具有强阻尼的非自治 Kirchhoff 型波方程的时间依赖拉回吸引子 The Pullback Attractors forNon-Autonomous Kirchhoff-TypeWave Equation with Strong Damping

doi:10.12677/AAM.2022.117483

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AdvancesinAppliedMathematicsA^êÆ?Ð,2022,11(7),4562-4577

PublishedOnlineJuly2022inHans.http://www.hanspub.org/journal/aam

https://doi.org/10.12677/aam.2022.117483

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ThePullbackAttractorsfor

Non-AutonomousKirchhoﬀ-Type

WaveEquationwithStrong

Damping

KaihongTian,XuanWang

∗

CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu

Received:Jun.11

,2021;accepted:Jul.7

,2022;published:Jul.14

,2022

∗ÏÕŠö"

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11(7):4562-4577.DOI:10.12677/aam.2022.117483

Xpõ§à

Abstract

Inthispaper,westudytheasymptoticbehaviorofsolutionsofNon-autonomous

Kirchhoﬀ-type wave equations with strongdamping and nonlinear perturbations.The

existenceoftime-dependentpullbackattractorsisveriﬁedbyusingcontractionfunc-

tionandasymptoticpriorestimation.

Keywords

ContractionFunction,TheTimeDependentPullbackAttractors,StrongDamping,

KirchhoﬀWaveEquation

2022byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).

http://creativecommons.org/licenses/by/4.0/

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(t) <09h5Ÿ, k

γt

(t)−e

γs

(s)

6−

γζ

h∇(u

),∇ω(ζ)ih∇(u

),∇ω

(ζ)idζ+γ

γζ

(ζ)dζ

−

Ω

γζ

|∇ω

(ζ)|

dxdζ−

Ω

γζ

(f(u

)−f(u

))ω

(ζ)dxdζ.(3.3)

DOI:10.12677/aam.2022.1174834570A^êÆ?Ð

Xpõ§à

2é(3.3)ª'us3[t−τ,t] þÈ©,k

τe

γt

(t)−

t−τ

γs

(s)ds

6+γ

t−τ

γζ

(ζ)dζds−

t−τ

Ω

γζ

(f(u

)−f(u

))ω

(ζ)dxdζds

−

t−τ

γζ

h∇(u

),∇ω(ζ)ih∇(u

),∇ω

(ζ)idζds

−

t−τ

Ω

γζ

k∇ω

(ζ)k

dxdζds.(3.4)

,,‰(3.1)ª¦±e

γt

ω(t),=

γt

(ε(t)kω

(t)k

k∇ω(t)k

)]+

γt

(k∇u

+k∇u

)k∇ω(t)k

= −

γt

h∇(u

),∇ω(t)i

+ε

(t)ω

(t)ω(t)−hh(u

)−h(u

),e

γt

ω(t)i

−e

γt

hf(u

)−f(u

),ω(t)i+γe

γt

[ε(t)kω

(t)k

k∇ω(t)k

].(3.5)

éþª3[s,t]×ΩþÈ©,¿dh5Ÿ,k

Ω

γζ

(k∇u

+k∇u

)|∇ω(ζ)|

dxdζ+

Ω

γζ

(ε(t)|ω

(ζ)|

|∇ω(ζ)|

)dxdζ

6−

Ω

γt

(ε(t)ω

(t)ω(t)+

|∇ω(t)|

)dx+

Ω

γs

(ε(s)ω

(s)ω(s)+

|∇ω(s)|

)dx

−

Ω

γζ

h∇(u

),∇ωi

dxdζ−

Ω

γζ

(f(u

)−f(u

))ω(ζ)dxdζ

+γ

Ω

γζ

(ε(t)|ω

(ζ)|

|∇ω(ζ)|

)dxdζ+

Ω

(ζ)e

γζ

(ζ)ω(ζ)dxdζ.(3.6)

