On the well posedness and large-time behavior of higher order Boussinesq system
aa r X i v : . [ m a t h . A P ] J u l ON THE WELL POSEDNESS AND LARGE-TIME BEHAVIOR OFHIGHER ORDER BOUSSINESQ SYSTEM
ROBERTO A. CAPISTRANO–FILHO, FERNANDO A. GALLEGO, AND ADEMIR F. PAZOTO
Abstract.
A family of Boussinesq systems has been proposed to describe the bi-directionalpropagation of small amplitude long waves on the surface of shallow water. In this pa-per, we investigate the well-posedness and boundary stabilization of the generalized higherorder Boussinesq systems of Korteweg-de Vries–type posed on a interval. We design atwo-parameter family of feedback laws for which the system is locally well-posed and thesolutions of the linearized system are exponentially decreasing in time. Introduction
Presentation of the problem.
J. L. Boussinesq introduced in [8] several simple non-linear systems of PDEs, including the Korteweg-de Vries (KdV) equation, to explain certainphysical observations concerning the water waves, e.g. the emergence and stability of soli-tons. Unfortunately, several systems derived by Boussinesq proved to be ill-posed, so thatthere was a need to propose other systems similar to Boussinesq’s ones but with bettermathematical properties. In this spirit, an evolutionary version of the Boussinesq systemswas proposed in [14, Eqs. (4.7)-(4.8), page 283]:(1.1) η t + u x + β (3 θ − u xxx + β (25 θ − θ + 1) u xxxxx + α ( ηu ) x + αβ ( θ − ηu xx ) x = 0 u t + η x + β (cid:2) (1 − θ ) − τ (cid:3) η xxx + β (cid:2) ( θ − θ + 5) + τ ( θ − (cid:3) η xxxxx αuu x + αβ [( ηη xx ) x + (2 − θ ) u x u xx ] = 0 , where η and u are real function of the real variables x , t . The small parameters α > β > τ represents a dimensionless surface tension coefficient, with τ = 0 corresponding to the case of no surface tension and the velocity potential at height0 ≤ θ ≤
1. For further discussions on the model and different modelling possibilities, see,e.g. [4, 5, 7, 12, 14, 23].The goal of this paper is to investigate two problems that appear on the mathematicaltheory when we consider the study of PDEs. The first one is the global well-posedness, intime, of system (1.1), which is so-called fifth order KdV–type system . Another problem isconcerned with boundary stabilization of the linearized system associated to (1.1).
Date : 2018-08-27-a.2010
Mathematics Subject Classification.
Primary: 93B05, 93D15, 35Q53.
Key words and phrases.
Boussinesq system of higher order, Stabilization, M¨obius transform, Criticallength, Fifth order KdV–type system.
First, we consider the following system, carefully derived by (1.1) in a short Appendixat the end of this paper,(1.2) η t + u x − au xxx + a ( ηu ) x + a ( ηu xx ) x + bu xxxxx = 0 , in (0 , L ) × (0 , ∞ ) ,u t + η x − aη xxx + a uu x + a ( ηη xx ) x + a u x u xx + bη xxxxx = 0 , in (0 , L ) × (0 , ∞ ) ,η ( x,
0) = η ( x ) , u ( x,
0) = u ( x ) , in (0 , L ) , where a > b > a = b , a > a < a > a >
0, with the following boundaryconditions(1.3) η (0 , t ) = η ( L, t ) = η x (0 , t ) = η x ( L, t ) = 0 , in (0 , ∞ ) ,u (0 , t ) = u ( L, t ) = u x (0 , t ) = u x ( L, t ) = 0 , in (0 , ∞ ) ,u xx (0 , t ) + α η xx (0 , t ) = 0 , u xx ( L, t ) − α η xx ( L, t ) = 0 , in (0 , ∞ ) , for α , α ∈ R + ∗ .The energy associated to the model is given by(1.4) E ( t ) := 12 Z L ( η ( x, t ) + u ( x, t )) dx, and, at least formally, we can verify that E ( t ) satisfies(1.5) ddt E ( t ) = − α b | η xx (0 , t ) | − α b | η xx ( L, t ) | − a Z L η u x dx − a Z L η u xxx dx + a Z L ηη xx u x dx + a Z L u x dx. Indeed, if we multiply the first equation of (1.2) by η , the second one by u and integrate byparts over (0 , L ), we obtain (1.5), by using the boundary conditions (1.3). This indicates that E ( t ) does not have a definite sign, but the boundary conditions play the role of a feedbackdamping mechanism for the linearized system, namely,(1.6) η t + u x − au xxx + bu xxxxx = 0 , in (0 , L ) × (0 , ∞ ) ,u t + η x − aη xxx + bη xxxxx = 0 , in (0 , L ) × (0 , ∞ ) ,η ( x,
0) = η ( x ) , u ( x,
0) = u ( x ) , in (0 , L ) , with the boundary conditions given by (1.3).Then, the following questions arise: Problem A . Does E ( t ) → 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 dissipativenature. Moreover, in the context of coupled systems, in order to achieve the desired decayproperty, the damping mechanism has to be designed in an appropriate way to capture allthe components of the system.Before presenting an answer for Problem A , it is necessary to investigate the globalwell-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 datain H s (0 , L ) , for some s ∈ R + ? IGHER ORDER BOUSSINESQ SYSTEM AND ITS PROPERTIES 3
Some previous results.
It is by now well know that mathematicians are interested inthe well-posedness of dispersive equation which depends on smoothing effects associated todatum (initial value or boundary value). The well-posedness of the initial value problem forsingle KdV equation and single fifth order KdV equation was deeply investigated. For anextensive 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 considerablyless advanced than the theory for single KdV–type equations [4, 5, 21, 22, 23]. The same istrue for the stabilization properties.In additional, other interesting problem, as mentioned previously, in the stabilizationproblem. Problem A was first addressed in [18] for a Boussinesq system of KdV-KdV type(1.7) η t + u x + ( ηu ) x + u xxx = 0 in (0 , L ) × (0 , T ), u t + η x + uu x + η xxx = 0 in (0 , L ) × (0 , T ), η ( x,
0) = η ( x ), u ( x,
0) = u ( x ) in (0 , L ) , with the boundary conditions(1.8) u (0 , t ) = u xx (0 , t ) = 0 , in (0 , T ), u x (0 , t ) = α η x (0 , t ) , u x ( L, t ) = − α η x ( L, t ) in (0 , T ), u ( L, t ) = α η ( L, t ) , u xx ( L, t ) = − α η xx ( L, t ) in (0 , T ) , where α ≥ α > α >
0. Note that, with boundary conditions (1.8), we have thefollowing identity ddt E ( t ) = − α | η ( L, t ) | − α | η x ( L, t ) | − α | η x (0 , t ) | − u ( L, t ) − Z L ( ηu ) x ηdx, which does not have a definite sign. In this case, first the authors studied the linearizedsystem to derive some a priori estimates and the exponential decay in the L –norm. It isestablished the Kato smoothing effect by means of the multiplier method, while the expo-nential decay is obtained with the aid of some compactness arguments that reduce the issueto prove a unique continuation property for a spectral problem associated to the space oper-ator (see, for instance, [2, 19]). The exponential decay estimate of the linear system is thencombined with the contraction mapping theorem in a convenient weighted space to provethe global well-posedness together with the exponential stability property of the nonlinearsystem (1.7)-(1.8) with small data.Recently, in [10], the authors studied a similar boundary stabilization problem for thesystem (1.7) with less amount of damping. More precisely, the following boundary conditionswas considered (cid:26) η (0 , t ) = 0 , η ( L, t ) = 0 , η x (0 , t ) = 0 , in (0 , T ), u (0 , t ) = 0 , u ( L, t ) = 0 , u x ( L, t ) = − αη x ( L, t ) in (0 , T ),with α >
0. In this case, it follows that ddt E ( t ) = − α | η x ( L, t ) | − u ( L, t ) − Z L ( ηu ) x ηdx. Proceeding as in [18] the local exponential decay is also obtained for solution issuing fromsmall data. However, due to the lack of dissipation, the unique continuation issue for thelinearized system can not be obtained by standard methods. In order to overcome thisdifficult, 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, theproblem was reduced to check for which values of
L >
CAPISTRANO–FILHO, GALLEGO, AND PAZOTO parameters. Then, the authors concluded that the stabilization properties holds if and onlyif the length L does not belong to the following critical set N := { π r k + kl + l k, l ∈ N ∗ } . We point out that the same set was obtaned by Rosier, [19] while studying the boundarycontrollability of the KdV equation with a single control in L (0 , T ) acting on the Neu-mann boundary condition. This shows that the linearized Boussinesq system inherits someinteresting properties initially observed for the KdV equation.1.3. Main results and comments.
In the present work, we address the problems describedin the previous subsection and our main results provide a partial positive answer for theProblems A and B . In order to give an answer for Problem B , we apply the ideas suggestedin [9, 10], therefore, let us consider X = { ( η, u ) ∈ [ H (0 , L ) ∩ H (0 , L )] ; η xx (0) = v xx ( L ) = 0 } . With this notation, one of the main result of this article can be read as follows:
Theorem 1.1.
