Decay rates of magnetoelastic waves in an unbounded conductive medium

Transcription

1 Decay rates of magnetoelastic waves in an unbounded conductive medium by Ruy Coimbra Charão 1 Jáuber Cavalcanti Oliveira G. Perla Menzala 3 Abstract We study the uniform decay of the total energy of solutions of the system of magnetoelasticity with localized damping near infinity in an exterior 3-D domoin. Using appropiatte multipliers and recent work due to R. Charão and R. Ikekata (Nonlinear Analysis, 7) we conclude that the energy decays like (1 + t) 1 as t +. Mathematics Subject Classifications: Primary 35B4, 35Q6; Secondary 36Q99 Key words and phrases: magnetoelastic waves, exterior domain, localized damping, energy decay. 1 Department of Mathematics, Federal University of Santa Catarina, CEP 884-9, Florianópolis, SC, Brasil, ruycharao@gmail.com Department of Mathematics, Federal University of Santa Catarina, CEP 884-9, Florianópolis, SC, Brasil, jauber@mtm.ufsc.br 3 National Laboratory of Scientific Computation (LNCC/MCT), Ave. Getulio Vargas 333, Quitandinha, Petropolis, RJ, CEP , Brasil and Federal University of Rio de Janeiro, Institute of Mathematics, P.O. Box 6853, Rio de Janeiro, RJ, Brasil, perla@lncc.br

2 1 Introduction The model we consider in this work is motivated by a phenomenon which appears frequently in nature: The interaction between the strain and electromagnetic fields in an elastic body. In the middle 195 s L. Knopoff [9] investigated the propagation of elastic waves in the presence of Earth s magnetic field. The correspondig model nowdays is part of the theory of magnetoelastic waves. More complete models were studied by J. Dunkin and A. Eringen [5] where they considered the propagation of elastic waves under the influence of both, a magnetic and an electric field. Further recent results on the subject can be found in [], [1], [15] and the references therein. The system under consideration in this work may be viewed as a coupling between the hyperbolic system of elastic waves and a parabolic system for the magnetic field. Somehow, the coupled system under consideration has the same structure as the classical isotropic thermoelastic system (see [13], [16] and references therein). When we try to study the asymptotic behavior of the total energy as t + for the model we are considering in this work, then, similar difficulties as for the thermoelastic system (in n = or 3 dimensions) will appear as clearly were pointed out in [6], [1] or [14]. In [] the model of magnetoelasticity was considered in a bounded region of R 3 with an extra localized dissipation ρ(x,u t ) effective only on a little piece of. The conclusion in [] was that the total energy decays uniformly as t + provided ρ(x,u t ) satisfied suitable conditions. The decay was exponential if ρ behaved almost linear in u t and polynomially if ρ had an almost polynomial growth. In this work we consider the magnetoelastic system in the exterior of a compact body and we replace ρ(x,u t ) by a localized dissipation near infinity. The final result is that the total energy associated to the system decays like (1 + t) 1 as t +. Let us describe the model we will consider in this work: Consider a compact set O of R 3 which is star-shaped with respect to the origem (,, ) O, that is η(x) x for all x O where η(x) denotes the unit normal vector at x pointing the exterior of O. Along the next sections we will assume the following hypotheses: (H1). Let = R 3 \ O be the exterior domain and α: R + be a function in L () which is effective only near infinity, that is, there exist constant a > and L >> 1 such that α(x) a > for all x with x L (Thus α(x) could be zero for x < L). (H). The boundary of denoted by is smooth, say of class C. Although, for several estimates is enough to assume that is Lipschitz continuous (see [8]). The model we consider

