Asymptotic dynamics of the solutions for a system of N-coupled fractional nonlinear Schrödinger equations

Abstract

In the current issue, we consider a system of N-coupled weakly dissipative fractional Schrödinger equations with cubic nonlinearities. We will prove that the asymptotic dynamics of the solutions will be described by the existence of a regular compact global attractor with finite fractal dimension.

Keywords

Schrödinger equation asymptotic behavior global attractor fractal dimension

1. Introduction

This article is devoted to study the regularity of the asymptotic dynamics for an infinite dimensional dynamical system generated by N-coupled nonlinear dispersive fractional Schrödinger type equations with cubic nonlinearities that reads $\begin{array}{l} (1) & \frac{\partial u_{k}}{\partial t} - i D^{α} u_{k} + i \sum_{l = 1}^{N} | u_{l} |^{2} u_{k} + γ u_{k} = f_{k} \\ (2) & u_{k} (t = 0) = u_{k 0}, 1 ⩽ k ⩽ N \end{array}$ where the fixed integer $N \in N^{*}$ , $α \in (1, 2)$ and D denotes for the sake of simplicity $\sqrt{- \frac{\partial^{2}}{\partial x^{2}}}$ . For every $1 ⩽ k ⩽ N$ , the unknown $u_{k} = u_{k} (t, x)$ maps $R_{+} \times R$ into $C$ . The initial data at $t = 0$ , $u_{k 0}$ , belongs to the fractional Sobolev space $H^{\frac{α}{2}} (R)$ that will be specified in the sequel. The reel $γ > 0$ is the damping parameter and, for a fixed $1 ⩽ k ⩽ N$ , $f_{k} \in L^{2} (R)$ is a given source term that is independent of time.

In recent years, the investigation of multi-component nonlinear systems involving evolution PDE’s has attracted much attention for many researchers. As typical examples we cite the nonlinear coupling of dispersive waves with different polarizations, the nonlinear coupling of beam modes in a beam-plasma system and the interaction of many modes in deep water (see [6,22,35] and the references therein). Systems of coupled nonlinear Schrödinger type equations in mathematical and physical aspects are of comparative interest as they appear in different sides of applications to various branches of physics such as electromagnetism, hydrodynamics, Bose–Einstein condensates at zero temperature, plasma physics and they can be derived from optical communication, nonlinear optics and large-scale Rossby waves. In optics, systems of coupled nonlinear Schrödinger equations are used to describe the propagation of light along birefringent optical fibers (see [22,35] and [36]). In hydrodynamics, when studying the modulation instability of gravity waves in fluids in great depths (see [6]) a coupling NLS system was considered. As in the study of electromagnetic waves (see [24]), coupling NLS systems arise also when studying either Feshbach resonances in atomic Bose–Einstein condensate systems (see [30]) or atmospheric gravity waves (see [23]).

Lately, interest in dissipative systems increased even more because of intensive elaboration of strange attractors. For a given dissipative dynamical system, the first question that arises is determining the existence of the global attractor, a bounded (or even compact) attracting set that contains much of the relevant information about the flow, on which one may reduce the qualitative study of the flow. We refer the reader to [7,26,29] and [25] for general frameworks of this theory.

In the literature, the initial value problem for (1), in its conservative case, was recently addressed by several authors. When $N = 1$ the system (1) reduces to the fractional nonlinear Schrödinger type equation $\begin{matrix} (3) & u_{t} - i {(- Δ)}^{\frac{α}{2}} u + i | u |^{2} u = 0, α \in (1, 2) . \end{matrix}$ This equation, introduced initially by Laskin [18,19], was extensively studied by several authors. The initial value problem for (3) was considered for initial data belonging to $L^{2} (R^{n})$ by B. Guo and Z. Huo [14] for $α \in (0, 2)$ . A more general result was given by Y. Hong and Y. Sire in [16] where the authors established local well-posedness and ill-posedness of (3) in Sobolev spaces for power-type nonlinearities.

While in the dissipative case where, in some physical contexts, an external forcing term and some damping effects have been taken into account, the following fractional NLS that reads $\begin{matrix} (4) & u_{t} - i {(- Δ)}^{\frac{α}{2}} u + i | u |^{2} u + γ u = f, α \in (1, 2), x \in R \end{matrix}$ was only studied, to the best of our knowledge, by O. Goubet and E. Zahrouni in [12] where they have proved the existence of a regular compact global attractor with finite fractal dimension in $H^{α}$ under suitable assumption on the forcing term f.

Now let us get back to the general framework for which $N ⩾ 2$ ,

Amongst the various research works on (1) like systems, we shall mention that existence and uniqueness of global smooth solutions to periodic boundary value problem of a two-coupled nonlinear fractional Schrödinger equations, with power type linearities, were considered in [15,17] and [33] in one dimensional bounded domain and for $1 < α < 2$ . For related topics, the existence and stability of standing waves was considered in [28]. While for the existence and uniqueness of ground states for N-coupled fractional NLS equations we refer the reader to [5].

When $α = 2$ , a coupled nonlinear Schrödinger equations in inhomogeneous fibers was considered and in which interactions of vector anti-dark solitons were considered in [36] as well as some numerical results were given. Similar studies have been carried out in [35] for a nonlinear optical system in two dimension space. Moreover, interesting numerical analysis for N-coupled NLS equations in one space dimension can be found in [3].

However, on to the matter at hand, G. Li and C. Zhu have considered in [20] a class of coupled NLS system with zero order dissipation in one dimensional bounded domain that states as follows $\begin{array}{l} (5) & i u_{t} + α_{1} u_{x x} + (| u |^{2} + | v |^{2}) u + i η_{1} u = f \\ (6) & i v_{t} + α_{2} v_{x x} + (| u |^{2} + | v |^{2}) v + i η_{2} v = g \end{array}$ and they have established the existence of a compact global attractor with finite fractal dimension. In 2013, M. Cheng studied in [4] the asymptotic dynamics for the following fractional dissipative NLS system $\begin{array}{l} (7) & \partial_{t} u_{k} - i {(- Δ)}^{\frac{α}{2}} u_{k} + i \sum_{j = 1}^{N} | u_{j} |^{q + 1} | u_{k} |^{q - 1} u_{k} + γ u_{k} = f_{k} \\ (8) & u_{k} (t = 0) = u_{k} (0) \in H^{α} (R), 1 ⩽ k ⩽ N \end{array}$ where $α \in (1, 2)$ and $1 ⩽ q < 2 α$ . He proved the well posedness of the Cauchy problem as well as the existence of a compact global attractor in the phase space.

It is important to note that in the limiting case ( $α = 1$ ) for the system (1)–(2), the contraction mapping principle to solve the well-posedness of the initial value problem does not apply. Hence, more refined analysis of the problem will be needed and this will be the subject of a forthcoming paper.

1.1. Notations and main results

Before giving the main results and the layout of this article, we introduce briefly some definitions and notations.

To begin with, we recall that in what follows we use the notation $(\cdot, \cdot)$ for the usual scalar product in $L^{2} (R)$ defined by $\begin{matrix} (u, v) = ℜ e \int_{R} u (x) \overline{v (x)} d x . \end{matrix}$ Our convention for the one dimensional space Fourier transform is $\begin{matrix} F (u) (ξ) = \int_{R} u (x) e^{- i x ξ} d x . \end{matrix}$ For a given fractional exponent $s \in (0, 1)$ , we recall (see [9] and [27]) that ${(- Δ)}^{s}$ is considered as the homogeneous fractional pseudo-differential operator defined, for $u \in S (R)$ , by $\begin{matrix} {(- Δ)}^{s} u (x) = p . v . \int_{R^{2}} \frac{u (x) - u (y)}{| x - y |^{1 + 2 s}} d y = F^{- 1} (| ξ |^{2 s} F (u)) (x), \end{matrix}$ where $S (R)$ denotes the Schwarz class and “p.v.” for principal value.

The fractional Sobolev space $H^{s} (R)$ defined as follows $\begin{matrix} H^{s} (R) = {u \in L^{2} (R) such that \int_{R} (1 + | ξ |^{2 s}) {| F (u) (ξ) |}^{2} d ξ < + \infty} \end{matrix}$ as an intermediary Banach space between $L^{2} (R)$ and $H^{1} (R)$ is a Hilbert space endowed, via the Fourier transform approach, with the norm $\begin{matrix} ‖ u ‖_{H^{s} (R)} = {(‖ u ‖_{L^{2}}^{2} + {‖ D^{s} u ‖}_{L^{2}}^{2})}^{\frac{1}{2}} \end{matrix}$ and the associate scalar product denoted by $\begin{matrix} {(u, v)}_{H^{s} (R)} = (u, v) + (D^{s} u, D^{s} v), u, v \in H^{s} (R) . \end{matrix}$ For the sake of simplicity and in order to remove any ambiguity in what follows, we will extensively use the following vectorial notations.