Let
T > . Then, there exists ρ = ρ ( T ) > such that, for every ( η , u ) ∈ X satisfying k ( η , u ) k X < ρ, there exists a unique solution ( η, u ) ∈ C ([0 , T ]; X ) of (1.2) - (1.3) . Moreover k ( η, u ) k C ([0 ,T ]; X ) ≤ C k ( η , u ) k X for some positive constant C = C ( T ) . In order to prove Theorem 1.1 we first analyze the linearized model by using a semi-group approach. Moreover, by using multiplier techniques, we also obtain the so-called Katosmoothing effect, which is crucial to study the stabilization problem. In what concerns thefull 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 neededto 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 thecorresponding adjoint system. Then, the existence and uniqueness can be proved by usingthe Riesz-representation theorem that gives, at first, a solution which is not continuous intime, only L ∞ . The continuity is then obtained with the aid of what is known as hiddenregularity of the boundary terms of the adjoint system. In fact, we prove that such systemhas 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 existenceof a global (in time) solution in the energy space. The same lack of a priori bounds occurswhen higher order Sobolev norms are considered (e.g. H s -norm).With the damping mechanism proposed in (1.3), the stabilization of the linearized higherorder Boussinesq system (1.6) holds for any length of the domain. Thus, the second mainresult of this paper is the following: Theorem 1.2.
Assume that α > , α > and L > . Then, there exist some constants C , µ > , such that, for any ( η , u ) ∈ X := [ L (0 , L )] , system (1.6)-(1.3) admits aunique solution ( η, u ) ∈ C ([0 , T ] ; X ) ∩ L (cid:0) , T ; [ H (0 , L )] (cid:1) satisfying k ( η ( t ) , u ( t )) k X ≤ C e − µ t k ( η , u ) k X , ∀ t ≥ . IGHER ORDER BOUSSINESQ SYSTEM AND ITS PROPERTIES 5
In order to prove Theorem 1.2 we proceed as in [10, 18], i.e, combining multipliers andcompactness arguments which reduces the problem to show a unique continuation result forthe state operator. To prove this result, we extend the solution under consideration by zeroin 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 usethe 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 ofthe proof. After to take Fourier transform, the issue is to establish when a certain quotient ofentire functions still turn out to be an entire function. We then pick a polynomial q : C → C and a family of functions N α : C × (0 , ∞ ) → C , with α ∈ C \ { } , whose restriction N α ( · , L ) is entire for each L >
0. Next, we consider afamily of functions f α ( · , L ), defined by f α ( µ, L ) = N α ( µ, L ) q ( µ ) , in its maximal domain. The problem is then reduced to determine L > α ∈ C \ { } such that f α ( · , L ) is entire. In contrast with the analysis developedin [10], this approach does not provide us an explicit characterization of a critical set, if itexists, only ensure that the roots of f have a relations with the M¨obius transform (see theproof of Theorem 1.2 above).The remaining part of this paper is organized as follows: In Section 2, we establish thewell-posedness of the linearized system. We also derived a series of linear estimates for aconservative linear Boussinesq system which will we used to prove the well-posedness forthe full system (1.2)-(1.3). Section 3, is then devoted to prove the well-posedness for thenonlinear 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 samesection. Finally, some additional comments and open problems are proposed in the Section5. We also include an Appendix with a detailed derivation of the system (1.2).2. Well-posedness: Linear system
The goal of the section is to prove the well-posedness of the linearized system. In order todo that, we use the semigroup theory and multiplier techniques, which allow us to derived so-called Kato smoothing effect. We also use the same approach to study a similar conservativelinear 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 linearhomogeneous system associated to (1.6), namely(2.1) η t + u x − au xxx + bu xxxxx = 0 , in (0 , L ) × (0 , T ) ,u t + η x − aη xxx + bη xxxxx = 0 , in (0 , L ) × (0 , T ) ,η (0 , t ) = η ( L, t ) = η x (0 , t ) = η x ( L, t ) = 0 , in (0 , T ) ,u (0 , t ) = u ( L, t ) = u x (0 , t ) = u x ( L, t ) = 0 , in (0 , T ) ,u xx (0 , t ) + α η xx (0 , t ) = 0 , in (0 , T ) ,u xx ( L, t ) − α η xx ( L, t ) = 0 , in (0 , T ) ,η ( x,
0) = η ( x ) , u ( x,
0) = u ( x ) , in (0 , L ) . CAPISTRANO–FILHO, GALLEGO, AND PAZOTO
We consider X with the usual inner product and the operator A : D ( A ) ⊂ X → X with domain D ( A ) = { ( η, u ) ∈ [ H (0 , L ) ∩ H (0 , L )] : u xx (0) + α η xx (0) = 0 , u xx ( L ) − α η xx ( L ) = 0 } , defined by A ( η, u ) = ( − u x + au xxx − bu xxxxx , − η x + aη xxx − bη xxxxx ) . Let us denote X = D ( A ). Moreover, we introduce the Hilbert space X θ := [ X , X ] [ θ ] , for 0 < θ < , where [ X , X ] [ θ ] denote the the Banach space obtained by the complex interpolation method(see, e.g., [3]).Then, the following result holds: Proposition 2.1. If α i ≥ , i = 1 , , then A generates a C -semigroup of contraction ( S ( t )) t ≥ in X .Proof. Clearly, A is densely defined and closed, so we are done if we prove that A and itsadjoint A ∗ are both dissipative in X . It is easy to see that A ∗ : D ( A ∗ ) ⊂ X −→ X is given by A ∗ ( ϕ, ψ ) = ( ψ x − aψ xxx + bψ xxxxx , ϕ x − aϕ xxx + bϕ xxxxx ) with domain D ( A ∗ ) = { ( ϕ, ψ ) ∈ X : ϕ (0) = ϕ ( L ) = ϕ x (0) = ϕ x ( L ) = 0 ,ψ (0) = ψ ( L ) = ψ x (0) = ψ x ( L ) = 0 ,ψ xx (0) − α ϕ xx (0) = 0 , ψ xx ( L ) + α ϕ xx ( L ) = 0 } . 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( A ( η, u ) , ( η, u )) X = − α bη xx ( L ) − α bη xx (0) ≤ , which demonstrates that A is a dissipative operator in X . Analogously, we can deduce that,for any ( ϕ, ψ ) ∈ D ( A ∗ ),( A ∗ ( ϕ, ψ ) , ( ϕ, ψ )) X = − α bϕ xx ( L ) − α bϕ xx (0) ≤ , so that A ∗ is dissipative, as well. Thus, the proof is complete. (cid:3) As a direct consequence of Proposition 2.1 and the general theory of evolution equation,we have the following existence and uniqueness result:
Proposition 2.2.
Let ( η , u ) ∈ X . There exists a unique mild solution ( η, u ) = S ( · )( η , u ) of (2.1) such that ( η, u ) ∈ C ([0 , T ]; X ) . Moreover, if ( η , u ) ∈ D ( A ) , then (2.1) has aunique (classical) solution ( η, u ) such that ( η, u ) ∈ C ([0 , T ]; D ( A )) ∩ C (0 , T ; X ) . The following proposition provides useful estimates for the standard energy and theKato smoothing effect for the mild solutions of (2.1).
Proposition 2.3.
Let ( η , u ) ∈ X and ( η ( t ) , u ( t )) = S ( t )( η , u ) . Then, for any T > ,we have that (2.2) k ( η ( x ) , u ( x )) k X − k ( η ( x, T ) , u ( x, T )) k X = Z T (cid:0) α b | η xx ( L, t ) | + α b | η xx (0 , t ) | (cid:1) dt IGHER ORDER BOUSSINESQ SYSTEM AND ITS PROPERTIES 7 and T k ( η ( x ) , u ( x )) k X = 12 k ( η ( x, t ) , u ( x, t )) k L (0 ,T ; X ) + α b Z T ( T − t ) | η xx ( L, t ) | dt + α b Z T ( T − t ) | η xx (0 , t ) | dt. (2.3) Furthermore, ( η, u ) ∈ L (0 , T ; X ) and (2.4) k ( η, u ) k L (0 ,T ; X ) ≤ C k ( η , u ) k X , where C = C ( a, b, T ) is a positive constant.Proof. We obtain the estimates (2.2)-(2.4) using multiplier techniques. Pick any ( η , u ) ∈ D ( A ). Multiplying the first equation in (2.1) by η , the second one by u , adding the resultingequations 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 , u ) ∈ X by a density argument.Moreover, multiplying the first equation in (2.1) by ( T − t ) η , the second by ( T − t ) u andintegrating 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 oneby xη and integrate over (0 , L ) × (0 , T ). Adding the obtained equations we get that Z T Z L x ( ηu ) t dxdt + Z T Z L x | η | + | u | ) x dxdt − a Z T Z L x ( ηη xxx + uu xxx ) dxdt + b Z T Z L x ( ηη xxxxx + uu xxxxx ) dxdt = 0 . After some integration by parts, it follows that Z T Z L x ( ηu ) t dxdt − Z T Z L ( | η | + | u | ) dxdt − a Z T Z L ( | η x | + | u x | ) dxdt − b Z T Z L ( | η xx | + | u xx | ) dxdt + bL Z T ( | η xx ( L, t ) | + | u xx ( L, t ) | ) dt = 0 , hence, 12 Z T Z L ( | η | + | u | ) dxdt + 3 a Z T Z L ( | η x | + | u x | ) dxdt + 5 b Z T Z L ( | η xx | + | u xx | ) dxdt ≤ L Z L ( η ( x, T ) u ( x, T ) − η ( x ) u ( x )) dx + bL (1 + α )2 Z T | η xx ( L, t ) | dt. (2.5) CAPISTRANO–FILHO, GALLEGO, AND PAZOTO
By using (2.2) and Young inequality in the first integral of the right hand side in (2.5), wehave that12 Z T Z L ( | η | + | u | ) dxdt + 3 a Z T Z L ( | η x | + | u x | ) dxdt + 5 b Z T Z L ( | η xx | + | u xx | ) dxdt ≤ L Z L ( | η ( x, T ) | + | u ( x, T ) | ) dx + L (cid:18) α α (cid:19) Z L ( | η ( x ) | + | u ( x ) | d ) dx. Clearly, (2.2) implies that E ( T ) ≤ E (0), thus Z T Z L ( | η | + | u | ) dxdt + 3 a Z T Z L ( | η x | + | u x | ) dxdt + 5 b Z T Z L ( | η xx | + | u xx | ) dxdt ≤ L (cid:18) α α (cid:19) Z L ( | η ( x ) | + | u ( x ) | d ) dx. Then, (2.4) holds. (cid:3)
Well-posedness: a conservative linear system.