3 in our discussion is the following. u tt a u (b a ) div u µ [curl h] H + α(x)u t = γh t + curl curlh γ curl(u t H) = div h = (1.1) in (, + ) with boundary conditions u = h η = [curl h] η = and initial conditions on (, + ) on (, + ) on (, + ) (1.) u(x, ) = u (x), u t (x, ) = u 1 (x), h(x, ) = h (x) in. (1.3) In (1.1), the vector field u = (u 1,u,u 3 ) denotes the displacement while h = (h 1,h,h 3 ) denotes the magnetic field. A known magnetic field taken along the x 3 -axis is denoted by H. By simplicity we write H = (,, 1). All other parameters stand as follows: a and b (both strictly positive) are the Lamé constants with b > a, µ is the magnetic permeability (µ > ) and γ is a parameter which is proportional to the electric conductivity (γ > ). A justification for the modelling of the coupled system (1.1) can be found in [11], Chapter. In (1.1) and (1.) we use the standard notation, u = ( u 1, u, u 3 ), u tt = ( u 1, u, ) u 3 t t t, denotes the Laplacian operator, is the gradient operator and div the (spatial) divergence, x denotes the usual vector product and curl is the rotational operator. where The total energy E(t) associated to system (1.1) (1.3) is given by E(t) = 1 [ ut + a u + (b a )(div u) + µ h ] dx (1.4) u t = j=1 3 u j t, u = 3 u j and h = j=1 Formally, by taking the inner product of the first equation in (1.1) by u t and the second by h, adding the resulting relations and integrating over we can verify the identity de dt = µ curlh dx α(x) u t dx (1.5) γ 3 3 j=1 h j

4 which says that the energy E(t) decreases along trajectories. This paper is devoted to study the large time behavior of the total energy under suitable assumptions on the localized extra damping coefficient α(x). Our main result says that E(t) decays like O((1 + t) 1 ) as t +. This is a new result for the coupled system of magnetoelasticity for exterior domains in R 3. There are several articles dealing with the asymptotic behavior of the total energy associated with system (1.1) (1.3) in bounded domains (see [], [6], [1], [14], [15] and the references therein). As far as we know, the only article on system (1.1) in unbounded domain is due to E. Andreou and G. Dassios [1] where they studied the Cauchy problem in = R 3 (and α(x) ). They proved that smooth solutions which vanish as x + do decay (in time) polynomially as t +. In Section we briefly prove the well posedness of problem (1.1) (1.3) in appropiatte Hilbert spaces and finally in Section 3 we prove the main result concerning the polynomial decay of the total energy. Our main tools are the use of multipliers and recent ideas introduced by R. Charão and R. Ikekata [3] while dealing with semilinear elastic waves in exterior domains. Our notations are standard and follow the books [1] and [4]. Well posedness: Functional setting With the notations given in Section 1 we consider the unbounded operator A given as I A = A 1 A 3 B (.1) C A where A 1 = a (b a ) div, A = 1 γ curl curl, A 3 = α(x)i, B = µ [curl( )] H and C = µ curl( H). Let H 1 () denote the usual Sobolev space and consider the Hilbert space H = [H 1 ()] 3 [L ()] 3 W where W is the closure of {f [C ()] 3 such that div f = in } in [L ()] 3. The norm in [L ()] 3 and W is given by f = f dx, while in [H()] 1 3 the norm is f [H = [a f + (b a ) div f ]dx. 1()]3 By Korn s inequality this norm is equivalent to the one induced by [H 1 ()] 3. 4

