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Diffusiverelaxationtoequilibriaforanextended
reaction-diffusionsystemontherealline
ThierryGallayandSiniˇsaSlijepˇcevi´c
June29,2021
Abstract
Westudythelong-timebehaviorofthesolutionsofatwo-componentreaction-diffusion
systemontherealline,whichdescribesthebasicchemicalreactionA−⇀↽−⇀↽−
thattheinitialdensitiesofthespeciesA,Bareboundedandnonnegative,weprovethatthe
solutionconvergesuniformlyoncompactsetstothemanifoldEofallspatiallyhomogeneous
,but
onextendeddissipativesystems[18],ourapproachreliesonlocalizedenergyestimates,and
providesanexplicitboundforthetimeneededtoreachaneighborhoodofthemanifoldE
asingleequilibriumast→+∞,buttheyarealwaysquasiconvergentinthesensethattheir
ω-limitsetsconsistofchemicalequilibria.
1Introduction
Reaction-diffusionsystemssatisfyingadetailedorcomplexbalanceconditionprovideinteresting
examplesofevolutionequationswherethequalitativebehaviorofthesolutionscanbestudied
k
α1A1+···+αnAn−−↽−−↽−−⇀β1A1+···+βnAn,()
k′
whereA1,...,Andenotethereactantandproductspecies,k,k′>0arethereactionrates,and
arXiv:[]29Jun2021thenonnegativeintegersαi,βi(i=1,...,n)
thelawofmassaction,theconcentrationci(x,t)ofthespeciesAisatisflesthereaction-diffusion
equation
YnYn
αj′βj
∂tci=di∆ci+(βi−αi)kcj−kcj,i=1,...,n,()
j=1j=1
where∆istheLaplaceoperatoractingonthespacevariablex,anddi>0denotesthediffusion
[25,41,8,29]foramoredetailedmathematical
modelingofchemicalreactions,includingtherealisticsituationwhereseveralreactionsoccurat
,thereisanotionofdetailedbalance,whichasserts
thatallreactionsarereversibleandindividuallyinbalanceateachequilibriumstate,anda
weakernotionofcomplexbalance,whichonlyrequiresthateachreactantorproductcomplexis
,wefocuson
aparticularexampleofthesingle-reactionsystem(),forwhichthedetailedbalancecondition
isautomaticallysatisfled.
1:.
Inrecentyears,manyauthorsinvestigatedthelong-timebehaviorofsolutionstoreaction-
diffusionsystemswithcomplexordetailedbalance,assumingthatthereactiontakesplaceina
boundeddomainΩ⊂RNandusinganentropymethodthatwebrieflyexplaininthecaseof
system()withk=k′.Ifc(t)=(c(t),...,c(t))isasolutionof()inΩsatisfyingno-flux
1n
boundaryconditionson∂Ω,wehavetheentropydissipationlawdΦ(c(t))=−D(c(t)),where
dt
Φistheentropyfunctiondeflnedby
XnZ
Φ(c)=φci(x)dx,φ(z)=zlog(z)−z+1,()
i=1Ω
andDistheentropydissipation
XnZ2Z
|∇ci(x)|B(x)
D(c)=didx+klogB(x)−A(x)dx,()
i=1Ωci(x)ΩA(x)
QαjQβj
whereA(x)=cj(x),B(x)=cj(x).Itisclearfrom()thattheentropydissipation
D(c)isnonnegativeandvanishesifandonlyiftheconcentrationsciarespatiallyhomogeneous
(∇ci=0)andthesystemisatchemicalequilibrium(A=B).TheentropyisthereforeaLya-
punovfunctionfor(),andusingLaSalle’sinvarianceprincipleonededucesthatallbounded
solutionsconvergetohomogeneouschemicalequilibriaast→+∞[21,40].Inaddition,under
appropriateassumptions,theentropydissipationD(c)canbeboundedfrombelowbyamulti-
pleoftheentropyΦ(c),ormorepreciselyoftherelativeentropyΦ(c|c∗)withrespecttosome
equilibriumc∗.Suchalowerboundcanbeestablishedusingacompactnessargument[20,22],or
invokingfunctionalinequalitiessuchathelogarithmicSobolevinequality[6,7,8,13,14,29,35].
