数理解析研究所講究録 1261 巻 2002 年 46-55

46

On the Wellposedness of the Cauchv Problem
for Weakly Hyperbo1ic Equations of Higher Order
By Piero D’ANCONA and木下保(Tamotu KINOSHITA)

筑波大学 数学系
Dipartimento di Matematica
Institute of Mathematics
Universit\‘a “La Sapienza” di Roma
University of Tsukuba
Piazzale Aldo Moro, 2Tsukuba, Toaraki
1-00185 Roma, Italy
305-8571, Japane-mail : dancona@mat.unir0ma1.it
kinosita@math.tsukuba.ac.jp\S 1. Introduction
We consider here the Cauchy problem on $[0, T]$(1)$\cross \mathrm{R}_{x}^{n}$$\{\begin{array}{l}D_{t}^{m}u=\Sigma c_{j_{\prime}\alpha}(t)D_{t}^{j}D_{x}^{\alpha}u+\Sigma c_{j,\alpha}(t)D_{t}^{j}D_{l}^{\alpha}u+f(\mathrm{t},x)j+|\alpha|=mj+|\alpha|\leq dD_{t}^{j}u(0,x)=u_{j}(x)(j=0,\cdots,m-1)\end{array}$where
in short$D_{t}=-:\partial_{t}$, $D_{x}=-$ :$(\partial_{x_{1}}, \cdots,\partial_{x_{n}})$, and $0\leq d\leq m-1$ . We shall write$p(t, \tau,\xi)=\tau^{m}-\sum_{j+|\alpha|=m}c_{j,\alpha}(t)\tau^{j}\xi^{\alpha}$

for the principal part and$p_{d}(t,\tau,\xi)=$$\sum$$c_{\mathrm{j},\alpha}(t)\tau^{j}\xi^{\alpha}$$j+|\alpha|\leq d$for the lower order terms. We shall assume that the principal part is hyperbolicwith respect to , thatis, for any,
the roots in of the algebraic.
equation $p(t, \tau,\xi)=$ 0 are all real. We name them, according to the rule$p$$t\in \mathrm{R}_{4}$$\tau$$\tau$$\xi\in \mathrm{R}_{\xi}^{n}$$\lambda_{j}(t,\xi)$$\lambda_{1}(t,\xi)\geq\lambda_{2}(t,\xi)\geq\cdots\geq\lambda_{m}(t,\xi)$,thus $p(t, \tau,\xi)$ can be written$p(t, \tau,\xi)=\prod_{k=1}^{m}(\tau-\lambda_{k}(t,\xi))$

We recall that the functions$\lambda_{j}(t,\xi)$.

are homogeneous of degree 1in .
$\xi$

47
wellposedness we
There are many results on this problem. As to themention that T. Nishitani [N1] considered the case when the multiplicity of thecharacteristic roots is at most double. F. Colombini and N. Orru’ [CO] assumed$C^{\infty}$that the characteristic roots vanish of finite order at$t^{2} \sum_{k,j=1,k\neq\dot{g}}^{m}\frac{|\lambda_{k}’(t,\xi)|^{2}+|\lambda_{j}’(t,\xi)|^{2}}{|\lambda_{k}(t,\xi)-\lambda_{j}(t,\xi)|^{2}}<\infty$

$t$

$=0$

and satisfy

near $t=0$ .

Moreover, K. Kajitani, S. Wakabayashi and K. Yagdjian [KWY] dealt with thecase of characteristic roots vanishing of infinite order. Concerning the Gevreywellposedness, F. Colombini and T. Kinoshita [CK] considered the Cauchy prob-

