一类带记忆和非经典耗散项的发展方程的适定性 The Well-Posedness of a Memory-Type Evolution Equation with Nonclassical Dissipation

doi:10.12677/PM.2021.118173

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PureMathematicsnØêÆ,2021,11(8),1546-1558

PublishedOnlineAugust2021inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2021.118173

˜a‘PÁÚš²;ÑÑ‘uÐ•§

·½5

444ÜÜÜ†††

∗

§§§444&&&§§§ÜÜÜôôô¥¥¥

•ânóŒÆ,êÆ†ÚOÆ,H•â

ÂvFÏµ2021c715F¶¹^FÏµ2021c818F¶uÙFÏµ2021c825F

Á‡

©Ì‡?Ø‘š²;ÑÑPÁ.uÐ•§·½5¯K,·‚$^š²;Galerkin•{9

©ÛE|f)•35,Óžy²)•˜5ÚéÐŠëY•65"

'…c

uÐ•§§PÁ‘§š²;ÑÑ§·½5

TheWell-PosednessofaMemory-Type

EvolutionEquationwithNonclassical

Dissipation

XimengLiu

∗

,DiLiu,JiangweiZhang

SchoolofMathematicsandStatistics,ChangshaUniversityofScienceandTechnology,Changsha

Hunan

Received:Jul.15

,2021;accepted:Aug.18

,2021;published:Aug.25

,2021

∗ÏÕŠö"

©ÙÚ^:4Ü†,4&,Üô¥.˜a‘PÁÚš²;ÑÑ‘uÐ•§·½5[J].nØêÆ,2021,11(8):

1546-1558.DOI:10.12677/pm.2021.118173

4Ü†

Abstract

Inthispaper,wemainlydiscussthewell-posednessproblemofaMemory-typeEvolu-

tionEquationwithnonclassicaldissipation.Theexistenceofweaksolutionisobtained

byusingtheGalerkin’smethodandanalyticaltechniques.Also,weprovetheunique-

nessofthesolutionandthecontinuousdependenceoninitialvalue.

Keywords

EvolutionEquation,Memory,NonclassicalDissipation,Well-Posedness

2021byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).

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

1.Úó

·‚•ÄXe‘kš²;ÑÑ‘ÚPÁ‘uÐ•§·½5¯K:











−M(|∇u|

)∆u−K

∆u+

−∞

µ(t−s)∆u(s)ds+h(u

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

u(x,t)|

∂Ω×R

= 0,u(x,0) = u

(x),u

(x,0) = u

(x),x∈Ω.

(1)

Ù¥Ω ⊂R

(N≥3)´äk1w>.∂Ω k.«•;M(r) =a+br(r= |·|

L«L

(Ω)‰ê),

…a,b>0;d,h(u

)=αu

−βu

•š²;ÑÑ‘,g∈L

(Ω).•ïÄþã¯K,·‚Ú\PÁ

Cþ,½ÂXe:

(x,s) = u(x,t)−u(x,t−s),(x,s) ∈Ω×R

,t≥0.

(2)

t= 0ž,·‚k

(x,s) = u

(x,0)−u

(x,−s), (x,s) ∈Ω×R

(3)

AO/,b

+∞

µ(s)ds,(4)

DOI:10.12677/pm.2021.1181731547nØêÆ

4Ü†

K•§(1)C•:











−M(|∇u|

)∆u−

+∞

µ(s)∆η

(s)ds+h(u

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

= −η

(5)

‘kÐ>Š^‡:

(

u(x,t)|

∂Ω×R

= 0,η

(x,s)|

∂Ω×R

= 0,

u(x,t)|

t=0

= u

(x),u

(x,t)|

t=0

= u

(x),η

(x,t)|

t=0

= η

(x),x∈Ω.

(6)

)'uš‚5‘fb:f: R→R,

|f(u)−f(v)|≤C

(1+|u|

+|v|

)|u−v|,∀u,v∈R.

