Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term $\chi_{t

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aASYMPTOTICBEHAVIOROFANONISOTHERMALVISCOUSCAHN-HILLIARDEQUATIONWITHINERTIALTERMMAURIZIOGRASSELLI ,HANAPETZELTOVA´ ,GIULIOSCHIMPERNA Abstract.Weconsideradi erentialmodeldescribingnonisothermalfastphasesepara-tionprocessestakingplaceinathree-dimensionalboundeddomain.ThismodelconsistsofaviscousCahn-Hilliardequationcharacterizedbythepresenceofaninertialtermχtt,χbeingtheorderparameter,whichislinearlycoupledwithanevolutionequationforthe(relative)temperature .ThelattercanbeofhyperbolictypeiftheCattaneo-Maxwellheatconductionlawisassumed.ThestatevariablesandthechemicalpotentialaresubjecttothehomogeneousNeumannboundaryconditions.We rstprovidecondi-tionswhichensurethewell-posednessoftheinitialandboundaryvalueproblem.Then,weprovethatthecorrespondingdynamicalsystemisdissipativeandpossessesaglobalattractor.Moreover,assumingthatthenonlinearpotentialisrealanalytic,weestab-lishthateachtrajectoryconvergestoasinglesteadystatebyusingasuitableversionoftheL ojasiewicz-Simoninequality.Wealsoobtainanestimateofthedecayratetoequilibrium.1.IntroductionConsideraboundeddomain R3withsmoothboundary whichcontains,foranytimet≥0,atwo-phasesystemsubjecttononisothermalphaseseparation.Awell-knownevolutionsystemwhichdescribes(1.1) thiskindofprocessis(see[8],cf.also[7])( +χ)t =0,χt ( χ+φ(χ) )=0,in ×(0,∞).Here denotesthe(relative)temperaturearoundagivencriticalone,χrepresentstheorderparameter(orphase- eld)andφisthederivativeofasuitablesmoothdoublewellpotential(e.g.,φ(r)=r3 ar,a>0).Forthesakeofsimplicity,all

theconstantshavebeensetequaltoone.

Intheisothermalcase,thefollowingsingularperturbationofCahn-Hilliardequationhasbeenexaminedinseveralpapers(see[6,13,20,21,52,53]andreferencestherein)(1.2)εχtt+χt ( χ+αχt+φ(χ))=0,

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term $\chi_{t

2´G.SCHIMPERNAM.GRASSELLI,H.PETZELTOVA,

whereε>0isasmallinertialparameterandα≥0isaviscositycoe cient.Theinertialtermεχttaccountsforfastphaseseparationprocesses(see,e.g.,[19]),whilethemotivationsforintroducingtheviscoustermαχtaredetailedin[38].Theabovequotedworksareconcernedwiththeanalysisofthein nite-dimensionaldissipativedynamicalsystemgeneratedby(1.2)endowedwithsuitableboundaryconditions.Werecallthatthecaseα=0hasbeenanalyzedsofarinonespatialdimensiononly,sinceintwoandthreedimensions,uniquenessandsmoothnessofsolutionsarestillopenissues(seehowever

[45]).

Inthispaperweconsiderequation(1.2)inthenonisothermalcase,namely, ( +χ)t+ ·q=0,

(1.3)σqt+q= , εχ+χ ( χ+αχ+φ(χ) )=0,tttt

whereσ∈[0,1].ObservethatthestandardFourierlawisobtainedwhenσ=0.Other-wise,wehavetheso-calledMaxwell-Cattaneoheatconductionlawwhichentailsthat propagatesat nitespeed(see,e.g.,[25,26,27]andtheirreferences).

System(1.3)issubjecttotheinitialconditions

(1.4)

(1.5) (0)= 0,σq(0)=σq0,χ(0)=χ0,χt(0)=χ1,in ,andtotheno- uxboundaryconditionsq·n= χ·n= ( χ)·n=0,on ×(0,∞),

wherenstandsfortheoutwardnormalderivativeand·indicatestheusualEuclideanscalarproduct.Observethat(1.3)reducesto(1.1)whenε=α=0.Moreover,notethat(1.5)areequivalenttoassumethe rsttwoconditionsand u·n=0,whereu= χ+αχt+φ(χ) istheso-calledchemicalpotential.

Herewewanttodemonstrate rstthatproblem(1.3)-(1.5)iswellposed.ThuswecanconstructastronglycontinuoussemigroupSσ(t)onanappropriatephase-space.Thissemigrouppossessesaboundedabsorbingsetwhichiscompactinthephase-spaceifσ=0,otherwiseweshowtheexistenceofacompactexponentiallyattractingsetwhichentailstheasymptoticcompactnessofSσ(t).Thelatterresultisbasedonarecentdecompositionofthesolutionsemigroupdevisedin[39].Therefore,foranyσ≥0,wededucethatSσ(t)possessesa(smooth)globalattractor.Takingadvantageoftheseresults,wecanalsodeducethatanytrajectoryoriginatingfromthephase-spaceisprecompact.Then,wecanproceedtoanalyzetheasymptoticbehaviorofasingletrajectory.Moreprecisely,weshowthatifφisrealanalytic,thenany(weak)solution( (t),σq(t),χ(t))converges,astgoesto∞,toasingleequilibrium,namely,toatriplet( ∞,0,χ∞),where ∞andχ∞satisfy ∞=| | 1( 0 εχ1), χ∞=(εχ1+χ0),(1.6) ( χ∞+φ(χ∞))=0,in , χ∞·n= ( χ∞)·n=0,on .

