Energy Decay of some boundary coupled systems involving wave\backslash Euler-Bernoulli beam with one locally singular fractional Kelvin-Voigt damping
aa r X i v : . [ m a t h . A P ] F e b ENERGY DECAY OF SOME BOUNDARY COUPLED SYSTEMS INVOLVING WAVE \ EULER-BERNOULLI BEAM WITH ONE LOCALLY SINGULAR FRACTIONALKELVIN-VOIGT DAMPING
MOHAMMAD AKIL , IBTISSAM ISSA , , AND ALI WEHBE Abstract.
In this paper, we investigate the stability of five models of systems. In the first model, we considera Euler-Bernoulli beam and a wave equations coupled via boundary connections. The localized non-smoothfractional Kelvin-Voigt damping acts through wave equation only, its effect is transmitted to the other equationthrough the coupling by boundary connections. In this model, we reformulate the system into an augmentedmodel and using a general criteria of Arendt-Batty, we show that the system is strongly stable. By usingfrequency domain approach, combined with multiplier technique we prove that the energy decays polynomiallywith rate t − − α . For the second model, we consider two wave equations coupled through boundary connectionswith localized non-regular fractional Kelvin-Voigt damping acting on one of the two equations. We prove usingthe same technique that we have polyniomial stability with energy decay rate of type t − − α . For the thirdmodel, we consider coupled Euler-Bernoulli beam and wave equations through boundary connections with thesame damping, the dissipation acts through the beam equation. We prove using the same technique as forthe first model combined with some interpolation inequalities and by solving ordinary differential equationsof order 4, that the energy has a polynomial decay rate of type t − − α . In the fourth model, we consider anEuler-Bernoulli beam with a localized non-regular fractional Kelvin-Voigt damping. We show that the energyhas a polynomial decay rate of type t − − α . Finally, in the fifth model, we study the polynomial stability of twoEuler-Bernoulli beam equations coupled through boundary connection with a localized non-regular fractionalKelvin-Voigt damping acting on one of the two equations. We establish a polynomial energy decay rate of type t − − α , where α ∈ (0 , Contents
1. Introduction 11.1. Literature 11.2. Physical interpretation of the models 31.3. Description of the paper 72. (EBB)-W
F KV
Model 82.1. Well-Posedness and Strong Stability 82.1.1. Augmented model and Well-Posedness. 82.1.2. Strong Stability 132.2. Polynomial Stability in the case η >
F KV
Model 254. W-(EBB)
F KV
Model 28 Universit´e Savoie Mont Blanc - Chamb´ery - France, Laboratoire LAMA Lebanese University, Faculty of sciences 1, Khawarizmi Laboratory of Mathematics and Applications-KALMA,Hadath-Beirut, Lebanon. Universit´e Aix-Marseilles - Marseille - France, Laboratoire I2M
E-mail addresses : [email protected], [email protected], [email protected] . Key words and phrases.
Wave equation; Euler-Bernoulli beam; fractional Kelvin-Voigt damping; Semigroup; Polynomialstability. i .1. Well-Posedness and Strong Stability 284.2. Polynomial Stability in the case η > F KV
Model 426. (EBB)-(EBB)
F KV
Model 447. Appendix 488. Conclusion 48Acknowledgements 49References 49ii.
Introduction
Literature.
In recent years, many researches showed interest in studying the stability and controlla-bility of certain system. The wave equation with different kinds of damping was studied extensively. Thewave is created when a vibrating source disturbs the medium. In order to restrain those vibrations, severaldampings can be added such as Kelvin-Voigt damping. Many researchers were interested in problems involv-ing this kind of damping (local or global) where different types of stability have been showed. We refer to[33, 25, 17, 42, 43, 62, 7, 45, 10] and the rich references therein.The beam, or flexural member, is frequently encountered in structures and machines, and its elementarystress analysis constitutes one of the most interesting facts of mechanics of materials. For beams, there was anextensive studying, since 80’s, on the stabilization of the beam equations (see [24, 36] for the one dimensionalsystem, and [18] for n-dimensional system). Also, the studies considered the linear and nonlinear boundaryfeedback acting through shear forces and moments [37, 39, 38] and the case control by moment has been studiedin [35].The studying of the beam equation with different types of damping was extensively considered. In 1998, theauthor in [43] considered the longitudinal and transversal vibrations of the Euler-Bernoulli beam with Kelvin-Voigt damping distributed locally on any subinterval of the region occupied by the beam. It was shown thatwhen the viscoelastic damping is distributed only on a subinterval in the interior of the domain, the exponentialstability holds for the transversal but not for the longitudinal motion. In [56], they considered a transmissionproblem for the longitudinal displacement of a Euler-Bernoulli beam, where one small part of the beam is madeof a viscoelastic material with Kelvin-Voigt constitutive relation and they proved that the semigroup associatedto the system is exponentially stable.Another type of damping which was studied extensively in the past few years is the fractional damping. Itis widely applied in the domain of science. The fractional-order type is not only important from the theoreticalpoint of view but also for applications. They naturally arise in physical, chemical, biological, and ecologicalphenomena see for example [49], and the rich references therein. They are used to describe memory and hered-itary properties of various materials and processes. For example, in viscoelasticity, due to the nature of thematerial microstructure, both elastic solid and viscous fluid-like response qualities are involved. Using Boltz-mann’s assumption, we end up with a stress strain relationship defined by a time convolution. The viscoelasticresponse occurs in a variety of materials, such as soils, concrete, rubber, cartilage, biological tissue, glasses, andpolymers (see [15, 59, 16, 47]). Fractional computing in modeling can improve the capturing of the complexdynamics of natural systems, and controls of fractional order type can improve performance not achievablebefore using controls of integer-order type. For example, systems in many quantum mechanics, nuclear physicsand biological phenomena such as fluid flow are indeed fractional (see for example [60, 48, 54]).Fractional calculus includes various extensions of the usual definition of derivative from integer to real order,including the Hadamard, Erdelyi-Kober, Riemann-Liouville, Riesz, Weyl, Gr¨unwald-Letnikov, Jumarie and theCaputo representation. A thorough analysis of fractional dynamical systems is necessary to achieve an appro-priate definition of the fractional derivative. For example, the Riemann-Liouville definition entails physicallyunacceptable initial conditions (fractional order initial conditions); conversely, for the Caputo representation,which is introduced by Michele Caputo [22] in 1967, the initial conditions are expressed in terms of integer-order derivatives having direct physical significance; this definition is mainly used to include memory effects.Recently, in [23] a new definition of the fractional derivative was presented without a singular kernel; thisderivative possesses very interesting properties, for instance the possibility to describe fluctuations and struc-tures with different scales. The case of wave equation with boundary fractional damping have been treatedin [50, 51] where they proved the strong stability and the lack of uniform stabilization. However, the caseof the plate equation or the beam equation with boundary fractional damping was treated in [1] where theyshowed that the energy is polynomially stable. In [4], they considered a multidimensional wave equation withboundary fractional damping acting on a part of the boundary of the domain. They established a polynomialenergy decay rate for smooth solutions, under some geometric conditions. Ammari et al., in [8], studied thestabilization for a class of evolution systems with fractional-damping. They proved the polynomial stability of1he system.Over the past few years, the coupled systems received a vast attention due to their potential applications.The coupled systems have many applications in the modeling and control of engineering, such as: aircraft,satellite antennas and road traffic(see [20] for example). Most of the work in the coupled system considers thestability of the system with various coupling, damping locations, and damping types. Many researches studiedcoupling systems with a Kelvin-Voigt damping such as wave-wave system, heat-wave system, Timoshinko (see[61, 63, 5]). In 2012, Tebou in [58] considered the Euler-Bernoulli equation coupled with a wave equation in abounded domain. The frictional damping is distributed everywhere in the domain and acts through one of theequations only. For the case where the dissipation acts through the Euler-Bernoulli equation he showed thatthe system is not exponentially stable and that the energy decays polynomially was proved. For the case wherethe damping acts through the wave equation polynomial stability was proved.Benaissa et al., in [6], considered the large time behavior of one dimensional coupled wave equations withfractional control applied at the coupled point. They showed an optimal decay result.In [31], Hassine considered a beam and a wave equations coupled on an elastic beam through transmissionconditions where the locally distributed damping acts through one of the two equations only. The systems aredescribed as follows (1.1) u tt − ( u x + D a u xt ) x = 0 , in Ω ,y tt + y xxxx = 0 , in Ω ,u ( ℓ, t ) = y ( ℓ, t ) , t > ,y x ( ℓ, t ) = 0 , t > ,u x ( ℓ, t ) + y xxx ( ℓ, t ) = 0 , t > ,u (0 , t ) = y ( L, t ) = y x ( L, t ) = 0 , t > ,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 ∈ ( ℓ, L ) and y tt − ( y xx + D b y xxt ) xx = 0 , in Ω ,u tt − u xx = 0 , in Ω ,u ( ℓ, t ) = y ( ℓ, t ) , t > ,y x ( ℓ, t ) = 0 , t > ,u x ( ℓ, t ) + y xxx ( ℓ, t ) = 0 , t > ,u (0 , t ) = y ( L, t ) = y x ( L, t ) = 0 , t > ,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 ∈ ( ℓ, L ) where Ω = (0 , ℓ ) × R + , Ω = ( ℓ, L ) × R + , D a = a ( x ) χ ( e,f ) and D b = b ( x ) χ ( e,f ) with 0 < e < f < ℓ < L and a ( x ) , b ( x ) ≥ c > e, f ). The author proved that for the case when the dissipation acts through the wave,the energy of this coupled system decays polynomially as the time variable goes to infinity. Also, for the casewhere the damping acts through the beam equation polynomial stability was proved.The case of a Euller-Bernoulli beam and a wave equations coupled via the interface by transmission conditionswas considered by Hassine in [32], where he supposed that the beam equation is stabilized by a localizeddistributed feedback. He reached that sufficiently smooth solutions decay logarithmically at infinity even thefeedback dissipation affects an arbitrarily small open subset of the interior.In [27], the authors studied the stabilization system of a coupled wave and a Euler-Bernoulli plate equationwhere only one equation is supposed to be damped with a frictional damping in the multidimensional case.Under some assumption about the damping and the coupling terms, they showed that sufficiently smoothsolutions of the system decay logarithmically at infinity without any geometric conditions on the effectivedamping domain.In [12], Ammari and Nicaise, considered the stabilization problem for coupling the damped wave equationwith a damped Kirchhoff plate equation. They proved an exponential stability result under some geometriccondition. In 2018, the authors considered in [41], a system of 1-d coupled string-beam. They obtained twokinds of energy decay rates of the string-beam system with different locations of the frictional damping. Onone hand, if the frictional damping is only actuated in the beam part, the system lacks exponential decay.Specifically, the optimal polynomial decay rate t − is obtained under smooth initial conditions. On the otherhand, if the frictional damping is only effective in the string part, the exponential decay of energy is presented.In 2020, the authors in [29], considered a system of two-dimensional coupled wave-plate with local frictionaldamping in a bounded domain. The frictional damping is only distributed in the part of the plate’s or wave’sdomain, and the other is stabilized by the transmission through the interface of the plate’s and wave’s domains.They showed that the energy of the system decays polynomially under some geometric condition when thefrictional damping only acts on the part of the plate, and the energy of the system is exponentially stable whenthe frictional damping acts only on the other part of the wave.In 2018, Guo and Ren in [28], studied the stabilization for a hyperbolic-hyperbolic coupled system consistingof Euler-Bernoulli beam and wave equations, where the structural damping of the wave equation is taken into2ccount. The coupling is actuated through boundary weak connection. The system is described as follows(1.2) w tt + w xxxx = 0 , ( x, t ) ∈ (0 , × R + ,u tt − u xx − su xxt = 0 , ( x, t ) ∈ (0 , × R + ,w (1 , t ) = w xx (1 , t ) = w (0 , t ) = 0 , t > ,w xx (0 , t ) = ru t (0 , t ) , u (1 , t ) = 0 , x ∈ (0 , ,su xt (0 , t ) + u x (0 , t ) = − rw xt (0 , t ) , x ∈ (0 , ,w ( x,
0) = w ( x ) , w t ( x,
0) = w ( x ) , x ∈ (0 , ,u ( x,
0) = u ( x ) , u t ( x,
0) = u ( x ) , x ∈ (0 , , where ( w , w , u , u ) is the initial state and r = 0 , s > Physical interpretation of the models.
In the first model (EBB)-W
F KV , we investigate the stabilityof coupled Euler-Bernoulli beam and wave equations. The coupling is via boundary connections with localizednon-regular fractional Kelvin-Voigt damping, where the damping acts through the wave equation only (seeFigure 1). The system that describes this model is as follows((EBB)-W
F KV ) u tt − ( au x + d ( x ) ∂ α,ηt u x ) x = 0 , ( x, t ) ∈ (0 , L ) × (0 , ∞ ) ,y tt + by xxxx = 0 , ( x, t ) ∈ ( − L, × (0 , ∞ ) ,u ( L, t ) = y ( − L, t ) = y x ( − L, t ) = 0 , t ∈ (0 , ∞ ) ,au x (0 , t ) + by xxx (0 , t ) = 0 , y xx (0 , t ) = 0 , t ∈ (0 , ∞ ) ,u (0 , t ) = y (0 , t ) , t ∈ (0 , ∞ ) ,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 ∈ ( − L, . The coefficients a, b are strictly positive constant numbers, α ∈ (0 ,
1) and η ≥
0. We suppose that there exists0 < l < l < L and a strictly positive constant d , such that(1.3) d ( x ) = ( d , x ∈ ( l , l )0 , x ∈ (0 , l ) ∪ ( l , L ) . ∂ α,ηt of order α ∈ (0 ,
1) with respect to time variable t defined by(1.4) [ D α,η ω ]( t ) = ∂ α,ηt ω ( t ) = 1Γ(1 − α ) Z t ( t − s ) − α e − η ( t − s ) dωds ( s ) ds, where Γ denotes the Gamma function. l Ll − L FKV-dampingBeam Part Wave Part
Figure 1. (EBB)-W
F KV
ModelIn the second model W-W
F KV , we investigate the stability of coupled wave equations coupled through boundaryconnections with localized non-regular fractional Kelvin-Voigt damping acting through one wave equation only(see Figure 2). The system that describes this model is as follows(W-W
F KV ) u tt − ( au x + d ( x ) ∂ α,ηt u x ) x = 0 , ( x, t ) ∈ (0 , L ) × (0 , ∞ ) ,y tt − by xx = 0 , ( x, t ) ∈ ( − L, × (0 , ∞ ) ,u ( L, t ) = y ( − L, t ) = 0 , t ∈ (0 , ∞ ) ,au x (0 , t ) = by x (0 , t ) , t ∈ (0 , ∞ ) ,u (0 , t ) = y (0 , t ) , t ∈ (0 , ∞ ) ,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 ∈ ( − L, . l Ll − L FKV-dampingWave Part Wave Part
Figure 2.
W-W
F KV
ModelIn the third model W-(EBB)
F KV , we consider a system of coupled Euler-Bernouli beam and wave equations.These two equations are coupled through boundary connections. In this case the localized non-smooth fractionalKelvin-Voigt damping acts only on the Euler-Bernoulli beam (see Figure 3). The system that represents this4odel is as follows(W-(EBB)
F KV ) u tt − au xx = 0 , ( x, t ) ∈ ( − L, × (0 , ∞ ) ,y tt + ( by xx + d ( x ) ∂ α,ηt y xx ) xx = 0 , ( x, t ) ∈ (0 , L ) × (0 , ∞ ) ,u ( − L, t ) = y ( L, t ) = y x ( L, t ) = 0 , t ∈ (0 , ∞ ) ,au x (0 , t ) + by xxx (0 , t ) = 0 , y xx (0 , t ) = 0 , t ∈ (0 , ∞ ) ,u (0 , t ) = y (0 , t ) , t ∈ (0 , ∞ ) ,u ( x,
0) = u ( x ) , u t ( x,
0) = u ( x ) , x ∈ ( − L, ,y ( x,
0) = y ( x ) , y t ( x,
0) = y ( x ) , x ∈ (0 , L ) . l Ll − L FKV-dampingWave Part Beam Part
Figure 3.
W-(EBB)
F KV
ModelIn the fourth model ((EBB)
F KV ), we study a system of Euler-Bernoulli beam with a non-regular localizedfractional Kelvin-Voigt damping (see Figure 4). The system is as follows((EBB)
F KV ) y tt + ( by xx + d ( x ) ∂ α,ηt y xx ) xx = 0 , ( x, t ) ∈ (0 , L ) × (0 , ∞ ) ,y (0 , t ) = y x (0 , t ) = y xx ( L, t ) = y xxx ( L, t ) = 0 , t ∈ (0 , ∞ ) ,y ( x,
0) = y ( x ) , y t ( x,
0) = y ( x ) , x ∈ (0 , L ) . l Ll Figure 4. (EBB)
F KV
ModelIn the last model (EBB)-(EBB)
F KV , we consider a system of two Euler-Bernouli beam equations coupledthrough boundary connections. The localized non-smooth fractional Kelvin-Voigt damping acts only on one of5he two equations (see Figure 5). The system that represents this model is as follows((EBB)-(EBB)
F KV ) u tt + au xxxx = 0 , ( x, t ) ∈ ( − L, × (0 , ∞ ) ,y tt + ( by xx + d ( x ) ∂ α,ηt y xx ) xx = 0 , ( x, t ) ∈ (0 , L ) × (0 , ∞ ) ,u ( − L, t ) = u x ( − L, t ) = y ( L, t ) = y x ( L, t ) = 0 , t ∈ (0 , ∞ ) ,au xxx (0 , t ) − by xxx (0 , t ) = 0 , u xx (0) = y xx (0 , t ) = 0 , t ∈ (0 , ∞ ) ,u (0 , t ) = y (0 , t ) , t ∈ (0 , ∞ ) ,u ( x,
0) = u ( x ) , u t ( x,
0) = u ( x ) , x ∈ ( − L, ,y ( x,
0) = y ( x ) , y t ( x,
0) = y ( x ) , x ∈ (0 , L ) . l L l − L FKV-dampingBeam Part Beam Part
Figure 5. (EBB)-(EBB)
F KV
We give the physical meaning of the following variables.(1.5) y = vertical displacement , u x = the stress of the wave .y x = rotation ,y xx = Bending moment ,y xxx = Shear Force ,y xxxx = Loading . The way the beam supported is translated into conditions on the function y and its derivatives. These conditionsare collectively referred to as boundary conditions. They are meaningful in physics and engineering. Theboundary conditions in the model ((EBB) F KV ) signifies the following • y (0 , t ) = 0: This signifies that the beam is pinned to its support, which means that the beam cannotexperience any deflection at x = 0. • y x (0 , t ) = 0: It signifies that the rotation at the pinned support is zero. • y xx ( L, t ) = 0: It means that there is no bending moment at the free end of the beam. • y xxx ( L, t ) = 0: This boundary conditions gives the assumption that there is no shearing force acting at thefree end of the beam.This kind of models, supported at one end with the other end free, described in the above four conditions canbe referred to as a cantilever beam. A good example on the cantilever beam is a balcony, it is supported atone end only, the rest of the beam extends over the open space. Other examples are a cantilever roof in a busshelter, car park or railway station. We give some of the advantages and disadvantages of the cantilever beam.Advantages: • Cantilever beams do not require support on the opposite side. • The negative bending moment created in cantilever beams helps to counteract the positive bendingmoments created. • Cantilever beams can be easily constructed. • These beam enables erection with little disturbance in navigation.Disadvantages: • Cantilever beams are subjected to large deflections. • Cantilever beams are subjected to larger moments. • A strong fixed support or a backspan is necessary to keep the structure stable.6n a cantilever beam, the bending moment at the free end always vanishes. In fact, if we connect the beamwith a wave (see ((EBB)-W
F KV ), (W-(EBB)
F KV )) at the free end the bending moment will be zero and thisinduces a shear force on the end of the beam. Consequently, the fourth boundary condition above is no longervalid, and it is replaced by au x (0 , t ) + by xxx (0 , t ) = 0. This condition signifies that the shear force of the beamand the stress force of the wave are such that one cancels the other.1.3. Description of the paper.
