Uniform Stabilization of a family of Boussinesq systems

Transcription

1 Uniform Stabilization of a family of Boussinesq systems Ademir Pazoto Instituto de Matemática Universidade Federal do Rio de Janeiro (UFRJ) ademir@im.ufrj.br In collaboration with Sorin Micu - University of Craiova (Romania) Supported by Agence Universitaire de la Francophonie

2 Outline Description of the model: a family of Boussinesq systems Setting of the problem: stabilization of a coupled system of two Benjamin-Bona-Mahony (BBM) equations Main results Main Idea of the proofs Remarks and open problems

3 The Benjamin-Bona-Mahony (BBM) equation The BBM equation u t + u x u xxt + uu x =, (1) was proposed as an alternative model for the Korteweg-de Vries equation (KdV) u t + u x + u xxx + uu x =, (2) to describe the propagation of one-dimensional, unidirectional small amplitude long waves in nonlinear dispersive media. u(x, t) is a real-valued functions of the real variables x and t. In the context of shallow-water waves, u(x, t) represents the displacement of the water surface at location x and time t.

4 The Boussinesq system J. L. Bona, M. Chen, J.-C. Saut - J. Nonlinear Sci. 12 (22). { ηt + w x + (ηw) x + aw xxx bη xxt = w t + η x + ww x + cη xxx dw xxt =, (3) The model describes the motion of small-amplitude long waves on the surface of an ideal fluid under the gravity force and in situations where the motion is sensibly two dimensional. 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; a, b, c, d, are parameters required to fulfill the relations a + b = 1 ( θ 2 1 ), c + d = (1 θ2 ), where θ [, 1] specifies which velocity the variable w represents.

5 The Boussinesq system J. L. Bona, M. Chen, J.-C. Saut - J. Nonlinear Sci. 12 (22). { ηt + w x + (ηw) x + aw xxx bη xxt = w t + η x + ww x + cη xxx dw xxt =, (3) The model describes the motion of small-amplitude long waves on the surface of an ideal fluid under the gravity force and in situations where the motion is sensibly two dimensional. 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; a, b, c, d, are parameters required to fulfill the relations a + b = 1 ( θ 2 1 ), c + d = (1 θ2 ), where θ [, 1] specifies which velocity the variable w represents.

6 The Boussinesq system J. L. Bona, M. Chen, J.-C. Saut - J. Nonlinear Sci. 12 (22). { ηt + w x + (ηw) x + aw xxx bη xxt = w t + η x + ww x + cη xxx dw xxt =, (3) The model describes the motion of small-amplitude long waves on the surface of an ideal fluid under the gravity force and in situations where the motion is sensibly two dimensional. 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; a, b, c, d, are parameters required to fulfill the relations a + b = 1 ( θ 2 1 ), c + d = (1 θ2 ), where θ [, 1] specifies which velocity the variable w represents.

7 Stabilization Results: E(t) ce()e ωt, ω >, c > The Boussinesq system posed on a bounded interval: A. Pazoto and L. Rosier, Stabilization of a Boussinesq system of KdV-KdV type, System and Control Letters 57 (28), R. Capistrano Filho, A. Pazoto and L. Rosier, Control of Boussinesq system of KdV-KdV type on a bounded domain, Preprint. The Boussinesq system posed on the whole real axis: ( η xx, w xx ) M. Chen and O. Goubet, Long-time asymptotic behavior of dissipative Boussinesq systems, Discrete Contin. Dyn. Syst. Ser. 17 (27), The Boussinesq system posed on a periodic domain: S. Micu, J. H. Ortega, L. Rosier and B.-Y. Zhang, Control and stabilization of a family of Boussinesq systems, Discrete Contin. Dyn. Syst. 24 (29),

8 Stabilization Results: E(t) ce()e ωt, ω >, c > The Boussinesq system posed on a bounded interval: A. Pazoto and L. Rosier, Stabilization of a Boussinesq system of KdV-KdV type, System and Control Letters 57 (28), R. Capistrano Filho, A. Pazoto and L. Rosier, Control of Boussinesq system of KdV-KdV type on a bounded domain, Preprint. The Boussinesq system posed on the whole real axis: ( η xx, w xx ) M. Chen and O. Goubet, Long-time asymptotic behavior of dissipative Boussinesq systems, Discrete Contin. Dyn. Syst. Ser. 17 (27), The Boussinesq system posed on a periodic domain: S. Micu, J. H. Ortega, L. Rosier and B.-Y. Zhang, Control and stabilization of a family of Boussinesq systems, Discrete Contin. Dyn. Syst. 24 (29),

9 Stabilization Results: E(t) ce()e ωt, ω >, c > The Boussinesq system posed on a bounded interval: A. Pazoto and L. Rosier, Stabilization of a Boussinesq system of KdV-KdV type, System and Control Letters 57 (28), R. Capistrano Filho, A. Pazoto and L. Rosier, Control of Boussinesq system of KdV-KdV type on a bounded domain, Preprint. The Boussinesq system posed on the whole real axis: ( η xx, w xx ) M. Chen and O. Goubet, Long-time asymptotic behavior of dissipative Boussinesq systems, Discrete Contin. Dyn. Syst. Ser. 17 (27), The Boussinesq system posed on a periodic domain: S. Micu, J. H. Ortega, L. Rosier and B.-Y. Zhang, Control and stabilization of a family of Boussinesq systems, Discrete Contin. Dyn. Syst. 24 (29),

10 Controllability and Stabilization S. Micu, J. H. Ortega, L. Rosier, B.-Y. Zhang - Discrete Contin. Dyn. Syst. 24 (29). b, d, a, c or b, d, a = c >. { ηt + w x + (ηw) x + aw xxx bη xxt = f(x, t) w t + η x + ww x + cη xxx dw xxt = g(x, t) where < x < 2π and t >, with boundary conditions r η x r (, t) = r η (2π, t), xr and initial conditions η(x, ) = η (x), r w x r (, t) = r w (2π, t) xr w(x, ) = w (x). f and g are locally supported forces.

