Stability results of an elastic/viscoelastic transmission problem of locally coupled waves with non smooth coefficients
aa r X i v : . [ m a t h . A P ] A p r STABILITY RESULTS OF AN ELASTIC/VISCOELASTIC TRANSMISSION PROBLEMOF LOCALLY COUPLED WAVES WITH NON SMOOTH COEFFICIENTS
MOHAMMAD AKIL , IBTISSAM ISSA , , AND ALI WEHBE Abstract.
We investigate the stabilization of a locally coupled wave equations with only one internal vis-coelastic damping of Kelvin-Voigt type (see System (1.2)-(1.4)). The main novelty in this paper is that boththe damping and the coupling coefficients are non smooth (see (1.5)). First, using a general criteria of Arendt-Batty, combined with an uniqueness result, we prove that our system is strongly stable. Next, using a spectrumapproach, we prove the non-exponential (uniform) stability of the system. Finally, using a frequency domainapproach, combined with a piecewise multiplier technique and the construction of a new multiplier satisfy-ing some ordinary differential equations, we show that the energy of smooth solutions of the system decayspolynomially of type t − . Contents
1. Introduction 11.1. Motivation and aims 11.2. Literature 21.3. Description of the paper 32. Well-Posedness and Strong Stability 32.1. Well-Posedness 32.2. Strong Stability 53. Lack of the exponential Stability 113.1. Lack of exponential stability with global Kelvin-Voigt damping. 113.2. Lack of exponential stability with Local Kelvin-Voigt damping. 124. Polynomial Stability 205. Appendix 255.1. Exponential stability of locally coupled wave equations with non-smooth coefficients 255.2. Definitions and Theorems 316. Conclusion 32Acknowledgments 32References 32 Lebanese University, Faculty of sciences 1, Khawarizmi Laboratory of Mathematics and Applications-KALMA,Hadath-Beirut, Lebanon. Aix-Marseille University, I2M, Marseille-France.
E-mail addresses : [email protected], [email protected], [email protected] . Key words and phrases.
Wave equation; Kelvin-Voigt damping; Semigroup; Stability. i . Introduction
Motivation and aims.
There are several mathematical models representing physical damping. The mostoften encountered type of damping in vibration studies are linear viscous damping and Kelvin-Voigt dampingwhich are special cases of proportional damping. Viscous damping usually models external friction forcessuch as air resistance acting on the vibrating structures and is thus called ”external damping”, while Kelvin-Voigt damping originates from the internal friction of the material of the vibrating structures and thus called”internal damping”. In 1988, F. Huang in [16] considered a wave equation with globally distributed Kelvin-Voigt damping, i.e. the damping coefficient is strictly positive on the entire spatial domain. He proved that thecorresponding semigroup is not only exponentially stable, but also is analytic (see Definition 5.10, Theorem 5.12and Theorem 5.14 below). Thus, Kelvin-Voigt damping is stronger than the viscous damping when globallydistributed. Indeed, it was proved that the semigroup corresponding to the system of wave equations withglobal viscous damping is exponentially stable but not analytic (see [11] for the one dimensional system and[8] for the higher dimensional system). However, the exponential stability of a wave equation is still true evenif the viscous damping is localized, via a smooth or a non smooth damping coefficient, in a suitable subdomainsatisfying some geometric conditions (see [8]). Nevertheless, when viscoelastic damping is distributed locally,the situation is more delicate and such comparison between viscous and viscoelastic damping is not validanymore. Indeed, the stabilization of the wave equation with local Kelvin-Voigt damping is greatly influencedby the smoothness of the damping coefficient and the region where the damping is localized (near or farawayfrom the boundary) even in the one-dimensional case. So, the stabilization of systems (simple or coupled)with local Kelvin-Voigt damping has attracted the attention of many authors (see the Literature below for thehistory of this kind of damping). From a mathematical point of view, it is important to study the stability ofa system coupling a locally damped wave equation with a conservative one. Moreover, the study of this kindof systems is also motivated by several physical considerations and occurs in many applications in engineeringand mechanics. In this direction, recently in 2019, Hassine and Souayeh in [15], studied the stabilization of asystem of global coupled wave equations with one localized Kelvin-Voigt damping. The system is described by(1.1) u tt − ( u x + b ( x ) u tx ) x + v t = 0 , ( x, t ) ∈ ( − , × R + ,v tt − cv xx − u t = 0 , ( x, t ) ∈ ( − , × R + ,u (0 , t ) = v (0 , t ) = 0 , u (1 , t ) = v (1 , t ) = 0 , t > ,u ( x,
0) = u ( x ) , u t ( x,
0) = u ( x ) , x ∈ ( − , ,v ( x,
0) = v ( x ) , v t ( x,
0) = v ( x ) , x ∈ ( − , , where c >
0, and b ∈ L ∞ ( − ,
1) is a non-negative function. They assumed that the damping coefficient isgiven by b ( x ) = d [0 , ( x ), where d is a strictly positive constant. The Kelvin-Voigt damping ( b ( x ) u tx ) x isapplied at the first equation and the second equation is indirectly damped through the coupling between thetwo equations. Under the two conditions that the Kelvin-Voigt damping is localized near the boundary andthe two waves are globally coupled, they obtained a polynomial energy decay rate of type t − . Then thestabilization of System (1.1) in the case where the Kelvin-Voigt damping is localized in an arbitrary subintervalof ( − , +1) and the two waves are locally coupled has been left as an open problem. In addition, we believe thatthe energy decay rate obtained in [15] can be improved. So, we are interested in studying this open problem.The main aim of this paper is to study the stabilization of a system of localized coupled wave equationswith only one Kelvin-Voigt damping localized via non-smooth coefficient in a subinterval of the domain. Thesystem is described by u tt − ( au x + b ( x ) u tx ) x + c ( x ) y t = 0 , ( x, t ) ∈ (0 , L ) × R + , (1.2) y tt − y xx − c ( x ) u t = 0 , ( x, t ) ∈ (0 , L ) × R + , (1.3)with fully Dirichlet boundary conditions,(1.4) u (0 , t ) = u ( L, t ) = y (0 , t ) = y ( L, t ) = 0 , ∀ t ∈ R + , b ( x ) = (cid:26) b if x ∈ ( α , α )0 otherwise and c ( x ) = (cid:26) c if x ∈ ( α , α )0 otherwiseand a > , b > c >
0, and where we consider 0 < α < α < α < α < L . This system is consideredwith the following initial data(1.6) u ( · ,
0) = u ( · ) , u t ( · ,
0) = u ( · ) , y ( · ,
0) = y ( · ) and y t ( · ,
0) = y ( · ) . α α α α L b c Literature.
The wave is created when a vibrating source disturbs the medium. In order to restrain thosevibrations, several dampings can be added such as Kelvin-Voigt damping which is originated from the extensionor compression of the vibrating particles. This damping is a viscoelastic structure having properties of bothelasticity and viscosity. In the recent years, many researchers showed interest in problems involving this kindof damping (local or global) where different types of stability have been showed. In particular, in the onedimensional case, it was proved that the smoothness of the damping coefficient affects critically the studying ofthe stability and regularity of the solution of the system. Indeed, in the one dimensional case we can considerthe following system(1.7) u tt − ( u x + b ( x ) u tx ) x = 0 , − ≤ x ≤ , t > ,u (1 , t ) = u ( − , t ) = 0 , t > ,u ( x,
0) = u ( x ) , u t ( x,
0) = u ( x ) , − ≤ x ≤ , with b ∈ L ∞ ( − ,
1) and(1.8) b ( x ) = (cid:26) x ∈ (0 , ,a ( x ) if x ∈ ( − , , where the function a ( x ) is non-negative. The case of local Kelvin-Voigt damping was first studied in 1998[19, 20], it was proved that the semigroup loses exponential stability and smooth property when the damping islocal and a = 1 or b ( · ) is the characteristic function of any subinterval of the domain. This surprising resultinitiated the study of an elastic system with local Kelvin-Voigt damping. In 2002, K. Liu and Z. Liu proved thatsystem (1.7) is exponentially stable if b ′ ( . ) ∈ C , ([ − , b wasweakened to b ( · ) ∈ C ([ − , a was taken. In 2004, Renardy’s results [32] hinted that thesolution of the system (1.7) may be exponentially stable under smoother conditions on the damping coefficient.This result was confirmed by K. Liu, Z. Liu and Q. Zhang in [26]. On the other hand, Liu and Rao in 2005(see [23]) proved that the semigroup corresponding to system (1.7) is polynomially stable of order almost 2 if a ( . ) ∈ C (0 ,
1) and a ( x ) ≥ a ≥ t − . In 2016, under the assumption that the damping coefficient has asingularity at the interface of the damped and undamped regions and behaves like x α near the interface, it wasproven by Liu and Zhang [24] that the semigroup corresponding to the system is polynomially or exponentiallystable and the decay rate depends on the parameter α ∈ (0 , t − . In [13], Hassine considered a beam and a wave equation coupled on an elastic beamthrough transmission conditions with locally distributed Kelvin-Voigt damping that acts through one of thetwo equations only. He proved a polynomial energy decay rate of type t − for both cases where the dissipationacts through the beam equation and through the wave equation. In 2016, Oquendo and Sanez studied the waveequation with internal coupled terms where the Kelvin-Voigt damping is global in one equation and the secondequation is conservative. They showed that the semigroup loses speed and decays with the rate t − and theyproved that this decay rate is optimal (see [30]).Let us mention some of the results that have been established for the case of wave equation with Kelvin-Voigtdamping in the multi-dimensional setting. In [16], the author proved that when the Kelvin-Voigt dampingdiv( d ( x ) ∇ u t ) is globally distributed, i.e. d ( x ) ≥ d > x ∈ Ω, the wave equation generatesan analytic semi-group. In [22], the authors considered the wave equation with local visco-elastic dampingdistributed around the boundary of Ω. They proved that the energy of the system decays exponentially tozero as t goes to infinity for all usual initial data under the assumption that the damping coefficient satisfies: d ∈ C , (Ω), ∆ d ∈ L ∞ (Ω) and |∇ d ( x ) | ≤ M d ( x ) for almost every x in Ω where M is a positive constant. Onthe other hand, in [33], the author studied the stabilization of the wave equation with Kelvin-Voigt damping.He established a polynomial energy decay rate of type t − provided that the damping region is localized in aneighborhood of a part of the boundary and verifies certain geometric condition. Also in [28], under the sameassumptions on d , the authors established the exponential stability of the wave equation with local Kelvin-Voigt damping localized around a part of the boundary and an extra boundary with time delay where theyadded an appropriate geometric condition. Later on, in [3], the wave equation with Kelvin-Voigt dampinglocalized in a subdomain ω far away from the boundary without any geometric conditions was considered.The authors established a logarithmic energy decay rate for smooth initial data. Further more, in [27], theauthors investigate the stabilization of the wave equation with Kelvin-Voigt damping localized via non smoothcoefficient in a suitable sub-domain of the whole bounded domain. They proved a polynomial stability resultin any space dimension, provided that the damping region satisfies some geometric conditions.1.3. Description of the paper.
This paper is organized as follows: In Subsection 2.1, we reformulate thesystem (1.2)-(1.6) into an evolution system and we prove the well-posedness of our system by semigroupapproach. In Subsection 2.2, using a general criteria of Arendt and Batty, we show the strong stability ofour system in the absence of the compactness of the resolvent. In Section 3, we prove that the system lacksexponential stability using two different approaches. The first case is by taking the damping and the couplingterms to be globally defined, i.e b ( x ) = b > c ( x ) = c > t − . 2. Well-Posedness and Strong Stability
In this section, we study the strong stability of System (1.2)-(1.6). First, using a semigroup approach, weestablish well-posedness result of our system.2.1.
Well-Posedness.
Firstly, we reformulate System (1.2)-(1.6) into an evolution problem in an appropriateHilbert state space.The energy of System (1.2)-(1.6) is given by E ( t ) = 12 Z L (cid:0) | u t | + a | u x | + | y t | + | y x | (cid:1) dx. u, u t , y, y t ) be a regular solution of (1.2)-(1.6). Multiplying (1.2), (1.3) by u t , y t , respectively, then usingthe boundary conditions (1.4), we get E ′ ( t ) = − Z L b ( x ) | u tx | dx, using the definition of the function b ( x ), we get E ′ ( t ) ≤
0. Thus, System (1.2)-(1.6) is dissipative in the sensethat its energy is a non-increasing function with respect to the time variable t . Let us define the energy space H by H = ( H (0 , L ) × L (0 , L )) . The energy space H is equipped with the inner product defined by h U, U i H = Z L vv dx + a Z L u x ( u ) x dx + Z L zz dx + Z L y x ( y ) x dx, for all U = ( u, v, y, z ) and U = ( u , v , y , z ) in H . We use k U k H to denote the corresponding norm. Wedefine the unbounded linear operator A : D ( A ) ⊂ H −→ H by D ( A ) = U = ( u, v, y, z ) ∈ H ; y ∈ H (0 , L ) ∩ H (0 , L ) v, z ∈ H (0 , L ) , ( au x + b ( x ) v x ) x ∈ L (0 , L ) and for all U = ( u, v, y, z ) ∈ D ( A ), A ( u, v, y, z ) = ( v, ( au x + b ( x ) v x ) x − c ( x ) z, z, y xx + c ( x ) v ) ⊤ . If U = ( u, u t , y, y t ) is the state of System (1.2)-(1.6), then this system is transformed into the first orderevolution equation on the Hilbert space H given by(2.1) U t = A U, U (0) = U , where U = ( u , u , y , y ). Proposition 2.1.
The unbounded linear operator A is m-dissipative in the energy space H . Proof.
