INVARIANTMANIFOLDTHEORYANDITSAPPLICATIONSTONONLINEARPARTIAL
DIFFERENTIALEQUATIONS
By
JiayinJin
ADISSERTATION
Submittedto
MichiganStateUniversity
inpartialoftherequirements
forthedegreeof
Mathematics-DoctorofPhilosophy
2015
ABSTRACT
INVARIANTMANIFOLDTHEORYANDITSAPPLICATIONSTONONLINEAR
PARTIALDIFFERENTIALEQUATIONS
By
JiayinJin
Thetheoryofinvariantmanifoldsandfoliationsprovidesindispensabletoolsforthestudyof
dynamicsofnonlinearsystemsinordimensionalspace.Asisthecasehere,invariant
manifoldscanbeusedtocapturecomplexdynamicsandthelongtermbehaviorofsolutionsand
toreducehighdimensionalproblemstotheanalysisoflowerdimensionalstructures.Invariant
manifoldswithinvariantfoliationsprovideacoordinatesysteminwhichsystemsofdi

erential
equationsmaybedecoupledandnormalformsderived.Theseplayanimportantroleinthestudy
ofstructuralstabilityofdynamicalsystemsor,whenadegeneracyoccurs,inunderstandingthe
natureofbifurcations.Thisthesisisdevotedtothestudyoftheconstructionofinvariantmanifolds
ofsolutionswithcertainspatialstructurestosomenonlinearparabolicpartialdi

erentialequations.
Iapproachtheseproblemsintwosteps:thestepistoconstructamanifoldofstatesthatis
approximatelyinvariant,thesecondstepistoshowtheexistenceofatrulyinvariantmanifoldof
thesestatesneartheapproximatelyinvariantone,andtodeterminethedynamicsonthismanifold.
Sincethisapproachmaybeappliedtomanydi

erentsystems,Ialsodevelopitinanabstractor
generalway,extendingearlierresultsof[19].
Mythesisconsistsoftwoprojects,intheproject,weconsiderthetwo-dimensionalmass-
conservingAllen-CahnEquation,
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
˚
t
(
x
;
t
)
=
"
2

˚
(
x
;
t
)

f
(
˚
(
x
;
t
))
+
>

f
(
˚
(

;
t
))
;
x
2

;
t
>
0
;
@
n
˚
(
x
;
t
)
=
0
;
x
2
@

;
t
>
0
;
(0.0.1)
where

ˆ
R
2
isaedboundeddomainwithsmoothboundary
@

,
@
n
istheexteriornormal
derivativeto
@

,and
>

=
1
j

j
R

meanstheaverageover

.Here
f
isthederivativeofadouble
wellpotential
W
.Weassumethefollowingconditionsfor
f
:
f
(

1)
=
0
;
f
0
(

1)
>
0
;
Z
s

1
f
=
Z
s
1
f
>
0forall
s
2
(

1
;
1)
:
(0.0.2)
Weestablishtheexistenceofaglobalinvariantmanifoldofbubblestatesforthisequationandgive
thedynamicsforthecenterofthebubble.
Inthesecondproject,weconsidertheexistence,inforwardandbackwardtime,ofdynamical
interiormulti-spikestatesdrivenbythenonlinearCahn-Hilliardequation:
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
u
t
=


(
"
2

u

f
(
u
))in


(0
;
1
)
;
@
n

u
=
@
n
u
=
0on
@


(0
;
1
)
;
(0.0.3)
where

ˆ
R
n
isaedboundeddomainwithsmoothboundary
@

and
f
isthederivativeof
adoublewellpotential
W
,thatis,
xf
(
x
)
>
0for
j
x
j
largeenoughand
f
hastwozeros
a
and
b
suchthat
f
0
(
a
)
;
f
0
(
b
)
>
0.Weconstructinvariantmanifoldsofinteriormulti-spikestatesfor
thenonlinearCahn-Hilliardequationandtheninvestigatethedynamicsonit.Anequationfor
themotionofthespikesisalsoderived.Itturnsoutthatthedynamicsofinteriorspikeshasa
globalcharacterandeachspikeinteractswithalltheothersandwiththeboundary.Moreover,we
showthatthespeedoftheinteriorspikesissuperslow,whichindicatesthelongtimeexistenceof
dynamicalmulti-spikesolutionsinbothpositiveandnegativetime.
Copyrightby
JIAYINJIN
2015
ACKNOWLEDGMENTS
Iwouldliketoexpressthedeepestappreciationtomyadvisor,Dr.PeterBates,forhisguidance,
assistance,encouragement,andheartysupportinallthephasesofmydoctoralprogramatMichi-
ganStateUniversity.Thisdissertationwouldnothavebeenpossiblewithouthisguidanceand
persistenthelp.
Iwouldliketothankmycommitteemembers,Dr.KeithPromislow,Dr.RussellSchwab,Dr.
BaishengYan,Dr.ZhengfangZhoufortakingtimetoserveonmydissertationcommitteeandfor
theirusefulcommentsandsuggestions.
IamthankfultoDr.ChongchunZengandDr.GiorgioFuscoforvaluablediscussions,during
whichtheyprovidedmewithinspiringsuggestionsaboutmyresearch.
IamgratefultoYinCao,HongliGao,YangGao,JiLi,YuLiang,QiliangWuandmanyothers
fortheirfriendshipatMichiganStateUniversity.
IalsowouldliketotakethisopportunitytoexpressmygratitudetoDr.PeterBates,Dr.Giorgio
Fusco,Dr.KeningLu,Dr.ChongchunZengandDr.ZhengfangZhouforsupportingmypostdoc
application.
Atlast,butbynomeanstheleast,myheart-feltappreciationsgotomydearestparentsandgirl
friend,XiaoyingHanfortheircontinuousandtime-invariantsupport.Withouttheirunderstanding,
thisworkcannotbedone.
v
TABLEOFCONTENTS
LISTOFFIGURES
......................................
vii
Chapter1Introduction
..................................
1
Chapter2Preliminaries:Approximatelyinvariantmanifolds
............
6
Chapter3Globalinvariantmanifoldsofboundarydropletsforthe2-Dmass-conserving
Allen-Cahnequation
.............................
11
3.1Introduction......................................11
3.2Approximatebubblesolutionforthemass-conservingAllen-Cahnequation....14
3.3Invariantmanifoldsofboundarydroplets.......................19
3.3.1Constructionof
M
"
..............................19
3.3.2
M
"
isapproximatelyinvariant........................20
3.3.3Splittingalongthemanifold
M
"
.......................21
3.3.4
M
"
isapproximatelynormallyhyperbolic..................22
3.4Dynamicalbubblesolution..............................30
3.4.1Motionon
f
M
"
................................32
3.4.2Equilibria...................................34
Chapter4Invariantmanifoldsofinteriormulti-spikestatesfortheCahn-Hilliard
equationinhigherspacedimensions
....................
36
4.1Introduction......................................36
4.2Approximatelystationaryinvariantmanifoldswithboundary............41
4.3Invariantmanifoldsofinteriormulti-spikestates...................51
4.3.1Constructionofthemanifold
M
"
.......................51
4.3.2SpectralanalysisofthelinearizedAllen-Cahnoperator...........58
4.3.3SpectralanalysisofthelinearizedCahn-Hilliardoperator..........61
4.3.4
M
"
isapproximatelystationary.......................65
4.3.5Transformationofthephasespace......................67
4.3.6Splittingspace
X
]
alongthemanifold
M
]
"
..................70
4.3.7Trichotomy..................................72
4.3.8ofvector..........................78
4.4Longtimedynamicson
f
M
"
..............................83
BIBLIOGRAPHY
.......................................
98
vi
LISTOFFIGURES
Figure2.1:Approximatelynormallyhyperbolicinvariantmanifolds..........7
Figure3.1:Graphofthefunctionf...........................12
Figure3.2:Fourstagesintheevolutionforatwo-dimensionaldomain

.Thelast
stageistheobjectofstudyinthispaper...................13
Figure3.3:Geometryofthebubble...........................15
Figure4.1:Interiormulti-spikestate..........................37
vii
Chapter1
Introduction
Indynamicalsystems,aninvariantmanifoldisamanifoldthatisinvariantunderamaporaw
(orw).Forinstance,aedpointorperiodicorbitofanordinarydi

erentialequationis
aninvariantmanifoldforthewgeneratedbythatODE.Thetheoryofinvariantmanifoldsfor
discreteandcontinuousdynamicalsystemshasalongandrichhistory.Numerousapplicationscan
befoundwhereanswerstothefollowingquestionsareneeded:1)Assumingthatadynamicalsys-
temhasaninvariantmanifold,doesaperturbationofthissystemalsohaveaninvariantmanifold?
2)Whenadynamicalsystemhasaninvariantmanifold,howdoesoneconstructlocallyinvariant
structuressuchasthecenter-stable,center-unstablemanifold,andcentermanifoldoftheoriginal
invariantmanifoldandinvariantfoliationsofthese,whichessentiallydecouplethedynamics.
Forthecaseoftheinvariantmanifoldconsistingofasingleedpoint,Hadamard[49]con-
structedtheunstablemanifoldofahyperbolicedpointofadi

eomorphismoftheplaneby
iteratingthemappingappliedtoacurveintheplane,thereforeobtainingaconvergentsequence
ofcurves.Thelimitofthesequenceofcurvesgivestheunstablemanifold.Peoplenowcall
thisgeometricapproachHadamard'sgraphtransform.Lyapunov[65]andPerron[78,79,80]
constructedtheunstablemanifoldofanequilibriumpointbyformulatingtheproblemasanin-
tegralequation.ThismethodisanalyticratherthangeometricandnowiscalledLyapunov-
Perronmethod.Thereisanextensiveliteratureonthestable,unstable,center,center-stable,
andcenter-unstablemanifoldsofequilibriumpointsforbothanddimensionaldy-
namicalsystems.Thegeneraltheoryfordimensionaldynamicalsystemsmaybefoundin
1
[26,29,54,52,56,58,60,67,81,83,84,86,88].Fordimensionaldynamicalsystemswe
referthereaderto[10,16,35,51,68,87,93].MostoftheseworksusetheapproachofLyapunov-
Perron.AgoodtreatmentofcentermanifoldtheoryforODE'susingtheLyapunov-Perronmethod
canbefoundinthemonographbyCarr[29],whereseveralapplicationsarealsosetforth.Cer-
taindimensionalsettingsarealsotreated.VanderbauwhedeandVanGils[88]alsousethe
Lyapunov-Perronmethodtoobtainsmoothcentermanifoldsbutwithsomeimportantdi

erences
intechnique.Ball[10]usedtheLyapunov-Perronapproachtoobtainlocalstable,unstableand
centermanifoldsforequilibriumpointsofdynamicalsystemsinBanachspace,withapplication
tothebeamequation.Henry[51]developedthetheoryforsemilinearparabolicequations.Later,
ChowandLu[35]usedthisapproachtoprovetheexistenceofsmoothcenter-unstablemanifolds
withapplicationtothedampedwaveequation.Formoreoncentermanifoldtheoryinthe
dimensionalsetting,usingtheLyapunov-Perronmethod,see[87].Thetheoryofinvariantmani-
foldsforanequilibriumpointofdimensionaldynamicalsystemsusingHadamard'sapproach
maybefoundin[54].Fordimensionaldynamicalsystems,wereferto[16],whereapplica-
tionsaregivendemonstratingthestabilityofapulsesolutiontotheFitzHugh-Nagumoequations
andtheinstabilityofstationarysolutionstothenonlinearKlein-Gordonequation.
Chow,LiuandYi[34]constructedcentermanifoldsforsmoothinvariantmanifoldsforsmooth
wsindimensionalspacesbyusingthemethodofHadamard'sgraphtransform.Krylov
andBogoliubov[24]studiedtime-periodicordinarydi

erentialequationsarisingfromthestudy
ofnonlinearoscillations.Undertheassumptionthattheaveragedequationhasanasymptotically
stableequilibriumpoint,theyprovedtheexistenceofperiodicintegralmanifolds,whichgivesthe
existenceofasymptoticallystableperiodicorbitsforaclassofequations.Anintegralmanifoldis
aninvariantmanifoldintheproductspaceoftimeandphasespace.Theaboveresultandmany
generalizationsandrelatedworkincollectedinthemonographofBigoliubovandMitropolsky
2
[25].
Levinson[62]studiedperiodicperturbationsofanautonomousordinarydi

erentialequation
possessinganasymptoticallystableperiodicorbit.Heprovedthatiftheperturbationwassu

-
cientlysmall,thentheperturbedequationhasaperiodicintegralmanifold,whichmaybeviewed
asatwo-dimensionaltorus.Levinson'sresultswereextendedtoperiodicsurfacesbyDiliberto
[40],Hu

ord[55],andKyner[61].Hale[50]establishedageneraltheoryofintegralmanifolds
fornonautonomousordinarydi

erentialequationsandobtainedmoregeneralresultsthanthose
justmentionedabove.AnextensionofHale'sintegralmanifoldtheorytoalargerclassofnonau-
tonomousordinarydi

erentialequationswasobtainedin[100].
Thepersistenceunderperturbationofacompactnormallyhyperbolicinvariantmanifoldsfor
adimensionaldynamicalsystemwasindependentlyobtainedbyHirsch,PughandShub
[52,53]andFenichel[41,42,43].Theyprovedthepersistenceofnormallyhyperbolicinvariant
manifolds,andtheexistenceofthecenter-stableandcenter-unstablemanifoldsandtheirinvariant
foliations.PlissandSell[82]studiedthepersistenceofhyperbolicattractorsforordinarydi

eren-
tialequations.Bates,LuandZengin[11]provedthepersistenceofcompactnormallyhyperbolic
invariantmanifolds,andtheexistenceofthecenter-stableandcenter-unstablemanifoldsandtheir
invariantfoliationsforwsinspacesandthenextendedtheirresults
withoutassumingcompactnessin[17,18].
Ma
Ÿ
n
´
e[66]provedthatnormalhyperbolicityin[52]isanecessaryconditionforthe
persistenceofaninvariantmanifoldunderperturbationofadimensionaldynamicalsystem.
Henry[51]extendedHale'stheoryofintegralmanifoldstogeneralnonautonomousabstract
semilinearequationswhoselinearpartgeneratesananalyticsemigroup.Henryalsostudiedcom-
pactnormallyhyperbolicinvariantmanifoldswithtrivialnormalbundleforsemilinearparabolic
equationsandobtainedacoordinatetransformationwhichleadtothesettingofintegralmanifolds
3
andpersistenceresults.
Inmanysingularperturbationproblemsforevolutionarypartialdi

erentialequations,people
areinterestedinsolutionswhichhavecertainqualitativefeatures,suchasinteriororboundarylay-
ersorlocalizedspikes,andmotionsoftheselayersandspikes,includingthelocationofstationary
layers.Thecanonicalshapeofsuchsolutions,inaneighborhoodoftheabruptspatialdisturbance
(layerorspike),canusuallybedeterminedbyarescalingorblow-upprocedure.Thus,areasonable
approximationtotheshapeofasolutionisfoundquiteeasilybyconsideringtheequationonthe
wholespaceandapproximatelyinvariantmanifoldsmadeupoftheseapproximatesolutionshave
beenconstructedbymanyauthors.Theapproach,involvingtheconstructionofanapproximately
invariantmanifoldofstateshavingacertainspatialstructure,waspioneeredmorethanthirtyyears
agoinpapersofG.FuscoandJ.Halein[45]andbyJ.CarrandR.Pegoin[31].Inthosepapers
theauthorswereinterestedintheslowdynamicsofinterfacesinsolutionstotheone-dimensional
Allen-CahnEquation.Thesameapproachwasalsotakentoobtainsimilarresultsfortheone-
dimensionalCahn-Hilliardequationin[4],[21]and[22],andtorigorouslyestablishtheslow
motionofflbubblefl-likesolutions[6,7]andmultipeakedstationarysolutionstotheCahn-Hilliard
equation[14,96]inmulti-dimensionaldomains.Theapproachwasalsousedtoproducespike-like
stationarysolutionstotheshadowGierer-Meinhardtsystemofbiologicalpatternformation[59].
Inmostofthesepapers,thequalitativeshapeofstationarysolutionswasthepointofinterestand
soatrueinvariantmanifoldwasnotshowntoexist,althoughthatwasdoneinasubsequentpaper
byCarrandPegoin[32]andalsoin[22].
Itisnaturaltoaskhowtodeducetheexistenceofatrueinvariantmanifoldinasmallneighbor-
hoodoftheapproximatelyinvariantmanifoldconstructedbyhand,asdescribedabove.In[19],the
authorsestablishedasystematicwaytoatrueinvariantmanifoldassumingtheapproximate
oneisgoodenoughandtheyfoundaninvariantmanifoldofboundaryspikestatesforaclassof
4
parabolicequations.Inthisthesis,weapplytheabstractresultsin[19]toconstructaninvari-
antmanifoldofboundarydropletsforthe2-Dmass-conservingAllen-Cahnequation.Thenwe
extendtheabstractresultsin[19]tomanifoldswithboundary,consistingofapproximatelystation-
arystatesandconstructinvariantmanifoldsofdynamicinterior-spikestatesfortheCahn-Hilliard
equationinhigherspacedimensions.
5
Chapter2
Preliminaries:Approximatelyinvariant
manifolds
Here,westatesomeresultsfrom[19],whichprovideatoolforobtaininganinvariantmanifold
whenagoodapproximationisavailable.Roughlyspeaking,ifanimmersedmanifold
 
(
M
)is
approximatelywinginvariantundermap
T
,andif
 
(
M
)isapproximatelynormallyhyperbolic,
thenonecanatrulylocallyinvariantmanifold
W
cs
,itscenter-stablemanifold,undermap
T
.Furthermoreif
 
(
M
)isapproximatelyovwinginvariantundermap
T
andapproximately
normallyhyperbolic,thenwecanacenter-unstablemanifold
W
cu
undermap
T
.If
W
cs
and
W
cu
intersecttransversally,thentheirintersectionisthetrulyinvariantmanifoldweseekasagraph
over
 
(
M
).Now,wegivethepreciseandstatementsofthetheorems.
Let
X
beaBanachspaceand
T
2
C
J
(
X
;
X
)
;
J

1with
M
aconnected
C
1
Banachmanifold
and
 
:
M
!
X
animmersion.
2.0.1.
 
(
M
)issaidtobeapproximatelywinginvariantunder
T
ifthefollowing
conditionshold
(1)Thereexists
>
0and
u
2
C
0
(
M
;
M
)suchthat
j
T
(
 
(
m
))

 
(
u
(
m
))
j
<
,forall
m
2
M
;
(2)Thereexists
r
0
2
(0
;
1)suchthat
 
(
B
c
(
m
0
;
r
0
))isclosedin
X
forany
m
0
2
u
(
M
),where
B
c
(
m
0
;
r
0
)istheconnectedcomponentof
 

1
(
B
(
 
(
m
0
)
;
r
0
))containing
m
0
.
6
Figure2.1:Approximatelynormallyhyperbolicinvariantmanifolds
Condition(1)meansthat
 
(
M
)isapproximatelyinvariantunder
T
and
u
on
M
isanapproxi-
mationof
T
on
 
(
M
).Condition(2)essentiallystatesthatthe`distance'betweentheprojectionof
T
(
 
(
M
))into
 
(
M
)andtheboundaryof
 
(
M
)isboundedfrombelow.
2.0.2.
Wesaythatanapproximatelywinginvariantmanifold
 
(
M
)isapproxi-
matelynormallyhyperbolic,ifconditions(H1)-(H3)hold:
(H1)Foreach
m
2
M
,thereisadecomposition
X
=
X
c
m

X
s
m

X
u
m
ofclosedsubspaceswith
projections

c
m
;

s
m
;

u
m
.
(H2)Forany
m
2
M
,

c
m
isanisomorphismfrom
D
 
(
m
)
T
m
M
to
X
c
m
.Furthermorethereexist
constants
B
;
L
,and
˜
2
(0
;
1
=
2),suchthatforany
m
0
2
M
and
m
1
;
m
2
2
B
c
(
m
0
;
r
0
),with
m
1
,
m
2
,for

=
c
;
u
;
s
,
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
:
k


m
0
k
B
,
k


m
1



m
2
k
L
j
 
(
m
1
)

 
(
m
2
)
j
;
j
 
(
m
1
)

 
(
m
2
)


c
m
0
(
 
(
m
1
)

 
(
m
2
))
j
j
 
(
m
1
)

 
(
m
2
)
j

˜:
(2.0.1)
(H3)Thereexist
˙;
2
(0
;
1)suchthat,forany
m
0
2
M
,if
m
1
=
u
(
m
0
),and

2f
c
;
s
g
,

2f
c
;
s
;
u
g
,
with

,

,then
7
(a)
k


m
1
DT
(
 
(
m
0
))
j
X

m
0
k
˙
,
(b)
k

s
m
1
DT
(
 
(
m
0
))
j
X
s
m
0
k

,
(c)

k
(

u
m
1
DT
(
 
(
m
0
))
j
X
u
m
0
)

1
k

1
>
max
f
1
;
k

c
m
1
DT
(
 
(
m
0
))
j
X
c
m
0
k
J
g
.
(H4)Thereexists
B
1
,suchthat
k
D
j
T
j
B
(
 
(
M
)
;
r
0
)
k
B
1
for1

j

J
.When
J
=
1,wewillneed
thefollowingfunction
A
(

)
=
sup
fk
DT
(
x
1
)

DT
(
x
2
)
k
:
x
1
;
x
2
2
B
(
 
(
M
)
;
)
;
j
x
1

x
2
j
<
g
,
andrequireittobesmallenough.
Hypothesis(H3)thedi

erentgrowthratesof
DT
,thelinearizationof
T
,indi

erent
directions.Condition(a)representstheapproximateinvarianceofthebundles
X
c
and
X
s
under
DT
.Di

erentratesintheunstableandcenter-stabledirectionsareassumedin(b)and(c).Hy-
pothesis(H4)isatechnicalassumptionon
T
,whichholdsautomaticallyif
 
(
M
)isprecompact.
For

=
c
;
u
;
s
,let
X

m
(
"
)
=
f
x
2
X

m
:
j
x
j
<"
g
and
X

(
"
)
=
f
(
m
;
x
):
m
2
M
;
x
2
X

m
(
"
)
g
.
Theorem2.0.3.
Assumethat(H1)-(H4)hold.Dependingonr
0
;
B
;
B
1
;
L
;
when
˜;˙;
inf
A
(

)
aresu

cientlysmall,thereexistsaC
J
positivelyinvariantmanifoldW
cs
,whichisgivenasthe
imageofamap
h
:
X
s
(

0
)
!
X
;
forsome

0
>
0
.Themappinghalso
h
(
m
;
x
s
)

 
(
m
)

x
s
2
X
u
m
(

0
)
;
andsocanbeviewedasagraphoverthebundleX
s
(

0
.
Furthermore,itholdsthat,foranym
0
2
M,thereexists
Ÿ
h
:
X
c
m
0
(

0
)

X
s
m
0
(

0
)
!
X
u
m
0
(

0
)
,so
8
that
f
h
(
m
;
x
s
):
m
2
B
c
(
m
0
;
r
0
)
\
 

1
(
B
(
 
(
m
0
)
;

0
4
))
;
x
s
2
X
s
m
(

0
4
)
g
ˆ
 
(
m
0
)
+
graph

Ÿ
h
j
X
c
m
0
(

0
2
)

X
s
m
0
(

0
2
)

ˆf
h
(
m
;
x
s
):
m
2
B
c
(
m
0
;
r
0
)
\
 

1
(
B
(
 
(
m
0
)
;
0
))
;
x
s
2
X
s
m
(

0
)
g
:
Remark
2.0.4
.
Theorem2.0.3isabriefstatementoftheresultin[19].Inapplications,wewilluse
Theorem4.2in[19],whichisapreciseandrigorousversion.
IfwewanttoextendTheorem2.0.3tothecaseofaw
T
t
,thenweneedfurtherassump-
tions[19]:
(H5)
(1)Conditions(H1)-(H4)holdfor
 
(
M
)and
T
t
0
forsome
t
0
,
(2)Thereexistsaninteger
k

0,suchthatforany
>
0,thereexists
>
0,suchthatforany
x
2
B
(
 
(
M
)
;
r
)and
t
2
[
kt
0
;
kt
0
+

],wehave
j
T
t
(
x
)

T
kt
0
(
x
)
j
<
.
Thenextconceptisthatofapproximatelynormallyhyperbolicovwinginvariantmanifold.
Theresultsarebasicallyparalleltothecaseofapproximatelywinginvariantmanifolds.
2.0.5.
Animmersedmanifold
 
(
M
)issaidtobeapproximatelyovwinginvariant
under
T
ifthefollowingconditionshold:
1Thereexistsarelativelyopensubset
M
1
ˆ
M
,ahomeomorphism
v
:
M
!
M
1
,and
>
0such
that
j
T
(
 
(
v
(
m
)))

 
(
m
)
j
<
,forall
m
2
M
;
2Thereexists
r
0
2
(0
;
1)suchthat
 
(
B
c
(
m
0
;
r
0
))isclosedin
X
forany
m
0
2
v
(
M
),where
B
c
(
m
0
;
r
0
)istheconnectedcomponentof
 

1
(
B
(
 
(
m
0
)
;
r
0
))containing
m
0
.
9
Inadditionto(H1)and(H2),insteadof(H3),weassumethefollowingapproximatenormal
hyperbolicityconditions.
(C3)Thereexist
a
;
2
(0
;
1)suchthat,forany
m
1
2
M
,if
m
0
=
v
(
m
1
),and

2f
c
;
u
g
,

2f
c
;
s
;
u
g
,
with

,

,then
1.
k


m
1
DT
(
 
(
m
0
))
j
X

m
0
k
˙
,
k
(

c
m
1
DT
(
 
(
m
0
))
j
X
c
m
0
)

1
k

1
>
a
2.

k
(

u
m
1
DT
(
 
(
m
0
))
j
X
u
m
0
)

1
k

1
>
1,
3.
k

s
m
1
DT
(
 
(
m
0
))
j
X
s
m
0
k
<
min
f
1
;
k
(

c
m
1
DT
(
 
(
m
0
))
j
X
c
m
0
)

1
k

J
g
.
Theorem2.0.6.
Assumethat(H1),(H2),(C3),and(H4)hold.Dependingonr
0
;
B
;
B
1
;
L
;
when
˜;˙;
inf
A
(

)
aresu

cientlysmall,thereexistsaC
J
negativelyinvariantmanifoldW
cu
,which
isgivenastheimageofamap
h
:
X
u
(

0
)
!
X
;
forsome

0
>
0
.Themappinghalso
h
(
m
;
x
u
)

 
(
m
)

x
u
2
X
s
m
(

0
)
:
10
Chapter3
Globalinvariantmanifoldsofboundary
dropletsforthe2-Dmass-conserving
Allen-Cahnequation
3.1Introduction
Weconsiderthetwo-dimensionalmassconservingAllen-Cahnequation,
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
˚
‹
"
t
(
y
;
t
)
=
‹
"
2

y
˚
‹
"
(
y
;
t
)

f
(
˚
‹
"
(
y
;
t
))
+
>

f
(
˚
‹
"
(

;
t
))
;
y
2

;
t
>
0
;
@
n
˚
‹
"
(
y
;
t
)
=
0
;
y
2
@

;
t
>
0
;
(3.1.1)
where

ˆ
R
2
isaedboundeddomainwithsmoothboundary
@

,
@
n
istheexteriornormal
derivativeto
@

,

y
representstheLaplacianwithrespecttoy,and
>

=
1
j

j
R

meanstheaverage
over

.Here
f
isthederivativeofadoublewellpotential
W
.Weassumethefollowingconditions
for
f
2
C
1
(
R
):
f
(

