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Global Existence Theorem for Nonlinear Wave Equation in Exterior Domain
L e c t u r e N o t e s in Num. Appl. Anal., 6, 155-196 (1983) Recent Topics in Nonlinear PDE, Hiroshima, 1983
Global Existence Theorem for Nonlinear Wave Equation in Exterior Domain Yoshihiro SHIBATA* and Yoshio TSUTSUMI**
*
*Department of Mathematics, University of Tsukuba Ibaraki 306, Japan **Department of Pure and Applied Sciences, College of General Education, University of Tokyo Tokyo 113, Japan Supported in part by the Sakkokai Foundation.
81.
Introduction.
The g l o b a l e x i s t e n c e o f s o l u t i o n s f o r t h e n o n l i n e a r wave e q u a t i o n has been extensively studied. improvement r e c e n t l y .
F o r t h e Cauchy problem Klainerman [ Z ] has made a remarkable That i s , he showed t h a t i f t h e s p a t i a l dimension i s n o t
s m a l l e r than 6 and i n i t i a l d a t a a r e s m a l l and smooth, then t h e Cauchy problem f o r t h e f u l l y n o n l i n e a r wave e q u a t i o n has a unique c l a s s i c a l g l o b a l s o l u t i o n .
On t h e o t h e r hand i t i s i m p o r t a n t t o c o n s i d e r t h e i n i t i a l boundary v a l u e problem f o r t h e n o n l i n e a r wave e q u a t i o n i n an e x t e r i o r domain i n o r d e r t o s t u d y s c a t t e r i n g o f a r e f l e c t i n g o b j e c t f o r t h e n o n l i n e a r wave equation.
I n t h e p r e s e n t paper we
s h a l l prove t h a t i f t h e s p a t i a l dimension i s n o t s m a l l e r than 3 and i n i t i a l data a r e small and smooth, then we have t h e g l o b a l unique e x i s t e n c e theorem o f c l a s s i c a l solutions f o r
a
l a r g e c l a s s o f n o n l i n e a r wave equations i n e x t e r i o r domains
w i t h t h e homogeneous O i r i c h l e t boundary c o n d i t i o n , which i n c l u d e s t h e n o n l i n e a r v i b r a t i o n equation.
n
Let
be an unbounded domain i n lRn
,n2
3, w i t h Cm and compact boundary an.
We denote a t i m e v a r i a b l e by t o r xo and a space v a r i a b l e by x = (x1,.-.,xn), respectively. o r ,a,
(*)
a . and ,;a J
We s h a l l a b b r e v i a t e a / a t , a/ax. and (a/axl)al...(a/axn)an
J
r e s p e c t i v e l y , where
a
i s a multi-index w i t h
Supported i n p a r t by t h e Sakkokai Foundation. 155
101
to a t
= al+***+a
n
Yoshihiro SHIBATA and Yashio TSUTSUMI
166
.
and j = l,...,n
We s h a l l consider t h e f o l l o w i n g problem:
u = o
on [OP)
~ ( 0 ~ =x 1$o(X), where
2
t11= at
-
A =
a t2
x
an,
(atu)(o,x)
- j.11n a Jz. and
= $,(XI
hu = (aiu,
i n n,
a J. a ku, j,k=O,.-.,n).
i=O,.-.,n;
Before we s t a t e assumptions and t h e main theorem, we s h a l l g i v e n o t a t i o n s .
For any i n t e g e r N 2 0 we w r i t e
L e t 6 be an a r b i t r a r y open s e t i n Rn
.
F o r any p w i t h 1 5 p 5
standard Lp space d e f i n e d on p and i t s norm by Lp(S) and
ml
II.llB,py
we denote t h e respectively.
For a v e c t o r v a l u e d f u n c t i o n h = ( h l y . - * , h s ~ we p u t 2 l h I 2 = l h 1 I 2 + - . * + l h s l , ~ ~ h ~= / 1 ~ y~ p~ h j ~ ~ ~ , p j=l We a l s o w r i t e
We s e t HP~ ( c - ) = t f
E
L’’(F)
;
IIfll,,p,N
<
m
1. Note t h a t Ho(e)= Lp(&) p a r t i c u l a r l y .
L e t robe a f i x e d p o s i t i v e constant such t h a t an C I x r > ro we denote t h e subset I x
E
n ; 1x1
< r 1 by
nr.
E
P R n ; 1x1 < ro I .
For
F o r any r > ro and any
i n t e g e r k 2 1 we p u t 2 ~,(n) = I u
E
~ ~ ( ;n supp ) uct x
E
R” ; 1x1 5 r I 1,
Ok Hr(n)
E
k Hz(n) ; supp u
E
R n ; 1x1 5 r 1,
I u
c( x
“0 2 kle s h a l l sometimes use Hr(n) = lr(n). (u,v),
and t h e D i r i c h l e t norm
llullD by
a:ulas2 = 0 (la1 L k-111.
We d e f i n e t h e D i r i c h l e t i n n e r p r o d u c t
Nonlinear Wave Equation in Exterior 1)wnain
157
we denote t h e completion o f $(n) i n t h e D i r i c h l e t norm. By ,uN(&) we N denote t h e s e t o f C ( b ) - f u n c t i o n s having a l l d e r i v a t i v e s o f o r d e r 5 N bounded i n E . F o r two Banach spaces X and Y we denote t h e Banach space c o n s i s t i n g o f a l l
By H,(Q)
bounded l i n e a r o p e r a t o r s from X t o Y and i t s norm by B(X,Y) and II-II ]B(X,Y) ' 1 r e s p e c t i v e l y . For an i n t e r v a l I(.- R and a Banach space X we denote t h e s e t
o f m-times c o n t i n u o u s l y d i f f e r e n t i a b l e X-valued f u n c t i o n on I by Cm(I;X).
We s e t
For 1 ;< p 5 =, a nonnegative number k and a nonnegative i n t e g e r N we w r i t e
hL =
I
U
c IELr\CL-l([O,-);H,(s2));
L atu(O,x)
= 0 1.
F o r s i m p l i c i t y we a l s o use t h e a b b r e v i a t i o n s :
f(c)
= f(cl,-*-.~n)
where xc = x
1 1
v . = (v~,.**,vJ)
=
+...+xncn.
IRn
e x p ( - a x c ) f ( x ) dx,
For p o s i t i v e integers
5,
i, v e c t o r s u = (u, . * . . , u s ) ,
( 1 5 j 2 i) and a s c a l a r f u n c t i o n H(t.x,u)
J .
(dLH)(t,x,u)(vl
1
,*-*.vi)
by
(V
E
IRs) we d e f i n e
Yoshihiro SHIRATA and Yoshio TSUTSUMI
158
We s h a l l make t h e f o l l o w i n g assumptions. Assumption 1.1. (2) m
The s p a t i a l dimension n 2 3.
(1)
The n o n l i n e a r mapping F i s a r e a l - v a l u e d f u n c t i o n belonging t o
([0,-)
x
sl
x
{ A
E
R 2(n+1)
.> I XI
5 - 1 I).
(3) F(t,x,A)
O ( 1 ~ 1 ~ ) near
A =
0(1~1~)
x
near
0,
= 0,
if n 1 6 , if 3 5 n
5.
The e x t e r i o r domain n i s "non-trapping" i n t h e f o l l o w i n g sense:
(4)
G(t,x,y)
=
Let
be t h e Green f u n c t i o n f o r t h e f o l l o w i n g problem
2 (at
-
A ~ ) G= 0
i n ( 0 , ~ ) x n,
where y i s an a r b i t r a r y p o i n t i n n and ax i s t h e Laplace o p e r a t o r w i t h r e s p e c t t o x.
L e t a and b be a r b i t r a r y p o s i t i v e constants such t h a t b 2 a 2 ro. F o r
f o r any v
E
L:(n),
Remark 1.1.
where To depends o n l y on n, a, b and n.
I t is w e l l known t h a t i f t h e complement of B i s convex, t h e n
Assumption 1.1(4) i s s a t i s f i e d (see, e.g.,
Melrose [5]).
1.io
Nonlinear Wave Equation i n Exterior Dxnain
Now we s h a l l s t a t e t h e main theorem.
Theorem 1.1.
L e t m be an a r b i t r a r y i n t e g e r w i t h m
(Existence).
G.
Let
Assumption 1.1 be a l l s a t i s f i e d . 1)
P u t m = 2max(4[n/2]+7,
m+l) t 4[n/2]
4-
1. I f n 2 6, then t h e r e e x i s t
p o s i t i v e constants a and 6 o having t h e f o l l o w i n g p r o p e r t i e s : $1 E.6 2h[n/21+2(5)
and
f
€3:
2m+[n/21+1([0,-)
x
i)
If
@o
, €1,.
2m+[n/2]+3(;)
s a t i s f y f o r some 6 w i t h 0.
6 ~6~
and t h e c o m p a t i b i l i t y c o n d i t i o n o f o r d e r m , then Problem (M.P) has a s o l u t i o n
u
E
Cm +2
([o,-)
x
IAu12,0,m 2)
+
ii)
satisfying
lAu14,(n-1)/4,m
6’
Put m = 2max(3[n/2]+6,
m+l) + 3[n/2]
+ 7.
If 4
n 5 5, then t h e r e e x i s t
p o s i t i v e constants a and 6 o having t h e f o l l o w i n g p r o p e r t i e s :
@1 E J ~ ‘ ~ ’ ( ; )
and f
Il@OIlm,2iT;+2
+
E F2Fn ([0,-)
1141 Ilm,2iii+l
x
+
5)
s a t i s f y f o r some 6 w i t h 0
Iflm,o,2in‘
If
o0
< 6
E),
2m+ 2
(E),
A0
2 a6
and t h e c o m p a t i b i l i t y c o n d i t i o n of o r d e r ‘m, then Problem (M.P) has a s o l u t i o n
u
E
c ~ + ~ ( [ o x, i) ~ )satisfying IAU12,0,m
3)
$2
Let
+
E
IAUl-,(n-1
)/z.in
= <
6.
be a p o s i t i v e c o n s t a n t w i t h 0 <
3$+(3m +
7 ) ~+ ] 3[n/2]
+ 6.
E
5
1
, and
in
an i n t e g e r w i t h
I f n = 3, then t h e r e e x i s t p o s i t i v e constants
,
Yoshihiro SHIBATA and Yoshio TSUTSUMI
160
a and A0 having t h e f o l l o w i n g p r o p e r t i e s :
f
2%
~2
([I),-)x
5 ) s a t i s f y f o r some
I f $o
6 with 0 < 6
,i
E,,
5
2$+2
(z), o1
E$,*~'(;)
and
60
and t h e c o m p a t i b i l i t y c o n d i t i o n o f o r d e r m, t h e Problem (M.P) has a s o l u t i o n
u
E
Cm " ([G,-)
x
$) s a t i s f y i n g
(Uniqueness). C3([0,-) I*ulm,o,o
i) a r e
x
= <
6 1 and
Remark 1.2.
There e x i s t s a small c o n s t a n t 6 , > 0 such t h a t i f u, v
E
two s o l u t i o n s o f Problem (M.P) f o r t h e same data k i t h
I"lm,o,o (1)
2 1, then u
= v.
For t h e c o m p a t i b i l i t y c o n d i t i o n , see 54.2 and Mizohata
[6l. (2)
Since t h e n o n l i n e a r f u n c t i o n F i s d e f i n e d o n l y i n [ 0 , m ) ; Ihl
x
5
I
A E
1 I , we always assume t h a t I A U ~ ~ = ,< ~1, , when ~ we c o n s i d e r a
s o l u t i o n u o f Problem (M.P).
One o f t h e d i f f i c u l t i e s i n t h e p r o o f i s t h a t the l o s s o f d e r i v a t i v e s occurs a t each s t e p i n t h e i t e r a t i o n .
E s p e c i a l l y n o t e t h a t t h e n o n l i n e a r term F a l s o
depends on t h e d e r i v a t i v e of o r d e r 2 w i t h r e s p e c t t o time t i n our problem.
For
t h e Cauchy problem we can overcome such a d i f f i c u l t y by reducing a f u l l y n o n l i n e a r e q u a t i o n t o a q u a s i l i n e a r equation, f o l l o w i n g Oionne [I] (see a l s o Klainerman and Ponce [3] and Shatah [ l o ] ) .
For t h e i n i t i a l boundary v a l u e problem, however,
such methods a r e n o t a p p l i c a b l e . case n = 3.
Furthermore, t h e l o s s of decay occurs i n t h e
I n o r d e r t o overcome such d i f f i c u l t i e s , we s h a l l make use o f t h e
s o - c a l l e d Nash-Moser technique.
Our s t r a t e g y f o l l o w s Klainerman [ 2 ] and Shibata
Nonlinear Wave Equation in Exterior Dnmain
161
[12] (see a l s o Rabinowitz [9]).
A u n i f o r m decay e s t i m a t e and an L 2 - e s t i m a t e f o r a l i n e a r i z e d problem w i l l p l a y an i m p o r t a n t r o l e i n t h e p r o o f .
I n p a r t i c u l a r , t h e r e s u l t s o f decay estimates
a r e new and a r e proved i n t h e same way as Shibata [12] and Tsutsumi [13].
Tools
used i n a p p l y i n g t h e Nash-Moser technique, such as an i n t e r p o l a t i o n i n e q u a l i t y between a f a m i l y o f c e r t a i n semi-norms and a proper smoothing o p e r a t o r , a r e t h e same as those used i n Shibata [ll,121. Now we g i v e a well-known example, i.e.,
" t h e n o n l i n e a r v i b r a t i o n equation":
Example.
I n t h e course o f t h e p r o o f below a l l constants w i l l be s i m p l y denoted by C. In particular, C =
C(*,.--,*) w i l l denote a c o n s t a n t depending on t h e q u a n t i t j e s
appearing i n parentheses.
52. Uniform Decay Estimate.
In t h i s s e c t i o n we s h a l l
show a u n i f o r m decay e s t i m a t e o f s o l u t i o n s f o r t h e
f o l l o w i n g l i n e a r problem:
(2.1)
t_:iu
= f
u = o
in
[O,m)
x
R,
an
[a,-)
x
an,
Throughout S e c t i o n 2 we always assume t h a t t h e data $o, $1 and f o f t h e e q u a t i o n ( 2 . 1 ) a r e so n i c e f u n c t i o n s t h a t a l l t h e i r norms and semi-norms appearing below a r e bounded.
We d e f i n e u j ( x ) ( j 2 0 ) s u c c e s i v e l y by
162
Yoshihiro SHIBATA and Yoshio TSUTSUMI
i
q x ) =
U l ( X ) = *,(x),
$O(X)*
We s h a l l say t h a t t h e data $o, $1 and f o f t h e equation (2.1) s a t i s f y t h e c o m p a t i b i l i t y c o n d i t i o n o f o r d e r i n f i n i t y i f u . ( x ) = 0 on an ( j = 0, 1, 2,e.e).
J
I t i s known t h a t i f q0
E
C"(C),
q1
E
Cm(T2) and f
E
Cm([O,-)
x
C)
satisfy the
c o m p a t i b i l i t y c o n d i t i o n o f o r d e r i n f i n i t y , then Problem (2.1) has a unique solution u
E
Cm([O,-)
x
5 ) (see Mizohata [ 6 ] ) .
F o r t h a t s o l u t i o n we s h a l l show
t h e f o l l o w i n g u n i f o r m decay estimate:
Theorem 2.1.
L e t n 2 3.
1 . 1 ( 4 ) i s s a t i s f i e d f o r n.
Assume t h a t R i s "non-trapping",
L e t q0
E
Cm(E),
a1
E
Cm(E) and f
s a t i s f y the compatibility condition o f order i n f i n i t y .
E
i.e., Assumption Cm([O,-)
(1) i f n
p and p' a r e p o s i t i v e numbers (p may be i n f i n i t y ) such t h a t
2 '-'(l--) T P
Cm([O,-)
1 + P P
1,=
x
E)
Then, t h e s o l u t i o n u ( t , x )
5 ) o f (2.1) s a t i s f i e s t h e f o l l o w i n g estimates:
E
x
4 and >
1 and
1, then f o r each nonnegative i n t e g e r N
( 2 ) i f n 2 3 and p and p' a r e p o s i t i v e numbers ( p may be i n f i n i y ) such t h a t 1-1 1 -(1-12 = 1 and -1 + , = 1, then f o r any s u f f i c i e n t l y small a > 0 and each 2 P P P onnegative i n t e g e r N
Nonlinear Wave Equation in Exterior Domain
163
We s h a l l d i v i d e t h e p r o o f o f Theorem 2.1 i n t o s e v e r a l steps.
The s t r a t e g y
o f t h e p r o o f f o l l o w s Shibata [12] (see a l s o Tsutsumi [13]).
