FINITE DIMENSIONAL GLOBAL ATTRACTOR FOR A DAMPED FRACTIONAL ANISOTROPIC SCHR¨ODINGER TYPE EQUATION WITH HARMONIC POTENTIAL

. We study the long time behaviour of the solutions for a class of nonlinear damped fractional Schr¨odinger type equation with anisotropic disper- sion and in presence of a quadratic potential in a two dimensional unbounded domain. We prove that this behaviour is characterized by the existence of regular compact global attractor with ﬁnite fractal dimension.


1.
Introduction. Dispersive wave equations provide excellent examples of infinite dimensional dynamical systems which are either conservative or exhibit some dissipation. In the last case, one can hope to reduce the study of the flow to a bounded (or even compact) attracting set or global attractor that contains much of the relevant information about the flow. Once the global attractor is obtained, the question arises if it has special regularity properties or if it has finite-dimensional character. Hence both existence of attractors and bounds on their dimensions are of a great interest. We refer the reader to R. Temam [43], J. Robinson [40], I. Chueshov [14] and G. Raugel [39] for general frameworks of this theory.
The aims of this paper is to study the asymptotic dynamics for a dynamical system generated by a nonlinear dispersive fractional anisotropic Schrödinger type equation with harmonic trapping potential that reads where ∂ 2 s denotes for the sake of simplicity ∂ 2 ∂s 2 . The unknown u = u(t, x, y) maps R + ×Ω into C with Ω = R×[0, 1] and α ∈ (1, 2). The equation (1.1) is supplemented with the Dirichlet boundary conditions u(t, x, 0) = u(t, x, 1) = 0 and with initial data at t = 0 u(0) = u 0 , (1.2) that belongs to the anisotropic phase space H α that will be specified in the sequel.
The fractional Schrödinger equations is a fundamental equation of fractional quantum mechanics which has been discovered as a result of expending the Feynman path integral from the Brownian-like to Lévy-like quantum mechanical paths.
In some physical problems, the presence of many particles leads one to consider nonlinear terms which stimulate the interaction effect among them. The nonlinear fractional Schrödinger equations, formulated by N. Laskin [31,32], have typically the conservative form: where α ∈ (0, 2), ψ(t, x) is the unknown complex-valued wave function and V (x) denotes a real-valued external potential. This prototypal equation has been widely used in many branches of applied sciences arising in nonlinear optics and beam propagation, in condensed matter physics, in deep water wave dynamics, in plasma physics, in wave turbulence and in dynamics of boson stars (see for instance [18,27,31,32,35,37,42]).
The mathematical literature for the conservative NLS equations is so huge that we do not even try to collect here a detailed bibliography. On the contrary, to the best of our knowledge, the literature for fractional Schrödinger equations is still expending and rather young. If α = 2, (1.3) becomes the standard Gross-Pitaevskii equation which has been extensively studied as a fundamental equation in modern mathematical physics specially for Bose-Einstein condensates (see [10,4,34] for instance). In the special case when V = 0 we refer the reader; for well-posedness results and existence of traveling waves for the resulting conservative fractional NLS; to [5,25,26,21,13,45,7] and the references therein. Now going back to the case of the equation (1.3) with F (|ψ| 2 ) = |ψ| 2σ , M. Cheng prove in [12] the existence of ground state by Lagrange multiplier method as well as the standing wave with prescribed frequency by Nehari's manifold approach. Under the same heading, in addition of proving the existence of standing waves, numerical results about the dynamics of (1.3) with harmonic potential were given in [17]. In [29], K. Kirkpatrick and Y. Zhang have also studied the following equation i∂ t ψ(t, x) = 1 2 (−∆) α 2 + |x| 2 ψ(t, x) + |ψ(t, x)| 2 ψ(t, x) , t ∈ R , x ∈ R d with considerable numerical analysis on soliton dynamics. For the non conservative case, O. Goubet and E. Zahrouni have studied the following dissipative one dimensional fractional NLS that reads ∂ t u(t, x) − i(−∂ 2 x ) α 2 u(t, x) + i|u(t, x)| 2 u(t, x) + γu(t, x) = f , t ∈ R , x ∈ R where α ∈ (1, 2). They prove in [23] the existence of a regular compact global attractor in H 2α (R) with finite fractal dimension (under suitable assumption on f ). Furthermore, the two dimensional Bose-Einstein equation which is none other then (1.1) with α = 2, was studied first of all in [1] with critical nonlinearity (cubic nonlinearity) in a thin strip. Then, with subcritical nonlinearities in [3] and [2] where the authors have prove in both cases the existence of a compact global attractor with finite fractal dimension. Finally, we point out that the case in which α = 1 in (1.1) is very interesting and rises many questions which will be the subject of a forthcoming paper. Now let us return to the matter at hand. The linear part of (1.1) is balanced between a skew-symmetric operator iA α = i(− ∂ 2 ∂x 2 + x 2 + D α y ) , and a zero-order dissipation term γu where γ > 0 is the damping parameter. f ∈ L 2 (Ω) is a given source term that is independent of time and the nonlinearity g is a C ∞ mapping from R + into R. For convenience use, we consider subcritical smooth nonlinearity, i.e. g that satisfies the following growth condition: for ξ ≥ 0 ξ 2 |g (ξ)| + ξ|g (ξ)| + |g(ξ)| ≤ c 1 ξ σ , (1.5) for a given σ ∈ (0, 2α 2+α ). For later use, we also assume that the derivatives g (k) are bounded for k ≥ 3, moreover we infer from (1.5) that for ξ ≥ 0, and (1.7) Before giving the layout of this article, we recall briefly some definitions and notations. First, denoting D y = −∂ 2 y , we recall (see [43,41]) that, for a given s ≥ 0, the operator D s y = ( −∂ 2 y ) s with Dirichlet boundary conditions is an unbounded non-negative self adjoint operator whose domain in where u k = √ 2 1 0 u(y) sin(kπ y) dy. Moreover, it is well known (see [16,41]) that for a fixed fractional exponent s ∈ (0, 1), the fractional Sobolev space H s y ([0, 1]) defined as follows is endowed with the natural norm which is equivalent to ||D s y u|| L 2 ([0,1]) thanks to the following Poincaré type inequality whose proof is rather straightforward.
. The Hilbert space L 2 = L 2 (Ω) is equipped with the usual scalar product denoted by (u, v) = e Ω u(x, y) v(x, y)dx dy. We will be regularly referring to the mixed space-time Lebesgue space denoted by L q t L p x L 2 y equipped with the semi-norm (for a fixed T > 0) we extensively use the notation L q x L p y for anisotropic Lebesgue spaces with obvious changes if q or p is infinity.

