_{R,N}(x)\equiv\phi(x):=\int_{-\infty}^x\eta(x'-R)-\eta(x'-N))\,dx'$.
Then $\phi_{R,N}=1$ in $[R+1,N]$,
 $\operatorname{supp}\phi_{R,N}\subseteq [R,N+1]$ and
 $|\phi^{(j)}_{R,N}|\leq c_j$ with $c_j$ independent of $R$ and $N$.
We will apply Theorem \ref{car} to the function $v_{R,N}\equiv v:=\phi_{R,N}\,w$.
From \eqref{Fjj}  with $p=n-2$, $v$ satisfies
\begin{equation}
 \begin{aligned}
&\partial_tv+(-1)^{k+1}\partial_x^nv\\
&=\phi(\partial_tw+(-1)^{k+1}\partial^n_xw)
 +\sum_{j=1}^nc_{n,j}\phi^{(j)}\partial_x^{n-j}w \\
&=-\sum_{j=0}^p\phi F_j\partial_x^jw
 +\sum_{j=1}^nc_{n,j}\phi^{(j)}\partial_x^{n-j} w \\
&=-\sum_{j=0}^pF_j\partial_x^j(\phi  w)
+\sum_{j=1}^pF_j\sum_{r=1}^j c_{j,r}\phi^{(r)}\partial^{j-r}_xw
+\sum_{j=1}^nc_{n,j}\phi^{(j)}\partial_x^{n-j} w.
\end{aligned}\label{t1}
\end{equation}


For $\lambda>2$ we now apply together \eqref{car1} and \eqref{car2}
in Theorem \ref{car} to $v$. We use \eqref{t1}, H\"older's inequality,
and the fact that $\|\cdot\|_{L^2_T L^2_x}\leq\|\cdot\|_{L_T^\infty L_x^2}$ and
$\|\cdot\|_{L^1_T L^2_x}\leq\|\cdot\|_{L^2_T L^2_x}$, and  take into account that $\phi$
is supported in $[R,N+1]$ and its derivatives $\phi^{(j)}$ are supported
in $[R,R+1]\cup[N,N+1]$,  to obtain
\begin{equation}
\begin{aligned}
&\| e^{\lambda x}\phi w\|_{L^2_T L^2_x} +\sum_{j=1}^{n-1}\|e^{\lambda x}
 \partial_x^j(\phi w)\|_{L^\infty_x L^2_T}    \\
&\leq C\lambda^{n-1}\| |J^{n}( e^{\lambda x}\phi w(1)) |
 +|J^{n}( e^{\lambda x}\phi w(0))|\|_L^2(\mathbb{R}   \\
&\quad +C\|e^{\lambda x}
 (\partial_t+(-1)^{k+1}\partial_x^n)v\|_{L^1_T L^2_x}{}_\cap{}_{L^1_x L^2_T}  \\
&\leq C\lambda^{2n-1}\sum_{j=0}^1\| e^{\lambda x}(|w(j)|
 +|\partial^n_xw(j)|)\|_{L^2([R,\infty))}    \\
&\quad +C\|F_0\|_{L^2_TL^\infty_{x\geq R}}\| e^{\lambda x}\phi w\|_{L^2_T L^2_x}
 +C\|F_0\|_{L^2_{x\geq R}L^\infty_T}\| e^{\lambda x}\phi w\|_{L^2_x L^2_T}   \\ % \label{IIa}
&\quad +C\sum_{j=1}^p\|F_j\|_{L^2_{x\geq R}L^\infty_T}\|
  e^{\lambda x}\partial_x^j(\phi w)\|_{L^\infty_x L^2_T} \\
&\quad +C\sum_{j=1}^p\|F_j\|_{L^1_{x\geq R}L^\infty_T}\|
  e^{\lambda x}\partial_x^j(\phi w)\|_{L^\infty_x L^2_T}     \\ %\label{IIIa}
&\quad +{C}e^{\lambda (R+1)}\sum_{j=0}^{n-1}\|\partial_x^jw\|_{L_T^\infty L_x^2}
 + {C}\sum_{j=0}^{n-1}e^{\lambda{(N+1)}}
 \|\partial_x^jw\|_{L^\infty_TL^2_{x\geq N}}   \\ %\label{IVa}
&=:\text{I}+\text{II}+\text{III}+\text{IV}.
\end{aligned} \label{F0}
\end{equation}
where ${C}$  does not depend on $\lambda$, $N$, and $R$.

