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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 85, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/85\hfil Energy estimate for wave equations]
{Energy estimate for wave equations with  coefficients in some Besov
type class}

\author[S. Tarama\hfil EJDE-2007/85\hfilneg]
{Shigeo Tarama}

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

\thanks{Submitted February 14, 2007. Published June 6, 2007.}
\subjclass[2000]{35L05, 16D10}
\keywords{Wave equation; energy estimate; non regular coefficients}

\begin{abstract}
 In this paper, we obtain an energy estimate for wave equations with
 coefficients satisfying Besov  type  conditions.
 We give an example of a wave equation with  continuous and nowhere
 differentiable  coefficients for which the $L^2$ estimate holds.
\end{abstract}

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

\section{Introduction}

Consider a wave equation on $[0,T]\times\mathbb{R}$:
\begin{equation}
Lu=\partial_t^2u-a(t)\partial_x^2u
\end{equation}
with a positive coefficient $a(t)\ge \delta_0$ with $\delta_0>0$.
It is well known that, if $a(t)$ is Lipschitz continuous, then we
have the energy estimate
\begin{equation}\label{est-1}
\sum_{0\le j+k\le 1}\|\partial_t^j\partial_x^{k}u(t,\cdot )\|
\le C\Bigl(\sum_{0\le j+k\le 1}\|\partial_t^j\partial_x^{k}u(0,\cdot )\|
+\int_0^t\|Lu(s,\cdot)\|\,ds
\Bigr)
\end{equation}
(see for example \cite[Ch. IX]{H}). Here $\|\cdot\|$ denotes $L^2$ norm.


 Colombini,  De Giorgi and  Spagnolo \cite{CDS} (see also \cite{DST})
 have shown that the estimate \eqref{est-1} is still valid if the
coefficient $a(t)$  has a bounded variation, that is, in the integral form,
there exists a constant $C\ge 0$ such that we have
\begin{equation}\label{bv}
\int_0^{T-\varepsilon}|a(t+\varepsilon)-a(t)|\,dt\le C\varepsilon\quad
 (0< \varepsilon\le T/2).
\end{equation}
Furthermore, in the same paper, they have shown that
if $a(t)$ satisfies
\begin{equation}\label{loglip}
\int_0^{T-\varepsilon}|a(t+\varepsilon)-a(t)|\,dt
\le C\varepsilon(|\log \varepsilon|+1)\quad (0< \varepsilon\le T/2)
\end{equation}
with a constant $C\ge 0$, then the Cauchy problem for $L$ is
$C^{\infty}$ well posed.

According to Yamazaki \cite{YZ}, we have  the estimate \eqref{est-1}
when $a(t)\in C^2((0,T])$  satisfies $|a(t)|+|ta'(t)|+|t^2a''(t)|\le C$
on $(0,T]$ (see also \cite{HRR}).
Then we see  that the estimate \eqref{est-1} is valid for $L$ with some
 coefficient $a(t)$ whose total variation is not finite, for example
$a(t)=2+\sin(\log t)$.

In this paper we introduce  an integral version of the  condition
$|a(t)|+|ta'(t)|+|t^2a''(t)|\le C$  so that the  estimate  \eqref{est-1}
holds still for $L$ with  the coefficient $a(t)$ satisfying such a condition.
Namely we show the following.  When   the coefficient $a(t)$ is a bounded
measurable function on  $[0,T]$ and satisfies:  there exists a constant
$C\ge 0$ such that we have
\begin{equation}\label{cond1}
\int_{\varepsilon}^{T-\varepsilon}|a(t+\varepsilon)
+a(t-\varepsilon)-2a(t)|\,dt\le C\varepsilon\quad (0< \varepsilon\le T/2),
\end{equation}
then  the estimate \eqref{est-1} holds.
Using the same method, we show also the following.
The Cauchy problem for $L$ is $C^{\infty}$ well posed
if  the coefficient $a(t)$ is a bounded measurable function on  $[0,T]$
and satisfies the following:  There exists a constant $C\ge 0$ such that
\begin{equation}\label{cond2}
\int_{\varepsilon}^{T-\varepsilon}|a(t+\varepsilon)
+a(t-\varepsilon)-2a(t)|\,dt\le C\varepsilon(|\log \varepsilon|+1)\quad
(0< \varepsilon\le T/2).
\end{equation}
Note that  the boundedness of $a(t)$ and  the estimate \eqref{cond1}
 imply
\begin{equation}\notag%\label{cond11}
\int_{0}^{T-\varepsilon}|a(t+\varepsilon)-a(t)|^2\,dt\le C\varepsilon\quad
(0<\varepsilon\le T/2)
\end{equation}
with some constant $C$. While from the boundedness of $a(t)$ and
\eqref{cond2} we obtain
\begin{equation}\notag%\label{cond21}
\int_{0}^{T-\varepsilon}|a(t+\varepsilon)-a(t)|^2\,dt
\le C\varepsilon(|\log \varepsilon|+1)\quad (0<\varepsilon\le T/2)
\end{equation}  with some constant $C$
(see the next section).

We remark that Colombini, Del Santo and Reissig \cite{CDR}
(see also  \cite{HR1} and \cite{HRR}) have  shown that the Cauchy
problem for $L$ is $C^{\infty}$ well posed when $a(t)$
satisfies $|a(t)|+|(t\log t) a'(t)|+|(t\log t)^2a''(t)|\le C$ on $(0,T]$.
For example the Cauchy problem for $L$ with $a(t)=2+\sin(|\log t|^2)$
is $C^{\infty}$ well posed but this function $a(t)$ does not satisfy
the condition \eqref{cond2}. Nonetheless  we can find some positive
function  $a(t)$ which satisfies the estimate \eqref{cond2} with
 the right hand  side replaced with
$ C\varepsilon(|\log \varepsilon|+1)^{1+\delta}$ ($\delta>0$),
so that the Cauchy problem for $L$ is not $C^{\infty}$ well posed.
Indeed Colombini and Lerner \cite{CL} have given an example of a positive
function $a(t)$ such that $a(t)$ satisfies
$\sup_{\varepsilon\in(0,1],t\in[0,1]}|
a(t+\varepsilon)-a(t)|/(\varepsilon(|\log \varepsilon|+1)^{1+\delta})<\infty$
(for any $\delta>0$) but the Cauchy problem on $[0,1]\times\mathbb{R}$ for
$\partial_t^2-a(t)\partial_x^2$ is not $C^{\infty}$ well posed.

In the next section, in order to study properties of  bounded functions
that satisfying \eqref{cond1} or \eqref{cond2}, we define
the function spaces  $Z_{\gamma}(I)$ and show some properties of functions
in such spaces.  Some properties of examples are discussed in the appendix.
In the third section, we state and prove the main theorems.