2.1. Local Energy Decay,
Theorem 2.2.
Here we s h a l l show t h e f o l l o w i n g theorem.
L e t n 2 3, Assume t h a t Assumption 1.1(4) holds.
L e t a and
b be a r b i t r a r y p o s i t i v e constants w i t h a, b 2 ro. L e t t h e d a t a J I ~J I, ~and f be smooth f u n c t i o n s s a t i s f y i n g t h e c o m p a t i b i l i t y c o n d i t i o n o f o r d e r i n f i n i t y such . R ( j = 0, 1) and supp f -. R i x Then, f o r any q w i t h J - a q 2 n-1 and each nonnegative i n t e g e r N t h e smooth s o l u t i o n u o f (2.1)
t h a t supp
0
$J.
s a t i s f i e s t h e f o l 1owing e s t i m a t e :
Following
We s h a l l f i r s t s t a t e t h e theorem needed f o r t h e proof o f Theorem 2 . 2 . Lax and P h i l l i p s C41, we s e t t h e H i l b e r t s p a c e 3 = { f = (fl,f2)
f2
E
2 L ( 8 ) 1 w i t h the inner product (f,g)
g = (g1,g2)),
= (fl,gl)D
; fl
+ (f2,g2)L2(n) 2
where ( - , - ) L 2 ( 8 ) i s t h e i n n e r p r o d u c t i n L ( 8 ) .
H,(n),
E
( f = (fl,f2),
F o r f = (fl,f2)
~ j !
we d e f i n e t h e l i n e a r o p e r a t o r A by
Then, i t f o l l o w s t h a t A i s a skew a d j o i n t o p e r a t o r ori
2 L ( 8 ) n H,(n)
& H 2 ( 8 ) n H,(n).
generated by A.
Theorem 2.3.
(1)
w i t h t h e domain D(A) =
L e t I U ( t ) 1 be t h e one parameter u n i t a r y group
F o r U ( t ) we have t h e f o l l o w i n g theorem.
L e t a and b be a r b i t r a r y p o s i t i v e constants w i t h a, b
Assume t h a t Assumption 1.1(4) holds.
I, 2).
;.
L e t f = (fl,f2) E X w i t h supp f . c J
> Qa
ro. (j =
Then,
i f n i s odd and n 2 3, then t h e r e e x i s t two constants C, 6
0 such t h a t
Yoshihiro SHIBATA and Ycshio TSUTSUMI
164 l u ( t ) f \ p , (b) = <
c
11 fllD
e-6t
where C and 6 depend o n l y on a,.b,
i f n i s even and n
2)
+
I I ~ ~I,I I t~ 2 0,
n and n;
4, then t h e r e e x i s t two constants C, 6 > 0 such
that
Remark 2.1.
Theorem 2.3(1) i s a l r e a d y w e l l known.
When n i s even and
n 2 4, t h e decay r a t e i n Theorem 2.3(2) seems t o be sharper than t h a t of a l r e a d y Melrose [5!).
known r e s u l t s (see, e.g.,
We d e f i n e 0- = I k
Sketch o f t h e p r o o f o f Theorem 2.3. and =
1 (1" {
k
, E
U(k.)f =
Q*'
;
-
3
n < arg k <
lom
e x p ( - m kt) U(t)f d t ,
Then we have (A
+ &T
k)G(k)f =
Hence, we have f o r k
(2.2)
; Im k
i f n i s odd and n 5 3,
5 I,
if
We define t h e Laplace t r a n s f o r m o f U ( t ) by ..I*
E C1
E
fl f 0-
F ( k ) f = (A + fl k ) - ' f
k
E
0-.
n i s even and n 2 4.
<
0 )
Nonlinear Wave Equation in Exterior Domain
165
2 1 2 where ( A + k ) - g denotes t h e s o l u t i o n u o f ( A + k ) u = g i n a, u = 0 on aa. Taking t h e i n v e r s e Laplace t r a n s f o r m o f (2.2), we o b t a i n (2.3)
U(t)f =
Z a L i
im-"" - m - & i o
e x p ( m k t ) (A + fl k ) - ' f
dk,
Thus, Theorem 2.3 f o l l o w s from a r o u t i n e c a l c u l u s i f we p r o p e r l y
f o r any 6 > 0.
s h i f t t h e contour o f t h e i n t e g r a l (2.3) by t h e r e l a t i o n ( A + k 2 ) - ' = k - ' -
+ k2)-'A and t h e f o l l o w i n g t h r e e lemmas ( f o r d e t a i l s , see Vainberg [17]
k-'(,
and Tsutsumi [14]):
Lemma 2.4.
(Vainberg [15]).
Let n
3.
w i t h a, b > ro. The r e s o l v e n t ( A + k 2 ) - l ( k to
D
2 2 as a B ( L a ( a ) , H ( n b ) ) - v a l u e d
function,
L e t a and b a r e p o s i t i v e constants L
D-) adntits a meromorphic e x t e n s i o n
Furthermore, t h e s e t o f a l l
p o l e s o f t h e meromorphic e x t e n s i o n has no l i m i t p o i n t i n 0 and does n o t l i e i n
D- i' ( R 1\
{
0
1).
Below we a l s o denote t h e meroniorphic e x t e n s i o n by ( A
Lemma 2.5. and n 2 3.
(Vainberg [17]).
+
kL)-'.
L e t a and b be p o s i t i v e constants w i t h a, b
Assume t h a t Assumption 1.1(4) holds.
ro
>
Then t h e r e e x i s t p o s i t i v e
constants a , B , C and T such t h a t f o r i n t e g e r s 0 5 s
1 and 0 2 j 5 2
i n the region V = I k s D ;
Lemma 2.6.
Vainberg [ 5, 161 and Tsutsumi [14]).
constants w i t h a, b > ro and n 2 3. such t h a t :
Ikl
(1)
(2)
Then t h e r e e x i s t s a p o s i t i v e c o n s t a n t y
i f n i s odd, ( A + k2)-'
Y j;
i f n i s even,
L e t a and b be p o s i t i v e
i s holomorphic i n t h e r e g i o n W = { k
E
0;
Yoshihiro SHIBATA and Yoshio TSUTSUMI
166
+ k 2 ) - l = Bl(k)
(A
i n t h e r e g i o n W' = I k
t k"'(1og
E
k)B2 t kn-2B3(k),
D ; Ikl
< y
2 2 B ( L a ( n ) , H ( 0, ) ) - v a l u e d f u n c t i o n , B2 i n W ' as a lB(Lg(i?),H2(~,))
I , where Bl(k) i s holomorphic i n W' as a 2 2 B(La(a),H (n,))
E
,
and B3(k) i s continuous
-valued function.
Now we s h a l l s t a t e t h e p r o o f o f Theorem 2.2.
'
L e t \r be t h e s o l u t i o n o f
Proof o f Theorem 2.2.
i
where vo
& V = A V
tf,
-
m
<
t
<
t
m
,
V(0) = VO' E
D(A),
f
E
V ( t ) = U(t)Vo + (see, e.g.,
c
C1(R1 ;:It), U(t
Mizohata [6]).
-
As i s w e l l known, we have t h e r e p r e s e n t a t i o n S ) f ( S ) ds
Therefore, we see from Theorem 2.3 t h a t f o r t h e data
Q0, 9, and f s a t i s f y i n g t h e assumptions o f Theorem 2.2 t h e s o l u t i o n u o f (2.1)
satisfies
i f n i s odd and n 2 3,
e-6t, t
t ) -n+l ,
if n i s even and n 2 4.
Here we have used the inequal it y
j
t
P(t
-
5)
1 t s ) - ~ds
C(q) ( 1 + t)-',
q > 0,
Nonlinear Wave Equation in Exterior Domain
167
A t l a s t Theorem 2.2 f o l l o w s from an i n d u c t i v e argument, (2.4) and t h e f o l l o w i n g we1 1-known e l 1i p t i c e s t i m a t e :
Lemma 2.7.
L e t a and b be a r b i t r a r y p o s i t i v e constants w i t h a
L e t a f u n c t i o n u s a t i s f y au = g i n na and u = 0 on an.
b > r 0'
>
Then, f o r each i n t e g e r
N 2 0, u s a t i s f i e s
2.2. Space.
Uniform Decay E s t i m a t e f o r S o l u t i o n s t o Wave Equation i n t h e F r e e I n t h i s s e c t i o n we s h a l l summarize t h e r e s u l t s concerning t h e decay o f
t h e s o l u t i o n t o t h e problem (2.5)
1x11=
i n [o,-)
f
U(O,X) = $,(XI,
x
(atu)(o,x)
R", i n R".
= q(x)
F o r g E Y ( R ~ we ) d e f i n e T ( t ) by a l i n e a r o p e r a t o r which naps g i n t o a s o l u t i o n o f t h e problem ( 2 . 5 ) w i t h
I),=,0,
$1 = g and f = 0.
Taking t h e
F o u r i e r t r a n s f o r m o f T ( t ) , we have
By u s i n g t h e above r e p r e s e n t a t i o n and t h e i n t e r p o l a t i o n technique we have t h e f o l l o w i n g w e l l known lemma (see, e.g.,
Lemma 2.8.
(2.6)
von Wahl [18] and Shatah [ l o ] ) :
F o r each i n t e g e r N 2 0 and any p w i t h 2
N+ 1 [ID T(t)gll;; 2 C(p,N,n)
n-1 --(1--) t
p
2 Ilgllp',N+[n/2]+2'
m
we have
Yoshihiro SHIBATA and Yoshio TSUTSUMI
168
1 1 f o r a l l t > 0, where p’ i s a r e a l number w i t h - + -,= 1. P P From Lemma 2.8 we have t h e f o l l o w i n g theorem:
Theorem 2.9.
L e t n 2 3.
L e t u ( t , x ) be t h e smooth s o l u t i o n o f (2.5) w i t h
, c + ( R n ) and f t h e data $o E Y ( R ~ ) $1
Then, f o r each i n t e g e r N
2 0,
Proof o f Theorem 2.9.
Cm([O,-)
R’)
bounded i n a l l norms 1 1 =l. below. L e t p and p’ be p o s i t i v e numbers such t h a t Y ( 1 - i ) ~1 and - + P P F o r s u f f i c i e n t l y small a > 0 we p u t E
x
-.
u s a t i s f i e s t h e f o l l o w i n g estimates:
u ( t , x ) can be represented as
u ( t ) = zd T ( t ) $ o + T ( t ) $ 1 +
:1
T(t
-
5)
f ( S ) ds.
Therefore, we o b t a i n Theorem 2.9 by u s i n g (2.6) o r (2.7) for t > 1 and (2.8) for 0 < t
<
1.
(Q. E. 0.)
Nonlinear Wave Equation in Exterior Domain
2.3.
P r o o f o f Theorem 2.1.
169
The p r o o f o f Theorem 2.1 i s e s s e n t i a l l y t h e
same as t h a t o f S h i b a t a [12] and Tsutsumi [13]. By t h e Seely technique we extend q o ( x ) , $ , ( X I and f(.,x)
C“-functions.
We denote t h e extended f u n c t i o n s by j b ( x ) , Tl(x)
respectively.
L e t u,(t,x)
from n t o Rn as and F(.,x),
be t h e smooth s o l u t i o n of t h e problem
F o r any i n t e g e r N 2 0 Theorem 2.9 g i v e s
where
(q(1-i)
,
1+ a
(a >
i f -(I--) n-1 2
2 P
>
1,
n l 2 if Z ( l - - ) 2 P
o),
= 1.
Next l e t y ( x ) be a f u n c t i o n b e l o n g i n g t o C i ( R n ) such t h a t ~ ( x =) 1 f o r 1x1 2 ro +1 and ~ ( x =) 0 f o r 1x1 2 ro + 2. (2.11)
u2(t,x)
= u(t,x)
-
(1
-
y(x))ul(t,x),
where u ( t , x ) i s t h e s o l u t i o n o f ( 2 . 1 ) . (2.12)
u u 2 =
Y f
+
9
Put
Then u2 s a t i s f i e s
i n [O,-)
x 0,
Yoshihiro SHIBATA and Yoshio TSUTSUMI
170
where g = 2
n
1
j=1
a .y a .u J
J 1
t AY u1
.
From (2.10) and (2.11) we have o n l y t o
e v a l u a t e u2 i n o r d e r t o o b t a i n t h e e s t i m a t e o f u. Applying Theorem 2.2 t o (2.12) w i t h b = ro+ 5, we have f o r any i n t e g e r N z 1
By t h e d e f i n i t i o n o f g and (2.10) we have f o r any i n t e g e r
N 20
where b = ro + 5. We s h a l l n e x t e v a l u a t e u2 f o r 1x1 > ro + 5.
Let
U(X)
be a Cm-function
such t h a t ~ ( x =) 1 f o r 1x1 2 ro t 3 and u ( x ) = 0 f o r 1x1 2 ro + 4. (2.14)
u ( ( 1 - u ) u 2 ) = (1-u)(yf t 9) + h
where h = 2
n
1 j=l
a
u
j
a u
j 2
+
AU
u2.
in
[o,-) x R" ,
Applying Theorem 2.9 t o (2.14), we have by
(2.13)'
' If1p',q,N+2[n/2]t3 where
Then
' If12,q,Nt2[n/2]t2
Nonlinear Wave Equation in Exterior Domain
Therefore, we o b t a i n Theorem 2.1 by (2.10),
(2.13),
171
(2.15) and t h e Sobolev
imbedding Theorem.
(Q. E. D.) Some Estimates f o r S o l u t i o n s of L i n e a r i z e d Problem.
53.
I n t h i s s e c t i o n we s h a l l show an L 2- e s t i m a t e and a u n i f o r m decay e s t i m a t e o f s o l u t i o n s f o r t h e f o l l o w i n g l i n e a r problem: (3.1)
2
0
= (1 + a (t,x))atu
,f,u
-
n
1 (&ij i,j=l
u = o u(0,x) where 6.
1j
+
n ’ 1 aJ(t,x)a.a u j=l J t
+
t aij(t,x))a.a.u
1 J
on [0,m)
. bJ(t,x)a.u = f ( t , x ) J
an,
x
i n R,
= (atu)(oyx) = 0
= 1 i f i = j and 6ij
n
1 j=o
= 0 if
iC j
We make t h e f o l l o w i n g assumptions:
Assumption 3.1. = (aJ(t,x),
(1)
g=UO,-)
Put j = O,..-,n;
A l l components o f & x
a
ij
(t.x),
i,j = l,-.-,n;
bJ(t,x),
j = O,....n)
are real-valued functions belonging t o
5).
(2)
aij(t,x)
(3)
F o r a l l 6 = (
= aji(t,x)
for a l l (t,x) E
E
Lo,-)
Rn and a l l ( t , x )
x E
5.
[O,-)
x ?i,
.
Yoshihiro SHIRATA and Yoshio TSUTSUMI
172
Then i f f o l l o w s t h a t f
E
d.i s
a s t r i c t l y hyperbolic operator.
EL-' (an i n t e g e r L 2 2), we have a unique s o l u t i o n u
[6]).
E
Thus, i f
FL (see Mizohata
By u s i n g Theorem Ap. 1 i n Appendix we can prove t h e f o l l o w i n g lemma
concerning an LL-estimate i n t h e same way as Shibata [ l l , 121.
L e t n 2 3.
Lemma 3.1.
Assumption 3.1 holds.
L e t L be any i n t e g e r w i t h L 2 0.
Then t h e r e e x i s t s a p o s i t i v e c o n s t a n t d depending o n l y
on n and n such t h a t i f f o r some n > 0 f
E
EL'',
Assume t h a t
t h e unique s o l u t i o n u
E
l,4
ZLt2
Im, +n,l
i 1,
Iw~,,o,o
5 d and
o f (3.1) s a t i s f i e s the following L
2
-
as t ima te:
Considering a s o l u t i o n u o f (3.1) as a s o l u t i o n o f
u(0,x) = (atu)(o,x)
= 0
i n n,
2 we have t h e f o l l o w i n g theorem concerning a decay e s t i m a t e and an L - e s t i m a t e by Theorem 2.1,
Lemna 3.1 and Theorem Ap.1 i n Appendix.
Theorem 3.2. Lemma 3.1.
L e t n 2 3.
L e t d be t h e p o s i t i v e constant described i n
Assume t h a t Assumption 3.1 holds.
K = L + 3[n/2]
+ 6.
Assume t h a t a l l norms of f and
t h e unique s o l u t i o n u
E
gK o f
an i n t e g e r L 2 0, p u t
For
A
(3.1) w i t h t h e data f
E
below a r e bounded.
