Characteristic Cauchy problems for some non$\cdot ... - Project Euclid

J. Math. Soc. Japan

Vol. 45, No. 3, 1993

Characteristic Cauchy problems for some non Fuchsian partial differential operators

$\cdot$

By Takeshi MANDAI (Received Dec. 2, 1991) (Revised July 30, 1992)

\S 0. Introduction. We consider the characteristic Cauchy problems for some non-Fuchsian partial differential operators with real-analytic coefficients. We give some theorems that are similar to the Cauchy-Kovalevskaya theorem and the Holmgren theorem. As a corollary, we obtain some results on the non-existence of null-

solutions. First, we give the definition of (essentially) Fuchsian operators and null-solutions. Consider a partial differential operator $P=t^{k} \partial_{t}^{m}+\sum_{mj+|a|\lessgtr m.j0$

$\mu\in S,$

$\mu\in S$

.

we put

,

$:=$

$C_{\mu}(\lambda;x)$

a

$\Sigma$

$0(0, x)\lambda^{j}$ $j,$

.

$(j, 0)\in I_{\mu}$

If

$O\in S$

,

then we put $C_{0}(\lambda;x)$ $:= \sum_{(j,0)\in I_{0}}$

a

$j.0(0, x)\lambda(\lambda-1)\cdots(\lambda-J+1)$

.

, then we put If . The polynomial is called the indicial polynomial of P. (M. S. Baouendi and C. Goulaouic ([1]) called this poly, in the case of nomial “the cbaracteristic polynomial associated with $C_{0}(\lambda;x)=\tilde{a}_{0.0}(0, x)$

$O\not\in S$

$C_{0}$

$P’$

Fuchsian (6)

operators.)

If

$O\in S$

we

, then

$\Lambda(P):=\sup$

If

$O\not\in S$

, then

we put

put

{

${\rm Re}\lambda\in R;C_{0}(\lambda;x)=0$

$\Lambda(P)=-\infty$

for some

$x\in\Omega$

}.

.

By the use of these notions, Fuchsian operators are characterized as follows.

PROPOSITION 1.2. and there exist no

is Fuchsian The oPerator such that . $P$

$(], \alpha)\in I_{0}$

NOW, (A-1)

we assume the following conditions. , there exist no For all $(j, 0)\in\hat{V}$

, then

and

$\mu\in S$

${\rm Re}\lambda0$

$h(P)\leqq 0,$

$S=\{0\}$

,

$\alpha\neq 0$

. all non-zero

such that

$\alpha\neq 0$

.

$a_{j,0}(0,0)\neq 0$

, then

roots

$\lambda$

of

$C_{\mu}(\lambda_{j}0)=0$

satisfy

.

$h(P)\leqq 0$

.

$C_{0}(\lambda;0)\neq 0$

for

$\lambda=|h(P)|,$

$|h(P)|+1,$ $|h(P)|+2,$

$\cdots$

REMARK 1.3. (1) The assumption (A-1) implies that the principal part $P$ is essentially Fuchsian. of under the assumption (A-1). , then (2) Note that if of non-degeneracy at $x=0$ . For an arbiThus, the condition (A-2) is a , put trarily fixed $P_{m}$

$(J, 0)\in\hat{V}$

$aj.0(0, x)\not\equiv O$

$k\cdot nd$

$y\in\Omega$

Characteristic Cauchy problems $P_{y}$

515

,

$:= \sum_{J+|\alpha_{|\leqq m}}a_{j,\alpha}(t, y)\partial_{t}^{f}\partial_{x}^{a}$

which is a differential operator with coefficients depending only on . Then, the condition (A-2) is equivalent to that in a neighborfor any $(0,0)$ hood of . (3) The condition (A-4) implies that for , .. , $|h(P)|-1$ . In fact, if $j-|h(P)|=h(P)$ and hence $(J0)$ . Further, if $|h(P)|\geqq 1$ then $\Lambda(P)\geqq|h(P)|-1$ . (4) If , tben $h(P)\geqq 0$ and hence (A-4) means that $h(P)=0$ . Further, (A-5) follows from (A-2). (5) We do not exclude Fuchsian operators. Fuchsian operators always satisfy the conditions $(A-1)-(A-4)$ . $t$

$\Delta(P_{y})=\Delta(P)$

$y$

$C_{0}(\lambda;x)\equiv 0$

$\lambda=0,1$

$\cdot$

$\not\in I_{0}$

$O\not\in S$

EXAMPLE 1.4.

Consider an operator $P=t^{\kappa}\partial_{t}^{2}-t^{\nu}\partial_{x}^{2}+at^{p}\partial_{t}+b$

on

$R^{2}$

, where

$\kappa,$

$\nu,$

$p\in N,$

$b\in C$

$a,$

,

. We assume that

$\kappa>2p,$

$p\geqq 1$

and

,

$b\neq 0$

. We

have (A-1)

$\Leftrightarrow\nu>\kappa-2$

.

Under this condition (A-1), there holds that $S=\{\kappa-p-1, p-1\}$ . The conditions (A-2) and (A-4) are always satisfied. (Note that $h(P)=0.$ ) Further, we have $(A-3)\Leftarrow\ni{\rm Re} a>0$ and if $p>1$ then ${\rm Re}(b/a)>0$ , (A-5)

are

then

$\epsilonarrow Ifp=1$

$b/a\not\in\{0, -1, -2, \cdots\}$

Under the above assumptions, we have the the main results of this article.

$follow_{1}ng$

.

two theorems, which

THEOREM 1.5. Assume the conditions $(A-1)-(A-5)$ . Then, there exist a posiand a neighborhood tive for which the following holds: of in $g_{j}\in O(\Omega)(0\leqq_{J}\leqq|h(P)|-1)$ , there exists $f\in C^{\infty}([0, T] ; O(\Omega))$ and any For any a unique $u\in C^{\infty}([0, T_{0}];o(\Omega_{0}))$ satisfying $\Omega_{0}$

$T_{0}$

$0$

$Pu=f(t, x)$

(CP)

If

$\partial_{t}^{j}u|_{t=0}=0(0\leqq_{J}\leqq|h(P)|-1)$

of

$[0, T_{0}]\cross\Omega_{0}$

on

,

$(0,0)$

$\Omega_{0}(0\leqq j\leqq|h(P)|-1)$

.

. Let $L\in NandL>\Lambda(P)$ . neighborhood in a for of $(0,0)$ , and in a neighborhood of , then $u=0$ for $t>0$ in a

Assume the conditions

$u\in C^{L}([0, T];9’(\Omega\cap R^{n}))$

neighborhood

in

$\{$

$\partial_{t}^{f}u|_{t=0}=g_{f}(x)$

THEOREM 1.6.

$C^{n}$

, $Pu=0$

$(A-1)-(A-5)$ $t>0$

$0$

.

