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On the well posedness and large-time behavior of higher order Boussinesq system

A family of Boussinesq systems has been proposed to describe the bi-directional propagation of small amplitude long waves on the surface of shallow water. In this paper, we investigate the well-posedness and boundary stabilization of the generalized higher order Boussinesq systems of Korteweg-de Vries--type posed on a interval. We design a two-parameter family of feedback laws for which the system is locally well-posed and the solutions of the linearized system are exponentially decreasing in time.

The goal of this paper is to investigate two problems that appear on the mathematical theory when we consider the study of PDEs. The first one is the global well-posedness, in time, of system (1.1), which is so-called fifth order KdV-type system. Another problem is concerned with boundary stabilization of the linearized system associated to (1.1).
Then, the following questions arise: Problem A. Does E(t) → 0 as t → ∞? If it is the case, can we find a decay rate of E(t)?
The problem might be easy to solve when the underlying model has a intrinsic dissipative nature. Moreover, in the context of coupled systems, in order to achieve the desired decay property, the damping mechanism has to be designed in an appropriate way to capture all the components of the system.
Before presenting an answer for Problem A, it is necessary to investigate the global well-posedeness of the full system (1.2)-(1.3). Thus, the following issue appears naturally: Problem B. Is the fifth order KdV-type system globally well-posed in time, with initial data in H s (0, L), for some s ∈ R + ? 1.2. Some previous results. It is by now well know that mathematicians are interested in the well-posedness of dispersive equation which depends on smoothing effects associated to datum (initial value or boundary value). The well-posedness of the initial value problem for single KdV equation and single fifth order KdV equation was deeply investigated. For an extensive reading on the subject see, for instance, [6,11,13,15,24] and the reference therein. In contrast, the well-posedness theory for the coupled system of KdV-type is considerably less advanced than the theory for single KdV-type equations [4,5,21,22,23]. The same is true for the stabilization properties.
In additional, other interesting problem, as mentioned previously, in the stabilization problem. Problem A was first addressed in [18] for a Boussinesq system of KdV-KdV type with the boundary conditions where α 0 ≥ 0, α 1 > 0 and α 2 > 0. Note that, with boundary conditions (1.8), we have the following identity which does not have a definite sign. In this case, first the authors studied the linearized system to derive some a priori estimates and the exponential decay in the L 2 -norm. It is established the Kato smoothing effect by means of the multiplier method, while the exponential decay is obtained with the aid of some compactness arguments that reduce the issue to prove a unique continuation property for a spectral problem associated to the space operator (see, for instance, [2,19]). The exponential decay estimate of the linear system is then combined with the contraction mapping theorem in a convenient weighted space to prove the global well-posedness together with the exponential stability property of the nonlinear system (1.7)-(1.8) with small data. Recently, in [10], the authors studied a similar boundary stabilization problem for the system (1.7) with less amount of damping. More precisely, the following boundary conditions was considered with α > 0. In this case, it follows that Proceeding as in [18] the local exponential decay is also obtained for solution issuing from small data. However, due to the lack of dissipation, the unique continuation issue for the linearized system can not be obtained by standard methods. In order to overcome this difficult, the spectral problem was then solved by extending the function (η, u) by 0 outside (0, L), by taking its Fourier transform and by using Paley-Wiener theorem. Finally, the problem was reduced to check for which values of L > 0 two functions are entire for a set of parameters. Then, the authors concluded that the stabilization properties holds if and only if the length L does not belong to the following critical set We point out that the same set was obtaned by Rosier, [19] while studying the boundary controllability of the KdV equation with a single control in L 2 (0, T ) acting on the Neumann boundary condition. This shows that the linearized Boussinesq system inherits some interesting properties initially observed for the KdV equation.
