Long-time behavior of solutions of a BBM equation with generalized damping

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1 Long-time behavior of solutions of a BBM equation with generalized damping Jean-Paul Chehab, Pierre Garnier, Youcef Mammeri To cite this version: Jean-Paul Chehab, Pierre Garnier, Youcef Mammeri Long-time behavior of solutions of a BBM equation with generalized damping <hal-9983> HAL Id: hal Submitted on Feb HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not The documents may come from teaching and research institutions in France or abroad, or from public or private research centers L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés

2 Long-time behavior of solutions of a BBM equation with generalized damping J-P Chehab, P Garnier, Y Mammeri Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, CNRS UMR 735, Université de Picardie Jules Verne, 869 Amiens, France Corresponding author: pierregarnier@u-picardiefr Abstract We study the long-time behavior of the solution of a damped BBM equation u t +u x u xxt +uu x +L γ(u) = The proposed dampings L γ generalize standards ones, as parabolic (L γ(u) = u) or weak damping (L γ(u) = γu) and allows us to consider a greater range After establish the local well-posedness in the energy space, we investigate some numerical properties Keywords BBM equation, dispersion, dissipation MS Codes 35B, 35Q53, 76B3, 76B5 Introduction The one-way propagation of small amplitude, long wavelength, gravity waves in shallow water can be described by the Korteweg-de Vries equation (KdV) [3] u t + u x + u xxx + uu x = or its regularized version, the Benjamin-Bona-Mahony (BBM) equation u t + u x u xxt + uu x = Many works related to the damped KdV equation can be found in the literature ([,, 5, 6, 8, 9,,,, 6, 7, 5] and references therein) These account for a wide variety of different results, such as regularizing effect of the damping, asymptotic behavior, existence of attractor or numerical computations Few literatures are concerned the damped BBM equation [, 8] This paper consists in establishing theoretical and numerical results for the solution of a generalized damped BBM equation We introduce the following equation, called dbbm, for x T(, L), L > and t R u t + u x u xxt + uu x + L γ (u) =,

3 where the operator L γ is defined by its Fourier symbol L γ (u)(k) := γ k û k Here û k is the k th Fourier coefficient of u and (γ k ) are positive real numbers chosen such that u(x)l γ (u)dµ(x) = γ k û k T Standard dampings are included choosing γ k = k as parabolic damping or γ k = γ as weak damping The proposed sequence (γ k ) allows us to consider a greater range of damping In particular, one may wonder if it is necessary to absorb all frequencies of a long wavelength gravity wave like the solution of KdV or BBM To obtain this kind of results, we can consider sequences γ k when k + or even γ k = for large k In comparison with the standard BBM equation for which the H norm is preserved, we notice that the H -norm decreases for the solution u of the damped equation dbbm More precisely, we have for all time t R d dt u(t) H = u(t) γ, and the natural space to study the well-posedness is the space H γ (T), defined as H γ (T) := u L (T); γ k û k < +, equipped with the norm u γ := γ k û k In order to simplify the writings, k can denote either an integer as an index, or the value πk L When there is no ambiguity, we denote C the different constants appearing in the following results The paper is organised as follows We establish in Section the local wellposedness of the dbbm equation We prove some important estimates about the space H γ (T) Some qualitative properties of the solution are also given In Section 3, numerical schemes are presented to solve the damped equation and to preserve the qualitative properties of the continuous solution Section deals with the numerical results In particular, we manage to build a family of dampings weaker than the standard ones These, for example taking lim k + γ k =, yet provide dampen solutions