5 Using the above notation, problem (1.1) (1.3) can be rewritten as du dt = AU U() = U (.) where U = (u,u 1,µ h ) H and U(t) = (u,u t,µ h). The operators which appear in (.1) have their domain defined as follows D(A 1 ) = [H ()] 3 [H()] 1 3 D(A ) = {h [H ()] 3 W such that [curlh] η = on } D(A 3 ) = [H()] 1 3 D(B) = {h W such that [curlh] H [L ()] 3 } D(C) = {v [L ()] 3 such that curl(v H) W }. We set D(A) = [H ()] 3 [H 1 ()] 3 [H 1 ()] 3 D(A ). Clearly D(A) is dense in H. It is convenient to consider the operator D(A) and given by I à = A 1 I A 3 B. C A à with domain D(Ã) = Lemma.1. With the above notations, the operator à is dissipative and Image (I Ã) = H. Proof: Let us consider the natural inner product in [H 1 ()] 3 u,v [H 1 ()] 3 = a ( u, v) + (b a )(div u, div v) + (u,v) where ( u, v) = 3 ( u i, v i ), u = (U 1,u,u 3 ) and (, ) denotes the inner product in L (). i=1 Whenever U = (u,v,h) and V = (ũ,ṽ, h) belong to H we consider the inner product given by U,V H = u,ũ [H 1 ()] 3 + (v, ṽ) + (h, h). 5

6 Let U = (u,v,h) D(Ã), we want to calculate ÃU,U H. Using a result due to D. Sheen [17] we know that (Cv,h) = (Bh,v) holds whenever (v,h) [H 1 ()] 3 D(A ) therefore ÃU,U = v,u [H 1 ()] 3 + ( A 1u u + A 3 v + Bh,v) + (Cv A h,h) = v,u [H 1 ()] 3 (u,v) + ( A 1u + A 3 v,v) + ( A h,h) = v,u [H 1 ()] 3 (u,v) + a ( u,v) + (b a )( div u,v) (α(x)v,v) 1 (curl curlh,h). γ Using the definition of inner product in H followed by integration by parts we obtain ÃU,U H = αv 1 (curl curlh,h) γ = αv 1 γ curl h because if h D(A ) then (h, curl curlh) = curlh (see D. Sheen [17]). Let F = (f 1,f,f 3 ) any element in H. We need to prove the existence of U = (u,v,h) D(Ã) such that U ÃU = F which means that u v = f 1 A 1 u + u + (I A 3 )v Bh = f Cv + (I + A )h = f 3. (.3) Since v = u f 1, (.3) reduces to solve the system A 1 u + u A 3 u Bh = g 1 Cu + h + A h = g (.4) where g 1 = f + (I A 3 )f 1 and g = f 3 Cf 1. Observe that Cf 1 belongs to W because f 1 [H 1 ()] 3. Furthermore, since [H 1 ()] 3 is continuously embedded into the domain of C it follows that g W. Thus the right hand side of system (.4) belongs to [L ()] 3 W. Let us consider the space H = [H 1 ()] 3 V where V is given by V = {v W, curl v [L ()] 3 }. The inner produt in given as follows: For any v and ṽ belonging to V then (v, ṽ) V = [v ṽ + 1 curlv curlṽ]dx. γ We consider the bilinear form a(, ) on [H()] 1 3 [H()] 1 3 given by a(u,ũ) = [( + α(x))u ũ + a u ũ + (b a ) div u div ũ]dx. (.5) 6

7 Next, we define the bilinear form B(, ) on the space H, given by B((u,h), (ũ, h)) = a(u,ũ) + (h, h) V (Bh,ũ) (Cu, h) (.6) where B and C are the operators defined at the begining of the section. We clain that B is continuous and coercive on H. In fact B((u,h), (ũ, h)) a(u,ũ) + (h, h) V + (Bh,ũ) + (Cu, h) K 1 u H 1 ũ H 1 + h V h V + Bh L ũ H 1 + Cu L h V K 1 u H 1 ũ H 1 + h V h V + µ curlh L ũ H 1 + K Cu L h V K 3 (u,h) H (ũ, h) H (.7) because curlh L h V. We used the notation H 1 instead of [H 1 ()] 3 and L instead of [L ()] 3 in order to simplify excess of symbols. In the above inequalities K 1,K and K 3 are positive constant which depend on the Lamé coefficients and/or α( ) L. By (.7) it follows that B is continuous. finally we observe that B((u,h), (u,h)) = a(u,u) + (h,h) V K 4 [ u [H 1 ] + 3 h V ] where K 4 > is a positive constant. Again, we used the identity (Cu,h) = (Bh,u). It follows by Lax-Milgram s lemma that there exists a unique (u,h) H such that B((u,h)(ũ, h)) = (g 1,ũ) + (g, h) (.8) for any (ũ, h) H. It is easy to see that (u,h) in (.8) is a weak solution of (.4). Furthermore, since h V we know that Bh [L ()] 3. Thus, u is a weak solution of A 1 u + u = A 3 u + Bh + g 1 [L ()] 3. (.9) Since A 1 is a strongly elliptic operator it follows from (,9) and elliptic regularity in an exterior domain that u D(A 1 ) = [H ()] 3 [H()] 1 3. Remains to prove that h D(A ). Similar procedure as above shows that h [H ()] 3 W. We claim that curlh η = on. In fact, taking ũ in (.8) and using (.6) we obtain the identity (h, h) V (Cu, h) = (g, h) for all h V. 7