1.3. Main results and comments. In the present work, we address the problems described in the previous subsection and our main results provide a partial positive answer for the Problems A and B. In order to give an answer for Problem B, we apply the ideas suggested in [9,10], therefore, let us consider With this notation, one of the main result of this article can be read as follows: In order to prove Theorem 1.1 we first analyze the linearized model by using a semigroup approach. Moreover, by using multiplier techniques, we also obtain the so-called Kato smoothing effect, which is crucial to study the stabilization problem. In what concerns the full system, the idea is to combine the linear theory and a fixed point argument. However, the linear theory described above seems to be unable to provide the a priori bounds needed to use a fixed point argument. To overcome this difficult, we consider solutions obtained via transposition method, which leads to consider a duality argument and the solutions of the corresponding adjoint system. Then, the existence and uniqueness can be proved by using the Riesz-representation theorem that gives, at first, a solution which is not continuous in time, only L ∞ . The continuity is then obtained with the aid of what is known as hidden regularity of the boundary terms of the adjoint system. In fact, we prove that such system has a class of solutions which belong to appropriate spaces possessing boundary regularity. On the other hand, it is also important to note that identity (1.5) does not provide any global (in time) a priori bounds for the solutions. Consequently, it does not lead to the existence of a global (in time) solution in the energy space. The same lack of a priori bounds occurs when higher order Sobolev norms are considered (e.g. H s -norm).
With the damping mechanism proposed in (1.3), the stabilization of the linearized higher order Boussinesq system (1.6) holds for any length of the domain. Thus, the second main result of this paper is the following: Theorem 1.2. Assume that α 1 > 0, α 2 > 0 and L > 0. Then, there exist some constants C 0 , µ 0 > 0, such that, for any (η 0 , u 0 ) ∈ X 0 := [L 2 (0, L)] 2 In order to prove Theorem 1.2 we proceed as in [10,18], i.e, combining multipliers and compactness arguments which reduces the problem to show a unique continuation result for the state operator. To prove this result, we extend the solution under consideration by zero in R \ [0, L] and take the Fourier transform. However, due to the complexity of the system, after taking the Fourier transform of the extended solution (η, u) it is not possible to use the same techniques used in [10]. Thus, to prove our main result we proceed as Santos et al. [20].
For a better understanding we will introduce a general framework to explain the idea of the proof. After to take Fourier transform, the issue is to establish when a certain quotient of entire functions still turn out to be an entire function. We then pick a polynomial q : C → C and a family of functions with α ∈ C 4 \ {0}, whose restriction N α (·, L) is entire for each L > 0. Next, we consider a family of functions f α (·, L), defined by in its maximal domain. The problem is then reduced to determine L > 0 for which there exists α ∈ C 4 \ {0} such that f α (·, L) is entire. In contrast with the analysis developed in [10], this approach does not provide us an explicit characterization of a critical set, if it exists, only ensure that the roots of f have a relations with the Möbius transform (see the proof of Theorem 1.2 above). The remaining part of this paper is organized as follows: In Section 2, we establish the well-posedness of the linearized system. We also derived a series of linear estimates for a conservative linear Boussinesq system which will we used to prove the well-posedness for the full system (1.2)-(1.3). Section 3, is then devoted to prove the well-posedness for the nonlinear system. In section 4, we prove an observability inequality associated to (1.6)-(1.3), which plays a crucial role to get second result of this paper, Theorem 1.2, proved in the same section. Finally, some additional comments and open problems are proposed in the Section 5. We also include an Appendix with a detailed derivation of the system (1.2).

Well-posedness: Linear system
The goal of the section is to prove the well-posedness of the linearized system. In order to do that, we use the semigroup theory and multiplier techniques, which allow us to derived socalled Kato smoothing effect. We also use the same approach to study a similar conservative linear Boussinesq system that will be used to study the full system (see, Definition 3.1).
2.1. Well-posedness: linear system. We will study the existence of solutions of the linear homogeneous system associated to (1.6), namely in (0, L).
We consider X 0 with the usual inner product and the operator A : . Moreover, we introduce the Hilbert space where [X 0 , X 5 ] [θ] denote the the Banach space obtained by the complex interpolation method (see, e.g., [3]).
