Upper Semicontinuity of Pullback Attractors for the 3D Nonautonomous Benjamin-Bona-Mahony Equations

We will study the upper semicontinuity of pullback attractors for the 3D nonautonomouss Benjamin-Bona-Mahony equations with external force perturbation terms. Under some regular assumptions, we can prove the pullback attractors 𝒜 ε (t) of equation ut-Δut-νΔu+∇·F→(u)=ɛg(x,t) , x ∈ Ω, converge to the global attractor 𝒜 of the above-mentioned equation with ε = 0 for any t ∈ ℝ.

The Benjamin-Bona-Mahony (BBM) equation is a wellknown model in physical applications which incorporates dispersive effects for long waves in shallow water that was introduced by Benjamin et al. [1] as an improvement of the Korteweg-de Vries equation (KdV equation) for modeling long waves of small amplitude in two dimensions. Contrasting with the KdV equation, the BBM equation is unstable in its high wave number components. Further, while the KdV equation has an infinite number of integrals of motion, the BBM equation only has three. Both KdV and BBM equations cover cases of surface waves of long wavelength in liquids, acoustic-gravity waves in compressible fluid, hydromagnetic waves in cold plasma, and acoustic waves in harmonic crystals.
For the well-posedness of global solutions for BBM equation, we can refer to [2][3][4][5][6][7]. For the long-time behavior, such as the existence of global attractor and its structure and the dimension of the attractors, we will discuss the known results in details.
Biler [8] investigated the long-time behavior of 2D generalized BBM equation in R 2 , ∈ R. Here ̸ = 0, ∈ R 2 , and ≥ 3 is an integer. The author proved the supremum norms of the solutions with small initial data decay to zero like −2/3 as tends to infinity.

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The Scientific World Journal By energy equation and weak continuous method, Wang [9] and Wang and Yang [10] investigated the finite-dimensional behavior of solutions and derived the global weak attractor and the strong attractors for BBM equation: with period boundary value condition in 2 per (Ω) and 1 per (Ω), respectively. Moreover, Wang et al. [11] got the existence of global attractor for the above BBM equation defined in a three-dimensional channel; the asymptotic compactness of the solution operator is obtained by the uniform estimates on the tails of solutions.
By the decomposition of the semigroup, Wang [12] studied the regularity of attractors for the BBM equation He proved that the global attractor is smooth if the forcing term is smooth. In addition, Wang [13] also obtained the approximate inertial manifolds to the global attractors for the generalized BBM equations. Wang [14] considered the stochastic BBM equations on unbounded domains and concluded the existence of random attractor in 1 0 under certain assumptions, here is the two-sided real-valued Wiener process on a probability space. He also proved the random attractor is invariant and attracts every pulled-back tempered random set under the forward flow. The asymptotic compactness of the random dynamical system is established by a tail-estimates method, which shows that the solutions are uniformly asymptotically small when space and time variables approach infinity.
Stanislavova et al. [15] first provided a sufficient condition to verify the asymptotic compactness of an evolution equation defined in an unbounded domain, which involves the Littlewood-Paley projection operators, then they proved the existence of an attractor for the Benjamin-Bona-Mahony equation in the phase space 1 (R 3 ) by showing the solutions are point dissipative and asymptotic compact for ∈ 2 (R 3 ) and ( ) = + (1/2) 2 . Stanislavova [16] investigated the existence of global attractors of (8) in two dimension. By the method of orthogonal decomposition, Zhu [17,18] obtained the asymptotic attractor, global attractor, and its Hausdorff dimension of the damped BBM equations with periodic boundary conditions in homogeneous periodic spacė1 per (Ω) which overcome difficulty coming from the precision of approximate inertial manifolds. Zhu and Mu [19] deduced the exponential decay estimates of solutions for time-delayed BBM equations. J. Park and S. Park [20] studied the pullback attractors for the nonautonomous BBM equations in unbounded domains by weak continuous method and some priori estimates in 1 0 (Ω). Qin et al. [21] derived the existence of pullback attractor of (10) in 2 0 (Ω) by weak continuous method. Zhao et al. [22] investigated the convergence of corresponding uniform attractors between averaging BBM and state BBM equations.
Moreover, Ç elebi et al. [23] deduced the existence of attractors with a finite fractal dimension and the existence of the exponential attractor for the corresponding asymptotically compact semigroup for the periodic initial-boundary value problem of a generalized BBM equation. Chueshov et al. [24] studied the regularity of global attractor for a generalized BBM equation.
To our knowledge, there are less results on the upper semicontinuity of pullback attractors for the 3D nonautonomous BBM equations with the nonautonomous perturbation; we will pay attention to this issue in the sequel. This paper is organized as following. In Section 2, we will recall some fundamental theory of pullback attractors for nonautonomous dynamical systems and give a method to verify the upper semicontinuity of pullback attractors. In Section 3, the upper semicontinuity of pullback attractors for the problems (1)-(3) will be proved.

