具有结构阻尼的扩展型梁方程的时间依赖吸引子 The Time-Dependent Global Attractors for an Extensible Beam Equation with StructuralDamping

doi:10.12677/AAM.2023.125238

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AdvancesinAppliedMathematicsA^êÆ?Ð,2023,12(5),2340-2363

PublishedOnlineMay2023inHans.https://www.hanspub.org/journal/aam

https://doi.org/10.12677/aam.2023.125238

äk({Z*Ð.ù•§

žm•6áÚf

HHHaaa§§§ààà

∗

Ü“‰ŒÆêÆ†ÚOÆ§[‹=²

ÂvFÏµ2023c428F¶¹^FÏµ2023c521F¶uÙFÏµ2023c529F

Á‡

©ïÄ‘k ({Z*Ð. ù•§3˜mH

¥)•žmÄåÆ1•"Äky²š‚ 5

‘f(u)3g.^‡e•§)·½5§2|^Â ¼ê•{y)L§{U(t,τ)}ìC

;5§•yžm•6áÚf•35"

'…c

({Z§Â ¼ê§žm•6áÚf§*Ð.ù•§

TheTime-DependentGlobal

AttractorsforanExtensible

BeamEquationwithStructural

Damping

RuiGuo,XuanWang

∗

CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu

Received:Apr.28

,2023;accepted:May21

,2023;published:May29

,2023

∗ÏÕŠö"

©ÙÚ^:Ha,à.äk({Z*Ð.ù•§žm•6áÚf[J].A^êÆ?Ð,2023,12(5):2340-2363.

DOI:10.12677/aam.2023.125238

Ha§à

Abstract

Thispaperstudieslongtimebehaviorofsolutionsforanextensiblebeamequation

withstructuraldampingintheH

space.Intheﬁrstinstance,we showthattheglobal

well-posednessofsolutionsfortheequationwithnonlinearityf(u)undersubcritical

condition.Moreover,theasymptoticcompactnessofthesolutionprocess{U(t,τ)}

isprovedbyusingthemethodofcontractionfunction.Finally,theexistenceofthe

time-dependentglobalattractorisgained.

Keywords

StructuralDamping,ContractionFunction,TheTime-DependentAttractors,

ExtensibleBeamEquation

2023byauthor(s)andHansPublishersInc.

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

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

1.Úó

Ω´R

(N>5) ¥äk1w>.k.•,•ÄXe‘k({Z*Ð.ù•§











ε(t)∂

u+∆

u+γ(−∆)

∂

u+f(u) = g(x),x∈Ω,t>τ,

u(x,t) = 0,x∈∂Ω,t>τ,

u(x,τ) = u

(x),∂

u(x,τ) = u

(x),x∈Ω,

(1.1)

Ù¥θ∈(

,1),γ>0 ,f(u)´š‚5‘, g(x) ´å‘.

31950c,Woinowsky−Krieger 3[1]¥JÑXeù•§:

∂

u+∆

u−M(

Ω

|∇u|

dx)∆u= F(x,u,∂

u,∆∂

u).(1.2)

3ù•§$ÄL§¥, duXÚgÏÚÜƒpŠ^ÚåXÚUþÅìÑÑy–¡

•{Z, { Z˜„Œ©•r{Z, f{Z†0uüöƒm({ Z. éu•§(1.1) ¥ε(t)≡C

DOI:10.12677/aam.2023.1252382341A^êÆ?Ð

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©[5]Ñ0 <θ61 ž,f(u) äk•`g.O•,ÙO••ê÷v:1 6p<p

N+4θ

(N−4)

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f, •êáÚfÚ. £áÚfØUéÐ/•x¯K(1.1)•žmÄåÆ1•. u´Conti JÑ

žm•6˜mgŽ,ddiåïÄžm•6áÚf9Œ.©[6]ïáXe.:

ε(t)∂

u−∆u+γ(−∆)

∂

u+f(u) = g(x),(1.3)

ùpα∈(0,

). „Ñ1 6p6p

∗∗

N+2

N−2

(N>3) ž)´•3, 1 6p6p

∗

N+4α

N−2

(N>

3) ž)´[-½, ?Ñžm•6áÚf•35. ©[7]Ú©[8]©O?Øù•§ÚPlate

•§žm•6áÚf•35, 8cÿ™kéäk({Z*Ð.ù•§žm•6áÚf

?1ïÄ. É±þ©zéu, ©Äk^%C•{y²¯K(1.1) )·½5, 2|^Â ¼

ê•{yéAL§ìC;5,•žm•6áÚf•35.

2.ý•£

Äk,•{',·‚¦^±e 

= L

(Ω),H

= H

(Ω),V

= H

(Ω),

= H

∩H

,k·k= k·k

,k·k

= k·k

Ù¥p>1. Kki\V

→→L

→V

−2

,½ÂŽfA: V

→V

−2

hAu,υi= h∆u,∆υi, ∀u,υ∈V

KA´L

þgŠŽf…3V

þî‚, ·‚•Œ±½ÂA˜A

.FËA˜mV

= D(A

)

äk±eSÈ†‰ê:

hu,υi

= hA

u,A

υi,kuk

= kA

uk, s∈R.

K¯K(1.1)Œ±¤e¡Žf/ª:











ε(t)∂

u+Au+γA

∂

u+f(u) = g(x),x∈Ω,t>τ,

u(x,t) = 0,x∈∂Ω,t>τ,

u(x,τ) = u

(x),∂

u(x,τ) = u

(x),x∈Ω.