Then, the following result holds: Proof. Clearly, A is densely defined and closed, so we are done if we prove that A and its adjoint A * are both dissipative in X 0 . It is easy to see that Pick any (η, u) ∈ D(A). Multiplying the first equation of (2.1) by η, the second one by u and integrating by parts, we obtain Analogously, we can deduce that, for any (ϕ, ψ) ∈ D(A * ), Thus, the proof is complete.
As a direct consequence of Proposition 2.1 and the general theory of evolution equation, we have the following existence and uniqueness result: The following proposition provides useful estimates for the standard energy and the Kato smoothing effect for the mild solutions of (2.1).
Proof. We obtain the estimates (2.2)-(2.4) using multiplier techniques. Pick any (η 0 , u 0 ) ∈ D(A). Multiplying the first equation in (2.1) by η, the second one by u, adding the resulting equations and integrating over (0, L)×(0, T ), we obtain (2.2) after some integration by parts. The identity may be extended to any initial state (η 0 , u 0 ) ∈ X 0 by a density argument. Moreover, multiplying the first equation in (2.1) by (T − t)η, the second by (T − t)u and integrating over (0, L) × (0, T ) we derive (2.3) in a similar way. Let us proceed to the proof of (2.4). Multiply the first equation by xu, the second one by xη and integrate over (0, L) × (0, T ). Adding the obtained equations we get that After some integration by parts, it follows that (2.5) By using (2.2) and Young inequality in the first integral of the right hand side in (2.5), we have that Then, (2.4) holds.

2.2.
Well-posedness: a conservative linear system. This subsection is devoted to analyze a conservative linear model that will be used to derived the nonlinear theory. Let us starting by introducing the spaces where [X 0 , X 5 ] [θ] denote the the Banach space obtained by the complex interpolation method (see, e.g., [3]).It is easily seen that On the other hand, we shall use at some place below the following space endowed with its natural norm. The space ., . X −s ,Xs stands for the duality between X −s and X s . Now, we turn our attention to the well-posedness of the system associated to the differential operator A, given by with domain, D( A) = X 5 . More precisely, we consider the following system Proposition 2.4. The operator A is skew-adjoint in X 0 , and thus it generates a group of isometries (e tA ) t∈R in X 0 .
, we have after some integration by parts, Thus, it follows that (2.9) Taking η ∈ C ∞ c (0, L) and u = 0, we deduce from (2.9) that v ∈ H 5 (0, L). Similarly, we obtain that θ ∈ H 5 (0, L). Integrating by parts in the left hand side of (2.9), we obtain that for all (η, u) ∈ D(A). Then, it follows that Corollary 2.5. For any (η 0 , u 0 ) ∈ X 0 , system (2.8) admits a unique solution (η, u) ∈ C(R; X 0 ), which satisfies (η(t), u(t)) X 0 = (η 0 , u 0 ) X 0 for all t ∈ R. If, in addition, Using the above Corollary combined with some interpolation argument between X 0 and X 5 , we can deduce that, for any s ∈ (0, 5), there exists a constant C s > 0 such that, for any (η 0 , u 0 ) ∈ X s , the solution (η, u) of (2.8) satisfies (η, u) ∈ C(R; X s ) and Now, we put our attention in the existence of traces. Indeed, we know that the traces vanish. Thus, we have a look at the other traces η xx (0, t) and u xx (L, t).

Well-posedness: Nonlinear system
In this section we prove the well-posedness for the nonlinear system with a > 0, b > 0, a = b, a 1 > 0, a 2 < 0, a 3 > 0 and a 4 > 0, with the following boundary conditions Before presenting the proof of the main theorem of this section, it is necessary to establish some definition to show how the solution of the problem (3.1)-(3.2) is obtained.