1)
=
0
;
f
0
(

1)
>
0
;
Z
s

1
f
=
Z
s
1
f
>
0forall
s
2
(

1
;
1)
:
(3.1.2)
11
Figure3.1:Graphofthefunctionf
(3.1.1)canbeconsideredastheassociated
L
2
gradientwofthefunctional
J
‹
"
(
u
)
=
Z

(
‹
"
2
2
jr
u
j
2
+
W
(
u
))
dx
;
u
2f
v
2
H
1
(

):
?

vdx
=
m
g
:
(3.1.3)
Thisfunctionalhasbeeninvestigatedbyseveralauthors,forexample,[5,9,27,28,30,33,38,69,
70,77,85].
In[5],N.D.Alikakosetal.constructedanapproximatelyinvariantmanifoldfor(3.1.1)using
acarefullydevisedasymptoticexpansion.Eachelementofthemanifoldisaso-calleddroplet,
orbubble,thatis,astatehavingaroughlysemicircularinterfaceattachedtotheboundaryofthe
domain,theinterfaceseparatingregionswherethesolutiontakesontwodi

erentalmostconstant
values.Thesedropletsmoveslowlytowardstheincreasinglycurvedregion,whilemaintaining
theirshape.ThemotionofthecenterofthebubblecanbedeterminedbythefollowingODE,
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
d
‹
˘
dt
=

4
‹
˘
2

3
ˇ
K
0

(
‹
˘
(
t
))
+
O
(
‹
˘
2

2
)
;
‹
˘
(0)
=
‹
˘
0
;
(3.1.4)
where
‹
˘
isthearc-lengthparameterof
@

whichrepresentsthecenterofthebubble.Moredetails
canbefoundinSection3.2.Moreover,theyprovedthatthebubbleshapeisstable,thatis,ifwe
12
startfromasmall
H
1
-neighborhoodofsize
O
(
"
)ofthebubblesolution,thenthewwillstay,for
positivetime,inasmallneighborhoodofthemanifoldofbubblestates,inthe
H
1
sense.Because
ofthedi

cultyinhandlingthecontactwiththeboundaryofthedomaininhigherdimensions,in
[5],theauthorsconsideredonlythetwo-dimensionalcase.Inhigherdimension,therearesome
resultsfortheinteriorbubbles,see[6,7].Inthosepapers,N.AlikakosandG.Fuscoconsidered
bubblesolutionsforCahn-Hilliardequation(themass-conservingAllen-Cahnequationwillpro-
ducesimilardynamics).Roughlyspeaking,theinterfaceofthebubblehasconstantcurvatureand
itmovestowardstheboundaryataexponentiallysmallspeed,retainingitsshapeuntilitgetsclose
totheboundary.Onceneartheboundary,itisconjecturedthatthebubblequicklyadheretothe
boundary,itsenergyroughlydroppingbyhalf,andthenfollowingthedynamicsdiscussedhere.
Figure3.2:Fourstagesintheevolutionforatwo-dimensionaldomain

.Thelaststageisthe
objectofstudyinthispaper.
Inthisproject,weusetheframeworkof[19]toconstructatrueinvariantmanifoldfor(3.1.1),
whichisclosetotheapproximatelyinvariantmanifoldgivenin[5].Theofapproxi-
mateinvariancyin[5]isdi

erentfromthein[19].In[5],amanifoldisapproximately
invariantiftheequation,evaluatedatapoint(i.e.,function)ofthemanifold,isuptoa
smallerror.In[19],approximateinvariancymeansthemanifoldisapproximatelyinvariantunder
13
thesolutionmapforaedtime(seeSection2.1formoredetails).Hence,ourmaintaskisto
provethattheapproximatelyinvariantmanifoldconstructedin[5]theconditionsinthe
givenin[19].Notethatoncewehaveobtainedtheglobalinvariantmanifold,bubble
solutionsonitexistgloballyintime,forwardandbackward,beingeitherstationaryorconnecting
stableandunstableequilibria.
Thischapterisorganizedasfollows,inSection3.2wegivesomebackgroundontheconstruc-
tionoftheapproximatebubblesolution.InSection3.3,weprovetheexistenceofatrueglobal
invariantmanifoldofbubblestatesfor(3.1.1).Finally,wewilldiscussthedynamicsofthebub-
ble,whichincludesthemotionofthebubbleinforwardandbackwardtime,andthelocationof
equilibriumbubblestates.
3.2Approximatebubblesolutionforthemass-conservingAllen-
Cahnequation
Inthissection,wewillintroducetheapproximatebubblesolutionsof(3.1.1),whichwerecon-
structedbyN.D.Alikakosetal.in[5].Roughlyspeaking,eachhasasemicircularinterface

,
whichisthezerolevelset,withsmallradius

.Thesolutionisalmost

1insidetheinterface,
andalmost
+
1outside.Thisstatethenmovesalongtheboundaryofthedomainaccordingtoa
one-dimensionaldynamicalsystem.Nowwegiveamoredetaileddescription.First,weintroduce
achangeofvariablesthatesthesizeofthebubblewhile

isvaried.
y
=

x
;
‹
"
=
;
u
"
(
x
;
t
)
=
˚
‹
"
(
y
;
t
)
;


=


1

:
=
f
x
;

x
2

g
:
(3.2.1)
Thenwecanwrite(3.1.1)as
14
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
u
"
t
(
x
;
t
)
=
"
2

u
"
(
x
;
t
)

f
(
u
"
(
x
;
t
))
+
>


f
(
u
"
(

;
t
))
;
x
2


;
t
>
0
;
@
n
u
"
(
x
;
t
)
=
0
;
x
2
@


;
t
>
0
:
(3.2.2)
Weparameterize
@


by
z

(
˘
),where
˘
isthearc-lengthof
@


measuredfromsomeed
pointof
@


.Weareseekinganinvariantmanifold
Ÿ
M
consistingofbubble-likefunctions
u
(

;˘;"
),
parameterizedby
˘
,whichisthecenteroftheapproximatelysemicircularinterface.Obviously,
Ÿ
M
isone-dimensional.Theinvariancemeansthatthevectoristangenttothemanifold,sofor
u
2
Ÿ
M
wecanwrite(3.2.2)analyticallyintheform:
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
:

"
2

u
+
f
(
u
)
+
"
2
cu
˘
+
"˙
=
0
;
x
2


;
t
>
0
;˘
2
R
1
@
n
u
(
x
;˘;"
)
=
0
;
x
2
@


;
t
>
0
;
R


u
(

;˘;"
)
=
j


j
ˇ:
(3.2.3)
Here
˙
=
˙
(
˘;"
)and
c
=
c
(
˘;"
)areconstantsin
x
andfollowing[5]wehavemultipliedbypowers
of
"
inanticipationoftheirsize.Wecall
c
thespeedofthedroplet,and
"˙
=
>


f
(
u
(

;˘;"
))
dx
adjustsforthemassconstraint.Themotionofthebubblecanberepresentedbythemotionofthe
center
˘
,which
d
˘
(
t
;"
)
dt
=
"
2
c
(
˘;"
)
:
(3.2.4)
Figure3.3:Geometryofthebubble
15
Consideringthefactthatourequationisthemass-conservinggradientwoftheenergyfunc-
tional(3.1.3),heuristicallywecanseethatasymptotically,solutionsshouldbealmostconstant
takingonvalues

1everywhereexceptforane

cienttransitionbetweenthosevalues,dictated
bythepredeterminedaveragevalue.Tomakethetransitione

cient,itshouldtakeplacealonga
minimalcurveenclosingagivenareaat
@


,thatis,acirculararcintersectingthedomainbound-
aryorthogonally.Furthermore,thetransitionshouldhavewidth
O
(
"
)sothatthegradientandbulk
partsof(3.1.3)arealmostequal.Sothatthedynamicsofthebubblestatearedeterminedlocally,
werequirethebubbletohavesmallradius
<<
1inoriginalcoordinatedsand1inourexpanded
coordinates,thusweourmasstobe
j


j
ˇ
.Inordertorigorouslyperformtheasymptotic
analysisin
"
,oneneeds0
<"<<
.
Byperforminganouterexpansion,aninnerexpansion,andacornerexpansion(wherethe
interfacemeetstheboundaryofthedomain),andpatchingthesetogether,in[5]theauthorscon-
structedanapproximatesolutiontosystem(3.2.3)havingbubble-likestructure.Thissolutionis
parameterizedby
c
;˙
,thelengthoftheinterface,
j

j
,thecurvatureoftheinterface,
K
,andthe
arc-lengthfromthecenter,
z

(
˘
),to
f
p

g
theintersectionof

and
@


(seeFigure4.).Invariance
ofthefamilywithrespecttothenonlocalparabolicequationandthemassconstraintuptoaspec-
orderdictatedcertainsolvableequationsforthesegeometricparameters.Thustheyfoundan
approximatesolutionfor(3.2.3)givenbythefollowing:
Theorem3.2.1.
[5]Assumethat

and
"
aresmallparameterssatisfying
"

1
2
C

1

2
,whereC

1
is
aconstantby(2.65)in[5].Thenforanypositiveintergerk,if
"
issu

cientlysmall,there
16
existu
=
u
(
x
;˘;"
)
;˙
=
˙
(
˘;"
)
;
c
=
c
(
˘;"
)
suchthat
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
"
2

u

f
(
u
)
+
"˙
=
"
2
cu
˘
+
O
(
"
k
)
in


;
@
n
u
=
0
on
@


;
R


u
(

;˘;"
)
=
j


j
ˇ;
"˙
=
>


f
(
u
(

;˘;"
)
dx
:
(3.2.5)
Remark
3.2.2
.
1.In[5],Theorem3.2.1requiresthat
"


m
forsome
m

2.Bycarefully
checkingtheproof,wefoundthatthisconditionisnotnecessary.
2.Infactwhen

issmall,therearetwoquasi-steadystates,onebeingthedropletwithan
interfaceseparatingregionswhereitisapproximately
+
1and

1,theotherbeingaspike
state.Theformershapeisstableandthelaterisunstable.As

becomes
O
(
"
),thedroplet
andspikemergeandcausetoexistforsmaller

.
3.Fortheinnerexpansion,neartheinterface,theauthorsusethecoordinates(
r
;
s
),where
r
isthesigneddistancefromtheinterface,whichispositiveoutsidethebubbleandnegative
inside,and
s
isthearc-lengthalongtheinterface.Theleadingorderoftheinteriorexpansion
istheheteroclinicsolutionto
¨
U

f
(
U
)
=
0
;
U
(

)
=

1
;
Z
1

R

U
2
(
R
)
dR
=
0
;
(3.2.6)
inthestretchedvariable
R
=
r

:
4.Theleadingtermoftheouterexpansionis

1,andthecornerexpansionis
O
(
"
)andis
exponentiallydecayingawayfromtheinterface.
17
5.Theleadingtermof
j

j
is
ˇ
,andtheleadingtermofthecurvature,
K
,of

is1,whichimplies
thattheinterfaceisapproximatelyasemicircle.
Now,westatesomeresultsofthespectralanalysisfortheoperatorobtainedbylinearizingat
oneofthesebubblestates.Considerthefollowingeigenvalueproblem:
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
:
L
¯
˚
:
=
"
2

¯
˚

f
0
(
u
)
¯
˚
+
>


f
0
(
u
)
¯
˚
=
¯

¯
˚
in


;
@
n
¯
˚
=
0on
@


;
R


¯
˚
=
0
;
(3.2.7)
where
u
isthesolutionto(3.2.5).Thelargesteigenvalueisoftheorder

2
,whichisveryclose
tozerosincebothfactorsaresmall,andthecorrespondingeigenfunctioniscloseto
u
˘
.More
precisely,thelargesteigenvalueis
4
"
2
3
ˇ
d
2
K


d
˘
2
(
˘
)
+
O
(
"
2

4
),where
K


isthecurvatureofthe
boundary
@


.Therestofthespectrumisnegativebutisonly
O
(

2
)awayfromzero.Theprecise
estimateisgivenbythefollowingtheoremfrom[5].
Theorem3.2.3.
[5]Letu
=
u
(
x
;˘;"
)
bethesolutionto(3.2.5).IfforlargeconstantC

,

2
>
C

"
holds,thenforanyv
2
H
1
(


)
satisfying
Z


v
=
0
;
Z


vu
˘
=
0
;
(3.2.8)
wehave
h
Lv
;
v
i
2
"
2
ˇ
2
j

j
2
Z


v
2
:
(3.2.9)
Thus,thereisagapbetweenthelargesteigenvalueandtheothersbecausethelargesteigenvalue
is
4
"
2
3
ˇ
d
2
K


d
˘
2
(
˘
)
+
O
(
"
2

4
),where
d
2
K


d
˘
2
(
˘
)isoforder

,and

isverysmall.
18
3.3Invariantmanifoldsofboundarydroplets
3.3.1Constructionof
M
"
Wechoosethespace
X
as
H
1
(


)withnormgivenby
j
u
j
2
X
=
j
"
r
u
j
2
L
2
+
j
u
j
2
L
2
.First,wemodifythe
function
f
tomakesurethattheevolutionawgloballyintime.Thus,weconsider
u
"
t
(
x
;
t
)
=
"
2

u
"
(
x
;
t
)

Ÿ
f
(
u
"
(
x
;
t
))
+
?


Ÿ
f
(
u
"
(

;
t
))
;
(3.3.1)
where
Ÿ
f
(
u
)
=

(
u
)
f
(
u
).Here,

(
s
)

0isa
C
1
bumpfunctionsatisfying

(
s
)
=
1
;
j
s
j
2;

(
s
)
=
0
;
j
s
j
4
:
(3.3.2)
Notethat
j
Ÿ
f
j
C
m
(
R
)
<
1
.Thisdoesnota

ectthebubblesolutionweseekbecause
thatsolutionhasitsrangein[

1
;
1].Forconvenience,wekeepthenotation
f
,insteadof
Ÿ
f
.
Let
W
(
x
;˘;"
)bethesecondorderapproximationofthesolutionto(3.2.5)givenbyTheorem
3.2.1,whichmeansthat
W

8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
"
2

W

f
(
W
)
+
"˙
=
"
2
cW
˘
+
O
(
"
3
)in


;
@
n
W
=
0on
@


;
R


W
(

;˘;"
)
=
j


j
ˇ;
"˙
=
>


W
(

;˘;"
)
:
(3.3.3)

 
"
:
@


!
X
as
 
"
(
˘
)
=
W
(
x
;˘;"
).Let
M
"
=
 
"
(
@


).Wewillprovethat
M
"
isan
approximatelynormallyhyperbolicinvariantmanifold,sothatwecanapplytheTheorems2.0.3
19
and2.0.6.
3.3.2
M
"
isapproximatelyinvariant
Onemayexpectthatthewof(3.2.2)startingfrom
W
(
x
;˘;"
)willstaycloseto
W
(
x
;˘
(
t
)
;"
),
where
˘
(
t
)(3.2.4).Infact,wehave
Lemma3.3.1.
ThereexistsC
>
0
suchthat,foranysmall
"
andz

(
˘
)
2
@

,thesolutionu
(
t
;
x
;"
)
of(3.3.1)withinitialdatau
(0
;
x
;"
)
=
 
"
(
˘
)

j
u
(
t
;
x
;"
)

 
"
(
˘
(
t
))
j
X

C
"
3
e
Ct
:
(3.3.4)
Proof.
Let
v
=
u

 
"
,then
v

8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
v
t
=
"
2

v

(
f
(
 
"
+
v
)

f
(
 
"
))
+
>


[
f
(
 
"
+
v
)

f
(
 
"
)]
dx
+
O
(
"
3
)
;
v
(0
;

)
=
0
:
(3.3.5)
Let
g
(
v
)

f
(
 
"
+
v
)

f
(
 
"
).Rewrite(3.3.5)as
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
v
t
=
L
"
v

g
(
v
)
+
>


g
(
v
)
dx
+
O
(
"
3
)
;
v
(0
;

)
=
0
(3.3.6)
where
L
"
=
"
2

,withhomogeneousNeumannboundarycondition.
20
Usingthevariationofconstantsformula,wehave
v
=
Z
t
0
e
L
"
(
t

s
)


g
(
v
)
+
?


g
(
v
)
dx
+
O
(
"
3
)

ds
:
(3.3.7)
Itiswellknownthat
L
"
generatesacontractionsemigroup,andrecallthat
f
hasbeenso
wehave
j
g
(
v
)
j
C
j
v
j
and
jr
g
(
v
)
j
=
j
g
0
(
v
)
r
v
j
C
jr
v
j
,whichimplies
j
v
(

;
t
)
j
X

Z
t
0

C
j
v
j
X
+
O
(
"
3
)

ds
:
(3.3.8)
ApplyingGronwall'sinequalityto
j
v
j
X
,wehave
j
v
j
X

C
"
3
e
Ct
:
(3.3.9)

Lemma3.3.1impliesthat
M
"
isanapproximatelyovwinginvariantmanifoldfor
T
being
thetime
t
0
solutionoperator,bytakingthefunction
v
in2.0.5as
˘
(

t
0
;

).
3.3.3Splittingalongthemanifold
M
"
FromthespectralanalysismentionedinSection2.2,wecansplitthespace
X
along
M
"
naturally.
Forany
z

(
˘
)
2
@


,let
X
c
";˘
=
T
 
"
(
˘
)
M
"
=
span
f
W
˘
(
x
;˘;"
)
g
;
X
s
";˘
=
f
v
2
X
:
Z


v
Ÿ
v
=
0
;
forallŸ
v
2
X
c
";˘
g
:
(3.3.10)
Let


";˘
;
=
c
;
s
betheprojectionsassociatedwiththissplitting.
21
Lemma3.3.2.
k


";˘
k
isuniformlyboundedandsmoothlydependson
˘
.
Proof.
Forany
x
2
X
,wecanwrite
x
=
x
c
+
x
s
,where
x

2
X

";˘
.Since
x
c
?
L
2
x
s
,
j
x
j
2
L
2
=
j
x
c
j
2
L
2
+
j
x
s
j
2
L
2
.Since
X
c
";˘
isdimensional,wehave
C
j
x
c
j
X
j
x
c
j
L
2
j
x
j
L
2
j
x
j
X
;
and
j
x
j
X
j
x
s
j
X
j
x
c
j
X
j
x
s
j
X

1
C
j
x
j
X
(3.3.11)
forsomeconstant
C
independentof
"
,becauseofourchoiceofnormon
X
.
Bycarefullycheckingtheconstructionoftheapproximatebubblesolutionsin[5],wethat
everytermintheasymptoticexpansioncomesfromsolvingacertainellipticequationwhichgives
thesolutionhighregularity.Likewise,
˙
and
c
arealsosmoothfunctionsof
˘
(seesection2.2and
2.3of[5]andtheappendixof[8]).Thisimpliesthat
 
"
(
@

)isatleasta
C
2
smoothmanifold.
Thentheuniformboundednessfollowsfromtheusualcompactnessargumentandthesmooth
dependencefollowsfromthesmoothnessof
 
"
(
˘
).

Thethirdinequalityin(H2)isautomaticallyforcompactmanifolds(theproofcan
befoundin[11]).Sofar(H1)and(H2)in2.0.2havebeenestablished.Weneedto
provethat
M
"
isapproximatelynormallyhyperbolicasanapproximatelyovwinginvariant
manifold,thatis,(C3)holds.
3.3.4
M
"
isapproximatelynormallyhyperbolic
Fromthesplittingofthespace,wecanseethatthewholespace
X
isthecenter-stablemanifold
W
cs
,becausethereisnounstablesubspace.Hence,weonlyneedtothecenter-unstableman-
22
ifold,whichisactuallyjustthecentermanifold.Sinceweneedtostudythelinearizedw,we
considerthelinearizedsystem.
Let
L
"
v
=
"
2

v

f
0
(
u
)
v
+
>


f
0
(
u
)
vdx
,where
u
isthesolutionto(3.2.2)withinitialdata
W
(
x
;˘;"
)
from(3.3.3)forsomeed
˘
.
Let
W
(
t
;

)bethesolutionto
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
W
t
=
L
"
W
;
W
(0
;

)
=
W
(

;˘;"
)
:
(3.3.12)
Let
e
L
"
v
=
"
2

v

f
0
(
 
"
(
˘
(
t
)))
v
+
>


f
0
(
 
"
(
˘
))
vdx
,and
e
W
(
t
;

)bethesolutionto
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
e
W
t
=
e
L
"
e
W
;
e
W
(0
;

)
=
W
(

;˘;"
)
:
(3.3.13)
Lemma3.3.3.
j
W
(
t
;

)

e
W
(
t
;

)
j
X

C
"
3
j
W
(

;˘;"
)
j
X
e
Ct
,forsomeconstantC
Proof.
Thedi

erence
v
=
W

e
W

8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
v
t
=
"
2

v

f
0
(
 
"
(
˘
(
t
))
v
+
g
(
u

 
"
(
˘
(
t
))
W
+
>



f
0
(
 
"
(
˘
))
v

g
(
u

 
"
(
˘
(
t
))
W

dx
;
v
(0
;

)
=
0
;
(3.3.14)
where
g
=
f
0
(
u
)

f
0
(
 
"
(
˘
(
t
))).
23
ByanargumentsimilartotheproofofLemma3.3.1,weget
j
W
(
t
;

)
j
X
j
W
(

;˘;"
)
j
X
e
Ct
:
(3.3.15)
Usingthevariationofconstantsformula,wehave
v
=
Z
t
0
e
L
"
(
t

s
)
[

f
0
(
 
"
(
˘
(
t
))
v
+
g
(
u

 
"
(
˘
(
t
))
W
+
?


(
f
0
(
 
"
(
˘
))
v

g
(
u

 
"
(
˘
(
t
))
W
)
dx
]
ds
:
(3.3.16)
UsingLemma3.3.1and(3.3.15),wehave
j
v
(

;
t
)
j
X

Z
t
0

C
j
v
(

;
s
)
j
X
+
C
"
3
j
W
(

;˘;"
)
j
X
e
Cs

ds
:
(3.3.17)
ApplyingGronwall'sinequalitygivesthedesiredresult.

Nextwestudythebehaviorof
e
W
(
t
;

)incenterandstabledirections.Write
W
(

;˘;"
)as
W
(

;˘;"
)
=
a
(0)
W
˘
(
x
;˘;"
)
+
W
s
(
x
;˘;"
)andsimilarly,
e
W
(
t
;

)
=
a
(
t
)
W
˘
(
x
;˘
(
t
)
;"
)
+
W
s
(
x
;˘
(
t
)
;"
),
where
W
s
(
x
;˘
(
t
)
;"
)
2
X
s
";˘
(
t
)
.
Lemma3.3.4.
Ifa
(0)
=
0
,i.e.,W
(

;˘;"
)
2
X
s
";˘
,then
j
a
(
t
)
jj
W
˘
(
x
;˘
(
t
)
;"
)
j
X

C
"
1
=
2
e
Ct
j
W
(

;˘;"
)
j
X
:
(3.3.18)
Proof.
Bydi

erentiatingwithrespectto
˘
in(3.3.3),wehave
e
L
"
W
˘
(
x
;˘
(
t
)
;"
)
=
"
2
cW
˘˘
(
x
;˘
(
t
)
;"
)
+
"
2
c
˘
W
˘
(
x
;˘
(
t
)
;"
)
+
O
(
"
2
)
:
(3.3.19)
Onethattheresidual
O
(
"
3
)in(3.3.3)becomes
O
(
"
2
),becausetakingaderivativenearthe
24
interfacegeneratesafactorof
1
"
.Itiseasytoseethatneartheinterface
W
˘
isoforder
1
"
and
W
˘˘
isoforder
1
"
2
.Alsonotethatthewidthofthelayeris
O
(
"
),so
j
W
˘
j
L
2

C
"

1
=
2
and
j
W
˘˘
j
L
2

C
"

3
=
2
,whichimpliesthat
j
e
L
"
W
˘
(
x
;˘
(
t
)
;"
)
j
L
2

C
"
1
=
2
:
(3.3.20)
Foradetailedproofofthesefacts,usetheexpressionfor
u
˘
fromp.294of[5]andthederivative
boundsin[5],Sections2.2and2.3.ByanargumentsimilartotheproofofLemma3.3.1,wehave
j
e
W
j
L
2

Ce
Ct
j
W
(

;˘;"
)
j
L
2
forsomeconstantC,whichimpliesthat
j
a
(
t
)
jj
W
˘
j
L
2
;
j
W
s
(
x
;˘
(
t
)
;"
)
j
L
2

Ce
Ct
j
W
(

;˘;"
)
j
L
2
:
(3.3.21)
Since
e
L
"
isself-adjointand
e
L
"
e
W
=
e
W
t
,wehave
h
e
W
(
t
;

)
;
e
L
"
a
(
t
)
W
˘
(
x
;˘
(
t
)
;"
)
i
=
h
e
W
t
(
t
;

)
;
a
(
t
)
W
˘
(
x
;˘
(
t
)
;"
)
i
=
h
a
0
(
t
)
W
˘
(
x
;˘
(
t
)
;"
)
+
a
(
t
)
W
˘˘
(
x
;˘
(
t
)
;"
)

˘;
a
(
t
)
W
˘
i
+
h
W
s
t
;
a
(
t
)
W
˘
i
=
a
(
t
)
a
0
(
t
)
h
W
˘
;
W
˘
i
+
1
2
a
2
(
t
)
d
dt
h
W
˘
;
W
˘

a
(
t
)
h
W
s
(
x
;˘
(
t
)
;"
)
;
W
˘˘


˘
i
=
d
dt
(
a
2
(
t
)
2
j
W
˘
j
2
L
2
)

a
(
t
)
h
W
s
(
x
;˘
(
t
)
;"
)
;
W
˘˘


˘
i
(3.3.22)
Forthethirdidentity,weusethefactthat
h
W
˘
(
x
;˘
(
t
)
;"
)
;
W
s
(
x
;˘
(
t
)
;"
)
i
=
0.Notethat,fromp.294
of[5],
j
W
˘
j
L
2

C

1
"

1
=
2
,whichcombinedwith(3.3.21),gives
j
a
(
t
)
j
C
"
1
=
2
e
Ct
j
W
(

;˘;"
)
j
L
2
:
(3.3.23)
25
Combine(3.3.23),(3.3.22),(3.3.21),(3.3.20)anduse

˘
=
"
2
c
toobtain
d
dt
(
a
(
t
)
2
j
W
˘
j
2
L
2
)

4
C
2
"
e
2
Ct
j
W
(

;˘;"
)
j
2
L
2
;
(3.3.24)
whichimpliesthat
a
2
(
t
)
j
W
˘
(
x
;˘
(
t
)
;"
)
j
2
L
2

a
2
(0)
j
W
˘
(
x
;˘
(0)
;"
)
j
2
L
2

2
C
"
(
e
2
Ct

1)
j
W
(

;˘;"
)
j
2
L
2
:
(3.3.25)
Thus,if
a
(0)
=
0,i.e.,
W
(

;˘;"
)
2
X
s
";˘
,thenwehave
j
a
(
t
)
jj
W
˘
(
x
;˘
(
t
)
;"
)
j
L
2

C
"
1
=
2
e
Ct
j
W
(

;˘;"
)
j
L
2
:
(3.3.26)
Since
X
c
";˘
isdimensional,wehave
j
a
(
t
)
jj
W
˘
(
x
;˘
(
t
)
;"
)
j
X

