conditions Kelvin–Voigt type damping under optimal geometric Stabilization of the wave equations with localized C.R. Acad. Sci. Paris, Ser.I

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Partial differential equations/Optimal control

Stabilization of the wave equations with localized Kelvin–Voigt type damping under optimal geometric conditions

Stabilisation d’une équation des ondes avec un amortissement de type Kelvin–Voigt localisé sous conditions géométriques optimales

Rayan Nasser

^a,c

, Nahla Noun

^b

, Ali Wehbe

^b

aUniversitélibanaise,EDST&Hadath,Beyrouth,Liban

bUniversitélibanaise,Facultédessciences1etEDST&Hadath,Beyrouth,Liban cUniversitédeBretagneoccidentale,France

a rt i c l e i n f o a b s t ra c t

Articlehistory:

Received6October2018

Acceptedafterrevision21January2019 Availableonline21March2019 PresentedbytheEditorialBoard

The purpose of this note is to investigate the stabilization ofthe waveequation with Kelvin–Voigt damping in a bounded domain. Damping is localized via a non-smooth coefficientinasuitable subdomain.Weproveapolynomial stabilityresult inanyspace dimension, provided that the damping regionsatisfies some geometric conditions. The mainnovelty ofthisnote isthat the geometricsituations coveredhere are richerthan thatconsideredin[25],[22],[16] andincludeinparticularanexamplewherethedamping regionisnotlocalizedinaneighborhoodofthewholeorapartoftheboundary.

©^{2019Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

ré s u m é

Nous nous intéressons à l’étude de la stabilisation d’une équation des ondes avec un amortissementdetypeKelvin–Voigtdansundomaineborné.L’amortissementestlocalisé viauncoefficientsingulier dans unepartie du domaine.Nous montrons unrésultat de stabilisationpolynomialeentoutedimensiond’espacedèsquelarégiond’amortissement satisfaitcertainesconditionsgéométriques. Laprincipalenouveautédecettenoteestque lessituationsgéométriquescouvertesicisontplusrichesquecellesconsidéréesdans[25], [22], [16] et incluent notamment un exemple où la région d’amortissement n’est pas localiséedansunvoisinagedelatotalitéoud’unepartiedelafrontière.

©^{2019Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

E-mailaddresses:[email protected](R. Nasser),[email protected](N. Noun),[email protected](A. Wehbe).

https://doi.org/10.1016/j.crma.2019.01.005

1631-073X/©^{2019Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

1. Introduction

Localviscoelasticdampingisanaturalphenomenonof bodiesarisingfromasolidthathaveonepartmadeofviscoelastic material,andtheothermadeofelasticmaterial.Let⊂R^{N beanonemptyboundedopensetwithboundary}ofclassC². WeconsiderthewaveequationwithlocallydistributedKelvin–Voigt-typedampinggiveninthefollowingequation:

⎧ ⎪

⎨

⎪ ⎩

ρ (

x

)

utt

(

x

,

t

) =

^div

(

a

(

x

) ∇

^u

+

^b

(

x

) ∇

^ut

)

in

× R

⁺

,

u

(

x

,

t

) =

^{0 on}

× R

⁺

,

u

(

x

,

0

) =

^u0

(

x

),

ut

(

x

,

0

) =

^u1

(

x

)

in

,

(1)

whereweassumethatthecoefficientfunctions

ρ

,a,b∈^L∞()and

ρ

(x)≥

ρ

0>0,a(x)≥^a0>0 andb(x)≥^{0 forallx}∈. In1988,F.HuangprovedthatwhentheKelvin–Voigtdampingdiv(b(x)∇^ut)isgloballydistributed,i.e. b(x)≥^b0>0 for almostevery x in,thecorresponding semigroupofsystem(1) isnot onlyexponentiallystable, butalsoisanalytic (see [8]).Thus, Kelvin–Voigtdampingisstronger thanthe viscous dampingb(x)u_t in thiscase.Indeed, in[10], itwas proved that the semigroup corresponding to the system of wave equations withglobal viscous damping is exponentially stable butnotanalytic. However,thisresultisstilltrueifviscous dampingislocalized;viaa smoothora non-smoothdamping coefficient,ina suitablesubdomain satisfyingtheGeometric ControlCondition(GCCinshort)introduced byC. Bardos,G.