BRAHIM ALOUINI
We use the notation = √ 1 + x 2 , x ∈ R and, in a general context, for a given weight w = w(x), we set From now on, A 0 = −∂ 2 x + x 2 denotes the one dimensional harmonic oscillator. It is well known that A 0 is an unbounded non-negative self adjoint operator which has a compact inverse. Moreover, the family of eigenfunctions defined by constitutes an orthonormal basis in L 2 (R), where we denote by the Hermite orthogonal polynomials (see [20] for more details), and satisfies the following spectral property: The phase space H α , α ∈ (1, 2), defined as the completion of C ∞ 0 (Ω)(the space of smooth functions with compact support in Ω with the norm || . || Hα (defined below) is the Hilbert space that is endowed with the scalar product defined by y v ; u, v ∈ H α and the associated norm, which is equivalent, thanks to Lemma 1.1, to It is well known that A α is an unbounded non-negative self adjoint operator whose domain in L 2 is, The operator A α has a compact inverse. Moreover, there exists an orthonormal family in L 2 (Ω) , such that A α ψ n,k = λ n,k ψ n,k and λ n,k = 1 + 2n + (kπ) α , (n, k) ∈ N × N * . (1.12) Hence, thanks to the spectral representation of A α , it leads that for all s ≥ 0, the operator A s α is an unbounded non-negative self adjoint operator defined by A s α u(x, y) = n≥0,k≥1 λ s n,k (u, ψ n,k )ψ n,k (x, y) , whose domain in L 2 is the Hilbert space (u, ψ n,k )ψ n,k (x, y) such that n≥0,k≥1 λ 2s n,k |(u, ψ n,k )| 2 < +∞