(7)

Ù¥C

>0 •~ê,0 <p≤

N−2

.d,b

f(u)u≥

f(u) ≥0,∀u∈R.

(8)

ùp

f(u) =

f(s)ds.

)'uPÁØ¼êb:

1)µ(s) ∈C

(R)∩L

(R),∀s∈R

,µ(s) ≥0,µ

(s) ≤0;

2)•3~êK

,δ>0,¦é∀s∈R

,µ(s)÷v(4) ªÚµ

(s)+δµ(s) ≤0.

d2)Œ•,0 ≤µ(s) ≤µ(0)e

−δs

(µ(0) 6= ∞),lim

s→∞

µ(s) = 0.

¯¤±•,3•§(1) ¥,h(u

) =0 ž,ÙÌ‡^u£ãg,.¥ˆ«ÅÄy–(•)Ô

N$Äœ¹ÚÔn5Æ).Cc5,éaqu(1)•§, ®ŒõêÆö¤•Ä(„[1–5]9Ùë

•©z).2012 c,3[6]¥,Lazo•Ä[‚5•§u

−M(k∇uk

)∆u+

h(t−τ)∆u(τ)dτ= 0,

Ù^Faedo-Galerkin•{ƒ'·Ü¯K•3˜‡fÛ).

2013 c,3[4]¥,Zhang<ïÄ˜‡äkš‚5ÛÜ{ZÚ„Ýƒ'á— Ýš‚5Ê

5•§:

−M(k∇uk

)∆u+α∆

u−β∆u

−γ∆u

g(t−s)∆u(s)ds+a(x)u

+bu|u|

= 0,

¦‚Ì‡|^Faedo-Galerkin•{f)Û•35.

2018 c,3[5]¥,Nadia<•Ä˜aäkžCò´š‚5Ê5•§,¿|^Uþ{

)Û•35.2020c,3[1]¥,Ü<ïÄXeäkPÁ‘[‚5ÅÄ•§:

−M(k∇uk

)∆u−K

∆u−∆u

−∞

µ(t−s)∆u(s)ds−α∆u

+f(u) = h,

DOI:10.12677/pm.2021.1181731548nØêÆ

4Ü†

¨‚y²N f)•3•˜5ÚNáÚf•35.Óc,3[2]¥,Li<•Ä3ÛÜ

Þ{ZŠ^eÊ5ÅÄ•§,¿ïÄ)•žmÄåÆ1•.

nþ¤ã,<‚Ì‡•Ä‘k²;ÑÑ‘(rÑÑ−∆u

½fÑÑu

)ÅÄ.•§.•§

(1)‘kš²;ÑÑ‘(h(u

)=αu

−βu

),…da•§Ì‡Ñy3R5(XÚ¥,Xˆ«Œ.

ì(XÊUœ1ì!Œ.EÍ!Œ.gÄzÅ:!]¢x ±9Ä1Ú^”).1993 c,

3[7]¥,Wang<®²•Ä‘kaq{Z‘ÅÄ•§:

−c

= f(ω

Ù¥f(ω

) =ω

−ω

.aqïÄ½„©z[8].AO/,äkR5(Œ.ìlŸþ5`E

,´ëYXÚ,…äkÃ¡‘gdÝ.Ïd,ïÄ‘kØ½.{Zš²;ÑÑXÚN)•3

5,´äk¢S¿Â.

©òÌ‡¦^š²;Faedo-Galerkin•{(Ük O•{,y²)•3•˜5ÚÙé

ÐŠëY•65.

2.ý•£

••Bå„,H9©,^C,C

L«?¿~ê,Ù3ØÓ1$–Ó˜1ÑŒØƒÓ, …C

L«T~ê•6uT.-|·|L«ýéŠ½LebesgueÿÝ,|·|

•L

(Ω)(p≥2) ‰ê,|∇·|

L«

(Ω)þ‰ê.