This subsection is devoted to an-alyze a conservative linear model that will be used to derived the nonlinear theory.Let us starting by introducing the spaces X := X := L (0 , L ) × L (0 , L ) , (2.6) X = { ( ϕ, ψ ) ∈ [ H (0 , L ) ∩ H (0 , L )] : ϕ xx (0) = ψ xx ( L ) = 0 } , and X θ := [ X , X ] [ θ ] , for 0 < θ < , where [ X , X ] [ θ ] denote the the Banach space obtained by the complex interpolation method(see, e.g., [3]).It is easily seen that X = H (0 , L ) × H (0 , L ) ,X = { ( η, v ) ∈ [ H (0 , L ) ∩ H (0 , L )] ; η x ( L ) = v x (0) = 0 } .X = { ( η, v ) ∈ [ H (0 , L ) ∩ H (0 , L )] ; η xx (0) = v xx ( L ) = 0 } . On the other hand, we shall use at some place below the following space X := { ( η, v ) ∈ [ H (0 , L ) ∩ H (0 , L )] ; η xx (0) = v xx ( L ) = 0 , − aη x ( L ) + bη x ( L ) = − av x ( L ) + bv x ( L ) = 0 , − aη x (0) + bη x (0) = − av x (0) + bv x (0) = 0 , − aη x (0) + bη x (0) = − av x ( L ) + bv x ( L ) = 0 } , endowed with its natural norm. The space X − s = ( X s ) ′ denotes the dual of X s with respect to the pivot space X = L (0 , L ) × L (0 , L ). The bracket h ., . i X − s ,X s 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 differ-ential operator e A , given by(2.7) e A ( ϕ, ψ ) = ( − ψ x + aψ xxx − bψ xxxxx , − ϕ x + aϕ xxx − bϕ xxxxx ) , IGHER ORDER BOUSSINESQ SYSTEM AND ITS PROPERTIES 9 with domain, D ( e A ) = X . More precisely, we consider the following system(2.8) η t + u x − au xxx + bu xxxxx = 0 , in (0 , L ) × (0 , T ) ,u t + η x − aη xxx + bη xxxxx = 0 , in (0 , L ) × (0 , T ) ,η (0 , t ) = η ( L, t ) = η x (0 , t ) = η x ( L, t ) = η xx (0 , t ) = 0 , on (0 , T ) ,u (0 , t ) = u ( L, t ) = u x (0 , t ) = u x ( L, t ) = u xx ( L, t ) = 0 , on (0 , T ) ,η ( x,
0) = η ( x ) , u ( x,
0) = u ( x ) , on (0 , L ) . Proposition 2.4.
The operator e A is skew-adjoint in X , and thus it generates a group ofisometries ( e tA ) t ∈ R in X .Proof. We show that e A ∗ = − e A . First, we prove that − e A ⊂ e A ∗ . Indeed, for any ( η, u ) , ( θ, v ) ∈ D ( e A ), we have after some integration by parts, (cid:16) ( θ, v ) , e A ( η, u ) (cid:17) X = − Z L [ θ ( u x − au xxx + bu xxxxx ) + v ( η x − aη xxx + bη xxxxx )] dx = Z L [ u ( θ x − aθ xxx + bθ xxxxx ) + η ( v x − av xxx + bv xxxxx )] dx = (cid:16) e A ( θ, v ) , ( η, u ) (cid:17) X . Now, we prove that e A ∗ ⊂ − e A . Pick any ( θ, v ) ∈ D ( e A ∗ ). Then, for some positive constant C , we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) ( θ, v ) , e A ( η, u ) (cid:17) X (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k ( η, u ) k X , ∀ ( η, u ) ∈ D ( A ) . Thus, it follows that (cid:12)(cid:12)(cid:12)(cid:12)Z L [ θ ( u x − au xxx + bu xxxxx ) + v ( η x − aη xxx + bη xxxxx )] dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18)Z L ( η + u ) dx (cid:19) , ∀ ( η, u ) ∈ D ( A ) . (2.9)Taking η ∈ C ∞ c (0 , L ) and u = 0, we deduce from (2.9) that v ∈ H (0 , L ). Similarly, weobtain that θ ∈ H (0 , L ). Integrating by parts in the left hand side of (2.9), we obtain that | aθ (0) u xx (0) − bθ x ( L ) u xxx ( L ) + bθ ( L ) u xxxx ( L ) − bθ xx (0) u xx (0) + bθ x (0) u xxx (0) − bθ (0) u xxxx (0) + av ( L ) η xx ( L ) − bv xx ( L ) η xx ( L ) + bv x ( L ) η xxx ( L ) − bv ( L ) η xxxx ( L ) − bv x (0) η xxx (0) − bv (0) η xxxx (0) | ≤ C (cid:18)Z L ( η + u ) dx (cid:19) , for all ( η, u ) ∈ D ( A ). Then, it follows that ( θ (0) = θ ( L ) = θ x (0) = θ x ( L ) = θ xx (0) = 0 ,v (0) = v ( L ) = v x (0) = v x ( L ) = v xx ( L ) = 0 . Hence ( θ, v ) ∈ D ( e A ) = D ( − e A ). Thus, D ( e A ∗ ) = D ( − e A ) and e A ∗ = − e A. (cid:3) Corollary 2.5.
For any ( η , u ) ∈ X , system (2.8) admits a unique solution ( η, u ) ∈ C ( R ; X ) , which satisfies k ( η ( t ) , u ( t )) k X = k ( η , u ) k X for all t ∈ R . If, in addition, ( η , u ) ∈ X , then ( η, u ) ∈ C ( R ; X ) with k ( η, u ) k X := k ( η, u ) k X + k e A ( η, u ) k X constant. Using the above Corollary combined with some interpolation argument between X and X , we can deduce that, for any s ∈ (0 , C s > η , u ) ∈ X s , the solution ( η, u ) of (2.8) satisfies ( η, u ) ∈ C ( R ; X s ) and(2.10) k ( η ( t ) , u ( t )) k X s ≤ C s k ( η , u ) k X s , ∀ t ∈ R . Now, we put our attention in the existence of traces. Indeed, we know that the traces η (0 , t ) , η ( L, t ) , η x (0 , t ) , η x ( L, t ) , η xx (0 , t ) ,u (0 , t ) , u ( L, t ) , u x (0 , t ) , u x ( L, t ) , u xx ( L, t ) , vanish. Thus, we have a look at the other traces η xx (0 , t ) and u xx ( L, t ). Proposition 2.6.
Let ( η , u ) ∈ X and let ( η, u ) denote the solution of (2.8) . Pick any T > . Then η xx ( L, t ) , u xx (0 , t ) ∈ L (0 , T ) with (2.11) Z T (cid:0) | η xx ( L, t ) | + | u xx (0 , t ) | (cid:1) dt ≤ C k ( η , u ) k X for some constant C = C ( L, T, a, b ) .Proof. Assume that ( η , u ) ∈ X , so that ( η, u ) ∈ C ([0 , T ]; X ) ∩ L ([0 , T ]; X ). We multiplythe first (resp. second) equation in (2.8) by xu (resp. xη ), integrate over (0 , T ) × (0 , L ),integrate by parts and add the two obtained equations to get(2.12) − b Z T Z L [ η xx + u xx ] dxdt − a Z T Z L [ η x + u x ] dxdt − Z T Z L [ η + u ] dxdt + (cid:20)Z L [ xηu ] dx (cid:21) T + bL Z T ( | η xx ( L, t ) | + | u xx ( L, t ) | ) dt = 0 . Since R T k ( η, u ) k X dt ≤ C k ( η , u ) k X , this yields Z T | η xx ( L, t ) | ( t, L ) dt ≤ C k ( η , u ) k X . By symmetry, using now as multipliers ( L − x ) u and ( L − x ) η , we infer that Z T | u xx ( t, | dt ≤ C k ( η , u ) k X . Thus, (2.11) is established when ( η , u ) ∈ X . Since X is dense in X , the result holds aswell for ( η , u ) ∈ X . (cid:3) Proposition 2.7.
