数理解析研究所講究録 1208 巻 2001 年 52-68

52
ON EXISTENCE OF SCATTERING
SOLUTIONS FOR DISSIPATIVE SYSTEMSMITSUTERU KADOWAKI

(門脇

光輝)

In this report we shall give two frameworks (Theorem 1and 3) for the existenceof scattering solutions of dissipative systems and apply these to some dissipative
wave equations.
. This no rl1l is
Let $H$ be aseparable Hilbert space with inner productandbe acontraction semi- group. Let
denoted by
and aunitary group in ??, respectively. We denote these generators by andandumptions oll( $V(t)=e^{-itA}$ and $U_{0}(t)=e^{-itA_{0}}$ ). We make the following$\langle\cdot, \cdot\rangle_{\mathcal{H}}$$||\cdot$$||_{\mathcal{H}}$$\{U_{0}(t)\}_{t\in \mathrm{R}}$$\{V(t)\}_{t\geqq 0}$

$A$

$A$

$\mathrm{a}\mathrm{s}\mathrm{s}$$A_{0}$$A_{0}$

.

.
or
(A1)
(A2) $(A-i)^{-1}-(A_{0}-i)^{-1}$ defined as aform is extended to acompact operator$K$ in.
andsuch that $P_{+}+I_{-}’=Id$(A3) There exist non-zero projection operators$\sigma(A_{0})=\sigma_{ac}(A_{0})=\mathrm{R}$$[0, \infty)$$\mathcal{H}$$P_{-}$$P_{+}$

and,(A3.2)$||KU_{0}(t)\psi(A_{0})P_{+}||\in L^{1}(\mathrm{R}_{+})$(A3.2)$||K^{*}U_{0}(t)\psi(A_{0})P_{+}||\in L^{1}(\mathrm{R}_{+})$(A3.3)$||K^{*}U_{0}(-t)\psi(A_{0})P_{-}||\in L^{1}(\mathrm{R}_{+})$(A3.4),,$\mathrm{w}-\lim_{tarrow+\infty}U_{0}(-t)\psi(A_{0})P_{-}f_{t}.=0$

satisfying
andfor each
operator norm of bounded operator from $H$ to .
be the space generated by the eignvectors ofLet$\psi$$\in C_{0}^{\infty}(\mathrm{R}\backslash 0)$$\{f_{t}\}_{t\in \mathrm{R}}$$\mathrm{s}\mathrm{t}$,$\mathrm{u}\mathrm{p}_{t\in \mathrm{R}}||f_{t}.||_{H}<\infty$, where$||\cdot||$

is tllc$\mathcal{H}$$H_{b}$

Theorem 1. Assume that $(Al)\sim(A\mathit{3})$ . For any$A$with real eigenvalues.

$f\in H_{b}^{[perp]}$$Wf= \lim_{tarrow\propto}U_{0}(-t)V(t)f$, the wave opearior.

eists. Moreover $W$ is not zero as an operator in $H$ .
To prove Theorem 1we shall use the following facts (see [17] and [14]):.
(F1) $\{(A-i)^{-2}Af\cdot\in \mathcal{H} : f\in D(A)\cap H_{b}^{1}\}$ is dense insuch that(F2) There exists asequence$H_{b}^{[perp]}$$\{t_{n}\}$$\lim_{narrow\infty}t_{n}=\infty$

andTypeset by$A\mathcal{M}S-\Psi X$

53
MITSUTERU KADOWAKI$\mathrm{w}-\lim_{narrow\infty}V(tn)/=0$

for any,$f\in H_{b}^{[perp]}$.

Theorem 1implies that there exists scattering states of $\frac{dV(t)f}{dt}=-iAV(t)f.$, $f\in$$D(A)$ as follows:
Corollary 2. Assume that $(Al)\sim(A\mathit{3})$ . Then there exist non-trivial initial data
. $\in H$
satisfying $\Re\zeta 0>0$
, and
and $f_{+}.\in H$ such that for any $k=0,1,2$ ,$\zeta 0\in \mathrm{C}$$\cdots$$f$.

$\lim_{tarrow\infty}||V(t)(A-\zeta_{0})^{-k}f\cdot-U_{0}(t)(A_{0}-\zeta_{0})^{-k}f_{+}||_{\mathcal{H}}=0$

Theorem 1is proven by using Enss’s approach [3] and [17]. Examples of Theorem
1contain scattering problem for elastic wave equation with dissipative boundarly(cf. [2]). To show (A3) we use the Mellin transformacondition ill ahalf space oftion (cf.[13]). Theorem 1is not applied to acoustic wave equations with dissipative
terms ill stratifed media(cf. [19]). Since generalized eigenfunctions of acoustic wave
in stratifed media are not smooth at thresholds, the key estimateshave not been obtained in the neighborhood of each threshold. So
wc consider tllc following assu mptions to deal with such equations .
bc non-negative operator.
Lct
.
is
(A4)
can [ extended to abouneded.
Lef belong to$>0$ , there exist positive
satisfies that for any
operatorand such that$\mathrm{R}^{3}$$\mathrm{P}^{1\mathrm{O}}.\mathrm{p}\mathrm{a}\mathrm{g}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}11$$(\mathrm{A}3.1)\sim(\mathrm{A}3.3)$$B_{0}$$B_{0}$$(\mathrm{A}_{\iota}\ulcorner))$$A_{0}- \mathrm{c}()1\mathrm{n}\mathrm{p}_{\dot{\mathrm{c}}}\iota \mathrm{c}\mathrm{t}$$\mathrm{C}\backslash \mathrm{R}$$\langle$$Q(\zeta)$$\mathrm{c}()1\mathrm{l}\mathrm{s}\mathrm{t}(11\mathrm{l}\mathrm{t}\mathrm{s}$$\sqrt{B}\mathrm{o}(A_{0}-\zeta)^{-1}\sqrt{B}0$$C_{\alpha,\beta}$$)\mathrm{e}$$\beta>\alpha$$\mathrm{w}1_{1}\mathrm{i}\mathrm{c}1_{1}$

$7l$

Sllp

$||Q(\zeta)||\leqq C_{\alpha,\beta}$.

$\alpha\leqq|Re\zeta|\leqq\beta,0<|Im\zeta|<?l$

We reset $A=A_{0}-iB_{0}$ , $D(A)=D(A_{0})$ . Then [15] (see Theorem X-50) implies
that generates acontraction sclni-gr llp , $\{V(t)\}_{t\geqq 0}(V(t)=e^{-itA})$ .
Wc have the following theoreln.
$A$

$()$

Theorem 3. Assume that (Al), (A4) and (A5). Then
) alucs.
(1) A lias no real
(2) The wave operator$e^{J}ige7bl$$W=s$ $-1\mathrm{i}111U_{0}(-t)V(t)tarrow\propto$

exists. $Morco\iota$) . $W$ is not zero as an operator in $H$ .
$cr$Corollary 4. Assume that (Al), (A4) and (A5). Then we have the same conclu.
of.
$si_{\mathit{0}7l}$$C_{/}orollar\uparrow_{/}\mathit{2}$’Fo prove$\ulcorner\Gamma 1\mathrm{l}\mathrm{C}()\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{l}$$3$wc shall used)$\mathrm{M}\mathrm{e}$$\mathrm{c}1_{1}\mathrm{i}_{\mathrm{A}11}^{\mathrm{r}}\mathrm{k}\mathrm{i}’ \mathrm{s}$idea [12] due to Kato’s smooth.
pert urbation fllc
to elastic wave equation with dissipative
In \S 4 we shall apply ourand acoustic wave equation with dissipaboundary condition in ahalf space of
there is little literature concerning suchthattive term in stratifed media. It seemdissipative systems (cf. [2]).
$()\mathrm{r}\mathrm{y}[8]$$\mathrm{f}\mathrm{l}\cdot c1\mathrm{l}\mathrm{r}\mathrm{l}\mathrm{c}\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k}\mathrm{s}$$\mathrm{R}^{3}$$\mathrm{s}$54
ON EXISTENCE OF SCATTERING SOLUTIONS

2. Proof of Theorem 1and Corollary 2.
In this section wc deal thc case $\sigma(A_{0})=\sigma_{ac}(A_{0})=\mathrm{R}$ only.
another casecan be dealt in the same way. We set
and $W(t)=U_{0}(-t)V(t)$ .
.
In this section $C$ is used as positive
Below wc shall give the proof of Theorem 1. First we prove tllc existence of $W$
by refering to [3], [17], [10], [13], [4], [18] and [14]. But
omit to notethe above references.
$\ulcorner 1^{\urcorner}11\mathrm{C}$$F(\lambda)=(\lambda-i,)^{-\mathit{2}}\lambda$$\mathrm{c}\mathrm{o}11\mathrm{S}\mathrm{t}_{1(}.\iota 11\mathrm{t}\mathrm{s}$$\mathrm{w}\mathrm{c}\iota \mathrm{b}^{1}()1\mathrm{n}\mathrm{c}\mathrm{t},\mathrm{i}111\mathrm{C}\mathrm{S}$proof of the existence of $W$ . For any.

