DECAY RATES OF MAGNETOELASTIC WAVES IN AN UNBOUNDED CONDUCTIVE MEDIUM

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1 Electronic Journal of Differential Equations, Vol. 11 (11), No. 17, pp ISSN: URL: or ftp eje.math.txstate.eu (login: ftp) DECAY RATES OF MAGNETOELASTIC WAVES IN AN UNBOUNDED CONDUCTIVE MEDIUM RUY COIMBRA CHARÃO, JÁUBER CAVALCANTE OLIVEIRA, GUSTAVO PERLA MENZALA Abstract. We stuy the uniform ecay of the total energy of solutions for a system in magnetoelasticity with localize amping near infinity in an exterior 3-D omain. Using appropriate multipliers an recent work by Charão an Ikekata [3], we conclue that the energy ecays at the same rate as (1 + t) 1 when t Introuction The moel we consier in this work is motivate by a phenomenon which appears frequently in nature: The interaction between the strain an electromagnetic fiels in an elastic boy (Eringen-Maugin [7]). In the mile 195 s Knopoff [9] investigate the propagation of elastic waves in the presence of Earth s magnetic fiel. The corresponing moel nowaays is part of the theory of magnetoelastic waves. More complete moels were stuie by Dunkin an Eringen [5] where they consiere the propagation of elastic waves uner the influence of both, a magnetic an an electric fiel. Further recent results on the subject can be foun in [, 1, 15] an the references therein. The system uner consieration in this work may be viewe as a coupling between the hyperbolic system of elastic waves an a parabolic system for the magnetic fiel. Somehow, the couple system uner consieration has the same structure as the classical isotropic thermoelastic system (see [13, 16] an references therein). When we try to stuy the asymptotic behavior of the total energy as t + for the moel we are consiering in this work, then, similar ifficulties as for the thermoelastic system (in n = or 3 imensions) will appear as clearly were pointe out in [6, 1, 14]. In [], the moel of magnetoelasticity was consiere in a boune region of R 3 with an extra localize issipation ρ(x, u t ) effective only on a little piece of. The conclusion in [] was that the total energy ecays uniformly as t + provie ρ(x, u t ) satisfie suitable conitions. The ecay was exponential if ρ behave almost linear in u t an polynomially if ρ ha an almost polynomial growth. Mathematics Subject Classification. 35B4, 35Q6, 36Q99. Key wors an phrases. Magnetoelastic waves; exterior omain; localize amping; energy ecay. c 11 Texas State University - San Marcos. Submitte September 3, 11. Publishe October 11, 11. 1

2 R. C. CHARÃO, J. C. OLIVEIRA, G. PERLA M. EJDE-11/17 In this work we consier the magnetoelastic system in the exterior of a compact boy an we replace ρ(x, u t ) by a localize issipation near infinity. The final result is that the total energy associate to the system ecays like (1 + t) 1 as t +. Let us escribe the moel we will consier in this work: Consier a compact set O of R 3 which is star-shape with respect to the origin (,, ) O, that is η(x) x for all x O where η(x) enotes 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 omain an α: R + be a function in L () which is effective only near infinity, that is, there exist constant a > an L >> 1 such that α(x) a > for all x with x L (Thus α(x) coul be zero for x < L). (H) The bounary of enote by is smooth, say of class C. Although, for several estimates is enough to assume that is Lipschitz continuous; see Girault-Raviart [8]. In our iscussion we consier the moel u tt a u (b a ) iv u µ [curl curl h] H + α(x)u t = h t + curl h curl(u t H) = iv h = in (, + ) with bounary conitions an initial conitions u = [curl h] η = on (, + ) on (, + ) (1.1) (1.) u(x, ) = u (x), u t (x, ) = u 1 (x), h(x, ) = h (x) in. (1.3) In (1.1), the vector fiel u = (u 1, u, u 3 ) enotes the isplacement while h = (h 1, h, h 3 ) enotes the magnetic fiel. A known magnetic fiel taken along the x 3 -axis is enote by H. By simplicity we write H = (,, 1). The other parameters stan as follows: a an b (both strictly positive) are relate to the Lamé constants with b > a, µ is the magnetic permeability (µ > ) an is a parameter which is proportional to the electric conuctivity ( > ). A justification for the moelling of the couple system (1.1) can be foun in [11, chapter ]. In (1.1) an (1.) we use the stanar notation, u = ( u 1, u, u 3 ), u t = ( u1 t, u t, u3 t ), enotes the Laplacian operator, is the graient operator an iv the (spatial) ivergence, enotes the usual vector prouct an curl is the rotational operator. The total energy E(t) associate to system (1.1) (1.3) is E(t) = 1 [ ut + a u + (b a )(iv u) + µ h ] x (1.4) where u t = 3 j=1 u j t, u = 3 u j, h = Formally, by taking the inner prouct of the first equation in (1.1) by u t an the secon by h, aing the resulting relations an integrating over we can verify, j=1 3 j=1 h j

