by a Klein-Gordon equation in spae dimension 2
byMathieuColin
UniversiteParis-Sud,UMRdeMathematiques
Bat425,91405OrsayCedex,FRANCE
Mathieu.Colinmath.u-psud.fr
tel. 01.69.15.57.85
Abstrat.Inthisartile,westudythenonlinearplasmawaveequation
"
2
2
u
"
t 2
+2i u
"
t +u
"
=
1
p
1+ju
"
j 2
1
!
u
"
+ (
p
1+ju
"
j 2
)
p
1+ju
"
j 2
u
"
withinitial datau
"
(;0)=u
"
0 ()2H
8
(R 2
),
t u
"
(;0)=u
"
1 ()2H
7
(R 2
). WeshowthattheCauhyproblem
isloally well-posedonaninterval[0;T℄where thetimeT isindependentof"ifu
"
1
issmallenough. Then,
wedemonstratethestrongonvergeneofu
"
towardsthesolutionuofanonlinearrelativistiShrodinger
equationas"goesto0.
1 Introdution
ThenonlinearShrodinger equationstatedin R 3
2i u
t +
? u
u
p
1+juj 2
? (
p
1+juj 2
)+(1
1
p
1+juj 2
)u=0: (1)
is known to desribe properly the self-hanneling of a high power ultra-short laser pulse in matter (see
for example [3℄, [4℄, [5℄, [6℄, [13℄ and [7℄ for more referenes). Here, we denote
?
=
2
x 2
+
2
y 2
and
u=u(x;y;z;t). Theaimofthispaperisto explainoneoftheapproximationswhihleadtothisequation.
FirstfromMaxwellsystemdesribingthepropagationofhighpowerultra-shortlaseringaseousmediumwe
anderivethefollowingwaveequation(see[7℄)
A=k 2
p
1+k 2
p
!
A; (2)
where A isthe eletromagnetivetoreld, = p
1+jAj 2
isthe Lorentz fatorand =
2
t 2
. Then,
aordingto physial observations (see [4℄),equation (2) anbe simplied ifweuse theapproximationin
whihtheomplexamplitudeAisslowlyvaryingoverdistanesontheorderofawavelengthinthediretion
ofpropagationzandovertimesontheorderoftheperiodofthehigh-frequenyeldosillations. Itisthen
naturaltodene
A=A
"
(x;y;z;t)=u
"
(x;y;"z;"t)e i(k z !t)
where
"=
!
p;0
!
;
dependsontheparametersoftheplasma(!
p;0
denotestheunperturbedplasmafrequeny).
? u
"
+"
2
2
u
"
z 2
"
2
2
u
"
t 2
+2i p
1 "
2 u
"
z +2i
u
"
t
=
1
"
1
u
"
+ (
? +"
2
2
z )
"
"
u
"
(3)
where
"
= p
1+ju
"
j 2
. Intheapproximationofunderdenseplasmas,oneanassumethat "1.
Then,takingformally "=0in (3)and performing ahangeofvariables givethenonlinearShrodinger
equation(1). It isthennaturaltothink thatthesolutionsu
"
of (3)ouldonvergetowardsthesolutionsu
ofequation(1).
This kind of questions onstitute an ative domain of researh in Mathematis. For example, in [2℄,
T. Colinand L.Bergeinvestigated theso-alled Langmuir waveenvelopeapproximationwhih onsists in
takingthelimit" !0inthenonlinearplasmawaveequation
"
2
2
E
"
t 2
2i E
"
t E
"
=f(jE
"
j 2
)E
"
:
Under spei assumptions on f, the authors show the onvergene of E
"
towards the solution E of the
orrespondingShrodingerequation. In[1℄,theauthorsprovethesamekindofresultsforasetofequations
desribingtheinertialregimeofthestrongLangmuirturbulene,namely
"
2
2
E
"
t 2
2i E
"
t E
"
= n
"
E
"
1
2
2
n
"
t 2
n
"
=jE
"
j 2
:
In[14℄, theauthorsshowthatthesolutionsofthenonlinearKlein-Gordonequationanbedesribedby
usingasystemoftwoouplednonlinearShrodingerequationsas tendstoinnity.
In[9℄,T.GallayandG.ShneiderobtaintheKPequation
2
2
A
xt +
2
A 2
x 2
+
4
A
x 4
+
2
A
y 2
=0
asanapproximationfortheBoussinesqequation
2
t 2
=+(
2
)+
2
t 2
:
Inordertojustifyrigorouslythederivationof(1)from(3),wehavetondasequeneu
"
solutionof(3)
whihsatisesthefollowing:
u
"
existsonaninterval[0;T℄whereT isindependentof"
u
"
isbounded inH s
(R 3
)withabound independentof"
theinitial valuesdonottendto zerowith".
Unfortunately,thisprogramistooambitiousforthemoment. That iswhywewillnotstudyhereequation
(3). Indeed,thepreseneof"
2
infrontof 2
z
givesrisetomanyproblems. Forexample,inordertoderivean
energyestimateonequation(3),wehavetoonsiderthenormjj(1
"
) s
2
jj
L 2
(R 3
)
where
"
= 2
x +
2
y +"
2
2
z ,
whihpreventsfrominjetingthefuntionsu
"
inL 1
(R 3
)withaonstantindependentof".
where allthetermsdependingonz havebeendropped
"
2
2
u
"
t 2
+2i u
"
t +u
"
=
1
"
1
u
"
+
"
"
u
"
: (4)
Here
"
= p
1+ju
"
j 2
. Formally,taking"=0in (4)givestherelativistiShrodingerequation(1) .
WerstsolvetheCauhyproblem(4)withinitial ondition
u
"
(x;y;0)=u
"
0
(x;y);
t u
"
(x;y;0)=u
"
1
(x;y); (5)
keepingin viewthethree assertionsof ourprogramstatedabove. Arst approahouldbeto treat(4)as
anonlinearKlein-Gordonequation. Indeed,ifwedenote
v
"
(x;y;t)=u
"
(x;y;t)e i
1
"
2 t
aneasyalulationshowsthat v
"
isasolutionof
"
2
2
v
"
t 2
+v
"
1
"
2 v
"
=
1
"
1
v
"
+
"
"
v
"
(6)
withinitial ondition
v
"
(x;y;0)=u
"
0 (x;y)
t v
"
(x;y;0)=u
"
1
(x;y;z)+i 1
"
2 u
"
0 (x;y):
Equation (6) satises the hypothesis of Klainerman and Pone [12℄. We an dedue that if the initial
data are small enough in an adequate spae, the Cauhy problem (6) admits a unique global solution
v
"
2C([0;+1[;H s
(R 2
))\C 1
([0;+1[;H s 2
(R 2
))where s is an integersuÆiently large. Bak to u
"
, we
anshowthat (4)admitsauniquesolutionu
"
2C([0;+1[;H s
(R 2
))\C 1
([0;+1[;H s 2
(R 2
))iftheinitial
data issmallenough. Theproblemisthat,in doingso,it iseasytosee thatthepreseneof theterm 1
"
2 v
"
imposestov
"
andsotou
"
totendto0inthespaeL 1
(0;T;H s
(R 2
))as"goesto0whihisinontradition
withourprogram. Thus,wehavetosolvetheCauhyproblem(4)withinitialondition(5)inanotherway.
