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
AdvancesinAppliedMathematicsA^êÆ?Ð,2021,10(7),2442-2456
PublishedOnlineJuly2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.107256
šg£BoissonadeXÚ)•žm5•x
www
∗
§§§¤¤¤²²²
þ°“‰ŒÆ§þ°
ÂvFϵ2021c619F¶¹^Fϵ2021c711F¶uÙFϵ2021c722F
Á‡
ùŸØ©Ì‡•)±eü‡•¡:Äky²g£BoissonadeXÚf)•˜5.Ï•g
£BoissonadeXÚg‘´uvØ´u
2
,¤±3y²f)•˜5ž†˜„•{k¤O,Ï
d,©‰Ñy²f)•˜5äN•{.•,Šâ˜— áÚf•3¿©7‡^‡y²š
g£BoissonadeXÚ˜—áÚf3E¥•35"
'…c
šg£ÄåXÚ§˜—áÚf§ëYL§§ÛáÚf§²£k.
LongTimeCharacterizationof
SolutionsofNonautonomous
BoissonadeSystems
ZhenqiongCui
∗
,ChengmingYang
ShanghaiNormalUniversity,Shanghai
Received:Jun.19
th
,2021;accepted:Jul.11
th
,2021;published:Jul.22
nd
,2021
∗ÏÕŠö"
©ÙÚ^:w ,¤².šg£BoissonadeXÚ)•žm5•x[J].A^êÆ?Ð,2021,10(7):2442-2456.
DOI:10.12677/aam.2021.107256
w §¤²
Abstract
This paper mainly includesthe following twoaspects:Firstly, we prove theuniqueness
ofweaksolutionofautonomousBoissonadesystem.Becausethequadratictermof
theautonomousBoissonadesystemisuvinsteadofu
2
,itisdifferentfromthegeneral
methodwhenprovingtheuniquenessofweaksolution.Therefore,thispapergivesa
specificmethodtoprovetheuniquenessofweaksolution.Finally,accordingtothe
sufficientandnecessary conditionsfortheexistenceofuniform attractor,the existence
ofuniformattractorinEofnon-autonomousBoissonadesystemisproved.
Keywords
Non-AutonomousDynamicalSystems,UniformAttractor,ContinuousProcess,
GlobalAttractor,TranslationBounded
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
g20-V60c“ ±5,)Û·½5!ì?5±9†ƒƒ'á‘ÄåXÚïÄ®²¤
•š‚5uЕ§+•¥•-‡•¡.3á‘ÄåXÚïÄ¥,éug£XÚ,·‚̇ïÄ
ÛáÚf•35nض̇(JŒ±ëw©z[1].éušg£XÚ§·‚ïĘ—áÚfÚ
;,áÚf•35nØ.
ÄkHaraux[2]•Äšg£ÄåXÚ˜AÏa.,¿‰Ñ†g£XÚÛáÚfƒ
²1,•Ò´˜—áÚfVg.‘,ChepyzhovandVishik[3]JÑšg£ÄåXÚ˜—á
Úf•35˜„•{,ÙØ%gŽ´3ÿÐƒ˜mþEÈ6.Ma[4]|^š;5ÿÝ5
Ÿ‰Ñšg£Ã¡‘ÄåXÚ˜—áÚf•3¿©7‡^‡,‘ ,Zhong[5]|^ù‡¿©7
‡^‡y²šg£2DNavier-Stokes•§3V¥•3˜—áÚf.
ÉZhong[5]éu,©Ì‡•Äešg£BoissonadeXÚ