lem in the case when the characteristic roots are H\"older continuous in . F.
Colombini, H. Ishida [CI] and H. Ishida, K. Yagdjian [IY] assumed that thecharacteristic roots vanish of infinite order at $t=0$ and satisfy for some$t$$\overline{s}>1$$\Phi_{1}(t)^{2\overline{s}/(\overline{s}-1)}\phi_{1}(t)^{2}\sum_{k,j=1,k\neq j}^{m}\frac{|\lambda_{k}’(t,\xi)|^{2}+|\lambda_{j}’(t,\xi)|^{2}}{|\lambda_{k}(t,\xi)-\lambda_{j}(t,\xi)|^{2}}<\circ \mathrm{p}$near $t=0$ ,,,are real-valued functions such that
and
where
, $m$ .
(i) $\phi_{k}(0)=\phi_{k}’(0)=0$ , $\phi_{k}’(t)>0$ if $t\in(0, T]$ for any $k=1$ ,for $\in[0, T]$ .
(ii) $i (t)\geq $i(t).
, $m$ andfor $k=1$ ,(iii).
for $k<j$ and.
Then they showed the welposedness in the Gevrey classes of order
We see that in most results concerning the higher order case $m>2$ the roots
are assumed to eoincide only at isolated points, and then aprecise behaviour is$\Phi_{1}(t)=\int_{0}^{t}\phi_{1}dt$$\phi_{1}(t)$$\phi_{m}(t)$$\cdots$$\cdots$$\geq\cdots\geq\phi_{m}(t)$$t$$|\lambda_{k}(t, \xi)|\leq C_{k}\phi_{k}(t)|\xi|(^{\exists}C_{k}>0)$$(\mathrm{i})$$(t, \xi)\in[0, T]\cross \mathrm{R}_{\xi}^{n}\backslash 0$$\cdots$$|\lambda_{k}(t,\xi)-\lambda_{j}(t,\xi)|\geq c\phi_{k}(t)|\xi|(^{\exists}c>0)$$(t,\xi)\in[0, T]\cross \mathrm{R}_{\xi}^{n}\backslash 0$$1\leq s<\overline{s}$assumed at those points. In this paper we try to give aglobal assumption valid
in more general cases, even when this happens at an arbitrary set of points (also,defined as follows:
infinite or dense). To this end we introduce the sets
, $m-1$ ,for any $0<\sigma<1$ , $k–1$ ,$\Omega_{\sigma}^{k}$$\Omega_{\sigma}$$\ldots$$\Omega_{\sigma}^{k}(\xi)=\{t\in[0, T] : |\lambda_{k}(t,\xi)-\lambda_{k+1}(t, \xi)|\leq\sigma\}$and$\Omega_{\sigma}(\xi)=\cup m-1k=1\Omega_{\sigma}^{k}(\xi)$.

48
These sets enclose, for each , the points where the roots coincide; thus we
can regard the measure, which is afunction of , (, as ameasure of thedefect of strict hyperbolicity of . Here $\mu(A)$ is the Lebesgue measure inof
the set. We denote by $AC([0, T])$ the space of absolutely continuousfunctions on $[0, T]$ and by
the space of Gevrey functions $g(x)$ satisfying
for any compact set,.
Our first result is the following:
$t$

$\xi$

$\mu(\Omega_{\sigma})$$\sigma$$p$$\mathrm{R}_{t}$$A\subseteq[0, T]$$G^{s}(\mathrm{R}^{l}’)$$\sup_{x\in K}|D_{x}^{\alpha}g(x)|\leq C_{K}\rho_{K}^{|\alpha|}|\alpha|!^{s}$$K\subset \mathrm{R}^{n}$$\alpha\in \mathrm{N}^{n}$

THEOREM 1. (Gevrey-wellposedness). Assume that the coefficientsof$C^{0}([0,T])$,belong to
and the characteristic roots of the principal part, ,belong to $AC([0,T])$ and that there exist constants $C>0$ , $a\geq 0$$md$ $b>0$ such that for any $0<\sigma<1$ , $|\xi|=1$ , $k=1$ ,, $m-1$$c_{j,\alpha}(t)$

$p$

$Pd$

$\lambda_{1}$$\cdots$$\lambda_{m}$$\cdots$(2)$\mu(\Omega_{\sigma}(\xi))\leq C\sigma^{0}$(3),$\int_{[0,T]\backslash \Omega_{\sigma}^{k}(\xi)}\frac{|\lambda_{k}’(t,\xi)|+|\lambda_{k+1}’(t,\xi)|}{|\lambda_{k}(t,\xi)-\lambda_{k+1}(t,\xi)|}dt\leq C\sigma^{-b}$Then, when the degree$d$.

of the lower order terms satis $ies$$0 \leq d\leq\frac{m(a+b)}{a+b+1}$,

the Cauchy problem (1) is wellposed in the Gevrey classes of order(4)$1 \leq s<1+\frac{a+1}{b}$i.e., for any data$u_{j}\in G^{s}(\mathrm{R}^{n})$(1) has aunique solution$d$,and $f\in C^{0}([0,T];G^{s}(\mathrm{R}^{n}))$ the Cauchy problem$u\in C^{m}([0,T];G^{\epsilon}(\mathrm{R}^{n}))$. Moreover, when the degreeof the lower order terms satisfies$d> \frac{m(a+b)}{a+b+1}$,

then the problem is wellposed for(5)$1 \leq s<\frac{m}{d+a(d-m)}$.

