For , we investigate the convergence of corresponding uniform attractors of the 3D non-autonomous Benjamin-Bona-Mahony equation with singularly oscillating force contrast with the averaged Benjamin-Bona-Mahony equation (corresponding to the limiting case ). Under suitable assumptions on the external force, we shall obtain the uniform boundedness and convergence of the related uniform attractors as .
MSC: 35B40, 35Q99, 80A22.
Keywords:Benjamin-Bona-Mahony equation; singularly oscillating forces; uniform attractors; translational bounded functions
Let be a fixed parameter, be a bounded domain with sufficiently smooth boundary ∂Ω. We investigate the long-time behavior for the non-autonomous 3D Benjamin-Bona-Mahony (BBM) equation with singularly oscillating forces:
Along with (1.1)-(1.3), we consider the averaged Benjamin-Bona-Mahony equation
as a straightforward consequence of (1.7), we have
The BBM equation is a well-known model for long waves in shallow water which was introduced by Benjamin, Bona, and Mahony (, 1972) 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 high wavenumber components. Further, while the KdV equation has an infinite number of integrals of motion, the BBM equation only has three. For more results on the wellposedness and infinite dimensional dynamical systems for BBM equations, we can refer to [2-7].
In this paper, firstly, we shall study the asymptotic behavior of the non-autonomous BBM equation depending on the small parameter ε, which reflects the rate of fast time oscillations in the term with amplitude of order , then we shall consider the boundedness and convergence of corresponding uniform attractors of (1.1)-(1.3) in contrast to (1.4)-(1.6).
Throughout this paper, () is the generic Lebesgue space, is the Sobolev space. We set , H, V, W is the closure of the set E in the topology of , , respectively. ‘⇀’ stands for the weak convergence of sequences.
Proof See, e.g., . □
Proof See, e.g., . □
For the non-autonomous general Benjamin-Bona-Mahony (BBM) equation,
Similar to , by the Galerkin method and a priori estimate, we easily derive the existence of a global weak solution and a uniform attractor which shall be stated in the following theorems.
3 Some lemmas
Proof As a similar argument in Section 2, we choose in Theorem 2.4, since and are translational bounded in , then for any fixed , is translational bounded in and we can easily deduce the existence of uniformly compact attractors . □
We can briefly describe the structure of the uniform attractor as follows: if the functions and are translational bounded, problem (1.1)-(1.3) generates the dynamical processes acting on V which is defined by , , where is the solution to (1.1)-(1.3). The processes have a uniformly (w.r.t. ) absorbing set
For every external force , equation (3.3) generates a class of processes on V, which shares similar properties to those of the processes , corresponding to the original equation (1.1) with the external force . Moreover, the map
Considering the linear equation
we get the following lemma.
Moreover, the following inequalities
Proof Firstly, using the Galerkin approximation method, we can deduce the existence of a global solution for (4.1), here we omit the details.
Then multiplying (4.1) by Y and AY respectively, we get
By the Gronwall inequality and Poincaré inequality, we can easily prove the lemma. □
Lemma 4.2Assume that the formula
Moreover, we also have
Proof Noting that
we can derive the following estimates from (4.8):
From Lemma 2.1, we have
Hence, from the Poincaré inequality, combining (4.12) and (4.4)-(4.5), we conclude that
such that from (4.13) and (4.14), we can derive
Hence, we conclude
The proof is finished. □
and assume that
From Lemma 4.2, we have the estimate
which satisfies the problem
Taking the scalar product of (4.28) with w, we obtain
Using the inequality
where λ is the first eigenvalue of −Δ.
Moreover, notice that
and inserting (4.29)-(4.30) into (4.28), we have
which implies that
The main result of the paper reads as follows.
satisfies the estimate
fulfills the Cauchy problem
Taking an inner product of equation (5.8) with q in H, we obtain
thus, it follows from (5.9) and (5.10) that
Moreover, we can derive the following formulas:
where R is a positive constant. The proof is finished. □
Lemma 5.3The inequality
holds, hereDandRare defined in Lemma 5.2.
Proof As the similar discussion in the proof of Lemma 5.2, replacing , and by , and , respectively, noting that (5.1) still holds for , and the family (), is -continuous, using (5.18) in place of (4.23), we can finally complete the proof of the lemma. □
From the equality
The authors declare that they have no competing interests.
The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
All authors give their thanks to the reviewer’s suggestions, XY was in part supported by the Innovational Scientists and Technicians Troop Construction Projects of Henan Province (No. 114200510011) and the Young Teacher Research Fund of Henan Normal University (qd12104).
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