4 Analysis of the problem After studying the space H γ (T), we establish the well-posedness of the Cauchy problem associated with the dbbm equation Proper energy space H γ (T) Here we take γ k >, k Z We first state some properties of injection Proposition Assume that γ k < + Then there exists a constant C > such that u C u γ The injection H γ (T) L (T) is continuous Proof Let u H γ (T) We notice that u(x) = û(k)e ikx Then u(x) û(k) = γk û(k) γk We assumed that γ k > Hence, the Cauchy-Schwarz inequality implies for all x T : u(x) γ k û(k) γ k This completes the proof = γ k u γ Remark The above result is also a condition to have û l C u γ Proposition 3 We assume that for all k Z we have γ k > β k Let β ρ N = max k k N γ k Then lim ρ N = if and only if the continuous injection N + H γ (T) H β (T) is compact Proof The condition is necessary, indeed if there exists α > such that ρ N > α, N, then the norms u β and u γ are equivalent, the injection cannot be compact Let us prove now that the condition is sufficient First we have for u H γ (T) : u β = β k û k γ k û k = u γ This shows that the injection is continuous Now we prove that the injection is compact We use finite rank operators and we take the limit Let I N 3

5 be the orthogonal operator on the polynomials of frequencies k such that N k N We have I N u = û k e ikx Hence (Id I N )u β = k N k N+ k N+ ρ N+ u γ β k û k, β k γ k γ k û k, N + Therefore Id is a compact operator and consequently the injection is compact Now we give some conditions on the sequence (γ k ) so that H γ (T) is an algebra Proposition Let u and v be two functions in H γ (T) Assume that there exists a constant C > such that ( γk j γ k C + ) γ j for all k, j Z Then we have the following inequality: Moreover if uv γ C ( u γ ˆv l + v γ û l ), γ k < + then H γ (T) is an algebra Proof Let u, v H γ (T) We have uv γ = γ k ûv(k) We remind that ûv(k) = û ˆv(k) We use the inequality γk C ( γk j + γ j ) We obtain for all k Z γk ûv(k) C γk j û(k j)ˆv(j) + γj û(k j)ˆv(j) j Z j Z Hence uv γ C γk j û(k j)ˆv(j) + γj û(k j)ˆv(j), j Z j Z C γk j û(k j)ˆv(j) + γj û(k j)ˆv(j), j Z j Z ( C ( ( γ k û ) ˆv l + û γk )) ˆv l

6 We remind that for two functions f and g defined from N to R such that f l and g l, we have Thus f g l g l f l uv γ C ( u γ ˆv l + v γ û l ) We know there exists a constant c > such that û l c u γ if + Then, there exists C > such that uv γ C u γ v γ γ k < Remark 5 With γ k = k s we find the standard Sobolev injection Indeed, we have the inequality ( γk j γ k C + ) γ j and if s > then + Local well-posedness γ k < We can study the well-posedness of problem dbbm We consider the Cauchy problem u t + u x u xxt + uu x + L γ (u) =, x T, t [, T ] () u(x, t = ) = u (x) () Theorem 6 Assume that γ k+j C( γ k + γ j ) for all k and j in Z, γ k < + and u H γ (T) Then there exists T = C u > and there γ exists a unique u C ([, T ], H γ (T)) solution of the Cauchy problem ()-() Moreover, for all M > with u γ M and v γ M, there exists a constant C > such that the solutions u and v, associated with the initial data u and v respectively, satisfy for all t C M u(, t) v(, t) γ C u v γ Proof Let write the initial value problem ()-() as where t u = Au + f(u), (3) u(t = ) = u (x), () Au = ( xx ) ( x u + L γ (u)), ( ) f(u) = ( xx ) u x 5

7 According to the Duhamel s formula, u is solution of the Cauchy problem (3)-() if and only if u(t) = φ (u(t)) = S t u + S t τ f (u(τ)) dτ, where S t v = e ikx e (ik+γ k ) +k tˆv(k) The purpose is to show that u is the unique fixed point of φ We define the closed ball B(T ) = {u C ([, T ]; H γ (T)) ; u(t) u γ 3 u γ } Our purpose is to apply the Banach fixed point theorem Let u B(T ) We show that φ (u(t)) B(T ) From the triangle inequality, we have: On the one hand, since γ k and φ (u(t)) γ S t u γ + S t τ (f(u)) γ dτ S t u γ = γ k S t u (k), γ k e γ k +k t û (k), u γ, S t τ f (u(τ)) γ f (u(τ)) γ On the other hand, we need to upper bound the term f (u(τ)) γ We have However, since ( ) f (u(τ)) γ = ( xx) u x (τ) γ, = ( ( )) γ k F ( xx ) u x (τ), = ik û γ k + k k +k and H γ (T) is an algebra, it gets f (u(τ)) γ C u γ 6