˜m½Â:

M= L



(Ω)





ξ: R

→H

(Ω)



∞

|∇ξ(s)|

ds<∞



.(1)

¿D±SÈÚ‰ê

hξ,ζi

∞

µ(s)h∇ξ,∇ζids,

kξk

∞

µ(s)|∇ξ|

ds.

Ù¥h·,·iL«L

(Ω)þSÈ.

©¤ïÄ•§ƒ˜m•:

H= H

(Ω)×L

(Ω)×M,

ÙD±Xe‰ê:

kz(t)k

= k(u(t),u

(t),η

= |∇u(t)|

+|u

(t)|

+kη

(2)

DOI:10.12677/pm.2021.1181731549nØêÆ

4Ü†

3.·½5

½Â3.1(f)½Â)b^‡(H

)-(H

) ¤á,z(t)= (u(t),u

(t),η

) ´• §(1) Nf

),…ÐŠz(0) = z

= (u

,η

) ∈H.Ké∀t∈[0,T],ez(t) ÷v•§(1) ¿…

u∈L

([0,T];H

(Ω)),

∈L

∞

([0,T];L

(Ω))∩L

([0,T];L

(Ω)),

∈L

([0,T];M).

(3)

,,é?¿ω∈C

∞

(Ω),ξ∈Mk:











−M(|∇u|

)∆u−

+∞

µ(s)∆η

(s)ds+αu

−βu

+f(u)−g(x),ωi= 0

hη

,ξi

= h−η

,ξi

+hu

,ξi

(4)

éut∈RA??¤á.

3.1.)•35

Äk,ÀS{ψ

}•eãDirechlet¯K

−∆ψ

= λ

∂Ω

= 0.

ƒAuAŠλ

A¼ê.K3∂Ω1w^‡ek{ψ

}

+∞

j=1

•˜mH

(Ω)˜|

Ä,…ψ

∈C

∞

(Ω).e5ÏL½Â˜mL

)¥˜|IOÄ{l

}

+∞

k=1

,·‚Œ±

{ϑ

}

+∞

j=1

= {l

}

+∞

k,j=1

¤L

(Ω))þ˜|Ä.

•§(1)•3XeCq):

(x,t) =

j=1

(t)ψ

(x),

(x,t) = ∂

(x,t) =

j=1

∂

(t)ψ

(x),

(x,t) =

j=1

(t)ϑ

(x).

DOI:10.12677/pm.2021.1181731550nØêÆ

4Ü†

Ù¥ϕ

(t) = hu,ψ

i,κ

= hη

,ϑ

,…z

= (u

,η

)÷v











mtt

−M(|∇u

)∆u

−

∞

µ(s)∆η

(s)ds+αu

−βu

+f(u

) = g

(x),

= −η

,η

∂Ω×R

= (0,0),

(0) =

j=1

(x)ξ

(0) =

j=1

,η

(0) =

j=1

(5)

AO/,m→∞ž,k

j=1

,ψ

)ψ

(x) =

j=1

(x) →u

∈H

(Ω),

j=1

,ψ

)ψ

(x) =

j=1

(x) →u

∈L

(Ω),

j=1

hη

,ϑ

(x) =

j=1

(x) →η

∈M.

Ï•{ψ

(x)}

+∞

j=1

´3þã˜m¥È—5,Œ•þãÂñ´¤á.

Ún3.2b^‡(H

)-(H

) ¤á,K•3~êC

>0,¦(5) ª)

(t) = (u

(t),u

(t),η

),÷v

+|∇u



kη

ds+

(s)|

ds≤C

(6)

y²:^u

¦±(5)ª,¿3ΩþÈ©,Œ

|∇u

M(s)ds+





,η



+α|u

−β|u

Ω

f(u

)dx−

Ω

(x)u

dx= 0.

(7)

E(t) = |u

|∇u

M(s)ds+



Ω

f(u

)dx−2

Ω

(x)u

dx.

(8)

¤±(7)ªŒz•

E(t)+α|u

−β|u



,η



= 0.