Aseondapproahouldbeto applytheusual energyestimates(see [12℄)toequation(4). Indeed,itis
possibleto provethetwofollowingassertions
The Cauhy problem (4) is loally well posed in L 1
(0;T
"
;H s
(R 3
)) for any initial data u
"
0 , u
"
1 with
T
"
=
"
2
(jju
"
0 jj
H s
(R 3
) +jju
"
1 jj
H s 1
(R 3
) )
.
The Cauhy problem (4) is loally well posed in L 1
(0;T
"
;H s
(R 3
)) with T = 1 if jju
"
0 jj
H s
(R 3
) +
jju
"
1 jj
H s 1
(R 3
)
h(") whereh(")tendstozerowith".
Thesetwoassertionsarenotveryusefultoourproblem. Indeed,itislearthatintherstone,thetimeT
"
tendsto0with"whereasin theseondone,theinitialdatau
"
0
tendsto zerowith".
Then,werstprovethefollowingtheorem
Theorem1.1 Assume that u
"
0 2 H
8
(R 2
) and u
"
1 2 H
7
(R 2
). Then there exists Æ
"
> 0 (Æ
"
2 1
"
2
) and
M>0independent of "anddepending onlyonthe initial datas suhthatif jju
"
1 jj
H 7
(R 2
) Æ
"
,there existsa
timeT independentof "suhthat (4)admits auniquesolutionu
"
satisfying
u
"
2L 1
(0;T;H 8
(R 2
))\C([0;T℄;L 2
(R 2
)):
Furthermore, wehave
sup
t2[0;T℄
jju
"
jj 2
H 8
(R 2
) +"
2
jj
t u
"
jj 2
H 6
(R 2
) +"
2
jj 2
t u
"
jj 2
H 6
(R 2
)
M:
We have to notie that u
1
tends to 0 very rapidly with ". In order to get estimates on u
"
whih
are independent of ", we will onsider (4) not asa wave equation but as a perturbation of a quasilinear
Shrodinger equation. Thismeans that in equation (1), theterm"
2
2
t u
"
will betreatedasaperturbative
term. Forthatpurpose,wewillusethemethodintroduedin[8℄,namelywerewriteequation(4)asasystem
in (u
"
;u
"
) t
. Then,wedierentiatethe equationwithrespetto spaeand timeto obtainanewsystemin
u
0 ,..,u
5
(wedropforonvenienetheindex ")where
u
0
=u
"
; u
j
=
j u
"
81j 2; u
3
=e q(ju
0 j
2
)
u
"
; u
4
=
t u
"
; u
5
= 2
t u
"
:
Thefuntionqplaystheroleofagaugetransformwhihallowstotreattheequationssatisedbyu
0 ,u
1 ,u
2 ,
andu
3
exatlyaswehavetreatedequations2:9, 2:10and 2:11of[8℄. Conerningu
4 andu
5
wedonotuse
gaugetransformsbeausetheirrespetiveequationswillbetreatedinadierentway,whihdoesnotsuppose
to integratebyparts thetermsinludingtherstderivativesofu
4 andu
5
. Indeed,to estimateu
4 andu
5 ,
weusethelassialenergyestimatesonwaveequations, namelywemultiplytheequations respetivelyby
(u
4 )
t and(u
5 )
t
andweintegratetherealpartoverR 2
. Finally,theproofofTheorem1:1willfollowfrom a
lassialxed-pointTheorem.
EquippedwithTheorem(1.1),weanstudytheonvergeneofu
"
toasolutionuof(1)withinitialdata
u(x;y;0)=u
0
(x;y)under theonditionthat
u
"
0
!
"!0 u
0
in H 6
(R 2
) and u
"
1
!
"!0
0 in H 7
(R 2
)
Morepreisely,wehavethefollowing
Theorem1.2 Let(u
"
0
)and(u
"
1
)betwoboundedsequenesrespetively inH 8
(R 2
)andH 7
(R 2
)satisfying
u
"
0
!
"!0 u
0
in H 6
(R 2
) and u
"
1
!
"!0
0 in H 7
(R 2
):
Wesupposefurthermorethatjju
"
1 jj
H 7
(R 2
) 2
1
"
2
. Letu
"
bethesolutionof(4)withinitialonditionu
"
(;0)=
u
"
0 (),
t u
"
(;0)=u
"
1
()given by Theorem 1:1. Letuthe solution of(1) withinitial ondition u(;0)=u
0 ()
given by Theorem 1:1 of [8 ℄Then, thereexists atime T >0independent of " suh that u
"
andu existon
[0;T℄for any" and
u
"
u !
"!0
0 in L 1
(0;T;H 6
(R 2
)):
Theproofusesanenergyestimatewhihreadsexatlyastheoneof[8℄.
Notationand funtion spaes:Asusual,wedenotebyL p
(R 2
)theLebesguespae
L p
(R 2
)= n
u2S 0
(R 2
) = jjujj
p
<+1 o
where
jjujj
p
= Z
R N
ju(x)j p
dx
1
p
if1p<+1
and
jjujj
1
=ess.sup
ju(x)j;x2R N
:
WedenetheSobolevspaeH s
(R 2
)asfollows
H s
(R 2
)=
u2S 0
(R 2
) = jjujj 2
H s
(R 2
)
= Z
R 2
(1+jj 2
) s
jbu()j 2
d <+1
where bu()istheFouriertransformofu. LetC(I;E)bethespaeofontinuousfuntionsfromaninterval
I of R to aBanah spae E. For 1 j 2, we set
j
=
x
j
and by onvention
0
u = u. Fork 2 N 2
,
k =(k
1
;k
2
),wedenote k
u= 1
1
2
2
uandjkj=k
1 +k
2
. As usual, the oeÆientsofamatrix Dwill be
denoted by D ij
. The notation D(D r
V) means that the matrix D depends on k
V for jkj r. Dierent
positiveonstantsmightbedenotedbythesameletterC. WealsodenotebyRe(u)andIm(u)therealpart
andtheimaginarypartofuand[a℄isused todenotetheintegerpartofa.
Insetion2,wewilltransformequation(4)intoasystemtowhihweanapplytheappropriateenergy
method. Then,in setion3,weproveTheorem 1:1onerningtheCauhyproblem (4). Finally, insetion
4,weproveTheorem1:2.