For all $U = {(u_{1}, \dots, u_{N})}^{t}$ , $V = {(v_{1}, \dots, v_{N})}^{t} \in C^{N}$ we denote $\begin{matrix} U . V = \sum_{k = 1}^{N} u_{k} v_{k} and | U |^{2} = U . \overline{U} \end{matrix}$ Now for a given $p \in [2, + \infty)$ , the space denoted $\begin{matrix} L^{p} = \prod_{k = 1}^{N} L^{p} (R) endowed with the norm ‖ U ‖_{L^{p}} = {(\sum_{k = 1}^{N} ‖ u_{k} ‖_{L^{p} (R)}^{p})}^{\frac{1}{p}} \end{matrix}$ is a Banach space. In the special case $p = 2$ , $L^{2}$ is a Hilbert space endowed with the usual scalar product denoted $\begin{matrix} ((U, V)) = \sum_{k = 1}^{N} (u_{k}, v_{k}), \forall U, V \in L^{2} . \end{matrix}$ Whereas for the limiting case $p = + \infty$ , we denote $\begin{matrix} ‖ U ‖_{L^{\infty}} = max_{1 ⩽ k ⩽ N} ‖ u_{k} ‖_{L^{\infty} (R)} . \end{matrix}$ The Hilbert space $H^{s}$ defined by $\begin{matrix} H^{s} = \prod_{k = 1}^{N} H^{s} (R), s ⩾ 0 \end{matrix}$ is endowed with the naturel norm $\begin{matrix} ‖ U ‖_{H^{s}}^{2} = ‖ U ‖_{L^{2}}^{2} + {‖ D^{s} U ‖}_{L^{2}}^{2} . \end{matrix}$

Hence, using the previous notations, the system (1)–(2) will be equivalently written as follows $\begin{array}{l} (9) & U_{t} - i D^{α} U + i | U |^{2} U + γ U = F \in L^{2} \\ (10) & U (t = 0) = U_{0} \in H^{\frac{α}{2}} . \end{array}$ where $U = {(u_{1}, \dots, u_{N})}^{t}$ and $F = {(f_{1}, \dots, f_{N})}^{t}$ .

First we recall the following existing result given in [4].

Theorem 1.
Let $F \in L^{2}$ . Then the Cauchy problem ( 9 )–( 10 ) is well posed in $H^{\frac{α}{2}}$ . Moreover, the problem ( 9 )–( 10 ) defines an infinite dimensional dissipative dynamical system that possesses a compact global attractor $A_{α}$ in $H^{\frac{α}{2}}$ .

Once the global attractor is obtained, the question arises if it has special regularity properties or if it has finite-dimensional character. Our main results for (9)–(10) state as follows:
Theorem 2.
Let $α \in (1, 2)$ and $F \in L^{2}$ . Then the global attractor $A_{α}$ associated to the dynamical system generated by ( 9 )–( 10 ) is a compact subset of $H^{α}$ . Moreover, without any additional condition on F, $A_{α}$ has a finite fractal dimension in $H^{\frac{α}{2}}$ .

This article is organized as follows: in Section 2, we establish some helpful results that play a crucial role in what follows. In Section 3 we give a rigorous proof of the existence of a compact global attractor in $H^{\frac{α}{2}}$ sightly different from that in [4] using a decomposition of the associated semigroup. The regularity of the global attractor $A_{α}$ for (9)–(10) will be discussed in Section 4. Finally, in Section 5 we use a new idea developed recently in [1] to bound from above the fractal dimension of $A_{α}$ without the need of supplementary assumptions on F.

In the end of this section it should be noted that throughout this article, the constants Cs are numerical positive constants that vary from one line to another and $A ≲_{β} B$ means the existence of $C (β) > 0$ such that $A ⩽ C (β) B$ .
2. Preliminary results

To begin with, we shall briefly introduce some tools from harmonic analysis that will be used extensively in the sequel. We recall a fractional Gagliardo–Nirenberg type inequality (see [10] for instance)

Lemma 3.
Let $α > 1$ . Then for every $p \in [2, + \infty]$ , there exists $C = C (α, p) > 0$ such that $\begin{matrix} ‖ u ‖_{L^{p} (R)} ⩽ C ‖ u ‖_{L^{2}}^{1 - \frac{2}{α} (\frac{1}{2} - \frac{1}{p})} {‖ D^{\frac{α}{2}} u ‖}_{L^{2}}^{\frac{2}{α} (\frac{1}{2} - \frac{1}{p})}, \forall u \in H^{\frac{α}{2}} . \end{matrix}$
Proof of Lemma 3.
For the sake of completeness, we give a simple proof. Using the Hausdorff-Young inequality and the Hölder inequality, it leads that for $p > 2$ $\begin{matrix} ‖ u ‖_{L^{p}} ≲ {‖ F (u) ‖}_{L^{p^{'}}} ≲ {(\int_{R} (1 + | ξ |^{α}) {| \hat{f} (ξ) |}^{2} d ξ)}^{\frac{1}{2}} {(\int_{R} \frac{d ξ}{{(1 + | ξ |^{α})}^{\frac{p^{'}}{2 - p^{'}}}})}^{\frac{2 - p^{'}}{2 p^{'}}} \end{matrix}$ where $p^{'}$ denotes the conjugate exponent of p.

Hence $\begin{matrix} (11) & ‖ u ‖_{L^{p} (R)} ⩽ C (α, p) (‖ u ‖_{L^{2} (R)} + {‖ D^{\frac{α}{2}} u ‖}_{L^{2} (R)}) . \end{matrix}$ Replacing u by $x ⟼ u (λ x), λ \in R$ in (11), it leads that $\begin{matrix} ‖ u ‖_{L^{p} (R)} ⩽ C (α, p) (\frac{1}{λ^{\frac{1}{2} - \frac{1}{p}}} ‖ u ‖_{L^{2} (R)} + λ^{\frac{α}{2} - (\frac{1}{2} - \frac{1}{p})} {‖ D^{\frac{α}{2}} u ‖}_{L^{2} (R)}) . \end{matrix}$ Minimizing the right hand side of the previous equation with respect to λ achieves the proof. □

For later use we recall the following result
Lemma 4.
Let $s \in [1, 2]$ . Then there exists $C > 0$ such that for all $p \in [2, + \infty)$ , $\begin{matrix} ‖ u ‖_{L^{p}} ⩽ C \sqrt{p} ‖ u ‖_{H^{s}}, \forall u \in H^{s} . \end{matrix}$
Proof of Lemma 4.
The proof is standard and follows by interpolation argument between $H^{\frac{1}{2}}$ and $H^{1}$ . We refer the reader to [21] (Theorem 8.5) for more details. □

We give now a commutator estimate that states as follows
Lemma 5.
Let $u \in S (R)$ and $v \in H^{α}$ . Then there exists $C > 0$ such that $\begin{matrix} {‖ D^{α} (u v) - u D^{α} v ‖}_{L^{2}} ⩽ C ‖ v ‖_{L^{2}} {| | | ξ |^{α} F (u) | |}_{L^{1}} . \end{matrix}$
Proof of Lemma 5.
Thanks to the Fourier transform, one has $\begin{array}{l} | F (D^{α} (u v)) (ξ) - F (u D^{α} v) (ξ) | \\ ⩽ \int_{R} | | ξ - η |^{α} - | ξ |^{α} | | F (u) (η) | | F (v) (ξ - η) | d η \\ ≲ \int_{R} | η |^{α} | F (u) (η) | | F (v) (ξ - η) | d η \end{array}$ and the desired estimate yields thanks to the Minkowski inequality and the Plancherel Theorem. □

We now introduce, for a given level $M > 0$ , the orthogonal projector $P_{M}$ acting in $L^{2} (R)$ by setting $\begin{matrix} (12) & P_{M} (u) = \int_{R} χ (\frac{ξ}{N}) F (u) (ξ) e^{i ξ x} d ξ = F^{- 1} (ξ \mapsto χ (\frac{ξ}{N}) F (u) (ξ)), \end{matrix}$ where χ is the characteristic function of the interval $[- 1, 1]$ . Actually, $P_{M}$ is the projector onto the low-frequencies modes of a given function, at level N. Clearly, $D^{α} P_{M} = P_{M} D^{α}$ . Moreover, $P_{M}$ and $Q_{M} = I d - P_{M}$ are bounded operators from $H^{s}$ , $s ⩾ 0$ , into itself and satisfy the following inequalities that state as follows
Lemma 6.
Let $0 ⩽ s_{1} ⩽ s_{2}$ . Then there exists $C > 0$ , that does not depends on M, such that
For all $u \in H^{s_{1}}$ , $\begin{matrix} (13) & {‖ D^{s_{2}} P_{M} (u) ‖}_{L^{2}} ⩽ C M^{s_{2} - s_{1}} {‖ D^{s_{1}} P_{M} (u) ‖}_{L^{2}} . \end{matrix}$

For all $u \in H^{s_{2}}$ , $\begin{matrix} (14) & {‖ D^{s_{1}} Q_{M} (u) ‖}_{L^{2}} ⩽ C M^{s_{1} - s_{2}} {‖ D^{s_{2}} Q_{M} (u) ‖}_{L^{2}} . \end{matrix}$

Proof of Lemma 6.
The proof follows merely from the very definition of $P_{M}$ and some well known properties from Fourier analysis (see [13]). □

Moreover, $P_{M}$ extends to a uniformly (in M) bounded operator in $L^{p} (R)$ . More precisely we have Lemma 7.
For $1 < p < + \infty$ , $P_{M}$ extends to a bounded operator from $L^{p} (R)$ into itself whose norm does not depends on M.
Proof of Lemma 7.
See [34] for instance for a complete proof. □

3. Existence of the global attractor

For the sake of completeness and clarity we recall the definition of the global attractor (see for instance [7] and [29]).

Definition 8.
Let $(H, {(S (t))}_{t ⩾ 0})$ be a continuous dynamical system where $H$ denotes the phase space (complete metric space). We say that a nonempty subset $A$ of $H$ is a global or universal attractor for the semigroup ${(S (t))}_{t ⩾ 0}$ if and only if $A$ enjoys the following properties:
$A$ is compact.

$A$ is invariant i.e: $S (t) A = A$ , $\forall t ⩾ 0$ .

