GLOBAL CLASSICAL SOLUTIONS
TO
THE CAUCHY PROBLEM
FOR A NONLINEAR WAVE
EQUATIONby
HAROLDO R.CLARK UniversidadeFederalFluminense
InstitutodeMatemAtica
GAN
Rua S.
Paulo, 3024.040110Niter6i,
RJ
Brasil email:(Received August 24, 1995 and in revised form January 3, 1997)
ABSTRACT. In
thispaperweconsidertheCauchy problemu" +
M
(IA1/2u[
2)
Au
0 in\
u(0)
uo,u’(0)
where
u’
is the derivativein the senseofdistributions andIA
1/2ul
istheHnorm
ofA
1/2u.We
prove the existence and uniqueness ofglobalclassical solution.KEY WORDS AND PHRASES:
Nonlinear, waveequation,globalclassical solution 1995AMS
SUBJECT CLASSIFICATION CODES:
35D10; 35005; 35105.1.
INTRODUCTION
In
this work we prove the existence and uniqueness of globalclassical solution to theCauchy problem
(1.1)
0"PM
(fn
[Vu(x,
t)l=dx)
/xu 0(, 0)
0(),
o(,
0)
(x)
wherefis aboundedorunbounded openset of Fi
n,
M(f)
is alocally Lipschitzfunction on0
[0, +o[,
AY’],=I
istheLaplaceoperator and]Vu(x,
t)l
2,=1
u77,
"2
The equation
(1.1)1
to modelthe smallvibrations ofanelasticstringwhere we admitTheCauchyproblem
(1.1)
canbewritteninanabstractframework,
iftheLaplace operatorisreplaced byaselfadjoint positive operatorA
inareal Hilbert spaceH.
Representingby
A1/2
the squarerootofA,
the problem(1.1)
have the followingabstract framework(1.2)
u"
+
M
(]A1/2ul
2)
Au
0,(o)
,o,,’(o)
u,in
]0, T[
where
u’
isthe derivative inthesenseof distributions andIA
1/2ul
istheHnorm
ofA
1/2u.Severalauthors havebeen studing theproblem
(1.1),
amongthem,
we canmention thefollowingones: ArosioSpagnolo
[1],
Dickey[3],
Ebiharaet.al.[4],
Lions[8],
Matos
[9],
Medeiros and Milla Miranda[10],
Menzala[11],
Nishihara[12],
Pohozaev[13],
Yamada[14].
In
Dickey[3]
theproblem(1.1)
wasstudiedfor thecasen 1, whenfisthe positive real line. His result wasgeneralized forf’
byMenzala[11].
These tworesultswereobtainedbythe methodof Fourier transforms and the results provedbyDickey
[3]
and Menzala[11]
was generalized byMatos
[9],
as an application ofthe DiagonalizationTheorem byVon
NeumannDixmier, e.g.Huet
[5]
andLionsMagenes[6].
To treatthe abstract case, when one does not havecompactness, Lions
[7][8]
proposedtostudytheCauchy problem
(1.2)
by makinguseof theDiagonalizationTheorem. Therefore,themainobjective of this work is tostudytheproblem
(1.2),
independent of compactness.To
obtain theglobalclassical solutionswesuppose that theinitialdataareanalytictype. With this choiceforu0andul,wefollow the ideas ofArosioSpagnolo[1]
that prove theglobalexistence ofasolution tothe Cauchy problem
(1.2)
when the domainD(A)
of the operatorA
havecompactimersiononthe real Hilbert spaceH
and theM()
isanonnegativelocallyLipschitz function.
Thearguments developedinthe present workstudythe Cauchy problem
(1.2)
and can be summarizedas follows.We
apply formally a diagonalization operatorL/ in the problem(1.2)
to obtain(1.3)
{
(0)
o,(0)
where
A
is apositive realparameter, e.g., Section 2. The solution of theCauchy problemfor2.
TERMINOLOGY
A
fieldof the Hilbert spaces is,by definition,isamappingA
7(A),
thatfor eachisassociatedaHilbert space
7(A).
A
vectorfieldonreal number is amappingA
definedon,
suchthatu(A)
6/(A).
We
representby"
thereal vector space of all vector fields on andbyua positive real measure.A
field of Hilbert spacesA ,
H
(A)
issaid tobepmeasurablewhenthereexistsasubspaceM
of
satisfying thefollowingconditions:The mapping
A
+Ilu(A)ltn(.x)
ispmeasurableforallu6]; Ifu6"
andA
,(u(A),
v(A))n()
isPmeasurablefor allv6,
thenu6.;
There existsin
Az
asequence(u,,)n6v
such that(un(A)),6v
is totalon/(A),
for eachA6R.