Lebeau,andJ.Rauchin[2] (see also[10]).Nevertheless,whenviscoelastic dampingisdistributedlocally, thesituationis moredelicateandsuchcomparisonbetweenviscousandviscoelasticdampingisnotvalidanymore.Infact,in1998,K.Liu andZ.Liuconsidereda one-dimensionalwave equation withKelvin–Voigtdampingdistributedlocallyon anysubinterval oftheregion occupiedbythe beam,wherethe dampingcoefficientisthe characteristicfunctionofthe subinterval.They proved that the semigroup associated withthe equation for the transversalmotion of thebeam is exponentially stable, althoughthesemigroupassociatedwiththeequationforthelongitudinalmotionofthebeamisnot(see[13]).Thisshows thatKelvin–VoigtdampingdoesnotobeytheGCC.Thissurprisingresult,duetothediscontinuityofthematerialsandthe unboundednessofviscoelasticdamping,motivatedthestudyofelasticsystemswithlocalKelvin–Voigtdamping.Later,inthe one-dimensionalcase,itwasfoundthatthesmoothnessofthedampingcoefficientattheinterfaceisacriticalfactorforthe stabilityandtheregularityofthesolutions(see[7,14,15,17,18,23]).However,thereareonlyasmall numberofpublications onthecorresponding N-dimensionalcase. In2006, K.LiuandB.Rao consideredthisproblemintheN-dimensionalspace where the damping region is a neighborhood (in ) of the entire boundary (see [16]). They proved that the energy of the system goes exponentially to zero as t goes to infinity for all usual initial data by assuming that the damping coefficient b satisfies b∈^C1,1(), b∈^L∞() and|∇^b(x)|²≤^M0b(x) for almost every x in , where M0 is a positive constant. Also in [19], under the same assumption on b, S. Nicaise andC. Pignotti established the exponential stability of the wave equation with localKelvin–Voigt damping localized around a part of the boundary and an extra boundary dampingwithtimedelaywheretheyaddedanappropriategeometric condition(section3.2(Q4)).Lateron,M.Cavalcanti, V. Cavalcanti and L. Tebou showed the exponential decay of the energy of a wave equation with two types of locally distributedmechanisms;africtionaldampingandaKelvin–Voigt-typedampingwherethelocationofeachdampingissuch that noneofthem aloneisableto exponentiallystabilizethesystem(see [6]).Underanappropriate geometric condition (PMGC) on a subset

ω

of⊂R^{N where the} dissipation is effective, they proved that the energyof the systemdecays polynomially astype ¹_t intheabsence ofregularityofthe Kelvin–Voigtdampingcoefficientb. However,they established exponential stability when thiscoefficient is smooth. In [1], K.Ammari, F. Hassine,and L.Robbianio considered a wave equationwithKelvin–Voigtdampinglocalizedinasubdomain

ω

farawayfromtheboundarywithoutgeometricconditions.

Theyestablisheda logarithmic energydecayrateforsmooth initial data.Ontheother hand,in[22] L.Teboustudiedthe stabilizationofthewaveequationwithKelvin–Voigtdamping.Heestablishedpolynomialenergydecayoftype ¹_t provided that the damping region is localized in a neighborhood of a partof the boundary and verifies the Piecewise Multiplier Geometric Condition(PMGCinshort) introducedbyK.Liu[12].Moreover,Q. Zhangin[25] consideredthewave equation withKelvin–Voigtdampinginanonemptyboundedconvexdomain withpartition =1∪2 wheretheviscoelastic dampingislocalizedin1.Undertheconditionthatthedampingcoefficientbisnon-smooth,sheestablishedapolynomial energydecayrateoftype ¹_t forsmoothinitialdatainthefollowingtwocases:(1)thedampingregion1isaneighborhood oftheentireboundary of ; (2)⊂R^N (N=2 or 3),∂1 and∂2 are eitherconvexcurvilinearpolygons orcurved plane polyhedra, thedampingregion 1 isa neighborhoodofa part1 = ∅ ^{oftheboundary} andm(x)·

ν

2≤^{0 where} m(x)=^x−^x0 forx0fixedinR^N(N=²,3)forallx∈2=\1.So,severalimportantgeometricsituationsarenotcovered bytheprevious papers(see forinstanceFig.1-c,Fig.2,Fig.3)andtheproblemoftheenergydecayrateisstillopen. So, ouraimistoanswerthisopenproblem.