$A$ attracts the bounded sets of $H$ ; i.e: $d_{h} (S (t) B, A) \to 0$ as $t \to + \infty$ , $\forall B$ bounded set in $H$ with $d_{h}$ stands for the Hausdorff semi-distance.

Denoting ${(S_{α} (t))}_{t \in R_{+}}$ the semigroup associated to (9)–(10), we have
Proposition 9.
The semigroup ${(S_{α} (t))}_{t \in R_{+}}$ possesses a bounded absorbing ball $B_{α}$ in $H^{\frac{α}{2}}$ . i.e: for any bounded subset $B \subseteq H^{\frac{α}{2}}$ there exists $t (B) > 0$ such that $\begin{matrix} S_{α} (t) B \subseteq B_{α}, \forall t ⩾ t (B) . \end{matrix}$
Proof of Proposition 9.
On the one hand, the scalar product of (9) by U, leads to $\begin{matrix} \frac{d}{d t} {‖ U (t) ‖}_{L^{2}}^{2} + γ {‖ U (t) ‖}_{L^{2}}^{2} = ((F, U)) . \end{matrix}$ Hence, applying the Cauchy–Schwarz and Young’s inequalities, it can be deduced by the Gronwall’s lemma that $\begin{matrix} (15) & {‖ U (t) ‖}_{L^{2}}^{2} ⩽ ‖ U_{0} ‖_{L^{2}}^{2} e^{- γ t} + \frac{‖ F ‖_{L^{2}}^{2}}{2 γ^{2}} (1 - e^{- γ t}) . \end{matrix}$ On the other hand, the scalar product of (9) by $- i (U + γ U)$ leads to the following energy equation that reads $\begin{matrix} (16) & \frac{1}{2} \frac{d}{d t} J (U (t)) + γ J (U (t)) = K (U (t)) \end{matrix}$ where $\begin{array}{l} (17) & J (U (t)) = {‖ D^{\frac{α}{2}} U (t) ‖}_{L^{2}}^{2} - \frac{1}{2} ({| U (t) |}^{2}, {| U (t) |}^{2}) - 2 ((i F, U (t))) \\ (18) & K (U (t)) = \frac{γ}{2} ({| U (t) |}^{2}, {| U (t) |}^{2}) - γ ((i F, U (t))) \end{array}$ Thanks to the Cauchy–Schwarz inequality, Lemma 3 and (16), one easily has the existence of a non negative reel constant C that depends only on $‖ F ‖_{L^{2}}$ , γ and N such that $\begin{array}{l} (19) & \frac{1}{2} {‖ D^{\frac{α}{2}} U (t) ‖}_{L^{2}}^{2} - C ⩽ J (U (t)) ⩽ {‖ D^{\frac{α}{2}} U (t) ‖}_{L^{2}}^{2} + C \\ (20) & K (U (t)) ⩽ \frac{γ}{4} {‖ D^{\frac{α}{2}} U (t) ‖}_{L^{2}}^{2} + C \end{array}$ Consequently, gathering (19), (20) and (16) we obtain $\begin{matrix} \frac{d}{d t} J (U (t)) + γ J (U (t)) ⩽ C . \end{matrix}$ This concludes the proof thanks to the Gronwall lemma and (19). □

Under a slightly different method then that appears in [4], we claim now to prove the existence of the global attractor $A_{α}$ by the use of a decomposition of the semigroup $S_{α} (t)$ into $S_{1} (t) + S_{2} (t)$ . To do this, one only has, thanks to Theorem 1.1 and Remark 1.4 in [29] and Theorem 5.1 in [7], to prove the asymptotic compactness of the semi-group ${(S_{α} (t))}_{t \in R_{+}}$ , that is Proposition 10.
The semi-group ${(S_{α} (t))}_{t \in R_{+}}$ is asymptotically compact in $H^{\frac{α}{2}}$ i.e., for every bounded sequence ${(U_{k})}_{k}$ in $H^{\frac{α}{2}}$ and every sequence $t_{k} ⟶ + \infty$ , ${(S_{α} (t_{k}) U_{k})}_{k}$ is relatively compact in $H^{\frac{α}{2}}$ .
Proof of Proposition 10.
The proof is divided into three steps. As a first step we state, Lemma 11.
The semi-group ${(S_{α} (t))}_{t \in R_{+}}$ is continuous on bounded subsets of $H^{\frac{α}{2}}$ for the strong topology of $L^{2}$ .
Proof of Lemma 11.
Let ${(Z_{n})}_{n}$ be a bounded sequence in $H^{\frac{α}{2}}$ that converges towards $Z \in H^{\frac{α}{2}}$ for the strong topology of $L^{2}$ . We denote $\begin{matrix} W (t) = S_{α} (t) Z_{n} - S_{α} (t) Z = U_{n} (t) - U (t) \end{matrix}$ the difference of two solutions of (9)–(10) issued respectively from $Z_{n}$ and Z. Then $W (t)$ satisfies $\begin{matrix} (21) & \frac{1}{2} \frac{d}{d t} {‖ W (t) ‖}_{L^{2}}^{2} + γ {‖ W (t) ‖}_{L^{2}}^{2} ⩽ C_{0} \int_{R} ({| U_{n} (x) |}^{2} + {| U (x) |}^{2}) {| W (x) |}^{2} d x . \end{matrix}$ Using an argument due to M. Vladimirov (see [31]) let $p \in (2, + \infty)$ . Thanks to the Hölder inequality $\begin{matrix} \int_{R} ({| U_{n} (x) |}^{2} + {| U (x) |}^{2}) {| W (x) |}^{2} d x ⩽ (‖ U_{n} ‖_{L^{4 p}}^{2} + ‖ U ‖_{L^{4 p}}^{2}) ‖ W ‖_{L^{4}}^{\frac{2}{p}} ‖ W ‖_{L^{2}}^{\frac{2 p - 2}{p}} \end{matrix}$ Since $U_{n}$ and U remain uniformly bounded in $H^{\frac{α}{2}}$ , it may be deduced, in accordance with (21), Lemma 3 and Lemma 4, that $\begin{matrix} \frac{d}{d t} {‖ W (t) ‖}_{L^{2}}^{2} ⩽ C_{0} C_{1}^{\frac{1}{α p}} p ‖ W ‖_{L^{2}}^{2 (1 - \frac{1}{2 p α})}, \end{matrix}$ which is equivalent to $\begin{matrix} (22) & \frac{d}{d t} {‖ W (t) ‖}_{L^{2}}^{\frac{1}{α p}} ⩽ C_{α} C_{1}^{\frac{1}{α p}} . \end{matrix}$ Integrating (22) on $[0, T^{}]$ for a chosen $T^{} > 0$ such that $C_{α} T^{} < 1$ , leads to $\begin{matrix} (23) & sup_{t \in [0, T^{}]} {‖ S_{α} (t) Z_{n} - S_{α} (t) Z ‖}_{L^{2}} ⩽ {(C_{α} C_{1}^{\frac{1}{α p}} T^{} + ‖ Z_{n} - Z ‖_{L^{2}}^{\frac{1}{2 p α}})}^{p α} . \end{matrix}$ Therefore $\begin{matrix} (24) & \underset{n \to + \infty}{lim sup} sup_{t \in [0, T^{}]} {‖ S_{α} (t) Z_{n} - S_{α} (t) Z ‖}_{L^{2}} ⩽ C_{1} {(C_{α} T^{})}^{α p} . \end{matrix}$ Letting $p ⟶ + \infty$ leads to the desired result for $t \in [0, T^{}]$ . Remark that for $t \in [T^{}, 2 T^{}]$ , we similarly deduce in accordance with (24) that $\begin{matrix} (25) & \underset{n \to + \infty}{lim sup} sup_{t \in [T^{}, 2 T^{}]} {‖ S_{α} (t) Z_{n} - S_{α} (t) Z ‖}_{L^{2}} ⩽ C_{1} {(C_{α} T^{})}^{α p} ⟶ 0 as p \to + \infty . \end{matrix}$ By induction, for arbitrary $t > 0$ there exists $m_{0} \in N^{}$ such that $t \in [m_{0} T^{}, (m_{0} + 1) T^{}]$ . Hence, integrating (22) on $[m_{0} T^{}, t]$ and assuming that ${lim sup}_{n \to + \infty} ‖ S_{α} (m_{0} T^{}) Z_{n} - S_{α} (m_{0} T^{}) Z ‖_{L^{2}} = 0$ ensure, by similar argument, that $\begin{matrix} \underset{n \to + \infty}{lim sup} {‖ S_{α} (t) Z_{n} - S_{α} (t) Z ‖}_{L^{2}} = 0 \end{matrix}$ and the proof is therefore achieved. □