For all U = ( u, v, y, z ) ∈ D ( A ), we have ℜ ( hA U, U i H ) = − Z L b ( x ) | v x | dx = − Z α α b | v x | dx ≤ , which implies that A is dissipative. Here ℜ is used to denote the real part of a complex number. Now, let F = ( f , f , f , f ), we prove the existence of U = ( u, v, y, z ) ∈ D ( A ), solution of the equation(2.2) − A U = F. Equivalently, one must consider the system given by − v = f , (2.3) − ( au x + b ( x ) v x ) x + c ( x ) z = f , (2.4) − z = f , (2.5) − y xx − c ( x ) v = f , (2.6)with the boundary conditions(2.7) u (0) = u ( L ) = 0 , and y (0) = y ( L ) = 0 . Let ( ϕ, ψ ) ∈ H (0 , L ) × H (0 , L ). Multiplying Equations (2.4) and (2.6) by ϕ and ψ respectively, integrateover (0 , L ), we obtain Z L ( au x + b ( x ) v x ) ϕ x dx + Z L c ( x ) zϕdx = Z L f ϕdx, (2.8) Z L y x ψ x dx − Z L c ( x ) vψdx = Z L f ψdx. (2.9) 4nserting Equations (2.3) and (2.5) into (2.8) and (2.9), we get Z L au x ϕ x dx = Z L f ϕdx + Z L b ( x )( f ) x ϕ x dx + Z L c ( x ) f ϕdx, (2.10) Z L y x ψ x dx = Z L f ψdx − Z L c ( x ) f ψdx. (2.11)Adding Equations (2.10) and (2.11), we obtain(2.12) a (( u, y ) , ( ϕ, ψ )) = L ( ϕ, ψ ) , ∀ ( ϕ, ψ ) ∈ H (0 , L ) × H (0 , L ) , where(2.13) a (( u, y ) , ( ϕ, ψ )) = a Z L u x ϕ x dx + Z L y x ψ x dx and(2.14) L ( ϕ, ψ ) = Z L f ϕdx + Z L b ( x )( f ) x ϕ x dx + Z L c ( x ) f ϕdx + Z L f ψdx − Z L c ( x ) f ψdx. Thanks to (2.13), (2.14) , we have that a is a bilinear continuous coercive form on (cid:0) H (0 , L ) × H (0 , L ) (cid:1) , and L is a linear continuous form on H (0 , L ) × H (0 , L ). Then, using Lax-Milgram theorem, we deduce that thereexists ( u, y ) ∈ H (0 , L ) × H (0 , L ) unique solution of the variational problem (2.12). Applying the classicalelliptic regularity we deduce that U = ( u, v, y, z ) ∈ D ( A ) is the unique solution of (2.2). The proof is thuscomplete. (cid:3) From Proposition 2.1, the operator A is m-dissipative on H and consequently, generates a C − semigroupof contractions (cid:0) e t A (cid:1) t ≥ following Lummer-Phillips theorem (see in [25] and [29]). Then the solution of theevolution Equation (2.1) admits the following representation U ( t ) = e t A U , t ≥ , which leads to the well-posedness of (2.1). Hence, we have the following result. Theorem 2.2.
Let U ∈ H then, problem (2.1) admits a unique weak solution U satisfies U ( t ) ∈ C (cid:0) R + , H (cid:1) . Moreover, if U ∈ D ( A ) then, problem (2.1) admits a unique strong solution U satisfies U ( t ) ∈ C (cid:0) R + , H (cid:1) ∩ C ( R + , D ( A )) . Strong Stability.
This part is devoted for the proof of the strong stability of the C -semigroup (cid:0) e t A (cid:1) t ≥ .To obtain strong stability of the C -semigroup (cid:0) e t A (cid:1) t ≥ we use the theorem of Arendt and Batty in [6] (seeTheorem 5.11 in Appendix). Theorem 2.3.
The C − semigroup of contractions (cid:0) e t A (cid:1) t ≥ is strongly stable in H ; i.e. for all U ∈ H , thesolution of (2.1) satisfies lim t → + ∞ k e t A U k H = 0 . For the proof of Theorem 2.3, according to Theorem 5.11, we need to prove that the operator A has no pureimaginary eigenvalues and σ ( A ) ∩ i R contains only a countable number of continuous spectrum of A . Theargument for Theorem 2.3 relies on the subsequent lemmas. Lemma 2.4.
For λ ∈ R , we have iλI − A is injective i.e. ker ( iλI − A ) = { } , ∀ λ ∈ R . Proof.
From Proposition 2.1, we have 0 ∈ ρ ( A ). We still need to show the result for λ ∈ R ∗ . Suppose thatthere exists a real number λ = 0 and U = ( u, v, y, z ) ∈ D ( A ), such that A U = iλU. v = iλu, (2.15) ( au x + b ( x ) v x ) x − c ( x ) z = iλv, (2.16) z = iλy, (2.17) y xx + c ( x ) v = iλz. (2.18)Next, a straightforward computation gives0 = ℜ h iλU, U i H = ℜ hA
U, U i H = − Z L b ( x ) | v x | dx = − Z α α b | v x | dx, consequently, we deduce that(2.19) b ( x ) v x = 0 in (0 , L ) and v x = 0 in ( α , α ) . It follows, from Equation (2.15), that(2.20) u x = 0 in ( α , α ) . Using Equations (2.16), (2.17), (2.19), (2.20) and the definition of c ( x ), we obtain(2.21) y x = 0 in ( α , α ) . Substituting Equations (2.15), (2.17) in Equations (2.16), (2.18), and using Equation (2.19) and the definitionof b ( x ) in (1.5), we get λ u + au xx − iλc ( x ) y = 0 , in (0 , L )(2.22) λ y + y xx + iλc ( x ) u = 0 , in (0 , L )(2.23)with the boundary conditions(2.24) u (0) = u ( L ) = y (0) = y ( L ) = 0 . Our goal is to prove that u = y = 0 on (0 , L ). For simplicity, we divide the proof into three steps. Step 1.
The aim of this step is to show that u = y = 0 on (0 , α ). so, using Equation (2.20), we have u x = 0 in ( α , α ) . Using the above equation and Equation (2.22) and the fact that c ( x ) = 0 on ( α , α ), we obtain(2.25) u = 0 in ( α , α ) . In fact, system (2.22)-(2.24) admits a unique solution ( u, y ) ∈ C ([0 , L ]), then(2.26) u ( α ) = u x ( α ) = 0 . Then, from Equations (2.22) and (2.26) and the fact that c ( x ) = 0 on (0 , α ), we get(2.27) u = 0 in (0 , α ) . Using Equations (2.20) and (2.25) and the fact that u ∈ C ([0 , L ]), we get(2.28) u = 0 in ( α , α ) . Now, using Equations (2.20), (2.21) and the fact that c ( x ) = c on ( α , α ) in Equations (2.22), (2.23) , weobtain(2.29) u = ic λ y in ( α , α ) . Using Equation (2.28) in Equation (2.29), we obtain(2.30) u = y = 0 in ( α , α ) . Since y ∈ C ([0 , L ]), then(2.31) y ( α ) = y x ( α ) = 0 . So, from Equations (2.23) and (2.31) and the fact that c ( x ) = 0 on ( α , α ), we obtain(2.32) y = 0 in ( α , α ) . , α ), we get(2.33) y = 0 in (0 , α ) . Hence, from Equations (2.25), (2.27), (2.28), (2.30), (2.32) and (2.33), we obtain u = y = 0 on (0 , α ).Consequently, we obtain U = 0 in (0 , α ) . Step 2.
The aim of this step is to show that u = y = 0 on ( α , α ). Using Equation (2.30), and the fact that( u, y ) ∈ C ([0 , L ]), we obtain the boundary conditions(2.34) u ( α ) = u x ( α ) = y ( α ) = y x ( α ) = 0 . Combining Equations (2.22), (2.23), and the fact that c ( x ) = c on ( α , α ), we get(2.35) au xxxx + ( a + 1) λ u xx + λ (cid:0) λ − c (cid:1) u = 0 . The characteristic equation of system (2.35) is P ( r ) := ar + ( a + 1) λ r + λ (cid:0) λ − c (cid:1) . Setting P ( m ) := am + ( a + 1) λ m + λ (cid:0) λ − c (cid:1) . The polynomial P has two distinct real roots m and m given by: m = − λ ( a + 1) − p λ ( a − + 4 ac λ a and m = − λ ( a + 1) + p λ ( a − + 4 ac λ a . It is clear that m < m depends on the value of λ with respect to c . We distinguish thefollowing three cases: λ < c , λ = c and λ > c . Case 1. If λ < c , then m >
0. Setting r = √− m and r = √ m . Then P has four simple roots ir , − ir , r and − r , and hence the general solution of system (2.22), (2.23), isgiven by u ( x ) = c sin( r x ) + c cos( r x ) + c cosh( r x ) + c sinh( r x ) ,y ( x ) = ( λ − ar ) iλc ( c sin( r x ) + c cos( r x )) + ( λ + ar ) iλc ( c cosh( r x ) + c sinh( r x )) , where c j ∈ C , j = 1 , · · · ,
4. In this case, the boundary condition in Equation (2.34), can be expressed by M c c c c = 0 , where M = sin( r α ) cos( r α ) cosh( r α ) sinh( r α ) r cos( r α ) − r sin( r α ) r sinh( r α ) r cosh( r α )( λ − ar ) iλc sin( r α ) ( λ − ar ) iλc cos( r α ) ( λ + ar ) iλc cosh( r α ) ( λ + ar ) iλc sinh( r α )( λ − ar ) iλc r cos( r α ) − ( λ − ar ) iλc r sin( r α ) ( λ + ar ) iλc r sinh( r α ) ( λ + ar ) iλc r cosh( r α ) . The determinant of M is given by det( M ) = r r a (cid:0) r + r (cid:1) λ c . System (2.22), (2.23) with the boundary conditions (2.34), admits only a trivial solution u = y = 0 if and onlyif det( M ) = 0, i.e. M is invertible. Since, r + r = m − m = 0, then det( M ) = 0. Consequently, if7 < c , we obtain u = y = 0 on ( α , α ). Case 2. If λ = c , then m = 0. Setting r = √− m = r ( a + 1) c a . Then P has two simple roots ir , − ir and 0 is a double root. Hence the general solution of System (2.22),(2.23) is given by u ( x ) = c sin( r x ) + c cos( r x ) + c x + c ,y ( x ) = ( λ − ar ) iλc ( c sin( r x ) + c cos( r x )) + λic ( c x + c ) , where c j ∈ C , for j = 1 , · · · ,
4. Also, the boundary condition in Equation (2.34), can be expressed by M c c c c = 0 , where M = sin( r α ) cos( r α ) α r cos( r α ) − r sin( r α ) 1 0( λ − ar ) iλc sin( r α ) ( λ − ar ) iλc cos( r α ) λα ic λic ( λ − ar ) iλc r cos( r α ) − ( λ − ar ) iλc r sin( r α ) λic . The determinant of M is given by det( M ) = − a r λ c . Since r = √− m = 0, then det( M ) = 0. Thus, System (2.22), (2.23) with the boundary conditions (2.34),admits only a trivial solution u = y = 0 on ( α , α ). Case 3. If λ > c , then m <
0. Setting r = √− m and r = √− m . Then P has four simple roots ir , − ir , ir and − ir , and hence the general solution of System (2.22), (2.23)is given by u ( x ) = c sin( r x ) + c cos( r x ) + c sin( r x ) + c cos( r x ) ,y ( x ) = ( λ − ar ) iλc ( c sin( r x ) + c cos( r x )) + ( λ − ar ) iλc ( c sin( r x ) + c cos( r x )) , where c j ∈ C , for j = 1 , · · · ,
4. Also, the boundary condition in Equation (2.34), can be expressed by M c c c c = 0 , where M = sin( r α ) cos( r α ) sin( r α ) cos( r α ) r cos( r α ) − r sin( r α ) r cos( r α ) − r sin( r α )( λ − ar ) iλc sin( r α ) ( λ − ar ) iλc cos( r α ) ( λ − ar ) iλc sin( r α ) ( λ + ar ) iλc cos( r α )( λ − ar ) iλc r cos( r α ) − ( λ − ar ) iλc r sin( r α ) ( λ − ar ) iλc r cos( r α ) − ( λ − ar ) iλc r sin( r α ) . M is given by det( M ) = − r r a ( r − r ) λc . Since r − r = m − m = 0, then det( M ) = 0. Thus, System (2.22)-(2.23) with the boundary condition(2.34), admits only a trivial solution u = y = 0 on ( α , α ). Consequently, we obtain U = 0 on ( α , α ). Step 3.
The aim of this step is to show that u = y = 0 on ( α , L ). From Equations (2.22), (2.23) and the factthat c ( x ) = 0 on ( α , L ), we obtain the following system(2.36) (cid:26) λ u + au xx = 0 over ( α , L ) λ y + y xx = 0 over ( α , L ) . Since ( u, y ) ∈ C ([0 , L ]) and the fact that u = y = 0 on ( α , α ), we get(2.37) u ( α ) = u x ( α ) = y ( α ) = y x ( α ) = 0 . Finally, it is easy to see that System (2.36) admits only a trivial solution on ( α , L ) under the boundarycondition (2.37).Consequently, we proved that U = 0 on (0 , L ). The proof is thus complete. (cid:3) Lemma 2.5.
For all λ ∈ R , we have R ( iλI − A ) = H . Proof.