The objects of
A
are called pmeasurable vector fields.In
the following,represents a pmeasurable field of Hilbert spaces and all the vector fields considered are
measurable.
Nextwedefine the space
7"lo
fe
7l(A)du(A)
in thefollowingway: a vectorfieldA
u(A)
is in7o
ifonlyifTwo
vector fields that are equala.e., in H0,relative to the measureu,will be identified.We
define in/o
thescalar product:(2.1)
(u,
v)0
=/R(u(A),
v(A))(A)du(A),
for allu,With the scalarproduct
(2.1)
the vector space/o
turns out tobea Hilbertspace which is called the hilbertianintegral(or,
as called by otherauthors, measurablehilbertian sum; e.g.Lions and
Magenes
[6])
of the fieldA ,
/(A).
Givenarealnumber
,
denote by/a
the Hilbertspaceofthe vector fieldsusuch that thefieldA

Aau(2)
is in/o.In
/a wedefinethe followingnormI()d(),
ue
Let
usfixaseparableHilbert spaceH
withscalar product(,)
andnormI,
I.
We
consider inH
aselfadjoint operatorA
such thatwhere
D(A)
isthe domainofA.
With this hypothesis, the operatorA
satisfiesthe conditions of theDiagonalizationTheorem,
e.g.[6].
It
then follows that thereexistsa HilbertianintegralH0
fo
H(A)dv(A),
wherevis apositiveRadonmeasure with support in]Ao, /c[,
0<
Ao
<
/
(where/
isthe constant of(2.3),
andaunitaryoperator/g fromH
onto/0,suchthat(2.4)
bl(Aau)
Aab/(u),
forall ue D(A’),
a>_
0(2.5)
/d is anisomorphismfromD(A
’)
ontowhere
D(A)
isequipped withthegraph norm,i.e.,lul(a
[u[
/[Aul
2,
for all uD(A’).
Observe thatwiththe
supp(v)
C]Ao,
+oc[,
wehave fora_>
3,aand3
realnumber, that(2.6)
(2.7)
lu[
_<
c(Ao)lull,
for alluwhere
c(Ao) >
0and 0<
Ao
<
B.
3.
GLOBAL SOLUTIONS: EXISTENCE AND UNIQUENESS
In
order to obtainglobalexistence and uniqueness for solutionof theCauchy problem(1.2)
wewill introducethefollowing hypotheses:(3.1)
(Au,
u)
lul
,
for alluD(A)
and3
>
0,(3.2)
M
islocallylipschitz function on[0,
+oo[
andM() _>
m0>
0, for all_>
0,(3.3)
andfor somer/>0, wheree2"xn
exp(2At)
andH
isthe unitary operatorof(2.4)
and(2.5).
We
havefrom the hypothesis(3.1)
thatsupp(v)
C]Ao, +c[,
0<
Ao
<
/.
Thus, let us definethe space of thefsuactions
that satisfies(3.3)
asfollows(3.4)
W
wH;
e2"’7[Ib[(w)ll2dv(A
<
+oo,
for someH
isthe Euclideanspace/R,the operator//is the Fourier transform operator"
andu(x)
isthefunction
u(x)
_{+11.,}ar/+6
>
0,thenubelongs
toW,
because.Tu(x)
v/e
’lxl The convolutionw(x)
(u
v)(x)
with w 6L2(),
are also example of functions ofW. Similarlyweobtainfunctionsini",n_>
2.The vector space
V
isidentified with the Hilbert space:(3.5)
V
{v
6D(A’);
IAvl
<_
CA
m!,
forsomeA >
0, c>
0, m6SV}.
The characterization of space
W
with the spaceV
isgiven by:PROPOSITION
3.1: IfW
andV
arespaces defined in(3.4)
and(3.5)
respectively, thenV=W.
PROOF. We
haveby(2.5)
(Diagonalization
Theorem),
inparticular, that/2
D
(A)
7
is aisomorphism andThus,if u6
W,
then from the inequality (2x)"_{m’}_< e2X
wehave.X"
<
m!
eTMand
,x"llU(u)ll2d,(,)
<_
m!
e2llU(u)ll2d,(,)
<
+.
Therefore, W

V.Reciprocaly, ifu6
V,
wehaveor
Therefore,
Thus
V , W.
Oe
IlU(u)ll2d(X)
< +.
3.1.
THE MAIN
RESULTTHEOREM
3.1:We
fixT
>
0 an arbitrary real number. Wehavethat if the hypothesis(3.1)(3.3)
arevalid,then problem(1.2)
hasanuniqueglobalclassical solutionu:[0,
T[
H,
satisfying
(3.6)
u6C2([O,T];
V),
where Visthe Hilbert space defined in
(3.5).