(9)

Ï•

β|u

= β

Ω

≤β(

Ω

)

·(

Ω

≤β(|u

)·|Ω|

≤

2α

|Ω|.

(10)

DOI:10.12677/pm.2021.1181731551nØêÆ

4Ü†

db^‡(H

)Œ•:



,η



∞

µ(s)

Ω

∇η

(x,s)∇η

(x,s)dxds=

∞

µ(s)



∇η

(x,s)



= 0−

∞

(s)



∇η

(x,s)



ds≥

kη

(11)

2dM(r) b±9^‡(H

)Œ•:

|∇u

M(s)ds≥a|∇u

(12)

Ω

f(u

)dx≥0,

(13)

,,|^YoungØªÚPoinc´areØª,k

Ω

(x)u

dx≤

|∇u

4ε

(14)

Ù¥λ

´Žf−∆1˜AŠ.

é(9)ª'ut3[0,T]þÈ©,¿(Ü(10)-(14),Œ:



−

2ε



|∇u



+δ

kη

ds+α

(s)|

≤

|Ω|+|u



−

2ε



|∇u

+kη

(15)

εvž,-λ= min{1,M

−

2ε

,δ,α}>0,K•3~êC

>0 Œ¦eª¤á:

+|∇u



kη

ds+

(s)|

ds≤C

(16)

y.œ

½n3.3b^‡(H

)–(H

) ¤á,…z

= (u

,η

) ∈H.K•§•3Nf)

z= (u,u

,η

) ÷vu∈L

([0,T];H

(Ω)),u

∈L

∞

([0,T];L

(Ω))∩L

([0,T];L

(Ω))Ú

∈L

([0,T];L

(Ω))).

y²µdÚn3.2,Œ•











([0,T];H

(Ω))¥˜—k.,

∞

([0,T];L

(Ω))∩L

([0,T];L

(Ω))¥˜—k.,

([0,T];L

(Ω)))¥˜—k..

(17)

DOI:10.12677/pm.2021.1181731552nØêÆ

4Ü†

Ïd,•3{u

}f(EP•{u

}),¦











→u3L

([0,T];H

(Ω))¥fÂñ,

→u

([0,T];L

(Ω))¥fÂñ,

→u

∞

([0,T];L

(Ω))¥f*Âñ,

*η

([0,T];L

(Ω)))¥fÂñ.

(18)

duu

(t)3L

([0,T];H

(Ω))¥˜—k.,u

(t)3L

∞

([0,T];L

(Ω))∩L

([0,T];

(Ω))⊂L

([0,T];L

(Ω))¥˜—k.,Œ•u

(t)•H

)¥k.8.qH

);i\

),K•3{u

(t)}f(EP•{u

}),÷v

*u3L

)¥rÂñ.

qd(H

)ÚÚn3.2 Œ

Ω

|f(u

dxdt≤C

(1+|u

(t)|

)dt

≤C

(1+|u

(t)|

2p+2

)dt

≤C

(1+|∇u

(t)|

)dt

≤ρ

Ù¥q•péóê,ρ

>0 •~ê.

K•âþãf(u

)˜—k.5,Œ

f(u

) *χ3L

([0,T];Ω)¥fÂñ.

d,•âu

3L

) ¥rÂñ5,Œ•u

→u(m→∞) 3Q

þA? ?Âñ.2df

ëY5,Œ

f(u

(x,t)) *f(u(x,t)) 3Q

þA??Âñ.

) ⊂H

−1

),KdLebesgueÅ‘È©½nŒ•,•3φ∈L

(0,T;H

(Ω)),¦

lim

m→∞

Ω

f(u

)φdxdt=

Ω

f(u)φdxdt.

Œf(u

) →f(u) 3H

−1

)¥fÂñ,df4••˜5,:χ= f(u).