2 Transformation of equation (4)
Inthissetion, wetransformequation(4)into asystemto whih weanapply ourenergymethod. Wex
a
"
0 2H
8
(R 2
)anda
"
1 2H
7
(R 2
). Following[8℄, equation(4)anberewrittenasasystemin(u
"
;u
"
) t
"
2
(u
"
)
tt
(u
"
)
tt
+2i
(u
"
)
t
(u
"
)
t
+A(u
"
)
u
"
u
"
u"
2
"
jru
"
j 2 u"
4 4
"
jrju
"
j 2
j 2
+u
"
g(ju
"
j 2
)
u"
2
"
jru
"
j 2
+ u"
4 4
"
jrju
"
j 2
j 2
u
"
g(ju
"
j 2
)
!
=0
(7)
where
g(s)= 1
p
1+s
1;
"
= p
1+ju
"
j 2
;
A(u
"
)= 1
2 2
"
2+ju
"
j 2
u 2
"
u
"
2
(2+ju
"
j 2
)
:
We nowdrop for onveniene theparameter". We set u
0
=u
"
, u
1
=
1 u
"
, u
2
=
2 u
"
, u
3
=e q(ju0j
2
)
u
"
,
u
4
=
t u
"
and u
5
= 2
t u
"
. Here, u
0
doesnotdenote aninitial ondition. The funtion q will playthe
role ofagaugetransformand willbehosenlater(see [8℄ formoredetails). We alsosetU
=(u
j )
2
j=0 and
U =(u
j )
5
j=0
. Then,equation(7)anberewrittenas
2i
t u
0
t u
0
+A(u
0 )
u
0
u
0
"
2
1
u
5
1
u
5
+F
0 (U
)=0 (8)
where
F
0 (U
)= 0
B
B
B
B
u
0
2
(u
0 )
( 2
X
k =1 ju
k j
2
)+ u
0
4 4
(u
0 )
( 2
X
k =1 (u
0 u
k +u
0 u
k )
2
)+u
0 g(ju
0 j
2
)
u
0
2
(u
0 )
( 2
X
k =1 ju
k j
2
) u
0
4 4
(u
0 )
( 2
X
k =1 (u
0 u
k +u
0 u
k )
2
) u
0 g(ju
0 j
2
) 1
C
C
C
C
A :
Dierentiatingequation(7)withrespettox
j
forj=1;2,weobtain
2i
(u
j )
t
(u
j )
t
+A(u
0 )
u
j
u
j
+ 2
X
k =1 B(u
0
;u
k )
T
k j u
3
T
k j u
3
+C(u
0
;u
j )
(e q(ju
0 j
2
)
u
3 )
(e q(ju
0 j
2
)
u
3 )
!
"
2
j
1
u
5
j
1
u
5
+
F(U
;u
j )
F(U
;u
j )
=0
B(v;w)= v
4 4
(v) (2jvj
2
w+2v 2
w) v
2
(v) w
v
4 4
(v) (2jvj
2
w+2v 2
w) v
2
(v) w
v
4 4
(v) (2jvj
2
w+2v 2
w)+ v
2
(v) w
v
4 4
(v) (2jvj
2
w+2v 2
w)+ v
2
(v) w
!
C(v;w)= (
1
2 2
(v)
2+jvj 2
2 4
(v)
)(vw+vw) vw
2
(v) +
v 2
4 4
(v)
(vw+vw)
vw
2
(v) v
2
2 4
(v)
(vw+vw) ( 1
2 2
(v) +
2+jvj 2
2 4
(v)
)(vw+vw)
!
F(U
;w)=
u
0 w+u
0 w
4
(u
0 )
w
2
(u
0 )
N
X
k =1 ju
k j
2
!
+u
0 g
0
(ju
0 j
2
)(u
0 w+u
0 w)
+
w
4 4
(u
0 )
(u
0 w+u
0 w)
2 6
(u
0 )
N
X
k =1 (u
0 u
k +u
0 u
k )
2
+
!
+ u
0
4 4
(u
0 )
2(u
0 w+u
0 w)(
N
X
k =1 ju
k j
2
)+2u
0 w(
N
X
k =1 u
2
k )
+2u
0 w(
N
X
k =1 u
2
k )
!
+wg(ju
0 j
2
)
andfori;j=1;2,T
i;j
isthefollowingoperator oforder0
T
i;j v=
i
j
1
(e q(ju0j
2
)
v):
Furthermore,thisoperatorisontinuousfromL 2
toL 2
withanormindependentof". Weanrewritethese
equationsasfollows
2i
(u
j )
t
(u
j )
t
+A(u
0 )
u
j
u
j
"
2
j
1
u
5
j
1
u
5
+F
j (U
;u
3
;Tu
3
)=0 (9)
where
F
j (U
;u
3
;Tu
3 )=
F(U
;u
j )
F(U
;u
j )
+C(u
0
;u
j )
(e q(ju
0 j
2
)
u
3 )
(e q(ju
0 j
2
)
u
3 )
!
+ 2
X
k =1 B(u
0
;u
k )
T
k j u
3
T
k j u
3
:
Inordertoderiveanequationforu
3
,wedierentiateequation(7) twiewithrespetto x
j
toobtain
2i
( 2
j u
0 )
t
( 2
j u
0 )
t
+A(u
0 )
2
j u
0
2
j u
0
+ 2
X
k =1 B(u
0
;u
k )
k
2
j u
0
k
2
j u
0
+2C(u
0
;u
j )
j u
0
j u
0
+
j C(u
0
;u
j )
u
0
u
0
"
2
( 2
j u
0 )
tt
( 2
j u
0 )
tt
+ 2
X
k =1
j B(u
0
;u
k )
k
j u
0
k
j u
0
+
j F(U
;u
j )
j F(U
;u
j )
=0:
(10)
Summing thetwoequationsandmultiplyingbyA (u
0
),weobtain
2iA 1
(u
0 )
(u
0 )
t
(u
0 )
t
+
u
0
u
0
+ 2
X
k =1 A
1
(u
0 )B(u
0
;u
k )
k u
0
k u
0
+2 2
X
j=1 A
1
(u
0 )C(u
0
;u
j )
j u
0
j u
0
+ 2
X
j=1 A
1
(u
0 )
j C(u
0
;u
j )
u
0
u
0
"
2
A 1
(u
0 )
(u
0 )
tt
(u
0 )
tt
+ 2
X
j=1 2
X
k =1 A
1
(u
0 )
j B(u
0
;u
k )
k
j u
0
k
j u
0
+ 2
X
j=1 A
1
(u
0 )
j F(U
;u
j )
j F(U
;u
j )
=0:
(11)
whihanberewritten,afterthemultipliationbye q(ju0j
2
)
,in thefollowingway
2i
(u
3 )
t
(u
3 )
t
+A(u
0 )
u
3
u
3
+ 2
X
k =1 E(u