GLOBAL ATTRACTOR FOR A FRACTIONAL NLS EQUATION WITH POTENTIAL 4549
endowed with the classical norm Finally, for any positive A and B, A B means that that exists c > 0 such that A ≤ c B and we recall that throughout this article the constants Cs are numerical constants that vary from one line to another.
Our main result of this paper is stated as follows Theorem 1.2 (Main Theorem). Let f ∈ L 2 (Ω) and α ∈ (1, 2). Then the equation (1.1) defines a dissipative dynamical system in H α that possesses a regular global attractor A α that is a compact subset of D(A α ). Moreover, the compact global attractor A α has a finite fractal dimension in H α .
It should be emphasized that, to the best of our knowledge, our results are new and will open the way to consider other class of fractional Schödinger equations.
The plan of the present paper is as follows. In section 2, we establish some anisotropic inequalities and Strichartz type estimates. The well-posedness of the Cauchy problem for (1.1) as well as the existence of the global attractor A α in H α for the associated semigroup (S α (t)) t∈R+ will be the subject of section 3. The issue of regularity of the global attractor A α will be discussed in Section 4. In the last section we establish the finite dimensionality of this global attractor.
2. Preliminary results. Given the complexity of the current issue, it becomes immediately apparent that the first recourse must be providing tools, namely some helpful anisotropic Sobolev inequalities. Then there exists C p > 0 that depends only on p such that for all u ∈ H 1 x L 2 y ∩ L 2 (R, L 2 y ), and H 1 x L 2 y stands for the usual Sobolev space H 1 (R, L 2 y ). Proof of Lemma 2.1. The case p ∈ [2, +∞] is none other than an application of the one dimensional Gagliardo-Nirenberg inequality, Minkowski's and Hölder's inequalities. For the case p ∈ [1, 2], we consider R > 0 and then by using Hölder's and the Cauchy-Schwarz inequalities Hence,

BRAHIM ALOUINI
This conclude the proof of the lemma by minimizing the last inequality with respect to R > 0.
Then there exists a reel constant C α,p > 0 that depends only on α and p such that for all u ∈ H α , where we recall that Proof of Lemma 2.2. Following the same idea used in [23], we consider R > 0. Thanks to the Cauchy-Schwarz inequality, where u k = 1 0 √ 2 u(x, y) sin(kπy) dy. By minimizing this bound with respect to R > 0, yields to Thus, thanks to the Hölder inequality, it follows that for all p ∈ ( 2α 2α−1 , 2α], (2.4) Combining (2.4) and Lemma 2.1 achieves the proof.
Lemma 2.3. Let α ∈ (1, 2) and q ∈ [2,4]. Then there exists a reel constant C α > 0 that depends on α and q such that for all u ∈ H α , Proof of Lemma 2.3. Thanks to the Agmon and Minkowski inequalities, one has hence, applying Hölder's and Minkowski's inequalities lead to (2.5) In the case 2 ≤ q < 4, we deduce from the one dimensional fractional Gagliardo Nirenberg type inequality (see [19]) that Proof of Lemma 2.4. As an application of the Hölder, Gagliardo-Nirenberg and Minkowski inequalities, we have for 2 ≤ p ≤ 6, Thus the proof is achieved.

Strichartz type estimates.
Definition 2.6. Following standard notations, we say that a pair (q, p) is admissible if 2 ≤ q, p ≤ +∞ and 2 In order to seek vector valued Strichartz estimates i.e. estimates in L q t L p x (L 2 y ([0, 1])) using the real axis Strichartz estimates in L q t L p x spaces, let us introduce the one dimensional Hermite operator A 0 = − d 2 dx 2 +x 2 , x ∈ R (the harmonic oscillator). We recall from [10], thanks to the Mehler's formula, that for u 0 ∈ L 2 Now, we state the following scalar Strichartz type estimates Proposition 2.7. Let T be fixed small enough, say 0 < T < π 4 and let (q, p) be an admissible pair. Then, there exists C = C(q) > 0 such that for all u 0 ∈ L 2 where δ and ρ denote respectively the conjugate exponent of δ and ρ.
Proof of Proposition 2.7. As a consequence of (2.11), it follows by interpolation that for 0 < t < T , T small enough (say 0 < T < π 4 ) and for p, p ≥ 2 such that Hence, thanks to Hardy-Littlewood-Sobolev inequality, it is standard to prove (2.12) and (2.13) (see [11,28]). We then skip the details and the proof is completed.
Proposition 2.8. Let T be fixed small enough, say 0 < T < π 4 . Then the following properties hold: (i) For every admissible pair (q, p), there exists C 1 > 0, depending only on q such (ii) Let (ρ, γ) be an admissible pair. Then for every admissible pair (q, p), there exists C 2 > 0 that depends only on ρ and q such that Moreover, for all admissible pair (ρ, γ) and for all ϕ (2.18)

GLOBAL ATTRACTOR FOR A FRACTIONAL NLS EQUATION WITH POTENTIAL 4553
In particular, for s > Moreover, for all admissible pair (ρ, γ) and for all ϕ (2.20) Remark 2. It is important, for the sequel, to point out that the statements of Proposition 2.8 as well as Corollary 1 and Corollary 2 are still valid if one replace A α by Λ α = A α + iγ (see [28]).
3. The initial value problem and the existence of the global attractor.