(2.1)

½Âžm•6˜m•:

= V

×L

DOI:10.12677/aam.2023.1252382342A^êÆ?Ð

Ha§à

¿…ƒéA‰ê•:

k(u,∂

u)k

= kuk

+(t)k∂

-C•~ê,e©¥Ñy3ØÓªf¥z˜‡CL«´éA~êŠ,·‚•^

,i∈N5L«Ù¦~ê. (t),f(u),g÷ve^‡:

)¼êε(t) ∈C

(R)´˜‡üN4~k.¼ê,¿÷v

lim

t→+∞

ε(t) = 0,(2.2)

…•3~êL>0,¦

sup

t∈R

[|ε(t)|+|ε

(t)|] 6L.(2.3)

)f∈C

(R),s∈R÷v:

= liminf

|s|→∞

f(s)

>−λ

,(2.4)

Ù¥λ

>0•ŽfA1˜AŠ. …N>5žk

(s)|6C(1+|s|

p−1

),1 6p<p

N+2θ

N−4

.(2.5)

)g∈L

(Ω),(u

) ∈H

,…kk(u

6R.

51d(2.4)Œ,•3~êη,0 <λ

−η1 ž,¦

F(s) >−

−C,f(s)s>−ηs

−C.(2.6)

Ù¥F(u) =

f(r)dr.

52N64 ž,w,kV

→L

∞

,¤±·‚X-?ØN>5 ž(J.

Ún2.1[2]X,BÚY´n‡Banach˜m, ÷vi\X→→B→Y,

W= {u∈L

(0,T;X)|u

∈L

(0,T;Y)},1 6p<∞,

= {u∈L

∞

(0,T;X)|u

∈L

(0,T;Y)},r>1.

W→→L

(0,T;B),W

→→C([0,T];B).

½n2.1[9]U(·,·) ´{H

}

t∈R

þ˜‡L§, ¿…k˜‡.£áÂxB= {B

}

t∈R

. d,b

é∀>0 ,•3T() 6t, Φ

∈C(B

),¦

DOI:10.12677/aam.2023.1252382343A^êÆ?Ð

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kU(t,T)x−U(t,T)yk6+Φ

(x,y), ∀x,y∈B

é?¿½t∈R, KU(·,·)´.£ìC;.

½Â2.1 [9]{H

}

t∈R

´˜xBanach˜m, C= {C

}

t∈R

´{H

}

t∈R

˜x˜—k.f8. ·

‚¡½Â3H

×H

þ¼êΦ

(·,·) •C

×C

þÂ ¼ê, XJé?¿½t∈R, ?¿S

{x

}

∞

n=1

⊂C

,•3˜‡fS{x

}

∞

k=1

⊂{x

}

∞

n=1

,¦

lim

k→∞

lim

l→∞

) = 0, ∀τ6t.

·‚^C(C

)L«C

×C

Â ¼ê8Ü.

Ún2.2[10]Φ,Ψ,Λ´šKëY¼ê, …÷v‡©Øª

(t) 6Φ(t)Ψ(t)+Λ(t),t>0,

Ψ(t) 6Ψ(0)e

Φ(s)ds

Λ(s)e

Φ(τ)dτ

ds.

XJØª

(t)+βΦ(t) 6Λ(t),

¤á,Ù¥β>0 .K

Ψ(t) 6e

−βt

Ψ(0)+

−β(t−s)

Λ(s)ds

Ún2.3[10]1 6p

6∞,0 6s<l(s,l∈Z

).e

ϑ=

−

−s

−

−l

÷v

6ϑ<1.K

kuk

s,p

6Ckuk

1−ϑ

0,p

kuk

l,p

½n2.2 [11](Banach−Alaoglu½n)X´˜‡g‡Banach˜m, eB⊂X´k.,

KB3fÿÀ˜mX¥ƒé;.

½Â2.2[12]{H

}

t∈R

´˜xD‰˜m, VëêŽfx{U(t,τ): H

→H

,t>τ,τ∈R}÷

v±e5Ÿ:

1)é?¿τ∈R,U(τ,τ) = Id´H

þðŽf;

2)éz‡σ∈RÚ?¿t>τ>σ,U(t,τ)U(τ,σ) = U(t,σ).

K¡U(t,τ)´˜‡L§.

½Â2.3[12]XJé?¿t∈R, •3˜‡~êR>0, ¦C

⊂{z∈H

:kzk

6R}=

DOI:10.12677/aam.2023.1252382344A^êÆ?Ð

Ha§à

(R), K¡k.8C

⊂H

8ÜxC= {C

}

t∈R

´˜—k..

½Â2.4 [12] XJé?¿R>0,•3~êt

(t,R) 6t,¦τ6t−t

⇒U(t,τ)B

(R) ⊂B

K¡˜—k.8xB= {B

}

t∈R

´L§U(t,τ)žm•6áÂ8.

½Â2.5[12]XJB={B

}

t∈R

˜—k., …é?¿R>0, •3~êt

(t,R)6t, ¦

τ6t

⇒U(t,τ)B

(R) ⊂B

,K¡B´.£áÂ.

½Â2.6[12]L§U(t,τ) žm•6áÚf´÷vXe5Ÿ•xA= {A

}

t∈R

1)z‡A

¥´;;

2)A´.£áÚ,=éz‡˜—k.xC= {C

}

t∈R

,4•

lim

τ→−∞

(U(t,τ)C

) = 0

¤á,Ù¥

(B,C)=sup

x∈B

inf

y∈C

kx−yk

L«8ÜBÚCHausdorﬀŒål.

½n2.3[12]XJL§U(t,τ)ìC;, =8ÜK = {K= {K

}

t∈R

: K

⊂H

•;8, K•.£

áÚ}´š˜;,Kžm•6áÚfA•3…•˜.

½Â2.7[12]XJ∀t>τ,U(t,τ)A

= A

,K¡žm•6áÚfA= {A

}

t∈R

´ØC.