8 Next using the second equation in (.3) in the above identity give us (h, h) V (Cu, h) = ( Cu + h + A h, h) for all h V (.1) Using Green s formula we integrate by parts the term (A h, h) and replace in (.1) to obtain (h, h) V = (h, h) + 1 γ (curlh, curl h) + 1 (η curl h) γ h ds for all h V. However, the inner product in V is given by Therefore (h, h) V = (h, h) + 1 (curlh, curl h). γ (η curlh) h ds = for all h V, which proves our claim. Since v = u f 1 [H 1 ()] 3, the proof of Lemma.1 is now complete. Theorem.. Let and α(x) satisfying hypothesis (H1) and (H) in Section 1 with µ and γ being positive constants. Let (u,u 1,µ h ) H. Then, there exists a unique (weak) solution {u,h} of system (1.1) (1.3) in the class u C([, + ); [H 1 ()] 3 ) C 1 ([, + ); [L ()] 3 ) and h C([, + );W). Moreover, if (u,u 1,µ h ) D(A), then, problem (1.1), (1.), (1.3) has a unique solution (u,h) in the class u C([, + ); [H () H()] 1 3 ) C 1 ([, + ); [H()] 1 3 ) C ([, + ); [L ()] 3 ) and h C([, + ); D(A )) C 1 ([, + );W). Proof: By Lemma.1 the operator à is maximal dissipative with D(Ã) dense in H, then, by Lumer Phillips theorem it follows that à is the infinitesimal generator of a strongly continuous semigroup of contractions. Obviously, this implies in particular that the operator A is also an infinitesimal generator of a strongly continuous semigroup {S(t)} t on H. Both conclusions of Theorem. follow from the theory of semigroups. 8

9 3 Asymptotic Behavior In this section we prove the main result of this article, namely the uniform decay rate of the total energy E(t) given by (1.4) and initial data (u,u 1,µ h ) H. Since (1.5) is valid we integrate over [,t] to obtain for all t. E(t) + µ γ t curl h dxds + t α(x) u t dxds = E() (3.1) Lemma 3.1 (Identities from multipliers). Let (u,h) be the solution of system (1.1) (1.3). Let ϕ: R 3 be (an auxiliary) a continuous function with ϕ x j L () (j = 1,, 3). Then, the following identities are valid d dt (u t,u) u t + a u + (b a ) div u µ u (curlh H)dx + 1 d α(x) u dx = dt d u t (ϕ : u)dx + 1 div ϕ[ u t a u (b a )(div u) ]dx dt 3 [ + a ϕ j u k u k + (b a ) ϕ ] j u k u i dx i,j,k=1 x i x i x j x i x k x j 1 { (ϕ η) a u η + (b a )(div u) }ds + α(x)u t (ϕ : u)dx µ (curlh H) (ϕ : u)dx = (3.) (3.3) Furthermore, if ψ: R + R + is given by a if x L ψ(x) = La if x L x 9