3 EJDE-11/17 DECAY RATES OF MAGNETOELASTIC WAVES 3 using the bounary conitions, the ientity E t = µ curl h x α(x) u t x (1.5) which says that the energy E(t) ecreases along trajectories. This article is evote to stuy the large time behavior of the total energy uner suitable assumptions on the localize extra amping coefficient α(x). Our main result says that E(t) ecays like O((1 + t) 1 ) as t +. This is a new result for the couple system of magnetoelasticity for exterior omains in R 3. There are several articles ealing with the asymptotic behavior of the total energy associate with system (1.1) (1.3) in boune omains (see [, 6, 1, 14, 15] an the references therein). As far as we know, the only article on system (1.1) in unboune omain is ue to Anreou an Dassios [1] where they stuie the Cauchy problem in = R 3 (an α(x) ). They prove that smooth solutions which vanish as x + o ecay (in time) polynomially as t +. In Section we briefly prove the well poseness of problem (1.1) (1.3) in appropriate Hilbert spaces an finally in Section 3 we prove the main result concerning the polynomial ecay of the total energy. Our main tools are the use of multipliers an recent ieas introuce by Charão an Ikekata [3] while ealing with semilinear elastic waves in exterior omains. Our notation is stanar an follow the books [4, 1].. Well poseness: Functional setting With the notation given in Section 1 we consier the unboune operator I A = A 1 A 3 B (.1) C A where A 1 = a (b a ) iv, A = 1 curl curl, A 3 = α(x)i, B = µ [curl( )] H an C = curl( H). Let H 1 () enote the usual Sobolev space an consier the Hilbert space H = [H 1 ()] 3 [L ()] 3 W where W is the closure of {f [C ()] 3 such that iv f = in } in [L ()] 3. The norm in [L ()] 3 an W is given by f = f x, while in [H 1 ()] 3 the norm is f [H = [a f + (b a ) iv f + f ]x. 1()]3 Using the above notation, problem (1.1) (1.3) can be rewritten as U t = AU U() = U (.) where U = (u, u 1, h ) H an U(t) = (u, u t, h).

4 4 R. C. CHARÃO, J. C. OLIVEIRA, G. PERLA M. EJDE-11/17 The operators which appear in (.1) have their omains efine as follows: We set D(A 1 ) = [H ()] 3 [H 1 ()] 3 D(A ) = {h [H ()] 3 W such that [curl h] η = on } D(A 3 ) = [H 1 ()] 3 D(B) = {h W such that [curl h] H [L ()] 3 } D(C) = {v [L ()] 3 such that curl(v H) W }. D(A) = [H ()] 3 [H 1 ()] 3 [H 1 ()] 3 D(A ). Clearly D(A) is ense in H. It is convenient to consier the operator à with omain D(Ã) = D(A) an given by I à = A 1 I A 3 B. C A Lemma.1. With the above notation, the operator à is issipative an Image(I Ã) = H. Proof. Let us consier the natural inner prouct in [H 1 ()] 3, u, v [H 1 ()] 3 = a ( u, v) + (b a )(iv u, iv v) + (u, v) where ( u, v) = 3 ( u i, v i ), u = (u 1, u, u 3 ), v = (v 1, v, v 3 ) an (, ) enotes i=1 the inner prouct in L (). Whenever U = (u, v, h) an V = (ũ, ṽ, h) belong to H we consier the inner prouct in H given by U, V H = u, ũ [H 1 ()] 3 + (v, ṽ) + (h, h). Let U = (u, v, h) D(Ã), we want to calculate ÃU, U H. Using a result ue to Sheen [17] we know that (Cv, h) = (Bh, v) hols 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 )( iv u, v) (α(x)v, v) 1 (curl curl h, h). Using the efinition of inner prouct in H followe by integration by parts we obtain ÃU, U H = αv 1 (curl curl h, h) = αv 1 curl h because if h D(A ) then (h, curl curl h) = curl h (see Sheen [17]).

5 EJDE-11/17 DECAY RATES OF MAGNETOELASTIC WAVES 5 Let F = (f 1, f, f 3 ) any element in H. We nee 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. Since v = u f 1, (.3) reuces to solve the system (.3) 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 an 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 embee into the omain of C it follows that g W. Thus the right han sie of system (.4) belongs to [L ()] 3 W. Let us consier the space H = [H 1 ()] 3 V where V is given by V = {v W, curl v [L ()] 3 }. The inner prouct in V is given as follows: For any v an ṽ belonging to V we efine (v, ṽ) V = [v ṽ + 1 curl v curl ṽ] x. We consier 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 ) iv u iv ũ] x. (.5) Next, we efine 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 an C are the operators efine at the beginning of this section. We claim that B is continuous an 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 + curl h L ũ H 1 + curl u L h V K (u, h) H (ũ, h) H (.7) because curl h L h V an curl u L Const. u H 1. We use the notation H 1 instea of [H 1 ()] 3 an L instea of [L ()] 3 in orer to simplify excess of symbols. In the above inequalities K 1 an K are positive constants which epen on 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 u [H 1 ] 3 + h V. Again, we use the ientity (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)

6 6 R. C. CHARÃO, J. C. OLIVEIRA, G. PERLA M. EJDE-11/17 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) an elliptic regularity in an exterior omain of C class that u D(A 1 ) = [H ()] 3 [H 1 ()] 3. Remains to prove that h D(A ). Similar proceure as above shows that h [H ()] 3 W. We claim that curl h η = on. In fact, taking ũ = u in (.8) an using the fact that u an h are solution of (.4) combine with (.6), we obtain the ientity (h, h) V (Cu, h) = (g, h) for all h V. Next using the secon equation in (.3) in the above ientity give us (h, h) V (Cu, h) = ( Cu + h + A h, h) for all h V (.1) Using Green s formula, Sheen [17], (curl u, v) = (u, curl v) + η u, v we integrate by parts the term (A h, h) an replace in (.1) to obtain (h, h) V = (h, h) + 1 (curl h, curl h) + 1 (η curl h) h S for all h V. However, the inner prouct in V is given by Therefore, (h, h) V = (h, h) + 1 (curl h, curl h). (η curl h) h S = for all h V, which proves our claim. Since v = u f 1 [H 1 ()] 3, the proof is complete. Theorem.. Let an α(x) satisfying hypotheses (H1) an (H) with µ an being positive constants. Let (u, u 1, h ) H. Then there exists a unique (weak) solution {u, h} of system (1.1) (1.3) with u C([, + ); [H 1 ()] 3 ) C 1 ([, + ); [L ()] 3 ), h C([, + ); W ). Moreover, if (u, u 1, h ) D(A), then, problem (1.1), (1.), (1.3) has a unique solution (u, h) with u C([, + ); [H () H 1 ()] 3 ) C 1 ([, + ); [H 1 ()] 3 ) C ([, + ); [L ()] 3 ), h C([, + ); D(A )) C 1 ([, + ); W ). Proof. By Lemma.1 the operator à is maximal issipative with D(Ã) ense 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.

7 EJDE-11/17 DECAY RATES OF MAGNETOELASTIC WAVES 7 3. Asymptotic Behavior In this section we prove the main result of this article, namely the uniform ecay rate of the total energy E(t) given by (1.4) an initial ata (u, u 1, h ) H. Since (1.5) is vali we integrate over [, t] to obtain for all t. E(t) + µ t curl h x s + t α(x) u t x s = E() (3.1) Lemma 3.1 (Ientities 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 ientities are vali t (u t, u) u t + a u + (b a ) iv u µ u (curl h H)x + 1 α(x) u x =, t u t (ϕ : u)x + 1 iv ϕ[ u t a u (b a )(iv u) ]x t 3 [ + a ϕ j u k u k + (b a ) ϕ j u k u i ] x i,j,k=1 x i x i x j x i x k x j 1 (ϕ η) { a u η + (b a )(iv u) } S + α(x)u t (ϕ : u)x µ (curl h H) (ϕ : u)x = Furthermore, if ψ : R n R + is given by { a if x L ψ(x) = ψ( x ) = if x L La x where a an L are as in Hypotheses (H1), then ψ(r)u t (x : u)x + 1 { [3ψ + rψ r ] u t t a u (b a )(iv u) } x 1 [ (x η)ψ(r) a u + (b a )(iv u) ] S η [ + (b a ) ψ(r)(iv u) + ψ ] r r iv u(x : u) x x + a [ ψ(r) u + ψ r r x : u ] x + α(x)ψ(r)u t (x : u)x µ (curl h H)ψ(r) (x : u)x = (3.) (3.3) (3.4) where r = x, ψ r = ψ r = x r ψ an ϕ : u = (ϕ u 1, ϕ u, ϕ u 3 ) with u = (u 1, u, u 3 ).

8 8 R. C. CHARÃO, J. C. OLIVEIRA, G. PERLA M. EJDE-11/17 Proof. Taking the inner prouct of the first equation in (1.1) with u followe by integration over give us ientity (3.). Next we use the multiplier ϕ : u. We take the inner prouct of the first equation in (1.1) with ϕ : u. Integration of the resulting ientity over give us (3.3). Finally (3.4) is obtaine from (3.3) by choosing ϕ(x) = ψ(x)x = ψ(r)x. Lemma 3.. Let {u, h} be the solution of system (1.1)-(1.3). Consier positive numbers k an ε an efine G k (t) = ψu t (x : u)x + ε u t ux + ε α(x) u x + ke(t). Then, assuming hypothesis (H1), [ 3ψ + t G rψr k(t) + ε + k ] u α(x) t x + k α(x) u t x + a { ε 3ψ + rψ r [1 + ψ + rψ r + b ] } u x a + kµ curl h x { + (b a ) ε 3ψ + rψ r [1 + ψ + rψ r + b ] } (3.5) (iv u) x a α(x)ψu t (x : u)x + εµ u (curl h H)x + µ ψ curl h (x : u)x. Proof. We aapt to our case some ieas use by Charão an Ikehata [3] for the system of elastic waves in exterior omains. Let us multiply ientity (1.5) by k > an (3.) by ε >. Aing both resulting ientities with (3.4), using hypothesis (H1) an the fact that ϕ r we obtain the inequality [ 3ψ + rψr ε + kα(x) ] u t x + k t G k(t) + α(x) u t x [ + a ε 3ψ + rψ r + ψ(r) + rψ r ] u x + kµ [ curl h x + (b a ) ε + ψ 3ψ + rψ ] r (iv u) x + (b a ψ r ) iv u(x : u) xx r εµ u (curl h H)x α(x)ψu t (x : u)x + µ ψ(curl h H) (x : u)x. Let δ = 1 + b a. We use Young s inequality to obtain (3.6) (b a ) iv u u (b a ) δ iv u + 1 δ (b a ) u (b a) = a u + ( a + b) (b a )(iv u). a a (3.7)

9 EJDE-11/17 DECAY RATES OF MAGNETOELASTIC WAVES 9 Since (iv u)(x : u) x iv u x u an ψ r, multiplying both sies of (3.7) by ψ r r we obtain ψ r r (iv u)(x : u) x rψ r iv u u. (3.8) Thus, using (3.8) an (3.7) in the last integral of the left han sie integral of (3.6) we obtain (3.5). Lemma 3.3. Assume < a < b < 4a an let (u, h) be the solution of system (1.1) (1.3). If (H1) is vali then there exist positive constants ε an ε 1 such that t M E(s)s + εµ curl H ( k + L a ) t α a α(x) u t x s ε 1 ( kµ + G k (t) + µ L a ) t a curl h x s ε 1 G k () + εµ [h curl(u H)] H x where M = min{ε, ε 1 /4}, k is any positive number such that { L a k > max α a, L a µ ε 1 a, (ε 1 + ε) } ε 1 a an H(x, t) = t h(x, s)s. Proof. Let us efine E 1 (t) = 1 { u t + a u + (b a )(iv u) }x. (3.9) 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 an 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 a 4a ) (then (1 + b a ) < a (1 + b 4a ) = ε < 3 a because b < a). Then, ue to < a < b < a, we can choose ε 1 = a ( 1 b it is easy to verify that (3.1) an (3.11) hol for all x if k > (ε1+ε) a. Using (3.1), (3.11), an (3.5), we euce the estimate t G k(t) + ε 1 E 1 (t) + k α(x)ψu t (x : u)x + εµ + µ ψ curl h (x : u)x. α(x) u t x + kµ curl h x u (curl h H)x 4a ) an (3.1)

10 1 R. C. CHARÃO, J. C. OLIVEIRA, G. PERLA M. EJDE-11/17 Now, we will estimate two terms on the right han sie of (3.1). Since x ψ(x) La in R 3, for any ε > we obtain α(x)ψu t (x : u)x La α(x) u t u x La α(x) u t x + La (3.13) ε α(x) u x. ε 4 Let us choose ε = a ε 1 La α L to obtain from (3.13) the inequality α(x)ψu t (x : u)x L a α L a ε 1 α(x) u t x + a ε 1 4 Similarly for any δ >, we obtain µ ψ curl h (x : u)x µ La µ La δa curl h u x curl h x + µ La δa 4 u x. u x. (3.14) (3.15) Using (3.14) an (3.15) together with (3.1) an choosing δ = ε 1 /(µ La ), we obtain t G k(t) + A k α(x) u t x + B k curl h x + C u x + ε 1 u t x + ε 1 (b a ) (iv u) x (3.16) εµ u (curl h H)x where A k = k L a α L a ε 1, B k = kµ µ La δa = kµ µ L a ε 1 a, C = a ε 1 8. Clearly we are choosing k large enough so that A k an B k are non-negative. Finally we consier the fiel H(x, t) = t h(x, s)s. Since {u, h} is the solution of problem (1.1) (1.3) then we can easily verify that H(x, t) satisfies H t + curl curl H curl(u H) = [h curl(u H)]. (3.17) Taking the inner prouct of (3.17) by H t an integrating the result over [, t] we obtain t h s + 1 curl H t (3.18) = curl(u H) h x s + [h curl(u H)] H x.

11 EJDE-11/17 DECAY RATES OF MAGNETOELASTIC WAVES 11 Next, we integrate inequality (3.16) over [, t] an a the result with ientity (3.18) multiplie by εµ to obtain t εµ h s + εµ curl H + G k (t) t t + A k α(x) u t x s + B k curl h x s t + C u x s + ε t 1 u t x s + ε t 1 (b a ) (iv u) x s t εµ curl(u H) h x s t + εµ u (curl h H)x s + G k () + εµ [h curl(u H] H x = G k () + εµ [h curl(u H)] Hx (3.19) because u (curl h H) = curl(u H) h hols for any three vectors u, curl h an H. Now, since C = ε 1a we choose M = min{ε, ε 1 /4} in (3.19) to conclue 8 the proof. Lemma 3.4. Let < a < b < 4a an {u, h} be the solution of system (1.1) (1.3) with initial ata (u, u 1, h ) H. Suppose that (H1) an (H) are vali. Assuming the Hoge ecomposition (see [4, page 3]) Φ + curl ϕ of h W hols with ϕ [L ()] 3, then we can fin positive constants c 1, c an c 3 such that t M E(s)s + c 1 curl H ( k + L a ) t α a α(x) u t x s ε 1 ( kµ + G k (t) + µ La ) t a curl h x s ε 1 G k () + c ϕ + c 3 u where k an M an H are as in Lemma 3.3. Proof. We have the ientity h H x = Φ iv Hx ϕ curl Hx = ϕ curl H x because iv H = in. We can estimate for any δ > h H x δ ϕ + 1 4δ curl H. (3.)

12 1 R. C. CHARÃO, J. C. OLIVEIRA, G. PERLA M. EJDE-11/17 Thus εµ [h curl(u H) H x εµ δ ϕ + εµ 4δ curl H εµ curl(u H) H x. Since H = 1 we euce for the same δ > as above curl(u H) H x = (u H) curl H x + η (u H), H u curl H x δ u + 1 curl H 8δ because u =. Consequently, the last term on the right han sie of (3.9) can be boune by εµ δ ϕ + εµ δ u + 3εµ 8δ curl H. Choosing δ > 3/4, from (3.9) we obtain (3.) with c 1 = εµ ( 1 3 8δ ) >, c = εµ δ an c 3 = εµ δ. Lemma 3.5. Let < a < b < 4a an {u, h} be the solution of system (1.1) (1.3). Assuming hypothesis (H1) we have G k (t) for all t an k >> 1 sufficiently large, where G k is given as in Lemma 3.. Proof. First, we estimate the term ε(u t, u) in G k (t). Using (H) an 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 t x + u x x L a {x, x <L} ε εδ E(t) + α(x) u t x + δε δ a C L u x ( ε δ + εδ ) a C L E(t) + εδ α(x) u t x. a where C L > is the constant of Poincaré s Lemma in { x < L}. Next, we estimate ψu t (x : u)x x ψ u t u x La u t u x La [ ut + a a u ] x La E(t) for any t. a (3.1) (3.)

13 EJDE-11/17 DECAY RATES OF MAGNETOELASTIC WAVES 13 Next we choose δ > small such that δ/a < 1 an after wors k >> 1 large such that ε δ + εδ a C L + La a < k. Hence, using (3.1) an (3.) with our above choice of k an δ we obtain ψu t (x : u)x ε u t u x ε α(x) u t x + ke(t) for any t which proves that G k (t) for all t. Theorem 3.6 (Energy ecay). Assume hypotheses of Lemma 3.4. Then the total energy E(t) (see (1.4)) associate with problem (1.1) (1.3) satisfies E(t) C(1 + t) 1 where C is a positive constant which epens on the L -norms of the initial ata u, u, u 1, h an the coefficients of the system. Proof. Using Lemmas 3.4 an 3.5 we can write the inequality In particular, M t t + A k t E(s)s + B k curl h s α(x) u t x s + c 1 curl H G k () + c ϕ + c 3 u. t E(s)s C(u, u 1, u, h ) where the positive constant can be estimate explicitly. The constant in above inequality also epens on the coefficients of the system an a, L, k an α(.). Furthermore, since E(t) is non-increasing (by (1.5)) then for any t > the inequality hols. Thus (1 + t)e(t) E() + (1 + t)e(t) E() + t t E(s)s E(s)s E() + C. Therefore the conclusion of Theorem 3.6 follows where C is a positive constant which epens only on the initial ata an the coefficients of the system. Acknowlegements. The authors want to express their gratitue for the economic support receive: G. Perla M. from a Research Grant of CNPq (Proc /9-); R. C. Charão an G. Perla M. from Project PROSUL (Proc. 4939/8-) from CNPq (Brasil).

