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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 of equation , converge to the global attractor of the above-mentioned equation with for any .
In this paper, we will consider the upper semicontinuity of pullback attractors for the following 3D Benjamin-Bona-Mahony equation: Here is a bounded domain with sufficiently smooth boundary ; is the velocity vector field; is the kinematic viscosity; is a nonlinear vector function; is a small nonnegative parameter; the external force is locally square integrable in time for , that is, for any , , where , , and and are the inner product and norm of , respectively.
The Benjamin-Bona-Mahony (BBM) equation is a well-known model in physical applications which incorporates dispersive effects for long waves in shallow water that was introduced by Benjamin et al.  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 [2–7]. 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  investigated the long-time behavior of 2D generalized BBM equation in , . Here , , and is an integer. The author proved the supremum norms of the solutions with small initial data decay to zero like as tends to infinity.
By energy equation and weak continuous method, Wang  and Wang and Yang  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 and , respectively. Moreover, Wang et al.  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  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  also obtained the approximate inertial manifolds to the global attractors for the generalized BBM equations.
Wang  considered the stochastic BBM equations on unbounded domains and concluded the existence of random attractor in 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.  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 by showing the solutions are point dissipative and asymptotic compact for and . Stanislavova  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 space which overcome difficulty coming from the precision of approximate inertial manifolds. Zhu and Mu  deduced the exponential decay estimates of solutions for time-delayed BBM equations.
J. Park and S. Park  studied the pullback attractors for the nonautonomous BBM equations in unbounded domains by weak continuous method and some priori estimates in . Qin et al.  derived the existence of pullback attractor of (10) in by weak continuous method. Zhao et al.  investigated the convergence of corresponding uniform attractors between averaging BBM and state BBM equations.
Moreover, Çelebi et al.  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.  studied the regularity of global attractor for a generalized BBM equation.
For the upper semicontinuity of corresponding attractors between autonomous and perturb nonautonomous systems, we can refer to Bao , Hale and Raugel , Carvalho et al. , Caraballo and Langa , Caraballo et al. , Fitzgibbon et al. , Kloeden , Miyamoto , Wang and Qin , Younsi , Wang , and Zhou .
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.
2. Pullback Attractors of Nonautonomous Dynamical Systems
In this section, we will consider the relationship between pullback attractors for the perturbed nonautonomous system with and global attractor for the unperturbed autonomous system with of the following equation: If the global attractor is unique, then the global attractor is the pullback attractor when .
Let be a Banach space with norm . The Hausdorff semidistance in between and is defined by where denotes the distance between two points and .
For an autonomous system, () is a -semigroup defined on . If the global attractor for exists, then it has the following properties: (1) is an invariant, compact set; (2) attracts every bounded sets in , that is, 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.
Definition 1. A family of compact sets is said to be a pullback attractor for the continuous process if it satisfies the following:(i) is invariant for all .(ii) is pullback attracting, that is, for all bounded subsets .
Definition 2. The family of subsets is said to be pullback absorbing for the process , if for every and all bounded subsets , there exists a time , such that
Definition 3. Let be a family of subsets in . A process is said to be pullback -asymptotically compact in if for all , any sequences and ; the sequence is precompact in .
Theorem 4. Let the family of sets be pullback absorbing set for the process and is pullback -asymptotically compact in . Then, the family that is defined by is a pullback attractor for in for the process , where
In the following, we will characterize the pullback -asymptotic compactness in terms of the noncompact measure.
Definition 5. Let , be a family of sets in . A process is said to be pullback - contracting, if for any , , there exists a time , such that Here is the Kuratowski noncompact measure defined as
Lemma 6. Let , be two families of sets in and satisfy that for any , there exists a time , such that Then is pullback -asymptotically compact, if it is pullback - contracting.
Proof. See, for example, Wang and Qin .
Theorem 7. Assume that the assumptions in Lemma 6 hold. If the process is pullback - contracting and the family of sets is pullback absorbing for , then the process possesses a pullback attractor.
Proof. See, for example, Wang and Qin .
Theorem 8. Let be a family of sets in . Suppose satisfies(i)for any , where satisfies for each ;(ii)for any and , is bounded and is precompact in for any .Then the process is pullback - contracting in .
Proof. See, for example, Wang and Qin .