Next, we show that the semigroup ${(S_{α} (t))}_{t \in R_{+}}$ is asymptotically compact in $L^{2}$ on bounded subsets of $H^{\frac{α}{2}}$ . Lemma 12.
For every bounded sequence* ${(U_{j})}_{j}$ in $H^{\frac{α}{2}}$ and every nonnegative sequence $t_{j} ⟶ + \infty$ , ${(S_{α} {(t_{j})}_{α} U_{j})}_{j}$ is relatively compact in $L^{2}$ .
Proof of Lemma 12.
Let $U_{0} \in H^{\frac{α}{2}}$ and $U (t) = S_{α} (t) U_{0}$ . Consider a smooth cut-off function θ such that $θ (x) = 1$ if $| x | ⩽ 1$ and $θ (x) = 0$ if $| x | ⩾ 2$ , $x \in R$ . For a given $R > 0$ , we set $\begin{matrix} θ_{R} (x) = θ (\frac{x}{R}) and Λ_{R} (t) = \int_{R} {| U (t) |}^{2} {[1 - θ_{R} (x)]}^{2} d x . \end{matrix}$ Multiplying (9) by $(1 - θ_{R})$ then making the scalar product of the resulting equation by $U (1 - θ_{R})$ lead to $\begin{matrix} \frac{1}{2} \frac{d}{d t} Λ (t) + γ Λ (t) = ((i (1 - θ_{R}) D^{α} U, (1 - θ_{R}) U)) + (((1 - θ_{R}) F, (1 - θ_{R}) U)) \end{matrix}$ Hence, by the Cauchy–Schwarz inequality $\begin{matrix} (26) & \begin{matrix} \frac{1}{2} \frac{d}{d t} Λ (t) + γ Λ (t) \\ ⩽ ({‖ (1 - θ_{R}) F ‖}_{L^{2}} + {‖ (1 - θ_{R}) D^{α} U - D^{α} [(1 - θ_{R}) U] ‖}_{L^{2}}) \sqrt{Λ (t)} . \end{matrix} \end{matrix}$ Thanks to Lemma 5, $\begin{array}{l} {‖ (1 - θ_{R}) D^{α} U - D^{α} [(1 - θ_{R}) U] ‖}_{L^{2}} & = {‖ θ_{R} D^{α} U - D^{α} (θ_{R} U) ‖}_{L^{2}} \\ ⩽ ‖ U ‖_{L^{2}} {‖ | ξ |^{α} F (θ_{R}) ‖}_{L^{1} (R)} \end{array}$ since $| ξ |^{α} F (θ_{R}) (ξ) = R | ξ |^{α} F (θ) (R ξ)$ , then $\begin{matrix} {‖ | ξ |^{α} F (θ_{R}) ‖}_{L^{1} (R)} = \frac{‖ F (D^{α} θ) ‖_{L^{1} (R)}}{R^{α}} ⩽ \frac{C}{R^{α}} . \end{matrix}$ Hence $\begin{matrix} (27) & {‖ (1 - θ_{R}) D^{α} U - D^{α} [(1 - θ_{R}) U] ‖}_{L^{2}} ⩽ \frac{C}{R^{α}} ‖ U ‖_{L^{2}} ≲ \frac{1}{R^{α}} . \end{matrix}$ Gathering (26) and (27) then applying the Young inequality and the Gronwall lemma lead to $\begin{matrix} (28) & Λ (t) ⩽ ‖ U_{0} ‖_{L^{2} (| x | > R)}^{2} e^{- γ t} + C (‖ F ‖_{L^{2} (| x | > R)}^{2} + \frac{1}{R^{2 α}}) (1 - e^{- γ t}) . \end{matrix}$ Consider now a bounded sequence ${(U_{j})}_{j}$ in $H^{\frac{α}{2}}$ and $t_{j} ⟶ + \infty$ . By splitting $S_{α} (t_{j}) U_{j}$ as follows $\begin{array}{l} S_{α} (t_{j}) U_{j} & = S_{α} (t_{j}) U_{j} θ_{R} + S_{α} (t_{j}) U_{j} (1 - θ_{R}) \\ = W_{j} (t_{j}) + V_{j} (t_{j}) \end{array}$ we deduce from (28) that for a given $ϵ > 0$ , there exist $R_{0} > 0$ such that $‖ V_{j} (t_{j}) ‖_{L^{2}} ⩽ ϵ$ for all $j ⩾ j_{0}$ . Moreover, ${(W_{j} (t_{j}))}_{j}$ remains trapped in a compact set of $L^{2}$ which ensure by classical argument that ${(S_{α} (t_{j}) U_{j})}_{j}$ is relatively compact in $L^{2}$ and the proof is complete. □

In this last step we continue the proof of the Proposition 10.

By using the well known John Ball’s argument (see [2] and [32]). Let ${(U_{j})}_{j}$ be a bounded sequence in $H^{\frac{α}{2}}$ and $t_{j} ⟶ + \infty$ . Since ${(S_{α} (t_{j}) U_{j})}_{j}$ is bounded in $H^{\frac{α}{2}}$ and in accordance with Lemma 11 and Lemma 12, we may assume, up to a subsequence extraction, the existence of $U \in H^{\frac{α}{2}}$ such that $\begin{matrix} (29) & S_{α} (t_{j}) U_{j} ⇀ U weakly in H^{\frac{α}{2}} and S_{α} (t_{j}) U_{j} ⟶ U strongly in L^{2} . \end{matrix}$ Thanks to the energy equation (16), we deduce that, in the one hand $\begin{matrix} (30) & J (S_{α} (t_{j}) U_{j}) = J (S_{α} (t_{j} - t) U_{j}) e^{- 2 γ t} + 2 \int_{0}^{t} e^{- 2 γ (t - s)} K (S_{α} (t_{j} - t + s) U_{j}) d s \end{matrix}$ and in the other hand, $\begin{matrix} (31) & J (U) = J (S_{α} (- t) U) e^{- 2 γ t} + \int_{0}^{t} e^{- 2 γ (t - s)} K (S_{α} (s - t) U) d s, \end{matrix}$ where J and K are defined by (17) and (18).

Thanks to the dominated convergence theorem, (29), (30) and (31) one easily obtain that $\begin{matrix} \underset{j \to + \infty}{lim sup} J (S (t_{j}) U_{j}) ⩽ J (U) \end{matrix}$ from which we deduce, thanks to (19), that $\begin{matrix} {‖ D^{\frac{α}{2}} S (t_{j}) U_{j} ‖}_{L^{2}} ⟶ {‖ D^{\frac{α}{2}} U ‖}_{L^{2}} as j ⟶ + \infty . \end{matrix}$ This concludes the proof of the current proposition. □

4. Regularity of the global attractor

We follow here the strategy in [12]. Let $U (t)$ be the solution of (9)-(10) which takes values in $H^{\frac{α}{2}}$ . We may assume that $U (t)$ remains into the $H^{\frac{α}{2}}$ -absorbing set, $B_{α}$ , for $t ⩾ 0$ .

We aim to prove the first part of our main result, that is

Theorem 13.
The global attractor $A_{α}$ associated to the dynamical system $({(S_{α} (t))}_{t \in R_{+}}), H^{\frac{α}{2}})$ is a compact subset of $H^{α}$ .

4.1. The auxiliary problem

To highlight the regularity of the global attractor $A_{α}$ , we split the solution as $U (t) = P_{M} (U) + Q_{M} (U) = V (t, x) + W (t, x)$ and then, thanks to Lemma 6, the regularity of $U = {(u_{1}, \dots, u_{N})}^{t}$ depends only on the regularity of $W = {(Q_{M} (u_{1}), \dots, Q_{M} (u_{N}))}^{t}$ . Therefore, we shall focus on the long-time behavior of $W (t)$ and for that purpose we approximate W, solution for $Q_{M} (9)$ supplemented with initial data $W (0) = Q_{M} (U (0))$ , by a more regular function $Z = {(z_{1}, \dots, z_{N})}^{t}$ which solves the auxiliary system $\begin{array}{l} (32a) & z_{k t} - i D^{α} z_{k} + i Q_{M} [\sum_{l = 1}^{N} | v_{l} + z_{l} |^{2} (v_{k} + z_{k})] + γ z_{k} = Q_{M} (f_{k}) \\ (32b) & z_{k} (0) = 0, 1 ⩽ k ⩽ N \end{array}$ For the sake of simplicity, this system is equivalently written in vector sense as follows $\begin{array}{l} (33a) & Z_{t} - i D^{α} Z + i Q_{M} [| V + Z |^{2} (V + Z)] + γ Z = Q_{M} (F) \\ (33b) & Z (0) = 0 . \end{array}$ To begin with, we have the following result

Proposition 14.
There exists $M_{0} > 0$ large enough depending only on γ, α, F and N such that for any $M ⩾ M_{0}$ , the problem ( 33a )–( 33b ) has a unique local in time solution in $C (R_{+}, H^{α})$ that remains bounded, uniformly in M and t, in $H^{\frac{α}{2}}$ .
Proof of Proposition 14.
The proof is classic and then sketched for the sake of conciseness. We proceed into two steps:

∙ First step : existence of a local-in-time solution in $H^{α}$ .

Since $H^{α}$ is an algebra and the mapping $Φ : Z \mapsto Q_{M} (| Z + V |^{2} (Z + V))$ is locally lipschitz from $H^{α}$ into itself, it is standard to establish a local in time solution Z in $C ([0, T^{}), H^{α})$ for the problem (33a)–(33b) by a fixed point argument on its Duhamel’s form.

∙ Second step* : the solution is uniformly bounded in $H^{\frac{α}{2}}$ .