Remark 1. In the cases mentioned above, when,,vmish of infinite order, assumption (2) can be dropped (one is forced to choos$\lambda_{1}$$(t, \xi)$$\cdots$$\lambda_{m}(t,\xi)$49
Thus by Theorem 1we see that the Cauchy Problem (1) is wellposed$a=0)$ .

in the Gevrey classes of order$1 \leq s<\min\{1+\frac{1}{b}$,$\frac{m}{d}\}$.

Remark 2. M. D. Bronshtein [B], S. Wakabayashi [W] proved the Lipschitz(or H\"older) continuity in of the characteristic roots of hyperbolic polynomials
are smoothwith smooth coefficients (see also [M]). Thus if we assume that$t$$c_{j,\alpha}$

for $j+|\alpha|=m$ , we can drop the assumption that$\lambda_{j}$belong to $AC([0, T])$ .

Remark 3. It is well-known that the lower order terms do not influence-wellposedness for strictly hyperbolic equations (the multiplicy of thethecharacteristic roots is equal to 1) and the lower order terms of order $d=m-1$
give the Gevrey- wellposedness of order $1\leq s<m/(m-1)$ for weakly hyperbolic equations (the multiplicy of the characteristic roots is equal to $m$ ) (see [B],$C^{\infty}$in (2) becomes greater, thetype of approaches to strictly hyperbolic type. Especially, when $d=m-1$ , thesecond exponent in (4) is equal to $m/(m-1-a)$ . Taking $0\leq a<m-1$ , we can[C], [CDS], [CJS], [OT], etc.). As the parameter

$a$

$p$

obtain an interpolation between$C^{\infty}$

and the Gevrey classes of order $m/(m-1)$ .

Example A. When the characteristic roots are$\lambda_{k}(t, \xi)=kt^{h}\{1+\sin^{2}(\frac{1}{t^{h/\alpha-1}})\}\cdot\xi$, $m$ , we find thatfor some $0<\alpha\leq 1$ , $\alpha<h<\alpha/(1-\alpha)$ and $k=1$ ,and vanish of finite order, ,belong to $AC([0,T])$ and also
at $=0$ and satisfy (2) with $a=1/h$ and (3) with $b=1/\alpha-1/h$ , since$\cdots$$\lambda_{1}$$\cdots$$\lambda_{m}$$C^{\alpha}([0, T])$$t$$\mu(\Omega_{\sigma}(\xi))\leq C\int_{0}^{C\sigma^{1/h}}dt\leq C\sigma^{1/h}$,.

$\int_{[0,T]\backslash \Omega_{\sigma}^{k}(\xi)}\frac{|\lambda_{k}’(t,\xi)|+|\lambda_{k+1}’(t,\xi)|}{|\lambda_{k}(t,\xi)-\lambda_{k+1}(t,\xi)|}dt\leq C\int_{C\sigma^{1/h}}^{T}(\frac{1}{t^{h/\alpha-1}})’dt\leq C\sigma^{1/h-1/\alpha}$

Applying Theorem 1, we get the wellposedness in the Gevrey classes of order(6)$1 \leq s<\frac{h}{h-\alpha}(1+\alpha).\cdot$

50
According to [CK] or [OT], if the characteristic roots belong toCauchy problem (1) is in the wellposed in the Gevrey classes of order$C^{\alpha}([0, T])$

$1\leq s<1+\alpha$

, the

.