Immediately from Theorem 1.6,

we have the non-existence of sufficiently

516

T. MANDAI

smooth null-solutions. We also have the following stronger results on the nonexistence of null-solutions, assuming only the conditions $(A-1)-(A-3)$ .

THEOREM 1.7. Assume the conditions $(A-1)-(A-3)$ . null-solutions for then there exist no at $(0,0)$ .

If

$N\in N$

and

$N>\Lambda(P)$ ,

$P$

$C^{N}$

, then any 9’ THEOREM 1.8. Assume the conditions $(A-1)-(A-3)$ . If null-solution for $P$ at $(0,0)$ has a support included in the initial surface $\Sigma=\{t=0\}$ in a neighborhood of $(0,0)$ . Further, if $h(P)=0$ in addition, then there exist no 9’ null-solutions for $P$ at $(0,0)$ . $O\not\in S$

REMARK 1.9. (1) AS is well-known, if is non-Fuchsian, then the solution and the in Theorem 1.5 is not necessarily holomorphic in $(t, x)$ even if coefficients of are all holomorphic in $(t, x)$ . Also, we cannot replace $[0, T]$ and $[0, T_{0}]$ with $[-T, T]$ and $[-T_{0}, T_{0}]$ . (2) Consider the following condition, which is “opposite” to (A-3): (A-3) There exist . , and such that $\mu>0,$ and $P$

$f,$

$u$

$gj$

$P$

$\mu\in S$

$\lambda\in C$

${\rm Re}\lambda>0$

$C_{\mu}(\lambda;0)=0$

The author believes that the following holds:

CONJECTURE. If satisfies the conditions there exists a null-solution for at $(0,0)$ . $P$

(A-1), (A-2),

and

$(A- 3)^{\#}$

, then

$P$

$C^{\infty}$

.

If the coefficients , of $P$ are independent of lows from Theorem A (1) in [8]. $a_{j}$

$x$

,

then this conjecture fol-

The assumptions (A-4) and (A-5) have a different nature from $(A-1)-(A-3)$ . In Theorem 1.5, the assumptions (A-4) and (A-5) are used only to show that is uniquely the formal Taylor expansion of the solution with respect to determined from and , as we shall see in Section 4. In other words, the Cauchy problem (CP) is reduced to the flat Cauchy problem by the use of (A-4) and (A-5), and the flat Cauchy problem is uniquely solvable under the assumptions $(A-1)-(A-3)$ . As for Theorem 1.6, the situation is similar. In Section 4, we shall give some extended results to the reduced problems under the assumptions $(A-1)-(A-3)$ (Theorems 4.3, 4.4, 4.6). From these extended results, the four theorems above easily follow. In order to give these extended results, we need to consider generalized Newton polygons and some function spaces associated with generalized Newton polygons. These are given in Sections 2 and 3. Our assumptions make it possible to treat the operator $P$ as a kind of perturbation of an ordinary differential operator. In Sections 5 and 6, we investigate ordinary differential operators. Using the results in Section 6, we prove Theorem 4.4 in Section 7. Theorems 4.3 and 4.6 are proved by the use of scales of Banach spaces. In Section 8, we give the unique solvability of abstract equations in scales of Banach spaces. $u$

$f$

$g_{j}$

$t$

517

Characteristic Cauchy problems

Finally, we prove Theorems 4.3 and 4.6 in Section 9.

\S 2. Additive group of generalized Newton polygons. In this section, we define an additive group extended from the Newton polygons and show some basic properties of this group. We begin with a definition of Newton polygons.

DEFINITION 2.1. $[b, \infty)\subset R^{2}$

For a point

(1)

$(a, b)\in N\cross R$

, put

$\Delta(a, b):=(-\infty, a]\cross$

.

of A subset is called a Newton polygon, if for a finite number of points $(a_{j)}b_{j})\in N\cross R(]^{=1}, , M)$ . Note that is polygon. also a Newton (3) For a Newton polygon , put $s=s( \Delta):=\max\{u\in N;(u, v)\in\Delta$ for some $v\in R\}$ and $h=h( \Delta):=\min$ { $v\in R;(u,$ for some $u\in N$ }. We call the $L(j)=L_{\Delta}(j):= \min\{v\in R;(j, v)$ Further, put size of and the height of . , . This is called the side function of . A Newton for $j=0,1$ , polygon and side function is uniquely determined by its size . Note . and that that (2)

$R^{2}$

$\Delta$

$\Delta=ch(U_{j=1}^{M}\Delta(a_{j}, b_{j}))$

$\Delta(a, b)$

$\Delta$

$v)\in\Delta$

$\Delta$

$s(\Delta)$

$\Delta$

$h(\Delta)$

$\Delta$

$s(\Delta)$

$\in\Delta\}$

$\Delta$

$s(\Delta)$

$L_{\Delta}$

$h(\Delta)=L_{\Delta}(O)$

$\Delta=ch(\bigcup_{j=0}^{S(\Delta)}\Delta(j, L_{\Delta}(j)))$

Figure 2.

Newton polygon.

REMARK 2.2. Let be the side function of a Newton polygon $\kappa_{j}:=L(j)-L(j-1)(j=1, , s)$ . Then, there hold Put $L$

$s$

.

(2.1)

$0\leqq\kappa_{1}\leqq\ldots\leqq\kappa_{s}$

(2.2)

$L(])=h+ \sum_{t=1}^{j}\kappa_{l}C=0,1,$

Conversely, let

tion

$L$

on

$h\in R,$

$\{0,1, \cdot.

, s\}$

$\cdots$

, ),

with size

,

where

$s$

$\Delta$

$h=h(\Delta)=L(O)$

.

satisfy (2.1). If we define a func, let , by (2.2), then there exists a Newton polygon with

$s\in Nand$

$\kappa_{1}$

$\kappa_{s}$

$\Delta$

518

T. MANDAI

size

and side function

$s$

Next,

$L$

.

we define an addition of Newton polygons.

DEFINITION 2.3. For two subsets (2.3)

$D_{1}$

and

of

$D_{2}$

$R^{2}$

$D_{1}+D_{2}$ $:=\{(u_{1}+u_{2}, v_{1}+v_{2})\in R^{2} ; (u_{\iota}, v_{i})\in D_{t}(i=1,2)\}$

PROPOSITION 2.4. (Cf. [10, Definition 3.1].) For ) polygon with size and side function . Put the sum is a Newton Polygon with size where $L_{i}$

$s_{i}$

(2.4)

of

$\Delta_{\ell}$

. be a Newton . Then,

$j>s_{i}(i=1,2)$

function

$L_{1}\oplus L_{2}$

,

$:= \min\{L_{1}(k)+L_{2}(]-k) ; k\in N0\leqq k\leqq j\}$

for The unit element

for

let

and side

$s_{1}+s_{2}$

$(L_{1}\oplus L_{2})(j)$

$[0, \infty)$

$i=1,2$ ,

$=\infty$

$L_{i}C^{\cdot}$

$\Delta_{1}+\Delta_{2}$

$th2s$

$j\in N,$

addition

.