½n2.4[12]U(·,·) •Š^uBanach ˜mx{H

}

t∈R

L§, KU(·,·) kžm•6Ûá

ÚfU

∗

= {A

∗

}

t∈R

÷v

∗

s6t

[

τ6s

U(t,τ)B

…=

(i)U(·,·) •3žm•6áÂ8xB= {B

}

t∈R

;

(ii)U(·,·) ´ìC;.

3.·½5

½n3.1e^‡(C

)-(C

)¤á,K¯K(2.1)k˜‡f)u,…(u,∂

u) ∈L

∞

([τ,T];

),∂

u∈L

([τ,T];V

),…keª¤á

k(u,∂

u)k

k∂

u(s)k

ds6C

, ∀t>τ,(3.1)

Ù¥C

= C(R,δ,kgk).d, 1 6p<p

N+2θ

N−4

ž)„äk±e5Ÿ:

(i)é?¿t>s>τÚ(u,∂

u) ∈H

keUþðª¤á

DOI:10.12677/aam.2023.1252382345A^êÆ?Ð

Ha§à

E(u(s),∂

u(s))+

(r)k∂

u(r)k

dr = E(u(t),∂

u(t))+γ

k∂

u(r)k

dr,(3.2)

Ù¥

E(u(t),∂

u(t)) =

(kuk

+ε(t)k∂

Ω

F(u)dx−hg,ui.

(ii) z=u−v•¯K(2.1) éAuÐŠ(u

),(v

) ), KT)3˜mV

2−θ

×V

−θ

LipschitzëY,=éτ6t6T,k

k(z,∂

z)(t))k

2−θ

×V

−θ

k(z,∂

z)(τ)k

2−θ

×V

−θ

,(3.3)

Ù¥C

= max{C

}= max{{2γ+bδL+δ,L+bL},{

17δ

δL

+δbL,bδγ,b}},ùpδ,b¿

©,¿…z„÷veª

kz(t)k

2−θ

+ε(t)k∂

z(t)k

−θ

−bt

(kz(τ)k

2−θ

+ε(τ)k∂

z(τ)k

−θ

)

−b(t−s)

(k∂

z(s)k

+kz(s)k

)ds.(3.4)

(iii)é?¿τ<ka<a6t6T,(u,∂

u,∂

u) ∈L

∞

([a,T];V

2+θ

×V

2−θ

×V

−θ

)∩L

([a,T];V

×V

×L

(Ω)),keª¤á

kuk

2+θ

+k∂

2−θ

+ε(t)k∂

−θ

t+1

(ku(s)k

+k∂

u(s)k

+k∂

u(s)k

)ds

6(1+

7ν

)

(t−τ)

(t−ka)

1−θ

2−θ

νγ

7ν

)h(t)+

,(3.5)

Ù¥k>0,ν= min{

δ,γ−δL+δ},h(t) =

(ka−τ)

(t−ka)

(ka)

(t−ka)

2−θ

= C(R,δ,kgk,L) .

y²é•§(2.1) ¦±∂

E(u,∂

u)+γk∂

−

(t)k∂

= 0,(3.6)

Ù¥

E(u,∂

u) =

(kuk

+ε(t)k∂

Ω

F(u)dx−hg,ui.

ò(3.6)3[s,t]þÈ©Œ(3.2).d51, H¨olderØª,Young Øª±9Poincar´eØªŒ

Ω

F(u)dx>−

kuk

−C>−

2λ

kuk

−C(η).(3.7)

hg,ui6

4δ

kgk

kuk

.(3.8)

DOI:10.12677/aam.2023.1252382346A^êÆ?Ð

Ha§à

¤±k

E(u,∂

u) =

(ε(t)k∂

+kuk

Ω

F(u)dx−hg,ui

(ε(t)k∂

+(1−

)kuk

)−hg,ui−C(η)

>dk(u,∂

u)k

−C(δ,kgk),(3.9)

Ù¥d= min{

−

η+2δ

2λ

}.é(3.6)lτtÈ©Œ

E(u(t),∂

u(t))+γ

k∂

u(s)k

ds−

(s)k∂

u(s)k

ds= E(u(τ),∂

u(τ)).(3.10)

dε(t)4~5Œ•

E(u(t),∂

u(t)) 6E(u(τ),∂

u(τ)).

k(u,∂

u)k

6C(R,δ,kgk).(3.11)

Ï•γ>0,2(Ü(3.11), k

k∂

u(s)k

ds6C(R,δ,kgk).(3.12)

= C(R,δ,kgk),2(Ü(3.11) Ú(3.12) Œ(3.1).

(ii)u,v••§(2.1) éAuÐŠ(u

),(v

)),Kz= u−v÷veª

ε(t)∂

z+Az+γA

∂

z+f(u)−f(v) = 0.(3.13)

ò(3.13)†2A

−

∂

z+2δz‰SÈk

(z,∂

z)+2δkzk

+[2γ−2δε(t)]k∂

−ε

(t)k∂

−θ

−2δε

(t)h∂

z,zi

= −hf(u)−f(v),2A

−

∂

z+2δzi,(3.14)

Ù¥

(z,∂

z) = ε(t)k∂

−θ

+2δε(t)h∂

z,zi+kzk

2−θ

+γδkzk

Ï•V

2−θ

→V

,¤±

|ε(t)h∂

z,zi|6ε(t)kA

−

∂

zk·kA

ε(t)

k∂

−θ

kzk

2−θ

δ¿©ž,

(z,∂

z) ∼ε(t)k∂

−θ

+kzk

2−θ

.(3.15)

DOI:10.12677/aam.2023.1252382347A^êÆ?Ð

Ha§à

N=4 ž,é1<r<∞,kV

→L

; N>5ž,é0<δ1,k

N−2(2−δ)

N−4

,,k

→L

N−4

; Ï•16p<p

N+2θ

N−4

, ¤±k

N(p−1)

θ+2−δ

N−4

, ?kV

→L

N(p−1)