EK-l s a t i s f i e s t h e
Then
Nonlinear Wave Equation in Exterior Domain
173
f o l l o w i n g estimates: ( 1 ) Suppose t h a t n 2 4. L e t p and p * be p o s i t i v e numbers such t h a t 1 +1 , = 1. I f 14 Im,O,O = q ( l - $ ) > 1 and ,1 1, then < d and 1 4 m , n - 1 P P -+I-$
z
(2)
Suppose t h a t n 1 3 .
2 n-1 -(l--) = 1 and 2 P and 14 lm,l+a,l 5
Here, f o r p =
54. 4.1.
m
1 1 - t -* = 1. P P 13 then
we d e f i n e
P-2
L e t p and p * be p o s i t i v e numbers such t h a t L e t a > 0 be s u f f i c i e n t l y small.
2 n-1 = 2 , p' = 1 and $I--)
P
I f I ~ j l ~2 , d ~ , ~
n-1 = -
2 .
I t e r a t i o n Scheme. Smoothing Operator.
A l i n e a r operator J,(e)
(e
_L
1 ) having the
f o l l o w i n g p r o p e r t i e s was constructed i n Shibata [12].
Lemma 4.1.
L e t e , k and p be r e a l numbers w i t h e 2 1, k 2 0 and 1 5 p
-
Then there e x i s t s a l i n e a r operator Sl(e) following properties:
w i t h t h e parameter e having t h e
m.
Yoshihiro SHIBATA and Yoshio TSUTSUMI
174
(1)
f o r any i n t e g e r N 2 0 and any f u n c t i o n u
ISl(e)ulp,k,N
= <
C(Pik,N)
and f o r any i n t e g e r i 2 0
(2)
f o r any i n t e g e r
and ( a l u ) ( o , x ) = 0
IUlp,k,N
and any f u n c t i o n u
EpyN with Iu) p,k,N
p,k,N
<
<
( i = O,.**,N-l)
f o r It1 5 1 and y ( t ) = 0 f o r It1 2 2 . $(el,e2)u
E
(U\
= 0;
Furthermore, we choose a f u n c t i o n v ( t )
(4.1)
:p’N
I
(a$l(e)u)(o,x)
N 20
with
E
= v(tej’)
E
C;(R
1 ) such t h a t f o r y ( t ) = 1
For 81 => 1 and e 2 ~
1 we, p u t
S,(e,)u.
By Lemma 4.1 we have t h e f o l l o w i n g theorem.
Lemna 4.2.
(1)
Let 1 5 p 5
-, e , 1 1
and e2 2 1.
Then,
f o r any i n t e g e r N 2 0, any r e a l number k 2 0 and any f u n c t i o n u
with I u ( p,k,N
,EpSN
<
IS2(el~e2)Ulp,k,N= < C(pSkYN) IUlp,k,N and f o r any i n t e g e r i 2 0
(a:S2(el (2)
with
,e,)u)(o,x)
-
=
o
;
f o r any i n t e g e r N 2 0, a r e a l number k 2 0 and any f u n c t i o n u
IuI Pik,N
<
and (a;u)(O,x)
= 0
(i= O,...,N-l)
€EpYN
Nonlinear Wave Equation in Exterior Domain
(3)
175
f o r any i n t e g e r M, N w i t h M > N 2 0, any r e a l numbers k , m w i t h
€ 3pyN
k > m 2 0 and any f u n c t i o n u
w i t h IuI
p,m,N
<
-
and ( a & ) ( o , x )
= 0
( i = 0,
1 ,..*,N-l)
4.2.
Compatibility Condition.
S i n c e t h e n o n l i n e a r term F a l s o depends on
t h e d e r i v a t i v e of o r d e r 2 w i t h r e s p e c t t o t i m e t i n o u r problem, we have t o pay s p e c i a l a t t e n t i o n t o t h e c o m p a t i b i l i t y c o n d i t i o n .
I n t h i s s e c t i o n we s h a l l
i n t r o d u c e t h e c o m p a t i b i l i t y c o n d i t i o n , f o l l o w i n g S h i b a t a [11? 12). we l e t u
E
Cm([O,m)
x
For s i m p l i c i t y
E ) and p u t
f ( t , x ) = n u + F(t,x,hu),
By t h e i m p l i c i t f u n c t i o n theorem i t f o l l o w s t h a t t h e r e e x i s t s a s u f f i c i e n t l y small p o s i t i v e c o n s t a n t d’ such t h a t i f
then t h e r e e x i s t f u n c t i o n s (4.3)
V. E
J
uJ. ( x ) = v ~ ( x , ~J;@,(x),
4’:”
( j 2 2 ) w i t h v.(x,O)
J
= 0 and
~ J X - ’ @ ~ ( X( )6,J - 2 f ) ( 0 3 ~ ) )
f o r a l l i n t e g e r s j 2 2. Thus, we i n t r o d u c e t h e c o m p a t i b i l i t y c o n d i t i o n i n t h e f o l l o w i n g form.
L e t d’ and v . be t h e same as i n (4,2) and (4.31, J We s h a l l say t h a t t h e data @ o ( x ) y +,(x) and f ( t , x ) s a t i s f y t h e
D e f i n i t i o n 4.1. respectively.
Yoshihiro SHIRATA and Yashio TSUTSUMI
176
c o m p a t i b i l i t y c o n d i t i o n o f o r d e r N i f $o, $1 and f s a t i s f y t h e f o l l o w i n g two conditions :
4.3.
I t e r a t i o n Scheme.
Let
% be
a p o s i t i v e c o n s t a n t described i n Theorem
L e t t h e c o m p a t i b i l i t y c o n d i t i o n of o r d e r
1.1.
$1 and f o f (M.P).
It1 2
We choose a f u n c t i o n y ( t )
It( 2
1 and y ( t ) = 0 f o r
C;(R 1 ) such t h a t y ( t ) = 1 f o r
E
Put
u,(x) = $,(x) and u . ( x ) ( j 2 2) a r e f u n c t i o n s c o n s t r u c t e d
where u o ( x ) = $,(x), i n 84.2.
2.
be s a t i s f i e d f o r t h e d a t a 40,
J
q Yf
Note t h a t v i s determined o n l y by $o,
and F.
By D e f i n i t i o n 4.1
and (4.3) i t f o l l o w s t h a t
-
a:(f
(1Jv + F ( t , x , A v ) ) ) I t +
f o r j = 0, l,...,m solution (4.4)
u
-
=
= 0
2 and t h a t v = 0 on [0,-)
x
an.
Putting w = u
o f (M.P), we see t h a t w s a t i s f i e s
tJw +
G(t,x,Aw)
= g
w(0,x) = (atw)(o,x)
in
= 0
[O,m)
x
R,
i n n,
where (4.5)
G(t,x,Aw)
=
1,
1
(1
-
2 r)(dxF)(t,x,Av
+ rAw)(Aw,Aw) d r ,
-
v for a
Nonlinear Wave Equation in Exterior Domain g = f
-
177
w
( u v + F(t,x,Av))
E
Em-’
I
Thus, we d e s c r i b e o u r i t e r a t i o n scheme f o r s o l v i n g t h e problem (4.4), f o l l o w i n g Klainerman [2] and Shibata [ 1, 121.
F i r s t we d e f i n e wo by t h e s o l u t i o n
of i n [O,-)
i w o = g
i n R.
= 0
wo(O,x) = (atwo)(o,x) Put
NOW we s h a l l d e f i n e a l r e a d y determined.
i
are ( p 2 0). F o r t h e moment we assume t h a t wo,~..,w P P L e t B be a f i x e d c o n s t a n t w i t h B > 1. L e t E be t h e p o s i t i v e
c o n s t a n t d e f i n e d i n Theorem 1 . 1 ( 3 ) .
Put
r
s,(ej)u,
if n
$- ( e j y e g ) u ,
i f n = 3,
,=E
(4.6)
,
e . = BJ. J
We define t h e smoothing o p e r a t o r S . by J
s.u = J
(4.7)
s,
and
s,
operator
2
by
(4.8)
.j
where
P
w P
2
4,
a r e t h e l i n e a r o p e r a t o r s d e f i n e d i n 54.1.
=xw+
(daG)(t,x,S
Aw )Aw.
P
P
We d e f i n e e: and eg- ( j 2 0 ) by J
e’ = (dAG)ft,x,Aw.)Ai j
J
j
-
(d,G)(t,x,S.nw.)& J J
j’
W e define the l i n e a r
Yoshihiro SHIBATA and Yoshio TSUTSUMI
178 e
C A
-
= G(t,x,Awjtl)
j
G(t,x A .) ' J'
-
(d,G)(t,x,Awj)Aij.
Put
+
e = e:
(4.9)
e
-1
(j 2 0).
We d e f i n e E j (j 2 0) by
Put
gp =
-
(Sp
-
Sp-l)Ep-l
-
SP e P-1 - (Sp - Sp-l)G(t,x,Awo)
( P 2 1).
F i n a l l y we d e f i n e \j by t h e s o l u t i o n o f
P
i P (o,x)
= ( a \j )(o,x)
=
t P
o
i n Q.
Thus, we can s u c c e s i v e l y determine two f u n c t i o n sequences I w {
i P 1.
P
1 and
Note t h a t
fwPtl
(4.13)
+ G(t,x,Awptl)
= g t (1
For wo and
i J.
-
Sp)G(t,x,AWo)
+ (1
-
Sp)Ep
+
ep.
(j 2 0) we have t h e f o l l o w i n g i m p o r t a n t lemma, which w i l l be
proved i n t h e n e x t s e c t i o n .
Lemma 4.3.
Assume t h a t Assumption 1.1 holds.
Theorem 1.1 be s a t i s f i e d . i n Theorem 1.1.
L e t a,
Then, wo and
E,
E, m
i . (j 2 0) J
L e t a l l assumptions i n
and A0 be p o s i t i v e constants d e f i n e d
s a t i s f y t h e f o l l o w i n g estimates:
Nonlinear Wave Equation in Exterior Domain
(1)
Suppose t h a t n 2 6.
(accordingly then Awo, AGj
= E
L e t 0 = max (4[n/21+7,
P+
3[n/2] + 6 ) . IE /I CP ( [ 0 , m ) x 4
z)
179
m+l) and
f
= 20 t [n/2] t 2
I f a and h0 a r e chosen s u f f i c i e n t l y s m a l l ,
and
4 f o r 6 s a t i s f y i n g t h e i n e q u a l i t i e s o f t h e d a t a i n Theorem 1.1-1) w i t h 0 < 6
(2) and
Suppose t h a t 4 5 n 2 5.
T=26 + l ( a c c o r d i n g l y
s m a l l , then !two,
AGj
E
=
-T IE n C
L e t B = max (3[n/2]+6, mtl) = max (12, mtl)
3[n/2]
t
6 = 'i:+ 12).
([0,-)
x
6)
' t
If a and 6 o a r e s u f f i c i e n t l y
and
f o r 6 s a t i s f y i n g t h e i n e q u a l i t i e s o f t h e d a t a i n Theorem 1.1-2) w i t h 0 < 6
(3) Suppose t h a t n = 3. integer w i t h r
i I [ t+
Let
60.
IJ
= ~ / 7and B =
( 3 m + 7 ) ~ ]( a c c o r d i n g l y
Ifa and 6o a r e chosen s u f f i c i e n t l y s m a l l , then
71 +
%=
Awe,
t
(m + 2 ) ~ . L e t
3[n/2] E
t
6
= yt
4 P € ! f \ C ([0,m)
z 60.
be an
9). x
5)
and
f o r 6 s a t i s f y i n g t h e i n e q u a l i t i e s o f t h e d a t a i n Theorem 1.1-3) w i t h 0 < 6 2 do.
65. P r o o f o f L e m a 4.3 and Main Results. We s h a l l now p r o v e Lemma 4.3 by an i n d u c t i o n argument. same n o t a t i o n s as i n 64.
We s h a l l use t h e
- L e t L denote an i n t e g e r and k denote a r e a l number.
Yoshihiro SHIBATA and Yoshio TSUTSUMI
180
For
t h e moment we always assume t h e f o l l o w i n g assumptions: '
[A.5.1]
Awo
E
V
N
zLflCL([O,m)
E) and
x
i f n 2 6,
w
CA.5.21
ko,.-.,kp
[A.5.3]
i f Ihl
a r e a l r e a d y determined and Lemma 4.3 h o l d s f o r
ko,**-,
* P '
z s1 ,
then f o r any i n t e g e r s N and L w i t h N ' 0
and
%
O 5 L 5 L
ld,G(-Y*J)Im,O,L N where
y=
<
C(n,L,N)
<
m,
+ 4[n/2] + 7 if n 2 6 and ';J =
+ 3[n/2] + 6 i f 3 5 n
Let
We s h a l l f i r s t prepare several lenmas t o prove Lemma 4.3. s u f f i c i e n t l y small p o s i t i v e c o n s t a n t and e s p e c i a l l y Noting t h a t
~ - E - B -L T.
and -B+o'i:)
T
T
=
5 if n
5.
T
be a
= 3.
if n = 3, we can prove t h e f o l l o w i n g
lemna i n t h e same way as Klainerman [2] and Shibata [ll,121.
Lemma 5.1.
Assume t h a t [A.5.1
the following: (a)
hwj
(b)
if n
-
31 h o l d .
For w . ( j = O,l,...,p+l) J
we have
-'i L IE fl c ([(I,-) x 6) ; Iv
E
2 4, then
'"J12,O,L
+
I"jlb(n),c(n),L
= <
C6
f o r -8+L
lAwj1b(n),c(n),L
= <
C s e J.
-B+L
1AwJ12,0,L
+
-'I,
f o r -B+L &
T
-
and 0 5 L 2 L,
Nonlinear Wave Equation in Exterior Domain
ISjAwj12,0,L
+
IsjAwJ1 b(n),c(n),L
= <
181
for L >
C(L) a ejBtL
where b ( n ) = 4 and c ( n ) = ( n - 1 ) / 4 i f n 2 6, and b ( n ) =
m
T,
and c ( n ) = (n-1)/2 i f
Y
i f n = 3, then f o r -@+uL 2
< C6
2,0,L
=
m,k,L
=
<
ca
f o r k-B+sL
-T,
_i -1,
for - B t d 2
T
f o r k-B+oL 2 for L >
12,0,L
= <
C(L) 6 9 j B + O L
IsjAwj Im,k,L
= <
C(k,L) 6 e t - B t a L
ISj"j
rv
L 5 L, h
T,
0 2 L 5 L and 0 5 k 5 1-E,
1, >
-
1-E o r L > L ;
i f n 2 4, then
(d) I(1
for k
and 0 5
-
sj)Awj12,0,L
+
I(1
-
Sj)Awjlb(n),c(n),L = < Csey'+L J
for
o
L
N
L,
where b ( n ) and c ( n ) a r e t h e same as i n ( b ) ; i f n = 3, then
(e)
I
-
'j)"j
Im,k,L
= <
c 6 et-B+oL
for 0
z k 5 1-E and 0 5 L z r .
By choosing 6 s u f f i c i e n t l y small we assume t h a t :
rA.5.41 < C6
IAwjlm,o,o
5 C l A w j 12,0,[n/21+1
=
lA'jl-,o,o
5 Cl~'jl2,0,[n/21+1
= <
From Theorem Ap.1,
rA.5.1
-
z a1 1
Cs 5 %
( j = O,l,...,p+l), ( j = O,l,-..,p).
41 and Lemmas 4.1 and 4.2 we have the following
Yoshihiro SHIBATA and Yoshio TSUTSUMI
182
1emma.
Lemma 5 . 2 .
For e . ( j = O,--.,p) J
Assume t h a t Assumption 1.1 holds.
the following:
rACL([O,-)
we have
&
ej
(a)
E)
E
]E
lej12,k,L
5
cs
lejll,k,L
<
c s e k-(1t7E)B+oL
=
x
;
(b)
Proof.
n,
for
‘jk-36toL
j
By t h e d e f i n i t i o n o f e j
i n t h e case o f n = 3.
Since e
j
= e*
5 k 2 2 ( 1 - ~ ) and 0 5 L 5 L,
i f n = 3,
f o r 0 5 k 5 1-E and 0 jL 2 L,
if n = 3.
-
T+B
(a) i s clear.
+
e:-, j
So, we s h a l l prove ( b ) o n l y
we have o n l y t o prove t h a t ( b ) holds ~
f o r e: and e - - , r e s p e c t i v e l y . However, we s h a l l prove o n l y f o r e: because we J j J can prove f o r e’- i n t h e same way. 2 Since dG , ej =
j
t,x,O)
lo{I, 1
= 0, i t f o l l o w s t h a t
l (d,G)(t,x,r’(S.Aw. 3 x
By Lemma 5.1,
(S.AW. J J
+ r(l
J
J
+
r(1
-
-
S.)Awj))dr’ J
S.)AW.,(l
J
J
-
1
Sj)AW
j’
Aij) d r .