We use the following notation.
Let $L^2(\mathbb{R}^d)$ or $L^2$ denote the space of all square integrable functions
on $\mathbb{R}^d$ with  the norm $\|\cdot\|$  given by
$\|f(\cdot)\|^2=\int |f(x)|^2\,dx$. For $s\in\mathbb{R}$ let $H^s$  denote the
space that consists of  functions $f(x)$ on $\mathbb{R}^d$ satisfying
$\int (1+|\xi|^2)^s|\hat{f}(\xi)|^2\,d\xi<\infty$ where $\hat{f}(\xi)$
is the Fourier transform of $f(x)$ and  $\|\cdot\|_s$ be its norm, that is,
$\|f(\cdot)\|_s^2=\int (1+|\xi|^2)^{s}|\hat{f}(\xi)|^2\,d\xi$.
We set $H^{\infty}=\bigcap_{s\in \mathbb{R}}H^s$.  For   $X=H^s$, $H^{\infty}$
or $C^{\infty}(\mathbb{R}^d)$, the space of  indefinitely differentiable functions
on $\mathbb{R}^d$, and $T>0$, we denote by $L^1([0,T],X)$ the space of $X$-valued
integrable functions on $[0,T]$ and by  $C^{j}([0,T],X)$ with  an integer $j$
the space of $X$-valued $j$-times continuously differentiable functions
on $[0,T]$.  We use also the standard notation of multi-index.
We use $C$ or $C$ with some suffix in order to denote a non-negative
constant that may be different line by line.

\section{Space $Z_{\gamma}(I)$}

Let $I=(t_0, t_1)\subset \mathbb{R}$ with $t_0<t_1$ and $\gamma\ge0$.
We say $f(t)\in Z_{\gamma}(I)$ if $f(t)$ is a  bounded measurable
function on  the interval $I$ and satisfies, with a constant $C\ge 0$,
\begin{equation}\label{bs}
\int_{t_0+\varepsilon }^{t_1-\varepsilon }|f(t+\varepsilon )+
f(t-\varepsilon )-2f(t)|\,dt\le C\varepsilon(\log (\varepsilon^{-1}+1)+1+\gamma)^{\gamma}
\end{equation}for any $\varepsilon\in (0,(t_1-t_0)/2)$.

Here we remark that, when $\gamma=0$, \eqref{bs} corresponds to the
Besov $B_{1,\infty}^1$ estimate.
We remark also that $s(\log (s^{-1}+1)+1+\gamma)^{\gamma}$ is  increasing
on $(0,\infty)$ when $\gamma\ge0$.
We set
\begin{align*}
\|f\|_{Z_{\gamma}(I)}&=\|f\|_{L^{\infty}(I)}
+\sup_{0<\varepsilon< d/2}\frac{1}{\varepsilon(\log (\varepsilon^{-1}+1)+1
+\gamma)^{\gamma}}\\
&\quad\times \int_{t_0+\varepsilon }^{t_1-\varepsilon }|f(t+\varepsilon )
+f(t-\varepsilon )-2f(t)|\,dt
\end{align*}
where $I=(t_0,t_1)$ and $d=t_1-t_0$.


In the following we assume that functions in $Z_{\gamma}(I)$ are real valued.
But we see that the properties discussed below are valid also for complex
valued functions by considering the real part and the imaginary part
separately.

From the boundedness of $f(t)$, we see that $f(t)\in Z_{\gamma}(I)$ satisfies
\begin{equation}
\begin{aligned}\label{bs-1}
&\int_{t_0 +\varepsilon }^{t_1-\varepsilon }(|f(t+\varepsilon )
 -f(t)|^2+|f(t-\varepsilon )-f(t)|^2)\,dt\\
&\le C\varepsilon(\log (\varepsilon^{-1}+1)+1+\gamma)^{\gamma}\quad
  (0<\varepsilon\le (t_1-t_0)/2).
\end{aligned}
\end{equation}
 with the constant $C$ depending only on $\|f(\cdot)\|_{Z_{\gamma}(I)}$.
Indeed, since
$$
|f(t+\varepsilon )-f(t)|^2=(f(t+\varepsilon )-f(t))
f(t+\varepsilon )-(f(t+\varepsilon )-f(t))f(t),
$$
 we see that
\begin{align*}
J&=\int_{t_0 +\varepsilon }^{t_1-\varepsilon }|f(t+\varepsilon )-f(t)|^2\,dt\\
&= \int_{t_0+2\varepsilon }^{t_1 }(f(t )-f(t-\varepsilon))f(t )\,dt
  -\int_{t_0+\varepsilon}^{t_1-\varepsilon }(f(t+\varepsilon )-f(t))f(t)\,dt.
\end{align*}
Then we see that
\[J=-
\int_{t_0+2\varepsilon}^{t_1-\varepsilon }(f(t+\varepsilon )-2f(t)
+f(t-\varepsilon))f(t)\,dt+R,
\]
where
\[
R=\int_{t_1-\varepsilon}^{t_1}(f(t)-f(t-\varepsilon ))f(t)\,dt
-\int_{t_0+\varepsilon}^{t_0+2\varepsilon}(f(t+\varepsilon )-f(t))f(t)\,dt,
\]
from which, taking account of \eqref{bs} and the boundedness of $f(t)$,
we obtain
$|J|\le C\varepsilon(\log (\varepsilon^{-1}+1)+1+\gamma)^{\gamma}$.
Similarly we obtain the estimate for the integral of second term.
Hence we have \eqref{bs-1}.
Since
\begin{align*}
&f(t+\varepsilon)g(t+\varepsilon)-2f(t)g(t)+f(t-\varepsilon)g(t-\varepsilon)\\
&=(f(t+\varepsilon)-2f(t)+f(t-\varepsilon))g(t)+f(t)(g(t+\varepsilon)-2g(t)
+g(t-\varepsilon))\\
&\quad +(f(t+\varepsilon)-f(t))(g(t+\varepsilon)-g(t))+(f(t-\varepsilon)
-f(t))(g(t-\varepsilon)-g(t)),
\end{align*}
we see from \eqref{bs-1} and Schwarz's inequality that
$f(t),g(t)\in Z_{\gamma}(I)$ implies that $f(t)g(t)\in Z_{\gamma}(I)$.

For $f(t)\in Z_{\gamma}(I)$, we consider an extension of $f(t)$ on $\mathbb{R}$
as a bounded measurable function so that its $L^{\infty}$-norm is equal
to $\|f\|_{L^{\infty}(I)}$. We still denote by $f(t)$ such an extension.
Let $I=(t_0,t_1)$.
Then for any $\varepsilon>0$, we have
\begin{gather}
\int_{t_0}^{t_1}|f(t+\varepsilon)-2f(t)+f(t-\varepsilon)|\,dt
\le C\varepsilon (\log (\varepsilon^{-1}+1)+1+\gamma)^{\gamma}\label{bs-2}\\
\int_{t_0}^{t_1}(|f(t+\varepsilon)-f(t)|^2+|f(t)-f(t-\varepsilon)|^2)\,dt
\le C\varepsilon(\log (\varepsilon^{-1}+1)+1+\gamma)^{\gamma}.\label{bs-3}
\end{gather}
where the constant $C$ depends only on $\|f\|_{Z_{\gamma}(I)}$.
Indeed if $\varepsilon\ge (t_1-t_0)/2$, we see that the right hand side
of \eqref{bs-2} is not larger than $8\varepsilon\|f\|_{L^{\infty}(I)}$.
While, in the case of  $\varepsilon< (t_1-t_0)/2$, we see that,
 on the right hand side of \eqref{bs-2},  the integral on the
interval  $[t_0+\varepsilon,t_1-\varepsilon]$ is not larger than
$\varepsilon(\log (\varepsilon^{-1}+1)+1+\gamma)^{\gamma}\|f\|_{Z_{\gamma}(I)}$
and the integral on the remainder part
is not larger than $8\varepsilon\|f\|_{L^{\infty}(I)}$.
Hence we have \eqref{bs-2}. Similarly we obtain \eqref{bs-3}.