DOI:10.12677/aam.2021.1072562443A^êÆ?Ð
w §¤²
∂u
∂t
= d
1
∆u+u−αv+γuv−u
3
+h
1
(t,x),(1.1)
∂v
∂t
= d
2
∆v+u−βv+h
1
(t,x),(1.2)
u(t,x) = v(t,x) = 0,t>τ,x∈∂Ω,(1.3)
u(τ,x) = u
τ
(x),v(τ,x) = v
τ
(x),x∈Ω.(1.4)
˜—áÚf•35.Ù¥(t,x) ∈(τ,∞)×Ω,Ω ⊂R
n
(n≤3)´äkÛÜLipschitz1w>.k
.«•§τ∈R
+
,Xêd
1
, d
2
, α, βÑ´~ê.Pf
1
(u,v) = −u+αv−γuv+u
3
, f
2
(u,v) = −u+βv§
@o•3~êC
i
>0,k
i
>0,i= 1···,4,¦
C
1
|u|
4
−k
1
≤f
1
(u,v)u≤C
2
|u|
4
+k
2
,(1.5)
C
3
|v|
2
−k
3
≤f
2
(u,v)v≤C
4
|v|
2
+k
4
.(1.6)
3©¥,½Â¦È˜mXe:
H= L
2
(Ω)×L
2
(Ω),E= H
1
0
(Ω)×H
1
0
(Ω),Π = H
1
0
(Ω)∩H
2
×H
1
0
(Ω)∩H
2
.
^k·kÚ(·,·)©OL«H½öL
2
(Ω)‰ê†SÈ,XJp6=2,@oÒ^k·k
L
p
5L«L
p
(Ω)‰
ê,LebesgueÿݽöEuclid˜m•þ‰êÑP•|·|,dPoincar´eØª·‚••3˜‡~
êη>0,¦
k5ξk
2
≥ηkξk
2
,∀ξ∈H
1
0
(Ω).(1.7)
¤±E¥ƒξ‰êkξk
E
Ò^k5ξk5L«.
2.ý•£
ù˜!̇0šg£ÄåXÚ˜ÄVg.
½Â2.1Σ´˜‡ëê8,X´˜‡Banach˜m,X J¹kü‡ëêNxU
σ
(t,τ):E→E
÷v
U
σ
(t,s)U
σ
(s,τ) = U
σ
(t,τ),∀t≥s≥τ,τ∈R
+
,(2.1)
U
σ
(τ,τ) = I
d
,τ∈R
+
.(2.2)
@o¡U
σ(t,τ)
´˜‡ŒL§.
XJéz˜‡σ∈Σ,U
σ
(t,τ)Ñ´˜‡ŒL§§@o¡U
σ
(t,τ),t≥τ,τ∈R
+
,σ∈Σ´Banach˜
mEþL§x.Ù¥Σ¡•ÎÒ˜m,σ¡•ÎÒ.
DOI:10.12677/aam.2021.1072562444A^êÆ?Ð
w §¤²
½Â2.2•3E¥˜‡8ÜB
0
,XJéu?¿τ ∈R
+
,E¥?¿k.8B,Ñ•3T
0
=
T
0
(B) ≥τ,¦t≥T
0
ž,k
∪
σ∈Σ
U
σ
(t,τ)B⊂B
0
.
@o·‚¡B
0
´ŒL§{U
σ
(t,τ)},σ∈Σ3E¥˜—áÂ8.
½Â2.3e¼êϕ∈L
2
Loc
(R);H÷v
kϕk
2
L
2
b
= kϕk
2
L
2
b
(R;H)
= sup
t∈R
Z
t+1
t
kϕk
2
H
ds≤∞,
K¡ϕ´²£k..
½Â2.4XJé?¿τ∈R
+
,E¥k.8B,±9?¿ε>0,• 3t
0
=t
0
(τ,B,ε)≥τÚE
˜‡k•‘f˜mE
1
,¦
(i)P(∪
σ∈Σ
∪
t≥t
0
U
σ
(t,τ)B)k.,
(ii)k(I−p)(∪
σ∈Σ
∪
t≥t
0
U
σ
(t,τ)x)k≤ε,∀x∈B,
K¡L§xU
σ
(t,τ),σ∈Σ÷v˜—('uσ∈Σ)^‡(C).
Ù¥p: E→E
1
´k.Žf.
b2.1{T(h)|h≥0}´Š^3ΣþŽfx…÷v
(i)T(h)Σ = Σ,∀h∈R
+
,
(ii)(²£ØC/)U
σ
(t+h,τ+h) = U
T(h)σ
(t,τ),∀σ∈Σ,t≥τ,τ∈R
+
,h≥0.
½n2.13b2.1¤á^‡e,U
σ
(t,τ),σ∈Σ´½Â3Banach˜mEþ˜‡ŒL§§XJ§
÷v
(i)k˜‡˜—('uσ∈Σ)áÂ8,
(ii)÷v˜—('uσ∈Σ)^‡(C).
@oÒ¡ŒL§U
σ
(t,τ),σ∈Σ3E¥•3˜‡;˜—áÚf.
3.áÂ8•35
••B§·‚Ï~ru(x,t),v(x,t)¤u(t),v(t)½ö†¤u,v.
d)ÛŒ+)¤½n©z[6],Ð>Š¯K(1.1)−(1.4)Œ±¤e¡Њ¯K





