数理解析研究所講究録 1411 巻 2005 年 48-58

48$L^{\mathrm{p}}$The role of local energy decay in-estimates for the wave equation with
time-dependent dissipation\dagger
松山登喜夫 (Tokio Matsuyama)

東海大学・理学部

Department of Mathematics
Tokai University

1. INTRODUCTION
This paper is the r\’ae um\’e of papers $[4, 5]$ . Some of proofs of theorems,
propositions and lemmas are omitted.
Let
be an unbounded domain having the compact and smooth) 0 be star-shaped with respect to the origin
boundary an, and let
for some po $>0$ , where we set $B_{\rho 0}=\{x\in$such that. We consider -estimates and scattering rates for thefollowing initial-boundary value problem in odd space dimension with$n>3$ :$\Omega$$\mathbb{R}^{n}$$\mathbb{R}^{n}\backslash \Omega\subset B_{\rho 0}$$\mathbb{R}^{n};|x|$$<\rho_{0}\}$$L^{\mathrm{p}}$

$n$

(P)

$\{u_{tt}-\Delta u+a(x,\mathrm{t})u_{t}=0u(x,0)=u_{0}(x),u_{t}(x,0’)=u_{1}(x)u(x,t)=0,,x\in\Omega(x,\mathrm{t})\in,\Omega\cross(0,\infty)(x,t)\in\partial\Omega \mathrm{x}(0,\infty$$(x, t)\in\partial\Omega\cross(0,\infty)$.

We make the following assumption on $a(x,t)$ :Assumption A. (i) $a(x,t)$ is nonnegative on(ii) $a(x,t)$ belongs to ”.
(iii) the support of $a(x, t)$ is contained in a$|23$$\overline{\Omega}\cross[0, \infty)$$ti\sqrt e$1$(t)\equiv$$\alpha$-dependent domain$\{ x\in\overline{\Omega};|x|<(R+t)^{\alpha}\}$. If
and withfor some
support of $a(x,t)$ is contained uniformly in$R>\rho_{0}$.

$(\overline{\Omega}\cross [0, \infty))$$0< \alpha<\frac{1}{2}$centered at the origin with radius$R$.

$\alpha=0,$$\Omega$

we assume that the,
being the ball$\cap B_{R}$$B_{R}$

The conditionmeans that the support of $a(x,t)$ expands at a speed strictly less than the wave speed. The equation ofthis kind was first treated by Tamura (see [14]), and it was provedthat if the data have compact supports, then the local energy decays
exponentialy. Since then, there is no work of asymptotic behaviour
for the problem (P). The difficulty of analysis lies in the fact that$0 \leq\alpha<\frac{1}{2}$tThis research was in part supported by Grant-in-Aid for Scientific Research (C)(2) (No. 16540205), Japan Society for the Promotion of Science.

4

the coefficient in the dissipative term depends on space time variables. For example, Wirth (see [16]) has treated delicately the equa$=0(\mu>0)$ through the Fourier representation
tion $\square u+\mu(1+t)$
formulae and obtained $If- L^{q}$ estimates
Recently, the present author obtained IPZAestimates and scattering
rates for the problem (P) (see [4, 5]). In deriving $IP$ -estimates we used
the time dependent cut-0ff method, which gives an extention of Shibata(see [11]) and Shibata and Tsutsumi (see [12]). As is well-known, thelocal energy decay plays a crucial role in this cut-0ff method. We
provide this estimate in Proposition 2.2 which the integral region is, and apply it to cut-0ff method.
given by time-dependent domain
In order to state results we introduce the notation of Sobolev norms :for $s\geq 1,$ we set$a$$-1u_{t}$$\Omega(t)$$I_{s}=||u_{0}$ $||Hs(\Omega)$ $+||u_{1}$ $||H^{\mathrm{a}-1}(\Omega)$$I_{s}^{(e)}=||e|$”$\mathrm{j}0||\mathrm{H}\mathrm{s}(\mathrm{C}1)+||e^{1}" u_{1}||_{H^{\mathrm{a}-1}(\Omega)}$,$e=$ the Napier number,,$I_{s}^{(\gamma)}=||\langle\cdot\rangle^{\gamma}u_{0}||_{H^{s}(\Omega)}+||\langle\cdot\rangle^{\gamma}u_{1}||H^{\mathrm{s}-1}(\Omega)$

where
and

$\langle x\rangle=\sqrt{1+|x|^{2}}$$\overline{D}u=$,$H^{s}(\Omega)$,is the ffactional order Sobolev space,$(\theta_{t}^{\dot{f}}\nabla^{\mu}u;j+|\mu|\leq 1)$,$Du=$ $(\theta_{t}^{j}\nabla^{\mu}u;j+|\mu|=1)$ ,$\Omega(L)=\Omega\cap B_{L}$$(L>0)$ .