Now we consider the regularization of a function $f(t)$ in $Z_{\gamma}(I)$.
We take  the above mentioned extension $f(t)$. Let $\phi(s)$ be a smooth
function on $\mathbb{R}$ satisfying
 $\phi(-s)=\phi(s)$, $\phi(s)\ge 0$, $\phi(s)=0$ for
$|s|\ge1$ and $\int_{\mathbb{R}}\phi(s)\,ds=1$. We denote by $f_{\varepsilon}(t)$
with $\varepsilon>0$
the regularization of $f(t)$ given by
\[
f_{\varepsilon}(t)=\frac{1}{\varepsilon}\int_{\mathbb{R}}\phi(\frac{t-s}{\varepsilon})f(s)\,ds.
\]
Then we have the following result.

\begin{lemma}\label{lemma2-1}
\begin{gather}
\int_I|f_{\varepsilon}(t)-f(t)|\,dt
\le C_1\varepsilon(\log (\varepsilon^{-1}+1)+1+\gamma)^{\gamma}\label{bs-4}\\
\int_I(|f''_{\varepsilon}(t)|+|f'_{\varepsilon}(t)|^2)\,dt
\le C_2(\log (\varepsilon^{-1}+1)+1+\gamma)^{\gamma}/\varepsilon\label{bs-5}
\end{gather}
where the constants $C_1$ and $C_2$ depend  on $\|f\|_{Z_{\gamma}(I)}$ and
$\phi(s)$ but not on the length of the interval $I$.
Furthermore, for any  function $F\in C^2(\mathbb{R})$, setting 
$h(t)=F(f_{\varepsilon}(t))$, we have
\begin{equation}\label{bs-6}
\int_I(|h''(t)|+|h'(t)|^2)\,dt
\le C(\log (\varepsilon^{-1}+1)+1+\gamma)^{\gamma}/\varepsilon.
\end{equation}
Here the constant $C$ is also independent of the length of the interval $I$.
\end{lemma}

\begin{proof}
 Since $f_{\varepsilon}(t)-f(t)=\int_{\mathbb{R}} \phi(s)(f(t-\varepsilon s)-f(t))\,ds$
and $\phi(-s)=\phi(s)$, we have
\begin{align*}
|f_{\varepsilon}(t)-f(t)|
&=|\int_{\mathbb{R}} \frac{\phi(s)+\phi(-s)}{2}(f(t-\varepsilon s)-f(t))\,ds|\\
&=\frac{1}{2}|\int_{\mathbb{R}} \phi(s)(f(t+\varepsilon s)+f(t-\varepsilon s)-2f(t))\,ds|,
\end{align*}
from which and  from \eqref{bs-2}, we obtain
\[
\int_I|f_{\varepsilon}(t)-f(t)|\,dt\le C\int_{\mathbb{R}}\phi(s)|s|\varepsilon(\log((|s|\varepsilon)^{-1}+1)+1+\gamma)^{\gamma}\,ds.
\]
Since $s(\log (s^{-1}+1)+1+\gamma)^{\gamma}$ is increasing, the right hand
side of the estimate above is not larger than
$C\varepsilon(\log (\varepsilon^{-1}+1)+1+\gamma)^{\gamma}$.
Similarly,
$f''_{\varepsilon}(t)=\varepsilon^{-2}\int_{\mathbb{R}} \phi''(s)
(f(t-\varepsilon s)-f(t))\,ds$ and $\phi''(-s)=\phi''(s)$ imply
\[
\int_I|f''_{\varepsilon}(t)|\,dt\le C(\log(\varepsilon^{-1}+1)
+1+\gamma)^{\gamma}/\varepsilon.
\]
While it follows from
$f'_{\varepsilon}(t)=\varepsilon^{-1}\int_{\mathbb{R}} \phi'(s)(f(t-\varepsilon s)
-f(t))\,ds$, \eqref{bs-2} and Schwarz's inequality that
\[
|f'_{\varepsilon}(t)|^2\le \varepsilon^{-2}\|\phi'(\cdot)\|_{L^1}
\int_I|\phi'(s)||f(t-\varepsilon s)-f(t)|^2\,ds,
\]
from which and from \eqref{bs-3} we obtain the desired estimate of
$\int_I|f'_{\varepsilon}(t)|^2\,dt$.
Hence we have \eqref{bs-5}. We obtain \eqref{bs-6} from \eqref{bs-4}
and \eqref{bs-5}.
\end{proof}

\begin{example}\label{exa1} \rm
 If $f(t)\in C^2((0,1/2])$ satisfies
$|f(t)|+|f''(t)|t^2/|\log t|^{\gamma}\le C$ on $I=(0,1/2)$, then
$f(t)$ belongs to $Z_{\gamma}(I)$. Indeed,  if $\varepsilon<t<1-\varepsilon$,
 $f(t+\varepsilon)+f(t-\varepsilon)-2f(t)$ is equal to
$\varepsilon^2(f''(t+\theta\varepsilon)+
f''(t-\theta\varepsilon))/2$ with some $\theta\in(0,1)$. Then  we have
\[
|f(t+\varepsilon)+f(t-\varepsilon)-2f(t)|
\le C\varepsilon^2|\log(t-\varepsilon)|^{\gamma}/(t-\varepsilon)^2 \quad
(2\varepsilon\le t< 1/2-\varepsilon),
\]
from which we have
$$
\int_{2\varepsilon}^{1/2-\varepsilon}|f(t+\varepsilon)+f(t-\varepsilon)
-2f(t)|\,dt\le C\varepsilon|\log \varepsilon|^{\gamma}.
$$
Then noting $|f(t)|\le C$, we see $f(t)\in Z_{\gamma}(I)$.

For example,  let  $h_{\gamma}(t)=\sin(|\log t|^{\gamma+1})$ with $\gamma\ge 0$. Then $h_{\gamma}(t)$ belongs to $Z_{2\gamma}((0,1/2))$. Indeed we have $h_{\gamma}''(t)=-(\gamma+1)^2\sin(|\log t|^{\gamma+1})|\log t|^{2\gamma}/t^2+r(t)$ where $|r(t)|\le C|\log t|^{\gamma}/t^2$.   We see also that  $h_{\gamma}(t)\notin Z_{\sigma}((0,1/2))$ when $0\le \sigma<2\gamma$.
Furthermore we see that
\begin{equation}\label{example}
\limsup_{\varepsilon\rightarrow 0}
\frac{1}{\varepsilon|\log \varepsilon|^{1+\gamma}}
\int_{0}^{1/2-\varepsilon}|h_{\gamma}(t+\varepsilon)-h_{\gamma}(t)|\,dt>0
\end{equation}
(see the appendix for detail). Thus we see that $h_{1/2}(t)$ belongs
to $ Z_{1}((0,1/2))$ but does not satisfy \eqref{loglip} with $T=1/2$.
\end{example}