dg
dt
= Ag+f(g)+h(t),t>0.
g(0) = g
0
= (u
0
,v
0
) ∈H.
(3.1)
DOI:10.12677/aam.2021.1072562445A^êÆ?Ð
w §¤²
Ù¥
g=
u
v
!
,A=
d
1
∆0
0d
1
∆
!
,f(g) =
u−αv+γuv−u
3
u−βv
!
,h(t) =
h
1
(x,t)
h
2
(x,t)
!
.
äN[!Œ±ë•©z[7].
53.1éu¯K(3.1),éu‰½ÎÒh(t), ·‚ÒPÎÒ˜mΣ = H(g),Ù¥H(g)L«g3L
2
Loc
(R;H)¥
4•.•õ[!Œë•©z[8].
Ún3.1é?¿τ,T∈R,T>τ,XJu
τ
∈H,KЊ¯K(3.1)•3•˜ÛÜf))g(t)=
(u(t),v(t)),t∈(τ,T)÷v
g∈C([τ,T];H)∩L
2
(τ,T;E)).
y².Њ¯K(3.1))•35ÏLGalerkin%C•{Œ±,äN[!„©z.e¡·‚y
²f)•˜5.(u
1
,v
1
)Ú(u
2
,v
2
)Ñ´Ð>Š¯K(1.1) −(1.4)±(u
τ
,v
τ
)•ÐŠü‡),d
f)½Â§·‚k
<
∂(u
1
−u
2
)
∂t
,u
1
−u
2
>−(d
1
5(u
1
−u
2
),5(u
1
−u
2
))
= (u
1
−u
2
−αv
1
−αv
2
+γu
1
v
1
−γu
2
v
2
−u
3
1
+u
3
2
,u
1
−u
2
),
(3.2)
<
∂(v
1
−v
2
)
v
1
−v
2
,v
1
−v
2
>−(d
2
5(v
1
−v
2
),5(v
1
−v
2
))
= (u
1
−u
2
−βv
1
+βv
2
,v
1
−v
2
),
(3.3)
Ù¥<·,·>L«H
1
†H
−1
éóÈ.·‚kOŽe¡ù‡Øª
Z
Ω
γ(u
1
v
1
−u
2
v
2
)(u
1
−u
2
)dx=
Z
Ω
γ[v
1
(u
1
−u
2
)
2
+u
2
(v
1
−v
2
)(u
1
−u
2
)]dx
≤γ