Let ( η , u ) ∈ X and let ( η, u ) denote the solution of (2.8) . Then η xx ( L, t ) , u xx (0 , t ) ∈ H (0 , T ) with (2.13) k η xx ( L, · ) k H (0 ,T ) + k u xx (0 , · ) k H (0 ,T ) ≤ C k ( η , u ) k X for some constant C = C ( L, T, a, b ) .Proof. In direction to prove (2.13), we consider ( η , u ) ∈ X . By Proposition 2.4, e A generates a group of isometries. Thus, by semigroup properties (see [17]) we obtain that( η, u ) ∈ C ( R ; X ), so that(2.14) ( b η, b u ) = ( η t , u t ) = e A ( η, u ) ∈ C ( R ; X ) IGHER ORDER BOUSSINESQ SYSTEM AND ITS PROPERTIES 11 and it solves(2.15) ( ( b η, b u ) t = e A ( b η, b u ) , ( b η, b u )(0) = e A ( η , u ) ∈ X . The, from (2.11), we deduce that(2.16) k η xx ( · , L ) k H (0 ,T ) + k u xx ( · , k H (0 ,T ) ≤ k ( η , u ) k X . Since X = [ X , X ] , we infer from (2.11) and (2.16) that k η xx ( · , L ) k H (0 ,T ) + k u xx ( · , k H (0 ,T ) ≤ k ( η , u ) k X , for some constant C = C ( T ) and all ( η , u ) ∈ X . (cid:3) Well-posedness: Nonlinear system
In this section we prove the well-posedness for the nonlinear system(3.1) η t + u x − au xxx + a ( ηu ) x + a ( ηu xx ) x + bu xxxxx = 0 ,u t + η x − aη xxx + a uu x + a ( ηη xx ) x + a u x u xx + bη xxxxx = 0 ,η ( x,
0) = η ( x ) , u ( x,
0) = u ( x ) , with a > b > a = b , a > a < a > a >
0, with the following boundaryconditions(3.2) η (0 , t ) = η ( L, t ) = η x (0 , t ) = η x ( L, t ) = 0 ,u (0 , t ) = u ( L, t ) = u x (0 , t ) = u x ( L, t ) = 0 ,u xx (0 , t ) + α η xx (0 , t ) = 0 , u xx ( L, t ) − α η xx ( L, t ) = 0 , α , α > Definition . Given
T >
0, ( η , u ) ∈ X , ( h , h ) ∈ L (0 , T ; X − ) and f, g ∈ H − (0 , T ),consider the non-homogeneous system(3.3) η t + u x − au xxx + bu xxxxx = h , in (0 , L ) × (0 , T ) ,u t + η x − aη xxx + bη xxxxx = h , in (0 , L ) × (0 , T ) ,η (0 , t ) = η ( L, t ) = η x (0 , t ) = η x ( L, t ) = 0 , η xx (0 , t ) = f ( t ) , on (0 , T ) ,u (0 , t ) = u ( L, t ) = u x (0 , t ) = u x ( L, t ) = 0 , u xx ( L, t ) = g ( t ) , on (0 , T ) ,η ( x,
0) = η ( x ) , u ( x,
0) = u ( x ) , on (0 , L ) . A solution of the problem (3.3) is a function ( η, u ) in C ([0 , T ]; X ) such that, for any τ ∈ [0 , T ]and ( ϕ τ , ψ τ ) ∈ X , the following identity holds(( η ( τ ) , u ( τ )) , ( ϕ τ , ψ τ )) X = (( η , u ) , ( ϕ (0) , ψ (0))) X + (cid:10) f ( t ) , χ (0 ,τ ) ( t ) ψ xx (0 , t ) (cid:11) H − (0 ,T ) ,H (0 ,T ) + (cid:10) g ( t ) , χ (0 ,τ ) ( t ) ϕ xx ( L, t ) (cid:11) H − (0 ,T ) ,H (0 ,T ) + Z τ h ( h ( t ) , h ( t )) , ( ϕ ( t ) , ψ ( t )) i ( X − ,X ) dt, (3.4) where ( · , · ) X is the inner product of X , h· , ·i is the duality of two spaces, χ (0 ,τ ) ( · ) denotesthe characteristic function of the interval (0 , τ ) and ( ϕ, ψ ) is the solution of(3.5) ϕ t + ψ x − aψ xxx + bψ xxxxx = 0 , in (0 , L ) × (0 , τ ) ,ψ t + ϕ x − aϕ xxx + bϕ xxxxx = 0 in (0 , L ) × (0 , τ ) ,ϕ (0 , t ) = ϕ ( L, t ) = ϕ x (0 , t ) = ϕ x ( L, t ) = ϕ xx (0 , t ) = 0 , on (0 , τ ) ,ψ (0 , t ) = ψ ( L, t ) = ψ x (0 , t ) = ψ x ( L, t ) = ψ xx ( L, t ) = 0 , on (0 , τ ) ,ϕ ( x, τ ) = ϕ τ , ψ ( x, τ ) = ψ τ , on (0 , L ) . The well-posedness of (3.5) is guaranteed by Corollary 2.5 and (2.10).
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 (0 , τ ) , by Proposition 2.7. The fact that χ (0 ,τ ) ψ xx (0 , · ) and χ (0 ,τ ) ϕ xx ( L, · ) belong to H (0 , T ) , for any τ ∈ [0 , T ] , follows from [16, Theorem 11.4, p.60] . The next result borrowed from [9], with minor changes, gives us the existence anduniqueness of solution for system (3.3). Its proof is presented here for the sake of complete-ness.
Lemma 3.2.
Let
T > , ( η , u ) ∈ X , ( h , h ) ∈ L (0 , T ; X − ) and f, g ∈ H − (0 , T ) .There exists a unique solution ( η, u ) ∈ C ([0 , T ]; X ) of the system (3.3) . Moreover, thereexists a positive constant C T , such that k ( η ( τ ) , u ( τ )) k X ≤ C T (cid:16) k ( η , u ) k X + k f k H − (0 ,T ) + k g k H − (0 ,T ) + k ( h , h ) k L (0 ,T ; X − ) (cid:1) , (3.6) for all τ ∈ [0 , T ] .Proof. Let
T > τ ∈ [0 , T ]. From Proposition 2.4, e A defined by (2.7)-(2.6) is skewadjoint and generated a C − semigroup e S ( t ). Note that making the change of variable( x, t ) ( ϕ ( x, τ − t ) , ψ ( x, τ − t )) and taking ( ϕ τ , ψ τ ) ∈ X , we have that the solution of (3.5)is given by ( ϕ, ψ ) = e S ∗ ( τ − t )( ϕ τ , ψ τ ) = − e S ( τ − t )( ϕ τ , ψ τ ) . Moreover, (2.10) implies that ( ϕ, ψ ) ∈ C ( R ; X ) . In particular, there exists C T >
0, such that(3.7) k ( ϕ ( t ) , ψ ( t )) k X = k e S ∗ ( τ − t )( ϕ τ , ψ τ ) k X ≤ C T k ( ϕ τ , ψ τ ) k X , ∀ t ∈ [0 , τ ] . Let us define L a linear functional given by the right hand side of (3.4), that is, L ( ϕ τ , ψ τ ) = (cid:16) ( η , u ) , e S ∗ ( τ )( ϕ τ , ψ τ ) (cid:17) X + (cid:28) ( g ( t ) , f ( t )) , χ (0 ,τ ) ( t ) d dx ( e S ∗ ( τ − t )( ϕ τ , ψ τ )) (cid:12)(cid:12)(cid:12) L (cid:29) ( H − (0 ,T ) ,H (0 ,T )) + Z τ D ( h ( t ) , h ( t )) , e S ∗ ( τ − t )( ϕ τ , ψ τ ) E ( X − ,X ) dt. Claim. L belongs to L ( X ; R ) . IGHER ORDER BOUSSINESQ SYSTEM AND ITS PROPERTIES 13
