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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 168, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/168\hfil Dispersive equations on the line]
{$L^2$-well-posed Cauchy problem for fourth-order dispersive
equations on the line}

\author[S. Tarama\hfil EJDE-2011/168\hfilneg]
{Shigeo Tarama}

\address{Shigeo Tarama \newline
Lab. of Applied Mathematics, Faculty of Engineering,
Osaka City University, Osaka 558-8585, Japan}
\email{starama@mech.eng.osaka-cu.ac.jp}

\thanks{Submitted August 16, 2011. Published December 14, 2011.}
\subjclass[2000]{37L50, 16D10}
\keywords{Dispersive operators; Cauchy problem; well posed}

\begin{abstract}
 Mizuhara \cite{MZ} obtained conditions for the Cauchy
 problem of a fourth-order dispersive operator to be well
 posed in the $L^2$ sense. Two of those conditions
 were shown to be necessary under additional assumptions.
 In this article, we prove the necessity without the additional
 assumptions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

Let $L$ be a fourth-order dispersive operator  given by
\begin{equation}\label{1}
L=D_t-D_x^4-a(x)D_x^3-b(x)D_x^2-c(x)D_x-d(x)
\end{equation}
where $D_t=\frac{1}{i}\partial_t$,
$D_x=\frac{1}{i}\partial_x$.
We consider the Cauchy problem
\[ %\label{2}
Lu=f(x,t), \quad (x,t)\in \mathbb{R}^2
\]
with the initial data on the line $t=0$, $u(x,0)=g(x)$.

Mizuhara \cite{MZ}, extending the arguments on \cite{TA},
obtained the following result.
\begin{quote}
 The above Cauchy problem is $L^2$-well-posed if the
 coefficients $a(x)$, $b(x)$, $c(x)$ satisfy:
\begin{gather}\label{c-1}
\big|\int_{x_0}^{x_1}\Im a(y)\,dy\big|\le C,\\
\label{c-2}
\big|\int_{x_0}^{x_1}\Im (b(y)-3a(y)^2/8)\,dy\big|
 \le C|x_1-x_0|^{1/3},\\
\label{c-3}
\big|\int_{x_0}^{x_1}\Im(c(y)-2a(y)b(y)+a(y)^3/8)\,dy\big|
 \le C|x_1-x_0|^{2/3}
\end{gather}
for any $x_0,x_1\in\mathbb{R}$, where $\Im(\cdot)$ is the
imaginary part of a complex number.
\end{quote}

In the same article, it was shown that \eqref{c-1} is necessary
for the $L^2$-well-posedness.
While the necessity of conditions \eqref{c-2} and \eqref{c-3}
is shown under the additional assumption that there exist
a constant $\mu$ such that
\begin{equation} \label{e6}
\big|\int_{x_0}^{x_1}\Re (b(y)-3a(y)^2/8- \mu )\,dy\big|
\le C|x_1-x_0|^{1/2},
\end{equation}
where $\Re(\cdot)$ is the real part of a complex number.

In this article, we show that the conditions \eqref{c-2}
and \eqref{c-3} are necessary for the $L^2$-well-posedness,
without using the additional assumption \eqref{e6}.

The method of proof is almost same as that in \cite{MZ};
that is, under the assumption that the conditions are not satisfied,
we construct the sequences of oscillating solutions that are not
consistent with the estimates required to be $L^2$-well-posed.
In our construction, we use ``time independent'' phases.
We remark that the idea of the above method has its origin
in  Mizohata's works on Schr\"odinger type equations
(see for example \cite{Mo}).

To make our method clear, we consider dispersive operators
\[
L[u]=D_tu-D_x^ku-\sum_{j=1}^{k}a_j(x)D_x^{k-j}u
\]
with $k\ge3$.
In the next section we draw some necessary conditions
for $L^2$-well-posedness. As for the case $k=4$, we show
the necessity of  the conditions \eqref{c-2} and \eqref{c-3}.

In the following, we denote by $B^{\infty}(\mathbb{R})$
the space of infinitely differentiable functions on
$\mathbb{R}$ that are bounded  on $\mathbb{R}$
together with all their derivatives of any order.
We denote by $\|f(\cdot)\|$ $L^2$-norm of $f(x)$ given by
$\|f(\cdot)\|=\big(\int_{\mathbb{R}}|f(x)|^2\,dx\big)^{1/2}$.
We use $C$ or $C$ with some subindex  to denote
positive constants that may be different, line by line.


\section{Main Result}

Let $L$ be a dispersive operator given by
\begin{equation}\label{7}
L[u]=D_tu-D_x^ku-\sum_{j=1}^{k}a_j(x)D_x^{k-j}u
\end{equation}
with $k\ge 3$ and $a_j(x)\in B^{\infty}(\mathbb{R})$.

