QUASI-STABILITY PROPERTY AND ATTRACTORS FOR A SEMILINEAR TIMOSHENKO SYSTEM

. This paper is concerned with the classical Timoshenko system for vibrations of thin rods. It has been studied by many authors and most of known results are concerned with decay rates of the energy, controllability and numerical approximations. There are just a few references on the long-time dynamics of such systems. Motivated by this scenario we establish the existence of global and exponential attractors for a class of semilinear Timoshenko systems with linear frictional damping acting on the whole system and without assuming the well-known equal wave speeds condition.

Timoshenko systems are nowadays a major research subject in second order evolution problems. With respect to their asymptotic behavior the nature of the damping term is very important since problem (1)-(2) is conservative. In this direction, it seems there is a kind of dichotomy based on a "equal speeds" assumption k proposed by Soufyane [29]. The main result in [29] asserts that the Timoshenko system with a damping ψ t acting only in the second equation (2) is exponentially stable if and only if (3) holds. This means that condition (3) is sufficient to stabilize the system exponentially with only one damping term, yet in the equation for rotation angle.
By taking condition (3) as a starting assumption several generalizations and extensions were established, including elastic, viscoelastic and thermoelastic models, see e.g. [1,2,4,5,10,14,15,17,21,23,30] and the references therein. In addition, recently, Almeida Júnior et al. [3] proved a complementary result which asserts that the Timoshenko system under assumption (3) is also exponentially stable with a damping ϕ t acting only in the equation for the displacement (1).
The stability of Timoshenko systems can also be studied without assuming condition (3). In this case, as a consequence of [29] and [3], the exponential stability is only achieved by adding damping term in both equations (1)- (2). Early works include [12,20,27]. More recently Cavalcanti et al. [7] proved that exponential stability of system (1)-(2), without assumption (3), can be obtained adding two locally distributed nonlinear damping α 1 (x)g 1 (ϕ t ) and α 2 (x)g 2 (ψ t ) on the equations (1) and (2), respectively. Now, with respect to the dynamical systems generated by Timoshenko systems there are just a few published works. The only one we found is due to Grasselli, Pata and Prouse [16]. It establishes the existence of a uniform attractor for a nonautonomous viscoelastic Timoshenko system where the dissipation is given by two memory terms acting on both equations, for displacement and rotation angle.
The purpose of the present work is to complement and extend some early works on Timoshenko systems by establishing new results on the existence of attractors and their properties. We shall consider the following damped semilinear Timoshenko system where f 1 , f 2 are nonlinear source terms representing the elastic foundation and h 1 , h 2 are external forces. The hypotheses on these functions will be given later. To this system we consider initial conditions and Dirichlet boundary condition We observe that our problem has damping terms in both equations (4) and (5). Therefore we shall not assume the equal wave speeds assumption (3). As a matter of fact, there is a criticism on this assumption since it never occurs physically. See for instance [24,7].
The main features of our work are summarized as follows. (a) It is the first work addressing the existence of a finite-dimensional attractor with optimal regularity and fractal exponential attractor for these kind of systems. This is achieved by using the recent quasi-stability theory of Chueshov and Lasiecka [8,9]. (b) It is also the first work that explores the gradient structure of Timoshenko systems in order to establish geometrical properties of their attractors. Our main result is Theorem 3.1.
The rest of the paper is organized as follows. In Section 2 we present our assumptions and state the results on existence and global well-posedness to the system (4)- (7). In Section 3 we consider the corresponding dynamical system and state our main result concerning to long-time dynamics. Finally, Section 4 is dedicated to the proof of all results stated above.
2.1. Abstract Cauchy problem. Let us first rewrite system (4)-(7) as a first order abstract Cauchy problem. Denoting by U the vector-valued function we transform system (4)- (7) in the Cauchy problem where A is the linear differential operator and F(U ) is the following nonlinear operator In order to include boundary conditions in (7) we consider the Hilbert space equipped with the following inner product and norm (equivalent to the usual one) for any U = (ϕ, Φ, ψ, Ψ),Û = (φ,Φ,ψ,Ψ) ∈ H, where (·, ·) and · 2 stand for usual inner product and norm in L 2 (0, L), respectively. More generally, throughout this paper the notation · p will stand for usual norm in L p (0, L), 1 ≤ p ≤ ∞. Here, we use the simplified Poincaré inequality . Besides, it is easy to check that . Therefore the well-posedness of (4)-(7) is given through equivalent problem (8) with respect to mild and strong solutions. Let us now recall these concepts: • A function U ∈ C([0, T ), H), T > 0, satisfying the following integral equation is called a mild solution of the initial value problem (8) on [0, T ).

