ATTRACTOR FOR THE NON-AUTONOMOUS LONG WAVE-SHORT WAVE RESONANCE INTERACTION EQUATION WITH DAMPING∗

Abstract In this paper, the long wave-short wave resonance interaction equation with a nonlinear term in bounded domain was studied. When β ≥ 3 2 , we obtained the existence and uniqueness of the weak solution of system (1.1)(1.4) by Galërkin’s method, and further proved the existence of the compact uniform attractor for damped driven by the non-autonomous long wave-short wave resonance interaction equation.


Introduction
The long wave-short wave (LS) resonance equation appeared in a recent study of the interaction of surface waves with gravity and capillary modes, as well as the analysis of internal waves and Rosby waves [12]. In the plasma physics, the long wave-short wave resonance equation explains the high frequency electron plasma resonance and associated low frequency ion density perturbation [22]. A general theory on the interaction between short wave and electromagnetic wave was presented [6].
The long wave-short wave resonance equation has attracted extensive attention from many physicists and mathematicians, due to its rich physical and mathematical properties. For one-dimensional wave propagation, there were many studies on this interaction. Guo [7,13] verified the existence of global solutions for the long waveshort wave equation and the generalized long wave-short wave equation, respectively. In [14], Guo studied the orbital stability of the solitary waves of the long waveshort wave resonance equation. In [15], Guo studied the asymptotic behavior of the solutions of long wave-short wave equations with zero order dissipation in H 2 per × H 1 per . The approximate inertial manifolds of LS equation was in [16]. In [4,5,19,24,25,29], the well-posedness of Cauchy problem for long wave-short wave resonance equation was studied.
In recent years, the study of attractors in dynamics has attracted extensive attention [20]. We can get its global attractors in [10,19,28,29] for autonomous systems. But unlike autonomous systems, non-autonomous systems have special temporal correlation. So we have obtained the attractors of non-autonomous system, such as pullback attractors (see [2,18,23]) and uniform attractors (see [1,27]). In this paper, we prove the uniform attractor of the long wave-short wave resonance interaction equation with damping (1.1)-(1.3) by defining the relevant uniform attractor in [9].
In this paper, we consider the following long wave-short wave resonance interaction equation with damping: Under the following initial condition Non-autonomous terms f and g are time-dependent external forces. When β = 2 or 3, the well-posedness of the solution of the long wave-short wave resonance interaction equation has been studied by many people. When β = 3, Gao first studied the well-posedness of the solution of the non-autonomous long waveshort wave resonance interaction equation and obtained the existence of attractors in [11], and then Cui proved the existence of uniform attractors in [9]. In [21], the author has also proved the well-posedness of the solution for the long wave-short wave resonance interaction equation and studied the existence of global attractors for the equation when β = 2. In this paper, our aim here is, firstly, to get the well-posedness of solutions for problem (1.1)-(1.4) for β ≥ 3 2 and then to derive the existence of the compact uniform attractor.
The rest of this paper is organized as follows. In section 2, we introduce symbols and preliminary results, and recall some facts about the uniform attractor. In section 3, a priori estimate of the solution is obtained. In section 4, existence and uniqueness of the weak solution of the system (1.1)-(1.4) are proved. In section 5, the existence of the strong compact uniform attractor for (1.1)-(1.4) is obtained.

Preliminary
In this section, we introduce some notations and preliminary results used in this paper. Firstly, We have added the subscript "per" to the usual Sobolev space to represent the Sobolev space over the periodic region. Now, we denote some notations: where Ω = [−D, D] ⊂ R, D > 0. Especially, when p = 2, the first formula becomes the space L 2 per , and (·, ·), || · || denote the inner product and norm of L 2 per (Ω), which are defined as follows: and u denotes the conjugate complex quantity of u. Similarly, we denote the norm of L p per (Ω) for all by ∥ · ∥ p . And ∥ · ∥ H p denotes the norm of H p per (Ω), which is defined by ∥u∥ 2 For simplicity and convenience, the letter C represents a constant, which may vary in different lines. C(·, ·) represents the constant C represented by the parameters appearing in parentheses.
Next, we introduce Sobolev embedding theorem for one-dimensional domain used in the following section.
the constant C depending only on p and Ω.
is defined the hull of f in X, denoted by H(f ).
(i) f is said to be translation bounded in

Definition 2.2.
Suppose is a parameter set. If for each σ ∈ , the mapping U σ (t, τ ) : X → X satisfies where {U σ (t, τ ), t ≥ τ, τ ∈ R}, σ ∈ is said to be a family of processes in X.
Let B 0 , B ∈ B(E) be the set of bounded subsets of E. If for any τ ∈ R, there Then B 0 is said to be uniformly absorbing set for the family of processes {U σ∈ ∑ (t, τ )}, A set Y ⊂ E is called uniformly attracting for the family of process {U σ (t, τ )}, σ ∈ if, for each fixed τ ∈ R and every B ∈ B(E), it satisfies that

Definition 2.3.
A closed set A ∑ ⊂ X is called the uniform attractor of the family of processes {U σ (t, τ )} σ∈ ∑ if it is uniformly attracting (attracting property) and it is contained in any closed uniformly attracting set A ′ of the family of processes Definition 2.4. {U σ (t, τ )} σ∈ ∑ , a family of processes in X, is said to be (X× , X)continuous, if, for any fixed T and τ , (ii) translation identity: Note that if the family of processes {U σ∈ ∑ (T, τ )} is (E × , E)-continuous and it has a uniform compact attracting set, then the skew product flow corresponding to it has a global attractor A on E × . And the projection of A on , A ∑ , is the compact uniform attractor of {U σ∈ ∑ (T, τ )}.
Definition 2.6. Suppose that the symbol Y (x, t) belongs to the symbol space , defined by and the closure is taken in the sense of local quadratic mean convergence topology in the topological space Due to the conception of translation compact/boundedness, we remark that