We now perturb the nonautonomous term with a small parameter ; thus we obtain a nonautonomous dynamical system driven by the process .
For each , , and , we have uniformly on bounded sets of .
Theorem 9 (Caraballo et al. [28, 29]). Assume that holds, and for any , there exist pullback attractors for all . If there exists a compact set , such that Then and have the upper semicontinuity, that is,
In order to apply Theorem 9 to obtain the upper semicontinuity of pullback attractors and global attractor , we now present a technique to verify for the process generated by the nonautonomous dissipative system.
Lemma 10. Assume that the family is pullback absorbing for , and for each , is a family of compact sets in . Suppose satisfies(i)for any and any , where satisfies for each ;(ii)for any and any , is bounded, and for any , there exists a time , which is independent of , such that and there exists a compact set , such that Then for each , there exists a pullback attractor and holds.
Proof. See, for example, Wang and Qin .
3. 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 with domain 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 for any and the continuous embedding for all , .
Assume that , the external force . Also we assume that there exist constants , , and , such that which implies that Moreover, we assume that From (31), we can easily derive that the term is locally square integrable in time; that is, and satisfies for and any .
For the nonlinear vector function , we denote where Assume that () are smooth functions satisfying for all , where , , and are positive constants.
At last, we will state the main result and the proof of this paper as the following.
Theorem 11. Assume that (30)–(36) hold, and , then the pullback attractors for (29) (which is equivalent to (1)) with and the global attractor for (29) with satisfy The Hausdorff semidistance is defined on the Banach space .
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 and pullback attractors for nonautonomous system (1) with 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.
Lemma 13. Assume that (30)–(36) hold, and , then problem (29) possesses a unique global solution () satisfying Moreover, the process generated by the global solutions possess pullback attractors for all in .
Proof. See, for example, .
Now we decompose the solution of (29) with initial data as where solve the following problems: respectively.
Proof. We choose , and denote by the class of families such that for some , where denotes the closed ball in centered at zero with radius .
Let , , and be fixed, and denote Since , then for all , we derive that that is, which gives for all .
Let be given, choosing appropriate parameter , we easily get for all , .
Setting , then we denote the nonnegative number given for each by and consider the family of closed balls in defined by It is straightforward to check that and hence is the -pullback absorbing for the process .
Setting then we can check that family is pullback absorbing in easily. Moreover,
Lemma 15. Let , be given as above. For any , the solution of (41) satisfies for all and .
Proof. Multiplying equation in (41) with and integrating over , we derive
Here we use the property of operator and as
where is the outer unit normal vector.
Using Poincaré’s inequality, it follows where we set .
Integrating (56) from to , we get for all , which completes our proof.
Proof. Taking the inner product of equation in (42) with in , we derive
By Poincaré’s inequality, Lemma 15, (51), and (34)–(36), we obtain
Hence according to (59)–(60) and (31), we have
where the constant depends on , , and the first eigenvalue of the operator .
Integrating (61) from to , we conclude that for all . This completes the proof of desiring lemma.
Lemma 17. For any , any , if varies in bounded sets, then the solution of problem (1) converges to the solution of the unperturbed problem (1) with uniformly in as , which means where is a bounded subset in .
Proof. Denote then we can verify that satisfies Multiplying (65) by, using (34)–(36) and noting the boundary value condition (66), we have Using (34)–(36) and the Sobolev compact embedding theorem , we get Hence that is, Using Lemmas 13, 14, 15, 16, and (31), noting that , we know Applying the Gronwall inequality to (71) and noting that , using Lemmas 14, 15, and 16, we conclude as , which implies (63).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Xinguang Yang was in part supported by the NSFC (no. 11326154), the Innovational Scientists and Technicians Troop Construction Projects of Henan Province (no. 114200510011), the Key Scientific and Technological Project of Henan Province (no. 142102210448), and the Young Teacher Research Fund of Henan Normal University (qd12104). Juntao Li was in part supported by the NSFC (no. 61203293), the Foundation of Henan Educational Committee (no. 134120524), the Key Scientific and Technological Project of Henan Province (no. 122102210131), the College Young Teachers Program of Henan Province (no. 2012GGJS-063), and the Program for Science and Technology Innovation Talents in University of Henan Province (no. 13HASTIT040).
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