C
"
1
=
2
e
Ct
j
W
(

;˘;"
)
j
X
:
(3.3.27)

Notethat
˘
(
t
)isgivenbyanODE,sowecanconsider
e
W
(

t
;

)
=
a
(

t
)
W
˘
(
x
;˘
(

t
)
;"
)
+
W
s
(
x
;˘
(

t
)
;"
).FollowingananalagousargumenttothatinLemma3.3.4,wehaveif
W
s
(
x
;˘;"
)
=
0,then
a
2
(

t
)
j
W
˘
(
x
;˘
(

t
)
;"
)
j
2
L
2

a
2
(0)
j
W
˘
(
x
;˘;"
)
j
2
L
2

2
C
"
e
Ct
a
2
(0)
j
W
˘
(
x
;˘;"
)
j
2
L
2
;
(3.3.28)
26
whichimpliesthat
j
a
(
t
)
jj
W
˘
(
x
;˘
(
t
)
;"
)
j
X

(
C
1
+
C
"
e
Ct
)
1
=
2
j
a
(0)
jj
W
˘
(
x
;˘;"
)
j
X
:
(3.3.29)
Wealsohavetheestimate:
Lemma3.3.5.
Ifa
(0)
=
0
,i.e.,W
(

;˘;"
)
=
W
s
(
x
;˘;"
)
2
X
s
";˘
,then
j
W
s
(
x
;˘
(
t
)
;"
)
j
L
2

(
e

bt
+
C
"
e
Ct
)
j
W
s
(
x
;˘;"
)
j
L
2
;
(3.3.30)
and
j
"
r
W
s
j
L
2

C
(
e

bt
+
"
e
Ct
+
"
3
2
e
Ct
(
e

bt
+
C
"
e
Ct
)
1
2
)
j
W
s
(
x
;˘;"
)
j
L
2
;
(3.3.31)
whereb
=
2
ˇ
2
j

j
2
,comingfrom(3.2.9).
IfW
s
(0
;

)
=
0
,i.e.,W
(

;˘;"
)
=
a
(0)
W
˘
(
x
;˘;"
)
2
X
c
";˘
,then
j
W
s
(
x
;˘
(
t
)
;"
)
j
L
2

C
"
e
Ct
j
a
(0)
jj
W
˘
(
x
;˘;"
)
j
L
2
;
(3.3.32)
and
j
"
r
W
s
j
L
2

C
"
e
Ct
j
a
(0)
jj
W
˘
(
x
;˘;"
)
j
L
2
:
(3.3.33)
Proof.
Withthedecompositionof
e
W
givenpriortoLemma3.3.4,wewrite(3.3.13)as
a
0
(
t
)
W
˘
(
x
;˘
(
t
)
;"
)
+
a
(
t
)
W
˘˘
(
x
;˘
(
t
)
;"
)

˘
+
W
s
t
(
x
;˘
(
t
)
;"
)
=
e
L
"
(
a
(
t
)
W
˘
(
x
;˘
(
t
)
;"
)
+
W
s
(
x
;˘
(
t
)
;"
))
:
(3.3.34)
27
Using(3.3.19),weget
a
0
(
t
)
W
˘
(
x
;˘
(
t
)
;"
)
+
W
s
t
=
e
L
"
(
W
s
(
t
))
+
"
2
a
(
t
)
c
˘
W
˘
(
x
;˘
(
t
)
;"
)
+
a
(
t
)
O
(
"
2
)
:
(3.3.35)
Takingtheinnerproductwith
W
s
andusing
W
s
t
=
W
s
˘

˘
,wehave
d
d
˘
(
j
W
s
(
t
)
j
2
L
2
)
"
2
c
=
2
h
e
L
"
W
s
;
W
s
i
+
a
(
t
)
h
O
(
"
2
)
;
W
s
i
;
(3.3.36)
Notethatfromthecalculationsgivenin[5],the
O
(
"
2
)termisoforder
O
(
"
2
)neartheinterface,but
oforder
O
(
"
3
)awayfromtheinterface,soits
L
2
normis
O
(
"
5
2
).If
a
(0)
=
0,wemayuseTheorem
3.2.3,Lemma3.3.4,and(3.3.21)toobtainthatforsomepositiveconstants
b
and
C
(whichmay
changefromlinetoline),
d
d
˘
(
j
W
s
(
x
;˘
(
t
)
;"
)
j
2
L
2
)

b
j
W
s
(
x
;˘
(
t
)
;"
)
j
2
L
2
+
C
"
e
Ct
j
W
s
(
x
;˘;"
)
j
L
2
j
W
s
(
x
;˘
(
t
)
;"
)
j
L
2
;
(3.3.37)
whichimpliesthat
j
W
s
(
x
;˘
(
t
)
;"
)
j
L
2

(
e

bt
+
C
"
e
(
b
+
C
)
t
b
+
C
)
j
W
s
(
x
;˘;"
)
j
L
2
;

(
e

bt
+
C
"
e
Ct
)
j
W
s
(
x
;˘;"
)
j
L
2
:
(3.3.38)
If
W
s
(0
;

)
=
0,wemayuseTheorem3.2.3,(3.3.25)and(3.3.21)toobtainthatforsomeconstant
b
and
C
,
d
d
˘
(
j
W
s
(
x
;˘
(
t
)
;"
)
j
2
L
2
)

b
j
W
s
(
x
;˘
(
t
)
;"
)
j
2
L
2
+
C
"
e
Ct
j
a
(0)
jj
W
˘
(
x
;˘;"
)
j
L
2
j
W
s
(
x
;˘
(
t
)
;"
)
j
L
2
;
(3.3.39)
28
whichimpliesthat
j
W
s
(
x
;˘
(
t
)
;"
)
j
L
2

C
"
e
Ct
b
+
C
j
a
(0)
jj
W
˘
(
x
;˘;"
)
j
L
2
;

C
"
e
Ct
j
a
(0)
jj
W
˘
(
x
;˘;"
)
j
L
2
:
(3.3.40)
If
a
(0)
=
0,notethat
R


W
s
dx
=
0,(3.3.36)gives
2
h
e
L
"
W
s
;
W
s
i
=

d
dt
(
j
W
s
j
2
L
2
)
+
a
(
t
)
h
O
(
"
2
)
;
W
s
i
;

2
Z


"
2
jr
W
s
j
2
+
f
0
(
W
)
j
W
s
j
2
dx
=

d
dt
(
j
W
s
j
2
L
2
)
+
a
(
t
)
h
O
(
"
2
)
;
W
s
i
:
(3.3.41)
Using(3.3.37)and(3.3.38)in(3.3.41)gives
"
2
jr
W
s
j
2
L
2

Z


j
f
0
(
W
)
jj
W
s
j
2
dx
+
b
"
2
j
W
s
j
2
L
2
+
C
"
3
e
Ct
j
W
s
(
x
;˘;"
)
j
L
2
j
W
s
(
x
;˘
(
t
)
;"
)
j
L
2
;


(
C
+
b
"
2
)(
e

bt
+
C
"
e
Ct
)
2
+
C
"
3
e
Ct
(
e

bt
+
C
"
e
Ct
)

j
W
s
(
x
;˘;"
)
j
2
L
2
;
(3.3.42)
whichimpliesthat
j
"
r
W
s
j
L
2

C
(
e

bt
+
"
e
Ct
+
"
3
2
e
Ct
(
e

bt
+
C
"
e
Ct
)
1
2
)
j
W
s
(
x
;˘;"
)
j
L
2
:
(3.3.43)
If
W
s
(
x
;˘;"
)
=
0,wemaycombine(3.3.36),(3.3.39)and(3.3.40)toobtain
29
"
2
jr
W
s
j
2
L
2

Z


j
f
0
(
W
)
jj
W
s
j
2
dx
+
b
"
2
j
W
s
j
2
L
2
+
C
"
3
e
Ct
j
a
(0)
jj
W
˘
(
x
;˘;"
)
j
L
2
j
W
s
j
L
2
;

(
C
+
b
"
2
)

C
"
e
Ct

2
j
a
(0)
j
2
j
W
˘
(
x
;˘;"
)
j
2
L
2
+
C
"
3
e
Ct
C
"
e
Ct
j
a
(0)
j
2
j
W
˘
(
x
;˘;"
)
j
2
L
2
;
(3.3.44)
whichimpliesthat
j
"
r
W
s
j
L
2

C
"
e
Ct
j
a
(0)
jj
W
˘
(
x
;˘;"
)
j
L
2
:
(3.3.45)

NowcombiningLemma3.3.2,Lemma3.3.3,Lemma3.3.4,(3.3.29),andLemma3.3.5,gives
(C3)intheofnormalhyperbolicityfor
T
t
0
"
,thetime-
t
0
solutionmapof(3.3.1)pro-
vided
t
0
islargeandwith
"
chosensmallenough.Precisely,

=
C
"
3
e
Ct
0
,
˙
=
C
"
1
2
e
Ct
0
,
a
=
(
C
1
+
C
"
e
Ct
0
)
1
=
2
,and

=
C
(
e

bt
0
+
"
e
Ct
0
+
"
3
2
e
Ct
0
(
e

bt
0
+
C
"
e
Ct
0
)
1
2
).Forinstance,we
choose
t
0
suchthat
Ce

bt
0

1
2
,andchooseany
"

"
(
t
0
)tosatisfyalltheconditions(
C
3).
3.4Dynamicalbubblesolution
Sofarwehaveconstructedtheapproximatelynormallyhyperbolicinvariantmanifold
M
"
.Using
thesplitting
X
=
H
1
(


)
=
X
c
";˘

X
s
";˘
(3.4.1)
andtheirrelatedestimateswehaveestablishedapproximatenormalhyperbolicity.Hence,wemay
applyTheorem2.0.6forsu

cientlysmall
"
,tothetime-
t
0
map
T
t
0
"
ofthewby
(3.3.1)forsome
t
0
largeenough.Wehavethefollowing:
30
1.Forthemap
T
t
0
"
,thereexistsaunique
C
2
normallyhyperbolicinvariantmanifold
f
M
"
=

"
(
@


)
ˆ
X
,where

"


"
(
˘
)

 
"
(
˘
)
2
X
s
";˘
.
2.FromLemma3.3.1andTheorem2.0.6,wehavethat
j

"
(
˘
)

 
"
(
˘
)
j
C
0
(
@


;
X
)
!
0as
"
!
0.
Toseethat
f
M
"
isinvariantunderthew
T
t
"
generatedby(3.3.1),wejustneedtoverify
condition(H5)statedinSection2.1:Thereexistsaninteger
k

0,suchthatforany
>
0,there
exists
>
0,suchthatforany
x
2
B
(
 
(
M
)
;
r
)and
t
2
[
kt
0
;
kt
0
+

],wehave
j
T
t
(
x
)

T
kt
0
(
x
)
j
<
.
Actually,wecaneasilyprovethisbyletting
k
=
1andusingthevariationofconstantsformula.
Wehave
T
t
"
(
x
)

T
t
0
"
(
x
)
=
e
L
"
t
x

e
L
"
t
0
x
+
Z
t
0
e
L
"
(
t

s
)
r
(
T
s
"
(
x
))
ds

Z
t
0
0
e
L
"
(
t
0

s
)
r
(
T
s
"
(
x
))
ds
=
Z
t
t
0
e
L
"
(
t

s
)
L
"
e
L
"
t
0
xds
+
Z
t
t
0
e
L
"
(
t

s
)
r
(
T
s
"
(
x
))
ds
+
Z
t
0
0
(
e
L
"
(
t

s
)

e
L
"
(
t
0

s
)
)
r
(
T
s
"
(
x
))
ds
=
Z
t
t
0
e
L
"
(
t

s
)
L
"
e
L
"
t
0
xds
+
Z
t
t
0
e
L
"
(
t

s
)
L
"
Z
t
0
0
e
L
"
(
t
0

˝
)
r
(
T
˝
"
(
x
))
d
˝
ds
+
Z
t
t
0
e
L
";
P
(
t

s
)
r
(
T
s
"
(
x
))
ds
=
Z
t
t
0
e
L
"
(
t

s
)
[
L
"
T
t
0
"
(
x
)
+
r
(
T
s
"
(
x
))]
ds
;
(3.4.2)
where
r
(
u
)
=

f
(
u
)
+
?


f
(
u
)
:
(3.4.3)
Recallthatthefunction
f
hasbeencuto

,socondition(H5)followsfromthesmoothinge

ect
ofthesemigroupoperator.Therefore,themanifold
f
M
"
islocallyinvariantunder(3.3.1).Fur-
thermore,since
f
M
"
isinan
O
(
"
)neighborhoodof
M
"
in
H
1
(


),byaregularityargumentand
31
Sobolevinequality,wecanactuallyfollowthesameprooftogetthesameresultforthespace
X
=
W
1
;
q
foranylarge
q
.Alsotheinvariantmanifoldisindependentof
q
.Therefore,wehavethe
followingtheorem.
Theorem3.4.1.
Foreverysu

cientlysmall
"
,thereexistsagloballyinvariantmanifoldfor(3.2.2),
f
M
"
,inanO
(
"
)
neighborhoodof
M
"
inL
1
\
H
1
andbeingagraphover
M
"
.
Qualitatively,
f
M
"
consistsoffunctionseachofwhichhasaroughlysemicircularinterface
structureattachedtotheboundaryof


.Inthenextsection,wewillgivethedynamicson
f
M
"
.
3.4.1Motionon
f
M
"
Fix
˘
0
,let

"
(
˘
(
˝
(
t
)))bethesolutionstartingfrom

"
(
˘
0
).Here
˘
(

)isthemotionoftheapproxi-
matebubblesolution,which(3.2.4),i.e.,
d
˘
(
t
;"
)
dt
=
"
2
c
(
˘;"
)
:
(3.4.4)
Notethat
c
(
˘;"
)isdeterminedbythegeometricproblems,soitisaknownfunction.The
function
˝
(
t
)describesthemotionon
f
M
"
.
Theorem3.4.2.
˝
(
t
)
theequation
˝
0
=
O
(
"
)
+
c
(1
+
O
(
"
))
c
;
(3.4.5)
whichimpliesthattheleadingorderof
˝
(
t
)
ist.
Proof.
Since

"
(
˘
)

 
"
(
˘
)
2
X
s
";˘
,wewrite

"
(
˘
(
˝
(
t
)))
=
W
(
x
;˘
(
˝
(
t
))
;"
)
+
V
(
t
).Bytheinvariance
32
of

"
(
˘
(
˝
(
t
)))under(3.2.2),wehavethat
W
˘
(
x
;˘
(
˝
(
t
))
;"
)
˘
0
(
˝
(
t
))
˝
0
(
t
)
+
V
0
(
t
)
=
"
2

W

f
(
W
)
+
?
f
(
W
)
dx
+
"
2

V
+
N
(
W
;
V
)
;
(3.4.6)
where
N
(
W
;
V
)
=
f
(
W
)

f
(
W
+
V
)
+
>
f
(
W
+
V
)

f
(
W
)
dx
,whichisatleastquadraticin
V
.
Plugging(3.3.3)and(3.2.4)into(3.4.6),wehavethat
"
2
c
(
˘
(
˝
(
t
))
;"
)(
˝
0

1)
W
˘
(
x
;˘
(
˝
(
t
))
;"
)
+
V
0
(
t
)
=
"
2

V
+
N
(
W
;
V
)
+
O
(
"
3
)
:
(3.4.7)
Takingtheinnerproductwith
W
˘
(
x
;˘
(
˝
(
t
))
;"
),weget
"
2
c
(
˝
0

1)
j
W
˘
j
2
L
2
+
h
V
0
(
t
)
;
W
˘
i
=
h
N
(
W
;
V
)
;
W
˘
i
+
h
O
(
"
3
)
;
W
˘
i
:
(3.4.8)
Bytakingthederivativewithrespectto
t
in
h
V
(
t
)
;
W
˘
i
=
0,weget
h
V
0
(
t
)
;
W
˘
i
=
h
V
(
t
)
;"
2
c
˝
0
W
˘˘
i
.
Thenwehave
"
2
c
(
˝
0

1)
j
W
˘
j
2
L
2
+
"
2
c
˝
0
h
V
(
t
)
;
W
˘˘
i
=
h
N
(
W
;
V
)
;
W
˘
i
+
h
O
(
"
3
)
;
W
˘
i
:
(3.4.9)
Notethat
h
V
(
t
)
;
W
˘˘
i
=
O
(1)
=
O
(
"
)
j
W
˘
j
2
L
2
.Furthermore,since
N
(
W
;
V
)isatleastquadraticin
V
,
wehavethat
h
N
(
W
;
V
)
;
W
˘
i
=
O
(
"
2
)
j
W
˘
j
L
1
=
O
(
"
3
)
j
W
˘
j
2
L
2
and
h
O
(
"
3
)
;
W
˘
i
=
O
(
"
4
)
j
W
˘
j
2
L
2
,since
j
W
˘
j
L
1
=
O
(1).Usingthesein(3.4.9),weobtain
"
2
c
(
˝
0

1)
+
O
(
"
3
)
c
˝
0
=
O
(
"
3
)
+
O
(
"
4
)
;
(3.4.10)
whichimplies
33
c
(
˝
0

1)
+
O
(
"
)
c
˝
0
=
O
(
"
)
;
(3.4.11)
givingthedesiredresult.

3.4.2Equilibria
Theorem3.4.3.
Letz

(
˘
0
)
beapointon
@


wherethecurvatureof
@


experiencesastrict
extreme;namely:
K
0


=
0
;
K
00


,
0
:
(3.4.12)
Thenthereexists
˘

ina

neighborhoodof
˘
0
suchthat

"
(
˘

)
isanequilibriumof(3.2.2).If
inaddition,
K
00


(
˘
0
)
>
0
,i.e.,thecurvaturearchivesalocalminimum,thentheequilibriumis
unstable.If
K
00


(
˘
0
)
<
0
,i.e.,thecurvaturearchivesalocalmaximum,thentheequilibriumis
stable.
Proof.
Welet
Ÿ
˘
(
t
)
=
˘
(
˝
(
t
)),whichdescribesthemotionontheinvariantmanifold
Ÿ
M
"
.From
Theorem3.4.2,wehave
Ÿ
˘
0
(
t
)
=
˘
0
(
˝
(
t
))
˝
0
=
"
2
(1
+
O
(
"
))
c
(
˘
(
˝
(
t
))
;"
)
=
"
2
(1
+
O
(
"
))
c
(
˘
(
t
+
O
(
"
))
;"
)
=
"
2
(1
+
O
(
"
))(
c
(
˘
(
t
)
;"
)
+
O
(
"
3
))
:
(3.4.13)
Let
Ÿ
˘
(
t
;
Ÿ
˘
0
)bethewoftheODE
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
Ÿ
˘
0
(
t
)
=
"
2
(1
+
O
(
"
))(
c
(
˘
(
t
)
;"
)
+
O
(
"
3
))
Ÿ
˘
(0)
=
Ÿ
˘
0
:
(3.4.14)
34
In[5]theauthorsshowthat
c
=

4

2
3
ˇ
K
0

(

)
+
O
(

3
).Wenowassumethat
K
00


(
˘
0
)
>
0,then
thereexist
˘
1
<˘
0
<˘
2
,with
j
˘
i

˘
0
j
=
O
(

)suchthat
(
c
(
˘
1
;"
)
+
O
(
"
3
))
<
0
<
(
c
(
˘
2
;"
)
+
O
(
"
3
))
:
(3.4.15)
TheIntermediateValueTheoremgivestheexistenceofastationarysolutionandthepositivity
ofthederivativegivestheinstability.

35
Chapter4
Invariantmanifoldsofinteriormulti-spike
statesfortheCahn-Hilliardequationin
higherspacedimensions
4.1Introduction
Thischapterisconcernedwiththeexistence,inforwardandbackwardtime,ofdynamicinterior
multi-spikestates(seeFigure4.1foranillustrationofaninteriormulti-spikestate)drivenbythe
nonlinearCahn-Hilliardequation:
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
u
t
=


(
"
2

u

f
(
u
))in


(0
;
1
)
;
@

u
@
n
=
@
u
@
n
=
0on
@


(0
;
1
)
;
(4.1.1)
where

ˆ
R
n
isaboundeddomainwithsmoothboundary,
@
@
n
istheexteriornormalderivative
to
@

,0
<"<<
1isasmallparameterand
f
:
R
!
R
isassumedtobesmoothandsupports
anondegenerategroundstate
w
withasymptoticvalue0fortheequation



g
(

)
=
0in
R
n
,
g
(

)
=
f
(¯
m
+

)

f
(¯
m
)fored¯
m
with
f
0
(¯
m
)
>
0,thatisfor¯
m
inthemetastableregion.Atypical
exampleis
f
(
u
)
=
u
p

u
with1
<
p
<
n
+
2
n

2
;
if
n

3.TheusualchoicefortheCahn-Hilliard
equationis
f
(
u
)
=
u
3

u
.
36
Figure4.1:Interiormulti-spikestate
Weproveanabstractresultontheexistenceofaninvariantmanifoldwithboundaryfor
amapwhenoneonlyhasafamilyofapproximatelyinvariantmanifoldsthatareapproximately
normallyhyperbolic,eachconsistingofalmoststationarystates.Theabstractresultisanextension
ofresultsin[19]totacklethecaseofmanifoldswithboundary.Ourabstractresultisinstrumental
inprovingtheexistenceoflocallyinvariantmanifoldsofmulti-spikestatesfortheCahn-Hilliard
equation.Thoughwedonotsetupageneralframeworkforws,theproofinthecurrent
paperisquitegeneralandshouldbewidelyapplicable.
TheCahn-Hilliardequation(where
f
(
u
)
=
u
3

u
)isawidelyacceptedmodelforthecom-
plicatedpatterningofthelocalconcentrationsinabinaryalloycontainedinavessel

,asitis
rapidlyquenchedbelowthecurveofmiscibility.Abovethatcurve,thealloyisinahomogeneous
phasecorrespondingtothermodynamicequilibrium.Belowthecurve,thethermodynamicequi-
libriumcorrespondstotwoseparatedphases.Theseparationphenomenathatoriginateafterthe
rapidquenchingincludenucleation,spinodaldecompositionandtheformationanddynamicsof
fronts.Wereferthereadersto[28]andthereferencesthereinforthephysicalbackground.Fora
discussionofthestationaryproblemforthisequation,wereferreadersto[14].
Themulti-spikeequilibriaof(4.1.1)hasbeenstudiedbymanyauthors.In[14],theauthors
provedtheexistenceofstationaryinteriormulti-spikesolutionsto(4.1.1)byusinganinvariant
37
manifoldapproach.Theyconstructedquasi-invariantmanifoldsofinteriormulti-spikestatesand
estimatedthemotionofeachspike,thenprovedtheexistenceofstationaryinteriormulti-spike
states.Similarresultswerereportedindependentlyin[96,97,99],wheretheauthorsuseda
Lyapunov-Schmidtreductiontechnique.In[12]multipleboundaryspikesolutionswerefound.
TherichcollectionofsolutionstothestationaryproblemfortheAllen-Cahnequation
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
u
t
=
"
2

u

u
+
f
(
u
)
;
x
2

ˆˆ
R
n
@
u
@
N
=
0
;
x
2
@

:
(4.1.2)
hasalsobeenstudiedbymanyauthors,especiallyforthecasewhere
f
(
u
)
=
u
p
withsuperlin-
earbutsubcriticalgrowth.In[71],theauthorsinvestigatedtheGierer-Meinhardtsysteminthe
asymptoticlimitasthedi

usivityoftheinhibitorbecomesunbounded.Inthatlimit,oneisleadto
(4.1.2),referredtoasthe`shadowequation'.Theyshowedthatnonon-constantpositivestationary
solutionsexistwhen
"
islarge.For(4.1.2),itwasshownin[64]thatpositivesolutionsmusthave
peakswithexponentiallydecayingtailsas
"
#
0.Thepaper[72]studies(4.1.2)with
f
(
u
)
=
u
p
.
Theauthorsobtainedapositivesolutionthathasasinglepeak,theso-calledleastenergysolution,
byusingamountainpassargument.Theyfurthershowedthatthispeakmustactuallylocateon
@

andtheofthesolutionisaofthegroundstateon
R
n
,translatedto
@

andrescaledby
"
.Later,atopologicallowerboundonthenumberofsuchsolutionswasgiven
byZ-Q.Wangin[90].InfurtherworkW-MNiandI.Takagi,in[73],investigatedthelocationof
thepeaks,andtheyprovedthatthepeaklocationtended,as
"
!
0,tothepointof
@

wherethe
meancurvatureachieveditsmaximum.Otherpapersfollowed,providingforsolutionswithspikes
atanycollectionofnon-degenerate(insomecasesonlytopologicallynontrivial)criticalpointsof
themeancurvature,andevenmultiplespikesaccumulatingatlocalminimalpointsofthemean
38
curvature,orsolutionstoothersingularlyperturbedequationsandsystems(see,e.g.,[44],[89],
[94],[39],[75],[76],[63],[20],[37],[48],[46],[47]and[98]).
Likewise,theDirichletproblemhasalsoattractedsomeattention,withresultsprovidingde-
tailedinformationabouttheexistenceandlocationofastationarypeak(see,e.g.,[57],[74],and
[36]).
Thecaseofcriticalgrowthisquitedi

erent,duetoascaleinvarianceandrelatedlackof
compactness.Stilltherearesomeresultsandwereferthereadersto[91],[92],[2],[1],and[3],
forexample.
Fordynamicalspikesolutions,therearenotmanyresults.In[19],theauthorsfoundaninvari-
antmanifoldofboundaryspikesolutionstoequationsoftheform
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
u
t
=
"
2

u

u
+
g
(
u
)
;
x
2

;
@
u
@
n
=
0
;
x
2
@

;
(4.1.3)
andshowedrigorouslythatthemotionoftheboundaryspikeisdrivenbythemeancurvatureofthe
boundaryofthedomain.Arelateddynamicalproblemwasconsideredin[15],wheretheauthors
constructedaninvariantmanifoldofboundarydropletsolutionstothe2-dmass-conservingAllen-
Cahnequation
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
u
t
(
x
;
t
)
=
"
2

u
(
x
;
t
)

f
(
u
(
x
;
t
))
+
>

f
(
u
(

;
t
))
;
x
2

;
t
>
0
;
@
u
@
n
=
0
;
x
2
@

;
t
>
0
;
(4.1.4)
andalsoshowedthatthemotionofthecenterofthedropletisdrivenbythecurvatureofthe
boundaryofthedomain.
Inthischapter,weshowtheexistenceofdynamicalinteriormulti-spikesolutionsto(4.1.1)by
39
constructinginvariantmanifoldsofinteriormulti-spikestates.Roughlyspeaking,weuse
k
ground
states,translatedto
k
pointsof

being

-awayfromeachotherand

2
awayfromtheboundary
andscaledwith
"
tocompriseamanifold
M
"
parametrizedbythelocationofthecentersofthe
spikes.Thus
M
"
hasdimension
nk
andhasboundaryduetothenon-proximityconstraint.In
[19],theauthorsestablishedageneraltheoremwhichstatesthatinasmallneighborhoodofan
approximatelyinvariantandapproximatelynormallyhyperbolicmanifold,thereexistsatruein-
variantmanifoldbeingasmoothgraphovertheformermanifold.Intheirsetting,theyrequirethat
theapproximatelyinvariantmanifoldiswingorovwingatan
O
(1)rate.However,dueto
thesuperslowmotionoftheinteriorspikes,theresultsin[19]cannotbeappliedtoourproblem
directly,sothethingwedoistoextendtheresultsin[19]tomakethemapplicabletoour
case.Thisinvolvesafiblowupfltechnique.Anothertechnicaldi

cultyinourproblemisthatthe
linearizedoperatorobtainedbylinearizingtheCahn-Hilliardequationatamulti-spikestateisnot
self-adjoint,sowecannotsplitthespaceaccordingtothespectrumofthatlinearizedoperatorto
provethenormalhyperbolicity.Toovercomethisdi

culty,weturntodealwiththecorresponding
fiintegratedflequation.Thatis,wetransformtheCahn-Hilliardequationusing(


