Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay
aa r X i v : . [ m a t h . A P ] J u l STABILITY RESULTS OF A SINGULAR LOCAL INTERACTIONELASTIC/VISCOELASTIC COUPLED WAVE EQUATIONS WITH TIME DELAY
MOHAMMAD AKIL , HAIDAR BADAWI , AND ALI WEHBE Abstract.
The purpose of this paper is to investigate the stabilization of a one-dimensional coupled waveequations with non smooth localized viscoelastic damping of Kelvin-Voigt type and localized time delay. Usinga general criteria of Arendt-Batty, we show the strong stability of our system in the absence of the compactnessof the resolvent. Finally, using frequency domain approach combining with a multiplier method, we prove apolynomial energy decay rate of order t − . Contents
1. Introduction 21.1. Description of the paper 21.2. Previous Literature 21.2.1. Coupled wave equations with Kelvin-Voigt damping and without time delay 31.2.2. Wave equations with time delay and without Kelvin-Voigt damping 31.2.3. Wave equations with Kelvin-Voigt damping and time delay 42. Well-posedness of the System 53. Strong Stability 94. Polynomial Stability 195. Conclusion 28Appendix A. Some notions and theorems of stability has been used 28Acknowledgments 29References 29 Universit´e Savoie Mont Blanc, Laboratoire LAMA, Chamb´ery-France Universit´e Polytechnique Hauts-de-France Valenciennes, LAMAV, France Lebanese University, Faculty of sciences 1, Khawarizmi Laboratory of Mathematics and Applications-KALMA,Hadath-Beirut, Lebanon.
E-mail addresses : [email protected], [email protected],[email protected] . Key words and phrases.
Coupled wave equation; Kelvin-Voigt damping; Time delay; Strong stability; Polynomial stability;Frequency domain approach. . Introduction
Description of the paper.
In this paper, we investigate the stability of local coupled wave equationswith singular localized viscoelastic damping of Kelvin-Voigt type and localized time delay. More precisely, weconsider the following System:(1.1) u tt − [ au x + b ( x )( κ u tx + κ u tx ( x, t − τ ))] x + c ( x ) y t = 0 , ( x, t ) ∈ (0 , L ) × (0 , ∞ ) ,y tt − y xx − c ( x ) u t = 0 , ( x, t ) ∈ (0 , L ) × (0 , ∞ ) ,u (0 , t ) = u ( L, t ) = y (0 , t ) = y ( L, t ) = 0 , t > , ( u ( x, , u t ( x, u ( x ) , u ( x )) , x ∈ (0 , L ) , ( y ( x, , y t ( x, y ( x ) , y ( x )) , x ∈ (0 , L ) ,u t ( x, t ) = f ( x, t ) , ( x, t ) ∈ (0 , L ) × ( − τ, , where L, τ, a and κ are positive real numbers, κ is a non-zero real number and ( u , u , y , y , f ) belongs toa suitable space. We suppose that there exists 0 < α < β < γ < L and a positive constant c , such that b ( x ) = ( , x ∈ (0 , β ) , , x ∈ ( β, L ) , and c ( x ) = ( c , x ∈ ( α, γ ) , , x ∈ (0 , α ) ∪ ( γ, L ) . The Figure 1 describes system (1.1)
Viscoelastic region & time delayCoupling region α β γ L Figure 1.
Local Kelvin-Voigt damping and Local time delay feedback.System (1.1) consists of two wave equations with only one singular viscoelastic damping acting on the firstequation, the second one is indirectly damped via a singular coupling between the two equations. The notionof indirect damping mechanisms has been introduced by Russell in [49] and since then, it has attracted theattention of many authors (see for instance [3], [5], [6], [9], [16], [24], [37] and [54] ). The study of such systemsis also motivated by several physical considerations like Timoshenko and Bresse systems (see for instance [1],[2], [40] and [42]). In fact, there are few results concerning the stability of coupled wave equations with localKelvin-Voigt damping without time delay, especially in the absence of smoothness of the damping and couplingcoefficients (see Subsection 1.2.1 ). The last motivates our interest to study the stabilization of system (1.1) inthe present paper.1.2.
Previous Literature.
The wave is created when a vibrating source disturbs the medium. In order torestrain those vibrations, several damping can be added such as Kelvin-Voigt damping which is originatedfrom the extension or compression of the vibrating particles. This damping is a viscoelastic structure havingproperties of both elasticity and viscosity. In the recent years, many researchers showed interest in problemsinvolving this kind of damping where different types of stability, depend on the smoothness of the dampingcoefficients, has been showed (see [7], [8], [27], [28], [31], [35], [38], [45] and [48] ). However, time delays havebeen used in several applications such as in physical, chemical, biological, thermal phenomenas not only depend2n the present state but also on some past occurrences (see [23],[33]) . In the last years, the control of partialdifferential equations with time delays have become popular among scientists. In many cases the time delayinduce some instabilities see [17, 19, 20, 22].However, let us recall briefly some systems of wave equations with Kelvin-Voigt damping and time delayrepresented in previous literature.1.2.1.
Coupled wave equations with Kelvin-Voigt damping and without time delay.
In 2019, Hassine and Souayehin [29] studied the behavior of a system with coupled wave equations with a partial Kelvin-Voigt damping, byconsidering the following system(1.2) u tt − ( u x + b ( x ) u tx ) x + v t = 0 , ( x, t ) ∈ ( − , × (0 , ∞ ) ,v tt − cv xx − u t = 0 , ( x, t ) ∈ ( − , × (0 , ∞ ) ,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 ispiecewise function in particular they supposed that b ( x ) = d [0 , ( x ), where d is a strictly positive constant.So, they took the damping coefficient to be near the boundary with a global coupling coefficient. They showedthe lack of exponential stability and that the semigroup loses speed and it decays polynomially with a rate as t − . In 2020, Akil, Issa and Wehbe in [4] studied the localized coupled wave equations, by considering thefollowing system: u tt − ( au x + b ( x ) u tx ) x + c ( x ) y t = 0 , ( x, t ) ∈ (0 , L ) × (0 , ∞ ) ,y tt − y xx − c ( x ) u t = 0 , ( x, t ) ∈ (0 , L ) × (0 , ∞ ) ,u (0 , t ) = u ( L, t ) = y (0 , t ) = y ( L, t ) = 0 , t > , ( u ( x, , u t ( x, u ( x ) , u ( x )) , x ∈ (0 , L ) , ( y ( x, , y t ( x, y ( x ) , y ( x )) , x ∈ (0 , L )where b ( x ) = ( b , x ∈ ( α , α ) , , otherwise and c ( x ) = ( c , x ∈ ( α , α ) , , otherwiseand where a > b > c > < α < α < α < α < L . They generalized the results ofHassine and Souayeh in [29] by establishing a polynomial decay rate of type t − . In the same year, Hayek etal. in [30] studied the stabilization of a multi-dimensional system of weakly coupled wave equations with oneor two locally Kelvin-Voigt damping and non-smooth coefficient at the interface. They established differentstability results.1.2.2. Wave equations with time delay and without Kelvin-Voigt damping.
The delay equations of hyperbolictype is given by(1.3) u tt − ∆ u ( x, t − τ ) = 0 . with a delay parameter τ >
0. This system is not well posed since there exists a sequence of solutions tendingto infinity for any fixed t > u tt ( x, t ) − ∆ u ( x, t ) = 0 , ( x, t ) ∈ Ω × (0 , ∞ ) ,u ( x, t ) = 0 , ( x, t ) ∈ Γ D × (0 , ∞ ) , ∂u∂ν ( x, t ) = − µ u t ( x, t ) − µ u t ( x, t − τ ) , ( x, t ) ∈ Γ N × (0 , ∞ ) , ( u ( x, , u t ( x, u ( x ) , u ( x )) , x ∈ Ω ,u t ( x, t ) = f ( x, t ) , ( x, t ) ∈ Γ N × ( − τ, . The second case concerns a wave equation with an internal feedback and a delayed velocity term ( i.e. aninternal delay) and a mixed Dirichlet-Neumann boundary condition(1.5) u tt ( x, t ) − ∆ u ( x, t ) + µ u t ( x, t ) + µ u t ( x, t − τ ) = 0 , ( x, t ) ∈ Ω × (0 , ∞ ) ,u ( x, t ) = 0 , ( x, t ) ∈ Γ D × (0 , ∞ ) , ∂u∂ν ( x, t ) = 0 , ( x, t ) ∈ Γ N × (0 , ∞ ) , ( u ( x, , u t ( x, u ( x ) , u ( x )) , x ∈ Ω ,u t ( x, t ) = f ( x, t ) , ( x, t ) ∈ Ω × ( − τ, , where Ω is an open bounded domain of R N with a boundary Γ of class C and Γ = Γ D ∪ Γ N , such thatΓ D ∩ Γ N = ∅ . Under the assumption µ < µ , an exponential decay achieved for the both systems (1.4)-(1.5).If this assumption does not hold, they found a sequences of delays { τ k } k , τ k →
0, for which the correspondingsolutions have increasing energy. Furthermore, we refer to [14] for the Problem (1.5) in more general abstractsetting. In 2010, Ammari et al. (see [10]) studied the wave equation with interior delay damping and dissipativeundelayed boundary condition in an open domain Ω of R N , N ≥ . The system is described by:(1.6) u tt ( x, t ) − ∆ u ( x, t ) + au t ( x, t − τ ) = 0 , ( x, t ) ∈ Ω × (0 , ∞ ) ,u ( x, t ) = 0 , ( x, t ) ∈ Γ × (0 , ∞ ) , ∂u∂ν ( x, t ) = − κu t ( x, t ) , ( x, t ) ∈ Γ × (0 , ∞ ) , ( u ( x, , u t ( x, u ( x ) , u ( x )) , x ∈ Ω ,u t ( x, t ) = f ( x, t ) , ( x, t ) ∈ Ω × ( − τ, , where τ > a > κ >
0. Under the condition that Γ satisfies the Γ-condition introduced in [34], theyproved that system (1.6) is uniformly asymptotically stable whenever the delay coefficient is sufficiently small.In 2012, Pignotti in [47] considered the wave equation with internal distributed time delay and local dampingin a bounded and smooth domain Ω ⊂ R N , N ≥
1. The considered system is given by the following:(1.7) u tt ( x, t ) − ∆ u ( x, t ) + aχ ω u t ( x, t ) + κu t ( x, t − τ ) = 0 , ( x, t ) ∈ Ω × (0 , ∞ ) ,u ( x, t ) = 0 , ( x, t ) ∈ Γ × (0 , ∞ ) , ( u ( x, , u t ( x, u ( x ) , u ( x )) , x ∈ Ω ,u t ( x, t ) = f ( x, t ) , ( x, t ) ∈ Ω × ( − τ, , where κ ∈ R , τ > a > ω is the intersection between an open neighberhood of the set Γ = { x ∈ Ω; ( x − x ) · ν ( x ) > } and Ω. Moreover, χ ω is the characteristic function of ω . We remark that thedamping is localized and it acts on a neighberhood of a part of Ω. She showed an exponential stability resultsif the coefficients of the delay terms satisfy the following assumption | k | < k < a .Several researches was done on wave equation with time delay acting on the boundary see ([20],[18], [53], [26],[25], [50], [52], [51]) and different type of stability has been proved.1.2.3. Wave equations with Kelvin-Voigt damping and time delay.