8 The Duhamel s formula implies for t T φ (u(t)) γ u γ + u(τ) γ dτ, But u B(T ), so that implies u γ + C ( ) sup u(t) γ T t [,T ] u(t) γ u γ u(t) u γ 3 u γ, u(t) γ u γ We have φ (u(t)) B(T ) if φ (u(t)) u γ 3 u γ The inequality ( ) φ (u(t) u ) γ u γ + CT 6 u γ 3 u γ, is true if we choose < T u γ 6C u γ = ( ) 6C u γ Now we show that φ is a contraction mapping on B(T ) Let u, v B(T ) We have φ (u(t)) φ (v(t)) γ = S t τ (f(u) f(v)) dτ, γ u v (τ)dτ γ We notice that u v = (u v)(u + v) Since H γ (T) is an algebra, we have u v γ C u v γ u + v γ, ) C ( u γ + v γ u v γ And for u, v B(T ), We infer for all t [, T ] u v γ 8C u γ u v γ φ (u(t)) φ (v(t)) γ 8C u γ u v γ (τ)dτ, 8C u γ T sup (u v)(t) γ t [,T ] 7

9 Hence ( ) sup t [,T ] φ (u(t)) φ (v(t)) γ 8C u γ T sup t [,T ] u(t) v(t) γ Consequently φ is a contraction mapping if 8C u γ T <, ie, T < 8C u γ Finally, from the Banach fixed-point theorem, φ has a unique fixed-point solution of u(t) = φ (u(t)) So, there exists a unique solution of the Cauchy problem It remains to prove the continuity with the initial data Let u and v solutions of the Cauchy problem ()-() with initial data u and v respectively, such that u γ M and v γ M The Duhamel s formula gives for t [, T ], T C M u v γ u v γ + f(u) f(v) γ dτ, ) u v γ + C T ( u γ + v γ sup t [,T ] u v γ It implies u v γ C u v γ 3 Behavior of the solution In this part, we aim to get some estimations of the damping rate Actually in the case without damping, the H -norm of the solution is invariant during time Here this norm is decreasing We adapt the work done on KdV equation [6] Let us begin with the linear equation, that reads u t u txx + u x + L γ (u) = x T, t [, T ] (5) Let u be valued in L (T), t > We write u as a Fourier series and, due to the orthogonality of the trigonometric polynomials, we obtain Hence It follows that ( + k ) dû k(t) dt + (γ k + ik)û k (t) = û k (t) = e γ k +ik +k t û k () u H = ( + k ) û k (t) = ( + k )e γ k +k t û k () 8

10 Proposition 7 Let γ k >, k Z and u H δ (T) where δ k = (+k ) γ k Then the unique solution u of (5) verifies ( ) e u H min t u δ, u H, t > More generally, if u H βδ (T) then u β e t u βδ where β k = ( + k )β k Proof On one hand, the scalar product in L (T) of (5) with u provides Hence d dt u H + γ k û k (t) = d dt u H Consequently u H u H On the other hand, to obtain the second inequality, we write u H = ( + k ) û k, = ( + k )e γ k +k t û k (), = ( + k ) γ k + k e γ k +k t γ k û k () Since the function β βe βt is uniformly bounded by e t, we infer that u H e t u δ In the same way we obtain the third inequality from β k û k = β k e γ k +k t û k () = ( + k ) + k e γ k γ k +k t ( βk γ k û k () ) We can prove the two following results with similar arguments: Proposition 8 Assume that γ k [, ], k Z and u H δ (s), where δ (s) k = (+k ) s+ γk s Then for all s >, u the solution of (5) verifies t > ( ( ) s s ) u H min e s u t δ, u (s) H 9