In this paper, we investigate the stability results of four models of systemswith a non-smooth localized fractional Kelvin-Voigt damping where the coupling is made via boundary con-nections. In the first model ((EBB)-W
F KV ) we consider the coupled Euler-Bernoulli beam and wave equationwith the damping acts on the wave equation only. In Subsection 2.1, we reformulate ((EBB)-W
F KV ) into anaugmented model and we prove the well-posedness of the system by semigroup approach. Moreover, using ageneral criteria of Arendt and Batty, we show the strong stability of our system in the absence of the com-pactness of the resolvent. In section 2.2, using the semigroup theory of linear operators and a result obtainedby Borichev and Tomilov we show that the energy of the System ((EBB)-W
F KV ) has a polynomial decay rateof type t − − α . In the second model (W-W F KV ), we consider two wave equations coupled through boundaryconnections with a non-smooth localized fractional Kelvin-Voigt damping acting only on one of the two equa-tions. We establish a polynomial energy decay rate of type t − − α . In the third model (W-(EBB) F KV ), weconsider Euler-Bernoulli beam and wave equations coupled through boundary connections with the dampingto act through the Euler-Bernoulli beam equation only. For this model, we show that the energy of the System(W-(EBB)
F KV ) has a polynomial decay rate of type t − − α . For the model ((EBB) F KV ) we consider the Euler-Bernoulli beam with a non-smooth localized fractional Kelvin-Voigt damping. We prove that the energy of thesystem (5) decays polynomially with a decay rate t − − α . Finally, for the fourth model ((EBB)-(EBB) F KV ), westudy the polynomial stability of two Euler-Bernoulli beam equations coupled through boundary connectionswith damping acting only on one of the two equation. We establish a polynomial energy decay rate of type t − − α . The table below (Table 1) summarizes the decay rate of the energy for the five models. Also, it gives thedecay rate of the same four models but with Kelvin-Voigt damping (as α → Model Decay Rate α → F KV t − − α t − W-W
F KV t − − α t − W-(EBB)
F KV t − − α t − (EBB) F KV t − − α Exponential(EBB)-(EBB)
F KV t − − α t − Table 1.
Decay ResultsSome significant deductions on the energy deacy rate are given below: • From the decay rate of the models ((EBB)-W
F KV ) and (W-(EBB)
F KV ) we can deduce that if we want tochoose the place where the fractional Kelvin-Voigt damping acts it is better to choose the damping on thewave. Since the energy of this model decays faster compared with that of (W-(EBB)
F KV ) model. • For the model ((EBB)-W
F KV ), if we replace the condition of the null bending moment ( y xx (0) = 0) at theconnecting boundary by taking the rotation to be zero ( y x (0) = 0), we get the same decay rate. So, this resultimproves the work in [31] where they reached energy decay rate of type t − , however in our work we provedan energy decay rate of type t − ( as α → • For the model ((EBB)-W
F KV ), we established an energy decay rate of type t − (as α → t − . Wecan see that, by comparing the energy decay rate of these two system that it is better, when consideringKelvin-Voigt damping, to consider the coupling through the boundary connection rather through the velocity. Remark 1.1.
We note that in the upcoming sections, the letters used to denote the variables are independentfrom each other in each section. (EBB)-W F KV
Model
In this section, we consider the ((EBB)-W
F KV ) model, where we study the stability of the system a Euler-Bernoulli and wave equations coupled through boundary connection with a localized fractional Kelvin-Voigtdamping acting on the wave equation only.2.1.
Well-Posedness and Strong Stability.
In this subsection, we study the strong stability of the sys-tem ((EBB)-W
F KV ) in the absence of the compactness of the resolvent. First, we will study the existence,uniqueness and regularity of the solution of the system.2.1.1.
Augmented model and Well-Posedness.
In this part, using a semigroup approach, we establish well-posedness for the system ((EBB)-W
F KV ). First, we recall theorem 2 stated in [50, 3].
Theorem 2.1.
Let α ∈ (0 , , η ≥ and µ ( ξ ) = | ξ | α − be the function defined almost everywhere on R . Therelation between the ’input’ V and the ’output’ O of the following system ∂ t ω ( x, ξ, t ) + ( ξ + η ) ω ( x, ξ, t ) − V ( x, t ) | ξ | α − = 0 , ( x, ξ, t ) ∈ (0 , L ) × R × (0 , ∞ ) , (2.1) ω ( x, ξ,
0) = 0 , ( x, ξ ) ∈ (0 , L ) × R , (2.2) O ( x, t ) − κ ( α ) Z R | ξ | α − ω ( x, ξ, t ) dξ = 0 , ( x, t ) ∈ (0 , L ) × (0 , ∞ ) , (2.3) is given by (2.4) O = I − α,η V, where [ I α,η V ]( x, t ) = 1Γ( α ) Z t ( t − s ) α − e − η ( t − s ) V ( s ) ds and κ ( α ) = sin( απ ) π . In the above theorem, taking the input V ( x, t ) = p d ( x ) u xt ( x, t ), then using Equation (1.4), we get that theoutput O is given by O ( x, t ) = p d ( x ) I − α,η u xt ( x, t ) = p d ( x )Γ(1 − α ) Z t ( t − s ) − α e − η ( t − s ) ∂ s u x ( x, s ) ds = p d ( x ) ∂ α,ηt u x ( x, t ) . Therefore, by taking the input V ( x, t ) = p d ( x ) u xt ( x, t ) in Theorem 2.1 and using the above equation, we get(2.5) ∂ t ω ( x, ξ, t ) + ( ξ + η ) ω ( x, ξ, t ) − p d ( x ) u xt ( x, t ) | ξ | α − = 0 , ( x, ξ, t ) ∈ (0 , L ) × R × (0 , ∞ ) ,ω ( x, ξ,
0) = 0 , ( x, ξ ) ∈ (0 , L ) × R , p d ( x ) ∂ α,ηt u x ( x, t ) − κ ( α ) Z R | ξ | α − ω ( x, ξ, t ) dξ = 0 , ( x, t ) ∈ (0 , L ) × (0 , ∞ ) . From system (2.5), we deduce that system ((EBB)-W
F KV ) can be recast into the following augmented model(2.6) u tt − (cid:18) au x + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ, t ) dξ (cid:19) x = 0 , ( x, t ) ∈ (0 , L ) × R + ∗ ,y tt + by xxxx = 0 , ( x, t ) ∈ ( − L, × × R + ∗ ,ω t ( x, ξ, t ) + (cid:0) | ξ | + η (cid:1) ω ( x, ξ, t ) − p d ( x ) u xt ( x, t ) | ξ | α − = 0 , ( x, ξ, t ) ∈ (0 , L ) × R × R + ∗ , with the following transmission and boundary conditions(2.7) u ( L, t ) = y ( − L, t ) = y x ( − L, t ) = 0 , t ∈ (0 , ∞ ) ,au x (0 , t ) + by xxx (0 , t ) = 0 , y xx (0 , t ) = 0 , t ∈ (0 , ∞ ) ,u (0 , t ) = y (0 , t ) , t ∈ (0 , ∞ ) , and with the following initial conditions(2.8) u ( x,
0) = u ( x ) , u t ( x,
0) = u ( x ) , ω ( x, ξ,
0) = 0 x ∈ (0 , L ) , ξ ∈ R ,y ( x,
0) = y ( x ) , y t ( x,
0) = y ( x ) , x ∈ ( − L, . E ( t ) = 12 Z L (cid:0) | u t | + a | u x | (cid:1) dx + 12 Z − L (cid:0) | y t | + b | y xx | (cid:1) dx + κ ( α )2 Z L Z R | ω ( x, ξ, t ) | dξdx. Lemma 2.2.
Let U = ( u, u t , y, y t , ω ) be a regular solution of the System (2.6) - (2.8) . Then, the energy E ( t ) satisfies the following estimation (2.9) ddt E ( t ) = − κ ( α ) Z L Z R ( ξ + η ) | ω ( x, ξ, t ) | dξdx. Proof.
First, multiplying the first and the second equations of (2.6) by u t and y t respectively, integrating over(0 , L ) and ( − L,
0) respectively, using integration by parts with (2.7) and taking the real part ℜ , we get(2.10) 12 ddt Z L | u t | + a | u x | ! dx + 12 ddt Z − L (cid:0) | y t | + b | y xx | (cid:1) dx + ℜ κ ( α ) Z L p d ( x )¯ u tx (cid:18)Z R | ξ | α − ω ( x, ξ, t ) dξ (cid:19) dx ! = 0 . Now, multiplying the third equation of (2.6) by κ ( α )¯ ω , integrating over (0 , L ) × R , then taking the real part,we get(2.11) κ ( α )2 ddt Z L Z R | ω ( x, ξ, t ) | dξdx + κ ( α ) Z L Z R (cid:0) ξ + η (cid:1) | ω ( x, ξ, t ) | dξdx = ℜ κ ( α ) Z L p d ( x ) u xt (cid:18)Z R | ξ | α − ω ( x, ξ, t ) dξ (cid:19) dx ! . Finally, by adding (2.10) and (2.11), we obtain (2.9). The proof is thus complete. (cid:3)
Since α ∈ (0 , κ ( α ) >
0, and therefore ddt E ( t ) ≤
0. Thus, system (2.6)-(2.8) is dissipative in the sensethat its energy is a non-increasing function with respect to time variable t . Now, we define the following Hilbertenergy space H by H = (cid:8) ( u, v, y, z, ω ) ∈ H R (0 , L ) × L (0 , L ) × H L ( − L, × L ( − L, × W ; u (0) = y (0) (cid:9) , where W = L ((0 , L ) × R ) and(2.12) ( H R (0 , L ) = { u ∈ H (0 , L ); u ( L ) = 0 } ,H L ( − L,
0) = { y ∈ H ( − L, y ( − L ) = y x ( − L ) = 0 } . We note that the space H is a closed subspace of H R (0 , L ) × L (0 , L ) × H L ( − L, × L ( − L, × W .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 + b Z − L y xx ( y ) xx dx + κ ( α ) Z L Z R ω ( x, ξ ) ω ( x, ξ ) dξdx, for all U = ( u, v, y, z, ω ) and U = ( u , v , y , z , ω ) in H . We use k U k H to denote the corresponding norm.We define the unbounded linear operator A : D ( A ) ⊂ H → H by D ( A ) = U = ( u, v, y, z, ω ) ∈ H ; ( v, z ) ∈ H R (0 , L ) × H L ( − L, , y ∈ H ( − L, , (cid:18) au x + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ ) dξ (cid:19) x ∈ L (0 , L ) , − (cid:0) | ξ | + η (cid:1) ω ( x, ξ ) + p d ( x ) v x | ξ | α − , | ξ | ω ( x, ξ ) ∈ W,au x (0) + by xxx (0) = 0 , y xx (0) = 0 , and v (0) = z (0) , U = ( u, v, y, z, ω ) ∈ D ( A ), A ( u, v, y, z, ω ) ⊤ = v (cid:18) au x + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ ) dξ (cid:19) x z − by xxxx − (cid:0) | ξ | + η (cid:1) ω ( x, ξ ) + p d ( x ) v x | ξ | α − . Remark 2.3.
The condition | ξ | ω ( x, ξ ) ∈ W is imposed to insure the existence of Z L Z R ( ξ + η ) | ω ( x, ξ ) | dξdx in (2.9) and p d ( x ) Z R | ξ | α − ω ( x, ξ ) dξ ∈ L (0 , L ) . If U = ( u, u t , y, y t , ω ) is a regular solution of system (2.6)-(2.8), then the system can be rewritten as evolutionequation on the Hilbert space H given by(2.13) U t = A U, U (0) = U , where U = ( u , u , y , y , Lemma 2.1.
Let α ∈ (0 , η ≥
0, then the following integrals(2.14) I ( η, α ) = κ ( α ) Z R | ξ | α − ξ + η dξ , I ( η, α ) = Z R | ξ | α − (1 + ξ + η ) dξ and I ( η, α ) = Z + ∞ ξ α +1 (1 + ξ + η ) dξ are well defined. Proof.
First, I ( η, α ) can be written as(2.15) I ( η, α ) = 2 κ ( α )1 + η Z + ∞ ξ α − ξ η dξ. Thus equation (2.15) can be simplified by defining a new variable y = 1+ ξ η . Substituting ξ by ( y − (1+ η ) in equation (2.15) , we get I ( η, α ) = κ ( α )(1 + η ) − α Z + ∞ y ( y − − α dy. Using the fact that α ∈ (0 , y − ( y − α − ∈ L (1 , + ∞ ), therefore I ( η, α ) is well defined.Now, for I ( η, α ), using η ≥ α ∈ (0 , I ( η, α ) < Z R | ξ | α − ξ + η dξ = I ( η, α ) κ ( α ) < + ∞ . Then, I ( η, α ) is well-defined.Now, for the integral I ( η, α ), since ξ α +1 (1 + ξ + η ) ∼ ξ α +1 (1 + η ) and ξ α +1 (1 + ξ + η ) ∼ + ∞ ξ − α , and the fact that α ∈ (0 , I ( η, α ) is well-defined.The proof is thus complete. (cid:3) Proposition 2.4.
The unbounded linear operator A is m-dissipative in the energy space H . roof. For all U = ( u, v, y, z, ω ) ∈ D ( A ), one has ℜ (cid:0) hA U, U i H (cid:1) = − κ ( α ) Z L Z R ( ξ + η ) | ω ( x, ξ ) | dξdx ≤ , which implies that A is dissipative. Now, let F = ( f , f , f , f , f ) ∈ H , we prove the existence of U =( u, v, y, z, ω ) ∈ D ( A ), solution of the equation ( I − A ) U = F. Equivalently, one must consider the system given by u − v = f , (2.16) v − (cid:18) au x + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ ) dξ (cid:19) x = f , (2.17) y − z = f , (2.18) z + by xxxx = f , (2.19) (1 + ξ + η ) ω ( x, ξ ) − p d ( x ) v x | ξ | α − = f ( x, ξ ) . (2.20)Using Equations (2.16), (2.20) and the fact that η ≥
0, we get ω ( x, ξ ) = f ( x, ξ )1 + ξ + η + p d ( x ) u x | ξ | α − ξ + η − p d ( x )( f ) x | ξ | α − ξ + η . Inserting the above equation and (2.16) in (2.17) and (2.18) in (2.19), we get u − au x + d ( x ) I ( η, α ) u x + d ( x ) I ( η, α )( f ) x − p d ( x ) κ ( α ) Z R | ξ | α − f ( x, ξ )1 + ξ + η dξ ! x = F , (2.21) y + by xxxx = F (2.22)where I ( η, α ) is defined in Equation (2.14), F = f + f and F = f + f . And with the following boundaryconditions(2.23) u ( L ) = y ( − L ) = y x ( − L ) = y xx (0) = 0 , au x (0) + by xxx (0) = 0 , and u (0) = y (0) . Now, we define V = (cid:8) ( ϕ, ψ ) ∈ H R (0 , L ) × H L ( − L, ϕ (0) = ψ (0) (cid:9) . The space V is equipped with the following inner product h ( ϕ, ψ ) , ( ϕ , ψ ) i V = a Z L ϕ x ϕ x dx + b Z − L ψ xx ( ψ ) xx dx. Let ( ϕ, ψ ) ∈ V . Multiplying equations (2.21) and (2.22) by ¯ ϕ and ¯ ψ , and integrating respectively on (0 , L ) and( − L, a (( u, y ) , ( ϕ, ψ )) = L ( ϕ, ψ ) ∀ ( ϕ, ψ ) ∈ V, where a (( u, y ) , ( ϕ, ψ )) = Z L uϕdx + a Z L u x ϕ x dx + Z − L yψdx + b Z − L y xx ψ xx dx + I ( η, α ) Z L d ( x ) u x ϕ x dx and L ( ϕ, ψ ) = Z L F ϕdx + I ( η, α ) Z L d ( x )( f ) x ϕ x dx − κ ( α ) Z L p d ( x ) ϕ x Z R | ξ | α − f ( x, ξ )1 + ξ + η dξ ! dx + Z − L F ψdx. Using the fact that I ( η, α ) >
0, we get a is a bilinear, continuous coercive form on V × V . Next, by usingCauchy-Schwartz inequality and the definition of d ( x ), we get(2.25) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z L p d ( x ) ¯ ϕ x Z R | ξ | α − f ( x, ξ )1 + ξ + η dξ ! dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ s d κ ( α ) p I ( η, α ) k ϕ x k L ( l ,l ) k f k W , I ( η, α ) is defined in Equation (2.14). Hence, L is a linear continuous form on V . Then, using Lax-Milgram theorem, we deduce that there exists unique ( u, y ) ∈ V solution of the variational problem (2.24).Applying the classical elliptic regularity, we deduce that y ∈ H ( − L, (cid:18) au x + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ, t ) dξ (cid:19) x ∈ L (0 , L ) . Defining(2.26) v := u − f , z := y − f and ω ( x, ξ ) = f ( x, ξ )1 + ξ + η + p d ( x ) u x | ξ | α − ξ + η − p d ( x )( f ) x | ξ | α − ξ + η . It is easy to see that ( v, z ) ∈ H R (0 , L ) × H L ( − L,
0) and v (0) = z (0).In order to complete the existence of U ∈ D ( A ), we need to prove ω ( x, ξ ) and | ξ | ω ( x, ξ ) ∈ W . From equation(2.26), we obtain Z L Z R | ω ( x, ξ ) | dx ≤ Z L Z R | f ( x, ξ ) | (1 + ξ + η ) dξdx + 3 d I ( η, α ) Z l l (cid:0) | u x | + | ( f ) x | (cid:1) dx. Using Lemma 2.1, the fact that ( u, f ) ∈ H R (0 , L ) × H R (0 , L ), we obtain I ( η, α ) Z l l (cid:0) | u x | + | ( f ) x | (cid:1) dx < ∞ . On the other hand, using the fact that f ∈ W , we get Z L Z R | f ( x, ξ ) | (1 + ξ + η ) dξdx ≤ η ) Z L Z R | f ( x, ξ ) | dξdx < + ∞ . It follows that ω ( x, ξ ) ∈ W . Next, using equation (2.26), we get Z L Z R | ξ ω ( x, ξ ) | dξdx ≤ Z L Z R ξ | f ( x, ξ ) | (1 + ξ + η ) dξdx + 6 d I ( η, α ) Z l l (cid:0) | u x | + | ( f ) x | (cid:1) dx ! , where I ( η, α ) = Z + ∞ ξ α +1 (1 + ξ + η ) dξ . Using Lemma 2.1 we get that I ( η, α ) is well-defined.Now, using the fact that f ( x, ξ ) ∈ W andmax ξ ∈ R ξ (1 + ξ + η ) = 14 (1 + η ) < , we get Z L Z R ξ | f ( x, ξ ) | (1 + ξ + η ) dξdx ≤ max ξ ∈ R ξ (1 + ξ + η ) Z L Z R | f ( x, ξ ) | dξdx < Z L Z R | f ( x, ξ ) | dξdx < + ∞ . It follows that | ξ | ω ∈ W . Finally, since ω, f ∈ W , we get − (cid:0) | ξ | + η (cid:1) ω ( x, ξ ) + p d ( x ) v x | ξ | α − = ω ( x, ξ ) − f ( x, ξ ) ∈ W. Therefore, there exists U := ( u, v, y, z, ω ) ∈ D ( A ) solution ( I − A ) U = F . The proof is thus complete. (cid:3) From proposition 2.4, the operator A is m-dissipative on H , consequently it generates a C -semigroup ofcontractions ( e t A ) t ≥ following Lummer-Phillips theorem (see in [53] and [46]). Then the solution of theevolution Equation (2.13) admits the following representation U ( t ) = e t A U , t ≥ , which leads to the well-posedness of (2.13). Hence, we have the following result. Theorem 2.5.