11 Periodic boundary conditions For b, d > and β 1, β 2, α 1, α 2, we consider the system η t + w x bη txx + (ηw) x + β 1 M α1 η =, w t + η x dw txx + ww x + β 2 M α2 w =, (4) with periodic boundary conditions and initial conditions η(, t) = η(2π, t); η x (, t) = η x (2π, t), w(, t) = w(2π, t); w x (, t) = w x (2π, t), η(x, ) = η (x), w(x, ) = w (x). In (4), M αj are Fourier multiplier operators given by ( ) M αj v k e ikx = (1 + k 2 ) α j 2 vk e ikx. k Z k Z

12 The energy associated to the model is given by E(t) = 1 2 2π and we can (formally) deduce that (η 2 + bη 2 x + w 2 + dw 2 x)dx (5) d dt E(t) = β 1 2π Since β 1, β 2 and 2π 2π (M α1 η) η dx β 2 (M α2 w) w dx (ηw) x η dx. (6) (M αj v, v) L 2 (,2π), j = 1, 2, the terms M α1 η and M α2 w play the role of feedback damping mechanisms, at least for the linearized system.

13 Assumptions on the Dissipation: T M αi ϕ(x)ϕ(x)dx Applications and study of asymptotic behavior os solutions: - J. L. Bona and J. Wu, M2AN Math. Model. Numer. Anal. (2). - J.-P. Chehab, P. Garnier and Y. Mammeri, J. Math. Chem. (21). - D. Dix, Comm. PDE (1992). - C. J. Amick, J. L. Bona and M. Schonbek, Jr. Diff. Eq. (1989). - P. Biler, Bull. Polish. Acad. Sci. Math. (1984). - J.-C. Saut, J. Math. Pures et Appl. (1979). Fractional derivative (Weyl fractional derivative operator): h(x) = a k e ikx Wx α (h)(x) = k Z k Z(ik) α a k e ikx, α (, 1).

14 Assumptions on the Dissipation: T M αi ϕ(x)ϕ(x)dx Applications and study of asymptotic behavior os solutions: - J. L. Bona and J. Wu, M2AN Math. Model. Numer. Anal. (2). - J.-P. Chehab, P. Garnier and Y. Mammeri, J. Math. Chem. (21). - D. Dix, Comm. PDE (1992). - C. J. Amick, J. L. Bona and M. Schonbek, Jr. Diff. Eq. (1989). - P. Biler, Bull. Polish. Acad. Sci. Math. (1984). - J.-C. Saut, J. Math. Pures et Appl. (1979). Fractional derivative (Weyl fractional derivative operator): h(x) = a k e ikx Wx α (h)(x) = k Z k Z(ik) α a k e ikx, α (, 1).

15 Main results The energy E(t) satisfies de dt = β 1 2π 2π (M α1 η) η dx β 2 (M α2 w) w dx 2π (ηw) x η dx, where M αj v = k Z (1 + k 2 ) α j 2 vk e ikx. Firstly, we analyze the linearized system: α 1 = α 2 = 2 and β 1, β 2 > = the exponential decay of solutions in the H s -setting, for any s R. max{α 1, α 2 } (, 2), β 1, β 2 and β β 2 2 > = polynomial decay rate of solutions in the H s -setting, by considering more regular initial data. Exponential decay estimate and contraction mapping argument = global well-posedness and the exponential stability property of the nonlinear system.

16 Main results The energy E(t) satisfies de dt = β 1 2π 2π (M α1 η) η dx β 2 (M α2 w) w dx 2π (ηw) x η dx, where M αj v = k Z (1 + k 2 ) α j 2 vk e ikx. Firstly, we analyze the linearized system: α 1 = α 2 = 2 and β 1, β 2 > = the exponential decay of solutions in the H s -setting, for any s R. max{α 1, α 2 } (, 2), β 1, β 2 and β β 2 2 > = polynomial decay rate of solutions in the H s -setting, by considering more regular initial data. Exponential decay estimate and contraction mapping argument = global well-posedness and the exponential stability property of the nonlinear system.

17 Main results The energy E(t) satisfies de dt = β 1 2π 2π (M α1 η) η dx β 2 (M α2 w) w dx 2π (ηw) x η dx, where M αj v = k Z (1 + k 2 ) α j 2 vk e ikx. Firstly, we analyze the linearized system: α 1 = α 2 = 2 and β 1, β 2 > = the exponential decay of solutions in the H s -setting, for any s R. max{α 1, α 2 } (, 2), β 1, β 2 and β β 2 2 > = polynomial decay rate of solutions in the H s -setting, by considering more regular initial data. Exponential decay estimate and contraction mapping argument = global well-posedness and the exponential stability property of the nonlinear system.