Taking into account the specific form of the polynomial $P$
in \eqref{poli1},  and since $\sum_{j=0}^{n-1}F_j\partial_x^jw=P(z_1)-P(z_2)$,
we see from  \eqref{P} and \eqref{Ad} that
each $F_j$ is a polynomial in $\partial_x^{j_1}u_1$ and $\partial_x^{j_2}u_2$
with $j_1,j_2\leq n-3$, except for $F_0$ which has a term
$\partial_x^{n-2}u_1$ coming from the quadratic term $w\partial_x^{n-2}u_1$
in  $A_2(z_1)-A_2(z_2)$ (see \eqref{Idos}).
 From \eqref{alpfa1} it follows that
\begin{equation}
\|(1+x_+)^{ \alpha(1-\frac {l+1}{n+1})}\partial_x^l u_i(t)
\|_{L^\infty(\mathbb{R})}\leq C\,,\quad\text{for all }t\in[0,1], \;
l=0,\dots,n. \label{interinter}
\end{equation}
For $l\leq n-3$, $\alpha(1-\frac {j+1}{n+1})>\frac{n+1}3(1-\frac{n-2}{n+1})=1$.
Therefore, $\|(1+x_+)^{1^+}F_j\|_{L^\infty_TL^\infty_x}<\infty$ for
$j=1,\dots,p$. In a similar way, with $l=n-2$ in \eqref{interinter},
 we see that  $\|(1+x_+)^{2/3}F_0\|_{L^\infty_TL^\infty_x}<\infty$.
From this decay of the functions $F_j$ we conclude that, by taking $R$
sufficiently large, the norms involving these functions in  II and III
of \eqref{F0} can be made small in such a way that II and III can be
absorbed by the left-hand side of \eqref{F0}.

After we perform this absorbtion, and taking into account that
$\phi\equiv 1$ in $[4R,4R+1]$,  we replace the left-hand side of \eqref{F0}
by a smaller amount to obtain
\begin{equation}
\begin{aligned}
&\|e^{ \lambda x}w\|_{L^2([4R,4R+1]\times[0,1])}
 +\sum_{j=1}^{n-1}\|e^{\lambda x}\partial_x^jw\|_{L^2([4R,4R+1]\times[0,1])} \\
&\leq C\lambda^{2n-1}\sum_{j=0}^1\| e^{\lambda x}(|w(j)|
 +|\partial^n_xw(j))|\|_{L^2([R,\infty))} \\
&\quad + {C}e^{\lambda (R+1)}\sum_{j=0}^{n-1}\|\partial_x^jw\|_{L_T^\infty L_x^2}
 +  {C}e^{\lambda{(N+1)}}\|\partial_x^jw\|_{L^\infty_TL^2_{x\geq N}}\\
&=:\text{I}+\text{IV}.
\end{aligned}  \label{F1}
\end{equation}