é(3.6)ª'us3[t−τ,t] þÈ©¿¦±γ, k

t−τ

Ω

(

γδ

γζ

(k∇u

+k∇u

)|∇ω(ζ)|

+γe

γζ

(ε(t)|ω

(ζ)|

|∇ω(ζ)|

))dxdζds

6−

δγ

t−τ

γζ

h∇(u

),∇ωi

dζds−γ

t−τ

Ω

γζ

(f(u

)−f(u

))ω(ζ)dxdζds

−γτ

Ω

γζ

(ε(t)ω

(t)ω(t)+

|∇ω(t)|

)dx+γ

t−τ

Ω

(ζ)ω

(ζ)ω(ζ)dxdζds

+γ

t−τ

Ω

γζ

(ε(t)|ω

(ζ)|

|∇ω(ζ)|

)dxdζds

+γ

t−τ

Ω

γζ

(ε(s)ω

(s)ω(s)+

|∇ω(s)|

)dxds.(3.7)

DOI:10.12677/aam.2022.1174834571A^êÆ?Ð

Xpõ§à

ò(3.7)ª“\(3.4)ª,k

γt

(t)−

t−τ

γs

(s)ds

6−

δγ

t−τ

γζ

h∇(u

),∇ωi

dζds−

Ω

γζ

(f(u

)−f(u

))ω

(ζ)dxdζ

−

γτ

Ω

γζ

(ε(t)ω

(t)ω(t)+

|∇ω(t)|

)dx+

t−τ

Ω

(ζ)ω

(ζ)ω(ζ)dxdζds

−

Ω

γζ

|∇ω

(ζ)|

dxdζ−

t−τ

Ω

γζ

(f(u

)−f(u

))ω(ζ)dxdζds

−

γζ

h∇(u

),∇ω(ζ)ih∇(u

),∇ω

(ζ)idζ

t−τ

Ω

γζ

(ε(t)|ω

(ζ)|

|∇ω(ζ)|

)dxdζds

t−τ

Ω

γζ

(ε(s)ω

(s)ω(s)+

|∇ω(s)|

)dxds.(3.6)

3[t−τ,t] þÈ©(3.5) ª,¿dh5Ÿ, k

t−τ

Ω

γs

(k∇u

+k∇u

)|∇ω(s)|

dxds

6−

t−τ

Ω

γs

h∇(u

),∇ωi

dxds−

t−τ

Ω

γs

(f(u

)−f(u

))ω(s)dxds

+γ

t−τ

Ω

γs

(ε(s)|ω

(s)|

|∇ω(s)|

)dxds−

t−τ

Ω

γs

ε(s)|ω

(s)|

dxds

−

Ω

γt

(ε(t)ω

(t)ω(t)+

|∇ω(t)|

)dx+

t−τ

Ω

(s)ω

(s)ω(s)dxds

Ω

γ(t−τ)

(ε(t−τ)ω

(t−τ)ω(t−τ)+

|∇ω(t−τ)|

)dx.(3.7)

ò(3.7)ª“\(3.6)ª,k

γt

(t)+

t−τ

γs

(s)ds

6−

γτ

Ω

γζ

(ε(t)ω

(t)ω(t)+

|∇ω(t)|

)dx+

t−τ

Ω

(ζ)ω

(ζ)ω(ζ)dxdζds

−

δγ

t−τ

γζ

h∇(u

),∇ωi

dζds−

Ω

γζ

(f(u

)−f(u

))ω

(ζ)dxdζ

−

t−τ

Ω

γs

h∇(u

),∇ωi

dxds−

t−τ

Ω

γs

(f(u

)−f(u

))ω(s)dxds

+γ

t−τ

Ω

γs

(ε(s)|ω

(s)|

|∇ω(s)|

)dxds−

t−τ

Ω

γs

ε(s)|ω

(s)|

dxds

−

Ω

γζ

|∇ω

(ζ)|

dxdζ−

t−τ

Ω

γζ

(f(u

)−f(u

))ω(ζ)dxdζds

DOI:10.12677/aam.2022.1174834572A^êÆ?Ð

Xpõ§à

−

Ω

γt

(ε(t)ω

(t)ω(t)+

|∇ω(t)|

)dx+

t−τ

Ω

(s)ω

(s)ω(s)dxds

Ω

γ(t−τ)

