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PureMathematicsnØêÆ,2023,13(3),423-427
PublishedOnlineMarch2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.133046
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∂
t
u= ∇·(D(u)∇u)−∇·(S(u)∇v)+∇·(R(u)∇w),x∈Ω,t>0,
∂
t
v= ∆v+f(u,v),x∈Ω,t>0,
∂
t
w= ∆w+g(u,w),x∈Ω,t>0,
∂u
∂ν
=
∂v
∂ν
=
∂w
∂ν
= 0,x∈∂Ω,t>0,
u(x,0) = u
0
(x),v(x,0) = v
0
(x),w(x,0) = w
0
(x),x∈Ω,
(0.1)
ùpΩ•R
n
(n≥1)¥k.1wà«•"©ÏL©ÛT. ¿(Ü®k ©z§3˜½^‡
e‰Ñùaªz.)ÛÜ•35"
'…c
ªz5§Keller-Segel.§ÛÜ•35
LocalExistenceofSolutionsforaClass
ofChemotaxisModel
ShutingChen
SchoolofMathematicsandInformationScience,GuangzhouUniversity,GuangzhouGuangdong
©ÙÚ^:•Ôx.˜aªz.)ÛÜ•35[J].nØêÆ,2023,13(3):423-427.
DOI:10.12677/pm.2023.133046
•Ôx
Received:Feb.6
th
,2023;accepted:Mar.6
th
,2023;published:Mar.14
th
,2023
Abstract
Inthispaper,weconsiderthefollowingparabolic-parabolic-parabolicChemotaxis
Model:





















∂
t
u= ∇·(D(u)∇u)−∇·(S(u)∇v)+∇·(R(u)∇w),x∈Ω,t>0,
∂
t
v= ∆v+f(u,v),x∈Ω,t>0,
∂
t
w= ∆w+g(u,w),x∈Ω,t>0,
∂u
∂ν
=
∂v
∂ν
=
∂w
∂ν
= 0,x∈∂Ω,t>0,
u(x,0) = u
0
(x),v(x,0) = v
0
(x),w(x,0) = w
0
(x),x∈Ω,
(0.2)
inaboundedconvexdomainΩ⊂R
n
,n≥1,withsmoothboundary.Basedonthe
knownresultsinthereferences,byanalyzingthemodel,weobtainthelocalexistence
ofsolutionsforthiskindofchemotaxismodelundercertainconditions.
Keywords
Chemotaxis,Keller-SegelModel,LocalExistence
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CC BY4.0).
http://creativecommons.org/licenses/by/4.0/
1.0
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(
u
t
= ∆u−χ∇·(u∇v),x∈Ω,t>0
v
t
= ∆v−v+u,x∈Ω,t>0.
(1.1)
éu.(1.1)9ÙCN©Û̇8¥3)k.5Ú»¯Kþ§¿…®2•ïÄ"
DOI:10.12677/pm.2023.133046424nØêÆ
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ªz.
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u
t
= ∇·(D(u,v)∇u)−∇·(S(u,v)∇v),x∈Ω,t>0
v
t
= ∆v+G(u,v),x∈Ω,t>0
(1.2)
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3XÚ(1.2)¥§XJG(u,v) = −v+u§@o·‚kXeXÚ
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u
t
= ∇·(D(u,v)∇u)−∇·(S(u,v)∇v),x∈Ω,t>0
v
t
= ∆v−v+u,x∈Ω,t>0
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TXÚ®²2•'5(ë„[2–5])"
©ïÄ˜a[‚5Ô-Ô-ÔªzXÚ
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∂
t
u= ∇·(D(u)∇u)−∇·(S(u)∇v)+∇·(R(u)∇w),x∈Ω,t>0,
∂
t
v= ∆v+f(u,v),x∈Ω,t>0,
∂
t
w= ∆w+g(u,w),x∈Ω,t>0,
∂u
∂ν
=
∂v
∂ν
=
∂w
∂ν
= 0,x∈∂Ω,t>0,
u(x,0) = u
0
(x),v(x,0) = v
0
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(x),x∈Ω,
(1.4)
ùpΩ•R
n
(n≥1)¥k.1wà«•,Ù¥D,SÚR´3[0,∞)þ5½¼ê,ÙŠ©O
3(0,∞),[0,∞)Ú(0,∞)þ,u(x,t)L«[œ—Ý,v(x,t)Úw(x,t)©OL«zÆáÚÚzÆü
½ßÝ.
2.^‡½
b½¼êD(s),S(s),R(s)÷vXe^‡µ
(A1)bD,SÚR÷v