$0\leqq j\leqq s_{1}+s_{2}$

for Newton

polygons is

$\Delta(0,0)=(-\infty, 0]\cross$

.

PROOF. $\kappa_{1}^{(2)},$

, put

,

$\cdots$

$\kappa_{s_{2}}^{(2)}$

Put

as

(2.5)

$\kappa_{j}^{(l)}:=L_{i}(j)-L_{i}(j-1)(1\leqq j\leqq s_{\iota} ; i=1,2)$

$0\leqq\kappa_{1}\leqq\cdots\leqq\kappa_{s_{1}+s_{2}}$

. Reorder

$\kappa_{1}^{(1)},$

$\cdots$

,

$\kappa_{s_{1}}^{(1)}$

,

. It is easy to show that

$(L_{1} \oplus L_{2})(])=L_{1}(0)+L_{2}(0)+\sum_{l=1}^{j}\kappa_{l}$

$(0\leqq_{J}\leqq s_{1}+s_{2})$

.

with size a Newton polygon and side function 2.2. First, we show . Take an arbitrary , then . If . Assume that . Since there holds trivially that is convex, we may assume that $u\in N$. We have and since $v\geqq(L_{1}\oplus L_{2})(u)=L_{1}(k)+L_{2}(u-k)$ for some by (2.4). Put $v_{2}=v-v_{1}\geqq L_{2}(u-k)$ . , . Hence, we We have

Thus, there exists , by Remark

$\Delta$

$s_{1}+s_{2}$

$L_{1}\oplus L_{2}$

$(u, v)\in\Delta$

$\Delta\subset\Delta_{1}+\Delta_{2}$

$u\geqq 0$

$(u, v)\in\Delta_{1}+\Delta_{2}$

$u\leqq 0$

$\Delta=ch(\Delta\cap$

$\Delta_{1}+\Delta_{2}$

$(Z\cross R))$

$k$

$0\leqq u\leqq s_{1}+s_{2},$

. have Next, we show there holds trivially that $=(u_{1}+u_{2}, v_{1}+v_{2})$ , where

$v_{1}=$

$(u-k, v_{2})\in\Delta_{2}$

$(k, v_{1})\in\Delta_{1}$

$L_{1}(k),$

$(u, v)\in\Delta_{1}+\Delta_{2}$

$\Delta_{1}+\Delta_{2}\subset\Delta$

gon, there exist

. Take an arbitrary . Assume that

$(u, v)\in\Delta$

, then . If . We can write as $(u, v)$ . Since is a Newton poly$(u, v)\in\Delta_{1}+\Delta_{2}$

$u\leqq 0$

$u\geqq 0$

$\pi_{\iota}:=(u_{i}, v_{t})\in\Delta_{\iota}(i=1,2)$

$\Delta_{i}$

such that the point belongs to the compact and . The point belongs to the closed line segment connecting $\pi_{i}^{(\pm)}(i=1,2)$ . Since is convex, if these four parallelogram with the vertices , belong belongs then also to to . Thus, we have only to vertices show that $(u, v)\in\Delta$ under the assumption that $u_{x}\in N(i=1,2)$ . In this case, $\pi_{i}^{(\pm)}\in\Delta_{i}\cap(N\cross R)$

$\pi_{i}^{(+)}$

$\pi_{i}$

$\pi_{i}^{(-)}$

$\pi_{1}+\pi_{2}$

$\Delta$

$\Delta$

since

$0\leqq u_{i}\leqq s_{i}$

and

$\Delta$

$\pi_{1}+\pi_{2}$

$v_{i}\geqq L_{i}(u_{i})(i=1,2)$

$(L_{1}\oplus L_{2})(u_{1}+u_{2})=(L_{1}\oplus L_{2})(u)$

.

Thus,

,

we have .

$v=v_{1}+v_{2}\geqq L_{1}(u_{1})+L_{2}(u_{2})\geqq$

$(u, v)\in\Delta$

The following two propositions give basic meanings of this addition.

$\blacksquare$

Characteristic Cauchy problems

519

PROPOSITION 2.5. Let be a Newton polygon with size and side function . For a non-negative real number , let L. Put $k_{j}:=L(])-LC-1)$ for be a Newton polygon with size 1 and stde function $L[k]$ , where $L[k](0)=0$ and $L[k](1)=k$ . Then, there holds $\Delta=\Delta(0, h)+\Delta[k_{1}]+\cdots+\Delta[k_{s}]$ , where is $\Delta$

$s$

$k$

$1\leqq j\leqq s$

$\Delta[k]$

$h$

of .

the height

$\Delta$

for some $h\in R$ . Let PROOF. It is trivial that if $s=0$ then . be the Newton polygon with size $s-1$ Put $L’(j):=L(j)(0\leqq j\leqq s-1)$ and let $L’$ $(L’\oplus L[k_{s}])(O)=L’(0)=L(0)$ . . We have and side function For $1\leqq j\leqq s-1$ , we bave $(L’ \oplus L[k_{s}])(])=\min\{L’(]), L’(]-1)+k_{s}\}=\min\{L(j), L(j-1)+k_{s}\}=$ $L(j)$ , since $L(j)-L(j-l)$ $ $L(s)$ $L(s-- l)=k_{s}$ . We also have $(L’\oplus L[k_{s}])(s)=$ Thus, we obtain $L=L’\oplus L[k_{s}]$ , and hence $L’(s-1)+k_{s}=L(s-1)+k_{s}=L(s)$ . . By iterating this argument, we get the desired result. $\Delta=\Delta(0, h)$

$s\geqq 1$

$\Delta’$

–

$\Delta=\Delta’+\Delta[k_{s}]$

$\blacksquare$

PROPOSITION 2.6. (Cf. [7].) form (1.1), then there holds

If

and

$P_{1}$

$P_{2}$

$\Delta(P_{1}P_{2})=\Delta(P_{1})+\Delta(P_{2})$

are differential .

$oPerators$

of the

satisfying that if $(u, v)\in D$ and PROOF. Let $D$ be a subset of then $(u, v’)\in D$ . For a partial differential operator $R$ , we say $R=O(D)$ if can be written in the form: $R^{2}$

“

(2.6)

$\zeta i,$

$a$

)

$?)’\geqq v$

$a_{j.\alpha}(t, x)t^{j}\partial_{t}^{j}\partial_{x}^{a}$

$R= \sum_{j+|a_{I}\leqq m}t$

,

$R$

, ( $j+|\alpha|$

$(]+|\alpha|, v(j, \alpha))\in D,\overline{a}_{f,a}\in C^{\infty}([-T, T];O(\Omega))$

SS

$m$

).