Inthisnote, weconsiderthestabilizationofthewaveequation withKelvin–Voigtdampinginaboundeddomain of classC²withnon-smoothdampingcoefficient.Thesystemisgivenby(1).Weestablishapolynomialenergydecayestimate of type ¹_t for smooth initial data provided that the damping coefficient b satisfies the localization condition (LA) (see below)andthedampingregion

ω

satisfiesoneofthegeometricconditions(A1)or(A2)(seebelow).Thefrequencydomain approachandthepiecewisemultiplier methodareused.Toourknowledge,theresultofTheorem 3.5isnew.Indeed,the geometricsituationscoveredbythistheoremarericherthanthatconsideredin[25],[22],[16] andincludeinparticularan exampleofdampingregionfarawayfromtheboundary.

2. Well-posednessandstrongstability

LetusdefinetheenergyspaceH=^H₀¹()×^L2()equippedwiththefollowinginnerproduct

((

u

,

v

), (˜

^u

,

^v

˜ ))

_H

=

(

a

∇

^u

·∇ ˜

^u

+ ρ

v^v

˜ )

dx

.

Let(u,u_t)bearegularsolutiontothesystem(1).Itsassociatedenergyisdefinedby E

(

t

) =

(

a

|∇

^u

|

²

+ ρ _|

ut

|

²

)

dx

.

(2)

Astraightforwardcomputation givesthat E

(

t

) = −

b

(

x

)|∇

^ut

|

^2dx

≤

⁰

.

(3)

Consequently,system(1) isdissipativeinthesensethatitsenergyisnon-increasingwithrespecttot.SettingU=(u,u_t), system (1) maybe recastas: U=^{AU in} (0,+∞),U(0)=(u0,u1),where theunbounded operator A:^D(A)−→H ^is givenby

D

(

A

) =

(

u

,

v

) ∈

H

:

^v

∈

^H₀¹

(),

div

(

a

∇

^u

+

^b

∇

^v

) ∈

^L2

()

,

A

(

u

,

v

) =

^v

,

¹

ρ

^div

⁽

^a

^∇

^u

⁺

^b

^∇

^v

⁾

.

Notingthatduetothefactthatb(x)≥^{0,theoperator}A^ism-dissipativeinH^{andgeneratesaC}0-semigroupofcontractions e^tA.So,system(1) iswell-posedinH^(see[16]).

Inaddition,if

ω

= ∅^{andb satisfiesthefollowing}localizationassumption

∃

^b0

>

0

:

^b

(

x

) ≥

^b0

∀

^x

∈ ω ,

(LA)

thensystem(1) isstronglystable(see[1],Theorem2.2)i.e.

t→+∞lim

^etA

(

u₀

,

u₁

) =

⁰

, ∀ (

u₀

,

u₁

) ∈

H

.

So,ouraimistostudytheenergydecayrate.

3. Polynomialenergydecayrate

Q.Zhangprovedin[24] thatsystem(1) isnotuniformly(exponentially)stableinanygeometry.So,itisnaturaltohope forapolynomialstabilityundersomeconsiderationsthatrepresentthemaingoalofthisnote.

Beforestatingourresults,werecalltheGeometricControlCondition(GCCinshort)introducedbyJ.RauchandM.Taylor in[21] formanifoldswithoutboundariesandbyC.Bardos,G.LebeauandJ.Rauchin[2] (seealso[10])fordomainswith boundaries.

Definition3.1.Fora subset

ω

of and T>0,we shallsaythat (

ω

,T) satisfiestheGeometric ControlConditionifevery geodesictravelingatspeedoneinmeets

ω

intimet<T.

Wealsointroducethefollowinggeometriccondition:

Definition3.2.Forasubset

ω

of,weshallsaythat

ω

satisfiesStrictlytheGeometricControlCondition(SGCC inshort)if thereexistsanopensubset

ω

includedstrictlyin

ω

(i.e.

ω

_⊂

ω

)andsatisfyingthe GCC.

Forthestudyoftheenergydecayrate,weneedthefollowinggeometricassumptions:

(A1) theopensubset

ω

verifiestheGCCandmeas(

ω

_∩)>0, (A2) theopensubset

ω

verifiestheSGCC.