10 where a and L are as in Hypotheses (H1). Then, d ψ(x)u t (x : u)dx + 1 { [3ψ + rψ r ] u t dt a u (b a )(div u) }dx 1 [ (x η)ψ(r) a u + (b a )(div u) ]ds η [ + (b a ) ψ(r)(div u) + ψ ] r r div u(x : u) x dx [ + a ψ(r) u + ψ ] r r x : u dx + αψu t (x : u)dx µ (curlh H)ψ(r) (x : u)dx = (3.4) where r = x, ψ r = dψ dr = x r ψ and ϕ : u = (ϕ u 1,ϕ u,ϕ u 3 ) with u = (u 1,u,u 3 ). Proof: Taking the inner product of the first equation in (1.1) with u followed by integration over give us identity (3.). Next we use the multiplier ϕ : u. We take the inner product of the first equation in (1.1) with ϕ : u. Integration of the resulting identity over give us (3.3). Finally (3.4) is obtained from (3.3) by choosing ϕ(x) = ψ(r)x. Lemma 3.. Let {u, h} be the solution of system (1.1) (1.3). Consider positive numbers k and ε and define G k (t) = ψu t (x : u)dx + ε u t udx + ε α(x) u dx + ke(t). Then, the following estimate d [ 3ψ + dt G rψr k(t) + ε + k ] u α(x) t dx + k { α(x) u t dx + a ε 3ψ + rψ [ r 1 + ψ + rψ t + b ]} u dx a + kµ curl h dx γ { + (b a ) ε 3ψ + rψ [ r 1 + ψ + rψ r + b ]} (div u) dx a α(x)ψu t (x : u)dx + εµ u (curlh H)dx + µ ψ curlh (x : u)dx holds. 1 (3.5)

11 Proof: We adapt to our case some ideas used by Charao and Ikehata [3] for the system of elastic waves in exterior domains. Let us multiply identity (1.5) by k > and (3.) by ε >. Adding both resulting identities with (3.4), using hypothesis (H1) and the fact that ϕ r we obtain the following inequality: d [ 3ψ + dt G rψr k(t) + ε + kα [ + a ε 3ψ + rψ r + ψ(r) + rψ r ] u dx + kµ [ curlh dx + (b a ) γ + (b a ψ r ) div u(x : u) xdx r εµ u (curlh H)dx αψu t (x : u)dx + µ ψ(curlh H) (x : u)dx. Let δ = 1 + b. We use Young s inequality to obtain a ] u t dx + k α u t dx ε + ψ 3ψ + rψ r ] (div u) dx (3.6) (b a ) div u u (b a ) δ u + 1 δ div u (b a ) = a (b a) ( a + b ) u + (b a )(div u). a a (3.7) Since (div u)(x : u) x div u x u and ψ r. Then multiplying both sides by ψr we r obtain ψ r r div u(x : u) x rψ r div u u. (3.8) Thus, using (3.8) and (3.7) in the last integral of the left hand side integral of (3.6) we obtain (3.5). Lemma 3.3. Assume < a < b < 4a and let (u,h) be the solution of system (1.1) (1.3). If (H1) is valid then there exist positive constants ε and ε 1 such that t M E(s)ds + εµ curlh γ ( k + L a ) t α α(x) u a t dxds ε 1 ( kµ + G k (t) + γ µ L a ) t curl h dxds a ε 1 G k () + εµ [h curl(u H)] H dx (3.9) 11