18 Main results The energy E(t) satisfies de dt = β 1 2π 2π (M α1 η) η dx β 2 (M α2 w) w dx 2π (ηw) x η dx, where M αj v = k Z (1 + k 2 ) α j 2 vk e ikx. Firstly, we analyze the linearized system: α 1 = α 2 = 2 and β 1, β 2 > = the exponential decay of solutions in the H s -setting, for any s R. max{α 1, α 2 } (, 2), β 1, β 2 and β β 2 2 > = polynomial decay rate of solutions in the H s -setting, by considering more regular initial data. Exponential decay estimate and contraction mapping argument = global well-posedness and the exponential stability property of the nonlinear system.

19 For any k Z, we denote by v k the k Fourier coefficient of v, v k = 1 2π v(x)e ikx dx, 2π and, for any s R, we define the space { Hp(, s 2π) = v = } v k e ikx H s (, 2π) v k 2 (1 + k 2 ) s <, k Z k Z which is a Hilbert space with the inner product defined by (v, w) s = k Z v k ŵ k (1 + k 2 ) s. (7) Then, M αj : H α j p (, 2π) L 2 (, 2π). M αj v = k Z (1 + k 2 ) α j 2 vk e ikx, j = 1, 2.

20 The Linearized System Since the linear system can be written as (I b 2 x)η t + w x + β 1 M 1 η =, (I d 2 x)w t + η x + β 2 M 2 η =, U t + AU =, U() = U, where A is given by ( ) β 1 I b 2 1 x Mα1 A = ( ) I d 2 1 ( ) x x β 2 I b 2 1 x Mα2 ( I b 2 x ) 1 x. (8) For α >, the operator (I α x) 2 1 is defined in the following way: v αv xx = ϕ in (, 2π), (I α x) 2 1 ϕ = v v() = v(2π), v x () = v x (2π).

21 Spectral Analysis If we assume that (η, w ) = k Z( η k, ŵ k )eikx, the solution can be written as (η, ω)(x, t) = k Z( η k (t), ω k (t))e ikx, where the pair ( η k (t), ŵ k (t)) fulfills where t (, T ). (1 + bk 2 )( η k ) t + ikŵ k + β 1 (1 + k 2 ) α 1 2 η k =, (1 + dk 2 )(ŵ k ) t + ik η k + β 2 (1 + k 2 ) α 2 2 ŵ k =, (9) η k () = η k, ŵ k() = ŵk,

22 We set A(k) = β 1 (1+k 2 ) α bk 2 Then system (9) is equivalent to η k ŵ k t ik 1+bk 2 ik β 2 (1+k 2 ) 1+dk 2 1+dk 2 (t) + A(k) η k ŵ k () = α 2 2. η k (t) =, η k. ŵ k ŵ k

23 Lemma The eigenvalues of the matrix A are given by λ ± k = 1 β 1(1 + k 2 ) α bk 2 + β 2(1 + k 2 α 2 ) 2 2 k e 2 k 1 + dk 2 ± 1, (1 + bk 2 )(1 + dk 2 ) where e k = 1 2k and k Z. Observe that ( ) β 1 (1 + k 2 ) α dk bk 2 β 2(1 + k 2 ) α2 1 + bk dk 2, λ ± k = λ± k. If e k < 1, the eigenvalues λ ± k C. If e k 1, the eigenvalues λ ± k R.

24 Lemma The solution ( η k (t), ŵ k (t)) of (9) is given by η k (t) = 1 1 ζ 2 k [ ) ( η k + iα k ζ k ŵk e λ + k t ( ] ζk 2 η k + iα kζ k ŵk) e λ k t, [ (iθk ŵ k (t) = 1 ζ 1 ζk k η 2 k ) ζ2 kŵ k e λ + k t ( ] iθ k ζ k η k k) ŵ e λ k t, if e k 1 and k, [( η k (t) = 1 ŵ k (t) = ) kζ k t η (1+bk2 )(1+dk 2 ) k ikt 1+bk ŵ 2 k [ ikt 1+dk 2 η k + (1 + if e k = 1 and k, and finally, ) kζ k t ŵ (1+bk 2 )(1+dk 2 ) k ] e λ+ k t, η (t) = η e β1t, ŵ (t) = ŵe β2t. 1+dk Here, α k = 1+bk 1+bk, θ 2 k = 1+dk 2 and ζ k = e k e 2 k 1. ] e λ+ k t,

25 The case s = For any t and k Z, we have that b η k (t) 2 + d ŵ k (t) 2 M ( b η k 2 + d ŵ k 2) e 2t min{ R(λ+ k ), R(λ k ) }, where min{ R(λ + k ), R(λ k ) } D >, and D is a positive number, depending on the parameters β 1, β 2, α 1, α 2, b and d. Moreover, If β 1 β 2 =, then R(λ ± k ), as k, and we cannot expect uniform exponential decay of the solutions. The fact that the decay of the solutions is not exponential is equivalent to the non uniform decay rate: given any non increasing positive function Θ, there is an initial data ( η, w ) such that the H s p H s p norm of the corresponding solution decays slower that Θ.