3.1.
Well-Posedness of the Cauchy problem. Within this framework, the current subsection aims to establish the first part of Theorem 1.2 that is Then, under the assumption (1.5) and for every u 0 ∈ H α , there exists T * > 0 and a unique solution to the problem where C ([0, T * ), H α ) stands for the space of continuous functions which take values in H α and C is a nonnegative reel constant that only depends on ||u 0 || Hα , ||f || L 2 and γ. Moreover, the maps S(t) : u 0 → u(t) are continuous on H α .
Proof of Proposition 3.1. Let T > 0 small enough as in Proposition 2.8 and R > 0 be a fixed reels that will be chosen later. We introduce the set stands for the space of continuous functions which take values in H α . In the following [7] for instance). We will be regularly referred to the following identity that reads We proceed in two steps Step 1: A local-in-time solution. We apply Duhamel's formula to (1.1) − (1.2), where F (u) = f − ig(|u| 2 )u and Λ α = A α + iγ and then we introduce for a fixed For the sake of simplicity we denote L r ([0, T )) = L r T , r ∈ [1, +∞]. To begin with, Lemma 3.2. ψ is a Lipschitz mapping on bounded subsets of X α .
Independently, from Corollary 2, Remark 2 and (3.1) it follows that Firstly, under assumption (1.5) and thanks to the Kato-Ponce inequality (see [24]) Hence, thanks to the Hölder inequality Similarly, using an analogous relation to (3.1) and under assumption (1.7), we obtain that D Thus, the Kato-Ponce inequality and Hölder's inequality imply that Gathering (3.5) and (3.7), the estimates (3.3) and (3.2) imply that for a given R > 0 and for u, v ∈ B Xα (0, R),

GLOBAL ATTRACTOR FOR A FRACTIONAL NLS EQUATION WITH POTENTIAL 4555
By similar computations, thanks to Corollary 2 and Proposition 2.8, we deduce that 8) and the proof is completed.
To conclude the first step, it is enough to prove the following lemma that reads Proof of Lemma 3.3. Observe that in accordance with Lemma 2.5, for a chosen 0 < T ≤ 1. This implies, thanks to (3.8), that for φ ∈ B Xα (0, R) where we recall that 0 < σ < 2α α+2 < 1, achieves the proof since Thanks to Lemmas 3.2 and 3.3, the existence of a unique local-in-time solution for (1.1) − (1.2) is obtained by a fixed point argument.
Step 2: Global (in time) existence in H α . The proof is classical and then details are omitted. Roughly speaking, the global existence is classical and easily deduced from the following energy equation that reads 1 2 where with G(s) = s 0 g(r) dr. In fact, one only has to write Hence, under assumption (1.5) and thanks to Lemma 2.4 This concludes the second step, knowing that the scalar product of (1.1) by u leads to ||u|| L 2 xy ≤ C, and then the proof of the proposition is achieved.
The semigroup (S α (t)) t∈R+ associated to (1.1) is well defined. At the beginning we highlight its dissipation.
Proof of Proposition 3.4. The proof is very standard and follows from (3.12) then we omit it and we refer the reader to [33] for a complete proof.