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



D∈C
2
([0,∞))
S∈C
2
([0,∞))
R∈C
2
([0,∞)).
(2.1)
Ù¥D´,SÚR´šK…S(0) = 0,R(0) = 0.
(A2)Щ^‡u
0
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0
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0
÷v±eb
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u
0
∈W
1,∞
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≥0
v
0
∈W
1,∞
(Ω)v
0
≥0
w
0
∈W
1,∞
(Ω)w
0
≥0.
(2.2)
Ù¥u
0
,v
0
,w
0
Ñ3ΩS.
DOI:10.12677/pm.2023.133046425nØêÆ
•Ôx
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t∆
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t≥0
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e
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L
p
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n
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(
1
q
−
1
p
)
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e
−λ
1
t
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L
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Ω
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(Ω)k:
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L
p
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1+t
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−
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(
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q
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p
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e
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t
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t
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L
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,
DOI:10.12677/pm.2023.133046426nØêÆ
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n
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
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(
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q
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p
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
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t
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L
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ë•©z
[1]Keller, E.F.and Segel,L.A.(1970) Initiationof SlimeMoldAggregationViewedasanInstabili-
ty. JournalofTheoreticalBiology, 26, 399-415. https://doi.org/10.1016/0022-5193(70)90092-5
[2]Winkler,M.(2013)Finite-TimeBlow-UpintheHigher-DimensionalParabolic-Parabolic
Keller-SegelSystem.JournaldeMath´ematiquesPuresetAppliqu´ees,100,748-767.
https://doi.org/10.1016/j.matpur.2013.01.020
[3]Osaki,K.andYagi, A.(2001)FiniteDimensionalAttractor forOne-DimensionalKeller-Segel
Equations.FunkcialajEkvacioj,44,441-469.
[4]Nagai,T.,Senba,T.and Yoshida, K.(1997) Application ofthe Trudinger-Moser Inequality to
aParabolicSystemofChemotaxis.FunkcialajEkvacioj,40,411-433.
[5]Cao, X.(2015) GlobalBoundedSolutionsoftheHigher-DimensionalKeller-SegelSystem under
SmallnessConditionsinOptimalSpaces.DiscreteandContinuousDynamicalSystems,35,
1891-1904.https://doi.org/10.3934/dcds.2015.35.1891
[6]Cie´slak, T. (2007)QuasilinearNonuniformly ParabolicSystem Modelling Chemotaxis.Journal
ofMathematicalAnalysisandApplications,326,1410-1426.
https://doi.org/10.1016/j.jmaa.2006.03.080
[7]Horstmann,D.andWinkler,M.(2005)Boundednessvs.Blow-UpinaChemotaxisSystem.
JournalofDifferentialEquations,215,52-107.https://doi.org/10.1016/j.jde.2004.10.022
[8]Tao,Y.andWinkler,M.(2011)AChemotaxis-HaptotaxisModel:TheRolesofNonlinear
DiffusionandLogisticSource.SIAMJournalonMathematicalAnalysis,43,685-704.
https://doi.org/10.1137/100802943
[9]Wrzosek,D.(2004)GlobalAttractorforaChemotaxisModelwithPreventionofOvercrowding.
NonlinearAnalysis,59,1293-1310.https://doi.org/10.1016/j.na.2004.08.015
[10]Winkler,M.(2010)Aggregationvs.GlobalDiffusiveBehaviorintheHigher-Dimensional
Keller-SegelModel.JournalofDifferentialEquations,248,2889-2905.
https://doi.org/10.1016/j.jde.2010.02.008
[11]Zhang, Q.and Li,Y.(2015) Stabilization andConvergence Rate in aChemotaxis System with
ConsumptionofChemoattractant.JournalofMathematicalPhysics,56,ArticleID:081506.
https://doi.org/10.1063/1.4929658
DOI:10.12677/pm.2023.133046427nØêÆ

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