It is easy to show that (2.7)

$(t^{v_{1}}a_{1}(t, x)t^{j_{1}}\partial_{t}^{j_{1}}\partial_{x^{1}}^{a})(t^{v_{2}}a_{2}(t, x)t^{j_{2}}\partial_{t}^{j_{2}}\partial_{x^{2}}^{\alpha})$

$=t^{v_{1}+v_{2}}a_{1}(t, x)a_{2}(t, x)t^{j_{1}+j_{2}}\partial_{t}^{J_{1}+J_{2}}\partial_{x}^{\alpha_{1}+\alpha_{2}}$

$+O(\Delta_{\backslash }’j_{1}+j_{2}+|\alpha_{1}+\alpha_{2}|, v_{1}+v_{2})\backslash \{(]_{1}+J_{2}+|\alpha_{1}+\alpha_{2}|, v_{1}+v_{2})\})$

$=O(\Delta(_{J_{1}^{\backslash }}+j_{2}+|\alpha_{1}+\alpha_{2}|, v_{1}+v_{2}))$

Thus, it is almost trivial that part, In order to show belong to .

$\Delta(P_{1}P_{2})\subset\Delta(P_{1})+\Delta(P_{2})$

$\supset$

LEMMA 2.7.

If

, then there exist unique is a vertex of is a such that $(u, v)=(u_{1}, v_{1})+(u_{2}, v_{2})$ . Further,

$\Delta(P_{i})(i=1,2)$

PROOF.

.

vertex of

$\Delta(P_{1})+\Delta(P_{2})$

$(u_{i}, v_{i})$

.

Assume that

$(u_{2}, v_{2})=(u_{1}’, v_{1}’)+(u_{2}’, v_{2}’)$

$v_{2}’=v_{2}-\sigma$

we have only to show that all the vertices of

$(u, v)$

$(u_{i}, v_{i})\in\Delta(P_{i})(i=1,2)$

of

.

$\Delta(P_{1}P_{2})$

$\Delta(P_{1})+\Delta(P_{2})$

vertex

.

.

$(u_{i}, v_{i}),$

$(u_{i}’, v_{i}’)\in\Delta(P_{\mathfrak{i}})(i=1,2)$

We can represent

$u_{1}’=u_{1}+\rho$

,

and

$u_{2}^{l}=u_{2}-\rho,$

Hence, there holds $(u\pm\rho, v\pm\sigma)\in\Delta(P_{1})+\Delta(P_{2})$ . , this implies $\rho=\sigma=0$ . Thus,

$\Delta(P_{1})+\Delta(P_{2})$

$(u, v)=(u_{1}, v_{1})+$

$(u_{t}, v_{i})$

$v_{1}’=v_{1}+\sigma$

,

Since $(u, v)$ is a are unique.

T. MANDAI

520

is not a vertex of . Since

.

There exist and such , we have $(u\pm\rho, v\pm\sigma)\in\Delta(P_{1})+$ that . . This contradicts the assumption that $(u, v)$ is a vertex of Thus, is a vertex of . way, is a vertex of . the we show tbat can same Just

Assume that

$(u_{1}, v_{1})$

$(u_{1}\pm\rho, v_{1}\pm\sigma)\in\Delta(P_{1})$

$\Delta(P_{1})$

$\rho\neq 0$

$\sigma$

$(u_{2}, v_{2})\in\Delta(P_{2})$

$\Delta(P_{1})+\Delta(P_{2})$

$\Delta(P_{2})$

$(u_{1}, v_{1})$

$\Delta(P_{1})$

$a(P_{2})$

$(u_{2}, v_{2})$

$\blacksquare$

NOW, we return to the proof of Proposition 2.6. Assume that $(u, v)$ is a in the lemma above. We decompose and take vertex of as the operator $\Delta(P_{1})+\Delta(P_{2})$

$(u_{i}, v_{i})$

$P_{\dot{t}}$

(2.8)

$P_{i}= \sum_{|j+\alpha|=u_{i}}t^{v_{i}}aj_{a}^{i)}(t, x)t^{j}\partial_{t}^{j}\partial_{x}^{a}+\sum_{j+t^{\alpha}I\neq u_{i}}a_{j.\alpha}^{(i)}(t, x)t^{j}\partial_{t}^{j}\partial_{x}^{a}=P_{i}’+P_{i}’’$

By (2.7),

we have

(2.9)

$P_{1}’P_{2}’= r^{v}\sum_{J+|\alpha|=u}(\sum_{J_{1}+1\alpha_{1}|=u_{1}}a4_{1^{\alpha_{1}}}^{1)}(t, x)aj_{-j_{1}.\alpha-\alpha_{1}}^{2)}(t, x))t^{j}\partial_{t}^{j}\partial_{x}^{\alpha}$

+0

$(\Delta(u, v)\backslash \{(u, v)\})$

On the other hand, from (2.10)

$=u_{i}$

.

Since and

.

$P_{i}’’\in O(\Delta(P_{i})\backslash \{(u_{i}, v_{i})\})$

we have

,

$P_{1}’P_{2}’’+P_{1}’’P_{2}’+P_{1}’’P_{2}’’\in O((\Delta(P_{1})+\Delta(P_{2}))\backslash \{(u, v)\})$

is a vertex of

$(u_{i}, v_{i})$

$\tilde{a}_{j_{i^{a}i}}^{(i)}(0, x)\not\cong O(i=1,2)$

(2.11)

.

$\Delta(P_{i})$

exists

, there

Hence,

$(j_{i}, \alpha_{i})$

such that

$j_{i}+|\alpha_{i}|$

we have

$\sum_{j+|\alpha|=u}(\sum_{j_{1}+|\alpha_{1}|=u_{1}}a_{j_{1},\alpha_{1}}^{(1)}(0, x)aj_{-j_{1}.\alpha-\alpha_{1}}^{2)}(0, x))\lambda^{j}\xi^{\alpha}$

$=(_{j_{1}+|\alpha_{1}} \sum_{|=u_{1}}aj_{1}^{1)}\alpha_{1}(0, X)\lambda^{j_{1}}\xi^{a_{1}})t_{2^{+|\alpha}2|\Rightarrow u_{2}}^{a_{j_{2}.\alpha_{2}}^{(2)}(0},$

$\not\equiv 0$

$x)\lambda^{j_{2}}\xi^{\alpha_{2}})$

.

This implies that there exists

$(J, \alpha)$

such that

$j+|\alpha|=u$

$\sum_{j_{1}+|\alpha_{1}|=u_{1}}a_{j_{1}.a_{1}}^{(1}(o, x)a_{f-j_{1}.\alpha-\alpha_{1}}^{(2)}(0, x)\not\equiv 0$

Thus, from

.

$P_{1}P_{2}=P_{1}’P_{2}’+P_{1}’P_{2}’’+P_{1}’’P_{2}’+P_{1}’’P_{2}’’$

,

we have

and

.

$(u, v)\in\Delta(P_{1}P_{2})$

.

$\blacksquare$

REMARK 2.8. For a complete locally convex topological vector space

$X$

,

put (2.12)

$\mathscr{F}([0, T];X):=\{v\in C^{0}([0, T];X);v(s^{q})\in C^{\infty}([O, T^{1/q}];X)$

for some

$q\in N\backslash \{0\}\}$

.