A priori estimates
In order to obtain the existence and uniqueness of weak solutions in the next section, in the following, we establish some uniform a priori estimate of the solutions both in time t and in symbol space (Y ∈ Σ).
Proof. Taking the inner product of (1.1) with u and taking the imaginary part, we get By using Young's inequality, we get And then by Gronwall's inequality, we can complete the proof.
Proof. Taking the inner product of (1.1) with u and taking the real part, we get that Taking the inner product of (1.1) with u t and taking the real part, we get that (3.6) can be written as follows Combining (3.5) and (3.10), we get Note that, by (1.1), (3.12) So (3.11) can be written that Taking the inner product of (1.2) with v, we get that Note that, by (1.1) Without loss of generality, we may assume that 2γ ≥ α. Applying (3.13)-(3.15), we get and then (3.16) can be written in the following form We choose some suitable ε i (i = 1, 2, 3) and use Young's inequality to obtain that where C = C(ε 1 , ε 2 , ε 3 , ∥u∥, ∥f ∥, ∥g∥). So (3.17) can be written as follows By using the Gronwall inequality as follows (3.20) By using Hölder's inequality, we get Now we estimate the value of φ(u, v), where

24)
where Proof. Taking the inner product of (1.1) with u xx and taking the real part, we get that Taking the inner product of (1.1) with u xxt and taking the real part, we get that

From (1.2) we can get
Re Inserting the above two equalities into (3.31) we obtain We differentiate (1.2) with respect to x and take the inner product with v x to get

by (1.1), we get
Re through integrating by parts yields, we have and then, we obtain that

Therefore, we have
Re Inserting the above equalities into (3.33) we obtain By some basic calculation from (3.32) and (3.34), we have Next, Gagliardo-Nirenberg inequality, Young's inequality and Holder's inequality are used to estimate ϕ 1 (u, v).
From the above estimates, we can get So (3.35) can be written as follows: (3.40) where C = C(α, β, γ, ∥u∥ H 1 , ∥v∥ H , ∥f ∥ H 1 , ∥g∥ H 1 ). So using Gronwall inequality, we can get that Since 2β − 3 ≥ 0 for β ≥ 3 2 , using the formula of integration by parts, we have Re And then, we have Re Ω uvu xx dx (3.43) By using the above estimates and choosing the appropriate ε i (i = 1, 2, 3), we get And from the discussion of (3.41) and (3.44), we get where

Existence and Uniqueness of the Solution
In this section, we will prove that the system (1.1)-(1.4) has a unique global weak solution. Since a prior estimate of the solution has been established in section 3, the existence of the solution can be easily obtained by Galërkin's method(see [16,17,26,29]). In this section, we will show the unique existence theorem and give a simple proof.  1)-(1.4) has a unique global weak solution W (x, t) ∈ L ∞ (τ, t; E 0 ), ∀T > τ.
Proof. We will prove the existence and uniqueness of a global solution respectively.
Step 1. The existence of solution Using Galërkin's method, we have the following approximate solution to approach the solution of (1.1)-(1.4): where {η j } l j=1 is a orthogonal basis of H(Ω), and (u l , v l ) satisfies We get that (4.2) is an initial-boundary value problem of ordinary differential equations. According to the standard existence of ordinary differential equations and the priori estimates in Section 3, we obtain the unique solution for (4.2). Similar to [13,26], we get where * ⇀ means weak star convergence.
Step 2. The uniqueness of solution.

Uniform Absorbing Set and Uniform Attractor
In this section, we will prove the existence of the strong compact uniform attractor of problem (1.1)-(1.4) applying Ball et al.'s idea (see [3,26]). Firstly, we construct a bounded uniformly absorbing set. Next, we prove the the existence of weakly compact uniform attractor of the system. Lastly, we derive that the weak uniform attractor is actually the strong one. In this section, "⇀" represents weak convergence, " * ⇀" represents weak star convergence, and {U σ∈ ∑ (t, τ )} represents a family of processes in E 0 satisfying definition 2.2. Proof. We prove this theorem by three steps.
By Definition 2.6, we know that ∥Y ∥ L 2 Then we get the set B 0 is a bounded uniformly absorbing set of {U σ∈ ∑ (t, τ )}.
Step 2. We show the weak compact uniform attractor A ∑ in E 0 .
Step 3. We show the weakly compact uniform attractor A ∑ is actually the strong one. From the proof of Lemma 3.3 , we know each solution for problem (1.1)-(1.4) satisfies By the uniform boundedness and the compactness embedding, we have that F , G, and F 1 , G 1 are all weakly continuous in E 0 × . From step 2, we know that the point (ω, m) ∈ A if and only if there exist two sequences {ω 0 k , m 0 k } k∈N and {t k } k∈N such that for all σ(t) ∈ , it uniformly satisfies that where t k → ∞ as k → ∞. If the weak convergence implies strong one, we obtain A ∑ is the strong compact attractor. For each fixed h > τ , because of t k → ∞, we consider it as h < t k , k ∈ N + . By Lemma 3.3 and Theorem 4.1, U σ (t k − h, τ )(ω 0 k l , m 0 k l ) is bounded in E 0 . Then there exists a subsequence U σ (t k l − h, τ )(ω 0 k l , m 0 k l ) of U σ (t k − h, τ )(ω 0 k l , m 0 k l ) and a point (n, p) ∈ E 0 , such that U σ (t k l − h, τ )(ω 0 k l , m 0 k l ) ⇀ (n, p) in E 0 . (5.32)