Then we proved
Theorem 1 ([4]). Assume that Assumption is satisfied with $0<$
being
. Let $m$ be a nonnegative integer and set $M=[ \frac{n}{2}]+1,$
satisfy
the integer part of . Let the data ,$A$$[ \frac{n}{2}]$$\alpha<\frac{1}{2}$$u_{0}$$\frac{n}{2}$$u_{0}\in H^{3M+m}(\Omega)$,$u_{1}$$u_{1}\in H^{3M+m-1}(\Omega)$,$I_{3M+m}^{(e)}<\infty$,

and the compatibility condition of order $3M+m-1.$ Then the solutionof problem (P) satisfies the folloing estimates :Let be a number
with $2\leq p\leq\infty$ . Then there exists a constant such that
$p$

$u$

$C$

$||\overline{D}$u(t)$||1\mathrm{y}m,\mathrm{p}(\Omega)$$\leq CI^{(e)}(1+t)^{-\frac{n-1}{2}(1-\frac{2}{\mathrm{p}})}3M-\frac{2(M-1)}{\mathrm{p}}+m$.

Theorem 1 imposes the exponential weight on the initial data ,. This condition is too restrictive. For the case of $\alpha=0,$ i.e., the, we can relax it to
support of $a(x, t)$ is contained uniformly in$u_{0}$$u_{1}$$\Omega\cap B_{R}$$\theta$

50

polynomially weighted condition.

Our result reads as follows :Theorem 2 ([5]). Assume that Assumption is satisfied with $ae=0.$
satisfy
Let $m$ be a nonnegative integer. Let the data Uo,$A$$u_{1}$$4\in H^{3M+m}(\Omega)$,$u_{1}\in H^{3M+m-1}(\Omega)$,$I_{3M+m}^{(\gamma)}<$oowith $\gamma>n-1$ , and the compatibility condition of order$3M+m-1.$ Then the solution of problem (P) satisfies the following
estimates :Let be a number with $2\leq p\leq\infty$ . Then there eists
constant such thatfor some$\mathrm{y}$

$u$

$a$

$p$

$C$

tz(t)

$||\overline{D}$$||W^{m_{:}\mathrm{p}}(\Omega)$$\leq CI_{3M-\frac{2(M-1)}{\mathrm{p}}+m}^{(\gamma)}(1+t)^{-\frac{n-1}{2}(1-;)}$.

Based on Theorems 1 and 2, we can argue the existence of scattering
states and determine its asymptotic rates.
Theorem 3 ([4, 5]). Let tz be the solution in Theorems 1 and 2. Thenin with finite energy such that
there exists a free wave$w^{+}$$\Omega$$t$ $- \frac{n-2}{2}+\frac{\alpha(n-\delta)}{2})$$||\mathrm{z}\mathrm{c}(t)$$-w^{+}(t)||_{E}=\mathit{1}\mathit{6}$$OOO\{$ $t^{-\frac{\mathfrak{n}-2}{2}}\log^{\frac{1}{2}}(2+t^{a}))$$\mathrm{t}^{-\frac{n-2}{2}})$as$tarrow\infty$, where

$||E$

$||$

$||u(t)$$||\mathrm{K}$$w$(x, ) $=w_{0}(x)$ ,$0$$w(x,t)=0,$$\delta=n,$$\delta>n_{l}$is energy norm defined by$=$$\mathrm{z}$$\int_{\Omega}(|\nabla u(t)|^{2}+u_{\mathrm{t}}(t)^{2})dx$

if$\Omega$$w^{+}$.

satisfies the folloing$(x, t)\in l$ $\cross(0, \infty)$$w_{u}-\Delta w=0,$
$\{$

,

,

Here, we say thatis free wave ininitial-boundary value problem :$w^{+}$

if $1<\delta<n,$
if
if

,

$w_{t}(x,0)=w_{1}(x)$,$x\in\Omega$,$(x, t)\in$an,$\cross(0, \infty)$.