In 2016, Messaoudi et al. in [41] consideredthe stabilization of the following wave equation with strong time delay u tt ( x, t ) − ∆ u ( x, t ) − µ ∆ u t ( x, t ) − µ ∆ u t ( x, t − τ ) = 0 , ( x, t ) ∈ Ω × (0 , ∞ ) ,u ( x, t ) = 0 , ( x, t ) ∈ Γ × (0 , ∞ ) , ( u ( x, , u t ( x, u ( x ) , u ( x )) , x ∈ Ω ,u t ( x, t ) = f ( x, t ) , ( x, t ) ∈ Ω × ( − τ, , µ > µ is a non zero real number. Under the assumption that | µ | < µ , they obtained anexponential stability result. In 2016, Nicaise et al. in [44] studied the multidimensional wave equation withlocalized Kelvin-Voigt damping and mixed boundary condition with time delay(1.8) u tt ( x, t ) − ∆ u ( x, t ) − div( a ( x ) ∇ u t ) = 0 , ( x, t ) ∈ Ω × (0 , ∞ ) ,u ( x, t ) = 0 , ( x, t ) ∈ Γ × (0 , ∞ ) , ∂u∂ν ( x, t ) = − a ( x ) ∂u t ∂ν ( x, t ) − κu t ( x, t − τ ) , ( x, t ) ∈ Γ × (0 , ∞ ) , ( u ( x, , u t ( x, u ( x ) , u ( x )) , x ∈ Ω ,u t ( x, t ) = f ( x, t ) , ( x, t ) ∈ Γ × ( − τ, , where τ > κ ∈ R , a ( x ) ∈ L ∞ (Ω) and a ( x ) ≥ a > ω such that ω ⊂ Ω is an open neighborhoodof Γ . Under an appropriate geometric condition on Γ and assuming that a ∈ C , (Ω), ∆ a ∈ L ∞ (Ω), theyproved an exponential decay of the energy of system (1.8). In 2019, Anikushyn and al. in [21] considered aninitial boundary value problem for a viscoelastic wave equation subjected to a strong time localized delay in aKelvin-Voigt type. The system is given by the following: u tt ( x, t ) − c ∆ u ( x, t ) − c ∆ u ( x, t − τ ) − d ∆ u t ( x, t ) − d ∆ u t ( x, t − τ ) = 0 , ( x, t ) ∈ Ω × (0 , ∞ ) ,u ( x, t ) = 0 , ( x, t ) ∈ Γ × (0 , ∞ ) , ∂u∂ν ( x, t ) = 0 , ( x, t ) ∈ Γ × (0 , ∞ ) , ( u ( x, , u t ( x, u ( x ) , u ( x )) , x ∈ Ω ,u ( x, t ) = f ( x, t ) , ( x, t ) ∈ Ω × ( − τ, . Under appropriate conditions on the coefficients, a global exponential decay rate is obtained. In 2015, Ammari and al. in [11] considered the stabilization problem for an abstract equation with delay and a Kelvin-Voigtdamping. The system is given by the following: u tt ( t ) + a BB ∗ u t ( t ) + BB ∗ u ( t − τ ) , t ∈ (0 , ∞ ) , ( u (0) , u t (0)) = ( u , u ) , B ∗ u ( t ) = f ( t ) , t ∈ ( − τ, , for an appropriate class of operator B and a > . Using the frequency domain approach, they obtained anexponential stability result.Thus, to the best of our knowledge, it seems to us that there is no result in the existing literature concerningthe case of coupled wave equations with localized Kelvin-Voigt damping and localized time delay, especially inthe absence of smoothness of the damping and coupling coefficients. The goal of the present paper is to fill thisgap by studying the stability of system (1.1).The paper is organized as follows: In Section 2, we prove the well-posedness of our system by using semigroupapproach. In Section 3, by using a general criteria of Arendt Batty, we show the strong stability of our systemin the absence of the compactness of the resolvent. Next, in Section 4, by using frequency domain approachcombining with a specific multiplier method, we prove a polynomial energy decay rate of order t − .2. Well-posedness of the System
In this section, we will establish the well-posedness of system (1.1) by using semigroup approach. For this aim,as in [43], we introduce the following auxiliary change of variable(2.1) η ( x, ρ, t ) := u t ( x, t − ρτ ) , x ∈ (0 , L ) , ρ ∈ (0 , , t > . Then, system (1.1) becomes u tt − [ au x + b ( x )( κ u tx + κ η x ( x, , t ))] x + c ( x ) y t = 0 , ( x, t ) ∈ (0 , L ) × (0 , ∞ ) , (2.2) y tt − y xx − c ( x ) u t = 0 , ( x, t ) ∈ (0 , L ) × (0 , ∞ ) , (2.3) τ η t ( x, ρ, t ) + η ρ ( x, ρ, t ) = 0 , ( x, ρ, t ) ∈ (0 , L ) × (0 , × (0 , ∞ ) , (2.4) 5ith the following boundary conditions(2.5) ( u (0 , t ) = u ( L, t ) = y (0 , t ) = y ( L, t ) = 0 , t > ,η (0 , ρ, t ) = 0 , ( ρ, t ) ∈ (0 , × (0 , ∞ ) , and the following initial conditions(2.6) u ( x,
0) = u ( x ) , u t ( x,
0) = u ( x ) , x ∈ (0 , L ) ,y ( x,
0) = y ( x ) , y t ( x,
0) = y ( x ) , x ∈ (0 , L ) ,η ( x, ρ,
0) = f ( x, − ρτ ) , ( x, ρ ) ∈ (0 , L ) × (0 , . The energy of system (2.2)-(2.6) is given by(2.7) E ( t ) = E ( t ) + E ( t ) + E ( t ) , where E ( t ) = 12 Z L (cid:16) | u t | + a | u x | (cid:17) dx, E ( t ) = 12 Z L (cid:16) | y t | + | y x | (cid:17) dx and E ( t ) = τ | κ | Z β Z | η x ( · , ρ, t ) | dρdx. Lemma 2.1.
Let U = ( u, u t , y, y t , η ) be a regular solution of system (2.2)-(2.6). Then, the energy E ( t ) satisfiesthe following estimation(2.8) ddt E ( t ) ≤ − ( κ − | κ | ) Z β | η x ( · , , t ) | dx. Proof.
First, multiplying (2.2) by u t , integrating over (0 , L ), using integration by parts with (2.5), then usingthe definition of b ( · ), c ( · ) and taking the real part, we obtain(2.9) ddt E ( t ) = − κ Z β | η x ( · , , t ) | dx − ℜ ( κ Z β η x ( · , , t ) η x ( · , , t ) dx ) − ℜ (cid:26) c Z γα y t u t dx (cid:27) . Using Young’s inequality in (2.9), we get(2.10) ddt E ( t ) ≤ − (cid:18) κ − | κ | (cid:19) Z β | η x ( · , , t ) | dx + | κ | Z β | η x ( · , , t ) | dx − ℜ (cid:26) c Z γα y t u t dx (cid:27) . Now, multiplying (2.3) by y t , integrating over (0 , L ), using the definition of c ( · ), then taking the real part, weget(2.11) ddt E ( t ) = ℜ (cid:26) c Z γα u t y t dx (cid:27) . Deriving (2.4) with respect to x , we obtain(2.12) τ η xt ( · , ρ, t ) + η xρ ( · , ρ, t ) = 0 . Multiplying (2.12) by | κ | η x ( · , ρ, t ), integrating over (0 , β ) × (0 , ddt E ( t ) = − | κ | Z β (cid:16) | η x ( · , , t ) | − | η x ( · , , t ) | (cid:17) dx. Finally, by adding (2.10), (2.11) and (2.13), we obtain (2.8). The proof is thus complete. (cid:3)
In the sequel, the assumption on κ and κ will ensure that(H) κ > , κ ∈ R ∗ and | κ | < κ . Under the hypothesis (H) and from Lemma 2.1, the system (2.2)-(2.6) is dissipative in the sense that its energyis non-increasing with respect to time (i.e. E ′ ( t ) ≤ H by H := (cid:0) H (0 , L ) × L (0 , L ) (cid:1) × W , where W := L ((0 , H L (0 , β )) and H L (0 , β ) := (cid:8)e η ∈ H (0 , β ) | e η (0) = 0 (cid:9) . W is an Hilbert space of H L (0 , β )-valued functions on (0 , η , η ) W := Z β Z η x η x dρdx, ∀ η , η ∈ W . The Hilbert space H is equipped with the following inner product(2.14) (cid:0) U, U (cid:1) H = Z L (cid:16) au x u x + vv + y x y x + zz (cid:17) dx + τ | κ | Z β Z η x ( · , ρ ) η x ( · , ρ ) dρdx, where U = ( u, v, y, z, η ( · , ρ )) ⊤ , U = ( u , v , y , z , η ( · , ρ )) ⊤ ∈ H . Now, we define the linear unboundedoperator A : D ( A ) ⊂ H 7−→ H by:(2.15) D ( A ) = U = ( u, v, y, z, η ( · , ρ )) ⊤ ∈ H | y ∈ H (0 , L ) ∩ H (0 , L ) , v, z ∈ H (0 , L )( S b ( u, v, η )) x ∈ L (0 , L ) , η ρ ( · , ρ ) ∈ W , η ( · ,
0) = v ( · ) and(2.16) A uvyzη ( · , ρ ) = v ( S b ( u, v, η )) x − c ( · ) zzy xx + c ( · ) v − τ − η ρ ( · , ρ ) , where S b ( u, v, η ) := au x + b ( · ) ( κ v x + κ η x ( · , b ( · ), we have(2.17) S b ( u, v, η ) = ( S ( u, v, η ) , x ∈ (0 , β ) ,au x , x ∈ ( β, L ) , where S ( u, v, η ) := au x + κ v x + κ η x ( · , . Now, if U = ( u, u t , y, y t , η ( · , ρ )) ⊤ , then system (2.2)-(2.6) can bewritten as the following first order evolution equation(2.18) U t = A U, U (0) = U , where U = ( u , u , y , y , f ( · , − ρτ )) ⊤ ∈ H . Remark 2.1.
The linear unbounded operator A is injective (i.e. ker( A ) = { } ). Indeed, if U ∈ D ( A )such that A U = 0, then v = z = η ρ ( · , ρ ) = 0 and since η ( · ,
0) = v ( · ), we get η ( · , ρ ) = 0. Consequently,( S b ( u, v, η )) x = u xx = 0 and y xx = 0. Finally, since u (0) = u ( L ) = y (0) = y ( L ) = 0, then u = y = 0. Thus, U = ( u, v, y, z, η ( · , ρ )) ⊤ = 0. (cid:3) Proposition 2.1.
Under the hypothesis (H), the unbounded linear operator A is m-dissipative in the energyspace H . Proof.