Let U ∈ H , then problem (2.13) admits a unique weak solution U satisfies U ( t ) ∈ C (cid:0) R + , H (cid:1) . Moreover, if U ∈ D ( A ) , then problem (2.13) admits a unique strong solution U satisfies U ( t ) ∈ C (cid:0) R + , H (cid:1) ∩ C (cid:0) R + , D ( A ) (cid:1) . Strong Stability.
This part is devoted to study the strong stability of the system. It is easy to see thatthe resolvent of A is not compact. For this aim, we use a general criteria of Arendt-Battay in [14] (see Theorem7.2) to obtain the strong stability of the C -semigroup ( e t A ) t ≥ . Our main result in this part is the followingtheorem. Theorem 2.6.
Assume that η ≥ , then the C − semigroup of contractions e t A is strongly stable on H inthe sense that lim t → + ∞ (cid:13)(cid:13) e t A U (cid:13)(cid:13) H = 0 ∀ U ∈ H . In order to proof Theorem 2.6 we need to prove that the operator A has no pure imaginary eigenvalues and σ ( A ) ∩ i R is countable, where σ ( A ) denotes the spectrum of A . For clarity, we divide the proof into severallemmas. Lemma 2.2.
Let α ∈ (0 , η ≥ λ ∈ R and f ∈ W . For ( η > λ ∈ R ) or ( η = 0 and λ ∈ R ∗ ), we have I ( λ, η, α ) = iλκ ( α ) Z R | ξ | α − iλ + ξ + η dξ < ∞ , I ( λ, η, α ) = κ ( α ) Z R | ξ | α − iλ + ξ + η dξ < ∞ , and I ( x, λ, η, α ) := κ ( α ) Z R | ξ | α − f ( x, ξ ) iλ + ξ + η dξ ∈ L (0 , L ) Proof.
The integrals I and I can be written in the following form I ( λ, η, α ) = λ I ( λ, η, α ) + iλ I ( λ, η, α ) , and I ( λ, η, α ) = − iλ I ( λ, η, α ) + I ( λ, η, α )where I ( λ, η, α ) = κ ( α ) Z R | ξ | α − λ + ( ξ + η ) dξ, and I ( λ, η, α ) = κ ( α ) Z R | ξ | α − (cid:0) ξ + η (cid:1) λ + ( ξ + η ) dξ. So, we need to prove that I ( λ, η, α ) , I ( λ, η, α ) are well defined.First, we have I ( λ, η, α ) = 2 κ ( α ) Z + ∞ ξ α − λ + ( ξ + η ) dξ = 2 κ ( α ) Z ξ α − λ + ( ξ + η ) dξ + 2 κ ( α ) Z + ∞ ξ α − λ + ( ξ + η ) dξ. Hence in the both cases where ( η > λ ∈ R ) or ( η = 0 and λ ∈ R ∗ ), we have ξ α − λ + ( ξ + η ) ∼ ξ α − λ + η and ξ α − λ + ( ξ + η ) ∼ + ∞ ξ − α . Since 0 < α < I ( λ, η, α ) is well-defined. Now, we have I ( λ, η, α ) = 2 κ ( α ) Z + ∞ ξ α − ( ξ + η ) λ + ( ξ + η ) dξ = 2 κ ( α ) Z ξ α − ( ξ + η ) λ + ( ξ + η ) dξ + 2 κ ( α ) Z + ∞ ξ α − ( ξ + η ) λ + ( ξ + η ) dξ. Similar to I , in the both cases where ( η > λ ∈ R ) or ( η = 0 and λ ∈ R ∗ ), we have ξ α − ( ξ + η ) λ + ( ξ + η ) ∼ ξ α − ( ξ + η ) λ + η and ξ α − ( ξ + η ) λ + ( ξ + η ) ∼ + ∞ ξ − α . Since 0 < α <
1, then I ( λ, η, α ) is well-defined. For I , using Cauchy-Schwarz inequality and the fact that f ∈ W and that I < ∞ , we get Z L | I ( x, λ, η, α ) | dx = κ ( α ) Z L (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R | ξ | α − f ( x, ξ ) iλ + ξ + η dξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ≤ κ ( α ) (cid:18)Z R | ξ | α − λ + ( ξ + η ) dξ (cid:19) Z L Z R | f ( x, ξ ) | dξdx < + ∞ . The proof is thus complete. (cid:3)
Lemma 2.3.
Let α ∈ (0 , η ≥ λ ∈ R . For ( η > λ ∈ R ) or ( η = 0 and λ ∈ R ∗ ), we have I ( λ, η, α ) = Z R | ξ | α − p λ + ( ξ + η ) dξ and I ( λ, η, α ) = Z R | ξ | α +1 λ + ( ξ + η ) dξ are well-defined. 13 roof. We have I ( λ, η, α ) = 2 Z ξ α − p λ + ( ξ + η ) dξ + 2 Z + ∞ ξ α − p λ + ( ξ + η ) dξ Hence in the both cases where ( η > λ ∈ R ) or ( η = 0 and λ ∈ R ∗ ), we have ξ α − p λ + ( ξ + η ) ∼ ξ α − p λ + η and ξ α − p λ + ( ξ + η ) ∼ + ∞ ξ − α . Since 0 < α < I ( λ, η, α ) is well-defined. Now, I ( λ, η, α ) = 2 Z + ∞ ξ α +1 λ + ( ξ + η ) dξ = 2 Z ξ α − ( ξ + η ) λ + ( ξ + η ) dξ + 2 Z + ∞ ξ α − ( ξ + η ) λ + ( ξ + η ) dξ. In a similar way, in the both cases where ( η > λ ∈ R ) or ( η = 0 and λ ∈ R ∗ ), we have ξ α +1 λ + ( ξ + η ) ∼ ξ α +1 λ + η and ξ α +1 λ + ( ξ + η ) ∼ + ∞ ξ − α . Since 0 < α <
1, then I ( λ, η, α ) is well-defined. The proof is thus complete. (cid:3) Lemma 2.4.
Assume that η ≥
0. Then, for all λ ∈ R , we have iλI − A is injective, i.e.ker ( iλI − A ) = { } . Proof.
Let λ ∈ R , such that iλ be an eigenvalue of the operator A and U = ( u, v, y, z, ω ) ∈ D ( A ) acorresponding eigenvector. Therefore, we have(2.27) A U = iλU. Equivalently, we have v = iλu, (2.28) (cid:18) au x + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ ) dξ (cid:19) x = iλv, (2.29) z = iλy, (2.30) − by xxxx = iλz, (2.31) (cid:0) iλ + | ξ | + η (cid:1) ω ( x, ξ ) = p d ( x ) v x | ξ | α − . (2.32)with the boundary conditions(2.33) ( u ( L ) = y ( − L ) = y x ( − L ) = 0 ,y xx (0) = 0 , and with the continuity transmission conditions(2.34) ( u (0) = y (0) ,au x (0) = − by xxx (0) . A straightforward calculation gives0 = ℜ (cid:0) h iλU, U i H (cid:1) = ℜ (cid:0) hA U, U i H (cid:1) = − κ ( α ) Z L Z R ( ξ + η ) | ω ( x, ξ ) | dξdx. Consequently, we deduce that(2.35) ω ( x, ξ ) = 0 a.e. in (0 , L ) × R . Inserting Equation (2.35) into (2.32) and using the definition of d ( x ), we get(2.36) v x = 0 in ( l , l ) . It follows, from Equation (2.28), that(2.37) λu x = 0 in ( l , l ) . Case 1. If λ = 0:From (2.28) and (2.30) we get v = z = 0 on (0 , L ) . Using Equations (2.29), (2.31) and (2.35) we get u xx = y xxxx = 0 . Using the boundary conditions in (2.33) we can write u and y as u = c ( x − L ) and y = c (cid:18) x − L x − L (cid:19) where c , c are constant numbers to be determined. Now, using conditions in (2.34) we get(2.38) c = L c ,ac = − bc . Then, c ( a L b ) = 0. Since a, b >
0, we deduce that c = c = 0. Then we get u = y = 0. Hence, U = 0. Inthis case the proof is complete. Case 2. If λ = 0:From Equation (2.37), we get(2.39) u x = 0 in ( l , l ) . Using Equations (2.35) and (2.39) in (2.29), and using Equation (2.28) we get(2.40) u = 0 in ( l , l ) . Substituting equations (2.28) and (2.30) into Equations (2.29) and (2.31) and using Equation (2.35), we get λ u + au xx = 0 , over (0 , L ) , (2.41) λ y − by xxxx = 0 , over ( − L, , (2.42)From Equation (2.41) and (2.40), and using the unique continuation theorem (see [40]) we get(2.43) u = 0 in (0 , L ) . From Equation (2.42), (2.33)-(2.34), and using (2.43) we get the following system(2.44) λ y − by xxxx = 0 , over ( − L, y (0) = y xx (0) = y ( xxx ) (0) = 0 ,y ( − L ) = y x ( − L ) = 0 . It’s easy to see that y = 0 is the unique solution of (2.44). Hence U = 0. The proof is thus completed. (cid:3) Lemma 2.5.
Assume that η = 0. Then, the operator −A is not invertible and consequently 0 ∈ σ ( A ). Proof.
Let F = (cid:16) cos (cid:16) πx L (cid:17) , , , , (cid:17) ∈ H and assume that there exists U = ( u, v, y, z, ω ) ∈ D ( A ) such that −A U = F . It follows that v x = − π L sin (cid:16) πx L (cid:17) in (0 , L ) and ξ ω ( x, ξ ) + π L p d ( x ) sin (cid:16) πx L (cid:17) | ξ | α − = 0 . From the above equation, we deduce that ω ( x, ξ ) = − π L | ξ | α − p d ( x ) cos (cid:16) πxL (cid:17) / ∈ W , therefore the assumptionof the existence of U is false and consequently the operator −A is not invertible. The proof is thus complete. (cid:3) Lemma 2.6.
If ( η > λ ∈ R ) or ( η = 0 and λ ∈ R ∗ ), then iλI − A is surjective.15 roof. Let F = ( f , f , f , f , f ) ∈ H , we look for U = ( u, v, y, z, ω ) ∈ D ( A ) solution of(2.45) ( iλI − A ) U = F . Equivalently, we have iλu − v = f , (2.46) iλv − (cid:18) au x + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ ) dξ (cid:19) x = f , (2.47) iλy − z = f , (2.48) iλz + by xxxx = f , (2.49) (cid:0) iλ + ξ + η (cid:1) ω ( x, ξ ) − p d ( x ) v x | ξ | α − = f ( x, ξ ) . (2.50)Using Equations (2.46) and (2.50) and that fact that η ≥ ω ( x, ξ ) = f ( x, ξ ) iλ + ξ + η + p d ( x ) iλu x | ξ | α − iλ + ξ + η − p d ( x )( f ) x | ξ | α − iλ + ξ + η . Substituting v and z in (2.46) and (2.48) into Equations (2.47) and (2.49), and using (2.51) we get λ u + ( au x + d ( x ) I ( λ, η, α ) u x − g ( x, λ, η, α )) x = f, (2.52) λ y − by xxxx = F , (2.53)such that f = − ( f + iλf ) ∈ L (0 , L ) ,g ( x, λ, η, α ) = I ( λ, η, α ) d ( x )( f ) x − p d ( x ) I ( x, λ, η, α ) ,F = − ( f + iλf ) ∈ L ( − L, , and I ( λ, η, α ), I ( λ, η, α ) and I ( λ, η, α ) are defined in Lemma 2.2.Now, we distinguish two cases: Case 1: η > λ = 0, then System (2.52)-(2.53) becomes( au x − g ( x, , η, α )) x = − f ,by xxxx = f . By applying Lax-Milgram theorem, and using Lemma 2.2 it is easy to see that the above system has a uniquestrong solution ( u, y ) ∈ V . Case 2: η ≥ λ ∈ R ∗ . The system (2.52)-(2.53) becomes λ u + ( au x + d ( x ) I ( λ, η, α ) u x ) x = G, (2.54) λ y − by xxxx = F , (2.55)such that G = f + g x ( x, λ, η, α ) . We first define the linear unbounded operator L : H := H R (0 , L ) × H L ( − L, H ′ where H ′ is the dualspace of H by L U = − ( au x + d ( x ) I ( λ, η, α ) u x ) x by xxxx ! , ∀ U ∈ H . Thanks to Lax-Milgram theorem, it is easy to see that L is isomorphism. The system (2.54)-(2.55) is equivalentto(2.56) (cid:0) λ L − − I (cid:1) U = L − F , where U = ( u, y ) ⊤ and F = ( G, F ) ⊤ . Since the operator L − is isomorphism and I is a compact operator from H to H ′ . Then, L − is compactoperator from H to H . Consequently, by Fredholm’s alternative, proving the existence of U solution of (2.56)16educes to proving ker (cid:0) λ L − − I (cid:1) = { } . Indeed, if (˜ u, ˜ y ) ∈ ker (cid:0) λ L − − I (cid:1) , then λ (˜ u, ˜ y ) − L (˜ u, ˜ y ) = 0. Itfollows that, λ ˜ u + ( a ˜ u x + d ( x ) I ( λ, η, α )˜ u x ) x = 0 , (2.57) λ ˜ y − b ˜ y xxxx = 0 , (2.58) ˜ u ( L ) = ˜ y ( − L ) = ˜ y x ( − L ) = ˜ y xx (0) = 0 , (2.59) a ˜ u x (0) + b ˜ y xxx (0) = 0 , ˜ u (0) = ˜ y (0) . (2.60)Multiplying (2.57) and (2.58) by ˜ u and ˜ y respectively, integrating over (0 , L ) and ( − L,
0) respectively and takingthe sum, then using by parts integration and the boundary conditions (2.59)-(2.60), and take the imaginarypart we get d ℑ ( I ( λ, α, η )) Z l l | ˜ u x | dx = 0 . From Lemma 2.2 we have ℑ ( I ( λ, α, η )) = λ I ( λ, η, α ) = 0, we get ˜ u x = 0 in ( l , l ).Then, system (2.57)-(2.59) becomes λ ˜ u + a ˜ u xx = 0 over (0 , L ) , (2.61) λ ˜ y − b ˜ y xxxx = 0 over ( − L, , (2.62) ˜ u x = 0 over ( l , l ) . (2.63)It is now easy to see that if (˜ u, ˜ y ) is a solution of system (2.61)-(2.63), then the vector e U defined by e U :=(˜ u, iλ ˜ u, ˜ y, iλ ˜ y,
0) belongs to D ( A ), and iλ e U − A e U = 0.Therefore, e U ∈ ker ( iλI − A ), then by using Lemma 2.4, we get e U = 0. This implies that system (2.56) admitsa unique solution due to Fredholm’s alternative, hence (2.56) admits a unique solution in V . Thus, we define v := iλu − f , z := iλy − f and(2.64) ω ( x, ξ ) = f ( x, ξ ) iλ + ξ + η + p d ( x ) iλu x | ξ | α − iλ + ξ + η − p d ( x )( f ) x | ξ | α − iλ + ξ + η . Since F ∈ H , it is easy to see that v ∈ H R (0 , L ), z ∈ H L ( − L, v (0) = z (0) and (cid:18) au x + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ, t ) dξ (cid:19) x ∈ L (0 , L ) . It is left to prove that ω and | ξ | ω ∈ W (for the both cases). From equation (2.64), we get Z L Z R | ω ( x, ξ ) | dξdx ≤ Z L Z R | f ( x, ξ ) | λ + ( ξ + η ) dξdx + 3 a Z l l ( | λu x | + | ( f ) x | ) dx ! I ( λ, η, α ) . Using the fact that f ∈ W and ( η > λ ∈ R ) or ( η = 0 and λ ∈ R ∗ ), we obtain Z L Z R | f ( x, ξ ) | λ + ( ξ + η ) dξ ≤ λ + η Z L Z R | f ( x, ξ ) | dξdx < + ∞ . Using Lemma 2.2, it follows that ω ∈ W . Next, using equation (2.64), we get Z L Z R | ξω | dξ ≤ Z L Z R ξ | f ( x, ξ ) | λ + ( ξ + η ) dξdx + 3 a Z L (cid:0) | λu x | + | ( f ) x | (cid:1) I ( λ, η, α ) , where I ( λ, η, α ) = Z R | ξ | α +1 λ + ( ξ + η ) dξ < + ∞ by using Lemma 2.3. Now, using the fact that f ∈ W andmax ξ ∈ R ξ λ + ( ξ + η ) = p η + λ λ + (cid:16)p η + λ + η (cid:17) = C ( λ, η ) , we get Z L Z R ξ | f ( x, ξ ) | λ + ( ξ + η ) dξdx ≤ Z L max ξ ∈ R ξ λ + ( ξ + η ) Z R | f ( x, ξ ) | dξ = C ( λ, η ) Z L Z R | f ( x, ξ ) | dξdx < + ∞ .
17t follows that | ξ | ω ∈ W . Finally, since ω ∈ W , we get − ( ξ + η ) ω ( x, ξ ) + p d ( x ) v x | ξ | α − = iλω ( x, ξ ) − f ( x, ξ ) ∈ W. Thus, we. obtain U = ( u, v, y, z, ω ) ∈ D ( A ) solution of ( iλI − A ) U = F. The proof is thus. complete. (cid:3) Proof of Theorem 2.6.
First, using Lemma 2.4, we directly deduce that A has no pure imaginary eigenvalues.Next, using Lemmas 2.5, 2.6 and with the help of the closed graph theorem of Banach, we deduce that σ ( A ) ∩ i R = {∅} if η > σ ( A ) ∩ i R = { } if η = 0. Thus, we get the conclusion by Applying theorem7.2 of Arendt Batty.2.2. Polynomial Stability in the case η > . In this section, we study the polynomial stability of the system(2.6)-(2.8) in the case η >
0. For this purpose, we will use a frequency domain approach method, namely wewill use Theorem 7.4. Our main result in this section is the following theorem.
Theorem 2.7.