It is well-known in Mochizuki $[7, 8]$ (cf. Mochizuki and Naffizawa [9])that the energy does in general not decay. Thus the scattering problem
is meaningful. The proof of the last theorem depends deeply on
estimate. For the proof, see $[4, 5]$ .
$L^{\infty}-$51

2. Local ENERGY DECAY AND$L^{2}$-BOUNDLet
be fixed, and let $v(x, t;\tau)$ be a finite energy solution of
the following problem$\tau\geq 0$1$v(x,t\cdot,\tau)=0v_{u}-\Delta v+a(,x,\mathrm{t})v_{t}=0$,$(x,t)\in\Omega\cross(\tau,\infty)$$(x,t)$,(2.1)$\in\partial\Omega \mathrm{x}(\tau,\infty)$.
,of compact supports inwith the initial data$0<\alpha<\frac{1}{2}$[8,
Tamura]andfor
If we reconsider the proof of [14,Mochizuki] for $\alpha=0$ carefully, the folowing proposition can be obtained.
$f_{1}(x, \tau)$$f_{2}(x,\tau)$$\Omega(\tau)$

Let $v(x, t;\tau)$
Proposition 2.1. Suppose that Assumption is
be the finite energy solution of problem (2.1), and let , $L>\rho_{0}$ , be fixed.
Then there exist constants $C>0$ and ) $>0,$ independent of , and, such that for,$A$$sati\mathit{8}fied$

$L$

$R$

$L$

$t\geq\tau$$\tau$(2.2)$|\mathrm{t})v(t;\tau)||_{L^{2}(\Omega(L))}^{2}\leq Ce^{\lambda(L^{\beta}+(R+\tau)^{\alpha\beta})}e^{-\lambda(t-\tau)^{\beta}}||f(\tau)||_{E}$whereshould be replaced by$f(\tau)=\{f_{1}(\cdot, \tau), f_{2}(\cdot,\tau)\}$. If$\alpha=0,$

the right-hand side of (2.2)$Ce^{\lambda(L+R)}e^{-\lambda(t-\tau)}||f(\tau)||_{E}$Remark. The constant$\beta$,.

can be taken so that $\beta=(p+1)^{-1}$ with

, where $\alpha(2+\gamma)<1$ and $0<\gamma\leq 1.$ For.
details, see [14]. Therefore we must assume that $0\leq 0t$$p\geq\alpha(2+\gamma)(1-\alpha(2+\gamma))$$< \frac{1}{2}$Based on Proposition 2.1, we consider the following problem with
forcing term :$(\mathrm{P})_{f}\{u_{tt}-\Delta u+a(x,t)u_{t}=f(x,t)u(x,0)=u_{0}(x),u_{t}(x,0)=u_{1}’(x)u(x,t)=0,,x\in\Omega(x,t)\in,\Omega\cross(0,\infty)(x,\mathrm{t})\in\partial\Omega \mathrm{x}(0,\infty,)$

Proposition 2.2. Suppose that Assumption is satisfied. Let $m$ bewitha nonnegative integer, and let be the solution of problem$\{u_{0},u_{1},f(\cdot,t)\}\inH^{m+1}(\Omega)\crossH^{m}(\Omega)\crossH^{m}(\Omega)$,u1}
such thatdata
satisfy the compatibility condition of order $m$ and the supports of, respectively, whereand,are contained inand$\{ x \in\overline{\Omega}; |x|<(R+t)^{\alpha}+1\}$ .
should be replaced
If$\Omega(R+1)$. Assume
that
by
$A$

$u$

$(\mathrm{P})_{f}$$u_{0}$$f(\cdot, 0)$$u_{0}$$u_{1}$$\Omega(R)$$f(\cdot, t)$$\tilde{\Omega}(t)$$\alpha=0,\overline{\Omega}(t)$$\tilde{\Omega}(t)=$$fir\mathcal{H}her$$\sum_{\mathrm{j}=0}^{m}||\mathrm{n}f(t)||_{H^{m-\mathrm{j}}}(\Omega)\leq\{\begin{array}{l}\Lambda e^{-\lambda t^{\beta}}\Lambda(1+t)^{-\gamma}\end{array}$$if0< \alpha<\frac{1}{2}if\alpha=0$’52Then there exist constants ,andfor some constants, such that, independent of the diameters of supports of , ,for$\Lambda>0$$C$$\gamma>0.$$f(\cdot, 0)$$u_{1}$$u_{0}$$\lambda$$t\geq 0,$

if
if$||Du(t)||_{H^{m}(\tilde{\Omega}(t))}\leq Ce^{2\lambda R^{\alpha\beta}}I_{m+1}e^{-\lambda’t^{\beta}}+C\Lambda e^{2\lambda R^{\alpha\beta}}e^{-\lambda’t^{\beta}}$$\{$$|\mathrm{p})\mathrm{v}\mathrm{z}(t)||H^{m}(\Omega(R+1))$-”$\leq e^{2\lambda R}I_{m+1}e^{-\lambda t}+C\mathrm{y}e^{2\lambda R}(1+?)$$0< \alpha<\frac{1}{2}$$\alpha=0.$