3.2.
Existence of the global attractor. The existence of an absorbing set was the first step toward the existence of the global attractor.
Theorem 3.5. The semigroup (S α (t)) t∈R+ associated to the dynamical system defined by (1.1) possesses a compact global attractor A α in H α .
Thanks to Theorem 1.1 and Remark 1.4 in [43] and Theorem 5.1 in [14], one only has to prove the asymptotic compactness of the semi-group (S α (t)) t∈R+ , that is Lemma 3.6. The semi-group (S α (t)) t∈R+ is asymptotically compact in H α i.e., for every bounded sequence (x k ) k in H α and every sequence t k −→ +∞, (S α (t k )x k ) k is relatively compact in H α .
Proof of Lemma 3.6. the proof is very standard, we sketch it for the sake of conciseness. By using the well known John Ball's argument (see [8] and [44]), let (u j ) j be a bounded sequence in H α and t j −→ +∞. Since (S α (t j )u j ) j is bounded in H α and in accordance with the compact embedding of H α into L p , p ∈ [1, 6], we may assume, up to a subsequence extraction, the existence of a ∈ H α such that S(t j )u j a weakly in H α and S(t j )u j −→ a strongly in L 2 . Thanks to the energy equation (3.12), we deduce that and where we recall that J and K are defined by (3.13) and (3.14).
Thanks to the dominated convergence Theorem, (3.16) and (3.17), from which we deduce, thanks again to (3.15), that ||S α (t j )u j || Hα −→ ||a|| Hα as j −→ +∞. This concludes the proof of the current lemma as well as the proof of Theorem 3.5.
4. The regularity of the global attractor. We claim now to prove an asymptotic smoothing effect of the dynamical system defined by (1.1) thought the study of the regularity of the global attractor A α . This main result is stated as follows