11 Proposition 9 We assume that there exist three positive constants α, β and C such that u the solution of (5) verifies: i û k Cγk δ with δ = α + β ii ( + k ) α+ γ β < + Then k ( ) α α u H Ce a ( + k ) α+ γ β k t ( ) = O t α We can now deal with the non-linear equation We can find similar kind of decreasing but less explicit than in the linear case We remind the equation u t u txx + u x + uu x + L (u) = (6) The scalar product in L of (6) with u yields Proposition We assume that i γ k > k Z, d dt u H + u γ = ii u H γ (T) H (T) with L u (x)dx =, Then lim u H t + = Proof Because of γ k >, we have d u H dt and then the function t u H is decreasing and consequently lim t + u(t) H = C We notice that we also have u H u H Since u H H γ, we deduce that lim t + u(t) γ = Finally, since γ k >, we have lim t + ûk = and then u =, which means that C = L Lemme For all function v smooth enough, we set v = L v(x, t)dx Let u be the solution of dbbm valued in H γ (T) H (T) Then ū(t) = e γ tū() Proof We integrate the equation in space on the interval [, L] and we obtain L ( ) u t + L γ(u) dx =

12 But L Hence we have Therefore L γ (u)(x, t)dx = d dt L L L We deduce the desired result γ k û k e iπkx L u(x, t)dx + L L dū dt + γ ū = L dx = Lγ û = γ L γ (u)(x, t)dx =, u dx The solution converges to in H with a rate depending on the H γ -norm Thus, as in [6], we introduce the following ratio function G : (u, t) G(u, t) = u γ γ k û k = u H ( + k ) û k (7) To simplify the writings, we use G(t) instead of G(u, t) when no confusion is possible Proposition Let u be the unique solution of dbbm valued in H γ (T) H (T) We assume that G is C in time Then we have the following equalities i u(t) H = e ii u(t) γ = G (t)e In particular, G (s)ds u H, G (s)ds u H lim t + u H = if and only if t G(t) / L t (, + ) Proof As previously, we take the scalar product in L (T) of the equation dbbm with u and we obtain Since u γ = G (t) u H, we have d dt u H + u γ = d dt u H + G (t) u H = thus u(t) H = e G (s)ds u H

13 We find lim t + u H = if and only if t G(t) / L t (, + ) Then, we have u(t) γ = G (t)e The proposition involves that lim t + G (t)e G (s)ds u H lim u γ = and consequently t + G (s)ds u H = Remark 3 We can establish similar results for the forced equation u t u txx + u x + uu x + L γ (u) = f Proposition Assume that f H /γ (T) and u H (T) H γ (T) Then the solution u of the forced equation satisfies: u(t) H e G L (s)ds u H + e Proof The scalar product of dbbm with u gives But d dt u H + u γ =< f, u > s G (τ)dτ f /γ ds f, u f /γ u γ and u γ = G (t) u H From the Young inequality, we have d dt u H + G (t) u H f /γ + u γ It involves that d dt u H + G (t) u H f /γ Using the Gronwall s lemma, we obtain u(t) H (T) e G L (s)ds u H + Multiplying each term with G (t), it gets u(t) γ(t) G (t)e G L (s)ds u H + e e s s G (τ)dτ f /γ ds G (τ)dτ G (t) f /γ ds