$f$$\in H_{b}^{[perp]}\cap D(A)$and , $.9>t_{n}$ , note (F1) and$t$$||(W(t)-W(s))F(A)^{2}f\cdot||_{\mathcal{H}}$$\leqq||(W(t)-W(t_{n}))F(A)^{2}f\cdot||_{H}+||(W(.\mathrm{s}.)-W(t_{n}))F(A)^{2}f\cdot||_{H}$Thus in order to prove(2.1)$\mathrm{t},\mathrm{h}\mathrm{e}$.

existence of $W$ , it is sufficent to Below$narrow\infty tarrow\infty 1\mathrm{i}111\overline{1\mathrm{i}1\mathrm{I}1}||(W(t)-W(t_{n},))F(A)^{2}f\cdot||_{\mathcal{H}}=0$(cf. [4])We estimate. follows (cf. [17]):$||(W(t)-W(t_{n}))F(A)^{2}f\cdot||_{H}\dot{‘}\mathrm{k}[searrow]$$||(W(t)-W(t_{n}))F(A)^{2}f||_{H}$$=||U_{0}(-t)(V(t-t_{n},)-U_{0}(t, -t_{n}))F(A)^{2}V(t_{n})f||_{\mathcal{H}}$$\leqq\sum_{j=1}^{5}||7_{j}^{1}||_{H}$

,

where
$T_{1}=$$(V(t, -t_{n})-U_{0}(t.

-t_{n}))(F(A)^{2}-F(A_{0})^{2})V(t_{n})f$,,$7_{2}^{\tau}=(V(t-t_{n})-U_{0}(t-t_{n}))(Id-\psi_{M}(A_{0}))F(A_{0})^{2}V(t_{n})f$,$?_{3}^{1}=(V(t-t_{n})-U_{0}(t-t_{n}))(\psi_{M}\Gamma.)(A_{0})P_{+}.F(A_{0})V(t_{n})f$$7_{4}^{1}=(V(t-t_{n})-U_{0}(t-t_{n}))(\psi_{M}\Gamma^{t})(A_{0})P_{-}F(A_{0})(I_{d}-\psi_{\mathrm{A}f}(A_{0}))V(t_{n})f\cdot$,$7_{5}^{1}=(V(t-t_{n})-U_{0}(t-t_{n},))(\psi_{M}\Gamma^{2})(A_{0})P_{-}(\psi_{M}\Gamma^{l})(A_{0})V(t_{n})f$, $\psi_{M}(\lambda)=0(|\lambda|<1/2M, |\lambda|>2M)$
andsatisfies$\psi_{M}(\lambda)=1(1/M<|\lambda|<M)$.
and, there exists $M>()$ such thatFirst, we note that for$0\leqq\psi_{M}(\lambda)\leqq 1$$\psi_{M}(\lambda)\in C_{0}^{\infty}(\mathrm{R})$$\epsilon$$\mathrm{c}\Re \mathrm{l}\mathrm{y}$$||T_{j}||_{H}\leqq C||(1-\psi_{M})\Gamma^{t}||_{L^{\infty}(\mathrm{R})}<\epsilon$$(j^{l}=2,4)$

Therefore once the limits
(2.2)

,

$\lim_{narrow\infty}\varlimsup_{tarrow\infty}||T_{j}||_{H}=0$$(j=1,3,5)$

are proved, we obtain (2.1). Below we shall show (2.2). For $j=1$ (A2) implies that
$F(A)^{2}-F(A_{0})^{2}$ is acompact operator in $H$ . Using (F2) we have$||T_{1}||_{?\mathrm{t}}\leqq C||(F(A)^{2}-F(A_{0})^{2})V(t_{n},)f\cdot||_{H}arrow 0$$(narrow\infty)$

55
MITSUTERU KADOWAKI

For$=3$ ,$.j$wc decompose$T_{3}$

as follows$T_{3}=T_{31}+T_{32}+T_{33}$

,

where
$T_{31}=V(t-t_{n})(F(A_{0})-F(A))(\psi_{M}F)(A_{0})P_{+}F(A_{0})V(t_{n})f$.

$T_{32}=(F(A)-F(A_{0}))U_{0}(t-t_{n})(\psi_{M}F)(A_{0})P_{+}F(A_{0})V(t_{n})f$$7_{33}^{1}=F(A)(V(t-t_{n})-U_{0}(t-t_{n}))\psi_{M}(A_{0})P_{+}F(A_{0})V(t_{n})f$$\mathrm{S}_{(}\backslash 111\mathrm{C}$argument$\acute{(}\mathrm{L}\mathrm{S}$ill the proof of$T_{1}$

implies$\lim_{narrow\infty}\varlimsup_{tarrow\infty}||T_{31}||_{H}=0$.

We have by (A1)$\mathrm{w}-\lim_{tarrow\infty}U_{0}(t-t_{n})f=0$

Tllllbk (A2).

$\mathrm{i}_{111}\mathrm{p}1\mathrm{i}\mathrm{e}\mathrm{s}$$\lim_{tarrow\infty}||T_{32}||_{H}=0$.

To esti mate 733, wc use Cook-Kuroda method. We have by (A2)$\langle?_{33}\urcorner, .t;\rangle_{H}$$=-i.[_{0}^{t-t_{n}}\langle V(t-t_{n}-s)A(A-i)^{-1}KU_{0}(s)\tilde{\psi}_{M}(A_{0})P_{+}F(A_{0})f_{n}., g\rangle_{k\ell}ds$

where $g\in?t$ , $f_{7l}=V(t_{n})f$ and
Therefore we obtain$\tilde{\psi}_{M}(\lambda)--(\lambda-i)\psi_{M}(\lambda)$.

$||T_{33}||_{H} \leqq C\int_{0}^{\infty}||KU_{0}(s)\tilde{\psi}_{M}(A_{0})P_{+}F(A_{0})f_{n}||ds$

For each.s$\geqq 0$

wc have by (F2) alld (A2) ,.

$narrow\infty 1\mathrm{i}\mathrm{r}\mathrm{n}||KU_{0}(s)\tilde{\psi}_{M}(A_{0})P+F(A_{0})f_{n}||_{\mathcal{H}}=0$

Therefore (A3.1) and Lebesgue’s theorem imply$\lim_{narrow\infty}\varlimsup_{tarrow\infty}||T_{33}||_{H}=0$

Now.

$\mathrm{w}\mathrm{c}\mathrm{o}\mathrm{b}\mathrm{t}_{C}^{\tau}\dot{\mathrm{u}}\mathrm{n}$$\lim_{narrow\infty}\varlimsup_{tarrow\infty}||T_{3}||_{\mathcal{H}}=0$

We estimate$T_{5}$.

as follows :$||T_{5}||_{\mathcal{H}}^{2}\leqq C||P_{-}(F\psi_{M})(A_{0})V(t_{n})f||_{H}^{2}$$=C, \sum_{j=1}^{3}T_{5j}$

,

.

56
ON EXISTENCE OF SCATTERING SOLUTIONS

where$T_{51}=\ \mathrm{M}(\mathrm{A}\mathrm{O})\mathrm{P}-\mathrm{h}\mathrm{n},$$(\mathrm{F}(\mathrm{A}\mathrm{q})-F(A))V(t_{n})f\rangle\tau\{$$T_{52}=\ \mathrm{M}(\mathrm{A}\mathrm{O})\mathrm{P}-\mathrm{h}\mathrm{n},$$(V(t_{n})-U_{0}(t_{n}))F(A)f\cdot\rangle_{H}$$T_{53}=\langle U_{0}(-t_{n})\psi_{M}(A_{0})P_{-}l\iota_{n}, F(A)f\cdot\rangle_{H}$and $h_{n}=(F\psi_{M})(A_{0})V(t_{n})f$ .
(A2) and (F2) imply$\lim_{narrow\infty}T_{51}=0$.

$\lim_{narrow\infty}T_{53}=0$.

(A3.4) implies

To estimate$T_{52}$, again we use Cook-Kuroda method. Note that$|T_{52}| \leqq C||f||_{H}\int_{0}^{\infty-}||K^{*}U_{0}(-s)\tilde{\psi}_{M}(A_{0})P_{-}l_{l_{n}}||_{\mathcal{H}}ds$

Using (A2), (F2) alld (A3.2) we have by Lebesgue’s theore$\lim_{narrow\infty}7_{52}^{1}=0$.

Now we obatin$\lim_{narrow\infty}\varlimsup_{tarrow\infty}||7_{5}^{1}||_{H}=0$.

Therefore the proof of the existence of $W$ is completed.
To show$W\not\equiv \mathrm{O}$, we introduce asubspace of$D=\{f\in??

Since

A $f=\lambda f$ ,:$\mathcal{H}$$\Rightarrow A^{*}f=\lambda f$(see Lemma 1.1.5 of [14]), we can easily show.

We prepare the following proposition without thc proof.

Proposition 2.1. Assume that$\mu_{b}^{[perp]}\circ D=\{0\}$.

Then one has
(2.3)

.
.