Theorems Ap.1 and Ap.2 we have f o r k w i t h k-6 2
T
and k 5 1-E
183
Next by L e w a 5.
le;ll
,k,L
= c
, Theorems
Ap
c I(' [Awjlm,o,L
-
+ Ihw. 2,0,LI(1 J
sJ.
Thus, t h e l a s t i n e q u a l i t y i n ( b ) i s proved.
(Q. E. 0.)
By L e n a s 4.1,
Lemma 5.3. (a)
Ep
E
4.2, 5.2 and
ej
=
BJ we have t h e f o l l o w i n g lemma.
Assume t h a t Assumption 1.1 h o l d s . ' U r v
EL/7CL([0,-)
x
5)
Then,
;
(b 1 JEpl~,n-l,t
2
' /Eplq,n-l,L 3-4-
= c
c5
2
for -&+L
5
-T,
if n 2 6 ,
Yoshihiro SHIBATA and Yoshio TSUTSUMI
184
IEpl2,n-1 ,L
+
< C 63
IEp12,k,L
=
IEpll,k,L
= <
IEpll,k,L
lEp11 ,C,L 2
C6
3
< C 63 =
z -T,
for - 2 ~ t L
< C 63
=
for
T+B
k and k-38toL 2 -'I, i f
f o r k - ( 1 + 7 ~ ) ~ + o2L
ek-(1t7E)B+oL p
-T,
if 4
n
zn 2 5,
= 3,
if n = 3 ;
f o r k - ( 1 + 7 ~ ) @ + o L2
T,
0 2 k
z 1-E
and
'u
O ~ L L L , i f n = 3.
N o t i n g t h a t l - ~ - ( l t 7 ~2 )T,~ 2 ( 1 - ~ ) - 3 8 2 Lemmas 4.1,
N
T
and -3BtoL 2
4.2 and 5.3 the f o l l o w i n g lemma.
L e m a 5.4.
Assume t h a t Assumption 1.1 holds.
Then,
T~
we have by
Nonlinear Wave Equation in Exterior Domain
3 k-3B+aL ' ~ ) ~ p l 2 , k , L= < C(k,L) 6 ep+l
1('p+l
I(sp+l
-
Sp)Epll,k,L
By Assumption 1.1,
Lemna 5.5. (a)
= <
3 k-(1+7~)4+0L C(k,L) 6 ep+,
[A.5.1]
and [A.5.3]
E
'rl - i C ir ([0,-)
E
,
x
f o r k 2 T+B and L 2 0, i f n = 3, f o r k 2 0, and L 2 0, i f n = 3.
we have t h e f o l l o w i n g lemma.
Assume t h a t Assumption 1.1 holds.
G(t,x,Awo)
185
Then,
;
(b)
By Lemmas 4.1, 4.2 and 5.5 and Theorems Ap.1 and Ap.2 we have t h e f o l l o w i n g 1emma.
Yoshihiro SHIBATA and Yoshio TSUTSUMI
186
Combining Lemmas 5.2, fact
eo
=
5.4,
5.5 and 5.6 and u s i n g Lemmas 4.1,
4.2 and t h e
1, we have t h e f o l l o w i n g lemma.
Lemma 5.7.
Assume t h a t Assumption 1.1 holds.
90' gp+l
(a)
E
E"nC"([O,-)
x
5)
Then,
;
(b)
I n o r d e r t o use Theorem 3.2, we have t o e v a l u a t e t h e c o e f f i c i e n t s o f t h e
:.tj
operator
d e f i n e d i n (4.8).
A . = (d,F)(t,x,Av) 3
+
Noting t h a t (d,F)(t,x,Av)
Put
(d,G)(t,x,SjAwj). = 0 for
It1 2 2,
we have t h e f o l l o w i n g lemma by
Nonlinear Wave Equation in Exterior Domain
Lemma 5.1,
[A.5.1
Lemma 5.8. $o,
-
41, Theorems Ap.1, Ap.2 and Ap.3.
c
L e t L be a p o s i t i v e c o n s t a n t d e f i n e d i n [A.5.2].
$1 and f s a t i s f y a l l assumptions i n Theorem 1.1.
holds.
< C6
=
f o r -B+L 2
-T
and 0 5
L e t t h e data
Assume t h a t Assumption 1.1
Then we have t h e f o l l o w i n g :
IA014,d,L 4
187
L 5 L,
Yoshihiro SHIBATA and Yoshio TSUTSUMI
188
cy
[ A p t 1 12,1tE,L
~
62 el+E-B+oL P+l
l.
f o r 0 5 L 5 L.
I n p a r t i c u l a r , choosing 6 s u f f i c i e n t l y small, we have t h e f o l l o w i n g :
Here d i s a p o s i t i v e constant g i v e n by Lemma 3.1.
Proof.
We s h a l l g i v e t h e sketch o f t h e proof for,%
P + l o n l y i n t h e case
o f n = 3. Since (dhG)(t,x,O) we have by L e m a 5.1,
= 0,
2 (dAG)(t,x,O)
Theorems Ap.1,
= 0 and f o r It1 2 2
Ap.2 and Ap.3
( d X F ) ( t r x , h v ) = 0,
Nonlinear Wave Equation in Exterior Domain
1+E -8 Noting t h a t 7
-T
, we
have f o r
l+E 2 -8toL
2
189
T
F i n a l l y we have by Lemma 5.1
From the above lemmas we can complete the p r o o f o f Lemma 4.3.
Proof of Lemma 4.3. First
We s h a l l prove Lemma 4.3 by an i n d u c t i o n argument.
we assume f o r the moment t h a t r A . 5 . 1
have by Lemas 5.7, 5.8 and Theorem 3.2 t h a t
- 41 hold.
Then, i f n 2 6 , we
Yoshihiro SHIBATA and Yoshio TSUTSUMI
190
I"ptl12,O,L
I"ptl14,n-JL
4
2
t 6 max(1,
We have used t h e f a c t
2 4[n/2]
e -~+L+3[n/21+6) P+l
t
7 a t the l a s t i n e q u a l i t y .
by choosing 6 so small t h a t max { C(L)s ; 0 holds f o r
I n t h e same iptl,
eply 1
IL
zT 1 -5 1 we
I n particular,
see t h a t Lemma 4.31)
way i t i s c l e a r t h a t under t h e assumptions CA.5.11
and EA.5.31 Lemma 4.3(1) h o l d s f o r
i0,By
t h e way, we see by t h e assumption on
t h e data i n Theorem 1.1, Theorem 3.2, Theorems Ap.2 and Ap.3 t h a t [A.5.1]
5 . 31 h o l d .
and [A.
Therefore, an i n d u c t i o n argument g i v e s Lemma 4.3(1).
i n t h e same way we o b t a i n Lemma 4.3(2) f o r 4 2 n 5 5. F i n a l l y , f o r n = 3 we s h a l l v e r i f y t h a t under t h e assumptions [A.5.1 Lemma 4.3(3) h o l d s f o r 1 +
E
2 B +
T
iptl. By Theorem
3.2,
Lemmas 5.7,
41
5.8 and t h e f a c t t h a t
we have
We have used t h e f a c t s t h a t 1+€ -B+o 5
-T
and t h a t I + E - ( ~ - E ) B + C5 J 0 a t the
second i n e q u a l i t y and t h e l a s t i n e q u a l i t y , r e s p e c t i v e l y .
Thus, we have
I n p a r t i c u l a r , by choosing 6 so small t h a t max { C(L) & 2 ; 0 2 L obtain
-
-.I
L 1 5 1 we
Nonlinear Wave Equation in Exterior Domain
191
Next, by Theorem 3.2, Lemmas 5.5 and 5.8 we have
("ptl
<
L,l-€,L
=
+
,2,1+~-6 P+l
6'1
t
c ( L ) 6 3 [ e l + ~ - ( 1 + 7 ~ ) 8 + o ( L + 3 [ n / 2+4 1 1 P+l el+~-36+o(L+3[n/2]+4) P+1
eAl7-6 max (1, e
I+~-B+o(L+3[n/2]+6) P+ 1
+
<
-
qL1 &3[
( 1+E ) / 2 P+ 1
- 6+0 ( L+3 [n/ 2]+4) 1
1
1+~-36
ep+l
el+~-(1+7~)6+o(L+3[n/2]t4) P+l
+ e 2( 1+~)-46+o(L+3[n/2]+6) P+l
-6+0 5 - -T a t t h e f i r s t
We have used t h e f a c t s t h a t 1 + ~2 6+r and t h a t inequality. <
2 By t h e way, s i n c e z ~ + o ( 3 [ n / 2 ] + 4 ) - & 6
<
Oand 1 + 3 ~ - ( 3 - ~ ) 6 3+ ~ ( 3 [ n / 2 ] + 6 )
0, we have
( b 7 ~6+0 ) (L+3 [n/2]+4) 5 -1 2( 1 + )-46+a ~ (L+3[n/2]+6)
-
E-
- ( 1- E ) 5
( 1+E ) B+oL
-
Thus, i t f o l l o w s t h a t
By t h e Sobolev imbedding theorem and ( 5 . 1 ) we have
Therefore, by i n t e r p o l a t i n g between (5.2) and (5.3) we have
for 0
z k 2 1-E
and 0
L
IT.
Thus, s i n c e -.~6+o([n/2]+1) 2 0, we o b t a i n by
(5.4) for 0
(5.5) Ifwe choose 6 so s m a l l t h a t max
{
C(L) '6
;0
k 5 1-E and 0 5
5L
zy
cu
L 5 1.
15 1, t h e n (5.5) and
Yoshihiro SHIBATA and Yoshio TSUTSUMI
192
(5.1)’ give
L e n a 4.3(3) f o r ;p+l.
Since we can prove i n t h e same way as t h e
case o f n 2 6 t h a t Lemma 4.3(3) h o l d s f o r
wo
and t h a t [A.5.1]
and [A.5.3
-
41
T h i s completes t h e p r o o f o f
hold, an i n d u c t i o n argument g i v e s Lemma 4 . 3 ( 3 ) . Lemma 4.3.
(9. E. D.) P r o o f o f main r e s u l t s .
Put
m
Then, from Lemnas 4.1
-
3, Lemmas 5.1
-
6 and (4.6) we e a s i l y see t h a t u = v + w
i s t h e d e s i r e d s o l u t i o n o f (M.P) ( f o r d e t a i l s , see Klainerman [2] and Shibata
[ll,121).
Furthermore, we can prove the uniqueness o f t h e s o l u t i o n o f (M.P)
by t h e energy method i n t h e same way as Shibata [12].
(Q. E. D.)
Concluding Remarks.
(1)
When n = 3, we used t h e c u t - o f f f u n c t i o n i n time.
The authors do n o t know whether we can prove w i t h o u t i t f o r n = 3 i n t h e same way as Klainerman and Ponce [ 3 ] and Shatah (2)
[lo].
We can a l s o o b t a i n t h e analogous r e s u l t s f o r t h e mixed problems o f t h e
n o n l i n e a r Klein-Gordon equation and t h e n o n l i n e a r Schrodinger e q u a t i o n i n t h e same way (see, e.g.,
TsuTsumi [13]).
56. Appendix. I n t h i s s e c t i o n we s h a l l s t a t e several theorems which p l a y an i m p o r t a n t r o l e i n the p r e s e n t paper.
Theorem Ap.1.
Let
For t h e i r p r o o f , see Shibata [ll,121.
p = Rn o r a. L e t
and f and g be f u n c t i o n s from [O,-) @,
x
and
JI
be f u n c t i o n s f r o m ,g t o
R1
& to IR1 . Assume t h a t a l l semi-norms o f
$I,f and g appearing below a r e bounded.
k and
@
L e t M and
N be nonnegative i n t e g e r s ,
m be nonnegative numbers and p and q be r e a l numbers w i t h 1 5 p, q 2
m.
Nonlinear Wave Equation in Exterior Domain
Then,
Furthermore, i f F(t,x,O)
Theorem Ap.3.
Let
= 0, then
o0,
$1 and f be t h e data o f (M.P)
such t h a t a l l semi-
193
YoRhihiro SHIBATA and Yoshio TSUTSUMI
194
Let
norms appearing below a r e bounded. i n 54.
L e t H(t,x,x)
E Wm([O,m)
x
5
x
% and {
1x1
v ( t , x ) be t h e same as those d e f i n e d
2 1 1).
I f H(t,x,O)
= 0, then
References P. Dionne, Sur l e s p r o b l i m e de Cauchy hyperboliques b i e n poses, J . Analyse Math., c21
10 (1962). 1-90.
S. Klainerman, Global e x i s t e n c e f o r n o n l i n e a r wave equations, Corn. Pure
Appl. Math,, 33 (1980), 43-101.
S. Klainerman and G. Ponce, Global s m a l l amplitude s o l u t i o n s t o n o n l i n e a r e v o l u t i o n equations, Comm. Pure Appl. Math., 36 (1983), 133-141.
P. Lax and R. P h i l l i p s , S c a t t e r i n g Theory, Acad Press, 1967.
R. B. Melrose, S i n g u l a r i t i e s and energy decay i n a c o u s t i c a l s c a t t e r i n g , Duke Math. J . , 46 (1979), 43-59. r61
S. Mizohata, Quelque problemes au bord, du t y p e rnixte, pour des equations
hyperboliques, S h i n a i r s u r l e s Gquations aux derivees p a r t i e l l e s , C o l l i g e de France, (1966/67), 23-60. [71
A. Moser, A r a p i d l y convergent i t e r a t i o n method and n o n - l i n e a r d i f f e r e n t i a l equations, Ann. Scu. Norm. Pisa, 20(3) (1966), 265-315, 499-535.
r 81
J . Nash, The embedding problem f o r Riemannian m a n i f o l d s , Ann. Math., 63 (1965), 20-63.
r91
P. H. Rabinowitz, P e r i o d i c s o l u t i o n s o f n o n l i n e a r h y p e r b o l i c p a t i a l d i f f e r e n t i a l equations
II , Corn.
Pure Appl. Math.
, 22
(1969), 15-39.
J . Shatah, Global e x i s t e n c e o f small s o l u t i o n s t o n o n l i n e a r e v o l u t i o n equations, J . D i f f e r e n t i a l Eqs.
, 46
(19821, 409-425.
Y. Shibata, On t h e g l o b a l e x i s t e n c e o f c l a s s i c a l s o l u t i o n s o f mixed
Nonliiiear Wave Equation i n Exterior Domain
195
problem f o r some second o r d e r n o n - l i n e a r h y p e r b o l i c o p e r a t o r s w i t h d i s s i p a t i v e term i n t h e i n t e r i o r domain, Funk. Ekva., [12]
25 (1982), 303-345.
Y . Shibata, On t h e g l o b a l e x i s t e n c e o f c l a s s i c a l s o l u t i o n s o f second o r d e r
f u l l y n o n l i n e a r h y p e r b o l i c equations w i t h f i r s t o r d e r d i s s i p a t i o n i n t h e e x t e r i o r domain, [13]
Tsukuba J . Math., 7 ( 1 ) (1983), 1-68.
Y . Tsutsumi, Global s o l u t i o n s o f t h e n o n l i n e a r Schrodinger e q u a t i o n i n
e x t e r i o r domains, t o appear i n Corn. P. 0. E. [14]
Y . Tsutsumi, Local energy decay o f s o l u t i o n s t o t h e f r e e Schrodinger
e q u a t i o n i n e x t e r i o r domains, t o appear i n 3. Fac. S c i . Univ. Tokyo, Sect.
IA, Math. [15]
B. R. Vainberg, On t h e a n a l y t i c a l p r o p e r t i e s o f t h e r e s o l v e n t f o r a c e r t a i n c l a s s o f o p e r a t o r - p e n c i l s , Math. USSR Sbornik, 6 ( 2 ) (1968), 241-273.
[16]
B. R. Vainberg, On e x t e r i o r e l l i p t i c problems p o l y n o m i a l l y depending on a s p e c t r a l parameters and t h e a s y m p t o t i c behaviour f o r l a r g e t i m e o f non s t a t i o n a r y problems, Wath. USSR Sbornik, 21(2) (1973), 221-239.
[17]
B.
-
R. Vainberg, On t h e s h o r t wave asymptotic behaviour of s o l u t i o n s o f
s t a t i o n a r y problems and t h e a s y m p t o t i c behaviour as t
-+
o f solutions o f
n o n s t a t i o n a r y problems, Russian Math., Surveys, 30(2) (1975), 1-58.
[la]
W. von Wahl, LP-decay r a t e s f o r homogeneous wave equations, Math. Z.,
120 (1971), 93-106.