From Proposition 2.1, we have 0 ∈ ρ ( A ). We still need to show the result for λ ∈ R ∗ . Set F =( f , f , f , f ) ∈ H , we look for U = ( u, v, y, z ) ∈ D ( A ) solution of(2.38) ( iλI − A ) U = F. Equivalently, we have v = iλu − f , (2.39) iλv − ( au x + b ( x ) v x ) x + c ( x ) z = f , (2.40) z = iλy − f , (2.41) iλz − y xx − c ( x ) v = f . (2.42)Let ( ϕ, ψ ) ∈ H (0 , L ) × H (0 , L ), multiplying Equations (2.40) and (2.42) by ¯ ϕ and ¯ ψ respectively and integrateover (0 , L ), we obtain Z L iλv ¯ ϕdx + Z L au x ¯ ϕ x dx + Z L b ( x ) v x ¯ ϕ x dx + Z L c ( x ) z ¯ ϕdx = Z L f ¯ ϕdx, (2.43) Z L iλz ¯ ψdx + Z L y x ¯ ψ x dx − Z L c ( x ) v ¯ ψdx = Z L f ¯ ψdx. (2.44)Substituting v and z by iλu − f and iλy − f respectively in Equations (2.43)-(2.44) and taking the sum, weobtain(2.45) a (( u, y ) , ( ϕ, ψ )) = L( ϕ, ψ ) , ∀ ( ϕ, ψ ) ∈ H (0 , L ) × H (0 , L ) , where a (( u, y ) , ( ϕ, ψ )) = a (( u, y ) , ( ϕ, ψ )) + a (( u, y ) , ( ϕ, ψ ))with a (( u, y ) , ( ϕ, ψ )) = Z L (cid:0) au x ¯ ϕ x + y x ¯ ψ x (cid:1) dx + iλ Z L b ( x ) u x ¯ ϕ x dx,a (( u, y ) , ( ϕ, ψ )) = − λ Z L (cid:0) u ¯ ϕ + y ¯ ψ (cid:1) dx + iλ Z L c ( x ) (cid:0) y ¯ ϕ − u ¯ ψ (cid:1) dx, and L( ϕ, ψ ) = Z L ( f + c ( x ) f + iλf ) ¯ ϕdx + Z L ( f − c ( x ) f + iλf ) ¯ ψdx + Z L b ( x ) ( f ) x ¯ ϕ x dx. V = H (0 , L ) × H (0 , L ) and V ′ = H − (0 , L ) × H − (0 , L ) the dual space of V . Let us consider the followingoperators, (cid:26) A : V → V ′ ( u, y ) → A( u, y ) (cid:26) A : V → V ′ ( u, y ) → A ( u, y ) (cid:26) A : V → V ′ ( u, y ) → A ( u, y )such that(2.46) (A( u, y )) ( ϕ, ψ ) = a (( u, y ) , ( ϕ, ψ )) , ∀ ( ϕ, ψ ) ∈ H (0 , L ) × H (0 , L ) , (A ( u, y )) ( ϕ, ψ ) = a (( u, y ) , ( ϕ, ψ )) , ∀ ( ϕ, ψ ) ∈ H (0 , L ) × H (0 , L ) , (A ( u, y )) ( ϕ, ψ ) = a (( u, y ) , ( ϕ, ψ )) , ∀ ( ϕ, ψ ) ∈ H (0 , L ) × H (0 , L ) . Our goal is to prove that A is an isomorphism operator. For this aim, we divide the proof into three steps.
Step 1.
In this step, we prove that the operator A is an isomorphism operator. For this goal, following the sec-ond equation of (2.46) we can easily verify that a is a bilinear continuous coercive form on H (0 , L ) × H (0 , L ).Then, by Lax-Milgram Lemma, the operator A is an isomorphism. Step 2.
In this step, we prove that the operator A is compact. According to the third equation of (2.46), wehave | a (( u, y ) , ( ϕ, ψ )) | ≤ C k ( u, y ) k L (0 ,L ) k ( ϕ, ψ ) k L (0 ,L ) . Finally, using the compactness embedding from H (0 , L ) to L (0 , L ) and the continuous embedding from L (0 , L ) into H − (0 , L ) we deduce that A is compact.From steps 1 and 2, we get that the operator A = A + A is a Fredholm operator of index zero. Consequently,by Fredholm alternative, to prove that operator A is an isomorphism it is enough to prove that A is injective,i.e. ker { A } = { } . Step 3.
In this step, we prove that ker { A } = { } . For this aim, let (˜ u, ˜ y ) ∈ ker { A } , i.e. a ((˜ u, ˜ y ) , ( ϕ, ψ )) = 0 , ∀ ( ϕ, ψ ) ∈ H (0 , L ) × H (0 , L ) . Equivalently, we have − λ Z L (cid:0) ˜ u ¯ ϕ + ˜ y ¯ ψ (cid:1) dx + iλ Z L c ( x ) (cid:0) ˜ y ¯ ϕ − ˜ u ¯ ψ (cid:1) dx + Z L (cid:0) a ˜ u x ¯ ϕ x + ˜ y x ¯ ψ x (cid:1) dx + iλ Z L b ( x )˜ u x ¯ ϕ x dx = 0 . (2.47)Taking ϕ = ˜ u and ψ = ˜ y in equation (2.47), we get − λ Z L | ˜ u | dx − λ Z L | ˜ y | dx + a Z L | ˜ u x | dx + Z L | ˜ y x | dx − λ ℑ Z L c ( x )˜ y ¯˜ udx ! + iλ Z L b ( x ) | ˜ u x | dx = 0 . Taking the imaginary part of the above equality, we get0 = Z L b ( x ) | ˜ u x | dx, we get,(2.48) ˜ u x = 0 , in ( α , α ) . Then, we find that − λ ˜ u − a ˜ u xx + iλc ( x )˜ y = 0 , in (0 , L ) − λ ˜ y − a ˜ y xx − iλc ( x )˜ u = 0 , in (0 , L )˜ u x = ˜ y x = 0 . in ( α , α )Therefore, the vector ˜ U defined by ˜ U = (˜ u, iλ ˜ u, ˜ y, iλ ˜ y )10elongs to D ( A ) and we have iλ ˜ U − A ˜ U = 0 . Hence, ˜ U ∈ ker ( iλI − A ), then by Lemma 2.4, we get ˜ U = 0, this implies that ˜ u = ˜ y = 0. Consequently,ker { A } = { } .Therefore, from step 3 and Fredholm alternative, we get that the operator A is an isomorphism. It is easy tosee that the operator L is continuous from V to L (0 , L ) × L (0 , L ). Consequently, Equation (2.45) admits aunique solution ( u, y ) ∈ H ( L ) × H (0 , L ). Thus, using v = iλu − f , z = iλy − f and using the classicalregularity arguments, we conclude that Equation (2.38) admits a unique solution U ∈ D ( A ). The proof is thuscomplete. (cid:3) Proof of Theorem 2.3.
Using Lemma 2.4, we have that A has non pure imaginary eigenvalues. Accordingto Lemmas 2.4, 2.5 and with the help of the closed graph theorem of Banach, we deduce that σ ( A ) ∩ i R = ∅ .Thus, we get the conclusion by applying Theorem 5.11 of Arendt Batty (see Appendix). The proof of thetheorem is thus complete. 3. Lack of the exponential Stability
In this section, our goal is to show that system (1.2)-(1.6) in not exponentially stable.3.1.
Lack of exponential stability with global Kelvin-Voigt damping.
In this part, assume that(3.1) b ( x ) = b > c ( x ) = c , ∀ x ∈ (0 , L ) . We introduce the following theorem.
Theorem 3.1.
Under hypothesis (3.1), for ε > t − ε for all initial data U ∈ D ( A ) and for all t > . Proof.
Following Huang and Pr u ss [17, 31] (see also Theorem 5.12 in the Appendix) it is sufficient to show theexistence of a real sequences ( λ n ) n with λ n → + ∞ , ( U n ) n ∈ D ( A ), and ( F n ) n ⊂ H such that ( iλ n I − A ) U n = F n is bounded in H and λ − εn k U n k → + ∞ . For this aim, take F n = (cid:16) , , , sin (cid:16) nπxL (cid:17)(cid:17) , U n = (cid:16) A n sin (cid:16) nπxL (cid:17) , iλ n A n sin (cid:16) nπxL (cid:17) , B n sin (cid:16) nπxL (cid:17) , iλ n B n sin (cid:16) nπxL (cid:17)(cid:17) , where λ n = nπL , A n = iLc nπ , B n = − inb πc L − a − c . Clearly that U n ∈ D ( A ), and F n is bounded in H . Let us show that ( iλ n I −A ) U n = F n . Detailing ( iλ n I −A ) U n ,we get ( iλ n I − A ) U n = (cid:16) , D ,n sin (cid:16) nπxL (cid:17) , , D ,n sin (cid:16) nπxL (cid:17)(cid:17) , where(3.2) D ,n = − (cid:0) L λ n − an π − iπ b λ n n (cid:1) A n L + iB n c λ n , and D ,n = − iA n c λ n + B n (cid:0) π n − L λ n (cid:1) L . Inserting λ n , A n , B n in D ,n and D ,n , we get D ,n = 0 and D ,n = 1. Hence we obtain( iλ n I − A ) U n = (cid:16) , , , sin (cid:16) nπxL (cid:17)(cid:17) = F n . Now, we have k U n k H ≥ Z L (cid:12)(cid:12)(cid:12) iλ n B n sin (cid:16) nπxL (cid:17)(cid:12)(cid:12)(cid:12) dx = Lλ n | B n | ∼ λ n . Therefore, for ε > λ − εn k U n k H ∼ λ εn → + ∞ . Then, we cannot expect the energy decay rate t − ε . (cid:3) Lack of exponential stability with Local Kelvin-Voigt damping.
In this part, under the equalspeed wave propagation condition (i.e. a = 1), we use the classical method developed by Littman and Markusin [18] (see also [12]), to show that system (1.2)-(1.6) with Local Kelvin-Voigt damping and global coupling isnot exponentially stable. For this aim, assume that(3.3) a = 1 , b ( x ) = (cid:26) < x ≤ , < x ≤ . , and c ( x ) = c ∈ R . Our main result in this part is following theorem.
Theorem 3.2.
Under condition (3.3) . The semigroup of contractions (cid:0) e t A (cid:1) t ≥ generated by the operator A isnot exponentially stable in the energy space H . For the proof of Theorem 3.2, we recall the following definitions: the growth bound ω ( A ) and the the spectralbound s ( A ) of A are defined respectively as ω ( A ) = inf n ω ∈ R : there exists a constant M ω such that ∀ t ≥ , (cid:13)(cid:13) e t A (cid:13)(cid:13) L ( H ) ≤ M ω e ωt o and s ( A ) = sup {ℜ ( λ ) : λ ∈ σ ( A ) } . Then, according to Theorem 2.1.6 and Lemma 2.1.11 in [12], one has that s ( A ) ≤ ω ( A ) . By the previous results, one clearly has that s ( A ) ≤ A whose realparts tend to zero.Since A is dissipative, we fix α > λ of A in the strip S = { λ ∈ C : − α ≤ Re( λ ) ≤ } . First, we determine the characteristic equation satisfied by the eigenvalues of A . For this aim, let λ ∈ C ∗ be aneigenvalue of A and let U = ( u, λu, y, λy ) ∈ D ( A ) be an associated eigenvector. Then, the eigenvalue problemis given by λ u − (1 + λ ) u xx + cλy = 0 , x ∈ (0 , , (3.4) λ y − y xx − cλu = 0 , x ∈ (0 , , (3.5)with the boundary conditions u (0) = u (1) = y (0) = y (1) = 0 . We define (cid:26) u − ( x ) := u ( x ) , y − ( x ) := y ( x ) x ∈ (0 , ) ,u + ( x ) := u ( x ) , y + ( x ) := y ( x ) x ∈ [ , . Then, system (3.4)-(3.5) becomes λ u − − u − xx + cλy − = 0 , x ∈ (0 , / , (3.6) λ y − − y − xx − cλu − = 0 , x ∈ (0 , / , (3.7) λ u + − (1 + λ ) u + xx + cλy + = 0 , x ∈ [1 / , , (3.8) λ y + − y + xx − cλu + = 0 , x ∈ [1 / , , (3.9)with the boundary conditions u − (0) = y − (0) = 0 , (3.10) u + (1) = y + (1) = 0 , (3.11)and the continuity conditions u − (1 /
2) = u + (1 / , (3.12) u − x (1 /
2) = (1 + λ ) u + x (1 / , (3.13) y − (1 /
2) = y + (1 / , (3.14) y − x (1 /
2) = y + x (1 / . (3.15) 12ere and below, in order to handle, in the case where z is a non zero non-real number, we denote by √ z thesquare root of z ; i.e., the unique complex number whose square is equal to z , that is defined by √ z = r | z | + ℜ ( z )2 + i sign( ℑ ( z )) r | z | − ℜ ( z )2 . Our aim is to study the asymptotic behavior of the largest eigenvalues λ of A in S . By taking λ large enough,the general solution of System (3.6)-(3.7) with boundary condition (3.10) is given by u − ( x ) = d λ − r c λ sinh( r x ) + d λ − r c λ sinh( r x ) ,y − ( x ) = d sinh( r x ) + d sinh( r x ) , and the general solution of System (3.6)-(3.7) with boundary condition (3.11) is given by u + ( x ) = − D λ − s c λ sinh( s (1 − x )) − D λ − s c λ sinh( s (1 − x )) ,y + ( x ) = − D sinh( s (1 − x )) − D sinh( s (1 − x )) , where d , d , D , D ∈ C ,(3.16) r = λ r icλ , r = λ r − icλ and(3.17) s = λ vuut λ + q − c λ − c λ (cid:0) λ (cid:1) , s = √ λ vuut λ + 2 − λ q − c λ − c λ (cid:0) λ (cid:1) . The boundary conditions in (3.12)-(3.15), can be expressed by M ( d d D D ) ⊤ = 0, where M = sinh( r ) sinh( r ) sinh( s ) sinh( s ) r cosh( r ) r cosh( r ) − s cosh( s ) − s cosh( s ) r sinh( r ) r sinh( r ) s sinh( s ) s sinh( s ) r cosh( r ) r cosh( r ) − s ( s − λ ( λ − s )) cosh( s ) − s ( s − λ ( λ − s )) cosh( s ) System (3.6)-(3.15) admits a non trivial solution if and only if det ( M ) = 0. Using Gaussian elimination, det ( M ) = 0 is equivalent to det ( M ) = 0, where M is given by M = sinh( r ) sinh( r ) sinh( s ) 1 − e − s r cosh( r ) r cosh( r ) − s cosh( s ) − s (1 + e − s ) r sinh( r ) r sinh( r ) s sinh( s ) s (1 − e − s ) r cosh( r ) r cosh( r ) − s ( s − λ ( λ − s )) cosh( s ) − s ( s − λ ( λ − s ))(1 + e − s ) . Then, we get(3.18) det ( M ) = F + F e − s , where F = − s s (cid:0) r − r (cid:1) (cid:0) s − s (cid:1) ( λ + 1) sinh (cid:16) r (cid:17) sinh (cid:16) r (cid:17) cosh (cid:16) s (cid:17) + r s (cid:0) r − s (cid:1) (cid:0) ( λ − s ) λ + r − s (cid:1) cosh (cid:16) r (cid:17) sinh (cid:16) r (cid:17) sinh (cid:16) s (cid:17) − r s (cid:0) r − s (cid:1) (cid:0) ( λ − s ) λ + r − s (cid:1) sinh (cid:16) r (cid:17) cosh (cid:16) r (cid:17) sinh (cid:16) s (cid:17) − r r (cid:0) r − r (cid:1) (cid:0) s − s (cid:1) cosh (cid:16) r (cid:17) cosh (cid:16) r (cid:17) sinh (cid:16) s (cid:17) + r s (cid:0) r − s (cid:1) (cid:0) ( λ − s ) λ + r − s (cid:1) sinh (cid:16) r (cid:17) cosh (cid:16) r (cid:17) cosh (cid:16) s (cid:17) − r s (cid:0) r − s (cid:1) (cid:0) ( λ − s ) λ + r − s (cid:1) cosh (cid:16) r (cid:17) sinh (cid:16) r (cid:17) cosh (cid:16) s (cid:17) F = − s s (cid:0) r − r (cid:1) (cid:0) s − s (cid:1) ( λ + 1) sinh (cid:16) r (cid:17) sinh (cid:16) r (cid:17) cosh (cid:16) s (cid:17) + r s (cid:0) r − s (cid:1) (cid:0) ( λ − s ) λ + r − s (cid:1) cosh (cid:16) r (cid:17) sinh (cid:16) r (cid:17) sinh (cid:16) s (cid:17) − r s (cid:0) r − s (cid:1) (cid:0) ( λ − s ) λ + r − s (cid:1) sinh (cid:16) r (cid:17) cosh (cid:16) r (cid:17) sinh (cid:16) s (cid:17) + r r (cid:0) r − r (cid:1) (cid:0) s − s (cid:1) cosh (cid:16) r (cid:17) cosh (cid:16) r (cid:17) sinh (cid:16) s (cid:17) − r s (cid:0) r − s (cid:1) (cid:0) ( λ − s ) λ + r − s (cid:1) sinh (cid:16) r (cid:17) cosh (cid:16) r (cid:17) cosh (cid:16) s (cid:17) + r s (cid:0) r − s (cid:1) (cid:0) ( λ − s ) λ + r − s (cid:1) cosh (cid:16) r (cid:17) sinh (cid:16) r (cid:17) cosh (cid:16) s (cid:17) . Lemma 3.3.