Thisresultisobtainedbyexploitingawell-knowntechniqueoriginatedfromsomeworksofS.L ojasiewicz[35,36]andthenre nedbyL.Simon[46].Werecallthat,inmorethanone

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term $\chi_{t

ASYMPTOTICBEHAVIOROFANONISOTHERMALCAHN-HILLIARDEQUATION3spatialdimension,thestructureofthesetofsolutionsto(1.6)maycontainacontinuumofsolutionsif isaballoranannulus(cf.,e.g.,[29]andreferencestherein).Ifthisisthecase,itisnontrivialtodecidewhetherornotagiventrajectoryconvergestoasinglestationarystate.Moreover,thismightnothappenevenfor nite-dimensionaldynamicalsystems(cf.[5])andtherearenegativeresultsforsemilinearparabolicequationswithsmoothnonlinearities(see[40,41]).

Duringthelastyears,theL ojasiewicz-Simontechniquehasbeenmodi edandusedbymanyauthors(cf.,e.g.,[9,10,12,17,30,31,32,33,34,50])toinvestigateanumberofparabolicandhyperbolicsemilinearequationswithvariationalstructure.Morerecently,thistechniquehasalsobeenusedforproblemswithonlyapartialvariationalstructure,likethephase- eldsystems.Moreprecisely,nonconservedmodels(withorwithoutmem-orye ects)havebeenanalyzedin[1,2,16,22,51],whilethecaseofahyperbolicdynamicsfortheorderparameterhasbeenexaminedin[23,48].Therearealsoresultsfornonlocalmodels(see[15,24]).ConcerningthestandardCahn-Hilliardequation,convergencetostationarystateshasbeenexaminedin[11,18,42,44,49],whilethenonconstanttem-peraturecase,namely(1.1)with(1.5),hasbeen rstanalyzedin[14]andthenin[43]inthecaseofdynamicboundaryconditions.Thememorye ectsintheheat uxhavebeentreatedin[3,4]fortheColeman-Gurtinlawand,recently,in[37]forageneralizationoftheMaxwell-Cattaneolaw.Asweshallsee,hereweneedaparticularL ojasiewicz-Simontypeinequalitywhichisare nementoftheoneprovedin[18](seeLemma4.1anditsproofinAppendix).

2.Well-posednessanduniformbounds

LetH=L2( )andH=(L2( ))3.Thesespacesareendowedwiththenaturalinnerproduct ·,· andtheinducednorm · .Forthesakeofsimplicity,wewillassume| |=1andε=1.Then,wesetV=H1( ),V=(H1( ))3andW=H2( ),bothendowedwiththeirstandardinnerproducts,andwede nethesubspaceofHofthenullmeanfunctions

H0={v∈H: v,1 =0}.

WealsointroducethelinearnonnegativeoperatorA= :D(A) H→H0withdomain

D(A)={v∈W: v·n=0,on },

anddenotebyA0itsrestrictiontoH0.NotethatA0isapositivelinearoperator;hence,r/2foranyr∈R,wecande neitspowersArand,consequently,setV0r=D(A0)endowedwiththeinnerproduct

r/2r/2 v1,v2 V0r= A0v1,A0v2 .

Clearly,wehaveV00≡H0.Inaddition,weneedtousetheHilbertspaces

V0={v∈V:v·n=0

and

Hσ=H×H×V×V ,Vσ=V×V0×D(A)×H,

endowedwiththefollowingnorms,respectively,

12223242 (z1,z2,z3,z4) 2Hσ= z +σ z + z V+ z V ,

42223212 (z1,z2,z3,z4) 2Vσ= z V+σ z V+ z W+ z ,on },

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term $\chi_{t

4´G.SCHIMPERNAM.GRASSELLI,H.PETZELTOVA,

ifσ>0.Otherwise,wesimplyset

H0=H×V×V ,V0=V×D(A)×H.

OurassumptionsonthefunctionφandonthepotentialΦ,de nedby y

Φ(y)=φ(ξ)dξ, y∈R,

arethefollowing

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)Φ∈C3(R)suchthatΦ(y)≥ c0,|φ′′(y)|≤c1(1+|y|),|φ(y)|≤ Φ(y)+c , y∈R; y∈R; y∈R; y∈R; >0,thereexistsc >0suchthat ∈R,thereexistc2>0andc3≥0suchthat(y )φ(y)≥c2Φ(y) c3,

φ′(y)≥ c4, y∈R;

forsomepositiveconstantsc0,c1,c4.Herec2andc3continuouslydependon .

Wenowrewritesystem(1.3)togetherwith(1.5)inthefollowingform in(0,∞), ( +χ)t,v q, v =0,

(2.6) σqt+q,v = , ·v ,in(0,∞), χ+χ,w + Aχ+φ(χ)+αχ ,Aw =0,in(0,∞),tttt

forallv∈V,v∈V0,andw∈D(A),endowedwithinitialconditions(1.4).