(ε(t−τ)ω

(t−τ)ω(t−τ)+

|∇ω(t−τ)|

)dx

−

γζ

h∇(u

),∇ω(ζ)ih∇(u

),∇ω

(ζ)idζ

t−τ

Ω

γζ

(ε(t)|ω

(ζ)|

|∇ω(ζ)|

)dxdζds

t−τ

Ω

γζ

(ε(s)ω

(s)ω(s)+

|∇ω(s)|

)dxds.(3.8)

é(3.2)ª3[t−τ,t] þÈ©,¿dε(t),h(u

)5Ÿ,Œ

−

t−τ

γs

(s)ds

6−

γt

(t)+

γ(t−τ)

(t−τ)−

t−τ

Ω

γs

(f(u

)−f(u

))ω

(s)dxds

−

2γ

t−τ

γs

h∇(u

),∇ω(s)ih∇(u

),∇ω

(s)ids.(3.9)

ò(3.9)ª‘\(3.8)ª,k

τe

γt

(t)6−

γτ

Ω

γζ

(ε(t)ω

(t)ω(t)+

|∇ω(t)|

)dx+

t−τ

Ω

(ζ)ω

(ζ)ω(ζ)dxdζds

−

δγ

t−τ

γζ

h∇(u

),∇ωi

dζds−

Ω

γζ

(f(u

)−f(u

))ω

(ζ)dxdζ

−

t−τ

Ω

γs

h∇(u

),∇ωi

dxds−

t−τ

Ω

γs

(f(u

)−f(u

))ω(s)dxds

+γ

t−τ

Ω

γs

(ε(s)|ω

(s)|

|∇ω(s)|

)dxds−

t−τ

Ω

γs

ε(s)|ω

(s)|

dxds

−

γt

(t)+

γ(t−τ)

(t−τ)−

t−τ

Ω

γs

(f(u

)−f(u

))ω

(s)dxds

−

Ω

γζ

|∇ω

(ζ)|

dxdζ−

t−τ

Ω

γζ

(f(u

)−f(u

))ω(ζ)dxdζds

−

Ω

γt

(ε(t)ω

(t)ω(t)+

|∇ω(t)|

)dx+

t−τ

Ω

(s)ω

(s)ω(s)dxds

Ω

γ(t−τ)

(ε(t−τ)ω

(t−τ)ω(t−τ)+

|∇ω(t−τ)|

)dx

−

2γ

t−τ

γs

h∇(u

),∇ω(s)ih∇(u

),∇ω

(s)ids

−

γζ

h∇(u

),∇ω(ζ)ih∇(u

),∇ω

(ζ)idζ

t−τ

Ω

γζ

(ε(t)|ω

(ζ)|

|∇ω(ζ)|

)dxdζds

t−τ

Ω

γζ

(ε(s)ω

(s)ω(s)+

|∇ω(s)|

)dxds.(3.10)

DOI:10.12677/aam.2022.1174834573A^êÆ?Ð

Xpõ§à

(Ü(3.8)ª-(3.10)ª,Œ

t,τ

((u

),(u

))

= −(

2τ

)

Ω

γζ

(ε(t)ω

(t)ω(t)+

|∇ω(t)|

)dx+

2τ

t−τ

Ω

(ζ)ω

(ζ)ω(ζ)dxdζds

2τ

t−τ

Ω

γζ

(ε(s)ω

(s)ω(s)+

|∇ω(s)|

)dxds+

t−τ

Ω

(s)ω

(s)ω(s)dxds

−

δγ

4τ

t−τ

γζ

h∇(u

),∇ωi

dζds−

Ω

γζ

(f(u

)−f(u

))ω

(ζ)dxdζ

t−τ

Ω

γs

(ε(s)|ω

(s)|

|∇ω(s)|

)dxds−

t−τ

Ω

γs

ε(s)|ω

(s)|

dxds

−

Ω

γζ

|∇ω

(ζ)|

dxdζ−

t−τ

Ω

γζ

(f(u

)−f(u

))ω(ζ)dxdζds

−

2τ

t−τ

γs

h∇(u

),∇ωi

ds−

t−τ

Ω

γs

(f(u

)−f(u

))ω(s)dxds

−

2γτ

t−τ

γs

h∇(u

),∇ω(s)ih∇(u

),∇ω

(s)ids

−

2τ

γζ

h∇(u

),∇ω(ζ)ih∇(u

),∇ω

(ζ)idζ

−

τγ

γt

(t)−

γτ

t−τ

Ω

γs

(f(u

)−f(u

))ω

(s)dxds

2τ

t−τ

Ω

γζ

(ε(t)|ω

(ζ)|

|∇ω(ζ)|

)dxdζds.(3.11)

l

(t) 6C

−βτ

t−τ

+Φ

t,τ

((u

)(v

),(u

)(v

)),(3.12)