, $f\in H$$D\subset?\{_{b}^{[perp]}$$\square$, $D$ , as follows :tarrow\infty 1\mathrm{i}\ln V(t)f\cdot=0\}$$\lambda\in \mathrm{R}$$\mathrm{m}$$w-1\mathrm{i}\mathrm{I}\mathrm{n}U_{0}(-t)V(t)f\cdot=\mathrm{t}1tarrow\infty$

57
MITSUTERU KADOWAKI

for any.

$f\in \mathcal{H}$

Below we shall show

proof of

$W\not\equiv \mathrm{O}$(2.6)$W\not\equiv \mathrm{O}(\mathrm{c}\mathrm{f}.

. For any.

$f$$\epsilon_{-}H$

and[12]\S 3)$.

$g\in \mathcal{H}$, note that$\langle U_{0}(-t)V(t)(A-i)^{-1}f, (A_{0}+i)^{-1}g\rangle_{H}$$= \langle(A-i)^{-1}f, (A_{0}+i)^{-1}g\rangle_{H}+i\int_{0}^{t}\langle V(\tau)f., K^{*}U_{0}(\tau)g, \rangle_{H}d\tau$

We assume that $W\equiv 0$ ,(2.7)$\mathrm{i}.\mathrm{e}$, for any.

$\in H_{b}^{[perp]}$

$f$

.

,

$||Wf||_{\mathcal{H}}= \lim_{tarrow\infty}||V(t)f||_{?\{}=0$(2.7) means$\mathcal{H}_{b}^{[perp]}\ominus D=\{0\}$.

.

Hence Proposition 2.1 and (2.6) imply.

$\langle(A-i)^{-1}f, (A_{0}+i)^{-1}g\rangle_{\mathcal{H}}=-i\int_{0}^{\infty}\langle V(\tau)f\cdot, K^{*}U_{0}(\tau)g, \rangle_{\mathcal{H}}d\tau$Putting$f\cdot=(A_{0}-i,)U_{0}(.\sigma.\grave{)}\psi_{M}(A_{0})P_{+}l?$for anv$\in \mathcal{H}$$\mathit{1}\iota$

and$g=(A_{0}+i)U_{0}(.\mathrm{s})\psi_{M}(A_{0})P_{+}h$. wc have$|_{1}^{1}\phi_{\Lambda P}(A_{0})I_{+}’h||_{\gamma\{}^{2}\leqq||h||_{H}(||((_{\backslash }A-i)^{-1}-(A_{0}-i)^{-1})U_{0}(.\mathrm{s})\tilde{\psi}_{M}(A_{0})P_{+}h||_{?t}$$+C_{M}/.

.

\int_{0}^{\infty}||K^{*}l_{0}^{\tau},(\tau+.\mathrm{s}^{1})\tilde{\psi_{\Lambda’I}’}(A_{0})P_{+}l\iota||_{H}d\tau)$(A 1) and (A2) imply$sarrow\infty 1\mathrm{i}_{111}||((A-i)^{-1}-(A_{0}-i)^{-1})U_{0}(s)\uparrow\tilde{l_{M}’}(A_{0})P_{+}l_{l}||_{H}=0$

and (A3.2) implies$sarrow\propto 1\mathrm{i}_{\mathrm{l}}\mathrm{n}.[_{0}^{\mathrm{x}}||K^{*}U_{0}(\tau+s)\tilde{\psi}_{M}(A_{0})P_{+}||d\tau=0$.

Thereforc we have
(2.8)$||\mathrm{t}^{[},’ M(A_{0})P_{+}l_{7},||_{\mathcal{H}}=0$$M>()$ .
andfor any(2.8) means $P_{+}\equiv 0$ . This is a$li$.

$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}’ \mathrm{f}$of$\in H_{0}$$W\not\equiv \mathrm{O}$,$\dot{\mathrm{c}}\mathrm{t}\mathrm{l}\mathrm{l}\mathrm{V}$.

$\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{t},\mathrm{r}\acute{‘}\iota \mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{l}\mathrm{l}$with (A3). Now we$\mathrm{c}\mathrm{o}$mplete the$\square$give abrief sketch of the proof of Corollary 2.
we have the case . $=0$ by
is unitary in
proof of Corollary J. Noting hat. It follows from the case $k=0$ and (A1) that tlle case $k=1$ .
induction.
bycan show tlrc cases A $=2,3,4$ ,$\mathrm{W}\sigma^{1}$$\mathrm{t}$$k$$\mathcal{H}$$L^{\Gamma_{0}}(t)$$\ulcorner 1^{arrow}11\mathrm{C}()1^{\cdot}\mathrm{C}\ln 1$$\backslash 4^{\gamma}\mathrm{t}^{1}$$\cdots$$\mathrm{t}_{1}11\mathrm{C}$$\square$

58
ON EXISTENCE OF SCATTERING SOLUTIONS

3. Proof of Theorem 3and Corollary 4.
For tlle sake of simplicity, we shall also rest .
only.
be the spectral fiunily of . Then wc haveLct$1^{\cdot}\mathrm{i}\mathrm{c}.\mathrm{t}$$()11\mathrm{r}_{\iota}\backslash ^{\mathrm{r}}\mathrm{e}1\mathrm{v}\mathrm{e}\mathrm{s}$to thc case $\sigma(A_{0})=$$\sigma_{ac}(A_{0})=\mathrm{R}$$E(\lambda)$$A_{0}$.

$A_{0}= \int_{-\propto}^{\infty}\lambda dE(\lambda)$.
For $\beta>\alpha>0$ , we denote $E((-\beta, -\alpha)\cup(\alpha, \beta))$ by
is -sinootb, i.e. for any $.(/\in H$(A3) means that
$E_{\alpha,\beta}(A_{0})$$\sqrt{B}\mathrm{o}E_{\alpha,\beta}(A_{0})$(3.1)$A_{0}$$\int_{-\infty}^{\infty}||\sqrt{B}0U_{0}(t)E_{\alpha,\beta}(A_{0})g||_{H}^{2}dt\leqq\tilde{C}_{\alpha,\beta}||.\iota/||_{H}^{2}$(cf. [8] or [16]), whereis apositive constant which depends on, $f\in D(A)$ :following identity
Moreover we note$\tilde{C}_{\alpha.\beta}$$\alpha\dot{\mathrm{t}}\mathrm{u}\mathrm{l}\mathrm{c}\mathrm{l}$$\beta$only.

$()\mathrm{f}V(t)f$$\mathrm{t},\mathrm{h}\mathrm{e}$

,

(3.2)

$||V(t)f \cdot||_{H}^{2}+2\int_{0}^{t}||\sqrt{B}0V(\tau)f\cdot||_{\mathcal{H}}^{2}cl\tau=||f\cdot||_{H}^{2}$

Using (3.1) and (3.2) we prove tllc following lermna.

Lemma 3.1. Let $\beta>\alpha>0$ . Then

$fo7^{\cdot}$

any $f\cdot\in D(A)$ one has$\lim_{t,sarrow\iota \mathrm{X}}||E_{\alpha,\theta}(A_{0})(U_{0}(-t)V(t)-U_{0}(-,\mathrm{s})V(s))f\cdot||_{\mathcal{H}}=0$.

proof. See [12] \S 3.
By Lemma 3.1 and (A1) we have the following lcuuna.

Lemma 3.2. One has$w- \lim_{tarrow\infty}V(t)=0$.

Using Lemma 3.2 wc prove Theorem $3(1)$ as follows.

. Ass umc that there exists $f\cdot\in D(A)$ , A
proof of Theorem. TllcIl we haveA$f^{\iota}(\mathit{1})$$f=\lambda f.$$\langle V(t,)f, f\rangle_{\mathcal{H}}=e^{-:t\lambda}|_{1}^{\mathrm{I}}f||_{H}^{2}$This yields acontradiction with Lemma 3.2.

L)

Theorem $3(1)$ and (F1) imply that
(3.4)$\{(A-i,)^{-2}Af.

Below we prove Theorem $3(2)$ .

\in H :f\in D(A)\}$is dense in

7#.

$\in \mathrm{R}$such$\mathrm{t}1_{1}\dot{‘}\iota \mathrm{t}$59
MITSUTERU KADGWAKI. First we show the existence of $W$ . Set $F(\lambda)=(\lambda-i)$
proof of Theorem
By (2.6) it is sufficient to show that $\{U_{0}(-t)V(t)F(A)f\}_{t\geqq 0}$ is Cauchy in.
, where $\in D(A)$ . We estimate as follows (cf. [17]) :$\mathit{3}(\mathit{2})$

’

$t$

$f$

$arrow\infty$$||(U_{0}(-t)V(t)-U_{0}(-s)V(s))F(A)f \cdot||_{H}\leqq\sum_{j=1}^{4}||T_{j}||_{H}$

,

where
.

$7_{1}^{1}=U_{0}(-t)(F(A)-F(A_{0}))V(t)f$$?_{2}^{\tau}=U_{0}(-s)(F(A)-F(A_{0}))V(s)f$$7_{3}^{1}=F(A_{0})(I_{d}-E_{1/M,M}(A_{0}))(U_{0}(-t)V(t)-U_{0}(-s)V(s))f$and$T_{4}=F(A_{0})E_{1/M,M}(A_{0})(U_{0}(-t)V(t)-U_{0}(-s)V(s))f$.