We can extend Definition 1.1 for the operators whose coefficients belong to . For such operators, the proposition above is also valid. $\mathscr{F}([0, T];\mathcal{O}(\Omega))$

Characteristic Cauchy problems

521

Let 92 be the set of all Newton polygons, which is a commutative semigroup with the addition above. By the following proposition, we can embed this additive semigroup su in an additive group.

PROPOSITION 2.9. Let be three Newton polygons. If then . (The converse is trivially valid.) EsPecially, if $\Delta,$

$\Delta_{1},$

$\Delta_{2}$

$\Delta_{1}=\Delta_{2}$

.

$\Delta_{1}+\Delta\subset\Delta_{2}+\Delta$

$a_{1}+\Delta=\Delta_{2}+\Delta$

$\Delta_{1}\subset\Delta_{2}$

(This

,

, then

is called the cancellation law.)

PROOF. By Proposition 2.5, we have only to give a proof in the case of , it is trivial. or . In the case of , where Assume that is a nonnegative real number. (resp. ) be the size (resp. side function) of $\Delta_{i}(i=1,2)$ . By Let , we have $s_{1}+1\leqq s_{2}+1$ , hence . Further, we have

$\Delta=\Delta(0, h)$

$\Delta(0, h)$

$\Delta=\Delta[k]$

$\Delta_{1}+\Delta[k]c\Delta_{2}+\Delta[k]$

$k$

$\Delta_{1}+\Delta[k]$

$L_{i}$

$s_{i}$

$\subset\Delta_{2}+\Delta[k]$

$s_{1}\leqq s_{2}$

(2.13)

$(L_{1}\oplus L[k])(])\geqq(L_{2}\oplus L[k])(j)$

$(0\leqq_{J}\leqq s_{1}+1)$

.

. Now, assume tbat From this inequality for $j=0$ , we have $0\leqq j\leqq d(0\leqq d\leqq s_{1}-1)$ . From (2.13) for $j=d+1$ , we have holds for $L_{1}(d+1) \geqq\min\{L_{2}(d+1), L_{2}(d)+k\}$ . If $L_{2}(d+1)\leqq L_{2}(d)+k$ , then we have $L_{1}(d+1)\geqq L_{2}(d+1)$ . If $L_{2}(d+1)>L_{2}(d)+k$ , then we obtain $L_{2}(d+2)>L_{2}(d+1)$ Hence, from (2.13) for $j=d+2$ , we have $L_{1}(d+1)+k\geqq L_{2}(d+1)$ $+k$ using (2.1). $+k$ . Thus, we have $L_{1}(d+1)\geqq L_{2}(d+1)$ . By the induction, we get $L_{1}(0)\geqq L_{2}(0)$

$L_{1}(])$

$\geqq L_{2}(j)$

$L_{1}(j)\geqq L_{2}(])$

for

$0\leqq j\leqq s_{1}$

.

$\blacksquare$

DEFINITION 2.10. (1) By the cancellation law, the additive semigroup can be embedded in an additive group St so that the addition in su is preserved is a difference of two elements of . and any in up to isomorphism. An element of This group Si is uniquely determined by , there holds St is called a generalized Newton Polygon. For . if and only if , the integer (2) For is called the is called the height of . It is and the real number size of almost trivial that these are well-defined. (3) For two elements , if in S72, we say $\Delta_{i}’\in\Re(i=1,2)$ . , where It follows from Proposition 2.9 and that this is a well-defined order relation in Si. $\Re$

$\overline{\Delta}\in\pi$

$5i$

$\Re:\overline{\Delta}=\Delta-\Delta’(\Delta, \Delta’\in\Re)$

$yj$

$\Delta_{1},$

$\Delta_{1}-\Delta_{1}’=\Delta_{2}-\Delta_{2}’$

$\Delta_{2},$

$\Delta_{2}’\in\Re$

$\Delta_{1}+\Delta_{2}’=\Delta_{2}+\Delta_{1}’$

$\overline{\Delta}=\Delta-\Delta’\in\pi(\Delta, \Delta’\in\Re)$

$s(\overline{\Delta})=s(\Delta)-s(\Delta’)$

$h(\overline{\Delta})=h(\Delta)-h(\Delta’)$

$\overline{\Delta}$

$a_{1},\overline{\Delta}_{2}$

$\Delta_{1}+\Delta_{2}’\subset\Delta_{2}+\Delta_{1}’$

$\Delta_{1}’,$

$\overline{\Delta}$

$\overline{\Delta}_{1}\subset\overline{\Delta}_{2}$

$\overline{\Delta}_{i}=\Delta_{i}-\Delta_{i}’(i=1,2)$

$\Delta_{i},$

\S 3. Function spaces associated with generalized Newton polygons. In this section, we define some function spaces associated with generalized Newton polygons and study some basic properties of these spaces. Let $T$ be a positive number and $X$ be a complete locally convex topological vector space. All the spaces in this section are considered as subspaces of $9’((0, T);X)$ .

522

T. MANDAI

Thus, for example, $u\in C^{0}([O, continuously extendable onto $[0,

DEFINITION 3.1. (1) Let function . Put

$\Delta$

T] ; X)$

T]$

means that

$u\in C^{0}((0, T)$

;

$X$

)

and

is

$u$

.

be a Newton polygon with size

$s$

and side

$L$

(3.1)

$F_{T}[\Delta:X]$ $:=\{u\in C^{S}((0, T] ; X) ; t^{L(j)}\theta^{f}(u)\in C^{0}([0, T] ; X)(0\leqq j\leqq s)\}$

We can endow this space with a natural topology. Especially, if space, then this space is also a Banach space with the norm (3.2)

$||u||_{\tau.\Delta.X}:= \sum_{J=0}^{s}\sup_{0\leqq t\leq T}||t^{L(j)}\theta^{j}(u)(t)||_{X}$

Let

(2)

$\Delta’$

be another Newton polygon with size

.

is a Banach

$X$

. and side function

$s’$

$L’$

.

Put (3.3)

$F_{T}[ \Delta, \Delta’ :

X]:=\{u\in 9’((0, T);X);u=\sum_{j=0}^{\iota’}t^{L’(f)}\theta^{j}(v_{j})$

for some Obviously, we have

we define a norm $||u||_{\tau.\Delta.\Delta’.X}$

(When

$F_{T}[\Delta, \Delta(0,0) :

$||\cdot||_{\tau.\Delta.\Delta’.X}$

$(0\leqq]:

$v_{j}\in F_{T}[\Delta:X]$

X]=F_{T}[\Delta:X]$

. When

$X$

s’)\}$

.

is a Banach space,

by

$:= \inf\{\sum_{j=0}^{s’}||v_{j}||_{T.\Delta.X}$

;

$v_{j}\in F_{T}[\Delta:X](0\leqq];s’),$

$u= \sum_{j=0}^{s’}t^{L(j)}\theta^{j}(v_{j})\}$

.

is not a Banach space, we do not consider any topology for sim-

$X$

plicity, though we can.)