Assume that η > . The C − semigroup ( e t A ) t ≥ is polynomially stable; i.e. there existsconstant C > such that for every U ∈ D ( A ) , we have (2.65) E ( t ) ≤ C t − α k U k D ( A ) , t > , ∀ U ∈ D ( A ) . According to Theorem 7.4, by taking ℓ = 1 − α , the polynomial energy decay (2.65) holds if the followingconditions(H ) i R ⊂ ρ ( A ) , and(H ) sup λ ∈ R (cid:13)(cid:13) ( iλI − A ) − (cid:13)(cid:13) L ( H ) = O (cid:0) | λ | − α (cid:1) are satisfied. Since Condition (H ) is already proved in Lemma 2.4. We will prove condition (H ) by an argumentof contradiction. For this purpose, suppose that (H ) is false, then there exists (cid:8)(cid:0) λ n , U n := ( u n , v n , y n , z n , ω n ( · , ξ )) ⊤ (cid:1)(cid:9) ⊂ R ∗ × D ( A ) with(2.66) | λ n | → + ∞ and k U n k H = k ( u n , v n , y n , z n , ω n ( · , ξ )) k H = 1 , such that(2.67) ( λ n ) − α ( iλ n I − A ) U n = F n := ( f ,n , f ,n , f ,n , f ,n , f ,n ( · , ξ )) ⊤ → H . For simplicity, we drop the index n . Equivalently, from (2.67), we have iλu − v = f λ − α in H R (0 , L ) , (2.68) iλv − ( S d ) x = f λ − α in L (0 , L ) , (2.69) iλy − z = f λ − α in H L ( − L, , (2.70) iλz + by xxxx = f λ − α in L ( − L, , (2.71) ( iλ + ξ + η ) ω ( x, ξ ) − p d ( x ) v x | ξ | α − = f ( x, ξ ) λ − α in W, (2.72)where S d = au x + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ ) dξ .Here we will check the condition (H ) by finding a contradiction with (2.66) by showing k U k H = o (1). Forclarity, we divide the proof into several Lemmas. 18 emma 2.7. Let α ∈ (0 , η > λ ∈ R , then I ( λ, η, α ) = Z R | ξ | α + ( | λ | + ξ + η ) dξ = c ( | λ | + η ) α − , I ( λ, η ) = (cid:18)Z R | λ | + ξ + η ) dξ (cid:19) = r π | λ | + η ) , I ( λ, η ) = Z R ξ ( | λ | + ξ + η ) dξ ! = √ π | λ | + η ) where c = Z ∞ ( y − α − y dy . Proof. I can be written as(2.73) I ( λ, η, α ) = 2( λ + η ) Z ∞ ξ α + (cid:16) ξ | λ | + η (cid:17) dξ. Thus, equation (2.73) may be simplified by defining a new variable y = 1+ ξ λ + η . Substituting ξ by ( y − ( λ + η ) in equation (2.73), we get I ( λ, η, α ) = ( λ + η ) α − Z ∞ ( y − α − y dy. Using the fact that α ∈ ]0 , y − ( y − α − ∈ L (1 , + ∞ ). Hence, the last integral in theabove equation is well defined. Now, I ( λ, η ) can be written as( I ( λ, η )) = 2( λ + η ) Z ∞ (cid:18) (cid:16) ξ √ λ + η (cid:17) (cid:19) dξ = 2( λ + η ) Z ∞ s ) ds = 2( λ + η ) × π π λ + η ) , Therefore, I ( λ, η ) = r π λ + η ) . Finally, I ( λ, η ) can be written as( I ( λ, η )) = 2( λ + η ) Z ∞ ξ (cid:18) (cid:16) ξ √ λ + η (cid:17) (cid:19) dξ = 2( λ + η ) Z ∞ s (1 + s ) ds = 2( λ + η ) × π . Then I ( λ, η ) = √ π λ + η ) . The proof has been completed. (cid:3) Lemma 2.8.
Assume that η >
0. Then, the solution ( u, v, y, z, ω ) ∈ D ( A ) of system (2.68)-(2.72) satisfiesthe following asymptotic behavior estimations(2.74) Z L Z R (cid:0) | ξ | + η (cid:1) | ω ( x, ξ ) | dξdx = o (cid:0) λ − α (cid:1) , Z l l | v x | dx = o (cid:0) λ − α (cid:1) and Z l l | u x | dx = o (cid:0) λ − − α (cid:1) . Proof.
For clarity, we divide the proof into several steps.
Step 1.
Taking the inner product of F with U in H , then using (2.66) and the fact that U is uniformlybounded in H , we get κ ( α ) Z L Z R (cid:0) ξ + η (cid:1) | ω ( x, ξ ) | dξdx = −ℜ (cid:0) hA U, U i H (cid:1) = ℜ (cid:0) h ( iλI − A ) U, U i H (cid:1) = o (cid:0) λ − α (cid:1) . Step 2.
Our aim here is to prove the second estimation in (2.74).From (2.72), we get p d ( x ) | ξ | α − | v x | ≤ (cid:0) | λ | + ξ + η (cid:1) | ω ( x, ξ ) | + | λ | − α | f ( x, ξ ) | . (cid:0) | λ | + ξ + η (cid:1) − | ξ | , integrate over R , we get(2.75) p d ( x ) I ( λ, η, α ) | v x | ≤ I ( λ, η ) (cid:18)Z R | ξω ( x, ξ ) | dξ (cid:19) + | λ | − α I ( λ, η ) (cid:18)Z R | f ( x, ξ ) | dξ (cid:19) , where I ( λ, η, α ) , I ( λ, η ) and I ( λ, η ) are defined in Lemma 2.7. Using Young’s inequality and the definitionof the function d ( x ) in (2.75), we get Z l l | v x | dx ≤ I I o (1) | λ | − α + 2 I I o (1) | λ | − α . It follows from Lemma 2.7 that(2.76) Z l l | v x | dx ≤ c ( | λ | + η ) α − o (1) | λ | − α + √ π c ( | λ | + η ) α o (1) | λ | − α . Since α ∈ (0 , α ,
2) = α , hence from the above equation, we get the second desired estimationin (2.74). Step 3.
From Equation (2.68) we have iλu x = v x − λ − α ( f ) x . It follows that k λu x k L ( l ,l ) ≤ k v x k L ( l ,l ) + | λ | − α k ( f ) x k L ( l ,l ) ≤ o (1) λ α + o (1) λ − α . Since α ∈ (0 , (cid:0) α , − α (cid:1) = 1 + α , hence from the above equation, we get Z l l | u x | dx = o (1) λ α . The proof is thus completed. (cid:3)
Lemma 2.9.
Let 0 < α < η >
0. Then, the solution ( u, v, y, z, ω ) ∈ D ( A ) of system (2.68)-(2.72)satisfies the following asymptotic behavior(2.77) Z l l | S d | dx = o (1) λ − α . Proof.
Using the fact that | P + Q | ≤ P + 2 Q , we obtain Z l l | S d | dx = Z l l (cid:12)(cid:12)(cid:12)(cid:12) au x + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ ) dξ (cid:12)(cid:12)(cid:12)(cid:12) dx ≤ a Z l l | u x | dx + 2 d κ ( α ) Z l l Z R | ξ | α − p ξ + η p ξ + η ω ( x, ξ ) dξ ! dx ≤ a Z l l | u x | dx + c Z l l Z R ( ξ + η ) | ω ( x, ξ ) | dξdx where c = d κ ( α ) I ( α, η ) and I ( α, η ) = Z R | ξ | α − | ξ | + η dξ . We have | ξ | α − | ξ | + η ∼ | ξ | α − η and | ξ | α − | ξ | + η ∼ + ∞ | ξ | − α . Since 0 < α < η >
0, then I ( α, η ) is well defined. Using the first and the third estimations in (2.74),we get our desired result. (cid:3) Lemma 2.10.
Assume that η >
0. Let g ∈ C ([ l , l ]) such that g ( l ) = − g ( l ) = 1 , max x ∈ ( l ,l ) | g ( x ) | = m g and max x ∈ ( l ,l ) | g ′ ( x ) | = m g ′ where m g and m g ′ are strictly positive constant numbers. Then, the solution ( u, v, y, z, ω ) ∈ D ( A ) of system(2.68)-(2.72) satisfies the following asymptotic behavior(2.78) | v ( l ) | + | v ( l ) | ≤ (cid:18) λ − α m g ′ (cid:19) Z l ℓ | v | dx + o (1) λ | S d ( l ) | + | S d ( l ) | ≤ λ α Z l l | v | dx + o (1) . Proof.
First we will prove Equation (2.78). From Equation (2.68), we have(2.80) v x = iλu x − λ − α ( f ) x . Multiply Equation (2.80) by 2 g ¯ v and integrate over ( l , l ), we get(2.81) | v ( l ) | + | v ( l ) | = Z l l g ′ | v | dx + ℜ iλ Z l l u x g ¯ vdx ! − ℜ λ − α Z l l ( f ) x g ¯ vdx ! . Then,(2.82) | v ( l ) | + | v ( l ) | ≤ m g ′ Z l l | v | dx + 2 λm g Z l l | u x || ¯ v | dx + 2 m g λ − α Z l l | ( f ) x || ¯ v | dx. Using Young’s inequality we have(2.83) 2 λm g | u x || ¯ v | ≤ λ − α | v | + 2 λ α m g | u x | and 2 m g λ − α | ( f ) x || ¯ v | ≤ m g ′ | v | + m g m g ′ λ − α | ( f ) x | . Using Equation (2.83), then Equation (2.82) becomes(2.84) | v ( l ) | + | v ( l ) | ≤ (cid:18) λ − α m g ′ (cid:19) Z l l | v | dx + 2 λ α m g Z l l | u x | dx + m g m g ′ λ − α Z l l | ( f ) x | . Using the third estimation in Equation (2.74) and the fact that k ( f ) x k L ( l ,l ) = o (1), we obtain(2.85) | v ( l ) | + | v ( l ) | ≤ (cid:18) λ − α m g ′ (cid:19) Z l l | v | dx + o (1) λ + o (1) λ − α . Since α ∈ (0 , | v ( l ) | + | v ( l ) | ≤ (cid:18) λ − α m g ′ (cid:19) Z l l | v | dx + o (1) λ . Now, we will prove (2.79). For this aim, multiply Equation (2.69) by − g ¯ S d and integrate over ( l , l ), we get(2.87) | S d ( l ) | + | S d ( l ) | = Z l l g ′ | S d | dx + ℜ iλ Z l l vg ¯ S d dx ! − ℜ λ − α Z l l f g ¯ S d dx ! . Then,(2.88) | S d ( l ) | + | S d ( l ) | ≤ m g ′ Z l l | S d | dx + 2 λm g Z l l | v || S d | dx + 2 m g λ − α Z l l | f || S d | dx. Using Young’s inequality and Equation (2.77) we obtain(2.89) 2 λm g | v || ¯ S d | ≤ λ α | v | + 2 m g λ − α | S d | ≤ λ α | v | + o (1) . Using Cauchy-Schwarz inequality, Equation (2.77) and the fact that k f k L ( l ,l ) = o (1), we obtain(2.90) 2 m g λ − α Z l l | f || S d | dx ≤ λ − α k f k L ( l ,l ) k S k L ( l ,l ) = o (1) λ − α . Using Equations (2.77), (2.89) and (2.90) in (2.88), and using the fact that α ∈ (0 ,
1) we get(2.91) | S d ( l ) | + | S d ( l ) | ≤ λ α Z l l | v | dx + o (1) . (cid:3) emma 2.11. Let 0 < α < η >
0. Then, the solution ( u, v, y, z, ω ) ∈ D ( A ) of system (2.68)-(2.72)satisfies the following asymptotic behavior(2.92) Z l l | v | dx = o (1) λ α . Proof.
Multiply Equation (2.69) by − iλ − ¯ v and integrate over ( l , l ), we get(2.93) Z l l | v | dx = ℜ iλ − Z l l S d ¯ v x dx ! − (cid:2) iλ − S d ¯ v (cid:3) l l + ℜ iλ − α Z l l f ¯ vdx ! . Estimation of the term ℜ iλ − Z l l S d ¯ v x dx ! . Using Cauchy-Schwarz inequality, the second estimation in(2.74) and the estimation in (2.77), we get(2.94) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℜ iλ − Z l l S d ¯ v x dx !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ λ Z l l | S d | dx ! Z l l | v x | dx ! = o (1) λ . Estimation for the term ℜ iλ − α Z l l f ¯ vdx ! . Using Cauchy-Schwarz inequality, v is uniformly bounded in L ( l , l ) and k f k L ( l ,l ) = o (1), we get(2.95) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℜ iλ − α Z l l f ¯ vdx !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ λ − α Z l l | f | dx ! Z l l | v | dx ! = o (1) λ − α . Inserting Equations (2.94) and (2.95) in (2.93), using the fact that min( , − α ) = , and using Young’sinequality on the second term of (2.93) we get(2.96) Z l l | v | dx ≤ λ − α (cid:2) | v ( l ) | + | v ( l ) | (cid:3) + λ − − α (cid:2) | S d ( l ) | + | S d ( l ) | (cid:3) + o (1) λ . Now, inserting (2.78) and (2.79) in (2.96), we get(2.97) Z l l | v | dx ≤ (cid:18)
12 + m g ′ λ − α (cid:19) Z l l | v | dx + o (1) λ α + o (1) λ − α . Since α ∈ (0 ,
1) then min(1 + α , − α ) = 1 + α , then Equation (2.97) becomes(2.98) (cid:18) − m g ′ λ − α (cid:19) Z l l | v | dx ≤ o (1) λ α . Using the fact that | λ | → + ∞ , we can take λ ≥ − α m − α g ′ , and we get our desired result. The proof is thuscomplete. (cid:3) Lemma 2.12.
Assume that η >
0. Let h ∈ C ([0 , L ]) and ϕ ∈ C ([ − L, u, v, y, z, ω ) ∈ D ( A ) of system (2.68)-(2.72) satisfies the following estimation(2.99) Z L h ′ (cid:0) | v | + a − | S d | (cid:1) dx + Z − L ϕ ′ (cid:0) | z | + 3 b | y xx | (cid:1) dx + 2 b Z − L y xx ϕ ′′ ¯ y x dx + bϕ ( − L ) | y xx ( − L ) | + h (0) | v (0) | − ah ( L ) | u x ( L ) | + ah (0) | u x (0) | + ℜ (2 by xxx (0) ϕ (0)¯ y x (0)) − ϕ (0) | z (0) | = o (1) . Proof.
The proof is divided into several steps.
Step 1.
Multiplying Equation (2.69) by 2 a − h ¯ S d and integrating over (0 , L ), we get(2.100) ℜ a − iλ Z L vh ¯ S d dx ! + a − Z L h ′ | S | dx − a − h ( L ) | S d ( L ) | + a − h (0) | S d (0) | = ℜ a − λ − α Z L f h ¯ S d dx ! . iλ ¯ u x = − ¯ v x − λ − α ( ¯ f ) x . Then(2.101) iλa − ¯ S d = − ¯ v x − λ − α ( ¯ f ) x + iλa − p d ( x ) κ ( α ) Z R | ξ | α − ¯ ω ( x, ξ ) dξ. Then the first term of (2.100), become(2.102) ℜ a − iλ Z L vh ¯ S d dx ! = Z L h ′ | v | dx − h ( L ) | v ( L ) | + h (0) | v (0) | − ℜ λ − α Z L vh ( ¯ f ) x dx ! + ℜ iλa − κ ( α ) Z L hv p d ( x ) (cid:18)Z R | ξ | α − ¯ ω ( x, ξ ) dξ (cid:19) dx ! . Inserting Equation (2.102) into (2.100), and using the fact that v and S d are uniformly bounded in L (0 , L )and k f k L (0 ,L ) = o (1) and k f k H L (0 ,L ) = o (1), we obtain(2.103) Z L h ′ (cid:0) | v | + a − | S d | (cid:1) dx + h (0) | v (0) | − h ( L ) | u x ( L ) | + h (0) a | u x (0) | + ℜ iλa − κ ( α ) Z L hv p d ( x ) (cid:18)Z R | ξ | α − ¯ ω ( x, ξ ) dξ (cid:19) dx ! = o (1) λ − α . Estimation of the term ℜ iλa − κ ( α ) Z L hv p d ( x ) (cid:18)Z R | ξ | α − ¯ ω ( x, ξ ) dξ (cid:19) dx ! . Using the definition of d ( x )and Cauchy-Schwarz inequality, the fact that 0 < α < η >
0, and using the first estimation in (2.74) andEquation (2.92), we obtain(2.104) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℜ iλa − κ ( α ) Z L hv p d ( x ) (cid:18)Z R | ξ | α − ¯ ω ( x, ξ ) dξ (cid:19) dx !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = o (1) . Inserting Equation (2.104) in Equation (2.103), and using the fact that α ∈ (0 ,
1) we obtain(2.105) Z L h ′ (cid:0) | v | + a − | S d | (cid:1) dx + h (0) | v (0) | − ah ( L ) | u x ( L ) | + ah (0) | u x (0) | = o (1) . Step 2.
Multiplying Equation (2.72) by 2 ϕ ¯ y x and integrating over ( − L, ℜ (cid:18) iλ Z − L zϕ ¯ y x dx (cid:19) − b Z − L y xxx ϕ ′ ¯ y x dx − b Z − L y xxx ϕ ¯ y xx dx + ℜ (cid:16) [2 by xxx ϕ ¯ y x ] − L (cid:17) = ℜ (cid:18) λ − α Z − L f ϕ ¯ y x dx (cid:19) . Integrating by parts the second and third terms of the above equation we get(2.107) ℜ (cid:18) iλ Z − L zϕ ¯ y x dx (cid:19) + 2 b Z − L ϕ ′ | y xx | dx + 2 b Z − L y xx ϕ ′′ ¯ y x dx − [2 by xx ϕ ′ ¯ y x ] − L + b Z − L ϕ ′ | y xx | dx − (cid:2) bϕ | y xx | (cid:3) − L + ℜ (2 by xxx (0) ϕ (0)¯ y x (0)) = ℜ (cid:18) λ − α Z − L f ϕ ¯ y x dx (cid:19) . From Equation (2.70), we have(2.108) iλ ¯ y x = − ¯ z x − λ − α ( ¯ f ) x . By inserting Equation (2.108) into the first term of (2.107), we get(2.109) ℜ (cid:18) iλ Z − L ϕz ¯ y x dx (cid:19) = ℜ (cid:18) − Z − L ϕz ¯ z x dx − λ − α Z − L ( ¯ f ) x ϕzdx (cid:19) = Z − L ϕ ′ | z | dx − ϕ (0) | z (0) | − ℜ (cid:18) λ − α Z − L ( ¯ f ) x ϕzdx (cid:19) . Z − L ϕ ′ (cid:0) | z | + 3 b | y xx | (cid:1) dx + 2 b Z − L ϕ ′′ y xx ¯ y x dx + bϕ ( − L ) | y xx ( − L ) | + ℜ (2 bϕ (0) y xxx (0)¯ y x (0)) − ϕ (0) | z (0) | = ℜ (cid:18) λ − α Z − L ϕf ¯ y x dx (cid:19) + ℜ (cid:18) λ − α Z − L ϕ ( ¯ f ) x zdx (cid:19) . Estimation of the term ℜ (cid:18) λ − α Z − L ϕf ¯ y x dx (cid:19) . Using Poincare inequality, Cauchy-Schwarz inequality, thedefinition of ϕ , and y xx is uniformly bounded in L ( − L, k f k L ( − L, = o (1), we get(2.111) (cid:12)(cid:12)(cid:12)(cid:12) ℜ (cid:18) λ − α Z − L ϕf ¯ y x dx (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ λ − α k y x k L (0 ,L ) k f k L (0 ,L ) ≤ λ − α c p k y xx k L (0 ,L ) k f k L (0 ,L ) = o (1) λ − α . Estimation of the term ℜ (cid:18) λ − α Z − L ϕ ( ¯ f ) x zdx (cid:19) . Using Cauchy-Schwarz inequality, the definition of ϕ , and z is bounded in L ( − L, k ( f ) x k L ( − L, = o (1) we get(2.112) (cid:12)(cid:12)(cid:12)(cid:12) ℜ (cid:18) λ − α Z − L ϕ ( ¯ f ) x zdx (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ λ − α Z L | z | dx ! Z L | ( f ) x | dx ! = o (1) λ − α . Inserting (2.111) and (2.112) into (2.110), we get(2.113) Z − L ϕ ′ (cid:0) | z | + 3 b | y xx | (cid:1) dx + 2 b Z − L ϕ ′′ y xx ¯ y x dx + bϕ ( − L ) | y xx ( − L ) | + ℜ (2 bϕ (0) y xxx (0)¯ y x (0)) − ϕ (0) | z (0) | = o (1) λ − α . Now, summing Equations (2.105) and (2.113), we get our desired result. (cid:3)
Lemma 2.13.