Remark 1. Note that the right hand side of (3.4) is well defined for all τ ∈ [0, T ], since ψ xx (0, ·) and ϕ xx (L, ·) belong to H The next result borrowed from [9], with minor changes, gives us the existence and uniqueness of solution for system (3.3). Its proof is presented here for the sake of completeness.
In particular, there exists C T > 0, such that Let us define L a linear functional given by the right hand side of (3.4), that is, Claim. L belongs to L(X 3 ; R).
Indeed, from the fact X 3 ⊂ X 2 and Proposition 2.7, we obtain that where in the last inequality we use (3.7). Then, from Riesz representation Theorem, there exist one and only one (η τ , u τ ) ∈ X 3 such that and the uniqueness of the solution to the problem (3.3) holds.
We prove now that the solution of the system . In order to prove that the solution (η, u) belongs to C([0, T ]; X 3 ), let τ ∈ [0, T ] and {τ n } n∈N be a sequence such that τ n −→ τ, as n → ∞.
The next result establishes the well-posedness of the non-homogeneous feedback linear system associated to (3.3). (3.21) Let 0 < β ≤ T that will be determinate later. For each (η 0 , u 0 ) ∈ X 3 , consider the map where, (w, v) is the solution of the system (3. 3) with f (t) = −α 1 η xx (0, t) and g(t) = α 2 η xx (L, t). By Lemma 3.2 and (3.21), the linear map Γ is well defined. Furthermore, there exists a positive constant C β , such that Then, .
We are now in position to prove one of the main result of this article.

Exponential stability for the linearized system
Let us now to prove Theorem 1.2 concerning of exponential stability for the linear system (2.1).
Proof of Theorem 1.2. Theorem 1.2 is a consequence of the following claim: There exists a constant C > 0, such that where (η, u) is the solution of (2.1) given by Proposition 2.2.
Indeed, if (4.1) is true, we get where E(t) is defined by (1.4). This implies that Thus, which gives Theorem 1.2 by using semigroup properties associated to the model.
We will divide the proof of the observability inequality (4.1) in three steps as follows: Proof of (4.1).
Moreover, from (4.3), we obtain that Thus, we have that (η, u) is the solution of the IBVP (2.1) with initial data (η 0 , w 0 ) which satisfies, additionally, Notice that (4.9) implies that the solution (η, u) can not be identically zero. However, from lemma bellow, one can conclude that (η, u) = (0, 0), which drive us to a contradiction.
Proof. The proof uses the same arguments as those given in [10,Theorem 3.7]. If N T = {0}, the map (η 0 , u 0 ) ∈ CN T → A(N T ) ⊂ CN T (where CN T denote the complexification of N T ) has (at least) one eigenvalue. Hence, there exists λ ∈ C and η 0 , To obtain the contradiction, it remains to prove that a triple (λ, η 0 , u 0 ) as above does not exist.
Proof. Firstly, let us consider the following change of variable ϕ(x) = η(x) ± u(x), thus we have the problem in only one equation: Note that, if we multiply the equation in (4.10) by ϕ and integrate in [0, L], it is easy to see that λ is purely imaginary, i.e., λ = ir, for r ∈ R. Now, we extend the function ϕ to R by setting ϕ(x) = 0 for x ∈ [0, L]. The extended function satisfies where δ ζ denotes the Dirac measure at x = ζ and the derivatives ϕ ′′′′ (0), ϕ ′′′′ (L), ϕ ′′′ (0) and ϕ ′′′ (L) are those of the function ϕ when restricted to [0, L]. Taking the Fourier transform of each term in the above system and integrating by parts, we obtain Setting λ = −ir and f α (ξ, L) = iφ(ξ), from the equation above it follows that with N α (·, L) defined by where α i , for i = 1, 2, 3, 4, are the traces of bϕ ′′′ and bϕ ′′′′ . For each r ∈ R and α ∈ C 4 \ {0} let F αr be the set of L > 0 values, for which the function f α (·, L) is entire. We introduce the following statements, which are equivalent: A1. f α (·, L) is entire; A2. all zeros, taking the respective multiplicities into account, of the polynomial q are zeros of N α (·, L); A3. the maximal domain of f α (·, L) is C.
