Optimal Polynomial Decay to Coupled Wave Equations and Its Numerical Properties

whereΩ is an open bounded set ofR with smooth boundary Γ. Here α is a small real positive constant. This problem is motivated by an analogous problem in ordinary differential equations for coupled oscillators and has potential application in isolation of objects from outside disturbances. As an example in engineering, rubber-like materials are used to absorb vibration or shield structures from vibration.Modeling of structures such as beam or plates sandwiched with rubber or similar materials will lead to equations similar to those (1)-(2). The exponential stability of coupled system of wave equations has been studied by Najafi et al. [1] in the case of two linear boundary feedback and by Komornik Rao [2] in the case of two nonlinear boundary feedback. We should mention that this initial boundary value problem (1)–(4) has been studied by many authors. For example, in [3] Alabau Boussoira proved that the energy of associated coupled weakly dissipative system decays polynomially as t and this decay rate can improve since the initial data aremore regular. In [4] Alabau Boussoira et al. showed the decay rates for an abstract coupled weakly dissipative system of using the same ideas introduced in [3]. For the case of the system (1)– (4) we can also cite the work of Santos et al. [5] where the authors proved t polynomial decay and this decay rate can improve since the initial data are more regular. The question we are interested in is to study optimal polynomial decay rates to the system (1)–(4). That is, we prove that the associated semigroup decays at a rate t and it is optimal. This work improves the results in [3–5]. Our result on the polynomial stability is based onTheorem 2.4 in [6]. Naturally, the system (1)–(4) is dissipative. Indeed, the energy of the solutions defined by


Introduction
This paper is concerned with the polynomial stability and the optimal rate of decay of the  0 -semigroup of the coupled system of two wave equations: ( (, 0) , V (, 0)) = ( 0 , V 0 ) , where Ω is an open bounded set of R  with smooth boundary Γ.Here  is a small real positive constant.This problem is motivated by an analogous problem in ordinary differential equations for coupled oscillators and has potential application in isolation of objects from outside disturbances.As an example in engineering, rubber-like materials are used to absorb vibration or shield structures from vibration.Modeling of structures such as beam or plates sandwiched with rubber or similar materials will lead to equations similar to those (1)- (2).
The exponential stability of coupled system of wave equations has been studied by Najafi et al. [1] in the case of two linear boundary feedback and by Komornik Rao [2] in the case of two nonlinear boundary feedback.
We should mention that this initial boundary value problem (1)-( 4) has been studied by many authors.For example, in [3] Alabau Boussoira proved that the energy of associated coupled weakly dissipative system decays polynomially as  −1 and this decay rate can improve since the initial data are more regular.In [4] Alabau Boussoira et al. showed the decay rates for an abstract coupled weakly dissipative system of using the same ideas introduced in [3].For the case of the system (1)- (4) we can also cite the work of Santos et al. [5] where the authors proved  −1 polynomial decay and this decay rate can improve since the initial data are more regular.The question we are interested in is to study optimal polynomial decay rates to the system (1)- (4).That is, we prove that the associated semigroup decays at a rate  −1/2 and it is optimal.This work improves the results in [3][4][5].Our result on the polynomial stability is based on Theorem 2.4 in [6].

Journal of Applied Mathematics
To verify the asymptotic behavior of systems like (1)-( 4), only the work of Najafi [7] treated numerically the problem of lack of exponential decay to a system of two coupled wave equations.Here, we use two numerical approaches: one is in finite differences and in the other one we use the spectral methods.In particular, we explore the properties of numerical energy associated with a particular discretization in finite differences.
We conclude this introduction with an outline of the paper.In Section 2, we show the existence of solutions for the system (1)-( 4).In the Section 3 we showed that the system (1)-( 4) is polynomially stable with optimal decay rate.Section 4 is dedicated with the numerical aspects.We derive a numerical energy and we show that it preserves the conservation and dissipation laws.We use also spectral methods to see the behavior of the numerical eigenvalues when the damping is weak.Finally, in Section 5, we finished this work with a conclusion.

The Semigroup Setting
In this section we will study the existence and uniqueness of strong and global solutions for the system (1)-( 4) using the semigroup techniques.