0
;u
k )
k u
3
k u
3
"
2
u
5
u
5
+I(U;Tu
3
; 1
u
4 )=0
(12)
where
E(u
0
;u
k )=B(u
0
;u
k
)+2C(u
0
;u
k )
2A(u
0 )q
0
(ju
0 j
2
)
u
0 u
k +u
0 u
k
0
0 u
0 u
k +u
0 u
k
andI isamatrixdependingonU,T
ij u
3
fori;j =1;2and 1
u
4
inthefollowingway
I(U;Tu
3
; 1
u
4 )=
2
X
m=1 e
q(ju0j 2
)
m F(U
;u
j )
m F(U
;u
j )
2iq 0
(ju
0 j
2
)
u
0 (
1
u
4 )u
3 +u
0 (
1
u
4 )u
3
u
0 (
1
u
4 )u
3 +u
0 (
1
u
4 )u
3
2
X
k =1 (2C(u
0
;u
k
)+B(u
0
;u
k ))q
0
(ju
0 j
2
)
(u
0 u
k +u
0 u
k )u
3
(u
0 u
k +u
0 u
k )u
3
+ 2
X
k =1
k C(u
0
;u
k )
u
3
u
3
+ 2
X
j=1 2
X
k =1 e
q(ju
0 j
2
)
k B(u
0
;u
k )
T
k j u
3
T
k j u
3
+A(u
0 )
0
B
B
B
B
(q
0
2
(ju
0 j
2
) q 00
(ju
0 j
2
))(
2
X
k =1 (u
0 u
k +u
0 u
k )
2
)u
3
(q 0
2
(ju
0 j
2
) q 00
(ju
0 j
2
))(
2
X
k =1 (u
0 u
k +u
0 u
k )
2
)u
3 1
C
C
C
C
A
A(u
0 )
0
B
B
B
B
q
0
(ju
0 j
2
)(2 2
X
k =1 ju
k j
2
+u
0 (
2
X
k =1 T
k k u
3 )u
3
q 0
(ju
0 j
2
)(2 2
X
k =1 ju
k j
2
+u
0 (
2
X
k =1 T
k k u
3 ))u
3 1
C
C
C
C
A A(u
0 )
0
B
B
B
B
q
0
(ju
0 j
2
)u
0 (
2
X
k =1 T
k k u
3 )u
3
q 0
(ju
0 j
2
)u
0 (
2
X
k =1 T
k k u
3 )u
3 1
C
C
C
C
A
We have to notie here that the matrix I depends on 1
u
4
, whih is due to the gauge transformu
3
=
e q(ju0j
2
)
u
0
. Indeed,inequation(11),wehavetoreplaeu
0 bye
q(ju0j 2
)
u
3
intheterm(u
0 )
t
whih gives
theontributione q(ju0j
2
)
(q(ju
0 j
2
))
t u
3
. Then,reallingthat(u
0 )
t
= 1
u
4
,weinlude thislastterminI.
"
2
A 1
(u
0 )
(u
4 )
tt
(u
4 )
tt
"
2
t A
1
(u
0 )
u
5
u
5
+2iA 1
(u
0 )
(u
4 )
t
(u
4 )
t
+2i
t A
1
(u
0 )
u
4
u
4
+
u
4
u
4
+ 2
X
k =1 A
1
(u
0 )B(u
0
;u
k )
k u
4
k u
4
+ 2
X
k =1
t A
1
(u
0 )B(u
0
;u
k )
k (e
q(ju0j 2
)
u
3 )
k (e
q(ju0j 2
)
u
3 )
!
+2 2
X
k =1 A
1
(u
0 )C(u
0
;u
k )
k u
4
k u
4
+2 2
X
j=1
t A
1
(u
0 )C(u
0
;u
j )
j (e
q(ju0j 2
)
u
3 )
j (e
q(ju
0 j
2
)
u
3 )
!
+ 2
X
j=1 A
1
(u
0 )
j C(u
0
;u
j )
u
4
u
4
+ 2
X
j=1
t A
1
(u
0 )
j C(u
0
;u
j )
e q(ju0j
2
)
u
3
e q(ju
0 j
2
)
u
3
!
+ 2
X
j=1 2
X
k =1 A
1
(u
0 )
j B(u
0
;u
k )
k
j
1
u
4
k
j
1
u
4
+ 2
X
j=1 2
X
k =1
t A
1
(u
0 )
j B(u
0
;u
k )
T
k j u
3
T
k j u
3
+ 2
X
j=1
t
A 1
(u
0 )
j F(U
;u
j )
j F(U
;u
j )
=0:
(13)
Keepingin viewthat wehave
t u
0
= 1
u
4
,weanrewritethelastequationin thefollowingway
"
2
A 1
(u
0 )
(u
4 )
tt
(u
4 )
tt
+2iA 1
(u
0 )
(u
4 )
t
(u
4 )
t
+
u
4
u
4
+ 2
X
k =1 D
k (u
0
;u
k )
k u
4
k u
4
+G(DU;Tu
3 )
u
4
u
4
+ 2
X
j=0 2
X
k =0 H
jk
(DU;Tu
3 )
j
k
1
u
4
j
k
1
u
4
=0
(14)
where fork=1;2
D
k (u
0
;u
k )=A
1
(u
0 )(B(u
0
;u
k
)+2C(u
0
;u
k ));
andforj;k=1;2,H
jk
isamatrixdependingonU anditsrstderivativesandonT
mn u
3
(m;n=1;2). We
havetonotieherethatinequation(14),allthetermsofequation(13)whihontains
t u
0
anditsrstand
seond spaederivativesareinludedinthelastterms
2
X
j=0 2
X
k =0 H
jk
(DU;Tu
3 )
j
k
1
u
4
j
k
1
u
4
:
Indeed,thissuminludethevaluesj=0andk=0. Reallingthatwehaveadoptedthenotation
0 u=u,
thenitispossibleto writeforj=0;::;2andk=0;::;2
t
j
k u
0
=
j
k
1
u
4 :
Finally, in orderto obtainanequationsatisedbyu
5
,wedierentiatetwie withrespetto tequation
"
2
A 1
(u
0 )
(u
5 )
tt
(u
5 )
tt
2"
2
t A
1
(u
0 )
(u
5 )
t
(u
5 )
t
"
2
2
t A
1
(u
0 )
u
5
u
5
+2iA 1
(u
0 )
(u
5 )
t
(u
5 )
t
+4i
t A
1
(u
0 )
u
5
u
5
+2i 2
t A
1
(u
0 )
u
4
u
4
+
u
5
u
5
+ 2
X
k =1 A
1
(u
0 )B(u
0
;u
k )
k u
5
k u
5
+2 2
X
k =1
t A
1
(u
0 )B(u
0
;u
k )
k u
4
k u
4
+ 2
X
k =1
2
t A
1
(u
0 )B(u
0
;u
k )
k (e
q(ju0j 2
)
u
3 )
k (e
q(ju0j 2
)
u
3 )
!