\begin{example}\label{exa2} \rm
Here we show that the Weierstrass function
\[
w_{\gamma}(t)=\sum_{n=1}^{\infty}2^{-n}n^{\gamma}\cos{2^nt}
\]
with $\gamma\ge0$, that is continuous and nowhere differentiable
(see for example \cite{K} ), belongs to $Z_{\gamma}((0,2\pi))$. Indeed,
for any $\varepsilon \in (0,1/2)$ we have
$w_{\gamma}(t)=w_{\gamma,1,\varepsilon}(t)+w_{\gamma,2,\varepsilon}(t) $
where
$$
w_{\gamma,1,\varepsilon}(t)=\sum_{1\le n
\le \frac{|\log \varepsilon|}{\log2}}2^{-n}n^{\gamma}\cos{2^nt}
\quad\text{and}\quad
w_{\gamma,2,\varepsilon}(t)=\sum_{ n> \frac{|\log \varepsilon|}
{\log2}}2^{-n}n^{\gamma}\cos{2^nt}.
$$
Since $|w''_{\gamma,1,\varepsilon}(t)|\le C\varepsilon^{-1}|
\log \varepsilon|^{\gamma}$ and $|w_{\gamma,2, \varepsilon}(t)|
\le C \varepsilon|\log \varepsilon|^{\gamma}$, then we see
$|w_{\gamma}(t+\varepsilon)+w_{\gamma}(t-\varepsilon)-2w_{\gamma}(t)|
\le C|\log \varepsilon|^{\gamma}\varepsilon$.
Hence $w_{\gamma}(t)\in Z_{\gamma}((0,2\pi))$.

We remark that $w_0(t)$ satisfies \eqref{loglip}.  Indeed, in the
expression above $w_0(t)=w_{0,1,\varepsilon}(t)+w_{0,2,\varepsilon}(t) $
we have $|w'_{0,1,\varepsilon}(t)|\le C|\log \varepsilon|$ and
$|w_{0,2, \varepsilon}(t)|\le C \varepsilon$.
Then we see $|w_0(t+\varepsilon)-w_0(t)|\le C|\log \varepsilon|\varepsilon$.
\end{example}

\section{Main results}

Let $a_{jk}(t)$ ($j,k=1,\dots,d$) be a real-valued bounded
measurable function on $(0,T)$ with $T>0$ satisfying $a_{kj}(t)=a_{jk}(t)$
and
\begin{equation}\label{hyp}
\sum_{j,k=1}^d a_{jk}(t)\xi_j\xi_k
\ge C_0|\xi|^2 \text{ for }\xi\in \mathbb{R}^d\text{ and }t\in(0,T)\end{equation}
with some positive constant $C_0>0$.
Set
\begin{equation}
P_2(t,\partial_t,\xi)=\partial_t^2+\sum_{j,k=1}^d a_{jk}(t)\xi_j\xi_k
\end{equation}
where $\xi\in \mathbb{R}^d$.
Then we have the following result.

\begin{theorem}\label{thm1}
Assume that $a_{jk}(t)\in Z_{\gamma}((0,T))$ $(j,k=1,\dots,d)$ with $\gamma\ge0$. Let $\xi\in \mathbb{R}^d$.
If $u(t)\in C^1([0,T])$ satisfies  $P_2(t,\partial_t,\xi)u=f(t)$ on
$(0,T)$ with $f(t)\in L^1([0,T])$, then we have
\begin{equation}\label{thm3-1}
\begin{aligned}
&(|\partial_tu(t_2)|^2+|\xi|^2|u(t_2)|^2)^{1/2}\\
&\le C_1e^{C_2(\log(|\xi|+1)+1+\gamma)^{\gamma}}
\bigl((|\partial_tu(t_1)|^2+|\xi|^2|u(t_1)|^2)^{1/2}+\int_{t_1}^{t_2}
|f(t)|\,dt\bigr)
\end{aligned}
\end{equation}
for any $0\le t_1\le t_2\le T$. Here constants $C_1$ and $C_2$ depend
on $C_0$ of \eqref{hyp} and $Z_{\gamma}$-norm of coefficients $a_{jk}(t)$
but not on the length of the interval $[0,T]$.
\end{theorem}

Before presenting the proof of Theorem above, we remark the following
well known result. Let $L=\partial^2_t+a^2(t)\rho^2$ where $a(t)$
is smooth and positive and $\rho>0$.  Noting that
$(\partial_t-ia(t)\rho-\frac{a'(t)}{2a(t)})
(\partial_t+i a(t)\rho+\frac{a'(t)}{2a(t)})$ and
$(\partial_t+i a(t)\rho-\frac{a'(t)}{2a(t)})
(\partial_t- ia(t)\rho+\frac{a'(t)}{2a(t)})$ are equal to
$$
L-(\frac{a'(t)}{2a(t)})^2+(\frac{a'(t)}{2a(t)})',
$$
we consider   the energy
 $$
 E(u)=\frac{1}{a(t)}|\partial_t u+\frac{a'(t)}{2a(t)}u|^2+a(t)\rho^2|u|^2.
$$
 Then we have
\begin{equation}\label{energ}
\frac{d}{dt}E(u)=\frac{2}{a(t)}{\rm Re}\bigl(\overline{\ (\partial_t u
+\frac{a'(t)}{2a(t)}u)\ }(Lu-Ru)\bigr)
\end{equation}
where $R=(\frac{a'(t)}{2a(t)})^2-(\frac{a'(t)}{2a(t)})'$.

\begin{proof}[Proof of Theorem \ref{thm1}]
If $\xi=0$, $P_2u=f(t)$ is equal to $\partial_t^2u=f(t)$.
Then we have immediately \eqref{thm3-1}. In the following,
 we assume $\xi\ne0$. First we extend the coefficients $a_{jk}(t)$ on
$\mathbb{R}$ so that
$\|a_{jk}(t)\|_{L^{\infty}(\mathbb{R})}=
\|a_{jk}(t)\|_{L^{\infty}((0,T))}$
and the estimate \eqref{hyp} still holds for $t\in\mathbb{R}$. Then we consider
the regularization $a_{jk,\varepsilon}(t)$ of $a_{jk}(t)$ given by
$\int_{\mathbb{R}}\phi((t-s)/\varepsilon)a_{jk}(s)\,ds/\varepsilon$ with
$\varepsilon>0$ using a non-negative, even  and smooth function $\phi(s)$
as  described in the section 2. Then we see that \eqref{hyp} with
$a_{jk,\varepsilon}(t)$ in the place of $a_{jk}(t)$ holds.
Then we define $a(t,\xi,\varepsilon)$ by
\[
a(t,\xi,\varepsilon)=|\xi|^{-1}\bigl(
\sum_{j,k=1}^d a_{jk,\varepsilon}(t)\xi_j\xi_k\bigr)^{1/2} \quad\text{for }
\xi\in\mathbb{R}^d\setminus\{0\}.
\]
We have
\begin{equation}\label{est3-11}
C_1\ge a(t,\xi,\varepsilon)\ge \sqrt{C_0}
\end{equation}
with  constants  $C_0$ appearing in \eqref{hyp} and $C_1$ depending only on $\|a_{jk}(\cdot)\|_{L^{\infty}((0,T))}$.
We see from \eqref{bs-6}, that
\begin{equation}\label{est3-2}
\int_0^T(|\partial_ta(t,\xi,\varepsilon)|^2
+|\partial^2 _ta(t,\xi,\varepsilon)|)\,dt
\le C_1\varepsilon^{-1}(|\log(\varepsilon^{-1}+1)|+1+\gamma)^{\gamma}
\end{equation}
for any $\varepsilon>0$.
Furthermore Lemma \ref{lemma2-1} implies that
\begin{equation}\label{est3-3}
\int_0^T |a(t,\xi,\varepsilon)^2|\xi|^2-\sum_{j,k=1}^d a_{jk}(t)\xi_j\xi_k|\,dt
\le  C_2\varepsilon(|\log(\varepsilon^{-1}+1)|+1+\gamma)^{\gamma}|\xi|^2.
\end{equation}
Here the constants above $C_1$ and $C_2$ may depend on $Z_{\gamma}$-norm
of $a_{jk} (t)$ and the constant $C_0$ of \eqref{hyp} but not on
the length of interval $[0,T]$.