For the second order polynomial $P(t, \tau,\xi)\equiv\tau^{2}-A(t)\xi^{2}$ where $A(t)\geq 0$ , if
$A(t)$ belongs to $C^{2\alpha}([0,T])$ , we also know the Gevrey order (7) (see [CJS], [D1]and [N2] . We remark that (6) approaches to (7) as tends to infinity andcan be taken arbitrarily large as tends to (the characteristic roots oscillate
$h$

$)$

$h$

$s$

$\alpha$more slowly). This example implies that the oscillation and the degeneracy ofthe characteristic roots influence on the wellposedness independently of theirregulality.
Example B. [CI] and [IY] gave an example of the folowing kind:$\lambda_{k}(t,\xi)=\{^{k\mathrm{e}\mathrm{x}_{0}\mathrm{p}(-\frac{1}{t^{h}})\{1+\sin^{2}(\exp\frac{\gamma}{t^{h}})\}\cdot\xi}$for some $\gamma>0$ , $h>0$ and $k=1$ ,$\ldots$, $m$ . They proved the welposedness in

the Gevrey classes of order $1\leq s<1+1/\gamma$ . Notice that,,belong to $AC([0,T])$ and vanish of infinite order at $t=0$ (see Remark 1) and$\lambda_{1}(t,\xi)$satisfy (3) with

$b=\gamma$$\cdots$$\lambda_{m}(t,\xi)$;$\int_{[0,T]\backslash \Omega_{\sigma}^{k}(\xi)}\frac{|\lambda_{k}’(t,\xi)|+|\lambda_{k+1}’(t,\xi)|}{|\lambda_{k}(t,\xi)-\lambda_{k+1}(t,\xi)|}dt\leq C\int_{1/(\log\sigma^{-1}+C)^{1/h}}^{T}(\exp\frac{\gamma}{t^{h}})’dt\leq C\sigma^{-\gamma}$.

Thus we can apply Theorem 1and we get the same Gevrey order $1\leq s<1+1/\gamma$ .

Our theorems can be applied also when the vanishing order of characteristicroots is different from the order of contact between the roots. For instance, if
the characteristic polynomial is$p(t,$$\tau$

we easily obtain, $()$

where $0<\alpha<\beta$ ,$=\tau^{2}-2t^{\alpha}\tau\xi+(t^{2\alpha}-t^{2\beta})\xi^{2}$$\lambda_{1}(t,\xi)=(t^{\alpha}+t^{\beta})\xi$$|\lambda_{k}(t,\xi)|\leq 2t^{\alpha}|\xi|(k=1,2)$,

and$\lambda_{2}(t, \xi)=(t^{\alpha}-t^{\beta})\xi$$|\lambda_{1}(t,\xi)-\lambda_{2}(t,\xi)|\geq 2t^{\beta}|\xi|$forwhich implies that$(t,\xi)\in[0,T]\cross \mathrm{R}_{\xi}$

.

Since
and
satisfy (2) with $a=1/\beta$ and (3) $b=1-\alpha/\beta$ , applying
Theorem 1we have welposedness in the Gevrey classes of order$\lambda_{1}(t, \xi)$$\lambda_{2}(t,\xi)$$1 \leq s<1+\frac{\beta+1}{\beta-\alpha}$.

In the favourable case of analytic characteristic roots, more generally from

Theorem 1we also obtain the folowing results

51
COROLLARY 2. (Gevrey-wellposedness). Assume that the coefficients
belong to $C^{0}([0, T])$ and the characteristic roots of the principal partof ,,are analytic in and vanish at $=0$ and that there exist,
Aiconstants $C>0$ , $c>0$ and $0<\alpha<\beta$ such that for any$cj,\alpha(t)$$p$$p_{d}$$(\mathrm{t}, \xi)$$\cdots$$t$$t$$\lambda_{m}(t, \xi)$$(t,\xi)\in[0,T]\cross \mathrm{R}_{\xi}^{n}$$|\lambda \mathrm{i}(t,\xi)|\leq Ct^{\alpha}|\xi|$for$|\lambda_{k+1}(t,\xi)-\lambda_{k}(t, \xi)|\geq ct^{\beta}|\xi|$Then, when the degree$d$$k=1$ ,

for$\cdots$$k=1$ ,

, $m$ ,$\cdots$, $m-1$ .

of the lower order terms satisfies$0 \leq d\leq\frac{m(\beta-\alpha+1)}{2\beta-\alpha+1}$,

the Cauchy probiem (1) is weiiposed in the Gevrey classes of order$1 \leq s<1+\frac{\beta+1}{\beta-\alpha}$

Moreover, when the degree$d$.

of the lower order terms satisfies$d> \frac{m(\beta-\alpha+1)}{2\beta-\alpha+1}$,

then the wellposedness holds for$1 \leq s<\frac{\beta m}{\beta d+d-m}$.