Z
Ω
v
4
1
dx

1
4

Z
Ω
(u
1
−u
2
)
2
dx

1
2

Z
Ω
(u
1
−u
2
)
4
dx

1
4
+γ

Z
Ω
u
2
2
dx

1
2

Z
Ω
(v
1
−v
2
)
4
dx

1
4
(
Z
Ω
(u
1
−u
2
)
4
dx)
1
4
≤γkv
1
k
L
4
ku
1
−u
2
kku
1
−u
2
k
L
4
+γku
2
kkv
1
−v
2
k
L
4
ku
1
−u
2
k
L
4
≤C
2
γkv
1
k
H
1
0
ku
1
−u
2
kku
1
−u
2
k
H
1
0
+C
2
γku
2
kkv
1
−v
2
k
H
1
0
ku
1
−u
2
k
H
1
0
≤
C
4
γ
2
d
1
kv
1
k
2
H
1
0
ku
1
−u
2
k
2
+
d
1
4
ku
1
−u
2
k
2
H
1
0
+
C
4
M
2
γ
2
d
1
kv
1
−v
2
k
2
H
1
0
+
d
1
4
ku
1
−u
2
k
2
H
1
0
=
C
4
γ
2
d
1
kv
1
k
2
H
1
0
ku
1
−u
2
k
2
+
C
4
M
2
γ
2
d
1
kv
1
−v
2
k
2
H
1
0
+
d
1
2
ku
1
−u
2
k
2
H
1
0
,
(3.4)
DOI:10.12677/aam.2021.1072562446A^êÆ?Ð
w §¤²
Ù¥^e¡i\Øª
kϕk
L
4
≤Ckϕk
H
1
0
,forϕ∈H
1
0
.
(3.2)ªŒ±¤
1
2
d
dt
ku
1
−u
2
k
2
+d
1
ku
1
−u
2
k
H
1
0
=
Z
Ω
[(u
1
−u
2
)
2
−α(v
1
−v
2
)(u
1
−u
2
)+γ(u
1
v
1
−u
2
v
2
)(u
1
−u
2
)
−(u
3
1
−u
3
2
)(u
1
−u
2
)]dx
≤
Z
Ω
[(u
1
−u
2
)
2
+
α
2
(v
1
−v
2
)
2
+
α
2
(u
1
−u
2
)
2
+γ(u
1
v
1
−u
2
v
2
)(u
1
−u
2
)]dx,
(3.5)
r(3.4)“\(3.5)2¦±2
d
dt
ku
1
−u
2
k
2
+d
1
ku
1
−u
2
k
H
1
0
≤2ku
1
−u
2
k
2
+αku
1
−u
2
k
2
+αkv
1
−v
2
k
2
+
2C
4
γ
2
d
1
kv
1
k
2
H
1
0
kv
1
−v
2
k
2
+
2C
4
M
2
γ
2
d
1
kv
1
−v
2
k
2
.
(3.6)
(3.3)ª•Œ±¤
d
dt
kv
1
−v
2
k
2
+2d
2
kv
1
−v
2
k
2
H
1
0
=
Z
Ω
[(u
1
−u
2
)(v
1
−v
2
)−β(v
1
−v
2
)(v
1
−v
2
)]dx
≤ku
1
−u
2
k
2
+kv
1
−v
2
k
2
+2βkv
1
−v
2
k
2
.
(3.7)
(3.7)ª¦±
3C
4
M
2
γ
2
2d
1
d
2