Assume that η >
0. The solution ( u, v, y, z, ω ) ∈ D ( A ) of system (2.68)-(2.72) satisfies thefollowing estimation(2.114) k U k H = o (1) . Proof.
The proof of this Lemma is divided into several steps.
Step 1.
In this step we will prove that k v k L ( ,L ) = o (1) and k u x k L ( ,L ) = o (1) . Taking h ( x ) = xθ ( x ) + ( x − L ) θ ( x ) and ϕ ( x ) = 0 in Equation (2.99), where θ , θ ∈ C ([0 , L ]) are defined asfollows(2.115) θ ( x ) = x ∈ [0 , l ] , x ∈ [ l , L ] , ≤ θ ≤ elsewhere, and(2.116) θ ( x ) = x ∈ [ l , L ] , x ∈ [0 , l ] , ≤ θ ≤ elsewhere. we get(2.117) Z L ( θ + θ ) (cid:0) | v | + a − | S d | (cid:1) dx + Z L ( xθ ′ + ( x − L ) θ ′ ) (cid:0) | v | + a − | S | (cid:1) dx = o (1) . Using Equations (2.77), (2.92) and the definition of θ and θ , we get(2.118) Z L ( θ + θ ) (cid:0) | v | + a − | S d | (cid:1) dx = o (1) . Z l | v | dx = o (1) and a Z l | u x | dx = o (1)and(2.120) Z Ll | v | dx = o (1) and a Z Ll | u x | dx = o (1) . Using (2.119), (2.120), (2.77) and (2.92), we get the desired result of Step 1.
Step 2.
Taking h ( x ) = x − L and ϕ ( x ) = 0 in Equation (2.99), we get(2.121) Z L (cid:0) | v | + a − | S | (cid:1) dx − L | v (0) | − aL | u x (0) | = o (1) . By using Step 1, we get(2.122) | v (0) | = o (1) and | u x (0) | = o (1) . Step 3.
The aim of this step is to prove that k z k L ( − L, = o (1) and k y xx k L ( − L, = o (1).Taking h ( x ) = 0 and ϕ ( x ) = x + L in Equation (2.99), we get(2.123) Z − L (cid:0) | z | + 3 b | y xx | (cid:1) dx + ℜ (2 Lby xxx (0)¯ y x (0)) − L | z (0) | = o (1) . Using (2.122) and the transmission conditions we have(2.124) b | y xxx (0) | = a | u x (0) | = o (1) and | z (0) | = | v (0) | = o (1) . Inserting Equation (2.124) in Equation (2.123) and using the fact that | y x (0) | ≤ √ L k y xx k L ( − L, = O (1) weget(2.125) Z − L | z | dx = o (1) and b Z − L | y xx | dx = o (1) . Finally, using Equations (2.11), (2.74), (3.20), and Step 1., we get that k U k H = o (1). (cid:3) Proof of Theorem 2.7.
From Lemma 2.13 we get that k U k H = o (1), which contradicts (2.66). Consequently,condition (H ) holds. This implies, from Theorem 7.4, the energy decay estimation (2.65). The proof is thuscomplete. 3. W-W
F KV
Model
In this section, we consider the (W-W
F KV ) model, where we study the stability of a system of two waveequations coupled through boundary connections with a localized fractional Kelvin-Voigt damping acting onone equation only.By taking the input V ( x, t ) = p d ( x ) u xt ( x, t ) in Theorem 2.1, we get that the output O is given by O ( x, t ) = p d ( x ) I − α,η u xt ( x, t ) = p d ( x )Γ(1 − α ) Z t ( t − s ) − α e − η ( t − s ) ∂ s u x ( x, s ) ds = p d ( x ) ∂ α,ηt u x ( x, t ) . Then system (W-W
F KV ) can be recast into the following augmented model(3.1) u tt − (cid:18) au x + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ, t ) dξ (cid:19) x = 0 , ( x, t ) ∈ (0 , L ) × R + ∗ ,y tt − by xx = 0 , ( x, t ) ∈ ( − L, × × R + ∗ ,ω t ( x, ξ, t ) + (cid:0) | ξ | + η (cid:1) ω ( x, ξ, t ) − p d ( x ) u xt ( x, t ) | ξ | α − = 0 , ( x, ξ, t ) ∈ (0 , L ) × R × R + ∗ , with the following transmission and boundary conditions(3.2) u ( L, t ) = y ( − L, t ) = 0 , t ∈ (0 , ∞ ) ,au x (0 , t ) = by x (0 , t ) , t ∈ (0 , ∞ ) ,u (0 , t ) = y (0 , t ) , t ∈ (0 , ∞ ) , u ( x,
0) = u ( x ) , u t ( x,
0) = u ( x ) , ω ( x, ξ,
0) = 0 x ∈ (0 , L ) , ξ ∈ R ,y ( x,
0) = y ( x ) , y t ( x,
0) = y ( x ) , x ∈ ( − L, . The energy of the system (3.1)-(3.3) is given by E ( t ) = 12 Z L (cid:0) | u t | + a | u x | (cid:1) dx + 12 Z − L (cid:0) | y t | + b | y x | (cid:1) dx + κ ( α )2 Z L Z R | ω ( x, ξ, t ) | dξdx. Using similar computations to Lemma 2.2, we obtain(3.4) ddt E ( t ) = − κ ( α ) Z L Z R ( ξ + η ) | ω ( x, ξ, t ) | dξdx. Since α ∈ (0 , κ ( α ) >
0, and therefore ddt E ( t ) ≤
0. Thus, system (3.1)-(3.3) is dissipative in the sensethat its energy is a non-increasing function with respect to time variable t . Now, we define the following Hilbertenergy space H by H = (cid:8) ( u, v, y, z, ω ) ∈ H R (0 , L ) × L (0 , L ) × H L ( − L, × L ( − L, × W ; u (0) = y (0) (cid:9) , where W = L ((0 , L ) × R ) and(3.5) ( H R (0 , L ) = { u ∈ H (0 , L ); u ( L ) = 0 } ,H L ( − L,
0) = { y ∈ H ( − L, y ( − L ) = 0 } . 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 + b Z − L y x ( y ) x dx + κ ( α ) Z L Z R ω ( x, ξ ) ω ( x, ξ ) dξdx, for all U = ( u, v, y, z, ω ) and U = ( u , v , y , z , ω ) in H . We use k U k H to denote the corresponding norm.We define the unbounded linear operator A : D ( A ) ⊂ H → H by D ( A ) = U = ( u, v, y, z, ω ) ∈ H ; ( v, z ) ∈ H R (0 , L ) × H L ( − L, , y ∈ H ( − L, , (cid:18) au x + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ ) dξ (cid:19) x ∈ L (0 , L ) , − (cid:0) | ξ | + η (cid:1) ω ( x, ξ ) + p d ( x ) v x | ξ | α − , | ξ | ω ( x, ξ ) ∈ W,au x (0) = by x (0) , and v (0) = z (0) , and for all U = ( u, v, y, z, ω ) ∈ D ( A ), A ( u, v, y, z, ω ) ⊤ = v (cid:18) au x + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ ) dξ (cid:19) x zby xx − (cid:0) | ξ | + η (cid:1) ω ( x, ξ ) + p d ( x ) v x | ξ | α − . If U = ( u, u t , y, y t , ω ) is a regular solution of system (3.1)-(3.3), then the system can be rewritten as evolutionequation on the Hilbert space H given by(3.6) U t = A U, U (0) = U , where U = ( u , u , y , y , A is m-dissipative in theenergy space H . Also, the C -semigroup of contractions e t A is strongly stable on H in the sense thatlim t → + ∞ (cid:13)(cid:13) e t A U (cid:13)(cid:13) H = 0. 26 heorem 3.1. Assume that η > . The C − semigroup ( e t A ) t ≥ is polynomially stable; i.e. there existsconstant C > such that for every U ∈ D ( A ) , we have (3.7) E ( t ) ≤ C t − α k U k D ( A ) , t > , ∀ U ∈ D ( A ) . According to Theorem 7.4, by taking ℓ = 1 − α , the polynomial energy decay (3.7) holds if the followingconditions(G ) i R ⊂ ρ ( A ) , and(G ) sup λ ∈ R (cid:13)(cid:13) ( iλI − A ) − (cid:13)(cid:13) L ( H ) = O (cid:0) | λ | − α (cid:1) are satisfied. Since Condition (G ) is already proved. We will prove condition (G ) by an argument of contra-diction. For this purpose, suppose that (G ) is false, then there exists (cid:8)(cid:0) λ n , U n := ( u n , v n , y n , z n , ω n ( · , ξ )) ⊤ (cid:1)(cid:9) ⊂ R ∗ × D ( A ) with(3.8) | λ n | → + ∞ and k U n k H = k ( u n , v n , y n , z n , ω n ( · , ξ )) k H = 1 , such that(3.9) ( λ n ) − α ( iλ n I − A ) U n = F n := ( f ,n , f ,n , f ,n , f ,n , f ,n ( · , ξ )) ⊤ → H . For simplicity, we drop the index n . Equivalently, from (3.9), we have iλu − v = f λ − α in H R (0 , L ) , (3.10) iλv − ( S d ) x = f λ − α in L (0 , L ) , (3.11) iλy − z = f λ − α in H L ( − L, , (3.12) iλz − by xx = f λ − α in L ( − L, , (3.13) ( iλ + ξ + η ) ω ( x, ξ ) − p d ( x ) v x | ξ | α − = f ( x, ξ ) λ − α in W, (3.14)where S d = au x + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ ) dξ .Here we will check the condition (G ) by finding a contradiction with (3.8) by showing k U k H = o (1). In orderto get this contradiction, we follow similar arguments as in Section 2.2. We get the same results as in Lemmas2.7, 2.8, 2.9, 2.10, 2.11.Using the same computations as in Lemma 2.12, we get that the solution ( u, v, y, z, ω ) ∈ D ( A ) of system(3.10)-(3.14) satisfies the following estimation(3.15) Z L h ′ (cid:0) | v | + a − | S d | (cid:1) dx + Z − L ϕ ′ (cid:0) | z | + b | y x | (cid:1) dx + h (0) | v (0) | − ah ( L ) | u x ( L ) | + ah (0) | u x (0) | − ϕ (0) | z (0) | − bϕ (0) | y x (0) | + bϕ ( − L ) | y x ( − L ) | = o (1) . We procced in a similar way to Lemma 2.13.
Step 1.
Taking h ( x ) = xθ ( x ) + ( x − L ) θ ( x ) and ϕ ( x ) = 0 in Equation (3.15), where θ , θ ∈ C ([0 , L ]) aredefined in Lemma 2.13, yields k v k L ( ,L ) = o (1) and k u x k L ( ,L ) = o (1) . Step 2.
Taking h ( x ) = x − L and ϕ ( x ) = 0 in Equation (3.15), we obtain(3.16) Z L (cid:0) | v | + a − | S d | (cid:1) dx − L | v (0) | − aL | u x (0) | = o (1) . By using Step 1, we get(3.17) | v (0) | = | u x (0) | = o (1) . tep 3. Taking h ( x ) = 0 and ϕ ( x ) = x + L in Equation (3.15), we get(3.18) Z − L (cid:0) | z | + b | y x | (cid:1) dx − L | z (0) | − bL | y x (0) | = o (1) . Using (3.17) and the transmission conditions we have(3.19) b | y x (0) | = a | u x (0) | = o (1) and | z (0) | = | v (0) | = o (1) . Inserting Equation (3.19) in Equation (3.18), we get(3.20) Z − L | z | dx = o (1) and b Z − L | y x | dx = o (1) . Proof of Theorem 3.1.
From Lemma Step 1. and Staep 3. we deduce that k U k H = o (1), which contradicts(3.8). Consequently, condition (G ) holds. This implies, from Theorem 7.4, the energy decay estimation (3.7).The proof is thus complete. 4. W-(EBB)
F KV
Model
This section is devoted to study the stability of the model (W-(EBB)
F KV ), where we consider the Euler-Bernoulli beam and wave equations coupled through boundary connection. We take the fractional Kelvin-Voigtdamping to be a localized internal damping acting on the Euler-Bernoulli beam only.4.1.
Well-Posedness and Strong Stability.
In this section, we give the strong stability results of thesystem (W-(EBB)
F KV ). First, using a semigroup approach, we establish well-posedness result for the system(W-(EBB)
F KV ).In Theorem 2.1, taking the input V ( x, t ) = p d ( x ) y xxt ( x, t ), then using (1.4), we get the output O is given by O ( x, t ) = p d ( x ) I − α,η y xxt ( x, t ) = p d ( x )Γ(1 − α ) Z t ( t − s ) − α e − η ( t − s ) ∂ s y xx ( x, s ) ds = p d ( x ) ∂ α,ηt y xx ( x, t ) . Therefore, by taking the input V ( x, t ) = p d ( x ) y xxt ( x, t ) in Theorem 2.1 and using the above equation, we get(4.1) ∂ t ω ( x, ξ, t ) + ( ξ + η ) ω ( x, ξ, t ) − p d ( x ) y xxt ( x, t ) | ξ | α − = 0 , ( x, ξ, t ) ∈ (0 , L ) × R × (0 , ∞ ) ,ω ( ξ,
0) = 0 , ( x, ξ ) ∈ (0 , L ) × R , p d ( x ) ∂ α,ηt y xx ( x, t ) − κ ( α ) Z R | ξ | α − ω ( x, ξ, t ) dξ = 0 , ( x, t ) ∈ (0 , L ) × (0 , ∞ ) . From system (4.1), we deduce that system (W-(EBB)
F KV ) can be recast into the following augmented model(4.2) u tt − au xx = 0 , ( x, t ) ∈ ( − L, × R + ∗ ,y tt + (cid:18) by xx + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ, t ) dξ (cid:19) xx = 0 , ( x, t ) ∈ (0 , L ) × R + ∗ ,ω t ( x, ξ, t ) + (cid:0) | ξ | + η (cid:1) ω ( x, ξ, t ) − p d ( x ) y xxt ( x, t ) | ξ | α − = 0 , ( x, ξ, t ) ∈ (0 , L ) × R × R + ∗ , with the following transmission and boundary conditions(4.3) u ( − L, t ) = y ( L, t ) = y x ( L, t ) = 0 , t ∈ (0 , ∞ ) ,au x (0 , t ) + by xxx (0 , t ) = 0 , y xx (0 , t ) = 0 , t ∈ (0 , ∞ ) ,u (0 , t ) = y (0 , t ) , t ∈ (0 , ∞ ) , and with the following initial conditions(4.4) u ( x,
0) = u ( x ) , u t ( x,
0) = u ( x ) x ∈ ( − L, y ( x,
0) = y ( x ) , y t ( x,
0) = y ( x ) , ω ( x, ξ,
0) = 0 x ∈ (0 , L ) , ξ ∈ R . The energy of the system (4.2)-(4.4) is given by E ( t ) = 12 Z − L (cid:0) | u t | + a | u x | (cid:1) dx + 12 Z L (cid:0) | y t | + b | y xx | (cid:1) dx + κ ( α )2 Z L Z R | ω ( x, ξ, t ) | dξdx.
28y similar computation to Lemma 2.2, it is easy to see that the energy E ( t ) satisfies the following estimation(4.5) ddt E ( t ) = − κ ( α ) Z L Z R ( ξ + η ) | ω ( x, ξ, t ) | dξdx. Since α ∈ (0 , κ ( α ) >
0, and therefore ddt E ( t ) ≤
0. Thus, system (4.2)-(4.4) is dissipative in the sensethat its energy is a non-increasing function with respect to time variable t . Now, we define the following Hilbertenergy space H by H = (cid:8) ( u, v, y, z, ω ) ∈ H L ( − L, × L ( − L, × H R (0 , L ) × L (0 , L ) × W ; u (0) = y (0) (cid:9) , where W = L ((0 , L ) × R ) and(4.6) ( H L ( − L,
0) = { u ∈ H ( − L, u ( − L ) = 0 } ,H R (0 , L ) = { y ∈ H (0 , L ); y ( L ) = y x ( L ) = 0 } . The energy space H is equipped with the inner product defined by h U, U i H = Z − L v ¯ v dx + a Z − L u x ( u ) x dx + Z L z ¯ z dx + b Z L y xx ( y ) xx dx + κ ( α ) Z L Z R ω ( x, ξ ) ω ( x, ξ ) dξdx, for all U = ( u, v, y, z, ω ) and U = ( u , v , y , z , ω ) in H . We use k U k H to denote the corresponding norm.We define the unbounded linear operator A : D ( A ) ⊂ H → H by D ( A ) = U = ( u, v, y, z, ω ) ∈ H ; ( v, z ) ∈ H L ( − L, × H R (0 , L ) , u ∈ H ( − L, , (cid:18) by xx + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ ) dξ (cid:19) xx ∈ L (0 , L ) , − (cid:0) | ξ | + η (cid:1) ω ( x, ξ ) + p d ( x ) z xx | ξ | α − , | ξ | ω ( x, ξ ) ∈ W,au x (0) + by xxx (0) = 0 , y xx (0) = 0 , and v (0) = z (0) , and for all U = ( u, v, y, z, ω ) ∈ D ( A ), A ( u, v, y, z, ω ) ⊤ = vau xx z − (cid:18) by xx + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ ) dξ (cid:19) xx − (cid:0) | ξ | + η (cid:1) ω ( x, ξ ) + p d ( x ) z xx | ξ | α − . If U = ( u, u t , y, y t , ω ) is a regular solution of system (4.2)-(4.4), then the system can be rewritten as evolutionequation on the Hilbert space H given by(4.7) U t = A U, U (0) = U , where U = ( u , u , y , y , A is m-dissipative on H , consequently it generates a C -semigroupof contractions ( e t A ) t ≥ following Lummer-Phillips theorem (see in [46] and [53]). Then the solution of theevolution Equation (4.7) admits the following representation U ( t ) = e t A U , t ≥ , which leads to the well-posedness of (4.7). Hence, we have the following result. Theorem 4.1.
Let U ∈ H , then problem (4.7) admits a unique weak solution U satisfies U ( t ) ∈ C (cid:0) R + , H (cid:1) . Moreover, if U ∈ D ( A ) , then problem (4.7) admits a unique strong solution U satisfies U ( t ) ∈ C (cid:0) R + , H (cid:1) ∩ C (cid:0) R + , D ( A ) (cid:1) . heorem 4.2. Assume that η ≥ , then the C − semigroup of contractions e t A is strongly stable on H inthe sense that lim t → + ∞ (cid:13)(cid:13) e t A U (cid:13)(cid:13) H = 0 . Proof of Theorem 4.2.
The proof of this theorem follows by proceeding with similar arguments as inSubsection 2.1, and using the Arendt Batty Theorem (see Theorem 7.2 in Appendix).4.2.