Assume that $u(t)\in C^1([0,T])$ satisfies
$\partial_t^2u+\sum_{j,k=1}^d a_{jk}(t)\xi_j\xi_ku=f(t)$ on $(0,T)$
with $\xi\in \mathbb{R}^d\setminus\{0\}$ and $f(t)\in L^1([0,T])$. Let
\begin{equation}
E_{\varepsilon}(t)=\frac{1}{a(t,\xi,\varepsilon)}
|\partial_t u+\frac{\partial_ta(t,\xi,\varepsilon)}{2a(t,\xi,\varepsilon)}u|^2
+a(t,\xi,\varepsilon)|\xi|^2|u|^2.
\end{equation}
Then it follows from \eqref{energ} that
\begin{equation}\label{est3-4}
\frac{d}{dt}E_{\varepsilon}(t)=\frac{2}{a(t,\xi,\varepsilon)}{\rm Re}
\bigl(\overline{\ (\partial_t u+\frac{\partial_ta(t,\xi,\varepsilon)}
{2a(t,\xi,\varepsilon)}u)\ }\ (L_{\varepsilon}u-R_{\varepsilon}u)\bigr)
\end{equation}
where $L_{\varepsilon}u=\partial_t^2-a(t,\xi,\varepsilon)^2|\xi|^2u$ and
$R_{\varepsilon}=(\frac{\partial_ta(t,\xi,
\varepsilon)}{2a(t,\xi,\varepsilon)})^2-\partial_t(\frac{\partial_t
a(t,\xi,\varepsilon)}{2a(t,\xi,\varepsilon)})$.
Note that
\[
|L_{\varepsilon}u|\le |a(t,\xi,\varepsilon)^2|\xi|^2-\sum_{j,k=1}^d
a_{jk}(t)\xi_j\xi_k||u|+|f(t)|
\]
and
\[
|R_{\varepsilon}u|\le C(|\partial_ta(t,\xi,\varepsilon)|^2
+|\partial^2_ta(t,\xi,\varepsilon)|).
\]
Since
$$
|(\partial_t u+\frac{\partial_ta(t,\xi,\varepsilon)}{2
a(t,\xi,\varepsilon)}u)||u|\le
\frac{1}{2|\xi|}E_{\varepsilon}(t),
$$
we see that
\begin{equation}
\left|\frac{d}{dt}E_{\varepsilon}(t) \right| \le
2 C(t,\xi,\varepsilon)E_{\varepsilon}(t)+E_{\varepsilon}(t)^{1/2}2C_0^{-1/4}|f(t)|
\end{equation}
where
\begin{align*}
C(t,\xi,\varepsilon)
&=\frac{1}{2} C_0^{-1/2}\Bigl(
|\bigl(a(t,\xi,\varepsilon)^2| \xi|^2-\sum_{j,k=1}^d a_{jk}(t)\xi_j\xi_k\bigr)|\\
&\quad +
C|\partial_ta(t,\xi,\varepsilon)|^2+|\partial^2_ta(t,\xi,\varepsilon)|
\Bigr)|\xi|^{-1}.
\end{align*}
Hence for any positive constant $\delta>0$, we have
\[
\big|\frac{d}{dt}\bigl(E_{\varepsilon}(t)+\delta\bigr) \big|
\le 2 C(t,\xi,\varepsilon)\bigl(E_{\varepsilon}(t)+\delta\bigr)
+\bigl(E_{\varepsilon}(t)+\delta\bigr)^{1/2}2C_0^{-1/4}|f(t)|,
\]
from which we obtain
\[
\big|\frac{d}{dt}\bigl(E_{\varepsilon}(t)+\delta\bigr)^{1/2} \big|
\le  C(t,\xi,\varepsilon)\bigl(E_{\varepsilon}(t)+\delta\bigr)^{1/2}
+C_0^{-1/4}|f(t)|.
\]
Then we see that, for $0\le t_1\le t_2\le T$,
\begin{equation*}
(E_{\varepsilon}(t_2)+\delta\bigr)^{1/2}\le e^{\int_{t_1}^{t_2}C(t,\xi,\varepsilon)
\,dt}(E_{\varepsilon}(t_1)+\delta\bigr)^{1/2}+\int_{t_1}^{t_2}e^{
\int_t^{t_2}C(s,\xi,\varepsilon)\,ds}
C_0^{-1/4}|f(t)|\,dt.
\end{equation*}
It follows from \eqref{est3-2} and \eqref{est3-3} that
\[
\int_0^TC(t,\xi,\varepsilon)
\,dt\le C(\varepsilon |\xi|+\frac{1}{\varepsilon|\xi|})(\log(\varepsilon^{-1}+1)+1+\gamma)^{\gamma}.
\]
Now picking $\varepsilon=1/|\xi|$, we obtain
 \begin{equation}
\bigl(E_{1/|\xi|}(t_2)+\delta\bigr)^{1/2}\le e^{C(\log(|\xi|+1)+1+\gamma)^{\gamma}
}\Bigl((\bigl(E_{1/|\xi|}(t_1)+\delta\bigr)^{1/2}+\int_{t_1}^{t_2}C_0^{-1/4}|f(t)|\,dt\Bigr).
\end{equation}
By taking $\delta\rightarrow 0$, we obtain
\[
\bigl(E_{1/|\xi|}(t_2)\bigr)^{1/2}\le e^{C(\log(|\xi|+1)+1+\gamma)^{\gamma}
}\Bigl((\bigl(E_{1/|\xi|}(t_2))^{1/2}+\int_{t_1}^{t_2}C_0^{-1/4}|f(t)|\,dt\Bigr).
\]
Since $|\partial_ta_{jk,\varepsilon}(t)|\le 
C\varepsilon^{-1}\|a_{jk}(\cdot)\|_{L^{\infty}((0,T))}$ and 
$\varepsilon=1/|\xi|$, we see from \eqref{est3-11} that there exists a 
constant $C>0$ such that
\[
C(|\partial_tu(t)|^2+|\xi|^2|u(t)|^2)\le
E_{1/|\xi|}(t)
 \le C^{-1}(|\partial_tu(t)|^2+|\xi|^2|u(t)|^2)
\]
for any $t\in[0,T]$ and any $\xi\in \mathbb{R}^d\setminus\{0\}$.
Then we obtain the desired estimate \eqref{thm3-1}.
\end{proof}

Since $u(t_2)=u(t_1)+i\int_{t_1}^{t_2}\partial_tu(t)\,dt$,
from \eqref{thm3-1} we obtain
\begin{equation}\label{thm3-2}
\begin{aligned}
&(|\partial_tu(t_2)|^2+(|\xi|^2+1)|u(t_2)|^2)^{1/2}\\
&\le C_Te^{C_2(\log(|\xi|+1)+1+\gamma)^{\gamma}}
\bigl((|\partial_tu(t_1)|^2+(|\xi|^2+1)|u(t_1)|^2)^{1/2}
+\int_{t_1}^{t_2}|f(t)|\,dt\bigr)
\end{aligned}
\end{equation}
where the constant $C_T$ may depend on the length of the interval $[0,T]$.