196
Yoshihiro SHIBATA and Yoshio TSUTSUMI
Global Existence Theorem for Nonlinear Wave Equation in Exterior Domain Yoshihiro SHIBATA* and Yoshio TSUTSUMI**
*
*Department of Mathematics, University of Tsukuba Ibaraki 306, Japan **Department of Pure and Applied Sciences, College of General Education, University of Tokyo Tokyo 113, Japan Supported in part by the Sakkokai Foundation.
81.
Introduction.
The g l o b a l e x i s t e n c e o f s o l u t i o n s f o r t h e n o n l i n e a r wave e q u a t i o n has been extensively studied. improvement r e c e n t l y .
F o r t h e Cauchy problem Klainerman [ Z ] has made a remarkable That i s , he showed t h a t i f t h e s p a t i a l dimension i s n o t
s m a l l e r than 6 and i n i t i a l d a t a a r e s m a l l and smooth, then t h e Cauchy problem f o r t h e f u l l y n o n l i n e a r wave e q u a t i o n has a unique c l a s s i c a l g l o b a l s o l u t i o n .
On t h e o t h e r hand i t i s i m p o r t a n t t o c o n s i d e r t h e i n i t i a l boundary v a l u e problem f o r t h e n o n l i n e a r wave e q u a t i o n i n an e x t e r i o r domain i n o r d e r t o s t u d y s c a t t e r i n g o f a r e f l e c t i n g o b j e c t f o r t h e n o n l i n e a r wave equation.
I n t h e p r e s e n t paper we
s h a l l prove t h a t i f t h e s p a t i a l dimension i s n o t s m a l l e r than 3 and i n i t i a l data a r e small and smooth, then we have t h e g l o b a l unique e x i s t e n c e theorem o f c l a s s i c a l solutions f o r
a
l a r g e c l a s s o f n o n l i n e a r wave equations i n e x t e r i o r domains
w i t h t h e homogeneous O i r i c h l e t boundary c o n d i t i o n , which i n c l u d e s t h e n o n l i n e a r v i b r a t i o n equation.
n
Let
be an unbounded domain i n lRn
,n2
3, w i t h Cm and compact boundary an.
We denote a t i m e v a r i a b l e by t o r xo and a space v a r i a b l e by x = (x1,.-.,xn), respectively. o r ,a,
(*)
a . and ,;a J
We s h a l l a b b r e v i a t e a / a t , a/ax. and (a/axl)al...(a/axn)an
J
r e s p e c t i v e l y , where
a
i s a multi-index w i t h
Supported i n p a r t by t h e Sakkokai Foundation. 155
101
to a t
= al+***+a
n
Yoshihiro SHIBATA and Yashio TSUTSUMI
166
.
and j = l,...,n
We s h a l l consider t h e f o l l o w i n g problem:
u = o
on [OP)
~ ( 0 ~ =x 1$o(X), where
2
t11= at
-
A =
a t2
x
an,
(atu)(o,x)
- j.11n a Jz. and
= $,(XI
hu = (aiu,
i n n,
a J. a ku, j,k=O,.-.,n).
i=O,.-.,n;
Before we s t a t e assumptions and t h e main theorem, we s h a l l g i v e n o t a t i o n s .
For any i n t e g e r N 2 0 we w r i t e
L e t 6 be an a r b i t r a r y open s e t i n Rn
.
F o r any p w i t h 1 5 p 5
standard Lp space d e f i n e d on p and i t s norm by Lp(S) and
ml
II.llB,py
we denote t h e respectively.
For a v e c t o r v a l u e d f u n c t i o n h = ( h l y . - * , h s ~ we p u t 2 l h I 2 = l h 1 I 2 + - . * + l h s l , ~ ~ h ~= / 1 ~ y~ p~ h j ~ ~ ~ , p j=l We a l s o w r i t e
We s e t HP~ ( c - ) = t f
E
L’’(F)
;
IIfll,,p,N
<
m
1. Note t h a t Ho(e)= Lp(&) p a r t i c u l a r l y .
L e t robe a f i x e d p o s i t i v e constant such t h a t an C I x r > ro we denote t h e subset I x
E
n ; 1x1
< r 1 by
nr.
E
P R n ; 1x1 < ro I .
For
F o r any r > ro and any
i n t e g e r k 2 1 we p u t 2 ~,(n) = I u
E
~ ~ ( ;n supp ) uct x
E
R” ; 1x1 5 r I 1,
Ok Hr(n)
E
k Hz(n) ; supp u
E
R n ; 1x1 5 r 1,
I u
c( x
“0 2 kle s h a l l sometimes use Hr(n) = lr(n). (u,v),
and t h e D i r i c h l e t norm
llullD by
a:ulas2 = 0 (la1 L k-111.
We d e f i n e t h e D i r i c h l e t i n n e r p r o d u c t
Nonlinear Wave Equation in Exterior 1)wnain
157
we denote t h e completion o f $(n) i n t h e D i r i c h l e t norm. By ,uN(&) we N denote t h e s e t o f C ( b ) - f u n c t i o n s having a l l d e r i v a t i v e s o f o r d e r 5 N bounded i n E . F o r two Banach spaces X and Y we denote t h e Banach space c o n s i s t i n g o f a l l
By H,(Q)
bounded l i n e a r o p e r a t o r s from X t o Y and i t s norm by B(X,Y) and II-II ]B(X,Y) ' 1 r e s p e c t i v e l y . For an i n t e r v a l I(.- R and a Banach space X we denote t h e s e t
o f m-times c o n t i n u o u s l y d i f f e r e n t i a b l e X-valued f u n c t i o n on I by Cm(I;X).
We s e t
For 1 ;< p 5 =, a nonnegative number k and a nonnegative i n t e g e r N we w r i t e
hL =
I
U
c IELr\CL-l([O,-);H,(s2));
L atu(O,x)
= 0 1.
F o r s i m p l i c i t y we a l s o use t h e a b b r e v i a t i o n s :
f(c)
= f(cl,-*-.~n)
where xc = x
1 1
v . = (v~,.**,vJ)
=
+...+xncn.
IRn
e x p ( - a x c ) f ( x ) dx,
For p o s i t i v e integers
5,
i, v e c t o r s u = (u, . * . . , u s ) ,
( 1 5 j 2 i) and a s c a l a r f u n c t i o n H(t.x,u)
J .
(dLH)(t,x,u)(vl
1
,*-*.vi)
by
(V
E
IRs) we d e f i n e
Yoshihiro SHIRATA and Yoshio TSUTSUMI
158
We s h a l l make t h e f o l l o w i n g assumptions. Assumption 1.1. (2) m
The s p a t i a l dimension n 2 3.
(1)
The n o n l i n e a r mapping F i s a r e a l - v a l u e d f u n c t i o n belonging t o
([0,-)
x
sl
x
{ A
E
R 2(n+1)
.> I XI
5 - 1 I).
(3) F(t,x,A)
O ( 1 ~ 1 ~ ) near
A =
0(1~1~)
x
near
0,
= 0,
if n 1 6 , if 3 5 n
5.
The e x t e r i o r domain n i s "non-trapping" i n t h e f o l l o w i n g sense:
(4)
G(t,x,y)
=
Let
be t h e Green f u n c t i o n f o r t h e f o l l o w i n g problem
2 (at
-
A ~ ) G= 0
i n ( 0 , ~ ) x n,
where y i s an a r b i t r a r y p o i n t i n n and ax i s t h e Laplace o p e r a t o r w i t h r e s p e c t t o x.
L e t a and b be a r b i t r a r y p o s i t i v e constants such t h a t b 2 a 2 ro. F o r
f o r any v
E
L:(n),
Remark 1.1.
where To depends o n l y on n, a, b and n.
I t is w e l l known t h a t i f t h e complement of B i s convex, t h e n
Assumption 1.1(4) i s s a t i s f i e d (see, e.g.,
Melrose [5]).
1.io
Nonlinear Wave Equation i n Exterior Dxnain
Now we s h a l l s t a t e t h e main theorem.
Theorem 1.1.
L e t m be an a r b i t r a r y i n t e g e r w i t h m
(Existence).
G.
Let
Assumption 1.1 be a l l s a t i s f i e d . 1)
P u t m = 2max(4[n/2]+7,
m+l) t 4[n/2]
4-
1. I f n 2 6, then t h e r e e x i s t
p o s i t i v e constants a and 6 o having t h e f o l l o w i n g p r o p e r t i e s : $1 E.6 2h[n/21+2(5)
and
f
€3:
2m+[n/21+1([0,-)
x
i)
If
@o
, €1,.
2m+[n/2]+3(;)
s a t i s f y f o r some 6 w i t h 0.
6 ~6~
and t h e c o m p a t i b i l i t y c o n d i t i o n o f o r d e r m , then Problem (M.P) has a s o l u t i o n
u
E
Cm +2
([o,-)
x
IAu12,0,m 2)
+
ii)
satisfying
lAu14,(n-1)/4,m
6’
Put m = 2max(3[n/2]+6,
m+l) + 3[n/2]
+ 7.
If 4
n 5 5, then t h e r e e x i s t
p o s i t i v e constants a and 6 o having t h e f o l l o w i n g p r o p e r t i e s :
@1 E J ~ ‘ ~ ’ ( ; )
and f
Il@OIlm,2iT;+2
+
E F2Fn ([0,-)
1141 Ilm,2iii+l
x
+
5)
s a t i s f y f o r some 6 w i t h 0
Iflm,o,2in‘
If
o0
< 6
E),
2m+ 2
(E),
A0
2 a6
and t h e c o m p a t i b i l i t y c o n d i t i o n of o r d e r ‘m, then Problem (M.P) has a s o l u t i o n
u
E
c ~ + ~ ( [ o x, i) ~ )satisfying IAU12,0,m
3)
$2
Let
+
E
IAUl-,(n-1
)/z.in
= <
6.
be a p o s i t i v e c o n s t a n t w i t h 0 <
3$+(3m +
7 ) ~+ ] 3[n/2]
+ 6.
E
5
1
, and
in
an i n t e g e r w i t h
I f n = 3, then t h e r e e x i s t p o s i t i v e constants
,
Yoshihiro SHIBATA and Yoshio TSUTSUMI
160
a and A0 having t h e f o l l o w i n g p r o p e r t i e s :
f
2%
~2
([I),-)x
5 ) s a t i s f y f o r some
I f $o
6 with 0 < 6
,i
E,,
5
2$+2
(z), o1
E$,*~'(;)
and
60
and t h e c o m p a t i b i l i t y c o n d i t i o n o f o r d e r m, t h e Problem (M.P) has a s o l u t i o n
u
E
Cm " ([G,-)
x
$) s a t i s f y i n g
(Uniqueness). C3([0,-) I*ulm,o,o
i) a r e
x
= <
6 1 and
Remark 1.2.
There e x i s t s a small c o n s t a n t 6 , > 0 such t h a t i f u, v
E
two s o l u t i o n s o f Problem (M.P) f o r t h e same data k i t h
I"lm,o,o (1)
2 1, then u
= v.
For t h e c o m p a t i b i l i t y c o n d i t i o n , see 54.2 and Mizohata
[6l. (2)
Since t h e n o n l i n e a r f u n c t i o n F i s d e f i n e d o n l y i n [ 0 , m ) ; Ihl
x
5
I
A E
1 I , we always assume t h a t I A U ~ ~ = ,< ~1, , when ~ we c o n s i d e r a
s o l u t i o n u o f Problem (M.P).
One o f t h e d i f f i c u l t i e s i n t h e p r o o f i s t h a t the l o s s o f d e r i v a t i v e s occurs a t each s t e p i n t h e i t e r a t i o n .
E s p e c i a l l y n o t e t h a t t h e n o n l i n e a r term F a l s o
depends on t h e d e r i v a t i v e of o r d e r 2 w i t h r e s p e c t t o time t i n our problem.
For
t h e Cauchy problem we can overcome such a d i f f i c u l t y by reducing a f u l l y n o n l i n e a r e q u a t i o n t o a q u a s i l i n e a r equation, f o l l o w i n g Oionne [I] (see a l s o Klainerman and Ponce [3] and Shatah [ l o ] ) .
For t h e i n i t i a l boundary v a l u e problem, however,
such methods a r e n o t a p p l i c a b l e . case n = 3.
Furthermore, t h e l o s s of decay occurs i n t h e
I n o r d e r t o overcome such d i f f i c u l t i e s , we s h a l l make use o f t h e
s o - c a l l e d Nash-Moser technique.
Our s t r a t e g y f o l l o w s Klainerman [ 2 ] and Shibata
Nonlinear Wave Equation in Exterior Dnmain
161
[12] (see a l s o Rabinowitz [9]).
A u n i f o r m decay e s t i m a t e and an L 2 - e s t i m a t e f o r a l i n e a r i z e d problem w i l l p l a y an i m p o r t a n t r o l e i n t h e p r o o f .
I n p a r t i c u l a r , t h e r e s u l t s o f decay estimates
a r e new and a r e proved i n t h e same way as Shibata [12] and Tsutsumi [13].
Tools
used i n a p p l y i n g t h e Nash-Moser technique, such as an i n t e r p o l a t i o n i n e q u a l i t y between a f a m i l y o f c e r t a i n semi-norms and a proper smoothing o p e r a t o r , a r e t h e same as those used i n Shibata [ll,121. Now we g i v e a well-known example, i.e.,
" t h e n o n l i n e a r v i b r a t i o n equation":
Example.
I n t h e course o f t h e p r o o f below a l l constants w i l l be s i m p l y denoted by C. In particular, C =
C(*,.--,*) w i l l denote a c o n s t a n t depending on t h e q u a n t i t j e s
appearing i n parentheses.
52. Uniform Decay Estimate.
In t h i s s e c t i o n we s h a l l
show a u n i f o r m decay e s t i m a t e o f s o l u t i o n s f o r t h e
f o l l o w i n g l i n e a r problem:
(2.1)
t_:iu
= f
u = o
in
[O,m)
x
R,
an
[a,-)
x
an,
Throughout S e c t i o n 2 we always assume t h a t t h e data $o, $1 and f o f t h e e q u a t i o n ( 2 . 1 ) a r e so n i c e f u n c t i o n s t h a t a l l t h e i r norms and semi-norms appearing below a r e bounded.
We d e f i n e u j ( x ) ( j 2 0 ) s u c c e s i v e l y by
162
Yoshihiro SHIBATA and Yoshio TSUTSUMI
i
q x ) =
U l ( X ) = *,(x),
$O(X)*
We s h a l l say t h a t t h e data $o, $1 and f o f t h e equation (2.1) s a t i s f y t h e c o m p a t i b i l i t y c o n d i t i o n o f o r d e r i n f i n i t y i f u . ( x ) = 0 on an ( j = 0, 1, 2,e.e).
J
I t i s known t h a t i f q0
E
C"(C),
q1
E
Cm(T2) and f
E
Cm([O,-)
x
C)
satisfy the
c o m p a t i b i l i t y c o n d i t i o n o f o r d e r i n f i n i t y , then Problem (2.1) has a unique solution u
E
Cm([O,-)
x
5 ) (see Mizohata [ 6 ] ) .
F o r t h a t s o l u t i o n we s h a l l show
t h e f o l l o w i n g u n i f o r m decay estimate:
Theorem 2.1.
L e t n 2 3.
1 . 1 ( 4 ) i s s a t i s f i e d f o r n.
Assume t h a t R i s "non-trapping",
L e t q0
E
Cm(E),
a1
E
Cm(E) and f
s a t i s f y the compatibility condition o f order i n f i n i t y .
E
i.e., Assumption Cm([O,-)
(1) i f n
p and p' a r e p o s i t i v e numbers (p may be i n f i n i t y ) such t h a t
2 '-'(l--) T P
Cm([O,-)
1 + P P
1,=
x
E)
Then, t h e s o l u t i o n u ( t , x )
5 ) o f (2.1) s a t i s f i e s t h e f o l l o w i n g estimates:
E
x
4 and >
1 and
1, then f o r each nonnegative i n t e g e r N
( 2 ) i f n 2 3 and p and p' a r e p o s i t i v e numbers ( p may be i n f i n i y ) such t h a t 1-1 1 -(1-12 = 1 and -1 + , = 1, then f o r any s u f f i c i e n t l y small a > 0 and each 2 P P P onnegative i n t e g e r N
Nonlinear Wave Equation in Exterior Domain
163
We s h a l l d i v i d e t h e p r o o f o f Theorem 2.1 i n t o s e v e r a l steps.
The s t r a t e g y
o f t h e p r o o f f o l l o w s Shibata [12] (see a l s o Tsutsumi [13]).
2.1. Local Energy Decay,
Theorem 2.2.
