In this work we consider a coupled system of two weakly dissipative wave equations. We show that the solution of this system decays polynomially and the decay rate is optimal. Computational experiments are conducted in the one-dimensional case in order to show that the energies properties are preserved. In particular, we use finite differences and also spectral methods.

1. Introduction

This paper is concerned with the polynomial stability and the optimal rate of decay of the -semigroup of the coupled system of two wave equations: where is an open bounded set of with smooth boundary . 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 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 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 and it is optimal. This work improves the results in [35]. Our result on the polynomial stability is based on Theorem 2.4 in [6].

Naturally, the system (1)–(4) is dissipative. Indeed, the energy of the solutions defined by is decreasing in , since

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.

2. 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.

Let us denote by the Hilbert space with internal product given by where and .

Let us consider the unbounded operator in the energy space with defined by

Putting and , (1) and (2) can be written as the following initial value problem: with and . Using the internal product (7), we obtain Thus, is a dissipative operator and therefore, using the Lumer-Phillips Theorem (see [8], Theorem 4.3), the operator generates a -semigroup of contractions on .

Thus, we have the following result.

Theorem 1. Let one assume that ; then there exists a unique solution to system (1)–(4) satisfying

3. Polynomial Decay and Optimal Result

To study the asymptotic behavior of the semigroup associated with (1)–(4), let us consider the spectral problem: where

Theorem 2. Let be -semigroup of contractions generated by . Then it follows that is not exponentially stable.

Proof. Our main tool is to use Prüss’ result [9], which states that a semigroup is exponentially stable if and only if the following conditions hold: This way, let us consider the resolvent equation that is,
Let us take and . We look for solution of the form , , , and , with . From (19)–(21), we get and satisfying
Now, choosing , and using the above equation, we obtain , so we have
Therefore, we have
Now we claim that
In fact, using (7) and (24) and noting that 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 of with the resolvent equation of ; 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 , respectively, integrating on , and summing up the result we get
Substituting given in (18) into and given in (20) into , 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 large enough.

Proof. Multiplying equality (21) by and integrating on , we get Using equality (20) and Poincaré and Young inequalities, we have where is a positive constant and 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 , we have which is equivalent to
Then using Theorem 2.4 in [6], we obtain
Since , it follows that is onto over ; then taking , 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 can be improved; for example, the rate is for some . 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 with and for such that is bounded in and Then, we can consider, for each , and , where, due to the boundary conditions, are in the form , , , and .
Then following the same steps as in the proof of Theorem 2 we can conclude that Therefore the rate cannot be improved. The proof is now complete.

4. 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 one-dimensional case. Then we consider the following one-dimensional system:

4.1. Explicit Time Integration Method

For our purpose we define a computational mesh defining and for and the nets given by where and for and .

We consider the following finite-difference discretization of (48)-(49): for all , . Such scheme is in fact explicit and its implementation is straightforward. The boundary conditions are and the initial conditions are given by

The discretizations above are consistent with truncation error of second order in and . Therefore, it is immediate by Lax Lemma [11] that (53)-(54) converges. For convenience, we use the stability criterion (taking into account the conservative case) due to Courant-Friedrichs-Lewy; namely, .

4.2. 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 be a solution of the finite difference scheme (53)–(57). Then for all and , the discrete rate of change of energy of the numerical scheme (53)–(57) at the instant of time is given by for all .

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 exponential decay by means of several numerical simulations, we can consider two damping and where . Then, making go to zero, we can see a numerical sequence of eigenvalues converging to zero.

We consider and we take going to zero. Figures 1, 2, 3, and 4 show that when , , the real part of complex roots approaches negative asymptotes , .

On the other hand, in Figures 5 and 6, we can see a sequence of eigenvalues converging to 0 as .

Next, we show results of numerical simulations of scheme (53)–(57) and its energy . Here we use the stability criterion .

4.4. Undamped Case

Here, we consider , , , , , and . At continuous level we obtain that ≈ 180,4029.

Figures 7, 8, 9, and 10 show the conservative behavior of numerical energy 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 .

A simple analysis, by using absolute error for each simulation, shows us the precision of the numerical energy. Table 1 shows the evolution of this error for each simulation.

4.5. 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 . 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 do not go to the state of rest. See Figures 16 and 18.

5. 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


The authors would like to acknowledge the support of the following corporations: UFPA, PROPESP, and FADESP through PAPQ. The authors thank the anonymous referees for their valuable comments as well as reference suggestions that helped to improve the presentation of this paper.