For all U = ( u, v, y, z, η ( · , ρ )) ⊤ ∈ D ( A ), from (2.14) and (2.16), we have ℜ ( A U, U ) H = ℜ (Z L av x u x dx ) + ℜ (Z L ( S b ( u, v, η )) x vdx ) + ℜ (Z L z x y x dx ) + ℜ (Z L y xx zdx ) − ℜ ( | κ | Z β Z η xρ ( · , ρ ) η x ( · , ρ ) dρdx ) . Using integration by parts to the second and fourth terms in the above equation, then using the definition of S b ( u, v, η ) and the fact that U ∈ D ( A ), we get ℜ ( A U, U ) H = − κ Z β | v x | dx − ℜ ( κ Z β η x ( · , v x dx ) − | κ | Z β Z ddρ | η x ( · , ρ ) | dρdx, the fact that η ( · ,
0) = v ( · ), implies that ℜ ( A U, U ) H = − (cid:18) κ − | κ | (cid:19) Z β | v x | dx − | κ | Z β | η x ( · , | dx − ℜ ( κ Z β η x ( · , v x dx ) . ℜ ( A U, U ) H ≤ − ( κ − | κ | ) Z β | v x | dx ≤ , which implies that A is dissipative. Now, let us prove that A is maximal. For this aim, let F = ( f , f , f , f , f ( · , ρ )) ⊤ ∈H , we look for U = ( u, v, y, z, η ( · , ρ )) ⊤ ∈ D ( A ) unique solution of(2.20) − A U = F. Equivalently, we have the following system − v = f , (2.21) − ( S b ( u, v, η )) x + c ( · ) z = f , (2.22) − z = f , (2.23) − y xx − c ( · ) v = f , (2.24) τ − η ρ ( · , ρ ) = f ( · , ρ ) , (2.25)with the following boundary conditions(2.26) u (0) = u ( L ) = y (0) = y ( L ) = 0 , η (0 , ρ ) = 0 and η ( · ,
0) = v ( · ) . From (2.21), (2.25) and (2.26), we get(2.27) η ( x, ρ ) = τ Z ρ f ( x, s ) ds − f , ( x, ρ ) ∈ (0 , L ) × (0 , . Since, f ∈ H (0 , L ) and f ( · , ρ ) ∈ W . Then, from (2.25) and (2.27), we get η ρ ( · , ρ ) , η ( · , ρ ) ∈ W . Now, see thedefinition of S b ( u, v, η ), substituting (2.21), (2.23) and (2.27) in (2.22) and (2.24), we get the following system (cid:20) au x + b ( · ) (cid:18) − κ f x + τ κ Z f x ( · , s ) ds − κ f x (cid:19)(cid:21) x + c ( · ) f = − f , (2.28) y xx − c ( · ) f = − f , (2.29) u (0) = u ( L ) = y (0) = y ( L ) = 0 . (2.30)Let ( φ, ψ ) ∈ H (0 , L ) × H (0 , L ). Multiplying (2.28) and (2.29) by φ and ψ respectively, integrating over (0 , L ),then using integrations by parts, we obtain(2.31) a Z L u x φ x dx = Z L f φdx + c Z γα f φdx + ( κ + κ ) Z β f x φ x dx − τ κ Z β (cid:18)Z f x ( · , s ) ds (cid:19) φ x dx and(2.32) Z L y x ψ x dx = Z L f ψdx − c Z γα f ψdx. Adding (2.31) and (2.32), we obtain(2.33) B (( u, y ) , ( φ, ψ )) = L ( φ, ψ ) , ∀ ( φ, ψ ) ∈ H (0 , L ) × H (0 , L ) , where B (( u, y ) , ( φ, ψ )) = a Z L u x φ x dx + Z L y x ψ x dx and L ( φ, ψ ) = Z L (cid:0) f φ + f ψ (cid:1) dx + c Z γα (cid:0) f φ − f ψ (cid:1) dx − τ κ Z β (cid:18)Z f x ( · , s ) ds (cid:19) φ x dx + ( κ + κ ) Z β f x φ x dx. It is easy to see that, B is a sesquilinear, continuous and coercive form on (cid:0) H (0 , L ) × H (0 , L ) (cid:1) and L is alinear and continuous form on H (0 , L ) × H (0 , L ). Then, it follows by Lax-Milgram theorem that (2.33) admitsa unique solution ( u, y ) ∈ H (0 , L ) × H (0 , L ). By using the classical elliptic regularity, we deduce that sys-tem (2.28)-(2.30) admits a unique solution ( u, y ) ∈ H (0 , L ) × (cid:0) H (0 , L ) ∩ H (0 , L ) (cid:1) such that ( S b ( u, v, η )) x ∈ L (0 , L ) and since ker( A ) = { } (see Remark 2.1), we get U = (cid:18) u, − f , y, − f , τ Z ρ f ( · , s ) ds − f (cid:19) ⊤ ∈ D ( A )8s a unique solution of (2.20). Then, A is an isomorphism and since ρ ( A ) is open set of C (see Theorem 6.7(Chapter III) in [32]), we easily get R ( λI − A ) = H for a sufficiently small λ >
0. This, together with thedissipativeness of A , imply that D ( A ) is dense in H and that A is m-dissipative in H (see Theorems 4.5, 4.6in [46]). The proof is thus complete. (cid:3) According to Lumer-Philips theorem (see [46]), Proposition 2.1 implies that the operator A generates a C -semigroup of contractions e t A in H which gives the well-posedness of (2.18). Then, we have the followingresult: Theorem 2.1.
Under hypothesis (H), for all U ∈ H , System (2.18) admits a unique weak solution U ( x, ρ, t ) = e t A U ( x, ρ ) ∈ C ( R + , H ) . Moreover, if U ∈ D ( A ), then the system (2.18) admits a unique strong solution U ( x, ρ, t ) = e t A U ( x, ρ ) ∈ C ( R + , D ( A )) ∩ C ( R + , H ) . Strong Stability
In this section, we will prove the strong stability of system (2.2)-(2.6). The main result of this section is thefollowing theorem.
Theorem 3.1.
Assume that (H) is true. Then, the C − semigroup of contraction (cid:0) e t A (cid:1) t ≥ is strongly stablein H ; i.e., for all U ∈ H , the solution of (2.18) satisfieslim t → + ∞ k e t A U k H = 0 . According to Theorem A.2, to prove Theorem 3.1, we need to prove that the operator A has no pure imaginaryeigenvalues and σ ( A ) ∩ i R is countable. The proof of Theorem 3.1 will be achieved from the following proposition. Proposition 3.1.
Under the hypothesis (H), we have(3.1) i R ⊂ ρ ( A ) . We will prove Proposition 3.1 by contradiction argument. Remark that, it has been proved in Proposition 2.1that 0 ∈ ρ ( A ). Now, suppose that (3.1) is false, then there exists ω ∈ R ∗ such that iω / ∈ ρ ( A ). According toRemark A.3, let (cid:8) ( λ n , U n := ( u n , v n , y n , z n , η n ( · , ρ )) ⊤ ) (cid:9) n ≥ ⊂ R ∗ × D ( A ), with(3.2) λ n → ω as n → ∞ and | λ n | < | ω | and(3.3) k U n k H = (cid:13)(cid:13) ( u n , v n , y n , z n , η n ( · , ρ )) ⊤ (cid:13)(cid:13) H = 1 , such that(3.4) ( iλ n I − A ) U n = F n := ( f ,n , f ,n , f ,n , f ,n , f ,n ( · , ρ )) ⊤ → H . Equivalently, we have iλ n u n − v n = f ,n → H (0 , L ) , (3.5) iλ n v n − ( S b ( u n , v n , η n )) x + c ( · ) z n = f ,n → L (0 , L ) , (3.6) iλ n y n − z n = f ,n → H (0 , L ) , (3.7) iλ n z n − y nxx − c ( · ) v n = f ,n → L (0 , L ) , (3.8) iλ n η n ( ., ρ ) + τ − η nρ ( · , ρ ) = f ,n ( · , ρ ) → W . (3.9)Then, we will proof condition (3.1) by finding a contradiction with (3.3) such as k U n k H →
0. The proof ofproposition 3.1 has been divided into several Lemmas.9 emma 3.1.
Under the hypothesis (H), the solution U n = ( u n , v n , y n , z n , η n ( · , ρ )) ⊤ ∈ D ( A ) of system (3.5)-(3.9) satisfies the following limits lim n →∞ Z β | v nx | dx = 0 , (3.10) lim n →∞ Z β | v n | dx = 0 , (3.11) lim n →∞ Z β | u nx | dx = 0 , (3.12) lim n →∞ Z β Z | η nx ( · , ρ ) | dρdx = 0 , (3.13) lim n →∞ Z β | η nx ( · , | dx = 0 , (3.14) lim n →∞ Z β | S ( u n , v n , η n ) | dx = 0 . (3.15) Proof.
First, taking the inner product of (3.4) with U n in H and using (2.19) with the help of hypothesis (H),we obtain(3.16) Z β | v nx | dx ≤ − κ − | κ | ℜ ( A U n , U n ) H = 1 κ − | κ | ℜ ( F n , U n ) H ≤ κ − | κ | k F n k H k U n k H . Then, by passing to the limit in (3.16) and by using the fact that k U n k H = 1 and k F n k H →
0, we obtain(3.10). Now, since v n ∈ H (0 , L ), then it follows from Poincar´e inequality that there exists a constant C p > k v n k L (0 ,β ) ≤ C p k v nx k L (0 ,β ) . Thus, From (3.10) and (3.17), we obtain (3.11). Next, from (3.5) and the fact that Z β | f ,nx | dx ≤ Z L | f ,nx | dx ≤ a − k F n k H , we deduce that(3.18) Z β | u nx | dx ≤ λ n ) Z β | v nx | dx + 2( λ n ) Z β | f ,nx | dx ≤ λ n ) Z β | v nx | dx + 2 a ( λ n ) k F n k H . Therefore, by passing to the limit in (3.18) and by using (3.2), (3.10) and the fact that k F n k H →
0, we obtain(3.12). Moreover, from (3.9) and the fact that η n ( · ,
0) = v n ( · ), we deduce that(3.19) η n ( x, ρ ) = v n e − iλ n τρ + τ Z ρ e iλ n τ ( s − ρ ) f ,n ( x, s ) ds, ( x, ρ ) ∈ (0 , L ) × (0 , . From (3.19) and the fact that Z β Z | f ,nx ( · , s ) | dsdx ≤ τ − | κ | − k F n k H , we obtain(3.20) Z β Z | η nx ( · , ρ ) | dρdx ≤ Z β | v nx | dx + 2 τ Z β Z ρ Z ρ | f ,nx ( · , s ) | dρdsdx ≤ Z β | v nx | dx + τ Z β Z | f ,nx ( · , s ) | dsdx ≤ Z β | v nx | dx + τ | κ | − k F n k H . Thus, by passing to the limit in (3.20) and by using (3.10) with the fact that k F n k H →
0, we obtain (3.13).On the other hand, from (3.19), we have η nx ( · ,
1) = v nx e − iλ n τ + τ Z e iλ n τ ( s − f ,nx ( · , s ) ds, Z β | S ( u n , v n , η n ) | dx = Z β | au nx + κ v nx + κ η nx ( · , | dx ≤ a Z β | u nx | dx +3 κ Z β | v nx | dx +3 κ Z β | η nx ( · , | dx. Finally, passing to the limit in the above estimation, then using (3.10), (3.12) and (3.14), we obtain (3.15).The proof is thus complete. (cid:3)
Now we fix a function g ∈ C ([ α, β ]) such that(3.21) g ( α ) = − g ( β ) = 1 and set max x ∈ [ α,β ] | g ( x ) | = M g and max x ∈ [ α,β ] | g ′ ( x ) | = M g ′ . Remark 3.1.
To prove the existence of a function g , we need to find an example. For this aim, we can take g ( x ) = 1 + 2( α − x ) β − α , then g ∈ C ([ α, β ]), g ( α ) = − g ( β ) = 1, M g = 1 and M g ′ = 2 β − α . Also, we can take g ( x ) = cos (cid:18) ( α − x ) πα − β (cid:19) . (cid:3) Lemma 3.2.
Under the hypothesis (H), the solution U n = ( u n , v n , y n , z n , η n ( · , ρ )) ⊤ ∈ D ( A ) of system (3.5)-(3.9) satisfies the following inequalities | z n ( β ) | + | z n ( α ) | ≤ M g ′ Z βα | z n | dx + 2 | λ n | M g Z βα | z n | dx ! + 2 M g k F n k H , (3.22) | y nx ( β ) | + | y nx ( α ) | ≤ M g ′ Z βα | y nx | dx + 2( | λ n | + c ) M g Z βα | y nx | dx ! + 2 M g k F n k H (3.23)and the following limits(3.24) lim n →∞ (cid:16) | v n ( β ) | + | v n ( α ) | (cid:17) = 0 , (3.25) lim n →∞ (cid:16)(cid:12)(cid:12) ( S ( u n , v n , η n )) ( β − ) (cid:12)(cid:12) + | ( S ( u n , v n , η n )) ( α ) | (cid:17) = 0 . Proof.
