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Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain

In this paper we are concerned with a Boussinesq system for small-amplitude long waves arising in nonlinear dispersive media. Considerations will be given for the global well-posedness and the time decay rates of solutions when the model is posed on a periodic domain and a general class of damping operator acts in each equation. By means of spectral analysis and Fourier expansion, we prove that the solutions of the linearized system decay uniformly or not to zero, depending on the parameters of the damping operators. In the uniform decay case, the result is extended for the full system.

1. Introduction. In recent years, Boussinesq equations have attracted a great deal of attention from all aspects of wave dynamic researchers due to the wide range of practical applications, specially for simulating waves propagation in coastal zones. Theoretical models of water waves are often derived under application driven assumptions facilitating analysis and numerical computation. The hope is that these models are accurate enough for the intended applications. There are numerous models because no single model can capture all the phenomena associated with shallow water waves. For example, the family of Korteweg-de Vries equations (KdV) describes the uni-directional propagation of shallow water waves, whereas the family of Boussinesq equations describes the bi-directional propagation of such waves. Each model within each family has its own range of applicability.
Considered herein is a variant of the classical Boussinesq system and its higherorder generalizations proposed by J. J. Bona, M. Chen and J.-C. Saut in [3,4]. Such equations were first derived by Boussinesq to describe the two-way propagation of small-amplitude, long wavelength, gravity waves on the surface of water in a canal, but they arise also when modeling the propagation of long-crested waves on large lakes or the ocean: (1) In the system above, η is the elevation of the fluid surface from the equilibrium position, w = w θ is the horizontal velocity in the flow at height θh, where h is the undisturbed depth of the liquid. The parameters a, b, c, d, a 2 , c 2 , b 2 , d 2 are required to fulfill the relations where θ ∈ [0, 1]. Observe that this class of systems contain some of the well-known systems, such as the classical Boussinesq system (a = b = c = a 2 = c 2 = b 2 = d 2 = 0, d = 1 3 ) [7]. Despite of the developments obtained for Boussinesq systems, there are many issues still open that deserves further attention, specially when dissipative mechanisms are incorporated to the models. In real physical situations, dissipative effects are often as important as nonlinear and dispersive effects (see, for instance, [5,6,10]) and this fact has given currency to the study of water wave model in nonlinear dispersive media. Indeed, it was clear from the experimental outcomes that damping effects must be accounted in addition to those of nonlinearity and dispersion for good quantitative agreement with model predictions. When suitable dissipation was added, it appeared that these long-wave models provided an accurate description of reality for a reasonably wide range of amplitudes and frequencies. This in itself presents an interesting challenge.
In this paper, our purpose is to investigate such questions by considering a general class of damping operator with nonnegative symbol. Considerations will be given to the Boussinesq system, posed on a periodic domain, when the parameters in (2) are such that a 2 = c 2 = 0. More precisely, we consider the model where 0 ≤ r, q ≤ 3, The operators M αj are, in some sense, similar to fractional derivative operators. Indeed, for a periodic function h(x) = k∈Z * a k e ikx , the Weyl fractional derivative operator of order α ≥ 0 applied to h is defined by (see Samko et al. [17]) Consequently, the Fourier coefficients of M α h and W α t h behave in the same manner for large k.
A natural question arises as to whether dissipative effects overcome the nonlineardispersive interaction and leads to the decay of solutions, and allows to obtain at which rate they decay. It is to this and related questions that the present work is directed. But before to address the issue, a global (in time) existence result is necessary, i. e., we need to show that system (3) has unique solutions corresponding to reasonably smooth initial data. Therefore, in order to make more precise the idea we have in mind, let us define with the operator H given in the following way where w 1 = 1−ak 2 1+bk 2 +b2k 4 and w 2 = 1−ck 2 1+dk 2 +d2k 4 . Then, we obtain a positive constant C > 0, depending only on β 1 , β 2 , a and c, such that (see (61)-(64)) for any t ≥ 0, where N (η, w) denotes the nonlinear terms on the right hand side of the equations in (3). The estimate above indicates that the terms M α1 and M α2 play the role of feedback damping mechanisms, at least for the linearized system, but E[η, w] does not have a definite sign. Moreover, it is not easy to ascertain that a solution of (3) defined locally in time has a global extension if it remains bounded in a suitable norm on bounded time intervals. Indeed, the standard energy techniques seem unable to establish the a priori bounds needed to guarantee global existence. Thus, the decay of solutions, as well as, the question of smoothing and an associated well-posedness theory set in certain function classes are interesting issues.