Suppose
(3.1)
thenby Diagonalization theorem,there exists an unitaryoperator//fromH
onto7/o suchthat/d is anisomorphism fromD(A’)
onto7/a, foraE/R,andll(Au)
A//(u),
forallue
D(A).
Moreover,
uis asolution of theproblem(1.2)
ifand onlyifv=/d(u)
issolution of:(3.7)
{
"
/M
(lvl )
Av
O,v(0)
v0t>O
,’(o)
u().
In
suchaproblem,v0andvl belongtospaceW.Now,
our goalis toprove theexistenceand uniqueness ofglobalsolutionv for(3.7)
in the followingclass(3.8)
where
T
>
0anda.
3.2.
TRUNCATED PROBLEM
(LOCAL
SOLUTION)
Let
k SV anddenoteby7/o,k
thesubspace of7/0
of the vector fieldsv(A)
such thatv(A)
0,va.e, on[k,
+o[.
It
followsthat7/0,k
equipped with the norm of7/0is a Hilbert space.For
each vector fieldv 7/,aR,
wedenotebyv
the truncated field associated tov,defined in thefollowingway:
fv,
ua.e,on]Ao,
k[,0<A0<fl
VkO, ua.e,
on[k,+c[,
wherefl
>
0is’
the constant of the(3.1).
The truncated problem
vk"
[0, Tk]
7lO,k
such thatcorresponding to
(3.7)(3.8)
consists of finding(3.9)
Vke
C
([0,
T]; 7/0,k),
T >
0,satisfying
(3.10)
{
vk+
M
(
Ivkl1/2
v(O)
vo;v(O)
v
t>O
Let
Vk
v
then(3.10)
it isequivalent to:(3.11)
Vk
F(Vk),
y(o)
Vo,
t>O
where
[
]
andyo
k[?)Ok]
[Vlk
As M
islocallyLipschitzfunction, thenF
isalsolocally Lipschitz.Thereforeby CauchyLipschitzPicardTheorem follows that thereexists0
< Tk <
T
and anunique local solutionVk
of(3.11)
inthe classCI([O,
Tk];
7/o,k
x7/0,).
Hence,
there exist anuniquevk"[0, T]

7/0,k
solution of(3.10),
satisfying(3.12)
v
6C
([0,
Our
next objective will be to obtainestimatesinorder to extend the solutionsv
toall interval[0, T[,
T >
0, and to prove thatvk converge uniformlytovsolutionof(3.7).
Estimate
"A
Priori"Let
usconsider the linear equation(3.13)
v_{k}+
a(t)Avk
O,
>_
O,
where
a(t)
is areal continuous and nonnegative functionon[0,
T[.
We
willassume that all solutions ofproblem(3.10)
also satisfy the linear equation(3.13).
Let
usintroduce thefollowingfunction:(3.14)
Ek(t)
{111
+
1111=},
LEMMA
3.1: Ifvk is asolutionof equation(3.13)
on[0, T[,
thenEk
satisfy:(3.15)
Ek(t) <_
c(T,)Ek(O)exp(A6),
for any di>
0.PROOF.
Let
uspc(t)
a smoothness net withsupp(pe)
C]0, e[,
then one definesake(t)
(k
*Pe) (t)
+
ae, whereak, if
t[0, T]
ak0, if
>
T;
(dk
*pe)(t)
"k(s)pe(t
s)ds
andIn
thisconditions,wehaveak
C1([0,
T]),
ake>
0, ake+ak uniformlyon[0,
T]
and using the inequalityT
lake
ak*Pel
_{dt}+
T
lake
akl
_{dt}<
_{dr,}wehaveby Lebesgue’sDoninated
Convergence
Theorem that:T
ake
akl
dt 0 when
(3.16)
ay/_d
e+O+.
Let
usdefinethe auxiliarity efunctions:Eke(t)
{ll,vll
+
allvll2},
Differentiating
Eke
and using(3.13)
weobtain dEke
llvll
ak__+
((Vk,
Vk))[a
a],
d
a,hence, by using
Eke
and standard inequalitywehaveddEke<
___
lael
+
A
lakeakl]
Eke.
By
Gronwall’sinequality,weconclude(3.17)
Moreover,
d
isbounded on
[0, T],
therefore Tlae___
_{dt}<_
C
C(T,e).
ake
Hence,
from(3.17)
and(3.18)
weinferthat(3.19)
Ek(t) <_
c(T,e)Ek(O)exp
A
Using
(3.16)
in(3.19)
itfollows:E}(t) <_
c(T,
5)E(O)
exp(A),
for all>
0and
>
0.As
required.The next stepisto obtainaestimateoftheVk in
H1/2.