First, from (3.7), we deduce that(3.26) iλ n y nx − z nx = f ,nx . Multiplying (3.26) and (3.8) by 2 gz n and 2 gy nx respectively, integrating over ( α, β ), using the definition of c ( · ),then taking the real part, we get(3.27) ℜ ( iλ n Z βα gy nx z n dx ) − Z βα g (cid:16) | z n | (cid:17) x dx = ℜ ( Z βα gf ,nx z n dx ) and(3.28) ℜ ( iλ n Z βα gz n y nx dx ) − Z βα g (cid:16) | y nx | (cid:17) x dx − ℜ ( c Z βα gv n y nx dx ) = ℜ ( Z βα gf ,n y nx dx ) . Using integration by parts in (3.27) and (3.28), we obtain h − g | z n | i βα = − Z βα g ′ | z n | dx − ℜ ( iλ n Z βα gy nx z n dx ) + ℜ ( Z βα gf ,nx z n dx ) and h − g | y nx | i βα = − Z βα g ′ | y nx | dx − ℜ ( iλ n Z βα gz n y nx dx ) + ℜ ( c Z βα gv n y nx ) + ℜ ( Z βα gf ,n y nx dx ) . g and Cauchy-Schwarz inequality in the above equations, we obtain(3.29) | z n ( β ) | + | z n ( α ) | ≤ M g ′ Z βα | z n | dx + 2 | λ n | M g Z βα | y nx | dx ! Z βα | z n | dx ! + 2 M g Z βα | f ,nx | dx ! Z βα | z n | dx ! and(3.30) | y nx ( β ) | + | y nx ( α ) | ≤ M g ′ Z βα | y nx | dx + 2 | λ n | M g Z βα | y nx | dx ! Z βα | z n | dx ! + 2 c M g Z βα | y nx | dx ! Z βα | v n | dx ! + 2 M g Z βα | f ,n | dx ! Z βα | y nx | dx ! . Therefore, from (3.29), (3.30) and the fact that Z βα | ξ n | dx ≤ Z L | ξ n | dx ≤ k U n k H = 1 with ξ n ∈ { v n , y nx , z n } and Z βα | ξ n | dx ≤ Z L | ξ n | dx ≤ k F n k H with ξ n ∈ { f ,nx , f ,n } , we obtain (3.22) and (3.23). On the otherhand, from (3.5), we deduce that(3.31) iλ n u nx − v nx = f ,nx . Multiplying (3.31) and (3.6) by 2 gv n and 2 gS ( u n , v n , η n ) respectively, integrating over ( α, β ), using the defi-nition of c ( · ) and S b ( u n , v n , η n ), then taking the real part, we get(3.32) ℜ ( iλ n Z βα gu nx v n dx ) − Z βα g ( | v n | ) x dx = ℜ ( Z βα gf ,nx v n dx ) and(3.33) ℜ ( iλ n Z βα gv n S ( u n , v n , η n ) dx ) − Z βα g (cid:16) | S ( u n , v n , η n ) | (cid:17) x dx + ℜ ( c Z βα gz n S ( u n , v n , η n ) dx ) = ℜ ( Z βα gf ,n S ( u n , v n , η n ) dx ) . Using integration by parts in (3.32) and (3.33), we get h − g | v n | i βα = − Z βα g ′ | v n | dx − ℜ ( iλ n Z βα gu nx v n dx ) + ℜ ( Z βα gf ,nx v n dx ) and h − g | S ( u n , v n , η n ) | i βα = − Z βα g ′ | S ( u n , v n , η n ) | dx − ℜ ( iλ n Z βα gv n S ( u n , v n , η n ) dx ) − ℜ ( c Z βα gz n S ( u n , v n , η n ) dx ) + ℜ ( Z βα gf ,n S ( u n , v n , η n ) dx ) . g and Cauchy-Schwarz inequality in the above equations, then using the fact that Z βα | z n | dx ≤ Z L | z n | dx ≤ k U n k H = 1 , Z βα | f ,nx | dx ≤ Z L | f ,nx | dx ≤ a − k F n k H and Z βα | f ,n | dx ≤ Z L | f ,n | dx ≤ k F n k H , we obtain(3.34) | v n ( β ) | + | v n ( α ) | ≤ M g ′ Z βα | v n | dx + 2 | λ n | M g Z βα | u nx | dx ! Z βα | v n | dx ! + 2 √ a M g Z βα | v n | dx ! k F n k H and(3.35) (cid:12)(cid:12) ( S ( u n , v n , η n )) ( β − ) (cid:12)(cid:12) + | ( S ( u n , v n , η n )) ( α ) | ≤ M g ′ Z βα | S ( u n , v n , η n ) | dx + 2 | λ n | M g Z βα | S ( u n , v n , η n ) | dx ! Z βα | v n | dx ! + 2 c M g Z βα | S ( u n , v n , η n ) | dx ! + 2 M g Z βα | S ( u n , v n , η n ) | dx ! k F n k H . Finally, passing to limit in (3.34) and (3.35), then using (3.2), Lemma 3.1 and the fact that k F n k H →
0, weobtain (3.24) and (3.25). The proof is thus complete. (cid:3)
Lemma 3.3.
Under the hypothesis (H), the solution U n = ( u n , v n , y n , z n , η n ( · , ρ )) ⊤ ∈ D ( A ) of system (3.5)-(3.8) satisfies the following limits(3.36) lim n →∞ Z βα | z n | dx = 0 and lim n →∞ Z βα | y nx | dx = 0 . Proof.
First, multiplying (3.6) by z n , integrating over ( α, β ), using the definition of c ( · ) and S b ( u n , v n , η n ),then taking the real part, we get(3.37) ℜ ( iλ n Z βα v n z n dx ) − ℜ (Z βα ( S ( u n , v n , η n )) x z n dx ) + c Z βα | z n | dx = ℜ (Z βα f ,n z n dx ) . From (3.7), we deduce that(3.38) z nx = − iλ n y nx − f ,nx . Using integration by parts to the second term in (3.37), then using (3.38), we get c Z βα | z n | dx = ℜ ( iλ n Z βα S ( u n , v n , η n ) y nx dx ) + ℜ (Z βα S ( u n , v n , η n ) f ,nx dx ) + ℜ n [ S ( u n , v n , η n ) z n ] βα o + ℜ (Z βα f ,n z n dx ) − ℜ ( iλ n Z βα v n z n dx ) . Z βα | ξ n | dx ≤ Z L | ξ n | dx ≤ k U n k H =1 with ξ n ∈ { y nx , z n } and Z βα | ξ n | dx ≤ Z L | ξ n | dx ≤ k F n k H with ξ n ∈ { f ,n , f ,nx } , we obtain(3.39) c Z βα | z n | dx ≤ ( | λ n | + k F n k H ) Z βα | S ( u n , v n , η n ) | dx ! + | λ n | Z βα | v n | dx ! + k F n k H + (cid:12)(cid:12) ( S ( u n , v n , η n )) ( β − ) (cid:12)(cid:12) | z n ( β ) | + | ( S ( u n , v n , η n )) ( α ) | | z n ( α ) | . Now, using the fact that Z βα | z n | dx ≤ Z L | z n | dx ≤ k U n k H = 1 in (3.22), we get(3.40) | z n ( x ) | ≤ ( M g ′ + 2 | λ n | M g + 2 M g k F n k H ) for x ∈ { α, β } . Inserting (3.40) in (3.39), we obtain c Z βα | z n | dx ≤ ( | λ n | + k F n k H ) Z βα | S ( u n , v n , η n ) | dx ! + | λ n | Z βα | v n | dx ! + k F n k H , + ( M g ′ + 2 | λ n | M g + 2 M g k F n k H ) (cid:0)(cid:12)(cid:12) ( S ( u n , v n , η n )) ( β − ) (cid:12)(cid:12) + | ( S ( u n , v n , η n )) ( α ) | (cid:1) . Therefore, by passing to the limit in the above inequality and by using (3.2), (3.25), Lemma 3.1 and the factthat k F n k H →
0, we obtain the first limit in (3.36). On the other hand, multiplying (3.8) by − z n ( λ n ) − ,integrating over ( α, β ), using the definition of c ( · ), then taking the imaginary part, we get − Z βα | z n | dx + ℑ ( ( λ n ) − Z βα y nxx z n dx ) + ℑ ( c ( λ n ) − Z βα v n z n dx ) = −ℑ ( ( λ n ) − Z βα f ,n z n dx ) . Using integration by parts to the second term in the above equation, then using (3.38), we obtain Z βα | y nx | dx = Z βα | z n | dx − ℑ ( ( λ n ) − Z βα f ,nx y nx dx ) − ℑ n ( λ n ) − [ y nx z n ] βα o − ℑ ( c ( λ n ) − Z βα v n z n dx ) − ℑ ( ( λ n ) − Z βα f ,n z n dx ) . Using Cauchy-Schwarz inequality in the above equation and the fact that k U n k H = 1, we get(3.41) Z βα | y nx | dx ≤ Z βα | z n | dx + c | λ n | − Z βα | v n | dx ! + 2 | λ n | − k F n k H + | λ n | − | y nx ( β ) || z n ( β ) | + | λ | − | y nx ( α ) || z n ( α ) | . Moreover, using the fact that Z βα | y nx | dx ≤ Z L | y nx | dx ≤ k U n k H = 1 in (3.23), we get(3.42) | y nx ( x ) | ≤ ( M g ′ + 2( | λ n | + c ) M g + 2 M g k F n k H ) for x ∈ { α, β } . Inserting (3.42) in (3.41), we obtain(3.43) Z βα | y nx | dx ≤ Z βα | z n | dx + c | λ n | − Z βα | v n | dx ! + 2 | λ n | − k F n k H + | λ n | − ( M g ′ + 2( | λ n | + c ) M g + 2 M g k F n k H ) ( | z n ( β ) | + | z n ( α ) | ) . Now, passing to the limit in inequality (3.22), then using (3.2), the first limit in (3.36) and the fact that k F n k H →
0, we get(3.44) lim n →∞ (cid:16) | z n ( β ) | + | z n ( α ) | (cid:17) = 0 . k F n k H →
0, we obtain the second limit in (3.36). The proof is thus complete. (cid:3)
Lemma 3.4.
Under the hypothesis (H), the solution U n = ( u n , v n , y n , z n , η n ( · , ρ )) ⊤ ∈ D ( A ) of system (3.5)-(3.9) satisfies the following estimations(3.45) lim n →∞ | u n ( β ) | = 0 and lim n →∞ | y n ( β ) | = 0 , (3.46) lim n →∞ | u nx ( β + ) | = 0 and lim n →∞ | y nx ( β ) | = 0 , (3.47) lim n →∞ (cid:18)Z γβ | u n | dx + Z γβ | u nx | dx + Z γβ | y n | dx + Z γβ | y nx | dx (cid:19) = 0 , (3.48) lim n →∞ Z γβ | v n | dx = 0 and lim n →∞ Z γβ | z n | dx = 0 . Proof.