Here we s h a l l show t h e f o l l o w i n g theorem.
L e t n 2 3, Assume t h a t Assumption 1.1(4) holds.
L e t a and
b be a r b i t r a r y p o s i t i v e constants w i t h a, b 2 ro. L e t t h e d a t a J I ~J I, ~and f be smooth f u n c t i o n s s a t i s f y i n g t h e c o m p a t i b i l i t y c o n d i t i o n o f o r d e r i n f i n i t y such . R ( j = 0, 1) and supp f -. R i x Then, f o r any q w i t h J - a q 2 n-1 and each nonnegative i n t e g e r N t h e smooth s o l u t i o n u o f (2.1)
t h a t supp
0
$J.
s a t i s f i e s t h e f o l 1owing e s t i m a t e :
Following
We s h a l l f i r s t s t a t e t h e theorem needed f o r t h e proof o f Theorem 2 . 2 . Lax and P h i l l i p s C41, we s e t t h e H i l b e r t s p a c e 3 = { f = (fl,f2)
f2
E
2 L ( 8 ) 1 w i t h the inner product (f,g)
g = (g1,g2)),
= (fl,gl)D
; fl
+ (f2,g2)L2(n) 2
where ( - , - ) L 2 ( 8 ) i s t h e i n n e r p r o d u c t i n L ( 8 ) .
H,(n),
E
( f = (fl,f2),
F o r f = (fl,f2)
~ j !
we d e f i n e t h e l i n e a r o p e r a t o r A by
Then, i t f o l l o w s t h a t A i s a skew a d j o i n t o p e r a t o r ori
2 L ( 8 ) n H,(n)
& H 2 ( 8 ) n H,(n).
generated by A.
Theorem 2.3.
(1)
w i t h t h e domain D(A) =
L e t I U ( t ) 1 be t h e one parameter u n i t a r y group
F o r U ( t ) we have t h e f o l l o w i n g theorem.
L e t a and b be a r b i t r a r y p o s i t i v e constants w i t h a, b
Assume t h a t Assumption 1.1(4) holds.
I, 2).
;.
L e t f = (fl,f2) E X w i t h supp f . c J
> Qa
ro. (j =
Then,
i f n i s odd and n 2 3, then t h e r e e x i s t two constants C, 6
0 such t h a t
Yoshihiro SHIBATA and Ycshio TSUTSUMI
164 l u ( t ) f \ p , (b) = <
c
11 fllD
e-6t
where C and 6 depend o n l y on a,.b,
i f n i s even and n
2)
+
I I ~ ~I,I I t~ 2 0,
n and n;
4, then t h e r e e x i s t two constants C, 6 > 0 such
that
Remark 2.1.
Theorem 2.3(1) i s a l r e a d y w e l l known.
When n i s even and
n 2 4, t h e decay r a t e i n Theorem 2.3(2) seems t o be sharper than t h a t of a l r e a d y Melrose [5!).
known r e s u l t s (see, e.g.,
We d e f i n e 0- = I k
Sketch o f t h e p r o o f o f Theorem 2.3. and =
1 (1" {
k
, E
U(k.)f =
Q*'
;
-
3
n < arg k <
lom
e x p ( - m kt) U(t)f d t ,
Then we have (A
+ &T
k)G(k)f =
Hence, we have f o r k
(2.2)
; Im k
i f n i s odd and n 5 3,
5 I,
if
We define t h e Laplace t r a n s f o r m o f U ( t ) by ..I*
E C1
E
fl f 0-
F ( k ) f = (A + fl k ) - ' f
k
E
0-.
n i s even and n 2 4.
<
0 )
Nonlinear Wave Equation in Exterior Domain
165
2 1 2 where ( A + k ) - g denotes t h e s o l u t i o n u o f ( A + k ) u = g i n a, u = 0 on aa. Taking t h e i n v e r s e Laplace t r a n s f o r m o f (2.2), we o b t a i n (2.3)
U(t)f =
Z a L i
im-"" - m - & i o
e x p ( m k t ) (A + fl k ) - ' f
dk,
Thus, Theorem 2.3 f o l l o w s from a r o u t i n e c a l c u l u s i f we p r o p e r l y
f o r any 6 > 0.
s h i f t t h e contour o f t h e i n t e g r a l (2.3) by t h e r e l a t i o n ( A + k 2 ) - ' = k - ' -
+ k2)-'A and t h e f o l l o w i n g t h r e e lemmas ( f o r d e t a i l s , see Vainberg [17]
k-'(,
and Tsutsumi [14]):
Lemma 2.4.
(Vainberg [15]).
Let n
3.
w i t h a, b > ro. The r e s o l v e n t ( A + k 2 ) - l ( k to
D
2 2 as a B ( L a ( a ) , H ( n b ) ) - v a l u e d
function,
L e t a and b a r e p o s i t i v e constants L
D-) adntits a meromorphic e x t e n s i o n
Furthermore, t h e s e t o f a l l
p o l e s o f t h e meromorphic e x t e n s i o n has no l i m i t p o i n t i n 0 and does n o t l i e i n
D- i' ( R 1\
{
0
1).
Below we a l s o denote t h e meroniorphic e x t e n s i o n by ( A
Lemma 2.5. and n 2 3.
(Vainberg [17]).
+
kL)-'.
L e t a and b be p o s i t i v e constants w i t h a, b
Assume t h a t Assumption 1.1(4) holds.
ro
>
Then t h e r e e x i s t p o s i t i v e
constants a , B , C and T such t h a t f o r i n t e g e r s 0 5 s
1 and 0 2 j 5 2
i n the region V = I k s D ;
Lemma 2.6.
Vainberg [ 5, 161 and Tsutsumi [14]).
constants w i t h a, b > ro and n 2 3. such t h a t :
Ikl
(1)
(2)
Then t h e r e e x i s t s a p o s i t i v e c o n s t a n t y
i f n i s odd, ( A + k2)-'
Y j;
i f n i s even,
L e t a and b be p o s i t i v e
i s holomorphic i n t h e r e g i o n W = { k
E
0;
Yoshihiro SHIBATA and Yoshio TSUTSUMI
166
+ k 2 ) - l = Bl(k)
(A
i n t h e r e g i o n W' = I k
t k"'(1og
E
k)B2 t kn-2B3(k),
D ; Ikl
< y
2 2 B ( L a ( n ) , H ( 0, ) ) - v a l u e d f u n c t i o n , B2 i n W ' as a lB(Lg(i?),H2(~,))
I , where Bl(k) i s holomorphic i n W' as a 2 2 B(La(a),H (n,))
E
,
and B3(k) i s continuous
-valued function.
Now we s h a l l s t a t e t h e p r o o f o f Theorem 2.2.
'
L e t \r be t h e s o l u t i o n o f
Proof o f Theorem 2.2.
i
where vo
& V = A V
tf,
-
m
<
t
<
t
m
,
V(0) = VO' E
D(A),
f
E
V ( t ) = U(t)Vo + (see, e.g.,
c
C1(R1 ;:It), U(t
Mizohata [6]).
-
As i s w e l l known, we have t h e r e p r e s e n t a t i o n S ) f ( S ) ds
Therefore, we see from Theorem 2.3 t h a t f o r t h e data
Q0, 9, and f s a t i s f y i n g t h e assumptions o f Theorem 2.2 t h e s o l u t i o n u o f (2.1)
satisfies
i f n i s odd and n 2 3,
e-6t, t
t ) -n+l ,
if n i s even and n 2 4.
Here we have used the inequal it y
j
t
P(t
-
5)
1 t s ) - ~ds
C(q) ( 1 + t)-',
q > 0,
Nonlinear Wave Equation in Exterior Domain
167
A t l a s t Theorem 2.2 f o l l o w s from an i n d u c t i v e argument, (2.4) and t h e f o l l o w i n g we1 1-known e l 1i p t i c e s t i m a t e :
Lemma 2.7.
L e t a and b be a r b i t r a r y p o s i t i v e constants w i t h a
L e t a f u n c t i o n u s a t i s f y au = g i n na and u = 0 on an.
b > r 0'
>
Then, f o r each i n t e g e r
N 2 0, u s a t i s f i e s
2.2. Space.
Uniform Decay E s t i m a t e f o r S o l u t i o n s t o Wave Equation i n t h e F r e e I n t h i s s e c t i o n we s h a l l summarize t h e r e s u l t s concerning t h e decay o f
t h e s o l u t i o n t o t h e problem (2.5)
1x11=
i n [o,-)
f
U(O,X) = $,(XI,
x
(atu)(o,x)
R", i n R".
= q(x)
F o r g E Y ( R ~ we ) d e f i n e T ( t ) by a l i n e a r o p e r a t o r which naps g i n t o a s o l u t i o n o f t h e problem ( 2 . 5 ) w i t h
I),=,0,
$1 = g and f = 0.
Taking t h e
F o u r i e r t r a n s f o r m o f T ( t ) , we have
By u s i n g t h e above r e p r e s e n t a t i o n and t h e i n t e r p o l a t i o n technique we have t h e f o l l o w i n g w e l l known lemma (see, e.g.,
Lemma 2.8.
(2.6)
von Wahl [18] and Shatah [ l o ] ) :
F o r each i n t e g e r N 2 0 and any p w i t h 2
N+ 1 [ID T(t)gll;; 2 C(p,N,n)
n-1 --(1--) t
p
2 Ilgllp',N+[n/2]+2'
m
we have
Yoshihiro SHIBATA and Yoshio TSUTSUMI
168
1 1 f o r a l l t > 0, where p’ i s a r e a l number w i t h - + -,= 1. P P From Lemma 2.8 we have t h e f o l l o w i n g theorem:
Theorem 2.9.
L e t n 2 3.
L e t u ( t , x ) be t h e smooth s o l u t i o n o f (2.5) w i t h
, c + ( R n ) and f t h e data $o E Y ( R ~ ) $1
Then, f o r each i n t e g e r N
2 0,
Proof o f Theorem 2.9.
Cm([O,-)
R’)
bounded i n a l l norms 1 1 =l. below. L e t p and p’ be p o s i t i v e numbers such t h a t Y ( 1 - i ) ~1 and - + P P F o r s u f f i c i e n t l y small a > 0 we p u t E
x
-.
u s a t i s f i e s t h e f o l l o w i n g estimates:
u ( t , x ) can be represented as
u ( t ) = zd T ( t ) $ o + T ( t ) $ 1 +
:1
T(t
-
5)
f ( S ) ds.
Therefore, we o b t a i n Theorem 2.9 by u s i n g (2.6) o r (2.7) for t > 1 and (2.8) for 0 < t
<
1.
(Q. E. 0.)
Nonlinear Wave Equation in Exterior Domain
2.3.
P r o o f o f Theorem 2.1.
169
The p r o o f o f Theorem 2.1 i s e s s e n t i a l l y t h e
same as t h a t o f S h i b a t a [12] and Tsutsumi [13]. By t h e Seely technique we extend q o ( x ) , $ , ( X I and f(.,x)
C“-functions.
We denote t h e extended f u n c t i o n s by j b ( x ) , Tl(x)
respectively.
L e t u,(t,x)
from n t o Rn as and F(.,x),
be t h e smooth s o l u t i o n of t h e problem
F o r any i n t e g e r N 2 0 Theorem 2.9 g i v e s
where
(q(1-i)
,
1+ a
(a >
i f -(I--) n-1 2
2 P
>
1,
n l 2 if Z ( l - - ) 2 P
o),
= 1.
Next l e t y ( x ) be a f u n c t i o n b e l o n g i n g t o C i ( R n ) such t h a t ~ ( x =) 1 f o r 1x1 2 ro +1 and ~ ( x =) 0 f o r 1x1 2 ro + 2. (2.11)
u2(t,x)
= u(t,x)
-
(1
-
y(x))ul(t,x),
where u ( t , x ) i s t h e s o l u t i o n o f ( 2 . 1 ) . (2.12)
u u 2 =
Y f
+
9
Put
Then u2 s a t i s f i e s
i n [O,-)
x 0,
Yoshihiro SHIBATA and Yoshio TSUTSUMI
170
where g = 2
n
1
j=1
a .y a .u J
J 1
t AY u1
.
From (2.10) and (2.11) we have o n l y t o
e v a l u a t e u2 i n o r d e r t o o b t a i n t h e e s t i m a t e o f u. Applying Theorem 2.2 t o (2.12) w i t h b = ro+ 5, we have f o r any i n t e g e r N z 1
By t h e d e f i n i t i o n o f g and (2.10) we have f o r any i n t e g e r
N 20
where b = ro + 5. We s h a l l n e x t e v a l u a t e u2 f o r 1x1 > ro + 5.
Let
U(X)
be a Cm-function
such t h a t ~ ( x =) 1 f o r 1x1 2 ro t 3 and u ( x ) = 0 f o r 1x1 2 ro + 4. (2.14)
u ( ( 1 - u ) u 2 ) = (1-u)(yf t 9) + h
where h = 2
n
1 j=l
a
u
j
a u
j 2
+
AU
u2.
in
[o,-) x R" ,
Applying Theorem 2.9 t o (2.14), we have by
(2.13)'
' If1p',q,N+2[n/2]t3 where
Then
' If12,q,Nt2[n/2]t2
Nonlinear Wave Equation in Exterior Domain
Therefore, we o b t a i n Theorem 2.1 by (2.10),
(2.13),
171
(2.15) and t h e Sobolev
imbedding Theorem.
(Q. E. D.) Some Estimates f o r S o l u t i o n s of L i n e a r i z e d Problem.
53.
I n t h i s s e c t i o n we s h a l l show an L 2- e s t i m a t e and a u n i f o r m decay e s t i m a t e o f s o l u t i o n s f o r t h e f o l l o w i n g l i n e a r problem: (3.1)
2
0
= (1 + a (t,x))atu
,f,u
-
n
1 (&ij i,j=l
u = o u(0,x) where 6.
1j
+
n ’ 1 aJ(t,x)a.a u j=l J t
+
t aij(t,x))a.a.u
1 J
on [0,m)
. bJ(t,x)a.u = f ( t , x ) J
an,
x
i n R,
= (atu)(oyx) = 0
= 1 i f i = j and 6ij
n
1 j=o
= 0 if
iC j
We make t h e f o l l o w i n g assumptions:
Assumption 3.1. = (aJ(t,x),
(1)
g=UO,-)
Put j = O,..-,n;
A l l components o f & x
a
ij
(t.x),
i,j = l,-.-,n;
bJ(t,x),
j = O,....n)
are real-valued functions belonging t o
5).
(2)
aij(t,x)
(3)
F o r a l l 6 = (
= aji(t,x)
for a l l (t,x) E
E
Lo,-)
Rn and a l l ( t , x )
x E
5.
[O,-)
x ?i,
.
Yoshihiro SHIRATA and Yoshio TSUTSUMI
172
Then i f f o l l o w s t h a t f
E
d.i s
a s t r i c t l y hyperbolic operator.
EL-' (an i n t e g e r L 2 2), we have a unique s o l u t i o n u
[6]).
E
Thus, i f
FL (see Mizohata
By u s i n g Theorem Ap. 1 i n Appendix we can prove t h e f o l l o w i n g lemma
concerning an LL-estimate i n t h e same way as Shibata [ l l , 121.
L e t n 2 3.
Lemma 3.1.
Assumption 3.1 holds.
L e t L be any i n t e g e r w i t h L 2 0.
Then t h e r e e x i s t s a p o s i t i v e c o n s t a n t d depending o n l y
on n and n such t h a t i f f o r some n > 0 f
E
EL'',
Assume t h a t
t h e unique s o l u t i o n u
E
l,4
ZLt2
Im, +n,l
i 1,
Iw~,,o,o
5 d and
o f (3.1) s a t i s f i e s the following L
2
-
as t ima te:
Considering a s o l u t i o n u o f (3.1) as a s o l u t i o n o f
u(0,x) = (atu)(o,x)
= 0
i n n,
2 we have t h e f o l l o w i n g theorem concerning a decay e s t i m a t e and an L - e s t i m a t e by Theorem 2.1,
Lemna 3.1 and Theorem Ap.1 i n Appendix.
Theorem 3.2. Lemma 3.1.
L e t n 2 3.
L e t d be t h e p o s i t i v e constant described i n
Assume t h a t Assumption 3.1 holds.
K = L + 3[n/2]
+ 6.
Assume t h a t a l l norms of f and
t h e unique s o l u t i o n u
E
gK o f
an i n t e g e r L 2 0, p u t
For
A
(3.1) w i t h t h e data f
E
below a r e bounded.