26 The case s = For any t and k Z, we have that b η k (t) 2 + d ŵ k (t) 2 M ( b η k 2 + d ŵ k 2) e 2t min{ R(λ+ k ), R(λ k ) }, where min{ R(λ + k ), R(λ k ) } D >, and D is a positive number, depending on the parameters β 1, β 2, α 1, α 2, b and d. Moreover, If β 1 β 2 =, then R(λ ± k ), as k, and we cannot expect uniform exponential decay of the solutions. The fact that the decay of the solutions is not exponential is equivalent to the non uniform decay rate: given any non increasing positive function Θ, there is an initial data ( η, w ) such that the H s p H s p norm of the corresponding solution decays slower that Θ.

27 The case s = For any t and k Z, we have that b η k (t) 2 + d ŵ k (t) 2 M ( b η k 2 + d ŵ k 2) e 2t min{ R(λ+ k ), R(λ k ) }, where min{ R(λ + k ), R(λ k ) } D >, and D is a positive number, depending on the parameters β 1, β 2, α 1, α 2, b and d. Moreover, If β 1 β 2 =, then R(λ ± k ), as k, and we cannot expect uniform exponential decay of the solutions. The fact that the decay of the solutions is not exponential is equivalent to the non uniform decay rate: given any non increasing positive function Θ, there is an initial data ( η, w ) such that the H s p H s p norm of the corresponding solution decays slower that Θ.

28 Let us introduce the space V s = Hp(, s 2π) Hp(, s 2π). Then, the following holds: Theorem (Micu, P., Preprint, 216) The family of linear operators {S(t)} t defined by S(t)(η, w ) = k Z( η k (t), ŵ k (t))e ikx, (η, w ) V s, (1) is an analytic semigroup in V s and verifies the following estimate S(t)(η, w ) V s C (η, w ) V s, (11) where C is a positive constant. Moreover, its infinitesimal generator is the compact operator (D(A), A), where D(A) = V s and A is given by ( ) β 1 I b 2 1 ( ) x Mα1 I b 2 1 x x A =. (12) ( ) I d 2 1 ( ) x x β 2 I b 2 1 x Mα2

29 Definition The solutions decay exponentially in V s if there exist two positive constants M and µ, such that S(t)(η, w ) V s Me µt (η, w ) V s, (13) t and (η, w ) V s. We have the following result: Theorem (Micu, P., Preprint, 216) The solutions decay exponentially in V s if and only if α 1 = α 2 = 2 and β 1, β 2 >. Moreover, µ from (13) is given by { µ = inf R(λ + k Z k ), R(λ k ) }, (14) where λ ± k are the eigenvalues of the operator A.

30 Definition The solutions decay exponentially in V s if there exist two positive constants M and µ, such that S(t)(η, w ) V s Me µt (η, w ) V s, (13) t and (η, w ) V s. We have the following result: Theorem (Micu, P., Preprint, 216) The solutions decay exponentially in V s if and only if α 1 = α 2 = 2 and β 1, β 2 >. Moreover, µ from (13) is given by { µ = inf R(λ + k Z k ), R(λ k ) }, (14) where λ ± k are the eigenvalues of the operator A.

31 Theorem (Micu, P., Preprint, 216) Suppose that β 1, β 2, β β2 2 > and min{α 1, α 2 } [, 2). Then, there exists δ and M >, such that S(t)(η, w ) V s M (1 + t) 1 δ (q 1 2 ) (η, w ) V s+q, t >, where s R and q > 1 2. Moreover, δ > is defined by 2 max{α 1, α 2 } if α 1 + α 2 2, max{α 1, α 2 } 1, δ = max{α 1, α 2 } if α 1 + α 2 2, max{α 1, α 2 } > 1, 2 min{α 1, α 2 } if α 1 + α 2 > 2. Remark: If α 1 = α 2 = 2 and β 1 = or β 2 =, then δ = 2.

32 The nonlinear problem Theorem (Micu, P., Preprint, 216) Let s and suppose that β 1, β 2 > and α 1 = α 2 = 2. There exist r >, C > and µ >, such that, for any (η, w ) V s, satisfying (η, w ) V s r, the system admits a unique solution (η, w) C([, ); V s ) which verifies (η(t), w(t)) V s Ce µt (η, w ) V s, t. Moreover, µ may be taken as in the linearized problem. The energy E(t) satisfies de dt = β 1 2π 2π (M α1 η) η dx β 2 (M α2 w) w dx 2π (ηw) x η dx.

33 We define the space Y s,µ = {(η, w) C b (R + ; V s ) : e µt (η, w) C b (R + ; V s )}, with the norm (η, w) Ys,µ := and the function Γ : Y s,µ Y s,µ by Γ(η, w)(t) = S(t)(η, w ) sup e µt (η, w)(t) V s, t< t S(t τ)n(η, w)(τ) dτ, where N(η, w) = ( (I b 2 x) 1 (ηw) x, (I d 2 x) 1 ww x ) and {S(t)} t is the semigroup associated to the linearized system.

34 Combining the estimates obtained for the linearized system we have Γ(η, w)(t) V s Me µt (η, w ) V s + MCe µt sup τ t e µτ (η, w) V s, for any t and some positive constants M and C. If we take (η, w) B R () Y s,µ, the following estimate holds Γ(η, w) Ys,µ M (η, w ) V s+mc (η, w) 2 Y s,µ MR+MCR 2. A similar calculations shows that, Γ(η 1, w 1 ) Γ(η 2, w 2 ) Ys,µ 2RMC (η 1, w 1 ) (η 2, w 2 ) Ys,µ, for any (η 1, w 1 ), (η 2, w 2 ) B R (). A suitable choice of R guarantees that Γ is a contraction.