Let $T$ be a positive number. Consider the Cauchy problem
forward and backward for $L$;
\begin{equation}\label{2-cp1}
L[u]=f(x,t) \quad  (x,t)\in \mathbb{R}\times (-T,T)
\end{equation}
with the initial condition
\begin{equation}\label{2-cp2}
u(x,0)=g(x)\quad  x\in \mathbb{R}.
\end{equation}
We say that the Cauchy problem \eqref{2-cp1}--\eqref{2-cp2}
is $L^2$-well-posed, if for any $f(x,t)\in L^1([-T,T],L^2(\mathbb{R}))$
and any $g(x)\in L^2(\mathbb{R})$,
there exists one and only one solution $u(x,t)$ in
 $ C^0([-T,T],L^2(\mathbb{R}))$ to the above problem
satisfying the following two estimates:
for any $t\in [0,T]$,
\begin{gather}\label{2-est+}
\|u(\cdot,t)\|\le C\Big(\|g(\cdot)\|+\int_{0}^{t}\|f(\cdot,s)\|
\,ds\Big),\\
\label{2-est-}
\|u(\cdot,-t)\|\le C\Big(\|g(\cdot)\|+\int_{-t}^{0}\|f(\cdot,s)\|
\,ds\Big),
\end{gather}
where the constant $C$ does not depend on $t$, $f(x,t)$, or $g(x)$.

We consider the behaviour of the  oscillating solution
$u(x,t)=e^{i(\xi x+\xi^kt)}U(x,t,\xi)$ to the equation $L[u]=0$.

Define the operator $L_0$ by
\[ %\label{12}
L_0[U]=e^{-i(\xi x+\xi^kt)}L[e^{i(\xi x+\xi^kt)}U].
\]
Then we see that
\[ %\label{13}
L_0=D_t-\xi^{k-1}(kD_x+a_1(x))-\sum_{j=2}^{k}\xi^{k-j}
\Big(\binom{k}{k-j}D_x^j+\sum_{l=1}^ja_l(x)\binom{k-l}{k-j}D_x^{j-l}
\Big).
\]
Setting $d_1(x)=-a_1(x)/k$ and multiplying $e^{iS_1(x)}$
with $S_1(x)=\int_{x_0}^xd_1(y)\,dy$, we eliminate the
term $-\xi^{k-1}a_1(x)$ from $L_0$. That is, defining the
operator $L_1$ by
\[ %\label{14}
L_1[U]=e^{-iS_1(x)}L_0[e^{iS_1(x)}U],
\]
 we obtain
\[ %\label{15}
L_1=D_t-\xi^{k-1}kD_x-\sum_{j=2}^{k}\xi^{k-j}P_{1,j}(x,D_x)
\]
where
\[ %\label{16}
P_{1,j}(x,D_x)=\sum_{l=0}^jb_{j,l}(x)D_x^l.
\]
Next, we eliminate the term $-\xi^{k-2}b_{2,0}(x)$ from $L_1$
by multiplying $e^{iS_2(x)/\xi}$ with
$S_2(x)=\int_{x_0}^xd_2(y)\,dy$
with $d_2(x)=-b_{2,0}(x)/k$.
That is, defining the operator $L_2$ by
\[ %\label{17}
L_2[U]=e^{-iS_2(x)/\xi}L_1[e^{iS_2(x)/\xi}U],
\]
 we see that $L_2$ satisfies
\[ %\label{18}
L_2=D_t-\xi^{k-1}kD_x-\sum_{j=2}^{2k}\xi^{k-j}P_{2,j}(x,D_x)
\]
where
\[ %\label{19}
P_{2,2}(x,D_x)=\sum_{l=1}^2c_{2,l}(x)D_x^l.
\]
and, for $j>2$
\[ %\label{20}
P_{2,j}(x,D_x)=\sum_{l=0}^{\min\{j,k\}}c_{j,l}(x)D_x^l.
\]
Repeating this process, we obtain the following result.

\begin{proposition}\label{prop-1}
There exist the functions
$d_1(x),d_2(x),\dots,d_{k}(x)\in  B^{\infty}(\mathbb{R})$, such that
with $S(x,x_0,\xi)$ defined by
\[ %\label{21}
S(x,x_0,\xi)=\sum_{j=1}^{k}\frac{1}{\xi^{j-1}}\int_{x_0}^xd_j(y)\,dy
\]
the operator $L_{00}$ defined by
\[ %\label{22}
L_{00}[U]=e^{-iS(x,x_0,\xi)}L_0[e^{iS(x,x_0,\xi)}U],
\]
which has the  form
\begin{equation}\label{23}
L_{00}=D_t-\xi^{k-1}kD_x-\sum_{j=2}^{k+k(k-1)}\xi^{k-j}P_{j}(x,D_x)
\end{equation}
where $P_j(x,D_x)$ is a differential operator of order at most $k$.
In particular for  $j=2,\dots,k$,
\begin{equation}\label{e2-10}
P_{j}(x,D_x)=\sum_{q=1}^jp_{j,q}(x)D_x^q\,.
\end{equation}
Here the functions  $d_j(x)$ are uniquely determined by
the coefficients of $L$.
\end{proposition}

\begin{remark} \label{rmk2.2} \rm
We see from \eqref{23} and \eqref{e2-10} that
 $L_{00}[1]=\sum_{j=1}^{k(k-1)}\xi^{-j}r_j(x)$ with some $r_j(x)$.
\end{remark}

\begin{proof}[Proof of Proposition \ref{prop-1}]
We have  to show only the uniqueness.
Assume that there exist some $\tilde{d}_j(x)$ ($1\le j\le k$)
such that the operator $\tilde{L}_{00}$ given by
\[
\tilde{L}_{00}[U]=e^{-i\tilde{S}(x,x_0,\xi)}
 L_0[e^{i\tilde{S}(x,x_0,\xi)}U],
\]
where $\tilde{S}(x,x_0,\xi)
=\sum_{j=1}^k\frac{1}{\xi^{j-1}}\int_{x_0}^x\tilde{d}_j(y)\,dy$,
 has the form similar to $L_{00}$, that is,
$\tilde{L}_{00}[1]=\sum_{j=1}^{k(k-1)}\xi^{-j}\tilde{r}_j(x)$
with some $\tilde{r}_j(x)$.