θ+2−δ

. dH¨olderØ

ª,Young Øª±9¥Š½nŒ

|hf(u)−f(v),2A

−

∂

z+2δzi|

Ω

(1+|u|

p−1

+|v|

p−1

)|z|(|A

−

∂

z|+δ|z|)dx

6C(1+kuk

p−1

N(p−1)

θ+2−δ

+kvk

p−1

N(p−1)

θ+2−δ

)kzk

N−2(2−δ)

(kA

−

∂

N−2θ

+δkzk

N−2θ

)

6C(1+kuk

p−1

+kvk

p−1

)kzk

2−δ

(kA

−

∂

+δkzk

)

kzk

(k∂

zk+δkzk

)

6δk∂

kzk

,(3.16)

Ù¥^

θ+2−δ

N−2(2−δ)

N−2θ

= 1,2−δ<2,θ<2.¤±é¿©b>0,ò(3.16) “\

(3.14)Œ

(z,∂

z)+bH

(z,∂

6[2−2δε(t)]k∂

+[ε

(t)+bε(t)]k∂

−θ

+δk∂

+bkzk

2−θ

+2δkzk

+bδkzk

+2δε

(t)h∂

z,zi+2δbε(t)h∂

z,zi+

kzk

.(3.17)



|2δε

(t)h∂

z,zi|62δLk∂

zkkzk

62δLk∂

δL

kzk

.(3.18)

|2δbε(t)h∂

z,zi|6δbL(k∂

+kzk

).(3.19)

(Ü(3.17)-(3.19),C

= max{2γ+bδL+δ,L+bL},C

= max{

17δ

δL

+δbL,bδγ,b},Kk

|(2γ−2δε(t))|k∂

+|(ε

(t)+bε(t))|k∂

−θ

+(2δL+δbL)k∂

+δk∂

6(2γ+bδL+δ)k∂

+(L+bL)k∂

−θ

k∂

.(3.20)

2δkzk

+bδγkzk

+bkzk

2−θ

δL

kzk

+δbLkzk

kzk

17δ

δL

+δbL)kzk

+bδγkzk

+bkzk

2−θ

kzk

.(3.21)

DOI:10.12677/aam.2023.1252382348A^êÆ?Ð

Ha§à

ò(3.20)-(3.21)“\(3.17)

(z,∂

z)+bH

(z,∂

z) 6C

(k∂

+kzk

),(3.22)

Ù¥C

= max{C

}.é(3.22)3[τ,t]þ$^Gronwall ÚnŒ(3.4).

(iii)ò•§(2.1)'ut¦,Œv= ∂

u÷v•§

ε(t)∂

v+ε

(t)∂

v+Av+γA

∂

v+f

(u)v= 0.(3.23)

−

∂

v+δv†(3.23) Š^Œ

(v,∂

v)+δkvk

(t)k∂

−θ

+[γ−δε(t)]k∂

+hf

(u)v,A

−

∂

v+δvi

= 0,(3.24)

Ù¥

(v,∂

v) =

[ε(t)k∂

−θ

+kvk

2−θ

]+δε(t)h∂

v,vi+

δγkvk

Ï•V

2−θ

→V

,¤±

|ε(t)h∂

v,vi|6ε(t)kA

−

∂

vkkA

ε(t)

k∂

−θ

kvk

2−θ

δ¿©ž,

(v,∂

v) ∼ε(t)k∂

−θ

+kvk

2−θ

.(3.25)

†(3.16)Oaq,k

|hf

(u)v,A

−

∂

v+δvi|

Ω

(1+|u|

p−1

)|v|(|A

−

∂

v|+δ|v|)dx

6C(1+kuk

p−1

N(p−1)

θ+2−δ

)kvk

N−2(2−δ)

(kA

−

∂

N−2θ

+δkvk

N−2θ

)

6C(1+kuk

p−1

)kvk

2−δ

(kA

−

∂

+δkvk

)

kvk

(k∂

vk+δkvk

)

6δk∂

kvk

.(3.26)

DOI:10.12677/aam.2023.1252382349A^êÆ?Ð

Ha§à

ò(3.26)“\(3.25)

(v,∂

v)+

δkvk

+[γ−δε(t)+δ]k∂

kvk

2−θ

−

(t)k∂

−θ

dε(t)4~5Œ,é¿©δÚν= min{

δ,γ−δL+δ},k

(v,∂

v)+ν(kvk

+k∂

) 6C

(v,∂

v)+C

.(3.27)

¦±(3.27)Œ

(v,∂

v))+νt

(kvk

+k∂

)

(v,∂

v)+C

+Ct

2−θ

(ε(t)k∂

−θ

+kvk

2−θ

), τ∈R.(3.28)

→→L

→V

−2

Ú(3.1)Œ

kf(u)k

−2

6Ckf(u)k6C(kuk+kuk

) 6C

.(3.29)

ε(t)k∂

−2

= ε(t)k∂

−2

6kAuk

−2

+γkA

∂

−2

+kf(u)k

−2

+kgk

−2

6kuk

+γk∂

2θ−2

+kf(u)k

−2

+kgk

.(3.30)

(Ü(3.30)ÚŠ½nŒ

2−θ

ε(t)k∂

−θ

2−θ

k∂

vkε(t)k∂

−2

k∂

2−θ

kvk

2−θ

kvk

2−θ

kvk6

kvk

Ù¥^

−1 <

.ò±þ(J“\(3.28),¿|^V

→→L



(v,∂

v))+

(kvk

+k∂

) 6C

(v,∂

v)+C

.(3.31)

−C

(t−τ)

¦±(3.31),¿3[τ,t]þÈ©Œ

ε(t)k∂

−θ

+kvk

2−θ

(t−τ)

.(3.32)