Remark3.3.Itiseasy toseethat,if

ω

verifiesthe SGCC,thenitverifiesthe GCC.Theconverseofthisimplicationisfalse (seeFig.1-c).

Thereareseveralgeometriesthatverifythepreviousassumptions.Forexample:

Fig. 1.Elastic–viscoelastic waves interaction model satisfying the assumption (A1).

Fig. 2.A model satisfying assumption (A2).

Fig. 3.A model satisfying both (A1) and (A2).

Remark3.4.The PMGC introduced in [12] is a generalization of the -condition introduced in [11] and is much more restrictivethanthe GCC.Forexample,inFig.1,weconsiderthecasewhereisadiskandwedrawthreedifferentsubsets in.The-conditionisonlysatisfiedby

ω

0.The PMGC issatisfiedby

ω

0 and

ω

1.However,

ω

2doesnotsatisfyeitherthe PMGC orthe-condition.Finally,the GCC issatisfiedbythethreedifferentsubsetsof.

Now,weareinpositiontostateourmainresult.

Theorem3.5(Polynomialdecayrate).Assumethatcondition(L A)holds.Assumealsothatassumption(A1)orassumption(A2) holds.Then,forallinitialdataU0∈^D(A),thereexistsaconstantC>0independentofU0suchthattheenergyofsystem(1) satisfies thefollowingestimation

E

(

t

,

U

) ≤

C1

t

||

U0

||

²_D(A)

, ∀

t

>

0

.

(4)

Remark3.6.i)The resultofTheorem 3.5generalizesthat of[16],[22],and[25].Indeed,thegeometric situationscovered bythistheoremarericherthanthatconsideredinthepreviousreferences.Inaddition,unliketheresultofTheorem4.1in [25],ourresultholdsforallN≥^{2 andfornon-convexdomains.}

ii)Itisunknownwhetherthepolynomialdecayrateobtainedin(4) isoptimalinthesensethat,forany

ε

>0,wecan notexpect thedecayrateoftype _t1¹+ε forall initialdataU0∈^D(A).From ourpoint ofview,theenergydecayrate(4) is notoptimal,andweconjectureanoptimaldecayoftype _t¹₂.

IdeasforprovingTheorem3.5undertheassumption(A1). For the proof of Theorem 3.5, we use a frequency-domain approach;namely,weuseTheorem2.4of[5] (seealso[3,4,17])thatwepartiallyrecall.

Theorem3.7.Let(T(t))t≥⁰beaboundedC0-semigrouponaHilbertspaceH withgeneratorA suchthatiR⊂

ρ

(A).Thenforafixed >0thefollowingconditionsareequivalent

(

^is

−

^A

)

⁻¹

=

^O

(|

^s

|

),

s

→ ∞,

⁽⁵⁾

T

(

t

)

A⁻¹

=

O

(

t^−1/

),

t

→ ∞.

(6)

Since the resolvent ofthe operator A ^{isnot compact in the energyspace} H^{(see [16])and0}∈

ρ

(A)^{, then to prove} iR⊂

ρ

(A) isequivalent toprove that (iβI−A) isbijectivein theenergyspaceH ^forall β∈R^{.Thislast isproven in} [1] accordingto aunique continuationtheoremandFredholm’salternative. Then,theproof ofTheorem 3.5isreducedto show thatcondition (5) holdswith=^{2.Thisischeckedby usinga}contradictionargument.Indeed,assumethat itdoes nothold,thenthereexistasequenceβn∈R^andasequence(un,vn)∈^D(A)suchthat

| β

_n

| → +∞ , || (

u_n

,

v_n

) ||

H

=

¹

,

(7)

β

_n²

(

i

β

nI

−

A

)(

un

,

vn

) = (

fn

,

gn

) →

^{0 in}H

.

(8)

Ouraimistoshowthat||(^un,v_n)||H→^{0.Thisconditionpermitstoconcludea}contradictionwith(7).Theproofisdivided intoseveralsteps.

Step1.(Localasymptoticestimationofβnv_n).First,using(7) and(8),wehavethefollowingestimation

|β

nu_n

|

^2dx

=

^O

(

1

).

(9)

Multiplying(8) byUn=(un,vn)inH,weget Re

(β

_n²

(

i

β

nI

−

A

)

U_n

,

U_n

)

_H

= −

b

|β

n

∇

^vn

|

²

=

^o

(

1

).