Polynomial Decay and Optimal Result
To study the asymptotic behavior of the semigroup associated with ( 1)-( 4), let us consider the spectral problem: where lim Theorem 2. Let () be  0 -semigroup of contractions generated by A. Then it follows that () is not exponentially stable.
Proof.Our main tool is to use Prüss' result [9], which states that a semigroup  A is exponentially stable if and only if the following conditions hold: This way, let us consider the resolvent equation that is, Let us take  1 =  3 = 0 and  2 =  4 =  ] .We look for solution of the form  =  ] , V =  ] ,  =  ] , and  =  ] , with , , ,  ∈ C. From ( 19)-(21), we get  and  satisfying Now, choosing  = √ ] + , and using the above equation, we obtain  = 2, so we have Therefore, we have Now we claim that In fact, using ( 7) and ( 24) and noting that lim we conclude that Recalling that it follows from ( 27) and ( 16) that () is not exponentially stable.
In order to show the polynomial decay from semigroup associated with the system (1)-( 4), first let us consider the product in H of  = (, , V, )  ∈  (A) with the resolvent equation of A; that is, Now, noting that we get Then taking the real part, we obtain Lemma 3. The strong solution of the system (1)-( 4) given by Theorem 1 satisfies where  is a positive constant.
Proof.Multiplying equalities ( 19) and ( 21) by  and V, respectively, integrating on Ω, and summing up the result we get Substituting  given in (18) into  1 and V given in (20) into  2 , we have Using (32) our conclusion follows.
Lemma 4. The strong solution of the system (1)-( 4) given by Theorem 1 satisfies where  is a positive constant and || > 1 large enough.
Proof.Multiplying equality (21) by  and integrating on Ω, we get Using equality (20) and Poincaré and Young inequalities, we have where  is a positive constant and || > 1 is large enough.The proof is now complete.Now, we are in the position to prove the main result of this paper.
Theorem 5.The semigroup associated with the system (1)-( 4) is polynomially stable and Moreover, this result is optimal.
Proof.From Lemmas 3 and 4 and for || > 1, we have which is equivalent to Then using Theorem 2.4 in [6], we obtain Since 0 ∈ (A), it follows that A is onto over H; then taking A 0 = , we get Therefore the solution decays polynomially.
To prove that the rate of decay is optimal, we will argue by contradiction.Suppose that the rate  −1/2 can be improved; for example, the rate is  −1/(2−) for some 0 <  < 2. From Theorem 5.3 in [10], the operator should be limited, but this does not happen.For this, let us suppose that there exists a sequence Therefore the rate cannot be improved.The proof is now complete.

Numerical Approaches
In this section, we give a number of numerical experiments to certify the analytical results (see Figure 15).We use two numerical approaches to this aim: the spectral method to compute the numerical eigenvalues and the explicit time integration method to compute the numerical solutions and to derive the associated numerical energy.In particular, the numerical energy preserves the conservation/dissipation of energy.Moreover, by means of numerical simulations we can note a lack of numerical exponential decay considering only damping (see Figure 17).Our numerical simulations are realized in the onedimensional case.Then we consider the following onedimensional system: ∀ ∈ (0, ) . (51)

Energy Conserving Scheme.
In this section we will build the numerical energy to system (53)-(57).Following the techniques performed, for example, in [12,13], this energy is given by We note that   is the discrete version of the continuous energy (5) in one-dimensional setting.Moreover, one can show that   decreases.Instead of computing the time derivative of the energy we can use the summation by parts.The discrete energy   is an important numerical instrument to certify our analytical results concerning the polynomial stabilization of system (53)-(57).
We have the following result.
Theorem 6 (discrete energy).Let (   , V   ) be a solution of the finite difference scheme (53)-(57).Then for all Δ and Δ, the discrete rate of change of energy of the numerical scheme (53)-(57) at the   instant of time is given by for all  = 1, . . ., ,  + 1.
Proof.We shortened the proof.It is identical to the proof of Proposition 1 in [12].Then, we can write from where we have Considering the conservative case we get 4.3.Discussion on Numerical Results.In this section, we consider two numerical approaches to certify the analytical results established in this work: the pseudospectral method to compute numerically the eigenvalues and also using the numerical energy   in (58).
For spectral analysis we apply the spectral method according to Trefethen's book [14].Using this numerical technique, we can see the numerical behavior of eigenvalues of ( 48)-(49).To see how this system is affected by a lack of

Conservative energy
No damping exponential decay by means of several numerical simulations, we can consider two damping  1   and  2 V  where  1 ,  2 ≥ 0.Then, making  2 go to zero, we can see a numerical sequence of eigenvalues converging to zero.We consider  1 = 1 and we take  2 going to zero.Figures 1, 2, 3, and 4 show that when   ̸ = 0,  = 1, 2, the real part of complex roots approaches negative asymptotes −  /2,  = 1, 2.
On the other hand, in Figures 5 and 6, we can see a sequence of eigenvalues converging to 0 as  2 → 0.
Next, we show results of numerical simulations of scheme (53)-(57) and its energy   .Here we use the stability criterion Δ ≤ Δ.A simple analysis, by using absolute error  := |(0) −  0 | for each simulation, shows us the precision of the numerical energy.Table 1 shows the evolution of this error for each simulation.

Full and Partial Damping.
It is clear that considering two damping to the system (53)-(54) its numerical energy   decays exponentially.See Figures 11 and 13.Therefore, in this case the energy   is controllable by an exponential of type  −  for some  > 0. On the other hand, taking into account damping only one dissipative mechanism we observe a slow decay from value ≈180 to value ≈90 (see Figures 12 and 14).We can note a lack of 50 by cent in terms of exponential decay.Note that the numerical solutions of the function V do not go to the state of rest.See Figures 16 and 18.

Conclusions
In this work we considered a coupled system of two weakly dissipative wave equations.That is to say, the damping mechanism acts partially on the system and it is well known that this system is not exponentially stable.Our main contribution is twofold: the optimality of the polynomial decay in accordance with a recent theoretical contribution due to Borichev and Tomilov [6] and its numerical certification by using finite differences method.In particular, we derive a numerical energy and we show the lack of exponential decay by means of several numerical experiments.The analytical and numerical techniques performed in this work can be applied to systems where there exists a lack of exponential decay.See, for example, the work of Santos et al. for examples of systems with the property of lack of exponential decay and polynomial decay.

Table 1 𝜇
to undamped system.We can see the conservative behavior of the energy   in agreement with Proposition 4.1.Moreover, it is clear that the accuracy of the approximations is achieved by means of refinements of type  := Δ/Δ ≤ 1.