DOI:10.12677/aam.2023.1252382350A^êÆ?Ð

Ha§à

é?¿τ<a6t,‰(3.31) ¦±e

−C

(t−a)

,¿3[a,t]þÈ©Œ

(kvk

+k∂

)ds6

(a−τ)

(t−a)

−C

(t−τ)

¦±(3.31)¿3[t,t+1] þÈ©k

(v(t+1),∂

v(t+1))e

−C

(t+1−τ)

t+1

−C

(s−τ)

(kvk

+k∂

)ds

(v(t),∂

v(t))e

−C

(t−τ)

.(3.33)

Ïdk

t+1

(kvk

+k∂

)ds6

(t−τ)

.(3.34)

(Ü(3.32)Ú(3.34)k

ε(t)k∂

−θ

+kvk

2−θ

t+1

(kvk

+k∂

)ds

6(1+

)

(t−τ)

.(3.35)

^Au†(2.1)‰SÈ,Œ

kuk

2+θ

+kuk

= hg−f(u)−ε(t)∂

u,Aui.(3.36)

Ï•p+1 <

2N+(2θ−4)

N−4

,¤±kV

→L

p+1

,2|^Young Øª,H¨olderØªÚŠ½nŒ

|hf(u),Aui|6kf(u)kkAuk

6C(1+kuk

p−1

p+1

)kuk

p+1

kuk

(3.37)

|hg−ε(t)∂

u,Aui|6(kgk+ε(t)k∂

uk)kuk

6C(kgk

k∂

kuk

(3.38)

ò(3.37)-(3.38)“\(3.36)Œ

kuk

2+θ

+kuk

62C

(kgk

k∂

).(3.39)

DOI:10.12677/aam.2023.1252382351A^êÆ?Ð

Ha§à

a>0,0 <k<1;½a<0,k>1 ž,é?¿τ<ka<a6t,é(3.39)¦±(t−ka)

2−θ

Œ

[(t−ka)

2−θ

kuk

2+θ

(t−ka)

2−θ

kuk

(t−ka)

2−θ

(kgk

k∂

2−θ

(t−ka)

θ−1

2−θ

kuk

2+θ

.(3.40)

dŠ½nÚYoung Øªk

2−θ

(t−ka)

θ−1

2−θ

kuk

2+θ

2−θ

(t−ka)

θ−1

2−θ

kuk

2−θ

kuk

(t−ka)

2−θ

kuk

+Ckuk

.(3.41)

¤±k

[(t−ka)

2−θ

kuk

2+θ

] 62C

(t−ka)

2−θ

(kgk

+kuk

k∂

).(3.42)

é(3.42)3[ka,t]È©,2|^c¡®(Jk

γkuk

2+θ

(ku(s)k

k∂

u(s)k

)ds

62C

(t−ka)

1−θ

2−θ

h(t),(3.43)

Ù¥

h(t) =

(ka−τ)

(t−ka)

(ka)

(t−ka)

2−θ

é(3.39)3[a,t]þÈ©Œ

kuk

ds6

8γ

ku(a)k

2+θ

(ku(s)k

k∂

u(s)k

)ds

16C

(a−ka)

1−θ

2−θ

(t−a)+

7ν

h(a)+

7ν

(a−τ)

(t−a)

,(3.44)

Ù¥C

= C(R,δ,kgk,L).

é(3.39)3[t,t+1] È©Œ

t+1

ku(s)k

ds6

16C

(t−ka)

1−θ

2−θ

7ν

h(t)+

7ν

(t−τ)

(3.45)

(Ü(3.43)-(3.45)Œ

kuk

2+θ

t+1

ku(s)k

(t−ka)

1−θ

2−θ

νγ

7ν

)h(t)+

7ν

(t−τ)

.(3.46)

(Ü(3.35)Ú(3.46)k(3.5)¤á.

DOI:10.12677/aam.2023.1252382352A^êÆ?Ð

Ha§à

-z

=(u

,∂

) ´¯K(2.1)éACq•§), ´•(3.1) éGalerkin CqSz

•´

¤á.Ïd,•3(u,∂

u) ∈L

∞

([τ,T];H

),∂

u∈L

([τ,T];V

¦

,∂

)f∗Âñu(u,∂

u)3L

∞

([τ,T];H

)¥,

∂

fÂñu∂

u3L

([τ,T];V

)¥.

A^Ún2.1Œ

→u3L

([τ,T];V

),∂

→∂

u3L

([τ,T];L

(Ω)),

,∂

) →(u,∂

u)3C([τ,T];V

2−δ

×V

−δ

),(3.47)

f(u

(t)) →f(u(t))3L

¥fÂñ,t∈[τ,T],

(x,t)3Ω×[τ,T] ¥A??Âñuu(x,t).

→L

p+1

Ú(2.5)Œ,é?¿ζ∈C

∞

(Ω)

hf(u

)−f(u),ζidt

Ω

(1+|u

p−1

+|u|

p−1

)|u

−u||ζ|dxdt

(1+ku

p−1

p+1

+kuk

p−1

p+1

)ku

−uk

p+1

|ζ|

p+1

(1+ku

p−1

+kuk

p−1

)ku

−uk

kζk

−uk

([τ,T];V

)

→0.

¤±z= (u,∂

u)´÷v(3.1)¯K(2.1)f).

é?¿t∈[τ,T], d(3.2) Ú(3.47) Œ

lim

s→t

E(u(s),∂

u(s)) = E(u(t),∂

u(t)),

x∈Ω ž,u(x,s) →u(x,t)(s→t)A??,

(u,∂

u) ∈C([τ,T];V

2−δ

×V

−δ

)∩L

∞

([τ,T];H

dFatouÚnÚ51Œ

lim

s→t

hg,u(s)i= hg,u(t)i,

k(u(t)),∂

u(t)k

6liminf

s→t

k(u(s)),∂

u(s)k

DOI:10.12677/aam.2023.1252382353A^êÆ?Ð

Ha§à

Ω

(F(u(t))+

|u(t)|

+C)dx

6liminf

s→t

Ω

(F(u(s))+

|u(s)|

+C)dx

6liminf

s→t

Ω

F(u(s))dx+

ku(t)k

+C|Ω|.