(10)

Itfollowsfromthelocalizationassumption(LA)that

ω

|β

n

∇

vn

|

²dx

=

o

(

1

).

(11)

Sinceassumption(A1)holds,thenusingPoincaré’sinequalityandequation(11),weobtain

ω

| β

_nv_n

|

^2dx

=

^o

(

1

).

(12)

Theaimofstep1isachieved.

Now,writingequation(8) inadetailedform:

i

β

nu_n

−

^vn

=

^fn

β

_n²

→

^{0 inH1}₀

(),

(13)

i

β

nvn

−

¹

ρ

^div

⁽

^a

^∇

^un

⁺

^b

^∇

^vn

^{) =}

g_n

β

_n²

→

^{0 inL2}

().

(14)

Step2.(Localasymptoticestimationofβnun).Multiplyingequation (13) byiβnun,integratingover

ω

andusingestimation (12),weget

ω

|β

nun

|

²

=

o

(

1

).

(15)

Step3.(Themultiplier

ϕ

n).Now,forallβn∈R^,let

ϕ

n∈^H2()∩^H1₀()bethesolutiontothefollowingsystem

ρ β

_n²

ϕ

n

+

^div

(

a

∇ ϕ

n

) −

ⁱ

β

n

( 1

ωb

)(

x

) ϕ

n

=

^un

,

in

,

ϕ

n

=

0

,

on

⁽¹⁶⁾

where (un,vn) is solution to (13)–(14). Since

ω

satisfies the GCC, then the wave equation with local viscous damping (1ωb)(x)

ϕ

tisexponentiallystable(see[10])and,followingHuang[9] andPruss[20],theresolventofitsassociatedoperator Aaux:^D(Aaux)−→^H1₀()×^L2()definedby:

D

(

Aaux

) = (

H²

() ∩

^H10

()) ×

^H10

(),

Aaux

( ϕ , ψ ) = ψ,

¹

ρ ⁽

^div

⁽

^a

^∇ ϕ ) − ( 1

ωb

)(

x

)ψ )

isuniformlyboundedontheimaginaryaxis.ThisimpliesthatthereexistsM>0 independentofnsuchthat

||β

n

ϕ

n

||

_L²₍₎

+ ||∇ ϕ

n

||

_L²₍₎

≤

M

||

un

||

_L²₍₎

.

(17) Step4. (Globalasymptoticestimations).Multiplying equations(13) and (14) byiβ_n³

ρϕ

_n andβ_n²

ρϕ

_n respectively, applying Green’sformulaandaddingtheresultingequations,weget

−

| β

_nu_n

|

^2dx

+

ω

i

β

_n³b

ϕ

_nu_ndx

=

^o

(

1

).

(18)

Itfollows,from(15) and(17),that

| β

nun

|

^2dx

=

^o

(

1

).

(19)

Thisimplies,byusingthemultiplierun that

|∇

un

|

²dx

=

o

(

1

).

(20)

Proof of Theorem3.5. Addingestimations(19) and(20),wededucethat||^Un||_H=^o(1),whichgivesthedesiredcontradic- tion.Thedemonstrationisthusachieved. 2

Acknowledgements

The authors wouldlike tothank the refereesandthe editor fortheir valuablecomments. MmeRayan Nassertient à remercierl’Université libanaise,l’AUFetle CNRS-Lpourleurs soutiensfinanciers danslecadre d’unebourse de doctorat.

ProfAliWehbetientàremercierl’AUFpoursonsoutiendanslecadreduprojetdeperfectionnementàlarecherche.

References

[1]K.Ammari,F.Hassine,L.Robbiano,StabilizationforthewaveequationwithsingularKelvin–Voigtdamping,arXiv:1805.10430v1 [math.AP],052018.

[2]C.Bardos,G.Lebeau,J.Rauch,Sharpsufficientconditionsfortheobservation,control,andstabilizationofwavesfromtheboundary,SIAMJ.Control Optim.30 (5)(1992)1024–1065.

[3]A.Bátkai,K.-J.Engel,J.Prüss,R.Shnaubelt,Polynomialstabilityofoperatorsemigroup,Math.Nachr.279(2006)1425–1440.

[4]C.J.K.Batty,T.Duyckaerts,Non-uniformstabilityforboundedsemi-groupsonBanachspaces,J.Evol.Equ.8 (4)(2008)765–780.