For the proof see $[4, 5]$ .
The following local energy decay estimate of free waves in odd spacedimensions plays an important role in later discussion.
Proposition 2.3. Let v be the smooth solution of the following Cauchy
problem in odd space dimension n $=2p+1$ (p $=$1,2, \ldots ) :,$\{$$(x,t)\in \mathbb{R}^{n}\cross(0,\infty)$$v(x,0)=v_{0}(x)v_{tt}-\Delta v=0$,$v_{t}(x,0)=v_{1}(x)$,$x\in \mathbb{R}^{n}$.

,

Let $L>0$ be fied and $m$ a nonnegative integer. Then we have thefollowing assrtions :(i) There exist constants , and , independentof , such that for $t\geq 0,$$\lambda$$C$$\lambda’$

$L$

$||v$

(t)

$||Dv$,$||_{H^{m}(|x|<L)}\leq Ce^{\lambda’L}\mathcal{J}_{2M+m-1}^{(\epsilon)}e^{-\lambda t}$(t)$||_{H^{m}(|x|<L)}\leq Ce^{\lambda’L}\mathcal{J}_{2M+m}^{(e)}e^{-\lambda t}$

,

where
$J_{2M+m-1}^{(e)}=.

,\cdot\sum_{=00}^{1}\sum_{\leq|\mu|\leq m}\sum_{k=0}^{p-}\dot{.}||e^{|\cdot|}\partial_{r}^{k}\nabla^{\mu}v_{\dot{*}}(\cdot)||_{H^{M}(\mathrm{B}^{n})}$$J \mathrm{z}\%_{+m}=\sum 1$$:=\mathrm{Q}$$( \sum||e^{|\cdot|}\partial_{f}^{k}\nabla^{\mu}\nabla v_{*}.(\cdot)||_{H^{M}(\mathrm{R}^{n})}\mathrm{p}-\dot{l}$$\sum$$0\leq|\mu|\leq m$$+-$ $\sum_{k=0}^{p+1-}.\cdot||e$. ,$\nabla$$\partial_{t}=\frac{x}{|x|}$

’

$k=0$

$|\mathrm{c}?\mathrm{j}\nabla’ \mathrm{t}\mathrm{z}_{\mathrm{i}}()|\mathrm{L}M(\mathrm{C})$$)$.

(ii) Let be a constant with $\gamma>n-1$ . Then there exist a constant$C(L)$ depending onsuch that for $t\geq 0,$$\mathrm{y}$$L$$||v4t)||H^{m}(|x|<L)$$\leq C(L)J_{2M+m-1}^{(\gamma)}(1+t)^{-\gamma+\frac{n-1}{2}}$,.

$|\mathrm{I})v(t)||_{H^{m}(|x|<L)}\leq C(L)J_{2M+m}^{(\gamma)}(1+t)^{-\gamma+\frac{\mathfrak{n}-1}{2}}’$

,

53

where
$J_{2M+m-1}^{(\gamma)}= \sum 1$$\sum$$\sum||p-i$$($.

$)^{\gamma}\partial$

rv$\mu v_{i}(\cdot)||_{H^{M}(1\mathrm{B}^{n})}$,$:=00\leq|\mu|\leq mk=0$$J_{2}^{(\mathrm{m}_{+m}=\sum}1$$( \sum^{p-}.\cdot||\langle\cdot\rangle^{\gamma}\partial_{r}^{k}\nabla^{\mu}\nabla v_{i}(\cdot)||_{H^{M}(\mathrm{R}^{n})}$$\sum$$=00\leq|\mu|\leq m$$+ \sum_{k=0}^{\mathrm{p}+1-:}||$$k=0$( ’(9”7’t7: $()$$\cdot\rangle$$||_{H^{M}(\mathrm{R}^{n})}$).

For the proof see $[4, 5]$ ,The final proposition is concerned with an
Proposition 2.4. Let
problem :$v$$L^{2}$-estimate.

be the smooth solution of the following Cauchy$(x,t)\in \mathrm{R}^{n}\mathrm{x}(0,\infty)$$\{$$v_{tt}-\Delta \mathrm{n}=0v(x,0)v_{0}(x’)$,$v_{t}(x,0)=v_{1}(x)$,$x\in \mathrm{R}^{n}$.