In the present paper, we first study the linearized system. Through a careful spectral analysis of the associated differential operator and an explicit Fourier series expansion of the solution in terms of the eigenvectors, we show that the solutions of the linearized system goes to zero, as the time t tends to infinity. Moreover, if α 1 = α 2 = 4 and β 1 , β 2 > 0, it is possible to prove that they decay exponentially to zero in the H s − setting, for any s ∈ R, whenever (η 0 , w 0 ) ∈ (H s p (0, 2π)) 2 . On the other hand, if min{α 1 , α 2 } ∈ [0, 4), β 1 , β 2 ≥ 0 and β 2 1 + β 2 2 > 0, we derive a polynomial decay rate, also in the H s − setting, but considering more regular data. From this second case we can conclude that the exponential decay does not holds when damping term acts in only one equation. In both cases, the same spectral approach combined with the semigroup theory allow to prove the well-posedness, also in the H s − setting. However, the linear well-posedness does not by itself imply that the nonlinear problem will be well-posed. The same is true for the stabilization property, i. e., the decay of solutions of linearized system does not assure the decay of the solutions of the full system. However, by combining the well-posedness and the exponential decay estimate obtained for the linear problem we prove the global well-posedness together with the exponential stability of the solutions (3) issued from small initial data in a convenient weighted space.
Our analysis were inspired by an earlier work on the subject [15] and aims to add considerably to the conclusions drawn therein (see also [2]). By using the same approach the authors derived similar results when the parameters given in (2) We note that, in the absence of the damping, the model considered in [15] is the so-called Boussinesq system of Benjamin-Bona-Mahony type (BBM-BBM), also derived in [3,4]. When the model is posed on a bounded interval, the previous work [14] addresses the stabilization problem for the linearized Benjamin-Bona-Mahony system (N ≡ 0), with a localized damping term that acts in one equation only. By considering Dirichlet boundary conditions, it was proved that the energy associated to the model converges to zero as time goes to infinity. A similar problem posed on the whole real axis was studied by Chen and Goubet [11], where they prove the exponential decay when the damping is active in both equations. Finally, let us mention that the study of the controllability and stability properties for model (3) was initiated in [13], when a = c = 0, N (η, w) = (−ηw) x , −ww x ) and a periodic domain is considered. The space of the controllable data for the associated linear system is determined for each value of the four parameters. Then, as an application of the newly established exact controllability results, some simple feedback controls are constructed for some particular choice of the parameters, such that the resulting closed-loop systems are exponentially stable. Later on, the stabilization problem was studied in [8,16] for Boussinesq system of KdV-KdV type (b = d = 0) posed on a bounded interval. In any case, depending on the values of its parameters, system (3) couples two equations that may be of KdV-KdV or BBM-BBM types. It is therefore interesting to see to which extent the stability properties of each model are maintained and/or improved.
The plan of the present article is as follows: Section 2 is devoted the study of the well-posedness of the linearized problem. In Section 3 we obtain the decay rates for the associated linear semigroup. Finally, in Section 4 we give the asymptotic behavior of the nonlinear system (3) for the case in which the linearized system has an exponential decay rate.
2. The linearized system. The aim of this section is to study the main properties of the linearized model corresponding to (3). Its well-posedness in the H s p − setting will be investigated in subsection 2.1. In order to do that, we make a careful spectral analysis of the state operator and Fourier expansion. Then, having the well-posedness result in hands, the stabilization will be established in subsection 2.2. We consider the following system The next steps are devoted to prove the well-posedness and stabilization results.