¤±k

Ω

F(u(t))dx6liminf

s→t

Ω

F(u(s))dx.(3.48)

d±þOªŒ

liminf

s→t

(

ε(s)k∂

u(s)k

ku(s)k

)+liminf

s→t

Ω

F(u(s))dx

6lim

s→t

(

ε(s)k∂

u(s)k

ku(s)k

Ω

F(u(s))dx)

ε(t)k∂

u(t)k

ku(t)k

Ω

F(u(t))dx

6liminf

s→t

(

ε(s)k∂

u(s)k

ku(s)k

)+liminf

s→t

Ω

F(u(s))dx.

k

ε(t)k∂

u(t)k

= liminf

s→t

ε(s)k∂

u(s)k

,(3.49)

ku(t)k

= liminf

s→t

ku(s)k

.(3.50)

(Ü(3.49),(3.50),˜mH

˜—à5Ú(u,∂

u) ∈C

([τ,T];H

),Œ(u,∂

u) ∈C([τ,T];

4.žm•6áÂ8

½n4.1^‡(C

)-(C

)¤á,é?¿τ<T,¯K(2.1))ëY•6ÐŠ.=XJ

= (u

,∂

),z

= (u

,∂

) ´¯K(2.1) 'uÐŠkz

(τ)k

6R,kz

(τ)k

6R), ùp

R>0 ´˜‡~ê,K

(t)−z

(t)k

(t−τ)

(τ)−kz

(τ)k

,∀t∈[τ,T].(4.1)

y²¯z(t) = {¯u(t),∂

¯u(t)}= z

(t)−z

(t),K¯z(t) ÷v•§

ε(t)∂

¯u(t)+A¯u+γA

∂

¯u(t)+f(u

)−f(u

) = 0.(4.2)

^2∂

¯u(t) †(4.2) ‰SÈ,Œ

DOI:10.12677/aam.2023.1252382354A^êÆ?Ð

Ha§à

(¯u,∂

¯u(t))+2γk∂

¯u(t)k

= −2hf(u

)−f(u

),∂

¯u(t)i+ε

(t)k∂

¯u(t)k

,(4.3)

Ù¥

(¯u,∂

¯u(t)) = ε(t)k∂

¯u(t)k

+k¯uk

dε(t)4~5Œ

(¯u,∂

¯u(t))+2γk∂

¯u(t)k

6−2hf(u

)−f(u

),∂

¯u(t)i.(4.4)

|−2hf(u

)−f(u

),∂

¯u(t)i|6C

Ω

(1+|u

p−1

+|u

p−1

)|¯u||∂

¯u(t)|dx

6C(1+ku

p−1

N(p−1)

θ+2−δ

+ku

p−1

N(p−1)

θ+2−δ

)k¯uk

N−2(2−δ)

k∂

¯u(t)k

N−2θ

6C(1+ku

p−1

+ku

p−1

)k¯uk

2−δ

k∂

¯u(t)k

k¯uk

2−δ

k∂

¯u(t)k

k¯uk

+γk∂

¯u(t)k

.(4.5)

ò(4.5)“\(4.4)Œ

(¯u,∂

¯u(t))+γk∂

¯u(t)k

(k¯uk

+ε(t)k∂

¯u(t)k

)

?k

(¯u,∂

¯u(t)) 6C

(¯u,∂

¯u(t)).(4.6)

‰(4.6)¦±e

−C

(t−τ)

,¿3[τ,t]þÈ©Œ

(t)−z

(t)k

(t−τ)

(τ)−kz

(τ)k

d½n3.1,4.1Œ•,1 6p<p

ž,·‚Œ±½Â¯K(2.1)L§U(t,τ) : H

→H

U(t,τ)(u

) = (u(t),∂

u(t)), t>τ,

Ù¥u(t)´¯K(2.1)ƒéuÐŠ(u

) ∈H

•˜),¿…TL§dH

N\H

´ëY.

½n4.2XJ^‡(C

)-(C

)¤á, PB

(R) = {z∈H

: kzk

6R},@o•3R

>0,¦

B= {B

)}

t∈R

´L§{U(t,τ)}žm•6áÂ8, …éS

,÷v

sup

∈B

)

{kU(t,τ)z

+∞

k∂

u(y)kdy}6S

.(4.7)

y²d½n4.1 9©[7],[13], ´y(4.7) ¤á.

DOI:10.12677/aam.2023.1252382355A^êÆ?Ð

Ha§à

5.žm•6áÚf

5.1.kO

•¯K(2.1)L§U(t,τ)ìC;5, ·‚k?1±eO.

(u

(t),∂

(t)),i= 1,2´•§(2.1)'uÐŠ(u

(τ),∂

(τ)) ∈B

)),P

w(t) = u

(t)−u

(t).Kw(t)÷v











ε(t)∂

w+Aw+γA

∂

w+f(u

)−f(u

) = 0,x∈Ω,t>τ,

w(x,t) = 0,x∈∂Ω,t>τ,

w(x,τ) = u

(τ)−u

(τ),∂

w(x,τ) = ∂

(τ)−∂

(τ),x∈Ω.