Indeed, from the fact X ⊂ X and Proposition 2.7, we obtain that | L ( ϕ τ , ψ τ ) | ≤ C T k ( η , u ) k X k ( ϕ τ , ψ τ )) k X + C T k ( ϕ τ , ψ τ ) k X k ( h , h ) k L (0 ,T ; X − ) + C T k ( f, g ) k ( H − (0 ,T )) k ( ϕ xx ( L ) , ψ xx (0) k ( H (0 ,τ )) ≤ C T (cid:16) k ( η , u ) k X + k ( f, g ) k ( H − (0 ,T )) + k ( h , h ) k L (0 ,T ; X − ) (cid:17) k ( ϕ τ , ψ τ )) k X , where in the last inequality we use (3.7). Then, from Riesz representation Theorem, thereexist one and only one ( η τ , u τ ) ∈ X such that(3.8) (( η τ , u τ ) , ( ϕ τ , ψ τ )) X = L ( ϕ τ , ψ τ ) , with k ( η τ , u τ ) k X = k L k L ( X ; R ) and the uniqueness of the solution to the problem (3.3) holds.We prove now that the solution of the system (3.3) satisfies (3.6). Let ( η, u ) : [0 , T ] → X be defined by(3.9) ( η ( τ ) , u ( τ )) := ( η τ , u τ ) , ∀ τ ∈ [0 , T ] . From (3.8) and (3.9), (3.4) follows and k ( η ( τ ) , u ( τ )) k X = k L k L ( X ; R ) ≤ C T (cid:16) k ( η , u ) k X + k f k ( H − (0 ,T )] + k g k ( H − (0 ,T )] + k ( h , h ) k L (0 ,T ; X − ) (cid:17) . In order to prove that the solution ( η, u ) belongs to C ([0 , T ]; X ), let τ ∈ [0 , T ] and { τ n } n ∈ N be a sequence such that τ n −→ τ, as n → ∞ .Consider ( ϕ τ , ψ τ ) ∈ X and { ( ϕ τ n , ψ τ n ) } n ∈ N be a sequence in X such that(3.10) ( ϕ τ n , ψ τ n ) → ( ϕ τ , ψ τ ) strongly in X , as n → ∞ .Note that lim n →∞ (cid:16) ( η , w ) , e S ∗ ( τ n )( ϕ τ n , ψ τ n ) (cid:17) X = (cid:16) ( η , w ) , e S ∗ ( τ )( ϕ τ , ψ τ ) (cid:17) X . (3.11)Indeed,lim n →∞ (cid:16) ( η , w ) , e S ∗ ( τ n )( ϕ τ n , ψ τ n ) (cid:17) X = lim n →∞ (cid:16) ( η , w ) , e S ∗ ( τ n ) (( ϕ τ n , ψ τ n ) − ( ϕ τ , ψ τ )) (cid:17) X + lim n →∞ (cid:16) ( η , w ) , e S ∗ ( τ n )( ϕ τ , ψ τ ) (cid:17) X . From (3.10) and since { e S ( t ) } t ≥ is a strongly continuous group of continuous linear operatorson X , we have lim n →∞ (cid:16) ( η , w ) , e S ∗ ( τ n ) (( ϕ τ n , ψ τ n ) − ( ϕ τ , ψ τ )) (cid:17) X = 0and consequently,lim n →∞ (cid:16) ( η , w ) , e S ∗ ( τ n )( ϕ τ , ψ τ ) (cid:17) X = (cid:16) ( η , w ) , e S ∗ ( τ )( ϕ τ , ψ τ ) (cid:17) X . Thus, (3.11) follows. Now, we have to analyze the following limits,lim n →∞ (cid:28) ( g ( t ) , f ( t )) , χ (0 ,τ n ) ( t ) d dx ( e S ∗ ( τ n − t )( ϕ τ n , ψ τ n )) (cid:12)(cid:12)(cid:12) L (cid:29) ( H − (0 ,T ) ,H (0 ,T )) (3.12) and lim n →∞ Z τ D ( h ( t ) , h ( t )) , e S ∗ ( τ n − t )( ϕ τ n , ψ τ n ) E ( X − ,X ) dt. (3.13)In fact, observe that, by group properties of e S ∗ and Proposition 2.7, we have that (cid:13)(cid:13)(cid:13)(cid:13) d dx e S ∗ ( τ − t )( ϕ τ , ψ τ ) (cid:12)(cid:12)(cid:12) L (cid:13)(cid:13)(cid:13)(cid:13) [ H (0 ,τ )] ≤ C k ( ϕ τ , ψ τ ) k X . Thus, the linear map ( ϕ τ , ψ τ ) ∈ X d dx ( e S ∗ ( τ − · )( ψ τ , ϕ τ )) (cid:12)(cid:12)(cid:12) L belongs to H (0 , τ ; R )and it is continuous. Moreover, as the natural extension by 0 outside (0 , τ ) is a continuousmapping from H (0 , τ ) into H (0 , T ) (cf. [16, Theorem 11.4, p. 60]), we obtain that themap ( ϕ τ , ψ τ ) ∈ X χ (0 ,τ n ) ( · ) d dx ( e S ∗ ( τ − · )( ψ τ , ϕ τ )) (cid:12)(cid:12)(cid:12) L belongs to H (0 , T ; R ) and it iscontinuous, as well. Since a continuous linear map between two Hilbert spaces is weaklycontinuous, (3.10) implies that(3.14) χ (0 ,τ n ) ( t ) d dx ( e S ∗ ( τ n − · )( ϕ τ n , ψ τ n )) (cid:12)(cid:12)(cid:12) L ⇀ χ (0 ,τ ) ( t ) d dx ( e S ∗ ( τ − · )( ϕ τ , ψ τ )) (cid:12)(cid:12)(cid:12) L , weakly in H ([0 , T ]; R ), as n → ∞ . Thus, by using (3.14), the limit (3.12) yields that(3.15) lim n →∞ (cid:28) ( g ( t ) , f ( t )) , χ (0 ,τ n ) ( t ) d dx ( e S ∗ ( τ n − t )( ϕ τ n , ψ τ n )) (cid:12)(cid:12)(cid:12) L (cid:29) ( H − (0 ,T ) ,H (0 ,T )) = (cid:28) ( g ( t ) , f ( t )) , χ (0 ,τ ) ( t ) d dx ( e S ∗ ( τ − t )( ϕ τ , ψ τ )) (cid:12)(cid:12)(cid:12) L (cid:29) ( H − (0 ,T ) ,H (0 ,T )) . On the other hand, extending by zero the functions h i , for i = 1 ,
2, we obtain elements of[ H ( − T, T )] ′ and L ( − T, T ; X − ), that is, h i ≡ − T, × (0 , L ) , and setting s = τ n − t , we have that(3.16) Z τ n D ( h ( t ) , h ( t )) , e S ∗ ( τ n − t )( ϕ τ n , ψ τ n ) E ( X − ,X ) dt = Z T χ (0 ,τ n ) ( s ) D ( h ( τ n − s ) , h ( τ n − s )) , e S ∗ ( s )( ϕ τ n , ψ τ n ) E ( X − ,X ) dt. Similarly, taking s = τ − t in (3.13), we get(3.17) Z τ D ( h ( t ) , h ( t )) , e S ∗ ( τ − t )( ϕ τ , ψ τ ) E ( X − ,X ) dt = Z T χ (0 ,τ ) ( s ) D ( h ( τ − s ) , h ( τ − s )) , e S ∗ ( s )( ϕ τ , ψ τ ) E ( X − ,X ) dt. Since the translation in time is continuous in L (0 , T ; X − ) and using the dominated conver-gence theorem, we obtain(3.18) χ n ( · )( h ( τ n − · , · ) , h ( τ n − · , · )) −→ χ ( · )( h ( τ − · , · ) , h ( τ − · , · )) , in L (0 , T ; X − ), as n → ∞ . Similarly, by the strong continuity of the group, it follows that e S ∗ ( · )( ϕ τ n , ψ τ n ) ⇀ e S ∗ ( · )( ϕ τ , ψ τ ) IGHER ORDER BOUSSINESQ SYSTEM AND ITS PROPERTIES 15 weakly in L ( − T, T ; X ), as n → ∞ . In particular, we obtain that(3.19) e S ∗ ( · )( ϕ τ n , ψ τ n ) ⇀ e S ∗ ( · )( ϕ τ , ψ τ ) , weakly in L ( − T, T ; X ), as n → ∞ . By using (3.16), (3.17), (3.18) and (3.19), the limit(3.13) yields that(3.20) lim n →∞ Z τ n D ( h ( t ) , h ( t )) , e S ∗ ( τ n − t )( ϕ τ n , ψ τ n ) E ( X − ,X ) dt = Z τ D ( h ( t ) , h ( t )) , e S ∗ ( τ − t )( ϕ τ , ψ τ ) E ( X − ,X ) dt. Finally, from (3.8), (3.9), (3.11), (3.15) and (3.20), one gets(( η ( τ n ) , w ( τ n )) , ( ϕ τ n , ψ τ n )) X −→ (( η ( τ ) , w ( τ )) , ( ϕ τ , ψ τ ))) X , as n → ∞ , which implies that( η ( τ n ) , w ( τ n )) −→ ( η ( τ ) , w ( τ )) in X , as n → ∞ . This concludes the proof. (cid:3)
The next result establishes the well-posedness of the non-homogeneous feedback linearsystem associated to (3.3).
Lemma 3.3.
Let
T > . Then, for every ( η , u ) in X and ( h , h ) in L (0 , T ; X − ) , thereexists a unique solution ( η, u ) of the system (3.3) such that ( η, u ) ∈ C ([0 , T ]; X ) , with f ( t ) := − α η xx (0 , t ) and g ( t ) := α η xx ( L, t ) , where α and α belong to R . Moreover,for some positive constant C = C ( T ) , we have k ( η ( t ) , u ( t )) k X ≤ C (cid:0) k ( η , u ) k X + k ( h , h ) k L (0 ,T ; X − ) (cid:1) , ∀ t ∈ [0 , T ] . Proof.