Let λ ∈ C be an eigenvalue of A . Then, we have ℜ ( λ ) is bounded . Proof.
Multiplying equations (3.6)-(3.9) by u − , y − , u + , y + respectively, then using the boundary conditions,we get(3.19) k λu − k + k u − x k + k λy − k + k y − x k + k λu + k + (1 + ℜ ( λ )) k u + x k + k λy + k + k y + x k = 0 . Since the operator A is dissipative then the real part of λ is negative. It is easy to see that u + x = 0, hence usingthe fact that k U k H = 1 in (3.19), we get that ℜ ( λ ) is bounded below. Therefore, there exists α >
0, such that − α ≤ ℜ ( λ ) < . (cid:3) Proposition 3.4.
Assume that the condition (3.3) holds. Then there exists n ∈ N sufficiently large and twosequences ( λ ,n ) | n |≥ n and ( λ ,n ) | n |≥ n of simple root of det ( M ) satisfying the following asymptotic behavior : Case 1.
If sin (cid:0) c (cid:1) = 0, then(3.20) λ ,n = 2 nπi + iπ − ( c )(1 − i sign ( n )) (cid:0) c ) (cid:1) p | n | π + O (cid:18) n (cid:19) and(3.21) λ ,n = 2 nπi + i arccos (cid:16) cos (cid:16) c (cid:17)(cid:17) − γ p | n | π + i sign ( n ) γ p | n | π + O (cid:18) n (cid:19) , where γ = (cid:18) cos( c ) sin (cid:18) arccos ( cos ( c ) ) (cid:19) + sin (cid:18) ( cos ( c ) ) (cid:19)(cid:19) q − cos (cid:0) c (cid:1) cos (cid:18) arccos ( cos ( c ) ) (cid:19) . Case 2.
If sin (cid:0) c (cid:1) = 0, then(3.22) λ ,n = 2 nπi + iπ + i c πn − (4 + iπ ) c π n + O (cid:18) | n | (cid:19) and(3.23) λ ,n = 2 nπi + O (cid:18) n (cid:19) . The proof of Proposition 3.4, is divided into two lemmas.
Lemma 3.5.
Assume that condition (3.3) holds. Let λ be largest eigenvalue of A , then λ is large root of thefollowing asymptotic behavior estimate(3.24) F ( λ ) := f ( λ ) + f ( λ ) λ / + f ( λ )8 λ + f ( λ )8 λ / + f ( λ )128 λ + O ( λ − / ) , f ( λ ) = cosh (cid:0) λ (cid:1) − cosh (cid:0) λ (cid:1) cos (cid:0) c (cid:1) ,f ( λ ) = sinh (cid:0) λ (cid:1) + sinh (cid:0) λ (cid:1) cos (cid:0) c (cid:1) ,f ( λ ) = c sinh (cid:0) λ (cid:1) − (cid:0) λ (cid:1) + 4 (cid:0) cosh (cid:0) λ (cid:1) cos (cid:0) c (cid:1) + c sinh (cid:0) λ (cid:1) sin (cid:0) c (cid:1)(cid:1) ,f ( λ ) = − (cid:0) λ (cid:1) + c cosh (cid:0) λ (cid:1) − c cosh (cid:0) λ (cid:1) sin (cid:0) c (cid:1) − (cid:0) λ (cid:1) cos (cid:0) c (cid:1) ,f ( λ ) = − c sinh (cid:0) λ (cid:1) + ( c + 72 c + 48) cosh (cid:0) λ (cid:1) + 32 c (cid:0) c cos (cid:0) c (cid:1) + 7 sin (cid:0) c (cid:1)(cid:1) sinh (cid:0) λ (cid:1) − (cid:0) c + 8 c sin (cid:0) c (cid:1) + 16(4 c + 3) cosh (cid:0) c (cid:1)(cid:1) cosh (cid:0) λ (cid:1) . Proof.
Let λ be a large eigenvalue of A , then λ is root of det ( M ). In this Lemma, we give an asymptoticdevelopment of the function det ( M ) for large λ . First, using the asymptotic expansion in (3.16)-(3.17), we get(3.26) r = λ + ic + c λ − ic λ + O ( λ − ) , r = λ − ic + c λ + ic λ + O ( λ − ) ,s = λ − c λ + O ( λ − ) , s = √ λ − √ λ + c +38 λ + O (cid:0) λ − / (cid:1) . From (3.26), we get(3.27) icλs s ( s − s )( λ + 1) = icλ / (cid:16) − λ + c λ + O ( λ − ) (cid:17) ,r s ( r − s ) (cid:0) ( λ − s ) λ + r − s (cid:1) = − icλ / (cid:16) − − i c λ + c +3+14 i c λ + O ( λ − ) (cid:17) ,r s ( r − s ) (cid:0) ( λ − s ) λ + r − s (cid:1) = icλ / (cid:16) − i c λ + c +3 − i c λ + O ( λ − ) (cid:17) , i cλr r (cid:0) s − s (cid:1) = icλ / (cid:16) √ λ − λ / + O ( λ − / ) (cid:17) ,r s ( r − s )(( λ − s ) λ + r − s ) = − icλ / (cid:16) √ λ − − ic λ / + O (cid:0) λ − / (cid:1)(cid:17) ,r s ( r − s )(( λ − s ) λ + r − s ) = icλ / (cid:16) √ λ − ic λ / + O ( λ − / ) (cid:17) . From equation (3.27) and using the fact that ℜ ( λ ) is bounded, we get (3.28) F icλ / = − (cid:20)(cid:18) − λ + 4 c + 34 λ (cid:19) sinh (cid:16) r (cid:17) sinh (cid:16) r (cid:17) cosh (cid:16) s (cid:17) + (cid:18) − λ + 5 c + 38 λ (cid:19) (cid:16) cosh (cid:16) r (cid:17) sinh (cid:16) r (cid:17) + sinh (cid:16) r (cid:17) cosh (cid:16) r (cid:17)(cid:17) sinh (cid:16) s (cid:17) + (cid:18) i c λ + 7 i c λ (cid:19) (cid:16) cosh (cid:16) r (cid:17) sinh (cid:16) r (cid:17) − sinh (cid:16) r (cid:17) cosh (cid:16) r (cid:17)(cid:17) sinh (cid:16) s (cid:17) + (cid:18) √ λ − λ / (cid:19) cosh (cid:16) r (cid:17) cosh (cid:16) r (cid:17) sinh (cid:16) s (cid:17) + (cid:18) √ λ − λ / (cid:19) (cid:16) sinh (cid:16) r (cid:17) cosh (cid:16) r (cid:17) + cosh (cid:16) r (cid:17) sinh (cid:16) r (cid:17)(cid:17) cosh (cid:16) s (cid:17) + (cid:18) i c λ / (cid:19) (cid:16) sinh (cid:16) r (cid:17) cosh (cid:16) r (cid:17) − cosh (cid:16) r (cid:17) sinh (cid:16) r (cid:17)(cid:17) cosh (cid:16) s (cid:17) + O (cid:16) λ − / (cid:17) (cid:21) . From equation (3.27) and using the fact that ℜ ( λ ) is bounded, we get(3.29) F = − i c λ / (cid:20) (cid:16) r (cid:17) sinh (cid:16) r (cid:17) cosh (cid:16) s (cid:17) + (cid:16) cosh (cid:16) r (cid:17) sinh (cid:16) r (cid:17) + sinh (cid:16) r (cid:17) cosh (cid:16) r (cid:17)(cid:17) sinh (cid:16) s (cid:17) + O (cid:16) λ − / (cid:17) (cid:21) . Since the real part of √ λ is positive, then lim | λ |→∞ λ − / e −√ λ = 0 , e −√ λ = o ( λ − / ) , then,(3.31) F e − s = − icλ / (cid:16) o ( λ − / ) (cid:17) . Inserting (3.28) and (3.31), in (3.18), we getdet(M ) = − ic λ / F ( λ ) , where,(3.32) F ( λ ) = (cid:18) − λ + 4 c + 38 λ (cid:19) (cid:18) cosh (cid:18) r + r (cid:19) − cosh (cid:18) r − r (cid:19)(cid:19) cosh (cid:16) s (cid:17) + (cid:18) − λ + 5 c + 38 λ (cid:19) sinh (cid:18) r + r (cid:19) sinh (cid:16) s (cid:17) − (cid:18) i c λ + 7 i c λ (cid:19) sinh (cid:18) r − r (cid:19) sinh (cid:16) s (cid:17) + (cid:18) √ λ − λ / (cid:19) (cid:18) cosh (cid:18) r + r (cid:19) + cosh (cid:18) r − r (cid:19)(cid:19) sinh (cid:16) s (cid:17) + (cid:18) √ λ − λ / (cid:19) sinh (cid:18) r + r (cid:19) cosh (cid:16) s (cid:17) + (cid:18) i c λ / (cid:19) sinh (cid:18) r − r (cid:19) cosh (cid:16) s (cid:17) + O (cid:16) λ − / (cid:17) . Therefore, system (3.10)-(3.15) admits a non trivial solution if and only if det(M ) = 0, if and only if theeigenvalues of A are roots of the function F . Next, from (3.26) and the fact that real λ is bounded, we get(3.33) cosh (cid:0) r + r (cid:1) = cosh( λ ) + c sinh( λ )8 λ + c cosh( λ )128 λ + O ( λ − ) , cosh (cid:0) r − r (cid:1) = cos (cid:0) c (cid:1) + c sin ( c ) λ + O ( λ − ) , sinh (cid:0) r + r (cid:1) = sinh( λ ) + c cosh( λ )8 λ + c sinh( λ )128 λ + O ( λ − ) , sinh (cid:0) r − r (cid:1) = i sin (cid:0) c (cid:1) − i c cos ( c ) λ + O ( λ − ) , sinh (cid:0) s (cid:1) = sinh( λ ) − c cosh ( λ ) λ + O ( λ − ) , cosh( s ) = cosh (cid:0) λ (cid:1) − c sinh ( λ ) λ + O ( λ − ) . Inserting (3.33) in (3.32), we get (3.24). (cid:3)
Lemma 3.6.
Under condition (3.3) , there exists n ∈ N sufficiently large and two sequences ( λ ,n ) | n |≥ n and ( λ ,n ) | n |≥ n of simple roots of F satisfying the following asymptotic behavior (3.34) λ ,n = 2 inπ + iπ + ǫ ,n where lim | n |→ + ∞ ǫ ,n = 0 and (3.35) λ ,n = 2 nπi + i arccos (cid:16) cos (cid:16) c (cid:17)(cid:17) + ǫ ,n where lim | n |→ + ∞ ǫ ,n = 0 . Proof.