14 3 Numerical schemes In this part, we study and present some numerical schemes well suited for the long-time behavior of the solution of dbbm equation We also give details on their implementation 3 Numerical schemes Let start with time discretization We denote by D the operator of derivation in space, by D the operator of second derivation in space and by u n the approximation of u at time t n = n t We first introduce the fully bacward Euler method We have the following discretization: (Id D ) un+ u n t + Du n+ + D ((u n+ ) ) + L γ u n+ = f (8) Proposition 3 If u = u H (T) and f H /γ (T) then the sequence (u n ) n N generated by the fully backward Euler method is well defined, in H (T) and u n+ H + un+ u n H + t Moreover, if f = then lim n + un H = Proof We use the following identity (Id D )(u n+ u n ), u n+ = γ k û n+ k u n H + t γ ˆf k k ( u n H un+ u n H un+ H ) We compute the scalar product of (8) with u n+ and we obtain u n+ H + un+ u n H un H + t L γu n+, u n+ = t u n+, f, where L γ u n+, u n+ = γ k û n+ k Hence we have u n+ u n H + un+ H + t The Young inequality on the right hand side provides u n+, f γ k û n+ k = u n H + un+, f γ k û n+ k + γ ˆf k k 3

15 We infer the expected result u n+ H + un+ u n H + t un+ γ un H + t If f =, we directly have the equality u n+ H + un+ u n H + t un+ γ = u n H γ k ˆf k We deduce that the sequence ( u n H ) n N is decreasing and lower bounded by Then it is convergent to a positive constant C It follows that lim γ k û n k = As γ k >, we have n + lim n + ûn k = and consequently C = Proposition 3 Let (u n ) n N be the sequence generated by the fully backward Euler method Assume that f = We set G (n) = un γ u n Then we H have n u n H + t(g (j) ) u H j= Moreover, if t is small enough and if (G (j) ) j Z / l then lim n + un H = Proof We compute the scalar product of (8) with u n+ and we obtain ) ( u n+ H t un H + un+ u n H + (G (n+) ) u n+ H = Hence Therefore ( + t(g (n+) ) ) u n+ H + un+ u n H un H We infer by induction that u n+ H u n H + t(g (n+) ) u n H n + t(g (j) ) u H j= Now if t is small enough, we have n n n log + t(g j= (j) ) = log( + t(g (j) ) ) t (G (j) ) j= j= If (G (j) ) / l then lim n + un H =

16 Remark 33 The forward Euler method can be applied to the BBM equation, contrarily to the KdV equation Moreover, it is more difficult to establish qualitative properties as Propositions 3 and 3 The scheme is written as (Id D ) un+ u n t + Du n + D ((u n ) ) + L γ u n = f Let introduce the Sanz-Serna scheme, given by (Id D ) un+ u n t ( u n+ + u n ) + D + ( D u n+ + u n ) ( u n+ + u n ) + L γ = f (9) Proposition 3 Assume that u H (T) and f L H /γ (T) Then the scheme (9) is stable in H (T) for all t > Proof We take the scalar product of (9) with un+ +u n We obtain u n+ H u n H t + un+ + u n γ = Using the duality and the Young s inequality, it implies u n+ H u n H t f, un+ + u n + un+ + u n γ f /γ + u n+ + u n Hence u n+ H + t un+ + u n γ un H + t f /γ Consequently we have the stability on every time interval [, T ] Remark 35 The Crank-Nicolson scheme is given by (Id D ) un+ u n δt + D ( ) u n+ + u n + D ( (u n+ ) + (u n ) ) + L γ u n+ + u n γ = f () We cannot have uniform H bounds for u n from this scheme because D ((u n+ ) + (u n ) ), un+ + u n 5