To the function f α (·, L) to be entire, due to the equivalence between statement A1 and A2, we must have where ξ i denotes the zeros of q(ξ), for i = 1, 2, 3, 4, 5. Let us define, for α ∈ C 4 \ {0}, the following discriminant Then, for α ∈ C 4 \ {0}, such that d(α) = 0 the Möbius transformations can be introduced by for each zero ξ i of the polynomial q(ξ).
The next claim analyzes the behavior of the roots of polynomial q(·): Claim 1. The polynomial q(·) has exactly one real root, with multiplicity 1 and two pairs of complex conjugate roots.
Proof of the Claim 1. Initially, we suppose that r = 0. Note that the derivative of q is given by q ′ (ξ) = 5bξ 4 + 3aξ 2 + 1, and its zeros are ±z 1 and ±z 2 , where and .
Second, we suppose that r = 0. Initially, note that from the derivation of the model (see the Appendix) we have that 4b > a 2 . Then, we obtain that q(ξ) = ξ(bξ 4 + aξ 2 + 1), whose roots are 0, ±ρ and ±k where (4.14) Thus, q(·) has two pairs of complex conjugate roots and one real root, proving Claim 1.
Let us finish the proof of Lemma 4.2. To do this, we need to consider two cases: i. d(α) = 0; ii. d(α) = 0, where d(α) was defined in (4.12).
In fact, if d(α) = 0, we can defined the Möbius transformation. Let us assume, by contradiction, that there exists L > 0 such that the function f a (·, L) is entire. Then, all roots of the polynomial q(·) must satisfy (4.13), i.e., there exists a Möbius transformation that takes each root ξ 0 of q(·) into e −iLξ 0 . However, this contradicts Lemma 4.4 and proves that if (N ) holds then F αr = ∅ for all r ∈ R. On the other hand, suppose that d(α) = 0 and note that by using the claim 1, we can conclude that the set of the imaginary parts of the polynomial q(·) has at least three elements, thus it follows from Lemma 4.3 that F αr = ∅ for all r ∈ R. Note that in both cases, we have that F αr = ∅, which implies that (N ) only has the trivial solution for any L > 0, and the proof of Lemma 4.2 is archived.
This proves (4.15) for s = 5. The fact that (4.15) is still valid for 0 < s < 5 follows from a standard interpolation argument, since X s = [X 0 , X 5 ] s/5 .

Further comments and open problems
In this section considerations will be done regarding the fifth order Boussinesq system (1.2)-(1.3). It is important to note that the classical energy estimate does not provide any global (in time) a priori bounds for the solutions of the corresponding nonlinear model. Consequently, it does not lead to the existence of a global (in time) solution in the energy space. Due to the structure of the nonlinear terms, the same lack of a priori bounds also occurs when higher order Sobolev norms are considered (e. g. H s −norm). Because to this strict requirement, we cannot proceed as in [10,18] and have only succeeded in deriving uniform decay results for the linear system. However, for the full system, we can find solutions -in a certain sense -globally in time in X s .
• Global well-posedness in time 3) with only one damping mechanism, that is, with α 1 or α 2 vanishing, we still have E(t), defined by (1.4), decreasing along the trajectories of the linearized system associated to the model. Thus, the following question can be formulated: Question C. Is still valid, with only one damping mechanism, the exponential stability for the linearized system associated to (1.2)-(1.3)?
• Exponential stability for the full system Due the lack of the classical energy estimate for the nonlinear model we are not able to prove, by using, e.g., [10,18], the exponential stability for the full model (1. Question E. Does the energy associated to the nonlinear system (1.2)-(1.3), with one or two damping mechanism, converges to zero, as t → ∞, for initial data on the energy space X 0 ?