First, we look at the roots of f . From (3.25), we deduce that f can be written as(3.36) f ( λ ) = 2 cosh (cid:18) λ (cid:19) (cid:16) cosh( λ ) − cos (cid:16) c (cid:17)(cid:17) . Then, the roots of f are given by (cid:26) µ ,n = 2 nπi + iπ, n ∈ Z ,µ ,n = 2 nπi + i arccos (cid:0) cos (cid:0) c (cid:1)(cid:1) , n ∈ Z . F are close to f . Let us start withthe first family µ ,n . Let B n = B ((2 n + 1) πi, r n ) be the ball of centrum (2 n + 1) πi and radius r n = | n | − and λ ∈ ∂ B n ; i.e. λ n = 2 nπi + iπ + r n e iθ , θ ∈ [0 , π [. Then(3.37) cosh (cid:18) λ (cid:19) = i ( − n r n e iθ O ( r n ) , and cosh( λ ) = − O ( r n ) . Inserting (3.37) in (3.36), we get f ( λ ) = − i ( − n r n e iθ (cid:16) (cid:16) c (cid:17) + O ( r n ) (cid:17) . It follows that there exists a positive constant C such that ∀ λ ∈ ∂ B n , | f ( λ ) | ≥ C r n = C | n | − . On the other hand, from (3.24), we deduce that | F ( λ ) − f ( λ ) | = O (cid:18) √ λ (cid:19) = O p | n | ! . It follows that, for | n | large enough ∀ λ ∈ ∂ B n , | F ( λ ) − f ( λ ) | < | f ( λ ) | . Hence, with the help of Rouch´e’s theorem, there exists n ∈ N ∗ large enough, such that ∀ | n | ≥ n , the firstbranch of roots of F denoted by λ ,n are close to µ ,n , that is(3.38) λ ,n = µ ,n + iπ + ǫ ,n where lim | n |→ + ∞ ǫ ,n = 0 . Passing to the second family µ ,n . Let ˜ B n = B ( µ ,n , r n ) be the ball of centrum µ ,n and radius r n := | n | if sin (cid:0) c (cid:1) = 0 , | n | if sin (cid:0) c (cid:1) = 0 , such that λ ∈ ∂ ˜ B n ; i.e. λ n = µ ,n + r n e iθ , θ ∈ [0 , π [. Then,cosh( λ ) − cos (cid:16) c (cid:17) = cosh (cid:16) nπi + i arccos (cid:16) cos (cid:16) c (cid:17) + r n e iθ (cid:17)(cid:17) − cos (cid:16) c (cid:17) . It follow that,(3.39) cosh( λ ) − cos (cid:16) c (cid:17) = i r n r − cos (cid:16) c (cid:17) e iθ + r n cos (cid:0) c (cid:1) e iθ O ( r n ) , and(3.40) cosh (cid:18) λ (cid:19) = ( − n cos arccos (cid:0) cos (cid:0) c (cid:1)(cid:1) ! + ir n e iθ ( − n arccos (cid:0) cos (cid:0) c (cid:1)(cid:1) ! + O ( r n ) . Inserting (3.39) and (3.40) in (3.36), we get(3.41) f ( λ ) = R e iθ r n + R e iθ r n + O ( r n ) , where(3.42) R = i ( − n q − cos (cid:0) c (cid:1) cos (cid:18) arccos ( cos ( c )) (cid:19) ,R = − ( − n q − cos (cid:0) c (cid:1) sin (cid:18) arccos ( cos ( c )) (cid:19) + ( − n cos (cid:0) c (cid:1) cos (cid:18) arccos(cos ( c ) )2 (cid:19) . We distinguish two cases:
Case 1.
If sin (cid:0) c (cid:1) = 0, then R = 0 and R = ( − n = 0 . It follows that there exists a positive constant C such that ∀ λ ∈ ∂ ˜ B n , | f ( λ ) | ≥ C r n = C | n | − . ase 2. If sin (cid:0) c (cid:1) = 0, then R = 0. It follows that, there exists a positive constant C such that ∀ λ ∈ ∂ ˜ B n , | f ( λ ) | ≥ C r n = C | n | − . On the other hand, from (3.24), we deduce that | F ( λ ) − f ( λ ) | = O (cid:18) √ λ (cid:19) = O p | n | ! . In both cases, for | n | large enough, we have ∀ λ ∈ ∂ ˜ B n , | F ( λ ) − f ( λ ) | < | f ( λ ) | . Hence, with the help of Rouch´e’s Theorem, there exists n ∈ N ∗ large enough, such that ∀| n | ≥ n , the secondbranch of roots of F , denoted by λ ,n are close to µ ,n that is defined in equation (3.35). The proof is thuscomplete. (cid:3) We are now in position to conclude the proof of Proposition 3.4.
Proof of Proposition 3.4.
The proof is divided into two steps.
Calculation of ǫ ,n . From (3.38), we have(3.43) cosh (cid:18) λ ,n (cid:19) = − i ( − n sinh (cid:18) ǫ ,n (cid:19) , sinh (cid:18) λ ,n (cid:19) = − i ( − n cosh (cid:18) ǫ ,n (cid:19) , cosh (cid:18) λ ,n (cid:19) = i ( − n sinh (cid:16) ǫ ,n (cid:17) , sinh (cid:18) λ ,n (cid:19) = i ( − n cosh (cid:16) ǫ ,n (cid:17) , λ ,n = − i πn + i πn + O (cid:0) ǫ ,n n − (cid:1) + O (cid:0) n − (cid:1) , λ ,n = − π n + O (cid:0) n − (cid:1) p λ ,n = 1 − i sign( n )2 p π | n | + i − sign( n )8 p π | n | + O (cid:16) ǫ ,n | n | − / (cid:17) + O (cid:16) | n | − / (cid:17) , q λ ,n = − − i sign( n )4 p π | n | + O (cid:16) | n | − / (cid:17) , q λ ,n = O (cid:16) | n | − / (cid:17) . On the other hand, since lim | n |→ + ∞ ǫ ,n = 0, we have the asymptotic expansion(3.44) sinh (cid:18) ǫ ,n (cid:19) = 3 ǫ ,n O ( ǫ ,n ) , cosh (cid:18) ǫ ,n (cid:19) = 1 + 9 ǫ ,n O ( ǫ ,n ) , sinh (cid:16) ǫ ,n (cid:17) = ǫ ,n O ( ǫ ,n ) , cosh (cid:16) ǫ ,n (cid:17) = 1 + ǫ ,n O ( ǫ ,n ) . Inserting (3.44) in (3.43), we get(3.45) cosh (cid:18) λ ,n (cid:19) = − i ( − n ǫ ,n O ( ǫ ,n ) , sinh (cid:18) λ ,n (cid:19) = − i ( − n − i ( − n ǫ ,n O ( ǫ ,n ) , cosh (cid:18) λ ,n (cid:19) = i ( − n ǫ ,n O ( ǫ ,n ) , sinh (cid:18) λ ,n (cid:19) = i ( − n + i ( − n ǫ ,n O ( ǫ ,n ) , λ ,n = − i πn + i πn + O (cid:0) ǫ ,n n − (cid:1) + O (cid:0) n − (cid:1) , λ ,n = − π n + O (cid:0) n − (cid:1) p λ ,n = 1 − i sign( n )2 p π | n | + i − sign( n )8 p π | n | + O (cid:16) ǫ ,n | n | − / (cid:17) + O (cid:16) | n | − / (cid:17) , q λ ,n = − − i sign( n )4 p π | n | + O (cid:16) | n | − / (cid:17) , q λ ,n = O (cid:16) | n | − / (cid:17) . ǫ ,n (cid:16) (cid:16) c (cid:17)(cid:17) (cid:18) i π n (cid:19) + (1 − i sign( n )) (cid:0) − cos (cid:0) c (cid:1)(cid:1) p π | n | + i c (cid:0) (cid:0) c (cid:1) − c (cid:1) πn − (2 + iπ ) (1 + i sign( n )) (cid:0) − cos (cid:0) c (cid:1)(cid:1) p π | n | + 4 c (7 − iπ ) sin (cid:0) c (cid:1) + c (2 iπ + 5 + 4 cos (cid:0) c (cid:1) )64 π n + O (cid:16) | n | − / (cid:17) + O (cid:16) ǫ ,n | n | − / (cid:17) + O (cid:16) ǫ ,n | n | − / (cid:17) + O (cid:0) ǫ ,n (cid:1) = 0 . We distinguish two cases.
Case 1.
If sin (cid:0) c (cid:1) = 0, then 1 − cos (cid:16) c (cid:17) = 2 sin (cid:16) c (cid:17) = 0, then from (3.46), we get ǫ ,n (cid:16) (cid:16) c (cid:17)(cid:17) + sin (cid:0) c (1 − i sign( n )) (cid:1)p | n | π + O ( ǫ ,n ) + O ( | n | − / ǫ ,n ) + O ( n − ) = 0 , hence, we get(3.47) ǫ ,n = − (cid:0) c (cid:1) (1 − i sign( n )) (cid:0) (cid:0) c (cid:1)(cid:1) + O ( n − ) . Inserting (3.47) in (3.38), we get (3.22).
Case 2.
If sin (cid:0) c (cid:1) = 0,1 − cos (cid:16) c (cid:17) = 2 sin (cid:16) c (cid:17) = 0 , sin (cid:16) c (cid:17) = 2 sin (cid:16) c (cid:17) cos (cid:16) c (cid:17) = 0 , then, from (3.46), we get2 ǫ ,n (cid:18) i πn (cid:19) − i c πn + c (2 iπ + 9)64 π n + O (cid:16) | n | − / (cid:17) + O (cid:16) ǫ ,n | n | − / (cid:17) + O (cid:16) ǫ ,n | n | − / (cid:17) + O (cid:0) ǫ ,n (cid:1) = 0 . (3.48)By a straightforward calculation in equation (3.48), we get(3.49) ǫ ,n = i c πn − (4 + i π ) c π n + O (cid:16) | n | − / (cid:17) . Inserting (3.49) in (3.38), we get (3.21).
Calculation of ǫ ,n . From (3.35), we have(3.50) 1 p λ ,n = 1 − i sign( n )2 p | n | π + O (cid:16) | n | − / (cid:17) and 1 λ ,n = O ( n − ) . Inserting (3.35) and (3.50) in (3.24), we get(3.51) cosh (cid:18) λ ,n (cid:19) (cid:16) cosh( λ ,n ) − cos (cid:16) c (cid:17)(cid:17) + (1 − i sign( n )) (cid:16) sinh (cid:16) λ ,n (cid:17) + sinh (cid:16) λ ,n (cid:17) cos (cid:0) c (cid:1)(cid:17) p | n | π + O ( n − ) = 0 . On the other hand, we have(3.52) cosh( λ ,n ) − cos (cid:0) c (cid:1) = cosh (cid:0) nπi + i arccos (cid:0) cos (cid:0) c (cid:1)(cid:1) + ǫ ,n (cid:1) − cos (cid:0) c (cid:1) = cos (cid:0) c (cid:1) cosh( ǫ ,n ) + i q − cos (cid:0) c (cid:1) sinh( ǫ ,n ) − cos (cid:0) c (cid:1) = i ǫ ,n q − cos (cid:0) c (cid:1) + O ( ǫ ,n ) , cosh (cid:16) λ ,n (cid:17) = ( − n cos (cid:18) arccos ( cos ( c )) (cid:19) + O ( ǫ ,n ) , sinh (cid:16) λ ,n (cid:17) = i ( − n sin (cid:18) arccos ( cos ( c )) (cid:19) + O ( ǫ ,n ) , sinh (cid:16) λ ,n (cid:17) = i ( − n sin (cid:18) ( cos ( c )) (cid:19) + O ( ǫ ,n ) . Inserting (3.52) and (3.53) in (3.51), we get(3.54) ǫ ,n cos (cid:18) arccos ( cos ( c )) (cid:19) q − cos (cid:0) c (cid:1) + O (cid:0) ǫ ,n n (cid:1) + O (cid:0) n (cid:1) + (1 − i sign( n )) (cid:18) sin (cid:18) arccos ( cos ( c )) (cid:19) + cos (cid:0) c (cid:1) sin (cid:18) arccos ( cos ( c )) (cid:19)(cid:19) p | n | π = 0 . We distinguish two cases.
Case 1.
If sin (cid:0) c (cid:1) = 0, then from (3.54), we get(3.55) ǫ ,n = − (cid:18) cos( c ) sin (cid:18) arccos ( cos ( c ) ) (cid:19) + sin (cid:18) ( cos ( c ) ) (cid:19)(cid:19) (1 − i sign( n ))4 q − cos (cid:0) c (cid:1) cos (cid:18) arccos ( cos ( c ) ) (cid:19) p π | n | + O ( n − ) . Inserting (3.55) in (3.35), we get (3.21).
Case 2.
If sin (cid:0) c (cid:1) = 0, we get(3.56) ǫ ,n = O ( n − ) . Inserting (3.56) in (3.35), we get (3.23). Thus, the proof is complete.
Proof of Theorem 3.2.
From Proposition 3.4, the operator A has two branches of eigenvalues such that thereal parts tending to zero. Then the energy corresponding to the first and second branch of eigenvalues is notexponentially decaying. Then the total energy of the wave equations with local Kelvin-Voigt damping withglobal coupling are not exponentially stable in the equal speed case.4. Polynomial Stability
From Section 3, System (1.2)-(1.6) is not uniformly (exponentially) stable, so we look for a polynomial decayrate. As the condition i R ⊂ ρ ( A ) is already checked in Lemma 2.4, following Theorem 5.13, it remains to provethat condition (5.39) holds. This is made with the help of a specific multiplier and by using the exponentialdecay of an auxiliary problem. Our main result in this section is the following theorem. Theorem 4.1.
There exists a constant c > U , such that the energy of system (1.2)-(1.6)satisfies the following estimation:(4.1) E ( t ) ≤ ct k U k D ( A ) , ∀ t > , ∀ U ∈ D ( A ) . According to Theorem 5.13, by taking ℓ = 2, the polynomial energy decay (4.1) holds if the following conditions(H1) i R ⊂ ρ ( A ) , and(H2) sup λ ∈ R (cid:13)(cid:13)(cid:13) ( iλI − A ) − (cid:13)(cid:13)(cid:13) L ( H ) = O (cid:0) | λ | (cid:1) , are satisfied. Condition (H1) is already proved in Lemma 2.4. We will prove condition (H2) using an argument ofcontradiction. For this purpose, suppose that (H2) is false, then there exists { ( λ n , U n = ( u n , v n , y n , z n )) } n ≥ ⊂ R × D ( A ) and(4.2) λ n → + ∞ , k U n k H = 1 , λ n ( iλ n U n − A U n ) = ( f ,n , g ,n , f ,n , g ,n ) := F n → H . For simplicity, we drop the index n . Detailing Equation (4.3), we obtain iλu − v = λ − f −→ H (0 , L ) , (4.4) iλv − ( au x + b ( x ) v x ) x + c ( x ) z = λ − g −→ L (0 , L ) , (4.5) iλy − z = λ − f −→ H (0 , L ) , (4.6) iλz − y xx − c ( x ) v = λ − g −→ L (0 , L ) . (4.7)Here we will check the condition (H2) by finding a contradiction with (4.2) such as k U k H = o (1). For clarity,we divide the proof into several lemmas. By taking the inner product of (4.3) with U in H , we remark that Z L b ( x ) | v x | dx = −ℜ ( hA U, U i H ) = ℜ ( h ( iλI − A ) U, U i H ) = o (cid:0) λ − (cid:1) . Then,(4.8) Z α α | v x | dx = o (cid:0) λ − (cid:1) . Remark 4.2.