Takingscalarproductof both sides in
(3.10)
withv
we have(v’, v)o +
M(Ivl)(Av,
v)0
0, for all 0_<
< Tk.
Taking
l(a)
f
M(s)ds,
weobtainwhere
E(t)
{Ivkl;.;
+/tr([v[)}.
Integrating from0 to
<_ Tk
itfollowsE(t)
E(0).
But,
byhypothesis(3.2),
l’I([vkl)
>
molvkl2_},
thus(3.20)
I,I,(t)lo
=
+
molv(t)l
<_ I,1o
+
(Ivol)
II,ll=d,(A)
+
M(s)ds.
and
eX’llvoll2dv(X)
<
+oo.
IVOkl
As M
islocallylipschitz,fo
M(s)ds < +o. Hence,
we conclude from(3.20)
that(3.21)
Iv(t)lo
/Iv(t)l
<
c, for all>
O,
wherec isapositive constant thatdepends onlyontheinitialdata.Multiplying
(3.15)
byA
_> A0 >
0and integrating in relation the measurevon]Ao, +oc[,
by usingthe hypothesis
(3.3)
we obtain(3.22)
Iv:l
+
Ivl
< c(T,(f)
e"Zllvl
2dv(,k)+
eZllvo,:lld,()
_<
c, for all[0,
T[.
REMARK
3.1:From
(3.21)
weconclude that the solutionvk of(3.10)
can be extended to the whole[0, T[,
for anyT >
0. Therefore,itfollows from(3.12)
thatvk satisfies(3.9).
[3Limit of the Truncated Solutions
In
order totake the limit in(3.10)
when k oit is necessary to prove thefollowingresult.
LEMMA
3.2: The sequence of the functions(v)et
istheCauchyin"Ra,
crEt.
PROOF.
Ifvkandv,
fork>
j aresolutions of the(3.13),
thenw
v
v satisfy(3.23)
w
+
aw
(a
a)
Associated tothe equation
(3.23),
letusdenotebyFk(t)
the functiongiven by1{
Fm(t)
5
Ilwk’ll2
2n()
+
llwl
It
isimmediatethat,(3.24)
F(0)
O,
for all kOur
nextgoal
isto provethatF(t)
+ 0 when k
oc.In’fact,
as inthe proof ofLemma
3.1,let
p(t)
asmoothness sequencewithsupp(p)
C]0,[
andak,(t)
(’d,
p)(t)
+
a,,, where"a,={
a,, ift[0,
T]
,
In
this conditions(e.g.
(3.10)),
wehave Tlake ael
_{dt}
0 when e
0+ for fix k.(3.25)
Associated to theake we definethe function: 1
Deriveting
F
withrespect to time,weobtainF
)allkl
+
Aa((wk,
wk’))
+ ((w’’,wk’)).
Usingthe equation(3.23),
wehave:It
follows thatFrom
(3.15)
we have that[[vii[
<_
c, for all j E 5r.
Thus,
integrating from 0to<_
T
andapplying Gronwallinequality,wegetin thesimilarway as
Lemma
3.1 thatFk(t) <_ c(T, 5)Fke(O)
exp(&5)
+
c(T, 5)
exp(&5)lak
_{}
;I(o,T)Using(3.18)
and(3.24)
itfollowsthatF(t) <
c(T,
5)
exp(AS)la
a.l
:_{L2(O,T)}By
hypothesis(3.3)
andfor5
>
0, smallenough,wehave2
eXF(t)
<_ c(T,
5)e2Xla
a]L2(O,T),
From
(3.21)
and(3.22)
itfollowsvk E
C
([0, T];
71/2).
Thus, the sequence of function
Ck(t)
[vk(t)[
is continuous in[0,
T].
On
the other hand, given s, E[0, T],
wegetICe(t)
(s)l
_< cl(t)
(s)l
<_
cI,()l]dn
<_
clt
sl
=
for allk and 0<
t,s<_
T.
Hence,
byArzel/scoli
Theorem there exists a subsequence(k)keV,
which we yetdenote by
(k)kev,
C([0,
T];
El)
such thatbk

,
uniformely inC([0,
T];).
As M
islocallylipschitzit follows thatM(Ivl)

M()
inC([0,
T]; E).
