一类具有对数非线性项的分数阶阻尼波方程的局部适定性 Local Well-Posedness for a Classof Fractional Damped Wave Equations with Logarithmic Nonlinearity

doi:10.12677/AAM.2023.124152

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AdvancesinAppliedMathematicsA^êÆ?Ð,2023,12(4),1474-1482

PublishedOnlineApril2023inHans.https://www.hanspub.org/journal/aam

https://doi.org/10.12677/aam.2023.124152

˜aäkéêš‚5‘©ê

{ZÅ•§ÛÜ·½5

 AAA

2²ŒÆêÆ†&E‰ÆÆ,2À2²

ÂvFÏµ2023c313F¶¹^FÏµ2023c49F¶uÙFÏµ2023c419F

Á‡

©Ì‡•Ääkéêš‚5‘©ê{ZÅÄ•§u

+(−∆)

u+(−∆)

= uln|u|Ð>Š

¯K§ Ù¥s∈(0,1)"Žf(−∆)

•©êLaplaceŽf§Cc5§TŽf¤•ÔnÆ!7Kê

Æ!6NÄåÆÆ‰+•¥ïÄ9:"©3?¿Ð©Uþe§|^Galerkin%C{ÚØ

Nn§y²T•§)ÛÜ·½5"

'…c

{ZÅÄ•§§©êLaplaceŽf§éêš‚5‘§ÛÜ·½5

LocalWell-PosednessforaClass

ofFractionalDampedWaveEquations

withLogarithmicNonlinearity

LingnaLin

SchoolofMathematicsandInformationScience,GuangzhouUniversity,GuangzhouGuangdong

Received:Mar.13

,2023;accepted:Apr.9

,2023;published:Apr.19

,2023

©ÙÚ^: A.˜aäkéêš‚5‘©ê{ZÅ•§ÛÜ·½5[J].A^êÆ?Ð,2023,12(4):

1474-1482.DOI:10.12677/aam.2023.124152

 A

Abstract

Inthispaper,wemainlydealwiththeinitial-boundaryvalueproblemforthefrac-

tionaldampedwaveequationsu

+ (−∆)

u+ (−∆)

=uln|u|,wheres∈(0,1).The

operator(−∆)

isthefractionalLaplaceoperator.Inrecentyears,thisoperatorhas

becomearesearchhotspotinphysics,ﬁnancialmathematics,ﬂuiddynamicsandoth-

erdisciplines.Atthearbitraryinitialenergylevels,thelocalwell-posednessofweak

solutionstoaboveproblemisprovedbyusingGalerkinapproximationmethodand

contractionmappingprincipleundersomecertainconditions.

Keywords

DampedWaveEquations,FractionalLaplaceOperator,LogarithmicNonlinearity,

LocalWell-Posedness

2023byauthor(s)andHansPublishersInc.

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

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

1.Úó

3Ln›c¥,©êuÐ•§®¤•ˆ‡‰Æ+•¥-‡g,.,Úå¯õÆö

'5.3êÆþ,©êuÐ•§Œ±^žm©êê5£ã,X*Ñ¯K!õš0Ÿ•§!tµ

y–¶•Œ±^˜m©êê5£ã,XþfåÆ!À/Ô*Ñ!%>DÂ.

Š˜J´,Cc5©ê.Ê.dŽf(−∆)

3ÔnÆ!7KêÆ!6NÄåÆ•¡k

•\2•A^.lêÆÝ5w,¹kùaŽf ‡©•§Œ^5£ãNõE,y–, ¿¼

´LnØ¤J(ë©z[1–3]9Ùë•©z).éêš‚5‘uln|u|3ÔnÆÓäkN

õkïÄ,§Œ±g,/^u‡é¡|Ø,þfåÆÚØÔn[4,5]¥.Galerkin•{3 ‡©•

§ïÄ¥A^š~2•[6–8],3•Ä ‡©•§)•35ž§·‚Ï ~¬|^d•{òE,

 ‡©•§=z•~‡©•§?1¦).Ø Nn3y²•§)•3•˜5ž§Óä

kÞv”-Š^.

©·‚•ÄXe‘kéêš‚5‘©ê{ZÅÄ•§Ð>Š¯K

DOI:10.12677/aam.2023.1241521475A^êÆ?Ð

 A







+(−∆)

u+(−∆)

= uln|u|,(x,t) ∈Ω×(0,T),

u(x,t) = 0,(x,t) ∈(R

\Ω)×[0,T),

u(x,0) = u

(x),u

(x,0) = u

(x),x∈Ω,

(1.1)

Ù¥Ω´R

(n≥2s)¥‘k1w>.∂Ωk .«•,s∈(0,1);©êLaplaceŽf(−∆)

dXeÛ

ÉÈ©½Â

(−∆)

u(x) = −

C(n,s)

u(x+y)+u(x−y)−2u(x)

|y|

n+2s

dy,x∈R

Ù¥C(n,s)=



1−cosξ

|ξ|

n+2s

dξ



−1

´˜‡8˜z~ê.3ùp,·‚•Ä½Â3Ωþ©ê

LaplaceŽf(−∆)

,´Ùéu¤½Â*Ð)|ŽX¯K,ù¿›X

(−∆)

u(x) = −

C(n,s)

u(x+y)+u(x−y)−2u(x)

|y|

n+2s

dy,x∈Ω.