(5.1)

‰(5.1)†mü>¦±w,¿3[τ,T]×Ω þÈ©Œ

Ω

ε(T)∂

w(T)w(T)dx−

Ω

ε(τ)∂

w(τ)w(τ)dx−

Ω

(s)∂

w(s)w(s)dxds

−

Ω

ε(s)|∂

w(s)|

dxds+

Ω

w(s)|

dxds+

Ω

w(T)|

Ω

(f(u

)−f(u

))w(s)dxds−

Ω

w(τ)|

dx= 0.(5.2)

¦±(5.1),¿3[s,T]×ΩþÈ©,k

Ω

ε(T)|∂

w(T)|

dx−

Ω

ε(s)|∂

w(s)|

dx+

Ω

w(T)|dx−

Ω

w(s)|dx

+γ

Ω

∂

w(t)|

dxdt−

Ω

(t)|∂

w(t)|

dxdt

Ω

(f(u

)−f(u

))∂

w(t)dxdt= 0.(5.3)

-G

(t) =

Ω

(ε(t)|∂

w(t)|

+|A

w(t)|)dx,Kd(5.3) Œ

(T)−G

(s)+

Ω

(f(u

)−f(u

))∂

w(t)dxds−

Ω

(t)|∂

w(t)|

dxdt

+γ

Ω

∂

w(t)|

dxdt= 0.(5.4)

dε(t)•4~¼êŒε

(t) 60,?k

Ω

(t)|∂

w(t)|

dxdt

(s)−

Ω

(f(u

)−f(u

))∂

w(t)dxdt−γ

Ω

∂

w(t)|

dxdt.

DOI:10.12677/aam.2023.1252382356A^êÆ?Ð

Ha§à

ε(t)|∂

w(t)|

Lε

(t)|∂

w(t)|

,¤±k

Ω

ε(t)|∂

w(t)|

dxdt

6LG

(s)−L

Ω

(f(u

)−f(u

))∂

w(t)dxdt−Lγ

Ω

∂

w(t)|

dxdt.(5.5)

(T)(T−τ)−

Ω

(t)|∂

w(t)|

dxdtds

(s)ds−

Ω

(f(u

)−f(u

))∂

w(t)dxdtds

−γ

Ω

∂

w(t)|

dxdtds.(5.6)

(Ü(5.5)Ú(5.2)k

(s)ds=

Ω

ε(s)|∂

w(s)|

dxds+

Ω

w(s)|

dxds

Ω

ε(s)|∂

w(s)|

dxds+

Ω

ε(τ)∂

w(τ)w(τ)dx+

Ω

w(τ)|

−

Ω

ε(T)∂

w(T)w(T)dx+

Ω

(s)∂

w(s)w(s)dxds

−

Ω

(f(u

)−f(u

))w(s)dxds−

Ω

w(T)|

6LG

(τ)−L

Ω

(f(u

)−f(u

))∂

w(s)dxds

−Lγ

Ω

∂

w(s)|

dxds+

Ω

ε(τ)∂

w(τ)w(τ)dx−

Ω

w(T)|

Ω

w(τ)|

dx−

Ω

(f(u

)−f(u

))w(s)dxds

Ω

L∂

w(s)w(s)dxds−

Ω

ε(T)∂

w(T)w(T)dx.(5.7)

ò(5.7)“\(5.6),2dε(t)4~5Œ

DOI:10.12677/aam.2023.1252382357A^êÆ?Ð

Ha§à

(T) 6

T−τ

(τ)+

2(T−τ)

Ω

ε(τ)∂

w(τ)w(τ)dx+

4(T−τ)

Ω

w(τ)|

−

(T−τ)

Ω

(f(u

)−f(u

))∂

w(s)dxds−

γL

(T−τ)

Ω

∂

w(s)|

dxds

−

(T−τ)

Ω

(f(u

)−f(u

))∂

w(t)dxdtds−

4(T−τ)

Ω

w(T)|

2(T−τ)

Ω

L∂

w(s)w(s)dxds−

(T−τ)

Ω

∂

w(t)|

dxdtds

−

2(T−τ)

Ω

ε(T)∂

w(T)w(T)dx−

2(T−τ)

Ω

(f(u

)−f(u

))w(s)dxds

= LG

(τ)+

Ω

ε(τ)∂

w(τ)w(τ)dx+

Ω

w(τ)|

dx.(5.8)

((u

(τ),∂

(τ)),(u

(τ),∂

(τ))) =

i=1

.(5.9)

Ù¥

= −

2(T−τ)

Ω

ε(T)∂

w(T)w(T)dx+

2(T−τ)

Ω

L∂

w(s)w(s)dxds,

(T−τ)

[−L

Ω

(f(u

)−f(u

))∂

w(s)dxds−

Ω

(f(u

)−f(u

))w(s)dxds

−

Ω

(f(u

)−f(u

))∂

w(t)dxdtds],

=−

(T−τ)

Ω

∂

w(t)|

dxdtds−

4(T−τ)

Ω

w(T)|

−

γL

(T−τ)

Ω

∂

w(s)|

dxds.

d(5.8),(5.9) Œ

(T) 6

(T−τ)

+Φ

((u

(τ),∂

(τ)),(u

(τ),∂

(τ)))

5.2.ìC;5

e¡·‚ò|^Â ¼ê•{5y²•§(2.1)éAL§´ìC;.

DOI:10.12677/aam.2023.1252382358A^êÆ?Ð

Ha§à

½n5.1XJ^‡(C

)-(C

)¤á, é?¿½t∈R, k.S{x

}

∞

n=1

∈H

,†?¿S

{τ

}

∞

n=1

∈[−∞,t)(n→∞,τ

→−∞)ž, KS{U(t,τ

}

∞

n=1

kÂñf.

y²(u

(t),∂

(t)) ´¯K(2.1) 'u ÐŠ(u

(τ),∂

(τ)) ), d½n3.1Œ•ku

ε(ξ)k∂

´k.,…ku

´k.;d^‡(C

)Œ•,éξ∈[τ,T],ε(ξ)

´k.,¤±k∂

´k..ŠâBanach−Alaoglu ½n,½n3.1, Ún2.1 k±e(J:

→u3L

∞

([τ,T];V

)¥f∗Âñ,(5.10)

∂

→u

([τ,T];V

)¥fÂñ,(5.11)

∂

→u

∞

([τ,T];L

)¥f∗Âñ,(5.12)

→u3L

([τ,T];V

),(5.13)

∂

→∂

u3L

([τ,T];L

),(5.14)

→u3L

([τ,T];V

2θ

),(5.15)

Ù¥^θ<2θ<2.