2.2.
Well-posedness result. In order to consider local existence and uniqueness of mild and strong solutions to Cauchy problem (8) we initially consider the following assumptions on f i , and h i , for each i = 1, 2.
(A2) f i : R 2 → R is locally Lipschitz continuous on each of its arguments, namely, there exist a constant γ i ≥ 1 and a continuous function σ i : R → R + such that for every (s j , r), (s, r j ) ∈ R 2 , j = 1, 2.
, then the corresponding mild solution U (t) is strong one. (iii) If U (t) and V (t) are two mild or strong solutions corresponding to initial data U 0 and V 0 , respectively, then Remark 1. The proof of Theorem 2.1 is made in Section 4. Now, in order to show that both mild and strong solutions are globally defined (i.e. T max = +∞) we additionally consider dissipative hypotheses on the source as follows.
(A3) There is a function F : for some constants Theorem 2.2 (Global Well-Posedness). Under assumptions (A1)-(A3), then problem (8), and consequently system (4)- (7), is global well posed with respect to strong and mild solutions. Remark 3. Given U 0 ∈ H and the (corresponding) unique mild solution U of (8) lying in C([0, ∞); H), then using standard density arguments it is possible to find a sequence of regular (strong) solutions U n such that Therefore, the above regularity is sufficient to justify any procedure on the calculations performed in this paper.
3.1. Generation of a dynamical system. From Theorem 2.2 we can define the dynamical system (H, S(t)), where H is introduced in (11), and the evolution operator S(t) : H → H is given by relation where , t ≥ 0, is the unique mild solution of (8).
Essential concepts within the theory of infinite-dimensional dynamical systems can be found e.g. in [6,8,9,11,18,31]. For the sake of the reader we present some of these preliminary concepts applied to the dynamical system (H, S(t)) given in (20), see e.g. Chueshov and Lasiecka [8,9]. More precisely, all properties stated in the next main result (see Theorem 3.1) follow the theoretical-lines introduced below.
• The dynamical system (H, S(t)) given in (20) is called quasi-stable on a set B ⊂ H (in accordance with [9, Definition 7.9.2]) if there exist a compact seminorm n X (·, ·) on X := H 1 0 (Ω) × H 1 0 (Ω) and nonnegative scalar functions a(t) and c(t) locally bounded in [0, ∞), and b(t) and for any U 1 , U 2 ∈ B, where we use the notation • A global minimal attractor for (H, S(t)) is a bounded closed set A min ⊂ H which is positively invariant (S(t)A min ⊆ A min ) and attracts uniformly every point, that is, and A min has no proper subsets possessing these two properties. • The fractal dimension of a compact set A ⊂ H is defined by where n(A, ε) is the minimal number of closed balls in H of radius ε which covers A. Since the Hausdorff dimension does not exceed the fractal one (see e.g. [18,Chapter 2]) it is enough to prove finiteness of the fractal dimension. • A compact set A exp ⊂ H is said to be a fractal exponential attractor of the dynamical system (H, S(t)) if A exp is a positively invariant set of finite fractal dimension in H and for every bounded set B ⊂ H there exist positive constants t B , C B and σ B such that If there exists an exponential attractor only having finite dimension in some extended space H ⊇ H, then this exponentially attracting set is called generalized fractal exponential attractor.
Remark 4. In order to construct a compact seminorm n X (·, ·) on X := H 1 0 (Ω) × H 1 0 (Ω), we need to rewrite the phase space H given (11) as H := X × Y with compact embedding X → Y. By isomorphism, we consider from now on that H can be expressed as whose solution trajectories (ϕ(t), ψ(t), ϕ t (t), ψ t (t)) are taken with inner product and norm (12). Such isomorphic phase space shall allow us to achieve the quasi-stability property according to Definition 7.9.2 in [9].

Main result.
Our main result is concerned with long-time behavior for the dynamical system (H, S(t)) defined in (20). It reads as follows. In particular, there exists a global minimal attractor A min given by A min = N . (iv) The attractor A has finite fractal and Hausdorff dimension dim f H A. (v) Every trajectory Γ = {(ϕ(t), ϕ t (t), ψ(t), ψ t (t)) ; t ∈ R} from the attractor A has the smoothness property Moreover, there exists a constant R > 0 such that (vi) The dynamical system (H, S(t)) possesses a generalized fractal exponential attractor A exp with finite dimension in the extended space which is seen as isomorphic to L 2 (0, L) × L 2 (0, L) × H −1 (0, L) × H −1 (0, L). In addition, from interpolation theorem, there exists a generalized fractal exponential attractor whose fractal dimension is finite in a smaller extended space Remark 5. The proof is made in second part of Section 4. Actually, we first prove several lemmas which provide sufficient tools to conclude all items of Theorem 3.1 as an application of abstract results from the general theory in dynamical systems.