,

Let $m$ be a nonnegative integer. Then there exists a constant $C>0$
such that for
$t$

(2.3)

$\geq 0,$$||\overline{D}v(t)||_{L^{\mathrm{z}}(\mathrm{i}^{n})}$$\leq C(||(\overline{D}v)(0)||_{L^{2}(\mathrm{R}^{n})}+||v_{1}||L$A$(\mathrm{R}^{\mathfrak{n}}))$For the proof see Ikehata [1] (cf. Dcehata and Matsuyama [2]).

3. $IP$ -ESTIMATES
In this section we give an outline of proof of Theorems 1 and 2. The
existence theorem is wel known in [6, Mizohata].
We use the cut-0ff method as in Shibata and Tsutsumi [12] andNakao [10]. Let be any fixed number with $L>R+1.$ Let us takeif $|x|\leq L+1$ and $=1$ if
so thata smooth function$|x|\geq L+2.$ Then the solution tz of problem (P) can be expressed$u=i$ +\^u, where and \^u are solutions of the following problems (P)$L$$/\mathrm{i}(\mathrm{x})$$\mathrm{f}\mathrm{i}(\mathrm{x})=0$$\underline{\mathrm{b}\mathrm{y}}$$\tilde{u}$and), respectively :$\overline{(\mathrm{P}}$$\overline{(\mathrm{P}})$$\{\begin{array}{l}\overline{u}_{ff}-\Delta\tilde{u}+a(x,t)\tilde{u}_{t}=0,(x,t)\in\Omega\cross(0,\infty)\tilde{u}(x,0)=\mu(x)u_{0}(x),\tilde{u}_{t}(x,0)=\mu(x)u_{1}(x),x\in \mathrm{l}\overline{u}(x,t)=0,(x,\mathrm{t})\in\partial\Omega\cross(0,\infty)\end{array}$

54

and
(P)

x

$\{\begin{array}{l}\hat{u}_{tt}-\Delta\hat{u}+a(x,t)\hat{u}_{t}=0,(x,t)\in\Omega\cross(0,\infty)\hat{u}(x,0)=(1-\mu(x))u_{0}(x),\hat{u}_{t}(x,0)=(1-\mu(x))u_{1}(x)\hat{u}(x,t)=0,(x,t)\in\partial\Omega \mathrm{x}(0,\infty)\end{array}$$\in\Omega$

We need the next estimates.
Proposition 3.1. For the solution iconstant C $>0$ such that for tof problem (P), there exists$a$$\geq 0,$$4^{\cdot}f0< \alpha<\frac{1}{2}$$||\overline{D}\tilde{u}(t)_{W^{m,\infty(\Omega)}}\leq CI_{3M+m}^{(e)}(1+t)||^{\frac{||}{D}}\tilde{u}(t)||_{H^{m}(\Omega)}\leq CI_{2M+m+1}^{(e)}$$- \frac{n-1}{2}\}$$||\overline{D}\mathrm{i}(t)||_{W^{m,\infty}(\Omega)}\leq CI_{3M+m}^{(\gamma)}(1+t)^{-\frac{n-1}{2}\}}$

i(t)lH$||\overline{D}$$m(\Omega)\leq CI\mathit{2}R+m$

if,$\alpha=0.$

a $+1$For the proof of Proposition 3.1, we consider the following Cauchy
problem :(CP)$v_{tt}-\Delta v=0,$$(x,t)\in \mathbb{R}^{n}\cross(0,\infty)$$\{$$v(x,0)=$$/\mathrm{J}(x)\mathrm{f}\mathrm{J}_{0}(x)$,,$\mathrm{f}7_{t}(x,0)=\mathrm{j}(x)\mathrm{t}\mathrm{z}_{1}(x)$,$x\in \mathrm{R}^{n}$.

Then it is known (cf. Klainerman [3] and W. von Wahl [15]) that if$n\geq 2,$ then the solution
to the problem (CP) satisfies
$v$

(3.1)
$||v(t)||L"(\mathrm{i}")$$\leq C(1+t)^{-\frac{n-1}{2}}(||v(0)||_{W^{M,1}(\mathrm{R}^{n})}+||v_{t}(0)||_{W}\mathrm{y}-1_{:^{1}(\mathrm{u}}n))$Now let us take a smooth function so that $\psi(x)=1$ if $|x|\geq R+1$
satisfies the following Cauchy problem :and $=0$ if $|x|\leq R.$ Then$\psi v$$(\psi v)_{u}-\Delta(\psi v)=g(x,t)$$\{$$(\psi v)(x,0)=\mu(x)u_{0}(x)$)
where we set, we see that$w=$
problem :$\mathrm{g}\{\mathrm{x},\mathrm{t}$$(\mathrm{P})_{w}\{$,$w$$(x,t)\in \mathrm{R}^{n}\mathrm{x}(0,\infty)$,$(\psi v)_{t}(x,0)=\mu(x)u_{1}(x)$.