Letusprove

0∈H,

σq0∈H,

χ0∈V,

χ1∈V ,

∈C0([0,∞),H)

σq∈C0([0,∞);H),

χ∈C0([0,∞),V),

χt∈C0([0,∞),V )∩L2(0,∞,V ),

αχt∈L2(0,∞,H).q∈L2(0,∞;H),Theorem2.1.Let(2.1)-(2.5)hold.Then,forany( 0,q0,χ0,χ1)suchthat(2.7)(2.8)(2.9)(2.10)(2.11)(2.12)(2.13)(2.14)(2.15)theCauchyproblem(2.6)-(1.4)hasa(weak)solution(θ,χ)withthefollowingproperties

andthereexistsapositiveconstantC,dependingonthenormsoftheinitialdataandonφ,suchthat,forallt≥0,

(2.16) ( (t),q(t),χ(t),χt(t)) 2Hσ ∞ 2222+ (τ) (τ),1 + q(τ) + χt(τ) V +α χt(τ) dτ≤C,

t

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term $\chi_{t

ASYMPTOTICBEHAVIOROFANONISOTHERMALCAHN-HILLIARDEQUATION5and

(2.17) ( +χ)(t),1 = 0+χ0,1 , χ(t),1 = χ0+χ1,1 χ1,e t .Ifα>0,thenthesolutionisuniqueandthefollowingboundholds t+1

(2.18)sup Aχ(τ) 2dτ≤C.t≥0t

Moreover,forany xedT>0,if( 0i,q0i,χ0i,χ1i)∈Hσ,i=1,2,thenthecorrespondingsolutions( i,qi,χi,χit)satisfy

(2.19) (( 1 2)(t),(q1 q2)(t),(χ1 χ2)(t),(χ1 χ2)t(t)) 2Hσ

≤C(R)eKT ( 01 02,q01 q02,χ01 χ02,χ11 χ12) 2Hσ,

forsomepositiveconstantsC(R)andK,bothindependentofT,where

( 0i,q0i,χ0i,χ1i) Hσ≤R,i=1,2. t∈[0,T],

Proof.We rstshowinequality(2.16)arguingformally.ThisargumentcanbemaderigorouswithinaFaedo-Galerkinschemeanditsu cestoprovetheexistenceofasolutionforallα≥0.FromnowonCwilldenoteagenericpositiveconstantwhichdependsonφandonthespatialaveragesoftheinitialdata,atmost.Ifasolutionexists,thenitiseasytoshowthevalidityof(2.17),duetotheboundaryconditions(1.5).Moreover,wehave

(2.20)

Letussetnow

(2.21) = ,1 , χ =χ χ,1 ,

andrewriteproblem(2.6)intheform +χ )t,v q, v =0,in(0,∞), (

·v ,(2.22) σqt+q,v = ,in(0,∞), χ Aw =0, tt+χ t,w + Aχ +φ(χ)+αχ t , χt(t),1 = χ1,1 e t.in(0,∞),

forallv∈V,v∈V0,andw∈D(A). inthe rstequation,v=qinthesecondequation,andw=Letustakev= 1A t+βχ ),whereβ>0willbechosensmallenough.Addingtogethertheresulting0(χ

identities,wegetd

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term $\chi_{t

6´G.SCHIMPERNAM.GRASSELLI,H.PETZELTOVA,

forsomeC1>0,while,onaccountof(2.3),weinfer(2.25)

φ

(

χ

)

, χt,1 = φ(χ),1 χt,1 ≥ βC1

2e t.

Hence,using(2.1),wehave

(2.26) 2 φ(χ), χt,1 +2β φ(χ),χ

≥βC1 Φ(χ),1 2βC2 cβe t≥ C(β+cβe t).

Then,taking(2.17)and(2.20)intoaccount,from(2.23)wededuce

d

(2.28)dt Aχ 2 χ t 2+2 Aχ 2+2 φ′(χ) χ, χ 2 , =0.

d 222 χ t,χ + χ +α χ Moreover,inthecaseσ>0,usingthe rsttwoequationsof(2.22),wehave

2

Then,letusintroducethefunctional

(2.32)

1/2 1/2 1/222 1 A0χ t κ21+σ q 2. 1/2 q,χ, 2+σ q 2+ A 1/2χΨσ( , χ t)= t 2+ χ 20+2β A0χ t,A0χ +β A0χ 2+αβ χ 2+2 Φ(χ),1 22+γ12 χ t,χ + χ +α χ

1 γ2 q, A 0 ,

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term $\chi_{t

ASYMPTOTICBEHAVIOROFANONISOTHERMALCAHN-HILLIARDEQUATION7and,recalling(2.30)and

(2.31),

let

us

choose,

in

turn,

γ

2

so

small

that

max

{

2

γ

2,γ2κ2(1+σ 1)}≤1,

andthenβandγ1sosmallthatβ≤1/2,γ1c4≤β/4,and(βκ1+γ1)≤γ2σ 1/4.Then,Ψσful llstheinequality

d

dtΨ0( , χ, χ t)+c A/2

0χ t 2+α χ t 2+β χ 2

≤C(β+cβe t). 2+ 1 +γ1 Aχ 2

Thenwecanargueasabove.