Since $L_0[U]=e^{iS(x,x_0,\xi)}L_{00}[e^{-iS(x,x_0,\xi)}U]$, we obtain
\[
\tilde{L}_{00}[U]=e^{-i(\tilde{S}(x,x_0,\xi)-S(x,x_0,\xi)}
L_{00}[e^{i(\tilde{S}(x,x_0,\xi)-S(x,x_0,\xi)}U].
\]
Then
\[
\sum_{j=1}^{k(k-1)}\xi^{-j}\tilde{r}_j(x)
=e^{-i(\tilde{S}(x,x_0,\xi)-S(x,x_0,\xi)}
L_{00}[e^{i(\tilde{S}(x,x_0,\xi)-S(x,x_0,\xi)}].
\]
Comparing the coefficient of $\xi^{k-j}$ ($j=1,2,\dots,k$), we see that
$\tilde{d}_j(x)=d_j(x)$ by the induction on $j$.
\end{proof}


Note that for the fourth-order operator in \eqref{1}, we have
the following: (see also \cite{MZ})
\begin{gather}
d_1(x) = \frac{-a(x)}{4} \label{ex1-1}\\
 d_2(x)= \frac{-1}{4}(b(x)-\frac{3}{8}a(x)^2
 -\frac{3}{2}D_xa(x) )\label{ex1-2}\\
 d_3(x)= \frac{-1}{4}\Bigl(c(x)+\frac{a(x)^3}{8}
 -\frac{a(x)b(x)}{2}+D_x(4D_xd_1(x)+6d_2(x)) \Bigr).\label{ex1-3}
\end{gather}


In this note, we show the following result.

\begin{theorem}\label{thm-1}
If the Cauchy problem \eqref{2-cp1}--\eqref{2-cp2} is
$L^2$-well-posed, then the functions $d_1(x),d_2(x),\dots,d_{k-1}(x)$
given in Proposition \ref{prop-1}, satisfy:
For $1\le j\le k-1$ and any $x_0,x_1\in\mathbb{R}$,
\begin{equation}\label{25}
\big|\int _{x_0}^{x_1}\Im d_j(y)\,dy\big|\le
C|x_1-x_0|^{\frac{j-1}{k-1}}.
\end{equation}
\end{theorem}

By Theorem \ref{thm-1}, it follows from \eqref{ex1-1}, \eqref{ex1-2}
and \eqref{ex1-3} that it is necessary that
\eqref{c-1}, \eqref{c-2} and \eqref{c-3} hold for
the Cauchy problem, for the operator given by \eqref{1},
to be $L^2$-well-posed.

To prove Theorem \ref{thm-1}, we prepare following propositions.

\begin{proposition}\label{prop-2}
If the Cauchy problem \eqref{2-cp1}-\eqref{2-cp2} is $L^2$-well-posed,
then we have
\begin{equation}\label{26}
\big|\int_{x_0}^{x_1}\Im d_1(y)\,dy\big|\le C
\end{equation}
for any $x_0,x_1\in\mathbb{R}$.
\end{proposition}

\begin{proof}
Assuming that $\int_{x_0}^{x_1}\Im d_1(y)\,dy$ is not bounded,
we construct the sequence of solutions $u_n(x,t)$ that are not
consistent with the estimates \eqref{2-est+} or \eqref{2-est-}.
Indeed,  if $\int_{x_0}^{x_1}\Im d_1(y)\,dy$ is not bounded,
for any positive integer $n$ we can find
$x_{0,n},x_{1,n}\in\mathbb{R}$  satisfying
\[
\big|\int_{x_{0,n}}^{x_{1,n}}\Im d_1(y)\,dy\big|> n.
\]
Here, we  may assume that
\[ %\label{27}
-\int_{x_{0,n}}^{x_{1,n}}\Im d_1(y)\,dy> n
\]
by exchanging  $x_{0,n}$ and $x_{1,n}$ if necessary.
Now we set $\xi_n=n|x_{1,n}-x_{0,n}|$.
We remark that the boundedness of $d_1(x)$ implies that
$|x_{1,n}-x_{0,n}|\to \infty$ as $n\to \infty$.
Hence $\xi_n\to \infty$ as $n\to \infty$.
We choose $t_n$ so that $x_{1,n}=x_{0,n}-kt_n\xi_n^{k-1}$. That is,
$t_n=-(x_{1,n}-x_{0,n})/(kn|x_{1,n}-x_{0,n}|\xi_n^{k-2})$.
We note that $|t_n\xi_n^{k-2}|=1/(kn)$ and
$t_n\to 0$ as $n\to \infty$.