Ïd,Ð>Š¯K(1.1))|ŽX êâ´3R

\ΩþŠ•NÈå‰Ñ,Ø==´3∂Ωþ,ù

†Žf(−∆)

šÛÜA˜—.

©‰ÑÐ>Š¯K(1.1)3?¿Ð©Uþe)ÛÜ·½5y²,ùò•?˜ÚïÄ¯

K(1.1))Û•35!ìC1•Ú»JøÄ:.

2.ý•£9Ì‡(J

3ù˜!,Äk•ã†Ð>Š¯K(1.1)ƒ'©êSobolev˜m½ÂÚ‰ê5Ÿ,•õ

[!Œ±ë©z[9].

·‚Q=R

\O,Ù¥O=(CΩ) ×(CΩ)⊂R

,CΩ=R

\Ω,K˜mX:R

→R´˜

‡LebesgueŒÿ‚5˜m,…÷v•›3Ωþu∈Xáu˜mL

(Ω)§=¦

|u(x)−u(y)|

|x−y|

n+2s

dxdy<∞,

Ù˜mX‰ê½Â•

kuk



Ω

|u(x)|

dx+

|u(x)−u(y)|

|x−y|

n+2s

dxdy



.(2.1)

?˜Ú§½Â©êSobolev˜mX

•

= {u∈X:u= 0 a.e.inR

\Ω}.

Dƒ‰ê

kuk



|u(x)−u(y)|

|x−y|

n+2s

dxdy



,(2.2)

DOI:10.12677/aam.2023.1241521476A^êÆ?Ð

 A

…SÈL«•

(u,v)

(u(x)−u(y))(v(x)−v(y))

|x−y|

n+2s

dxdy.

,,éu?¿u∈X

,·‚kXe'Xª¤á

((−∆)

u,v)

)



(−∆)

u,(−∆)



)

C(n,s)(u,v)

,(2.3)

v= už,

((−∆)

u,u)

)

= k(−∆)

)

C(n,s)kuk

.(2.4)

e5§·‚‰Ñ˜mX

→L

∗

Sobolevi\½n,ÙQãXe.

Ún2.1.s∈(0,1)…u∈X

.@o,•3˜‡†nÚsƒ'•Zi\~êB

n−2s

,¦

kuk

∗

(Ω)

= kuk

∗

)

≤B

n−2s

|u(x)−u(y)|

|x−y|

n+2s

dxdy= B

n−2s

kuk

,(2.5)

Ù¥2

∗

n−2s

©Ì‡(ØXe:

½n2.1.(ÛÜ·½5)eu

(x)∈X

(x)∈L

(Ω),K•3˜‡žmT>0,¦¯K(1.1)•3

˜‡•˜f)u,÷v

u∈L

∞

([0,T],X

)…u

∈L

∞

([0,T],L

)∩L

([0,T],X

3.ÛÜ·½5

ù˜!·‚ò3?¿Ð©Uþe§y²¯K(1.1)ÛÜ)•3•˜5.••Bå„,˜‡

½T>0,½ÂXe˜m

D:=



u∈L

∞

([0,T],X

)…u

∈L

∞

([0,T],L

)∩L

([0,T],X

)



,(3.1)

…Dƒ‰ê

kuk

=max

t∈[0,T]



C(n,s)ku(t)k

+ku

(t)k



.(3.2)

d,·‚‰ÑXe'…5Ún,§éy²!Ì‡(ØåXš~-‡Š^.

Ún3.1.éu∀T>0Úu∈D,•3•˜˜‡

z∈D…z

∈L

∞

([0,T],Y

(3.3)

DOI:10.12677/aam.2023.1241521477A^êÆ?Ð

 A

…÷vXe‚5Ð>Š¯K











+(−∆)

z+(−∆)

= uln|u|,(x,t) ∈Ω×[0,T),

z(x,t) = 0,(x,t) ∈(R

\Ω)×[0,T),

z(x,0) = u

(x),z

(x,0) = u

(x),x∈Ω,

(3.4)

Ù¥Y

´X

éó˜m,u

(x)Úu

(x)®3Ð>Š¯K(1.1)¥½Â.

y².Äk,·‚ò|^IOGalerkin•{,y²z•35.-{ω

(x)}

∞

k=1

´X

¥(−∆)