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term $\chi_{t

8M.GRASSELLI,H.PETZELTOVA,´G.SCHIMPERNA

Estimate(2.19)isstandard,providedthatα>0.Indeed,itsu cestowritedownproblem(2.6)forthedi erenceoftwosolutions( i,qi,χi),i=1,2,andthenmultiplythe rstequationby 1 2,thesecondonebyq1 q2,andthethirdonebyA 1 1 χ 2)tUsingtheGronwalllemmaandtaking(2.2)intoaccount,oneeasilygets0(χ.

thewanted

estimate(see,e.g.,[6]or[20]fortheisothermalcase). FromTheorem2.1anditsproofwededucethat,letting

Xσδ={(z1,z2,z3,z4)∈Hσ:| z1,1 |+| z3,1 |+| z4,1 |≤δ}

forsomeδ≥0,endowedwiththemetricinducedbythenormofHσ,andsetting

( (t),q(t),χ(t),χt(t))=:Sσ(t)( 0,q0,χ0,χ1), t≥0,

wehavethatSσ(t)isastronglycontinuoussemigrouponXδ

de neastronglycontinuousdissipativeσwithaboundedabsorbing

set.Similarly,wecansemigroupS0(t)onXδSummingup,wehave0.Corollary2.2.Let(2.1)-(2.5)hold.Foranygivenσ∈[0,1],thesemigroupSσ(t)actingonXσδhasaboundedabsorbingset.

3.Precompactnessoftrajectoriesandglobalattractor

Hereweprove

Theorem3.1.Let(2.1)-(2.5)holdandsupposeα>0.Ifσ∈(0,1]and( 0,q0,χ0,χ1)satis es(2.7)-(2.10),then,indicating

orbit

(t) by( ,q,χ)thecorrespondingsolutionto(2.6)-(1.4)givenbyTheorem2.1,thet≥0( (t),q(t),χ(t),χt(t))isprecompactinHσ.Moreover,thereholds(3.1) 0 χ1,1 →0,

(3.2) q(t) →0,

(3.3) χt(t) V →0,

astgoesto∞,andtheω-limitsetω( 0,q0,χ0,χ1)consistsonlyofequilibriumpointsoftheform( ∞,0,χ∞,0)where( ∞,χ∞)satis es(1.6).Similarresultsholdwhenσ=0.Proof.Onaccountof[39],observe rstthat,thanksto(2.2),(2.5),and(2.16),wecanchoose ≥c4largeenough,anddependingonthenormsoftheinitialdata,suchthat(3.4)1

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term $\chi_{t

ASYMPTOTICBEHAVIOROFANONISOTHERMALCAHN-HILLIARDEQUATION9and

(3.6) ( c+χc)t,v qc, v =0,in(0,∞), σqct+qc,v = c, ·v ,in(0,∞),

χc+χc,w + Aχc+ψ(χc)+αχc c

tttt,Aw = χ,Aw ,in(0,∞),

c(0)= 0,1 ,σqc(0)=0,χc(0)= χ0,1 ,χct(0)= χ1,1 ,in ,

forallv∈V,v∈V0,andw∈D(A).

Weshallprovethat( d(t),qd(t),χd(t),χd

c,qc,χc,χct(t))exponentiallydecaysat0inHσastgoes

to∞,while(t)isboundedinaspacewhichiscompactlyembeddedinHσ,uniformlyintime.

Letusprove rstthat,foranyt≥s≥0andevery >0,thereholds

(3.7)α t

χct(τ) 2dτ≤ (t s)+C

s

(3.8)dt0χ t + χ +2 Ψ(χ),1 2 χ,χ

+ c 2+σ qc 2+ A 1/2c2c2cc2 qc 2+2 A 1/2

0χ ct +2α χ ct 2

=2 ψ(χc), χt,1 2 χt,χ c .

HereΨisaprimitiveofψ.Observe rstthatitisnotdi culttorealizethatanestimatesimilarto(2.16)holdsfor( c,qc,χc,χc

>0andanyt≥0,t)aswell.Therefore,onaccountof(2.2)and(2.20),

wehave,forany

2 ψ(χc(t)), χt(t),1 2 χt(t),χ c(t) ≤2 +C

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term $\chi_{t

10M.GRASSELLI,H.PETZELTOVA,´G.SCHIMPERNA

obtain

d

qd, A 01 d

σ +γ

2 χd 2,

wehavethat,forβandγsmallenough,

(3.12)1

dtΛd+cβ,γΛd≤Cβ,γ

Thus,onaccountof(2.16)and(3.7),wecan χt 2+ χct 2apply[39,Lemma Λd.5]anddeducetheexpo-nentialdecayofΛd,sothat(cf.(2.20)and(3.12))

(3.13) ( d(t),qd(t),χd(t),χdt(t)) H ct

σ≤C(R)e,

providedthat ( 0,q0,χ0,χ1) dHσ≤R.Moreover,takingw=χinthethirdequationof(3.5),weobtain(cf.(2.28))d

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term $\chi_{t

ASYMPTOTICBEHAVIOROFANONISOTHERMALCAHN-HILLIARDEQUATION11whichyields,usingtheYounginequality,(2.23),and(3.13),

d

dt

Wealsohave(cf.(2.29))

d +σ ·q c2c2 cc2+2 χct,A +2 ·q =0.

(3.17)dt

+β χ +αβ χ +2 ψ(χ),Aχ 2c2cc χct + Aχ +2β χt,χ c2c2cc

2c2c2+2(1 β) χct +2α χt +2β Aχ

cc+2β ψ(χc),Aχc 2 c,Aχct 2β ,Aχ

ccc 2 ψ′(χc)χct,Aχ = χ, (χt+βχ) .