4.1.
Proof of local existence. The proof of Theorem 2.1 will follow as consequence of the next two lemmas.  Proof. Let us consider R > 0 and initial data U = (ϕ, Φ, ψ, Ψ), (10) and norm H we have Now let us estimate the right side terms in (26). Firstly we note that Also, since σ 1 is continuous and H 1 0 (0, L) → L ∞ (0, L), there exists a constant c ∞ > 0 such that ϕ ∞ , ϕ ∞ , ψ ∞ , ψ ∞ ≤ c ∞ R and so for some constant K 2 R > 0. We also note that u x Thus, there exists a constant K 3 R > 0 depending on R > 0 so that 2 Analogously, there exists a constant K 4 R > 0 such that 2 Finally, inserting (27)- (28) in (26), we conclude Therefore, (16) is obtained after applying Gronwall inequality. This completes the proof of Theorem 2.1.

4.2.
Proof of global well-posedness. The proof of Theorem 2.2 is obtained from Theorem 2.1 and Lemma 4.3 below.
Proof. The proof is first made for strong solutions. Then, it holds for mild solutions by taking standard density arguments. Equality (30) follows by taking the multipliers ϕ t in (4) and ψ t in (5), and adding the resulting expression. Now we define where From assumption (17) one has

4.3.
Proof of the main result. The proof of Theorem 3.1 will be given through next three lemmas.
We first recall that dynamical system (H, S(t)) given in (20) is said to be gradient if there exists a strict Lyapunov functional on H, namely, there exists a continuous functional Φ such that t → Φ(S(t)U 0 ) is non-increasing for any U 0 ∈ H, and the condition Φ(S(t)U 0 ) = Φ(U 0 ) for all t > 0 and some U 0 ∈ H implies that S(t)U 0 = U 0 for all t > 0.
In addition, from (36) it is obvious that Φ is bounded from above on bounded subsets of H. Also, from (31) and (36) we have , for any mild solution U (t) corresponding to U 0 ∈ H. This is enough to show that Φ R is a bounded set of H. The proof of Lemma 4.4 is complete. Given U = (ϕ, 0, ψ, 0) ∈ N and using (9)-(10) we have the stationary system Multiplying (37) by ϕ and (38) by ψ, integrating over (0, L), and adding the resulting expression results Applying assumptions (17)- (18) it follows that

Also, Hölder and Poincaré inequalities imply
Replacing these two last estimates in (39) and using (19) we obtain from where it follows that N is bounded in H. This concludes the proof of Lemma 4.5.
Now we consider a key result to conclude the existence of a global attractor as well as its properties for the dynamical system (H, S(t)). It brings up a stability inequality which allows us to conclude the quasi-stability property for (H, S(t)). Lemma 4.6. Let B be a bounded set of H. Given initial data U i = (ϕ i 0 , ϕ i 1 , ψ i 0 , ψ i 1 ) ∈ B, i = 1, 2, let us consider the mild solutions S(t)U i = (ϕ i (t), ϕ i t (t), ψ i (t), ψ i t (t)) of (8), t ≥ 0, respectively. Then there exist constants γ > 0, and K B > 0 depending on the size of B, such that for any t ≥ 0, where we denote u = ϕ 1 − ϕ 2 and v = ψ 1 − ψ 2 .
Proof. Let us start by denoting U (t) := S(t)U 1 − S(t)U 2 = (u(t), u t (t), v(t), v t (t)), t ≥ 0. Thus U solves the following problem in the mild sense with initial condition Taking the multipliers u t in (42) and v t in (43), and adding the resulting expression, we have 1 2 where we denote and, for each i = 1, 2, Let us estimate the right hand side of (44). From (30)-(31) there exists a constant (14)- (15) and Hölder inequality we get Applying Young inequality with = α 1 /4, there exists a constant K 4 B > 0 such that Similarly there exists a constant K 5 Replacing (46)-(47) in (44) and denoting K 6 Now let us define the perturbed functional where ε > 0 will be fixed later and First, it is easy to check that there exists a constant C > 0 such that Next, we show that there exist constants K 7 B > 0 and ε 1 > 0 so that Indeed, deriving I and J , and using equations (42)-(43), it follows where Now we estimate the terms I 1 and I 2 . Using analogous arguments to conclude (46) and (47), but replacing functions u t and v t by u and v, respectively, we infer Also, Hölder and Young inequalities lead us to Inserting these two last estimates in (51) and neglecting nonnegative terms we arrive at d dt with K 1 = 1 + L 2 2ρ1k + L 4 ρ1b + L 2 2bρ2 . Deriving L ε , utilizing (48), (52), and adding ε 2 L(t) in both sides of the resulting expression, we get Taking ε ≤ ε 1 := 2 K1+1 min 1 ρ1 , 1 ρ2 > 0, and K 7 B = εC 1 B + K 6 B , then (50) holds true. Now we fix ε 2 = min ε 1 , 1 2C > 0. So choosing ε ≤ ε 2 we derive from (49) that Further, from (50) we have Finally, combining (53) and (54), we conclude where we take γ = ε 3 > 0 and K B = K 7 B > 0. Regarding that L(t) = S(t)U 1 − S(t)U 2 2 H , t ≥ 0, then estimate (41) follows. The proof of Lemma 4.6 is complete. Now we are in conditions to conclude the proof of Theorem 3.1.