Global stability of travelling fronts for a damped wave equation with bistable

Thierry Gallay

Description

Global stability of travelling fronts for a damped wave equation with bistable nonlinearity Thierry GALLAY & Romain JOLY Institut Fourier, UMR CNRS 5582 Universite de Grenoble I B.P. 74 38402 Saint-Martin-d'Heres, France October 3, 2007 Abstract We consider the damped wave equation ?utt+ut = uxx?V ?(u) on the whole real line, where V is a bistable potential. This equation has travelling front solutions of the form u(x, t) = h(x?st) which describe a moving interface between two different steady states of the system, one of which being the global minimum of V . We show that, if the initial data are sufficiently close to the profile of a front for large |x|, the solution of the damped wave equation converges uniformly on R to a travelling front as t ? +∞. The proof of this global stability result is inspired by a recent work of E. Risler [38] and relies on the fact that our system has a Lyapunov function in any Galilean frame. Keywords: travelling front, global stability, damped wave equation, Lyapunov function. AMS classification codes (2000): 35B35, 35B40, 37L15, 37L70. 1 Introduction The aim of this paper is to describe the long-time behavior of a large class of solutions of the semilinear damped wave equation ?utt + ut = uxx ? V ?(u) , (1.1) where ? > 0 is a parameter, V : R? R is a smooth bistable

• monotone reaction-diffusion system


• time toward


• obtain global


• parabolic


• relaxation time


• any galilean frame


• lyapunov function


• without any


• local minima


• equation

Sujets

• Parabolic
• Relaxation (physics)
• Lyapunov function
• Maxima and minima
• Équation

Informations

• Publié par: pefav
• Nombre de lectures: 21
• Langue: English

Extrait

Globalstabilityoftravellingfrontsfora
dampedwaveequationwithbistable
nonlinearity

ThierryGALLAY&RomainJOLY
InstitutFourier,UMRCNRS5582
Universit´edeGrenobleI
B.P.74
38402Saint-Martin-d’He`res,France
October3,2007

Abstract
Weconsiderthedampedwaveequation
αu
tt
+
u
t
=
u
xx
−
V
0
(
u
)onthewholereal
line,where
V
isabistablepotential.Thisequationhastravellingfrontsolutionsof
theform
u
(
x,t
)=
h
(
x
−
st
)whichdescribeamovinginterfacebetweentwodifferent
steadystatesofthesystem,oneofwhichbeingtheglobalminimumof
V
.Weshow
that,iftheinitialdataaresufficientlyclosetotheprofileofafrontforlarge
|
x
|
,the
solutionofthedampedwaveequationconvergesuniformlyon
R
toatravellingfront
as
t
→
+
∞
.Theproofofthisglobalstabilityresultisinspiredbyarecentworkof
E.Risler[38]andreliesonthefactthatoursystemhasaLyapunovfunctioninany
Galileanframe.
Keywords:
travellingfront,globalstability,dampedwaveequation,Lyapunov
function.
AMSclassificationcodes(2000):
35B35,35B40,37L15,37L70.

1Introduction
Theaimofthispaperistodescribethelong-timebehaviorofalargeclassofsolutionsof
thesemilineardampedwaveequation
αu
tt
+
u
t
=
u
xx
−
V
0
(
u
)
,
(1.1)
where
α>
0isaparameter,
V
:
R
→
R
isasmoothbistablepotential,andtheunknown
u
=
u
(
x,t
)isareal-valuedfunctionof
x
∈
R
and
t
≥
0.Equationsofthisformappear
inmanydifferentcontexts,especiallyinphysicsandinbiology.Forinstance,Eq.(1.1)
describesthecontinuumlimitofaninfinitechainofcoupledoscillators,thepropagation
ofvoltagealonganonlineartransmissionline[4],andtheevolutionofaninteracting