)

1
2
,andby
sodoing,thecorrespondinglinearizedoperatorbecomesself-adjoint.Moredetailsabouthowto
makethistransformationcanbefoundinSection4.3.5.Thenweconstructalocally
invariantmanifoldoftransformed(by(


)

1
2
)interiormulti-spikestatesforthetransformedsemi-
w,andaftertransformingeverythingback,weobtainalocallyinvariantmanifold
f
M
"
ofinterior
multi-spikestatesfortheoriginalequation.Notethat,onceweobtainsuchaninvariantmanifold,
thesolutionof(4.1.1)startingfromanypointinthatmanifoldexistsforbothpositiveandnegative
time.Byreducing(4.1.1)on
f
M
"
,wederiveanequationwhichdeterminesthevelocityofeach
spikeanalytically.ItturnsoutthatthedynamicsofinteriorspikesfortheCahn-Hilliardequation
hasaglobalcharacterwherenotonlytheclosestspikesinteractbuteachspikeinteractswithall
40
theothersandwiththeboundary.Furthermore,thespeedoftheinteriorspikesisestimatedtobe
exponentiallysmall,whichindicatesthatthemulti-spikestatesexistforaverylongtime,forwards
andbackwards.
Thischapterisorganizedasfollows:inSection4.2,weextendtheresultsin[19]toslowman-
ifoldswithboundary.InSection4.3,weconstructapproximatelyinvariantmanifoldsofinterior
multi-spikestatesfor(4.1.1)andthenprovetheexistenceoftrulyinvariantmanifoldsofinterior
multi-spikestatesnearby.InSection4.4,weinvestigatethedynamicsofmulti-spikestatesand
giveanestimateofthespeedofthecentersofthespikes.
4.2Approximatelystationaryinvariantmanifoldswithbound-
ary
In2.0.1,atechnicalassumptionthatthedistancefrom
u
(
M
)totheboundaryoftheman-
ifold
M
isboundedbelowismade.Thisguaranteesthattheimageunder
T
ofagraphover
 
(
M
)
isstillagraphover
 
(
M
).Asimilarassumptionisalsomadein2.0.5forapproximately
ovwinginvariantmanifolds.However,sometimestheapproximatelyinvariantmanifoldisap-
proximatelystationary,sothatthemap
T
isapproximatedbytheidentitymap,andthereforethat
assumptioncannotbeThishappenswhenweareseekinganinvariantmanifoldofstates
withsuperslowmotion.Inthissection,weestablishaframeworktoobtaininvariantmanifolds
nearapproximatelystationaryandapproximatelynormallyhyperbolicmanifolds.Theideaisto
modifythemap
T
onlyneartheboundaryof
 
(
M
)togetanewmap
e
T
suchthat
 
(
M
)isapprox-
imatelywing(ovwing)invariantunder
e
T
and
e
T
=
T
whenappliedtothepointsaway
fromtheboundary.More,iftheflprojectionflof
T
(
x
)to
 
(
M
)isneartheboundaryof
 
(
M
),thenwemovetheflprojectionflinside(outside)
 
(
M
)andawayfromtheboundaryof
 
(
M
).
41
Moreover,themovementshouldbeveryslightforthepurposeofkeepingthenormalhyperbolicity.
Moreprecisestatementmaybefoundbelow.
Let
X
;
Y
betwoBanachspacesand
T
2
C
J
(
X
;
X
)with
J
>
1and
M
ˆ
Y
bea
C
J

dimensionalclosedmanifoldwithsmoothboundary
@
M
.Let
 
2
C
J
(
M
;
X
)beanembedding
1
satisfying
k
D
i
 
k
<
B
2
for1

i

J
and
k
D
 

1
k
<
B
3
forall
m
2
M
.Wedenotethemetricon
M
by
d
(

;

).Furthermore,wemakethefollowingassumptionontheboundaryof
M
.
With
B
(
@
M
;
r
0
)
=
f
m
2
M
:
d
(
m
;@
M
)

r
0
g
,weassumethatthereexists
r
>
0
;
1
;
2
>
0
and
˚
:
B
(
@
M
;
r
)
!
R
n
+
suchthat
˚
(
@
M
)
ˆ
@
R
n
+
,where
R
n
+
istheupperhalfspaceof
R
n
,and

1
k
D
˚

1
k

2
,
k
D
˚
k

3
;
k
D
2
˚

1
k

4
.Here
˚

1
meanstakingtheinverseforalocal
chart.Since
M
isassumedtobecompact,itispossibletoconstructsuchamap
˚
byusinga
partitionofunity.
4.2.1.
 
(
M
)issaidtobeapproximatelystationaryinvariantunder
T
ifthereexistssmall
>
0suchthat
j
T
(
 
(
m
))

 
(
m
)
j
<
,forall
m
2
M
.
4.2.2.
Wesaythatanapproximatelystationaryinvariantmanifold
 
(
M
)isapproxi-
matelynormallyhyperbolic,ifthefollowingconditionshold:
1.Foreach
m
2
M
,thereisadecomposition
X
=
X
c
m

X
s
m

X
u
m
ofclosedsubspaceswith
projections

c
m
;

s
m
;

u
m
varyingin
C
J
waywithrespectto
m
.
2.Forany
m
2
M
,

c
m
isanisomorphismfrom
D
 
(
m
)
T
m
M
to
X
c
m
and
T
m
M
=
X
c
m
+
m
(
X
c
m
),
where

m
2
L
(
X
c
m
;
X
s
m

X
u
m
).Furthermorethereexist
B
;
L
,and
˜
2
(0
;
1
=
2),suchthatfor
1
ThistheorycouldbedevelopedforimmersedBanachmanifolds,butinouropiniontheincreaseingeneralityis
notworththelossofclarityandincreaseofpages.
42
any
m
0
2
M
and
m
1
;
m
2
2
B
c
(
m
0
;
r
0
),with
m
1
,
m
2
,for

=
c
;
u
;
s
,
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
:
k


m
0
k
B
,
k


m
1



m
2
k
L
j
 
(
m
1
)

 
(
m
2
)
j
;
j
 
(
m
1
)

 
(
m
2
)


c
m
0
(
 
(
m
1
)

 
(
m
2
))
j
j
 
(
m
1
)

 
(
m
2
)
j

˜:
(4.2.1)
3.Thereexist
˙;
2
(0
;
1)and
a
suchthat,forany
m
0
2
M
,and

2f
c
;
s
;
u
g
,

2f
c
;
s
;
u
g
,with

,

,then
(a)
k


m
0
DT
(
 
(
m
0
))
j
X

m
0
k
˙
,
k
(

c
m
0
DT
(
 
(
m
0
))
j
X
c
m
0
)

1
k

1
>
a
,
(b)
k

s
m
0
DT
(
 
(
m
0
))
j
X
s
m
0
k

,

k
(

u
m
0
DT
(
 
(
m
0
))
j
X
u
m
0
)

1
k

1
>
1,
(c)

k
(

u
m
0
DT
(
 
(
m
0
))
j
X
u
m
0
)

1
k

1
>
max
f
1
;
k

c
m
0
DT
(
 
(
m
0
))
j
X
c
m
0
k
J
g
,
k

s
m
0
DT
(
 
(
m
0
))
j
X
s
m
0
k
<
min
f
1
;
k
(

c
m
0
DT
(
 
(
m
0
))
j
X
c
m
0
)

1
k

J
g
.
4.Thereexists
B
1
,suchthat
k
D
j
T
j
B
(
 
(
M
)
;
r
0
)
k
B
1
for1

j

J
.
Remark
4.2.3
.
(4.2.1)impliesthat
k

m
k
B
˜
,forany
m
2
M

X

m
(

):
=
f
x

2
X

m
:
j
X

m
j

g
;
=
s
;
u
;
and
N
(
M
;"
):
=
f
 
(
m
)
+
x
s
+
x
u
:
x
s
2
X
s
m
(
"
)
;
x
u
2
X
u
m
(
"
)
g
:
Lemma4.2.4.
If
"<
min
f
L
8
;
BL
8
g
,thenN
(
M
;"
)
isatubularneighborhoodofMsatisfying
1.Foranytwopoints
 
(
m
i
)
+
x
s
i
+
x
u
i
2
N
(
M
;"
)
,i
=
1
;
2
,if
 
(
m
1
)
+
x
s
1
+
x
u
1
=
 
(
m
2
)
+
x
s
2
+
x
u
2
;
43
then
m
1
=
m
2
;
x
s
1
=
x
s
2
;
x
u
1
=
x
u
2
:
2.Thereexistssmall

suchthatif
j
x

(
 
(
m
)
+
x
s
+
x
u
)
j

,where
(
 
(
m
)
+
x
s
+
x
u
)
2
N
(
M
;"
)
,
then
x
=
 
(
m

)
+
x
s

+
x
u

;
where
j
x
s

j
"
and
j
x
u

j
"
.
Proof.
WerefertotheproofofLemma3.6in[19].

Remark
4.2.5
.
Asaconsequenceoftheproof,wehavethat
m
(
x
)
;
x
s
(
x
)
;
x
u
(
x
)areallsmoothin
x
for
x
2
N
(
M
;"
).
Nowwestarttoconstructacenter-stablemanifoldfor
T
.Weseveralfunctionsthat
willbeusedlaterinthissection.Write
x
2
R
n
as(
x
0
;
x
n
),and
S

:
R
n
!
R
n
as
S

(
x
0
;
x
n
)
=
(
x
0
;
x
n
+

)
;
(4.2.2)
and
e
:
R
n
!
R
as
e
(
x
0
;
x
n
)
=
x
n
:
(4.2.3)
Let
b
(
z
)beasmoothmonotonefunctionsatisfying
b
(
z
)
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
:
=
1
;
z

0
;
2
(0
;
1)
;
z
2
(0
;
1)
;
=
0
;
z

1
;
(4.2.4)
44
thenforany
m
2
M
,we
e
m
=
˚

1

S

(
m
)

˚
(
m
)

;
(4.2.5)
where

(
m
)
=
b

e
(
˚
(
m
))

d
d


l
;
(4.2.6)
where
d
>
0isaconstantsatisfying
d
<

1
r
2
and
l
>
0isasmallconstanttobedetermined.
Remark
4.2.6
.
Notethat
k
D
˚

1
k

2
,so
k
D
˚
k
1

2
.Itfollowsimmediatelythatforany
m
satisfying2

2
d
<
d
(
m
;@
M
)
<
r
,wehave
e
(
˚
(
m
))
>
2
d
,whichimpliesthat

(
m
)
=
0.So
e
m
=
m
if
2

2
d
<
d
(
m
;@
M
)
<
r
.
For
x
2
N
(
m
;"
),byLemma4.2.4,wecanwrite
x
as
x
=
 
(
m
(
x
))
+
x
s
(
x
)
+
x
u
(
x
).Weconstruct
anewmap
e
T
as
e
T
(
x
)
=
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
T
(
x
)
;
m
(
x
)
2
M
n
B
(
@
M
;
r
)
;
T
(
x
)
+
 
(
g
m
(
x
))

 
(
m
(
x
))
;
m
(
x
)
2
B
(
@
M
;
r
))
:
(4.2.7)
ByRemark4.2.6,wehave
e
T
2
C
J
(
X
;
X
).
Let
u
(
m
)
=
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
m
;
for
m
2
M
n
B
(
@
M
;
r
)
;
e
m
;
for
m
2
B
(
@
M
;
r
)
:
(4.2.8)
ByRemark4.2.6,onecanseethat
u
iscontinuous.
Lemma4.2.7.
Foranym
2
M,d
(
u
(
m
)
;@
M
)


1
l.
Proof.
Wewrite
˚
(
m
)
=
(
x
0
;
0)andforany¯
m
2
@
M
,wewrite¯
m
=
(¯
x
0
;
0).Thenonecancheck
45
that
j
u
(
m
)

¯
m
j
=
j
˚

1
(
x
0
;
t
)

˚

1
(¯
x
0
;
0)
j

1
j
(
x
0
;
t
)

(¯
x
0
;
0)
j

1
t
:
Thus
d
(
u
(
m
)
;@
M
)
=
inf
¯
m
2
@
M
d
(
u
(
m
)
;
¯
m
)


1
t
.

Lemma4.2.8.
j
e
T
(
 
(
m
))

 
(
u
(
m
))
j
<
,foranym
2
M.
Proof.
Bydirectcomputation,wehave
e
T
(
 
(
m
))

 
(
u
(
m
))
=
T
(
 
(
m
))

 
(
m
)
;
whichimpliesthedesiredresult.

CombiningLemma4.2.7andLemma4.2.8,itisclearthat
 
(
M
)isanapproximatelywing
invariantmanifoldfor
e
T
.Themap
e
T
isalmostthesameas
T
,exceptthatitshiftsthefibase
pointsflof
x
on
 
(
M
)thatareneartheboundaryof
 
(
M
)alittle.Intuitively,onemayexpectthat
if
l
issmallenough,
 
(
M
)isalsoapproximatelynormallyhyperbolicfor
e
T
,sothatwecanapply
Theorem2.0.3toconcludetheexistenceofacenter-stablemanifold.However,onemaynoticethat
thedistancefrom
u
(
M
)totheboundaryof
M
dependson
l
and
l
isalsoinvolvedinthetrichotomy
propertyof
D
e
T
.AsweapplyTheorem2.0.3,thedistancefrom
u
(
M
)totheboundaryof
M
needs
tobeedthenwemaketheotherparameterssmall.Therefore,Theorem2.0.3cannotbe
applieddirectly.Toovercomethisproblem,weperformthefollowingblow-upanalysis.
46
Let
M
l
=
M
l
=
f
m
l
=
m
l
:
m
2
M
g
;
 
l
(
m
l
)
=
1
l
 
(
lm
l
)
;
T
l
(
x
)
=
1
l
T
(
lx
)
;
e
T
l
(
x
)
=
1
l
e
T
(
lx
)
;
u
l
(
m
l
)
=
1
l
u
(
lm
l
)
;
¯
X

m
l
=
X

lm
l
;
¯


m
l
=

lm
l
;
=
c
;
s
;
u
:
(4.2.9)
First,itiseasytocheckthat
j
e
T
l
(
 
l
(
m
l
))

 
l
(
u
l
(
m
l
))
j
<

l
(4.2.10)
forany
m
l
2
M
l
,and
d
(
u
(
M
l
)
;@
m
l
)


1
:
(4.2.11)
Thus,
 
l
(
M
l
)isanapproximatelywinginvariantmanifoldfor
e
T
l
"
.
Clearly,foreach
m
l
2
M
l
,
X
=
¯
X
c
m
l

¯
X
s
m
l

¯
X
u
m
l
ofclosedsubspaceswithprojections
¯

c
m
l
;
¯

s
m
l
;
¯

u
m
l
,andforany
m
l
0
2
M
l
and
m
l
1
;
m
l
2
2
B
c
(
m
l
0
;
r
0
l
),with
m
l
1
,
m
l
2
,for

=
c
;
u
;
s
,
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
k
¯


m
l
0
k
B
,
k
¯


m
l
1

¯


m
l
2
k
lL
j
 
l
(
m
l
1
)

 
l
(
m
l
2
)
j
;
j
 
l
(
m
l
1
)

 
l
(
m
l
2
)

¯

c
m
l
0
(
 
l
(
m
l
1
)

 
l
(
m
l
2
))
j
 
l
(
m
l
1
)

 
l
(
m
l
2
)

˜:
(4.2.12)
47
Notethat
¯


u
l
(
m
l
)
D
e
T
l
(
m
l
)
j
¯
X

m
l
=

u
(
m
)
D
e
T
(
m
)
j
X

m
;
thusinordertoshowthetrichotomypropertyfor
e
T
l
,wejustneedtoconsider


u
(
m
)
D
e
T
(
m
)
j
X

m
.
Furthermore,for
m
2
M
n
B
(
@
M
;
r
),wehave
e
T
=
T
and
u
(
m
)
=
m
,sowejustneedtodealwith
m
2
B
(
@
M
;
r
).
Since
DS

(
m
)
=
I
+
D

(
m
)

e
n
;
where
e
n
=
(0
;

;
1),and
D
˚

1
(
˚
(
x
))
D
˚
(
x
)
=
I
,wehave
D
 
(
g
m
(
x
))

D
 
(
m
(
x
))
=
D
 
(
g
m
(
x
))
D
e
m
(
x
)

D
 
(
m
(
x
))
Dm
(
x
)
=
D
 
(
g
m
(
x
))
D
(
˚

1
)(
S

(
m
)
˚
(
m
))(
I
+
D

(
m
(
x
))

e
n
)
D
˚
(
m
(
x
))
Dm
(
x
)

D
 
(
m
(
x
))
Dm
(
x
)
=
(
D
 
(
g
m
(
x
))

D
 
(
m
(
x
)))
D
(
˚

1
)(
S

(
m
)
˚
(
m
))(
I
+
D

(
m
(
x
))

e
n
)
D
˚
(
m
(
x
))
Dm
(
x
)
+
D
 
(
m
(
x
))
D
(
˚

1
)(
S

(
m
)
˚
(
m
))

D
(
˚

1
)(
˚
(
m
))(
I
+
D

(
m
(
x
))

e
n
)
D
˚
(
m
(
x
))
Dm
(
x
)
+
D
 
(
m
(
x
))
D
(
˚

1
)(
˚
(
m
))(
D

(
m
(
x
))

e
n
)
D
˚
(
m
(
x
))
Dm
(
x
)
:
(4.2.13)
Usingthefactthat
k
D
i
 
k
B
2
,
k
D
˚

1
k
<
2
,
k
D
˚
k
<
3
,
k
D

(
m
(
x
))
k
=
O
(
l
),
k
Dm
(
x
)
k
is
uniformlybounded,
j
e
m

m
j

2
t
and
k
D
˚

1
(
S

(
m
)
˚
(
m
))

D
˚

1
(
˚
(
m
))
k
=
O
(
l
),wehave
k
D
 
(
g
m
(
x
))

D
 
(
m
(
x
))
k
Cl
;
(4.2.14)
forsomeconstant
C
beingindependentof
x
.
48
Thuswehave
k


u
(
m
)
D
e
T
(
m
)
j
X

m
kk


u
(
m
)



m
kk
D
e
T
(
m
)
j
X

m
k
+
k


m
D
e
T
(
m
)
j
X

m
k
k


u
(
m
)



m
k
(
k
DT
(
m
)
j
X

m
k
+
k
D
 
(
g
m
(
x
))

D
 
(
m
(
x
))
k
)
+
k


m
DT
(
m
)
j
X

m
k
+
k


m
kk
D
 
(
g
m
(
x
))

D
 
(
m
(
x
))
k
k


m
DT
(
m
)
j
X

m
k
+
Cl
:
(4.2.15)
Similarly,onecanprove
k


u
(
m
)
D
e
T
(
m
)
j
X

m
kk


m
DT
(
m
)
j
X

m
k
Cl
:
(4.2.16)
Therefore,when
l
issmallenough,thetrichotomypropertiesaresothatTheorem
2.0.3canbeappliedto
e
T
l
and
M
l
toobtainacenter-stablemanifold
¯
W
cs
(
l
)for
e
T
l
.Notethatif
¯
W
cs
(
l
)isinvariantunder
e
T
l
,then
W
cs
(
l
)
=
l
¯
W
cs
isinvariantunder
e
T
.Since
e
T
(
x
)
=
T
(
x
)when
e
(
˚
(
m
(
x
)))
>
2
d
or
d
(
m
(
x
)
;@
M
)
>
r
,itisclearthat
W
cs
(
l
)islocallyinvariantunder
T
.Sowehave
thefollowingtheorem:
Theorem4.2.9.
Dependingonr
0
;
B
;
B
1
;
L
;
when
˜
and
˙
aresu

cientlysmall,foreveryl
su

cientlysmall,thereexistsaC
J
positivelylocallyinvariantmanifoldW
cs
(
l
)
forT,whichis
givenastheimageofamap
h
:
f
(
m
;
x
s
):
m
2
M
;
x
s
2
X
s
m
(

0
)
g!
X
;
forsome

0
>
0
.Alsoforanyx
2
W
cs
(
l
)
withe
(
˚
(
m
(
x
)))
>
2
dord
(
m
(
x
)
;@
M
)
>
r,wehave
T
(
x
)
2
W
cs
(
l
)
.
49
Remark
4.2.10
.
Theorem4.2.9essentiallysaysthat
W
cs
(
l
)isinvariantunder
T
ifthefibasepointfl
on
 
(
M
)isawayfromtheboundaryof
M
.
Similarly,wecanconstructcenter-unstablemanifoldsfor
T
.Let
e
m
=
˚

1

S


(
m
)

˚
(
m
)

;
(4.2.17)
thenweconstruct
e
T
and
v
respectivelyas
e
T
(
x
)
=
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
T
(
x
)
;
m
(
x
)
2
M
n
B
(
@
M
;
r
)
;
T
(
x
)
+
 
(
g
m
(
x
))

 
(
m
(
x
))
;
m
(
x
)
2
B
(
@
M
;
r
))
;
(4.2.18)
and
v
(
m
)
=
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
m
;
for
M
2
M
n
B
(
@
M
;
r
)
;
˚

1

S

(
m
)

˚
(
m
)

;
for
M
2
B
(
@
M
;
r
)
:
(4.2.19)
Bytheconstructionof
v
,onecaneasilyseethat
v
isahomeomorphism.Followingthesame
argumentasabove,weobtainthefollowtheorem:
Theorem4.2.11.
Dependingonr
0
;
B
;
B
1
;
L
;
when
˜
and
˙
aresu

cientlysmall,forevery
tsu

cientlysmall,thereexistsaC
J
negativelylocallyinvariantmanifoldW
cu
(
l
)
forT,whichis
givenastheimageofamap
h
:
f
(
m
;
x
u
):
m
2
M
;
x
u
2
X
u
m
(

0
)
g!
X
;
forsome

0
>
0
.Alsoforanyx
2
W
cu
(
l
)
withe
(
˚
(
m
(
x
)))
>
2
dord
(
m
(
x
)
;@
M
)
>
r,wehave
T
(
x
)
2
W
cu
(
l
)
.
50
4.3Invariantmanifoldsofinteriormulti-spikestates
4.3.1Constructionofthemanifold
M
"
Westartfromthefollowingequation,

"
2

u
+
f
(
u
)
=
˙;
(4.3.1)
whosesolutionsarestationarysolutionsto(4.1.1).
Let
˙
=
f
(¯
m
),where¯
m
isinametastableregion,whichmeansthat
f
0
(¯
m
)
>
0.Wemakethe
followingtransformation,
u
=
v
+
¯
m
;
g
(
v
)
=
˙

f
(
v
+
¯
m
)
:
(4.3.2)
Obviously
g
(0)
=
0.Let
g
0
(0)
=


,thenwecanwrite
g
(
v
)
=


v
+
h
(
v
)with
h
satisfying
h
(0)
=
h
0
(0)
=
0
:
Then(4.1.1)becomes
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
v
t
=


(
"
2

v
+
g
(
v
))in


(0
;
1
)
;
@

v
@
n
=
@
v
@
n
=
0on
@


(0
;
1
)
;
(4.3.3)
or
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
v
t
=


(
"
2

v


v
+
h
(
v
))in


(0
;
1
)
;
@

v
@
n
=
@
v
@
n
=
0on
@


(0
;
1
)
:
(4.3.4)
And(4.3.1)becomes
"
2

v
+
g
(
v
)
=
0
;
(4.3.5)
or
"
2

v


v
+
h
(
v
)
=
0
:
(4.3.6)
51
Bythemaintheoremof[23],thereexistsagroundstate
w
satisfying
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:

w
+
g
(
w
)
=
0in
R
n
;
w
(0)
=
max
w
(
x
)
;
w
>
0
;
w
(
x
)
=
w
(
j
x
j
)
;
w
(
x
)
!
0
;
j
x
j!1
:
(4.3.7)
Also,
w
(
x
)that
j
@
k
r
w
(
x
)
j
Ce


j
x
j
forany
x
2
R
n
and
k

0,where
@
r
isthederivativein
theradialdirection.Notethat
g
isassumedtobesmooth,so
w
issmooth.Furthermore,weassume
thegroundstate
w
isnon-degenerate,thatis,theoperatorobtainedbylinearizingat
w
has0asan
eigenvalueofmultiplicity
n
(when
f
=
u
3

u
,thegroundstateisnon-degenerate).
Wewillconstructanapproximatelyinvariantmanifoldofinteriormulti-spikestatesparametrized
bythelocationsofthespikesfor(4.1.1).Roughlyspeaking,wepatch
k
translatedand
"
-rescaled
groundstatestogether,requiringthat
k
centerpointsofthespikesareatleast
>
0awayfromeach
otherand

2
awayfromtheboundary.Heretheconstant

isnotnecessarily
O
(1),butmustsatisfy

"
!1
as
"
!
0,so

couldbe
O
(
j
"
ln
"
j
)or
O
(
p
"
),forexample.
Let
e

k
=

P
=
(
p
1
;
p
2
;

;
p
k
):
p
i
2

;
j
p
i

p
j
j
;
d
(
p
i
;@

)


2

;
(4.3.8)
obviously,
e

k
isaclosedsubmanifoldof



|
        
{z
        
}
k

copies
withboundary.
Thenwelet
W
";
P
(
x
)
=
k
X
i
=
1
w
";
p
i
;
(4.3.9)
where
P
=
(
p
1
;
p
2
;

;
p
k
)
2
Ÿ

k
and
w
";
p
i
=
w
(
x

p
i
"
).
Fromthispointon,tosimplifythenotation,wehavereplaced

2
by

intheexponentiallysmall
52
terms.
Lemma4.3.1.
W
";
P

"
2

W
";
P


W
";
P
+
h
(
W
";
P
)
=
R
";
P
(
x
)
;
(4.3.10)
where
j
R
";
P
(
x
)
j
C
m
(

)
=
O
(
e


"
)
foranym.
Proof.
First,notethat
W
";
P
and
h
aresmooth,so
R
";
P
issmooth.Forany
x
2f
x
2

:
d
(
x
;
p
i
)
<

2
g
,wehave
j
w
(
x

p
l
"
)
j
=
O
(
e


"
)for
l
,
i
.Usingthefactthat
h
(0)
=
0,wehave
j
R
";
P
(
x
)
j
=
j
h
(
W
";
P
)

h
(
w
(
x

p
i
"
))

X
l
,
i
h
(
w
(
x

p
l
"
))
j

C
X
l
,
i
j
w
(
x

p
l
"
)
j
=
O
(
e


"
)
:
Forany
x
2f
x
2

:
d
(
x
;
p
i
)
>

2
;
i
=
1
;

;
k
g
,wehave
j
w
(
x

p
i
"
)
j
=
O
(
e


"
)for
i
=
1
;

;
k
.
Thus
j
R
";
P
(
x
)
j
=
j
h
(
W
";
P
)

k
X
i
=
1
h
(
w
(
x

p
l
"
))
j
=
O
(
e


"
)
:
Therefore,
j
R
";
P
(
x
)
j
C
0
(

)
=
O
(
e


"
).Onecanfollowthesameargumenttoshowthatforany
m
,
j
R
";
P
(
x
)
j
C
m
(

)
=
O
(
e


"
)
:
Notethattakingaderivativeof
R
";
P
generates
1
"
whichisabsorbedby
e


"
bychanging

slightly.