3C
4
M
2
γ
2
2d
1
d
2
d
dt
kv
1
−v
2
k
2
+
3C
4
M
2
γ
2
d
1
kv
1
−v
2
k
2
H
1
0
≤
3C
4
M
2
γ
2
2d
1
d
2
ku
1
−u
2
k
2
+
3C
4
M
2
γ
2
2d
1
d
2
kv
1
−v
2
k
2
+
3C
4
M
2
γ
2
β
d
1
d
2
kv
1
−v
2
k
2
.
(3.8)
DOI:10.12677/aam.2021.1072562447A^êÆ?Ð
w §¤²
ò(3.6)†(3.8)üªƒ\
d
dt

ku
1
−u
2
k
2
+
3C
4
M
2
γ
2
2d
1
d
2
kv
1
−v
2
k
2

+d
1
ku
1
−u
2
k
H
1
0
+
C
4
M
2
γ
2
d
1
kv
1
−v
2
k
2
H
1
0
≤

2+α+
3C
4
M
2
γ
2
2d
1
d
2
+
2C
4
γ
2
d
1
kv
1
k
H
1
0

ku
1
−u
2
k
2
+

α+1+
3C
4
M
2
γ
2
β
d
1
d
2

2d
1
d
2
3C
4
M
2
γ
2
3C
4
M
2
γ
2
2d
1
d
2
kv
1
−v
2
k
2
≤

2+α+
3C
4
M
2
γ
2
2d
1
d
2
+
2C
4
γ
2
d
1
kv
1
k
H
1
0
+

α+1+
3C
4
M
2
γ
2
β
d
1
d
2

2d
1
d
2
3C
4
M
2
γ
2


ku
1
−u
2
k
2
+
3C
4
M
2
γ
2
2d
1
d
2
kv
1
−v
2
k
2

.
(3.9)
|^Gronwall.sØª§·‚k
ku
1
−u
2
k
2
+
3C
4
M
2
γ
2
2d
1
d
2
kv
1
−v
2
k
2
≤e
R
t
0
[2+α+
3C
4
M
2
γ
2
2d
1
d
2
+
2C
4
γ
2
d
1
kv
1
(s)k
H
1
0
+(α+1+
3C
4
M
2
γ
2
β
d
1
d
2
)
2d
1
d
2
3C
4
M
2
γ
2
]ds
(ku
10
−u
20
k
2
+
3C
4
M
2
γ
2
2d
1
d
2
kv
10
−v
20
k
2
)
= 0.
(3.10)
Ïd,·‚u
1
= u
2
,v
1
= v
2
,ùL²ÐŠ¯K(3.1)f)´•˜.l(3.10)¥·‚„Œ±
)éЊëY•65.
Ún3.2é?¿τ,T∈R,T>τ,XJu
τ
∈E,KЊ¯K(3.1)•3•˜r)g(t)=
(u(t),v(t)),t∈(τ,T)÷v
g∈C([τ,T];E)∩L
2
(τ,T;D(A)).
½n3.1é?¿τ,T∈R,T>τ,ifu
τ
∈E,šg£Boissonadeu Е§(3.1)•3•˜Nr
)g(t) = (u(t),v(t)),t∈(τ,∞)
d½n3.1•šg£BoissonadeuЕ§(3.1)Nr)½ÂEþ˜‡ŒL§:
{U
σ
(t,τ)}: U
σ
(t,τ)g
τ
= g(t) = (u(t),v(t)).
DOI:10.12677/aam.2021.1072562448A^êÆ?Ð
w §¤²
Ún3.3eh(t) ∈L
2
Loc
(R
τ
;H)´²£k.§K•3~êK
1
>0,¦4¥
B
1
= g{∈V: kgk
2
E
≤K
1
}
´ŒL§{U
σ
(t,τ)},σ∈Σ3V¥˜—('uσ∈Σ)áÂ8,=é?¿k.8B⊂H,•3ž
mT
B
>0,¦é?¿t≥T
B
,g
τ
∈B,σ∈Σ,k
kU
σ
(t,τ)g
τ
k
2
E
≤K
1
.
ù‡Úny²žë•©z[9]
Ún3.4eh(t)∈L
2
Loc
(R
τ
;H)´²£k.§K•3~êK
p
>0,¦é?¿k.8B⊂H,
•3žmt
B
>0,é?¿t≥t
B
,g
τ
∈B,σ∈Σ,k
kU
σ
(t,τ)g
τ
k
2
L
p
≤K
p
.
ù‡Úny²žë•©z[10]"
4.˜—áÚf•35
Ún4.1ϕ∈L
2
Loc
(R;L
2
(Ω)),XJé?¿ε≥0,•3~êη>0,¦
sup
t∈R
Z
t+η
t
kϕk
2
E
ds≤ε.
¤á,K¡ϕ´5.
L
2
Loc
(R;L
2
(Ω))¥¤k5¼ê8ÜP¤L
2
n
(R;L
2
(Ω)).
Ún4.2XJ˜‡¼êϕ
0
∈L
2
n
(R;E),@oé?¿τ∈R,k
lim
γ→∞
sup
t≥τ
Z
t
τ
e
−γ(t−s)
kϕ(s)k
2
E
ds= 0.
Ún4.3eh
1
(t,x),h
2
(x,t) ∈L
2
Loc
(R;L
2
(Ω))´5,Ké?¿ε≥0,Ú?¿k.8B⊂H,
(u
τ
,v
τ
) ⊂B,•3M≤0,t
0
≤0,η≤0¦
Z
t+η
t
Z
Ω(u≥M)
|u|
3
|(u−M)
+
|
3
dxds≤ε,t≥t
0
,σ∈Σ,(4.1)
Z
t+η
t
Z
Ω(u≤−M)
|u|
3
|(u+M)
−
|
3
dxds≤ε,t≥t
0
,σ∈Σ,(4.2)
Z
t+η
t
Z
Ω(v≥M)
|v||(v−M)
+
|dxds≤ε,t≥t
0
,σ∈Σ,(4.3)
DOI:10.12677/aam.2021.1072562449A^êÆ?Ð
w §¤²
Z
t+η
t
Z
Ω(v≤−M)
|v||(v+M)
−
|dxds≤ε,t≥t
0
,σ∈Σ,(4.4)
¤á,Ù¥
(u−M)
+
=