In Corollary 2and Examples Aand , the characteristic roots coincide only
or at afinite number of points. We give afinal example to emphasize
atthat our results allow the characteristic roots to coincide at an infinite number$\mathrm{B}$$t=\cdot 0$of points.
Example$\mathrm{C}$(see also Example$\mathrm{A}$). When the characteristic roots are$\lambda_{k}(t, \xi)=kt^{h}\sin^{h}(\frac{1}{t^{h-1}})\cdot\xi$are, ,, $m$ , we find thatfor some even number and $k=1$ ,absolutely continuous in , more precisely Lipschitz continuous in and vanish
at $=(\pi j)^{1/(1-h)}(j=1,2, \cdots)$ , they satisfy (2) with $a<1/h$ and (3) with$b>1-1/h$ . Applying Theorem 1, we get the wellposedness in the Gevrey$h$$\cdots$$t$$t$classes of order $1\leq s<2h/(h-1)$ (see (7))$\lambda_{1}(t, \xi)$$\lambda_{m}(t,()$$\cdots$$t$

52\S 2. Sketch of the proof
When $s=1$ , the Cauchy problem (1) is wellposed in the class of real
analytic functions. Therefore we can suppose that $s>1$ for the proof. By

Fourier transform with respect to , the Cauchy problem (1) turns into
$x$

(8)

$\{\begin{array}{l}p(t,D_{t},\xi)\hat{u}=\hat{f}(t,\xi)+p_{d}(t,D_{t},\xi)\hat{u}D_{t}^{j}\hat{u}(0,\xi)=\hat{u}_{j}(\xi)(j=0,\cdots,m-1)\end{array}$

Let $0<\sigma<1$ md$\varphi(r)\equiv 0$$\varphi(r)$for $|r|\geq 2$ andbe anon-negative fimction such that$\varphi(r)\equiv 1$$\mu_{k}(t,\xi)=\lambda_{k}(t,\xi)+ik\omega(t,\xi)$$q(t, \tau,\xi)$,

for $|r|\leq 1$ . We define$\omega(t,\xi)=\sigma|\xi|\sum_{l=1}^{m-1}\varphi(\sigma^{-1}\{\lambda_{l}$Moreover we denote by$\varphi\in C_{0}^{\infty}(\mathrm{R})$(,$t$$\frac{\xi}{|\xi|})-\lambda_{l+1}(t,$$\frac{\xi}{|\xi|})_{\backslash }\})$,

for $k=1$ , $\cdots,m$ .

the polynomial of degree $m$ in$q(t, \tau,\xi)=\prod_{k=1}^{m}(\tau-\mu_{k}(t,\xi))$$\tau$.

Now we set the energy density$E(t, \xi)=\frac{1}{2}\sum_{l=1}^{m}|q_{l}(t, D_{t},\xi)\hat{u}|^{2}$where$q_{l}(t, \tau,\xi)$is the polynomial of degree $m-1$ in,$\tau$

defined by$q_{l}(t, \tau, \xi)=\frac{q(t,\tau,\xi)}{\tau-\mu_{l}(t,\xi)}(=\prod_{k=1,k\neq l}^{m}(\tau-\mu_{k}(t,\xi)))$.

We denote by ’the derivative in . Differentiating $E(t,\xi)$ in and dividing
by $2\sqrt{E(t,\xi)}$, by (8) we have
$t$

$t$

.

$\sqrt{E}’\leq C(_{1\leq k\leq m-1}\max\frac{|\lambda_{k}’|+|\lambda_{k+1}’|+|\omega’|}{|\lambda_{k}-\lambda_{k+1}|+\omega}+\omega+\frac{|\xi|^{d}}{\prod_{k=1}^{m-1}|\lambda_{k}-\lambda_{k+1}|+\omega^{m-1}})\sqrt{E}+|\hat{f}|$Thus, Gronwedl’s inequalty yields the estimate$\sqrt{E(t,\xi)}\leq\exp\{C\int_{0}^{T}$($1 \leq k\leq\max\frac{|\lambda_{k}’|+|\lambda_{k+1}’|+|\omega’|}{|\lambda_{k}-\lambda_{k+1}|+\omega}+\omega m-1$$\dagger\frac{|\xi|^{d}}{\prod_{k=1}^{m-1}|\lambda_{k}-\lambda_{k+1}|+\omega^{m-1}}$$\cross\{\sqrt{E(0,\xi)}+\int_{0}^{T}|\hat{f}(t,\xi)|dt\}$.