Since v and z are uniformly bounded in L (0 , L ), then from equations (4.4) and (4.6), thesolution ( u, v, y, z ) ∈ D ( A ) of (4.4)-(4.7) satisfies the following asymptotic behavior estimation k u k = O (cid:0) λ − (cid:1) , (4.9) k y k = O (cid:0) λ − (cid:1) . (4.10)Using equation (4.4), and equation (4.8) we get(4.11) Z α α | u x | dx = o (cid:0) λ − (cid:1) . Lemma 4.3.
Let ε < α − α , the solution ( u, v, y, z ) ∈ D ( A ) of the system (4.4)-(4.7) satisfies the followingestimation(4.12) Z α − εα + ε | v | dx = o (1) and Z α − εα + ε | λu | dx = o (1) . Proof.
We define the function ρ ∈ C ∞ (0 , L ) by(4.13) ρ ( x ) = x ∈ ( α + ǫ, α − ǫ ) , x ∈ (0 , α ) ∪ ( α , L ) , ≤ ρ ≤ elsewhere. Multiply equation (4.5) by 1 λ ρ ¯ v , integrate over (0 , L ), using the fact that k g k L (0 ,L ) = o (1) and v is uniformlybounded in L (Ω), we get(4.14) Z L iρ | v | dx + 1 λ Z L ( au x + b ( x ) v x ) ( ρ ′ ¯ v + ρ ¯ v x ) dx + 1 λ Z L c ( x ) zρ ¯ vdx = o ( λ − ) . Using Equation (4.8), Remark 4.2 and the fact that v and z are uniformly bounded in L (Ω), we get(4.15) 1 λ Z L ( au x + b ( x ) v x ) ( ρ ′ ¯ v + ρ ¯ v x ) dx = o ( λ − ) and 1 λ Z L c ( x ) zρ ¯ vdx = o (1) . Inserting Equation (4.15) in Equation (4.14), we obtain(4.16) Z L iρ | v | dx = o (1) . λρ ¯ u integrateover (0 , L ) and using the fact that k f k H (Ω) = o (1) and Remark 4.2, we get Z L iρ | λu | dx − Z L ρλv ¯ udx = o ( λ − ) . Using Equation (4.16), we get Z L iρ | λu | dx = o (1) . Then, we obtain the desired second estimation in Equation (4.12). (cid:3)
Inserting equations (4.4) and (4.6) respectively in equations (4.5) and (4.7), we get λ u + ( au x + b ( x ) v x ) x − iλc ( x ) y = F , (4.17) λ y + y xx + iλc ( x ) u = F , (4.18)where(4.19) F = − λ − g − iλ − f − c ( x ) λ − f and F = − λ − g − iλ − f + c ( x ) λ − f . Lemma 4.4.
Let ε < α − α , the solution ( u, v, y, z ) ∈ D ( A ) of the system (4.4)-(4.7) satisfies the followingestimation(4.20) Z α − εα | λy | dx = o (1) and Z α − εα | z | dx = o (1) . Proof.
We define the function ζ ∈ C ∞ (0 , L ) by(4.21) ζ ( x ) = x ∈ ( α + 2 ε, α − ε ) , x ∈ (0 , α + ε ) ∪ ( α − ε, L ) , ≤ ζ ≤ elsewhere. Multiply equations (4.17) by λζ ¯ y and (4.18) by λζ ¯ u respectively, integrate over (0 , L ), using Remark 4.2 andthe fact that k F k H = k ( f , g , f , g ) k H = o (1), we get(4.22) Z L λ ζu ¯ ydx − Z L λ ( au x + b ( x ) v x ) ( ζ ′ ¯ y + ζ ¯ y x ) dx − i Z L c ( x ) ζ ( x ) | λy | dx = o ( λ − )and(4.23) Z L λ ζy ¯ udx − Z L λy x ζ ′ ¯ u x dx − Z L λy x ζ ¯ u x dx + i Z L c ( x ) ζ ( x ) | λu | dx = o ( λ − ) . Using Remark 4.2, Lemma 4.3 and the fact that y x is uniformly bounded in L (0 , L ), we get(4.24) Z L λ ( au x + b ( x ) v x ) ( ζ ′ ¯ y + ζ ¯ y x ) dx = o (1) , − Z L λy x ζ ′ ¯ u x dx = o (1) and Z L λy x ζ ¯ u x dx = o (1) . Using Lemma 4.3, we have that(4.25) Z L c ( x ) ζ | λu | dx = o (1) . Inserting Equations (4.24) and (4.25) in Equations (4.22) and (4.23), and summing the result by taking theimaginary part, and using the definition of the functions c and ζ , we get the first estimation of Equation (4.20).Now, multiplying equation (4.6) by ¯ z , integrating over ( α , α − ε ) and using the fact that k f k H (0 ,L ) = o (1)and z is uniformly bounded in L (0 , L ), in particular in L ( α , α − ε ), we get Z α − εα iλy ¯ zdx − Z α − εα | z | dx = o ( λ − ) . Then, using the first estimation of Equation (4.20), we get the second desired estimation of Equation (4.20). (cid:3)
Lemma 4.5.
Let 0 < α < α < α < α < L and suppose that ε < α − α , and c ( x ) the function defined inEquation (1.5). Then, for any λ ∈ R , the solution ( ϕ, ψ ) ∈ (( H (0 , L ) ∩ H (0 , L )) of system(4.26) λ ϕ + aϕ xx − iλ (cid:0) ( α ,α − ε ) (cid:1) ( x ) ϕ − iλc ( x ) ψ = u, x ∈ (0 , L ) λ ψ + ψ xx − iλ (cid:0) ( α ,α − ε ) (cid:1) ( x ) ψ + iλc ( x ) ϕ = y, x ∈ (0 , L ) ϕ (0) = ϕ ( L ) = 0 ,ψ (0) = ψ ( L ) = 0 , satisfies the following estimation(4.27) k λϕ k L (0 ,L ) + k ϕ x k L (0 ,L ) + k λψ k L (0 ,L ) + k ψ x k L (0 ,L ) ≤ M (cid:16) k u k L (0 ,L ) + k y k L (0 ,L ) (cid:17) . Proof.
Following Theorem 5.2, the exponential stability of System (5.1), proved in the Appendix, implies thatthe resolvent of the auxiliary operator A a defined by (5.2)-(5.3) is uniformly bounded on the imaginary axisi.e. there exists M > λ ∈ R k ( iλI − A a ) − k L ( H a ) ≤ M < + ∞ where H a = (cid:0) H (0 , L ) × L (0 , L ) (cid:1) . Now, since ( u, y ) ∈ H (0 , L ) × H (0 , L ), then (0 , − u, , − y ) belongs to H a , and from (4.28), there exists ( ϕ, η, ψ, ξ ) ∈ D ( A a ) such that ( iλI − A a ) ( ϕ, η, ψ, ξ ) = (0 , − u, , − y ) ⊤ i.e.iλϕ − η = 0 , (4.29) iλη − aϕ xx + (cid:0) ( α ,α − ε ) (cid:1) ( x ) η + c ( x ) ξ = − u, (4.30) iλψ − ξ = 0 , (4.31) iλξ − ψ xx + (cid:0) ( α ,α − ε ) (cid:1) ( x ) ξ − c ( x ) η = − y, (4.32)such that(4.33) k ( ϕ, η, ψ, ξ ) k H a ≤ M (cid:0) k u k L (0 ,L ) + k y k L (0 ,L ) (cid:1) . From equations (4.29)-(4.33), we deduce that ( ϕ, ψ ) is a solution of (4.26) and we have k λϕ k L (0 ,L ) + k ϕ x k L (0 ,L ) + k λψ k L (0 ,L ) + k ψ x k L (0 ,L ) ≤ M (cid:16) k u k L (0 ,L ) + k y k L (0 ,L ) (cid:17) . Then, we get our desired result. (cid:3)
Remark 4.6.
There was no reference found for the proof of the exponential stability of System (5.1) when thecoefficients of the damping and the coupling are both non smooth. For this, we give the proof of the exponentialstability of System (5.1) in Theorem 5.2 (see Subsection 5.1 in Appendix section).
Lemma 4.7.
Let ε < α − α . Then, the solution ( u, v, y, z ) ∈ D ( A ) of (4.4)-(4.7) satisfies the followingasymptotic behavior estimation(4.34) Z L | λu | dx = o (1) , and(4.35) Z L | λy | dx = o (1) . Proof.
The proof of this Lemma is divided into two steps.
Step 1. λ ¯ ϕ , integrate over (0 , L ), and using Equation (4.27) and the facts that u isuniformly bounded in L (0 , L ) and k F k H = k ( f , g , f , g ) k H = o (1), we get(4.36) Z L (cid:0) λ ¯ ϕ + a ¯ ϕ xx (cid:1) λ udx − Z L λ b ( x ) v x ¯ ϕ x dx − Z L iλc ( x ) y ¯ ϕdx = o ( λ − ) . Using Equations (4.8) and (4.27), we get(4.37) Z L λ b ( x ) v x ¯ ϕ x dx = o (1) . Combining Equations (4.36) and (4.37), we obtain(4.38) Z L (cid:0) λ ¯ ϕ + a ¯ ϕ xx (cid:1) λ udx − Z L iλ c ( x ) y ¯ ϕdx = o (1) . From System (4.26), we have(4.39) λ ¯ ϕ + a ¯ ϕ xx = − iλ (cid:0) ( α ,α − ε ) (cid:1) ( x ) ¯ ϕ − iλc ( x ) ¯ ψ + ¯ u. Substituting (4.39) in (4.38), we get(4.40) Z L | λu | dx − Z L iλ (cid:0) ( α ,α − ε ) (cid:1) ( x ) u ¯ ϕdx − Z L iλ c ( x ) ¯ ψudx − Z L iλ c ( x ) y ¯ ϕdx = o (1) . Using Remark 4.2, Lemma 4.3 and Equation (4.27), we obtain(4.41) Z L iλ (cid:0) ( α ,α − ε ) (cid:1) ( x ) u ¯ ϕdx = o (1) . Inserting Equation (4.41) in Equation (4.40), we get(4.42) Z L | λu | dx − Z L iλ c ( x ) ¯ ψudx − Z L iλ c ( x ) y ¯ ϕdx = o (1) . Step 2.
Multiplying equation (4.18) by λ ¯ ψ , integrate over (0 , L ), and using Equation (4.27) and the facts that y isuniformly bounded in L (0 , L ) and k F k H = k ( f , g , f , g ) k H = o (1), we get(4.43) Z L (cid:0) λ ¯ ψ + ¯ ψ xx (cid:1) λ ydx + Z L iλc ( x ) u ¯ ψdx = o ( λ − ) . From System (4.26), we have(4.44) λ ¯ ψ + a ¯ ψ xx = − i (cid:0) ( α ,α − ε ) (cid:1) ( x ) ¯ ψ + iλc ( x ) ¯ ϕ + ¯ y. Substituting (4.44) in (4.43), we get(4.45) Z L | λy | dx − Z L iλ (cid:0) ( α ,α − ε ) (cid:1) ( x ) y ¯ ψdx + Z L iλ c ( x ) ¯ ϕydx + Z L iλ c ( x ) u ¯ ψdx = o ( λ − ) . Using Remark 4.2, Lemma 4.4 and Equation (4.27), we obtain(4.46) Z L iλ (cid:0) ( α ,α − ε ) (cid:1) ( x ) y ¯ ψdx = o (1) . Inserting Equation (4.46) in Equation (4.45), we get(4.47) Z L | λy | dx + Z L iλ c ( x ) ¯ ϕydx + Z L iλ c ( x ) u ¯ ψdx = o (1) . Finally, summing up equations (4.42) and (4.47) we get Z L | λu | dx = o (1) and Z L | λy | dx = o (1) . Hence,(4.48) Z L | v | dx = o (1) and Z L | z | dx = o (1) . (cid:3) Lemma 4.8.
The solution ( u, v, y, z ) ∈ D ( A ) of the (4.4)-(4.7) satisfies the following asymptotic behaviorestimations(4.49) Z L | u x | dx = o (1) and Z L | y x | dx = o (1) . Proof.
Multiplying (4.17) by ¯ u integrate over (0 , L ), using the fact that k F k H = k ( f , g , f , g ) k H = o (1) and u is uniformly bounded in L (0 , L ), we get(4.50) Z L | λu | dx − Z L a | u x | dx − Z L b ( x ) v x ¯ u x dx − Z L iλc ( x ) y ¯ udx = o ( λ − ) . Using equations (4.8) and (4.34), we get Z L | u x | dx = o (1) . Similarly, multiply (4.18) by ¯ y and integrate, we get Z L | y x | dx = o (1) . The proof has been completed. (cid:3)
Proof of Theorem 4.1. . Consequently, from the results of Lemmas 4.7 and 4.8, we obtain Z L (cid:0) | v | + | z | + a | u x | + | y x | (cid:1) dx = o (1) . Hence k U k H = o (1), which contradicts (4.2). Consequently, condition (H2) holds. This implies, from Theorem5.13, the energy decay estimation (4.1). The proof is thus complete.5. Appendix
Exponential stability of locally coupled wave equations with non-smooth coefficients.