Wc note that for any . there exists $M>1$ such that$\epsilon$$||(1-\mathrm{t}(-M,-1/M)\cup(1/M,M))F||_{L^{\infty}(\mathrm{R})}<\epsilon$.

Thus wc have$||T_{3}||_{\mathcal{H}}<\epsilon||f\cdot||_{\mathcal{H}}$$(3..\ulcorner))$

By (A4),$\mathrm{F}(\mathrm{A})-\mathrm{F}(\mathrm{A}\mathrm{q})$.

is acompact operator. Hence Lemma 3.2 implies

.

(3.6)

$\lim_{tarrow\propto}||T_{1}||_{\mathcal{H}}=\lim_{sarrow\infty}||T_{2}||_{H}=0$

Le mma 3.1 implies(3.7)$\lim||T_{4}||_{H}=0$.

$t,sarrow\propto$(3.5), (3.6) and (3.7) imply the existence of .
(cf. [12]\S 3). Assume that $W\equiv 0$ i.e. for any $f\in H$
Next wc prove$\mathrm{M}^{\Gamma}$$W^{7}\not\equiv 0$(3.8)

Wc set$\lim_{tarrow \mathrm{x}}||V(t)f.||_{\mathcal{H}}=0$$\mathrm{G}(\mathrm{A})=(\lambda-i)^{-1}$.

$\ulcorner 1^{\tau}\mathrm{h}\mathrm{e}11$.

noting$\langle U_{0}(-t)V(t)G(A)f\cdot, G(A_{0})f\cdot\rangle_{\mathcal{H}}$$= \langle G(A4)f\cdot.G(A_{0})f\rangle_{\mathcal{H}}-\int_{0}^{t}\langle[I_{0}(-\tau)BV(\tau)G(A)f.G(A_{0})f\rangle_{H}d\tau$$\mathrm{W}\mathrm{G}$have$\mathrm{b}\mathrm{v}(3.8)$,

and Schwartz inequality(3.9)$|\langle G(A)f., G(A_{0})f\cdot\rangle_{?\{}|$.

$\leqq(\int_{0}^{\infty}||\sqrt{B}V(\tau)G(A)f\cdot||_{H}^{2}d\tau)^{\frac{1}{\prime\sim)}}\cross(\int_{0}^{\propto}||\sqrt{B}U_{0}(\tau)G(A_{0})f||_{H}^{2}d\tau)^{\frac{1}{2}}$60
ON EXISTENCE OF SCATTERING SOLUTIONS(3.2) and (3.8) imply

2 $\int_{0}^{\infty}||\sqrt{B}V(\tau)G(A)f||_{H}^{2}d\tau=||G(A)f||_{H}^{2}$ .

(3.10)

Hence we have by (3.9) and (3.10)$||G(A_{0})f||_{H}^{2} \leqq||f||_{H}\{||(G(A)-G(A_{0}))f||_{\mathcal{H}}+(\frac{1}{2}\int_{0}^{\infty}||\sqrt{B}U_{0}(\tau)G(A_{0})f\cdot||_{H}^{2}d\tau)^{\underline{\frac{1}{)}}}\}$.

Let fix $M>1$ . Put $f=U_{0}(s)g,g$ satisfying $E_{1/M,M}(A_{0})g=g$ . Then we have
(3.11)$||G(A_{0})g||_{H}^{2}\leqq||g||_{H}\{||(G(A)-G(A_{0}))U_{0}(s)g||_{H}$$+(. \frac{1}{2}\int_{s}^{\infty}||\sqrt{B}E_{1},/_{M,M}(A_{0})U_{0}(\tau)G(A_{0}).q||_{\mathcal{H}}^{2}d\tau)^{\underline{\frac{1}{\prime)}}}\}$.

(A1) and (A4) imply(3.12)$\lim_{sarrow\infty}||(G(A)-G(A_{0}))U_{0}(s).c/||_{H}=0$.

(3.1) implies(3.13)$\lim_{sarrow\infty}\int_{s}^{\infty}||\sqrt{B}E_{1/M,M}(A_{0})U_{0}(\tau)G(A_{0})g||_{H}^{2}d\tau=0$

Therefore it follows from (3.11), (3.12) and (3.13) that.
tion. Therefore we have
$W\not\equiv \mathrm{O}$$g\equiv 0$.

. This is acontradic-$\square$

To prove Corollary 4we should repeat the same way as in the proof of Corollary
2. Here we omit to do it.

4. Applications.
Application 1(Elastic wave equation with dissipative boundary condition).
in ahalf space ofWe shall apply Theorem 1. In this section we also use $C$ as positive constants.
satistying
and $\mu_{0}>0$ , $\rho_{0}>0$ ,Let$O3X3$$3\lambda_{0}+2\mu_{0}>0$ .
type,
and
as zero and unit matrix of 3
Wc userespectively.
We set$\mathrm{R}^{3}$$x=(\prime x_{1}.\prime x_{2}, \prime x_{3})=(y, x_{3})\subset-\mathrm{R}^{2}\cross \mathrm{R}_{+}$$\lambda_{0}\in \mathrm{R}$$\cross 3$$/3\mathrm{X}3$$\epsilon_{hj}(u(x))=\frac{1}{2}(\frac{\partial\tau\iota_{h}}{\partial x_{j}}+\cdot\frac{\partial u_{j}}{\partial x_{h}})$and$\sigma_{hj}(u(x))=\lambda_{0}(\nabla_{x}\cdot u)\delta_{hj}+2\mu_{0}\epsilon_{hj}(\tau\iota)$

Here $h,j=1,2,3$ ,$u(\prime x)=^{t}(u_{1}(x), u_{2}(\prime x),$ $u_{3}(\prime x))\in \mathrm{C}^{3}$and$\nabla_{x}=(\partial/\partial_{1}, \partial/\partial_{2}, \partial/\acute{c}J_{3}).|$

61
MITSUTERU KADOWAKI

We define operators$\tilde{L}_{0}$as$( \tilde{L}_{0}u)_{h}=-\sum_{j=1}^{3}\frac{1}{\rho_{0}}\frac{\partial\sigma_{hj}(u(x))}{\partial x_{j}}$$(h=1,2, 3)$ .

We consider two elastic wave equations as follows:,$\partial_{t}^{2}u(x, t)+\tilde{L}_{0}u(x, t)=0$(4.1)$(x, t)\in \mathrm{R}_{+}^{3}\cross[0, \infty)$

,

$\{$

${}^{t}(\sigma_{13}(u), \sigma_{23}(\prime u)$, $\sigma_{33}(u))|_{x_{3}=0}=B(y)\partial_{t}u|_{x_{3}=0}$$\iota \mathrm{l}\mathrm{r}\mathrm{l}\mathrm{l}\mathrm{d}$$\partial_{t}^{2}u(x_{\backslash }t)+\tilde{L}_{0}u(x, t)=0$

(4.2)

,

$(x, t)\in \mathrm{R}_{+}^{3}\cross \mathrm{R}$$\{$$\sigma_{i3}(u)|_{x_{3}=0}=0(i=1,2,3)$.

To set assumptions for $B(y)$ we introduce afunction space$B^{k}(\Omega)$$\Omega\subset \mathrm{R}^{n}$

(4.3)

.

that
belongs to$\mathrm{A}\mathrm{k}\mathrm{s}_{\iota}^{\urcorner}\mathrm{b}^{1}11111\mathrm{e}$$B(.\iota/)$$B^{1}(\mathrm{R}^{2}, \mathrm{M}_{3\cross 3})$and satisfies$O_{3\cross 3}\leqq B(y)\leqq\varphi(|y|)I_{3\cross 3}$,

is a11 11-incrcasing function and belongs to.
:in
The following operator

where

as follows :,$B^{k}( \Omega)=\{\iota\iota\in C^{k}(\Omega);\sum_{\backslash |\alpha|_{=}’karrow}||\partial^{\alpha}\tau\iota||_{L^{\infty}(\Omega)}<\infty\}$

where

,

$L^{1}(\mathrm{R}_{+}$ $)$$()$$\varphi(7^{\cdot})$-.