u−M,u≥M,
0,u≤M.
(u+M)
−
=





u+M,u≤−M,
0,u≥−M.
(v−M)
+
=





v−M,v≥M,
0,v≤M.
(v+M)
−
=





v+M,v≤−M,
0,v≥−M.
½n4.1XJh
1
(x,t),h
2
(x,t)∈L
2
Loc
(R
τ
;H)´5,f
1
(u,v),f
2
(u,v)÷v(1.5)Ú(1.6),@o
éAuЊ¯K(2.1)ŒL§U
σ
(t,τ)•3˜‡˜—('uσ∈Σ)áÚf.
y².dÚn3.2•§ŒL§U
σ
(t,τ)3V¥k˜‡˜—('uσ∈Σ)áÂ8B
1
.
e¡·‚y²˜—^‡(C).Ï•(−∆)
−1
´H¥ëY;Žf,d²;ÌnØ••3S{λ
j
}
∞
j=1
:
0 <λ
1
≤λ
2
≤···≤λ
i
≤···,λ
j
→∞,asj→∞,(4.5)
ÚD(−∆)¥3H¥˜x¼ê{ω
j
}
∞
j=1
,÷v
−∆ω
j
= λ
j
ω
j
,∀j∈N.
P
b
V
m
=span{ω
1
,ω
2
,···,ω
m
}⊂H
1
0
(Ω),P
m
:V→V
m
´Žf,Ù¥V
m
=
b
V
m
×
b
V
m
.é?¿g
∈D(−∆),¤
g= P
m
g+(I−P
m
)g= g
1
+g
2
Ù¥g
1
= (u
1
,v
1
),g
2
= (u
2
,v
2
).é?¿½
f
h
1
,
f
h
2
∈Σ,ò((1.1),−∆u
2
)Ú((1.2),−∆v
2
)ƒ\,
·‚
1
2
d
dt
(ku
2
k
2
H
1
0
+kv
2
k
2
H
1
0
)+d
1
k∆u
2
k
2
+d
2
k∆v
2
k
2
= (f
1
(u,v),∆u
2
)−(
f
h
1
(t),∆u
2
)
+(f
2
(u,v),∆v
2
)−(
f
h
2
(t),∆v
2
).
(4.6)
d(1.3)Υ
|f
1
(u,v)|≤C
1
(|u|
3
+1).
DOI:10.12677/aam.2021.1072562450A^êÆ?Ð
w §¤²
dYoung’sØª,·‚
|(f
1
(u,v),∆u
2
)|≤kf
1
(u,v)kk∆u
2
k
≤
d
1
4
k∆u
2
k
2
+
1
d
1
kf
1
(u,v)k
2
≤
d
1
4
k∆u
2
k
2
+
1
d
1
Z
Ω
C
2
1
(|u|
3
+1)
2
dx
≤
d
1
4
k∆u
2
k
2
+
2C
2
1
d
1
|Ω|+
2C
2
1
d
1
Z
Ω
|u|
6
dx
≤
d
1
4
k∆u
2
k
2
+
2C
2
1
d
1
|Ω|+
2C
2
1
d
1
Z
Ω(|u|≥M)
|u|
6
dx+
2C
2
1
d
1
Z
Ω(|u|≤M)
|u|
6
dx
≤
d
1
4
k∆u
2
k
2
+
2C
2
1
d
1
|Ω|+
2C
2
1
M
6
d
1
|Ω|+
2C
2
1
d
1
Z
Ω(|u|≥M)
|u|
6
dx,
(4.7)
±9
|(
f
h
1
(t),−∆u
2
)|= |
Z
Ω
f
h
1
(t)(−∆u
2
)dx|
≤k
f
h
1
(t)k
2
k∆u
2
k
2
≤
d
1
4
k∆u
2
k
2
+
1
d
1
k
f
h
1
(t)k
2
.
(4.8)
d(1.4)Υ
|f
2
(u,v)|≤C
2
(|u|+1).
dYoung’sØª,·‚
|f
2
(u,v),∆v
2
|= |
Z
Ω
f
2
(u,v)∆v
2
dx|
≤kf
2
(u,v)kk∆v
2
k
≤
d
2
4
k∆v
2
k
2
+
1
d
2
kf
2
(u,v)k
2
≤
d
2
4
k∆v
2
k
2
+
1
d
2
Z
Ω
C
2
2
(|v|+1)
2
dx
≤
d
2
4
k∆v
2
k
2
+
2C
2
2
d
2
|Ω|+
2C
2
2
d
2
Z
Ω
|v|
2
dx
≤
d
2
4
k∆v
2
k
2
+
2C
2
2
d
2
|Ω|+
2C
2
2
d
2
Z
Ω(|v|≥M)
|v|
2
dx+
2C
2
2
d
2
Z
Ω(|v|≤M)
|v|
2
dx
≤
d
2
4
k∆v
2
k
2
+
2C
2
2
d
2
|Ω|+
2C
2
2
M
2
d
2
|Ω|+
2C
2
2
d
2
Z
Ω(|v|≥M)
|v|
2
dx,
(4.9)
DOI:10.12677/aam.2021.1072562451A^êÆ?Ð
w §¤²
±9
|(
f
h
2
(t),−∆v
2
)|≤k
f
h
2
(t)k
2
k∆v
2
k
2
≤
d
2
4
k∆v
2
k
2
+
1
d
2
k
f
h
2
(t)k
2
.
(4.10)
ò(4.7)−(4.10)‘\(4.6),
d
dt
(ku
2
k
2
H
1
0
+kv
2
k
2
H
1
0
)+d
1
k∆u
2
k
2
+d
2
k∆v
2
k
2
≤
4C
2
1
d
1
|Ω|+
4C
2
1
M
6
d
1
|Ω|+
4C
2
2
d
2
|Ω|+
4C
2
2
M
2
d
2
|Ω|
+
4C
2
1
d
1
Z
Ω(|u|≥M)
|u|
6
dx+
4C
2
2
d
2
Z
Ω(|v|≥M)
|v|
2
dx+
2
d
1
k
f
h
1
(t)k
2
+
2
d
2
k
f
h
2
(t)k
2
.
Šâpoincar´eØª
kϕk≤
1
p
λ
m+1
kDuk
2
,∀ϕ∈H
1
0
,
-d= min{d
1
,d
2
},Òk
d
dt
(ku
2
k
2
H
1
0
+kv
2
k
2
H
1
0
)+λ
m+1
d(ku
2
k
2
H
1
0
+kv
2
k
2
H
1
0
)
≤
4C
2
1
d
1
|Ω|+
4C
2
1
M
6
d
1
|Ω|+
4C
2
2
d
2
|Ω|+
4C
2
2
M
2
d
2
|Ω|
+
4C
2
1
d
1
Z
Ω(|u|≥M)
|u|
6
dx+
4C
2
2
d
2
Z
Ω(|v|≥M)
|v|
2
dx+
2
d
1
k
f
h
1
(t)k
2
+
2
d
2
k
f
h
2
(t)k
2
.
(4.11)
-k>max{T
B
,t
B
},(4.11)ª^Gronwall’sØª,
ku
2
k
2
H
1
0
+kv
2
k
2
H
1
0
≤e
−λ
m+1
d(t−k)
(ku
2
(τ)k
2
H
1
0
+kv
2
(τ)k
2
H
1
0
)
+
Z
t
k
e
−λ
m+1
d(t−s)
(
4C
2
1
d
1
|Ω|+
4C
2
1
M
6
d
1
|Ω|+
4C
2
2
d
2
|Ω|+
4C
2
2
M
2
d
2
|Ω|)ds
+
4C
2
1
d
1
Z
t
k
e
−λ
m+1
d(t−s)
Z
Ω(|u|≥M)
|u(s)|
6
dxds+
4C
2
2
d
2
Z
t
k
e
−λ
m+1
d(t−s)
Z
Ω(|v|≥M)
|v(s)|
2
dxds
+
2
d
1
Z
t
k
e
−λ
m+1
d(t−s)
k
f
h
1
(s)k
2
+
2
d
2
Z
t
k
e
−λ
m+1
d(t−s)
f
h
2
(s)k
2
ds
≤ρ
2
V
e
−λ
m+1
d(t−k)
+
1
dλ
m+1