Now,weseethat
W
";
P
(
x
)approximately(4.3.6).Howeveritdoesnotsatisfythe
53
Neumannboundaryconditionsin(4.1.1),soweneedtomodifyitslightly.
Let
H
(
ˆ
)bethesolutionto
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
"
2

v


v
=
0in

;
@
v
@
n
=
@ˆ
@
n
on
@

;
(4.3.11)
andlet
H
(
ˆ
)bethesolutionto
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
"
2

v


v
=
"
2
˜

ˆ

"
2

H
(
ˆ
)in

;
@
v
@
n
=
0on
@

;
(4.3.12)
where
˜
(
x
)isasmoothcut-o

functionsatisfying
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
˜
(
x
)
=
0
;
x
2f
x
:
d
(
x
;@

)


4
g
;
˜
(
x
)
=
1
;
x
2f
x
:
d
(
x
;@

)


8
g
:
(4.3.13)
Onecaneasilycheckthat
H
(
W
";
P
)
=
P
1

i

k
H
(
w
";
p
i
)and
H
(
W
";
P
)
=
P
1

i

k
H
(
w
";
p
i
).To
estimate
H
(
W
";
P
)and
H
(
W
";
P
),weprovethefollowinglemma.Itmayhavebeenproved
elsewhere,butforthecompleteness,wegiveaproofhere.
Lemma4.3.2.
IfG
2
L
q
(

)
andH
2
L
q
(
@

)
forsomeq

2
,andifvisthesolutionto
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:

v


v
=
Gin

;
@
v
@
n
=
Hon
@

;
(4.3.14)
54
then
j
v
j
L
nq
n

2
(

)

C
(
j
G
j
L
q
(

)
+
j
H
j
L
q
(
@

)
)
Proof.
Z


v

v


v
2
dx
=
Z

Gvdx
:
ByGreen'sformula,wehave

Z

jr
v
j
2
dx
+
Z
@

vHds

Z


v
2
dx
=
Z

Gvdx
:
Then,wehave
Z

jr
v
j
2
dx
+
Z


v
2
dx

Z
@

j
vH
j
ds
+
Z

j
Gv
j
dx


Z
@

j
v
j
2
ds
+
1
4

Z
@

j
H
j
2
ds
+

Z

j
v
j
2
dx
+
1
4

Z

j
G
j
2
dx


C
(
Z

j
v
j
2
ds
+
Z

jr
v
j
2
dx
)
+
1
4

Z
@

j
H
j
2
ds
+

Z

j
v
j
2
dx
+
1
4

Z

j
G
j
2
dx
;
whichimpliesthat
(1


C
)
Z

jr
v
j
2
dx
+
(



C


)
Z

j
v
j
2
dx

1
4

Z

j
G
j
2
dx
+
1
4

Z
@

j
H
j
2
ds
:
Choosing

smallenough,wehave
j
v
j
W
1
;
2

C
(
Z

j
G
j
2
dx
+
Z
@

j
H
j
2
ds
)
:
55
For
q
>
2,andwith
v
+
=
max
f
v
;
0
g
,
Z


v

v
q

1
+
dx


Z

v
q
+
dx
=
Z

Gv
q

1
+
dx
:
ByGreen'sformula,weget

(
q

1)
Z

v
q

2
+
jr
v
+
j
2
dx
+
Z
@

v
q

1
+
Hds


Z

v
q
+
dx
=
Z

Gv
q

1
+
dx
:
Then
(
q

1)
Z

v
q

2
+
jr
v
+
j
2
dx
+

Z

v
q
+
dx

Z
@

v
q

1
+
j
H
j
ds
+
Z

j
G
j
v
q

1
+
dx
;
4(
q

1)
q
2
Z

jr
(
v
q
2
+
)
j
2
dx
+

Z

v
q
+
dx


Z
@

v
q
+
ds
+

(

)
Z
@

j
H
j
q
ds
+

Z

v
q
+
dx
+
Z

j
G
j
q
dx
;
whichindicatesthat
j
v
q
2
+
j
W
1
;
2
(

)

C
(
j
G
j
L
q
(

)
+
j
H
j
L
q
(
@

)
)
:
So
j
v
+
j
L
nq
n

2
(

)

C
(
j
G
j
L
q
(

)
+
j
H
j
L
q
(
@

)
)
:
Onecanprovethesameinequalityfor
v

,whichcompletestheproof.

UsingLemma4.3.2androutinebootstrapargument,wehavethatforany
m
,
jH
(
W
";
P
)
j
C
m
(

)
and
j
H
(
W
";
P
)
j
C
m
(

)
areboth
O
(
e


"
).Let
e
H
(
W
";
P
)
=
H
(
W
";
P
)
+
H
(
W
";
P
)
+
K
";
P
;
(4.3.15)
56
thenlet
e
W
";
P
=
W
";
P

e
H
(
W
";
P
)
;
(4.3.16)
where
K
";
P
isaconstantfored
P
with
K
";
P
=
0forsome
P
=
P

,suchthatthemass
R

e
W
";
P
dx
remainsconstantaswevary
P
2
e

k
.Onecaneasilycheckthat
@
e
W
";
P
@
n
=
@

e
W
";
P
@
n
=
0
:
Bytheexponentiallydecayingpropertyofgroundstatesandthefactthat
H
(
W
";
P
)
;
H
(
W
";
P
)
=
O
(
e


"
),onecaneasilyseethat
K
";
P
=
O
(
e


"
).Byusingthesymmetryofthegroundstates,one
alsocancheckthat
D
P
K
";
P
=
O
(
e


"
).Moreprecisely,bydirectcomputation,onehas
D
P
K
";
P
=
1
j

j
R

D
P
W
";
P


D
P
H
(
W
";
P
)
+
D
P
H
(
W
";
P
)

dx
.First,itiseasytoverifythat
D
P
H
(
W
";
P
)
=

x
P
i
H
(
W
";
p
i
)
=
O
(
e


"
),similarlywehave
D
P
H
(
W
";
P
)
=
O
(
e


"
).Itremainstoesti-
mate
R

D
P
W
";
P
dx
.Since
D
P
W
";
P
=

P
i
r
x
w
";
p
i
,itissu

cienttoshowthat
R

r
x
l
w
";
p
i
dx
=
O
(
e


"
).Notethat
r
x
l
w
isoddin
x
l
and
B
(
p
i
;
2)
ˆ

,sowehave
R
B
(
p
i

2)
r
x
l
w
";
p
i
dx
=
0.By
theexponentiallydecayingpropertyofthegroundstate,itisalsoclearthat
R

n
B
(
p
i

2)
r
x
l
w
";
p
i
dx
=
O
(
e


"
).Itfollowsimmediatelythat
D
P
K
";
P
=
O
(
e


"
).Moreover,since
K
";
P
isaconstantin
x
,
onethatforany
m
,
j
e
H
(
W
";
P
)
j
C
m
(

)
=
O
(
e


"
)
:
(4.3.17)

 
"
(
P
)
=
e
W
";
P
:
(4.3.18)
Thentheapproximatelyinvariantweconstructis
M
"
=
 
"
(
e

k
)
:
(4.3.19)
57
Remark
4.3.3
.
Everypointonthemanifold
M
"
hasthesamemass.Sinceweareonlybuildingan
approximatelyinvariantmanifold,thismayseemunnecessary.However,aswewillseeinSection
4.3.5,thisisimportantforthetransformedequation.
4.3.2SpectralanalysisofthelinearizedAllen-Cahnoperator
Let
L
0
v
:
=
v


v
+
h
0
(
w
)
v
;
H
2
(
R
n
)
!
L
2
(
R
n
)
;
e
L
"
0
v
:
=
"
2

v


v
+
h
0
(
]
W
";
P
)
v
;
H
2
(

)
!
L
2
(

)
;
(4.3.20)
and
B
u
(
v
;
v
)
=
Z

(
"
2
jr
v
j
2
+

v
2

h
0
(
u
)
v
2
)
dx
(4.3.21)
Recallthatthegroundstate
w
isassumedtobenon-degenerate,whichmeansthatthereexists
b
>
0
;
1
>
0,suchthat
˙
(
L
0
)
\
(

b
;
1
)
=
f
0
;
1
g
with0havingmultiplicity
n
.Infact

1
issimple,and
thecorrespondingeigenfunction
V
isradiallysymmetricandexponentiallydecayingwiththesame
rateasthatof
w
.Also,thecorrespondingeigenspacewithrespectto0is
span
f
@
w
@
x
i
;
i
=
1
;

;
n
g
.
Wewillusethespectrumof
L
0
toestimatethespectrumof
e
L
"
0
.Nowweconsidertheeigenvalue
problem
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
e
L
"
0
˚
=


"
˚;
in

;
@˚
@
n
=
0on
@

:
(4.3.22)
Firstwelet
W
ij
"
=
˜
i
w
x
j
(
x

p
i
"
)
;
V
i
=
˜
i
V
(
x

p
i
"
)
;
(4.3.23)
58
where
˜
i
isasmoothcut-o

functionsatisfying
˜
i
(
x
)
=
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
1
;
x
2

i
:
=
f
x
:
d
(
x
;
p
i
)

1
2
;
d
(
x
;@

)

1
4

g
;
0
;
x
2
Ÿ

i
:
=
f
x
:
d
(
x
;
p
i
)

3
4
;
d
(
x
;@

)

1
8

g
:
(4.3.24)
Lemma4.3.4.
B
e
W
";
P
(
W
ij
"
;
W
ij
"
)
h
W
ij
"
;
W
ij
"
i
=
O
(
e


"
)
;
B
e
W
";
P
(
V
i
;
V
i
)
h
V
i
;
V
i
i
=


1
+
O
(
e


"
)
;
(4.3.25)
Proof.
Here,wejustprovethestatement,theproofoftheotheroneissimilar.
B
e
W
";
P
(
W
ij
"
;
W
ij
"
)
=
B
w
";
p
i
(
W
ij
"
;
W
ij
"
)
+
Z

(
h
0
(
w
";
p
i
)

h
0
(
e
W
";
P
))(
W
ij
"
)
2
dx
=
Z

i
(
"
2
(
r
W
ij
"
)
2
+

(
W
ij
"
)
2

h
0
(
w
";
p
i
)(
W
ij
"
)
2
)
dx
+
Z

(
h
0
(
w
";
p
i
)

h
0
(
e
W
";
P
))(
W
ij
"
)
2
dx
+
Z

n
(

i
[
Ÿ

i
)
(
"
2
(
r
W
ij
"
)
2
+

(
W
ij
"
)
2

h
0
(
w
";
p
i
)(
W
ij
"
)
2
)
dx
=
I
+
II
+
III
:
(4.3.26)
Obviouly,
I
=
O
(
e


"
).Bydirectcomputation,wehave
j
II
j
Z

C
j
e
W
";
P

w
";
p
i
j
(
W
ij
"
)
2
dx

Z

n
Ÿ

i
(
C
j
X
l
,
i
w
";
p
l
j
+
O
(
e


"
))(
W
ij
"
)
2
dx

Ce


"
Z

(
W
ij
"
)
2
dx
:
(4.3.27)
59
Wearetheproofbyestimating
III
.Firstitiseasytocheckthat
W
ij
"
;"
r
W
ij
"
=
O
(
e


"
)
in

n
(

i
[
Ÿ

i
)and
j
"
r
˜
i
j
L
2
=
o
(1).Combiningwiththefactthat
h
0
(0)
=
0,wehave
j
III
j
O
(
e


"
)
h
W
ij
"
;
W
ij
"
i
.

Sofarweseethat
e
L
"
0
has
k
positiveeigenvaluesand
nk
eigenvaluesnear0.In[20],theauthors
usedascalingandlimitingprocesstoshowthatthereisan
O
(1)spectralgapbetweenthenegative
spectrumandthespectrumnearzeroforthelinearizedAllen-Cahnoperatorobtainedbylinearizing
ataboundarymulti-spikestate.In[95],theauthorsusedasimilartechniquetoshowsuchspectral
gapfortheoperatorobtainedbylinearizingatasingleinterior-spikestate.Thesameargument
adaptshere.Performingachangeofvariable
y
=
x

p
i
"
,where
p
i
isthecenterofoneofthespikes,
extendingtheeigenfunctionfor(4.3.22)to
R
n
,thenletting
"
tendtozero(see[20,72,73]for
moredetailsaboutthisprocess),onethattheeigenvalueproblem(4.3.22)convergesalonga
sequenceweaklyin
H
1
(
R
n
)tothelimitingeigenvalueproblem

˚
1


1
+
h
0
(
w
)
˚
1
=

1
˚
1
in
R
n
:
(4.3.28)
Ifwelet
y
=
x

z
"
for
z
2

and
z
<
f
p
i
g
k
i
=
1
andthenperformthesamelimitingprocess,wethe
limitingeigenvalueproblem

˚
1


1
=

1
˚
1
in
R
n
:
(4.3.29)
Inboth(4.3.28)and(4.3.29),thenegativespectrumisboundedawayfrom0.Therefore,for
"
su

cientlysmalltherestofthespectrumof
e
L
"
0
liesin(

;

C
]forsomeconstant
C
>
0.
60
4.3.3SpectralanalysisofthelinearizedCahn-Hilliardoperator
Considertheeigenvalueproblem:
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:


(
"
2

˚


+
h
0
(
e
W
";
P
)
˚
)
=

"
˚
in

;
@˚
@
n
=
@

˚
@
n
=
0on
@

:
(4.3.30)
First,werecallthevariationalcharacterizationoftheeigenvaluesofCahn-Hilliardequation
([13]Theorem5):


"
n
=
min
n
max
n
B
e
W
";
P
(
v
;
v
)
h
(


)

1
)
v
;
v
i
;
(4.3.31)
wheremax
n
isoverall
v
2
T
n
,andmin
n
isoverall
n
-dimensionalsubspaces
T
n
of
‹
H
1
(

):
=
f
v
:
v
2
H
1
(

)
;
R

vdx
=
0
g
.Also,(


)

1
ison
‹
H
1
(

)by(


)

1
v
=

for
v
2
‹
H
1
(

)ifand
onlyif

2
‹
H
1
(

)isthesolutionto
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:



=
v
in

;

@
n
=
0on
@

:
Now,weconstructatestfunctionfor(4.3.31)usingtheunstableeigenfunction
V
.Let

bea
functionsatisfying
supp
(

)
ˆf
x
:
3
8

j
x
j
1
2

g
;
and
Z
R
n

(
x
)
=
Z
R
n
V
(
x
)
;
thenlet


(
x
)
=

n

(

x
)
:
(4.3.32)
61
Itiseasytoverifythat
j


(
x
)
j
2
L
2
(
R
n
)
=

n
j

(
x
)
j
2
L
2
(
R
n
)
;
jr


(
x
)
j
2
L
2
(
R
n
)
=

n
+
2
jr

(
x
)
j
2
L
2
(
R
n
)
:
Welet
V
i
=

i
(
x
)
V
(
x

p
i
"
)


i

(
x
)
:
(4.3.33)
Here,

i
(
x
)isansmoothcut-o

functionsatisfying

i
(
x
)
=
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
1
;
x
2f
x
2

:
d
(
x
;
p
i
)

1
4

g
;
0
;
x
2f
x
2

:
d
(
x
;
p
i
)

3
8

g
;
(4.3.34)
and

i

(
x
)
=
˝
i


(
x

p
i
"
)
;
(4.3.35)
where
˝
i
isaconstantsuchthat
R

V
i
dx
=
0.Notethat
˝
i
=
O
(1)and
supp
(

i

)
ˆ

x
:
3

8


d
(
x
;
p
i
)


2


:
Soifwechoose

suchthat1
<
"

<
1
+
"
,then
supp
(

i

)and
supp
(

i
)aredisjoint,andfurthermore
supp
(

i

)iscontainedin

for
"
smallenough.Then,onehas
B
e
W
";
P
(
V
i
;
V
i
)
=
B
e
W
";
P
(

i
(
x
)
V
(
x

p
i
"
)
;
i
(
x
)
V
(
x

p
i
"
))
+
B
e
W
";
P
(

i

(
x
)
;
i

(
x
))
=
(


1
+
O
(
e


"
))
h

i
(
x
)
V
(
x

p
i
"
)
;
i
(
x
)
V
(
x

p
i
"
)
i
+
O
(

n
)
:
(4.3.36)
Itremainstoestimate
h
(


)

1
V
i
;
V
i
i
,whichisthepointofthefollowinglemma.
62
Lemma4.3.5.
0
<
h
(


)

1
v
;
v
i
C
h
v
;
v
i
,foranyv
2
‹
H
1
(

)
,v
,
0
forsomeconstantC
>
0
.
Proof.
(


)asaoperatoractingonfunctionswithmean-valuezeroandhomogenousNeumann
boundaryconditionisapositiveoperator,so
h
(


)

1
v
;
v
i
>
0.Let

bethemeanvaluezero
solutionoftheequation
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
(


)

(
x
)
=
v
(
x
)in

;

@
n
=
0on
@

:
(4.3.37)
Bydirectcalculation,wehave
h
(


)

1
v
;
v
i
=
Z

v

dx
=
Z

jr

j
2
dx
:
(4.3.38)
Forany
>
0,
Z

v

dx

1
4

Z

v
2
dx
+

Z


2
dx

1
4

Z

v
2
dx
+

C
Z

jr

j
2
dx
:
(4.3.39)
Bychoosing

smallenough,itfollowsimmediatelythat
Z

v

dx

C
Z

v
2
dx
:

Since
V
i
isconstructedbyscaling
V
and


by
"
,andsince
V
decaysexponentially,onehas
<
(


)

1
V
i
;
V
i
>
=
O
(
"
2
).OnemayarigorousproofofthisfactintheproofofLemma
4.4.3inthecurrentpaper.Combining(4.3.31),(4.3.36)andLemma4.3.5,wehavethatthereare
k
positiveeigenvalues

i
"
(1

i

k
)greaterthan
C

1
"
2
withcorrespondingeigenfunctionsdenotedby
63
V
i
"
forsomeconstant
C
.Similarly,onecanprovethatthereare
nk
eigenvalueswhichare
O
(
e


"
).
Ifthereareotherpositiveeigenvaluesawayfromzero,thenby(4.3.31)onecanseethattherewill
beextrapositiveeigenvaluesawayfrom0forthecorrespondinglinearizedAllen-Cahnoperator,
whichisacontradiction.Therefore,thereareexactly
k
positiveeigenvaluesboundedawayfrom0
for(4.3.30).Moreover,ifthereisanegativeeigenvaluefor(4.3.30)oforder
o
(
1
"
2
),thenby(4.3.31)
andLemma4.3.5,oneseesthattherewillbeanegativeeigenvalueapproachingzeroas
"
!
0for
thecorrespondinglinearizedAllen-Cahnoperator,whichisagainacontradiction.Thus,therestof
thespectrumisin(

;

¯
b
"
2
)forsome
¯
b
>
0.Forconvenience,wedenotethesizeofthisspectral
gapby
b
"
=
b
"
2
.
Notethat


(
"
2

e
W
ij
"


e
W
ij
"
+
h
0
(
e
W
";
P
)
e
W
ij
"
)
=
O
(
e


"
)
;
(4.3.40)
where
e
W
ij
"
=
D
j
(
w
";
p
i

e
H
(
w
";
p
i
))
;
(4.3.41)
and
e
W
ij
"
almosttheboundaryconditions.Thus,wemayuse
span
f
e
W
ij
"
,
i
=
1
;

;
k
;
j
=
1
;

;
n
g
toapproximatethecenterspacefor
L
";
P
,where
L
";
P
v
:
=


(
"
2

v


v
+
h
0
(
e
W
";
P
)
v
)with
thedomain

v
:
v
2
W
4
;
q
"
(

)
;
@
v
@
n
=
@

v
@
n
=
0

:
64
4.3.4
M
"
isapproximatelystationary
Firstofall,wearescaledSobolevspace
W
k
;
q
"
asthefamilyof
k

th
di

erentiablefunctions
(indistributionsense)endowedwiththerescalednorm
jj
W
k
;
q
"
=
X
0



2
j
"
j

j
D

(

)
j
L
q
(

;"

n
d

)
:
Thenwechooseourphasespace
X
:
=

u
2
W
2
;
q
"
:
@

u
@
n
=
@
u
@
n
=
0

withlarge
q
.Recallthatwe

L
";
P
as
L
";
P
v
:
=


(
"
2

v


v
+
h
0
(
e
W
";
P
)
v
)(4.3.42)
withthedomain

v
:
v
2
W
4
;
q
"
(

)
;
@
v
@
n
=
@

v
@
n
=
0

:
Itisclearthat
L
";
P
generatesananalyticsemigroup
e
tL
";
P
:
L
q
"
!
L
q
"
,where
L
q
"
=
W
0
;
q
"
.Byusing
thespectralpropertyof
L
";
P
andtheembeddingtheoremoffractionalpowerspaces(see[51]),we
have
j
e
tL
";
P
j
(
L
q
"
;
L
q
"
)

Ce
Ct
"
2
;
j
e
tL
";
P
j
(
L
q
"
;
W
2
;
q
"
)

C
(1
+
(
t
"
2
)

1
2
)
e
Ct
"
2
:
(4.3.43)
Wemodify
h
tomakesure(4.3.4)generatesawgloballyintime.Let
Ÿ
h
(
u
)
=

(
u
)
h
(
u
),where

(
s
)isasmoothcut-o

functionsatisfying

(
s
)
=
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
1
;
j
s
j
<
max
w
+
1
;
0
;
j
s
j
>
max
w
+
2
:
65
Wenowconsidertheequation
u
t
=


(
"
2

u


u
+
Ÿ
h
(
u
))(4.3.44)
withthesameNeumannboundaryconditions.Forconvenience,westillkeepthenotation
h
instead
of
Ÿ
h
.
Foranyed
P
2
e

k
,weconsidertheinitialvalueproblem
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
:
u
t
=


(
"
2

u


u
+
h
(
u
))in


(0
;
1
)
;
@

u
@
n
=
@
u
@
n
=
0on
@


(0
;
1
)
;
u
(

;
0)
=
e
W
";
P
in

:
(4.3.45)
Lemma4.3.6.
Letubethesolutionto(4.3.45).Thenu
j
u

e
W
";
P
j
X

C
(
t
+
"
t
1
2
)
e


"
e
Ct
"
2
:
(4.3.46)
Proof.
Let
v
bethedi

erence
u

e
W
";
P
,then
v

8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
v
t
=


(
"
2

v


v
+
h
0
(
e
W
";
P
)
v
)
+
(
"
2

e
H

e
H
+
h
(
e
W
";
P
+
e
H
)

h
(
e
W
";
P
+
v
)
+
h
0
(
e
W
";
P
)
v

R
";
P
)
=
L
";
P
v
+
r
(
v
)
;
@

v
@
n
=
@
v
@
n
=
0
;
v
(

;
0)
=
0
:
(4.3.47)
66
UsingLemma4.3.1,(4.3.17)andthefactthat
h
hasbeenweget
j
r
(
v
)
j
L
q
"

C
(
j
v
j
X
+
e


"
)
:
(4.3.48)
Usingthevariationofconstantsformula,wewritethesolution
v
as
v
=
Z
t
0
e
L
";
P
(
t

s
)
r
(
v
)
ds
:
(4.3.49)
Applying(4.3.43)to(4.3.49),wehave
j
v
j
X

Z
t
0
C
(1
+
(
t

s
"
2
)

1
2
)
e
C
(
t

s
)
"
2
(
j
v
j
X
+
e


"
)
ds
:
ThenitfollowsdirectlybyGronwall'sinequalitythat
j
v
j
X

C
(
t
+
"
t
1
2
)
e


"
e
Ct
"
2
:
(4.3.50)

4.3.5Transformationofthephasespace
ObservethatthelinearizedCahn-Hilliardoperator
L
";
P
isnotself-adjointin
L
2
,soitishardto
showthenormalhyperbolicitywhichleadstotheexistenceofantrulyinvariantmanifolddirectly.
Tohandlethis,weintroducethefollowingtransformation.Forany
˚
2
X
satisfying
R

˚
dx
=
0,
67
let
 
bethesolutiontotheequation
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
:


 
=
˚
in

;
@ 
@
n
=
0on

;
R

 
dx
=
0
:
(4.3.51)

A
:
‹
W
4
;
q
"
\f
u
2
W
4
;
q
"
:
@
u
@
n
=
0on

g!
‹
W
2
;
q
"
by
A
 
=
˚
,where
‹
W
k
;
q
"
=
W
k
;
q
"
\f
u
:
R

udx
=
0
g
.Forany
˚
2
‹
W
2
;
q
"
,let
˚
]
2
‹
W
3
;
q
"
satisfy
A
1
2
˚
]
=
˚
,i.e.
˚
]
=
A

1
2
˚
.Onecancheck
thatthespectrumof
A
liesin(0
;
1
)(Aisapositiveoperator),so
A

1
2
is([51]).Let
u
(
t
;

)bethesolutionto(4.3.44).SincetheCahn-Hilliardequationconservesthemass,onecan

u
]
=
A

1
2
(
u

q
(
u
)),where
q
(
u
)
=
1
j

j
R

udx
isaconstantin
t
and
x
.Itisclearthat
u
]

8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
((


)
1
2
u
]
)
t
=


(
"
2

((


)
1
2
u
]
)


((


)
1
2
u
]
)
+
h
((


)
1
2
u
]
+
q
(
u
)))in

;
@
(


)
1
2
u
]
@
n
=
@
(


)
3
2
u
]
@
n
=
0on
@

:
(4.3.52)
Notethatallpointsontheapproximatelyinvariantmanifold
M
"
havethesamemeanvalue,
whichwedenoteby
q
"
.Let
M
]
"
=
A

1
2
(
M
"

q
"
):
=
f
A

1
2
(
 
"
(
P
)

q
"
):
P
2
e

k
g
:
(4.3.53)
Since
D
P
R

 
"
(
P
)
dx
=
R

D
P
 
"
(
P
)
dx
=
0,thetangentspaceof
M
]
"
isalsoas
A

1
2
T
M
"
.Inordertoobtainatrulyinvariantmanifoldnear
M
"
forthew
u
(
t
;

)generated
by(4.3.44),weatrulyinvariantmanifoldnear
M
]
"
withzeromassfor
u
]
(
t
;

),thenbythe
68
injectivityof
A

1
2
,oneobtainsatrulyinvariantmanifoldfor
u
(
t
;