First, from (3.5) and (3.7), we get | u n ( β ) | ≤ λ n ) − | v n ( β ) | + 2( λ n ) − | f ,n ( β ) | and | y n ( β ) | ≤ λ n ) − | z n ( β ) | + 2( λ n ) − | f ,n ( β ) | . Using the fact that | f ,n ( β ) | ≤ β Z β | f ,nx | dx ≤ βa − k F n k H and | f ,n ( β ) | ≤ β Z β | f ,nx | dx ≤ β k F n k H in theabove inequalities, we obtain | u n ( β ) | ≤ λ n ) − | v n ( β ) | + 2 βa − ( λ n ) − k F n k H and | y n ( β ) | ≤ λ n ) − | z n ( β ) | + 2 β ( λ n ) − k F n k H . Passing to the limit in the above inequalities, then using (3.2), (3.24), (3.44) and the fact that k F n k H →
0, weobtain (3.45). Second, since S b ( u n , v n , η n ) ∈ H (0 , L ) ⊂ C ([0 , L ]), then we deduce that(3.49) (cid:12)(cid:12) ( S ( u n , v n , η n )) ( β − ) (cid:12)(cid:12) = | au nx ( β + ) | . Thus, from (3.25) and (3.49), we obtain the first limit in (3.46). Moreover, passing to the limit in inequality(3.23), then using (3.2), the second limit in (3.36) and the fact that k F n k H →
0, we obtain the second limit in(3.46). On the other hand, (3.5)-(3.8) can be written in ( β, γ ) as the following form( λ n ) u n + au nxx − iλ n c y n = G ,n in ( β, γ ) , (3.50) ( λ n ) y n + y nxx + iλ n c u n = G ,n in ( β, γ ) , (3.51)where(3.52) G ,n = − f ,n − iλ n f ,n − c f ,n and G ,n = − f ,n − iλ n f ,n + c f ,n . Let V n = ( u n , u nx , y n , y nx ) ⊤ , then (3.50)-(3.51) can be written as the following(3.53) V nx = B n V n + G n , where B n = − a − ( λ n ) a − iλ n c
00 0 0 1 − iλ n c − ( λ n ) = ( b ij ) ≤ i,j ≤ and G n = a − G ,n G ,n . The solution of the differential equation (3.53) is given by(3.54) V n ( x ) = e B n ( x − β ) V n ( β + ) + Z xβ e B n ( s − x ) G n ( s ) ds, where e B n ( x − β ) = ( c ij ) ≤ i,j ≤ and e B n ( s − x ) = ( d ij ) ≤ i,j ≤ are denoted by the exponential of the matrices B n ( x − β ) and B n ( s − x ) respectively. Now, from (3.2), the entries b ij are bounded for all 1 ≤ i, j ≤ b ij ( x − β ) and b ij ( s − x ) are bounded. In addition, from the definition of theexponential of a square matrix, we obtain e B n ζ = ∞ X k =0 ( B n ζ ) k k ! for ζ ∈ { x − β, s − x } . Therefore, the entries c ij and d ij are also bounded for all 1 ≤ i, j ≤ e B n ( x − β ) and e B n ( s − x ) are two bounded matrices. From (3.45) and (3.46), we directly obtain(3.55) V n ( β + ) → L ( β, γ )) , as n → ∞ . Moreover, from (3.52), we deduce that(3.56) Z γβ | G ,n | dx ≤ Z L | f ,n | dx + 3( λ n ) Z L | f ,n | dx + 3 c Z L | f ,n | dx and(3.57) Z γβ | G ,n | dx ≤ Z L | f ,n | dx + 3( λ n ) Z L | f ,n | dx + 3 c Z L | f ,n | dx. Now, since f ,n , f ,n ∈ H (0 , L ), then it follows by Poincar´e inequality that there exist two constants C > C > k f ,n k L (0 ,L ) ≤ C k f ,nx k L (0 ,L ) and k f ,n k L (0 ,L ) ≤ C k f ,nx k L (0 ,L ) . Consequently, from (3.56), (3.57) and (3.58), we get(3.59) Z γβ | G ,n | dx ≤ (cid:0) a − ( λ n C ) + ( c C ) (cid:1) k F n k H , and(3.60) Z γβ | G ,n | dx ≤ (cid:0) λ n C ) + a − ( c C ) (cid:1) k F n k H . Hence, from (3.2), (3.59), (3.60) and the fact that k F n k H →
0, we obtain(3.61) G n → L ( β, γ )) , as n → ∞ . Therefore, from (3.54), (3.55), (3.61) and as e B n ( x − β ) , e B n ( s − x ) are two bounded matrices, we get V n → L ( β, γ )) and consequently, we obtain (3.47). Next, from (3.5) , (3.7) and (3.58), we deduce that Z γβ | v n | dx ≤ λ n ) Z γβ | u n | dx + 2 Z γβ | f ,n | dx ≤ λ n ) Z γβ | u n | dx + 2 C a − k F n k H , Z γβ | z n | dx ≤ λ n ) Z γβ | y n | dx + 2 Z γβ | f ,n | dx ≤ λ n ) Z γβ | y n | dx + 2 C k F n k H . Finally, passing to the limit in the above inequalities, then using (3.2), (3.47) and the fact that k F n k H → (cid:3) Lemma 3.5.
Let h ∈ C ([0 , L ]) be a function. Under the hypothesis (H), the solution U n = ( u n , v n , y n , z n , η n ( · , ρ )) ⊤ ∈ D ( A ) of system (3.5)-(3.9) satisfies the following estimation Z L h ′ (cid:16) a − | S b ( u n , v n , η n ) | + | v n | + | z n | + | y nx | (cid:17) dx − h h (cid:16) a − | S b ( u n , v n , η n ) | + | y nx | (cid:17)i L − ℜ ( Z L c ( · ) hv n y nx dx ) + ℜ ( a Z L c ( · ) hz n S b ( u n , v n , η n ) dx ) + ℜ ( iλ n a Z L b ( · ) hv n ( κ v nx + κ η nx ( · , dx ) = ℜ ( Z L hf ,nx v n dx ) + ℜ ( a Z L hf ,n S b ( u n , v n , η n ) dx ) + ℜ ( Z L hf ,nx z n dx ) + ℜ ( Z L hf ,n y nx dx ) . roof. First, multiplying (3.6) and (3.8) by 2 a − hS b ( u n , v n , η n ) and 2 hy nx respectively, integrating over (0 , L ),then taking the real part, we get(3.62) ℜ ( iλ n a Z L hv n S b ( u n , v n , η n ) dx ) − a − Z L h (cid:16) | S b ( u n , v n , η n ) | (cid:17) x dx + ℜ ( a Z L c ( · ) hz n S b ( u n , v n , η n ) dx ) = ℜ ( a Z L hf ,n S b ( u n , v n , η n ) dx ) and(3.63) ℜ ( iλ n Z L hz n y nx dx ) − Z L h (cid:16) | y nx | (cid:17) x dx − ℜ ( Z L c ( · ) hv n y nx dx ) = ℜ ( Z L hf ,n y nx dx ) . From (3.5) and (3.7), we deduce that iλ n u nx = − v nx − f ,nx , (3.64) iλ n y nx = − z nx − f ,nx . (3.65)Consequently, from (3.64) and the definition S b ( u n , v n , η n ), we have(3.66) iλS b ( u n , v n , η n ) = − a (cid:16) v nx + f ,nx (cid:17) + iλb ( · ) ( κ v nx + κ η nx ( · , . Substituting (3.66) and (3.65) in (3.62) and (3.63) respectively, we obtain − Z L h (cid:16) | v n | + a − | S b ( u n , v n , η n ) | (cid:17) x dx + ℜ ( iλ n a Z L b ( · ) hv n ( κ v nx + κ η nx ( · , dx ) + ℜ ( a Z L c ( · ) hz n S b ( u n , v n , η n ) dx ) = ℜ ( Z L hf ,nx v n dx ) + ℜ ( a Z L hf ,n S b ( u n , v n , η n ) dx ) and − Z L h (cid:16) | z n | + | y nx | (cid:17) x dx − ℜ ( Z L c ( · ) hv n y nx dx ) = ℜ ( Z L hf ,n y nx dx ) + ℜ ( Z L hf ,nx z n dx ) . Finally, adding the above equations, then using integration by parts and the fact that v n (0) = v n ( L ) = 0 and z n (0) = z n ( L ) = 0, we obtain the desired result. The proof is thus complete. (cid:3) Now, we fix the cut-off functions χ , χ ∈ C ([0 , L ]) (see Figure 2) such that 0 ≤ χ ( x ) ≤
1, 0 ≤ χ ( x ) ≤
1, forall x ∈ [0 , L ] and χ ( x ) = (cid:26) x ∈ [0 , α ] , x ∈ [ β, L ] , and χ ( x ) = (cid:26) x ∈ [0 , β ] , x ∈ [ γ, L ] , and set max x ∈ [0 ,L ] | χ ′ ( x ) | = M χ ′ and max x ∈ [0 ,L ] | χ ′ ( x ) | = M χ ′ , α β γ L χ χ Figure 2.
Geometric description of the functions χ and χ .17 emma 3.6. Under the hypothesis (H), the solution U n = ( u n , v n , y n , z n , η n ( · , ρ )) ⊤ ∈ D ( A ) of system (3.5)-(3.9) satisfies the following limits lim n →∞ (cid:18)Z α | y nx | dx + Z α | z n | dx (cid:19) = 0 , (3.67) lim n →∞ a Z Lγ | u nx | dx + Z Lγ | v n | dx + Z Lγ | y nx | dx + Z Lγ | z n | dx ! = 0 . (3.68) Proof.
First, using the result of Lemma 3.5 with h = xχ , then using the definition of b ( · ), c ( · ), S b ( u n , v n , η n )and χ , we get Z α | y nx | dx + Z α | z n | dx = − Z α | v n | dx − a − Z α | S ( u n , v n , η n ) | dx − Z βα ( χ + xχ ′ ) (cid:16) a − | S ( u n , v n , η n ) | + | v n | + | y nx | + | z n | (cid:17) dx − ℜ ( c a Z βα xχ z n S ( u n , v n , η n ) dx ) + ℜ ( c Z βα xχ v n y nx dx ) − ℜ ( iλ n a Z β xχ v n ( κ v nx + κ η nx ( · , dx ) + ℜ ( a Z β xχ f ,n ( κ v nx + κ η nx ( · , dx ) + ℜ ( Z L xχ (cid:16) f ,nx v n + f ,n u nx + f ,nx z n + f ,n y nx (cid:17) dx ) . Using Cauchy-Schwarz inequality in the above Equation and the fact that k U n k H = k ( u n , v n , y n , z n , η n ( · , ρ )) ⊤ k H =1, we obtain Z α | y nx | dx + Z α | z n | dx ≤ Z α | v n | dx + a − Z α | S ( u n , v n , η n ) | dx + (cid:0) βM χ ′ (cid:1) Z βα (cid:16) a − | S ( u n , v n , η n ) | + | v n | + | z n | + | y nx | (cid:17) dx + 2 c βa Z βα | S ( u n , v n , η n ) | dx ! + 2 c β Z βα | v n | dx ! + 2 βa ( | λ n | + k F n k H ) κ Z β | v nx | dx ! + | κ | Z β | η nx ( · , | dx ! + 4 L (cid:18) √ a + 1 (cid:19) k F n k H . Therefore, by passing to the limit in the above inequality and by using (3.2), Lemmas 3.1, 3.3 and the factthat k F n k H →
0, we obtain (3.67). On the other hand, using the result of Lemma 3.5 with h = ( x − L ) χ ,then using Cauchy-Schwarz inequality and the fact that k U n k H = 1, we get a Z Lγ | u nx | + Z Lγ | v n | dx + Z Lγ | y nx | dx + Z Lγ | z n | dx ≤ (cid:0) L − β ) M χ ′ (cid:1) Z γβ (cid:16) a | u nx | + | v n | + | y nx | + | z n | (cid:17) dx + 2 c ( L − β ) (cid:18)Z γβ | v n | dx (cid:19) (cid:18)Z γβ | y nx | dx (cid:19) + 2 c ( L − β ) (cid:18)Z γβ | z n | dx (cid:19) (cid:18)Z γβ | u nx | dx (cid:19) + 4 L (cid:18) √ a + 1 (cid:19) k F n k H . Finally, passing to the limit in the above inequality, then using Lemma 3.4 and the fact that k F n k H →
0, weobtain (3.68). The proof is thus complete. (cid:3)
Proof of Proposition 3.1.