LEMMA 3.2. Assume that (1) There holds that

$\Delta_{1},$

$\Delta_{1}’,$

$\Delta_{2},$

$F_{T}[\Delta_{1}, \Delta_{1}’ :

(3.4)

$F_{T}[\overline{\Delta}:X]$

(2)

the

If

space

$X$

$\Delta_{2}’\in\Re$

and

X]=F_{T}[\Delta_{2}, \Delta_{2}’ :

$:=F_{T}[\Delta_{1}, \Delta_{1}’ :

. we can define

$\overline{\Delta}:=\Delta_{1}-\Delta_{1}’=\Delta_{2}-\Delta_{2}’$

X]$

X]$

.

Thus,

.

and is a Banach space, then the norms space Further, the other. to equivalent each are $||u||_{\tau.\Delta_{1}.\Delta_{1}’.X}$

$||u||_{\tau.\Delta_{2},\Delta_{2}’,X}$

$F_{T}[\overline{\Delta} :

$F_{T}[\overline{\Delta}:X]$

X]$

in is

a Banach space with these equivalent norms. TO prove this lemma, we use a result on ordinary differential operators. Hence, the proof is delayed until Section 5.

EXAMPLE 3.3. (1) There holds

$C^{0}([0, T];X)=F_{T}[\Delta(0,0):X]$

.

Moreover,

if we put (3.5)

$C_{flat}^{N}([0, T];X):=\{u\in C^{N}([0, T];X);\partial_{t}^{j}u|_{t=0}=0(0\leqq j\leqq N-1)\}$

then there holds (3.6)

$C_{flat}^{N}([0, T] ; X)=F_{T}[\Delta(N, -N) : X]$

.

,

Characteristic Cauchy problems

523

The space $F_{T}[\Delta(l, 0):X]$ is equal to the space $C_{\iota}^{0}([0, T], X)$ in [1]. The space $F_{T}[-\Delta(1,0):X]$ is equal to the space $C_{-l}^{0}([0, T], X)$ in [2]. (2)

REMARK 3.4. We can also define some function spaces similar to spaces instead of $C^{0}([0, based on other various spaces such as fact, J. Elschner ([3]) used weighted Sobolev spaces similar to , without referring to Newton polygons. $L_{p}$

$F_{T}[\overline{\Delta}:X]$

T];X)$

,

. In

$F_{T}[\Delta:C]$

$X_{p}^{\rho}$

$(\Delta\in\Re)$

We give some properties of the spaces $F_{T}[\Delta:X]$ and two lemmas are almost trivial and those proofs are omitted.

$F_{T}[\overline{\Delta}:X]$

be a Newton Polygon with size and side function T);X)$ , the following nine conditions are equivalent to each other.

LEMMA 3.5. For

$u\in 9’((0,

Let

$\Delta$

(2)

$t^{L(j)}\theta^{\tilde{j}}(u)\in C^{0}([0, T] ; X)(0\leqq j\leqq s)$

(3)

$t^{L(j)+j}\partial\{(u)\in C^{0}([O, T] ; X)(0\leqq j\leqq s)$

(4)

$S^{j}(t^{L(j)}u)\in C^{0}([0, T];X)(0\leqq j\leqq s)$

(5)

$\tilde{\theta}^{j}(t^{L(j)}u)\in C^{0}([0, T] ; X)$

(6)

$\partial_{t}^{j}(t^{L(j)+j}u)\in C^{0}([0, T] ; X)(0\leqq j\leqq s)$

. .

.

( $0\leqq]$ ;Ill ). $s$

For any posrtive integer , $+\rho_{r}\geqq L(j_{1}+\cdots+]_{r})$ , then (8) For any Posrtive integer , $+\rho_{r}\geqq L(j_{1}+\cdots+j_{r})$ , then (9) For any Positive integer , (7)

$r$

if

.

$j_{i}\geqq 0(0\leqq i\leqq r),$

$r$

if

$j_{i}\geqq 0(0\leqq i\leqq r),$

and

$\rho_{1}+$

$+]_{r}\leqq s$

and

$\rho_{1}+\cdot$

$+\rho_{r}\geqq L(]_{1}+\cdots+j_{r})+_{J_{1}}+\cdots+_{j_{r}}$

$F_{T}[\Delta, \Delta’ :

LEMMA 3.7. Especially, if

If

,

if

then

X](\Delta, \Delta’\in\Re)$

$\overline{\Delta}_{1},\overline{\Delta}_{2}\in ffi$

and

$j_{t}\geqq 0(0\leqq i\leqq r),$

$j_{1}+\cdots+_{J_{r}}\leqq s$

and

$\rho_{1}+\cdots$

,

then

..

.

$t^{\rho_{1}}\partial_{t}^{j_{1}}\cdots t^{\rho_{r}}\partial_{t}^{J_{r}}(u)\in C^{0}([0, T];X)$

.

we have a similar lemma, which we omit

$\overline{\Delta}_{1}\subset\overline{\Delta}_{g}$

, then

$\varphi\in \mathscr{F}([0, T];X(X, Y))$

$h(\overline{\Delta})\geqq 0$

$\cdot$

.

$j_{1}+$

$t^{\rho_{1}}\theta^{\tilde{i}_{1}}\cdots t^{\rho_{r}}\theta^{\tilde{J}_{r}}(u)\in C^{0}([0, T];X)$

$r$

..

$+]_{r}\leqq s$

$j_{1}+$

$t^{\rho_{1}}\theta^{j_{1}}\cdots t^{\rho_{r}}S^{j_{\gamma}}(u)\in C^{0}([0, T];X)$

If

.

.

$u\in F_{T}[\Delta:X]$

LEMMA 3.6.

$L$

$s$

(1)

Also for to state.

. The first

and

$F_{T}[\overline{\Delta}_{1} :

X]\supset F_{T}[\overline{\Delta}_{2} :

$u\in F_{T}[\overline{\Delta}:X]$

$\mathscr{F}([0, T];X)\subset F_{T}[\overline{\Delta}:X]$

, then

X]$

.

$\varphi u\in F_{T}[\overline{\Delta}:Y]$

.