Firstly, note that if ( η, u ) ∈ C ([0 , T ]; X ), then f ( t ) := − α η xx (0 , t ) and g ( t ) := α η xx ( L, t ) ∈ H − (0 , T ) . In fact, by using the continuous embedding L (0 , T ) ⊂ H − (0 , T ) and the trace theorem [1,Theorem 7.53], there exists a positive constant C := C ( L, α , α ) such that k α η xx (0 , · ) k H − (0 ,T ) + k α η xx ( L, · ) k H − (0 ,T ) ≤ C Z T (cid:0) α η xx (0 , t ) + α η xx ( L, t ) (cid:1) dt ≤ C Z T k η ( t ) k X dt ≤ T C k ( η, u ) k C ([0 ,T ]; X ) . (3.21)Let 0 < β ≤ T that will be determinate later. For each ( η , u ) ∈ X , consider the mapΓ : C ([0 , β ]; X ) −→ C ([0 , β ]; X )( η, u ) Γ( η, u ) = ( w, v )where, ( w, v ) is the solution of the system (3.3) with f ( t ) = − α η xx (0 , t ) and g ( t ) = α 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 k Γ( η, u ) k C ([0 ,β ]; X ) ≤ C β (cid:16) k ( η , u ) k X + k ( α η xx (0 , t ) , α η xx ( L, t )) k ( H − (0 ,β )) + k ( h , h ) k L (0 ,T ; X − ) (cid:17) . Then, k Γ( η, u ) k C ([0 ,β ]; X ) ≤ C T (cid:16) k ( η , u ) k X + k ( h , h ) k L (0 ,T ; X − ) (cid:17) + C T β k ( η, w ) k C ([0 ,β ]; X ) . Let ( η, u ) ∈ B R (0) := { ( η, u ) ∈ C ([0 , β ]; X ) : k ( η, u ) k C ([0 ,β ]; X ) ≤ R } , with R = 2 C T (cid:16) k ( η , u ) k X + k ( h , h ) k L (0 ,T ; X − ) (cid:17) . Choosing β such that C T β ≤ , it implies that k Γ( η, u ) k C ([0 ,β ]; X ) ≤ R , for all ( η, u ) ∈ B R (0), i.e, Γ maps B R (0) into B R (0).On the other hand, note that k Γ( η , u ) − Γ( η , u ) k C ([0 ,β ]; X ) ≤ C T β k ( η − η , u − u ) k C ([0 ,β ]; X ) ≤ k ( η − η , u − u ) k C ([0 ,β ]; X ) . Hence, Γ : B R (0) −→ B R (0) is a contraction and, by Banach fixed point theorem, we obtaina unique ( η, u ) ∈ B R (0), such that Γ( η, u ) = ( η, u ) and k ( η, u ) k C ([0 ,β ]; X ) ≤ C T (cid:16) k ( η , u ) k X + k ( h , h ) k L (0 ,T ; X − ) (cid:17) . Since the choice of β is independent of ( η , u ), the standard continuation extension argumentyields that the solution ( η, u ) belongs to C ([0 , β ]; X ), thus, the proof is complete. (cid:3) We are now in position to prove one of the main result of this article.3.1.
Proof of Theorem 1.1.
Let
T > k ( η , u ) k X < ρ , where ρ > η, u ) ∈ C ([0 , T ]; X ), there exists a positive constant C suchthat k ηu x k L (0 ,T ; L (0 ,L )) ≤ Z T k η ( t ) k L ∞ (0 ,L ) k u x ( t ) k L (0 ,L ) dt ≤ C ′ Z T k η ( t ) k H (0 ,L ) k u ( t ) k H (0 ,L ) dt ≤ C T k ( η, u ) k C ([0 ,T ]; X ) , (3.22) k ηu xx k L (0 ,T ; L (0 ,L )) ≤ Z T k η ( t ) k L ∞ (0 ,L ) k u xx ( t ) k L (0 ,L ) dt ≤ C ′ Z T k η ( t ) k H (0 ,L ) k u ( t ) k H (0 ,L ) dt ≤ C T k ( η, u ) k C ([0 ,T ]; X ) , (3.23) IGHER ORDER BOUSSINESQ SYSTEM AND ITS PROPERTIES 17 k η x u xx k L (0 ,T ; L (0 ,L )) ≤ Z T k η x ( t ) k L ∞ (0 ,L ) k u xx ( t ) k L (0 ,L ) dt ≤ C ′ Z T k η ( t ) k H (0 ,L ) k u ( t ) k H (0 ,L ) dt ≤ C T k ( η, u ) k C ([0 ,T ]; X ) , (3.24)and k ηu xxx k L (0 ,T ; L (0 ,L )) ≤ Z T k η ( t ) k L ∞ (0 ,L ) k u xxx ( t ) k L (0 ,L ) dt ≤ C ′ Z T k η ( t ) k H (0 ,L ) k u ( t ) k H (0 ,L ) dt ≤ C T k ( η, u ) k C ([0 ,T ]; X ) . (3.25)This implies that, for any ( η, u ) ∈ C ([0 , T ]; X ) and a i ∈ R , with i = 1 , , ,
4, we have that − a ( ηu ) x − a ( ηu xx ) , − a uu x − a ( ηη xx ) x − a u x u xx ∈ L (0 , T ; X ) ⊂ L (0 , T ; X − ) . Consider the following linear mapΓ : C ([0 , T ]; X ) −→ C ([0 , T ]; X )( η, u ) Γ( η, w ) = ( η, u ) , where ( η, u ) is the solution of the system (3.3) with( h , h ) = ( − a ( ηu ) x − a ( ηu xx ) , − a uu x − a ( ηη xx ) x − a u x u xx )in L (0 , T ; X − ), with f ( t ) := − α η xx (0 , t ) and g ( t ) := α η xx ( L, t ). Claim.
The map Γ is well-defined, maps B R (0) into itself and it is a contraction in a ball. Indeed, firstly note that Lemma 3.3 ensures that Γ is well-defined, moreover, usingLemma 3.2, there exists a positive constant C T , such that k Γ( η, u ) k C ([0 ,T ]; X ) ≤ C T (cid:16) k ( η , u ) k X + k a ( ηu ) x + a ( ηu xx ) k L (0 ,T ; X − ) + k a uu x + a ( ηη xx ) x + a u x u xx k L (0 ,T ; X − ) + k ww x k L (0 ,T ; X − ) (cid:17) . Then, equations (3.22), (3.23), (3.24) and (3.25) yield that k Γ( η, u ) k C ([0 ,T ]; X ) ≤ C T k ( η , u ) k X + (3 | α | + | a | + 2 | a | + | a | ) T / C T C k ( η, u ) k C ([0 ,T ]; X ) ≤ C T k ( η , u ) k X + 7 M T / C T C k ( η, u ) k C ([0 ,T ]; X ) , (3.26)where M = max {| a | , | a | , | a | , | a |} . Consider the ball B R (0) = n ( η, u ) ∈ C ([0 , T ]; X ) : k ( η, u ) k C ([0 ,T ]; X ) ≤ R o , where R = 2 C T k ( η , u ) k X . From the estimate (3.26) we get that k Γ( η, u ) k C ([0 ,T ]; X ) ≤ R M T / C T C R < R M T / C T C ρR, for all ( η, u ) ∈ B R (0). Consequently, if we choose ρ > M T / C T C ρ < , Γ maps the ball B R (0) into itself. Finally, note that k Γ( η , u ) − Γ( η , u ) k C ([0 ,T ]; X ) ≤ C T k a (( η u ) x − ( η u ) x ) k L (0 ,T ; X − ) + C T k a (( η u ,xx ) x − ( η u ,xx ) x ) k L (0 ,T ; X − ) + C T k a ( u u ,x − u u ,x ) k L (0 ,T ; X − ) + C T k a (( η η ,xx ) x − ( η η ,xx ) x ) k L (0 ,T ; X − ) + C T k a ( u ,x u ,xx − u u ,xx ) k L (0 ,T ; X − ) . Thus, we obtain k Γ( η , u ) − Γ( η , u ) k C ([0 ,T ]; X ) ≤ T / C T C M ( k η k C ([0 ,T ]; H (0 ,L )) + k η k C ([0 ,T ]; H (0 ,L )) ) k u − u k C ([0 ,T ]; H (0 ,L )) + 3 T / C T C M ( k u k C ([0 ,T ]; H (0 ,L )) + k u k C ([0 ,T ]; H (0 ,L )) ) k η − η k C ([0 ,T ]; H (0 ,L )) + 2 T / C T C M ( k u k C ([0 ,T ]; H (0 ,L )) + k u k C ([0 ,T ]; H (0 ,L )) ) k u − u k C ([0 ,T ]; H (0 ,L )) . Finally, it follows that k Γ( η , u ) − Γ( η , u ) k C ([0 ,T ]; X ) ≤ T / C T C M R k ( η − η , u − u ) k C ([0 ,T ]; X ) < T / C T C M ρ k ( η − η , u − u ) k C ([0 ,T ]; X ) Therefore, from (3.27), we get k Γ( η , u ) − Γ( η , u ) k C ([0 ,T ]; X ) ≤ k ( η − η , u − u ) k C ([0 ,T ]; X ) , for all ( η, u ) ∈ B T (0). Hence, Γ : B R (0) −→ B R (0) is a contraction and the claim is archived.Thanks to Banach fixed point theorem, we obtain a unique ( η, u ) ∈ B R , such thatΓ( η, u ) = ( η, u ) and k ( η, w ) k C ([0 ,T ]; X ) ≤ C T k ( η , u ) k X . Thus, the proof is archived. (cid:3)
Well-posedness in time.
Adapting the proof of Theorem 1.1, one can also prove,without any restriction over the initial data ( η , u ), that there exist T ∗ > η, u ) of (3.1)-(3.2), satisfying the initial condition η ( · ,
0) = η ( · ) and u ( · ,
0) = u ( · ). Moreprecisely, Theorem 3.4.