Observethat,onaccountof(2.16),

(3.18)

Therefore,setting ψ(χ′cc)χct,Aχ ≤C1+c≤C χct V Aχ . χc 2L6( ) c χct L6( ) Aχ

2c2ccΛc= c 2+σ ·qc 2+ χct + Aχ +2β χt,χ

+β χc 2+αβ χc 2+2 ψ(χc),Aχc γ qc, c ,

forsomeγ>0,usingtheYounginequality,wecanchooseβandγsmallenoughsothatd

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term $\chi_{t

12´G.SCHIMPERNAM.GRASSELLI,H.PETZELTOVA,

Thesecondequationof(3.6)cannowbewritteninthestrongform,namely,σq

c

t+qc= c,

sothata.e.in ×(0,∞),

σ( ×qc)t+ ×qc=0,a.e.in ×(0,∞),

and,since( ×qc)(0)=0,wehave( ×qc)(t)=0foranyt≥0.Consequently,thanksto(3.19), qc(t) Visuniformlyboundedaswell.

Summingup,wehaveshownthatagiventrajectoryoriginatingfromHσisasumofanexponentiallydecayingpartandatermwhichbelongstoaclosedboundedsubsetofVσ.ThereforethetrajectoryisprecompactinHσand,duetotheintegralcontrolsof(2.16)andto(2.17),weinfer(3.1)-(3.3).Finally,itisnotdi culttoprovethat

ω( 0,q0,χ0,χ1) {( ∞,0,χ∞,0):( ∞,χ∞)satis es(1.6)}.

Thecaseσ=0iseasier.Infact,arguingasintheisothermalcase(see[6]),wecanprovethebound

( (t),χ(t),χt(t)) 2V0≤C, t≥t1=t1(R)>0,

providedthat ( 0,χ0,χ1) H0≤R.HencethetrajectoryisprecompactinH0andwecanconcludeasabove. FromtheproofofTheorem3.1,wededucethatthesemigroupSσ(t)hasaboundedattractingsetinVσ,foranyσ∈(0,1],whileS0(t)hasacompactabsorbingset.Thereforewehave(see,e.g.,[28,47])

Corollary3.2.Foreachσ∈[0,1],thesemigroupSσ(t)hasaconnectedglobalattractorAσwhichisboundedinVσ.

Remark3.3.Theaboveresultisa rst,butessential,steptowardtheconstructionofafamilyofexponentialattractorswhichisstable(robust)withrespecttoσand,possibly,toε(see[20]fortheisothermalcase).Thiswillbethesubjectofafutureinvestigation.

4.Convergencetostationarystates

Letusset

E(v)=1

2),η>0,andapositiveconstantLsuchthat

(4.3) (v) φ (v),1 1,|E(v) E(v∞)|1 ρ≤L A0v+φV0

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term $\chi_{t

ASYMPTOTICBEHAVIOROFANONISOTHERMALCAHN-HILLIARDEQUATION13forallv∈V01suchthat v v∞ V01≤η.

Thenweprove

Theorem4.2.LettheassumptionsofLemma4.1holdand

let

α

>

and

σ

>

be

xed.

If

(

0,q0,χ0,χ1)satis es(2.7)-(2.10),thenthetrajectory( (t),q(t),χ(t),χt(t))originatedfrom( 0,q0,χ0,χ1)issuchthat

(4.4)

where( ∞,χ∞)satis esω( 0,q0,χ0,χ1)={( ∞,0,χ∞,0)}, 1 ( 0 χ1), =| | ∞

χ∞=(χ1+χ0), A(Aχ+φ(χ))=0.∞∞

t→∞(4.5)Moreover,(4.6)lim χ(t) χ∞ V=0,

ρandthereexistst >0andapositiveconstantCsuchthat(4.7) (t) ∞ V + χ(t) χ∞ V ≤Ct

dt

where

(4.11) (t),q(t),χL( (t),χ t(t))= q(t) 2 χ t(t) 2 t(t) 2+ h(χ (t)),χ t(t) ,V α χ (t),q(t),χL( (t),χ t(t))

1=

2 χ t(t) 2,

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term $\chi_{t

14´G.SCHIMPERNAM.GRASSELLI,H.PETZELTOVA,

usingalsotheYounginequality.Therefore,from(4.10)wededuce

d

(4.12)

χ

t(t) 2+Cαe 2t,2

forallt≥0.

Then,combining(2.29)with(4.10),weobtain

d

φ (χ )) ,t≥0,

where

G= A 01χ tt,A 01

dt(A0χ +φ (χ )

φ ′(χ )χ t)

= A 01χ t,A 01(A0χ +φ (χ )

φ (χ )) 2

α χ t,A 01(A0χ +φ (χ )

φ (χ ))

+ h(χ ),A 01(A0χ +φ (χ )

φ ′(χ)χ t) .

Observethat(cf.(2.2))

(4.15) A 01χ t,A 01(φ ′(χ )χ t

dtM+Cµ,νN2≤0,

forµ>0andν>0su cientlysmall,where

M(t)=1

φ (χ (t)) 2

V0 1,

forallt≥0.