.

for any . Hence, if PROOF. First, note that , where , then the lemma is almost trivial. Let . $L’$ function) ) be the size (resp. side (resp. of . Since Let , we have for some $v_{j}\in F_{T}[\Delta:X]$ . We prove by the induction on . that $s’=0$ , then If for some $h\in R$ . Hence, $\Delta-\Delta’=\{(u, v-h);(u, v)$ and $T[\Delta, \Delta’ : X]=F_{T}[\Delta-\Delta’ : X]$ . Thus, it is already proved. . Then, Assume that we have proved for $s’-1$ . Put by the induction hypothesis, we have . Now, we have $\theta^{j}(\varphi)\in \mathscr{F}([0, T] ; X(X, Y))$

$\overline{\Delta}=\Delta-\Delta’$

$\overline{\Delta}=\Delta\in\Re$

$\Delta’$

$s’$

$F_{T}[\Delta, \Delta’ :

$j$

$\Delta,$

$u\in F_{T}[\overline{\Delta}:X]=$

$u=\Sigma_{j’=0}^{s}t^{L(j)}\theta^{j}(v_{j})$

X]$

$\varphi u\in F_{T}[\Delta, \Delta’ :

Y]$

$s’$

$\Delta’=\Delta(0, h)$

$\in\Delta\}\in\Re$

$u’=\Sigma_{f=0}^{S’-1}t^{L’(j)}\theta^{j}(v_{f})$

$\varphi u’\in F_{T}[\Delta, \Delta’ :

Y]$

$\Delta’\in\Re$

524

T. MANDAI

(3.7)

$t^{L’(S’)} \theta^{s^{r}}(\varphi v_{s’})=t^{L’(S’)}\sum_{\iota=0}^{s’}(\begin{array}{l}s’l\end{array})\theta^{S’-l}(\varphi)\theta^{l}(v_{s’})$

.

$= \varphi t^{L’(S’)}\theta^{S’}(v_{S’})+\sum_{l=0}^{s’-1}(\begin{array}{l}s’l\end{array})\tau$

, we have Since by the induction hypothesis, we have $s’-1)$ . Hence, by (3.7), we have $\varphi v_{s’}\in F_{T}[\Delta:Y]$

$t^{L’(S’)}\theta^{S’}(\varphi v_{S’})\in F_{T}[\Delta, \Delta’ :

Y]$

. On the other

$"’-\iota(\varphi)t^{L’(s’)}\theta^{l}(v_{s’})\in F_{T}[\Delta, \Delta’ :

$\varphi t^{L’(S’)}\theta^{\theta’}(v_{s’})\in F_{T}[\Delta, \Delta’ :

$\varphi u\in F_{T}[\Delta, \Delta’ :

Y]$

LEMMA 3.8. $a\geqq-h(\overline{\Delta})$

Y]$

.

Thus,

hand,

Y](0\leqq l\leqq$

we have

.

$\blacksquare$

For JESt and

$a\in R$

, there holds

$t^{a}\in F_{T}[\overline{\Delta} :

X]$

if and

only

if

.

The if part is easy. We prove the only-if part. Suppose . Put $h=h(\Delta),$ $h’=h(\Delta’)$ , and $s’=s(\Delta’)$ . We may assume that $h’-h-1$ without loss of generality. Since , we and $F_{T}[\Delta, \Delta’ : X]\subset F_{T}[\Delta(0, h), \Delta(s’, h’):X]$ Hence, . we can write $t^{a}=$ have , we get , where $u_{j}\in C^{0}([0, T] ; X)$ . Using , where $v_{j}\in C^{0}([O, T];X)$ . For any $v\in C^{0}([0, T] ; X)$ , there . Hence, we get the equation exists $w\in C^{0}([0, T];X)$ such that for some $v\in C^{0}([0, T];X)$ . Solving this equation, we have $v=$ $(a-h’+h+1)^{-S}$ for some constants $a-h’+h\geqq 0$ , that is $v\in C^{0}([0, T] ; X)$ , we have . Since

PROOF.

$F_{T}[\Delta, \Delta’ :

$t^{a}\in$

X]$

$a\neq$

$\Delta\supset\Delta(0, h)$

$\Delta’\subset\Delta(s’, h’)$

$\theta^{j}\cdot t^{b}=t^{b}(\tilde{\theta}-1+b)^{j}$

$t^{h’}\Sigma_{j=0}^{S’}\theta^{j}(t^{-h}u_{j})$

$t^{a-h’+h}=\Sigma_{j=0}^{S’}\tilde{\theta}^{j}(v_{j})$

$\tilde{\theta}(w)=v$

$t^{a-\hslash’+h}$

$=\tilde{\theta}^{S’}(v)$

‘

$t^{a-h’+h}+\Sigma_{j=0}^{S’-1}C_{j}t^{-1}(\log t)^{j}$

$C_{j}(0\leqq_{J}\leqq s’-1)-\cdot$

$a\geqq-h(\overline{\Delta})$

Last,

we define another function space.

DEFINITION 3.9.

We put $\mathcal{G}_{\Lambda.T}(X)$

$:= \bigcup_{\Lambda a>,N\in N}F_{T}[-\Delta(N, a) :

X]$

.

REMARK 3.10. For and $a\in R$ , there holds the following: (1) , if and only if (2) if and only if . $\overline{\Delta}\in ffi$

$t^{a}\in \mathcal{G}_{\Lambda.T}(X)$

$F_{T}[\overline{\Delta}:X]\subset \mathcal{G}_{\Lambda.T}(X)$

$a>\Lambda$

$h(\overline{\Delta})0$ $u=0$ neighborhood in a then of $(0,0)$ . for $u\in t^{L}\cross C^{0}([0, T] ; 9’(\Omega\cap R^{n}))$

$t>0$

.

Let $L>\Lambda(P)$ . If in a neighborhood of $(0,0)$ ,

$(A-1)-(A-3)$

NOW, we give some extensions of these propositions in the form of the following three theorems. By the reduction argument made above, Theorems

1.5 and 1.6 follow from these three theorems. In the following, we assume that the coefficients of belong to . (See Remark 2.8.) We also assume the conditions (A-1), (A-2), and . The first theorem is the unique solvability in the class $P$

$\mathscr{F}([0, T]$

$o(\Omega))$

$\mathcal{G}_{\Lambda.T}(O(\Omega))$

;

(A-3).

526

T. MANDAI

THEOREM 4.3. There exist a positive and a neighborhood which the following holds: , there exists a unique For any $T_{0}$

for

$f\in \mathcal{G}_{\Lambda(P)+h(P).T}(\mathcal{O}(\Omega))$

ing $Pu=f$ on

$(0, T_{0})\cross\Omega_{0}$

of

$\Omega_{0}$

$0$

in

$C^{n}$

satisfy-

$u\in \mathcal{G}_{\Lambda(P),T_{0}}(O(\Omega_{0}))$

.

The second theorem is the regularity of solutions with respect to . $t$

THEOREM 4.4. There exists a neighborhood of in such that, for any $T’\in(O_{y}T$ ] and any subdomain of , the follouing holds: $h(\overline{\Delta})\Lambda(P)+h(P)$ ,

$N\in N,$

,

$C^{n}$

. and

$Pu\in C_{flat}^{N}([0, T’];\mathcal{O}(\Omega’))$

, then

where $N’= \min\{N, [N-h(P)]\}$ .

Note that the condition

$F_{T’}[\overline{\Delta}+\Delta(P):\mathcal{O}(\Omega’)]\subset \mathcal{G}_{\Lambda(P),T’}(\mathcal{O}(\Omega’))$

$h(\overline{\Delta})0$ in a neighbora neighborhood of $(0,0)$ .