Taking the inner product of (32a) by $- i (z_{k t} + γ z_{k})$ then summing up the resultant equation over $1 ⩽ k ⩽ N$ we obtain that $\begin{matrix} (34) & \frac{1}{2} \frac{d}{d t} J (Z (t)) + γ J (Z (t)) = K (Z (t)) \end{matrix}$ where we set $\begin{array}{l} (35) & J (Z) = {‖ D^{\frac{α}{2}} Z ‖}_{L^{2}}^{2} - \frac{1}{2} (| Z + V |^{2}, | Z + V |^{2}) - 2 ((i F, Z)) \\ (36) & K (Z) = \frac{γ}{2} (| Z + V |^{2}, | Z + V |^{2}) - γ ((i F, Z)) - γ ((| Z + V |^{2} (Z + V), V_{t} + γ V)) . \end{array}$ At this stage we need the following lemma Lemma 15.
There exists $C > 0$ depending only on γ, α, F and N such that for all $M ⩾ 0$ , $\begin{matrix} {‖ P_{M} (U (t)) ‖}_{H^{\frac{α}{2}}} + {‖ P_{M} (U_{t} (t)) ‖}_{H^{- \frac{α}{2}}} ⩽ C . \end{matrix}$
Proof of Lemma 15.
Thanks to Lemma 6, $P_{M}$ is uniformly (in M) bounded from $H^{- \frac{α}{2}}$ into itself. Moreover, in accordance with (9) one has, in vector sense, $\begin{matrix} P_{M} (U_{t} (t)) = i D^{α} P_{M} (U (t)) - γ P_{M} (U (t)) - i P_{M} [{| U (t) |}^{2} U (t)] + P_{M} (F) \end{matrix}$ from which, and the fact that $H^{\frac{α}{2}}$ is an algebra (Lemma 3), we deduce that $P_{M} (U_{t} (t))$ remains bounded in $H^{- \frac{α}{2}}$ and then the proof is completed. □

For the purpose of establishing an upper and a lower bounds for $J (Z)$ , it should be noted that due to Lemma 3 and Lemma 6 we have $\begin{array}{l} (37) & ‖ Z ‖_{L^{2}}^{2} ≲ \frac{1}{M^{α}} {‖ D^{\frac{α}{2}} Z ‖}_{L^{2}}^{2} \\ (38) & ‖ Z ‖_{L^{4}}^{4} ≲_{N} \frac{1}{M^{2 α - 1}} {‖ D^{\frac{α}{2}} Z ‖}_{L^{2}}^{4} . \end{array}$ Hence, thanks to the Cauchy–Schwarz inequality, (37), (38) and Lemma 15 we deduce, for M large enough, the existence of $C_{0}, C_{1} > 0$ that not depend on M such that $\begin{matrix} (39) & \frac{1}{2} {‖ D^{\frac{α}{2}} Z ‖}_{L^{2}}^{2} - C_{0} \frac{‖ D^{\frac{α}{2}} Z ‖_{L^{2}}^{4}}{M^{2 α - 1}} - C_{1} ⩽ J (Z) ⩽ \frac{3}{2} {‖ D^{\frac{α}{2}} Z ‖}_{L^{2}}^{2} + C_{1} \end{matrix}$ Now we derive an upper bound for $K (Z)$ . Thanks again to Lemma 3 and Lemma 6 $\begin{matrix} (40) & ‖ Z ‖_{L^{\infty}}^{2} ≲_{N} \frac{1}{M^{α - 1}} {‖ D^{\frac{α}{2}} Z ‖}_{L^{2}}^{2} . \end{matrix}$ Therefore, thanks to Lemma 15 $\begin{array}{l} (41) & | ((| Z + V |^{2} (Z + V), V_{t})) | & ⩽ {‖ | Z + V |^{2} (Z + V) ‖}_{H^{\frac{α}{2}}} ‖ V_{t} ‖_{H^{- \frac{α}{2}}} \\ (42) & ≲_{N} ‖ Z + V ‖_{L^{\infty}}^{2} ‖ Z + V ‖_{H^{\frac{α}{2}}} ‖ V_{t} ‖_{H^{- \frac{α}{2}}} \\ (43) & ≲_{N} (1 + \frac{‖ D^{\frac{α}{2}} Z ‖_{L^{2}}^{2}}{M^{α - 1}}) (1 + {‖ D^{\frac{α}{2}} Z ‖}_{L^{2}}) . \end{array}$ Gathering (37), (38) and (43) it may be deduced, in accordance with the young inequality, that $\begin{matrix} (44) & K (Z) ⩽ C_{0} + \frac{γ}{4} {‖ D^{\frac{α}{2}} Z ‖}_{L^{2}}^{2} + \frac{C_{1}}{M^{α - 1}} {‖ D^{\frac{α}{2}} Z ‖}_{L^{2}}^{3} + \frac{C_{2}}{M^{2 α - 1}} {‖ D^{\frac{α}{2}} Z ‖}_{L^{2}}^{4} . \end{matrix}$ In accordance with (39), (34), and (44), the Gronwall lemma implies that for M large enough $\begin{array}{l} J (Z (t)) ⩽ & C_{0} + \frac{C_{1}}{M^{α - 1}} \int_{0}^{t} e^{- γ (t - s)} {‖ D^{\frac{α}{2}} Z (s) ‖}_{L^{2}}^{3} d s + \frac{C_{2}}{M^{2 α - 1}} \int_{0}^{t} e^{- γ (t - s)} {‖ D^{\frac{α}{2}} Z (s) ‖}_{L^{2}}^{4} d s \end{array}$ As a result, thanks again to (39), setting $\begin{matrix} h (t) = sup_{s \in [0, t]} {‖ D^{\frac{α}{2}} Z (s) ‖}_{L^{2}}^{2} \end{matrix}$ it may be deduced the existence of nonnegative reel constants $K_{0}$ , $K_{1}$ and $K_{2} > 0$ independent of M such that $\begin{matrix} (45) & h (t) ⩽ K_{0} + \frac{K_{1}}{M^{α - 1}} h {(t)}^{\frac{3}{2}} + \frac{K_{2}}{M^{2 α - 1}} h {(t)}^{2}, t \in [0, T^{}) \end{matrix}$ Now we use a “mean value argument”. Introducing for $x ⩾ 0$ the function $\begin{matrix} Φ (x) = x - K_{0} + \frac{K_{1}}{M^{α - 1}} x^{\frac{3}{2}} + \frac{K_{2}}{M^{2 α - 1}} x^{2}, \end{matrix}$ we chose M large enough such that $Φ (2 K_{0}) > 0$ . If there exists $t_{1} > 0$ such that $h (t_{1}) > 2 K_{0}$ then due to the continuity and the monotonicity of h, taking into account that $h (0) = 0$ , it may be deduced the existence of $0 < t_{2} < t_{1}$ such that $h (t_{2}) = 2 K_{0}$ . Consequently, $Φ (h (t_{2})) = Φ (2 K_{0}) > 0$ and this contradicts the fact that $Φ (h (t)) ⩽ 0$ .

Hence $h (t) ⩽ 2 K_{0}$ . This implies on the one hand that $T^{} = + \infty$ and on the other, the solution Z remains uniformly bounded in $C (R_{+}, H^{\frac{α}{2}})$ . The proof of the proposition is therefore achieved. □

We now proceed to the next step in which we shall prove that Z remains bounded in $H^{α}$ with an upper bound that may depend on M. Proposition 16.
There exist a reel $K > 0$ and $M_{0} > 0$ large enough that depend only on γ, α, F and N such that for any $M ⩾ M_{0}$ , the following estimate holds true $\begin{matrix} sup_{t \in R_{+}} {‖ Z (t) ‖}_{H^{α}} ⩽ K M^{\frac{α}{2}} . \end{matrix}$
Proof of Proposition 16.
In order to prove that $D^{α} Z$ is bounded in $L^{2}$ we will prove equivalently that $Y = Z_{t} = {(y_{1}, \dots, y_{N})}^{t}$ is bounded in $L^{2}$ . To do this we will establish some a priori estimates on $‖ Y ‖_{H^{\frac{α}{2}}}$ which can be proved rigorously using some smooth approximation argument. For the sake of simplicity we denote $Λ = W + V = {(λ_{1}, \dots, λ_{M})}^{t}$ .

We differentiate the system (32a) with respect to t to have for $1 ⩽ k ⩽ N$ , $\begin{matrix} y_{k t} - i D^{α} y_{k} + γ y_{k} + i \sum_{l = 1}^{N} Q_{M} [2 ℜ e (\overline{λ_{l}} λ_{l t}) λ_{k} + | λ_{l} |^{2} λ_{k t}] = 0 \end{matrix}$ Taking the scalar product the previous equation by $- i (y_{k t} + γ y_{k})$ as a starting point then summing up the resultant equations lead, by mere computations, to $\begin{matrix} (46) & \frac{1}{2} \frac{d}{d t} Ψ (Y (t)) + γ Ψ (Y (t)) = Θ (t) \end{matrix}$ where $\begin{matrix} (47) & \begin{matrix} Ψ (Y) = & {| D^{\frac{α}{2}} Y |}_{L^{2}}^{2} - (| Λ |^{2}, | Y |^{2}) - 2 {| ℜ e (\overline{Λ} . Y) |}_{L^{2}}^{2} \\ - 2 (| Λ |^{2}, ℜ e (\overline{Y} . V_{t})) - 4 (ℜ e (\overline{Λ} . V_{t}), ℜ e (\overline{Λ} . Y)) \end{matrix} \end{matrix}$ and $\begin{matrix} (48) & \begin{matrix} Θ (Y) = & - 2 γ (ℜ e (\overline{Λ} . V_{t}), ℜ e (\overline{Λ} . Y)) - (1 + γ) (| Λ |^{2}, ℜ e (\overline{Y} . V_{t})) - 2 (ℜ e (\overline{Λ} . Y), ℜ e (\overline{Y} . Λ_{t})) \\ - (| Y |^{2}, ℜ e (\overline{Λ} . Λ_{t})) - 2 (ℜ e (\overline{Λ} . V_{t}), ℜ e (\overline{Y} . Λ_{t})) - 2 (ℜ e (\overline{Λ} . Y), ℜ e (\overline{V_{t}} . Λ_{t})) \\ - 2 (ℜ e (\overline{Λ} . Λ_{t}), ℜ e (\overline{Y} . V_{t})) - 2 (ℜ e (\overline{Λ} . V_{t t}), ℜ e (\overline{Λ} . Y)) . \end{matrix} \end{matrix}$ First of all, thanks to Lemma 3, Lemma 6 and since Λ is uniformly bounded in $H^{\frac{α}{2}}$ , the following estimates hold $\begin{array}{l} (49) & \begin{matrix} (| Λ |^{2}, | Y |^{2}) + {‖ ℜ e (\overline{Λ} . Y) ‖}_{L^{2}}^{2} & ≲_{N} ‖ Λ ‖_{L^{4}}^{2} ‖ Y ‖_{L^{4}}^{2} ≲_{α, N} \frac{1}{M^{α - \frac{1}{2}}} {‖ D^{\frac{α}{2}} Y ‖}_{L^{2}}^{2}, \end{matrix} \\ (50) & \begin{matrix} | (| Λ |^{2}, ℜ e (\overline{Y} . V_{t})) | + | (ℜ e (\overline{Λ} . V_{t}), ℜ e (\overline{Λ} . Y)) | & ≲_{N} ‖ Λ ‖_{L^{\infty}}^{2} ‖ Y ‖_{L^{2}} ‖ V_{t} ‖_{L^{2}} \\ ≲_{α, N} {‖ D^{\frac{α}{2}} Y ‖}_{L^{2}} . \end{matrix} \end{array}$ As a result of (49), (50) and (47) one easily obtain, for M large enough, that $\begin{matrix} (51) & \frac{1}{2} {‖ D^{\frac{α}{2}} Y ‖}_{L^{2}}^{2} - C ⩽ Ψ (Y) ⩽ \frac{3}{2} {‖ D^{\frac{α}{2}} Y ‖}_{L^{2}}^{2} + C \end{matrix}$ where $C > 0$ does not depends on M.