Thisleadstoaflrstorderdifferentialinequalityfortherelativeentropy,whichimpliesexponen-
,thisentropy-dissipationapproach
evenprovidesexplicitestimatesoftheconvergencerateandofthetimeneededtoreachaneigh-
borhoodoftheflnalequilibrium[6,7].Itisalsoworthmentioningthatthereaction-diffusion
system()isactuallythegradientflowoftheentropyfunction()withrespecttoanappro-
priatemetricbasedontheWassersteindistanceforthediffusionpartofthesystem[26,28,30].
Finally,weobservethatLyapunovfunctionssuchastheentropy()werealsousefultoprove
globalexistenceofsolutionstoreaction-diffusionsystems,see[3,4,12,15,23,33,34,42].
Muchlessisknownonthedynamicsofthereaction-diffusionsystem()inanunbounded
domainsuchasΩ=,theentropy()istypicallyinflnite,anditis
knownthat(),whichexistin
manyexamples,donotconvergetoequilibriaast→+∞,atleastnotinthetopologyofuniform
,thebestwecanhopeforingeneralisquasiconvergence,namely
uniformconvergenceoncompactsubsetsofΩtothefamilyofspatiallyhomogeneousequilibria.
Thatpropertyisnotautomaticatall,andhasbeenestablishedsofaronlyforrelativelysimple
scalarequationswherethemaximumprincipleisapplicable[10,27,31,32,36,37,38].Onthe
otherhand,itisimportanttomentionthatentropyisstilllocallydissipatedundertheevolution
deflnedby(),inthesensethattheentropydensitye(x,t),theentropyfluxf(x,t)andthe
entropydissipationd(x,t)satisfythelocalentropybalanceequation∂te=divf−
theexplicitexpressions
XnXn
e(x,t)=φci(x,t),f(x,t)=dilog(ci(x,t))∇ci(x,t),
i=1i=1
n()
X|∇ci(x,t)|2B(x,t)
d(x,t)=di+klogB(x,t)−A(x,t),
ci(x,t)A(x,t)
i=1
2:.
fromwhichwededucethepointwiseestimate|f|2≤Cedlog(2+e)forsomeconstantC>0.
Thispreciselymeansthatthereaction-diffusionsystem()isanextendeddissipativesystem
inthesenseofourpreviouswork[18].IfN≤2,theresultsof[18]showthatallbounded
solutionsof()inRNconvergeuniformlyoncompactsubsetstothefamilyofspatiallyho-
mogeneousequilibriafor“almostall”times,+ofzero
densityinthelimitwheret→+∞.Inparticular,theω-limitsetofanyboundedsolution,
withrespecttothetopologyofuniformconvergenceoncompactsets,alwayscontainsanequi-
,however,thatextendeddissipativesystemsinthesenseof[18]
mayhavenon-quasiconvergentsolutions,
thatpreventsquasiconvergenceisthecoarseningdynamicsthatisobserved,forinstance,inthe
one-dimensionalAllen-Cahnequation[11,36].
Inthepresentpaper,weconsideraverysimpleparticularcaseofthereaction-diffusion
system(),forwhichwecanprovethatallpositivesolutionsconvergeuniformlyoncompact
A,Bwhichparticipatetothesimplisticreaction
k
A−−↽−−↽−−⇀2B.()
k
Denotingbyu,vtheconcentrationsofA,B,respectively,weobtainthesystem
2
ut(x,t)=auxx(x,t)+kv(x,t)−u(x,t),
2()
vt(x,t)=bvxx(x,t)+2ku(x,t)−v(x,t),
whichisconsideredonthewholereallineΩ=
a,b>0andthereactionratek>0,butscalingargumentsrevealthattheratioa/bistheonly
,givenboundedandnonnegativeinitialdata
u0,v0,thesystem()hasauniqueglobalsolutionthatremainsboundedandnonnegativefor
allpositivetimes,
thelong-timebehaviorofthosesolutions,usingthelocalformoftheentropydissipationand
someadditionalpropertiesofthesystem.
Asawarm-upweconsiderthecaseofequaldiffusivitiesa=b,whichisconsiderablysimpler
becausethefunctionw=2u+vthensatisflestheone-dimensionalheatequationwt=awxx.
Usingthatobservation,itiseasytoprovethefollowingresult:
=banyboundednonnegativesolutionof()satisfles,forallt>0,
tku(t)k2∞+tkv(t)k2∞+(1+t)ku(t)−v(t)2k∞≤C,()
xLxLL
wheretheconstantonlydependsontheparametersa,kandonku0kL∞,kv0kL∞.