+2 2
X
j=1 A
1
(u
0 )C(u
0
;u
j )
j u
5
j u
5
+4 2
X
j=1
t A
1
(u
0 )C(u
0
;u
j )
j u
4
j u
4
+2 2
X
j=1
2
t A
1
(u
0 )C(u
0
;u
j )
j (e
q(ju0j 2
)
u
3 )
j (e
q(ju0j 2
)
u
3 )
!
+ 2
X
j=1 A
1
(u
0 )
j C(u
0
;u
j )
u
5
u
5
+2 2
X
j=1
t A
1
(u
0 )
j C(u
0
;u
j )
u
4
u
4
+ 2
X
j=1
2
t A
1
(u
0 )
j C(u
0
;u
j )
(e q(ju0j
2
)
u
3 )
(e q(ju0j
2
)
u
3 )
!
+ 2
X
j=1 2
X
k =1 A
1
(u
0 )
j B(u
0
;u
k )
k
j
1
u
5
k
j
1
u
5
+2 2
X
j=1 2
X
k =1
t A
1
(u
0 )
j B(u
0
;u
k )
k
j
1
u
4
k
j
1
u
4
+ 2
X
j=1 2
X
k =1
2
t A
1
(u
0 )
j B(u
0
;u
k )
T
k j u
3
T
k j u
3
+ 2
t 0
2
X
j=1 A
1
(u
0 )
j F(U
;u
j )
j F(U
;u
j )
1
A
=0:
Thisequationtakestheform
"
2
A 1
(u
0 )
(u
5 )
tt
(u
5 )
tt
+C(u
0
; 1
u
4 )
(u
5 )
t
(u
5 )
t
+
u
5
u
5
+ 2
X
k =1 D
k (u
0
;u
k )
k u
5
k u
5
+I(DU;Tu
3 )
u
5
u
5
+ 2
X
j=0 2
X
k =0 J
jk
(DU;Tu
3 )
j
k
1
u
5
j
k
1
u
5
+ 2
X
j=0 2
X
k =0 K
jk
(DU;Tu
3
;D 2
1
u
4 )
j
k
1
u
4
j
k
1
u
4
=0:
(15)
where
C(u
0
; 1
u
4
)=2iA 1
(u
0 ) 2"
2
t A
1
(u
0 )
:
Forj;k=0;::;2,thematriesJ
jk
dependonj,k,onU anditsrstderivativesandonT
mn u
3
form;m=1;2.
Forj;k=0;::;2,thematriesK
jk
dependonj, k,onU anditsrstderivatives,onT
mn u
3
form;m=1;2
andon 1
u
4
anditsrstandseond derivatives. Inorder tosimplifythenotations,wenowintroduefor
j;k=0;::;2theoperatorR
jk
R
jk
=
j
k
1
:
Oneagain,wehaveinludedthetermswhihontain
t u
0
anditsrstandseondspaederivativesin
2
X
j=0 2
X
k =0 K
jk
(DU;Tu
3
;D 2
1
u
4 )
j
k
1
u
4
j
k
1
u
4
andthetermswhihontain
t u
0
anditsrstandseondspaederivativesin
2
X
j=0 2
X
k =0 J
jk
(DU;Tu
3 )
j
k
1
u
5
j
k
1
u
5
:
Inonlusion,wehavetransformedequation(7)intothefollowingsystem
2i
t u
0
t u
0
+A(u
0 )
u
0
u
0
"
2
1
u
5
1
u
5
+F
0 (U
)=0 (16)
forj=1;2,
2i
(u
j )
t
(u
j )
t
+A(u
0 )
u
j
u
j
"
2
j
1
u
5
j
1
u
5
+F
j (U
;u
3
;Tu
3
)=0 (17)
2i
(u
3 )
t
(u
3 )
t
+A(u
0 )
u
3
u
3
+ 2
X
k =1 E(u
0
;u
k )
k u
3
k u
3
"
2
u
5
u
5
+I(U;Tu
3
; 1
u
4 )=0
(18)
"
2
A 1
(u
0 )
(u
4 )
tt
(u
4 )
tt
+2iA 1
(u
0 )
(u
4 )
t
(u
4 )
t
+
u
4
u
4
+ 2
X
k =1 D
k (u
0
;u
k )
k u
4
k u
4
+ 2
X
j=0 2
X
k =0 H
jk
(DU;Tu
3 )
R
jk u
4
R
jk u
4
+G(DU;Tu
3 )
u
4
u
4
=0
(19)
"
2
A 1
(u
0 )
(u
5 )
tt
(u
5 )
tt
+C(u
0
; 1
u
4 )
(u
5 )
t
(u
5 )
t
+
u
5
u
5
+ 2
X
k =1 D
k (u
0
;u
k )
k u
5
k u
5
+ 2
X
j=0 2
X
k =0 J
jk
(DU;Tu
3 )
R
jk u
5
R
jk u
5
+ 2
X
j=0 2
X
k =0 K
jk
(DU;Tu
3
;D 2
1
u
4 )
R
jk u
4
R
jk u
4
+I(DU;Tu
3 )
u
5
u
5
=0:
(20)
3 Proof of Theorem 1:1
Inthis setion,wegiveaproofofTheorem 1:1. Westudythesystem(16),(17), (18), (19)and(20)inthe
followingfuntion spae
X
T
= 8
>
<
>
:
U =(u
j )
5
j=0 :u
j
2C([0;T℄;L 2
(R 2
))\L 1
(0;T;H 4
(R 2
));
jjUjj
XT
= 5
X
j=0 sup
0tT jju
j (t)jj
H 4
(R 2
)
<1
9
>
=
>
;
ForM =(m
j )
5
j=0 2R
6
+
andr2R
+
,wedenote
X
T
(M;r)=
U =(u
j )
5
j=0 2X
T
:8j=0;::;5 jju
j jj
L 1
(0;T;H 4
(R 2
))
m
j
jj(u
0 )
t jj
L 1
(0;T;H 2
(R 2
))
r andu
0
(;0)=u
"
0 ()
andweletV =(v
j )
j=0 2X
T
(M;r). WealsosetV =(v
j )
j=0
. Considerthelinearizedversionofequations
(16),(17),(18),(19)and(20)
2i
t u
0
t u
0
+A(v
0 )
u
0
u
0
"
2
1
v
5
1
v
5
+F
0 (V
)=0 (21)
forj=1;2,
2i
(u
j )
t
(u
j )
t
+A(v
0 )
u
j
u
j
"
2
j
1
v
5
j
1
v
5
+F
j (V
;v
3
;Tv
3
)=0 (22)
2i
(u
3 )
t
(u
3 )
t
+A(v
0 )
u
3
u
3
+ 2
X
k =1 E(v
0
