Research Article | Open Access

# Weakly Compact Uniform Attractor for the Nonautonomous Long-Short Wave Equations

**Academic Editor:**Lucas JÃ³dar

#### Abstract

Solutions and weakly compact uniform attractor for the nonautonomous long-short wave equations with translation compact forces were studied in a bounded domain. We first established the existence and the uniqueness of the solution to the system by using Galerkin method and then obtained the uniform absorbing set and the weakly compact uniform attractor of the problem by applying techniques of constructing skew product flow in the extended phase space.

#### 1. Introduction

The long wave-short wave (LS) resonance equations arise in the study of the interaction of the surface waves with both gravity and capillary modes presence and also in the analysis of internal waves, as well as Rossby wave [1]. In the plasma physics they describe the resonance of the high-frequency electron plasma oscillation and the associated low-frequency ion density perturbation [2]. Benney [3] presents a general theory for the interaction between the short wave and the long wave.

Due to their rich physical and mathematical properties the long wave-short wave resonance equations have drawn much attention of many physicists and mathematicians. For one-dimensional propagation of waves, there are many studies on this interaction. Guo [4, 5] obtains the existence of global solution for long-short wave equations and generalized long-short wave equations, respectively. The existence of global attractor was studied in [6â€“8]. The orbital stability of solitary waves for this system has been studied in [9]. In [10], Guo investigated the asymptotic behavior of solutions for the long-short wave equations with zero order dissipation in . The approximation inertial manifolds for LS type equations have been studied in [11]. The well posedness of the Cauchy problem for the long wave-short wave resonance equations was studied in [8, 12â€“17].

In this paper, nonautonomous LS equations with translation compact forces were studied. The essential difference between nonautonomous systems and autonomous ones is that the former get much influenced by the time-depended external forces, which breaks semigroup property of the flow or semiflow created by autonomous systems. Also, attractors of nonautonomous systems are no longer invariable; they change with the changing of the initial time. This makes it impossible for us to consider nonautonomous systems completely in the same way of autonomous ones. Fortunately, Chepyzhov and Vishik [18, 19] developed techniques by which skills in the study of autonomous systems can be used in dealing with nonautonomous problems. Their central idea is that constructing skew product flow in extended phase space is obtained by where is a family of processes, is a translation semigroup, and the flow can be proved to be a semigroup under some preconditions, such as the translation identity and -continuity of , and more importantly, the compactness of the symbol space . By this means, we can get the uniform attractor by projecting the global attractor of to the phase space if the latter exists. We consider the following nonautonomous dissipative generalized long-short wave equations: with the initial conditions and the boundary value conditions where . . Nonautonomous terms and are time-depended external forces, which are supposed to be translation compact (cf. [18] or Assumption 1). Nonlinear terms and are given smooth and real, satisfying where , , , and constants , , and are given in for .

Our aim here is, firstly, to get the unique existence of solutions for problem (2)~(5) and then to derive the existence of weakly compact uniform attractor for it with the above-mentioned method. Here, and throughout this paper, uniform means uniform about symbols () in symbol space () unless there is special explanation. In fact, it is the same if we say uniform about the initial time, since the translation identity and the -continuity of hold in our case (cf. [20]).

Throughout this paper, we denote by the norm of with usual inner product , denote by the norm of for all , and denote by the norm of a usual Sobolev space for all . And we denote different constants by a same letter , and represents that the constant relies only on the parameters appearing in the brackets.

This paper is organized as follows. In Section 2, we recall some facts about the nonautonomous system. In Section 3, we provide the uniform a priori estimates in time. In Section 4, we obtain the unique existence of the solutions for problem (2)~(5) by Galerkin method. Section 5 contains the weakly compact uniform attractor for the nonautonomous system (2)~(5), and in the proof of Theorem 13, the -continuity of is proved.

#### 2. Preliminary Results

Let be a topological space, and let be a function. The set
is called the *hull* of in , denoted by . is *translation compact* if is compact in .

We denote all the translation compact functions in by , where is a Banach space. Apparently, implies that is translation bounded; that is,

Let be a Banach space, and let a family of two-parameter mappings act in . We also need the following definitions and lemma (cf. [19, 20]).