Weconsider the following auxiliary problem,(5.1) ϕ tt − aϕ xx + (cid:0) ( α ,α − ε ) (cid:1) ( x ) ϕ t + c ( x ) ψ t = 0 , ( x, t ) ∈ (0 , L ) × R + ,ψ tt − ψ xx + (cid:0) ( α ,α − ε ) (cid:1) ( x ) ψ t − c ( x ) ϕ t = 0 , ( x, t ) ∈ (0 , L ) × R + ,ϕ (0 , t ) = ϕ ( L, t ) = 0 , t > ,ψ (0 , t ) = ψ ( L, t ) = 0 , t > . Since, we have a system of coupled wave equations with two interior damping acting on a part of the interval(0 , L ), then system (5.1) is exponentially stable in the associated energy space H a = (cid:0) H (0 , L ) × L (0 , L ) (cid:1) .In this section, our aim is to show that the auxiliary problem (5.1) is uniformly stable. The energy of System(5.1) is given by E a ( t ) = 12 Z L | ϕ t | + a | ϕ x | + | ψ t | + | ψ x | dx ! and by a straightforward calculation, we have ddt E a ( t ) = − Z L (cid:0) ( α ,α − ε ) (cid:1) ( x ) | ϕ t | dx − Z L (cid:0) ( α ,α − ε ) (cid:1) ( x ) | ψ t | dx ≤ . Thus, System (5.1) is dissipative in the sense that its energy is a non-increasing function with respect to thetime variable t . The auxiliary energy Hilbert space of Problem (5.1) is given by H a = (cid:0) H (0 , L ) × L (0 , L ) (cid:1) . We denote by η = ϕ t and ξ = ψ t . The auxiliary energy space H a is endowed with the following norm k Φ k H a = k η k + a k ϕ x k + k ξ k + k ψ x k , k · k denotes the norm of L (0 , L ). We define the unbounded linear operator A a by(5.2) D ( A a ) = (cid:0) ( H (0 , L ) ∩ H (0 , L )) × H (0 , L ) (cid:1) , and(5.3) A a ( ϕ, η, ψ, ξ ) = ( η, aϕ xx − (cid:0) ( α ,α − ε ) (cid:1) ( x ) η − c ( x ) ξ, ξ, ψ xx − (cid:0) ( α ,α − ε ) (cid:1) ( x ) ξ + c ( x ) η ) ⊤ . If Φ = ( ϕ, ψ, η, ξ ) is the state of System (5.1), then this system is tranformed into a first order evolutionequation on the auxiliary Hilbert space H a given byΦ t = A a Φ , Φ(0) = Φ , where Φ = ( ϕ , η , ψ , ξ ). It is easy to see that A a is m-dissipative and generates a C − semigroup ofcontractions (cid:0) e t A a (cid:1) t ≥ . Theorem 5.1.
The C − semigroup of contractions ( e t A a ) t ≥ is strongly stable on H a , i.e. for all U ∈H a , lim t → + ∞ k e t A a U k H a = 0 . Proof.
Following Arendt and Batty Theorem in [6], we have to prove the following two conditions1. A has no pure imaginary eigenvalues,2. σ ( A ) ∩ i R is countable.In order to prove these two conditions we proceed with the same argument of subsection 2.2 and we reach thedesired result. (cid:3) Now, we present the main result of this section
Theorem 5.2.
The C − semigroup of contractions (cid:0) e t A a (cid:1) t ≥ is exponentially stable, i.e. there exists constants M ≥ and τ > independent of Φ such that (cid:13)(cid:13) e t A a Φ (cid:13)(cid:13) H a ≤ M e − τt k Φ k H a , t ≥ . According to Huang [17] and Pruss [31], we have to check if the following conditions hold:(H3) i R ⊆ ρ ( A a )and(H4) sup λ ∈ R k ( iλI − A a ) − k L ( H a ) = O (1) . By using the same argument of Lemma 2.4, the operator A a has no pure imaginary eigenvalues. Then, condition(H3) holds. We will prove condition (H4) using an argument of contradiction. Indeed, suppose there exists { ( λ n , Φ n = ( ϕ n , η n , ψ n , ξ n )) } n ≥ ⊂ R ∗ + × D ( A a )such that(5.4) λ n → + ∞ and k Φ n k H a = 1and there exists a sequence F n = ( f ,n , f ,n , f ,n , f ,n ) ∈ H a such that(5.5) ( iλ n I − A a ) Φ n = F n → H a . Detailing (5.5), we get the following system iλϕ n − η n = f ,n in H (0 , L ) , (5.6) iλη n − a ( ϕ n ) xx + (cid:0) ( α ,α − ε ) (cid:1) ( x ) η n + c ( x ) ξ n = f ,n in L (0 , L ) , (5.7) iλψ n − ξ n = f ,n in H (0 , L ) , (5.8) iλξ n − ( ψ n ) xx + (cid:0) ( α ,α − ε ) (cid:1) ( x ) ξ n − c ( x ) η n = f ,n in L (0 , L ) . (5.9)In what follows, we will check the condition (H4) by finding a contradiction with (5.4) such as k Φ n k H a = o (1).For clarity, we divide the proof into several lemmas. From now on, for simplicity, we drop the index n. Lemma 5.3.
The solution ( ϕ, η, ψ, ξ ) ∈ D ( A a ) of Equations (5.6) - (5.9) satisfies the following asymptoticbehavior estimation Z α − εα | η | dx = o (1) and Z α − εα | ξ | dx = o (1) . roof. Taking the inner product of (5.5) with Φ in H a , then using the fact that Φ is uniformly bounded in H a , we get Z α − εα | η | dx + Z α − εα | ξ | dx = −ℜ hA a Φ , Φ i H a = ℜ h ( iλI − A a ) Φ , Φ i = o (1) . Thus, the proof of the Lemma is complete. (cid:3)
Substituting η and ξ by iλϕ − f and iλψ − f respectively in (5.7) and (5.9), we get the following system λ ϕ + aϕ xx − iλ (cid:0) ( α ,α − ε ) (cid:1) ( x ) ϕ − iλc ( x ) ψ = − iλf + (cid:0) ( α ,α − ε ) (cid:1) ( x ) f − f − c ( x ) f , (5.10) λ ψ + ψ xx − iλ (cid:0) ( α ,α − ε ) (cid:1) ( x ) ψ + iλc ( x ) ϕ = c ( x ) f − iλf − (cid:0) ( α ,α − ε ) (cid:1) ( x ) f − f . (5.11) Lemma 5.4.
Let < δ < α − ε − α . The solution ( ϕ, η, ψ, ξ ) ∈ D ( A a ) of Equations (5.5) - (5.8) satisfies thefollowing asymptotic behavior estimation Z α − ε − δα + δ | ϕ x | dx = o (1) and Z α − ε − δα + δ | ψ x | dx = o (1) . Proof.
First, we define the first cut-off function θ in C (0 , L ) by , defined by(5.12) 0 ≤ θ ≤ , θ = 1 on ( α + δ, α − ε − δ ) and θ = 0 on (0 , α ) ∪ ( α − ε, L ) . Multiplying Equations (5.10) and (5.11) by θ ¯ ϕ and θ ¯ ψ respectively, integrate over (0 , L ) and using the factthat λϕ and λψ are uniformly bounded in L (0 , L ) and k F k → H a and taking the real part, we get(5.13) Z L θ | λϕ | dx − a Z L θ | ϕ x | dx − a Z L θ ′ ¯ ϕϕ x dx − ℜ (cid:18) iλc Z α − εα θψ ¯ ϕdx (cid:19) = o (1)and(5.14) Z L θ | λψ | dx − Z L θ | ψ x | dx − Z L θ ′ ¯ ψψ x dx + ℜ (cid:18) iλc Z α − εα θϕ ¯ ψdx (cid:19) = o (1) . Using the fact that λϕ and λψ are uniformly bounded in L (0 , L ), in particular in L ( α , α − ε ), and thedefinition of θ , we get(5.15) ℜ (cid:18) iλc Z α − εα θψ ¯ ϕdx (cid:19) = o (1) and ℜ (cid:18) iλc Z α − εα θϕ ¯ ψdx (cid:19) = o (1) . On the other hand, using the fact that λϕ, λψ, ϕ x and ψ x are uniformly bounded in L (0 , L ), we get(5.16) a Z L θ ′ ¯ ϕϕ x dx = o (1) and Z L θ ′ ¯ ψψ x dx = o (1) . Furthermore, using Lemma 5.3, Equations (5.6), (5.8) and the definition of the function θ in Equation (5.12),we get(5.17) Z L θ | λϕ | dx = o (1) and Z L θ | λψ | dx = o (1) . Inserting Equations (5.15)-(5.17) in Equations (5.13) and (5.14), we get the desired results. Thus, the proof ofthis Lemma is complete . (cid:3)
From Lemma 5.3 and Lemma 5.4, we get k Φ k H a = o (1) on ( α + δ, α − ε − δ ). In order to complete the proof,we need to show that k Φ k H a on ( α + δ, α − ε − δ ) c . Lemma 5.5.
Let h ∈ C (0 , L ) . The solution ( ϕ, η, ψ, ξ ) ∈ D ( A a ) of Equations (5.6) - (5.9) satisfies the followingasymptotic behavior estimation Z L h ′ (cid:0) | η | + a | ϕ x | + | ξ | + | ψ x | (cid:1) dx − ℜ (cid:16)(cid:2) ah | ϕ x | (cid:3) L (cid:17) − ℜ (cid:16)(cid:2) h | ψ x | (cid:3) L (cid:17) + 2 ℜ Z L c ( x ) hξ ¯ ϕ x dx ! − ℜ Z L c ( x ) hη ¯ ψ x dx ! = 2 Z L h ¯ ϕ x f dx + 2 Z L hη ( ¯ f ) x dx + 2 Z L h ¯ ψ x f dx + 2 Z L hξ ( ¯ f ) x dx. (5.18) 27 roof. Multiplying Equations (5.7) and (5.9) by 2 h ¯ ϕ x and 2 h ¯ ψ x respectively, integrate over (0 , L ) and usingthe fact that ϕ x , ψ x are uniformly bounded in L (0 , L ) and k F k H a → Z L iλhη ¯ ϕ x dx − a Z L hϕ xx ¯ ϕ x dx + 2 Z L c ( x ) hξ ¯ ϕ x dx = 2 Z L h ¯ ϕ x f dx (5.19) 2 Z L iλhξ ¯ ψ x dx − Z L hψ xx ¯ ψ x dx − Z L c ( x ) hη ¯ ψ x dx = 2 Z L h ¯ ψ x f dx. (5.20)From Equations (5.6) and (5.8), we have − iλ ¯ ϕ x = ¯ η x + (cid:0) ¯ f (cid:1) x and − iλ ¯ ψ x = ¯ ξ x + (cid:0) ¯ f (cid:1) x . Inserting the above equations in Equations (5.19) and (5.20) and by taking the real part, we obtain − Z L h | η | x dx − a Z L h | ϕ x | x dx + 2 ℜ Z L c ( x ) hξ ¯ ϕ x dx ! = 2 Z L h ¯ ϕ x f dx + 2 Z L hη ( ¯ f ) x dx, (5.21) − Z L h | ξ | x dx − Z L h | ψ x | x dx − ℜ Z L c ( x ) ηh ¯ ψ x dx ! = 2 Z L h ¯ ψ x f dx + 2 Z L hξ ( ¯ f ) x dx. (5.22)Using by parts integration in Equations (5.21) and (5.22), we get the desired results. (cid:3) Lemma 5.6.
Let < δ < α − ε − α . The solution ( ϕ, η, ψ, ξ ) ∈ D ( A a ) of Equations (5.6) - (5.9) satisfies thefollowing asymptotic behavior estimation Z α + δ (cid:0) | η | + a | ϕ x | + | ξ | + | ψ x | (cid:1) dx = o (1) . Proof.
Define the cut-off function ˜ θ in C ([0 , L ]) by(5.23) 0 ≤ ˜ θ ≤ , ˜ θ = 1 on (0 , α + δ ) , ˜ θ = 0 on ( α − ε − δ, L ) . Take h = x ˜ θ ( x ) in Equation (5.18), we get(5.24) Z L h ′ (cid:0) | η | + a | ϕ x | + | ξ | + | ψ x | (cid:1) dx + 2 c ℜ Z α − ε − δα x ˜ θξ ¯ ϕ x dx ! − c ℜ Z α − ε − δα x ˜ θη ¯ ψ x dx ! = o (1) . Using Lemma (5.3) and ϕ x and ψ x are uniformly bounded in L (0 , L ) and in particular in L ( α , α − ε − δ ),we get 2 c ℜ Z α − ε − δα x ˜ θξ ¯ ϕ x dx ! = o (1) and 2 c ℜ Z α − ε − δα x ˜ θη ¯ ψ x dx ! = o (1) . Inserting the above equations in Equation (5.24), and using Lemmas (5.3)-(5.4) and the definition the function˜ θ , we get the desired result. (cid:3) From the preceded results of Lemmas 5.3, 5.4 and 5.6 , we deduce that k Φ k H a = o (1) on ( α + δ, α − ε − δ ) . Now, our goal is to prove that k Φ k H a = o (1) on ( α − ε − δ, L ). For this aims, let g ∈ C ([ α − ε − δ, α ])such that g ( α ) = − g ( α − ε − δ ) = 1 , max x ∈ [ α − ε,α ] | g ( x ) | = c g and max x ∈ [ α − ε,α ] | g ′ ( x ) | = c g ′ where c g and c g ′ are strictly positive constant numbers. Remark 5.7.
It is easy to see the existence of g ( x ) . For example, we can take g ( x ) = cos (cid:18) ( α − x ) πα − α + 2 ε + δ (cid:19) to get g ( α ) = − g ( α − ε − δ ) = 1 , g ∈ C ([ α − ε − δ, , | g ( x ) | ≤ and | g ′ ( x ) | ≤ πα − α +2 ε + δ . emma 5.8. Let < δ < α − ε − α . The solution ( ϕ, η, ψ, ξ ) ∈ D ( A a ) of Equations (5.5) - (5.8) satisfies thefollowing asymptotic behavior estimation | η ( α ) | = O (1) , | η ( α − ε − δ ) | = O (1) , | ξ ( α ) | = O (1) and | ξ ( α − ε − δ ) | = O (1) . Proof.