$\mathrm{M}_{3\cross 3}$

is the class$()\mathrm{f}3\cross 3111‘.\iota \mathrm{t}\mathrm{l}\cdot \mathrm{i}\mathrm{x}$$\mathcal{G}=L^{2}(\mathrm{R}_{+}^{3}, \mathrm{C}^{3}; /J_{0}\mathrm{r}lx)$$I_{\lrcorner}0$$L_{0}\iota\iota--\tilde{L}_{0}?\iota$$\dot{(}\iota 11\mathrm{d}$$\mathrm{D}\{\mathrm{L}\mathrm{O})=\{\iota\iota\in_{-}H^{1}(\mathrm{R}_{+}^{3}/.\mathrm{C}^{3});\tilde{L}_{0}\iota\iota \in \mathcal{G}t, \sigma h3(\tau\iota)|_{x_{3}=0}=()(l\iota=1,2,3)\}$$\mathrm{i}_{\iota}\mathrm{b}\subset.1$

Let

$H$

sclf-ael.i illf.
be Hilbert space with illllel product :$11\mathrm{O}11-11\iota^{\backslash }\mathrm{g}_{\dot{\mathrm{t}}\iota \mathrm{t}\mathrm{i}\backslash }\prime \mathrm{e}$$()$$\iota’ 1$)$\mathrm{t}^{\backslash }1^{\cdot}\mathrm{a}\mathrm{t}\mathrm{e}$)$\mathrm{r}$$\cdot$$/\backslash ^{f\cdot}.$,$.‘/ \rangle_{H}=\int_{\mathrm{R}_{1}^{2}}.(\sum_{h,j,k,l=1}^{3}.(\iota_{hjkl}\epsilon_{kl}(f_{1}\cdot)_{\hat{b}hj}\overline{(.(J1)}+f_{2}\cdot\overline{.q_{2}}\gamma\prime 0)(lx$$\mathrm{w}11\mathrm{C}1^{\cdot}\mathrm{C}‘\iota_{hjkl}=\lambda_{0}\delta_{hj}\delta_{kl}+/\iota_{0}(\delta_{hk}\delta_{jl}+\delta_{hl}\delta_{jk})\mathrm{c}‘ 11\mathrm{l}\mathrm{d}$$f\cdot=^{t}(f_{1}., f_{2}.)$,, $g=^{t}(.‘/1, g_{2})$ . By

Korn’s inequality (cf. [5]) wc note that $H$ is equivalent to.
as,, wllcrc
is the solution to (4.1) (resp. (4.2))
We sct. Then (4.1) (resp. (4.2)) can bewith ainitial data$\dot{H}^{1}(\mathrm{R}_{+}^{3}, \mathrm{C}^{3})\cross L^{2}(\mathrm{R}_{+}^{3}, \mathrm{C}^{3})$$\mathrm{B}\mathrm{a}11\dot{\mathrm{e}}\iota \mathrm{c}11\backslash i^{\mathrm{t}}\mathrm{I})_{\mathrm{c}}\tau \mathrm{c}\mathrm{e}$$f\cdot=$${}^{t}(n(r\cdot, t)$$’\{\iota_{t}(x.t))$$\mathit{1}^{\cdot}0=t$written as$\tau\iota(x, t)$$(’\iota\iota(.\iota\cdot.()), \iota_{t}(x\cdot.\mathrm{O}))\in H$

62
ON EXISTENCE OF SCATTERING SOLUTIONS$\partial_{t}’f\cdot=-iAf$.

(rcsp.

$\partial_{t}f\cdot=-iA_{0}f\cdot$

),

where
$A=i$

$(\begin{array}{ll}() I_{3\cross 3}-\tilde{L}_{0} ()\end{array})$,$A_{0}=i$$D(A)=\{f={}^{t}(f_{1}., f_{2}.)\in H;\tilde{L}_{0}f_{1}.\in L^{2}(\mathrm{R}_{+}^{3}, \mathrm{C}^{3})$${}^{t}(\sigma_{13}(f_{1}), \sigma_{23}(f_{1}.)$,,$(\begin{array}{ll}() I_{3\mathrm{x}3}-\tilde{L}_{0} ()\end{array})$,$f_{2}.\in H^{1}(\mathrm{R}_{+}^{3}, \mathrm{C}^{3})$,$\sigma_{33}(f_{1}.))|_{x_{\backslash }\tau=0}.=B(y)f_{2}.|_{x_{i}’=0}\}$and$D(A_{0})=\{f={}^{t}(f_{1}, f_{2})\in H;\tilde{L}_{0}f_{1}\in L^{2}(\mathrm{R}_{+}^{3}, \mathrm{C}^{3})$,$f_{2}.\in H^{1}(\mathrm{R}_{+}^{3}, \mathrm{C}^{3})$,$\sigma_{h3}(f_{1})|_{x_{3}=0}=\mathrm{t})(l_{1}, =1,2,3)\}$

According to P210-P211 of [11] or Corollary 1.1.4 of [14] wc$\mathrm{g}.\mathrm{e}^{\mathrm{r}},11\mathrm{C}1^{\cdot}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{s}$$H$$\acute{.}k^{\backslash }$. Using

it colltr$\dot{‘}\iota \mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}11\backslash _{\mathrm{k}}’ \mathrm{C}111\mathrm{i}- \mathrm{g}1^{\cdot}\mathrm{o}n1$$\{V(t)\}_{t\geqq 0}$(resp.

)) we solve$\partial_{t}f\cdot=-iAf$follows$f=V(t)f_{0}$Below we make a cllec.k on[2] mplies$\{U_{0}(t)\}_{t\in \mathrm{R}})$

$A$

ill
)

$\partial_{t}f\cdot=-\mathrm{i}\mathrm{A}\mathrm{f}f\cdot$.

and (A3). Iliercforc wc have (A1).

Next we show (A2). For , $.‘/\in H$ , we
$f$

(4.4)

show that$\mathrm{A}_{\iota^{1}}\backslash _{\mathrm{L}}\mathrm{b}.11\mathrm{I}1\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{o}11_{\mathrm{t}}\backslash ^{1}(\mathrm{A}\mathrm{I}),(\mathrm{A}2)$$\sigma(A_{0})=\sigma_{ac}(A_{0})=\mathrm{R}$$\mathrm{i}$$(\mathrm{r}\mathrm{c}_{\iota}\backslash \cdot \mathrm{p}.f\cdot=U_{0}(t)f_{0}.)$) $111$ ). (resp.

$\{V(t)\}_{t\geqq 0}(1^{\cdot}\mathrm{C}_{\mathrm{t}}[searrow].1).\dot{C}\iota 1111\mathrm{i}\{_{(\gamma 1}\cdot \mathrm{y}\mathrm{g}\mathrm{r}\mathrm{c}$$\{U_{0}(t,)\}_{t\in \mathrm{R}}$$\mathrm{c}\mathrm{u}\mathrm{l}$,$1\mathrm{l}\dot{\mathrm{c}}\iota \mathrm{v}\mathrm{e}$by easy cal culation$\langle((A-i)^{-1}-(A_{0}-i)^{-1})f\cdot,g\rangle_{H}$$=i$, $\int_{\mathrm{R}-}.\prime B(y)\Gamma_{0}((A_{0}-i)^{-1}f)_{2}\overline{\Gamma_{0}((A^{*}+i)^{-1}g)_{2}}dy$

where$\Gamma_{0}$,is atarce operator which is defined by$(\Gamma_{0}\cdot u)(y)=u(y, 0)$.

and
by Korn’s
Note thatbelong toinequality for any , $\in(1/2,1)$ . Since $B(y)\Gamma_{0}\Pi_{2}(A_{0}-i)^{-1}$ is acompact operator
by Rellich’s theorem, where $\Pi_{j}{}^{t}(f_{1}, f_{2})=f_{j}(j=1,2)$ , tlc
from to$(A-i)^{-1}-(A_{0}-i)^{-1}$form
can be extended to acompact )$i)^{-1})^{*}B(y)\Gamma_{0}\Pi_{2}(A_{0}-i)^{-1}$ , in $H$ .
,andTo show (A3) we sate aresult from [2]. There exist $Fru$ )ontowhich are partially isometric operators from, respectively. Defining theas follows :and$\Gamma_{0}((A^{*}+i)^{-1}f)_{2}$$\Gamma_{0}((A_{0}-\mathrm{i})^{-1}f)_{2}$$H^{1-s}(\mathrm{R}_{+}^{3}, \mathrm{C}^{3})$$\mathrm{s}$$\prime \mathcal{H}$$L^{2}(\mathrm{R}^{2}, \mathrm{C}^{3})$$\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r},(\Gamma_{0}\square _{2}(A^{*}+$$\mathrm{o}\mathrm{I}$$\Gamma\sqrt s?\iota$$\mathcal{G}=L^{2}(\mathrm{R}_{+}^{3}, \mathrm{C}^{3};\rho_{0}dx)$$L^{2}(\mathrm{R}^{2}, \mathrm{C}^{3})$$\Gamma^{\ell}$$\mathrm{e}_{\mathrm{j}}\mathrm{p}\mathrm{c}\mathrm{r}\mathrm{a}\mathrm{t},\mathrm{o}\mathrm{r}$$Fu=$ ($\Gamma_{P}^{l}u$

we have by Theorem 3.6 of [2], Fshu,$F^{1}SH’1l,$$FR’n$)

for$u\in \mathcal{G}$,$\Gamma\sqrt SHu$$\Gamma_{R}^{\mathrm{t}}$$L^{2}(\mathrm{R}_{+}^{3}, \mathrm{C}^{3})[$

63
MITSUTERU KADOWAKI

Lemma A.