*Definition 1. *Let be a parameter set. , is said to be a family of processes in Banach space , if for each , from to satisfies

*Definition 2. *, a family of processes in Banach space , is called -continuous, if for all fixed and , , â€‰projection â€‰is continuous from to .

A set is said to be uniformly absorbing set for the family of processes , if for any and which denotes the set of all bounded subsets of , there exists , such that for all . A set is said to be uniformly attracting for the family of process , if for any fixed and every ,

*Definition 3. *A closed set is called the uniform attractor of the family of process , if it is uniformly attracting (attracting property), and it is contained in any closed uniformly attracting set of the family of process , (minimality property).

Lemma 4. *Let be a compact metric space, and suppose is a family of operators acting on , satisfying the following:**(i)
**(ii) translation identity:
**
where is arbitrarily a process in compact metric space . Moreover, if the family of processes is continuous, and it has a uniform compact attracting set, then the skew product flow corresponding to it has a global attractor on , and the projection of on , is the compact uniform attractor of .*

*Remark 5. *Assumption (13) holds if the system has a unique solution.

For brevity, we rewrite system (2)~(5) in the vector form by introducing and . We denote by the space of vector functions with norm Similarly, we denote by the space of with norm Then system (2)~(5) can be considered as where is the symbol of (16).

*Assumption 1. *Assume that the symbol comes from the symbol space defined by
where and the closure is taken in the sense of local quadratic mean convergence topology in the topological space . Moreover, we suppose that .

*Remark 6. *By the conception of translation compact/boundedness we remark that(i),â€‰â€‰;(ii), where is an translation operator.

#### 3. Uniform a Priori Estimates in Time

In this section, we derive uniform a priori estimates in time which enable us to show the existence of solutions and the uniform attractor. First we recall the following interpolation inequality (cf. [21]).

Lemma 7. *Let , , such that , , . Then one has
**
for , where , , and .*

Lemma 8. *If and , then for the solutions of problem (2)~(5), one has
**
where , .*

*Proof. *Taking the inner product of (2) with in we get that
Taking the imaginary part of (20), we obtain that
By Young inequality and Remark 6 we have
And then by Gronwall lemma we can complete the proof.

In the following, we denote that , which will not cause confusions.

Lemma 9. *Under assumptions of (6), (7) and Assumption 1, if , solutions of problem (2)~(5) satisfy
**
where and .*

*Proof. *Taking the inner product of (2) with in and taking the real part, we get that
By (3) we know that
which shows that
where is introduced by
Taking the inner product of (2) with in and taking the real part, we get that
Multiply (28) by , and add the resulting identity to (26) to get
That is,
In the following, we denote by any constants depending only on the data , and means it depends not only on but also on parameters in the brackets. , when is sufficiently large, by (6) and Lemmas 7 and 8, we have
By (6) we deduce that
And then
By (30)~(35) we get that
Similarly we can also deduce that
Taking the inner product of (3) with in , we see that
By (2) we get that
It comes from (38)~(40) that
Deal with the right hand side of inequality (41), by Lemmas 7 and 8,
So
Analogously, we can also deduce that
Set , and
Then by (36), (47) and (37), (48) we can, respectively, get
which shows that if we set , we can deduce that
where . By Gronwall lemma we see that
Similar to (33), (34), (45), and (42), for we have
And then
where when . Then by (52) we infer that
where and . By (49), (53), and (55) we infer that
Choose ; then we have
which concludes the proof by using Lemma 8.

Lemma 10. *Under assumptions of Lemma 9, if , solutions of problem (2)~(5) satisfy
**
where and .*

*Proof. *Taking the real part of the inner product of (2) with in , we have
By (2) and (3), we have
Since
we see that
Multiplying (2) by and taking the real part, we find that
therefore,
Now we deal with (64) to get (70). Due to equalities
we deduce that
We take care of terms in (66) as follows:
It follows from (66)~(67) that
And then
From (64) and (69) we have
By (59), (60), (62), and (70) we conclude that
where .

For later purpose, we let
Then from (71) we have
or
By Lemma 9 and Agmon inequality we have
In the following, we denote by . By Lemma 7 and (76) we estimate the size of to get
Taking the inner product of (3) with in , we see that
Since
by (78) we can deduce that
From (2) we know that
Taking the real part of the inner product to (81) with in , we have
Because of
it holds that