Pullback Attractors of Nonautonomous Dynamical Systems
In this section, we will consider the relationship between pullback attractors A = { ( )} ∈R for the perturbed nonautonomous system with > 0 and global attractor A for the unperturbed autonomous system with = 0 of the following equation: If the global attractor is unique, then the global attractor is the pullback attractor when = 0. Let be a Banach space with norm ‖ ⋅ ‖ . The Hausdorff semidistance dist ( 1 , 2 ) in between 1 ⊆ and 2 ⊆ is defined by (12) where ( , ) denotes the distance between two points and .
The Scientific World Journal 3 For an autonomous system, ( ) : → ( ∈ R) is a 0 -semigroup defined on . If the global attractor A for ( ) exists, then it has the following properties: (1) A is an invariant, compact set; (2) A attracts every bounded sets in , that is, lim → +∞ dist( ( ) , A) = 0 for all bounded subsets ⊂ . For a nonautonomous system, the two-parameter mapping class { ( , )} ≥ is said to be a process in if Moreover, throughout the paper, we always assume that the process (⋅, ⋅) is continuous in . Now we will recall some definitions and framework on the existence theory of pullback attractors.
In the following, we will characterize the pullback Basymptotic compactness in terms of the noncompact measure.
We now perturb the nonautonomous term with a small parameter ∈ (0, 0 ]; thus we obtain a nonautonomous dynamical system driven by the process (⋅, ⋅).
For each ∈ R, ∈ R, and ∈ , we have uniformly on bounded sets of .
Theorem 9 (Caraballo et al. [28,29] Then A and A have the upper semicontinuity, that is, 4 The Scientific World Journal In order to apply Theorem 9 to obtain the upper semicontinuity of pullback attractors A and global attractor A, we now present a technique to verify ( 2 ) for the process generated by the nonautonomous dissipative system.

Upper Semicontinuity of Pullback Attractors
In this section, firstly, we recall some notations about the functional spaces which will be used later to discuss the regularity of pullback attracting set. The operator is denoted by = −Δ with domain ( ) = 2 (Ω) ⋂ 1 0 (Ω) and is the first eigenvalue of ; we consider the family of Hilbert spaces generated by the Laplacian operator with the Dirichlet boundary value conditions equipped with the standard inner product and norm respectively, then we have ( /2 ) → ( /2 ) for any > and the continuous embedding for all ∈ [0, 3/2), Then, applying the Helmholtz-Leray projector P to the systems (1)-(3), we obtain the following problem which is equivalent to the original problems (1) Assume that ∈ 1 0 (Ω), the external force ∈ 2 loc (R, ). Also we assume that there exist constants > 0, 0 ≤ < /2, and = 2]/((2/ ) + 2), such that which implies that Moreover, we assume that where Assume that ( = 1, 2, 3) are smooth functions satisfying for all ∈ R, where 1 , 2 , and are positive constants. At last, we will state the main result and the proof of this paper as the following.
In order to apply Theorem 9 and Lemma 10 to prove Theorem 11, we will introduce the existence of global attractor for autonomous system (1) with = 0 and pullback attractors for nonautonomous system (1) with > 0 in the following lemmas. Proof. Using similar technique as in [9-11, 17, 18], we only need to consider the Dirichlet boundary value condition instead of the periodic boundary value condition in these papers which investigated the existence of global attractors. This means that we can obtain our lemma easily, here we omit the details. respectively.

Lemma 14.
Suppose that (34)- (36) hold. For any bounded set ⊂ and ∈ R, there exists a time ( , ) > 0, such that Let ∈ R, ∈ R, and ∈ be fixed, and denote Since ∈ (( , ); ), then for all ∈ , we derive that that is, The Scientific World Journal which gives for all ∈ R. Let̂∈ D be given, choosing appropriate parameter , we easily get for all ∈ ( ), ≥ .
denote ( ) the nonnegative number given for each ∈ R by and consider the familŷof closed balls in defined by It is straightforward to check that̂∈ D and hencêis the D -pullback absorbing for the process { ( , , )}.
Proof. Multiplying equation in (41) with V and integrating over Ω, we derive Here we use the property of operator (⋅) and F (0) = 0 as where → is the outer unit normal vector.