Polynomial Stability in the case η > . The aim of this part is to study the polynomial stability ofsystem (4.2)-(4.4) in the case η >
0. As the condition i R ⊂ ρ ( A ) is already checked in the subsection 4.1,it remains to prove that condition (7.1) holds (see Theorem 7.4 in Appendix). This is established by usingspecific multipliers, some interpolation inequalities and by solving differential equations of order 4. Our mainresult in this part is the following theorem. Theorem 4.3.
Assume that η > . The C − semigroup ( e t A ) t ≥ is polynomially stable; i.e. there existsconstant C > such that for every U ∈ D ( A ) , we have (4.8) E ( t ) ≤ C t − α k U k D ( A ) , t > , ∀ U ∈ D ( A ) . According to Theorem 7.4, by taking ℓ = 2 − α , the polynomial energy decay (4.8) holds if the followingconditions(R ) i R ⊂ ρ ( A ) , and(R ) sup λ ∈ R (cid:13)(cid:13) ( iλI − A ) − (cid:13)(cid:13) L ( H ) = O (cid:0) | λ | − α (cid:1) are satisfied. Since condition (R )is already proved (see Subsection 4.1), we still need to prove condition(R ). For this purpose we will use an argument of contradiction. Suppose that (R ) is false, then there exists (cid:8)(cid:0) λ n , U n := ( u n , v n , y n , z n , ω n ( · , ξ )) ⊤ (cid:1)(cid:9) ⊂ R ∗ × D ( A ) with(4.9) | λ n | → + ∞ and k U n k H = k ( u n , v n , y n , z n , ω n ( · , ξ )) k H = 1 , such that(4.10) (cid:0) λ − αn (cid:1) ( iλ n I − A ) U n = F n := ( f ,n , f ,n , f ,n , f ,n , f ,n ( · , ξ )) ⊤ → H . For simplicity, we drop the index n . Equivalently, from (4.10), we have iλu − v = f λ − α in H L ( − L, , (4.11) iλv − au xx = f λ − α in L ( − L, , (4.12) iλy − z = f λ − α in H R (0 , L ) , (4.13) iλz + S xx = f λ − α in L (0 , L ) , (4.14) ( iλ + ξ + η ) ω ( x, ξ ) − p d ( x ) z xx | ξ | α − = f ( x, ξ ) λ − α in W, (4.15)where S = by xx + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ ) dξ . Here we will check the condition (R ) by finding a contra-diction with (4.9) by showing k U k H = o (1). For clarity, we divide the proof into several Lemmas. Lemma 4.1.
Assume that η >
0. Then, the solution ( u, v, y, z, ω ) ∈ D ( A ) of system (4.11)-(4.15) satisfiesthe following asymptotic behavior estimations(4.16) Z L Z R (cid:0) | ξ | + η (cid:1) | ω ( x, ξ ) | dξdx = o (1) λ − α , Z l l | z xx | dx = o (1) λ , Z l l | y xx | dx = o (1) λ and Z l l | S | dx = o (1) λ − α . roof. For the clarity of the proof, we divide the proof into several steps.
Step 1.
Taking the inner product of F with U in H , then using (4.9) and the fact that U is uniformly boundedin H , we get κ ( α ) Z L Z R (cid:0) ξ + η (cid:1) | ω ( x, ξ ) | dξdx = −ℜ (cid:0) hA U, U i H (cid:1) = ℜ (cid:0) h ( iλI − A ) U, U i H (cid:1) = o (cid:0) λ − α (cid:1) . Step 2.
Our aim here is to prove the second estimation in (4.16).From (4.15), we get p d ( x ) | ξ | α − | z xx | ≤ (cid:0) | λ | + ξ + η (cid:1) | ω ( x, ξ ) | + | λ | − α | f ( x, ξ ) | . Multiplying the above inequality by (cid:0) | λ | + ξ + η (cid:1) − | ξ | , integrating over R and proceeding in a similar way asin Lemma 2.8 (Section 2.2), we get the second desired estimation in (4.16). Step 3.
From Equation (4.13) we have that k λy xx k L ( l ,l ) ≤ k z xx k L ( l ,l ) + | λ | − α k ( f ) xx k L ( l ,l ) . Using Step 2, the fact that k f k H R (0 ,L ) = o (1), and that α ∈ (0 , Step 4.
Using the fact that | P + Q | ≤ P + 2 Q , we obtain Z l l | S | dx = Z l l (cid:12)(cid:12)(cid:12)(cid:12) by xx + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ ) dξ (cid:12)(cid:12)(cid:12)(cid:12) dx ≤ b Z l l | y xx | dx + 2 d κ ( α ) Z l l Z R | ξ | α − p ξ + η p ξ + η ω ( x, ξ ) dξ ! dx ≤ b Z l l | y xx | dx + c Z l l Z R ( ξ + η ) | ω ( x, ξ ) | dξdx where c = d κ ( α ) I ( α, η ) is defined in Lemma 2.9. Thus, we get the last estimation in (4.16). Hence, theproof is complete. (cid:3) Lemma 4.2.
Assume that η >
0. The solution ( u, v, y, z, ω ) ∈ D ( A ) of system (4.11)-(4.15) satisfies thefollowing asymptotic behavior(4.17) k z k H ( l ,l ) = o (1) λ and k y k H ( l ,l ) = o (1) λ . Proof.
The proof of this Lemma will be divided into two steps.
Step 1.
Let 0 < ε < l − l h ∈ C ∞ ([0 , L ]), 0 ≤ h ≤ , L ], h = 1 on ( l + ε, l − ε ), and h = 0on (0 , l ) ∪ ( l , L ). Also, we define max x ∈ [0 ,L ] | h ′ ( x ) | = m ′ h and max x ∈ [0 ,L ] | h ′′ ( x ) | = m ′′ h , where m ′ h and m ′′ h are strictlypositive constant numbers. Multiply equation (4.14) by − iλ − h ¯ z and integrate over ( l , l ), we get(4.18) Z l l h | z | dx = iλ − Z l l S ( h ′′ ¯ z + 2 h ′ z x + hz xx ) dx − iλ − − α Z l l hf ¯ zdx. Using Nirenberg inequality Theorem (see [52]), Equations (4.9) and (4.16), we have(4.19) k z x k L ( l ,l ) ≤ k z xx k / L ( l ,l ) k z k / L ( l ,l ) + k z k L ( l ,l ) ≤ O (1) . Estimation of the term iλ − Z l l h ′′ S ¯ z ! . Using Young’s inequality, last estimation in (4.16) and the fact that z is uniformly bounded in L (0 , L ) and that 0 < α <
1, we get(4.20) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) iλ − Z l l h ′′ S ¯ zdx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ λ Z l l | h ′′ || S || z | dx ≤ m ′′ h λ Z l l | S || z | dx ≤ m ′′ h Z l l | S | dx + m ′ h λ Z l l | z | dx = o (1) λ − α . iλ − Z l l hSz xx dx ! . Using Cauchy Schwarz inequality, the second and the lastestimations in (4.16), we get(4.21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) iλ − Z l l hS ¯ z xx dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ m ′ h λ Z l l | S | dx ! / Z l l | z xx | dx ! / = o (1) λ − α . Estimation of the term iλ − Z l l h ′ Sz x dx ! . Using Young’s inequality, the last estimation in (4.16), Equation(4.19) and the fact that 0 < α <
1, we get(4.22) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) iλ − Z l l h ′ Sz x dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ m ′ h Z l l | S | dx + m ′ h λ Z l l | z x | dx = o (1) λ − α . Estimation of the term iλ − − α Z l l hf ¯ zdx ! . Using Cauchy-Schwarz inequality, the fact that k f k L (0 ,L ) = o (1) and the fact that z is uniformly bounded in L (0 , L ), we get(4.23) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) iλ − − α Z l l hf ¯ zdx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = o (1) λ α . Thus, using Equations (4.20)-(4.23) in (4.18), we get(4.24) Z l l h | z | dx = o (1) λ − α and Z l − εl + ε | z | dx = o (1) λ − α . Step 2.
Applying the interpolation theorem involving compact subdomain ([2], Theorem 4.23), we obtain k z x k L ( l ,l ) ≤ k z xx k L ( l ,l ) + k z k L ( l + ε,l − ε ) . Then, by using (4.24) and that 0 < α <
1, we get(4.25) k z x k L ( l ,l ) = o (1) λ . Also, the same interpolation theorem yields that(4.26) k z k L ( l ,l ) = o (1) λ . Thus, using the first estimation in (4.16), (4.25) and (4.26), we obtain the first estimation in Lemma 4.2.Now, from Equation (4.13) we have iλy − z = λ − α f , then k y k H ( l ,l ) ≤ λ k z k H ( l ,l ) + 1 λ − α k f k H ( l ,l ) , using the fact that α ∈ (0 , k f k H ( l ,l ) = o (1), and the first estimation in Lemma 4.2, we obtain the secondestimation of Lemma 4.2. (cid:3) Remark 4.4.
It is easy to see the existence of h ( x ) . For example, we can take h ( x ) = cos (cid:18) π ( l − x ) l − l (cid:19) andwe get that h ( l ) = − h ( l ) = 1 , h ∈ C ∞ ([0 , L ]) , | g ( x ) | ≤ and | g ′ ( x ) | ≤ πl − l . Lemma 4.3.
Assume that η >
0. The solution ( u, v, y, z, ω ) ∈ D ( A ) of system (4.11)-(4.15) satisfies thefollowing asymptotic behavior(4.27) Z l l | S x | dx = O ( λ ) . roof. In order to prove this Lemma, we define the function γ ∈ C ∞ ([0 , L ]), 0 ≤ γ ≤ , L ], such that γ = 1 on [ l , l ], and γ = 0 on (0 , l − ǫ ) ∪ ( l + ǫ, L ) for all ǫ >
0. Multiply Equation (4.14) by − γ ¯ S andintegrate over (0 , L ), we get(4.28) − ℜ iλ Z L zγ ¯ Sdx ! + ℜ Z L γ ′ S x ¯ Sdx ! + Z L γ | S x | dx = −ℜ λ − ℓ Z L f γ ¯ Sdx ! . Integrating the second term of the above equation by parts, we get(4.29) Z L γ | S x | dx = ℜ iλ Z L zγ ¯ Sdx ! + 12 Z L γ ′′ | S | dx − ℜ λ − ℓ Z L f γ ¯ Sdx ! . For the term ℜ iλ Z L zγ ¯ Sdx ! , and using the fact that z and S are uniformly bounded in L (0 , L ), we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℜ iλ Z L zγ ¯ Sdx !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ O ( λ ) . Using the fact that k S k L (0 ,L ) = O (1) and that k f k L (0 ,L ) = o (1) in the last integral of Equation (4.29), weget our desired result. (cid:3) Lemma 4.4.
Assume that η >
0. The solution ( u, v, y, z, ω ) ∈ D ( A ) of system (4.11)-(4.15) satisfies thefollowing asymptotic behavior(4.30) | y ( l ) | = o (1) λ , | y x ( l ) | = o (1) λ , | y ( l ) | = o (1) λ , and | y x ( l ) | = o (1) λ . Moreover,(4.31) 1 λ | y xxx ( l − ) | = o (1) λ − α and 1 λ | y xxx ( l +1 ) | = o (1) λ − α . Proof.
For the proof of (4.30). Since y, z ∈ H (0 , L ), Sobolev embedding theorem implies that y, z ∈ C [0 , L ].Then, using the second estimation in Lemma 4.2 we get (4.30).Define(4.32) J ( z )( x ) = 1 λ − α Z Lx Z Ls z ( τ ) dτ ds and(4.33) X = 1 iλ [ − S + J ( f )] . From (4.13) and (4.33), we get(4.34) X xx = z. From (4.33), and using the fact that k f k L (0 ,L ) = o (1), we have that(4.35) k λX k L ( l ,l ) ≤ k S k L ( l ,l ) + k J ( f ) k L ( l ,l ) ≤ o (1) λ − α . It follows that,(4.36) k X k L ( l ,l ) = o (1) λ − α . Using Equation (4.34) and Lemma 4.2 we get(4.37) k X xx k L ( l ,l ) = o (1) λ , k X xxx k L ( l ,l ) = o (1) λ , and k X xxxx k L ( l ,l ) = o (1) λ . Now, using the interpolation inequality theorem [52], we get(4.38) k X x k L ( l ,l ) = o (1) λ − α .
33y using Equations (4.36)-(4.38), we get(4.39) k X k H ( l ,l ) = o (1) λ . From the interpolation inequality (see Theorem 4.17 in [2]), we have(4.40) k λ / X k H ( l ,l ) . k λ / X k H ( l ,l ) · k λ / X k L ( l ,l ) = o (1) λ − α . We note that S = by xx on (0 , l ) ∪ ( l , L ). Also, we have that y ∈ H (0 , l ) and y ∈ H ( l , L ).From (4.33), we have(4.41) by xxx ( l − ) = ( J ( f ) − iλX ) x ( l ) and by xxx ( l +1 ) = ( J ( f ) − iλX ) x ( l )Dividing Equation (4.41) by λ , and using Equation (4.40) and the fact that k f k L (0 ,L ) = o (1), we get Equation(4.31). Thus, the proof of the Lemma is complete. (cid:3) Lemma 4.5.
Assume that η >
0. The solution ( u, v, y, z, ω ) ∈ D ( A ) of system (4.11)-(4.15) satisfies thefollowing asymptotic behavior for every h ∈ C ([0 , L ]) and h (0) = h ( L ) = 0, and for every g ∈ C [ − L, Z L h ′ | z | dx + 2 b Z L h ′ | y xx | dx + b Z l h ′ | y xx | dx + b Z Ll h ′ | y xx | dx + ℜ Z L h ′′ S ¯ y x dx ! −ℜ Z l l hS x ¯ y xx dx ! − bh ( l ) | y xx ( l − ) | + bh ( l ) | y xx ( l +1 ) | = o (1) λ − α and(4.43) Z − L g ′ (cid:0) | v | + a | u x | dx (cid:1) − g (0) | v (0) | − ag (0) | u x (0) | + ag ( − L ) | u x ( − L ) | = o (1) λ − α . Proof.
Multiply Equation (4.14) by 2 h ¯ y x and integrate over (0 , L ) we get(4.44) ℜ iλ Z L zh ¯ y x dx ! + ℜ Z L hS xx ¯ y x dx ! = ℜ λ − α Z L f h ¯ y x dx ! . From Equation (4.13) we have iλ ¯ y x = − ¯ v x − λ − α ( ¯ f ) x . Then(4.45) ℜ iλ Z L zh ¯ y x dx ! = Z L h ′ | z | dx − ℜ λ − α Z L hz ( ¯ f ) x dx ! . Estimation of the term ℜ λ − α Z L hz ( ¯ f ) x dx ! . Using Cauchy-Schwarz inequality, the definition of h , thefact that k f k H (0 ,L ) = o (1), and that z is uniformly bounded in L (0 , L ), we get(4.46) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℜ λ − α Z L hz ( ¯ f ) x dx !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . | λ | − α Z L | z | dx ! / Z L | ( f ) x | dx ! / = o (1) λ − α . Then, using Equation (4.46) in Equation (4.45) we get(4.47) ℜ iλ Z L zh ¯ y x dx ! = Z L h ′ | z | dx + o (1) λ − α . Estimation of the term ℜ λ − α Z L f h ¯ y x dx ! . Using Cauchy-Schwarz inequality, the definition of h, thefact that k f k L (0 ,L ) = o (1), and that k y x k L (0 ,L ) . k y xx k L (0 ,L ) = O (1), we get(4.48) ℜ λ − α Z L f h ¯ y x dx ! . λ − α Z L | f | dx ! / Z L | y xx | dx ! / ≤ o (1) λ − α . ℜ Z L hS xx ¯ y x dx ! = ℜ Z L h ′′ S ¯ y x dx ! + ℜ Z L h ′ S ¯ y xx dx ! − ℜ Z L hS x ¯ y xx dx ! . For the term ℜ Z L h ′ S ¯ y xx dx ! , we have(4.50) ℜ Z L h ′ S ¯ y xx dx ! = 2 b Z L h ′ | y xx | dx + ℜ Z L h ′ ¯ y xx p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ ) dξdx ! . Estimation of the term ℜ Z L h ′ ¯ y xx p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ ) dξdx ! . Using Cauchy-Schwarz inequality,the definition of the functions d ( x ) and h , and using Lemma 4.1, we get(4.51) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℜ Z L h ′ ¯ y xx p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ ) dξdx !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = o (1) λ − α . This yields that,(4.52) ℜ Z L h ′ S ¯ y xx dx ! ≤ b Z L | y xx | dx + o (1) λ − α . For the term −ℜ Z L hS x ¯ y xx dx ! . We have(4.53) −ℜ Z L hS x ¯ y xx dx ! = −ℜ b Z l hy xxx ¯ y xx dx + 2 b Z Ll hy xxx ¯ y xx dx ! − ℜ Z l l hS x ¯ y xx dx ! . Integrating the above Equation by parts, we get(4.54) −ℜ Z L hS x ¯ y xx dx ! = b Z l h ′ | y xx | dx + b Z Ll h ′ | y xx | dx − bh ( l ) | y xx ( l ) | + bh ( l ) | y xx ( l ) | − ℜ Z l l hS x ¯ y xx dx ! . Then, substiting Equation (4.52) and (4.54) in Equation (4.49), we get(4.55) ℜ Z L hS xx ¯ y x dx ! = 2 b Z L | y xx | dx + b Z l h ′ | y xx | dx + b Z Ll h ′ | y xx | dx − bh ( l ) | y xx ( l ) | + bh ( l ) | y xx ( l ) | − ℜ Z l l hS x ¯ y xx dx ! + ℜ Z L h ′′ S ¯ y x dx ! + o (1) λ − α . Therefore, substituting Equations (4.47), (4.48), and (4.55) into (4.44), we get our desired result.Now, we will prove Equation (4.43). For this aim, multiply Equation (4.12) by 2 g ¯ u x and integrate over ( − L, ℜ (cid:18) iλ Z − L vg ¯ u x dx (cid:19) − ℜ (cid:18) a Z − L gu xx ¯ u x dx (cid:19) = ℜ (cid:18) λ − α Z − L f g ¯ u x dx (cid:19) . We have iλ ¯ u x = − ¯ v x − λ − α ( ¯ f ) x . Then,(4.57) ℜ (cid:18) iλ Z − L vg ¯ u x dx (cid:19) = Z − L g ′ | v | dx − g (0) | v (0) | − ℜ (cid:18) λ − α Z − L vg ( ¯ f ) x dx (cid:19) . ℜ (cid:18) λ − α Z − L vg ( ¯ f ) x dx (cid:19) . Using Cauchy-schwarz inequality, k f k H L ( − L, = o (1),and the fact that v is bounded in L (0 , L ), we get(4.58) (cid:12)(cid:12)(cid:12)(cid:12) ℜ (cid:18) λ − α Z − L vg ( ¯ f ) x dx (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ o (1) λ − α . Then, using the above equation, (4.57) becomes(4.59) ℜ (cid:18) iλ Z − L vg ¯ u x dx (cid:19) = Z − L g ′ | v | dx − g (0) | v (0) | + o (1) λ − α . Integrating the second term of Eqaution (4.56), we get(4.60) − ℜ (cid:18) a Z − L gu xx ¯ u x dx (cid:19) = a Z − L g ′ | u x | dx − g (0) | u x (0) | + ag ( − L ) | u x ( − L ) | . Estimation of the term ℜ (cid:18) λ − α Z − L f g ¯ u x dx (cid:19) . Using Cauchy-Schwarz inequality, k f k L ( − L, = o (1), andthe fact that u x is bounded in L ( − L, (cid:12)(cid:12)(cid:12)(cid:12) ℜ (cid:18) λ − α Z − L f g ¯ u x dx (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = o (1) λ − α . Thus, summing Equations (4.59), (4.60) and (4.61) we obtain our desired result. (cid:3)
Lemma 4.6.