Now consider $u(t,x)\in C^2([0,T],H^{\infty})$. Let
 $$
f(t,x)=\partial_t^2u(t,x)-\sum_{j,k=1}^d a_{jk}(t)\partial_{x_j}
\partial_{x_k}u(t,x).
$$
Then we have $P_2\hat{u}(t,\xi)=\hat{f}(t,\xi)$ where $\hat{u}(t,\xi)$ and
$\hat{f}(t,\xi)$  are the Fourier transform of $u(t,x)$ and $f(t,x)$
in variables $x$ respectively. Then  from \eqref{thm3-2}, we obtain
\begin{equation}\label{thm3-2-1}
\begin{aligned}
&(|\partial_t\hat{u}(t_2,\xi)|^2+(|\xi|^2+1)|\hat{u}(t_2,\xi)|^2)^{1/2}\\
&\le C_Te^{C_2(\log(|\xi|+1)+1+\gamma)^{\gamma}}
\bigl((|\partial_t\hat{u}(t_1,\xi)|^2+(|\xi|^2+1)|\hat{u}(t_1,\xi)|^2)^{1/2}
\\
&\quad +\int_{t_1}^{t_2}|\hat{f}(t,\xi)|\,dt\bigr) \quad
\text{for $0\le t_1<t_2\le T$}.
\end{aligned}
\end{equation}
Hence from the estimate
$\|\int_{t_1}^{t_2}g(t,\xi)\,dt\|_{L^2(\mathbb{R}^d_{\xi})}
\le \int_{t_1}^{t_2}\|g(t,\xi)\|_{L^2(\mathbb{R}^d_{\xi})}\,dt$,
which follows from the convexity of norm, and the Plancherel Theorem,
we obtain
\begin{equation} \label{thm3-3}
\begin{aligned}
&\|\partial_tu(t_2,\cdot)\|+\sum_{|\alpha|\le 1}\|
 \partial_x^{\alpha}u(t_2,\cdot)\| \\
&\le C\bigl(\|A_{\gamma}\partial_tu(t_1,\cdot)\|
+\sum_{|\alpha|\le 1}\|A_{\gamma}\partial_x^{\alpha}u(t_1,\cdot)\|
+\int_{t_1}^{t_2}\|A_{\gamma}f(t,\cdot)\|\,dt\bigr)
\end{aligned}
\end{equation}
where $A_{\gamma}$ is a Fourier multiplier given by
$$
A_{\gamma}v(x)=\int e^{i(x-y)\xi}
e^{\frac{1}{2}C_2(\log(|\xi|+1)+1+\gamma)^{\gamma}}v(y)\,d\xi dy/(2\pi)^d.
$$
Similarly multiplying \eqref{thm3-2-1} by $(|\xi|^2+1)^{s/2}$ with
$s\in\mathbb{R}$, we obtain
\begin{equation} \label{thm3-4}
\begin{aligned}
&\|\partial_tu(t_2,\cdot)\|_{s}+\|u(t_2,\cdot)\|_{s+1}\\
&\le C\bigl(\|A_{\gamma}\partial_tu(t_1,\cdot)\|_{s}+
\|A_{\gamma}u(t_1,\cdot)\|_{s+1}
+\int_{t_1}^{t_2}\|A_{\gamma}f(t,\cdot)\|_{s}\,dt\bigr).
\end{aligned}
\end{equation}
If $\gamma=0$, then $A_0v(x)=Cv(x)$ with $C=e^{C_2/2}$. Hence
\begin{equation}\label{a-est0}
\|A_0v(\cdot)\|\le C\|v(\cdot)\|\end{equation}
 for any $v\in L^2$, while $e^{C_2(\log(|\xi|+1)+2))/2}=C(|\xi|+1)^m$
with $m=C_2/2$ implies  that
 \begin{equation}\label{a-est1}
 \|A_1v(\cdot)\|_s\le C\|v(\cdot)\|_{s+m}
\end{equation}
  with some  $m\ge 0$ for any $s\in \mathbb{R}$ and any $v\in H^{s+m}$.
Then we have the following theorem.

\begin{theorem}\label{thm33}
 Let $a_{jk}(t)$ $(j,k=1,\dots,d)$ be a real-valued bounded
measurable function on $(0,T)$ with $T>0$ satisfying $a_{kj}(t)=a_{jk}(t)$ and
\eqref{hyp}. Let $L$ be a second order hyperbolic operator given by
\[
L=\partial_t^2-\sum_{j,k=1}^da_{jk}(t)\partial_{x_{j}}\partial_{x_{k}}.
\]

If $a_{jk}(t)\in Z_0((0,T))$ $(j,k=1,\dots,d)$, then we have the estimate
\begin{equation}\label{thm33-l2}
\sum_{l+|\alpha|\le 1}\|\partial^l_t\partial_x^{\alpha}u(t_2,\cdot)\|
\le C(\sum_{l+|\alpha|\le 1}\|\partial^l_t\partial_x^{\alpha}u(t_1\cdot)\|
+\int_{t_1}^{t_2}\|Lu(s,\cdot)\|\,ds)
\end{equation}
for any $0\le t_1\le t_2\le T$. Here
$u(t,x)\in \bigcap_{j=0}^1C^j([0,T],H^{1-j})$ satisfying
$Lu\in L^1([0,T],L^2)$.

 If  $a_{jk}(t)\in Z_1((0,T))$ $(1\le j,k\le d)$, then the Cauchy
problem for $L$ is $C^{\infty}$ well posed. Namely, for any
$u_0(x),u_1(x)\in C^{\infty}(\mathbb{R}^d)$ and
$f(t,x)\in L^1([0,T], C^{\infty}(\mathbb{R}^d))$, we have a unique solution
$u(t,x)\in C^{1}([0,T],C^{\infty}(\mathbb{R}^d))$ to the equation
$Lu=f(t,x)$ on $(0,T)\times \mathbb{R}^d$ with the initial conditions
$u(0,x)=u_0(x)$ and $\partial_tu(0,x)=u_1(x)$.
\end{theorem}