)$dt\}$

53
We remark that there exists C $>0$ such that for any (t,$\xi)\in[0, T]\cross \mathrm{R}_{\xi}^{n}\backslash 0$$C^{-1}( \sigma|\xi|)^{m-1}|\xi|^{-j}|D_{t}^{j}\hat{u}|\leq\sqrt{E(t,\xi)}\leq C\sum_{j=0}^{m-1}|\xi|^{m-1-j}|D_{t}^{j}\hat{u}|$.

belong to $AC([0, T])$, ,LEMMA 1. Let $b\geq 0$ . Assume thatand satisfy (3). Then there exists $C>0$ such that for any $0<\sigma<1$ , $|\xi|=1$, $m$
and $k=1$ ,$\lambda_{1}(t,\xi)$$\cdots$$\lambda_{m}(t,\xi)$$\cdots$

if $b\geq 1$$C$$\leq C\sigma^{1-b}$$\int_{\Omega_{\sigma}^{k}(\xi)\cup\Omega_{\sigma}^{k-1}(\xi)}|\lambda_{k}’(t,\xi)|dt\leq\{$

if $0\leq b<1$

$C\sigma^{1-b}$

where$\Omega_{\sigma}^{0}(\xi)=\Omega_{\sigma}^{m}(\xi)=\phi$and$\Omega_{\sigma}^{k}(\xi)$

for $k=1$ ,$\cdots$, $m-1$ are defined in \S .1.

LEMMA 2. Let $0\leq a<m-1$ . Assume that , \cdots ,exists $C>0$ such that for any $0<\sigma<1$ , $|\xi|=1$$\lambda_{1}$,$\lambda_{m}$satisfy (2). Then there

,

(21)

$\int_{[0,T]\backslash \Omega_{\sigma}(\xi)}\frac{dt}{\prod_{k=1}^{m-1}|\lambda_{k}(t,\xi)-\lambda_{k+1}(t,\xi)|}\leq C\sigma^{a+1-m}$

where$\Omega_{\sigma}(\xi)$is defined in \S .1.

Consequently, it follows that$\sum_{j=0}^{m-1}|\xi|^{-j}|D_{t}^{j}\hat{u}(t, \xi)|\leq C\sigma^{1-m}\exp\{C(\sigma^{-b}+\sigma^{a+1}|\xi|+\sigma^{a+1-m}|\xi|^{d+1-m})\}$.

$\cross\{\sum_{j=0}^{m-1}|\xi|^{-j}|\hat{u}_{j}|+\int_{0}^{T}|\xi|^{1-m}|\hat{f}(t,\xi)|dt\}$

When$d \leq\frac{m(a+b)}{a+b+1}$,

the third term is smaller and this choice gives immediately$|\xi|^{\gamma b}+|\xi|^{1-\gamma(a+1)}+|\xi|^{\gamma(m-a-1)+d+1-m}\leq 3|\xi|^{\frac{b}{\circ+b+1}}$

Hence, there exists $\rho>0$ such that for any.

$(t,\xi)\in[0,T]\cross \mathrm{R}_{\xi}^{n}\backslash 0$$\sum_{j=0}^{m-1}|\xi|^{-j}|D_{t}^{j}\hat{u}(t,\xi)|\leq C\exp\{\rho|\xi|^{\frac{b}{a+b+1}}\}\{\sum_{j=0}^{m-1}|\xi|^{\frac{m-1}{a+b+1}-j}|\hat{u}_{j}(\xi)|+\int_{0}^{T}|\xi|^{\frac{(1-m)(a+1)}{a+b+1}}|\hat{f}(t,\xi)|dt\}$