EK-l s a t i s f i e s t h e
Then
Nonlinear Wave Equation in Exterior Domain
173
f o l l o w i n g estimates: ( 1 ) Suppose t h a t n 2 4. L e t p and p * be p o s i t i v e numbers such t h a t 1 +1 , = 1. I f 14 Im,O,O = q ( l - $ ) > 1 and ,1 1, then < d and 1 4 m , n - 1 P P -+I-$
z
(2)
Suppose t h a t n 1 3 .
2 n-1 -(l--) = 1 and 2 P and 14 lm,l+a,l 5
Here, f o r p =
54. 4.1.
m
1 1 - t -* = 1. P P 13 then
we d e f i n e
P-2
L e t p and p * be p o s i t i v e numbers such t h a t L e t a > 0 be s u f f i c i e n t l y small.
2 n-1 = 2 , p' = 1 and $I--)
P
I f I ~ j l ~2 , d ~ , ~
n-1 = -
2 .
I t e r a t i o n Scheme. Smoothing Operator.
A l i n e a r operator J,(e)
(e
_L
1 ) having the
f o l l o w i n g p r o p e r t i e s was constructed i n Shibata [12].
Lemma 4.1.
L e t e , k and p be r e a l numbers w i t h e 2 1, k 2 0 and 1 5 p
-
Then there e x i s t s a l i n e a r operator Sl(e) following properties:
w i t h t h e parameter e having t h e
m.
Yoshihiro SHIBATA and Yoshio TSUTSUMI
174
(1)
f o r any i n t e g e r N 2 0 and any f u n c t i o n u
ISl(e)ulp,k,N
= <
C(Pik,N)
and f o r any i n t e g e r i 2 0
(2)
f o r any i n t e g e r
and ( a l u ) ( o , x ) = 0
IUlp,k,N
and any f u n c t i o n u
EpyN with Iu) p,k,N
p,k,N
<
<
( i = O,.**,N-l)
f o r It1 5 1 and y ( t ) = 0 f o r It1 2 2 . $(el,e2)u
E
(U\
= 0;
Furthermore, we choose a f u n c t i o n v ( t )
(4.1)
:p’N
I
(a$l(e)u)(o,x)
N 20
with
E
= v(tej’)
E
C;(R
1 ) such t h a t f o r y ( t ) = 1
For 81 => 1 and e 2 ~
1 we, p u t
S,(e,)u.
By Lemma 4.1 we have t h e f o l l o w i n g theorem.
Lemna 4.2.
(1)
Let 1 5 p 5
-, e , 1 1
and e2 2 1.
Then,
f o r any i n t e g e r N 2 0, any r e a l number k 2 0 and any f u n c t i o n u
with I u ( p,k,N
,EpSN
<
IS2(el~e2)Ulp,k,N= < C(pSkYN) IUlp,k,N and f o r any i n t e g e r i 2 0
(a:S2(el (2)
with
,e,)u)(o,x)
-
=
o
;
f o r any i n t e g e r N 2 0, a r e a l number k 2 0 and any f u n c t i o n u
IuI Pik,N
<
and (a;u)(O,x)
= 0
(i= O,...,N-l)
€EpYN
Nonlinear Wave Equation in Exterior Domain
(3)
175
f o r any i n t e g e r M, N w i t h M > N 2 0, any r e a l numbers k , m w i t h
€ 3pyN
k > m 2 0 and any f u n c t i o n u
w i t h IuI
p,m,N
<
-
and ( a & ) ( o , x )
= 0
( i = 0,
1 ,..*,N-l)
4.2.
Compatibility Condition.
S i n c e t h e n o n l i n e a r term F a l s o depends on
t h e d e r i v a t i v e of o r d e r 2 w i t h r e s p e c t t o t i m e t i n o u r problem, we have t o pay s p e c i a l a t t e n t i o n t o t h e c o m p a t i b i l i t y c o n d i t i o n .
I n t h i s s e c t i o n we s h a l l
i n t r o d u c e t h e c o m p a t i b i l i t y c o n d i t i o n , f o l l o w i n g S h i b a t a [11? 12). we l e t u
E
Cm([O,m)
x
For s i m p l i c i t y
E ) and p u t
f ( t , x ) = n u + F(t,x,hu),
By t h e i m p l i c i t f u n c t i o n theorem i t f o l l o w s t h a t t h e r e e x i s t s a s u f f i c i e n t l y small p o s i t i v e c o n s t a n t d’ such t h a t i f
then t h e r e e x i s t f u n c t i o n s (4.3)
V. E
J
uJ. ( x ) = v ~ ( x , ~J;@,(x),
4’:”
( j 2 2 ) w i t h v.(x,O)
J
= 0 and
~ J X - ’ @ ~ ( X( )6,J - 2 f ) ( 0 3 ~ ) )
f o r a l l i n t e g e r s j 2 2. Thus, we i n t r o d u c e t h e c o m p a t i b i l i t y c o n d i t i o n i n t h e f o l l o w i n g form.
L e t d’ and v . be t h e same as i n (4,2) and (4.31, J We s h a l l say t h a t t h e data @ o ( x ) y +,(x) and f ( t , x ) s a t i s f y t h e
D e f i n i t i o n 4.1. respectively.
Yoshihiro SHIRATA and Yashio TSUTSUMI
176
c o m p a t i b i l i t y c o n d i t i o n o f o r d e r N i f $o, $1 and f s a t i s f y t h e f o l l o w i n g two conditions :
4.3.
I t e r a t i o n Scheme.
Let
% be
a p o s i t i v e c o n s t a n t described i n Theorem
L e t t h e c o m p a t i b i l i t y c o n d i t i o n of o r d e r
1.1.
$1 and f o f (M.P).
It1 2
We choose a f u n c t i o n y ( t )
It( 2
1 and y ( t ) = 0 f o r
C;(R 1 ) such t h a t y ( t ) = 1 f o r
E
Put
u,(x) = $,(x) and u . ( x ) ( j 2 2) a r e f u n c t i o n s c o n s t r u c t e d
where u o ( x ) = $,(x), i n 84.2.
2.
be s a t i s f i e d f o r t h e d a t a 40,
J
q Yf
Note t h a t v i s determined o n l y by $o,
and F.
By D e f i n i t i o n 4.1
and (4.3) i t f o l l o w s t h a t
-
a:(f
(1Jv + F ( t , x , A v ) ) ) I t +
f o r j = 0, l,...,m solution (4.4)
u
-
=
= 0
2 and t h a t v = 0 on [0,-)
x
an.
Putting w = u
o f (M.P), we see t h a t w s a t i s f i e s
tJw +
G(t,x,Aw)
= g
w(0,x) = (atw)(o,x)
in
= 0
[O,m)
x
R,
i n n,
where (4.5)
G(t,x,Aw)
=
1,
1
(1
-
2 r)(dxF)(t,x,Av
+ rAw)(Aw,Aw) d r ,
-
v for a
Nonlinear Wave Equation in Exterior Domain g = f
-
177
w
( u v + F(t,x,Av))
E
Em-’
I
Thus, we d e s c r i b e o u r i t e r a t i o n scheme f o r s o l v i n g t h e problem (4.4), f o l l o w i n g Klainerman [2] and Shibata [ 1, 121.
F i r s t we d e f i n e wo by t h e s o l u t i o n
of i n [O,-)
i w o = g
i n R.
= 0
wo(O,x) = (atwo)(o,x) Put
NOW we s h a l l d e f i n e a l r e a d y determined.
i
are ( p 2 0). F o r t h e moment we assume t h a t wo,~..,w P P L e t B be a f i x e d c o n s t a n t w i t h B > 1. L e t E be t h e p o s i t i v e
c o n s t a n t d e f i n e d i n Theorem 1 . 1 ( 3 ) .
Put
r
s,(ej)u,
if n
$- ( e j y e g ) u ,
i f n = 3,
,=E
(4.6)
,
e . = BJ. J
We define t h e smoothing o p e r a t o r S . by J
s.u = J
(4.7)
s,
and
s,
operator
2
by
(4.8)
.j
where
P
w P
2
4,
a r e t h e l i n e a r o p e r a t o r s d e f i n e d i n 54.1.
=xw+
(daG)(t,x,S
Aw )Aw.
P
P
We d e f i n e e: and eg- ( j 2 0 ) by J
e’ = (dAG)ft,x,Aw.)Ai j
J
j
-
(d,G)(t,x,S.nw.)& J J
j’
W e define the l i n e a r
Yoshihiro SHIBATA and Yoshio TSUTSUMI
178 e
C A
-
= G(t,x,Awjtl)
j
G(t,x A .) ' J'
-
(d,G)(t,x,Awj)Aij.
Put
+
e = e:
(4.9)
e
-1
(j 2 0).
We d e f i n e E j (j 2 0) by
Put
gp =
-
(Sp
-
Sp-l)Ep-l
-
SP e P-1 - (Sp - Sp-l)G(t,x,Awo)
( P 2 1).
F i n a l l y we d e f i n e \j by t h e s o l u t i o n o f
P
i P (o,x)
= ( a \j )(o,x)
=
t P
o
i n Q.
Thus, we can s u c c e s i v e l y determine two f u n c t i o n sequences I w {
i P 1.
P
1 and
Note t h a t
fwPtl
(4.13)
+ G(t,x,Awptl)
= g t (1
For wo and
i J.
-
Sp)G(t,x,AWo)
+ (1
-
Sp)Ep
+
ep.
(j 2 0) we have t h e f o l l o w i n g i m p o r t a n t lemma, which w i l l be
proved i n t h e n e x t s e c t i o n .
Lemma 4.3.
Assume t h a t Assumption 1.1 holds.
Theorem 1.1 be s a t i s f i e d . i n Theorem 1.1.
L e t a,
Then, wo and
E,
E, m
i . (j 2 0) J
L e t a l l assumptions i n
and A0 be p o s i t i v e constants d e f i n e d
s a t i s f y t h e f o l l o w i n g estimates:
Nonlinear Wave Equation in Exterior Domain
(1)
Suppose t h a t n 2 6.
(accordingly then Awo, AGj
= E
L e t 0 = max (4[n/21+7,
P+
3[n/2] + 6 ) . IE /I CP ( [ 0 , m ) x 4
z)
179
m+l) and
f
= 20 t [n/2] t 2
I f a and h0 a r e chosen s u f f i c i e n t l y s m a l l ,
and
4 f o r 6 s a t i s f y i n g t h e i n e q u a l i t i e s o f t h e d a t a i n Theorem 1.1-1) w i t h 0 < 6
(2) and
Suppose t h a t 4 5 n 2 5.
T=26 + l ( a c c o r d i n g l y
s m a l l , then !two,
AGj
E
=
-T IE n C
L e t B = max (3[n/2]+6, mtl) = max (12, mtl)
3[n/2]
t
6 = 'i:+ 12).
([0,-)
x
6)
' t
If a and 6 o a r e s u f f i c i e n t l y
and
f o r 6 s a t i s f y i n g t h e i n e q u a l i t i e s o f t h e d a t a i n Theorem 1.1-2) w i t h 0 < 6
(3) Suppose t h a t n = 3. integer w i t h r
i I [ t+
Let
60.
IJ
= ~ / 7and B =
( 3 m + 7 ) ~ ]( a c c o r d i n g l y
Ifa and 6o a r e chosen s u f f i c i e n t l y s m a l l , then
71 +
%=
Awe,
t
(m + 2 ) ~ . L e t
3[n/2] E
t
6
= yt
4 P € ! f \ C ([0,m)
z 60.
be an
9). x
5)
and
f o r 6 s a t i s f y i n g t h e i n e q u a l i t i e s o f t h e d a t a i n Theorem 1.1-3) w i t h 0 < 6 2 do.
65. P r o o f o f L e m a 4.3 and Main Results. We s h a l l now p r o v e Lemma 4.3 by an i n d u c t i o n argument. same n o t a t i o n s as i n 64.
We s h a l l use t h e
- L e t L denote an i n t e g e r and k denote a r e a l number.
Yoshihiro SHIBATA and Yoshio TSUTSUMI
180
For
t h e moment we always assume t h e f o l l o w i n g assumptions: '
[A.5.1]
Awo
E
V
N
zLflCL([O,m)
E) and
x
i f n 2 6,
w
CA.5.21
ko,.-.,kp
[A.5.3]
i f Ihl
a r e a l r e a d y determined and Lemma 4.3 h o l d s f o r
ko,**-,
* P '
z s1 ,
then f o r any i n t e g e r s N and L w i t h N ' 0
and
%
O 5 L 5 L
ld,G(-Y*J)Im,O,L N where
y=
<
C(n,L,N)
<
m,
+ 4[n/2] + 7 if n 2 6 and ';J =
+ 3[n/2] + 6 i f 3 5 n
Let
We s h a l l f i r s t prepare several lenmas t o prove Lemma 4.3. s u f f i c i e n t l y small p o s i t i v e c o n s t a n t and e s p e c i a l l y Noting t h a t
~ - E - B -L T.
and -B+o'i:)
T
T
=
5 if n
5.
T
be a
= 3.
if n = 3, we can prove t h e f o l l o w i n g
lemna i n t h e same way as Klainerman [2] and Shibata [ll,121.
Lemma 5.1.
Assume t h a t [A.5.1
the following: (a)
hwj
(b)
if n
-
31 h o l d .
For w . ( j = O,l,...,p+l) J
we have
-'i L IE fl c ([(I,-) x 6) ; Iv
E
2 4, then
'"J12,O,L
+
I"jlb(n),c(n),L
= <
C6
f o r -8+L
lAwj1b(n),c(n),L
= <
C s e J.
-B+L
1AwJ12,0,L
+
-'I,
f o r -B+L &
T
-
and 0 5 L 2 L,
Nonlinear Wave Equation in Exterior Domain
ISjAwj12,0,L
+
IsjAwJ1 b(n),c(n),L
= <
181
for L >
C(L) a ejBtL
where b ( n ) = 4 and c ( n ) = ( n - 1 ) / 4 i f n 2 6, and b ( n ) =
m
T,
and c ( n ) = (n-1)/2 i f
Y
i f n = 3, then f o r -@+uL 2
< C6
2,0,L
=
m,k,L
=
<
ca
f o r k-B+sL
-T,
_i -1,
for - B t d 2
T
f o r k-B+oL 2 for L >
12,0,L
= <
C(L) 6 9 j B + O L
IsjAwj Im,k,L
= <
C(k,L) 6 e t - B t a L
ISj"j
rv
L 5 L, h
T,
0 2 L 5 L and 0 5 k 5 1-E,
1, >
-
1-E o r L > L ;
i f n 2 4, then
(d) I(1
for k
and 0 5
-
sj)Awj12,0,L
+
I(1
-
Sj)Awjlb(n),c(n),L = < Csey'+L J
for
o
L
N
L,
where b ( n ) and c ( n ) a r e t h e same as i n ( b ) ; i f n = 3, then
(e)
I
-
'j)"j
Im,k,L
= <
c 6 et-B+oL
for 0
z k 5 1-E and 0 5 L z r .
By choosing 6 s u f f i c i e n t l y small we assume t h a t :
rA.5.41 < C6
IAwjlm,o,o
5 C l A w j 12,0,[n/21+1
=
lA'jl-,o,o
5 Cl~'jl2,0,[n/21+1
= <
From Theorem Ap.1,
rA.5.1
-
z a1 1
Cs 5 %
( j = O,l,...,p+l), ( j = O,l,-..,p).
41 and Lemmas 4.1 and 4.2 we have the following
Yoshihiro SHIBATA and Yoshio TSUTSUMI
182
1emma.
Lemma 5 . 2 .
For e . ( j = O,--.,p) J
Assume t h a t Assumption 1.1 holds.
the following:
rACL([O,-)
we have
&
ej
(a)
E)
E
]E
lej12,k,L
5
cs
lejll,k,L
<
c s e k-(1t7E)B+oL
=
x
;
(b)
Proof.
n,
for
‘jk-36toL
j
By t h e d e f i n i t i o n o f e j
i n t h e case o f n = 3.
Since e
j
= e*
5 k 2 2 ( 1 - ~ ) and 0 5 L 5 L,
i f n = 3,
f o r 0 5 k 5 1-E and 0 jL 2 L,
if n = 3.
-
T+B
(a) i s clear.
+
e:-, j
So, we s h a l l prove ( b ) o n l y
we have o n l y t o prove t h a t ( b ) holds ~
f o r e: and e - - , r e s p e c t i v e l y . However, we s h a l l prove o n l y f o r e: because we J j J can prove f o r e’- i n t h e same way. 2 Since dG , ej =
j
t,x,O)
lo{I, 1
= 0, i t f o l l o w s t h a t
l (d,G)(t,x,r’(S.Aw. 3 x
By Lemma 5.1,
(S.AW. J J
+ r(l
J
J
+
r(1
-
-
S.)Awj))dr’ J
S.)AW.,(l
J
J
-
1
Sj)AW
j’
Aij) d r .