;v
k )
k u
3
k u
3
"
2
v
5
v
5
+I(V;Tv
3
; 1
v
4 )=0
(23)
"
2
A 1
(v
0 )
(u
4 )
tt
(u
4 )
tt
+2iA 1
(v
0 )
(u
4 )
t
(u
4 )
t
+
u
4
u
4
+ 2
X
k =1 D
k (v
0
;v
k )
k u
4
k u
4
+ 2
X
j=0 2
X
k =0 H
jk
(DV;Tv
3 )
R
jk u
4
R
jk u
4
+G(DV;Tv
3 )
u
4
u
4
=0
(24)
"
2
A 1
(v
0 )
(u
5 )
tt
(u
5 )
tt
+C(v
0
; 1
v
4 )
(u
5 )
t
(u
5 )
t
+
u
5
u
5
+ 2
X
k =1 D
k (v
0
;v
k )
k u
5
k u
5
+ 2
X
j=0 2
X
k =0 J
jk
(DV;Tv
3 )
R
jk u
5
R
jk u
5
+ 2
X
j=0 2
X
k =0 K
jk
(DV;Tv
3
;D 2
1
v
4 )
R
jk u
4
R
jk u
4
+I(DV;Tv
3 )
u
5
u
5
=0:
(25)
LetZ =
L 1
(0;T;H 4
(R 2
))\C([0;T℄;L 2
(R 2
))
6
. Then,it isnotdiÆult tosee that thelinearinhomoge-
neousCauhyproblem(21),(22),(23),(24)and(25)withinitialondition
u
0
(;0)=u
"
0 (); u
1
(;0)=
1 u
"
0 (); u
2
(;0)=
2 u
"
0 ();
u
3
(;0)=e q(ju
"
0 j
2
)
u
"
0 (); u
4
(;0)=u
"
1 (); u
5
(;0)=
t u
4 (;0);
t u
4 (;0)=
1
"
2
u
"
0 +2iu
"
1 (
1
p
1+ju
"
0 j
2 1)u
"
0
1
p
1+ju
"
0 j
2 (
p
1+ju
"
0 j
2
)u
"
0
;
t u
5 (;0)=
1
"
2
u
"
1 (
1
p
1+ju
"
0 j
2 1)u
"
1 +
u
"
0 u
"
1 +u
"
0 u
"
1
2(1+ju
"
0 j
2
) 3
2 u
"
0
1
p
1+ju
"
0 j
2 (
p
1+ju
"
0 j
2
)u
"
1 +
u
"
0 u
"
1 +u
"
0 u
"
1
2(1+ju
"
0 j
2
) 3
2 (
p
1+ju
"
0 j
2
)u
"
0
+ u
"
0
p
1+ju
"
0 j
2
u
"
0 u
"
1 +u
"
0 u
"
1
2(1+ju
"
0 j
2
) 3
2
+ 2i
"
2 (u
"
0 +2iu
"
1
( 1
p
1+ju
"
0 j
2 1)u
"
0 1
p
1+ju
"
0 j
2 (
p
1+ju
"
0 j
2
)u
"
0
:
(26)
denesamappingS
S : Z !Z
V 7 !U:
wehavetondatime T andonstantsM andrsuh thatS mapsthelosedballX
T
(M;r)intoitselfand
isaontratingmappinginthenorm
sup
t2[0;T℄
5
X
j=0 jjv
j jj
2 :
Therstthingto doisto estimateu
4
byusingequation (24). Aswehavealreadysaidin theintrodution
of thisartile,wewill treat(24)asawaveequation. Multiplying equation(24)byA(v
0
)andapplyingthe
operator(1 ),weobtaindenoting
4
=(1 )u
4
"
2
(
4 )
tt
(
4 )
tt
+2i
(
4 )
t
(
4 )
t
+A(v
0 )
4
4
+ 2
X
k =1 L(v
0
;v
k
;
k v
0 )
k
4
k
4
X
jj2 M
(D
h1 1
V;D h2 1
Tv
3 )
1
1
2
2 u
4
1
1
2
2 u
4
+ X
jj2 2
X
j;k =0 N
(D
h
2
V;D h
2 1
Tv
3 )
1
1
2
2 R
jk u
4
1
1
2
2 R
jk u
4
=0:
(27)
where thematriesM
dependonthemulti-index2N 2
,N
on,j andk
L(v
0
;v
k
;
k v
0
)=A(v
0 )D
k (v
0
;v
k )+2
k A(v
0 );
h
1
=3 ifjj=1;2; h
1
=4 ifjj=0;
h
2
=3 jj:
Finally,dierentiating(27)withrespettox
m
form=1;2anddenoting m
4
=
m
4
,weobtain
"
2
( m
4 )
tt
( m
4 )
tt
+2i
( m
4 )
t
( m
4 )
t
+A(v
0 )
m
4
m
4
+ 2
X
k =1 L(v
0
;v
k
;
k v
0 )
k
m
4
k
m
4
+ 2
X
k =1
m A(v
0 )
k
k
4
k
k
4
!
+ X
jj3
M
(D
h1
V;D h2
Tv
3 )
1
1
2
2 u
4
1
1
2
2 u
4
+ X
jj3 2
X
j;k =0
N
(D
h
2 +1
V;D h
2
Tv
3 )
1
1
2
2 R
jk u
4
1
1
2
2 R
jk u
4
=0:
(28)
where
h
1
=2; h
2
=0 ifjj=3:
Thematries
M
and
N
depend onforjj3. TheyalsodependonrespetivelyD h
1
V, D h
2
Tv
3 and
D h2+1
V,D h2
Tv
3
. Wehavetonotiethat in(27),thedierentiationof
A(v
0 )
4
4
withrespetto x
m
foraxedmgivestheterm
m (A(v
0 ))
4
4
whihisrewrittenin(28)as
2
X
k =1
m A(v
0 )
k
k
4
k
k
4
!
:
WealsoreallheretheexpressionofA (v
0 )
A 1
(v
0 )=
1
2
2+jv
0 j
2
v 2
0
v 2
0
(2+jv
0 j
2
)
:
Wemultiplyequation(28)byA 1
(v
0
)andweonsidertherstlineoftheresultingequation. Furthermore,
weaddtobothsideoftheequationtheterm m
4
. Weannowperformtheusualenergyestimateforwave
equations,namelywemultiplytheequationby( m
4 )
t
andweintegrateoverR 2
.