12 where M = min{εµ,ε 1 /4}, k is any positive number such that k > max and H(x,t) = t h(x,s)ds. Proof: Let us define E 1 (t) = 1 { u t + a u + (b a )(div u) }dx. { L a α a ε 1 }, L a µ γ a ε 1 Let us choose ε > such that a (1 + b a ) < ε < 3a which is possible because b < 4a. Next, we can choose ε 1 > such that and 3ψ + rψ r ε + kα(x) ε 1 > (3.1) ε 3ψ + rψ r + ψ + rψ r ( a + b a ) ε 1 >. (3.11) In fact, say for instance ε = a (1 + b ) (then a 4a (1 + b) < a a (1 + b ) = ε < 3a 4a because b < a). Then, we can choose ε 1 = a ( 1 b ) and it is easy to verify that (3.1) and (3.11) hold for all 4a x. Using (3.1) and (3.11) together with (3.5) we deduce the estimate d dt G k(t) + ε 1 E 1 (t) + k α(x) u t dx + kµ curlh dx γ α(x)ψu t (x : u)dx + εµ u (curlh H)dx + µ ψ curlh (x : u)dx. (3.1) Now, we will estimate two terms on the right hand side of (3.1). Since x ψ(x) La in R 3, we obtain for any ε > α(x)u t ψ (x : u)dx La α(x) u t u dx La α(x) u t dx + La ε ε 4 α(x) u dx. (3.13) Let us choose ε = a ε 1 La α L to obtain from (3.13) the inequality α(x)ψu t (x : u)dx L a α L a ε 1 α(x) u t dx + a ε 1 4 u dx. (3.14) 1

13 Similarly we obtain for any δ > µ ψ curlh (x : u)dx µ La curl h u dx µ La curlh dx + µ La δa δa 4 u dx. Using (3.14) and (3.15) together with (3.1) and choosing δ = ε 1 µ La we obtain d dt G k(t) + A k α(x) u t dx + B k curl h dx + C u dx + ε 1 u t dx + ε 1 (b a ) (div u) dx εµ u (curlh H)dx where (3.15) (3.16) A k = k L a α L a ε 1 >, B k = kµ γ µ La δa > and C = a ε 1 8. Clearly we are choosing k large enough so that A k and B k are non-negative. Finally we consider the field H(x,t) = t h(x,s)ds. Since {u,h} is the solution of problem (1.1) (1.3) then we can easily verify that H(x,t) satisfies γh t + curl curlh γ curl(u H) = γ[h curl(u H)]. (3.17) Taking the inner product of (3.17) by H t ad integrating the result over [,t] we obtain t h ds + 1 curlh γ curl(u H) h dxds + t = [h curl(u H)] H dx. (3.18) Next, we integrate inequality (3.16) over [, t] and add the result with indentity (3.18) multiplied 13

14 by εµ to obtain t εµ h ds + εµ γ curl H + G k (t) t t + A k α(x) u t dxds + B k curlh dxds t + C u dxds + ε t 1 u t dxds + ε t 1 (b a ) (div u) dxds t εµ curl(u H) h dxds t + εµ u (curlh H)dxds + G k () + εµ [h curl(u H] H dx = G k () + εµ [h curl(u H)] Hdx (3.19) because u (curlh H) = curl(u H) h holds for any three vectors u, curlh and H. Lemma 3.4. Let < a < b < 4a and {u,h} be the solution of system (1.1) (1.3). Assuming (H1) and (H) are valid and h W L (), then we can find positive constant c 1,c and c 3 such that t M E(s)ds + c 1 curlh ( ) k + L a t α α(x) u a t dxds ε 1 ( ) kµ + G k (t) + γ µ La t curlh dxds a ε 1 G k () + c ϕ + c 3 u where k and M and H are as in Lemma 3.3. (3.) Proof: Due to our assumptions on the initial data h, the Hodge decomposition h = Φ + curlϕ holds (see [4] page 3). Then, we have the identity h H dx = = Φ div Hdx ϕ curlh dx 14 ϕ curlhdx