,$x\in \mathrm{R}^{n}$,v. Then, setting
satisfies the following initial-boundazy value$=-2\nabla \mathrm{t}\mathrm{q}$$\mathrm{i}-tpv$,$\nabla v-(\Delta\psi)v+2\psi_{t}v_{t}t^{v}p_{tt}$$w_{tt}-\Delta w+a\{x,t)w_{t}=-g(x,t)$ ,$(x,t)\in\Omega\cross(0,\infty)$$w(x,0)=w_{t}(x,0)=0,$$x\in\Omega$$w(x,t)=0,$$(x,t)\in\partial\Omega\cross(0,\infty)$

,

,

.

and $\Omega(R+1)$ , combinFor the local energy of rp in the domains
ing Propositions 2.2, 2.3, 2.4 with Poincare’s inequality, we have thefollowing estimates.
$\tilde{\Omega}(t)$,55

Proposition 3.2. Let be a solution of problem. Then we havethe following assertions: (i) If, there exists a constant $C>0$
such that for)$(\mathrm{P})_{w}$$\mathrm{m}\mathrm{z}$$0< \alpha<\frac{1}{2}$$t\geq 0_{f}$$||\overline{D}$w(t)$||\overline{D}$where$\gamma>n-$$R${$\tilde{\Omega}(t)=$

$1$

. If

tp(t)

$\alpha=0,$w(t)$||\overline{D}$$\leq CI_{3M+m}^{(e)}e^{-\lambda’t^{\beta}}$”,$||_{H^{m}(\tilde{\Omega}(t))}\leq CI_{2M+m}^{(e)}e_{j}^{-\lambda’t^{\beta}}$(ff $+t)^{\alpha}+1$ } . (ii) Let be a number withthen there eists a constant $C(R)$ depending on$x\in\overline{\Omega};|x|<$such that for $t\geq 0,$$||\overline{D}$$||_{W^{m},(\mathrm{i}(t))}$$\gamma$,$||_{W^{\mathrm{m},\infty}(\Omega(R+1))} \leq C(R)I_{3M+m}^{(\gamma)}(1+t)^{-}\mathrm{y}+\frac{n-1}{2}$w(t)$||H^{m}(\Omega(R+1))$$\leq C(R)I_{2M+m}^{(\gamma)}(1+t)^{-\gamma+\frac{n-1}{2}}$We must estimate $w(t)$ outside of the domains il(t) and $\Omega(R+1)$ .
For this purpose, we set
Then satisfies the following Cauchy
problem :$\overline{w}=\psi w.$$\{$$\tilde{w}$$\tilde{w}(x,0)=\tilde{w}_{t}(x,0)=0\tilde{w}_{u}-\Delta\overline{w}=g(x,t)+\tilde{g,}(x,t)$,$\mathbb{R}^{n}\cross(0,\infty)$$x\in \mathbb{R}^{n}(x,\mathrm{t})\in$,,

where $\tilde{g}(x,t)=2\psi_{t}w_{t}+$ CatJ) $-2\nabla\psi$ $7w$ $-(\Delta\psi)w$ . It follows fromandDuhameFs principle and the decay estimate (3.1) that
norms ofare estimated as follows :$L^{2_{-}}$$L^{\infty}$$\overline{D}\tilde{w}(t)$Lemma 3.3. Let $m$ be a nonnegative integer. Then we have the folsuch, there exists a constantlowing assertions :(i) If$C$$0< \alpha<\frac{1}{2}$that$||\overline{D}\tilde{w}0)||_{W}\mathrm{v}m$$||\overline{D}\mathrm{J}$$(_{\dot{\mathrm{u}}})\backslash$

for

If

$\alpha=0,$, $\infty(2^{n})$ $\leq CI_{3M+m}^{(e)}(1+t)^{-\frac{n-1}{2}}$$(t)||_{H^{m}(\mathrm{i}^{n})}\leq CI_{2M+m+1}^{(e)}$,.

there exists a constant $C(R)$ depending on$R$such that$t\geq 0_{f}$$||\overline{D}\tilde{w}(t)||_{W^{m,\infty}}(\mathrm{i}^{n})$$\leq CI_{3M+m}^{(\gamma)}(1 +t)^{-\frac{n-1}{2}}$$||\overline{D}\tilde{w}(t)||Hm(^{\mathrm{g}n})$$\leq CI_{2M+m+1}^{(\gamma)}$,.