˜|Ä

¼ê,÷v

(−∆)

(x) = λ

(x),

Ù¥λ

´éAAŠ.·‚½Â

k=1



Ω



k=1



Ω



Ïd,k→∞ž,3X

þku

→u

,3L

(Ω)þku

→u

.éu¤kk≥1,·‚Œ±é˜¼

êd

,...,d

∈C

[0,T],¦Cq)z

(x,t)÷v¯K(3.4).z

(x,t)Œ¤:

(x,t) =

k=1

(t)ω

,(3.5)

…÷vXe‚5¯K







Ω

(¨z

+(−∆)

˙z

−uln|u|)ηdx= 0,(x,t) ∈Ω×[0,T),

(0) = u

,˙z

(0) = u

,x∈Ω,

(3.6)

Ù¥˙z

,η∈X

.3¯K(3.6)¥,-η= ω

,Œ

(¨z

,ω

) =

(t),(3.7)

((−∆)

,ω

) = λ

(t),(3.8)

((−∆)

˙z

,ω

) = λ

(t).(3.9)

ù,·‚Œ±˜‡äk™•¼êd

‚5~‡©•§ÐŠ¯Kµ







(t)+λ

(t) =

Ω

u(t)ln|u(t)|ω

dx,(x,t) ∈Ω×[0,T),

(0) =

Ω

dx,

(0) =

Ω

dx,x∈Ω.

(3.10)

DOI:10.12677/aam.2023.1241521478A^êÆ?Ð

 A

Ïd,þã¯Ké¤kkÑŒ˜‡•˜ÛÜ)d

∈C

[0,T],ùÒ¿›X•3•˜d

ªf(3.5)½Âz

÷v(3.6).e5,‰Ñ

Ω

(uln|u|)

dxO.ÏL†OŽ†Ún2.1¥

Sobolevi\,Œ

Ω

(uln|u|)

dx=

{x∈Ω;|u(x)|≤1}

(uln|u|)

dx+

{x∈Ω;|u(x)|>1}

(uln|u|)

≤e

−2

|Ω|+



n−2s



{x∈Ω;|u(x)|>1}

n−2s

≤e

−2

|Ω|+



n−2s



kuk

n−2s

≤e

−2

|Ω|+



n−2s



n−2s

kuk

n−2s

(3.11)

Ù¥B

n−2s

´Sobolevi\X

→L

n−2s

(Ω)•Z~ê.

?˜Ú/,éu¤km≥1,3(3.6)¥η=˙z

(t),¿…3[0,t]⊂[0,T]þ?1È©,(Üª

f(3.11),·‚Œ±

k˙z

C(n,s)kz

+C(n,s)

k˙z

dτ

≤C(n,s)ku

+ku

+C+

k˙z

dτ

≤C+



k˙z

C(n,s)kz

+C(n,s)

k˙z

dς



dτ.

(3.12)

|^GronwallØª,´

k˙z

C(n,s)kz

+C(n,s)

k˙z

dτ≤Ce

.(3.13)

,,ò(3.6)¥1˜‡•§ü>ÓžØ±kηk

<¨z

,η>

kηk

(uln|u|,η)−((−∆)

,η)−((−∆)

˙z

,η)

kηk

d(3.11),(3.13)ÚH¨olderØª,Œ

<¨z

,η>

kηk

≤C(T).(3.14)

éuη∈X

\{0},(3.14)ü>Óžþ.,k

k¨z

≤C(T).(3.15)

¤±,•3˜‡S{z

},¦

DOI:10.12677/aam.2023.1241521479A^êÆ?Ð

 A

}3L

∞

([0,T],X

)þk.;

{˙z

}3L

∞

([0,T],L

(Ω))∩L

([0,T],X

)þk.;

{¨z

}3L

∞

([0,T],Y

)þk..

Ïd,Œ•3)z∈D∩C

([0,T],Y

)÷v¯K(3.4).

òü‡•§ƒ~,¿…†v

−w

‰SÈ,Œ

−w

C(n,s)kv−wk

+C(n,s)

−w

dτ= 0,

ù`²v≡w.Ún3.1y²..

y3,·‚‰ÑÐ>Š¯K(1.1)ÛÜ·½5y².