From Lemmas 3.1-3.6, we obtain k U n k H → n → ∞ which contradicts k U n k H = 1. Thus, (3.1) is holds true. The proof is thus complete. (cid:3) roof of Theorem 3.1. From proposition 3.1, we have i R ⊂ ρ ( A ) and consequently σ ( A ) ∩ i R = ∅ . Therefore,according to Theorem A.2, we get that the C − semigroup of contraction ( e t A ) t ≥ is strongly stable. The proofis thus complete. (cid:3) Polynomial Stability
In this section, we will prove the polynomial stability of system (2.2)-(2.6). The main result of this section isthe following theorem.
Theorem 4.1.
Under the hypothesis (H), for all U ∈ D ( A ), there exists a constant C > U such that the energy of system (2.2)-(2.6) satisfies the following estimation E ( t ) ≤ Ct k U k D ( A ) , ∀ t > . According to Theorem A.4, to prove Theorem 4.1, we will prove the following two conditions(4.1) i R ⊂ ρ ( A )and(4.2) sup λ ∈ R | λ | (cid:13)(cid:13) ( iλI − A ) − (cid:13)(cid:13) L ( H ) < + ∞ . From proposition 3.1, we obtain condition (4.1). Next, we will prove condition (4.2) by a contradiction argu-ment. For this purpose, suppose that (4.2) is false, then there exists (cid:8) ( λ n , U n := ( u n , v n , y n , z n , η n ( · , ρ )) ⊤ ) (cid:9) n ≥ ⊂ R ∗ × D ( A ) with(4.3) | λ n | → ∞ and k U n k H = (cid:13)(cid:13) ( u n , v n , y n , z n , η n ( · , ρ )) ⊤ (cid:13)(cid:13) H = 1 , such that(4.4) ( λ n ) ( iλ n I − A ) U n = F n := ( f ,n , f ,n , f ,n , f ,n , f ,n ( · , ρ )) ⊤ → H . For simplicity, we drop the index n . Equivalently, from (4.4), we have iλu − v = λ − f → H (0 , L ) , (4.5) iλv − ( S b ( u, v, η )) x + c ( · ) z = λ − f → L (0 , L ) , (4.6) iλy − z = λ − f → H (0 , L ) , (4.7) iλz − y xx − c ( · ) v = λ − f → L (0 , L ) , (4.8) iλη ( ., ρ ) + τ − η ρ ( ., ρ ) = λ − f ( ., ρ ) → W . (4.9)Here we will check the condition (4.2) by finding a contradiction with (4.3) such as k U k H = o (1). For clarity,we divide the proof into several Lemmas. Lemma 4.1.
Under the hypothesis (H), the solution U = ( u, v, y, z, η ( · , ρ )) ⊤ ∈ D ( A ) of system (4.5)-(4.9)satisfies the following estimations Z β | v x | dx = o ( λ − ) , (4.10) Z β | u x | dx = o ( λ − ) , (4.11) Z β Z | η x ( · , ρ ) | dρdx = o ( λ − ) , (4.12) Z β | η x ( · , | dx = o ( λ − ) , (4.13) Z β | S ( u, v, η ) | dx = o ( λ − ) . (4.14) 19 roof. First, taking the inner product of (4.4) with U in H and using (2.19) with the help of hypothesis (H),we obtain(4.15) Z β | v x | dx ≤ − κ − | κ | ℜ ( A U, U ) H = λ − κ − | κ | ℜ ( F, U ) H ≤ λ − κ − | κ | k F k H k U k H . Thus, from (4.15) and the fact that k F k H = o (1) and k U k H = 1, we obtain (4.10). Now, from (4.5), we deducethat(4.16) Z β | u x | dx ≤ λ − Z β | v x | dx + 2 λ − Z β | f x | dx ≤ λ − Z β | v x | dx + 2 λ − Z L | f x | dx. Therefore, from (4.10), (4.16) and the fact that k f x k L (0 ,L ) = o (1), we obtain (4.11). Next, from (4.9) and thefact that η ( · ,
0) = v ( · ) , we get(4.17) η ( x, ρ ) = ve − iλτρ + τ λ − Z ρ e iλτ ( s − ρ ) f ( x, s ) ds, ( x, ρ ) ∈ (0 , L ) × (0 , . From (4.17), we deduce that(4.18) Z β Z | η x ( · , ρ ) | dρdx ≤ Z β | v x | dx + τ λ − Z β Z | f x ( · , s ) | dsdx. Thus, from (4.10), (4.18) and the fact that f ( · , ρ ) → W , we obtain (4.12). On the other hand, from(4.17), we have η x ( · ,
1) = v x e − iλτ + τ λ − Z e iλτ ( s − f x ( · , s ) ds, consequently, similar to the previous proof, we obtain (4.13). Next, it is clear to see that Z β | S ( u, v, η ) | dx = Z β | au x + κ v x + κ η x ( · , | dx ≤ a Z β | u x | dx + 3 κ Z β | v x | dx + 3 κ Z β | η x ( · , | dx. Finally, from (4.10), (4.11), (4.13) and the above estimation, we obtain (4.14). The proof is thus complete. (cid:3) ε ε α α + ε β − ε β − ε β − ε β γ L θ θ θ Figure 3.
Geometric description of the functions θ , θ and θ . Lemma 4.2.
Let 0 < ε < min (cid:16) α , β − α (cid:17) . Under the hypothesis (H), the solution U = ( u, v, y, z, η ( · , ρ )) ⊤ ∈ D ( A ) of system (4.5)-(4.9) satisfies the following estimation Z β − εε | v | dx = o (1) . (4.19) Proof.
First, we fix a cut-off function θ ∈ C ([0 , L ]) (see Figure 3) such that 0 ≤ θ ( x ) ≤
1, for all x ∈ [0 , L ]and θ ( x ) = ( x ∈ [ ε, β − ε ] , x ∈ { } ∪ [ β, L ] , and set max x ∈ [0 ,L ] | θ ′ ( x ) | = M θ ′ . λ − θ v , integrating over (0 , L ), then taking the imaginary part, we obtain Z L θ | v | dx − ℑ ( λ − Z L θ ( S b ( u, v, η )) x vdx ) + ℑ ( λ − Z L c ( · ) θ zvdx ) = ℑ ( λ − Z L θ f vdx ) . Using integration by parts in the above equation and the fact that v (0) = v ( L ) = 0, we get(4.20) Z L θ | v | dx = −ℑ ( λ Z L ( θ ′ v + θ v x ) S b ( u, v, η ) dx ) −ℑ ( λ Z L c ( · ) θ zvdx ) + ℑ ( λ Z L θ f vdx ) . Using the definition of c ( · ), S b ( u, v, η ) and θ , then, using Cauchy-Schwarz inequality, we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℑ ( λ − Z L ( θ ′ v + θ v x ) S b ( u, v, η ) dx )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℑ ( λ − Z β ( θ ′ v + θ v x ) S ( u, v, η ) dx )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | λ | − M θ ′ Z β | v | dx ! + Z β | v x | dx ! Z β | S ( u, v, η ) | dx ! and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℑ ( λ − Z L c ( · ) θ zvdx )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℑ ( c λ − Z βα θ zvdx )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c | λ | − Z βα | z | dx ! Z βα | v | dx ! . Thus, from the above inequalities, Lemma 4.1 and the fact that v and z are uniformly bounded in L (0 , L ), weobtain(4.21) −ℑ ( λ − Z L ( θ ′ v + θ v x ) S b ( u, v, η ) dx ) = o ( λ − ) and −ℑ ( λ − Z L c ( · ) θ zvdx ) = O ( | λ | − ) = o (1) . Inserting (4.21) in (4.20), then using the fact that v is uniformly bounded in L (0 , L ) and k f k L (0 ,L ) = o (1),we obtain Z L θ | v | dx = o (1) . Finally, from the above estimation and the definition of θ , we obtain (4.19). The proof is thus complete. (cid:3) Lemma 4.3.
Let 0 < ε < min (cid:16) α , β − α (cid:17) . Under the hypothesis (H), the solution U = ( u, v, y, z, η ( · , ρ )) ⊤ ∈ D ( A ) of system (4.5)-(4.9) satisfies the following estimations(4.22) Z β − εα | z | dx = o (1) and Z β − εα + ε | y x | dx = o (1) . Proof.
First, we fix a cut-off function θ ∈ C ([0 , L ]) (see figure 3) such that 0 ≤ θ ( x ) ≤
1, for all x ∈ [0 , L ]and θ ( x ) = ( x ∈ [0 , ε ] ∪ [ β − ε, L ] , x ∈ [2 ε, β − ε ] , and set max x ∈ [0 ,L ] | θ ′ ( x ) | = M θ ′ . Multiplying (4.6) and (4.8) by θ z and θ v respectively, integrating over (0 , L ), then taking the real part, weobtain(4.23) ℜ ( iλ Z L θ vzdx ) − ℜ (Z L θ ( S b ( u, v, η )) x zdx ) + Z L c ( · ) θ | z | dx = ℜ ( λ − Z L θ f zdx ) and(4.24) ℜ ( iλ Z L θ zvdx ) − ℜ (Z L θ y xx vdx ) − Z L c ( · ) θ | v | dx = ℜ ( λ − Z L θ f vdx ) . v (0) = v ( L ) = 0 and z (0) = z ( L ) = 0,we get(4.25) Z L c ( · ) θ | z | dx = Z L c ( · ) θ | v | dx − ℜ (Z L ( θ ′ z + θ z x ) S b ( u, v, η ) dx ) − ℜ (Z L ( θ ′ v + θ v x ) y x dx ) + ℜ ( λ − Z L θ f zdx ) + ℜ ( λ − Z L θ f vdx ) . From (4.7), we deduce that(4.26) z x = − iλy x − λ − f x . Using (4.26) and the definition of S b ( u, v, η ) and θ , then using Cauchy-Schwarz inequality, we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℜ (Z L ( θ ′ z + θ z x ) S b ( u, v, η ) dx )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℜ (Z β − εε h θ ′ z + θ ( − iλy x − λ − f x ) i S ( u, v, η ) dx )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M θ ′ Z β − εε | z | dx ! + | λ | Z β − εε | y x | dx ! + λ − Z β − εε | f x | dx ! Z β − εε | S ( u, v, η ) | dx ! and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℜ (Z L ( θ ′ v + θ v x ) y x dx )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℜ (Z β − εε ( θ ′ v + θ v x ) y x dx )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M θ ′ Z β − εε | v | dx ! + Z β − εε | v x | dx ! Z β − εε | y x | dx ! . Thus, from the above inequalities, Lemmas 4.1, 4.2 and the fact that y x , z are uniformly bounded in L (0 , L )and k f x k L (0 ,L ) = o (1), we obtain(4.27) − ℜ (Z L ( θ ′ z + θ z x ) S b ( u, v, η ) dx ) = o (1) and − ℜ (Z L ( θ ′ v + θ v x ) y x dx ) = o (1) . Inserting (4.27) in (4.25), then using the fact that v , z are uniformly bounded in L (0 , L ) and k f k L (0 ,L ) = o (1), k f k L (0 ,L ) = o (1), we obtain Z L c ( · ) θ | z | dx = Z L c ( · ) θ | v | dx + o (1) . Therefore, from the above estimation, Lemma 4.2 and the definition of c ( · ) and θ , we obtain the first estimationin (4.22). On the other hand, let us fix a cut-off function θ ∈ C ([0 , L ]) (see Figure 3) such that 0 ≤ θ ( x ) ≤ x ∈ [0 , L ] and θ ( x ) = ( x ∈ [0 , α ] ∪ [ β − ε, L ] , x ∈ [ α + ε, β − ε ] , Now, multiplying (4.8) by − λ − θ z , integrating over (0 , L ), then taking the imaginary part, we obtain − Z L θ | z | dx + ℑ ( λ − Z L θ y xx zdx ) + ℑ ( λ − Z L c ( · ) θ vzdx ) = −ℑ ( λ − Z L θ f zdx ) . Using integration by parts in the above equation and the fact that z (0) = z ( L ) = 0, then using (4.26), we get(4.28) Z L θ | y x | dx = Z L θ | z | dx + ℑ ( λ − Z L θ ′ y x zdx ) − ℑ ( λ − Z L c ( · ) θ vzdx ) − ℑ ( λ − Z L θ f x y x dx ) − ℑ ( λ − Z L θ f zdx ) . c ( · ) and θ , the first estimation of (4.22) and the fact that v and y x are uniformlybounded in L (0 , L ), we obtain(4.29) ℑ ( λ − Z L θ ′ y x zdx ) = ℑ ( λ − Z β − εα θ ′ y x zdx ) = o (cid:0) | λ | − (cid:1) , −ℑ ( λ − Z L c ( · ) θ vzdx ) = −ℑ ( c λ − Z β − εα θ vzdx ) = o (cid:0) | λ | − (cid:1) . Inserting (4.29) in (4.28), then using the fact that y x , z are uniformly bounded in L (0 , L ) and k f x k L (0 ,L ) = o (1), k f k L (0 ,L ) = o (1), we get Z L θ | y x | dx = Z L θ | z | dx + o ( | λ | − ) . Finally, from the above estimation, the first estimation of (4.22) and the definition of θ , we obtain the secondestimation in (4.22). The proof is thus complete. (cid:3) Lemma 4.4. < ε < min (cid:16) α , β − α (cid:17) . Under the hypothesis (H), the solution U = ( u, v, y, z, η ( · , ρ )) ⊤ ∈ D ( A )of system (4.5)-(4.9) satisfies the following estimations | v ( γ ) | + | v ( β − ε ) | + a | u x ( γ ) | + a − | ( S ( u, v, η )) ( β − ε ) | = O (1) , (4.30) | z ( γ ) | + | z ( β − ε ) | + | y x ( γ ) | + | y x ( β − ε ) | = O (1) . (4.31) Proof.