GLOBAL ATTRACTOR FOR A FRACTIONAL NLS EQUATION WITH POTENTIAL 4557
For that purpose, we shall briefly introduce some tools from harmonic analysis that will be used extensively in the sequel. Let 1 < p < +∞ and setting α ), we have the following equivalence between norms Proof of Lemma 4.2. the proof of the first equivalence is identical to that in [3], Theorem 3.2, and then details are omitted since both of the operators A 0 = −∂ 2 x +x 2 and D α y belongs to the class of bounded imaginary powers in L p x L 2 y for all p ∈ (1, +∞). The second one follows from Theorem 3.2 in [3] to which we refer the reader for a complete proof. Now, for a given positive integer N . we denote P N and Q N = id − P N the orthogonal projectors acting in L 2 x (R, H) by setting Actually, P N is the projector onto the low-frequencies modes of a given function, at level N . Clearly, A α P N = P N A α . Moreover, P N and Q N are bounded operators from D(A s α ), s ≥ 0, into itself and satisfy the following inequalities that state as follows Lemma 4.3. Let 0 ≤ s 1 ≤ s 2 . Then there exists C > 0, that does not depends on N , such that Proof of Lemma 4.3. The proof follows merely from the very definition of P N .
Besides, the following statement holds true Lemma 4.4. Let p ∈ ( 4 3 , 4). Then there exists C p > 0 depending only on p such that for any u ∈ L p x L 2 y , Proof of Lemma 4.4. the proof follows promptly from a well known result of R. Askey and S. Wainger (see [6] and [38]) that is, The orthogonal projector P N defined from L 2 x onto the finite dimensional subset generated by the N -first eigenfunctions of the operator A 0 = −∂ 2 x + x 2 , is a bounded operator from L p x into itself if and only if p ∈ ( 4 3 , 4). As a consequence of the previous lemma and the Minkowski inequality, we have the following space-time estimate that states as follows y and the proof is completed.
We now turn to establish an important mixed space-time estimate Proposition 4.6. Let T > 0 small enough as in Proposition 2.8 and (q, p) an admissible pair. Consider u(t), solution of (1.1)-(1.2), a complete trajectory included in H α . We denote by t 0 its time entrance in the bounded absorbing set B Hα , the ball with radius M > 0 and for the sake of simplicity we suppose t 0 = 0. Then there exist K 1 , K 2 > 0 that depend on T , γ, σ, M and f , such that for t ≥ 0, the following estimates hold where L denotes either id or P N .
We now proceed, similarly to u but with a slight difference, to prove (4.7) for P N (u).
Thanks to Lemma 4.5 and Lemma 2.2 it follows that for the case σ < α 4 , recalling that α ∈ (1, 2), choosing an admissible pair (δ, ρ) such that leads to (4.12) and then (4.7) follows. For the case σ ∈ [ α 4 , 2α α+2 ), we choose an admissible pair pair (δ, ρ) such that leads, in accordance with Lemma 4.5 and Lemma 2.2, to (4.13) and then (4.7) yields thanks to the Young inequality. This concludes the first part of the proposition. Now, for the second part, we consider an admissible pair (q, p). Then, thanks again to (4.9), Proposition 2.8 and Remark 1 we obtain the following estimate that reads α is a continuous operator from H α into L 2 and by using Lemma 4.2 and Remark 2 it follows from (4.17) that where ρ ∈ ( 4 3 , 2) when L = P N . Hence, either for the case L = id (and under assumption (4.11) or (4.14)), or for the case L = P N (and under assumption (4.15) or (4.16)), we obtain the desired estimate (4.8) thanks to the previous estimates (4.12) and (4.13). The proof of the Proposition 4.6 is therefore achieved.
An other important space-time estimate, for either u t or P N (u t ), states as follows Proposition 4.7. Under the same assumptions as Proposition 4.6, there exists K > 0 that depend on T , γ, σ and f , such that where L still denotes either id or P N .
Proof of Proposition 4.7. The proof is a straightforward consequence of Proposition 4.6, then details are omitted for sake of conciseness, thanks to Proposition 2.8, Lemma 4.3 and Remark 1 (for more details see [3] for instance).
For later use, we need the following straightforward application of the previous proposition. To highlight the regularity of the global attractor A α , we split the solution as u(t) = P N (u) + Q N (u) = v(t, x, y) + w(t, x, y) and then, thanks to Lemma 4.3, the regularity of u depends only on the regularity of w. Therefore, we shall focus on the long-time behavior of w(t) and for that purpose we approximate w, solution for Q N (1.1) supplemented with initial data w(0) = Q N (u(0)), by a more regular function W which solves the following auxiliary problem that reads C(T, γ, σ, f ) and κ = κ(σ, α) that do not depend on N 0 such that the following estimates hold Proof of Proposition 4.9. the proof is divided into three steps and sketched for the sake of conciseness. For convenience use, let us set Λ = v + W .
Step 1. Existence of a local (in-time) solution in Q N (D(A α )). Since, thanks to Lemma 2.5, D(A α ) is an algebra and then by applying standard fixed point argument (without recourse to use Strichartz type estimates) we show the existence of a local-in-time solution W for (4.19)-(4.20) which belongs to Moreover, this local-in-time solution; defined from a maximal interval [0, T * ); satisfies the following alternative either T * = +∞ or ||A α W (t)|| L 2 −→ +∞ as t → T * , t < T * .
Step 2. Existence of a global (in-time) solution in Q N (H α ). The scalar product of (4.19) by W then by −i(W t + γW ) lead to the following energy equation where we set with G(s) = s 0 g(r) dr. Thanks to Lemma 2.4 and Lemma 4.3, we easily obtain that for N large enough there exist C 1 , C 2 > 0 that depend only on γ, σ and f such that . Now we shall focus on the majorization of ψ(W ). For that purpose and in accordance with Lemmas 2.4 and 4.3, it will be enough to bound g(|Λ| 2 )Λ − g(|v| 2 )v, v t . Let p ∈ (1, +∞), then using the following identity In the light of the type estimates (3.4) and (3.6), we deduce from Lemma 4.2 and the Hölder inequality that for p ∈ (max(1, 2 2σ+1 ), 2), recalling that σ ∈ (0, 2α α+2 ), where we denote From the above it can be assumed, in accordance with (4.25), that for p ∈ (2, +∞) such that p ∈ max(1, 2 2σ + 1 ), 2 , (4.28) the following estimate holds Applying Young's inequality to the previous estimate leads to where (q, p) denotes an admissible pair satisfying the condition (4.28).