,uniformlyoncompactinter-
valsI⊂R,tothemanifoldofspatiallyhomogeneousequilibriadeflnedby
no
E=(¯u,v¯)∈R2;¯u=¯v2,()
+
,theω-limitsetofanysolution,
withrespecttothetopologyofuniformconvergenceoncompactsets,isentirelycontainedinE.
Theproofshowsthatthedecayratesgivenby(),
itisclearthattheω-limitsetisnotalwaysreducedtoasingleequilibrium,becauseexamples
ofnonconvergentsolutionscanbeconstructedevenforthelinearheatequationonR,see[5].
3:.
auxiliaryfunctionw=2u+v,
becomesmuchmorechallengingwhena6=b,becausesystem()doesnotreducetoascalar
,andcanbestatedasfollows.
()satisfles,forallt>0,
C2C
kux(t)kL∞+kvx(t)kL∞≤1/2log(2+t),ku(t)−v(t)kL∞≤1/2,()
t(1+t)
wheretheconstantonlydependsontheparametersa,b,kandonku0kL∞,kv0kL∞.
Thedecayratesofthederivativesux,vxin()agreewith()uptoalogarithmiccorrec-
tion,buttheestimateofthedifferenceu−v2,whichmeasuresthedistancetothelocalchemical
equilibrium,
,andthattheoptimal
estimates()remainvalidwhena6=,itisworthmentioningthatthebounds
()areactuallyderivedfromauniformlylocalestimatewhichfullyagreeswiththedecay
ratesgivenin().Indeed,weshallproveinSection4thatanyboundednonnegativesolution
to()satisfles,foranyt>0,
Z√
x0+t
sup|u(x,t)|2+|v(x,t)|2+ u(x,t)−v(x,t)2 dx≤Ct−1/2,()
√xx
x0∈Rx0−t
wheretheconstantdependsonlyontheparametersa,b,
that()implies(),buttheconverseisnotquitetrueandthebestwecouldobtainsofar
istheweakerestimate().
Asbefore,wecanconcludethatallsolutionsconvergeuniformlyoncompactsetstothe
manifoldEast→+∞.
,thesolutionof()satisfles,for
anytimet>0andanyboundedintervalI⊂R,
noC|I|
infku(t)−u¯kL∞(I)+kv(t)−v¯kL∞(I);(¯u,v¯)∈E≤1/2log(2+t),()
|I|+t
wheretheconstantonlydependsontheparametersa,b,kandonku0kL∞,kv0kL∞.
,thedynamicsofsystem()is
completelydifferentifweconsidersolutionsforwhichthesecondcomponentvmaytakenegative
,ifa=b=k=1,wecanlookforsolutionsoftheparticularform
3z(x,t)3z(x,t)
u(x,t)=1−,v(x,t)=−1+,
42
inwhichcase()reducestotheFisher-KPPequationzt=zxx+3z(1−z).Thatequationhas
apulse-likestationarysolutiongivenbytheexplicitformula
31
z¯(x)=1−2√,x∈R,
2cosh(3x/2)
whichprovidesanexampleofasteadystate(¯u,v¯)for()thatisnotspatiallyhomogeneous
noratchemicalequilibrium,inthesensethat¯u6=¯,foranyspeedc>0,the
4:.
Fisher-KPPequationhastravelingwavesolutionsoftheformz(x,t)=ϕ(x−ct)wherethe
waveproflleϕsatisflesϕ(−∞)=1andϕ(+∞)=(),
thequantitieskux(t)kL∞,kvx(t)kL∞,andku(x)−v(t)2kL∞areboundedawayfromzeroforall
times,insharpcontrastwith().
()isconsideredona
boundedintervalI=[0,L],withhomogeneousNeumannboundaryconditions,becausethe
solutionsu,vcanthenbeextendedtoevenand2RL-
thatcasethetotalmassM=L2u(x,t)+v(x,t)dxisaconservedquantity,andthesolution
0
necessarilyconvergestotheuniqueequilibrium(u∞,v∞)∈Esatisfying2u∞+v∞=M/
in()wehavethebound
CL
ku(t)−u∞kL∞(I)+kv(t)−v∞kL∞(I)≤1/2log(2+t),t≥0,
L+t
whichisfarfromoptimalbecause,inthatparticularcase,itisknownthatconvergenceoccursat
exponentialrate,see[6],
,thesecond
estimatein()showsthatthetimenee