)near
M
"
.Naturally,wechoose
thephasespace
X
]
for
u
]
tobe
‹
W
3
;
q
"
.Firstofall,onecaneasilycheckthat
M
]
"
isapproximately
stationaryfor
u
]
(
t
;

).Infact,byLemma4.3.6,onehas
j
u
]

t
;
A

1
2
(
e
W
";
P

q
"
)


A

1
2
(
e
W
";
P

q
"
)
j
W
3
;
q
"
=
j
A

1
2

u
(
t
;
e
W
";
P
)

e
W
";
P

j
W
3
;
q
"

C
j
u
(
t
;
e
W
";
P
)

e
W
";
P
j
X

C
(
t
+
"
t
1
2
)
e


"
e
Ct
"
2
(4.3.54)
Linearizingequation(4.3.52)at
A

1
2
(
e
W
";
P

q
"
),weobtain

u
]
t
=
(


)
1
2
(
"
2

(


)
1
2

u
]


(


)
1
2

u
]
+
h
0
(
e
W
";
P
)(


)
1
2

u
]
)
=
L
]
";
P

u
]
;
(4.3.55)
where
L
]
";
P
:
=
(


)
1
2

(
"
2



+
h
0
(
e
W
";
P
))

(


)
1
2
;
(4.3.56)
withthedomain
f
v
:
v
2
W
4
;
q
"
;
@
(


)
1
2
v
@
n
=
@
(


)
3
2
v
@
n
=
0
g
:
Bythemaintheoremsof[13],weknowthat
L
]
";
P
isself-adjointin
L
2
"
(aftertakingtheself-
adjointextension)andtheeigenvalueproblemfor
L
]
";
P
isequivalenttotheeigenvalueproblemfor
L
";
P
.Let
e
W
ij
;]
"
=
A

1
2
D
p
ij
e
W
ij
"
;
V
i
;]
"
=
A

1
2
V
i
"
:
(4.3.57)
69
Thesearesinceboth
D
p
ij
e
W
ij
"
and
V
i
"
havemean-valuezero.Also,onecancheck
L
]
";
P
e
W
ij
;]
"
=
O
(
e


"
)
;
(4.3.58)
and
L
]
";
P
V
i
;]
"
=

i
"
V
i
;]
"
:
(4.3.59)
Withoutlossofgenerality,weassumethat
j
e
W
ij
;]
"
j
L
2
"
=
j
V
i
;]
"
j
L
2
"
=
1.Furthermore,itisclearthat
L
]
";
P
generatesananalyticsemigroup
e
L
]
";
P
t
withsimilarpropertiesto
e
L
";
P
t
.
4.3.6Splittingspace
X
]
alongthemanifold
M
]
"
Spectralanalysisof
L
]
";
P
yieldsthattheapproximateunstablespaceof
L
]
";
P
is
span
f
V
i
;]
"
g
andthe
approximatecenterspaceof
L
]
";
P
is
span
f
e
W
ij
;]
"
:1

i

k
;
1

j

n
g˘
T
 
]
"
(
P
)
M
]
"
.Wewilluse
thistoconstructcenter-stableandcenter-unstablemanifoldsfor
M
]
"
.
Remark
4.3.7
.
e
W
ij
;]
"
doesnotsatisfytheboundaryconditions,butwecanmodifyitslightlywith
thesamewaythatwe
W
";
P
inSection4.3.1.Notethatthewillbe
O
(
e


"
).
Forconvenience,wewillkeepthenotation
e
W
ij
;]
"
forthefunction.
Since
L
]
";
p
isself-adjoint,wehave

l
"
h
e
W
ij
;]
"
;
V
l
;]
"
i
=
h
e
W
ij
;]
"
;
L
]
";
p
V
l
;]
"
i
=
h
L
]
";
p
e
W
ij
;]
"
;
V
l
;]
"
i
=
O
(
e


"
)
;
whichimpliesthat
h
e
W
ij
;]
"
;
V
l
;]
"
i
=
O
(
e


"
)
;
(4.3.60)
where
h
;
i
denotesthe
L
2
"
innerproduct,i.e.,
h
f
;
g
i
=
R

fg
"

n
dx
.Since
h
e
W
ij
;]
"
;
e
W
lm
;]
"
i
=
h
(


)

1
e
W
ij
"
;
e
W
lm
"
i
,
70
wehavethefollowinglemma
Lemma4.3.8.
h
e
W
uv
;]
"
;
e
W
lm
;]
"
i
=
o
(
"
)
if
(
l
;
m
)
,
(
u
;
v
)
:
(4.3.61)
Proof.
Wejuststatethislemmahere,theproofissimilartotheproofofLemma4.4.3inthelast
sectionofthecurrentpaper.

Let
¯
V
i
;]
"
bethecomponentof
V
i
;]
"
whichisorthogonal(inthesenseof
L
2
"
)to
span
f
e
W
ij
;]
"
g
.
Clearly
j
¯
V
i
;]
"

V
i
;]
"
j
X
]
=
O
(
e


"
).Thenwedecompose
X
]
as
X
]
=
X
c
";
P

X
s
";
P

X
u
";
P
,where
X
c
";
P
=
span
f
e
W
ij
;]
"
;
i
=
1
;

;
k
;
j
=
1
;

;
n
g
;
X
u
";
P
=
span
f
¯
V
i
;]
"
;
i
=
1
;

;
k
g
;
X
s
";
P
=
n
v
:
Z

v
Ÿ
vdx
=
0
;
foranyŸ
v
2
X
c
";
P

X
u
";
P
o
:
(4.3.62)
Denotetheassociatedprojectionmapsby


";
P
,

=
c
;
u
;
s
.Bythesmoothnessof
e
W
ij
;]
"
and
V
i
;]
"
,we
havethat


";
P
,

=
c
;
s
;
u
,aresmoothin
P
.Usingthe
L
2
"
-orthogonalityanddimensionality
ofcenterandunstablespaces,wehavetheboundednessoftheseprojections.Followingfrom
thecompactnessof
M
"
,weimmediatelyhavethat


";
P
areuniformlyboundedanduniformly
Lipschitzin
P
.Clearlyalltheboundsareindependentof
"
for
"
smallenough.Furthermore,we
havethatforsmallenough
"
,thereexists
B
>
0,independentof
"
,suchthat
j


";
P
j
C
m

e

k
;
L
(
x
)


B
;
foranypositiveinteger
m
:
(4.3.63)
71
4.3.7Trichotomy
Toinvestigatethelinearizedwatasolution
u
]
to(4.3.52)withinitialcondition
A

1
2
e
W
";
P
,we
consider
¯

u
]
t
=
(


)
1
2

(
"
2



+
h
0
((


)
1
2
u
]
))

(


)
1
2
¯

u
]
:
(4.3.64)
Togetestimatesfor
¯

u
]
,wealsoconsider

u
]
t
=
L
]
";
P

u
]
=
(


)
1
2

(
"
2



+
h
0
(
]
W
";
P
))

(


)
1
2

u
]
:
(4.3.65)
Lemma4.3.9.
If

]
u
(0)
=
¯

u
]
(0)
,then
j
¯

u
]


u
]
j
X
]

C
(
t
+
"
t
1
2
)(
t
+
"
3
2
t
1
4
)
e


"
e
Ct
"
2
j
¯

u
]
(0)
j
X
]
.
Proof.
Firstitiseasytoprovethat
j
¯

u
]
j
X
]
;
j

u
]
j
X
]

Ce
Ct
"
2
j
¯

u
]
(0)
j
X
]
:
(4.3.66)
Let
v
=
¯

u
]


u
]
.Then
v

8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
v
t
=
L
]
";
P
v

(


)
1
2
[(
h
0
(
e
W
";
P
)

h
0
(
u
))(


)
1
2
¯

u
]
]
;
v
(0)
=
0
:
Usingthevariationofconstantsformula,weget
v
=
Z
t
0
e
L
]
";
P
(
t

s
)
(


)
1
2
[

(
h
0
(
e
W
";
P
)

h
0
(
u
))

1
2
¯

u
]
]
ds
:
72
Notethat
j
e
W
";
P

u
j
W
2
;
q
"

C
(
t
+
"
t
1
2
)
e


"
e
Ct
"
2
,sofor
q
largeenough,wehave
j
e
W
";
P

u
j
C
1

C
(
t
+
"
t
1
2
)
e


"
e
Ct
"
2
;
whichimpliesthat
j
(


)
1
2
[(
h
0
(
e
W
";
P
)

h
0
(
u
))(


)
1
2
¯

u
]
]
j
L
q
"

C
(
t
+
"
t
1
2
)
e


"
e
Ct
"
2
j
¯

u
]
j
W
2
;
q
"
:
(4.3.67)
Clearly,
j
e
L
]
";
P
t
j
L
(
L
q
"
;
X
]
)

C
(1
+
(
t
"
2
)

3
4
)
e
Ct
"
2
:
Thenitfollowsimmediatelythat
j
v
j
X
]

C
(
t
+
t
1
2
"
)(
t
+
"
3
2
t
1
4
)
e


"
e
Ct
"
2
j
¯

u
]
(0)
j
X
]
:

Decompose

u
]
as

u
]
=
"
D
e
W
]
";
P

a
(
t
)
+
X
b
i
(
t
)
¯
V
i
;]
"
+
W
s
(
t
)
;
(4.3.68)
where
D
meansthegradientwithrespectto
P
.Itiseasytocheckthat
j

u
]
j
X
]

Ce
Ct
"
2
j

u
]
(0)
j
X
]
and
j

u
]
j
L
q
"

Ce
Ct
"
2
j

u
]
(0)
j
L
q
"
.Thenbytheboundednessoftheprojectionmaps,wehave
j
a
(
t
)
j
Ce
Ct
"
2
j

u
]
(0)
j
L
q
"
;
j
b
i
(
t
)
j
Ce
Ct
"
2
j

u
]
(0)
j
L
q
"
;
j
W
s
(
t
)
j
L
q
"

Ce
Ct
"
2
j

u
]
(0)
j
L
q
"
;
(4.3.69)
73
and
j
W
s
(
t
)
j
X
]

Ce
Ct
"
2
j

u
]
(0)
j
X
]
:
(4.3.70)
Takingtheinnerproductof(4.3.68)with
L
]
";
P
"
D
e
W
]
";
P

a
(
t
),wehave
h

u
]
;
L
]
";
P
"
D
e
W
]
";
P

a
(
t
)
i
=
h
L
]
";
P

u
]
;"
D
e
W
]
";
P

a
(
t
)
i
=
h

u
]
t
;"
D
e
W
]
";
P

a
(
t
)
i
=
h
"
D
e
W
]
";
P

a
0
(
t
)
;"
D
e
W
]
";
P

a
(
t
)
i
+
h
X
i
b
0
i
(
t
)
¯
V
i
;]
"
;"
D
e
W
]
";
P

a
(
t
)
i
:
(4.3.71)
Notethat
j

u
]
j
L
q
"

Ce
Ct
"
2
j

u
]
(0)
j
L
q
"
and
j
L
";
P
"
D
e
W
]
";
P

a
(
t
)
j
L
1

Ce


"
j
a
(
t
)
j
Ce


"
e
Ct
"
2
j

u
]
(0)
j
L
q
"
.
Combiningwith(4.3.69)andLemma4.3.8,weobtain
j
a
(
t
)

a
0
(
t
)
j
Ce


"
e
Ct
"
2
(
j

u
]
(0)
j
2
L
q
"
+
j

u
]
(0)
j
L
q
"
j
b
0
i
(
t
)
j
)
:
(4.3.72)
Since
d
dt
j
a
(
t
)
j
2
=
2
a
(
t
)

a
0
(
t
),byintegratingbothsidesof(4.3.72)weget



j
a
(
t
)
j
2
j
a
(0)
j
2




Ce


"
e
Ct
"
2
j

u
]
(0)
j
2
L
q
"
:
(4.3.73)
Similarly,takingtheinnerproductof(4.3.68)with
L
]
";
P
¯
V
i
;]
"
,wehave
j
b
i
(
t
)

e

i
"
t
b
i
(0)
j
Ce
Ct
"
2
e


"
j

u
]
(0)
j
L
q
"
:
(4.3.74)
74
Nowweconsiderthestabledirection.Since
L
]
";
P
isself-adjoint,onehas
jh
L
]
";
P
W
s
;"
D
e
W
";
P

˝
ij
=
jh
W
s
;
L
]
";
P
"
D
e
W
";
P

˝
ij
Ce


"
j
˝
jj
W
s
j
L
2
"
;
and
jh
L
]
";
P
W
s
;
¯
V
i
;]
"
ij
=
jh
W
s
;
L
]
";
P
¯
V
i
;]
"
i
Ce


"
j
W
s
j
L
2
"
;
(4.3.75)
whichimpliesthat
j

c
";
P
L
]
";
P
W
s
j
L
2
"
+
j

s
";
P
L
]
";
P
W
s
j
L
2
"

Ce


"
j
W
s
j
L
2
"
:
(4.3.76)
Usingthefactthatcenterandunstablebundlesaredimensional,wehave
j

c
";
P
L
]
";
P
W
s
j
X
]
+
j

s
";
P
L
]
";
P
W
s
j
X
]

Ce


"
j
W
s
j
X
]
:
(4.3.77)
Write(4.3.65)as
"
D
e
W
";
P

a
0
(
t
)
+
X
b
0
i
(
t
)
¯
V
i
;]
"
+
d
dt
W
s
(
t
)
=
L
]
";
P
"
D
e
W
";
P

a
(
t
)
+
X
b
i
(
t
)
L
]
";
P
¯
V
i
;]
"
+
L
]
";
P
W
s
(
t
)
:
(4.3.78)
Applying

s
";
P
tobothsides,weobtain
d
dt
W
s
(
t
)
=
s
";
P
L
]
";
P
W
s
+
s
";
P
(
L
]
";
P
"
D
e
W
";
P

a
(
t
)
+
X
b
i
(
t
)
L
]
";
P
¯
V
i
;]
"
)
:
(4.3.79)
By(4.3.77),weseethat

s
";
P
L
]
";
P
:
X
s
";
P
!
X
s
";
P
isasmallperturbationof
L
]
";
P
,so

s
";
P
L
]
";
P
generatesasemigroup.Thenbythespectralgapfor
L
]
";
P
andperturbationtheory,wehave
j
W
s
(
t
)
j
X

e

b
"
t
j
W
s
(0)
j
X
+
Ce


"
e
Ct
"
2
j

u
(0)
j
L
q
:
(4.3.80)
75
Denotethewgeneratedby(4.3.52)by
T
]
t
"
.Combining(4.3.73),(4.3.74),(4.3.80)and
Lemma4.3.9,weobtainthefollowingtrichotomyproperties:
Lemma4.3.10.
Ift
=
O
(
"
2
)
and
"
issmallenough,for

=
c
;
u
;
sand

,

,wehave
k


";
P
DT
]
t
"
j
X

";
P
k
Ce


"
e
Ct
"
2
;
k

s
";
P
DT
]
t
"
j
X
s
";
P
k
e

b
"
t
+
Ce


"
e
Ct
"
2
;
1

Ce


"
e
Ct
"
2
k
(

c
";
P
DT
]
t
"
j
X
c
";
P
)

1
k

1
k

c
";
P
DT
]
t
"
j
X
c
";
P
k
1
+
Ce


"
e
Ct
"
2
;
k
(

u
";
P
DT
]
t
"
j
X
u
";
P
)

1
k

1

e
Ÿ

"
t

Ce


"
e
Ct
"
2
;
(4.3.81)
where
Ÿ

"
isinsection4.3.2.
Proof.
If

c
";
P

u
]
(0)
=
0,i.e.,
a
(0)
=
0,by(4.3.73),wehave
j
a
(
t
)
j
Ce


"
e
Ct
"
2
j

u
]
(0)
j
L
q
"
;
whichimplies
j

c
";
P

u
]
j
L
q
"

Ce


"
e
Ct
"
2
j

u
]
(0)
j
L
q
"

Ce


"
e
Ct
"
2
j

u
(0)
j
X
:
Since
X
c
";
P
isdimensional,wehave
j

c
";
P

u
]
j
X
]

Ce


"
e
Ct
"
2
j

u
]
(0)
j
X
]
;
whichimplies
k

c
";
P
DT
]
t
"
j
X

";
P
k
Ce


"
e
Ct
"
2
,for

,
c
.
If

u
]
(0)
2
X
c
";
P
,i.e.,

u
]
(0)
=
"
D
e
W
]
";
P

a
(0),by(4.3.73),wehave
j
a
(
t
)
j
(1
+
Ce


"
e
Ct
"
2
)
j
a
(0)
j
;
(4.3.82)
76
whichimplies
j

c
";
P

u
]
j
L
q
"

(1
+
Ce


"
e
Ct
"
2
)
j

u
]
(0)
j
L
q
"
.Againweusethedimensionalityof
thecenterspacetoobtain
k

c
";
P
DT
]
t
"
j
X
c
";
P
k
1
+
Ce


"
e
Ct
"
2
:
(4.3.83)
Observethatwhen
t
=
O
(
"
2
)and
"
issmallenough,(4.3.72)combinedwithLemma4.3.8im-
pliesthat

c
";
P
DT
]
t
"
j
X
c
";
P
isasmallperturbationoftheidentitymap,so

c
";
P
DT
t
"
j
X
c
";
P
isalsoan
isomorphism.Onecanfollowasimilarargumenttoobtain
1

Ce


"
e
Ct
"
2
k
(

c
";
P
DT
]
t
"
j
X
c
";
P
)

1
k

1
:
(4.3.84)
Theotherinequalitiescanbeprovedsimilarly,thisbeinglefttothereaders.Theonlythingwe
wanttopointoutisthatby(4.3.74),

u
";
P
DT
]
t
"
j
X
u
";
P
isasmallperturbationofanisomorphismif
t
=
O
(
"
2
)and
"
issmallenough,whichindicatesthat

u
";
P
DT
]
t
"
j
X
u
";
P
isanisomorphism.

Recallthat
b
"
;
Ÿ

"
=
O
(
1
"
2
).Fored

,wechoose
t
o
=
"
2
K
withKlargeenoughtomake
e

b
"
t
0
smallenoughand
e
Ÿ

"
t
0
largeenough,thenwelet
"
besu

cientlysmall,sothat
M
"
is
anapproximatelystationaryinvariantandapproximatelynormallyhyperbolicmanifoldfor
T
]
t
0
"
.
ThenbythegeneraltheoremweestablishedinSection4.2,wehavealocallytrulyinvariantmani-
foldnear
M
"
for
T
]
t
0
"
byintersectingthecenter-stablemanifoldandthecenter-unstablemanifold
of
M
"
.However,duetothenon-uniquenessofcentermanifolds(di

erentgivedif-
ferentcentermanifolds),wecannotconcludethatthecentermanifoldfor
T
]
t
0
"
isinvariantunder
thew
T
]
t
"
.Thecorrectwayshouldbetomodifythevectortogeneratea
w,thenprovetheexistenceofatrulyinvariantmanifoldforthew.
77
4.3.8ofvector
Inthissection,wetake
t
0
=
"
2
K
suchthat
M
"
isanapproximatelystationaryinvariantandapprox-
imatelynormallyhyperbolicmanifoldfor
T
]
t
0
"
.Wealsoassumethattheboundaryof


theconditionslistedinSection4.2.FollowingthesetupinSection4.2,we
Ÿ
P
2
e

k
thesame
wayas
e
m
.Forany
x
2
N
(
M
]
"
;
r
),where
N
(
M
]
"
;
r
)isatubularneighborhoodof
M
]
"
,let
m
(
x
)bethe
projectionof
x
into
M
]
"
andlet
P
(
x
)
2
e

k
bethatpointsuchthat
 
]
"
(
P
(
x
))
=
m
(
x
),thenwewrite
x
as
x
=
 
]
"
(
P
(
x
))
+
x
s
(
x
)
+
x
u
(
x
).
Rewrite(4.3.52)as
u
]
t
=
L
]
";
P
u
]
+
r
P
(
u
]
)
;
(4.3.85)
where
r
P
(
u
]
)
=
(


)
1
2

h
((


)
1
2
u
]
)

h
0
(
 
"
(
P
))(


)
1
2
u
]

.
Usingthevariationofconstantsformula,wewrite(4.3.85)as
u
]
=
e
L
]
";
P
t
u
(0)
+
Z
t
0
e
L
]
";
P
(
t

s
)
r
P
(
u
]
)
ds
:
(4.3.86)
ByLemma4.3.6,wehavethatfor
"
smallenough,
 
]
"
(
P
)
=
e
L
]
";
P
t
 
]
"
(
P
)
+
Z
t
0
e
L
]
";
P
(
t

s
)
r
P
(
 
]
"
(
P
))
ds
+
O
(
e


"
)
e
Ct
"
2
;
(4.3.87)
where
O
(
e


"
)isinthe
X
]
sense.
Nowwemodifythevectortoobtainanewequation:
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
Ÿ
u
]
t
=
L
]
";
P
Ÿ
u
]
+
r
P
(Ÿ
u
]
)
+
1
t
0
e
L
]
";
P
t
 
]
"
(
Ÿ
P
)
+
r
P
(
 
]
"
(
Ÿ
P
))

[
1
t
0
e
L
]
";
P
t
 
]
"
(
P
)
+
r
P
(
 
]
"
(
P
))]
;
Ÿ
u
]
(0)
=
x
:
(4.3.88)
78
Usingthevariationofconstantsformula,wewrite(4.3.88)as
Ÿ
u
]
=
e
L
]
";
P
t
x
+
Z
t
0
e
L
]
";
P
(
t

s
)
[
r
P
(Ÿ
u
]
)
+
1
t
0
e
L
]
";
P
s
 
]
"
(
Ÿ
P
)
+
r
P
(
 
]
"
(
Ÿ
P
))

(
1
t
0
e
L
]
";
P
s
 
]
"
(
P
)
+
r
P
(
 
]
"
(
P
)))]
ds
=
e
L
]
";
P
t
x
+
Z
t
0
e
L
]
";
P
(
t

s
)
r
P
(Ÿ
u
]
)
ds
+
t
t
0
e
L
]
";
P
t
 
]
"
(
Ÿ
P
)
+
Z
t
0
e
L
]
";
P
(
t

s
)
r
P
(
 
]
"
(
Ÿ
P
))
ds

(
t
t
0
e
L
]
";
Ÿ
P
t
 
]
"
(
P
)
+
Z
t
0
e
L
]
";
P
(
t

s
)
r
P
(
 
]
"
(
P
))
ds
)
:
(4.3.89)
Denotethewgeneratedby(4.3.88)by
e
T
]
t
"
.Recallthat
T
]
t
"
(
x
)isthewby
(4.3.85).Then
T
]
t
"
(
x
)
=
e
L
]
";
P
t
x
+
R
t
0
e
L
]
";
P
(
t

s
)
r
P
(
T
]
s
"
(
x
))
ds
.Itfollowsfrom(4.3.89)that
e
T
]
t
"
(
x
)
=
T
]
t
"
(
x
)
+
Z
t
0
e
L
]
";
P
(
t

s
)
(
r
P
(
e
T
]
s
"
(
x
))

r
P
(
T
]
s
"
(
x
)))
ds
+
t
t
0
e
L
]
";
P
t
 
]
"
(
Ÿ
P
)
+
Z
t
0
e
L
]
";
P
(
t

s
)
r
P
(
 
]
"
(
Ÿ
P
))
ds

(
t
t
0
e
L
]
";
P
t
 
]
"
(
P
)
+
Z
t
0
e
L
]
";
P
(
t

s
)
r
P
(
 
]
"
(
P
))
ds
)
:
(4.3.90)
RecallthatinSection4.2,weuse
l
todenotehowmuchweshiftthebasepoint.Notethat
j
Ÿ
P

P
j
Cl
forany
P
,onecaneasilyprovebyusingGronwall'sinequalitythat
j
e
T
]
t
"
(
x
)

T
]
t
"
(
x
)
j
C
(
t
+
"
t
1
2
)
e
Ct
"
2
l
;
and
k
D
e
T
]
t
"
(
x
)

DT
]
t
"
(
x
)
k
C
(
t
+
"
t
1
2
)
e
Ct
"
2
l
:
(4.3.91)
For
x
2
W
3
;
q
"
,let¯
x
=
(


)
1
2
x
and(


)
1
2
e
T
]
t
"
(
x
)
=
e
T
t
"
(¯
x
).Thenbydirectcomputation,wehave
r
P
(
e
T
]
t
"
(
x
))

r
P
(
T
]
t
"
(
x
))
=

(


)
1
2

h
0
(
 
"
(
P
))

e
T
t
"
(¯
x
)

T
t
"
(¯
x
)



h
(
e
T
t
"
(¯
x
))

h
(
T
t
"
(¯
x
)


(4.3.92)
79
Wehave
h
(
e
T
t
"
(¯
x
))

h
(
T
t
"
(¯
x
))
=
h
0
(
T
t
"
(¯
x
))(
e
T
t
"
(¯
x
)

T
t
"
(¯
x
))
+
O
(
j
e
T
t
"
(¯
x
)

T
t
"
(¯
x
)
j
2
)
:
(4.3.93)
Recallthat
h
hasbeenbyusingLemma4.3.6,onecancheckthatfor
x
=
 
]
"
(
P
),we
have
j
r
P
(
e
T
]
t
"
(
x
))

r
P
(
T
]
t
"
(
x
))
j
W
1
;
q
"

C

(
t
+
"
t
1
2
)
e
Ct
"
2
e


"
l
+
((
t
+
"
t
1
2
)
e
Ct
"
2
l
)
2

:
(4.3.94)
Therefore,combiningwith(4.3.87),(4.3.90),(4.3.91)and(4.3.94),wehavethatforany
x
=
 
]
"
(
P
),
e
T
]
t
0
"
(
x
)
=
T
]
t
0
"
(
x
)
+
 
]
"
(
g
P
(
x
))

 
]
"
(
P
(
x
))
+
O
(
e


"
l
+
l
2
)
;
D
e
T
]
t
0
"
(
x
)
=
DT
]
t
0
"
(
x
)
+
D
 
]
"
(
g
P
(
x
))

D
 
]
"
(
P
(
x
))
+
O
(
e


"
l
+
l
2
)
:
(4.3.95)
FollowingtheestimatesinSection4.2.1,nowonecanapplyTheorem2.0.3toobtaina
C
m
(
m
dependsonlyonthesmoothnessofthenonlinearityandtheboundary
@

)center-stablemanifold
W
cs
]
"
near
M
]
"
for
e
T
]
t
0
"
.Toobtaintheinvarianceforthew
e
T
]
t
"
,weneedtoverifytheweak
uniformcontinuityin(H5).Infact,wewillprove

Forany
>
0,thereexists
>
0,suchthatforany
x
2
B
(
M
]
"
;
r
)and
t
2
[
t
0
;
t
0
+
"
2

],we
have
j
e
T
]
t
"
(
x
)

e
T
]
t
0
"
(
x
)
j
<
.
Animportantpointhereisthatwewant

and

tobeindependentof
"
.Firstwewrite(4.3.88)
as
Ÿ
u
]
t
=
L
]
";
P
Ÿ
u
]
+
Ÿ
r
P
(Ÿ
u
]
)
;
(4.3.96)
80
wherefor
x
2
X
]
,Ÿ
r
P
(
x
;
t
)
=
r
P
(
x
)
+
1
t
0
e
L
]
";
P
t
 