35 Combining the estimates obtained for the linearized system we have Γ(η, w)(t) V s Me µt (η, w ) V s + MCe µt sup τ t e µτ (η, w) V s, for any t and some positive constants M and C. If we take (η, w) B R () Y s,µ, the following estimate holds Γ(η, w) Ys,µ M (η, w ) V s+mc (η, w) 2 Y s,µ MR+MCR 2. A similar calculations shows that, Γ(η 1, w 1 ) Γ(η 2, w 2 ) Ys,µ 2RMC (η 1, w 1 ) (η 2, w 2 ) Ys,µ, for any (η 1, w 1 ), (η 2, w 2 ) B R (). A suitable choice of R guarantees that Γ is a contraction.

36 Dirichlet boundary conditions We consider the BBM-BBM system η t + w x bη txx + εa(x)η =, x (, 2π), t >, w t + η x dw txx =, x (, 2π), t >, with boundary conditions η(t, ) = η(t, 2π) = w(t, ) = w(t, 2π) =, t >, and initial conditions η(, x) = η (x), w(, x) = w (x), x (, 2π). We assume that b, d > and ε > are parameters. a = a(x) is a nonnegative real-valued function satisfying a(x) a >, in Ω (, 2π).

37 Dirichlet boundary conditions We consider the BBM-BBM system η t + w x bη txx + εa(x)η =, x (, 2π), t >, w t + η x dw txx =, x (, 2π), t >, with boundary conditions η(t, ) = η(t, 2π) = w(t, ) = w(t, 2π) =, t >, and initial conditions η(, x) = η (x), w(, x) = w (x), x (, 2π). We assume that b, d > and ε > are parameters. a = a(x) is a nonnegative real-valued function satisfying a(x) a >, in Ω (, 2π).

38 The energy associated to the model is given by E(t) = 1 2 2π and we can (formally) deduce that (η 2 + bη 2 x + w 2 + dw 2 x)dx (15) d 2π E(t) = ε a(x)η 2 (t, x)dx. (16) dt Does E(t) converges to zero as t? If yes, at which rates it decays?

39 The energy associated to the model is given by E(t) = 1 2 2π and we can (formally) deduce that (η 2 + bη 2 x + w 2 + dw 2 x)dx (15) d 2π E(t) = ε a(x)η 2 (t, x)dx. (16) dt Does E(t) converges to zero as t? If yes, at which rates it decays?

40 Lack of Compactness There exist T > and C > such that E() C T [ 2π ] εa(x)η 2 (x, t)dx dt, (17) for every finite energy solution. Indeed, from (17) and the energy dissipation law, we have that E(T ) C E(). (18) C + 1 Since E(t) E(kT ) γ k E(), for < γ < 1 and k >, E(t) 1 ln γ E()e T t, where γ = C γ C + 1. (19) (17) does not hold for the BBM-BBM model.

41 Lack of Compactness There exist T > and C > such that E() C T [ 2π ] εa(x)η 2 (x, t)dx dt, (17) for every finite energy solution. Indeed, from (17) and the energy dissipation law, we have that E(T ) C E(). (18) C + 1 Since E(t) E(kT ) γ k E(), for < γ < 1 and k >, E(t) 1 ln γ E()e T t, where γ = C γ C + 1. (19) (17) does not hold for the BBM-BBM model.

42 Lack of Compactness There exist T > and C > such that E() C T [ 2π ] εa(x)η 2 (x, t)dx dt, (17) for every finite energy solution. Indeed, from (17) and the energy dissipation law, we have that E(T ) C E(). (18) C + 1 Since E(t) E(kT ) γ k E(), for < γ < 1 and k >, E(t) 1 ln γ E()e T t, where γ = C γ C + 1. (19) (17) does not hold for the BBM-BBM model.

43 Main results We assume that a = a(x) is nonnegative and a(x) a >, in Ω (, 2π), a W 2, (, 2π), with a() = a () =. (2) Theorem (Micu, P., Journal d Analyse Mathématique) There exits ε, such that, for any ε (, ε ) and (η, w ) in (H 1 (, 2π))2, the solution (η, w) of the system verifies lim (η(t), w(t)) t (H 1 (,2π))2 =. Moreover, the decay of the energy is not exponential, i. e., there exists no positive constants M and ω, such that (η(t), w(t)) (H 1 (,2π)) 2 Me ωt, t.

44 Spectral analysis and eigenvectors expansion of solutions Since (I b x)η 2 t + w x + εa(x)η =, x (, 2π), t >, (I d x)w 2 t + η x =, x (, 2π), t >, the system can be written as U t + A ε U =, U() = U, where A ε : (H 1 (, 2π)) 2 (H 1 (, 2π)) 2 is given by ε ( I b 2 1 ( ) x) a( ) I I b 2 1 x x A ε =. (21) ( ) I d 2 1 x x We have that A ε L((H 1 (, 2π)) 2 ) and A ε is a compact operator.