17 3 Implementation The Fourier Transform in space is applied to the dbbm equation equation (6) becomes The ( + k ) dû k dt + (ik + γ k)û k + (ûu x ) k = We remind that the term L γ (u) is written in the Fourier space as γ k û k e ikx Then we use the schemes introduced in the previous subsection for the discretization in time and the FFT for the space discretization These are implicit methods and we apply a fixed-point method We solve equations which can be written as u n+ = Φ(u n, u n+ ) by Picard iterates Algorithm Picard iterates v = u n, m = while Φ(u n, u m ) u m > ǫ do v m+ = Φ(u n, v m ) m = m + end while u n+ = v m Unfortunately, even if the used scheme is unconditionally stable, this method presents instabilities The stability of the fixed-point iterate can be improved by applying extrapolation like technic as follows (see also []) The principle is to change the iterate v m+ = Φ(u n, v m ) by where k Φ vm = k j= v m+ = v n ( ) k α k m k Φv m, C k j ( )k j Φ (j) (u n, v m ), Φ (j) denotes the j-th composition of Φ with itself The parameter α k m is computed such as minimizing the Euclidean norm of the linearized part of the residual r m+ = v m+ Φ(u n, v m ) We have α k m = ( )k Φ vm, k+ Φ vm k+ Φ vm, k+ Φ vm In fact, we take k =, and we have the following algorithm 6

18 Algorithm Improved fixed-point algorithm v = u n, m = while Φ(u n, v m ) v m > ǫ do Φ vm = Φ(u n, v m ) v m Φ vm = Φ(u n, φ(u n, v m )) Φ(u n, v m ) + v m α m = Φ vm, Φ vm Φ vm, Φ vm v m+ = v m + α m Φ vm m = m + end while u n+ = v m Numerical results Let us begin by stating the parameters chosen for the simulations We present the results obtained with the Sanz-Serna scheme for the non-forced equation (f = ) on the interval [ L, L], with L = 5, discretized in points We obtain similar results with fully backaward Euler and Crank- Nicolson schemes The time step is t = Let us remind that k can denote either an integer as an index, or the value πk L 5 Solution at different times u t=5 t= t= t= x Figure : Solution at times t = ( ), t = 5 ( ), t = (- - -), t = ( ), t = 3 ( ) for γ k = k Here the initial datum is the soliton 7

19 We consider the soliton as initial datum ( u (x) = 3( c) cosh ± ) c (x d), c with c = 5 and d = L We begin with a laplacian damping in Figure The figure shows the effect of the damping on the solution Since γ =, the solution tends to the mean value of u Figure corresponds to the simulation of the following standard dampings the constant: γ k =, k, the laplacian: γ k = k, k, the bilaplacian: γ k = k, k 3 u H 9 8 G(t) constant= laplacian bilaplacian constant= laplacian bilaplacian t 5 5 t Figure : Evolution of u H and G(t) with respect to time for standard dampings, the constant ( ), the laplacian (- - -) and the bilaplacian ( ) Here the initial datum is the soliton For these dampings, we verify that the H -norm decreases as expected Moreover the solution converge to the norm value of u for the laplacian and bilpalacian dampings We can also classify these dampings Here the constant one is more efficient than the two other But the laplacian one is more efficient than the bilaplacian one These results are in agreement with the ones about the KdV equation [5, 8, 9,,, ] 8

20 We extend, in Figure 3, the simulations to the dampings such that γ k > with lim γ k = We consider here three "tend to " dampings, with k + polynomial or exponential decay as follows γ k = + k, γ k = (+ k ) α, α = 3 γ k = e k These dampings do not fit with the asymptions of the theorem 6 but they still damp down the soliton Indeed we notice that the H -norm is decreasing We can classify the dampings Precisely, the first one damps more than the two other and the third one is more efficient than the second one Since these dampings tends to, their effects are principally on low frequencies And for low frequencies, we have + k > e k > ( + k ) α Then the classification of dampings in Figure 3 seems licit 3 u H + k (+ k ) 3 e k 9 G(t) + k (+ k ) 3 e k t 3 5 t Figure 3: Evolution of u H and G(t) with respect to time for "tend to " dampings, γ k = + k ( ), γ k = ( ) and γ (+ k ) 3 k = e k (- - -) Here the initial datum is the soliton 9