Since $\xi_n=n|x_{1,n}-x_{0,n}|$, it follows that, if $j\ge 2$,
\[
\big|\frac{1}{\xi_n^{j-1}}\int_{x_{0,n}}^{x_{1,n}}d_j(y)\,dy\big|\le C.
\]
Then, by setting $x_0=x_{0,n}$ and $\xi=\xi_n$ in $S(x,x_0,\xi)$;
 that is,
$S(x,x_{0,n},\xi_n)=\sum_{j=1}^k\frac{1}{\xi_n^{j-1}}
\int_{x_{0,n}}^xd_j(y)\,dy$, we have, for large $n$, 
\[
|S(x_{1,0},x_{0,n},\xi_n)-\int_{x_{0,n}}^{x_{1,n}}d_1(y)\,dy|\le C,\quad
-\Im S(x_{1,n},x_{0,n},\xi_n)\ge \frac{n}{2}.
\]

Consider the case where there exist infinitely many $n$'s such
that $t_n>0$. Then, by choosing a subsequence,
we may assume $t_n>0$ for all $n>0$.
Let $s_n\in [0,t_n]$ be a number satisfying
\[ %\label{29}
-\Im S(x_{0,n}-ks_n\xi_n^{k-1},x_{0,n},\xi_n)
=\max_{0\le t\le t_n}-\Im S(x_{0,n}-kt\xi_n^{k-1},x_{0,n},\xi_n).
\]
Since $x_{0,n}-kt_n\xi_n^{k-1}=x_{1,n}$, we see that
$-\Im S(x_{0,n}-ks_n\xi_n^{k-1},x_{0,n},\xi_n)\ge n/2$.
Pick a non-negative function $g(x)\in C^\infty(\mathbb{R})$
satisfying:
\begin{gather}\label{2-g1}
g(x)=0 \quad \text{for }|x|\ge 1,\\
\label{2-g2}
\int_{\mathbb{R}}g(x)^2\,dx=1.
\end{gather}
Set
\[
u_n(x,t)=e^{i(x\xi_n+t\xi_n^k+S(x,x_{0,n},\xi_n))}
g(x+tk\xi_n^{k-1}-x_{0,n}).
\]
Then
\[
L[u_n(x,t)]=e^{i(x\xi_n+t\xi_n^k+S(x,x_{0,n},\xi_n))}
L_{00}[g(x+tk\xi_n^{k-1}-x_{0,n})].
\]
Noting $(D_t-k\xi_n^{k-1}D_x)g(x+tk\xi_n^{k-1}-x_{0,n})=0$,
we see that
\[
L_{00}[g(x+tk\xi_n^{k-1}-x_{0,n})]
=\sum_{0\le j\le k,\ 0\le q\le k^2-2}\xi_n^{k-2-q}
r_{q,j}(x)g^{(j)}(x+tk\xi_n^{k-1}-x_{0,n})
\]
and
\begin{align*}
L[u_n(x,t)]&=e^{i(x\xi_n+t\xi_n^k+S(x,x_{0,n},\xi_n))}\\
&\quad \times
\sum_{0\le j\le k,\ 0\le q\le k^2-2}\xi_n^{k-2-q}r_{q,j}(x)
g^{(j)}(x+tk\xi_n^{k-1}-x_{0,n}).
\end{align*}
On the support of $g^{(j)}(x+tk\xi_n^{k-1}-x_{0,n})$, where
$|x-(x_{0,n}-kt\xi_n^{k-1})|\le 1$, we have
\begin{equation}\label{2-2222}
|S(x,x_{0,n},\xi_n)-S(x_{0,n}-kt\xi_n^{k-1},x_{0,n},\xi_n)|\le C.
\end{equation}
By the definition of $s_n$, if $0\le t\le s_n$,
$-\Im S(x_{0,n}-kt\xi_n^{k-1},x_{0,n},\xi_n)\le
-\Im S(x_{0,n}-ks_n\xi_n^{k-1},x_{0,n},\xi_n)$.
Then, if $0\le t\le s_n$, we obtain
\[
|L[u_n(x,t)]|\le Ce^{-\Im S(x_{0,n}-ks_n\xi_n^{k-1},x_{0,n},\xi_n)}
\xi_n^{k-2}\sum_{j=0}^k|g^{(j)}(x+tk\xi_n^{k-1}-x_{0,n})|,
\]
from which we obtain
\begin{equation}\label{33}
\begin{split}
\int_0^{s_n}\|L[u_n(\cdot,t)]\|\,dt
&\le Cs_n\xi_n^{k-2}e^{-\Im S(x_{0,n}-ks_n\xi_n^{k-1},x_{0,n},\xi_n)}\\
&\le C\frac{1}{kn}e^{-\Im S(x_{0,n}-ks_n\xi_n^{k-1},x_{0,n},\xi_n)}.
\end{split}
\end{equation}
While we obtain
\begin{equation}\label{34}
\|u_n(\cdot,0)\|\le C
\end{equation}
from
\[
u_n(x,0)=e^{i(x\xi_n+S(x,x_{0,n},\xi_n))}g(x-x_{0,n})
\]
and \eqref{2-2222}.
Here we remark $S(x_{0,n},x_{0,n},\xi_n)=0$.