4C
2
1
d
1
|Ω|+
4C
2
1
M
6
d
1
|Ω|+
4C
2
2
d
2
|Ω|+
4C
2
2
M
2
d
2
|Ω|

+
4C
2
1
d
1
Z
t
k
e
−λ
m+1
d(t−s)
Z
Ω(|u|≥M)
|u(s)|
6
dxds+
4C
2
2
d
2
Z
t
k
e
−λ
m+1
d(t−s)
Z
Ω(|v|≥M)
|v(s)|
2
dxds
+
2
d
1
Z
t
k
e
−λ
m+1
d(t−s)
k
f
h
1
(s)k
2
+
2
d
2
Z
t
k
e
−λ
m+1
d(t−s)
f
h
2
(s)k
2
ds,
(4.12)
DOI:10.12677/aam.2021.1072562452A^êÆ?Ð
w §¤²
Ù¥T
B
´Ún3.3¥,t
B
´Ún3.4¥.
,˜•¡§·‚k
Z
Ω(|u|≥M)
|u|
6
dx=
Z
Ω(u≥M)
|u|
6
dx+
Z
Ω(u≤−M)
|u|
6
dx
=
Z
Ω(u≥M)
|u|
3
|u|
3
dx+
Z
Ω(u≤−M)
|u|
3
|u|
3
dx
=
Z
Ω(u≥M)
|u|
3
|u−M+M|
3
dx+
Z
Ω(u≤−M)
|u|
3
|u+M−M|
3
dx
≤4[
Z
Ω(u≥M)
|u|
3
(|(u−M)
+
|
3
+|M|
3
)dx+
Z
Ω(u≤−M)
|u|
3
(|(u+M)
−
|
3
+|M|
3
)dx]
≤4[
Z
Ω(u≥M)
|u|
3
|(u−M)
+
|
3
dx+
Z
Ω(u≤−M)
|u|
3
|(u+M)
−
|
3
dx]+4M
3
Z
Ω
(|u|≥M)|u|
3
dx
≤4[
Z
Ω(u≥M)
|u|
3
|(u−M)
+
|
3
dx+
Z
Ω(u≤−M)
|u|
3
|(u+M)
−
|
3
dx]+4M
2
Z
Ω(|u|≥M)
|u|
4
dx,
(4.13)
±9
Z
Ω(|v|≥M)
|v|
2
dx=
Z
Ω(v≥M)
|v|
2
dx+
Z
Ω(v≤−M)
|v|
2
dx
=
Z
Ω(v≥M)
|v||v|dx+
Z
Ω(v≤−M)
|v||v|dx
=
Z
Ω(v≥M)
|v||v−M+M|dx+
Z
Ω(v≤−M)
|v||v+M−M|dx
≤
Z
Ω(v≥M)
(|v||v−M|+|M|)dx+
Z
Ω(v≤−M)
|v|(|v+M|+|M|)dx
≤
Z
Ω(v≥M)
|v||(v−M)
+
|dx+
Z
Ω(v≤−M)
|v||(v+M)
−
|dx+M
Z
Ω(|v|≥M)
|v|dx
≤
Z
Ω(v≥M)
|v||(v−M)
+
|dx+
Z
Ω(v≤−M)
|v||(v+M)
−
|dx+
Z
Ω(|v|≥M)
|v|
2
dx.
(4.14)
dÚn5.1,·‚•é?¿k.8
e
B⊂L
2
(Ω),•3M¿0,t
0
>0,η>0¦
Z
t+η
t
Z
Ω(u≥M)
|u|
3
|(u−M)
+
|
3
dxds≤
d
1
ε
320C
2
1
,t≥t
0
,σ∈Σ,u
τ
∈
e
B,(4.15)
Z
t+η
t
Z
Ω(u≤−M)
|u|
3
|(u+M)
−
|
3
dxds≤
d
1
ε
320C
2
1
,t≥t
0
,σ∈Σ,u
τ
∈
e
B,(4.16)
Z
t+η
t
Z
Ω(v≥M)
|v||(v−M)
+
|dxds≤
d
2
ε
80C
2
2
,t≥t
0
,σ∈Σ,u
τ
∈
e
B.(4.17)
Z
t+η
t
Z
Ω(v≤−M)
|v||(v+M)
−
|dxds≤
d
2
ε
80C
2
2
,t≥t
0
,σ∈Σ,u
τ
∈
e
B.(4.18)
DOI:10.12677/aam.2021.1072562453A^êÆ?Ð
w §¤²
5¿
Z
t+η
t
e
−λ
m+1
d(t−s)
Z
Ω(u≥M)
|u|
3
|(u−M)
+
|
3
dxds
=

Z
t
t−η
+
Z
t−η
t−2η
+
Z
t−2η
t−3η
+···

e
−λ
m+1
d(t−s)
Z
Ω(u≥M)
|u|
3
|(u−M)
+
|
3
dxds
≤
Z
t
t−η
Z
Ω(u≥M)
|u|
3
|(u−M)
+
|
3
dxds+e
−λ
m+1
dη
Z
t−η
t−2η
Z
Ω(u≥M)
|u|
3
|(u−M)
+
|
3
dxds
+e
−2λ
m+1
dη
Z
t−2η
t−3η
Z
Ω(u≥M)
|u|
3
|(u−M)
+
|
3
dxds+···
≤
1
1−e
−dλ
m+1
η
d
1
ε
320C
2
1
,fork>t
0
.
(4.19)
Ó§·‚k
Z
t+η
t
e
−λ
m+1
d(t−s)
Z
Ω(u≤−M)
|u|
3
|(u−M)
+
|
3
dxds≤
1
1−e
−dλ
m+1
η
d
1
ε
320C
2
1
,fork>t
0
,(4.20)
Z
t+η
t
e
−λ
m+1
d(t−s)
Z
Ω(v≥M)
|v||(v+M)
−
|dxds≤
1
1−e
−dλ
m+1
η
d
2
ε
80C
2
2
,fork>t
0
,(4.21)
Z
t+η
t
e
−λ
m+1
d(t−s)
Z
Ω(v≤−M)
|v||(v−M)
+
|dxds≤
1
1−e
−dλ
m+1
η
d
2
ε
80C
2
2
,fork>t
0
.(4.22)
d(4.5),·‚Œ±m+1vŒ,¦
16C
2
1
d
1
Z
t
k
e
−λ
m+1
d(t−s)
Z
Ω(u≥M)
|u|
3
|(u−M)
+
|
3
dxds≤
ε
10
,(4.23)
16C
2
1
d
1
Z
t
k
e
−λ
m+1
d(t−s)
Z
Ω(u≤−M)
|u|
3
|(u+M)
−
|
3
dxds≤
ε
10
,(4.24)
4C
2
2
d
2
Z
t
k
e
−λ
m+1
d(t−s)
Z
Ω(v≥M)
|v||(v−M)
+
|dxds≤
ε
10
,(4.25)
4C
2
2
d
2
Z
t
k
e
−λ
m+1
d(t−s)
Z
Ω(v≤−M)
|v||(v+M)
−
|dxds≤
ε
10
,(4.26)
4C
2
1
d
1
Z
t
k
e
−λ
m+1
d(t−s)
Z
Ω(|u|≥M)
4M
2
|u(s)|
4
|dxds≤
ε
10
,(4.27)
4C
2
2
d
2
Z
t
k
e
−λ
m+1
d(t−s)
Z
Ω(|v|≥M)
|v(s)|
2
|dxds≤
ε
10
,(4.28)
1
dλ
m+1