45 The operator A ε has a family of eigenvalues (λ n ) n 1, such that: 1. R(λ n ) c n 2, n n, and R(λ n ) <, n. 2. The corresponding eigenfunctions (Φ n ) n 1 form a Riesz basis in (H 1 (, 2π))2. Then, (η(t), w(t)) = n 1 a n e λnt Φ n and c 1 a n 2 e 2R(λn)t (η(t), w(t)) 2 (H c 1 2 a n 2 e 2R(λn)t. (,2π))2 n n n 1

46 The operator A ε has a family of eigenvalues (λ n ) n 1, such that: 1. R(λ n ) c n 2, n n, and R(λ n ) <, n. 2. The corresponding eigenfunctions (Φ n ) n 1 form a Riesz basis in (H 1 (, 2π))2. Then, (η(t), w(t)) = n 1 a n e λnt Φ n and c 1 a n 2 e 2R(λn)t (η(t), w(t)) 2 (H c 1 2 a n 2 e 2R(λn)t. (,2π))2 n n n 1

47 The operator A ε has a family of eigenvalues (λ n ) n 1, such that: 1. R(λ n ) c n 2, n n, and R(λ n ) <, n. 2. The corresponding eigenfunctions (Φ n ) n 1 form a Riesz basis in (H 1 (, 2π))2. Then, (η(t), w(t)) = n 1 a n e λnt Φ n and c 1 a n 2 e 2R(λn)t (η(t), w(t)) 2 (H c 1 2 a n 2 e 2R(λn)t. (,2π))2 n n n 1

48 Theorem (Micu, P., Journal d Analyse Mathématique) Let (η, w) be a finite energy solution of the system with a. If there exist T > and an open set Ω (, 2π), such that η(x, t) =, (t, x) (, T ) Ω, (22) then η = w in R (, 2π). Idea of the proof: A ε is a compact operator in (H 1 (, 2π))2 analyticity in time of solutions property (22) holds for t R. Fourier decomposition of solutions. Unique continuation principle for each eigenfunction.

49 Theorem (Micu, P., Journal d Analyse Mathématique) Let (η, w) be a finite energy solution of the system with a. If there exist T > and an open set Ω (, 2π), such that η(x, t) =, (t, x) (, T ) Ω, (22) then η = w in R (, 2π). Idea of the proof: A ε is a compact operator in (H 1 (, 2π))2 analyticity in time of solutions property (22) holds for t R. Fourier decomposition of solutions. Unique continuation principle for each eigenfunction.

50 Theorem (Micu, P., Journal d Analyse Mathématique) Let (η, w) be a finite energy solution of the system with a. If there exist T > and an open set Ω (, 2π), such that η(x, t) =, (t, x) (, T ) Ω, (22) then η = w in R (, 2π). Idea of the proof: A ε is a compact operator in (H 1 (, 2π))2 analyticity in time of solutions property (22) holds for t R. Fourier decomposition of solutions. Unique continuation principle for each eigenfunction.

51 Theorem (Micu, P., Journal d Analyse Mathématique) Let (η, w) be a finite energy solution of the system with a. If there exist T > and an open set Ω (, 2π), such that η(x, t) =, (t, x) (, T ) Ω, (22) then η = w in R (, 2π). Idea of the proof: A ε is a compact operator in (H 1 (, 2π))2 analyticity in time of solutions property (22) holds for t R. Fourier decomposition of solutions. Unique continuation principle for each eigenfunction.

52 Main steps of the proof 1. The spectrum of the differential operator corresponding A ε is located in the left open half-plane of the complex plane. We also obtain the asymptotic behavior (of the spectrum). 2. There exists a Riesz basis (Φ m ) m 1 (H 1 (, 2π)) 2 consisting of generalized eigenvectors of the differential operator A ε. We obtain the asymptotic behavior of the high eigenfunctions and prove that they are quadratically close to a Riesz basis (Ψ m ) m 1 formed by eigenvectors of a well chosen dissipative differential operator with constant coefficients: m N+1 Φ m Ψ m 2 (H 1 (,2π))2 1 m 2. This is done by using less common two dimensional versions of the Shooting Method and Rouché s Theorem.

53 Main steps of the proof 1. The spectrum of the differential operator corresponding A ε is located in the left open half-plane of the complex plane. We also obtain the asymptotic behavior (of the spectrum). 2. There exists a Riesz basis (Φ m ) m 1 (H 1 (, 2π)) 2 consisting of generalized eigenvectors of the differential operator A ε. We obtain the asymptotic behavior of the high eigenfunctions and prove that they are quadratically close to a Riesz basis (Ψ m ) m 1 formed by eigenvectors of a well chosen dissipative differential operator with constant coefficients: m N+1 Φ m Ψ m 2 (H 1 (,2π))2 1 m 2. This is done by using less common two dimensional versions of the Shooting Method and Rouché s Theorem.

54 Main steps of the proof 1. The spectrum of the differential operator corresponding A ε is located in the left open half-plane of the complex plane. We also obtain the asymptotic behavior (of the spectrum). 2. There exists a Riesz basis (Φ m ) m 1 (H 1 (, 2π)) 2 consisting of generalized eigenvectors of the differential operator A ε. We obtain the asymptotic behavior of the high eigenfunctions and prove that they are quadratically close to a Riesz basis (Ψ m ) m 1 formed by eigenvectors of a well chosen dissipative differential operator with constant coefficients: m N+1 Φ m Ψ m 2 (H 1 (,2π))2 1 m 2. This is done by using less common two dimensional versions of the Shooting Method and Rouché s Theorem.