GLOBAL ATTRACTOR FOR A FRACTIONAL NLS EQUATION WITH POTENTIAL 4563
Thanks to Lemma 2.4 and Lemma 4.3, it may be concluded from (4.30) and (4.24) that for N large enough ||W (s)|| 2 where ν = 1 − σ( 1 α − 1 2 ) and ς 1 is defined by (4.27). At this stage we need the following lemma that reads Lemma 4.10. Let T > 0, say (0 < T < π 4 ), and t > T . Then there exist N 0 > 0 large enough and nonnegative reel constants (C i ) 0≤i≤3 that depend only on α, σ, f and T such that for all N ≥ N 0 and for all t 0 ∈ [0, t] the following estimate holds and (ν j ) 1≤j≤4 are nonnegative constants depending only on α and σ.
Proof of Lemma 4.10. By applying the Duhamel formula for (4.19) and (4.20), Corollary 2 and Lemma 4.5, it may be concluded that for every admissible pair (δ, ρ) such that ρ ∈ ( 4 3 , 2) and for a fixed s > 1 2 the following estimates hold Independently, Choosing a fixed reel 1 2 < s < α 2 (say s = α+1 4 for instance) then using interpolation argument and Lemma 4.3 we have Following the lines of the proof of Proposition 4.6, two several cases can be distinguished: • In cases where 0 < σ < α 4 , assuming that (δ, ρ) satisfies (4.15) i.e: leads, thanks to the Hölder inequality, Lemma 2.2, Lemma 4.3, Proposition 4.6 and in accordance with (4.35) and (4.34), to the following estimate that reads x Hα and then (4.33) follows thanks to Young's inequality.
• While in the other, where α 4 ≤ σ < 2α α+2 , choosing an admissible pair (δ, ρ) satisfying (4.16) i.e: allows us, thanks to (4.35), Lemma2.2, Proposition 4.6 and the Hölder inequality, to obtain Hence, in accordance with (4.34), (4.35) and Proposition 4.6 we obtain ). In the light of (4.37) the estimate (4.33) is deduced, thanks again to Young's inequality and the proof of the lemma is therefore achieved. and in accordance with Lemma 4.10, it can be assumed by classical argument that for a chosen N 0 > 0 large enough sup s∈[0,t] ||W (s)|| 2 Hα ≤ 2 C 0 where C 0 > 0 does not depends on N 0 and then W remains uniformly bounded in H α . This conclude the second step as well as the proof of (4.22) thanks again to Lemma 4.10.
Step 3. The solution is bounded in Q N (D(A α )). In order to prove that A α W is bounded in L 2 (Ω), we will prove equivalently that Z = W t remains bounded in L 2 (Ω) with an upper bound that may depends on N .

GLOBAL ATTRACTOR FOR A FRACTIONAL NLS EQUATION WITH POTENTIAL 4565
We differentiate (4.19) with respect to t, then we consider the scalar product of the resulting equation by −i(Z t + γZ). This leads to where we set and First, we establish the coercivity of Φ.
Lemma 4.11. Let N 0 be a fixed positive integer chosen large enough depending on γ, f and σ. Then there exists C > 0 such that for any N ≥ N 0 , Proof of Lemma 4.11. Since W remains uniformly bounded in H α , then (4.42) easily follows by application of the Hölder inequality, Lemma 2.4, Lemma 4.3 and the estimate ||v t || L 2 √ N . We omit the details for the sake of conciseness and the proof is completed.
We now proceed to bound Ψ(Z). In the one hand, thanks to the Hölder inequality, Lemma 2.4, Lemma 4.3 and the estimate ||v t || L 2 √ N we easily obtain Moreover, In the other hand, thanks again to the Hölder inequality, Lemma 2.4 and Lemma 4.3 Independently, since the following terms (g (|Λ| 2 ) e(ΛZ)Z, v t ), (g (|Λ| 2 )Λ|Z| 2 , v t ) and (g (|Λ| 2 ) e(ΛZ) 2 , v t ) are similar we just indicate how to handle the first one. Let p 1 ∈ (2, +∞) such that 1 < p 1 ≤ α. by applying Lemma 4.2 and the Hölder inequality it follows, and under assumption (1.5), that Thus, for a chosen reel p 1 ∈ (2, +∞) such that it leads from the estimate above, Lemma 2.4, Lemma 2.5 and Lemma 4.3 that there exists C > 0 depending only on T , γ, σ and f such that Now it only remains to handle the worst terms that contain v tt .
Lemma 4.12. Let p 2 ∈ (2, +∞) be a fixed reel number satisfying Then under assumption (1.5) there exists C > 0 depending only on T , γ, σ and f such that the following estimate holds (4.49) Proof of Lemma 4.12. Using the Cauchy-Schwarz then the Hölder inequalities, leads to As can be seen from above, it remains to bound ||v tt || L 2 . Differentiating P N (1.1) with respect to t leads to and then we obtain, thanks to Lemma 4.3 and Proposition 4.7, that Since the terms of the right hand side of (4.51) are similar, we indicate briefly how to bound one of them. Thanks to the Riesz-representation-Theorem and by a density argument it follows that for p ∈ (2, +∞) Now observe that choosing p ∈ (2, +∞) such that p < min(2, 1 σ ) and applying the Hölder inequality leads to (4.53) Consequently, since u remains uniformly bounded in H α and in accordance with the estimate (4.52), Lemma 2.1, Lemma 2.2 and Lemma 4.3 it follows from (4.53) that for a chosen reel p 2 ∈ (2, +∞) satisfying (4.54) Gathering (4.50), (4.51) and (4.54) achieves the proof of the lemma.
At this stage, we need the following lemma that is for W the analog of Proposition 4.6, Proposition 4.7 and Lemma 4.8 for u. Since the proof of the following statement is similar to the proofs of the previous ones, we omit it. Lemma 4.13. Let T > 0 small enough as in Proposition 2.8. Then there exists C > 0 depending only on T , γ, σ and f such that for all admissible pairs (q, p) the following estimate holds Moreover, for all t ≥ 0, Thanks to (4.43), (4.44), (4.45), (4.47), Lemma 4.11 and Lemma 4.12 it may be concluded, in accordance with (4.39), that for a chosen couple of admissible pairs (q 1 , p 1 ) and (q 2 , p 2 ) that satisfy respectively the conditions (4.46) and (4.48) we have for N large enough where κ denotes a nonnegative reel constant that depends only on α and σ.
This concludes the third step as well as the proof of the proposition.
Let us now turn to proving Theorem 4.1,

