\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 $$\label{1} L=D_t-D_x^4-a(x)D_x^3-b(x)D_x^2-c(x)D_x-d(x)$$ 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 $$\label{e6} \big|\int_{x_0}^{x_1}\Re (b(y)-3a(y)^2/8- \mu )\,dy\big| \le C|x_1-x_0|^{1/2},$$ 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 $$\label{7} L[u]=D_tu-D_x^ku-\sum_{j=1}^{k}a_j(x)D_x^{k-j}u$$ 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 $L$; $$\label{2-cp1} L[u]=f(x,t) \quad (x,t)\in \mathbb{R}\times (-T,T)$$ with the initial condition $$\label{2-cp2} u(x,0)=g(x)\quad x\in \mathbb{R}.$$ 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 $$\label{23} L_{00}=D_t-\xi^{k-1}kD_x-\sum_{j=2}^{k+k(k-1)}\xi^{k-j}P_{j}(x,D_x)$$ where $P_j(x,D_x)$ is a differential operator of order at most $k$. In particular for $j=2,\dots,k$, $$\label{e2-10} P_{j}(x,D_x)=\sum_{q=1}^jp_{j,q}(x)D_x^q\,.$$ 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 $x_0,x_1\in\mathbb{R}$, $$\label{25} \big|\int _{x_0}^{x_1}\Im d_j(y)\,dy\big|\le C|x_1-x_0|^{\frac{j-1}{k-1}}.$$ \end{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 $$\label{26} \big|\int_{x_0}^{x_1}\Im d_1(y)\,dy\big|\le C$$ 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 $$\label{2-2222} |S(x,x_{0,n},\xi_n)-S(x_{0,n}-kt\xi_n^{k-1},x_{0,n},\xi_n)|\le C.$$ 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 $$\label{33} \begin{split} \int_0^{s_n}\|L[u_n(\cdot,t)]\|\,dt &\le Cs_n\xi_n^{k-2}e^{-\Im S(x_{0,n}-ks_n\xi_n^{k-1},x_{0,n},\xi_n)}\\ &\le C\frac{1}{kn}e^{-\Im S(x_{0,n}-ks_n\xi_n^{k-1},x_{0,n},\xi_n)}. \end{split}$$ While we obtain $$\label{34} \|u_n(\cdot,0)\|\le C$$ 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 $$\label{2-34} \|u_n(\cdot,s_n)\|\ge C_0e^{-\Im S(x_{0,n}-ks_n\xi_n^{k-1},x_{0,n},\xi_n)}.$$ If the Cauchy problem is $L^2$-well-posed, we have estimate \eqref{2-est+}: $$\notag \|u_n(\cdot,s_n)\|\le C(\|u(\cdot,0)]\| +\int_0^{s_n}\|L[u(\cdot,t)]\|\,dt).$$ 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 $x,\xi\in\mathbb{R}$, $$\label{36} \big|\int_{x}^{x+\xi^{l} }\Im d_j(y)\,dy\big|\le C|\xi|^{j-1}.$$ If the Cauchy problem \eqref{2-cp1}--\eqref{2-cp2} is $L^2$-well-posed, then $$\label{prop2-3} \big|\sum_{j=1}^{l+1}\frac{1}{\xi^{j-1}} \int_{x}^{x+\xi^{l+1}}\Im d_j(y)\,dy\big|\le C$$ 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 $u_n(x,t)$, $$\label{38} |S(x,x_n,\xi_n)-\sum_{j=1}^{l+1}\frac{1}{\xi_n^{j-1}} \int_{x_n}^xd_j(y)\,dy|\le C$$ 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 $$\label{39} \big|\sum_{j=1}^{l+1}\frac{1}{\xi_n^{j-1}} \int_{x_n}^x\Im d_j(y)\,dy-\sum_{j=1}^{l+1}\frac{1}{\xi_n^{j-1}} \int_{x_n}^{x_n-kt\xi_n^{k-1}}\Im d_j(y)\,dy\big|\le C$$ 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 $$\label{2-c21} \begin{split} &\int_0^{s_n}\|L[u_n(x,t)]\|\,dt\le Cs_n|\xi_n|^{k-2-l} e^{-\Im S(x_n-ks_n\xi_n^{k-1},x_n,\xi_n)}\\ &\le \frac{C}{n}e^{-\Im S(x_n-ks_n\xi_n^{k-1},x_n,\xi_n)}. \end{split}$$ 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 $$\label{2-c22} \|u_n(x,0)\|\le C.$$ 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 $$\label{2-c23} \|u_n(x,s_n)\|\ge C_0e^{-\Im S(x_n-ks_n\xi_n^{k-1},x_n,\xi_n)}.$$ 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 $$\label{RHS-1} \begin{split} &\xi_n^{k-2}D_x^2g_{j,p}(x,t)\\ &= \xi_n^p\frac{-1}{k\xi_n}D_xr_{p,j}(x)g^{(j)} \big(\frac{x-(x_n-kt\xi_n^{k-1})}{\xi_n^l}\big)|\xi_n|^{-l/2}\\ &\quad +\xi_n^p\frac{2i}{k\xi_n^{1+l}}r_{p,j}(x) g^{(j+1)} \big(\frac{x-(x_n-kt\xi_n^{k-1})}{\xi_n^l}\big)|\xi_n|^{-l/2}\\ &\quad +\xi_n^p\frac{i\xi_n^{k-2-2l}}{k\xi_n^{k-1}} \int_{x_n}^xr_{p,j}(y)\,dy\, g^{(j+2)} \big(\frac{x-(x_n-kt\xi_n^{k-1})}{\xi_n^l}\big)|\xi_n|^{-l/2} \end{split}$$ and $$\label{RHS-2} \begin{split} &\xi_n^{k-3}D_xg_{j,p}(x,t)\\ &=\xi_n^p\frac{-1}{k\xi_n^2}r_{p,j}(x) g^{(j)} \big(\frac{x-(x_n-kt\xi_n^{k-1})}{\xi_n^l}\big)|\xi_n|^{-l/2}\\ &\quad +\xi_n^p\frac{-\xi_n^{k-3-l}}{k\xi_n^{k-1}} \int_{x_n}^xr_{p,j}(y)\,dy\, g^{(j+1)} \big(\frac{x-(x_n-kt\xi_n^{k-1})}{\xi_n^l}\big)|\xi_n|^{-l/2}. \end{split}$$ 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 $j=1,2,\dots,l+1$, $$\label{43} \big|\int_{x_0}^{x_1}\Im d_j(y)\,dy\big| \le C|x_1-x_0|^{(j-1)/(l+1)}.$$ \end{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 $$\label{prop2-31} \big|\sum_{j=1}^{l+1}\frac{2^{p(j-1)}}{\eta^{j-1}} \int_{y}^{y+\eta^{l+1}}\Im d_j(y)\,dy\big|\le C_p.$$ 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}