]
"
(
Ÿ
P
)
+
r
P
(
 
]
"
(
Ÿ
P
))

[
1
t
0
e
L
]
";
P
t
 
]
"
(
P
)
+
r
P
(
 
]
"
(
P
))].
Thenbydirectcalculation,wehave
e
T
]
t
"
(
x
)

e
T
]
t
0
"
(
x
)
=
e
L
]
";
P
t
x

e
L
]
";
P
t
0
x
+
Z
t
0
e
L
]
";
P
(
t

s
)
Ÿ
r
(
e
T
]
s
"
(
x
))
ds

Z
t
0
0
e
L
]
";
P
(
t
0

s
)
Ÿ
r
(
e
T
]
s
"
(
x
))
ds
=
Z
t
t
0
e
L
]
";
P
(
t

s
)
L
]
";
P
e
L
]
";
P
t
0
xds
+
Z
t
t
0
e
L
]
";
P
(
t

s
)
Ÿ
r
(
e
T
]
s
"
(
x
))
ds
+
Z
t
0
0
(
e
L
]
";
P
(
t

s
)

e
L
]
";
P
(
t
0

s
)
)Ÿ
r
(
e
T
]
s
"
(
x
))
ds
=
Z
t
t
0
e
L
]
";
P
(
t

s
)
L
]
";
P
e
L
]
";
P
t
0
xds
+
Z
t
t
0
e
L
]
";
P
(
t

s
)
L
]
";
P
Z
t
0
0
e
L
]
";
P
(
t
0

˝
)
Ÿ
r
(
e
T
]˝
"
(
x
))
d
˝
ds
+
Z
t
t
0
e
L
]
";
P
(
t

s
)
Ÿ
r
(
e
T
]
s
"
(
x
))
ds
=
Z
t
t
0
e
L
]
";
P
(
t

s
)
[
L
]
";
P
e
T
]
t
0
"
(
x
)
+
Ÿ
r
(
e
T
]
s
"
(
x
))]
ds
:
(4.3.97)
Firstnotethat
e
L
]
";
P
t
isananalyticsemigroup,so
k
(

L
]
";
P
)
1
2
e
L
]
";
P
t
0
k
C
(
t
0
"
2
)

1
2
e
Ct
0
"
2
=
CK

1
2
e
CK
:
Observethat
j
Ÿ
r
P
(

;
t
)
j
W
1
;
q
"
isuniformlyboundedonanyboundedsetin
X
]
foranytimeperiod,
therefore(

L
]
";
P
)
1
2
e
T
]
t
0
"
(

)isboundedforanyboundedset

ˆ
X
]
.Againweusethefactthat
k
(

L
]
";
P
)
1
2
e
L
]
";
P
(
t

s
)
k
C
(
t

s
"
2
)

1
2
e
C
(
t

s
)
"
2
toobtain
j
e
T
]
t
"
(
x
)

e
T
]
t
0
"
(
x
)
j
C

1
2
e
C

;
(4.3.98)
whichimpliestheweakuniformcontinuityrequiredin(H5).
81
Therefore
W
cs
]
"
islocallyinvariantunder
e
T
]
t
"
.Notethat
T
]
t
"
(
x
)
=
e
T
]
t
"
(
x
)ifthebasepoint
P
(
x
)is
l
-awayfromtheboundaryof
e

k
,whichimpliesthat
W
cs
]
"
islocallyinvariantunder
T
]
t
"
.
Similarly,wecanconstructacenter-unstablemanifold
W
cu
]
"
for
e
T
]
t
"
.Themanifolds
W
cs
]
"
and
W
cu
]
"
aregraphsoverthestablebundleandunstablebundle,respectively,of
M
]
"
,bothhaving
smallLipschitzconstant.Theintersectionofthesegives
f
M
]
"
,alocallyinvariantmanifoldfor
T
]
t
"
inforwardandbackwardtime.Also,since
f
M
]
"
isagraphover
M
]
"
,itmaybewrittenas

]
"
(
e

k
)
and

]
"
(
P
)

 
]
"
(
P
)
2
X
s
";
P

X
u
";
P
.Furthermore,sincetheoriginalmeasureofnon-invariance,

,
isoforder
O
(
e


"
),
f
M
]
"
isinan
O
(
e


"
)neighborhoodof
M
]
"
inthe
X
]
(
W
3
;
q
"
)topology.Then,
bytheinjectivityof
A
1
2
,thereexistsalocallyinvariantmanifold
f
M
"
=
"
(
e

k
)for
T
t
"
,and
f
M
"
isinan
O
(
e


"
)neighborhoodof
M
"
inthe
W
2
;
q
"
sense.Also,fromthefactthat
A
1
2
actingon
anyfunctioninitsdomaingivesafunctionwithmean-valuezero,themassofeachstatein

"
(
P
)
is
q
"
forany
P
2
e

k
.However,
T
t
"
isthewgeneratedby(4.3.44),wherewethe
nonlinearity
h
.Noticethatwhen
q
islarge,
W
2
;
q
"
isembeddedinto
C
0

,whichindicatesthat
f
M
"
isinan
O
(
e


"
)neighborhoodof
M
"
inthe
L
1
sense.Therefore,
f
M
"
islocallyinvariantunder
thewgeneratedbytheoriginalequation(4.3.4).Insummary,wehavethemainresult:
Theorem4.3.11.
For

,if
"
issu

cientlysmall,thenthereexistsaC
m
(
e

k
;
X
)
(monlydepends
onthesmoothnessofthenonlinearityandtheboundary
@

)manifold
f
M
"
=
"
(
e

k
)
whichis
locallyinvariantunderthegeneratedby(4.3.4).Furthermore,
f
M
"
liesinaO
(
e


"
)
neighborhoodof
M
"
intheL
1
\
W
2
;
q
"
sense.
Remark
4.3.12
.
1.TheproofofTheorem4.2andtheparallelversionofthecenter-unstable
manifoldsin[19]indicatesthat

"

 
"
2
C
m
(
e

k
;
X
)withbounded
C
m
norm,furthermore
j

"

 
"
j
C
0
(
e

k
;
X
)

Ce


"
;
lim
"
!
0
j

"

 
"
j
C
1
(
e

k
;
X
)
!
0
:
(4.3.99)
82
2.Roughlyspeaking,Theorem4.3.11yieldsthatmulti-spikestatesexistuntilthespikesattach
totheboundaryofthedomainortheycollidewitheachother.Wewillshowinthenext
sectionthatthemotionofeachspikeisexponentiallyslow,somulti-spikestatesexistfor
verylongpositiveandnegativetime.
4.4Longtimedynamicson
f
M
"
Sofar,weconstructedalocallyinvariantmanifold
f
M
"
ofinteriormulti-spikestatesfor(4.3.4),
suchthataninteriormulti-spikestatemaintainsuntilaspikeattachestotheboundaryofthedomain
oracollisionbetweenspikesoccurs.Nowweinvestigatethedynamicson
f
M
"
.Bytheinvariance
of
f
M
"
,wehavethatforany
P
2
e

k
,thereexists
˝
"
(
P
)suchthat
D

"
(
P
)


"˝
"
(
P
)

=



"
2

"
(
P
)



"
(
P
)
+
h
(

"
(
P
))

;
(4.4.1)
where
D
meansthederivativewithrespectto
P
and
"˝
"
(
P
)isthevelocityvectorofallspikes.Here
weincludeafactorof
"
with
˝
"
toeliminatethe
1
"
inourcalculationsgeneratedbydi

erentiating

"
.
Let
Ÿ
h
"
(
P
)
=
"
(
P
)

 
"
(
P
)
:
(4.4.2)
Notethat
j
Ÿ
h
"
(
P
)
j
C
0
(
e

k
;
X
)

Ce


"
;
andlim
"
!
0
j
Ÿ
h
"
(
P
)
j
C
1
(
e

k
;
X
)
!
0
:
(4.4.3)
Recallthat
 
"
(
P
)
=
W
";
P

e
H
(
W
";
P
),andrewrite(4.4.1)as
D

"
(
P
)


"˝
"
(
P
)

=

D
 
"
(
P
)
+
D
Ÿ
h
"
(
P
)



"˝
"
(
P
)

=
L
";
P
Ÿ
h
"
+

";
P
(
x
)
;
(4.4.4)
83
where

";
P
=

R
";
P
+
"
2

e
H
(
W
";
P
)


e
H
(
W
";
P
)

(
h
(
W
";
P

e
H
(
W
";
P
)
+
Ÿ
h
"
(
P
))

h
(
W
";
P
)

h
0
(
e
W
";
P
)
Ÿ
h
"
(
P
))

:
(4.4.5)
Itisclearthat
j

";
P
j
L
q
"
(

)

C
"
2
e


"
:
(4.4.6)
Sofar,weonlyknowlim
"
!
0
j
D
Ÿ
h
"
(
P
)
j
X
!
0.Inordertogetaestimatefor
j
D
Ÿ
h
"
(
P
)
j
X
,an
expressionfor
A

1
2
Ÿ
h
"
(
P
)shouldbeobtainedfrom(4.3.52)byusingthenormalhyperbolicity.
Reducingtheequation(4.3.52)totheinvariantmanifold
f
M
]
"
,weget
D

]
"
(
P
)


"˝
"
(
P
)

=
L
]
";
P
Ÿ
h
"
(
P
)
+

]
";
P
(
x
)
:
(4.4.7)
Clearly,
j

]
";
P
(
x
)
j
L
q
"
(

)

C
"
e


"
:
Ourpurposeistogetanexpressionfor
Ÿ
h
]
"
(
P
)withoutinvolving
D

]
"
(
P
)

(
"˝
"
(
P
)),sowe
considerthefollowingdecomposition
L
q
"
=
X
?
";
P

T

]
"
(
P
)
f
M
]
"
;
where
T

]
"
(
P
)
f
M
]
"
=
f
D

]
"
(
P
)

˝
:
˝
2
R
nk
g
and
X
?
";
P
=
f
v
2
L
q
"
(

):
R

vD

]
"
(
P
)

˝
dx
=
0forany
˝
2
R
nk
g
.Thecorrespondingprojectionmap

?
";
P
for
X
?
";
P
is

?
";
P
=
I

D

]
"
(
P
)

D
 
]
"
(
P
)
+
c
";
P
D
Ÿ
h
]
"
(
P
)


1

c
";
P
=

I


c
";
P

I

D
Ÿ
h
]
"
(
P
)

D
 
]
"
(
P
)
+
c
";
P
D
Ÿ
h
]
"
(
P
)


1

c
";
P

:
(4.4.8)
84
By(4.4.3),onecanverifythat

?
";
P
isanduniformlyboundedforany
P
andsmall
enough
"
.Since
L
]
";
p
almostleaveseachsubspace
X

";
P
,

=
c
;
s
;
u
invariant,and
T

]
"
(
P
)
f
M
]
"
is
verycloseto
X
c
";
P
,itisclearthatthereexists
C
>
0,independentof
P
and
"
,suchthat
j
(

?
";
P
L
]
";
p
j
X
s
";
P

X
u
";
P
\
D
(
L
]
";
p
)
)

1
j
L
(
X
?
";
P
;
W
4
;
q
"
(

)
\
D
(
L
]
";
P
))

C
"
2
:
(4.4.9)
Applying

?
";
P
to(4.4.7)andusingthefactthat
Ÿ
h
]
"
(
P
)
2
X
s
";
P

X
u
";
P
,wehave
Ÿ
h
]
"
(
P
)
=
(

?
";
P
L
]
";
p
j
X
s
";
P

X
u
";
P
\
D
(
L
]
";
p
)
)

1

?
";
P
(

]
";
P
)
:
(4.4.10)
Bydirectcomputation,onecancheckthatforsome
C
independentof
P
and
"
,
j
D

?
";
P
j
L
(
L
q
"
)

C
;
j
D

]
";
P
j
L
q
"

C
"
e


"
:
Therefore,itfollowsimmediatelyfrom(4.4.10)that
j
D
Ÿ
h
]
"
(
P
)
j
W
4
;
q
"

C
"
e


"
;
(4.4.11)
whichimpliesthat
j
D
Ÿ
h
"
(
P
)
j
W
3
;
q
"

C
"
e


"
:
(4.4.12)
Wenowderiveanequationfor
˝
";
P
.Since
f
M
"
isconstructedwithallpointshavingequal
mass,onehas
R

D

"
(
P
)
dx
=
0.Then(4.4.1)canberewrittenas

(


)

1

D
 
"
(
P
)
+
D
Ÿ
h
"
(
P
)




"˝
"
(
P
)

=
"
2

"
(
P
)



"
(
P
)
+
h


"
(
P
)

+
ˆ
"
=
";
P
Ÿ
h
"
(
P
)
+
¯

";
P
;
(4.4.13)
85
where
ˆ
"
=

1
j

j
R

[
"
2

"
(
P
)



"
(
P
)
+
h


"
(
P
)

]
dx
isaddedtomaketherighthandsideofmean
valuezeroasrequiredbyourof(


)

1
and

";
P
v
:
=
"
2

v


v
+
h
0
(
 
"
(
P
))
v
and¯

";
P
=

(
R
";
P
+
"
2

e
H
(
W
";
P
)


e
H
(
W
";
P
)

(
h
(
W
";
P

e
H
(
W
";
P
)
+
Ÿ
h
"
(
P
))

h
(
W
";
P
)

h
0
(
e
W
";
P
)
Ÿ
h
"
(
P
))
+
ˆ
"
.
AsinSection4.3.5,(


)

1
actingonamean-valuezerofunction
v
istobethemean-
valuezerosolutionof



=
v
withthehomogeneousNeumannboundarycondition.Notethat
"
2

"
(
P
)



"
(
P
)
+
h


"
(
P
)

=
O
(
e


"
),so
ˆ
"
=
O
(
e


"
).
Recallthat
 
"
(
P
)
=
P
1

i

k
w
";
p
i

e
H
,sodirectcomputationyields
D
 
"
(
P
)
=
(

w
";
p
1
;

;

w
";
p
k
)

D
e
H
;
where
r
meansthederivativeswithrespectto
x
.Write
P
=
(
p
1
;

;
p
k
)and
p
i
=
(
p
i
1
;

;
p
in
),
thenlet

lm
ij
=
h
(


)

1
"
D
p
ij

";
P
;

"
r
x
m
w
";
p
l
i
=
h
(


)

1
(
"
r
x
j
w
";
p
i

q
ij
"
)
;"
r
x
m
w
";
p
l
i
+
h
(


)

1
(
q
ij
"
+
"
D
p
ij
e
H
"
D
p
ij
Ÿ
h
"
)
;"
r
x
m
w
";
p
l
i
;
(4.4.14)
where
q
ij
"
=
1
j

j
R

"
r
x
j
w
";
p
i
dx
=

1
j

j
R

"
D
p
ij
e
H
"
D
p
ij
Ÿ
h
"
dx
.
Clearly,
q
ij
"
=
O
(
e


"
)since
R

"
D
ij
e
H
"
D
ij
Ÿ
h
"
dx
=
O
(
e


"
).Beforeweestimate

lm
ij
,we
provethefollowinglemma.Herewewanttoremindthereadersthat

;

isstillthe
L
2
"
inner
productas
h
f
;
g
i
=
R

fg
"

n
dx
.
Lemma4.4.1.
Iff
(
x
):
R
n
!
R

j
f
(
x
)
j
+
j
D
i
f
(
x
)
j
Ke

k
j
x
j
foranyx
2
R
n
,then
j
S
(
D
i
f
)(
x
)
j
:
=
j
c
n
R
R
n
(
D
i
f
)(
y
)
j
x

y
j
n

2
dy
j
C
1
+
j
x
j
n

1
foranyx
2
R
n
,wherec
n
=
1
n
(
n

2)
!
n
with
!
n
thevolumeoftheunitballin
R
n
.
86
Proof.
Let
B
r
betheballwithradius
r
>
0centeredatorigin.Wewrite
Z
R
n
(
D
i
f
)(
y
)
j
x

y
j
n

2
dy
=
Z
B
R
(
D
i
f
)(
y
)
j
x

y
j
n

2
dy
+
Z
R
n
n
B
R
(
D
i
f
)(
y
)
j
x

y
j
n

2
dy
=
I
R
+
J
R
:
(4.4.15)
Fored
x
andany
y
2
R
n
n
B
R
with
R
largeenough,wehave
j
y
jj
x

y
jj
x
j
;
j
x

y
j
R
j
x
j
:
Itfollowsthat
j
J
R
j
Ke
k
j
x
j
Z
R
n
n
B
R
e

x

y
j
j
x

y
j
n

2
dy

Ke
k
j
x
j
Z
1
R

x
j
e

kr
rdr
=
K
k
e
2
k
j
x
j
e

2
kR
(
1
k
+
R
j
x
j
)
;
(4.4.16)
whichindicatesthat
lim
R
!1
J
R
=
0
:
(4.4.17)
Write
y
=

+
y
i
e
i
where

=
P
j
,
i
y
j
e
j
and
e
j
isthecanonicalbasisof
R
n
.Then,withanintegration
byparts,wehave
I
R
=
Z
j

j
<
R
Z
q
R
2


j
2

q
R
2


j
2
(
D
i
f
)(

+
y
i
e
i
)
j

+
y
i
e
i

x
j
n

2
dy
i
d

=
Z
j

j
<
R
f
(

+
p
R
2
j

j
2
e
i
)
j

+
p
R
2
j

j
2
e
i

x
j
n

2

f
(


p
R
2
j

j
2
e
i
)
j


p
R
2
j

j
2
e
i

x
j
n

2
d

+
(
n

2)
Z
j

j
<
R
Z
q
R
2


j
2

q
R
2


j
2
f
(

+
y
i
e
i
)(

+
y
i
e
i

x
)

e
i
j

+
y
i
e
i

x
j
n
dy
i
=
I
a
R
+
I
b
R
:
(4.4.18)
87
Since
j
f
(


y
i
e
i
)
j
Ke

kR
and
j


y
i
e
i

x
jj
R
jj
x
j
,onehas
I
a
R

2
K
e

kR
(
R
j
x
j
)
n

2
!
n
R
n

1
;
(4.4.19)
andso
lim
R
!1
I
a
R
=
0
:
(4.4.20)
Bydirectcomputation,wehave
j
I
b
R
j
=
(
n

2)



Z
B
R
f
(
y
)(
y

x
)

e
i
j
y

x
j
n
dy




(
n

2)
Z
B
R
j
f
(
y
)
j
j
y

x
j
n

1
dy

(
n

2)
Z
R
n
j
f
(
y
)
j
j
y

x
j
n

1
dy
:
(4.4.21)
Combining(4.4.17),(4.4.20)and(4.4.21),wehave



Z
R
n
(
D
i
f
)(
y
)
j
x

y
j
n

2
dy




(
n

2)
Z
R
n
j
f
(
y
)
j
j
y

x
j
n

1
dy
:
(4.4.22)
Wenowestimate
R
R
n
j
f
(
y
)
j
j
y

x
j
n

1
dy
.Weconsiderthecase
j
x
j
1.Since
j
y
jj
x

y
jj
x
j
,wehave
Z
R
n
j
f
(
y
)
j
j
x

y
j
n

1
dy

K
Z
R
n
e

k
j
y
j
j
x

y
j
n

1
dy

Ke
k
j
x
j
Z
R
n
e

k
j
x

y
j
j
x

y
j
n

1
dy

Ke
k
Z
R
n
e

k
j
x

y
j
j
x

y
j
n

1
dy
=
C
0
;
(4.4.23)
where
C
0
=
Ke
k
R
R
n
e

k
j
y
j
j
y
j
n

1
dy
.Assumenow
j
x
j
>
1,let
B
r
(
x
)betheballcenteredat
x
withradius
88
r
,thenwrite
Z
R
n
j
f
(
y
)
j
j
x

y
j
n

1
dy
=
Z
B
j
x
j
=
2
(
x
)
j
f
(
y
)
j
j
x

y
j
n

1
dy
+
Z
R
n
n
B
j
x
j
=
2
(
x
)
j
f
(
y
)
j
j
x

y
j
n

1
dy
=
I
+
II
:
(4.4.24)
Weestimate
I

Ke

k
j
x
j
2
Z
B
j
x
j
=
2
(
x
)
1
j
x

y
j
n

1
dy
=
!
n
Ke

k
j
x
j
2
j
x
j
2
=
1
2
!
n
K
j
x
j
n
e

k
j
x
j
2
j
x
j

(
n

1)

C
j
x
j

(
n

1)
;
(4.4.25)
and
II

(
j
x
j
2
)

(
n

1)
Z
R
n
n
B
j
x
j
=
2
(
x
)
j
f
(
y
)
j
dy

(
j
x
j
2
)

(
n

1)
Z
R
n
j
f
(
y
)
j
dy

C
j
x
j

(
n

1)
:
(4.4.26)
Then,thedesiredresultfollowsdirectly.

Lemma4.4.2.
Assume

ˆ
R
n
isasmoothboundedconnectedopenset.Let
‹
L
2
(

)
bethesubset
ofL
2
(

)
ofthefunctionswithmean-valuezero.Then,foreachf
2
‹
L
2
,thereisauniquesolution
¯
uwithmean-valuezerooftheproblem
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:


u
=
f
(
x
)
;
in

;
@
u
@
n
=
0
;
on
@

:
Moreover,thereexitsaNeumannfunction(sometimescalledsecondtypeofGreen'sfunction)
89
N
:



nf
(
x
;
x
):
x
2

g!
R
suchthat
¯
u
(
x
)
=
((


)

1
f
)(
x
)
=
Z

N
(
x
;
y
)
f
(
y
)
dy
:
(4.4.27)
Forn

3
,theNeumannfunctionisoftheform
N
(
x
;
y
)
=
G
(
x
;
y
)
+
˚
(
x
;
y
)
;
(4.4.28)
whereG
(
x
;
y
)
=
j
x

y
j
2

n
n
(
n

2)
!
n
and
˚
(
x
;
y
):



!
R
issmooth.
Proof.
OnecantheproofofthisLemmain[85],wheretheyconstructanNeumannfunction
for
n
=
3.Thesameargumentisvalidingeneral.

Lemma4.4.3.

lm
ij
=
C

"
2
+
O
(
"
n
+
1
)
;
ifi
=
l
;
j
=
m
;

lm
ij
=
O
(
"
n
+
1
)
;
ifi
=
l
;
j
,
m
;

lm
ij
=
O
(
"
n
+
1

n

1
)
;
ifi
,
l
;
(4.4.29)
whereC

isapositiveconstantbelow.
Proof.
Since
N
(
x
;
y
)
˘j
x

y
j
2

n
,wehave
R

R

j
N
(
x
;
y
)
j
dydx

C
.Combiningthiswiththe
factthat
q
ij
"
;"
D
p
ij
e
H
;"
D
p
ij
Ÿ
h
"
)
=
O
(
e


"
),itfollowsthatwecanignorethecontributionofthese
terms.Thereforeitissu

cienttoestimate
Z

N
(
x
;
y
)
"
r
x
j
w
";
p
i
(
y
)
dy
"
r
x
m
w
";
p
l
(
x
)
dx
:
(4.4.30)
90
Let
r
i
w
(
x
)
=
@
w
@
x
i
,thenonehas
"
r
x
j
w
";
p
i
(
y
)
dy
=
r
j
w
(
x

p
i
"
).ByLemma4.4.2,wehave
Z

N
(
x
;
y
)
"
r
x
j
w
";
p
i
(
y
)
dy
=
Z

N
(
x
;
y
)
r
j
w
(
y

p
i
"
)
dy
=
c
n
Z

1
j
x

y
j
n

2
r
j
w
(
y

p
i
"
)
dy
+
Z

˚
(
x
;
y
)
r
j
w
(
y

p
i
"
)
dy
=
I
ij
+
J
ij
:
(4.4.31)
Changingvariablesbysetting
y
=
"
z
+
p
i
,wehave
I
ij
=
c
n
"
2
Z

i
"
r
j
w
(
z
)
j
x

p
i
"

z
j
n

2
dz
=
c
n
"
2
Z
R
n
r
j
w
(
z
)
j
x

p
i
"

z
j
n

2
dz

c
n
"
2
Z
R
n
n

i
"
r
j
w
(
z
)
j
x

p
i
"

z
j
n

2
dz
=
I
a
ij
+
I
b
ij
;
(4.4.32)
where

i
"
=
f
z
:
p
i
+
"
z
2

g
.Weestimate
j
I
b
ij
j
c
n
"
2
Z
R
n
n

i
"
\fj
x

p
i
"

z

1
g
jr
j
w
(
z
)
j
j
x

p
i
"

z
j
n

2
dz
+
c
n
"
2
Z
R
n
n

i
"
\fj
x

p
i
"

z
j
>
1
g
jr
j
w
(
z
)
j
j
x

p
i
"

z
j
n

2
dz

C
"
2
e


"
Z
j
˘

1
1
j
˘
j
n

2
d
˘
+
C
n
"
2
Z
j
z
j
>

2
"
jr
j
w
(
z
)
j
dz
=
O
(
e


"
)
:
(4.4.33)
91
Toestimate
J
ij
,wewrite
J
ij
=
Z
B

2
(
p
i
)
˚
(
x
;
y
)
r
j
w
(
y

p
i
"
)
dy
+
Z

n
B

2
(
p
i
)
˚
(
x
;
y
)
r
j
w
(
y

p
i
"
)
dy
=
J
a
ij
+
J
b
ij
:
(4.4.34)
Since
r
j
w
(
x
)isoddin
x
j
,wehave
R
B

2
(
p
i
)
r
j
w
(
y

p
i
"
)
dy
=
0.Itfollowsthat
j
J
a
ij
j
=
j
Z
B

2
(0)
(
˚
(
x
;
z
+
p
i
)

˚
(
x
;
p
i
))
r
j
w
(
z
"
)
dz
j

C
Z
B

2
(0)
j
z
j
e

C
j
z
j
"
dz
=
Cnw
n
"
n
+
1
Z

2
"
0
s
n
e

Cs
ds

C

"
n
+
1
:
(4.4.35)
Also,onecanestimate
j
J
b
ij
j
Ce


"
Z

j
˚
(
x
;
y
)
j
dy
=
O
(
e


"
)
:
(4.4.36)
Let
N
ij
(
x
)
=
R

N
(
x
;
y
)
"
r
x
j
w
";
p
i
(
y
)
dy
=
R

N
(
x
;
y
)
r
j
w
(
y

p
i
"
)
dy
.For
i
,
l
,recallthat
j
p
i

p
l
j
>
and
d
(
p
l
;@

)


2
,therefore,forany
x
2
B

2
(
p
l
),wehave
j
x

p
i
j

2
.FromthisandusingLemma
4.4.1,wehave
92
j
Z

N
ij
(
x
)
r
m
w
(
x

p
l
"
)
dx
j
j
Z
B

2
(
p
l
)
N
ij
(
x
)
r
m
w
(
x

p
l
"
)
dx
j
+
j
Z

n
B

2
(
p
l
)
N
ij
(
x
)
r
m
w
(
x

p
l
"
)
dx
j

C
"
2
(
"

)
n

1
Z
B

2
(
p
l
)
jr
m
w
(
x

p
l
"
)
j
dx
+
Ce


"
Z

n
B

2
(
p
l
)
j
N
ij
(
x
)
j
dx

C

n

1
"
2
n
+
1
+
Ce


"
(
Z
B

2
(
p
i
)
j
N
ij
(
x
)
j
dx
+
Z

n
(
B

2
(
p
l
)
[
B

2
(
p
i
)
j
N
ij
(
x
)
j
dx

C

n

1
"
2
n
+
1
+
Ce


"
"
n
+
1
+
C

n

1
e


"
"
n
+
1
=
O
(
"
2
n
+
1

n

1
)
:
(4.4.37)
For
i
=
l
;
j
,
m
,westartwith
j
Z

N
ij
(
x
)
r
m
w
(
x

p
l
"
)
dx
jj
Z

I
a
ij
r
m
w
(
x

p
l
"
)
dx
j
+
j
Z

(
I
b
ij
+
J
b
ij
)
r
m
w
(
x

p
l
"
)
dx
j
+
j
Z

J
a
ij
r
m
w
(
x

p
l
"
)
dx
j
:
Since
I
a
ij
(
x
+
p
i
)isoddin
x
j
and
r
m
w
isoddin
x
m
,wehave
j
Z

I
a
ij
r
m
w
(
x

p
l
"
)
dx
j
=
j
Z
B

2
(
p
i
)
I
a
ij
r
m
w
(
x

p
l
"
)
dx
+
Z

n
B

2
(
p
i
)
I
a
ij
r
m
w
(
x

p
l
"
)
dx
j
=
j
Z

n
B

2
(
p
i
)
I
a
ij
r
m
w
(
x

p
l
"
)
dx
j

C
"
n
+
1

n

1
e


"
:
(4.4.38)
93
Onecanalsoeasilycheckthat
j
Z

(
I
b
ij
+
J
b
ij
)
r
m
w
(
x

p
l
"
)
dx
j
Ce


"
"
n
;
j
Z

J
a
ij
r
m
w
(
x

p
l
"
)
dx
j
C
"
2
n
+
1
:
(4.4.39)
Therefore,wehavefor
i
=
l
;
j
,
m
,
Z

N
ij
(
x
)
r
m
w
(
x

p
l
"
)
dx
=
O
(
"
2
n
+
1
)
:
(4.4.40)
Finally,weconsiderthecase
i
=
l
;
j
=
m
.Wewrite
Z

N
ij
(
x
)
r
j
w
(
x

p
i
"
)
dx
=
Z

I
a
ij
r
j
w
(
x

p
i
"
)
dx
+
Z

(
I
b
ij
+
J
b
ij
)
r
j
w
(
x

p
i
"
)
dx
+
Z

J
a
ij
r
j
w
(
x

p
i
"
)
dx
:
(4.4.41)
Wecompute
Z

I
a
ij
r
j
w
(
x

p
i
"
)
dx
=
Z
B

2
(
p
i
)
I
a
ij
r
j
w
(
x

p
i
"
)
dx
+
Z

n
B

2
(
p
i
)
I
a
ij
r
j
w
(
x

p
i
"
)
dx
:
(4.4.42)
Onecancheckbydirectcomputationthat
Z
B

2
(
p
i
)
I
a
ij
r
j
w
(
x

p
i
"
)
dx
=
c
n
"
2
Z
B

2
(
p
i
)
Z
R
n
r
j
w
(
z
)
j
x

p
i
"

z
j
n

2
dz
r
j
w
(
x

p
i
"
)
dx
=
c
n
"
n
+
2
Z
B

2
"
(0)
Z
R
n
r
j
w
(
z
)
j
y

z
j
n

2
dz
r
j
w
(
y
)
dy
=
C

"
n
+
2
;
(4.4.43)
94
where
C

=
c
n
R
B

2
"
(0)
R
R
n
r
j
w
(
z
)
j
y

z
j
n

2
dz
r
j
w
(
y
)
dy
.Alsowehave
j
Z

(
I
b
ij
+
J
b
ij
)
r
j
w
(
x

p
i
"
)
dx
j
Ce


"
"
n
;
j
Z

J
a
ij
r
j
w
(
x

p
i
"
)
dx
j
C
"
2
n
+
1
:
(4.4.44)
Thenthedesiredresultfollowsdirectly.