55 Main steps of the proof 1. The spectrum of the differential operator corresponding A ε is located in the left open half-plane of the complex plane. We also obtain the asymptotic behavior (of the spectrum). 2. There exists a Riesz basis (Φ m ) m 1 (H 1 (, 2π)) 2 consisting of generalized eigenvectors of the differential operator A ε. We obtain the asymptotic behavior of the high eigenfunctions and prove that they are quadratically close to a Riesz basis (Ψ m ) m 1 formed by eigenvectors of a well chosen dissipative differential operator with constant coefficients: m N+1 Φ m Ψ m 2 (H 1 (,2π))2 1 m 2. This is done by using less common two dimensional versions of the Shooting Method and Rouché s Theorem.

56 To control the low frequencies we use a result originally proved for a unbounded operator: B. Z. Guo, Riesz basis approach to the stabilization of a flexible beam with a tip mass, SIAM J. Control Optim. 39 (21), B. Z. Guo and R. Yu, The Riesz basis property of discrete operators and application to a Euler-Bernoulli beam equation with boundary linear feedback control, IMA J. Math. Control Inform. 18 (21), It was extended to the bounded case: X. Zhang and E. Zuazua, Unique continuation for the linearized Benjamin-Bona-Mahony equation with space-dependent potential, Math. Ann. 325 (23),

57 We consider the spectral problem ( ) ( η η A ε = µ w w ), where A ε : (H 1 (, 2π)) 2 (H 1 (, 2π)) 2, which is equivalent to the BVP η bη xx + µw x + εa(x)µη = for x (, 2π) w dw xx + µη x = for x (, 2π) η() = η(2π) = w() = w(2π) =. (23) From (23) we obtain a family of eigenvalues and eigenfunctions...

58 A two dimensional shooting method For each (µ, γ) C 2, consider the IVP η bη xx + µw x + εa(x)µη = for x (, 2π) w dw xx + µη x = for x (, 2π) η() =, η x () = 1 w() =, w x () = γ. ( ) η(µ, γ, 2π) and the map F : C 2 C 2, given by F (µ, γ) =. w(µ, γ, 2π) Then, ( ) µ is an eigenvalue of (23) with corresponding eigenfunction η, if and only if, w (24) ( F (µ, γ) = ). The spectrum of the differential operator with variable potential is given by the zeros of the map F.

59 A two dimensional shooting method For each (µ, γ) C 2, consider the IVP η bη xx + µw x + εa(x)µη = for x (, 2π) w dw xx + µη x = for x (, 2π) η() =, η x () = 1 w() =, w x () = γ. ( ) η(µ, γ, 2π) and the map F : C 2 C 2, given by F (µ, γ) =. w(µ, γ, 2π) ( Then, ) µ is an eigenvalue of (23) with corresponding eigenfunction η, if and only if, w (24) ( F (µ, γ) = ). The spectrum of the differential operator with variable potential is given by the zeros of the map F.

60 A two dimensional shooting method For each (µ, γ) C 2, consider the IVP η bη xx + µw x + εa(x)µη = for x (, 2π) w dw xx + µη x = for x (, 2π) η() =, η x () = 1 w() =, w x () = γ. ( ) η(µ, γ, 2π) and the map F : C 2 C 2, given by F (µ, γ) =. w(µ, γ, 2π) ( Then, ) µ is an eigenvalue of (23) with corresponding eigenfunction η, if and only if, w (24) ( F (µ, γ) = ). The spectrum of the differential operator with variable potential is given by the zeros of the map F.

61 Next, we define the map G = G(σ, β), associated to a spectral problem with constant coefficients, for which the eigenfunctions form a Riesz basis: bψ xx + σu x + εa σψ = for x (, 2π) du xx + σψ x = for x (, 2π) (25) ψ() =, ψ x () = 1 u() =, u x () = β. ( ) ψ(σ, β, 2π) The map G : C 2 C 2 is given by G(σ, β) =. u(σ, β, 2π) The corresponding BVP (ψ() = u() = ψ(2π) = u(2π) = ) has a double indexed family of complex eigenvalues (σ j n) n Z, j {1,2} and the family of corresponding eigenfunctions (Ψ j n) n Z, j {1,2} forms a Riesz basis in (H 1 ) 2.

62 Next, we define the map G = G(σ, β), associated to a spectral problem with constant coefficients, for which the eigenfunctions form a Riesz basis: bψ xx + σu x + εa σψ = for x (, 2π) du xx + σψ x = for x (, 2π) (25) ψ() =, ψ x () = 1 u() =, u x () = β. ( ) ψ(σ, β, 2π) The map G : C 2 C 2 is given by G(σ, β) =. u(σ, β, 2π) The corresponding BVP (ψ() = u() = ψ(2π) = u(2π) = ) has a double indexed family of complex eigenvalues (σ j n) n Z, j {1,2} and the family of corresponding eigenfunctions (Ψ j n) n Z, j {1,2} forms a Riesz basis in (H 1 ) 2.