,$\Gamma$is unitary operator from$\mathcal{G}$to$\hat{H}=\oplus^{3}j=1L^{2}(\mathrm{R}_{+}^{3}, \mathrm{C}^{3})\oplus L^{2}(\mathrm{R}^{2}, \mathrm{C}^{3})$$nnd$

for cvery $u\in D(L_{0})$FLouwhere A$=(c_{P}^{2}|k.|^{2}\Gamma_{P}\sqrt u, c_{S}^{2}|k.|^{2}F_{S}u, c_{S}^{2}|k|^{2}F_{SH}u, c_{R}^{2}|p|^{2}F_{R}u)$,$=(p,p_{3})\in \mathrm{R}^{2}\cross \mathrm{R}+\cdot$

Using $\Gamma_{j}\sqrt(j=P, S, SH, R)$ as above, we construst(4.5)

follows:$F_{j}^{*}P_{\pm}^{(3)}I_{3\mathrm{x}3}F_{j}O_{3\mathrm{x}3})$$P_{\pm}=T^{-1} \{\sum_{j_{-}^{--}P,S,SH}(_{\mathit{0}_{3\mathrm{x}3}}^{F_{j}^{*}P_{\mp}^{(3)}I_{3\cross 3}F_{j}}$$+$$P\pm \mathrm{a}\mathrm{s}$($F_{R}^{*}P_{\pm}^{(2)}I_{3\cross 3}F_{R}O_{3\cross 3}$

)

$\}T$

where
$-iI_{3\cross 3}iI_{3\cross 3})$

$T= \frac{1}{\sqrt{2}}(_{L}^{L^{\frac{1}{\frac{\tilde{0_{1}}^{9}}{\frac{9}{0}}}}}$(resp.
andofions
projcct$P_{-}^{(3)}$$P_{+}^{(3)}$) alld$P_{-}^{(2)}$$D^{(3)}= \frac{1}{2i}(k\cdot\nabla_{k}+\nabla_{k}\cdot k)$(resp.

and$P_{+}^{(2)}$) are negative(resp. positive) spectral$D^{(2)}= \frac{1}{2i}(p\cdot\nabla_{p}+\nabla_{p}\cdot p)$,

respectively.

(see [2]) and theUsing the representation of the generalized eigenfunction of(cf. [13] and [6]). The Mellin transMellin transfor mation we show,
are given asfor$L_{0}$$(\mathrm{A}3.1)\sim(\mathrm{A}3.4)$$D^{(2)}$$D^{(3)}$$\mathrm{f}\mathrm{o})\mathrm{r}\mathrm{l}\mathrm{r}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{S}$$(M^{(3)}u)( \lambda, \omega)=(2\pi)^{-1/2}\int_{0}^{+\infty}r^{1/2-i\lambda}u(r\omega)dr$and$(M^{(2)}v)( \lambda, \nu)=(2\pi)^{-1/2}\int_{0}^{+\infty}r^{-i\lambda}v(r\nu)dr$

where
:
Then

$u(k.)\in C_{0}^{\propto}/(\mathrm{R}_{+}^{3}\backslash \{0\})$$\mathrm{S}^{2}$$\omega_{3}>0$$I_{J}^{2}(\mathrm{R}^{2}))$} and$M^{(3)}$

to$\nu\in \mathrm{S}$(resp.

,$v(p)\in C_{0}^{\infty}(\mathrm{R}^{2}\backslash \{0\})$$P_{\pm-}$$\omega$$\in \mathrm{S}_{+}^{2}=\{(\omega_{1},\omega_{2}, \omega_{3})=(\overline{\omega}, \omega_{3})\in 1$.

$M^{(2)}$) is extended to aunitary operator from(cf.[13] Lemma 2).

$L^{2}(\mathrm{R}\cross \mathrm{S}_{+}^{2})(\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}.

Proposition 4.1.

,,

L^{2}(\mathrm{R}\cross \mathrm{S}^{1}))$

as in (4.5) satisfy (A3).

To show Proposition 4.1 wc prepar$L^{2}(\mathrm{R}_{+}^{3})$(resp.

64
ON EXISTENCE OF SCATTERING SOLUTIONS

Lemma 4.2. Let
be same as in (A3) and $0<\delta<c_{R}$ (for , see Appendix).
, there exists a positive constant
Then for any positive integer $N$ andwhich is independent of such that$\psi(\lambda)$$c_{R}$$t\in \mathrm{R}_{\pm}$$C_{N,\psi}$

$t$

(4.6)

$||\nabla_{x}(e^{-itA_{0}}\psi(A_{0})P_{\pm}f)_{1}||_{L^{2}(\mathrm{R}_{+}^{3},\mathrm{C}^{3})}^{|x|\leqq\delta|t|}\leqq C_{N,\psi}’(1+|t|)^{-N}||f\cdot||_{\mathcal{H}}$(4.7)$||(e^{-itA_{\mathrm{O}}}\psi(A_{0})P_{\pm}f\cdot)_{2}||_{L^{\underline{9}}(\mathrm{R}_{+}^{3},\mathrm{C}^{3})}^{|x|\leqq\delta|t|}\leqq C_{N,\psi}(1+|t|)^{-N}||f\cdot||_{H}$

,

and
(4.8)

$||\Gamma_{0}(e^{-itA_{\mathrm{O}}}\psi(A_{0})P_{\pm}f\cdot)_{2}||_{L^{\underline{\mathrm{o}}}(\mathrm{R}^{\underline{9}},\mathrm{C}^{3})}^{|y|\leqq\delta|t|}\leqq C_{N,\psi}(1+|t|)^{-N}||f\cdot||_{\mathcal{H}}$for any$f\in H_{0}$, where$||u||_{L\sim(\mathrm{R}_{+}^{3},\mathrm{C}^{3})}^{B} \circ=(\int_{B}|u|^{2}dx)^{\frac{1}{2}}$and$||v||_{L\sim(\mathrm{R}^{\underline{9}},\mathrm{C}^{3})}^{B_{\circ}}=( \int_{B}|v|^{2}dy)^{\frac{1}{\prime\sim)}}$.

This lemma is the key lemma to show (A3). The proof is done by using,
and Lemma A. But we omit to prove (cf. [13] or [6]).

$M^{(3)}$$M^{(2)}$

proof of Proposition 4.1. Lecnrna Aof Appendix implies thatandjectioll operators and satisfy $P_{+}+P_{-}=Id$ in . Below wc show
For any , $g\in H$ we have by (4.4)$P_{+}$’$H$$P_{-}$

arc prO-$(\mathrm{A}3.1)\sim(\mathrm{A}3.4)$$f$$|\langle Ke^{-itA_{\mathrm{O}}}\psi(A_{0})P_{+}f\cdot,g\rangle_{H}|$$\leqq CI(t)\cross(||A^{*}(A^{*}+i)^{-1}g||_{H}+||(A^{*}+i)^{-1}g||_{\mathcal{H}})$

,

where
$I(t)=( \int_{\mathrm{R}\sim}\circ|B(y)\Gamma_{0}(e^{-itA_{0}}(A_{0}-i)^{-1}\psi(A_{0})f)_{2}|^{2}dy)^{\frac{1}{9\sim}}\cross$$\cross(||A^{*}(A^{*}+i)^{-1}g||_{\mathcal{H}}+||(A^{*}+i)^{-1}g||_{H})$.

Decomposing $I(t)$ as follows :$I(t) \leqq C\{(\int_{\mathrm{R}^{2}\cap\{|y|\leqq\delta t\}}|\Gamma_{0}(e^{-itA_{\mathrm{O}}}(A_{0}-i)^{-1}\psi(A_{0})P_{+}f)_{2}|^{2_{(}}ly)^{\underline{\frac{1}{9}}}$$+( \int_{\mathrm{R}-\cap\{|y|\geqq\delta t\}}.,\backslash |B(y)\Gamma_{0}(e^{-itA_{\mathrm{O}}}(A_{0}-i)^{-1}\psi(A_{0})P_{+}f)_{2}|^{2}cly)^{\underline{\frac{1}{\prime)}}}\}$

we have by (4.8) of Lemma 4.2 and (4.3)$I(t)\leqq C_{N,\psi}\{(1+t)^{-N}+\varphi(\delta t)\}||f||_{H}$Therefore (A3.1) is proven

.

,

.

65
MITSUTERU KADOWAKI

To prove (A3.2) and (A3.3) we note$\langle f., K^{*}g\rangle_{H}=\langle((A-i)^{-1}-(A_{0}-i)^{-1})f, \mathrm{L}q\rangle_{\mathcal{H}}$for allV , $g\in H$ .
By easy caluculation we have$f\cdot$(4.9)$\langle((A-\mathrm{i})^{-1}-(A_{0}-\mathrm{i})^{-1})f, g\rangle_{H\mathrm{o}}$$=i \int_{\mathrm{R}^{\underline{\supset}}}‘\Gamma_{0}((A-i)^{-1}f\cdot)_{2}\overline{B(y)\Gamma_{0}((A_{0}+i)^{-1}g)_{2}}dy$.

way in tllc proof of (A3.1), we obtain (A3.2) and
Then using (4.9) and the(A3.3). Here we omit thc detail.
Wc show (A3.4). Lem ma 4.2 implies$\dot{c}\lambda \mathrm{b}^{\backslash }$$\mathrm{s}_{\dot{\mathrm{e}}}\mathrm{u}\mathrm{n}\mathrm{c}$$|\langle e^{itA_{0}}\psi’(A_{0})P_{-}f_{t}., .c/\rangle_{\mathcal{H}}|$,$\leqq C_{N}/,\psi\{(1+t)^{-N}||g||_{H}+||\nabla_{x^{(/1}}.||_{L(\mathrm{R}_{\}}^{3},\mathrm{C}^{3})}^{|x|\geq\delta t}\underline{)}--+||g_{2}||_{L^{\underline{\mathrm{o}}}(\mathrm{R}_{+}^{3},\mathrm{C}^{3})}^{|x|\geqq\delta t}\}||f_{t}.||_{H}$for any ($/\in H$ and any$1$)$()\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}$. $N$ . TIIIIS, noting$\mathrm{i}_{11}\mathrm{t}\mathrm{e}\mathrm{g}_{\mathrm{C}1}$$\mathrm{S}11\mathrm{P}t\in \mathrm{R}||f_{t}||_{H}<\infty$

, we

$.$

$11_{(}^{\cdot}\iota \mathrm{v}\mathrm{G}$(A3.4).