On the other hand, from
\[
u_n(x,s_n)=e^{i(x\xi_n+S(x,x_{0,n},\xi_n))}g(x+ks_n\xi_n^{k-1}-x_{0,n})
\]
and \eqref{2-2222}, it follows that
\begin{equation}\label{2-34}
\|u_n(\cdot,s_n)\|\ge C_0e^{-\Im S(x_{0,n}-ks_n\xi_n^{k-1},x_{0,n},\xi_n)}.
\end{equation}

If the Cauchy problem  is $L^2$-well-posed, we have  estimate
\eqref{2-est+}:
\begin{equation}\notag
\|u_n(\cdot,s_n)\|\le C(\|u(\cdot,0)]\|
+\int_0^{s_n}\|L[u(\cdot,t)]\|\,dt).
\end{equation}
Hence  estimates \eqref{33},\eqref{34} and \eqref{2-34} imply
\[
e^{-\Im S(x_{0,n}-ks_n\xi_n^{k-1},x_{0,n},\xi_n)}
\le C_0^{-1}C(1+\frac{1}{n}e^{-\Im S(x_{0,n}-ks_n\xi_n^{k-1},
x_{0,n},\xi_n)}).
\]
But since $-\Im S(x_{0,n}-ks_n\xi_n^{k-1},x_{0,n},\xi_n)\to  \infty$     as $n\to \infty$, the above estimate is impossible for large $n$. Then \eqref{26} has to hold.
In the case where there exists an $N$ such that $t_n<0$ for $n>N$,
we can construct similarly to the previous case, a sequence of
functions $u_n(x,t)$ that are not consistent with
estimate \eqref{2-est-}.
\end{proof}

\begin{proposition}\label{prop-3}
Let  $l\in\{1,2,\dots,k-2\}$. Assume that, for any
$j\in\{1,2,\dots,l\}$ and any $x,\xi\in\mathbb{R}$,
\begin{equation}\label{36}
\big|\int_{x}^{x+\xi^{l} }\Im d_j(y)\,dy\big|\le C|\xi|^{j-1}.
\end{equation}
If the Cauchy problem \eqref{2-cp1}--\eqref{2-cp2} is
$L^2$-well-posed, then
\begin{equation}\label{prop2-3}
\big|\sum_{j=1}^{l+1}\frac{1}{\xi^{j-1}}
\int_{x}^{x+\xi^{l+1}}\Im d_j(y)\,dy\big|\le C
\end{equation}
for any $x,\xi\in\mathbb{R}$ with $\xi\ne0$.
\end{proposition}

\begin{proof}
Similarly to the proof of Proposition \ref{prop-2},
assuming that \eqref{prop2-3} is not valid,
we construct the sequence of solutions $u_n(x,t)$ that are
not consistent with the estimates \eqref{2-est+} or \eqref{2-est-}.
Indeed, if
$\sum_{j=1}^{l+1}\frac{1}{\xi^{j-1}}\int_{x}^{x+\xi^{l+1}}
\Im d_j(y)\,dy$ is not bounded, for any positive integer $n$
we can find  $x_{n}\in\mathbb{R}$ and
$\xi_n\in \mathbb{R}\setminus\{0\}$ such that
\[
\big|\sum_{j=1}^{l+1}\frac{1}{\xi_n^{j-1}}
\int_{x_{n}}^{x_{n}+\xi_n^{l+1}}\Im d_j(y)\,dy\big|>n^2.
\]
We note that the boundedness of $d_j(x)$ implies that
$|\xi_n|\to \infty$ as $n\to \infty$.
We set $y_p=x_{n}+\frac{p}{n}\xi_n^{l+1}$ ($p=0,1,2,\dots,n$).
Then, noting
\[
\sum_{p=1}^n\int_{y_{p-1}}^{y_p}d_j(y)\,dy
=\int_{x_{n}}^{x_{n}+\xi_n^{l+1}}d_j(y)\,dy,
\]
 we see that there exists some $p$ such that
\[
\big|\sum_{j=1}^{l+1}\frac{1}{\xi_n^{j-1}}
\int_{y_{p-1}}^{y_p}\Im d_j(y)\,dy\big|>n.
\]
Then, redefining $x_n$ by $x_n=y_{p-1}$, we have
\[
\big|\sum_{j=1}^{l+1}\frac{1}{\xi_n^{j-1}}
\int_{x_n}^{x_n+\frac{\xi_n^{l+1}}{n}}\Im d_j(y)\,dy\big|>n.
\]
First we consider the case where for infinitely many $n$, we have
\[
-\sum_{j=1}^{l+1}\frac{1}{\xi_n^{j-1}}
\int_{x_n}^{x_n+\frac{\xi_n^{l+1}}{n}}\Im d_j(y)\,dy>n.
\]
Then we consider only such $n$.