First, we fix a function g ∈ C ([ β − ε, γ ]) such that g ( β − ε ) = − g ( γ ) = 1 and set max x ∈ [ β − ε,γ ] | g ( x ) | = M g and max x ∈ [ β − ε,γ ] | g ′ ( x ) | = M g ′ . From (4.5), we deduce that(4.32) iλu x − v x = λ − f x . Multiplying (4.32) and (4.6) by 2 g v and 2 a − g S b ( u, v, η ) respectively, integrating over ( β − ε, γ ), using thedefinition of c ( · ) and S b ( u, v, η ), then taking the real part, we obtain ℜ (cid:26) iλ Z γβ − ε g u x vdx (cid:27) − Z γβ − ε g (cid:16) | v | (cid:17) x dx = ℜ (cid:26) λ − Z γβ − ε g f x vdx (cid:27) and ℜ (cid:26) iλ Z γβ − ε g vu x dx (cid:27) + ℜ ( iλa Z ββ − ε g v ( κ v x + κ η x ( · , dx ) − a − Z ββ − ε g (cid:16) | S ( u, v, η ) | (cid:17) x dx − a Z γβ g (cid:16) | u x | (cid:17) x dx + ℜ ( c a Z ββ − ε g zS ( u, v, η ) dx ) + ℜ (cid:26) c Z γβ g zu x dx (cid:27) = ℜ ( aλ Z ββ − ε g f S ( u, v, η ) dx ) + ℜ (cid:26) λ Z γβ g f u x dx (cid:27) . Adding the above Equations, then using integration by parts, we get h − g | v | i γβ − ε + h − a − g | S ( u, v, η ) | i ββ − ε + h − ag | u x | i γβ = − Z γβ − ε g ′ | v | dx − a − Z ββ − ε g ′ | S ( u, v, η ) | dx − a Z γβ g ′ | u x | dx − ℜ ( iλa Z ββ − ε g v ( κ v x + κ η x ( · , dx ) − ℜ ( c a Z ββ − ε g zS ( u, v, η ) dx ) − ℜ (cid:26) c Z γβ g zu x dx (cid:27) + ℜ (cid:26) λ Z γβ − ε g f x vdx (cid:27) + ℜ ( aλ Z ββ − ε g f S ( u, v, η ) dx ) + ℜ (cid:26) λ Z γβ g f u x dx (cid:27) . g and Cauchy-Schwarz inequality in the above Equation, we obtain | v ( γ ) | + | v ( β − ε ) | + a | u x ( γ ) | + a − | ( S ( u, v, η )) ( β − ε ) | + K ( β ) ≤ M g ′ "Z γβ − ε | v | dx + a − Z ββ − ε | S ( u, v, η ) | dx + a Z γβ | u x | dx + 2 | λ | M g a κ Z ββ − ε | v x | dx ! + | κ | Z ββ − ε | η x ( · , | dx ! Z ββ − ε | v | dx ! + 2 c M g a Z ββ − ε | S ( u, v, η ) | dx ! Z ββ − ε | z | dx ! + 2 c M g (cid:18)Z γβ | z | dx (cid:19) (cid:18)Z γβ | u x | dx (cid:19) + 2 M g λ (cid:18)Z γβ − ε | f x | dx (cid:19) (cid:18)Z γβ − ε | v | dx (cid:19) + 2 M g aλ Z ββ − ε | f | dx ! Z ββ − ε | S ( u, v, η ) | dx ! + 2 M g λ (cid:18)Z γβ | f | dx (cid:19) (cid:18)Z γβ | u x | dx (cid:19) . where K ( β ) = g ( β ) (cid:0) a | u x ( β + ) | − a − | ( S ( u, v, η )) ( β − ) | (cid:1) . Moreover, since S b ( u, v, η ) ∈ H (0 , L ) ⊂ C ([0 , L ]),then we obtain(4.33) | ( S ( u, v, η )) ( β − ) | = | au x ( β + ) | and consequently K ( β ) = 0 . Inserting (4.33) in the above inequality, then using Lemma 4.1 and the fact that u x , v , z are uniformly boundedin L (0 , L ) and k f x k L (0 ,L ) = o (1), k f k L (0 ,L ) = o (1), we obtain (4.30). Next, from (4.7), we deduce that(4.34) iλy x − z x = λ − f x . Multiplying Equations (4.34) and (4.8) by 2 g z and 2 g y x respectively, integrating over ( β − ε, γ ), using thedefinition of c ( · ), then taking the real part, we obtain(4.35) ℜ (cid:26) iλ Z γβ − ε g y x zdx (cid:27) − Z γβ − ε g (cid:16) | z | (cid:17) x dx = ℜ (cid:26) λ − Z γβ − ε g f x zdx (cid:27) and(4.36) ℜ (cid:26) iλ Z γβ − ε g zy x dx (cid:27) − Z γβ − ε g (cid:16) | y x | (cid:17) x dx − ℜ (cid:26) c Z γβ − ε g vy x dx (cid:27) = ℜ (cid:26) λ − Z γβ − ε g f y x dx (cid:27) . Adding Equations (4.35) and (4.36), then using integration by parts, we obtain h − g (cid:16) | z | + | y x | (cid:17)i γβ − ε = − Z γβ − ε g ′ ( | z | + | y x | ) dx + ℜ (cid:26) c Z γβ − ε gvy x dx (cid:27) + ℜ (cid:26) λ − Z γβ − ε g f x zdx (cid:27) + ℜ (cid:26) λ − Z γβ − ε g f y x dx (cid:27) . Using the definition of g and Cauchy-Schwarz inequality in the above Equation, we obtain | z ( γ ) | + | z ( β − ε ) | + | y x ( γ ) | + | y x ( β − ε ) | ≤ M g ′ Z γβ − ε (cid:16) | z | + | y x | (cid:17) dx + 2 c M g (cid:18)Z γβ − ε | v | dx (cid:19) (cid:18)Z γβ − ε | y x | dx (cid:19) + 2 λ − M g "(cid:18)Z γβ − ε | f x | dx (cid:19) (cid:18)Z γβ − ε | z | dx (cid:19) + (cid:18)Z γβ − ε | f | dx (cid:19) (cid:18)Z γβ − ε | y x | dx (cid:19) . Finally, from the above inequality, the fact that v , y x , z are uniformly bounded in L (0 , L ) and k f x k L (0 ,L ) = o (1), k f k L (0 ,L ) = o (1), we obtain (4.31). The proof is thus complete. (cid:3) emma 4.5. Let h ∈ C ([0 , L ]) be a function. Under the hypothesis (H), the solution U = ( u, v, y, z, η ( · , ρ )) ⊤ ∈ D ( A ) of system (4.5)-(4.9) satisfies the following estimation Z L h ′ (cid:16) a − | S b ( u, v, η ) | + | v | + | z | + | y x | (cid:17) dx − h h (cid:16) a − | S b ( u, v, η ) | + | y x | (cid:17)i L − ℜ ( Z L c ( · ) h vy x dx ) + ℜ ( a Z L c ( · ) h zS b ( u, v, η ) dx ) + ℜ ( iλa Z L b ( · ) hv n ( κ v x + κ η x ( · , dx ) = ℜ ( λ Z L h f x vdx ) + ℜ ( aλ Z L h f S b ( u, v, η ) dx ) + ℜ ( λ Z L h f x zdx ) + ℜ ( λ Z L h f y x dx ) . Proof.
See the proof of Lemma 3.5. (cid:3)
Let 0 < ε < min (cid:16) α , β − α (cid:17) , we fix the cut-off functions θ , θ ∈ C ([0 , L ]) (see Figure 4) such that 0 ≤ θ ( x ) ≤ ≤ θ ( x ) ≤
1, for all x ∈ [0 , L ] and θ ( x ) = ( x ∈ [0 , α + ε ] , x ∈ [ β − ε, L ] , and θ ( x ) = ( x ∈ [0 , α + ε ] , x ∈ [ β − ε, L ] , α α + ε β − ε β γ L θ θ Figure 4.
Geometric description of the functions θ and θ . Lemma 4.6.
Let 0 < ε < min (cid:16) α , β − α (cid:17) . Under the hypothesis (H), the solution U = ( u, v, y, z, η ( · , ρ )) ⊤ ∈ D ( A ) of the System (4.5)-(4.9) satisfies the following estimations Z α + ε | v | dx + Z α + ε | y x | dx + Z α + ε | z | dx = o (1) , (4.37) a Z Lβ | u x | dx + Z Lβ − ε | v | dx + Z Lβ − ε | y x | dx + Z Lβ − ε | z | dx = o (1) . (4.38) Proof.