$\square$Application 2(Acoustic wave equations with dissipative terms in stratified media ).
. First wc explain acoustic operator.
apply
We. Wc sct
and
Lct$r\Gamma 11\mathrm{t}^{\mathrm{Y}}\mathrm{t}$$\backslash ^{\tau}11\dot{\mathrm{c}}\iota 11$($\uparrow\iota\geqq 1$) $1^{\cdot}\mathrm{C}1113$$(.l\cdot, y)\in \mathrm{R}^{n}\cross \mathrm{R}$$(?/\geqq l\})$$\zeta’+$$(.\mathrm{o}(y)=\{$$(()<\mathrm{c}/<l\iota)$$(.h$$(y\leqq 0)$$c\cdot-$for

positive constants
Acoust ic operators are$l\iota$$\subset.\mathrm{u}101\mathrm{r}_{+_{\dot{J}}}\cdot c_{-}.$$\mathrm{k}\mathrm{b}.()11\mathrm{l}\mathrm{C}$

,

$()h$

.

.

$L_{0}=-‘.0(\tau/)^{2}$Is,$\mathrm{w}11(^{1}1^{\cdot}\mathrm{C}$$\triangle=\sum_{j=1}^{7l}\frac{\partial^{2}}{\partial.\iota_{j}^{2}}.+\cdot\frac{\partial^{2}}{\partial\tau/^{2}}$.

. wc find tllc guided waves (cf. [18] or [19]).
Considering tllC case (not restrict ourselves to sllcll cases.
But we. $D(L_{0})$is anon-negative self-adjoint operator.
of order .9 overbeing Sobolev
is given bywith tllc following dissipative wave equations :We$.h$$<111\mathrm{i}11((.+\backslash (_{-)}$$\mathrm{d}()$$\mathrm{i}_{11}\mathcal{G}=L^{2}(\mathrm{R}^{?\mathrm{z}+1} ; c.\circ(y)^{-2}dxdy)$$I_{\lrcorner}0$$H^{2}(\mathrm{R}^{7\iota+1})$.

$\mathrm{R}^{n+1}$$H^{s}(\mathrm{R}^{n+1})$$\mathrm{s}1)_{\dot{\subset}}\iota\epsilon\cdot \mathrm{e}$$\mathrm{e}1(^{\mathrm{Y}}j\iota 1$(4.10)$(.J_{t}^{\mathit{2}}.\iota\iota(.r.

.\iota/\cdot t)+l_{J(.I}\cdot, y)c‘?\iota^{1l(J1/\cdot t)+L_{0}\iota\iota(Ji}.,$, , $t)=()$$y$and(1.11)$\dot{(})_{t}^{2}\tau\iota(.\iota\cdot, y, t)+\langle\partial_{t^{\mathfrak{l}l}},.\varphi\rangle_{\mathcal{G}}\varphi(.r, y)+L_{0}\iota\iota(_{J?/}.,, t)=\mathrm{t})$,66
ON EXISTENCE OF SCATTERING SOLUTIONS

where
We assrunc that$(x, y, t)\in \mathrm{R}^{n}\cross \mathrm{R}\cross[\mathrm{t}1, \infty)$$b(.’\iota., y)$

and

and

$\varphi(.’\iota., y)$is thc inner-product of .
are measurable functions which satisfy$\mathcal{G}$$\langle\cdot, \cdot\rangle_{\mathcal{G}}$$0\leqq l)(x, y)\leqq C(1+|.\iota\cdot|^{2}+\uparrow/^{2})^{-}.\underline{‘}-$”and$\varphi(.\mathit{1}i, ?/)\in L^{2}(\mathrm{R}^{n+1} ; (1+|.\iota\cdot|^{2}+\uparrow/^{2})^{\underline{\frac{\theta}{\prime)}}}‘ l_{lil}.l\uparrow/)$

for so me $\theta>1$ (.uld $C>0$ .
existence ofWe shall show
are considered as tllc pertubed$\mathrm{t},11\mathrm{C}$$\mathrm{t},11\mathrm{C}$scattering states for (4.10) and (4.11) wllicllof$\mathrm{s}.\mathrm{y}\mathrm{s}\mathrm{t},\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{l}\mathrm{S}$(4.12)$\partial_{t}^{2}n(x, y, t)+L_{0}\tau\iota(x, \uparrow/, t)=()$,$(.\iota\cdot, \tau/, t)\in \mathrm{R}^{\iota}’\cross \mathrm{R}\cross \mathrm{R}$find local resolvent estim ates as
In [19], [1], ‘llld [21], wc$>\alpha>0$ , there exists positive constants
:ulel
that

$\mathrm{t}\dot{(}$$\mathrm{C}_{\dot{(}}111$$lf$$C_{\alpha,\beta}$$1\mathrm{l}\mathrm{t}$)$\mathrm{w}.\backslash \cdot$: for$‘.\iota 11\mathrm{V}$$?l\mathrm{s}\iota\iota \mathrm{c}1_{1}$’(4.13)).

$||X_{\underline{\frac{\prime}{)}}}.‘(L_{0}-\zeta^{2})^{-1}X.\underline{‘’}’||_{L-’(\mathrm{R}^{\iota|1})arrow L-(\mathrm{R}^{\prime|1})}-\cdot,.,,\leqq C_{\alpha,\beta}’$$\mathrm{s}n1)$$\alpha\leqq|\mathrm{R}\mathrm{c}\zeta|\leqq\beta,0<|{\rm Im}\zeta|<\eta$(llld,.is tllc
of tl
where
.
bounded operator in[12] llffi already dealt with thc case ( $=c:+=c_{-}.=1$ andof (4.10). His
pertubation theory [10]proof has been baesd on Kato’s
global resolvent)rclll
4.4.1)for (scc also [10]cstilnWc apply Theorem 3(Corollaty 4) to (4.10). We set. rcspcc$(4.10)$ can 1) written as
Then (4.12)
andtivcly, where$\zeta\in \mathrm{C}$$X_{\gamma}=(1+|\prime x|^{2}+y^{2})^{-f}\underline{.)}$$||\cdot||_{L^{t}(\mathrm{R}^{\iota|1})arrow L-(\mathrm{R}^{*|1})}\underline{)},.,$$11\mathrm{O}1^{\cdot}111$$1\mathrm{C}$$L^{2}(\mathrm{R}^{n+1})$$r\iota$$.h$$\iota\backslash \cdot 11\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{t}\mathrm{h}$$\mathrm{T}1_{1}\mathrm{c}\iota$$L_{0}$$‘ \mathrm{a}\mathrm{t}_{1}\mathrm{e}_{\iota}\mathrm{s}$$\geqq 2$$\dot{(}\mathrm{u}\mathrm{l}\mathrm{d}$$f(t)=(1\iota(t, .l,\cdot, \uparrow/),\dot{(})_{t’}\iota\iota(t, .\iota\cdot, y)).\mathrm{I}$$\partial_{t}f\cdot=-iA_{0}f^{\backslash }.$$\acute{(}\mathrm{u}\mathrm{l}\mathrm{d}$$\mathrm{e}$$A_{0}=\dot{\iota}$$(\begin{array}{ll}() \mathrm{l}-L_{0} 0\end{array})$,$A=\dot{\iota}$$\partial_{t}f\cdot=-iAf$$(\begin{array}{ll}() 1-L_{0} -b(.c,y)\end{array})$.