The following proposition is an immediate consequence of Lemma 3.3,
andand we can obtain estimates of $w(t)$ outside of the domains$\Omega(R+1)$ as follows:$\tilde{\Omega}(t)$Proposition 3.4. Let $m$ be a nonnegative integer. Then we have the, there exists a constant $C$ such
follo wing assertions :(i) If$0< \alpha<\frac{1}{2}$

56

that
$||\overline{D}\mathrm{J}$$(t)||_{W^{m,\infty}\mathrm{B}(t)^{\mathrm{c}})}\leq CI_{3M+m}^{(e)}(1+t)^{-\frac{n-1}{2}}$$||\overline{D}$i(t)$||_{H}$.,, $(\tilde{\Omega}(t)\cdot)\leq CI^{(e)}2M+m+1$ .

(ii) If $\alpha=0,$ there exists a constant $C(R)$ depending on $R$ such thatfor$t\geq 0,$$||\overline{D}\tilde{w}(t)||W^{m,\mathrm{o}\mathrm{o}}(\mathrm{O}(R+1)c)$$||\mathrm{j}$$\leq CI_{3M+m}^{(\gamma)}(1+t)^{-\frac{n-1}{2}}$$w(t)||_{H^{m}(\mathrm{p}(R+1)^{\mathrm{c}})}$

E$CI_{2M+m+1}^{(\gamma)}$,.

We are now in a position to prove Proposition 3.1.
Proof of Proposition 3.1. It suffices for our purpose to prove the, since the casecan be handled in a similar way.
case
It follows from Propositions 3.2 and 3.3 that$\mathrm{a}=0$$0< \alpha<\frac{1}{2}$$||\overline{D}w(t)||_{W^{m,\infty}(\Omega)}\leq||\overline{D}w(t)||_{W^{m,\infty}(\tilde{\Omega}(t))}+||\overline{D}w(t)||_{W^{m,\infty}(\tilde{\Omega}(t)^{\mathrm{C}})}$$(3.2)$$\leq CI_{3M+m}^{(e)}(1+t)^{-\frac{n-1}{2}}$,$||\overline{D}w(t)||_{H^{m}(\Omega)}\leq||\overline{D}w(t)||_{H^{m}(\tilde{\Omega}(t))}+||\overline{D}w(t)||_{H^{m}(\tilde{\Omega}(t)^{c})}$(3.3)$\leq CI_{2M+m+1}^{(e)}$.

Notice that$||/" 0||_{W\dagger}$”$m+1,1(\mathrm{R}n)$$+||/’ u_{1}$ $||\mathrm{s}^{M+}-,1(\mathrm{I}\mathrm{R}^{n})$$\leq CI_{M+m+1}^{(\mathrm{e})}$”$||/’ u1||L \neg n^{2n}\mathrm{z}_{(\mathrm{R}^{n})}\leq C(\int_{\Omega}e^{2|x|}u_{1}^{2}dx)^{\frac{1}{2}}=C||e|$,$u_{1}||L^{2}(\mathrm{O})$.

Then we see from the decay estimate (3.1) and Proposition 2.4 that$||D(\psi v)(t)|\mathrm{h}m,"(\mathrm{r})$(3.4)$\leq C(||\mu u_{0}||_{\%^{\mathrm{w}+m+1,1}}(\mathrm{R}^{n})+||\mu u_{1}||_{\mathrm{r}^{\mathrm{y}+m,1}(\mathrm{R}^{n})})$$\leq$(3.5)CIu)+m+lCl$+t)^{-\frac{n-1}{2}}$$(1+\mathrm{t})^{-\frac{n-1}{2}}$,$||\overline{D}(\psi v)(t)||_{L^{2}(\mathrm{B}^{n})}\leq C(I_{m+1}+||e^{\mathrm{j}}" u_{1}||_{L^{2}}(\Omega))$

Since $\tilde{u}=w+\psi v,$ we combine (3.2) and (3.3) with (3.4) and (3.5),
Clrespectively, to complete the proof of Proposition 3.1.
Next, we introduce the decay for the solution$i$of problem$\overline{(\mathrm{P})}$.

57

Proposition 3.5. For a nonnegative integer $m$ , there exists a constant$C>0$ such that for $t\geq 0,$$||\mathrm{Z}$\^u(t)$||W^{m}:"(*)$ $\leq CI_{M+m+1}(1+t)^{-\frac{n-1}{2}}$$||\overline{D}\hat{u}(t)||H^{m}(\mathrm{p})$$\leq CI_{M+m+1}$$||D\hat{u}(t)||_{H^{m}(\Omega)}\leq
CI_{M+m+1}$.