15 because div H = in. We can estimate for any δ > h H dx δ ϕ + 1 4δ curlh. Thus εµ [h curl(u H) H dx εµ δ ϕ + εµ 4δ curl H εµ curl(u H) H dx. Since H = 1 we deduce for any δ > curl(u H) H dx = (u H) curlh dx u curlh dx δ u + 1 8δ curlh. Consequently, the last term on the right hand side of (3.9) can be bounded by εµ δ ϕ + εµ δ u + 3εµ 8δ curl H. Choosing δ > 3γ 4 we obtain (3.) with c 1 = εµ ( 1 γ 3 8δ ) >, c = εµ δ and c 3 = εµ δ. Lemma 3.5. Let < a < b < 4a and {u,h} be the solution of system (1.1) (1.3). Assuming hypotheses (H1) and (H) we have G k (t) for all t and k >> 1 sufficiently large, where G k is given as in Lemma 3.. Proof: First, we estimate ε(u t,u) in G k (t). Using (H) and Poincaré s lemma we have for any δ > ε(u t,u) ε δ u t + εδ u ε δ E(t) + εδ { α(x) } u dx + u dx x L a {x, x <L} O ε εδ E(t) + α(x) u dx + δε δ a C L u dx ( ε δ + εδ ) C L E(t) + εδ α(x) u dx. a (3.1) 15

16 Next, we estimate ψu t (x : u)dx x ψ u t u dx La u t u dx La a La E(t) for any t. a [ ut ] + a u dx Next we choose δ > small such that δ/a < 1 and afterwords k >> 1 large such that ε δ + εδ C L + La a < k. Hence, using (3.1) and (3.) with our above choice of k and δ we obtain ψu t (x : u)dx ε u t udx ε α u dx + ke(t) for any t which proves that G k (t) for all t. (3.) Theorem 3.1. (Energy decay). Let < a < b < 4a and assume (H1) (H). Let (u,u 1,h ) H with h W L (). Then, the total energy E(t) (see (1.4)) associated with problem (1.1) (1.3) satisfies E(t) C(1 + t) 1 where C is a positive constant which depends on the L -norms of the initial data u, u,u 1 and h. Proof: Using Lemmas 3.4 and 3.5 we can write the inequality In particular t t M E(s)ds + B k curl h ds t + A k α(x) u t dxds + c 1 curlh G k () + c ϕ + c 3 u. t E(s)ds C(u,u 1, u,h ) (3.3) where the constant can be estimated explicitely. Furthermore, since E(t) is nonincreasing (by (1.5)) then for any t > the inequality (1 + t)e(t) E() + 16 t E(s)ds (3.4)

17 holds. Thus (1 + t)e(t) E() + t E(s)ds E() + C. Therefore the conclusion of Theorem 3.1 follows where C is a positive constant which depends only on the initial data. Adknowledgements: The co-author GPM was partially supported by a Research Grant of CNPq (Proc /9-). The co-author RCCh and GPM were also partially supported by Project PROSUL (Proc. 4939/8-) from CNPq (Brasil). They would like to express their gratitude for such important support. References [1] E. Andreou and G. Dassios, Dissipation of energy for magnetoelastic waves in a conductive medium, Quart. Appl. Math. 55 (1997), pp [] R. Charao, J. Oliveira and G. Perla Menzala, Energy decay rates of magnetoelastic waves in a bounded conductive medium, Discrete and Continuous dynamical Systems, Serie A, Vol 5 (9), pp [3] R. Charao and R. Ikehata, Decay of solutions for a semilinear system of elastic waves in an exterior domain with damping near infinity, Nonlinear Analysis 67 (7), pp [4] R. Dautray and J.L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol 3, Spectral Theory and Applications (with the collaboration of Michel Artola and Michel Cessenat), Springer-Verlag, (199). [5] J. Dunkin and A. Eringen, On the propagation of waves in an electromagnetic elastic solid, Internat. J. Engrg. Sci. 1 (1963), pp [6] Th. Duyckaerts, Stabilization haute fréquence d équations aux derivées partialles linéaires, These de Doctorat, Université Paris XI, Orsay, 4. [7] A.C. Eringen and G.A. Maugin, Electrodynamics of Continua, Springer, Berlin, (199). [8] V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Springer-Verlag, Berlin (1986). 17

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