We define $t_n$ by $kt_n\xi_n^{k-1}=-\frac{\xi_n^{l+1}}{n}$;
 that is, $t_n=\frac{-1}{n\xi_n^{k-2-l}}$. We see that $t_n\to 0$
as $n\to \infty$.
Similarly to the proof of Proposition \ref{prop-2}, using
the phase function
$S(x,x_n,\xi_n)=\sum_{j=1}^{k}\frac{1}{\xi_n^{j-1}}
\int_{x_n}^xd_j(y)\,dy$
 and a non-negative  function $g(x)\in C^{\infty}(\mathbb{R})$
satisfying \eqref{2-g1} and \eqref{2-g2}, we consider $u_n(x,t)$
given by
\[
u_n(x,t)=e^{i(\xi x+t\xi^k+S(x,x_n,\xi_n))}g(\frac{x+kt\xi_n^{k-1}-x_n}{\xi_n^{l}})|\xi_n|^{-l/2}.
\]
We note that, if $|x+kt\xi_n^{k-1}-x_n|\le |\xi_n|^{l}$ and
$|t|\le |t_n|$,
\[
|x-x_n|\le |\xi_n|^{l}+|kt\xi_n^{k-1}|\le |\xi_n|^{l}+|\xi_n^{l+1}/n|
\]
from which we obtain, on the support of $u_n(x,t)$,
\[
\big|\frac{1}{\xi_n^{j-1}}\int_{x_n}^xd_j(y)\,dy\big|\le C
\]
for $j\ge l+2$.
Hence, on the support of $u_n(x,t)$,
\begin{equation}\label{38}
|S(x,x_n,\xi_n)-\sum_{j=1}^{l+1}\frac{1}{\xi_n^{j-1}}
\int_{x_n}^xd_j(y)\,dy|\le C
\end{equation}
On the other hand, if $|x+kt\xi_n^{k-1}-x_n|\le |\xi_n|^{l}$,
the assumption \eqref{36} on $d_j(x)$
($j=1,\dots,l$)  of Proposition \ref{prop-3} implies that
\[
\big|\int_{x_n}^x\Im d_j(y)\,dy-\int_{x_n}^{x_n-kt\xi_n^{k-1}}
\Im d_j(y)\,dy\big|\le C |\xi_n|^{j-1}
\]
which implies that
\begin{equation}\label{39}
\big|\sum_{j=1}^{l+1}\frac{1}{\xi_n^{j-1}}
\int_{x_n}^x\Im d_j(y)\,dy-\sum_{j=1}^{l+1}\frac{1}{\xi_n^{j-1}}
\int_{x_n}^{x_n-kt\xi_n^{k-1}}\Im d_j(y)\,dy\big|\le C
\end{equation}
on the support of $u_n(x,t)$.

Similarly to the proof of Proposition \ref{prop-2}, we assume
$t_n>0$ and choose $s_n\in[0,t_n]$ so that
\[
- \sum_{j=1}^{l+1}\frac{1}{\xi_n^{j-1}}
\int_{x_n}^{x_n-ks_n\xi_n^{k-1}}\Im d_j(y)\,dy=
\max_{0\le t\le t_n}\Bigl(- \sum_{j=1}^{l+1}
\frac{1}{\xi_n^{j-1}} \int_{x_n}^{x_n-kt\xi_n^{k-1}}
\Im d_j(y)\,dy\Bigr).
\]
We have
\[
L[u_n(x,t)]=e^{i(\xi x+t\xi^{k}+S(x,x_n,\xi_n))}L_{00}
[g(\frac{x+kt\xi_n^{k-1}-x_n}{\xi_n^{l}})|\xi_n|^{-l/2}] .
\]
Note that
$(D_t-k\xi_n^{k-1}D_x)g(\frac{x+kt\xi_n^{k-1}-x_n}{\xi_n^{l}})=0$
and
\[
D_x^j g(\frac{x+kt\xi_n^{k-1}-x_n}{\xi_n^{l}})=g^{(j)}
(\frac{x+kt\xi_n^{k-1}-x_n}{\xi_n^{l}})\xi_n^{-jl}.
\]
 Then we see from \eqref{23}, \eqref{e2-10} and $l+2\le k$ that
\begin{align*}
&L_{00}[g(\frac{x+kt\xi_n^{k-1}-x_n}{\xi_n^{l}})|\xi_n|^{-l/2}]\\
&=\sum_{k\ge j\ge0,k^2-2-l\ge p\ge 0}
\xi_n^{k-2-l-p}r_{p,j}(x)g^{(j)}(\frac{x+kt\xi_n^{k-1}-x_n}{\xi_n^{l}})|\xi_n|^{-l/2}.
\end{align*}
On the support of $g(\frac{x+kt\xi_n^{k-1}-x_n}{\xi_n^{l}})$
with $0\le t\le s_n$, we have
\[
\big|\Im S(x,x_n,\xi_n) -\sum_{j=1}^{l+1}\frac{1}{\xi_n^{j-1}}
\int_{x_n} ^{x_n-kt\xi_n^{k-1}}\Im  d_j(y)\,dy\big|\le C
\]
Then if $0\le t\le s_n$, we have
\[
\|L[u_n(x,t)]\|\le C|\xi_n|^{k-2-l}
e^{-\Im S(x_n-ks_n\xi_n^{k-1},x_n,\xi_n)}.
\]
Hence
\begin{equation}\label{2-c21}
\begin{split}
&\int_0^{s_n}\|L[u_n(x,t)]\|\,dt\le Cs_n|\xi_n|^{k-2-l}
 e^{-\Im S(x_n-ks_n\xi_n^{k-1},x_n,\xi_n)}\\
&\le \frac{C}{n}e^{-\Im S(x_n-ks_n\xi_n^{k-1},x_n,\xi_n)}.
\end{split}
\end{equation}
Noting $u_n(x,0)=e^{i(\xi x+S(x,x_n,\xi_n))}
g(\frac{x-x_n}{\xi_n^l})|\xi_n|^{-l/2}$,
we obtain  $|\Im S(x,x_n,\xi_n)|\le C$ on the support of
$u_n(x,0)$ from \eqref{38} and  \eqref{39}.
Then we have
\begin{equation}\label{2-c22}
\|u_n(x,0)\|\le C.
\end{equation}
Finally we see from \eqref{38} and  \eqref{39} that,
on the support of $u_n(x,s_n)$,
\[
-\Im S(x,x_n,\xi_n)\ge -\Im S(x_n-ks_n\xi_n^{k-1},x_n,\xi_n)+C,
\]
from which we obtain
\begin{equation}\label{2-c23}
\|u_n(x,s_n)\|\ge C_0e^{-\Im S(x_n-ks_n\xi_n^{k-1},x_n,\xi_n)}.
\end{equation}