Ù¥

t−τ

Ω

γ(t−τ)

(ε(t−τ)ω

(t−τ)ω(t−τ)+

|∇ω(t−τ)|

)dx+

γ(t−τ)

(t−τ).

-(u

)•éAÐ©Š(u

) ∈

t−τ

).®•k∇u

´k..d(1.2)-(1.3)

ª•:éu?¿½t−τ

,ζ∈[t−τ

,t],ε(ζ)k.,ε(ζ)ku

k..Ïdk∇u

+ε(ζ)ku

k..ŠâAlaoglu½n,k±e(J:

→u3L

∞

(τ,T;H

(Ω))¥f∗Âñ,(3.13)

→u

∞

(τ,T;L

(Ω))¥f∗Âñ,(3.14)

→u3L

p+1

(τ,T;L

p+1

(Ω))¥rÂñ,(3.15)

→u3L

(τ,T;L

(Ω))¥rÂñ.(3.16)

e¡O(3.11)ª¥z˜‘.d(3.13)-(3.15)ª,k

lim

n→∞

lim

m→∞

Ω

γζ

(ε(t)(∂

−∂

)(u

−u

|∇(u

−u

)dx= 0.(3.17)

DOI:10.12677/aam.2022.1174834574A^êÆ?Ð

Xpõ§à

lim

n→∞

lim

m→∞

t−τ

Ω

γζ

L(∂

−∂

)(u

−u

)dxds= 0.(3.18)

lim

n→∞

lim

m→∞

t−τ

Ω

L(∂

−∂

)(u

−u

)dxdζds= 0.(3.19)

lim

n→∞

lim

m→∞

t−τ

Ω

γζ

(L(∂

−∂

)(u

−u

|∇(u

−u

)dxds= 0.(3.20)

lim

n→∞

lim

m→∞

t−τ

Ω

γζ

(L|∂

−∂

|∇ω(u

−u

)dxdζds= 0.(3.21)

lim

n→∞

lim

m→∞

t−τ

γs

h∇(u

),∇(u

−u

ds= 0.(3.22)

lim

n→∞

lim

m→∞

t−τ

γζ

h∇(u

),∇(u

−u

dζds= 0.(3.23)

lim

n→∞

lim

m→∞

γζ

h∇(u

),∇(u

−u

)ih∇(u

),∇(∂

−∂

)idζ= 0.(3.24)

lim

n→∞

lim

m→∞

t−τ

γs

h∇(u

),∇(u

−u

)ih∇(u

),∇(∂

−∂

)ids= 0.(3.25)

lim

n→∞

lim

m→∞

t−τ

Ω

γs

(L|(∂

−∂

|∇(u

−u

)dxds= 0.(3.26)

lim

n→∞

lim

m→∞

t−τ

Ω

γζ

(f(u

)−f(u

))(u

−u

)dxdζ= 0.(3.27)

lim

n→∞

lim

m→∞

t−τ

Ω

βζ

(f(u

)−f(u

))(u

−u

)dxdζds= 0.(3.28)

d(2.8)ª±9H¨olderØª,k

t−τ

Ω

βζ

(f(u

)−f(u

))·(u

−u

)dxdζ|

t−τ

Ω

βζ

C(1+|u

p−1

+|u

p−1

)·|u

−u

|·|u

−u

|dxdζ

t−τ

βζ

(C(1+ku

p−1

2(p+1)

(Ω)

+ku

p−1

2(p+1)

(Ω)

)

·ku

−u

p+1

(Ω)