φ∈C

∞

(0,T;C

∞

(Ω)),k

mtt

,φidτ=

hM(|∇u

)∆u

+∞

µ(s)∆η

(s)ds−h(u

)−f(u

)+g

,φidτ.(19)

DOI:10.12677/pm.2021.1181731553nØêÆ

4Ü†

•gOþªm>z˜‘,¿(ÜYoungØª,H¨olderØª,ÚÚn3.2,Œ

hM(|∇u

)∆u

,φidτ≤

M(|∇u

)|∇u

|∇φ|

dτ

≤C

k∇φk

(0,T;H

(Ω))

(20)

+∞

µ(s)h∆η

(s),φidsdτ≤

+∞

µ(s)|∇η

(s)|

|∇φ|

dsdτ

≤K

|∇φ|

kη

dτ

≤C

k∇φk

(0,T;H

(Ω))

(21)

h−αu

,φidτ≤α

|φ|

dτ≤C

k∇φk

(0,T;L

(Ω))

≤C

k∇φk

(0,T;L

(Ω))

(22)

hβu

,φidτ≤β

|φ|

dτ≤C

k∇φk

(0,T;L

(Ω))

(23)

hf(u

),φidτ≤

kf(u

−1

kφk

dτ≤C

kφk

(0,T;H

(Ω))

(24)

,φidτ≤

|φ|

dτ≤C

kφk

(0,T;L

(Ω))

(25)

¤±(Ü(20)-(25),Œ

mtt

,φidτ≤C

(kφk

(0,T;H

(Ω))

+kφk

(0,T;L

(Ω))

mtt

∈L

(0,T;H

−1

(Ω)).

(26)

•â(17)ªÚ(26) Œ•

→u

)¥rÂñ.

(27)

DOI:10.12677/pm.2021.1181731554nØêÆ

4Ü†

u

→u

¥A??Âñ.u´u

→u

¥A??¤á.qu

→χ

)¥fÂñ.df4••˜5•:u

= χ

,,dug

(x)∈L

(Ω),éu˜ƒx∈Ω,kg

→g(m→∞)´fÂñ,…

g(x) ∈L

(0,T;L

(Ω)).

nþ¤ã,m→∞ž,é?¿ω∈H

(Ω),ξ∈M,k:











,ωi−hM(|∇u|

)∇u,∇ωi−

∞

µ(s)h∇η

(s),∇ωids+αhu

,ωi−βhu

,ωi

+hf(u),ωi= hg(x),ωi.

hη

,ξi

= h−η

,ξi

+hu

,ξi

,η

)÷vÐ©^‡:

du∈L

(0,T;H

(Ω)) Úu

∈L

(0,T;L

(Ω)),¤±ku∈C(0,T;L

(Ω)).Ku

(0)*u(0)

(Ω)¥fÂñ,qdu

(0) *u

(Ω)¥,u(0) = u

(Ω)¤á.

ν(t) ∈C

mtt

,ν(t)idt−

hM(|∇u

)∆u

),νidt−

+∞

µ(s)∆η

(s)ds,ν(t)idt

+α

,ν(t)idt−β

,ν(t)idt+

hf(u

),ν(t)idt=

,ν(t)idt.

z{Œ:

−hu

(0),ν(0)i−

,ν(t)idt+

M(|∇u

)h∇u

,∇ν(t)idt

+∞

µ(s)h∇η

(s),∇ν(t)idsdt+α

,ν(t)idt

−β

,ν(t)idt+

hf(u

),ν(t)idt=

,ν(t)idt.

m→∞ž,k

−hu

,ν(0)i−

,ν(t)idt+

M(|∇u|

)h∇u,∇ν(t)idt

+∞

µ(s)h∇η

(s),∇ν(t)idsdt+α

,ν(t)idt

−β

,ν(t)idt+

hf(u),ν(t)idt=

hg,ν(t)idt.

(28)

DOI:10.12677/pm.2021.1181731555nØêÆ

4Ü†

−hu

(0),ν(0)i−

,ν(t)idt+

M(|∇u|

)h∇u,∇ν(t)idt

+∞

µ(s)h∇η

(s),∇ν(t)idsdt+α

,ν(t)idt

−β

,ν(t)idt+

hf(u),ν(t)idt=

hg,ν(t)idt.

