Internat. J. Math. & Math. Sci.
(1988) 335-342
335
V0L. ll NO. 2
ON A NONLINEAR WAVE EQUATION IN UNBOUNDED DOMAINS
CARLOS FREDERICO VASCONCELLOS
Institute de Mal;emtica
Universidade Fe(leral do Rio de Janeiro
P.O. Box 68530 CEP 21944, Rio de JaJeiro, RJ
Brazil
(Received October 16, 1986)
ABSTRACT.
We study existence and uniqeness of the nonlinear wave
equation
’-’ll +
M(x,
IvuCx,t)12dx
in unbounded domains.
E]Y
i.
Our teeImiques ivolve fixed point
combined wih She energy method.
WORDS AND PHR2SES.
1980
0
The above model describes nonlinear wave plle-
nomenon in non-homogeneous media.
armens
I-(x,)ldx)C-Au+.)
+
Nonlinear equation, unbotuded domain,
energy method, fixed point theorems.
AMS SUBJECT CLASSIFICATION CODE.
35L70.
INTRODUCTION
In this paper we prove the existence
and uniqueness of a local so-
lution for the following problem:
82u
8t
2
+
u(,o)
where
M:
nx , ,
Ilu(t)ll 2
II()II)
-(
"t (x,)
"o
Z
j=l
n
0
12dx
IK.(x,t)
J
Ul(X)
lu(,t)
+
2
dx,
t > 0
n
and
i] a 1)ondod
open sl)set of
[R
n.
]le
’od a weak local solt,i(,, lot tl,e
using G;lerkin met]od a;tl t]e (liscrc, tc" spectrm o J"
el,crater in bouded
C](."
l.tll;’i,t];
C. F. VASCONCELLOS
336
In he oher hand since ]a|, tle mappinl, M depends explicity on
not use Forier transform as was (lono for example, 1)y. G.P.
x
we can
([2])
Menzala
in which he studied t],e problem
is indel)ondent of
M
that is w]en the mapping
m
+
o
>
(x)
.o. in
0
nx
f()(x),
able with
]lero
(x,)
0
(X)
z
,m
W
(e n)
0
where
a e
L
{
0
W(x)
M(,X)
nd
+
is continuously differenti-
f:
(n):
e
n
[(x)
3
(n)
L
(n)
consider the usual Sobolev space
3=1
Rn
in
O.
For
Ilul[
x.
are described below.
M
Our assumptions about the mapping
(1.1)
We also
n}
j i
with the norm
lu(x)12dx.
dx +
n
Our main resul% in this paper will be:
There exis%s a unique local solution for problem
(1.1)
with the fol-
lowing properties:
We
Where i
>
0
eio
d
:
nX[o,w2]
wi
(.)
;2(n))
2
C([ O,W23 ;H() ).
<
For eh
C
Here
H2(n)
u(.,)
W
L
and
denoe
(.,e)
H
1
().
(.)
L2
2
wice con%inuously differen%iable in
(-]-)
H2(n)
Du
e u
[0,T2],
v 6 L
2(n).
ie= podu in
We
o noe y
the usual Sobolev space of order
The basic idea in order %o ob%ain our resu1% will be to use fixed
point argents together with the
ener
method in appropriate Banach
spaces.
I% is impor%ant %o observe that our main result holds also in any
n.
open subse% of
few
Before concluding this introduction we would like %o make a
coones on ehe Zieeraeure. J.. Lions ([3]) considered the problem:
(.)
u(,o)
where
()
Ni,i=
m
0
([’]),
>
u
0
(x),
O
and
pooo
ue(x,0)
z()
u
denotes a bounded open subset of
([]),
io
([.6]),
iboio
([?])
337
NONLINEAR WAVE EQUATIONS IN UNBOUNDED DOMAINS
A PRELIblINARY IESULT
2.