Now we handle the right hand side of (46).

Firstly, thanks to Lemma 15 and Proposition 14 one easily obtain the existence of $K_{0} > 0$ depending only on F, γ, α and N such that $\begin{matrix} (52) & sup_{t \in R_{+}} {‖ Λ (t) ‖}_{H^{- \frac{α}{2}}} ⩽ K_{0} . \end{matrix}$ Now by the use of (52), Lemma 3, Lemma 6 and in accordance with Proposition 14, we have on the one hand $\begin{array}{l} (53) & | (ℜ e (\overline{Λ} . Y), ℜ e (\overline{Y} . Λ_{t})) | + | (| Y |^{2}, ℜ e (\overline{Λ} . Λ_{t})) | & ≲_{N} {‖ | Y |^{2} Λ ‖}_{H^{\frac{α}{2}}} ‖ Λ_{t} ‖_{H^{- \frac{α}{2}}} \\ (54) & ≲_{α, N} ‖ Y ‖_{L^{\infty}} (‖ Y ‖_{L^{\infty}} + {‖ D^{\frac{α}{2}} Y ‖}_{L^{2}}) \\ (55) & ≲_{α, N} \frac{1}{M^{\frac{α - 1}{2}}} {‖ D^{\frac{α}{2}} Y ‖}_{L^{2}}^{2} . \end{array}$ On the other hand, recalling that $Λ_{t} = Y + V_{t}$ we easily check that $\begin{array}{l} (ℜ e (\overline{Λ} . V_{t}), ℜ e (\overline{Y} . Λ_{t})) + (ℜ e (\overline{Λ} . Y), ℜ e (\overline{V_{t}} . Λ_{t})) + (ℜ e (\overline{Λ} . Λ_{t}), ℜ e (\overline{Y} . V_{t})) \\ = (| Y |^{2}, ℜ e (\overline{Λ} . V_{t})) + 2 (ℜ e (\overline{Y} . V_{t}), ℜ e (\overline{Λ} . Y)) \\ (56) & + (| V_{t} |^{2}, ℜ e (\overline{Λ} . Y)) + 2 (ℜ e (\overline{Λ} . V_{t}), ℜ e (\overline{Y} . V_{t})) . \end{array}$ Independently, $\begin{array}{l} | (| V_{t} |^{2}, ℜ e (\overline{Λ} . Y)) | + | (ℜ e (\overline{Λ} . V_{t}), ℜ e (\overline{Y} . V_{t})) | & ≲_{N} ‖ Y ‖_{L^{\infty}} ‖ Λ ‖_{L^{\infty}} ‖ V_{t} ‖_{L^{2}}^{2} \\ (57) & ≲_{α, N} M^{\frac{α + 1}{2}} {‖ D^{\frac{α}{2}} Y ‖}_{L^{2}} \end{array}$ In the light of the above, we deduce from (55) and (56) that $\begin{array}{l} | (ℜ e (\overline{Λ} . V_{t}), ℜ e (\overline{Y} . Λ_{t})) + (ℜ e (\overline{Λ} . Y), ℜ e (\overline{V_{t}} . Λ_{t})) + (ℜ e (\overline{Λ} . Λ_{t}), ℜ e (\overline{Y} . V_{t})) | \\ (58) & ≲_{α, N} M^{\frac{α + 1}{2}} {‖ D^{\frac{α}{2}} Y ‖}_{L^{2}} + \frac{‖ D^{\frac{α}{2}} Y ‖_{L^{2}}^{2}}{M^{\frac{α - 1}{2}}} \end{array}$ Now it only remains to bound the worst term $(ℜ e (\overline{Λ} . V_{t t}), ℜ e (\overline{Λ} . Y))$ .

Knowing that $V_{t}$ satisfies $\begin{matrix} V_{t t} = i D^{α} V_{t} - γ V_{t} - i P_{M} [| U |^{2} U_{t} + 2 ℜ e (\overline{U} . U_{t}) U], \end{matrix}$ observe that due to the fact that $H^{\frac{α}{2}}$ is continuously embedded in $L^{\infty}$ , either $P_{M} [| U |^{2} U_{t}]$ or $P_{M} [ℜ e (\overline{U} . U_{t}) U]$ are uniformly bounded in $H^{- \frac{α}{2}}$ . Thus, thanks to Lemma 6 and Lemma 15 $\begin{matrix} (59) & ‖ V_{t t} ‖_{H^{- \frac{α}{2}}} ≲_{α, N} {‖ D^{α} V_{t} ‖}_{H^{- \frac{α}{2}}} + γ ‖ V_{t} ‖_{H^{- \frac{α}{2}}} ≲_{α, N} M^{α} \end{matrix}$ This will enable us to have $\begin{array}{l} (60) & | (ℜ e (\overline{Λ} . V_{t t}), ℜ e (\overline{Λ} . Y)) | & ≲_{N} {‖ | Λ |^{2} Y ‖}_{H^{\frac{α}{2}}} ‖ V_{t t} ‖_{H^{- \frac{α}{2}}} \\ (61) & ≲_{α, N} M^{α} {‖ D^{\frac{α}{2}} Y ‖}_{L^{2}} . \end{array}$ Gathering (49), (50), (55), (58) and (61), it may deduced in accordance with the Young inequality that for M large enough $\begin{matrix} Θ (Y) ⩽ \frac{γ}{4} {‖ D^{\frac{α}{2}} Y ‖}_{L^{2}}^{2} + C M^{2 α} \end{matrix}$ which, with (51), leads to $\begin{matrix} \frac{d}{d t} Ψ (Y (t)) + γ Ψ (Y (t)) ⩽ C M^{2 α} . \end{matrix}$ This completes the proof of the current proposition thanks to the Gronwall lemma, (51) and Lemma 6. □

4.2. Proof of Theorem 13

To begin with, we proceed to the large time comparison between $U = W + V$ and $Λ = Z + V$ .

Proposition 17.
There exists $C > 0$ depending only on γ, α and F such that for all $t ⩾ 0$ $\begin{matrix} {‖ U (t) - Λ (t) ‖}_{H^{\frac{α}{2}}} = {‖ W (t) - Z (t) ‖}_{H^{\frac{α}{2}}} ⩽ C e^{- γ t} . \end{matrix}$
Proof of Proposition 17.
As a starting point and for the sake of simplicity we shall denote $\begin{matrix} X (t) = U (t) - Λ (t) = W (t) - Z (t) = {(ξ_{1}, \dots, ξ_{N})}^{t} \end{matrix}$ solving the following system that reads $\begin{array}{l} (62) & ξ_{k t} - i D^{α} ξ_{k} + γ ξ_{k} + i \sum_{l = 1}^{N} Q_{M} [| u_{l} |^{2} u_{k} - | λ_{l} |^{2} λ_{k}] = 0 \\ (63) & ξ_{k} (0) = u_{k} (0), 1 ⩽ k ⩽ N . \end{array}$ using the following identity $\begin{matrix} | u_{l} |^{2} u_{k} - | λ_{l} |^{2} λ_{k} = | u_{l} |^{2} ξ_{k} + λ_{k} ℜ e [(\overline{u_{l} + λ_{l}}) ξ_{l}], \end{matrix}$ it can be deduced, using the scalar product of (62) by $- i (ξ_{k t} + γ ξ_{k})$ then summing the resultant equations over $1 ⩽ k ⩽ N$ , that $\begin{matrix} (64) & \frac{1}{2} \frac{d}{d t} Ξ (X (t)) + γ Ξ (X (t)) = Γ (X (t)) \end{matrix}$ where we denote $\begin{matrix} (65) & Ξ (X) = {‖ D^{\frac{α}{2}} X ‖}_{L^{2}}^{2} - (| U |^{2}, | X |^{2}) - 2 {‖ ℜ e (\overline{Λ} . X) ‖}_{L^{2}}^{2} \end{matrix}$ and $\begin{matrix} (66) & \begin{matrix} Γ (X (t)) = & γ (| X |^{2}, ℜ e (\overline{Λ} . X)) + (| X |^{2}, ℜ e (\overline{Λ} . X_{t})) \\ - (| X |^{2}, ℜ e (\overline{U} . U_{t})) - 2 (ℜ e (\overline{Λ} . X), ℜ e (\overline{X} . Λ_{t})) . \end{matrix} \end{matrix}$ Since $U_{t}$ and $Λ_{t}$ remain uniformly bounded in $H^{- \frac{α}{2}}$ , then applying Lemma 3 and Lemma 6 by means of which, using similar computations as (49) and (55), we conclude on the one hand that for M large enough $\begin{matrix} (67) & \frac{1}{2} {‖ D^{\frac{α}{2}} X ‖}_{L^{2}}^{2} ⩽ Ξ (X) ⩽ {‖ D^{\frac{α}{2}} X ‖}_{L^{2}}^{2} \end{matrix}$ and on the other hand one has $\begin{matrix} (68) & Γ (X) ≲_{α, N} \frac{‖ D^{\frac{α}{2}} X ‖_{L^{2}}^{2}}{M^{\frac{α - 1}{2}}} ⩽ \frac{γ}{4} {‖ D^{\frac{α}{2}} X ‖}_{L^{2}}^{2} . \end{matrix}$ Gathering (67), (68) and (64) completes the proof thanks to the Gronwall lemma. □