Theorems Ap.1 and Ap.2 we have f o r k w i t h k-6 2
T
and k 5 1-E
183
Next by L e w a 5.
le;ll
,k,L
= c
, Theorems
Ap
c I(' [Awjlm,o,L
-
+ Ihw. 2,0,LI(1 J
sJ.
Thus, t h e l a s t i n e q u a l i t y i n ( b ) i s proved.
(Q. E. 0.)
By L e n a s 4.1,
Lemma 5.3. (a)
Ep
E
4.2, 5.2 and
ej
=
BJ we have t h e f o l l o w i n g lemma.
Assume t h a t Assumption 1.1 h o l d s . ' U r v
EL/7CL([0,-)
x
5)
Then,
;
(b 1 JEpl~,n-l,t
2
' /Eplq,n-l,L 3-4-
= c
c5
2
for -&+L
5
-T,
if n 2 6 ,
Yoshihiro SHIBATA and Yoshio TSUTSUMI
184
IEpl2,n-1 ,L
+
< C 63
IEp12,k,L
=
IEpll,k,L
= <
IEpll,k,L
lEp11 ,C,L 2
C6
3
< C 63 =
z -T,
for - 2 ~ t L
< C 63
=
for
T+B
k and k-38toL 2 -'I, i f
f o r k - ( 1 + 7 ~ ) ~ + o2L
ek-(1t7E)B+oL p
-T,
if 4
n
zn 2 5,
= 3,
if n = 3 ;
f o r k - ( 1 + 7 ~ ) @ + o L2
T,
0 2 k
z 1-E
and
'u
O ~ L L L , i f n = 3.
N o t i n g t h a t l - ~ - ( l t 7 ~2 )T,~ 2 ( 1 - ~ ) - 3 8 2 Lemmas 4.1,
N
T
and -3BtoL 2
4.2 and 5.3 the f o l l o w i n g lemma.
L e m a 5.4.
Assume t h a t Assumption 1.1 holds.
Then,
T~
we have by
Nonlinear Wave Equation in Exterior Domain
3 k-3B+aL ' ~ ) ~ p l 2 , k , L= < C(k,L) 6 ep+l
1('p+l
I(sp+l
-
Sp)Epll,k,L
By Assumption 1.1,
Lemna 5.5. (a)
= <
3 k-(1+7~)4+0L C(k,L) 6 ep+,
[A.5.1]
and [A.5.3]
E
'rl - i C ir ([0,-)
E
,
x
f o r k 2 T+B and L 2 0, i f n = 3, f o r k 2 0, and L 2 0, i f n = 3.
we have t h e f o l l o w i n g lemma.
Assume t h a t Assumption 1.1 holds.
G(t,x,Awo)
185
Then,
;
(b)
By Lemmas 4.1, 4.2 and 5.5 and Theorems Ap.1 and Ap.2 we have t h e f o l l o w i n g 1emma.
Yoshihiro SHIBATA and Yoshio TSUTSUMI
186
Combining Lemmas 5.2, fact
eo
=
5.4,
5.5 and 5.6 and u s i n g Lemmas 4.1,
4.2 and t h e
1, we have t h e f o l l o w i n g lemma.
Lemma 5.7.
Assume t h a t Assumption 1.1 holds.
90' gp+l
(a)
E
E"nC"([O,-)
x
5)
Then,
;
(b)
I n o r d e r t o use Theorem 3.2, we have t o e v a l u a t e t h e c o e f f i c i e n t s o f t h e
:.tj
operator
d e f i n e d i n (4.8).
A . = (d,F)(t,x,Av) 3
+
Noting t h a t (d,F)(t,x,Av)
Put
(d,G)(t,x,SjAwj). = 0 for
It1 2 2,
we have t h e f o l l o w i n g lemma by
Nonlinear Wave Equation in Exterior Domain
Lemma 5.1,
[A.5.1
Lemma 5.8. $o,
-
41, Theorems Ap.1, Ap.2 and Ap.3.
c
L e t L be a p o s i t i v e c o n s t a n t d e f i n e d i n [A.5.2].
$1 and f s a t i s f y a l l assumptions i n Theorem 1.1.
holds.
< C6
=
f o r -B+L 2
-T
and 0 5
L e t t h e data
Assume t h a t Assumption 1.1
Then we have t h e f o l l o w i n g :
IA014,d,L 4
187
L 5 L,
Yoshihiro SHIBATA and Yoshio TSUTSUMI
188
cy
[ A p t 1 12,1tE,L
~
62 el+E-B+oL P+l
l.
f o r 0 5 L 5 L.
I n p a r t i c u l a r , choosing 6 s u f f i c i e n t l y small, we have t h e f o l l o w i n g :
Here d i s a p o s i t i v e constant g i v e n by Lemma 3.1.
Proof.
We s h a l l g i v e t h e sketch o f t h e proof for,%
P + l o n l y i n t h e case
o f n = 3. Since (dhG)(t,x,O) we have by L e m a 5.1,
= 0,
2 (dAG)(t,x,O)
Theorems Ap.1,
= 0 and f o r It1 2 2
Ap.2 and Ap.3
( d X F ) ( t r x , h v ) = 0,
Nonlinear Wave Equation in Exterior Domain
1+E -8 Noting t h a t 7
-T
, we
have f o r
l+E 2 -8toL
2
189
T
F i n a l l y we have by Lemma 5.1
From the above lemmas we can complete the p r o o f o f Lemma 4.3.
Proof of Lemma 4.3. First
We s h a l l prove Lemma 4.3 by an i n d u c t i o n argument.
we assume f o r the moment t h a t r A . 5 . 1
have by Lemas 5.7, 5.8 and Theorem 3.2 t h a t
- 41 hold.
Then, i f n 2 6 , we
Yoshihiro SHIBATA and Yoshio TSUTSUMI
190
I"ptl12,O,L
I"ptl14,n-JL
4
2
t 6 max(1,
We have used t h e f a c t
2 4[n/2]
e -~+L+3[n/21+6) P+l
t
7 a t the l a s t i n e q u a l i t y .
by choosing 6 so small t h a t max { C(L)s ; 0 holds f o r
I n t h e same iptl,
eply 1
IL
zT 1 -5 1 we
I n particular,
see t h a t Lemma 4.31)
way i t i s c l e a r t h a t under t h e assumptions CA.5.11
and EA.5.31 Lemma 4.3(1) h o l d s f o r
i0,By
t h e way, we see by t h e assumption on
t h e data i n Theorem 1.1, Theorem 3.2, Theorems Ap.2 and Ap.3 t h a t [A.5.1]
5 . 31 h o l d .
and [A.
Therefore, an i n d u c t i o n argument g i v e s Lemma 4.3(1).
i n t h e same way we o b t a i n Lemma 4.3(2) f o r 4 2 n 5 5. F i n a l l y , f o r n = 3 we s h a l l v e r i f y t h a t under t h e assumptions [A.5.1 Lemma 4.3(3) h o l d s f o r 1 +
E
2 B +
T
iptl. By Theorem
3.2,
Lemmas 5.7,
41
5.8 and t h e f a c t t h a t
we have
We have used t h e f a c t s t h a t 1+€ -B+o 5
-T
and t h a t I + E - ( ~ - E ) B + C5 J 0 a t the
second i n e q u a l i t y and t h e l a s t i n e q u a l i t y , r e s p e c t i v e l y .
Thus, we have
I n p a r t i c u l a r , by choosing 6 so small t h a t max { C(L) & 2 ; 0 2 L obtain
-
-.I
L 1 5 1 we
Nonlinear Wave Equation in Exterior Domain
191
Next, by Theorem 3.2, Lemmas 5.5 and 5.8 we have
("ptl
<
L,l-€,L
=
+
,2,1+~-6 P+l
6'1
t
c ( L ) 6 3 [ e l + ~ - ( 1 + 7 ~ ) 8 + o ( L + 3 [ n / 2+4 1 1 P+l el+~-36+o(L+3[n/2]+4) P+1
eAl7-6 max (1, e
I+~-B+o(L+3[n/2]+6) P+ 1
+
<
-
qL1 &3[
( 1+E ) / 2 P+ 1
- 6+0 ( L+3 [n/ 2]+4) 1
1
1+~-36
ep+l
el+~-(1+7~)6+o(L+3[n/2]t4) P+l
+ e 2( 1+~)-46+o(L+3[n/2]+6) P+l
-6+0 5 - -T a t t h e f i r s t
We have used t h e f a c t s t h a t 1 + ~2 6+r and t h a t inequality. <
2 By t h e way, s i n c e z ~ + o ( 3 [ n / 2 ] + 4 ) - & 6
<
Oand 1 + 3 ~ - ( 3 - ~ ) 6 3+ ~ ( 3 [ n / 2 ] + 6 )
0, we have
( b 7 ~6+0 ) (L+3 [n/2]+4) 5 -1 2( 1 + )-46+a ~ (L+3[n/2]+6)
-
E-
- ( 1- E ) 5
( 1+E ) B+oL
-
Thus, i t f o l l o w s t h a t
By t h e Sobolev imbedding theorem and ( 5 . 1 ) we have
Therefore, by i n t e r p o l a t i n g between (5.2) and (5.3) we have
for 0
z k 2 1-E
and 0
L
IT.
Thus, s i n c e -.~6+o([n/2]+1) 2 0, we o b t a i n by
(5.4) for 0
(5.5) Ifwe choose 6 so s m a l l t h a t max
{
C(L) '6
;0
k 5 1-E and 0 5
5L
zy
cu
L 5 1.
15 1, t h e n (5.5) and
Yoshihiro SHIBATA and Yoshio TSUTSUMI
192
(5.1)’ give
L e n a 4.3(3) f o r ;p+l.
Since we can prove i n t h e same way as t h e
case o f n 2 6 t h a t Lemma 4.3(3) h o l d s f o r
wo
and t h a t [A.5.1]
and [A.5.3
-
41
T h i s completes t h e p r o o f o f
hold, an i n d u c t i o n argument g i v e s Lemma 4 . 3 ( 3 ) . Lemma 4.3.
(9. E. D.) P r o o f o f main r e s u l t s .
Put
m
Then, from Lemnas 4.1
-
3, Lemmas 5.1
-
6 and (4.6) we e a s i l y see t h a t u = v + w
i s t h e d e s i r e d s o l u t i o n o f (M.P) ( f o r d e t a i l s , see Klainerman [2] and Shibata
[ll,121).
Furthermore, we can prove the uniqueness o f t h e s o l u t i o n o f (M.P)
by t h e energy method i n t h e same way as Shibata [12].
(Q. E. D.)
Concluding Remarks.
(1)
When n = 3, we used t h e c u t - o f f f u n c t i o n i n time.
The authors do n o t know whether we can prove w i t h o u t i t f o r n = 3 i n t h e same way as Klainerman and Ponce [ 3 ] and Shatah (2)
[lo].
We can a l s o o b t a i n t h e analogous r e s u l t s f o r t h e mixed problems o f t h e
n o n l i n e a r Klein-Gordon equation and t h e n o n l i n e a r Schrodinger e q u a t i o n i n t h e same way (see, e.g.,
TsuTsumi [13]).
56. Appendix. I n t h i s s e c t i o n we s h a l l s t a t e several theorems which p l a y an i m p o r t a n t r o l e i n the p r e s e n t paper.
Theorem Ap.1.
Let
For t h e i r p r o o f , see Shibata [ll,121.
p = Rn o r a. L e t
and f and g be f u n c t i o n s from [O,-) @,
x
and
JI
be f u n c t i o n s f r o m ,g t o
R1
& to IR1 . Assume t h a t a l l semi-norms o f
$I,f and g appearing below a r e bounded.
k and
@
L e t M and
N be nonnegative i n t e g e r s ,
m be nonnegative numbers and p and q be r e a l numbers w i t h 1 5 p, q 2
m.
Nonlinear Wave Equation in Exterior Domain
Then,
Furthermore, i f F(t,x,O)
Theorem Ap.3.
Let
= 0, then
o0,
$1 and f be t h e data o f (M.P)
such t h a t a l l semi-
193
YoRhihiro SHIBATA and Yoshio TSUTSUMI
194
Let
norms appearing below a r e bounded. i n 54.
L e t H(t,x,x)
E Wm([O,m)
x
5
x
% and {
1x1
v ( t , x ) be t h e same as those d e f i n e d
2 1 1).
I f H(t,x,O)
= 0, then
References P. Dionne, Sur l e s p r o b l i m e de Cauchy hyperboliques b i e n poses, J . Analyse Math., c21
10 (1962). 1-90.
S. Klainerman, Global e x i s t e n c e f o r n o n l i n e a r wave equations, Corn. Pure
Appl. Math,, 33 (1980), 43-101.
S. Klainerman and G. Ponce, Global s m a l l amplitude s o l u t i o n s t o n o n l i n e a r e v o l u t i o n equations, Comm. Pure Appl. Math., 36 (1983), 133-141.
P. Lax and R. P h i l l i p s , S c a t t e r i n g Theory, Acad Press, 1967.
R. B. Melrose, S i n g u l a r i t i e s and energy decay i n a c o u s t i c a l s c a t t e r i n g , Duke Math. J . , 46 (1979), 43-59. r61
S. Mizohata, Quelque problemes au bord, du t y p e rnixte, pour des equations
hyperboliques, S h i n a i r s u r l e s Gquations aux derivees p a r t i e l l e s , C o l l i g e de France, (1966/67), 23-60. [71
A. Moser, A r a p i d l y convergent i t e r a t i o n method and n o n - l i n e a r d i f f e r e n t i a l equations, Ann. Scu. Norm. Pisa, 20(3) (1966), 265-315, 499-535.
r 81
J . Nash, The embedding problem f o r Riemannian m a n i f o l d s , Ann. Math., 63 (1965), 20-63.
r91
P. H. Rabinowitz, P e r i o d i c s o l u t i o n s o f n o n l i n e a r h y p e r b o l i c p a t i a l d i f f e r e n t i a l equations
II , Corn.
Pure Appl. Math.
, 22
(1969), 15-39.
J . Shatah, Global e x i s t e n c e o f small s o l u t i o n s t o n o n l i n e a r e v o l u t i o n equations, J . D i f f e r e n t i a l Eqs.
, 46
(19821, 409-425.
Y. Shibata, On t h e g l o b a l e x i s t e n c e o f c l a s s i c a l s o l u t i o n s o f mixed
Nonliiiear Wave Equation i n Exterior Domain
195
problem f o r some second o r d e r n o n - l i n e a r h y p e r b o l i c o p e r a t o r s w i t h d i s s i p a t i v e term i n t h e i n t e r i o r domain, Funk. Ekva., [12]
25 (1982), 303-345.
Y . Shibata, On t h e g l o b a l e x i s t e n c e o f c l a s s i c a l s o l u t i o n s o f second o r d e r
f u l l y n o n l i n e a r h y p e r b o l i c equations w i t h f i r s t o r d e r d i s s i p a t i o n i n t h e e x t e r i o r domain, [13]
Tsukuba J . Math., 7 ( 1 ) (1983), 1-68.
Y . Tsutsumi, Global s o l u t i o n s o f t h e n o n l i n e a r Schrodinger e q u a t i o n i n
e x t e r i o r domains, t o appear i n Corn. P. 0. E. [14]
Y . Tsutsumi, Local energy decay o f s o l u t i o n s t o t h e f r e e Schrodinger
e q u a t i o n i n e x t e r i o r domains, t o appear i n 3. Fac. S c i . Univ. Tokyo, Sect.
IA, Math. [15]
B. R. Vainberg, On t h e a n a l y t i c a l p r o p e r t i e s o f t h e r e s o l v e n t f o r a c e r t a i n c l a s s o f o p e r a t o r - p e n c i l s , Math. USSR Sbornik, 6 ( 2 ) (1968), 241-273.
[16]
B. R. Vainberg, On e x t e r i o r e l l i p t i c problems p o l y n o m i a l l y depending on a s p e c t r a l parameters and t h e a s y m p t o t i c behaviour f o r l a r g e t i m e o f non s t a t i o n a r y problems, Wath. USSR Sbornik, 21(2) (1973), 221-239.
[17]
B.
-
R. Vainberg, On t h e s h o r t wave asymptotic behaviour of s o l u t i o n s o f
s t a t i o n a r y problems and t h e a s y m p t o t i c behaviour as t
-+
o f solutions o f
n o n s t a t i o n a r y problems, Russian Math., Surveys, 30(2) (1975), 1-58.
[la]
W. von Wahl, LP-decay r a t e s f o r homogeneous wave equations, Math. Z.,
120 (1971), 93-106.
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Yoshihiro SHIBATA and Yoshio TSUTSUMI