If the Cauchy problem is $L^2$-well-posed, we have the estimate
\eqref{2-est+}, to which we apply \eqref{2-c21}, \eqref{2-c22}
and \eqref{2-c23}. Then we obtain the inequality that
is not valid for large $n$. Hence the estimate \eqref{prop2-3}
has to hold.


In the case where there exists some integer $N>0$ such that
$t_n<0$ for $n>N$.
Then we can construct the series of functions $u_n(x,t)$ for
which the estimate
\eqref{2-est-} is not valid for large $n$.

If there exists some integer $N>0$ such that, for $n>N$,
$$
\sum_{j=1}^{l+1}-\frac{1}{\xi_n^{j-1}}
\int_{x_n}^{x_n+\xi_n^{l+1}}\Im  d_j(y)\,dy<-n,
$$
then, by setting, $y_n=x_n+\xi_n^{l+1}$, we have
 $$
\sum_{j=1}^{l+1}- \frac{1}{\xi_n^{j-1}}
\int_{y_n}^{y_n-\xi_n^{l+1}}\Im d_j(y)\,dy>n.
$$
By setting  $t_n=-\xi_n^{l-k}/n$, as the above argument,
we can construct the series of functions $u_n(x,t)$ which are
not consistent with the estimates \eqref{2-est+} or \eqref{2-est-}.
\end{proof}

\begin{remark} \label{rmk2.7} \rm
If the coefficient $a_1(x)$ of $L$ is zero, we can obtain the
oscillating solutions $u_n(x,t)$ having smaller $L[u_n(x,t)]$
in the power of $\xi_n$ by solving the transport equation.
We note that, if $a_1(x)=0$, the operator $P_{2}(x,D_x)$
appearing in \eqref{23} is
$P_{2}(x,D_x)=\binom{k}{2}D_x^2$.

Note that
$L_{00}[g(\frac{x-(x_n-kt\xi_n^{k-1})}{\xi_n^l})|\xi_n|^{-l/2}]$
is a sum of
\[
\xi_n^pr_{p,j}(x)g^{(j)}(\frac{x-(x_n-kt\xi_n^{k-1})}{\xi_n^l})
|\xi_n|^{-l/2}
\]
with $0\le j\le k$ and $-k(k-1)\le p\le k-2-l$.
We choose
\[
g_{j,p}(x,t)=\xi_n^p\frac{-i}{k\xi_n^{k-1}}
\int_{x_n}^xr_{p,j}(y)\,dy\, g^{(j)}
(\frac{x-(x_n-kt\xi_n^{k-1})}{\xi_n^l})|\xi_n|^{-l/2}
\]
as a solution of the transport equation
\[
D_tg-k\xi_n^{k-1}D_xg=\xi_n^pr_{p,j}(x)g^{(j)}
(\frac{x-(x_n-kt\xi_n^{k-1})}{\xi_n^l})|\xi_n|^{-l/2}.
\]
We have
\begin{equation}\label{RHS-1}
\begin{split}
&\xi_n^{k-2}D_x^2g_{j,p}(x,t)\\
&= \xi_n^p\frac{-1}{k\xi_n}D_xr_{p,j}(x)g^{(j)}
\big(\frac{x-(x_n-kt\xi_n^{k-1})}{\xi_n^l}\big)|\xi_n|^{-l/2}\\
&\quad +\xi_n^p\frac{2i}{k\xi_n^{1+l}}r_{p,j}(x) g^{(j+1)}
\big(\frac{x-(x_n-kt\xi_n^{k-1})}{\xi_n^l}\big)|\xi_n|^{-l/2}\\
&\quad +\xi_n^p\frac{i\xi_n^{k-2-2l}}{k\xi_n^{k-1}}
\int_{x_n}^xr_{p,j}(y)\,dy\,  g^{(j+2)}
\big(\frac{x-(x_n-kt\xi_n^{k-1})}{\xi_n^l}\big)|\xi_n|^{-l/2}
\end{split}
\end{equation}
and
\begin{equation}\label{RHS-2}
\begin{split}
&\xi_n^{k-3}D_xg_{j,p}(x,t)\\
&=\xi_n^p\frac{-1}{k\xi_n^2}r_{p,j}(x) g^{(j)}
\big(\frac{x-(x_n-kt\xi_n^{k-1})}{\xi_n^l}\big)|\xi_n|^{-l/2}\\
&\quad +\xi_n^p\frac{-\xi_n^{k-3-l}}{k\xi_n^{k-1}}
 \int_{x_n}^xr_{p,j}(y)\,dy\, g^{(j+1)}
 \big(\frac{x-(x_n-kt\xi_n^{k-1})}{\xi_n^l}\big)|\xi_n|^{-l/2}.
\end{split}
\end{equation}
Then  it follows from $1\le l\le k-2$,
$|x-x_n|\le |\xi_n|^l+k|t|\xi_n^{k-1}$
on the support of $g^{(j)}(\frac{x-(x_n-kt\xi_n^{k-1})}{\xi_n^l})$,
and $|s_n\xi_n^{k-2-l}|\le 1/n$,
that, if $0\le t\le s_n$ and $n$ is large, $L^2$ norm of
$L_{00}[g(x,t)-\sum g_{j,p}(x,t)]$ is  smaller than that
of $L_{00}[g(x,t)]$ where
$g(x,t)=g(\frac{x-(x_n-kt\xi_n^{k-1})}{\xi_n^l})|\xi_n|^{-l/2}$
and we assume  $s_n>0$. We see also that $L^2$ norm of $g_{j,p}(x,0)$
is smaller than that of $g(x,0)$ for large $n$.
 Taking into account of \eqref{RHS-1} and \eqref{RHS-2},
we see that $L_{00}[g(x,t)-\sum g_{j,p}(x,t)]$ is also a linear
combination of terms like:
$\xi_n^pr_{p,j}(x)g^{(j)}(\frac{x-(x_n-kt\xi_n^{k-1})}
{\xi_n^l})|\xi_n|^{-l/2}$. Then we can repeat this process.
\end{remark}