"
2 Z
R 2
(2+jv
0 j
2
)
2 (
m
4 )
tt (
m
4 )
t dx "
2 Z
R 2
v 2
0
2 (
m
4 )
tt (
m
4 )
t dx
i Z
R 2
(2+jv
0 j
2
)j(
m
4 )
t j
2
dx+i Z
R 2
v 2
0 (
m
4 )
t (
m
4 )
t dx
Z
R 2
m
4 (
m
4 )
t dx
2
X
k =1 Z
R 2
O 11
(v
0
;v
k
;
k v
0 )
k
m
4 (
m
4 )
t dx
2
X
k =1 Z
R 2
O 12
(v
0
;v
k
;
k v
0 )
k
m
4 (
m
4 )
t dx+
Z
R 2
m
4 (
m
4 )
t dx
2
X
k =1 Z
R 2
A 1
(v
0 )
m A(v
0 )
11
k
k
4 (
m
4 )
t dx
Z
R 2
m
4 (
m
4 )
t dx
2
X
k =1 Z
R 2
A 1
(v
0 )
m A(v
0 )
12
k
k
4 (
m
4 )
t dx
X
jj3 Z
R 2
P 11
(D
h1
V;D h2
Tv
3 )(
1
1
2
2 u
4 )(
m
4 )
t dx
X
jj3 Z
R 2
P 12
(D
h1
V;D h2
Tv
3 )(
1
1
2
2 u
4 )(
m
4 )
t dx
X
jj3 2
X
j;k =0 Z
R 2
Q 11
(D
h2+1
V;D h2
Tv
3 )
1
1
2
2 R
jk u
4 (
m
4 )
t dx
X
jj3 2
X
j;k =0 Z
R 2
Q 12
(D
h
2 +1
V;D h
2
Tv
3 )
1
1
2
2 R
jk u
4 (
m
4 )
t dx=0:
(29)
where O=A 1
(v
0 )L, P
=A 1
(v
0 )
M
and Q
=A 1
(v
0 )
N
.
Wenowtaketherealpartofequation(29). Therstlineoftheresultingexpressiongives
Re
"
2 Z
R 2
(2+jv
0 j
2
)
2 (
m
4 )
tt (
m
4 )
t dx "
2 Z
R 2
v 2
0
2 (
m
4 )
tt (
m
4 )
t dx
="
2 Z
R 2
(2+jv
0 j
2
)
4 j(
m
4 )
t j
2
t dx "
2 Z
R 2
v 2
0
8 d
dt (
m
4 )
2
t +
v 2
0
8 d
dt (
m
4 )
2
t
dx
= d
dt
"
2 Z
R 2
(2+jv
0 j
2
)
4 j(
m
4 )
t j
2
dx "
2 Z
R 2
v 2
0
8 (
m
4 )
2
t +
v 2
0
8 (
m
4 )
2
t
dx
"
2 Z
R 2
(2+jv
0 j
2
)
t
4
j(
m
4 )
t j
2
dx+"
2 Z
R 2
(v 2
0 )
t
8 (
m
4 )
2
t +
(v 2
0 )
t
8 (
m
4 )
2
t
dx:
Fortheseondline,sineV 2X
T
(M;r)wehave,usingtheontinuousembeddingofH 2
(R 2
)intoL 1
(R 2
)
i
Z
R 2
(2+jv
0 j
2
)(
m
4 )
t (
m
4 )
t dx+i
Z
R 2
v 2
0 (
m
4 )
t (
m
4 )
t dx
C(M) Z
R 2
j(
m
4 )
t j
2
dx:
Re Z
R 2
m
4 (
m
4 )
t dx
= 1
2 d
dt Z
R 2
jr m
4 j
2
dx:
UsingagainthefatthatV 2X
T
(M;r)andthatthematrixO
m
dependsonlyonV and
k v
0
,weobtain
byCauhy-Shwarzinequality
2
X
k =1 Z
R 2
O 11
(v
0
;v
k
;
k v
0 )
k
m
4 (
m
4 )
t dx+
Z
R 2
O 12
(v
0
;v
k
;
k v
0 )
k
m
4 (
m
4 )
t dx
C(M) 2
X
k =1 jj
k
m
4 jj
2 jj(
m
4 )
t jj
2
C(M) 2
X
k =1 jj
k
m
4 jj
2
2 +jj(
m
4 )
t jj
2
2
!
:
Inthesameway,wehave
2
X
k =1 Z
R 2
A 1
(v
0 )
m A(v
0 )
11
k
k
4 (
m
4 )
t dx
C(M) 2
X
k =1 jj
k
k
4 jj
2
2 +jj(
m
4 )
t jj
2
2
!
;
2
X
k =1 Z
R 2
A 1
(v
0 )
m A(v
0 )
12
k
k
4 (
m
4 )
t dx
C(M) 2
X
k =1 jj
k
k
4 jj
2
2 +jj(
m
4 )
t jj
2
2
!
:
Wenowtreatthetwotermsontaining m
4 (
m
4 )
t
bytwodierentmethods. Ononehand,oneanwrite
Re Z
R 2
m
4 (
m
4 )
t dx
= 1
2 d
dt Z
R 2
j m
4 j
2
dx:
Ontheotherhand,wederivefromCauhy-Shwarzinequality
Z
R 2
m
4 (
m
4 )
t dx
C(M) jj m
4 jj
2
2 +jj(
m
4 )
t jj
2
2
:
Forthenextterm,wehaveto bemoreareful. If jj=3,P
depends onderivativesof V oforder less
thanorequalto2andonTv
3
anditsrstorderderivatives. Thus,itanbeestimatedinL 1
(R 2
). Thetwo
other termsareestimated byCauhy-Shwarzinequality. Asaonsequene,one anndapositiveC(M)
suhthat
X
jj=3 Z
R 2
P 11
(D
h
1
V;D h
2
Tv
3 )(
1
1
2
2 u
4 )(
m
4 )
t dx
C(M) X
jj=3 jj
1
1
2
2 u
4 jj
2 jj(
m
4 )
t jj
2
C(M) 2
X
k =1 jj
k
4 jj
2
2 +jj(
m
4 )
t jj
2
2
!
:
Ifjj=0;::;2, P
depends onderivativesof V of order lessorequalto 4and onderivativesof Tv
3 of
order lessor equal to 3. Thus it has to be estimated in L 2
(R 2
) together with ( m
4 )
t
by Cauhy-Shwarz
inequalitywhereas
1
2 u
4
isestimated inL 1
(R 2
).
2
X
jj=0 Z
R 2
P 11
(D
h
1
V;D h
2
Tv
3 )(
1
1
2
2 u
4 )(
m
4 )
t dx
C(M) 2
X
jj=0 jj
1
1
2
2 u
4 jj
L 1
(R 2
) jj(
m
4 )
t jj
2
C(M) jj
4 jj
2
2 +
2
X
k =1 (jj
k
k
4 jj
2
2 +jj
k
4 jj
2
2 )+jj(
m
4 )
t jj
2
2
!
:
Indenitive,wehaveprovedthat
X
jj3 Z
R 2
P 11
(D
h1
V;D h2
Tv
3 )(
1
1
2
2 u
4 )(
m
4 )
t dx
C(M) jj
4 jj
2
2 +
2
X
k =1 (jj
k
4 jj
2
2 +jj
k
k
4 jj
2
2 )+jj(
m
4 )
t jj
2
2
!
:
Inthesameway,oneanalsoobtain
X
jj3 Z
R 2
P 12
(D
h1
V;D h2
Tv
3 )(
1
1
2
2 u
4 )(
m
4 )
t dx
C(M) jj
4 jj
2
2 +
2
X
k =1 (jj
k
4 jj
2
2 +jj
k
k
4 jj
2
2 )+jj(
m
4 )
t jj
2
2
!
