设为首页
加入收藏
期刊导航
网站地图
首页
期刊
数学与物理
地球与环境
信息通讯
经济与管理
生命科学
工程技术
医药卫生
人文社科
化学与材料
会议
合作
新闻
我们
招聘
千人智库
我要投稿
办刊
期刊菜单
●领域
●编委
●投稿须知
●最新文章
●检索
●投稿
文章导航
●Abstract
●Full-Text PDF
●Full-Text HTML
●Full-Text ePUB
●Linked References
●How to Cite this Article
PureMathematics
n
Ø
ê
Æ
,2023,13(3),423-427
PublishedOnlineMarch2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.133046
˜
a
ª
z
.
)
Û
Ü
•
3
5
•••
ÔÔÔ
xxx
2
²
Œ
Æ
ê
Æ
†
&
E
‰
ÆÆ
§
2
À
2
²
Â
v
F
Ï
µ
2023
c
2
6
F
¶
¹
^
F
Ï
µ
2023
c
3
6
F
¶
u
Ù
F
Ï
µ
2023
c
3
14
F
Á
‡
3
©
¥
§
·
‚
•
Ä
±
e
Ô
-
Ô
-
Ô
ª
z
.
:
∂
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)
ù
p
Ω
•
R
n
(
n
≥
1
)
¥
k
.
1
w
à
«
•
"
©
Ï
L
©
Û
T
.
¿
(
Ü
®
k
©
z
§
3
˜
½
^
‡
e
‰
Ñ
ù
a
ª
z
.
)
Û
Ü
•
3
5
"
'
…
c
ª
z
5
§
Keller-Segel
.
§
Û
Ü
•
3
5
LocalExistenceofSolutionsforaClass
ofChemotaxisModel
ShutingChen
SchoolofMathematicsandInformationScience,GuangzhouUniversity,GuangzhouGuangdong
©
Ù
Ú
^
:
•
Ô
x
.
˜
a
ª
z
.
)
Û
Ü
•
3
5
[J].
n
Ø
ê
Æ
,2023,13(3):423-427.
DOI:10.12677/pm.2023.133046
•
Ô
x
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)
inaboundedconvexdomain
Ω
⊂
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
Keller
Ú
Segel
3
1970
c
0
ª
z
²
;
ê
Æ
.
(
ë
„
[1])
§
£
ã
[
œ
Å
ÿ
•
•p
ß
Ý
z
Æ&
Ò
à
8
§
Ù
²
;
ª
z
.
X
e
(
u
t
= ∆
u
−
χ
∇·
(
u
∇
v
)
,x
∈
Ω
,t>
0
v
t
= ∆
v
−
v
+
u,x
∈
Ω
,t>
0
.
(1.1)
é
u
.
(1.1)
9
Ù
C
N
©
Û
Ì
‡
8
¥
3
)
k
.
5
Ú
»
¯
K
þ
§
¿
…
®
2
•
ï
Ä
"
DOI:10.12677/pm.2023.133046424
n
Ø
ê
Æ
•
Ô
x
•
•
Ð
/
)
X
Ú
(1.4)
u
Ð
§
·
‚
é
®
k
ó
Š
?
1
{
‡
0
"
é
u
[
‚
5
Keller-Segel
ª
z
.
(
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)
3
L
›
c
p
®Ú
å
I
S
Æ
ö
2
•
'
5
"
3
X
Ú
(1.2)
¥
§
X
J
G
(
u,v
) =
−
v
+
u
§
@
o
·
‚
k
X
eX
Ú
(
u
t
=
∇·
(
D
(
u,v
)
∇
u
)
−∇·
(
S
(
u,v
)
∇
v
)
,x
∈
Ω
,t>
0
v
t
= ∆
v
−
v
+
u,x
∈
Ω
,t>
0
(1.3)
T
X
Ú
®
²
2
•
'
5
(
ë
„
[2–5])
"
©
ï
Ä
˜
a
[
‚
5
Ô
-
Ô
-
Ô
ª
z
X
Ú
∂
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
,
(1.4)
ù
p
Ω
•
R
n
(
n
≥
1)
¥
k
.
1
w
à
«
•
,
Ù
¥
D
,
S
Ú
R
´
3
[0
,
∞
)
þ
5
½
¼
ê
,
Ù
Š
©
O
3
(0
,
∞
),[0
,
∞
)
Ú
(0
,
∞
)
þ
,
u
(
x,t
)
L
«
[
œ
—
Ý
,
v
(
x,t
)
Ú
w
(
x,t
)
©
OL
«
z
Æ
á
Ú
Ú
z
Æ
ü
½
ß
Ý
.
2.
^
‡
½
b
½
¼
ê
D
(
s
)
,S
(
s
)
,R
(
s
)
÷
v
X
e
^
‡
µ
(A1)
b
D
,
S
Ú
R
÷
v
D
∈
C
2
([0
,
∞
))
S
∈
C
2
([0
,
∞
))
R
∈
C
2
([0
,
∞
))
.
(2.1)
Ù
¥
D
´
,
S
Ú
R
´
š
K
…
S
(0) = 0
,R
(0) = 0.
(A2)
Ð
©
^
‡
u
0
,v
0
,w
0
÷
v
±
e
b
u
0
∈
W
1
,
∞
(Ω)
u
0
≥
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
Ñ
3
Ω
S
.