\begin{proposition}\label{prop-4}
Let  $l\in\{1,2,\dots,k-2\}$. Assume that
the estimate \eqref{prop2-3} holds
for any $x_0,\xi\in\mathbb{R}$ with $\xi\ne0$.

Then we see that, for $j=1,2,\dots,l+1$,
\begin{equation}\label{43}
\big|\int_{x_0}^{x_1}\Im d_j(y)\,dy\big|
\le C|x_1-x_0|^{(j-1)/(l+1)}.
\end{equation}
\end{proposition}

\begin{proof}
Indeed, for any integer $p\ge 1$, any $y\in \mathbb{R}$ and any
$\eta\in \mathbb{R}\setminus\{0\}$, we see that
\[
\frac{2^{p(j-1)}}{\eta^{j-1}}\int_y^{y+\eta^{l+1}}d_j(y)\,dy
= \sum_{q=1}^{2^{p(l+1)}}\frac{1}{(2^{-p}\eta)^{j-1}}
\int_{y+(2^{-p}\eta)^{l+1}(q-1)}^{(y+(2^{-p}\eta)^{l+1}(q-1))
 +(2^{-p}\eta)^{l+1}}d_j(y)\,dy.
\]
Then from \eqref{prop2-3} we obtain
\begin{equation}\label{prop2-31}
\big|\sum_{j=1}^{l+1}\frac{2^{p(j-1)}}{\eta^{j-1}}
\int_{y}^{y+\eta^{l+1}}\Im d_j(y)\,dy\big|\le C_p.
\end{equation}
Here the constant $C_p$ may depend on $p$, but not on $y$ or on
$\eta$. Hence, by setting
$X_j=\frac{1}{\eta^{j-1}}\int_{y}^{y+\eta^{l+1}}\Im d_j(y)\,dy$
($j=1,2,\dots,l+1$),  for $p=0,1,\dots,l$, we have
\[
\sum_{j=1}^{l+1}2^{p(j-1)}X_j=K_p
\]
with $|K_p|\le C_p$.

Since the $l+1$-th order matrix whose $(i,j)$ element is
$2^{(i-1)(j-1)}$ is invertible,
we see that
$X_j=\frac{1}{\eta^{j-1}}\int_{y}^{y+\eta^{l+1}}\Im d_j(y)\,dy$
is bounded on $\mathbb{R}_y\times\mathbb{R}_{\eta}\setminus\{0\}$.
Hence
\[
\big| \int_{y}^{y+\eta^{l+1}}\Im d_j(y)\,dy\big|\le C|\eta|^{j-1}
\]
which implies
\[
\big|\int_{y}^{w}\Im  d_j(y)\,dy\big|\le C|w-y|^{(j-1)/(l+1)}
\]
for any $y,w\in\mathbb{R}$, where $j=1,2,\dots,l+1$.
The proof  is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm-1}]
Using Proposition \ref{prop-2}, \ref{prop-3} and \ref{prop-4},
we see obviously that the assertion of Theorem \ref{thm-1} is valid.
\end{proof}

\begin{remark} \label{rmk} \rm
For the operator $L$,  defined in the Introduction,
 Mizuhara \cite{MZ} proved Proposition \ref{prop-2},
Proposition \ref{prop-3} in the case of $l=1$,  and
Proposition \ref{prop-4} in the case of $l=2$.
\end{remark}

\subsection*{Acknowledgements}
The author would like to thank  the anonymous referee for
his or her valuable comments.

\begin{thebibliography}{99}
\bibitem{Mo}  S. Mizohata;
\emph{On some Scr\"odinger type equations},
Proc. Japan Acad. Ser. A Math. Sci.(1981) 81--84

\bibitem{MZ}  R. Mizuhara;
\emph{The initial Value Problem for Third and Fourth Order
Dispersive Equations in One Space Dimension},
Funkcialaj Ekvacioj, 49(2006) 1--38.

\bibitem{TA}  S. Tarama;
\emph{Remarks on $L^2$-wellposed Cauchy problem for some dispersive
equations}, J. Math. Kyoto Univ.,  37(1997) 757--765.

\end{thebibliography}
\end{document}