:
Thelasttwotermsaretreatedexatlyin thesameway,theonlythingtonotieisthattheoperatorR
jk
isontinuousfromL 2
(R 2
)to L 2
(R 2
). Thusweanwrite
X
jj3 2
X
j;k =0 Z
R 2
Q 11
(D
h
2 +1
V;D h
2
Tv
3 )
1
1
2
2 R
jk u
4 (
m
4 )
t dx
jj
4 jj
2
2 +
2
X
k =1 (jj
k
4 jj
2
2 +jj
k
k
4 jj
2
2 )+jj(
m
4 )
t jj
2
2
!
;
X
jj3 2
X
j;k =0 Z
R 2
Q 12
(D
h2+1
V;D h2
Tv
3 )
1
1
2
2 R
jk u
4 (
m
4 )
t dx
jj
4 jj
2
2 +
2
X
k =1 (jj
k
4 jj
2
2 +jj
k
k
4 jj
2
2 )+jj(
m
4 )
t jj
2
2
!
:
d
dt
"
2 Z
R 2
(2+jv
0 j
2
)
4 j(
m
4 )
t j
2
dx "
2 Z
R 2
v 2
0
8 (
m
4 )
2
t +
v 2
0
8 (
m
4 )
2
t
dx+ 1
2 Z
R 2
jr m
4 j
2
dx
+ 1
2 d
dt Z
R 2
j m
4 j
2
dx
"
2 Z
R 2
jv
0 j
2
t
4 j(
m
4 )
t j
2
dx+"
2 Z
R 2
(v 2
0 )
t
8 (
m
4 )
2
t +
(v 2
0 )
t
8 (
m
4 )
2
t
dx
C(M) jj
4 jj
2
2 +
2
X
k =1 jj
k
k
4 jj
2
2 +jj
k
4 jj
2
2
+jj(
m
4 )
t jj
2
2
!
:
(30)
Sine
4
=(1 )u
4
,itisnotdiÆultto derivefrom(24)
jj
4 jj
2
2
C(M) 2
X
k =1 jj
k
k
4 jj
2
2 +jj
k
4 jj
2
2
+jj(
m
4 )
t jj
2
2
!
:
Then,integratinginequality(30)from 0tot,oneobtains
"
2
2 Z
R 2
j(
m
4 (t))
t j
2
dx+ 1
2 Z
R 2
jr m
4 (t)j
2
dx+
"
2
2 Z
R 2
jv
0 (t)j
2
2 j(
m
4 (t))
t j
2
dx
+ 1
2 Z
R 2
j m
4 j
2
dx
"
2
2 Z
R 2
v 2
0 (t)
4 (
m
4 (t))
2
t v
2
0 (t)
4 (
m
4 (t))
2
t
dx
"
2 Z
R 2
(2+jv
0 (0)j
2
)
4
j(
m
4 )
t (0)j
2
dx "
2 Z
R 2
v 2
0 (0)
8 (
m
4 )
2
t (0)+
v 2
0 (0)
8 (
m
4 )
2
t (0)
dx
+ 1
2 Z
R 2
jr m
4 (0)j
2
dx+ 1
2 Z
R 2
j m
4 (0)j
2
dx+"
2 Z
t
0 Z
R 2
(jv
0 (s)j
2
)
s
4 j(
m
4 (s))
s j
2
dxds
+"
2 Z
t
0 Z
R 2
j(v 2
0 (s))
s j
8 j(
m
4 (s))
2
s j+
j(v 2
0 (s))
s j
8 j(
m
4 (s))
2
s j
dxds
+C(M) Z
t
0 2
X
k =1 jj
k
k
4 (s)jj
2
2 +jj
k
4 (s)jj
2
2
+jj(
m
4 )
s (s)jj
2
2
!
ds:
(31)
Usingthefatforalls2[0;t℄, jv0(s)j
2
2 j(
4 (s))
s j
2 v
2
0 (s)
4 (
m
4 (s))
2
s v
2
0 (s)
4 (
m
4 (s))
2
s
0,thatjj(u
0 )
t jj
L 1
(R 2
)
r, anddenoting
f m
(0)= 1
2 Z
R 2
jr
"
m
4 (0)j
2
dx+ 1
2 Z
R 2
j m
4 (0)j
2
dx
+"
2 Z
R 2
(2+jv
0 (0)j
2
)
4
j(
m
4 )
t (0)j
2
dx
"
2 Z
R 2
v 2
0 (0)
8 (
m
4 )
2
t (0)+
v 2
0 (0)
8 (
m
4 )
2
t (0)
dx;
2
X
m=1
"
2
2 Z
R 2
j(
m
4 (t))
t j
2
dx+ 1
2 Z
R 2
jr m
4 (t)j
2
dx+ 1
2 Z
R 2
j m
4 (t)j
2
dx
2
X
m=1 f
m
(0)+C(M;r) 2
X
m=1 Z
t
0 2
X
k =1 jj
k
m
4 (s)jj
2
2 +jj
k
4 (s)jj
2
2
+jj(
m
4 (s))
s jj
2
2
!
ds:
(32)
Thisimplies,
2
X
m=1
"
2
2 Z
R 2
j(
m
4 (t))
t j
2
dx+ 2
X
k =1 Z
R 2
j
k
m
4 (t)j
2
dx+ Z
R 2
j m
4 (t)j
2
dx
!
2
X
m=1 f
m
(0)+C(M;r) 2
X
m=1 Z
t
0 2
X
k =1 jj
k
m
4 (s)jj
2
2 +jj
k
4 (s)jj
2
2
+jj(
m
4 (s))
s jj
2
2
!
ds:
(33)
ApplyingGronwalllemmato(33), weget
sup
t2[0;T℄
2
X
m=1
"
2
2 Z
R 2
j(
m
4 (t))
t j
2
dx+ 2
X
k =1 Z
R 2
j
k
m
4 (t)j
2
dx+ Z
R 2
j m
4 (t)j
2
dx
!
e C(M;r)
T
"
2 2
X
m=1 f
m
(0):
(34)
Then,hoosingC(M;r)T 1andu
4
(0)suhthat
2
X
m=1 f
m
(0)e T
"
2
m
4
(35)
gives
sup
t2[0;T℄
jju
4 (t)jj
H 4
(R 2
)
m
4
: (36)
It isimportanttonotieherethat(35)impliesthat
jju
4 (;0)jj
H 4
(R 2
) e
T
"
2
m
4 :
Furthermore,wealsohave
sup
t2[0;T℄
"jj
t u
4 (t)jj
H 3
(R 2
)
m
4
: (37)
Wenow studyequation (25)satisedbyu
5
. Thisequation isof thesametypeasequation (24)andit
willbetreatedwiththesamemethod; wemultiply(25)byA(v
0
)andweapplytheoperator
m
(1 )for