4.2.
Proof of Theorem 4.1. At first, we highlight the asymptotic smoothing effect of the semi-group by comparing W (t) to w(t) when t converges towards +∞.
Proposition 4.14. Let T > 0 small enough as in Proposition 2.8. There exists C > 0 depending only on γ, σ, T , f and ||u 0 || Hα and such that proof of Proposition 4.14.
Then χ satisfies supplemented with initial data Applying the scalar product of (4.56) by χ then by −i(χ t + γχ) we obtain that where we denote  On the other hand, in order to bound Υ(χ), it is worth noticing that the majorization of the terms of the right-hand side of (4.60) involving Θ t are similar we just handle the first one.
In the light of the type estimates (3.4) and (3.6) and in accordance with the uniform bound of u and Λ in H α , applying Lemma 2.2, Lemma 4.2 and the Hölder inequality, it can be concluded under assumption (1.5) that This completes the proof of the proposition thanks again to (4.61).
Propositions 4.14 and 4.9 provide, by classical arguments, that A α is a bounded subset of D(A α ) (see [2,3,22] for instance). Moreover, let N 0 be a fixed positive integer chosen large enough depending on T , γ, f and σ. Then there exist κ, C > 0 that not depend on N 0 such that ∀ N ≥ N 0 , (4.64) The proof of the compactness of A α into D(A α ) is standard and follows by using the well known J. Ball's argument [8]. we omit it for the sake of conciseness and we refer the reader to [2], Theorem 2.10, for a similar complete proof. Hence, the proof of the theorem is achieved.

5.
Finite fractal dimension of the global attractor. When studying dissipative dynamical systems, we attach a paramount importance to the fractal dimension of the global attractor. This is due to the fact that, in term of physical parameters, the fractal dimension while is finite implies that the infinite-dimensional dynamical system possesses an asymptotic behavior determined by a finite number degrees of freedom. Within this framework and for sake of completeness we recall a general result given in [15]. 2. There exist compact semi-norms n 1 and n 2 on X such that ∀ u 1 , u 2 ∈ M , ||V (u 1 ) − V (u 2 )|| X ≤ δ ||u 1 − u 2 || X + K[n 1 (u 1 − u 2 ) + n 2 (V (u 1 ) − V (u 2 ))] .
where 0 < δ < 1 and K > 0 are constants. Then M is a compact set in X of a finite fractal dimension. (A semi norm n on X is said to be compact if and only if for any bounded set B ⊂ X there exists a sequence (x n ) n ⊂ B such that n(x n − x m ) −→ 0 as n, m → +∞).
We now state the third part of the main result of this paper 4570 BRAHIM ALOUINI Theorem 5.2. Assume that the forcing term f belongs L 2 (Ω). Then the global attractor A α has a finite fractal dimension in H α .
With the objective of proving Theorem 5.2 by checking the assumptions in Theorem 5.1, one only has, thanks to Lemma 4.3, to prove the following result Proposition 5.3. There exist t * > 0 and N > 0 depending on γ, σ, f and T such that for all u 0 , v 0 ∈ A α , (Ω) , (5.1) where L = L(t * ) > 0 depends on t * and Q N = id − P N still denotes the orthogonal projector defined by (4.1).