Let ( η , u ) ∈ X . Then, there exists T ∗ > a unique solution ( η, u ) ∈ C ([0 , T ∗ ]; X ) of (3.1) - (3.2) . Moreover k ( η, u ) k C ([0 ,T ]; X ) ≤ C k ( η , u ) k X , for some positive constant C = C ( T ∗ ) . Observe that if ( η , u ) ∈ C ([0 , T ]; X ) and ( η , u ) ∈ C ([0 , T ]; X ) are the solutionsgiven by the Theorem 1.1 with initial data ( η , u ) and ( η ( T ) , u ( T )), respectively, thefunction ( η, u ) : [0 , T + T ] → X defined by( η ( t ) , u ( t )) = ( ( η ( t ) , u ( t )) if t ∈ [0 , T ] , ( η ( t − T ) , u ( t − T )) if t ∈ [ T , T + T ] , is the solution of the feedback system on interval [0 , T + T ] with initial data ( η , u ).This argument allows us to extend a local solution until a maximal interval, that is, for all0 < T < T max ≤ ∞ there exists a function ( η, u ) ∈ C ([0 , T ]; X ), solution of the feedbacksystem (3.1)-(3.2). The following proposition, easily holds: IGHER ORDER BOUSSINESQ SYSTEM AND ITS PROPERTIES 19
Proposition 3.5.
Let ( η , u ) ∈ X and ( η, w ) ∈ C ([0 , T ]; X ) solution of the feedbacksystem, for all < T < T max , with initial data ( η , u ) . Then, only one of the followingassertions hold: (i) T max = ∞ ; (ii) If T max < ∞ , then, lim t → T max k ( η ( t ) , w ( t )) k X = ∞ . 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 > , such that (4.1) k ( η , u ) k X ≤ C Z T (cid:0) | η xx ( L, t ) | + | η xx (0 , t ) | (cid:1) dt, where ( η, u ) is the solution of (2.1) given by Proposition 2.2. Indeed, if (4.1) is true, we get E ( T ) − E (0) ≤ − E (0) C , where E ( t ) is defined by (1.4). This implies that E ( T ) ≤ E (0) − E (0) C ≤ E (0) − E ( T ) C .
Thus, E ( T ) ≤ (cid:18) CC + 1 (cid:19) E (0) , which gives Theorem 1.2 by using semigroup properties associated to the model. (cid:3) We will divide the proof of the observability inequality (4.1) in three steps as follows:
Proof of (4.1) . Step 1:
Compactness-uniqueness argument
We argue by contradiction. Suppose that (4.1) does not hold, then there exists asequence { ( η ,n , u ,n ) } n ∈ N ∈ X , such that(4.2) 1 = k ( η ,n , u ,n ) k X > n Z T (cid:0) | η n,xx ( L, t ) | + | η n,xx (0 , t ) | (cid:1) dt, where ( η n ( t ) , u n ( t )) = S ( t )( η ,n , u ,n ). Thus, from (4.2) we obtain(4.3) lim n →∞ Z T (cid:0) | η n,xx ( L, t ) | + | η n,xx (0 , t ) | (cid:1) dt = 0 . Estimate (2.4) in Proposition 2.3, together with (4.2), imply that the sequence { ( η n , u n ) } n ∈ N is bounded in L (0 , T ; X ). Furthermore, by (2.1) we deduce that { ( η n,t , u n,t ) } n ∈ N is boundedin L (0 , T ; X − ). Thus, the compact embedding(4.4) X ֒ → X ֒ → X − , allows us to conclude that { ( η n , u n ) } n ∈ N is relatively compact in L (0 , T ; X ) and, conse-quently, we obtain a subsequence, still denoted by the same index n , satisfying(4.5) ( η n , u n ) → ( η, u ) in L (0 , T ; X ) , as n → ∞ . Moreover, using (2.3), (4.3) and (4.5), we obtain that { ( η ,n , u ,n ) } n ∈ N is a Cauchy sequencein X . Hence, there exists ( η , u ) ∈ X , such that(4.6) ( η ,n , u ,n ) → ( η , u ) in X , as n → ∞ , and, from (4.2) we get k ( η , u ) k X = 1. On the other hand, note that combining (2.2), (4.3)and (4.6), we obtain a subsequence { ( η n , u n ) } n ∈ N , such that(4.7) ( η n , u n ) → ( η, u ) in C ([0 , T ]; X ) , as n → ∞ . In particular, ( η (0) , u (0)) = lim n →∞ ( η n (0) , u n (0)) = lim n →∞ ( η ,n , u ,n ) = ( η , u ) . Consequently, passing to the weak limit, by Proposition 2.2, we obtain( η ( t ) , u ( t )) = S ( t )( η , u ) . Moreover, from (4.3), we obtain that Z T (cid:0) | η xx ( L, t ) | + | η xx (0 , t ) | (cid:1) dt ≤ lim inf n →∞ Z T (cid:0) | η n,xx ( L, t ) | + | η n,xx (0 , t ) | (cid:1) dt. Thus, we have that ( η, u ) is the solution of the IBVP (2.1) with initial data ( η , w ) whichsatisfies, additionally,(4.8) η xx ( L, t ) = η xx (0 , t ) = 0and(4.9) k ( η , u ) k X = 1 . Notice that (4.9) implies that the solution ( η, u ) can not be identically zero. However, fromlemma bellow, one can conclude that ( η, u ) = (0 , (cid:3) Step 2:
Reduction to a spectral problem
Lemma 4.1.
For any
T > , let N T denote the space of the initial states ( η , u ) ∈ X , suchthat the solution ( η ( t ) , u ( t )) = S ( t )( η , u ) of (2.1) satisfies (4.8) . Then, N T = { } .Proof. The proof uses the same arguments as those given in [10, Theorem 3.7]. If N T = { } ,the map ( η , u ) ∈ C N T → A ( N T ) ⊂ C N T (where C N T denote the complexification of N T )has (at least) one eigenvalue. Hence, there exists λ ∈ C and η , u ∈ H (0 , L ) \ { } , suchthat λη + u ′ − au ′′′ + bu ′′′′′ = 0 , in (0 , L ) ,λu + η ′ − aη ′′′ + bη ′′′′′ = 0 , in (0 , L ) ,η (0) = η ( L ) = η ′ (0) = η ′ ( L ) = η ′′ (0) = η ′′ ( L ) = 0 ,u (0) = u ( L ) = u ′ (0) = u ′ ( L ) = u ′′ (0) = u ′′ ( L ) = 0 . To obtain the contradiction, it remains to prove that a triple ( λ, η , u ) as above does notexist. (cid:3) Step 3:
M¨obius transformation
To simplify the notation, henceforth we denote ( η , u ) := ( η, u ). Moreover, the notation { , L } means that the function is applied to 0 and L , respectively. IGHER ORDER BOUSSINESQ SYSTEM AND ITS PROPERTIES 21
Lemma 4.2.
Let
L > and consider the assertion ( N ) : ∃ λ ∈ C , ∃ ( η, u ) ∈ ( H (0 , L ) ∩ H (0 , L )) such that λη + u ′ − au ′′′ + bu ′′′′′ = 0 , in (0 , L ) ,λu + η ′ − aη ′′′ + bη ′′′′′ = 0 , in (0 , L ) ,η ( x ) = η ′ ( x ) = η ′′ ( x ) = 0 , in { , L } ,u ( x ) = u ′ ( x ) = u ′′ ( x ) = 0 , in { , L } . Then, if ( λ, η, u ) ∈ C × ( H (0 , L ) ∩ H (0 , L )) is solution of ( N ) , then η = u = 0 . Proof.