é?¿>0 Ú½T>τ,¦T−τvŒ, Kk

T−τ

<.

d½n2.1Œ•, ·‚•Iy²é?¿½T,Φ

∈C(B

)) ¤á=Œ.•d, ·‚Å‘?n

(5.9).

Äk,d½n4.2Ú(5.10)-(5.13)Œ

lim

n→∞

lim

m→∞

Ω

−u

dx= 0.(5.16)

lim

n→∞

lim

m→∞

Ω

(∂

−∂

dxds= 0.(5.17)

Ï•V

→L

p+1

,2(Ü(5.10)Ú(5.13)Œ

lim

n→∞

lim

m→∞

Ω

ε(t)(∂

−∂

)(u

−u

)dx

6Clim

n→∞

lim

m→∞

Lk∂

−∂

−u

6Clim

n→∞

lim

m→∞

Lk∂

+∂

−u

= 0.(5.18)

lim

n→∞

lim

m→∞

Ω

L(∂

−∂

)(u

−u

)dxds

6Clim

n→∞

lim

m→∞

(

k∂

−∂

ds)

(

−u

ds)

= 0.(5.19)

DOI:10.12677/aam.2023.1252382359A^êÆ?Ð

Ha§à

lim

n→∞

lim

m→∞

Ω

(f(u

)−f(u

))(u

−u

)dxds

6Clim

n→∞

lim

m→∞

Ω

(1+|u

p−1

+|u

p−1

)|u

−u

dxds

6Clim

n→∞

lim

m→∞

(1+ku

p−1

p+1

+ku

p−1

p+1

)ku

−u

p+1

6Clim

n→∞

lim

m→∞

(1+ku

p−1

+ku

p−1

)ku

−u

lim

n→∞

lim

m→∞

−u

ds= 0.(5.20)

(Ü(5.16)-(5.17),±9(5.18)-(5.19),k

lim

n→∞

lim

m→∞

= 0,(5.21)

lim

n→∞

lim

m→∞

= 0.(5.22)

Ùg,Ï•

Ω

(f(u

)−f(u

))(∂

−∂

)dxds

Ω

f(u

)∂

dxds−

Ω

f(u

)∂

dxds−

Ω

f(u

)∂

dxds

Ω

f(u

)∂

dxds

Ω

F(u

(T))dx−

Ω

F(u

(τ))dx−

Ω

f(u

)∂

dxds−

Ω

f(u

)∂

dxds

Ω

F(u

(T))dx−

Ω

F(u

(τ))dx.(5.23)

d(2.5)ÚV

→L

p+1

,Œ

Ω

(F(u

(t))−F(u(t)))dx|6

Ω

|(F(u

(t))−F(u(t)))|dx

Ω

|f(u(t)+λ(u

(t)−u(t)))||u

(t)−u(t)|dx

Ω

(1+|u

(t)|

p−1

+|u(t)|

p−1

)|u

(t)−u(t)|

6C(1+ku

(t)k

p−1

p+1

+ku(t)k

p−1

p+1

)ku

(t)−u(t)k

p+1

6C(1+ku

(t)k

p−1

+ku(t)k

p−1

)ku

(t)−u(t)k

.(5.24)

DOI:10.12677/aam.2023.1252382360A^êÆ?Ð

Ha§à

d(5.14)Œ,n→∞,m→∞žk

lim

n→∞

lim

m→∞

hf(u

),∂

ids

=lim

n→∞

hf(u

),∂

uids

hf(u),∂

uids

Ω

F(u(T))dx−

Ω

F(u(τ))dx.(5.25)

aq/,k

lim

n→∞

lim

m→∞

hf(u

),∂

ids=

Ω

F(u(T))dx−

Ω

F(u(τ))dx.(5.26)

(Ü(5.23)-(5.26)Œ

lim

n→∞

lim

m→∞

Ω

(f(u

)−f(u

))(∂

−∂

)dxds= 0(5.27)

é?¿½T, |

Ω

(f(u

)−f(u

))(∂

−∂

)dxdt|´k., ?dLebesgue ››Âñ

½nŒ

lim

n→∞

lim

m→∞

Ω

(f(u

)−f(u

))(∂

−∂

)dxdtds

( lim

n→∞

lim

m→∞

Ω

(f(u

)−f(u

))(∂

−∂

)dxdt)ds

0ds= 0.(5.28)

(Ü(5.27)Ú(5.28)Œ

lim

n→∞

lim

m→∞

= 0.(5.29)

nþŒΦ

∈C(B

)).

½n5.2XJ^‡(C

)-(C

) ¤á, @o¯K(2.1)éAL§U(t,τ):H

→H

k˜‡ØC

žm•6áÚfA= {A

}

t∈R

y²d½n2.1, ½n3.1, ½n4.1, ½n5.1 ´y.

6.(Ø†Ð"

žmt¼ê, ùéL§;5O¬‘5Ÿþ(J, ^Â ¼ê•{éÐ/)ûù˜J

DOI:10.12677/aam.2023.1252382361A^êÆ?Ð

Ha§à

)•žmÄåÆ1•,38ïÄ¥Œ±•Ä)3.½ö‡.œ¹e5Ÿ.

Ä7‘8

I[g,‰ÆÄ7‘8(1OÒ:12061062;11961059).

ë•©z

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