63 Theorem (N. G. Lloyd, J. London Math. Soc. 2 (1979)) Let D be a bounded domain in C N and h, G holomorphic maps of D into C N such that h(z) < G(z) for z D. Then G has finitely many zeros in D, and G and h + G have the same number of zeros in D, counting multiplicity. Given a zero (σn, j βn) j of the map G, we define the domain { Dn(δ) j = (µ, γ) C 2 : µ σn j 2 + γ βn j 2 δ }. n If h = F G we obtain G(µ, γ) C n 2, F (µ, γ) G(µ, γ) C n 2, (µ, γ) Dj n(δ).

64 Theorem (N. G. Lloyd, J. London Math. Soc. 2 (1979)) Let D be a bounded domain in C N and h, G holomorphic maps of D into C N such that h(z) < G(z) for z D. Then G has finitely many zeros in D, and G and h + G have the same number of zeros in D, counting multiplicity. Given a zero (σn, j βn) j of the map G, we define the domain { Dn(δ) j = (µ, γ) C 2 : µ σn j 2 + γ βn j 2 δ }. n If h = F G we obtain G(µ, γ) C n 2, F (µ, γ) G(µ, γ) C n 2, (µ, γ) Dj n(δ).

65 ( ϕ(µ, γ, x) Finally, we obtain an ansatz z(µ, γ, x) the IVP More precisely, ϕ(µ, γ, x) = z(µ, γ, x) = ) for the solutions of η bη xx + µw x + εa(x)µη = for x (, 2π) w dw xx + µη x = for x (, 2π) η() =, η x () = 1 w() =, w x () = γ. ( η(µ, γ, x) w(µ, γ, x) and α(x) = µx bd + 1 d 2 b ) = ( ϕ(µ, γ, x) z(µ, γ, x) ) ( ) 1 + O µ 2, where bd γd sinh (α(x)) + µ µ cosh (α(x)) γd µ + da(x) b µ (cosh (α(x)) 1) + γ bd γd sinh (α(x)) + µ µ x a(s)ds. x a(s)ds,

66 F (µ, γ) G(µ, γ) ( ϕ(µγ, 2π) F (µ, γ) z(µ, γ, 2π) ) ( + ϕ(µγ, 2π) z(µ, γ, 2π) ) G(µ, γ) C 1 (from the ansatz property) µ 2 + C 2 µ 2 (by choosing conveniently the constant potential a = 1 2π C µ 2. 2π a(s)ds)

67 F (µ, γ) G(µ, γ) ( ϕ(µγ, 2π) F (µ, γ) z(µ, γ, 2π) ) ( + ϕ(µγ, 2π) z(µ, γ, 2π) ) G(µ, γ) C 1 (from the ansatz property) µ 2 + C 2 µ 2 (by choosing conveniently the constant potential a = 1 2π C µ 2. 2π a(s)ds)

68 F (µ, γ) G(µ, γ) ( ϕ(µγ, 2π) F (µ, γ) z(µ, γ, 2π) ) ( + ϕ(µγ, 2π) z(µ, γ, 2π) ) G(µ, γ) C 1 (from the ansatz property) µ 2 + C 2 µ 2 (by choosing conveniently the constant potential a = 1 2π C µ 2. 2π a(s)ds)

69 Similar strategies have been successfully used by S. Cox and E. Zuazua, The rate at which energy decays in a damped string, Comm. Partial Differential Equations 19 (1994), A. Benaddi and B. Rao, Energy decay rate of wave equations with indefinite damping, J. Differential Equations 161 (2), X. Zhang and E. Zuazua, Unique continuation for the linearized Benjamin-Bona-Mahony equation with space-dependent potential, Math. Ann. 325 (23),

70 Remarks and open problems Dirichlet boundary conditions: Less regularity for the potential a. Stabilization results for the nonlinear problem. Dissipative mechanisms, like [a(x)ϕ x ] x, ensures the uniform decay? The mixed KdV-BBM system is exponentially stabilizable? Periodic boundary conditions: The decay of solutions of a nonlinear problem with a linearized part that does not decay uniformly. Unique Continuation Property for the BBM-BBM system. Dissipative mechanisms, like a(x)ϕ or [a(x)ϕ x ] x, ensures the uniform decay? Boundary stabilization results. The 2-d Boussinesq system...

71 Remarks and open problems Dirichlet boundary conditions: Less regularity for the potential a. Stabilization results for the nonlinear problem. Dissipative mechanisms, like [a(x)ϕ x ] x, ensures the uniform decay? The mixed KdV-BBM system is exponentially stabilizable? Periodic boundary conditions: The decay of solutions of a nonlinear problem with a linearized part that does not decay uniformly. Unique Continuation Property for the BBM-BBM system. Dissipative mechanisms, like a(x)ϕ or [a(x)ϕ x ] x, ensures the uniform decay? Boundary stabilization results. The 2-d Boussinesq system...

72 Remarks and open problems Dirichlet boundary conditions: Less regularity for the potential a. Stabilization results for the nonlinear problem. Dissipative mechanisms, like [a(x)ϕ x ] x, ensures the uniform decay? The mixed KdV-BBM system is exponentially stabilizable? Periodic boundary conditions: The decay of solutions of a nonlinear problem with a linearized part that does not decay uniformly. Unique Continuation Property for the BBM-BBM system. Dissipative mechanisms, like a(x)ϕ or [a(x)ϕ x ] x, ensures the uniform decay? Boundary stabilization results. The 2-d Boussinesq system...