From the decay hypothesis \eqref{w} of $w$  and the exponential
decay preservation proved in Theorem \ref{decaimiento}, it follows that
 $\|e^{\lambda x}w(t)\|_L^2(\mathbb{R}\leq C_\lambda<\infty$, for all $\lambda>0$
and all $t\in[0,1]$. From an interpolation argument similar to
that in \eqref{inter}, we also have that
$\|e^{\lambda x}\partial_x^jw(t)\|_L^2(\mathbb{R}\leq C_\lambda$, for $j=1,\dots,n$.
Therefore,
\begin{align*}
\text{IV}&\leq   {C}e^{\lambda (R+1)}\sum_{j=0}^{n-1}\|\partial_x^jw\|_{L_T^\infty L_x^2}
+ {C}e^{\lambda (N+1)}e^{-2\lambda N}\|e^{2\lambda x}
\partial_x^jw\|_{L^\infty_TL^2_x}\\
&\leq   {C}e^{\lambda (R+1)}
+C_\lambda e^{-\lambda(N-1)}\,,%\label{IV}
\end{align*}
Then, as  $N\to\infty$, from \eqref{F1} we conclude that
\begin{equation}
\begin{aligned}
&e^{4R\lambda}\sum_{j=0}^{n-1}\|\partial_x^jw\|_{L^2([4R,4R+1]\times[0,1])}\\
&\leq C\lambda^{2n-1}\sum_{j=0}^1\| e^{\lambda x}(|w(j)|
 +|\partial^n_xw(j))|\|_{L^2([R,\infty))}+{C}e^{\lambda (R+1)}.
\end{aligned}\label{4R}
\end{equation}
For $a>0$  to be determined later, we take $\lambda=aR^{1/3+\epsilon/2}$.
Since   $\lambda x=aR^{1/3+\epsilon/2} x\leq ax^{4/3+\epsilon/2}$ for $x\geq R$,
 from \eqref{4R} we have
\begin{align*}
&e^{4aR^{4/3+\epsilon/2}}\sum_{j=0}^{n-1}\|\partial_x^jw\|_{L^2([4R,4R+1]
 \times[0,1])}\\
&\leq Ca^{2n-1}R^{(2n-1)(\frac13+\epsilon/2)}\sum_{j=0}^1
 \|e^{ax^{4/3+\epsilon/2}}(|w(j)|+|\partial^n_xw(j))|\|_{L^2(\mathbb{R})}\\
&\quad +{C}e^{aR^{1/3+\epsilon/2} (R+1)},
\end{align*}
and thus, from the definition of $A_R(w)$ given in \eqref{Aw}
and from \eqref{aa},
\begin{equation}
e^{4aR^{4/3+\epsilon/2}}A_R(w)
\leq C_aR^{(2n-1)(1/3+\epsilon/2)}+{C}e^{2a R^{4/3+\epsilon/2}}
\leq C_ae^{2a R^{4/3+\epsilon/2}}.
\label{5R}
\end{equation}
We now apply Theorem \ref{estinferior} with $p=n-2$ to obtain
\begin{align*}
\|w\|_{L^2(Q)}&\leq Ce^{C_*R^{4/3+\epsilon/2}}A_R(w) \\
&\leq Ce^{C_*R^{4/3+\epsilon/2}}C_ae^{-2aR^{4/3+\epsilon/2}}
=C_ae^{(C_*-2a)R^{4/3+\epsilon/2}},
\end{align*}
Where $C^*=C^*(r)$.
If we fix $a>C_*/2$, then by taking $R\to\infty$ we conclude that
 $\|w\|_{L^2(Q)}=0$, which contradicts the original assumption
$\|w\|_{L^2(Q)}\neq 0$. Then we conclude that $w\equiv0$, and
Theorem \ref{thmI} is proved.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thmII}]
From Remark \ref{remdec}, $w$ satisfies \eqref{tres1}.
 With $\beta=1$, after applying Gronwall's inequality and taking $N\to\infty$,
we can conclude that for $t_0\in[0,1]$,
 \begin{equation}
\int e^xw(t)^2\leq C\int e^xw(t_0)^2\quad \text{for all }t\in[t_0,1].
 \end{equation}
By making the change of variables $x\mapsto-x$ and $t\mapsto1-t$,
and taking into account that
$w\in C([0,1]; H^{n+1}(\mathbb{R})\cap L^2((1+x_-)^{2\alpha_0}\,dx)$
we can also see that
  \begin{equation}
\int e^{-x}w(t)^2\leq C\int e^{-x}w(t_0)^2\quad \text{for all }t\in[0,t_0].
 \end{equation}
Thus we can conclude that if $w(t_0)=0$, then $w\equiv0$.

We will find a constant $a>0$ such that if
$w(0), w(1)\in L^2(e^{ax^{n/(n-1)}})$, then $w\equiv 0$.
We reason by contradiction. Suppose that $w$ does not vanish identically
in $D:=\mathbb{R}\times[0,1]$. Then, by the uniqueness argument just given, $w$
does not vanish identically in $D_0:=\mathbb{R}\times[1/3,2/3]$. Therefore, there is
a rectangle $Q:=[x_0,x_0+1]\times[1/3,2/3]$ such that $\|w\|_{L^2(Q)}>0$.
By making a translation if necessary, we can suppose without loss of generality
that  $Q=[0,1]\times[1/3,2/3]$.
We now continue applying the same arguments used to prove Theorem \ref{thmI},
using $\lambda=aR^{1/(n-1)}$ instead of $aR^{1/3+\epsilon/2}$.
In this case we apply Theorem \ref{estinferior} with $p\leq k$,
and $C_*=C_*(1/3)$ and choose $a=\frac{C_*}2+1>\frac{C_*}2$,
which gives a value of $a$ which depends only on $n$.
\end{proof}


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\end{document}