Propositions 17, 14 and 16 enable us to prove, in identical manner as in [11], that $A_{α}$ is a bounded subset of $H^{α}$ . The proof of the compactness of the global attractor $A_{α}$ in $H^{α}$ , based on the famous J. Ball’s argument (see [2]), is standard and similar to that in [11] to which we refer the reader. We omit it for the sake of conciseness and the proof of Theorem 13 is therefore completed.
5. Fractal dimension of the global attractor

For the sake of completeness we start this section by recalling the definition of the fractal dimension (see [7] for instance).

Definition 18.
The fractal dimension of a compact subset $M$ of a metric space $H$ is defined by $d_{f} (M) = {lim}_{ϵ \to 0} \frac{ln N (M, ϵ)}{ln (\frac{1}{ϵ})}$ , where $N (M, ϵ)$ denotes the minimal number of closed balls of the radius ϵ which cover the set $M$ .

Now we shall focus on proving the second part of our main result. We state Theorem 19.
Let $α \in (1, 2)$ and $F \in L^{2}$ . Then the compact global attractor $A_{α}$ has finite fractal dimension in $H^{\frac{α}{2}}$ .

5.1. Some helpful tools

For the purpose of proving the finite dimensionality of the global attractor $A_{α}$ , we first recall the following result derived from that given in [8], Theorem 2.15.

Proposition 20.
Let X be a Banach space and M be a bounded closed set in X. Assume that there exists a mapping $V : M ⟶ X$ such that
$M \subseteq V (M)$ . Moreover, V is Lipschitz on M, i.e, there exists $L > 0$ such that for all $u_{1}, u_{2} \in M$ , $‖ V (u_{1}) - V (u_{2}) ‖_{X} ⩽ L ‖ u_{1} - u_{2} ‖_{X}$ .

There exist compact semi-norms n on X (i.e:for any bounded set $B \subset X$ there exists a sequence ${(x_{k})}_{k} \subset B$ such that $n (x_{k} - x_{l}) ⟶ 0$ as $k, l \to + \infty$ ) such that $\forall u_{1}, u_{2} \in M$ , $\begin{matrix} {‖ V (u_{1}) - V (u_{2}) ‖}_{X} ⩽ δ ‖ u_{1} - u_{2} ‖_{X} + K n (u_{1} - u_{2}) . \end{matrix}$ where $0 < δ < 1$ and $K > 0$ are constants.
Then M is a compact subset of X and has a finite fractal dimension. Moreover, the following estimate holds $\begin{matrix} (69) & {dim}_{f} (M) ⩽ {[ln \frac{2}{1 + δ}]}^{- 1} ln [m_{0} (\frac{4 K {(1 + L^{2})}^{\frac{1}{2}}}{1 - δ})], \end{matrix}$ where $m_{0} (R)$ is the maximal number of points $x_{i}$ in the ball of radius $R > 0$ satisfying $n (x_{i} - x_{j}) > 1$ , $i \neq j$ .

Now we establish the following statement (see [1]) that will also play a very important role in the sequel. Lemma 21.
Let $C$ be a compact subset of $L^{2}$ . Then for every $ϵ > 0$ there exists $R = R (ϵ, N) > 0$ such that for all $ϕ \in C$ and for all $U \in H^{\frac{α}{2}}$ , $α \in (1, 2)$ , the following estimate holds $\begin{matrix} \int_{R} | ϕ | | U |^{2} d x ⩽ ϵ ‖ u ‖_{H^{\frac{α}{2}}}^{2} + R ‖ U ‖_{L^{2} ([- R, R])}^{2} . \end{matrix}$
Proof of Lemma 21.
Let $R > 0$ . Writing $\begin{array}{l} \int_{R} | ϕ | | U |^{2} d x = & \int_{[- R, R] \cap {| ϕ | ⩽ R}} | ϕ | | U |^{2} d x \\ + \int_{[- R, R] \cap {| ϕ | > R}} | ϕ | | U |^{2} d x + \int_{{| x | > R}} | ϕ | | U |^{2} d x \end{array}$ we obtain by applying the Hölder inequality that $\begin{matrix} \int_{R} | ϕ | | U |^{2} d x ⩽ R ‖ U ‖_{L^{2} ([- R, R])}^{2} + (‖ ϕ ‖_{L^{2} ({| ϕ | > R})} + ‖ ϕ ‖_{L^{2} ({| x | > R})}) ‖ U ‖_{L^{4} (R)}^{2} \end{matrix}$ by means of which and the continuous embedding of $H^{\frac{α}{2}}$ in $L^{4}$ (Lemma 3) we deduce that $\begin{matrix} \int_{R} | ϕ | | U |^{2} d x ⩽ R ‖ U ‖_{L^{2} ([- R, R])}^{2} + C (N) (‖ ϕ ‖_{L^{2} ({| ϕ | > R})} + ‖ ϕ ‖_{L^{2} ({| x | > R})}) ‖ U ‖_{H^{\frac{α}{2}}}^{2} . \end{matrix}$ Knowing that $‖ ϕ ‖_{L^{2} ({| ϕ | > R})} + ‖ ϕ ‖_{L^{2} ({| x | > R})} ⟶ 0$ as $R ⟶ + \infty$ uniformly in $ϕ \in C$ achieves the proof of the lemma. □

5.2. Proof of Theorem 19

We now check the assumptions in Theorem 20 by proving the following statement

Proposition 22.

There exist $t^{*} > 0$ and $R^{*} > 0$ that depend on γ, α, F and N such that for all $U_{0}$ , $V_{0} \in A_{α}$ , $\begin{matrix} (70) & {‖ S_{α} (t^{*}) U_{0} - S_{α} (t^{*}) V_{0} ‖}_{H^{\frac{α}{2}}}^{2} ⩽ \frac{1}{2} ‖ U_{0} - V_{0} ‖_{H^{\frac{α}{2}}}^{2} + L ‖ U_{0} - V_{0} ‖_{L^{2} ([- 2 R^{*}, 2 R^{*}])}^{2}, \end{matrix}$ where $L > 0$ is a non negative reel constant depending only on $t^{*}$ and $R^{*}$ .

Proof of Proposition 22.

Firstly, let $U_{0}, V_{0} \in A_{α}$ and we denote for the sake of simplicity $\begin{matrix} Z (t) = S (t) U_{0} - S (t) V_{0} = U (t) - V (t) = {(z_{1}, \dots, z_{N})}^{t} \end{matrix}$ solving the following system that reads $\begin{array}{l} (71) & z_{k t} - i D^{α} z_{k} + γ z_{k} + i \sum_{l = 1}^{N} Q_{M} [| u_{l} |^{2} u_{k} - | v_{l} |^{2} v_{k}] = 0 \\ (72) & z_{k} (0) = u_{k} (0) - v_{k} (0), 1 ⩽ k ⩽ N . \end{array}$ Now we prove the following statement Lemma 23.

There exist $R_{0} > 0$ and $C_{1}, C_{2} > 0$ depending only on γ, α, N and $A_{α}$ such that $\begin{matrix} (73) & {‖ Z (t) ‖}_{H^{\frac{α}{2}}}^{2} ⩽ C_{1} {‖ Z (0) ‖}_{H^{\frac{α}{2}}}^{2} e^{- γ t} + C_{2} R_{0} \int_{0}^{t} e^{- γ (t - s)} {‖ Z (s) ‖}_{L^{2} ([- R_{0}, R_{0}])}^{2} d s . \end{matrix}$

Proof of Lemma 23.