4C
2
1
d
1
|Ω|+
4C
2
1
M
6
d
1
|Ω|+
4C
2
2
d
2
|Ω|+
4C
2
2
M
2
d
2
|Ω|

≤
ε
10
.(4.29)
DOI:10.12677/aam.2021.1072562454A^êÆ?Ð
w §¤²
-t
1
=
1
dλ
m+1
ln
6ρ
2
V
ε
+k,Kt≥t
1
žk
ρ
2
V
e
−λ
m+1
(t−k)
≤
ε
10
.(4.30)
dÚn(4.1),é?¿ε>0,Œ±m+1vŒ¦
Z
t
k
e
−dλ
m+1
(t−s)

f
h
1
(s)k
2
+
f
h
2
(s)k
2

ds≤
ε
10(
2
d
1
+
2
d
2
)
,∀σ∈Σ.(4.31)
ò(4.13)−(4.31)“\(4.12),
kg
2
(t)k
2
V
= ku
2
k
2
H
1
0
+kv
2
k
2
H
1
0
≤ε,t≥t
1
,σ∈Σ,g
τ
∈B.
ùL²U
σ
(t,τ),σ∈Σ3V¥÷v˜—('uσ∈Σ)^‡(C).d½n(2.1)Œ•(ؤá.
ë•©z
[1]Robinson,J.C.andPierre,C.(2003)Infinite-DimensionalDynamicalSystems:AnIntroduc-
tiontoDissipativeParabolicPDEsandtheTheoryofGlobalAttractors.CambridgeTextsin
AppliedMathematics.AppliedMechanicsReviews,56,B54-B55.
https://doi.org/10.1115/1.1579456
[2]Haraux,A. (1991)Syst`emesdynamiquesdissipatifsetapplications.In:Syst`emesDynamiques
DissipatifsetApplications,Masson,Paris.
[3]Chepyzhov,V.V.andVishik,M.I.(2002)AttractorsforEquationsofMathematicalPhysics.
Vol.49,AmericanMathematicalSociety,Providence,RI.https://doi.org/10.1090/coll/049
[4]Ma, Q.F.,Wang, S.H. and Zhong,C.K. (2002) Necessary and Sufficient Conditions for the Ex-
istence of GlobalAttractors for Semigroups and Applications.IndianaUniversityMathematics
Journal,51,1541-1559.
[5]Lu,S.S.,Wu,H.Q.andZhong,C.K.(2005)AttractorsforNonautonomous2DNavier-Stokes
EquationswithNormalExternalForces.DiscreteandContinuousDynamicalSystems,13,
701-719.https://doi.org/10.3934/dcds.2005.13.701
[6]Sell,G.R.andYou,Y.(2002)DynamicsofEvolutionaryEquations.Springer,NewYork.
[7]Tu,J.Y.(2015)GlobalAttractorsandRobustnessoftheBoissonadeSystem.JournalofDy-
namicsandDifferentialEquations,27,187-211.https://doi.org/10.1007/s10884-014-9396-8
[8]Song, H.T.,Ma, S. andZhong, C.K. (2009)Attractors of Non-Autonomous Reaction-Diffusion
Equations.Nonlinearity,22,667-681.https://doi.org/10.1088/0951-7715/22/3/008
[9]±•…,ë¯,•¡.šg£BoissonadeXÚ.£Ú˜—•êáÚf[J].¥I‰Æ(êÆ),
2017,47(12):1891-1906.https://doi.org/10.1360/N012017-00062
DOI:10.12677/aam.2021.1072562455A^êÆ?Ð
w §¤²
[10]Song,H.T.andZhong,C.K.(2008)AttractorsofNon-AutonomousReaction-DiffusionEqua-
tionsinL
p
.NonlinearAnalysis:Theory,MethodsandApplications,68,1890-1897.
https://doi.org/10.1016/j.na.2007.01.059
DOI:10.12677/aam.2021.1072562456A^êÆ?Ð

版权所有：汉斯出版社 (Hans Publishers) Copyright © 2021 Hans Publishers Inc. All rights reserved.