Assume that η >
0. The solution ( u, v, y, z, ω ) ∈ D ( A ) of system (4.11)-(4.15) satisfies thefollowing asymptotic behavior(4.62) | λy (0) | = O (1) , | y xx ( l − ) | = O (1) and | y xx ( l +1 ) | = O (1) . Proof.
Take g ( x ) = x + L in Equation (4.43), we get(4.63) Z − L (cid:0) | v | + a | u x | dx (cid:1) − L | v (0) | − aL | u x (0) | = o (1) λ − α . From the above Equation and using the fact that v and u x are uniformly bounded in L ( − L, | v (0) | = O (1) and | u x (0) | = O (1) . From Equation (4.13) we have iλu = v + λ − α f . Then, using (4.64) and the fact that k f k L ( − L, = o (1), we get(4.65) | λu (0) | ≤ | v (0) | + λ − α | f (0) | ≤ | v (0) | + λ − α √ L k f k L ( − L, = O (1) . From the continuity condition ( u (0) = y (0)), we deduce that | λy (0) | = O (1). In order to prove the second termin the Equation (4.62) we proceed as follows. From Equation (4.13) we have z = iλy − λ − α f . Substituting the above Equation into Equation (4.14), we get(4.66) − λ y + by xxxx = F on (0 , l ) ∪ ( l , L ) , where F = λ − α ( f + iλf ).Multiply Equation (4.66) by 2 ζ ¯ y x , where ζ ∈ C [0 , l ], ζ (0) = ζ ′ ( l ) = 0 and ζ ( l ) = 1, and integrate over(0 , l ), we get(4.67) − ℜ λ Z l ζy ¯ y x dx ! + ℜ b Z l ζy xxxx ¯ y x dx ! = ℜ λ − α Z l ( f + iλf ) ζ ¯ y x dx ! . ℜ λ Z l ζy ¯ y x dx ! . Integrating by parts and using Equation (4.30), we get(4.68) − ℜ λ Z l ζy ¯ y x dx ! = Z l ζ ′ | λy | dx + o (1) λ . Integrating by parts the second term in Equation (4.67), and using Lemma 4.4, we get(4.69) ℜ b Z l ζy xxxx ¯ y x dx ! = −ℜ b Z l ζ ′ y xxx ¯ y x dx ! − ℜ b Z l ζy xxx ¯ y xx dx ! + ℜ (cid:0) by xxx ( l − )¯ y x ( l ) (cid:1) = ℜ b Z l ζ ′′ y xx ¯ y x dx ! + 3 b Z l ζ ′ | y xx | dx − bζ ( l ) | y xx ( l ) | + o (1) λ − α . Integrating by parts the last term of Equation (4.67) and using (4.30) and the fact that | f ( l ) | . k f k H R (0 ,L ) = o (1), we get(4.70) ℜ λ − α Z l ( f + iλf ) ζ ¯ y x dx ! = ℜ λ − α Z l f ζ ¯ y x dx ! − ℜ iλ − α Z l f ζ ′ λ ¯ ydx ! −ℜ iλ − α Z l ( f ) x ζλ ¯ ydx ! + o (1) λ − α . Substituting Equations (4.68), (4.69), and (4.70) in Equation (4.67), and using the fact that k U k H = 1, k y x k L (0 ,L ) ≤ c p k y xx k L (0 ,L ) = O (1) and k f k H R (0 ,L ) = o (1), we obtain our desired term. For the last term inEquation (4.62), we proceed in a similar way as above and thus the proof of the Lemma is complete. (cid:3) Lemma 4.7.
Assume that η >
0. The solution ( u, v, y, z, ω ) ∈ D ( A ) of system (4.11)-(4.15) satisfies thefollowing asymptotic behavior(4.71) | y xx ( l − ) | = o (1) λ and | y xx ( l +1 ) | = o (1) λ . Proof.
Equation (4.66) can be written as(4.72) (cid:18) ∂∂x − iµ (cid:19) (cid:18) ∂∂x + iµ (cid:19) (cid:18) ∂ ∂ x − µ (cid:19) y = 1 b F , on (0 , l ) ∪ ( l , L ) . where µ = s λ √ b .On the interval (0 , l ):Let Y l = (cid:18) ∂∂x + iµ (cid:19) (cid:18) ∂ ∂x − µ (cid:19) y . Solving on the interval (0 , l ) the following Equation (cid:18) ∂∂x − iµ (cid:19) Y l = 1 b F we get(4.73) Y l = K e iµ ( x − l ) + 1 b Z xl e iµ ( x − z ) F ( z ) dz, where K = y xxx ( l − ) − µ y x ( l ) + iµy xx ( l − ) − iµ y ( l ) . Let Y l = (cid:18) ∂ ∂x − µ (cid:19) y . We will solve the following differential equation(4.74) (cid:18) ∂∂x + iµ (cid:19) Y l = Y l .
37y using the solution Y l , we obtain the solution of the differential equation (4.74)(4.75) Y l = K e − iµ ( x − l ) + K µ sin(( x − l ) µ ) + 1 bµ Z xl Z sl e iµ (2 s − x − z ) F ( z ) dzds. Integrating by parts the last term of the above Equation, we get1 bµ Z xl Z sl e iµ (2 s − x − z ) F ( z ) dzds = 1 bµ Z xl sin(( x − z ) µ ) F ( z ) dz. Inserting the above Equation in (4.75), we get(4.76) Y l = K e − iµ ( x − l ) + K µ sin(( x − l ) µ ) + 1 bµ Z xl sin(( x − z ) µ ) F ( z ) dz where K = y xx ( l − ) − µ y ( l ) . Let Y l = (cid:18) ∂∂x − µ (cid:19) y . The solution of the following differential equation(4.77) (cid:18) ∂∂x + µ (cid:19) Y l = Y l is given by(4.78) Y l = K e − µ ( x − l ) + K µ (1 − i ) h e − iµ ( x − l ) − e − µ ( x − l ) i + 1 bµ Z xl Z sl e µ ( s − x ) sin(( s − z ) µ ) F ( z ) dzds − K µ h cos(( x − l ) µ ) − sin(( x − l ) µ ) − e − µ ( x − l ) i where K = y x ( l ) − µy ( l ) . Take x = 0 in (4.78) and multiply the equation by µe − µl , we get(4.79) r | y xx ( l − ) | ≤ | µy x (0) | e − µl + | µ y (0) | e − µl + | y x ( l ) | + | µy ( l ) | + 12 µ | y xxx ( l ) | (cid:2) e − µl + 1 (cid:3) + 12 | µ y ( l ) | (cid:0) e − µl + 1 (cid:1) + 12 | µy x ( l ) | (cid:0) e − µl + 1 (cid:1) + 1 √ | µ y ( l ) | (cid:0) e − µl + 1 (cid:1) + (cid:18) √ (cid:19) | y xx ( l − ) | e − µl + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b Z l Z sl e ( s − l ) µ sin(( s − z ) µ ) F ( z ) dzds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . For the integral in the above Equation, we have(4.80) 1 b Z l Z sl e µ ( s − l ) sin(( s − z ) µ ) F ( z ) dzds = 1 bλ − α Z l Z sl e µ ( s − l ) sin(( s − z ) µ ) f ( z ) dzds + 1 bλ ℓ Z l Z sl e µ ( s − l ) sin(( s − z ) µ ) iλf ( z ) dzds. Estimation of the first integral in the right side of Equation (4.80).(4.81) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) bλ ℓ Z l Z sl e µ ( s − l ) sin(( s − z ) µ ) f ( z ) dzds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ l / λ − α Z l | f ( z ) | dz = o (1) λ − α . | f (0) | . k ( f ) x k L (0 ,L ) = o (1) and | f ( l ) | . k ( f ) x k L (0 ,L ) = o (1) and k f k L (0 ,L ) = o (1), we get (4.82) (cid:12)(cid:12)(cid:12)(cid:12) bλ ℓ Z l Z sl e µ ( s − l ) sin(( s − z ) µ ) iλf ( z ) dzds (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) iλe − µl bλ ℓ Z l (cid:18)Z z e µs sin(( s − z ) µ ) ds (cid:19) f ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) iλe − µl bµλ ℓ Z l (cos( µz ) + sin( µz ) − e µz ) f ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ λ be µl λ ℓ µ (cid:20) Z l | f ( z ) | dz + (cid:12)(cid:12)(cid:12)(cid:12)Z l e µz f ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) ≤ λ be µl λ ℓ µ (cid:20) Z l | f ( z ) | dz + 1 µ ( f (0) + f ( l ) e µl )+ e µl µ Z l | f ′ ( z ) | dz (cid:21) = o (1) λ − α . Hence,(4.83) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b Z l Z sl e µ ( s − l ) sin(( s − z ) µ ) F ( z ) dzds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = o (1) λ − α . Since e √ ζ ≥ ζ , and taking ζ = λl √ b ∀ λ ∈ R ∗ + , and using Lemmas 4.4 and 4.6, and Equation (4.83) in Equation(4.79), we get that | y xx ( l − ) | = o (1) λ . On the interval ( l , L ):Proceeding with a similar computation as on (0 , l ), we get(4.84) Y l = K e − iµ ( x − l ) + K µ sin(( x − l ) µ ) + 1 bµ Z xl sin(( x − z ) µ ) F ( z ) dz where K = y xxx ( l +1 ) − µ y x ( l ) + iµy xx ( l +1 ) − iµ y ( l )and K = y xx ( l +1 ) − µ y ( l ) . We will solve the following differential Equation(4.85) (cid:18) ∂∂x − µ (cid:19) Y l = Y l where Y l = (cid:18) ∂∂x + µ (cid:19) y. The solution of (4.85) is(4.86) Y l = K e µ ( x − l ) − K i h e − iµ ( x − l ) − e µ ( x − l ) i − K µ h cos(( x − l ) µ ) + sin(( x − l ) µ ) − e µ ( x − l ) i + 1 bµ Z xl Z sl e µ ( x − s ) sin(( s − z ) µ ) F ( z ) dzds where K = y x ( l ) + µy ( l ) . Taking x = L in Equation (4.86) and multiplying by µe − µ ( L − l ) , we get(4.87) 1 √ | y xx ( l +1 ) | ≤ | µy x ( l ) | + | µ y ( l ) | + e − µ ( L − l ) √ | y xx ( l +1 ) | + 1 √ | µ y x ( l ) | (cid:16) e − µ ( L − l ) + 1 (cid:17) + 1 µ y xxx ( l +1 ) (cid:16) e − µ ( L − l ) + 1 (cid:17) + | µy x ( l ) | (cid:16) e − µ ( L − l ) + 1 (cid:17) + 2 | y xx ( l +1 ) | e − µ ( L − l ) + | µ y ( l ) | (cid:16) e − µ ( L − l ) + 1 (cid:17) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b Z Ll Z sl e − µ ( s − l ) sin(( s − z ) µ ) F ( z ) dzds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . b Z xl Z sl e − µ ( s − l ) sin(( s − z ) µ ) F ( z ) dzds = o (1) λ − α . Since e √ λ ( L − l ) > λ ( L − l ) , for all λ ∈ R ∗ + and using Lemmas 4.4, 4.6, and Equation (4.88) in (4.87), we get | y xx ( l +1 ) | = o (1) λ . Thus the proof of this Lemma is complete. (cid:3)
Lemma 4.8.
Assume that η >
0. The solution ( u, v, y, z, ω ) ∈ D ( A ) of system (4.11)-(4.15) satisfies thefollowing asymptotic behavior(4.89) Z L | z | dx = o (1) λ and b Z L | y xx | dx = o (1) λ . Proof.
Taking h ( x ) = xθ ( x ) in Equation (4.42), where θ ∈ C ([0 , L ]) is defined as follows(4.90) θ ( x ) = x ∈ [0 , l ] , x ∈ [ l , L ] , ≤ θ ≤ elsewhere. Estimation of the term ℜ Z L h ′′ S ¯ y x dx ! . Using Cauchy-Schwarz inequality, the definition of h , Lemmas 4.1and 4.2, we get(4.91) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℜ Z L h ′′ S ¯ y x dx !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = o (1) λ − α . Estimation of the term ℜ Z l l hS x ¯ y xx dx ! . Using Cauchy-Schwarz inequality, the definition of h , Lemmas4.1 and 4.3, we get(4.92) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℜ Z l l hS x ¯ y xx dx !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z l l | S x | dx ! / Z l l | y xx | dx ! / = o (1) λ . Using Lemma 4.7, Equations (4.91) and (4.92) in (4.42), we get(4.93) Z L h ′ | z | dx + 2 b Z L h ′ | y xx | dx + b Z l h ′ | y xx | dx + b Z Ll h ′ | y xx | dx = o (1) λ . By using the definition of h , we obtain(4.94) Z l | z | dx = o (1) λ and b Z l | y xx | dx = o (1) λ . Now, taking h ( x ) = xθ ( x ) in Equation (4.42), where θ ∈ C ([0 , L ]) is defined as follows(4.95) θ ( x ) = x ∈ [ l , L ] , x ∈ [0 , l ] , ≤ θ ≤ elsewhere. Proceeding in a similar way as above we get(4.96) Z Ll | z | dx = o (1) λ and b Z Ll | y xx | dx = o (1) λ . Therefore, using the third estimation of (4.16), Lemma 4.2 and combining Equations (4.94), and (4.96) we getour desired result. (cid:3)
Lemma 4.9.
Assume that η >
0. The solution ( u, v, y, z, ω ) ∈ D ( A ) of system (4.11)-(4.15) satisfies thefollowing asymptotic behavior(4.97) | y xxx (0) | = o (1) . roof. From the interpolation inequality Theorem (see [52]), and using the fact that y ∈ H (0 , l ), k f k L (0 ,l ) = o (1), Equation (4.14) and Lemma 4.8, we get(4.98) k y xxx k L (0 ,l ) . k y xxxx k / L (0 ,l ) · k y xx k / L (0 ,l ) + k y xx k L (0 ,l ) ≤ (cid:20) λ k z k L (0 ,l ) + 1 λ − α k f k L (0 ,l ) (cid:21) k y xx k L (0 ,l ) + k y xx k L (0 ,l ) = o (1) . Since y xxx ∈ H (0 , l ), then | y xxx (0) | ≤ k y xxx k H (0 ,l ) = o (1) . Thus, we obtained our desired result. (cid:3)
Lemma 4.10.
Assume that η >
0. The solution ( u, v, y, z, ω ) ∈ D ( A ) of system (4.11)-(4.15) satisfies thefollowing asymptotic behavior(4.99) Z − L (cid:0) | v | + a | u x | (cid:1) dx = o (1) . Proof.
Using the transmission condition and Lemma 4.9, we have(4.100) | u x (0) | = | y xxx (0) | = o (1) . Now, let q ∈ C ([0 , l ]) such that q ( l ) = q x ( l ) = 0, q (0) = 1. Multiply Equation (4.66) by q ¯ y x and integrateover (0 , l ), and using the fact that k f k H R (0 ,L ) = o (1), k f k L (0 ,L ) = o (1), and k y x k L (0 ,l ) ≤ k y x k L (0 ,L ) . k y xx k L (0 ,L ) = O (1) , we get(4.101) 12 Z l q ′ (cid:0) | λy | + 3 | y xx | (cid:1) dx + b Z l q ′′ y xx ¯ y x dx + 12 | λy (0) | − ℜ ( by xxx (0)¯ y x (0)) = o (1) λ − α . By using Lemmas 4.8, 4.9 and the fact that | y x (0) | . k y xx k L (0 ,L ) = O (1) and k y x k L (0 ,l ) ≤ k y x k L (0 ,L ) . k y xx k L (0 ,L ) = O (1) in Equation (4.101), we obtain(4.102) | λy (0) | = o (1) . From the transmission conditions we have | λu (0) | = | λy (0) | = o (1) . It follows from the above Equation and Equation (4.11) that(4.103) | v (0) | = o (1) . Thus, by taking g ( x ) = x + L in Equation (4.43), and using Equations (4.100) and (4.103) we get our desiredresult. (cid:3) Proof of Theorem 2.7.
From Lemmas 4.8 and 4.10, we get that k U k H = o (1), which contradicts (4.9).Consequently, condition (R ) holds. This implies, from Theorem 7.4, the energy decay estimation (4.8). Theproof is thus complete. Remark 4.5.
In this Remark, we show that the result in [31] can be improved. In [31] , Fathi considereda Euler-Bernoulli beam and wave equations coupled through transmission conditions. The damping is locallydistributed and acts on the wave equation, and the rotation vanishes at the connecting point ( y x ( ℓ ) = 0) . Thesystem is given in the left side of Equation (1.1) . He proved the polynomial stability with energy decay rate oftype t − . By using similar computations as in Section 2 by taking α = 1 , and by solving the ordinary differentialequations in Section 4 we can reach that | y xx ( ℓ ) | = o (1) . Thus, with the same technique of the proof of Section2, we can reach that energy of the system (the left system in Equation (1.1) ) of the mentioned paper satisfiesthe decay rate t − . (EBB) F KV
Model
In this section, we consider the Euler-Bernoulli beam with localized fractional Kelvin-Voigt damping. Westudy the polynomial stability of the system ((EBB)
F KV ). In Theorem 2.1, taking the input V ( x, t ) = p d ( x ) y xxt ( x, t ), then using (1.4), we get the output O is given by O ( x, t ) = p d ( x ) I − α,η y xxt ( x, t ) = p d ( x )Γ(1 − α ) Z t ( t − s ) − α e − η ( t − s ) ∂ s y xx ( x, s ) ds = p d ( x ) ∂ α,ηt y xx ( x, t ) . Then, we deduce that system ((EBB)
F KV ) can be recast into the following augmented model(5.1) y tt + (cid:18) by xx + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ, t ) dξ (cid:19) xx = 0 , ( x, t ) ∈ (0 , L ) × × R + ∗ ,ω t ( x, ξ, t ) + (cid:0) | ξ | + η (cid:1) ω ( x, ξ, t ) − p d ( x ) y xxt ( x, t ) | ξ | α − = 0 , ( x, ξ, t ) ∈ (0 , L ) × R × R + ∗ , with the boundary conditions(5.2) y (0 , t ) = y x (0 , t ) = 0 , y xx ( L, t ) = 0 , y xxx ( L, t ) = 0 , t ∈ (0 , ∞ ) , and with the following initial conditions(5.3) y ( x,
0) = y ( x ) , y t ( x,
0) = y ( x ) , ω ( x, ξ,
0) = 0 , x ∈ (0 , L ) , ξ ∈ R . The energy of the system (5.1)-(5.3) is given by E ( t ) = 12 Z L (cid:0) | y t | + b | y xx | (cid:1) dx + κ ( α )2 Z L Z R | ω ( x, ξ, t ) | dξdx. Lemma 5.1.