54
In virtue of Paley-Wiener theorem, $\{D_{t}^{j}u(\cdot, t) ; t\in[0, T], j=0, \cdots, m-1\}$ isboimded in the Gevrey classes of order (5). Thus, taking into account that is
asolution of (1), we find $u\in C^{m}([0,T];G^{s}(\mathrm{R}^{n}))$ . This concludes the proof of
Theorem 1in the case when $d\leq m(a+b)/(a+b+1)$ .
On the other hand, when$u$$d> \frac{m(a+b)}{a+b+1}$,

the dominant terms in$|\xi|^{\gamma b}+|\xi|^{1-\gamma(a+1)}+|\xi|^{\gamma(m-a-1)+d+1-m}$

are the last two (the first one is smaller). In this case we choose$\gamma=\frac{m-r}{m}$and proceeding as above we conclude the proof of this case and we get (4).

REFERENCES[B]

M.D. Bronstein, The Cauchy problem for hyperbolic operators with characteristics of variable multiplicity, Ihdy Moskov. Mat.
. 41 (1980), 87-103(hans. Moscow Math. Soc, 1(1982), 87-103).
$Ok.\dot{\mathrm{c}}$[C]

M. Cicognani, On the strictly hyperbolic equations which are Holder continuouswith respect to time, Ital J. Pure AppL Math., 4(1998), 73-82.

[CDS]

F. Colombini, E. De Giorgi and S. Spagnolo, Sur les equations hyperboliques

avec des coefficients qui ne d\’ependent que du temps, Ann. Scuola Norm Sup.
Pisa, 6(1979), 511-559.
[CI]

F. Colombini and H. Ishida, Well-posedness of the Cauchy problem in Gevrey

classes for some weakly hyperbolic equations of higher order, preprint.
[CJS]

F. Colombini, E. Jannelli and S. Spagnolo, Wellposedness in the Gevrey classesof the Cauchy problem for anon strictly hyperbolic equation with coefficientsdepending on time, Ann. Scuola Nom Sup. Pisa, 10 (1983), 291-312

55

[CO]

CauchyF. Colombini and T. Kinoshita, On the Gevrey wellposedness of the
problem for weakly hyperbolic equations of higher order, preprint.
for some weakly hyperbolic
F. Colombini and N. Orrii, Well posedness in[D1]equations, J. Math. Kyoto. Univ., 39 (1999), 399-420.
P. D’Ancona, Gevrey well posedness of an abstract Cauchy problem of weakly

[CK]

[D2]

$C^{\infty}$

hyperbolic type, Publ RIMS Kyoto Univ.,24(1988), 433-449.
for aweakly hyperbolic second order
P. D’Ancona, Well posedness in$C^{\infty}$[KWY]

equation, Rend. Sem. Mat Univ. Padova, 91 (1994), 65-83.
-well posed Cauchy
K. Kajitani, S. Wakabayashi and K. Yagdjian, The
problem for hyperbolic operators with multiple characteristics vanishing with[I]

the different speeds, to appear in Osaka J. Math.
V. Ya. Ivrii, Cauchy problem conditions for hyperbolic operators with characteristics of variable multiplicity for Gevrey classes, Siberian. Math., 17 (1976),

[IY]
[M]
[N1]
[N2]

[OT]

[W]

$C^{\infty}$

921-931.
H. Ishida and K. Yagdjian, preprint.
T. Mandai, Smoothness of roots of hyperbolic polynomials with respect to onedimensional parameter, Bull. Fac. Gen. Ed. Gifu Univ., 21 (1985), 115-118.
T. Nishitani, The Cauchy problem for weakly hyperbolic equations of second
order, Comrn. P. D. E., 5(1980), 1273-1296.
T. Nishitani, Sur les equations hyperboliques \‘a coefficients holderiens en t et declasses de Gevrey en $x,Bull\sqrt$ . Sci. Math., 107 (1983), 113-138.
Y. Ohya and S. Tarama, Le probleme de Cauchy \‘a caract\’eristiquae multiples-coefficients holderiens en t-, (Proc. Taniguchi Inte m. Sympos. on Hyperbolic
Equations and Related Topics 1984), Kinokuniya, 1986, 273-306.

S. Wakabayashi, Remarks on hyperbolic polynomials, Tsukuba Journal of Mathematics, 10 (1986), 17-28