(29)

é'(28)ªÚ(29) ª,k

,ν(0)i= hu

(0),ν(0)i,

¤±h(u

−u

(0)),ν(0)i= 0,u

(0) = u

aqþãÐŠy,Œθ(t)∈C

(0,T;L

;M) (θ(T)=0),¿3˜mL

(0,T;L

(Ω))

+∞

µ(s)hη

(0)−η

,θ(0)ids= 0,

=hη

(0)−η

,θ(0)i= 0,η

t=0

= η

.y.œ

3.2.•˜5ÚëY5

½n3.4b^‡(H

)-(H

) ¤á,K¯K(5)-(6))´•˜…ÙëY•6uÐŠ.

y²:-ω

=(u,u

,η

)∈HÚω

=(v,v

,ξ

(0)=ω

,η

) ∈HÚω

(0) = ω

= (v

,ξ

) ∈Hü‡),…ω= ω

−ω

= (u−v,u

−v

,η

−ξ

ÙÐ©Š•ω

= (u

−v

,η

−ξ

),Kω÷v•§:











−a∆ω−

+∞

µ(s)∆ζ

(s)ds+αu

−αv

−βω

+f(u)−f(v) = b(|∇u|

−|∇v|

= −ζ

+ω

(30)

dM(r)=a+brŒ•M(|∇u|

)=a+b|∇u|

.?^ω

†•§(30) 3L

(Ω) þSÈ,2d

(11)Œ

|ω

|∇ω|



+α((|u

−|v

),ω

)

≤β|ω

+(f(v)−f(u),ω

)+b(|∇u|

−|∇v|

,ω

(31)

e¡é(31)Øªm>1‘?1?n,w,

2(p+1)

p+1

2(p+1)

= 1¤á,2Šâi\

DOI:10.12677/pm.2021.1181731556nØêÆ

4Ü†

½nÚYoungØª,Kk:

(|f(v)−f(u)|,ω

) ≤C

Ω

(1+|u|

+|v|

)|ω||ω

|dx

≤C[1+|u|

2(p+1)

+|v|

2(p+1)

]·|ω|

2(p+1)

·|ω

≤C[1+|∇u|

+|∇v|

]·|∇ω|

·|ω

≤C[|∇ω|

+|ω



(32)

e¡é(31)Øªm>1n‘?1?n,k:

(|∇u|

−|∇v|

,ω

) ≤≤(|∇u|

+|∇v

|)·|(|∇(u−v)|

)|·(

Ω

|ω

dx)

≤C

|∇ω|

·|ω

≤C

[|∇ω|

+|ω



(33)

duu

•üN4O¼ê,

Ω

(|u

−|v

)·ω

dx≥0.

nþ,(31)ªŒz•

[|ω

+a|∇ω|

+kζ

] ≤C[|ω

+a|∇ω|



eλ

= min{a,1},dGronwallÚn,k:

kω

−ω

≤Ce

kω

−ω

…=ω

= ω

ž,Ò¤á.¤±Òy²)•˜5,ÚéÐŠëY•65.y.œ

—

2018JJ2416)]Ï"

ë•©z

33(4):894-904.

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withLocallyDistributedFrictionalandViscoelasticDamping.CommunicationsinNonlinear

ScienceandNumericalSimulation,92,ArticleID:105472.

https://doi.org/10.1016/j.cnsns.2020.105472

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4Ü†

[3]Liao,M.,Guo,B.andZhu,X.(2020)BoundsforBlow-UpTimetoaViscoelasticHyperbolic

Equation ofKirchhoﬀType withVariable Sources.Acta ApplicandaeMathematicae, 170, 1-18.

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[6]Lazo,P.P.D.(2011)GlobalSolutionforaQuasilinearWaveEquationwithSingularMemory.

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DOI:10.12677/pm.2021.1181731558nØêÆ