Let U = ( y, y t , ω ) be a regular solution of the System (5.1) - (5.3) . Then, the energy E ( t ) satisfiesthe following estimation (5.4) ddt E ( t ) = − κ ( α ) Z L Z R ( ξ + η ) | ω ( x, ξ, t ) | dξdx. Since α ∈ (0 , κ ( α ) >
0, and therefore ddt E ( t ) ≤
0. Thus, system (5.1)-(5.3) is dissipative in the sensethat its energy is a non-increasing function with respect to time variable t . Now, we define the following Hilbertenergy space H by H = H L (0 , L ) × L (0 , L ) × W, where W = L ((0 , L ) × R ) and H L (0 , L ) = (cid:8) y ∈ H (0 , L ); y (0) = y x (0) = 0 (cid:9) .The energy space H is equipped with the inner product defined by h U, U i H = Z L z ¯ z dx + b Z L y xx ( y ) xx dx + κ ( α ) Z L Z R ω ( x, ξ ) ω ( x, ξ ) dξdx, for all U = ( y, z, ω ) and U = ( y , z , ω ) in H . We use k U k H to denote the corresponding norm. We definethe unbounded linear operator A : D ( A ) ⊂ H → H by D ( A ) = U = ( y, z, ω ) ∈ H ; z ∈ H L (0 , L ) , (cid:18) by xx + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ, t ) dξ (cid:19) xx ∈ L (0 , L ) , − (cid:0) | ξ | + η (cid:1) ω ( x, ξ ) + p d ( x ) z xx | ξ | α − , | ξ | ω ( x, ξ ) ∈ W,y xx ( L ) = 0 , and y xxx ( L ) = 0 . , and for all U = ( y, z, ω ) ∈ D ( A ), A ( y, z, ω ) ⊤ = z − (cid:18) by xx + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ, t ) dξ (cid:19) xx − (cid:0) | ξ | + η (cid:1) ω ( x, ξ ) + p d ( x ) z xx | ξ | α − . U = ( y, y t , ω ) is a regular solution of system (5.1)-(5.3), then the system can be rewritten as evolutionequation on the Hilbert space H given by(5.5) U t = A U, U (0) = U , where U = ( y , y , A is m-dissipative in theenergy space H . Also, the C -semigroup of contractions e t A is strongly stable on H in the sense thatlim t → + ∞ (cid:13)(cid:13) e t A U (cid:13)(cid:13) H = 0. Theorem 5.2.
Assume that η > . The C − semigroup ( e t A ) t ≥ is polynomially stable; i.e. there existsconstant C > such that for every U ∈ D ( A ) , we have (5.6) E ( t ) ≤ C t − α k U k D ( A ) , t > , ∀ U ∈ D ( A ) . According to Theorem 7.4, by taking ℓ = 1 − α , the polynomial energy decay (5.6) holds if the followingconditions(P ) i R ⊂ ρ ( A ) , and(P ) sup λ ∈ R (cid:13)(cid:13) ( iλI − A ) − (cid:13)(cid:13) L ( H ) = O (cid:0) | λ | − α (cid:1) are satisfied. Since condition (P )is already satisfied (similar way as in Subsection 4.1), we still need to provecondition (P ). For this purpose we will use an argument of contradiction. Suppose that (P ) is false, thenthere exists (cid:8)(cid:0) λ n , U n := ( y n , z n , ω n ( · , ξ )) ⊤ (cid:1)(cid:9) ⊂ R ∗ × D ( A ) with(5.7) | λ n | → + ∞ and k U n k H = 1 , such that(5.8) (cid:0) λ − αn (cid:1) ( iλ n I − A ) U n = F n := ( f ,n , f ,n , f ,n ( · , ξ )) ⊤ → H . For simplicity, we drop the index n . Equivalently, from (5.8), we have iλy − z = f λ − α in H L (0 , L ) , (5.9) iλz + S xx = f λ − α in L (0 , L ) , (5.10) ( iλ + ξ + η ) ω ( x, ξ ) − p d ( x ) z xx | ξ | α − = f ( x, ξ ) λ − α in W, (5.11)where S = by xx + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ ) dξ . Here we will check the condition (P ) by finding a contra-diction with (5.7) by showing k U k H = o (1). In order to reach this contradiction, we proceed in a similar wayas in Section 4.2. We give the estimation results directly considering that the proof of these results can followusing similar computations as in Section 4.2.Assume that η >
0. Then, the solution ( y, z, ω ) ∈ D ( A ) of system (5.1)-(5.3) satisfies the asymptotic behaviorestimations mentioned below.Similar to Lemma 4.1, we obtain(5.12) Z L Z R (cid:0) | ξ | + η (cid:1) | ω ( x, ξ ) | dξdx = o (1) λ − α , Z l l | z xx | dx = o (1) and Z l l | y xx | dx = o (1) λ . Similar to Lemma 4.1, we obtain(5.13) Z l l | S | dx = o (1) λ − α . Similar to Lemma 4.2, we get(5.14) k z k H ( l ,l ) = o (1) . | y ( l ) | = o (1) λ , | y x ( l ) | = o (1) λ , | y ( l ) | = o (1) λ , | y x ( l ) | = o (1) λ and(5.16) 1 λ | y xxx ( l − ) | = o (1) λ − α and 1 λ | y xxx ( l +1 ) | = o (1) λ − α . Similar to Lemma 4.7, we obtain(5.17) | y xx ( l − ) | = o (1) and | y xx ( l +1 ) | = o (1) . Finally, by proceeding in a similar way as in Lemma 4.8, we reach our desired result(5.18) Z L | z | dx = o (1) and b Z L | y xx | dx = o (1) Proof of Theorem 5.7.
From the first estimation in Equation (5.12) and Equation (5.18), we get that k U k H = o (1), which contradicts (5.7). Consequently, condition (P ) holds. This implies, from Theorem 7.4,the energy decay estimation (5.6). The proof is thus complete.6. (EBB)-(EBB) F KV
Model
In this section, we consider a system of two Euler-Bernoulli beam equation coupled via boundary connectionswith a localized non-regular fractional Kelvin-Voigt damping acting on one of the two equations only. In thispart, we study the polynomial stabilty of the system.In Theorem 2.1, taking the input V ( x, t ) = p d ( x ) y xxt ( x, t ), then using (1.4), we get the output O is given by O ( x, t ) = p d ( x ) I − α,η y xxt ( x, t ) = p d ( x )Γ(1 − α ) Z t ( t − s ) − α e − η ( t − s ) ∂ s y xx ( x, s ) ds = p d ( x ) ∂ α,ηt y xx ( x, t ) . Then, we deduce that system ((EBB)-(EBB)
F KV ) can be recast into the following augmented model(6.1) u tt + au xxxx = 0 , ( x, t ) ∈ ( − L, × R + ∗ ,y tt + (cid:18) by xx + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ, t ) dξ (cid:19) xx = 0 , ( x, t ) ∈ (0 , L ) × R + ∗ ,ω t ( x, ξ, t ) + (cid:0) | ξ | + η (cid:1) ω ( x, ξ, t ) − p d ( x ) y xxt ( x, t ) | ξ | α − = 0 , ( x, ξ, t ) ∈ (0 , L ) × R × R + ∗ , with the following transmission and boundary conditions(6.2) u ( − L, t ) = u x ( − L, t ) = y ( L, t ) = y x ( L, t ) = 0 , t ∈ (0 , ∞ ) ,au xxx (0 , t ) − by xxx (0 , t ) = 0 , u xx (0 , t ) = y xx (0 , t ) = 0 , t ∈ (0 , ∞ ) ,u (0 , t ) = y (0 , t ) , t ∈ (0 , ∞ ) , and with the following initial conditions(6.3) u ( x,
0) = u ( x ) , u t ( x,
0) = u ( x ) x ∈ ( − L, y ( x,
0) = y ( x ) , y t ( x,
0) = y ( x ) , ω ( x, ξ,
0) = 0 x ∈ (0 , L ) , ξ ∈ R . The energy of the system (6.1)-(6.3) is given by E ( t ) = 12 Z − L (cid:0) | u t | + a | u xx | (cid:1) dx + 12 Z L (cid:0) | y t | + b | y xx | (cid:1) dx + κ ( α )2 Z L Z R | ω ( x, ξ, t ) | dξdx. By similar computation to Lemma 2.2, it is easy to see that the energy E ( t ) satisfies the following estimation(6.4) ddt E ( t ) = − κ ( α ) Z L Z R ( ξ + η ) | ω ( x, ξ, t ) | dξdx. Since α ∈ (0 , κ ( α ) >
0, and therefore ddt E ( t ) ≤
0. Thus, system (6.1)-(6.3) is dissipative in the sense44hat its energy is a non-increasing function with respect to time variable t . Now, we define the following Hilbertenergy space H by H = (cid:8) ( u, v, y, z, ω ) ∈ H L ( − L, × L ( − L, × H R (0 , L ) × L (0 , L ) × W ; u (0) = y (0) (cid:9) , where W = L ((0 , L ) × R ) and(6.5) ( H L ( − L,
0) = { u ∈ H ( − L, u ( − L ) = u x ( − L ) = 0 } ,H R (0 , L ) = { y ∈ H (0 , L ); y ( L ) = y x ( L ) = 0 } . The energy space H is equipped with the inner product defined by h U, U i H = Z − L v ¯ v dx + a Z − L u xx ( u ) xx dx + Z L z ¯ z dx + b Z L y xx ( y ) xx dx + κ ( α ) Z L Z R ω ( x, ξ ) ω ( x, ξ ) dξdx, for all U = ( u, v, y, z, ω ) and U = ( u , v , y , z , ω ) in H . We use k U k H to denote the corresponding norm.We define the unbounded linear operator A : D ( A ) ⊂ H → H by D ( A ) = U = ( u, v, y, z, ω ) ∈ H ; ( v, z ) ∈ H L ( − L, × H R (0 , L ) , u ∈ H ( − L, , (cid:18) by xx + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ ) dξ (cid:19) xx ∈ L (0 , L ) , − (cid:0) | ξ | + η (cid:1) ω ( x, ξ ) + p d ( x ) z xx | ξ | α − , | ξ | ω ( x, ξ ) ∈ W,au xxx (0) − by xxx (0) = 0 , u xx (0) = y xx (0) = 0 , and v (0) = z (0) , and for all U = ( u, v, y, z, ω ) ∈ D ( A ), A ( u, v, y, z, ω ) ⊤ = v − au xxxx z − (cid:18) by xx + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ ) dξ (cid:19) xx − (cid:0) | ξ | + η (cid:1) ω ( x, ξ ) + p d ( x ) z xx | ξ | α − . If U = ( u, u t , y, y t , ω ) is a regular solution of system (6.1)-(6.3), then the system can be rewritten as evolutionequation on the Hilbert space H given by(6.6) U t = A U, U (0) = U , where U = ( u , u , y , y , A is m-dissipative on H , consequently it generates a C -semigroupof contractions ( e t A ) t ≥ following Lummer-Phillips theorem (see in [46] and [53]). Then the solution of theevolution Equation (6.6) admits the following representation U ( t ) = e t A U , t ≥ , which leads to the well-posedness of (6.6). Hence, we have the following result. Theorem 6.1.
Let U ∈ H , then problem (4.7) admits a unique weak solution U satisfies U ( t ) ∈ C (cid:0) R + , H (cid:1) . Moreover, if U ∈ D ( A ) , then problem (4.7) admits a unique strong solution U satisfies U ( t ) ∈ C (cid:0) R + , H (cid:1) ∩ C (cid:0) R + , D ( A ) (cid:1) . Theorem 6.2.
Assume that η > . The C − semigroup ( e t A ) t ≥ is polynomially stable; i.e. there existsconstant C > such that for every U ∈ D ( A ) , we have (6.7) E ( t ) ≤ C t − α k U k D ( A ) , t > , ∀ U ∈ D ( A ) . ℓ = 2 − α , the polynomial energy decay (6.7) holds if the followingconditions(Q ) i R ⊂ ρ ( A ) , and(Q ) sup λ ∈ R (cid:13)(cid:13) ( iλI − A ) − (cid:13)(cid:13) L ( H ) = O (cid:0) | λ | − α (cid:1) are satisfied. Since condition (Q )is already proved (see Subsection 4.1), we still need to prove condition(Q ). For this purpose we will use an argument of contradiction. Suppose that (Q ) is false, then there exists (cid:8)(cid:0) λ n , U n := ( u n , v n , y n , z n , ω n ( · , ξ )) ⊤ (cid:1)(cid:9) ⊂ R ∗ × D ( A ) with(6.8) | λ n | → + ∞ and k U n k H = k ( u n , v n , y n , z n , ω n ( · , ξ )) k H = 1 , such that(6.9) (cid:0) λ − αn (cid:1) ( iλ n I − A ) U n = F n := ( f ,n , f ,n , f ,n , f ,n , f ,n ( · , ξ )) ⊤ → H . For simplicity, we drop the index n . Equivalently, from (6.9), we have iλu − v = f λ − α in H L ( − L, , (6.10) iλv + au xxxx = f λ − α in L ( − L, , (6.11) iλy − z = f λ − α in H R (0 , L ) , (6.12) iλz + S xx = f λ − α in L (0 , L ) , (6.13) ( iλ + ξ + η ) ω ( x, ξ ) − p d ( x ) z xx | ξ | α − = f ( x, ξ ) λ − α in W, (6.14)where S = by xx + p d ( x ) κ ( α ) Z R | ξ | α − ω ( x, ξ ) dξ . Here we will check the condition (Q ) by finding a contra-diction with (6.8) by showing k U k H = o (1). We need to prove several asymptotic behavior estimations for thesolution to obtain this contradiction. Here, we give the estimations directly since the proof can be done in asmiliar way as in Subsection 4.2. Assume that η >
0. Similar to Lemma 4.1, the solution ( u, v, y, z, ω ) ∈ D ( A )of system (6.10)-(6.14) satisfies the following asymptotic behavior estimations(6.15) Z L Z R (cid:0) | ξ | + η (cid:1) | ω ( x, ξ ) | dξdx = o (1) λ − α , Z l l | z xx | dx = o (1) λ , Z l l | y xx | dx = o (1) λ and Z l l | S | dx = o (1) λ − α . Similar to Lemma 4.2, we have that the solution of the system (6.8)-(6.14) satisfies(6.16) k z k H ( l ,l ) = o (1) λ and k y k H ( l ,l ) = o (1) λ . Similar to Lemma 4.3, we obtain(6.17) Z l l | S x | dx = O ( λ ) . Similar to Lemma 4.4, we get that the solution ( u, v, y, z, ω ) ∈ D ( A ) of system (6.10)-(6.14) satisfies thefollowing asymptotic behavior(6.18) | y ( l ) | = o (1) λ , | y x ( l ) | = o (1) λ , | y ( l ) | = o (1) λ , and | y x ( l ) | = o (1) λ . Moreover,(6.19) 1 λ | y xxx ( l − ) | = o (1) λ − α and 1 λ | y xxx ( l +1 ) | = o (1) λ − α . h ¯ y x , where h ∈ C ([0 , L ]) and h (0) = h ( L ) = 0 andproceeding as Lemma 4.5, we obtain(6.20) Z L h ′ | z | dx + 2 b Z L h ′ | y xx | dx + b Z l h ′ | y xx | dx + b Z Ll h ′ | y xx | dx + ℜ Z L h ′′ S ¯ y x dx ! −ℜ Z l l hS x ¯ y xx dx ! − bh ( l ) | y xx ( l − ) | + bh ( l ) | y xx ( l +1 ) | = o (1) λ − α Similar to Lemma (4.6), we get that(6.21) | y xx ( l − ) | = O (1) and | y xx ( l +1 ) | = O (1) . Similar to Lemma 4.7, we obtain that the solution ( u, v, y, z, ω ) ∈ D ( A ) of system (6.10)-(6.14) satisfies thefollowing asymptotic behavior(6.22) | y xx ( l − ) | = o (1) λ and | y xx ( l +1 ) | = o (1) λ . Similar to Lemma 4.8, we have that the solution ( u, v, y, z, ω ) ∈ D ( A ) of system (6.10)-(6.14) satisfies thefollowing asymptotic behavior(6.23) Z L | z | dx = o (1) λ and b Z L | y xx | dx = o (1) λ . Similar to Lemma 4.9, we reach that(6.24) | y xxx (0) | = o (1) . Using the transmission condition ( au xxx (0) − by xxx (0) = 0) and Equation (6.24), we obtain(6.25) | u xxx (0) | = o (1) . Similar the computations in Equations (4.101) and using the fact that | y x (0) | . k y xx k L (0 ,L ) = O (1), we obtain(6.26) | λy (0) | = o (1) . By using the transmission conditions and Equation (6.26), we get(6.27) | λu (0) | = | λy (0) | = o (1) . Substituting v = iλu − λ − α into Equation (6.11), we get(6.28) − λ u + au xxxx = λ − α f + iλ − α f . Now, multiply Equation (6.28) by 2( x + L ) ¯ u x and integrate over ( − L, k f k H L ( − L, = o (1), k f k L (0 ,L ) = o (1), | u x (0) | ≤ k u xx k L ( − L, = O (1), k u x k L ( − L, ≤ c p k u xx k L ( − L, = O (1) , we get(6.29) Z − L (cid:0) | λu | + 3 | u xx | (cid:1) dx − L | λy (0) | + ℜ (2 aLu xxx (0)¯ u x (0)) = o (1) λ − α . By using equations (6.25) and (6.27), and the fact that | u x (0) | . k u xx k L ( − L, = O (1) in Equation (6.29), weobtain(6.30) Z − L | v | dx = o (1) and a Z − L | u xx | dx = o (1) . Proof of Theorem 6.8.
From the first estimation in Equation (6.15) and Equations (6.23) and (6.30), we getthat k U k H = o (1), which contradicts (6.8). Consequently, condition (Q ) holds. This implies, from Theorem7.4, the energy decay estimation (6.7). The proof is thus complete.47. Appendix
In this section, we introduce the notions of stability that we encounter in this work.
Definition 7.1.
Assume that A is the generator of a 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 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 [14]. Theorem 7.2.
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, 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 [34] and Pr¨uss [55]. Theorem 7.3.
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 [21] (see also [44] and [19]). Theorem 7.4.
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 > the following conditions are equivalent (7.1) 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) , (7.2) k e tA U k H ≤ Ct ℓ k U k D ( A ) , ∀ t > , U ∈ D ( A ) , for some C > . Conclusion
We have studied the stabilization of five models of systems. We considered a Euler-Bernoulli beam equationand a wave equation coupled through boundary connections with a localized non-regular fractional Kelvin-Voigt damping that acts through the wave equation only. We proved the strong stability of the system usingArendt-Batty criteria. In addition, we established a polynomial energy decay rate of type t − − α . Also, weconsidered two wave equations coupled via boundary connections with localized non-smooth fractional Kelvin-Voigt damping. We showed a polynomial energy decay rate of type t − − α . Moreover, we studied the systemof Euler-Bernoulli beam and wave equations coupled through boundary connections where the dissipation actsthrough the beam equation. We proved a polynomial energy decay rate of type t − − α . In addition, we considered48he Euler-Bernoulli beam alone with the same localized non-smooth damping. We established a polynomialenergy decay rate of type t − − α . In the last model, we studied the polynomial stability of a system of twoEuler-Bernoulli beam equations coupled through boundary conditions with a localized non-regular fractionalKelvin-Voigt damping acting only on one of the two equations. We reached a polynomial energy decay rate oftype t − − α . Acknowledgements
Mohammad Akil would like to thank LAMA laboratory of Mathematics of the Universit´e Savoie Mont Blancfor their supports.Ibtissam Issa would like to thank the Lebanese University for its funding and LAMA laboratory of Mathematicsof the Universit´e Savoie Mont Blanc for their 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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