2.1. Well-posedness. Given s ∈ R, let us introduce the Hilbert space endowed with the inner product defined by and the operator H defined in the following way where w 1 = 1−ak 2 1+bk 2 +b2k 4 and w 2 = 1−ck 2 1+dk 2 +d2k 4 . Let us remark that system (6) can be written in the following vectorial form where A is the linear compact operator in V s defined by We pass now to study the existence of solutions to (6). If we assume that the initial data in (6) are given by then, at least formally, the solution of (6) can be written as where ( η k (t), w k (t)) fulfills      The next result provides an explicit formula for the eigenvalues of the operator A, as well as, for the solution of (13).
Lemma 2.1. The eigenvalues of the operator A defined by (10) are given by and if |e k | = 1 and k ̸ = 0, and finally, Proof. It is easy to see that (13) Hence, the solution of (13) is given by The eigenvalues λ ± k of the matrix A(k) are k ∈ Z * , that can be rewritten as (14). In order to compute the solutions of (13), we can make use of the following result, whose proof can be found in [1]: Moreover, if λ 1 = λ 2 = λ 0 and Q = A − λ 0 I, then e At = (I + tQ) e λ0t .
We have that, It is straightforward to verify that On the other hand, from Proposition 1 we have that Thus, from (25) and (20) the solution of (13) is given by (16) and (17) in the respective cases. 2. Case |e k | = 1 and k ̸ = 0.

Remark 1.
Firstly, we note that λ ± k = λ ± −k and the following holds: • If e k < 1, then the eigenvalues λ ± k are complex numbers. • If e k ≥ 1, then the eigenvalues λ ± k are real numbers and λ + k ≥ λ − k . The next steps are devoted to analyze the eigenvalues λ ± k given by (14). In the sequel l, M and C denote generic positive constants which may change from one row to another.
We have the following result.

Lemma 2.2. With the notation introduced in Lemma 2.1, we have that:
1. There exists only a finite number of values k ∈ Z with the property that |e k | = 1. 2. There exists a subsequence (e km ) m≥1 of (e k ) k≥1 , such that lim km→∞ |e km | = 1 if and only if one of the following cases holds Proof. For the first part of the Lemma, let us suppose that we have an infinite number of different values (k m ) m≥1 ⊂ N, such that e km = 1. Without loss of generality, we may assume that lim m→∞ k m = ∞. We have the following cases: which implies that α 1 = 3 and β 1 = 2 acb2 d2 . Then, where, On the other hand, and, consequently, Thus , if α 2 ≥ 1, (36) implies that β 2 = θ 1 0 = 0. If α 2 < 1, from (36) we obtain θ 1 0 = 0. Then, from (35) we deduce that which implies that β 2 = θ 1 1 = 0. However, e km can be written as Therefore, e km = 1 is equivalent to a eighth order equation in k m which has at most eight solutions. We have obtain a contradiction and, thus, this case is not possible. • The case α 1 < α 2 may be treated as before, and we obtain the same conclusion.
Therefore, e km = 1 is equivalent to a fourteenth order equation in k m which has at most fourteen solutions. We have again obtained a contradiction. Hence, there exists only a finite number of values k ∈ Z with the property that |e k | = 1. The second part of the Lemma follows as before, by analyzing the similar three cases.
For the third part of Lemma, we consider the following cases: for |e k | ≥ 1, in the sequel we consider |ℜ(λ − k )| := |λ − k |. We have the following result. Theorem 2.3. There exists a constant M > 0, such that the solution ( η k (t), w k (t)) of (13) verifies the following estimate, Proof. We have to analyze two different cases.
We claim that there exists a constant M > 0, such that Assume that it was proved. Then, from (16)-(17) we obtain Therefore, from (54) and (57) we obtain the following estimate, Hence, the series ∞ n=0 d n dt n (η, w)(t 0 ) (t−t0) n n! is (absolutely) convergent and As a direct consequence of the Theorems 2.4 and 2.5 and the general theory of the evolution equations (see, for instance, [9]), we have the following existence and uniqueness result: