Stability and global attractors for thermoelastic Bresse system

In this article, we consider the stability properties for thermoelastic Bresse system which describes the motion of a linear planar, shearable thermoelastic beam. The system consists of three wave equations and two heat equations coupled in certain pattern. The two wave equations about the longitudinal displacement and shear angle displacement are effectively damped by the dissipation from the two heat equations. We use the multiplier techniques to prove the exponential stability result when the wave speed of the vertical displacement coincides with the wave speed of longitudinal or of the shear angle displacement. Moreover, the existence of the global attractor is first achieved.

In the present work, we consider system (1.1)-(1.5), That is, we use the multiplier techniques to prove the exponential stability result only for an equal wave speed case.

Equal wave speeds case: E^G
Here we state and prove a decay result in the case of equal wave-speeds propagation.
Define the state spaces where The associated energy term is given by By a straightforward calculation, we have From the semigroup theory [6,7], we have the following existence and regularity result, for the explicit proofs, we refer the reader to [12].
, 0 (x) 2 H, then the energy E(t) decays exponentially as time tends to infinity; that is, there exist two positive constants C and independent of the initial data and t, such that EðtÞ CEð0Þe Àt 8t 4 0: ð2:3Þ The proof of our result will be established through several lemmas. Let where f is the solution of LEMMA 2.2 Let w 1 , w 3 , 2 , 1 , 3 be a solution of (1.1)-(1.5), then we have, 8" 1 4 0, Applicable Analysis 3

R e t r a c t e d
By using the inequalities and Young's inequality, the assertion of the lemma follows. Let Proof Using Equations (1.4) and (1.1), we get The assertion of the Lemma then follows, using Young's and Poincare´'s inequalities. Let Then using Young's and Poincare´'s inequalities, we can obtain the assertion. Next, we set

R e t r a c t e d
Notice that E ¼ G, then then use Young's inequality, we can obtain the assertion. We set LEMMA 2.6 Let w 1 , w 3 , 2 , 1 , 3 be a solution of (1.1)-(1.5), then we have, 8" 5 4 0, Then, note that E ¼ G again, from the above two equalities and Young's inequality, we can obtain the assertion. Next, we set From (2.1) and (2.2), we have that, 9C 4 0, satisfy  Àk 3x k 2 þ Cð" 7 Þk 3xt k 2 þ " 7 k 2x k 2 : ð2:22Þ Proof Using Equations (1.5), we have Then using Young's and Poincare´'s inequalities, we can obtain the assertion. Now, let N, N 1 , N 2 , N 3 , N 4 , N 5 , N 6 , N 7 4 0, we define the Lyapunov functional F as follows Applicable Analysis 7 where At this point, we chose our constants very carefully and properly so that the existing ! 4 0, (2.24) takes the form We are now ready to prove Theorem 2.1.

Global attractors
In this section, we establish the existence of the global attractor for system (1.1)-(1.5).
Setting v ¼ w 1t , ' ¼ w 3t , ¼ 2t , ¼ 3t . Then, Equations (1.1)-(1.5) can be transformed into the system We consider the problem in the following Hilbert space Recall that the global attractor of S(t) acting on H is a compact set A & H enjoying the following properties: More details on the subject can be found in the books [17,20,21].
it is clear that R 0 is still a bounded absorbing set which is also invariant for S(t), that is, S(t)R 0 & R 0 for every t ! 0.
In the sequel, we define the operator A as Af ¼ Àf xx with Dirichlet boundary conditions. It is well known that A is a positive operator on L 2 with domain DðAÞ ¼ H 2 \ H 1 0 . Moreover, we can define the powers A s of A for s 2 R. The space V 2s ¼ D(A s ) turns out to be a Hilbert space with the inner product hu, vi V 2s ¼ hA s u, A s vi, where hÁi stands for L 2 -inner product on L 2 .

Applicable Analysis 9
In particular, The injection V s 1 ,! V s 2 is compact whenever s 1 4 s 2 . For further convenience, for s 2 R, introduce the Hilbert space Clearly, H 0 ¼ H. Now, let z 0 ¼ (u 0 , w 0 , 0 , 0 , v 0 , ' 0 , 0 , 0 , 0 ), where R 0 is the invariant, bounded absorbing set of S(t) given by Remark 3.1, take the inner product in H 0 of (3.1)-(3.9) and (A w 1 , A w 3 , : ð3:10Þ Here, the boundary term of integration by parts is neglected since we are working with more regular functions. We denote hA w 3t f dx, Then, introduce the functional Let R(t) be the ball of V 3/2 Â V 3/2 Â V 3/2 Â V 3/2 Â (V 1/2 ) 5 , from the compact is compact in H. Then, due to the compactness of R(t), for every fixed t ! 0 and every d 4 c 2 e Àc 1 t , there exist finitely many balls of H of radius d such that z(t) belongs to the union of such balls, for every z 0 2 R 0 . This implies that H ðSðtÞR 0 Þ c 2 e Àc 1 t 8t ! 0, ð3:13Þ where H is the Kuratowski measure of non-compactness, defined by H ðRÞ ¼ inffd : R has a finite cover of balls of H of diameter less than d g: Since the invariant, connected, bounded absorbing set R 0 fulfils (3.13), exploiting a classical result of the theory of attractors of semigroups (see, e.g. [22]), we conclude that the !-limit set of R 0 , that is, is a connected and compact global attractor of S(t). Therefore we have proved the following result.