\begin{proof}
Assume that $a_{jk}(t)\in Z_0((0,T))$ ($j,k=1,\dots,d$). If $u(t,x)$
belongs to $\bigcap_{j=0}^2C([0,T],H^{2-j})$,
the estimate \eqref{thm33-l2} follows from \eqref{thm3-3} with $\gamma=0$
and \eqref{a-est0}.  In the case where
$u(t,x)\in \bigcap_{j=0}^1C^j([0,T],H^{1-j})$ and
$f(t,x)=Lu\in L^1([0,T],L^2)$, we regularize $u(t,x)$ with respect to
$x$-variables by setting
$u_{\delta}(t,x)=\int e^{i(x-y)\xi}
(1+\delta |\xi|^2)^{-1}u(t,y)\,d\xi dy/(2\pi)^d$ with $\delta>0$.
We denote this by $(1-\delta \Delta)^{-1}u(t,x)$.
Then we regularize $u_{\delta}$  with respect to $t$-variable by setting
$u^{\varepsilon}_{\delta}(t,x)
=\int_{\mathbb{R}} \psi_{\varepsilon}(t-s)u_{\delta}(s,x)\,ds$ 
with $\varepsilon>0$ where  $\psi_{\varepsilon}(s)$ is given by
$\psi_{\varepsilon}(s)=\psi(s/\varepsilon)/\varepsilon$
with a smooth function $\psi(s)$ on $\mathbb{R}$ satisfying
\[
\int_{\mathbb{R}} \psi(s)\,ds=1 \text{ and }\psi(s)=0 \quad
\text{for $s\ge 0$ or $s\le -1$}.
\]
We denote this convolution by $\psi_{\varepsilon}*u_{\delta}(t,x)$.
 Then we see that 
 $Lu^{\varepsilon}_{\delta}(t,x)=F^{\varepsilon}_{\delta}(t,x)$
 for $t\in[0,T-\varepsilon]$ where
$F^{\varepsilon}_{\delta}(t,x)=f^{\varepsilon}_{\delta}(t,x)
+R^{\varepsilon}_{\delta}$
with $f^{\varepsilon}_{\delta}(t,x)=\psi_{\varepsilon}*
(1-\delta \Delta)^{-1}f(t,x)$ and
\[
R^{\varepsilon}_{\delta}=\sum_{j,k=1}^d [\psi_{\varepsilon}*,a_{jk}(t)]
\partial_{x_j}\partial_{x_k}u_{\delta}(t,x).
\]
Here $[\cdot,\cdot]$ denotes the commutator.
Since $u^{\varepsilon}_{\delta}(t,x)\in \bigcap_{j=0}^2 C^j
([0,T-\varepsilon],H^{2-j})$, the estimate \eqref{thm33-l2}
is valid for $u^{\varepsilon}_{\delta}(t,x)$ when $0\le t_1 \le t_2 \le T-
\varepsilon$.
Since $f(t,x)\in L^1([0,T],L^2)$, we see that, for $0\le t_1\le t_2<T$,
$\int_{t_1}^{t_2}\|f^{\varepsilon}_{\delta}(t,\cdot)\|\,dt$ converges to
$\int_{t_1}^{t_2}\|f_{\delta}(t,\cdot)\|\,dt$ as $\varepsilon$ tends to zero.
While $u(t,x)\in C^0([0,T],H^1)$ implies $u_{\delta}(t,x) \in C^0([0,T],H^2)$.
Then we have $\partial_{x_j}\partial_{x_k}u_{\delta}(t,x) \in C^0([0,T],L^2)$,
which implies that $\psi_{\varepsilon}*\partial_{x_j}\partial_{x_k}u_{\delta}$
and  $\psi_{\varepsilon}*(a_{jk}(t)\partial_{x_j}\partial_{x_k}u_{\delta})$
converge to $\partial_{x_j}\partial_{x_k}u_{\delta}$ and
$a_{jk}(t)\partial_{x_j}\partial_{x_k}u_{\delta}$  in $L^1([0,T],L^2)$
respectively as $\varepsilon$ tends to zero. Hence we see that
$\int_{t_1}^{t_2}\|R^{\varepsilon}_{\delta}\|\,dt\rightarrow 0$
as $\varepsilon$ tends to zero when $0\le t_1\le t_2<T$,
Then the estimate \eqref{thm33-l2} is valid for $u_{\delta}(t,x)$ when
$0\le t_1\le t_2< T$. Finally we obtain the desired estimate for $u(t,x)$
 by taking $\delta\rightarrow 0$.

  Now consider  the case where $a_{jk}(t)\in Z_1((0,T))$ ($j,k=1,\dots,d$).
The estimates  \eqref{thm3-4} with $\gamma=1$ and \eqref{a-est1} imply that
for any $u_0(x)\in H^{s+1}$,  $u_1(x)\in H^{s}$ and $f(t,x)\in L^1([0,T],H^s)$
with arbitrarily chosen $s\in\mathbb{R}$, there exist a solution
$u(t,x)\in \bigcap_{j=0,1}C^j([0,T],H^{s+1-j-m})$ with some positive $m$
independent of $s$ to the equation $Lu=f$ satisfying the initial condition
$u(0,x)=u_0(x)$ and $\partial_tu(0,x)=u_1(x)$. The uniqueness of solutions
follows from the existence of solutions to the adjoint Cauchy problem.
Then the Cauchy problem is $H^{\infty}$ well posed. Since in the
article \cite{CDS} one has shown the
existence of the finite propagation speed for $L$ with the coefficients
in more general function classes, we see that the Cauchy problem is
$C^{\infty}$ well posed.
 We see  also the existence of finite propagation speed for $L$ by
considering the wave operator
$L_{\varepsilon}=\partial_t^2-\sum_{j,k=1}^da_{jk,\varepsilon}(t)
\partial_{x_j}\partial_{x_k}$ where the  coefficients
$a_{jk,\varepsilon}(t)$ $(j,k=1\dots,d)$
are defined at the beginning of  the proof of Theorem \ref{thm1}
as the  regularization of  $a_{jk}(t)$ $(j,k=1\dots,d)$.
First remark that we see from \eqref{est3-11} that the propagation
speed for $L_{\varepsilon}$ is not larger than $C_1$. For any smooth
and  compactly supported initial data $u_0$, $u_1$ and $f(t)$,
solutions $u_{\varepsilon}$ $(0<\varepsilon<1)$ to the equation
$L_{\varepsilon}u_{\varepsilon}=f$ with  the initial condition
$u_{\varepsilon}(0,x)=u_0(x)$ and $\partial_tu_{\varepsilon}(0,x)=u_1(x)$
have the uniform estimate \eqref{thm3-4} with $\gamma=1$ and
 \eqref{a-est1}. Hence we see that the solution $u$ to the equation $Lu=f$
with the same initial condition $u(0,x)=u_0(x)$ and $\partial_tu(0,x)=u_1(x)$
can be obtained as a limit of a suitable subsequence
$\{u_{\varepsilon_n}(t,x)\}$ with $\varepsilon_n\rightarrow 0$.
Then we  see the existence of finite propagation speed for $L$.
\end{proof}

\begin{example} \label{exa3} \rm
From  example \ref{exa2} of the previous section and Theorem \ref{thm33},
we see that the $L^2$ estimate \eqref{est-1} for
$L=\partial_t^2-(2+w_0(t))\partial_x^2$ holds where
 $w_0(t)$ is a continuous and nowhere differentiable function given
by $w_0(t)=\sum_{n\ge 1}2^{-n}\cos2^n t$.
\end{example}