In this section
.
existence and uniqueness oF
[l}e
we ])rove
Let
of ]e following "linearized" problem:
C([O,] ;())
2u
2
t
u(,0)
o
()
(0,T]
t(x,0)
H
From now on we shall denote by
I1 =
consider the norm
H2(n).
D(A)
with
u()
A:
A
Clearly
L2( n)
the usual space
lu(xll2d
operator
us consider the linear
sol,ion
Let
T > O.
and ironer product
D(A)
H
c H
in whic] we
(-I-).
defined by A,
square root of
A,
V
n
!1" !1
defined in
bv
bu
Z
j=l
with norm
has domain
D(A I/2)
V
The
Hl(n).
is defined by:
(A1/2ulA1/2v)
July]
A I/2
denoted by
-A,+u,
is self-adjoill and satisfies:
All functions we consider in this paper will be real valued.
The inner product in
Let
JRn
ax.J x.
J
u(x)v(x)dx
dx +
n
i.
For each X 6
we define
Because of our assumptions on
M(x,k)
the operator
B(k)
has the fol-
lowing properties:
k 6
,
For each
),. ;
o
For each
k
O
For each
on
B(X)
is a linear bounded symmetric operator
H.
((x),l,)
B(k): V
-,
o
V
I,1
u6 H
is a linear continuous
bijective operator
V T
if
>
0
’q
T
>
XI, x=l
0
e.
such that
continuous operators on
M T > O
8T
>
O
(V)
Here
II(x_)-(xe)II.(v
is the space of linear
V
such that if
(u,v)
6
D(A)xV
and
Ix
T
(::.7)
C.F. VASCONCELLOS
338
(2./4) we
N(t)
By
PROOF:
symmetric we obtain
(2.8).
In the other hand if
that
and
t 6
[A/2.[
if{
B(llv (t)II2)
(2.i)
we obtain by
2
o
o b (e.8)
(e.o)
N(t)[N(0)3 -1 e
po
PROP0SITI0S 1.
u
H
mtd
Let
u
e
Then there is a
e
D(A)
u
D(A),
A I/2
m
o
(2.3)
and
u]
are
2
that
t]terefore
oiows.
Ne consider
u
sii,ce
(s(}}()]}2)/2.[Ai/2.)
(().)
zo
IIonco
H.
is
(2.9)
D(N(t))
can show t],at
the image of
ce(v)
que
T >
0
d t osd
(2.5)
then by
D(A3/2),
H3(n)
fction
u
1
e
H3( n}
u:
p
too,
we obtain,
H2(n)
and
such that:
c(e;,(A))
(e.)
(e.e)
u(0)
u’ (o)
Uo
u
By Lena 1 and a result due to J. Goldstein (see Theorem 2.2.
n) such that
[20]) there is a unique fction w: R
PROOF:
in
H2(
(o)
/e u o
w’ (o)
Let us consider
satisfies
(2.11)
and
A=l/2w(t)
u(t)
(2.12).
Therefore it follows that
which satisfies
PROPOSXTXON 2.
cl(0,T;V),
v
unique function
u
e
ce(o,wv)
e
then u:
is the ique solution of
u
(2.12)
(2.11).
Let
T
be a positive real nmber.
H3(n) D(A 3/2) and
u
H(n}
u(v): [0,W]
C(o,w;D(A))
u
v(t)
w(*)
cl(; V)
v’
if
(T) t-T) +v( t)
’ (o)t
+
(o)
and hence there is
Proposition I, in particular
: [ O,T]
Then given
e H2(n).
u
1
such that:
There
s
a
(e.)
u
0
T
t
il"
t > T
ir
t <
u
o
n
u(w):
satisfies
(2.15)
which satisfies tle
and
(2.16),
wit],
,(n).
Remains to prove the uniqueness.
ton
t
for
We
PROOF
w 6
(z’u)
/e u
or (e.6)
whoh .tr
Suppose that we have another so-
(e.5).