Takingtheinnerproductof(4.4.13)with

x
j
w
";
p
i
for
i
=
1
;

;
k
and
j
=
1
;

;
n
,wegeta
linearsystemconsistingof
nk
equationswhichcanbewrittenas

";
P

˝
";
P
=
";
P
;
(4.4.45)
where

";
P
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4

11
11


11
1
n


11
k
1


11
kn
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:

1
n
11


1
n
1
n


1
n
k
1


1
n
kn
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:

k
1
11


k
1
1
n


k
1
k
1


k
1
kn
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:

kn
11


kn
1
n


kn
k
1


kn
kn
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
;
95
and

";
P
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
h

";
P
Ÿ
h
"
(
P
)
+
¯

";
P
;

"
r
x
1
w
";
p
1
i
:
:
:
h

";
P
Ÿ
h
"
(
P
)
+
¯

";
P
;

"
r
x
n
w
";
p
1
i
:
:
:
:
:
:
h

";
P
Ÿ
h
"
(
P
)
+
¯

";
P
;

"
r
x
1
w
";
p
k
i
:
:
:
h

";
P
Ÿ
h
"
(
P
)
+
¯

";
P
;

"
r
x
n
w
";
p
k
i
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
:
Since

";
P
isself-adjointand¯

";
P
=
O
(
e


"
),oneimmediatelyhas

";
P
=
O
(
e


"
)
:
ByLemma(4.4.3),onethat

";
P
isadiagonallydominantmatrix,whichimpliesthat

";
P
isinvertible.Thus
˝
";
P
canbeexpressedas
˝
";
P
=
(

";
P
)

1

";
P
;
(4.4.46)
andso
j
˝
"
(
P
)
j
=
O

"

2
e


"

:
(4.4.47)
Remark
4.4.4
.
Oncethematrix

";
P
hasbeencomputed,equation(4.4.46)determinesthecom-
pletedynamicsofthespikesontheinvariantmanifold.Althoughentriesof

";
P
mayhavedi

erent
exponentialrates,eachentryofthematrix

";
P
isoforder
"
tosomepower,therefore

";
P
mixes
alltheentriesof

";
P
together,whichyieldsthattheinteriormulti-spikedynamicsoftheCahn-
96
Hilliardequationhasaglobalcharacterwherenotonlytheclosestspikesinteractbuteachspike
interactswithalltheothersandwiththeboundary.
97
BIBLIOGRAPHY
98
BIBLIOGRAPHY
[1]Adimurthi,G.Mancini,andS.L.Yadava,
Theroleofthemeancurvatureinsemilinear
Neumannprobleminvolvingcriticalexponent:Probleminvolvingcriticalexponent
,Com-
municationsinPartialDi

erentialEquations
20
(1995),no.3-4,591Œ631.
[2]Adimurthi,F.Pacella,andS.L.Yadava,
Interactionbetweenthegeometryoftheboundary
andpositivesolutionsofasemilinearneumannproblemwithcriticalnonlinearity
,Journal
ofFunctionalAnalysis
113
(1993),no.2,318Œ350.
[3]
,
CharacterizationofconcentrationpointsandL
1
-estimatesforsolutionsofasemi-
linearNeumannprobleminvolvingthecriticalSobolevexponent
,Di

erentialandIntegral
Equations
8
(1995),no.1,41Œ68.
[4]N.D.Alikakos,P.W.Bates,andG.Fusco,
SlowmotionfortheCahnŒHilliardequationin
onespacedimension
,JournalofDi

erentialEquations
90
(1991),no.1,81Œ135.
[5]N.D.Alikakos,X.Chen,andG.Fusco,
Motionofadropletbysurfacetensionalongthe
boundary
,CalculusofVariationsandPartialDi

erentialEquations
11
(2000),no.3,233Œ
305.
[6]N.D.AlikakosandG.Fusco,
SlowdynamicsfortheCahnŒHilliardequationinhigherspace
dimensionpartI:spectralestimates
,CommunicationsinPartialDi

erentialEquations
19
(1994),no.9-10,1397Œ1447.
[7]
,
SlowdynamicsfortheCahnŒHilliardequationinhigherspacedimensions:the
motionofbubbles
,ArchiveforRationalMechanicsandAnalysis
141
(1998),no.1,1Œ61.
[8]N.D.Alikakos,G.Fusco,andV.Stefanopoulos,
Criticalspectrumandstabilityofinterfaces
foraclassofreactionŒdi

usionequations
,JournalofDi

erentialEquations
126
(1996),
no.1,106Œ167.
[9]S.M.AllenandJ.W.Cahn,
Amicroscopictheoryforantiphaseboundarymotionandits
applicationtoantiphasedomaincoarsening
,ActaMetallurgica
27
(1979),no.6,1085Œ
1095.
[10]J.M.Ball,
Saddlepointanalysisforanordinarydi

erentialequationinabanachspace,and
anapplicationtodynamicbucklingofabeam
,Nonlinearelasticity(1973),93Œ160.
99
[11]PeterW.Bates,KeningLu,andChongchunZeng,
Existenceandpersistenceofinvariant
manifoldsforinBanachspace
,Mem.Amer.Math.Soc.
135
(1998),no.645,
viii
+
129.
[12]P.W.Bates,E.N.Dancer,andJ.Shi,
Multi-spikestationarysolutionsoftheCahnŒHilliard
equationinhigher-dimensionandinstability
,AdvancesinDi

erentialEquations
4
(1999),
no.1,1Œ69.
[13]P.W.BatesandP.C.Fife,
SpectralcomparisonprinciplesfortheCahnŒHilliardandphase-
equations,andtimescalesforcoarsening
,PhysicaD:NonlinearPhenomena
43
(1990),
no.2,335Œ348.
[14]P.W.BatesandG.Fusco,
EquilibriawithmanynucleifortheCahnŒHilliardequation
,Jour-
nalofDi

erentialEquations
160
(2000),no.2,283Œ356.
[15]P.W.BatesandJ.Jin,
Globaldynamicsofboundarydroplets
,DiscreteandContinuousDy-
namicalSystems
34
(2014),no.1,1Œ17.
[16]P.W.BatesandC.K.R.T.Jones,
Invariantmanifoldsforsemilinearpartialdi

erential
equations
,Dynamicsreported,Vol.2,Dynam.Report.Ser.Dynam.SystemsAppl.,vol.2,
Wiley,Chichester,1989,pp.1Œ38.
[17]P.W.Bates,K.Lu,andC.Zeng,
Persistenceofomanifoldsfor
,Com-
municationsonPureandAppliedMathematics
52
(1999),no.8,983Œ1046.
[18]
,
Invariantfoliationsnearnormallyhyperbolicinvariantmanifoldsfor
,
TransactionsoftheAmericanMathematicalSociety
352
(2000),no.10,4641Œ4676.
[19]
,
Approximatelyinvariantmanifoldsandglobaldynamicsofspikestates
,Inven-
tionesMathematicae
174
(2008),no.2,355Œ433.
[20]P.W.BatesandJ.Shi,
Existenceandinstabilityofspikelayersolutionstosingularpertur-
bationproblems
,JournalofFunctionalAnalysis
196
(2002),no.2,211Œ264.
[21]P.W.BatesandJ.Xun,
MetastablepatternsfortheCahnŒHilliardequation,partI
,Journal
ofDi

erentialEquations
111
(1994),no.2,421Œ457.
[22]
,
MetastablepatternsfortheCahnŒHilliardequation:PartII.Layerdynamicsand
slowinvariantmanifold
,JournalofDi

erentialEquations
117
(1995),no.1,165Œ216.
100
[23]H.BerestyckiandP-L.Lions,
Nonlinearscalarequations,Iexistenceofagroundstate
,
ArchiveforRationalMechanicsandAnalysis
82
(1983),no.4,313Œ345.
[24]N.BogoliubovandN.Krylov,
Theapplicationofmethodsofnonlinearmechanicsinthe
theoryofstationaryoscillations
,Publ.8oftheUkrainianAcad,Sci.Kiev(1934).
[25]N.N.BogoliubovandY.A.Mitropolsky,
Asymptoticmethodsinthetheoryofnon-linear
oscillations
,TranslatedfromthesecondrevisedRussianedition.InternationalMonographs
onAdvancedMathematicsandPhysics,HindustanPublishingCorp.,Delhi,Gordonand
BreachSciencePublishers,NewYork,1961.
[26]A.Burchard,B.Deng,andK.Lu,
Smoothconjugacyofcentremanifolds
,Proceedingsof
theRoyalSocietyofEdinburgh
120A
(1992),no.1-2,61Œ77.
[27]J.W.Cahn,
Onspinodaldecomposition
,ActaMetallurgica
9
(1961),no.9,795Œ801.
[28]J.W.CahnandJ.E.Hilliard,
Freeenergyofanonuniformsystem.I.Interfacialfreeenergy
,
TheJournalofChemicalPhysics
28
(1958),no.2,258Œ267.
[29]J.Carr,
Applicationsofcentremanifoldtheory
,AppliedMathematicalSciences,vol.35,
Springer-Verlag,NewYork-Berlin,1981.
[30]J.Carr,M.E.Gurtin,andM.Slemrod,
Structuredphasetransitionsonainterval
,
ArchiveforRationalMechanicsandAnalysis
86
(1984),no.4,317Œ351.
[31]J.CarrandR.L.Pego,
Metastablepatternsinsolutionsofu
t
=
"
2
u
xx

f
(
u
),Communica-
tionsonPureandAppliedMathematics
42
(1989),no.5,523Œ576.
[32]
,
Invariantmanifoldsformetastablepatternsinu
t
=
"
2
u
xx

f
(
u
),Proceedingsof
theRoyalSocietyofEdinburgh:SectionAMathematics
116
(1990),no.1-2,133Œ160.
[33]X.ChenandM.Kowalczyk,
ExixtanceofequilibriafortheCahn-Hilliardequationvialocal
minimizersoftheperimeter
,CommunicationsinPartialDi

erentialEquations
21
(1996),
no.7-8,1207Œ1233.
[34]S.-N.Chow,W.Liu,andY.Yi,
Centermanifoldsforsmoothinvariantmanifolds
,Trans.
Amer.Math.Soc.
352
(2000),no.11,5179Œ5211.
[35]S.-N.ChowandK.Lu,
Invariantmanifoldsforinbanachspaces
,JournalofDi

eren-
tialEquations
74
(1988),no.2,285Œ317.
101
[36]E.N.Dancer,
Somemountain-passsolutionsforsmalldi

usion
,Di

erentialandIntegral
Equations
16
(2003),no.8,1013Œ1024.
[37]E.N.DancerandS.Yan,
Asingularlyperturbedellipticprobleminboundeddomainswith
nontrivialtopology
,AdvancesinDi

erentialEquations
4
(1999),no.3,347Œ368.
[38]E.DeGiorgiandT.Franzoni,
Suuntipodiconvergenzavariazionale
,AttiAccad.Naz.
LinceiRend.Cl.Sci.Fis.Mat.Natur.
58
(1975),no.6,842Œ850.
[39]M.DelPino,P.L.Felmer,andJ.Wei,
Ontheroleofmeancurvatureinsomesingularly
perturbedneumannproblems
,SIAMJournalonMathematicalAnalysis
31
(1999),no.1,
63Œ79.
[40]S.P.Diliberto,
PerturbationtheoremsforperiodicsurfacesI-andmaintheorems
,
RendicontidelCircoloMatematicodiPalermo
9
(1960),no.3,265Œ299.
[41]N.Fenichel,
Persistenceandsmoothnessofinvariantmanifoldsfor
,IndianaUniversity
MathematicJournal
21
(1971),no.193-226,1972.
[42]
,
Asymptoticstabilitywithrateconditions
,IndianaUniversityMathematicJournal
23
(1973),no.1109-1137,74.
[43]
,
Asymptoticstabilitywithrateconditionsii
,IndianaUniversityMathematicsJour-
nal
26
(1977),no.1,81Œ93.
[44]A.FloerandA.Weinstein,
NonspreadingwavepacketsforthecubicSchr¨odingerequation
withaboundedpotential
,JournalofFunctionalanalysis
69
(1986),no.3,397Œ408.
[45]G.FuscoandJ.K.Hale,
Slow-motionmanifolds,dormantinstability,andsingularpertur-
bations
,JournalofDynamicsandDi

erentialEquations
1
(1989),no.1,75Œ94.
[46]C.GuiandJ.Wei,
MultipleinteriorpeaksolutionsforsomesingularlyperturbedNeumann
problems
,JournalofDi

erentialEquations
158
(1999),no.1,1Œ27.
[47]
,
Onmultiplemixedinteriorandboundarypeaksolutionsforsomesingularlyper-
turbedNeumannproblems
,CanadianJournalofMathematics
52
(2000),no.3,522Œ538.
[48]C.Gui,J.Wei,andM.Winter,
Multipleboundarypeaksolutionsforsomesingularlyper-
turbedNeumannproblems
,Annalesdel'InstitutHenriPoincare(C)NonLinearAnalysis,
vol.17,Elsevier,2000,pp.47Œ82.
102
[49]J.Hadamard,
Surl'it´erationetlessolutionsasymptotiquesdes´equationsdi

´erentielles
,
Bull.Soc.Math.France
29
(1901),224Œ228.
[50]J.K.Hale,
Integralmanifoldsofperturbeddi

erentialsystems
,Ann.ofMath.(2)
73
(1961),
496Œ531.
[51]D.Henry,
Geometrictheoryofsemilinearparabolicequations
,LectureNotesinMathemat-
ics,vol.840,Springer-Verlag,Berlin-NewYork,1981.
[52]M.W.Hirsch,C.C.Pugh,andM.Shub,
Invariantmanifolds
,LectureNotesinMathematics,
Vol.583,Springer-Verlag,Berlin-NewYork,1977.
[53]M.W.Hirsch,C.C.Pugh,andM.Shub,
Invariantmanifolds
,BulletinoftheAmerican
MathematicalSociety
76
(1970),no.5,1015Œ1019.
[54]M.W.HirschandC.C.Pugh,
Stablemanifoldsandhyperbolicsets
,GlobalAnalysis(Proc.
Sympos.PureMath.,Vol.XIV,Berkeley,Calif.,1968),Amer.Math.Soc.,Providence,R.I.,
1970,pp.133Œ163.
[55]G.A.Hu

ord,
Banachspacesandtheperturbationofordinarydi

erentialequations
,Ph.D.
thesis,PrincetonUniversity,1953.
[56]M.C.Irwin,
Onthestablemanifoldtheorem
,BulletinoftheLondonMathematicalSociety
2
(1970),no.2,196Œ198.
[57]J.Jang,
Onspikesolutionsofsingularlyperturbedsemilineardirichletproblem
,Journalof
Di

erentialEquations
114
(1994),no.2,370Œ395.
[58]A.Kelley,
Thestable,center-stable,center,center-unstable,unstablemanifolds
,Journalof
Di

erentialEquations
3
(1967),no.4,546Œ570.
[59]M.Kowalczyk,
MultiplespikelayersintheshadowGierer-Meinhardtsystem:existenceof
equilibriaandthequasi-invariantmanifold
,DukeMathematicalJournal
98
(1999),no.1,
59Œ111.
[60]J.Kurzweil,
Invariantmanifoldsfor
,Di

erentialEquationsandTheirApplications
(1967),89Œ92.
[61]W.T.Kyner,
Apointtheorem
,Contributionstothetheoryofnonlinearoscillations,
vol.3,AnnalsofMathematicsStudies,no.36,PrincetonUniversityPress,Princeton,N.J.,
1956,pp.197Œ205.
103
[62]N.Levinson,
Smallperiodicpertubationsofanautonomoussystemwithastableorbit
,
AnnalsofMathematics
52
(1950),no.3,727Œ738.
[63]Y.Li,
Onasingularlyperturbedequationwithneumannboundarycondition
,Communica-
tionsinPartialDi

erentialEquations
23
(1998),no.3-4,487Œ545.
[64]C.-S.Lin,W.-M.Ni,andI.Takagi,
Largeamplitudestationarysolutionstoachemotaxis
system
,JournalofDi

erentialEquations
72
(1988),no.1,1Œ27.
[65]A.Lyapunov,
Probl˚emeG´en´eraldelaStabilit´eduMouvement
,AnnalsofMathematics
Studies,no.17,PrincetonUniversityPress,Princeton,N.J.;OxfordUniversityPress,Lon-
don,1947.
[66]R.Ma
Ÿ
n
´
e,
Persistentmanifoldsarenormallyhyperbolic
,Trans.Amer.Math.Soc.
246
(1978),261Œ283.
[67]J.MarsdenandJ.Scheurle,
Theconstructionandsmoothnessofinvariantmanifoldsbythe
deformationmethod
,SIAMJournalonMathematicalAnalysis
18
(1987),no.5,1261Œ1274.
[68]AlexanderMielke,
HamiltonianandLagrangianoncentermanifolds
,LectureNotes
inMathematics,vol.1489,Springer-Verlag,Berlin,1991,Withapplicationstoellipticvari-
ationalproblems.
[69]L.Modica,
Gradienttheoryofphasetransitionandsingularperturbation
,ArchiveforRa-
tionalMechanicsandAnalysis
98
(1986),123Œ142.
[70]L.ModicaandS.Mortola,
Unesempiodi

-convergenza
,Boll.Un.Mat.Ital.B
14
(1977),
no.1,285Œ299.
[71]W.-M.NiandI.Takagi,
OntheNeumannproblemforsomesemilinearellipticequations
andsystemsofactivator-inhibitortype
,TransactionsoftheAmericanMathematicalSociety
297
(1986),no.1,351Œ368.
[72]W-MNiandI.Takagi,
Ontheshapeofleast-energysolutionstoasemilinearNeumann
problem
,CommunicationsonPureandAppliedMathematics
44
(1991),no.7,819Œ851.
[73]
,
Locatingthepeaksofleast-energysolutionstoasemilinearNeumannproblem
,
DukeMathematicalJournal
70
(1993),no.2,247Œ281.
104
[74]W.-M.NiandJ.Wei,
Onthelocationandprofspike-layersolutionstosingularlyper-
turbedsemilinearDirichletproblems
,CommunicationsonPureandAppliedMathematics
48
(1995),no.7,731Œ768.
[75]Y.-G.Oh,
ExistenceofsemiclassicalboundstatesofnonlinearSchr¨odingerequationswith
potentialsoftheclass
(
V
)
a
,CommunicationsinPartialDi

erentialEquations
13
(1988),
no.12,1499Œ1519.
[76]
,
Onpositivemulti-lumpboundstatesofnonlinearSchr¨odingerequationsunder
multiplewellpotential
,CommunicationsinMathematicalPhysics
131
(1990),no.2,223Œ
253.
[77]N.C.OwenandP.Sternberg,
Gradientandfrontpropagationwithboundarycontact
energy
,Proc.Roy.Soc.LondonSer.A
437
(1992),no.1901,715Œ728.
[78]O.Perron,
¨
Uberstabilit¨atundasymptotischesverhaltenderintegralevondi

erentialgle-
ichungssystemen
,MathematischeZeitschrift
29
(1929),no.1,129Œ160.
[79]
,
Diestabilit¨atsfragebeidi

erentialgleichungen
,MathematischeZeitschrift
32
(1930),no.1,703Œ728.
[80]OskarPerron,
¨
Uberstabilit¨atundasymptotischesVerhaltenderl¨osungeneinessystems
endlicherDi

erenzengleichungen
,Journalf
¨
urdiereineundangewandteMathematik
161
(1929),41Œ64.
[81]V.A.Pliss,
Areductionprincipleinthetheoryofstabilityofmotion
,IzvestiyaRossiiskoi
AkademiiNauk.SeriyaMatematicheskaya
28
(1964),no.6,1297Œ1324.
[82]V.A.PlissandG.R.Sell,
Perturbationsofattractorsofdi

erentialequations
,J.Di

erential
Equations
92
(1991),no.1,100Œ124.
[83]J.Sijbrand,
Propertiesofcentermanifolds
,TransactionsoftheAmericanMathematical
Society
289
(1985),no.2,431Œ469.
[84]S.Smale,
Di

erentiabledynamicalsystems
,BulletinoftheAmericanmathematicalSociety
73
(1967),no.6,747Œ817.
[85]P.Sternberg,
Thee

ectofasingularperturbationonnon-convexvariationalproblems
,
ArchiveforRationalMechanicsandAnalysis
101
(1988),no.3,209Œ260.
105
[86]A.Vanderbauwhede,
Centremanifolds,normalformsandelementarybifurcations
,Dynam-
icsreported,Vol.2,Dynam.Report.Ser.Dynam.SystemsAppl.,vol.2,Wiley,Chichester,
1989,pp.89Œ169.
[87]A.VanderbauwhedeandG.Iooss,
Centermanifoldtheoryindimensions
,Dynamics
reported:expositionsindynamicalsystems,Dynam.Report.ExpositionsDynam.Systems
(N.S.),vol.1,Springer,Berlin,1992,pp.125Œ163.
[88]A.VanderbauwhedeandS.VanGils,
Centermanifoldsandcontractionsonascaleofba-
nachspaces
,JournalofFunctionalAnalysis
72
(1987),no.2,209Œ224.
[89]X.Wang,
OnconcentrationofpositiveboundstatesofnonlinearSchr¨odingerequations
,
CommunicationsinMathematicalPhysics
153
(1993),no.2,229Œ244.
[90]Zhi-QiangWang,
Ontheexistenceofmultiple,single-peakedsolutionsforasemilinear
Neumannproblem
,ArchiveforRationalMechanicsandAnalysis
120
(1992),no.4,375Œ
399.
[91]Z.Q.Wang,
RemarksonanonlinearNeumannproblemwithcriticalexponent
,Houston
JournalofMathematics
120
(1992),no.4,375Œ399.
[92]
,
Thee

ectofthedomaingeometryonthenumberofpositivesolutionsofNeumann
problemswithcriticalexponents
,Di

erentialandIntegralEquations
8
(1995),no.6,1533Œ
1554.
[93]C.E.Wayne,
Invariantmanifoldsforparabolicpartialdi

erentialequationsonunbounded
domains
,ArchiveforRationalMechanicsandAnalysis
138
(1997),no.3,279Œ306.
[94]J.Wei,
OntheboundaryspikelayersolutionstoasingularlyperturbedNeumannproblem
,
JournalofDi

erentialEquations
134
(1997),no.1,104Œ133.
[95]
,
OnsingleinteriorspikesolutionsoftheGiererŒMeinhardtsystem:uniquenessand
spectrumestimates
,EuropeanJournalofAppliedMathematics
10
(1999),no.04,353Œ378.
[96]J.WeiandM.Winter,
OnthestationaryCahnŒHilliardequation:interiorspikesolutions
,
JournalofDi

erentialEquations
148
(1998),no.2,231Œ267.
[97]
,
StationarysolutionsfortheCahn-Hilliardequation
,Annalesdel'InstitutHenri
Poincare(C)NonLinearAnalysis,vol.15,Elsevier,1998,pp.459Œ492.
106
[98]
,
Multi-peaksolutionsforawideclassofsingularperturbationproblems
,Journal
oftheLondonMathematicalSociety
59
(1999),no.02,585Œ606.
[99]
,
Multi-interior-spikesolutionsfortheCahnŒHilliardequationwitharbitrarily
manypeaks
,CalculusofVariationsandPartialDi

erentialEquations
10
(2000),no.3,
249Œ289.
[100]Y.Yi,
Ageneralizedintegralmanifoldtheorem
,J.Di

erentialEquations
102
(1993),no.1,
153Œ187.
107