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term $\chi_{t

ASYMPTOTICBEHAVIOROFANONISOTHERMALCAHN-HILLIARDEQUATION15Letusintroducetheunboundedset

ηΣ=t

≥

:

χ

(t) χ ∞ V01≤

dt d|M(t)|ρsgnM(t)=ρ|M(t)|ρ 1

(4.21)

Therefore χ t(·) V isintegrableoverJand τ(t0)

0≤limsup(4.22) χ t(t) V dtt0∈Σ,t0→∞t0 ρρ 2(1 ρ)t0≤climsup|M(t0)|+|M(τ(t0))|+Ce=0.t0∈Σ,t0→∞wherewemeanthat|M(τ(t0))|=0ifτ(t0)=∞.Ontheotherhand,weeasilyget N(t)dt≤Ce 2(1 ρ)t0.J2 ρ|M(t)|sgnM(t)dtdt ρρ≤C|M(t0)|+|M(τ(t0))|,

Noticethat,foreveryt∈J,

(4.23) χ (t) χ ∞ V ≤

Supposenowthatτ(t0)<∞foranyt0∈Σ.Then,byde nition,

χ (τ(t0)) χ ∞ V01=η, t0∈Σ. tt0 (t0) χ ∞ V . χ t(s) V ds+ χ

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term $\chi_{t

16M.GRASSELLI,H.PETZELTOVA,´G.SCHIMPERNA

Consideranunboundedsequence{tn}n∈N Σwiththeproperty

nlim→∞ χ (tn) χ ∞ V1

0=0.

Bycompactness,wecan ndasubsequence{tnk}k∈Nandanelementv∞∈D(A)suchthat( ∞,0,v∞,0)∈ω( 0,q0,χ0,χ1), v ∞ χ ∞ V1

0=η,and

klim→∞ χ (τ(tnk)) v ∞ V1

0=0.

Then,owingto(4.22)and(4.23),wededucethecontradiction

0< v χ ∞ V ≤limsup τ(tnk)

∞ χ t(s) (tnk) χ ∞

k→∞V ds+ χV =0.

tnk

Hence,τ(t0)=∞forsomet0>0largeenoughand,recalling(2.20),wecandeducethat χt(·) V isindeedintegrableover(t0,∞).Thisyields(4.6)byprecompactness.Ontheotherhand,onaccountof(3.1)-(3.3),(4.4)holdsaswell.Finally,arguingasin[23],wecanprovethat

(4.24) ∞

N(τ)dτ≤Ct ρ

t

1 2ρ, t≥t .

Thuswehave

(4.25) χ(t) χ∞ V ≤Ct ρ

1 2ρ, t≥t .

Therefore,rateestimate(4.7)isaconsequenceof(4.25)-(4.27).Inthecaseσ=0wecanproceedinasimilar(actually,simpler)way,notingthatq= . Remark4.3.Thedecayestimate(4.7)for canbeslightlyimproved.Actually,usingthedecomposition

(4.28) (t) ∞ 2≤2 d(t) 2+2 c(t) ∞ 2,

weseethat,by(3.13),the rsttermdecaysexponentially.Concerningthelatter,onecanuse(3.19),(4.27)andtheinterpolationinequality v 2≤c v V v V

v∈V.Thus,(4.28)eventuallygives,holdingforall

(4.29) (t) ∞ ≤Ct ρ

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term $\chi_{t

ASYMPTOTICBEHAVIOROFANONISOTHERMALCAHN-HILLIARDEQUATION17

5.Appendix

ThissectionisdevotedtodemonstrateLemma4.1.

Let

us

introduce

the

functional

1E(v)=

(v∞)=0.φ

Moreover,v∞isacriticalpointofEonV01.Indeed,itiseasytocheckthat,owingtoourhypotheses,Eiscontinuouslydi erentiableonV01,and (v∞)hdx(5.2) E(v∞)h= v∞· h+φ

(v∞)h = v∞· h+φ

f V .

ConsiderthemappingF:V02→H0,F= E|V02de nedby

(v) F(v)=A0v+φ

(v)w.φ

Hence,inbothcases, 2E(v∞)canbeviewedasaboundedperturbationofA0restrictedtotherespectivespaces.ItfollowsthatKer 2E(v∞) V02anditsrangeisclosedin(V01) andH0,respectively.Moreover,thereholds

(V01) =Ker( 2E(v∞))⊕Ran( 2E(v∞)),H0=Ker( 2E(v∞))⊕Ran( F(v∞)).

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term $\chi_{t

18´G.SCHIMPERNAM.GRASSELLI,H.PETZELTOVA,

Now,wecanapply[9,Thm.3.10]and[9,Cor.3.11]toobtain

|E(v) E(v∞)|1 ρ≤L E(v) (V01) ,

and,consequently,(4.3).

References

[1]S.Aizicovici,E.Feireisl,Long-timestabilizationofsolutionstoaphase- eldmodelwithmemory,J.

Evol.Equ.1(2001),69–84.

[2]S.Aizicovici,E.Feireisl,F.Issard-Roch,Longtimeconvergenceofsolutionstoaphase- eldsystem,

Math.MethodsAppl.Sci.24(2001),277–287.

[3]S.Aizicovici,H.Petzeltov´a,Asymptoticbehaviorofsolutionsofaconservedphase- eldsystemwith

memory,J.IntegralEquationsAppl.15(2003),217–240.