Let 7{ be Hilbert spaces with illller product$\langle f, .q\rangle_{H}=\int_{\mathrm{R}^{n|1}}(\nabla f_{1}(\prime x, y)\overline{\nabla_{J1}‘(_{J\prime}\prime\cdot,y)}+f_{2}(.x., y)\overline{.(/2(.l.\cdot,y)}_{Ci_{0}}-2(.y))_{tl\prime}r\cdot cly$

and
The

$||\cdot||_{\mathcal{H}}$is$\mathrm{t}_{l}11\mathrm{C}$$\mathrm{d}()\mathrm{l}\mathrm{I}\mathrm{l}\mathrm{a}\mathrm{i}\mathrm{I}\mathrm{l}\mathrm{S}$,corresponding norm, wllere $f=^{t}(f_{1}, f_{2}.)$ , $.‘/=^{t}(/1 , J‘ 2)$ .
ofis$A_{0}$$D(A_{0})=\{f\in H; \triangle f_{1}\in L^{2}(\mathrm{R}^{n+1}), f_{2}\in H^{1}(\mathrm{R}^{n+1})\}$.

is aself-adjoint operator in $H$ and generates aunitary group
Then
(A5).
in 7{. Below wc rnakc achcak on (A1), (A4)
Note that$A_{0}$$\acute{‘}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{l}$$7_{0}’ A_{0}T_{0}^{-1}=$($-\sqrt{L_{0}}0$

where$7_{0}^{1}= \frac{1}{\sqrt{2}}(_{\sqrt{L_{0}}}^{\sqrt{L_{0}}}$$-ii)$),$\{U_{0}(t)\}_{t\in \mathrm{R}}$67
MITSUTERU KADOWAKI

andis aunitary operator fro$7_{0}^{7}$$\mathrm{m}H$

onto$\mathcal{G}\cross \mathcal{G}$. It follows from (4.13) that for any$\uparrow\iota\in \mathcal{G}$$\alpha\leqq|Re\zeta|\leqq\beta,0<|Im\zeta|<\eta\sup|{\rm Im}\langle(\pm\sqrt{L_{0}}-\zeta)^{-1}X_{\frac{\theta}{2}}u, X_{\underline{\frac{\theta}{\circ}}}u\rangle_{\mathcal{G}}|<\infty$.

Therefore wc have (A1) by [16] Theorem XIII-20.
$\mathrm{S}\mathrm{i}_{11\mathrm{C}\mathrm{C}}$$B_{0}=(\begin{array}{ll}0 00 b(x,y)\end{array})$)
by Rellich’s theorem, we have (A2). Therefore
isgenerates acon.
raction semi-group
In tltc sameas in [12]\S 3 we can show (A5) as follow. Let $g=(g_{1}, g_{2})\in$$H$ . Wc set$A_{0}- \mathrm{c}\mathrm{e}$$A$$1\mathrm{n}\mathrm{p}\acute{\mathrm{e}}1\mathrm{C}\mathrm{t}$$\{V(t)\}_{t\geqq 0}\mathrm{i}_{11}H$$\mathrm{t}$$\acute{\epsilon}\iota 1^{\cdot}\mathrm{g}\mathrm{l}\mathrm{u}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t}$$u=(\begin{array}{l}u_{1}u_{2}\end{array})$$=(A_{0}-\zeta)^{-1\sqrt{B}}0$$(\begin{array}{l}g_{1}g_{2}\end{array})$.

Then wc have$(L_{0}-(^{2})u_{2}=(\sqrt{b(x,y)}g_{2}$$\dot{(}\iota 11(1$$\sqrt{B_{0}}(A_{0}-\zeta)^{-1}\sqrt{B_{0}}g=\sqrt{B_{0}}u=^{t}(0_{\backslash }\sqrt{b(x,y)}u_{2})$.

Therefore we can calculate as follows:(4.14)$||\sqrt{B_{0}}(A_{0}-\zeta)^{-1}\sqrt{B_{0}}g||_{H}=|\zeta|||\sqrt{b(x,y)}(L_{0}-(^{2})^{-1}\sqrt{b(x,y)}g_{2}||_{\mathcal{G}0}$(A5). Thus we have the conclusion of Theorem(4.13)
4) for (4.10) and (4.12).
to (4.11). we set
Next we apply Theorem$\subset.111\epsilon 1(4.14)$$\mathrm{i}_{1}\mathrm{n}\mathrm{p}1\mathrm{y}$.

$3(\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{y}$$3(\mathrm{C}\mathrm{o}\mathrm{l}\cdot \mathrm{o}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{y}4)$$B_{0}=(\begin{array}{lll}0 00 \langle\cdot \varphi\rangle_{\mathcal{G}}\varphi\end{array})$

Then $B$ is acompact operator in $H$ . We shall show (A5). Note that
(4.15)

for ally$|\mathrm{I}_{\ln}\zeta|||\sqrt{B}(A_{0}-\zeta)^{-1}f\cdot||_{H}^{2}\leqq|{\rm Im}\zeta|||X_{\underline{\frac{\theta}{9}}}((A_{0}-\zeta)^{-1}f.)_{2}||_{\mathcal{G}}^{2}\cross||X_{-\underline{\frac{\theta}{\mathrm{Q}}}}\varphi||_{\mathcal{G}}^{2}$$f\in \mathrm{H}$. Wc set$B_{1}=(\begin{array}{ll}0 00 X_{\theta}\end{array})$

Then we.

$11\acute{\epsilon}\iota \mathrm{v}\mathrm{c}$$|\mathrm{I}_{111}(|||X_{\underline{\frac{\theta}{\supset}}}.((A_{0}-()^{-1}f\cdot)_{2}||_{\mathcal{G}}^{2}=|\mathrm{I}_{\mathrm{l}}\mathrm{n}(|||\sqrt{B_{1}}(A_{0}-()^{-1}f\cdot||_{?\{}^{2}$$\leqq||\sqrt{B_{1}}\{(A_{0}-\zeta)^{-1}-(A_{0}-\overline{\zeta})^{-1}\}\sqrt{B_{1}}||||f||_{H}^{2}$

Noting (4.14) which is(A5). Therefore

and
to
aud
tllc conclusion of Theore 111$\mathrm{c}11\dot{\mathrm{c}}\mathrm{u}1_{[mathring]_{8\supset}}^{\cdot}\mathrm{C}\mathrm{d}B_{0}$$\mathrm{w}\mathrm{c}1_{1\dot{\mathrm{e}}}\iota \mathrm{v}\mathrm{c}$$b(.\iota\cdot, y)$$B_{1}$$X_{\theta}$.

, respectively we getfor (4.11) and$3(\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{y} 4)$

68
ON EXISTENCE OF SCATTERJNG SOLUTIONSREFERENCES
appear1. –, Analyticity properties and estimates of resolvent kernels near thresholds, toin Comm.PDE.
of elastic waves in a perturbed isotropic half space
2. Y. Dermenjian-J.C. Guillot,
with a free boundary. The limiting absorption principle, Math. Meth. in the Appl. Sci. 10$Scatte\mathrm{r}\dot{\tau}ng$(1988), 87-124.

3. V. Enss, As ymptotic completeness for quantum mechanical potential scattering, Comm. Math.
Phys 61 (1978), 28521.
4. H.Isozaki-H.Kitada, Modified wave operators with time-independent modifiers, J.Fac.Sci.Univ.Tokyol
Sect.IA, Math. 32 (1985), 77-104.
H.Ito,
Extended Korn’s inequalities and the associated best possible constants, J. Elasticity
5.
24 (1990), 4378.
media,
6. M. Kadowaki, Asymptotic completeness for acoustic propagators in perturbed stratified
Integral Eq. Operator Th. 26 (1996), 432-459.
7. –, Low and high energy resolvent estimates for wave propagation in stratified media andtheir applications, to appear in J. Differential Equations.
8. T.Kato, Wave operator and similarity for some non-selfadjoint operators, Math. Annalen
162 (1966), 258279.
9. S. T. Kuroda, Spectral theory II, Iwanami, Tokyo, 1979. (Japanese)
51, Matematisk Insti10. –, An Introduction to Scattering Theory, Lecture Note Series
tut. Aarhus University, 1980.
11. P.D. Lax-R.S. Phillips, Scattering theory for dissipative systems, J.Funct. Anal. 14 (1973),$N^{o}$

172-235.

12. K. Mochizuki, Scattering theory for wave equations with dissipative terms, Publ. RIMS, Kyoto
Univ 12 (1976), 383390.
13. P. Perry, Mellin transforms and scattering theory I. Short range potentials, Duke Math. J47(1980), 187-193.

14. V. Petkov, Scattering theory for hyperbolic operators, North-Holland, Amsterdam, New york,

Oxford, Tokyo, 1989.
15. M. Reed -B. Simon, Methods of Modern Mathematical Physics, II. Fourier Analysis, SelfAdjointness, Academic Press, New York, San Francisco, London, 1975.
16. –, Methods of Modern Mathematical Physics, IX, Analysis of Operators, Academic
Press, New York, San Francisco, London, 1978.
17. B.Simon, Phase space analysis of simple scattering systems :extensions of some work ofEnss, Duke Math. J. 46 (1979), 11916.
tence of the scattering operator for dissipative hyperbolic sysP.
Stefanov-V. Georgiev,
18.
terns with variable multiplicities, J. Operator Theory 19 (1988), 217-241.
media,
19. R. Weder, Spectral and scattering theorry for wave propagation in perturbed stratified
1991.
Heidelberg,
Berlin,Applied Mathematical Sciences 87, Springer-Verlag, New york,
20. C. Wilcox, Sound propagation in stratified fluids, Applied Mathematical Sciences 50, SpringerVerlag, New york, Berlin, Heidelberg, 1984.
21. B. Zhang, Commutator estimate, Besov spaces and scattering problems for the acoustic wave
propagation in perturbed stratified fluids, Math. Proc. Camb. Phil. Soc. 128 (2000), 177-192.
$E\dot{m}$TOKYO METOROPOLITAN COLLEGE OF AERONAUTICAL EINGINEERING