339
NONLINEAR WAVE EQUATIONS IN UNBOUNDED DOMAINS
(t)
Then
g"
(t)
B(llv(t)ll2)Ag
+
o(o)
u(t)-z(t)
o
0
o’(o)
V x
in
[0,T]
(2.a7)
o
lO (t)]
(t)
We consider
(2.17)
s,tisfies
+ ((ll
(t)}}2 ) 1/2 oI /
then by
we obtain that
+
(e. )
,o,
((ll()l)i/oli/eo’)
[()l, (t)] (, ((t)l )i/i/
-((l,(t)ll)o ’)
’ (t)
d
+
(e.)
8T
(lo(t)l
’(t)
+
2
lo’(t)l 2
+
cI(t) e
+
wh e r e
C
Now, usin
(’
u,
T
0s tsT
(2.3)
since
T
l(*)’ (t))"
DT
we obtain that there exists
n(t),
’(t)
}{ence,
(ll(t))[z(-)
(0)
such that
te (o.].
(t)
it follows t]lt
0,
> 0
% t
0,
6
[O,T]
which
proves the Proposition 2.
comoLLamr i.
-
vi(0.W;V(e)), no e H(e)
e
Lt
Th hr i
,:
iq,
e
SOLUTION OF PROBLGN
H(en)
[O,W]
Ce([0,W] ;,i(n)) n ci([o, w]
H
2
(ran))
u=h
,d
tisrie
(e.o).
(I.I)
X
[u: [0,T]
T
H
is a Banach space with the
Now, we
2([Rn).
consider
We observe that given
which satisfies
(2.15)
and
0
0
0
0
wllere
’
<
2
2
T0 ,’11 x T
i’
(11 ,’ t)ll 2 )11 a: (,)
T
)"
0
z(t)
X
(2.3)
fore, ]’y
t
v 6
z(O)
exp(---r 2
Yc t),
0
t
g
T.
t
T.
Thus
C.F. VASCONCELLOS
340
o
(o)
o’(0)
If we define
by
1
y(t)
I’ (t) 2 + (B(llu(t)l 2 AI/ 2G
(2.6) and (3.U), tlen we obtain that:
’.)11 Io’ (’.)1 +
(,.)
+
d
alovo we ol)taln tlnxt
(t)
e
<
To
Let us consider
we obtain by
(.)
Now,
..aTI"o
r
+
me Ycr
1
F
(3.5)
IIllxT
,,here
e
+
)y(t)
T
+
+ 2a
1
In
C
To
r
(2.3), (2- 5)
r211u-VllYT IIo11 YT
O
2
(o 2F t
0
T1 <
)ilu-ll
rain[To,
o
I1oll
YTo YTomin(l,mo)r
i
0 <
log
(.
t < T
/4 a 2 r 2
To
I
II(t)ll) = e il u-vii
YT I I! oll YT I
aT r2 (e 2FTI I) min(l,mo)
_
and Lel,,vv,,, 2
o
Then, since that
that
If we repeat the proof for
(1o (t)l
l;]e,, l,y
,’,
2
we choose
+
I1’ (t)ll I^(*)I I1*’ (ll"(t)ll )ll()llo(t)ll :’-
=
II,,llxTo
1/2
s..,,llo(
’
lI(ll,,()ll)-(ll(e)ll::)ll()i^s()l Io’ ()1
+
-"
A
g
To
t
O,
(3.6)
o
I)
+
follows, by
o
y(O)
t
T
(3.6),
that:
(. )
NONLINEAR WAVE EQUATIONS IN UNBOUNDED DOMAINS
E
D(A).
T]le]l
341
t],ere
exists
(].8)
(3.o)
(3.Ii)
C. F. VASCONCELLOS
342
(in memoriam)
P.H.
This paper is dedicated to Professor
ACKNOWLEDGEMENTS.
friend and adviser.
Rivera
The author would like to tl,ank
Professor G.P. Menzala for his helpful suggestions.
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