21 We now consider band limited dampings, namely { if k N γ k = if k > N (I(N)) We define M := max(k), where k is the finite vector of frequencies used for the numerical tests in Figure To keep γ k >, k, we choose { if k N γ k = e a k N if k > N (EI(N)) or { if k N γ k = (PI(N)) a (+ k N ) if k > N We denote the classic band limited damping by I(N), the one corrected with exponential decay EI(N) and the third one corrected with polynomial decay PI(N) Here we set a = 3, we obtain γ k > with lim γ k = k + 3 u H I ( ) M PI ( ) M EI ( ) M I ( ) M 6 PI ( ) M 6 EI ( ) M G(t) I ( ) M PI ( ) M EI ( ) M I ( ) M 6 PI ( ) M 6 EI ( ) M t 3 5 t Figure : Evolution of u H ( ) and ( G(t) ) with respect ( to) time for ( band ) limited type dampings, I M ( ), PI M ( ), EI M ( ), I M 6 ( ), ( ) ( ) PI M 6 (- - -),EI M 6 ( ) Here the initial datum is the soliton The difference between these dampings is when k > N For these frequencies, since γ k, the dampings with exponential or polynomial decay should damp more than the ideal one (I(N)) It is the case when

22 N = M 6, moreover the damping with polynomial decay is more effective than the one with exponential decay But we observe similar damping when N = N We notice the damping depends on the band width It seems rightful because the equation considered is dispersive Indeed, for this kind of equation, high frequencies appear and these frequencies cannot be damped especially if the band width is too small Let us illustate this by considering now the gaussian as initial datum u (x) = e x σ Here the standard deviation is fixed equal to σ = max(k) Then, in order to see the effect of the damping on the low frequencies, we study the influence of the band limited damping size in Figure 5 We observe that the more N increases, the more efficient is the damping However, it seems that all dampings with N 3σ behave similarly Besides, for lower N, the different kind of dampings can be classified We notice that for all N and all dampings, the decreasing speed is similar at beginning and the difference appears after The difference appears when the high frequencies does 3 u H I ( ) 3σ PI ( ) 3σ EI ( ) 3σ I ( ) 3σ PI ( ) 3σ EI ( ) 3σ G(t) I ( ) 3σ PI ( ) 3σ EI ( ) 3σ I ( ) 3σ PI ( ) 3σ EI ( ) 3σ t 3 5 t Figure 5: Evolution of u H ( ) and G(t) ( ) with respect( to time ) for band ( limited ) type dampings, I 3σ ( ), PI 3σ (- - -), EI 3σ ( ), I 3σ ( ), ( ) ( ) PI 3σ ( ),EI 3σ ( ) Here the initial datum is the gaussian The influence of the domain is finally inspected It is known that the eigenvalues of the laplacian and of the bilapalcian depends on the length of

23 the domain L Here we choose the initial datum as ( ) πx u (x) = sin L Figure 6 presents results with L equal to π, π and 3π We notice that if L < π, the laplacian damping is less effective than the bilaplacian one It is the contrary if L > π But, the dampings are similar when L = π We also observe that for a fixed damping, the greater is L, the slower is the damping u H 8 G(t) 3 6 laplacian, L = π bilaplacian, L = π laplacian, L = π bilaplacian, L = π laplacian, L = 3π bilaplacian, L = 3π 8 laplacian, L = π bilaplacian, L = π laplacian, L = π 3 bilaplacian, L = π laplacian, L = 3π bilaplacian, L = 3π 3 5 t t Figure 6: Evolution of u H and G(t) with respect to time for L = π with γ k = k ( ) and γ k = k (- - -), L = π with γ k = k ( ) and γ k = k ( ), L = 3π with γ k = k ( ) and γ k = k ( ) Here the initial datum is a sine 5 Final comments The damping operator L γ considered in this article allows to have a large variety of dampings We first get back the standard dampings (for example u) But we also have less common dampings like band-limited ones or weaker ones, eg, such that lim γ k = Moreover, with this frequential k + approach, we can adjust the dampings by frequency bands This is interesting in order to build cheap and efficient dampings Using these dampings, the numerical observation shows a damping for energy norms like the H - norm This is consistent with the results about the asymptotic behavior

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