From (5.7) and (5.9), we have(5.25) iλϕ x − η x = ( f ) x and iλψ x − ξ x = ( f ) x . Multiplying the first equation and the second equation of (5.25) respectively by 2 g ( x )¯ η and 2 g ( x ) ¯ ξ , integrateover ( α − ε − δ, α ) and using the fact that k F k H a → η and ξ are uniformly bounded in L (0 , L ) inparticular in L ( α − ε − δ, α ), we get ℜ (cid:18) iλ Z α α − ε − δ gϕ x ¯ ηdx (cid:19) − Z α α − ε − δ g ( x ) (cid:0) | η | (cid:1) x dx = o (1) , (5.26) ℜ (cid:18) iλ Z α α − ε − δ gψ x ¯ ξdx (cid:19) − Z α α − ε − δ g ( x ) (cid:0) | ξ | (cid:1) x dx = o (1) . (5.27)Using integration by parts in Equations (5.26) and (5.27), we get Z α α − ε − δ g ′ ( x ) | η | dx + ℜ (cid:18) iλ Z α α − ε − δ gϕ x ¯ ηdx (cid:19) = | η ( α ) | + | η ( α − ε − δ ) | + o (1) , (5.28) Z α α − ε − δ g ′ ( x ) | ξ | dx + ℜ (cid:18) iλ Z α α − ε − δ gψ x ¯ ξdx (cid:19) = | ξ ( α ) | + | ξ ( α − ε − δ ) | + o (1) . (5.29)Multiplying Equations (5.7) and (5.9) by 2 g ( x ) ¯ ϕ x and 2 g ( x ) ¯ ψ x respectively , integrate over ( α − ε − δ, α ),using the fact k F k H a → ϕ x and ψ x are uniformly bounded in L (0 , L ) and Lemma 5.3 and taking the realpart, we get ℜ (cid:18) iλ Z α α − ε − δ g ( x ) η ¯ ϕ x dx (cid:19) − a Z α α − ε − δ g ( x ) (cid:0) | ϕ x | (cid:1) x dx + 2 ℜ (cid:18) c Z α α − ε − δ g ( x ) ξ ¯ ϕ x dx (cid:19) = o (1) , ℜ (cid:18) iλ Z α α − ε − δ g ( x ) ξ ¯ ψ x dx (cid:19) − Z α α − ε − δ g ( x ) (cid:0) | ψ x | (cid:1) x dx − ℜ (cid:18) c Z α α − ε − δ g ( x ) η ¯ ψ x dx (cid:19) = o (1) . Using integration by parts in the second terms of the above Equations, we obtain(5.30) ℜ (cid:18) iλ Z α α − ε − δ g ( x ) η ¯ ϕ x dx (cid:19) + a Z α α − ε − δ g ′ ( x ) | ϕ x | dx + 2 ℜ (cid:18) c Z α α − ε − δ g ( x ) ξ ¯ ϕ x dx (cid:19) = a | ϕ x ( α ) | + a | ϕ x ( α − ε − δ ) | + o (1)and(5.31) ℜ (cid:18) i Z α α − ε − δ g ( x ) ξ ¯ ψ x dx (cid:19) + Z α α − ε − δ g ′ ( x ) | ψ x | dx − ℜ (cid:18) c Z α α − ε − δ g ( x ) η ¯ ψ x dx (cid:19) = | ψ x ( α ) | + | ψ x ( α − ε − δ ) | + o (1) . Adding Equations (5.28)-(5.31), we get M ( α , α − ε − δ ) + N ( α , α − ε − δ ) = Z α α − ε − δ g ′ ( x ) (cid:0) | η | + a | ϕ x | + | ξ | + | ψ x | (cid:1) dx +2 ℜ (cid:18) c Z α α − ε − δ g ( x ) ξ ¯ ϕ x dx (cid:19) − ℜ (cid:18) c Z α α − ε − δ g ( x ) η ¯ ψ x dx (cid:19) + o (1)(5.32)where M ( α , α − ε − δ ) = | η ( α ) | + | η ( α − ε − δ ) | + a | ϕ x ( α ) | + a | ϕ x ( α − ε − δ ) | ,N ( α , α − ε − δ ) = | ξ ( α ) | + | ξ ( α − ε − δ ) | + | ψ x ( α ) | + | ψ x ( α − ε − δ ) | . From Equation (5.32), we get M ( α , α − ε − δ ) + N ( α , α − ε − δ ) ≤ c g ′ Z α α − ε − δ (cid:0) | η | + a | ϕ x | + | ξ | + | ψ x | (cid:1) dx + c c g k ξ k L (0 ,L ) k ϕ x k L (0 ,L ) + c c g k η k L (0 ,L ) k ψ x k L (0 ,L ) + o (1) . k Φ k is uniformly bounded in H a , we obtain the desired result. The proof of this Lemmahas been completed. (cid:3) Lemma 5.9.
Let < δ < α − ε − α . The solution ( ϕ, η, ψ, ξ ) ∈ D ( A a ) of Equations (5.6) - (5.9) satisfies thefollowing asymptotic behavior estimation Z Lα − ε − δ (cid:0) | η | + a | ϕ x | + | ξ | + | ψ x | (cid:1) dx = o (1) . Proof.
Define the cut-off function ˆ θ in C ([0 , L ]) by(5.33) 0 ≤ ˆ θ ≤ , ˆ θ = 1 on ( α − ε − δ, L ) , and ˆ θ = 0 on (0 , α + δ ) . Take h = ( x − L )ˆ θ in Equation (5.18), using Lemmas (5.3)-(5.4) and the definition of the function ˆ θ , we get(5.34) Z Lα − ε − δ (cid:0) | η | + a | ϕ x | + | ξ | + | ψ x | (cid:1) dx + 2 c ℜ (cid:18)Z α α − ε − δ ( x − L )ˆ θξ ¯ ϕ x dx (cid:19) − c ℜ (cid:18)Z α α − ε − δ ( x − L )ˆ θη ¯ ψ x dx (cid:19) = o (1) . Using the fact that ξ = iλψ − f and η = iλϕ − f in the second and third term of Equation (5.34) and that ϕ x , ψ x are uniformly bounded in L (0 , L ) and the fact that k F k H a →
0, we get2 c ℜ (cid:18)Z α α − ε − δ ( x − L )ˆ θξ ¯ ϕ x dx (cid:19) − c ℜ (cid:18)Z α α − ε − δ ( x − L )ˆ θη ¯ ψ x dx (cid:19) = 2 c ℜ (cid:18)Z α α − ε − δ iλ ( x − L )ˆ θψ ¯ ϕ x dx (cid:19) − c ℜ (cid:18)Z α α − ε − δ iλ ( x − L )ˆ θϕ ¯ ψ x dx (cid:19) + o (1) . Using integration by parts in the first term of the right hand side of the above equation and the fact that λϕ and λψ are uniformly bounded in L (Ω), we obtain(5.35) 2 c ℜ (cid:18)Z α α − ε − δ ( x − L )ˆ θξ ¯ ϕ x dx (cid:19) − c ℜ (cid:18)Z α α − ε − δ ( x − L )ˆ θη ¯ ψ x dx (cid:19) =2 c ℜ (cid:0) [ iλ ( x − L ) ψ ¯ ϕ ] α α − ε − δ (cid:1) + o (1) . Inserting Equation (5.35) in Equation (5.34), we obtain(5.36) Z Lα − ε − δ (cid:0) | η | + a | ϕ x | + | ξ | + | ψ x | (cid:1) dx = A ( α ) + B ( α − ε − δ ) + o (1) , where A ( α ) = 2 c ℜ ( iλ ( L − α ) ψ ( α ) ¯ ϕ ( α )) ,B ( α − ε − δ ) = 2 c ℜ ( iλ ( α − ε − δ − L ) ψ ( α − ε − δ ) ¯ ϕ ( α − ε − δ )) . On the other hand, from Equations (5.6) and (5.8), we have(5.37) | λϕ ( s ) | ≤ | η ( s ) | + | f ( s ) | and | λψ ( s ) | ≤ | ξ ( s ) | + | f ( s ) | for s ∈ { α − ε − δ, α } . Using the fact that | f ( s ) | ≤ s Z s | ( f ) x | dx ≤ sa − k F k H a and | f ( s ) | ≤ s Z s | ( f ) x | dx ≤ s k F k H a for all s ∈{ α − ε − δ, α } , and using Lemma 5.8 in Equation (5.37), we obtain | λϕ ( s ) | = O (1) and | λψ ( s ) | = O (1) , for s ∈ { α − ε − δ, α } . Its follow that(5.38) A ( α ) + B ( α − ε − δ ) = o (1) . Using Equation (5.37) in Equation (5.36), we obtain Z Lα − ε − δ (cid:0) | η | + a | ϕ x | + | ξ | + | ψ x | (cid:1) dx = o (1) . Thus, the proof has been completed. (cid:3) roof of Theorem 5.2 Using Lemmas 5.3, 5.4, 5.6 and 5.9, we get k Φ k H a = o (1) on [0 , L ], which contradictsEquation (5.4). Therefore, (H4) holds, by Huang [17] and Pruss [31] we deduce the exponential stability of theauxiliary problem (5.1).5.2. Definitions and Theorems.
We introduce here the notions of stability that we encounter in this work.
Definition 5.10.
Assume that A is the generator of a C -semigroup of contractions (cid:0) e tA (cid:1) t ≥ on a Hilbertspace H . The C -semigroup (cid:0) e tA (cid:1) t ≥ is said to be strongly stable if lim t → + ∞ k e tA x k H = 0 , ∀ x ∈ H ;2. exponentially (or uniformly) stable if there exist two positive constants M and ǫ such that k e tA x k H ≤ M e − ǫt k x k H , ∀ t > , ∀ x ∈ H ;3. polynomially stable if there exists two positive constants C and α such that k e tA x k H ≤ Ct − α k x k H , ∀ t > , ∀ x ∈ D ( A ) . In that case, one says that the semigroup (cid:0) e tA (cid:1) t ≥ decays at a rate t − α . The C -semigroup (cid:0) e tA (cid:1) t ≥ is said to be polynomially stable with optimal decay rate t − α (with α > ) if it is polynomially stablewith decay rate t − α and, for any ε > small enough, the semigroup (cid:0) e tA (cid:1) t ≥ does not decay at a rate t − ( α − ε ) . To show the strong stability of a C − semigroup of contraction ( e tA ) t ≥ we rely on the following result due toArendt-Batty [6]. Theorem 5.11.
Assume that A is the generator of a C − semigroup of contractions (cid:0) e tA (cid:1) t ≥ on a Hilbertspace H . If A has no pure imaginary eigenvalues, σ ( A ) ∩ i R is countable,where σ ( A ) denotes the spectrum of A , then the C − semigroup (cid:0) e tA (cid:1) t ≥ is strongly stable. Concerning the characterization of exponential stability of a C − semigroup of contraction ( e tA ) t ≥ we rely onthe following result due to Huang [17] and Pr u ss [31]. Theorem 5.12.
Let A : D ( A ) ⊂ H → H generate a C − semigroup of contractions (cid:0) e tA (cid:1) t ≥ on H . Assumethat iλ ∈ ρ ( A ) , ∀ λ ∈ R . Then, the C − semigroup (cid:0) e tA (cid:1) t ≥ is exponentially stable if and only if lim λ ∈ R , | λ |→ + ∞ k ( iλI − A ) − k L ( H ) < + ∞ . Also, concerning the characterization of polynomial stability of a C − semigroup of contraction ( e tA ) t ≥ we relyon the following result due to Borichev and Tomilov [10] (see also [23] and [9]). Theorem 5.13.
Assume that A is the generator of a strongly continuous semigroup of contractions (cid:0) e tA (cid:1) t ≥ on H . If i R ⊂ ρ ( A ) , then for a fixed ℓ > the following conditions are equivalent (5.39) sup λ ∈ R (cid:13)(cid:13)(cid:13) ( iλI − A ) − (cid:13)(cid:13)(cid:13) L ( H ) = O (cid:0) | λ | ℓ (cid:1) , (5.40) k e tA U k H ≤ Ct ℓ k U k D ( A ) , ∀ t > , U ∈ D ( A ) , for some C > . Finally, the analytic property of a C − semigroup of contraction ( e tA ) t ≥ is characterized in the followingtheorem due to Arendt, Batty and Hieber [7]. Theorem 5.14.
Let ( S ( t ) = e tA ) t ≥ be a C − semigroup of contractions in a Hilbert space. Assume that (A1) i R ⊂ ρ ( A ) . Then, ( e tA ) t ≥ is analytic if and only if (A2) lim sup λ ∈ R , | λ |→∞ | λ | − k ( iλ − A ) − k L ( H ) < ∞ . Conclusion
We have studied the stabilization of a system of locally coupled wave equations with only one internallocalized Kelvin-Voigt damping via non-smooth coefficients. We proved the strong stability of the systemusing Arendt-Batty criteria. Lack of exponential stability results has been proved in both cases: The case ofglobal Kelvin-Voigt damping and the case of localized Kelvin-Voigt damping, taking into consideration thatthe coupling is global. In addition, if both coupling and damping are localized internally via non-smoothcoefficients, we established a polynomial energy decay rate of type t − . We can conjecture that the energydecay rate t − is optimal. However, if the intersection between the supports of the domains of the damping andthe coupling coefficients is empty, the nature of the decay rate of the system will be unknown. This questionis still an open problem. Acknowledgments
The authors thanks professors Michel Mehrenberger and Kais Ammari for their valuable discussions and com-ments.Mohammad Akil would like to thank the Lebanese University for its support.Ibtissam Issa would like to thank the Lebanese University for its support.Ali Wehbe would like to thank the CNRS and the LAMA laboratory of Mathematics of the Universit´e SavoieMont Blanc for their supports.
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