Thanks again to the identity $\begin{matrix} (74) & | u_{l} |^{2} u_{k} - | v_{l} |^{2} v_{k} = | u_{l} |^{2} z_{k} + v_{k} ℜ e [(\overline{u_{l} + v_{l}}) z_{l}], \end{matrix}$ we deduce on the one hand, by the scalar product of (71) by $z_{k}$ then summing up, that $\begin{matrix} (75) & \frac{1}{2} \frac{d}{d t} ‖ Z ‖_{L^{2}}^{2} + γ ‖ Z ‖_{L^{2}}^{2} = ((ℜ e [(\overline{U + V}) . Z], ℑ m (V . \overline{Z}))) \end{matrix}$ and on the other hand, by the scalar product of (71) by $- i (z_{t} + γ z)$ then summing up the resultant equations, we obtain that $\begin{matrix} (76) & \frac{1}{2} \frac{d}{d t} J (Z) + γ J (Z) = K (Z) \end{matrix}$ where $\begin{matrix} (77) & J (Z) = {‖ D^{\frac{α}{2}} Z ‖}_{L^{2}}^{2} - (| U |^{2}, | Z |^{2}) - 2 {‖ ℜ e (\overline{V} . Z) ‖}_{L^{2}}^{2} \end{matrix}$ and $\begin{matrix} (78) & \begin{matrix} K (Z) = & γ (| Z |^{2}, ℜ e (\overline{V} . Z)) + (| Z |^{2}, ℜ e (\overline{V} . Z_{t})) \\ - (| Z |^{2}, ℜ e (\overline{U} . U_{t})) - 2 (ℜ e (\overline{V} . Z), ℜ e (\overline{Z} . Z_{t})) . \end{matrix} \end{matrix}$ since U an V are uniformly bounded in $H^{\frac{α}{2}}$ , we deduce from Lemma 3 the existence of $C_{α} = C (N, A_{α}) > 0$ such that $\begin{matrix} (79) & (| U |^{2}, | Z |^{2}) + {‖ ℜ e (\overline{V} . Z) ‖}_{L^{2}}^{2} ⩽ C_{α} ‖ Z ‖_{L^{2}}^{2} . \end{matrix}$ Introducing $\begin{array}{l} (80) & Ψ (Z) = J (Z) + (1 + C_{α}) ‖ Z ‖_{L^{2}}^{2} \\ (81) & Φ (Z) = K (Z) + (1 + C_{α}) ((ℜ e [(\overline{U + V}) . Z], ℑ m (V . \overline{Z}))) \end{array}$ that, in accordance with (76), satisfy $\begin{matrix} (82) & \frac{1}{2} \frac{d}{d t} Ψ (Z) + γ Ψ (Z) = Φ (Z) \end{matrix}$ we obtain that $\begin{matrix} (83) & ‖ Z ‖_{H^{\frac{α}{2}}}^{2} ⩽ Ψ (Z) ⩽ (1 + C_{α}) ‖ Z ‖_{H^{\frac{α}{2}}}^{2} . \end{matrix}$ Moreover, $\begin{matrix} (84) & Φ (Z) ⩽ C (N) \int_{R} (| U |^{2} + | V |^{2}) | Z |^{2} d x + \int_{R} (| U_{t} | + | V_{t} |) | Z |^{2} d x . \end{matrix}$ Thanks to Theorem 13 and Lemma 3, either ${| U |^{2}, U \in A_{α}}$ or ${U_{t}, U \in A_{α}}$ is a compact subset of $L^{2}$ . Hence, in accordance with (84) and by the use of Lemma 21 it may be deduced the existence of $R_{0} = R_{0} (N) > 0$ such that $\begin{matrix} | Φ (z) | ⩽ \frac{γ}{2} ‖ Z ‖_{H^{\frac{α}{2}}}^{2} + C R_{0} ‖ Z ‖_{L^{2} ([- R_{0}, R_{0}])}^{2} . \end{matrix}$ This implies, thanks to (83), that $\begin{matrix} (85) & Φ (z) ⩽ \frac{γ}{2} Ψ (z) + C R_{0} ‖ Z ‖_{L^{2} ([- R_{0}, R_{0}])}^{2} . \end{matrix}$ Gathering (82) and (85) achieves the proof thanks again to (83) and the Gronwall lemma. □

Next, we handle the right hand side of (73). Lemma 24.

Let $R > 0$ . Then there exist C, $c_{1}$ and $c_{2} > 0$ depending only on γ, α, N and $A_{α}$ such that $\begin{matrix} (86) & {‖ Z (t) ‖}_{L^{2} ([- R, R])}^{2} ⩽ {‖ Z (0) ‖}_{L^{2} ([- 2 R, 2 R])}^{2} e^{c_{1} t} + C \frac{R_{0}}{R^{2 α}} e^{c_{2} t} {‖ Z (0) ‖}_{H^{\frac{α}{2}}}^{2} \end{matrix}$ where $R_{0}$ still denotes the fixed nonnegative reel given by Lemma 23 .

Proof of Lemma 24.

Let φ be a cut-off smooth function such that $\begin{matrix} φ : x ⟼ φ (x) = \{\begin{matrix} 1 & for | x | ⩽ 1, \\ 0 & for | x | ⩾ 2, \end{matrix} x \in R . \end{matrix}$ Now consider for a given $R > 0$ , $φ_{R} : x \mapsto φ (\frac{x}{R})$ , $x \in R$ .

Multiplying (71) by $φ_{R}$ followed by the scalar product of the resultant equation by $φ z_{k}$ , $1 ⩽ k ⩽ N$ and finally summing up lead, thanks to (74), to $\begin{matrix} (87) & \begin{matrix} \frac{1}{2} \frac{d}{d t} ‖ φ_{R} Z ‖_{L^{2}}^{2} + γ ‖ φ_{R} Z ‖_{L^{2}}^{2} \\ = (ℜ e ([\overline{U + V}] φ_{R} Z), ℑ m [V . (\overline{φ_{R} Z})]) + ((i φ_{R} D^{α} Z, φ_{R} Z)])) \end{matrix} \end{matrix}$ Since U and V remain uniformly bounded in $H^{\frac{α}{2}} ↪ L^{\infty}$ , we infer from (87), Lemma 5 and the estimate $‖ | ξ |^{α} F (φ_{R}) ‖_{L^{1} (R)} ≲ \frac{1}{R^{α}}$ that $\begin{array}{l} \frac{1}{2} \frac{d}{d t} ‖ φ_{R} Z ‖_{L^{2}}^{2} + γ ‖ φ_{R} Z ‖_{L^{2}}^{2} \\ (88) & ⩽ C_{1} ‖ φ_{R} Z ‖_{L^{2}}^{2} + {‖ φ_{R} D^{α} Z - D^{α} (φ_{R} Z) ‖}_{L^{2}} ‖ φ_{R} Z ‖_{L^{2}} \\ (89) & ≲_{N} ‖ φ_{R} Z ‖_{L^{2}}^{2} + \frac{1}{R^{α}} ‖ Z ‖_{L^{2}} ‖ φ_{R} Z ‖_{L^{2}} . \end{array}$ Hence, thanks to the Young inequality, we obtain that $\begin{matrix} (90) & \frac{d}{d t} ‖ φ_{R} Z ‖_{L^{2}}^{2} ⩽ C_{1} ‖ φ_{R} Z ‖_{L^{2}}^{2} + \frac{C_{2}}{R^{2 α}} ‖ Z ‖_{H^{\frac{α}{2}}}^{2} . \end{matrix}$ Independently, we infer from the scalar product of (71) by $z_{k}$ , $1 ⩽ k ⩽ N$ , then summing up the resultant equations that $\begin{matrix} (91) & \frac{1}{2} \frac{d}{d t} ‖ Z ‖_{L^{2}}^{2} + γ ‖ Z ‖_{L^{2}}^{2} = ((ℜ e ([\overline{U + V}] Z), ℑ m [V . (\overline{Z})])) ⩽ C ‖ Z ‖_{L^{2}}^{2} \end{matrix}$ This leads, thanks to Gronwall’s lemma and in accordance with Lemma 23, to $\begin{matrix} (92) & {‖ Z (t) ‖}_{H^{\frac{α}{2}}}^{2} ⩽ C_{1} {‖ Z (0) ‖}_{H^{\frac{α}{2}}}^{2} e^{- γ t} + C_{2} R_{0} e^{C t} {‖ Z (0) ‖}_{L^{2}}^{2} . \end{matrix}$ Gathering (90) and (92) achieves the proof thanks to the Gronwall lemma. □

By means of Lemma 23 and Lemma 24, we can deduce that $\begin{matrix} (93) & {‖ Z (t) ‖}_{H^{\frac{α}{2}}}^{2} ⩽ (C_{1} e^{- γ t} + C_{2} \frac{R_{0}}{R^{2 α}} e^{C t}) {‖ Z (0) ‖}_{H^{\frac{α}{2}}}^{2} + C_{3} R_{0} e^{C_{0} t} {‖ Z (0) ‖}_{L^{2} ([- 2 R, 2 R])}^{2} . \end{matrix}$ Recalling that $R_{0}$ is given by Lemma 23, choosing $t^{*} > 0$ depending on N such that $\begin{matrix} C_{1} e^{- γ t^{*}} ⩽ \frac{1}{4} \end{matrix}$ then $R^{*} > max (1, R_{0})$ large enough such that $C_{2} \frac{R_{0}}{{(R^{*})}^{2 α}} e^{C t^{*}} ⩽ \frac{1}{4}$ achieves the proof of the current proposition. □

Thanks to Proposition 22, the Theorem 19 is therefore proved by applying Proposition 20 with $V = S_{α} (t^{*})$ , $M = A_{α}$ and $n = ‖ \cdot ‖_{L^{2} ([- 2 R^{*}, 2 R^{*}])}$ .

Remark 25.

In the end of this paper it is important to highlight that in the last section, for the purpose of bounding the fractal dimension of the global attractor $A_{α}$ , we have used a new idea recently introduced by B. Alouini in [1] and which enables us to get rid of additional assumption on the forcing term that appears, for instance, in [12]. This argument can be adapted for other models of dissipative Schrödinger type equations and this will be the subject of forthcoming works.

Footnotes

Acknowledgements

The author would like to express his deep gratitude to the referee for his careful reading as well as for his helpful comments and suggestions leading to the improvement of this work.

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