First, using the result of Lemma 4.5 with h = xθ , we obtain Z α + ε | v | dx + Z α + ε | y x | dx + Z α + ε | z | dx = − a − Z α + ε | S ( u, v, η ) | dx − Z β − εα + ε ( θ + xθ ′ ) (cid:0) a − | S ( u, v, η ) | + | v | + | y x | + | z | (cid:1) dx + ℜ ( Z L xc ( · ) θ vy x dx ) − ℜ ( a Z L xc ( · ) θ zS b ( u, v, η ) dx ) − ℜ ( iλa Z L xb ( · ) θ v ( κ v x + κ η x ( ., dx ) + ℜ ( λ Z L xθ f x vdx ) + ℜ ( aλ Z L xθ f S b ( u, v, η ) dx ) + ℜ ( λ Z L xθ f x zdx ) + ℜ ( λ Z L xθ f y x dx ) . v , y x , z are uniformly boundedin L (0 , L ) and k f x k L (0 ,L ) = o (1), k f x k L (0 ,L ) = o (1), k f k L (0 ,L ) = o (1), we obtain(4.39) Z α + ε | v | dx + Z α + ε | y x | dx + Z α + ε | z | dx = ℜ ( Z L xc ( · ) θ vy x dx ) − ℜ ( a Z L xc ( · ) θ zS b ( u, v, η ) dx ) + ℜ ( aλ Z L xθ f S b ( u, v, η ) dx ) − ℜ ( iλa Z L xb ( · ) θ v ( κ v x + κ η x ( ., dx ) + o (1) . Using the definition of b ( · ), c ( · ), S b ( u, v, η ), θ , then using Cauchy-Schwarz inequality, we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℜ ( Z L xc ( · ) θ vy x dx )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℜ ( c Z β − εα xθ vy x dx )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ( β − ε ) Z β − εα | v | dx ! Z β − εα | y x | dx ! , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℜ ( a Z L xc ( · ) θ zS b ( u, v, η ) dx )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℜ ( c a Z β − εα xθ zS ( u, v, η ) dx )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c a ( β − ε ) Z β − εα | z | dx ! Z β − εα | S ( u, v, η ) | dx ! , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℜ ( aλ Z L xθ f S b ( u, v, η ) dx )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℜ ( aλ Z β − ε xθ f S ( u, v, η ) dx )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ β − ε ) aλ Z β − ε | f | dx ! Z β − ε | S ( u, v, η ) | dx ! , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℜ ( iλa Z L xb ( · ) θ v ( κ v x + κ η x ( ., dx )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℜ ( iλa Z β − ε xθ v ( κ v x + κ η x ( ., dx )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | λ | ( β − ε ) a κ Z β − ε | v x | dx ! + | κ | Z β − ε | η x ( · , | dx ! Z β − ε | v | dx ! . Thus, from the above inequalities, Lemmas 4.1, 4.2 and the fact that u x , v , y x , z are uniformly bounded in L (0 , L ) and k f k L (0 ,L ) = o (1), we obtain(4.40) ℜ ( Z L xc ( · ) θ vy x dx ) = o (1) , −ℜ ( a Z L xc ( · ) θ zS b ( u, v, η ) dx ) = o ( | λ | − ) , ℜ ( aλ Z L xθ f S b ( u, v, η ) dx ) = o ( λ − ) , −ℜ ( iλa Z L xb ( · ) θ v ( κ v x + κ η x ( ., dx ) = o (1) . Therefore, by inserting (4.40) in (4.39), we obtain (4.37). On the other hand, using the result of Lemma 4.5with h = ( x − L ) θ , then using the definition of b ( · ), S b , θ and Lemmas 4.1, 4.2, 4.3 with the fact that u x , v , y x , z are uniformly bounded in L (0 , L ) and k f x k L (0 ,L ) = o (1), k f k L (0 ,L ) = o (1), k f x k L (0 ,L ) = o (1), k f k L (0 ,L ) = o (1), we obtain(4.41) a Z Lβ | u x | dx + Z Lβ − ε | v | dx + Z Lβ − ε | y x | dx + Z Lβ − ε | z | dx = ℜ ( Z L ( x − L ) c ( · ) θ vy x dx ) − ℜ ( a − Z L ( x − L ) c ( · ) θ zS b dx ) + o (1) . c ( · ), S b , θ and by using Lemmas 4.1, 4.2 with the fact that y x , z are uniformlybounded in L (0 , L ), we obtain(4.42) ℜ ( Z L ( x − L ) c ( · ) θ vy x dx ) − ℜ ( a − Z L ( x − L ) c ( · ) θ zS b dx ) = ℜ (cid:26) c Z γβ − ε ( x − L ) vy x dx (cid:27) − ℜ (cid:26) c Z γβ − ε ( x − L ) zu x dx (cid:27) + o (1) . From (4.5) and (4.7), we deduce that(4.43) u x = iλ − v x + iλ − f x and y x = iλ − z x + iλ − f x . Substituting (4.43) in the right hand side of (4.42), then using the fact that v , z are uniformly bounded in L (0 , L ) and k f x k L (0 ,L ) = o (1), k f x k L (0 ,L ) = o (1), we obtain ℜ ( Z L ( x − L ) c ( · ) θ vy x dx ) − ℜ ( a − Z L ( x − L ) c ( · ) θ zS b dx ) = ℜ (cid:26) c iλ Z γβ − ε ( x − L ) vz x dx (cid:27) − ℜ (cid:26) c iλ Z γβ − ε ( x − L ) zv x dx (cid:27) + o (1) . Using integration by parts to the second integral in the right hand side of the above equation, we obtain(4.44) ℜ ( Z L ( x − L ) c ( · ) θ vy x dx ) − ℜ ( a − Z L ( x − L ) c ( · ) θ zS b dx ) = ℜ (cid:26) c iλ Z γβ − ε zvdx (cid:27) − ℜ (cid:26) c iλ [( x − L ) zv ] γβ − ε (cid:27) + o (1) . Furthermore, by using Cauchy-Schwarz inequality, we get(4.45) (cid:12)(cid:12)(cid:12)(cid:12) ℜ (cid:26) c iλ Z γβ − ε zvdx (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c | λ | − (cid:18)Z γβ − ε | z | dx (cid:19) (cid:18)Z γβ − ε | v | dx (cid:19) and(4.46) (cid:12)(cid:12)(cid:12)(cid:12) ℜ (cid:26) c iλ [( x − L ) zv ] γβ − ε (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c | λ | − [( L − γ ) | z ( γ ) || v ( γ ) | + ( L − β + 3 ε ) | z ( β − ε ) | | v ( β − ε ) | ] . From Lemma 4.4, we deduce that(4.47) | v ( β − ε ) | = O (1) , | v ( γ ) | = O (1) , | z ( β − ε ) | = O (1) and | z ( γ ) | = O (1) . Using the fact that v , z are uniformly bounded in L (0 , L ) in (4.45) and inserting (4.47) in (4.46), we obtain(4.48) ℜ (cid:26) c iλ Z γβ − ε zvdx (cid:27) = O (cid:0) | λ | − (cid:1) = o (1) and − ℜ (cid:26) c iλ [( x − L ) zv ] γβ − ε (cid:27) = O (cid:0) | λ | − (cid:1) = o (1) . Inserting (4.48) in (4.44), we get(4.49) ℜ ( Z L ( x − L ) c ( · ) θ vy x dx ) − ℜ ( a − Z L ( x − L ) c ( · ) θ zS b dx ) = o (1) . Finally, inserting (4.49) in (4.41), we obtain (4.38). The proof is thus complete. (cid:3)
Proof of Theorem 4.1.
The proof of Theorem is divided into three steps.
Step 1.
From Lemmas 4.1-4.3, we obtain(4.50) Z β | u x | dx = o ( λ − ) , Z β Z | η x ( · , ρ ) | dρdx = o ( λ − ) , Z β − εε | v | dx = o (1) , Z β − εα | z | dx = o (1) and Z β − εα + ε | y x | dx = o (1) . tep 2. From Lemma 4.6 and (4.50), we deduce that Z ε | v | dx = o (1) , Z α + ε | y x | dx = o (1) , Z α | z | dx = o (1) , Z Lβ | u x | dx = o (1) , Z Lβ − ε | v | dx = o (1) , Z Lβ − ε | y x | dx = o (1) and Z Lβ − ε | z | dx = o (1) . According to
Step 1 and
Step 2 , we obtain k U k H = o (1) in (0 , L ), which contradicts (4.3). Thus, (4.2) isholds true. Next, since the conditions (4.1) and (4.2) are proved, then according to Theorem A.4, the proof ofTheorem 4.1 is achieved. The proof is thus complete. (cid:3) Conclusion
We have studied the stabilization of a one-dimensional coupled wave equations with non smooth localizedviscoelastic damping of Kelvin-Voigt type and localized time delay. We proved the strong stability of the systemby using Arendt-Batty criteria. Finally, we established a polynomial energy decay rate of order t − . Appendix A. Some notions and theorems of stability has been used
In order to make this paper more self-contained, we have introduced this short appendix that brings up thenotions of stability that we encounter in this work.
Definition A.1.
Assume that A is the generator of C − semigroup of contractions (cid:0) e tA (cid:1) t ≥ on a Hilbert space H . The C − semigroup (cid:0) e tA (cid:1) t ≥ is said to be(1) Strongly stable if lim t → + ∞ k e tA x k H = 0 , ∀ x ∈ H. (2) Exponentially (or uniformly) stable if there exists two positive constants M and ε such that k e tA x k H ≤ M e − εt 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 Ax k H , ∀ t > , ∀ x ∈ D ( A ) . (cid:3) For proving the strong stability of the C -semigroup (cid:0) e tA (cid:1) t ≥ , we will recall the result obtained by Arendt andBatty in [12]. Theorem A.2 (Arendt and Batty in [12]) . Assume that A is the generator of a C − semigroup of contractions (cid:0) e tA (cid:1) t ≥ on a Hilbert space H . If A has no pure imaginary eigenvalues and 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. (cid:3) There exist a second classical method based on Arendt and Batty theorem and the contradiction argument(see page 25 in [39]).
Remark A.3.
Assume that the unbounded linear operator A : D ( A ) ⊂ H H is the generator of aC − semigroup of contractions (cid:0) e tA (cid:1) t ≥ on a Hilbert space H and suppose that 0 ∈ ρ ( A ) . According to (page25 in [39]), in order to prove that(A.1) i R ≡ { iλ | λ ∈ R } ⊆ ρ ( A ) , we need the following steps:(i) It follows from the fact that 0 ∈ ρ ( A ) and the contraction mapping theorem that for any real number λ with | λ | < k A − k − , the operator iλI − A = A ( iλA − − I ) is invertible. Furthermore, k ( iλI − A ) − k is a continuous function of λ in the interval (cid:0) −k A − k − , k A − k − (cid:1) .28ii) If sup (cid:8) k ( iλI − A ) − k | | λ | < k A − k − (cid:9) = M < ∞ , then by the contraction mapping theorem, theoperator iλI − A = ( iλ I − A )( I + i ( λ − λ )( iλ I − A ) − ) with | λ | < k A − k − is invertible for | λ − λ | < M − . It turns out that by choosing | λ | as close to k A − k − as we can, we conclude that (cid:8) λ | | λ | < k A − k − + M − (cid:9) ⊂ ρ ( A ) and k ( iλI − A ) − k is a continuous function of λ in the interval (cid:0) −k A − k − − M − , k A − k − + M − (cid:1) . (iii) Thus it follows from the argument in (ii) that if (A.1) is false, then there is ω ∈ R with k A − k − ≤| ω | < ∞ such that { iλ | | λ | < | ω |} ⊂ ρ ( A ) and sup (cid:8) k ( iλ − A ) − k | | λ | < | ω | (cid:9) = ∞ . It turns out thatthere exists a sequence { ( λ n , U n ) } n ≥ ⊂ R × D ( A ) , with λ n → ω as n → ∞ , | λ n | < | ω | and k U n k H = 1,such that ( iλ n I − A ) U n = F n → H, as n → ∞ . Then, we will prove (A.1) by finding a contradiction with k U n k H = 1 such as k U n k H → . (cid:3) .Concerning the characterization of polynomial stability stability of a C − semigroup of contraction (cid:0) e tA (cid:1) t ≥ ,we rely on the following result due to Borichev and Tomilov [15] (see also [13] and [36]). Theorem A.4.
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 > 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) , (A.3) k e t A U k H ≤ Ct ℓ k U k D ( A ) , ∀ t > , U ∈ D ( A ) , for some C > . (cid:3) Acknowledgments
The authors thanks professor Serge Nicaise for his valuable discussions and comments.Mohammad Akil would like to thank the Lebanese University for its support.Haidar Badawi would like to thank the LAMAV laboratory of Mathematics of the Universit´e polytechniqueHauts-De-France Valenciennes for its support.Ali Wehbe would like to thank the CNRS for its support.
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