DOI:10.12677/pm.2023.133046425
n
Ø
ê
Æ
•
Ô
x
3.
)
Û
Ü
•
3
5
Ï
L
©
Û
T
.
¿
(
Ü
ë
•
©
z
[6–10].
3
D,S,R
÷
v
(A1)
§
…
u
0
,v
0
,w
0
÷
v
(2.2)
^
‡
e
,
·
‚
T
.
)
Û
Ü
•
3
5
,
¿
‰
Ñ
Œ
*
Ð
5
Ä
•
ã
.
ù
p
·
‚
¦
^
©
z
[11]
•
X
Ú
(1.2)
J
Ñ
•{
5
?
n
X
Ú
(1.4)
§
Ù
¥
f
(
u,v
)=
−
v
+
u
,
g
(
u,w
) =
−
uw
½
g
(
u,w
) =
−
u
+
w
.
é
u
ƒ
A
¯
K
f
(
u,v
) =
−
v
+
u
Ú
g
(
u,w
) =
−
uw
§
·
‚
k
∂
t
u
=
∇·
(
D
(
u
)
∇
u
)
−∇·
(
S
(
u
)
∇
v
)+
∇·
(
R
(
u
)
∇
w
)
,x
∈
Ω
,t>
0
,
∂
t
v
= ∆
v
−
v
+
u,x
∈
Ω
,t>
0
,
∂
t
w
= ∆
w
−
uw,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
.
(3.1)
½
n
3.1
D,S,R
÷
v
(A1)
§
…
u
0
,v
0
,w
0
÷
v
(2.2)
§
-
ϑ>n
,
K
•
3
T
max
∈
(0
,
∞
]
Ú
n
-
š
K
¼
ê
u
∈
C
0
(Ω
×
[0
,T
max
))
∩
C
2
,
1
¯
Ω
×
(0
,T
max
)
v
∈
C
0
(Ω
×
[0
,T
max
))
∩
C
2
,
1
(Ω
×
(0
,T
max
))
∩
L
∞
loc
[0
,T
max
);
W
1
,ϑ
(Ω)
w
∈
C
0
(Ω
×
[0
,T
max
))
∩
C
2
,
1
(Ω
×
(0
,T
max
))
∩
L
∞
loc
[0
,T
max
);
W
1
,ϑ
(Ω)
¦
(
u,v,w
)
3
Ω
×
(0
,T
max
)
¥
´
(3.1)
;
)
§
¿
…
¦
·
‚
Œ
±
À
J
T
max
=
∞
,
½
ö
limsup
t
→
T
max
k
u
(
·
,t
)
k
L
x
(Ω)
+
k
v
(
·
,t
)
k
W
1
,ϑ
(Ω)
+
k
w
(
·
,t
)
k
W
1
,ϑ
(Ω)
=
∞
.
(3.2)
½
n
3.2
e
t
∆
t
≥
0
•
Ω
¥
Neumann
9
Œ
+
§
λ
1
>
0
L
«
Neumann
>
.
^
‡
e
Ω
¥
−
∆
1
˜
š
"
A
Š
.
K
•
3
=
•
6
u
Ω
~
ê
M
1
,
···
,M
4
§
§
‚
ä
k
±
e
5
Ÿ
:
(1)
X
J
1
≤
q
≤
p
≤∞
,
@
o
é
¤
k
t>
0
k
:
e
t
∆
w
L
p
(Ω)
≤
M
1
1+
t
−
n
2
(
1
q
−
1
p
)
e
−
λ
1
t
k
w
k
L
a
(Ω)
,
¤
á
§
…
é
¤
k
w
∈
L
q
(Ω)
÷
v
R
Ω
w
= 0.
(2)
X
J
1
≤
q
≤
p
≤∞
,
@
o
é
¤
k
t>
0,
é
z
‡
w
∈
L
q
(Ω)
k
:
∇
e
t
∆
w
L
p
(Ω)
≤
M
2
1+
t
−
1
2
−
n
2
(
1
q
−
1
p
)
e
−
λ
1
t
k
w
k
L
q
(Ω)
,
(3)
X
J
2
≤
p<
∞
,
@
o
é
¤
k
t>
0
,w
∈
W
1
,p
(Ω)
k
:
∇
e
t
∆
w
L
p
(Ω)
≤
M
3
e
−
λ
1
t
k∇
w
k
L
p
(Ω)
,
DOI:10.12677/pm.2023.133046426
n
Ø
ê
Æ
•
Ô
x
(4)
-
1
<p
≤
p<
∞
,
@
o
é
¤
k
w
∈
(
C
∞
0
(Ω))
n
,
t>
0
k
:
e
t
∆
∇·
w
L
p
(Ω)
≤
M
4
1+
t
−
1
2
−
n
2
(
1
q
−
1
p
)
e
−
λ
1
t
k
w
k
L
q
(Ω)
,
(3.3)
¤
á
.
Ï
d
é
u
¤
k
t>
0
§
Ž
f
e
t
∆
∇
P k
˜
‡
•
˜
(
½
l
L
q
(Ω)
L
p
(Ω)
Ž
f
*
Ð
§
Ù
¥
‰
ê
U
(3.3)
›
›
"
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.2023.133046427
n
Ø
ê
Æ