14 14 R. C. CHARÃO, J. C. OLIVEIRA, G. PERLA M. EJDE-11/17 References [1] E. Anreou an G. Dassios; Dissipation of energy for magnetoelastic waves in a conuctive meium, Quart. Appl. Math. 55 (1997), pp [] R. Charao, J. Oliveira, G. Perla Menzala; Energy ecay rates of magnetoelastic waves in a boune conuctive meium, Discrete an Continuous ynamical Systems, Serie A, Vol. 5 (9), pp [3] R. Charao, R. Ikehata; Decay of solutions for a semilinear system of elastic waves in an exterior omain with amping near infinity, Nonlinear Analysis, 67 (7), pp [4] R. Dautray, J. L. Lions; Mathematical Analysis an Numerical Methos for Science an Technology, Vol. 3, Spectral Theory an Applications (with the collaboration of Michel Artola an Michel Cessenat), Springer-Verlag, (199). [5] J. Dunkin, A. Eringen; On the propagation of waves in an electromagnetic elastic soli, Internat. J. Engrg. Sci., 1 (1963), pp [6] Th. Duyckaerts; Stabilization haute fréquence équations aux erivés partialles linéaires, These e Doctorat, Université Paris XI, Orsay, 4. [7] A. C. Eringen, G. A. Maugin; Electroynamics of Continua, Springer, Berlin, (199). [8] V. Girault, P.A. Raviart; Finite Element Methos for Navier-Stokes Equations, Theory an Algorithms, Springer-Verlag, Berlin (1986). [9] L. Knopoff; The interaction between elastic wave motions an a magnetic fiel in electrical conuctors, J. Geophys Research 6 (1955), pp [1] J. L. Lions, E. Magenes; Non-homogeneous bounary value problems an applications, Springer-Verlag, Berlin-Heielberg, 197. [11] W. Nowacki; Electromagnetic interactions in elastic solis, Eite by H. Parkus, Int. Centre for Mech. Sci. Courses an Lectures, No. 57, Springer-Verlag, New York (1979). [1] G. Perla Menzala, E. Zuazua; Energy ecay of magnetoelastic waves in a boune conuctive meium, Asymptotic Analysis 18 (1998), pp [13] R. Racke; On the time-asymptotic behaviour of solutions in thermoelasticity, Proc. Royal Soc. Einburgh, 17A (1987), pp [14] R. Racke, J. Rivera; Magneto-Thermoelasticity, large time behavior for linear systems, Av. Diff. Equations, 6 (1), pp [15] J. Rivera, M. Santos; Polynomial stability to three imensional magnetoelastic waves, IMA, J. Appl. Math. 66 (1), pp [16] J. M. Rivera; Asymtototic behavior in inhomogeneous linear thermoelasticity, Applicable Analysis, Vol 53, Number 1 (1994), pp [17] D. Sheen; A generalize Green s Theorem, Appl. Math. Letters, Vol 5, No. 4 (199), pp Ruy Coimbra Charão Department of Mathematics, Feeral University of Santa Catarina, CEP 884-9, Florianópolis, SC, Brazil aress: ruycharao@gmail.com Jáuber Cavalcante Oliveira Department of Mathematics, Feeral University of Santa Catarina, CEP 884-9, Florianópolis, SC, Brazil aress: jauber@mtm.ufsc.br Gustavo Perla Menzala National Laboratory of Scientific Computation (LNCC/MCT), Ave. Getulio Vargas 333, Quitaninha, Petropolis, RJ, CEP , Brasil Feeral University of Rio e Janeiro, Institute of Mathematics, P.O. Box 6853, Rio e Janeiro, RJ, Brazil aress: perla@lncc.br