[4]S.Aizicovici,H.Petzeltov´a,Convergenceofsolutionsofphase- eldsystemswithanonconstantlatent

heat,Dynam.SystemsAppl.,14(2005),163–173.

[5]B.Aulbach,Continuousanddiscretedynamicsnearmanifoldsofequilibria,Springer,Berlin,1984.

[6]A.Bonfoh,Existenceandcontinuityofuniformexponentialattractorsforasingularperturbationof

ageneralizedCahn-Hilliardequation,Asymptot.Anal.43(2005),233–247.

[7]M.Brokate,J.Sprekels,HysteresisandPhaseTransitions,Springer,NewYork,1996.

[8]G.Caginalp,Thedynamicsofaconservedphase eldsystem:Stefan-like,Hele-Shaw,andCahn-

Hilliardmodelsasasymptoticlimits,IMAJ.Appl.Math.44(1990),77–94.

[9]R.Chill,OntheL ojasiewicz-Simongradientinequality,J.Funct.Anal.201(2003),572–601.

[10]R.Chill,E.Faˇsangov´a,Convergencetosteadystatesofsolutionsofsemilinearevolutionaryintegral

equations,Calc.Var.PartialDi erentialEquations22(2005),321–342.

[11]R.Chill,E.Faˇsangov´a,J.Pr¨uss,ConvergencetosteadystatesofsolutionsoftheCahn-Hilliard

equationwithdynamicboundaryconditions,Math.Nachr.(toappear).

[12]R.Chill,M.A.Jendoubi,Convergencetosteadystatesinasymptoticallyautonomoussemilinear

evolutionequations,NonlinearAnal.53(2003),1017–1039.

[13]A.Debussche,AsingularperturbationoftheCahn-Hilliardequation,AsymptoticAnal.4(1991),

161-185.

[14]E.Feireisl,F.Issard-Roch,H.Petzeltov´a,Long-timebehaviourandconvergencetowardsequilibria

foraconservedphase eldmodel,DiscreteContin.Dyn.Syst.10(2004),239–252.

[15]E.Feireisl,F.Issard-Roch,H.Petzeltov´a,Anon-smoothversionoftheL ojasiewicz-Simontheorem

withapplicationstonon-localphase- eldsystems,J.Di erentialEquations,199(2004),1–21.

[16]E.Feireisl,G.SchimpernaLargetimebehaviourofsolutionstoPenrose-Fifephasechangemodels,

Math.MethodsAppl.Sci.28(2005),2117–2132.

[17]E.Feireisl,F.Simondon,Convergenceforsemilineardegenerateparabolicequationsinseveralspace

dimensions,J.Dynam.Di erentialEquations12(2000),647–673.

[18]H.Gajewski,A.-J.Griepentrog,Adescentmethodforthefreeenergyofmulticomponentsystems,

DiscreteContin.Dyn.Syst.15(2006),505–528.

[19]P.Galenko,D.Jou,Di use-interfacemodelforrapidphasetransformationsinnonequilibriumsys-

tems,Phys.Rev.E71(2005),046125(13).

[20]S.Gatti,M.Grasselli,A.Miranville,V.Pata,HyperbolicrelaxationoftheviscousCahn-Hilliard

equationin3-D,Math.ModelsMethodsAppl.Sci.15(2005),165–198.

[21]S.Gatti,M.Grasselli,A.Miranville,V.Pata,Onthehyperbolicrelaxationoftheone-dimensional

Cahn-Hilliardequation,J.Math.Anal.Appl.312(2005),230–247.

[22]M.Grasselli,H.Petzeltov´a,G.Schimperna,LongtimebehaviorofsolutionstotheCaginalpsystem

withsingularpotential,Z.Anal.Anwend.25(2006),51–72.

[23]M.Grasselli,H.Petzeltov´a,G.Schimperna,Convergencetostationarysolutionsforaparabolic-

hyperbolicphase- eldsystem,Commun.PureAppl.Anal.5(2006),827–838.

[24]M.Grasselli,H.Petzeltov´a,G.Schimperna,Anonlocalphase- eldsystemwithinertialterm,Quart.

Appl.Math.(toappear).

[25]L.Herrera,D.P´avon,Hyperbolictheoriesofdissipation:Whyandwhendoweneedthem?,Phys.A

307(2002),121–130.

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term $\chi_{t

ASYMPTOTICBEHAVIOROFANONISOTHERMALCAHN-HILLIARDEQUATION19

[26]D.D.Joseph,L.Preziosi,Heatwaves,Rev.ModernPhys.61(1989),4173.

[27]D.D.Joseph,L.Preziosi,Addendumtothepaper:Heatwaves[Rev.ModernPhys.61(1989),no.1,

4173],Rev.ModernPhys.62(1990),375391.

[28]J.K.Hale,Asymptoticbehaviourofdissipativesystems,Amer.Math.Soc.,Providence,RI,1988.

[29]A.Haraux,Syst`emesdynamiquesdissipatifsetapplications,Masson,Paris,1991.

[30]A.Haraux,M.A.Jendoubi,Convergenceofboundedweaksolutionsofthewaveequationwithdissi-

pationandanalyticnonlinearity,Calc.Var.PartialDi erentialEquations9(1999),95–124.

[31]A.Haraux,M.A.Jendoubi,O.Kavian,Rateofdecaytoequilibriuminsomesemilinearparabolic

equations,J.Evol.Equ.3(2003),463–484.