1

populationifthespatialspreadoftheindividualsismodelledbyavelocityjumpprocess
insteadoftheusualBrownianmotion[18,21,24].
Aswasalreadyobservedbyseveralauthors,thelong-timeasymptoticsofthesolutions
ofthedampedwaveequation(1.1)arequitesimilartothoseofthecorrespondingreaction-
diffusionequation
u
t
=
u
xx
−
V
0
(
u
).Inparticular,if
V
0
(
u
)vanishesrapidlyenoughas
u
→
0,thesolutionsof(1.1)originatingfromsmallandlocalizedinitialdataconverge
as
t
→
+
∞
tothesameself-similarprofilesasintheparaboliccase[12,23,27,34,35].
Theanalogypersistsforsolutionswithnontriviallimitsas
x
→±∞
,inwhichcasethe
long-timeasymptoticsareoftendescribedbyuniformlytranslatingsolutionsoftheform
u
(
x,t
)=
h
(
x
−
st
),whichareusuallycalled
travellingfronts
.Existenceofsuchsolutions
forhyperbolicequationsoftheform(1.1)wasfirstprovedbyHadeler[19,20],andafew
stabilityresultsweresubsequentlyobtainedbyGallay&Raugel[10,11,13,14].
Whilelocalstabilityisanimportanttheoreticalissue,intheapplicationsoneisoften
interestedin
globalconvergenceresults
whichensurethat,foralargeclassofinitialdata
withaprescribedbehavioratinfinity,thesolutionsapproachtravellingfrontsas
t
→
+
∞
.Forthescalarparabolicequation
u
t
=
u
xx
−
V
0
(
u
),suchresultswereobtained
byKolmogorov,Petrovski&Piskunov[29],byKanel[25,26],andbyFife&McLeod
[8,9]undervariousassumptionsonthepotential.Alltheproofsuseinanessential
waycomparisontheoremsbasedonthemaximumprinciple.Thesetechniquesarevery
powerfultoobtainglobalinformationonthesolutions,andwerealsosuccessfullyapplied
tomonotoneparabolicsystems[44,41]andtoparabolicequationsoninfinitecylinders
[39,40].
However,unlikeitsparaboliccounterpart,thedampedwaveequation(1.1)hasno
maximumprincipleingeneral.Moreprecisely,solutionsof(1.1)takingtheirvaluesin
someinterval
I
⊂
R
obeyacomparisonprincipleonlyif
4
α
sup
V
00
(
u
)
≤
1
,
(1.2)
Iu∈see[37]or[11,AppendixA].Inphysicalterms,thisconditionmeansthattherelaxation
time
α
issmallcomparedtotheperiodofthenonlinearoscillations.Inparticular,if
I
isaneighborhoodofalocalminimum
u
¯of
V
,inequality(1.2)impliesthatthelinear
oscillator
αu
tt
+
u
t
+
V
00
(
u
¯)
u
=0isstronglydamped,sothatnooscillationsoccur.Itwas
shownin[11,13]thatthetravellingfrontsof(1.1)withamonostablenonlinearityare
stableagainstlargeperturbationsprovidedthattheparameter
α
issufficientlysmallso
thatthestrongdampingcondition(1.2)holdsforthesolutionsunderconsideration.In
otherwords,thebasinofattractionofthehyperbolictravellingfrontsbecomesarbitrarily
largeas
α
→
0,butif
α
isnotassumedtobesmallthereisnohopetouse“parabolic”
methodstoobtainglobalstabilityresultsforthetravellingfrontsofthedampedwave
equation(1.1).
Recently,however,adifferentapproachtothestabilityoftravellingfrontshasbeen
developpedbyRisler[15,38].Thenewmethodispurelyvariationalandistherefore
restrictedtosystemsthatpossessagradientstructure,butitsmaininterestliesinthe
factthatitdoesnotrelyonthemaximumprinciple.Thepowerofthisapproachis
demonstratedinthepioneeringwork[38]whereglobalconvergenceresultsareobtained
forthenon-monotonereaction-diffusionsystem
u
t
=
u
xx
−r
V
(
u
),with
u
∈
R
n
and
V
:
R
n
→
R
.TheaimofthepresentarticleistoshowthatRisler’smethodcanbe

2

adaptedtothedampedhyperbolicequation(1.1)andallowsinthiscontexttoprove
globalconvergenceresults
withoutanysmallnessassumption
ontheparameter
α
.This
willalsobeanopportunitytopresentthemainargumentsof[38]inanalternativeway,
althoughsomeimportantingredientsofourproofhavenocounterpartinRisler’swork.
Beforestatingourtheorem,weneedtospecifytheassumptionswemakeonthenon-
linearityin(1.1).Wesupposethat
V
∈C
3
(
R
),andthatthereexistpositiveconstants
a
,
b
suchthat
uV
0
(
u
)
≥
au
2
−
b,
forall
u
∈
R
.
(1.3)
Inparticular,
V
(
u
)
→
+
∞
as
|
u
|→∞
.Wealsoassume
V
(0)=0
,V
0
(0)=0
,V
00
(0)
>
0
,
(1.4)
V
(1)
<
0
,V
0
(1)=0
,V
00
(1)
>
0
.
(1.5)
Finallywesupposethat,exceptfor
V
(0)and
V
(1),allcriticalvaluesof
V
arepositive:
onu
∈
R

V
0
(
u
)=0
,V
(
u
)
≤
0=
{
0;1
}
.
(1.6)
Inotherwords
V
isasmooth,strictlycoercivefunctionwhichreachesitsglobalminimum
at
u
=1andhasinadditionalocalminimumat
u
=0.Wecall
V
a
bistable
potential
becauseboth
u
=0and
u
=1arestableequilibriaoftheone-dimensionaldynamical
system
u
˙=
−
V
0
(
u
).ThesimplestexampleofsuchapotentialisrepresentedinFig.1.
Notehoweverthat
V
isallowedtohavepositivecriticalvalues,includinglocalminima.

1(V)

)u(V

0

1u

Fig.1:
Thesimplestexampleofapotential
V
satisfyingassumptions(1.3)–(1.6).
Underassumptions(1.4)–(1.6),itiswell-knownthattheparabolicequation
u
t
=
u
xx
−
V
0
(
u
)hasafamilyoftravellingfrontsoftheform
u
(
x,t
)=
h
(
x
−
c
∗
t
−
x
0
)connecting
thestableequilibria
u
=1and
u
=0,seee.g.[2].Moreprecisely,thereexistsaunique
speed
c
∗
>
0suchthattheboundaryvalueproblem
h
00
(
y
)+
c
∗
h
0
(
y
)
−
V
0
(
h
(
y
))=0
,y
∈
R
,
)7.1(h
(
−∞
)=1
,h
(+
∞
)=0
,
hasasolution
h
:
R
→
(0
,
1),inwhichcasetheprofile
h
itselfisuniqueuptoatranslation.
Moreover
h
∈C
4
(
R
),
h
0
(
y
)
<
0forall
y
∈
R
,and
h
(
y
)convergesexponentiallytowardits
3

(u0(x)−1)2+u00(x)2+u1(x)2xd≤δ,(19.)limitsas
y
→±∞
.Aswasobservedin[11,19],forany
α>
0thedampedhyperbolic
equation(1.1)hasacorrespondingfamilyoftravellingfrontsgivenby
u
(
x,t
)=
h
(1+
αc
∗
2
x
−
c
∗
t
−
x
0
)
,x
0
ͧ