$u\in \mathcal{G}_{\Lambda(P).T}(9’(\Omega\cap R^{n}))$

$u=0$

for

$t>0\iota n$

We now prove Theorem

Theorem 1.7 easily follows from Theorem 4.6. 1.8 from Theorem 4.6.

. Also assume that $u\in$ THEOREM 1.8. Assume that $9’((-T, T)\cross(\Omega\cap R^{n}))$ in a neighborhood of $(0,0)$ and $suppu\subset$ satisfies . Since is a distribution of finite order in a smaller neighborhood of $(0,0)$ , we may assume that , we have . Since $\Lambda(P)=-\infty$ $(0, T_{1})xU_{1}$ $u=0$ implies in for some and hence Theorem 4.6 that $T_{1}>0$ and an open neighborhood . Thus, $suppu\cap((-T_{1}, T_{1})\cross U_{1})$ of PROOF

OF

$O\not\in S$

$P_{\mathcal{U}}=0$

$\{t\geqq 0\}$

$u$

$u_{It>0}\in \mathcal{G}_{-\infty.T}(9’(\Omega\cap R^{n}))$

$U_{1}$

$\subset\{t=0\}$

$O\not\in S$

$0\in R^{n}$

.

assume

in addition. Since $suppu\cap((-T_{1}, T_{1})\cross U_{1})\subset\{t=0\}$ , can be written as the distribution for some $P$ $M\in N$ and $a_{j}\in 9’(U_{1})(j=0,1, \cdots , M)$ . From (4.1), the operator can be $P=C_{0}(S;x)+tq(t, x;S, \partial_{x})$ , the indicial polynomial . Since written as $C_{0}(\lambda;x)=\alpha(x)$ is independent of and does not vanish in a neighborhood of $x=0$ by the assumption (A-2). Thus, there holds NOW,

$h(P)=0$

$u=\Sigma_{\dot{j}=0}^{1f}a_{j}(x)\delta^{(j)}(t)$

$u|_{(-T_{1},T_{1})\cross U_{1}}$

$O\not\in S$

$\lambda$

(4.4)

on

$\{\alpha(x)+tq(t, x;9, \partial_{x})\}\{\sum_{j=0}^{M}a_{j}(x)\delta^{(j)}(t)\}=0$

By this equation and tbe facts that (4.5)

$\theta^{l}(\delta^{\langle f)}(t))=(-]-1)^{l}\delta^{(j)}(t)$

,

$(-T_{1}, T_{1})\cross U_{1}$

.

Characteristic Cauchy problems

527

$(-])(-]+1\cdot\cdot-]$

(4.6)

$(_{J}\geqq l)$

,

$t^{f}\delta^{(j)}(t)=\{$ $0$

we can easily show that

$(_{j}0$

then

$\lambda_{j}(0)\neq 0$

.

The numbers , .. , Let be the side function of Then, there holds (2)

$h,$

$k_{1}$

$\cdot$

$\Gamma$

$h=h(9)_{y}$

Especially,

of

$S,$

$S=\{k_{1}, \cdots , k_{m}\}$

$C_{\mu}(\lambda)=0$

are

$\lambda_{j}(0)$

$k_{m},$

$\Delta(9)$

.

,

$\lambda_{m}(0)$

$\cdot$

.

Further, such that

, then the roots of Plicity. If repetitions according to multiplicity.

if

are determined as follows: . and as

$k’ s$

$k_{j}=\Gamma(j)-\Gamma(j-1)$

for 7‘

$O\in S$

.. , Renumber

$\lambda_{1}(0)$

$\lambda’ s$

$k_{1}\leqq k_{2}\leqq\ldots\leqq k_{m}$

$(1Sj\leqq m)$

.

and $\mu>0$ , then the nonzero roots , with repetitions according to multiare for such that $k_{j}=0$ , with

$\mu\in S$

$k_{f}=\mu$

$C_{0}(\lambda)=0$

$\lambda_{j}(0)$

$j$

By this proposition, the condition (A-3) means that “if $k_{j}>0$ , then , we have “if $k_{j}=0$ , then in (5.2). Further, by the definition of ${\rm Re}\lambda_{j}(0)$

$0,$ . For any $T’\in(0, T]$ and any complete locally convex topological vector space $X$ , there holds the following: (i) If satisfies that $Ru=0$ on $(0, T’)$ , then $u=0$ on $(0, T’)$ . $a\in R$ (ii) For any and $f\in t^{a}\cross C^{0}([O, T’] ; X)$ , there exists a unique $C^{0}([0, T’];X)$ satisfying $Ru=f$ on $(0, T’)$ . Further, for any seminorm of . $X$ , there holds the estimate

LEMMA 5.4.

$\lambda\in \mathscr{F}([0, T];C)$

${\rm Re}\lambda(0)0$

,

this is well-defined and

$S[f]\in C^{0}((0, T’]$

;

$X$

).

Further, it is

529

Characteristic Cauchy problems

easy to see that $RS[f](t)=f(t)$ on use the following estimate:

SUBLEMMA 5.5. For any following estimate holds: (5.5)

$(0, T’)$

$p\in R$ ,

. Put

there exists a constant

$e^{\Phi(t)} \int_{0}^{t}e^{-\Phi_{(S)}}s^{p-1}ds\leqq C_{p}t^{p+k}$

POOF. Since (5.6)

$\Phi’(t)=\frac{{\rm Re}\tilde{\lambda}(t)}{t^{k+1}}$

where

$\Phi(t)=Re\Lambda(t)/t^{k}$

$(t\in(O, T])$

$\overline{\lambda}(t)=\lambda(t)-bt^{k}$

,

.

We want to

$C_{p}$

for which

the

.

we have

$e^{\Phi_{(t)}} \int_{0}^{t}e^{-\Phi_{(S)}}s^{p-1}ds=e^{\Phi_{(t)}}\int_{0}^{t}e^{-\Phi_{(S)}}(-\Phi’(s))\frac{-1}{{\rm Re}\tilde{\lambda}(s)}s^{p+k}ds$

$=e^{\Phi(t)}([e^{-\Phi_{(S)}} \frac{-1}{{\rm Re}\tilde{J}(s)}s^{p+k}]_{0}^{t}+\int_{0}^{t}e^{-\Phi(s)}(\frac{1}{{\rm Re}\tilde{\lambda}(s)}s^{p+k})’ds)$

$\leqq\frac{t^{p+}}{|{\rm Re}}+Ce^{\Phi_{(t)}}\tilde{\lambda}\overline{(t)|}k\int_{0}^{t}e^{-\Phi_{(S)}}s^{p+k-1}ds$

in a neighborhood of

$t=0$

for some constant

$e^{\Phi(t)}e^{-\Phi(S)}\leqq C’$

for some constant

$C’$

,

$C$

. Because

$(0