Takingk,j

oc andremarkingthatak(t)
M(Iv])
itfollowsThus,
(3.26)
enF(t)

O,
forallt_>
O.Therefore,
(vk)kev
is aCauchysequence in"Ra,
aEt.
rnIntegrating
(3.26)
inrelation tothe measureu on]Ao, +oc[,
we obtain(3.27)
vk
v stronglyinC([0,
T];
As
M
islocallyLipschitz, follows from(3.27)
that(3.28)
M(lvl)
,
M(Ivl)
emC([0,
T];
Using
(3.27)
and(3.28)
in(3.10),
wehave(3.29)
v
+v"
stronglyinC([0,
T];
PROOF
OFTHEOREM
3.1Theoperator/d
D(A)

Na,
cr6i, isanisomorphism. Thus, from(3.7)
and(3.8)
we havethatthe vector function u
[0, T[
H
defined byu=/gl(v)
satisfies the problem(1.2)
and(3.30)
ue
C([0,
T]; D(A)).
Now,
let us verify that the vector fieldu(t)
belongs toHilbert spaceV
defined in(3.5).
Thus,we affirmthat the application
[[u(t)[[
e
Cg([0,
T]).
In
fact,from Proposition 3.1 issufficientto provethatu 6W,
is that,e"llZg(u)ll2dw(A)
<
4o.The function
E(t)
isgiven by1 1
]2
E(t)
{llv’ll
+ livll
2)
{IiU()’I
+
[lU(u)l
},
andfrom
Lemma
3.1, followse
II(u)ll2d()<
eZE(t)d(A)
< c(T, 1)
eTM{llU(u)ll
/IlU(uo)ll
2)
de(A)
c
c(U();
u(0)).
Therefore, u
In
order to accomplish theProofof Theorem3.1,letusprovethat the solutionofProblem(1.2)
is unique.Or
equivalently, that the solution ofProblem(3.7)
is unique.In fact,
ifVlandv2 aretwosolutions ofproblem
(3.7),
then w vl v2satisfy(3.31)
,2
(0)
’(0)
0From
(3.31)
letusdefine a functionDeriveting
Ge
(t),
we obtainG’e(t)
((w’, w"))
+ a’(t)AIIwll +
a(t)A((w’,
w)),
using the equation
(3.31)
1, weget(3.33)
((,,
’)) + ’(t)llll
.
As M
islocallyLipschitz,follows(3.34)
for allt[O,T].
By
the substitution of(3.34)
in(3.33)
and using(3.32),
weobtainae(t)
M(Ivl)]
Ge(t)+
la’(t)l
G’e(t)
<_
ae(t)
G,(t)
+
,Xll,xll
IIw’ll
Iwl1/2.
Taking
and multiplying
(3.35)
forefo’
r,(8)ds,
we havedt
G,(t)efo"(’)a"
< llv,
II
IIw’ll
111/2
fo
,()a
Integratingfrom 0 tot,
f0
Ge(t)e
fo
.(s)as<_
cA
Ilvxll
IIw’ll
Iwl1/2e
fo
or, stillGe(t) <
cAI111
IIw’ll
Iwl1/2d.
Integrating in relation to the measure on
]Ao,+c[
and denoting byJ(t)
fxo
Ge(t)d(A),
weobtain(3.36)
J(t)
<_
cIwl1/2
AIIvll
IIw’lld(A)
UsingtheHSlderinequality and the estimate
(3.21),
itfollows thatSubstitutingin
(3.36)
weconclude that(3.37)
J(t) <_
cfo
Iw[1/2
v/()
d
<_
"dfo
J()d.
Using theGronwall’sinequality, weget
J(t)
0, forall;_>
0.Therefore,
w(t)
O,
for all>
O.REMARK
3.2: Without thehypothesis(2.3),
i.e.,considering only(Au,
u) >_
O, Vu
qD(A),
our mainresult(Theorem
3.1)
can beproved.In
thiscase,wetake theoperatorAe
A
+
I,
where
I
represents the identity operator and>
0, which satisfies the conditions oftheDiagonalizationTheorem. Thesolution to
(1.2)
would be obtain asalimitofue
solution tou
+
M
A
ul
Au
O(0)
o,(0)
.
Iftheoperator
A
A
in"
wehave to considerAe
A
+
eI
toapplytheDiagonalizationTheorem.
Acknowledgement Theauthor is very gratefulto the referee for his valuablesuggestions which improved this paper.
REFERENCES
[1]
AROSIO,
A;
SPAGNOLO, S.
"Globalsolution of the Cauchy problemforanonlinearhyperbolic equation", Nonlinear Partial Differential Equation and their Applications,
Collge
deFrance
Seminar, Vol. 6(ed.
byH.
Brezis andJ.L.
Lions),
Pitman, London,(1984).
[2]
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