·ku

−u

(Ω)

dζ

t−τ

βξ

C(1+ku

(s)k

p−1

+ku

(s)k

p−1

)

·ku

(s)−u

(s)k

p+1

(Ω)

·ku

−u

(Ω)

dζ,



lim

n→∞

lim

m→∞

t−τ

Ω

βζ

(f(u

)−f(u

))(u

−u

)dxdζ= 0.(3.29)

DOI:10.12677/aam.2022.1174834575A^êÆ?Ð

Xpõ§à

aq/, éu½t, |

Ω

βζ

(f(u

(ξ)) −f(u

(ζ)))(u

(ζ) −u

(ζ))dxdζ|k., dV‚

››Âñ½nŒ

lim

n→∞

lim

m→∞

t−τ

Ω

βζ

(f(u

(ζ))−f(u

(ζ)))(u

(ζ)−u

(ζ))dxdζds

t−τ

( lim

n→∞

lim

m→∞

Ω

γζ

(f(u

(ζ))−f(u

(ζ)))(u

(ζ)−u

(ζ))dxdζ)ds

= 0.(3.30)

Ïd,Œ±Φ

t,τ

((u

),(u

))•

t−τ

þÂ ¼ê.

½n53ª(1.2)-(1.9) 9(1.11) ¤ábe, d¯K(1.10) )šg£ÄåXÚ(θ,φ) 3

¥•3žm•6.£áÚf.

y²:dÚn2 •, (3.11) ª½Â¼êΦ

t,τ

´˜‡Â ¼ê, (Ü½n2 Ú½n3Œ•¯K

(1.1))šg£ÄåXÚ(θ,φ) 3H

¥•3žm•6.£áÚf.

5.o(

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A^uó§ÔnÆ¥ïþxùÄ!ÚîåÆ!°(Æ!‰»Ôn!)ÔÉó¯K+•.100õ

c5, ‘X‰E uÐ, <‚éKirchhoﬀ •§A^+•Øä*Œ, Kirchhoﬀ •§Lˆª•3Ø

6Äšg £KirchhoﬀÅ•§, ÄkÏLIOFaedo−Galerkin •{)•3•˜5, Ù

gæ^kO•{)ûš‚5{Z‘!šg£å‘ÚKirchhoﬀ.šÛÜ‘3yáÂ

8ž‘5 (J, •$^Â ¼êÚìCkO•{y²žm•6.£áÚf•35.

ISNõÍ¶ÆöÑ—åuÃ¡‘ÄåXÚïÄ¿Œþ`D¤J, ¦‚ïá¿uÐáÚ

f•35nØ,¦Ã¡‘ÄåXÚnØ3êÆÔn•§ïÄ¥•2•A^.

Ä7‘8

I[g,‰ÆÄ7‘8(1OÒ:11761062;11961059;11661071).

ë•©z

[1]Kirchhoﬀ,G.(1883)Vorlesungen¨uberMechanik.Teubner,Sluttgart.

[2]Yang,L.andZhong,C.K.(2008)GlobalAttractorforPlateEquationwithNonlinearDamping.

NonlinearAnalysis,69,3802-3810.https://doi.org/10.1016/j.na.2007.10.016

[3]Yang,L.(2008)UniformAttractorforNon-AutonomousPlateEquationwithaLocalized

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1243-1254.https://doi.org/10.1016/j.jmaa.2007.06.011

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[4]Xiao,H.B.(2009)AsymptoticDynamicsofPlateEquationwithaCriticalExponentonUn-

boundedDomain.NonlinearAnalysis,70,1288-1301.

https://doi.org/10.1016/j.na.2008.02.012

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https://doi.org/10.1186/1687-2770-2014-75

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2014,31(3):306-314.

[7]Ma,H.L.,Wang,J.andXie,J.(2021)PullbackAttractorsforNonautonomousDegenerate

KirchhoﬀEquationswithStrongDamping.AdvancesinMathematicalPhysics,2021,Article

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[8]Conti,M., Pata, V.andTemam, R.(2013)AttractorsforProcessesonTime-DependentSpaces.

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[10]4ËË,ê|û.šg£Plate•§žm•6r.£áÚf•35[J].êÆcr:A6,2017,

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