\begin{remark} \label{rmk1} \rm
We assume  the boundedness of the coefficients in  the theorems above.
While  Colombini,  De Giorgi and  Spagnolo \cite{CDS} have shown that
the condition \eqref{loglip} without the assumption of boundedness is
sufficient for $C^{\infty}$ well posed.  But we see from the
example \ref{exa1} of the previous section that, even for bounded functions,
the condition   \eqref{bs} with $\gamma=1$ is still less restrictive than
that of \eqref{loglip}. For the related problem for  wave equations with
 unbounded coefficients having some special type of singularity see,
for example, \cite{HR1} or \cite{YZ}.
\end{remark}

\begin{remark} \label{rmk2} \rm
As mentioned in Theorem \ref{thm1}, the constants $C_1$ and $C_2$
in \eqref{thm3-1} are independent of the length of interval.
Then we obtain the following from \eqref{thm3-1} with $\gamma=0$  .
If $a_{jk}(t)$ ($j,k=1,\dots,d)$ belongs to $Z_0((0,\infty))$, that is,
$a_{jk}(t)$ is bounded measurable on $ (0,\infty)$ and satisfies
\[
\int_{\varepsilon}^{\infty}|a_{jk}(t+\varepsilon)+a_{jk}
(t-\varepsilon)-2a_{jk}(t)|\,dt\le C\varepsilon \quad
\text{for any $\varepsilon>0$},
\]
then under the condition \eqref{hyp} with $T=\infty$
we have  the following estimate for the homogeneous energy
$E_0(u)(t)=\|\partial_t u(t,\cdot)\|^2+\sum_{j=1}^d
\|\partial_{x_j}u(t,\cdot)\|^2$:
\[
E_0(u)(t_1)\le C E_0(u)(t_0) \quad \text{$\bigl(t_0,t_1\in [0,\infty)\bigr)$}
\]
for any   $u(t,x)\in \bigcap_{j=0,1}C^j([0,\infty),H^{1-j})$ satisfying $Lu=0$
 on $(0,\infty)\times \mathbb{R}^d$.
\end{remark}

\section{Appendix}

In this section we show \eqref{example}.
Let $h_{\gamma}(t)=\sin(|\log t|^{1+\gamma})$ with $\gamma>0$.
For any positive integer $n$, let $t_n,t_{n-},t_{n+}\in(0,1)$ be given by
\[
t_n=e^{-(2\pi n)^{1/(1+\gamma)}},\quad
 t_{n-}=e^{-(2\pi n-\pi/4)^{1/(1+\gamma)}},\quad
t_{n+}=e^{-(2\pi n+\pi/4)^{1/(1+\gamma)}}.
\]
We note $t_{n+}<t_n<t_{n-}$ and
\[
|\log t_n|^{1+\gamma}=2\pi n, \quad
|\log t_{n-}|^{1+\gamma}=2\pi n-\pi/4, \quad
|\log t_{n+}|^{1+\gamma}=2\pi n+\pi/4.
\]
We obtain   $t_{n-}-t_n>t_n-t_{n+}$ from $\frac{d}{ds}e^{-s^{1/(1+\gamma)}}<0$
and $\frac{d^2}{ds^2}e^{-s^{1/(1+\gamma)}}>0$.

Since $h_{\gamma}'(t)=-(1+\gamma)\cos(|\log t|^{1+\gamma})|\log t|^{\gamma}/t$
on $(0,1)$, we see that
\[
|h_{\gamma}'(t)|\ge C|\log t|^{\gamma}/t \quad \text{ for }t_{n+}\le  t\le t_{n-}.
\]
Since $Cn^{-\gamma/(1+\gamma)}\le |(2n\pm 1/4)^{1/(1+\gamma)}
-(2n)^{1/(1+\gamma)}|\le C^{-1}n^{-\gamma/(1+\gamma)}$, we see that
\[
t_n-t_{n+}\ge C e^{-(2\pi n)^{1/(1+\gamma)}}n^{-\gamma/(1+\gamma)}
\]
and $1\le t_{n-}/t_{n+}\le C$.
Then when $0<\varepsilon\le t_n-t_{n+}$, we have
\[
|h_{\gamma}(t+\varepsilon)-h_{\gamma}(t)|
\ge C\varepsilon(|\log t|^{\gamma}/t) \quad \text{for }t_{n+}\le  t\le t_n,
\]
 from which we have
\[
I_n=\int_{t_{n+}}^{t_n}|h_{\gamma}(t+\varepsilon)-h_{\gamma}(t)|\,dt\ge 
C\varepsilon(|\log t_{n+}|^{1+\gamma}-
|\log t_n|^{1+\gamma}.
)
\]
Then we have $I_n\ge C\varepsilon$ with some constant $C>0$ for any
positive integer $n$ and $\varepsilon\in[0,t_n-t_{n+}]$.
 We pick a large positive integer $n_0$ so that we have $t_{n-}\le 1/2$
for $n\ge n_0$.
For any large positive integer $N>n_0$, pick
$\varepsilon=t_N-t_{N+}$. Then we see that
\[
\int_{0}^{1/2-\varepsilon}|h_{\gamma}(t+\varepsilon)-h_{\gamma}(t)|\,dt
\ge \sum_{n=n_0}^N I_n\ge C(N-n_0+1)\varepsilon.
\]
Since $t_N-t_{N+}\ge Ce^{-(2\pi N)^{1/(1+\gamma)}}N^{-\gamma/(1+\gamma)}$,
we see that
$N\ge C|\log \varepsilon|^{1+\gamma}$ for large $N$. Then we see that
\[
\limsup_{\varepsilon\rightarrow 0}\frac{1}{\varepsilon|
\log \varepsilon|^{1+\gamma}}\int_{0}^{1/2-\varepsilon}|h_{\gamma}
(t+\varepsilon)-h_{\gamma}(t)|\,dt>0.
\]

By  a similar argument, we see that $h_{\gamma}(t)\notin Z_{\sigma}((0,1/2))$
when $0\le \sigma<2\gamma$.
Indeed noting that $h_{\gamma}''(t)
=-(\gamma+1)^2\sin(|\log t|^{\gamma+1})|\log t|^{2\gamma}/t^2+r(t)$
where $|r(t)|\le C|\log t|^{\gamma}/t^2$, we choose
$s_n$ and $s_{n\pm}$ in $(0,1)$ so that
$|\log s_n|^{\gamma+1}=2\pi n+\pi/2$ and
$|\log s_{n\pm}|^{\gamma+1}=2\pi n+\pi(1/2\pm1/4)$.
Then if  an integer $n$ is large and $\varepsilon \le (s_n-s_{n+})/2$, we have
\[
\int_{s_{n+}+\varepsilon}^{s_{n-}-\varepsilon} |h_{\gamma}(t+\varepsilon)
+h_{\gamma}(t-\varepsilon)-2h_{\gamma}(t)|\,dt\ge C\varepsilon^2
(s_n-s_{n+})|\log s_{n-}|^{2\gamma}/s_{n-}^2,
\]
from which and from the arguments similar to the above we see that
\[
\limsup _{\varepsilon\rightarrow 0}\frac{1}{\varepsilon(\log(\varepsilon^{-1}+1)
+1+\gamma)^{\gamma}}
\int_{\varepsilon}^{1/2-\varepsilon}|h_{\gamma}(t+\varepsilon)
+h_{\gamma}(t-\varepsilon)-2h_{\gamma}(t)|\,dt>0.
\]

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