The proof of this proposition can be given much easier than theprevious arguments, if we note that the supports of data S(0),are compact. Thus we may omit the details.
$u$$\hat{u}_{t}(0)$Proof of Theorems 1 and 2 completed. It follows from Proposi, thentions 3.1 and 3.5 that if$0< \alpha<\frac{1}{2}$(3.6)$||\overline{D}u(t)$ $||W^{m},"(\Omega)$

(3.7)

tj(t)

$||\overline{D}$Interpolating between
tain L2-estimateu(t)$||\overline{D}$$L^{\infty}$$||W^{\mathrm{v}\mathrm{n},\mathrm{p}}(\Omega)$$\leq 1$

$I_{3M+m}^{(e)}(1+t)^{-\frac{\mathfrak{n}-1}{2}}$$||_{H^{m}(\Omega)}\leq I_{2M+m}^{(e)}$-estimate (3.6) and,.

$L^{2}$-estimate (3.7), we ok$\leq CI_{3M-\frac{2(M-1)}{\mathrm{p}}+m}^{(e)}(1+t)^{-\frac{n-1}{2}(}$”.

For more details see [4]. As to the case $\alpha=0,$ we can prove thesame argument as above (see [5]). The proofs of Theorems 1 and 2 arecomplete.
REFERENCES[1] R. Ikehata, Small data global eistence of solutions for dissipative wave equations in an exterior domain, Funkcial. Ekvac. 45 (2002), 259- 269.
[2] R. Ikehata and T. Matsuyama, Remarks on the behaviour of solutions to thelinear wave equations in unbounded domains, Proc. Sch. Sci. Tokai Univ. 36(2001), 1–13.
[3] S. Klainerman, Global existence for nonlinear wave equations, Comm. Pure
Appl. Math. 33 (1980), 43- 101.
[4] T. Matsuyama, $IP$ -estirnates for the wave equation with time-dependent dissipation, to appear in Proc. Roy. Soc. Edinburgh Sect. A.
[5] T. Matsuyama, IP estimates and scattering rates for the wave equation with
time-dependent dissipation, Nonlinear Partial Differential Equations and Their
Applications (Shanghai, 2003), 159- 174, GAKUTO Internat. Ser. Math. Sci.
Appl., 20, Gakk\={o}tosho, Tokyo, 2004.
[6] S. Mizohata, Quelque problemes au bord, du type rnide, pour des iquations
hyperboliques, Seminair sur les \’equations aux deriv es partielles, College deFrance (1966/1967), 23 60.
[7] K. Mochizuki, Scattering theory for wave equations with dissipative tems,
Publ.
. Inst. Math. Sci. 12 (1976), 383 390.
[8] K. Mochizuki, Scattering theory for wave equations (in Japanese) , Kinokuniya,$\mathrm{e}$-${\rm Res}$

1984.

-

[9] K. Mochizuki and H. Nakazawa, Energy decay and asymptotic behavior of s0. Inst. Math.
lutions to the wave equations with linear dissipation, Publ.
Sci. 32 (1996), 401 414.
estimates for the linear wave equation and global eistence for[10] M. Nakao,
semilinear wave equations in an exterior domain, Math. Ann. 320 (2001), 11${\rm Res}$-$L^{\mathrm{p}}$-31.

[11] Y. 0hibata, On the global eistence of classical solutions of second order fully
hyperbolic equations with first 0 rder dissipation in the exterior d0$mai\eta$ Tsukuba J. Math. 7 (1983), 1-68.
[12] Y. Shibata and Y. Tsutsumi, Global existence theorem of nonlinear wave equations in the exterior domain, Lecture Notes in Num. Appl. Anal. 6 (1983), 155$non_{u\acute{l}}^{}’ iear$-Holland.
-196,[13] Y. Shibata and Y. Tsutsumi, On a global existence theorem of small amplitude
solutions for nonlinear wave equations in an exterior domain’ Math. Z. 191(1986), 165 199.
[14] H. Tamura, Local energy decays for wave equations with time-dependent coefficients, Nagoya Math. J. 71 (1978), 107- 123.
-decay rates for ho mogeneous wave equations, Math. Z. 120[15] W. von Wahl,(1971), 93 106.
[16] J. Wirth, Solution representations for a wave equation with weak dissipation,
Math. Methods Appl. Sci. 27 (2004), 101 124.
$\mathrm{K}\mathrm{i}\mathrm{n}\mathrm{o}\mathrm{k}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{y}\mathrm{a}/\mathrm{N}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{h}$-$L^{\mathrm{p}}$-–