[5]A.Borichev,Y.Tomilov,Optimalpolynomialdecayoffunctionsandoperatorsemigroups,Math.Ann.347 (2)(2010)455–478.

[6]M.M.Cavalcanti,V.DomingosCavalcanti,L.T.Tebou,StabilizationofthewaveequationwithlocalizedcompensatingfrictionalandKelvin–Voigtdissi- patingmechanisms,Electron.J.Differ.Equ.59 (2)(2017)1–18.

[7]S.Chen,K.Liu,Z.Liu,SpectrumandstabilityforelasticsystemswithglobalorlocalKelvin–Voigtdamping,SIAMJ.Appl.Math.59 (2)(1998)651–668.

[8]F.Huang,Onthemathematicalmodelforlinearelasticsystemswithanalyticdamping,SIAMJ.ControlOptim.26 (3)(1988)714–724.

[9]F.L.Huang,CharacteristicconditionsforexponentialstabilityoflineardynamicalsystemsinHilbertspaces,Ann.Differ.Equ.1 (1)(1985)43–56.

[10]G.Lebeau,Équationdesondesamorties,in:AlgebraicandGeometricMethodsinMathematicalPhysics,Kaciveli,1993,in:Math.Phys.Stud.,vol. 19, KluwerAcad.Publ.,Dordrecht,theNetherlands,1996,pp. 73–109.

[11]J.Lions,Contrôlabilitéexacte,perturbationsetstabilisationdesystèmesdistribués:Perturbations,Recherchesenmathématiquesappliquées,Masson, Paris,1988.

[12]K.Liu,Locallydistributedcontrolanddampingfortheconservativesystems,SIAMJ.ControlOptim.35 (5)(1997)1574–1590.

[13]K.Liu,Z.Liu,ExponentialdecayofenergyoftheEuler–BernoullibeamwithlocallydistributedKelvin–Voigtdamping,SIAMJ.ControlOptim.36 (3) (1998)1086–1098.

[14]K.Liu,Z.Liu,Exponentialdecayofenergyofvibratingstringswithlocalviscoelasticity,Z.Angew.Math.Phys.53 (2)(2002)265–280.

[15]K.Liu,Z.Liu,Q.Zhang,EventualdifferentiabilityofastringwithlocalKelvin–Voigtdamping,ESAIMControlOptim.Calc.Var.23 (2)(2017)443–454.

[16]K.Liu,B.Rao,ExponentialstabilityforthewaveequationswithlocalKelvin–Voigtdamping,Z.Angew.Math.Phys.57 (3)(2006)419–432.

[17]Z.Liu,B.Rao,Characterizationofpolynomialdecayrateforthesolutionoflinearevolutionequation,Z.Angew.Math.Phys.56 (4)(2005)630–644.

[18]Z.Liu,Q.Zhang,StabilityofastringwithlocalKelvin–Voigtdampingandnonsmoothcoefficientatinterface,SIAMJ.ControlOptim.54 (4)(2016) 1859–1871.

[19]S.Nicaise,C.Pignotti,StabilityofthewaveequationwithlocalizedKelvin–Voigtdampingandboundarydelayfeedback,DiscreteContin.Dyn.Syst., Ser.S9 (3)(2016)791–813.

[20]J.Prüss,OnthespectrumofC0-semigroups,Trans.Amer.Math.Soc.284 (2)(1984)847–857.

[21]J.Rauch,M.Taylor,R.Phillips,Exponentialdecayofsolutionstohyperbolicequationsinboundeddomains,IndianaUniv.Math.J.24 (1)(1974)79–86.

[22]L.Tebou,AconstructivemethodforthestabilizationofthewaveequationwithlocalizedKelvin–Voigtdamping,C.R.Acad.Sci.Paris,Ser.I350 (11) (2012)603–608.

[23]Q.Zhang,ExponentialstabilityofanelasticstringwithlocalKelvin–Voigtdamping,Z.Angew.Math.Phys.61 (6)(2010)1009–1015.

[24]Q.Zhang,Onthelackofexponentialstabilityforanelastic–viscoelasticwavesinteractionsystem,NonlinearAnal.,RealWorldAppl.37(2017)387–411.

[25]Q.Zhang,Polynomialdecayofanelastic/viscoelasticwavesinteractionsystem,Z.Angew.Math.Phys.69 (4)(2018)88.