½n2.1y².R

C(n,s)ku

+ku

,…

= {u∈D: u(x,0) = u

(x,0) = u

,kuk

≤R},∀T>0.(3.16)

ÏLÚn3.1,·‚•éu∀u∈B

,•3•˜)z¦¯K(3.4)¤á.3ùp,·‚òy² éu

˜‡·T≥0,Ψ´˜‡Ø N,=Ψ(B

) ⊂B

Äk,ò¯K(3.4)¥1˜‡•§3Ω×[0,t)þ†z

‰SÈ,

C(n,s)kzk

+C(n,s)

dτ

C(n,s)ku

+ku

Ω

uln|u|z

dxdτ,

(3.17)

Ù¥z= Ψ(u)´u∈B

½ž,¯K(3.4)éA).|^Cauchy-SchwarzØªÚYoungØª,

Ω

uln|u|z

(τ)dxdτ≤2

kuln|u|kkz

kdτ≤

(2kuln|u|k

)dτ,(3.18)

?˜Ú(Ü(3.11),´

Ω

uln|u|z

dxdτ≤CT(R

n−2s

+1)+

C(n,s)

dτ,∀t∈(0,T].(3.19)

d(3.17)Ú(3.19),3[0,T]þ•ŒŠ,Œ

kzk

≤

+CT(R

n−2s

+1).

(3.20)

Tvž,·‚kkzk

≤R,ù`²Ψ(B

) ⊂B

DOI:10.12677/aam.2023.1241521480A^êÆ?Ð

 A

bB

¥•3ü‡¼êw

Úw

.-z

= Ψ(w

),z

= Ψ(w

),¿…z= z

−z

.ò¯K(3.4)¥

1˜‡•§†z

idτ+

C(n,s)

,z)

dτ+

C(n,s)

)

dτ

Ω

ln|w

|−w

ln|w

|)z

dxdτ.

(3.21)

|^Lagrange½n,ÏL†OŽ,k

C(n,s)kzk

+C(n,s)

dτ

= 2

Ω

−w

)(|ξ|+1)z

dxdτ

≤C

−w



C(n,s)kzk

+C(n,s)

dτ



dτ,

(3.22)

Ù¥0 ≤|ξ|≤ln|w

|.dGronwallØª,Œ



C(n,s)kzk

+C(n,s)

dτ



≤CTkw

−w

Ïd,·‚uy

kΨ(w

)−Ψ(w

= kzk

≤C

−w

≤σkw

−w

(3.23)

ù§·‚Tvž§oŒ±(• 3σ<1.ŠâØ Nn,Œy²Ð>Š¯K(1.1)•

3•˜f).y..

4.o(†Ð"

{ZÅÄ•§u

+(−∆)

u+(−∆)

1•Ú»ƒ'5Ÿ.

ë•©z

[1]Bisci,G.M.,Radulescu,V.D.andServadei,R.(2016)VariationalMethodsforNonlocalFrac-

tionalProblems.In:EncyclopediaofMathematicsandItsApplications,Vol.162,Cambridge

UniversityPress,Cambridge.

DOI:10.12677/aam.2023.1241521481A^êÆ?Ð

 A

[2]Bucur,C.andValdinoci,E.(2016)NonlocalDiﬀusionandApplications.In:LectureNotesof

theUnioneMatematicaItaliana,Vol.20,Springer,Cham.

https://doi.org/10.1007/978-3-319-28739-3

[3]Dipierro,S.,Medina,M.andValdinoci,E.(2017)FractionalEllipticProblemswithCritical

Growthinthe Wholeof R

.In:Publications oftheScuola NormaleSuperiore, Vol.15, Edizioni

dellaNormale,Pisa.

[4]Barrow, J.D. andParsons, P. (1995)Inﬂationary Models with LogarithmicPotentials. Physical

ReviewD,52,5576-5587.https://doi.org/10.1103/PhysRevD.52.5576

[5]Enqvist,K.and McDonald,J. (1998)Q-Balls andBaryogenesis in the MSSM. PhysicsLetters,

425,309-321.https://doi.org/10.1016/S0370-2693(98)00271-8

[6]Liu,W.J.,Yu,J.Y.andLi,G.(2021)GlobalExistence,ExponentialDecayandBlow-Upof

Solutions for a Class ofFractional Pseudo-Parabolic Equations withLogarithmic Nonlinearity.

Discrete&ContinuousDynamicalSystemsS,14,4337-4366.

https://doi.org/10.3934/dcdss.2021121

[7]Lian,W.andXu,R.Z.(2019)GlobalWell-Posedness ofNonlinearWaveEquationwithWeak

andStrongDampingTermsandLogarithmicSourceTerm.AdvancesinNonlinearAnalysis,

9,613-632.https://doi.org/10.1515/anona-2020-0016

[8]Xu,R.Z.,Lian,W.,Kong,X.K.andYang,Y.B.(2019)FourthOrderWaveEquationwith

NonlinearStrain andLogarithmicNonlinearity.AppliedNumerical Mathematics, 141, 185-205.

https://doi.org/10.1016/j.apnum.2018.06.004

[9]DiNezza,E.,Palatucci,G.andValdinoci,E.(2012)Hitchhiker’sGuidetotheFractional

SobolevSpaces.BulletindesSciencesMath´ematiques,136,521-573.

https://doi.org/10.1016/j.bulsci.2011.12.004

DOI:10.12677/aam.2023.1241521482A^êÆ?Ð