Firstly, let us consider the following change of variable ϕ ( x ) = η ( x ) ± u ( x ), thus wehave the problem in only one equation:(4.10) ( λϕ + ϕ ′ − aϕ ′′′ + bϕ ′′′′′ = 0 , in (0 , L ) ,ϕ ( x ) = ϕ ′ ( x ) = ϕ ′′ ( x ) = 0 , in { , L } . Note that, if we multiply the equation in (4.10) by ϕ and integrate in [0 , L ], it is easy to seethat λ is purely imaginary, i.e., λ = ir , for r ∈ R . Now, we extend the function ϕ to R bysetting ϕ ( x ) = 0 for x [0 , L ]. The extended function satisfies λϕ + ϕ ′ − aϕ ′′′ + bϕ ′′′′′ = bϕ ′′′′ (0) δ ′ − bϕ ′′′′ ( L ) δ ′ L + bϕ ′′′ (0) δ − bϕ ′′′ ( L ) δ L , in S ′ ( R ), 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 Fouriertransform of each term in the above system and integrating by parts, we obtain λ ˆ ϕ ( ξ ) + iξ ˆ ϕ ( ξ ) − a ( iξ ) ˆ ϕ ( ξ ) + b ( iξ ) ˆ ϕ ( ξ ) = b ( iξ ) ϕ ′′′ (0) − b ( iξ ) ϕ ′′′ ( L ) e − iLξ + bϕ ′′′′ (0) − bϕ ′′′′ ( L ) e − iLξ . Setting λ = − ir and f α ( ξ, L ) = i ˆ ϕ ( ξ ), from the equation above it follows that f α ( ξ, L ) = N α ( ξ, L ) q ( ξ ) , with N α ( · , L ) defined by(4.11) N α ( ξ, L ) = α iξ − α iξe − iξL + α − α e − iξL and q ( ξ ) = bξ + aξ + ξ + r, where α i , for i = 1 , , ,
4, are the traces of bϕ ′′′ and bϕ ′′′′ .For each r ∈ R and α ∈ C \ { } let F αr be the set of L > 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 arezeros 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 α iξ i + α α iξ i + α = e − iLξ i , where ξ i denotes the zeros of q ( ξ ), for i = 1 , , , ,
5. Let us define, for α ∈ C \ { } , thefollowing discriminant(4.12) d ( α ) = α α − α α . Then, for α ∈ C \ { } , such that d ( α ) = 0 the M¨obius transformations can be introducedby(4.13) M ( ξ i ) = e − iLξ i , 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 and two pairs ofcomplex conjugate roots.Proof of the Claim 1. Initially, we suppose that r = 0. Note that the derivative of q is givenby q ′ ( ξ ) = 5 bξ + 3 aξ + 1 , and its zeros are ± z and ± z , where z = s − a − √ a − b b and z = s − a + √ a − b b . It is easy to see that z and z belong to C \ R . Hence, the polynomial q ( · ) does not havecritical points, which means that q ( · ) has exactly one real root. Suppose that ξ ∈ R is theroot of q ( · ) with multiplicity m ≤
5. Hence, q ( ξ ) = q ′ ( ξ ) = ... = q ( m − ( ξ ) = 0 . Consider the following cases:(i) If ξ has multiplicity 5, it follows that q ( ξ ) = 0 and q ′′′′ ( ξ ) = 120 bξ = 0, implyingthat ξ = 0 and r = 0.(ii) If ξ has multiplicity 4, it follows that q ′′′ ( ξ ) = 60 bξ + 6 a = 0, implying that ξ ∈ i R .(iii) If ξ has multiplicity 3, it follows that q ( ξ ) = 0 and q ′′ ( ξ ) = 20 bξ + 6 aξ = 0,implying that ξ = 0 and r = 0 or ξ ∈ i R .(iv) If ξ has multiplicity 4, it follows that q ′ ( ξ ) = 5 bξ + 3 aξ + 1 = 0, implying that ξ ∈ C \ R . In any case, we have a contradiction, since r = 0 and ξ ∈ R . Consequently, q ( · ) has exactlyone real root, with multiplicity 1. This means that this polynomial has two pairs of complexconjugate roots.Second, we suppose that r = 0. Initially, note that from the derivation of the model(see the Appendix) we have that 4 b > a . Then, we obtain that q ( ξ ) = ξ ( bξ + aξ + 1) , whose roots are 0 , ± ρ and ± k where(4.14) ρ = − a b + i √ b − a b and k = − a b − i √ b − a b = ρ = ρ . Thus, q ( · ) has two pairs of complex conjugate roots and one real root, proving Claim 1. (cid:3) Besides of the Claim 1 the following two auxiliary lemmas are necessary to conclude theproof of the Lemma 4.2. Their proofs can be found in [20, Lemmas 2.1 and 2.2], thus wewill omit them.
IGHER ORDER BOUSSINESQ SYSTEM AND ITS PROPERTIES 23
Lemma 4.3.
Let non null α ∈ C with d ( α ) = 0 and L > for d ( α ) defined in (4.12) . Then,the set of the imaginary parts of the zeros of N α ( · , L ) in (4.11) has at most two elements. Lemma 4.4.
For any
L > , there is no M¨obius transformation M , such that M ( ξ ) = e − iLξ , ξ ∈ { ξ , ξ , ¯ ξ , ¯ ξ } , with ξ , ξ , ¯ ξ , ¯ ξ all distinct in C . 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 M¨obius transformation. Let us assume, bycontradiction, that there exists L > f a ( · , L ) is entire. Then, allroots of the polynomial q ( · ) must satisfy (4.13), i.e., there exists a M¨obius transformationthat takes each root ξ of q ( · ) into e − iLξ . However, this contradicts Lemma 4.4 and provesthat if ( N ) holds then F αr = ∅ for all r ∈ R . On the other hand, suppose that d ( α ) = 0 andnote that by using the claim 1, we can conclude that the set of the imaginary parts of thepolynomial q ( · ) has at least three elements, thus it follows from Lemma 4.3 that F αr = ∅ forall r ∈ R . Note that in both cases, we have that F αr = ∅ , which implies that ( N ) only hasthe trivial solution for any L >
0, and the proof of Lemma 4.2 is archived. (cid:3)
To close this section we derive an exponential stability result in each space X s , for s ∈ [0 , s ∈ [0 , X s denote the collection of all the functions( η, u ) ∈ [ H s (0 , L )] := { ( η, u ) ∈ [ H s (0 , L )] : ( η, u ) ( j ) (0) = 0 , ( η, u ) ( j ) ( L ) = 0 } , for j = 0 , , ..., [ s ] , endowed with the Hilbertian norm k ( η, u ) k X s = k η k H s (0 ,L ) + k u k H s (0 ,L ) . Using Theorem 1.2 and some interpolation argument, we derive the following result.
Corollary 4.5.
Let α i , i = 1 , be as in Theorem 1.2. Then, for any s ∈ [0 , , there existsa constant C s > , such that, for any ( η , u ) ∈ X s , the solution ( η ( t ) , u ( t )) of (2.1) belongsto C ( R + ; X s ) and fulfills (4.15) k ( η ( t ) , u ( t )) k X s ≤ C e − µ t k ( η , u ) k X s , ∀ t ≥ . Proof. (4.15) was already been established for s = 0 in Theorem 1.2. Pick any U = ( η , u ) ∈ X = D ( A ) and write U ( t ) = ( η ( t ) , u ( t )) = S ( t ) U . Let V ( t ) = U t ( t ) = AU ( t ). Then V isthe mild solution of the system ( V t = AVV (0) = AU ∈ X , (4.16)hence, by using Theorem 1.2, estimate k V ( t ) k X ≤ C e − µ t k V k X , holds. Since V ( t ) = AU ( t ), V = AU , and the norms k U k X + k AU k X and k U k X are equivalent in X , weconclude that, for some constant C >
0, we have that k U ( t ) k X ≤ C e − µ t k U k X . This proves (4.15) for s = 5. The fact that (4.15) is still valid for 0 < s < X s = [ X , X ] s/ . (cid:3) For any real number s , [ s ] stands for its integer part. 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 anyglobal (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 energyspace. Due to the structure of the nonlinear terms, the same lack of a priori bounds alsooccurs when higher order Sobolev norms are considered (e. g. H s − norm). Because to thisstrict requirement, we cannot proceed as in [10, 18] and have only succeeded in derivinguniform decay results for the linear system. However, for the full system, we can findsolutions - in a certain sense - globally in time in X s . • Global well-posedness in time
Theorem 1.1 and Proposition 3.5 give us a positive answer to the well-posedness problem.However, the following questions are still open:
Question A . Is the nonlinear system (1.2) - (1.3) , global well-posedness in time? If yes,should we expect some restriction on the initial data? • One feedback on the boundary condition
If we consider in (1.2)-(1.3) with only one damping mechanism, that is, with α or α vanishing, we still have E ( t ), defined by (1.4), decreasing along the trajectories of thelinearized 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 forthe 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 toprove, by using, e.g., [10, 18], the exponential stability for the full model (1.2)-(1.3). Then,one natural question remains open:
Question E . Does the energy associated to the nonlinear system (1.2) - (1.3) , with one ortwo damping mechanism, converges to zero, as t → ∞ , for initial data on the energy space X ? Appendix
The following fifth-order Boussinesq system(6.1) ( η t + u x − au xxx + a ( ηu ) x + a ( ηu xx ) x + bu xxxxx = 0 ,u t + η x − aη xxx + a uu x + a ( ηη xx ) x + a u x u xx + bη xxxxx = 0 , with a > b > a = b , a > a < a > a >
0, can be derived from (1.1) witha carefully choice of the parameters θ , β and τ .Indeed, taking τ = − θ , we have that16 β (3 θ −
1) = β (cid:20)
12 (1 − θ ) − τ (cid:21) , IGHER ORDER BOUSSINESQ SYSTEM AND ITS PROPERTIES 25 and a = β (3 θ − θ = − √ ≈ , < and notingthat β >
0, it follows that(6.2) 5 (cid:18) θ − (cid:19) = ( θ − − θ )and a = 12 β (cid:18) θ − (cid:19) < . Note that (6.2) is equivalent to5 (cid:18) θ − (cid:19) = ( θ − (cid:18) − θ (cid:19) = ( θ − (cid:0) θ − τ (cid:1) = ( θ − θ −
1) + 12( θ − τ. Thus, 524 (cid:18) θ − (cid:19) = 124 ( θ − θ −
1) + 12 ( θ − τ ⇔ (cid:0) θ − θ + 1 (cid:1) = 124 ( θ − θ + 5) + 12 ( θ − τ, and b can define as b = β (cid:0) θ − θ + 1 (cid:1) = β (cid:20)
124 ( θ − θ + 5) + 12 ( θ − τ (cid:21) > . Finally, with the choice of a = α > , a = 12 αβ ( θ − < , a = αβ > a = αβ (2 − θ ) > , we obtain (6.1). Acknowledgments.
R. A. Capistrano–Filho was partially supported by CNPq (Brazil) bythe grants 306475/2017-0 and Propesq (UFPE) by Edital “Qualis A”. A. F. Pazoto waspartially supported by CNPq (Brazil). This work was carried out during two visits of thefirst author to the Federal University of Rio de Janeiro and one visit of the second author tothe Federal University of Pernambuco. The authors would like to thank both Universitiesfor its hospitality.
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