[32]S.-Z.Huang,P.Tak´aˇc,Convergenceingradient-likesystemswhichareasymptoticallyautonomous

andanalytic,NonlinearAnal.46(2001),675–698.

[33]M.A.Jendoubi,Asimpleuni edapproachtosomeconvergencetheoremsofL.Simon,J.Funct.

Anal.153(1998),187–202.

[34]M.A.Jendoubi,Convergenceofglobalandboundedsolutionsofthewaveequationwithlineardissi-

pationandanalyticnonlinearity,J.Di erentialEquations144(1998),302–312.

[35]S.L ojasiewicz,Unepropri´et´etopologiquedessous-ensemblesanalytiquesr´eels,in“Colloquesinter-

nationauxduC.N.R.S.117:Les´equationsauxd´eriv´eespartielles(Paris,1962)”pp.87–89,EditionsduC.N.R.S.,Paris,1963.

[36]S.L ojasiewicz,Ensemblessemi-analytiques,notes,I.H.E.S.,Bures-sur-Yvette,1965.

[37]G.Mola,Globalandexponentialattractorsforaconservedphase- eldsystemwithGurtin-Pipkin

heatconductionlaw,PhDthesis,PolitecnicodiMilano,Milan,2006.

[38]A.Novick-Cohen,OntheviscousCahn-Hilliardequation,in“Materialinstabilitiesincontinuum

mechanics(Edinburgh,1985–1986)”,pp.329–342,OxfordSci.Publ.,OxfordUniv.Press,NewYork,1988.

[39]V.Pata,S.Zelik,Aremarkonthedampedwaveequation,Commun.PureAppl.Anal.5(2006),

609-614.

[40]P.Pol´aˇcik,K.P.Rybakowski,Nonconvergentboundedtrajectoriesinsemilinearheatequations,J.

Di erentialEquations124(1996),472–494.

[41]P.Pol´aˇcik,F.Simondon,Nonconvergentboundedsolutionsofsemilinearheatequationsonarbitrary

domains,J.Di erentialEquations186(2002),586–610.

[42]J.Pr¨uss,R.Racke,S.Zheng,MaximalregularityandasymptoticbehaviorofsolutionsfortheCahn-

Hilliardequationwithdynamicboundaryconditions,AnnaliMat.PuraAppl.(4)185(2006),627–648.

[43]J.Pr¨uss,M.Wilke,MaximalLp-regularityfortheCahn-Hilliardequationwithnonconstanttemper-

atureanddynamicboundaryconditions,in“Partialdi erentialequationsandfunctionalanalysis”,Oper.TheoryAdv.Appl.168,pp.209–236,Birkh¨auser,Basel,2006.

[44]P.Rybka,K.-H.Ho mann,ConvergenceofsolutionstoCahn-Hilliardequation,Comm.Partial

Di erentialEquations24(1999),1055–1077.

[45]A.Segatti,OnthehyperbolicrelaxationoftheCahn-Hilliardequationin3-D:approximationand

longtimebehaviour,Math.ModelsMethodsAppl.Sci.(toappear).

[46]L.Simon,Asymptoticsforaclassofnon-linearevolutionequationswithapplicationstogeometric

problems,Ann.ofMath.(2)118(1983),525–571.

[47]R.Temam,In nite-DimensionalDynamicalSystemsinMechanicsandPhysics,Springer-Verlag,

NewYork,1997.

[48]H.Wu,M.Grasselli,S.Zheng,Convergencetoequilibriumforaparabolic-hyperbolicphase- eld

systemwithNeumannboundaryconditions,Math.ModelsMethodsAppl.Sci.(toappear).

[49]H.Wu,S.Zheng,ConvergencetoequilibriumfortheCahn-Hilliardequationwithdynamicboundary

condition,J.Di erentialEquations204(2004),511–531.

[50]H.Wu,S.Zheng,Convergencetoequilibriumforthedampedsemilinearwaveequationwithcritical

exponentanddissipativeboundarycondition,Quart.Appl.Math.64(2006),167–188.

[51]Z.Zhang,Asymptoticbehaviorofsolutionstothephase- eldequationswithNeumannboundary

conditions,Commun.PureAppl.Anal.4(2005),683–693.

[52]S.Zheng,A.Milani,Exponentialattractorsandinertialmanifoldsforsingularperturbationsofthe

Cahn-Hilliardequations,NonlinearAnal.57(2004),843–877.

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term $\chi_{t

20´G.SCHIMPERNAM.GRASSELLI,H.PETZELTOVA,

[53]S.Zheng,A.Milani,GlobalAttractorsforsingularperturbationsoftheCahn-Hilliardequations,J.

Di erentialEquations209(2005),101–139.

DipartimentodiMatematica“F.Brioschi”

PolitecnicodiMilano

ViaBonardi,9

I-20133Milano,Italy

E-mailaddress:maugra@mate.polimi.it

MathematicalInstituteASCRˇ´,25Zitna

CZ-11567Praha,CzechRepublic

E-mailaddress:petzelt@math.cas.cz

DipartimentodiMatematica“F.Casorati”

`degliStudidiPaviaUniversita

ViaFerrata,1

I-27100Pavia,Italy

E-mailaddress:giusch04@unipv.it

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