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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 830715, 17 pages
Uniform Attractor and Approximate Inertial Manifolds for Nonautonomous Long-Short Wave Equations
1School of Mathematics and Statistics Science, Ludong University, Yantai 264025, China
2School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
Received 28 August 2013; Revised 17 October 2013; Accepted 18 October 2013
Academic Editor: Carlo Bianca
Copyright © 2013 Hongyong Cui and Jie Xin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Nonautonomous long-short wave equations with quasiperiodic forces are studied. We prove the existence of the uniform attractor for the system by means of energy method, which is widely used to deal with problems who have no continuity (with respect to the initial data) property, as well as to those which Sobolev compact imbedding cannot be applied. Afterwards, we construct an approximate inertial manifold by means of extending phase space method and we estimated the size of the corresponding attracting neighborhood for this manifold.
In this paper, we investigate the long time behavior of solutions for the following nonautonomous generalized dissipative LS equations with quasiperiodic forces: with initial conditions and space-periodic boundary conditions as where and and are positive constants.
The long-wave short-wave (LS) resonance equations arise in many kinds of physical models (see [1–4]). Due to their rich physical and mathematical properties, the LS equations have drawn much attention. The autonomous situations, including the existence of solutions, the solitary waves and their stability, and the long time behaviors of the solutions, have been deeply researched (see [5–14]).
Recently, the nonautonomous case of LS equations with translation compact forces was studied in . Because of the nonlinear resonance of the equations, it is difficult to prove the continuity of the process generated by (1)–(4). Thus, it is hard to construct the uniform attractor directly by constructing a compact uniform absorbing set even if the forces are translation compact, and in  only the weakly compact uniform attractor for the system is obtained.
In this paper, we firstly investigate the compact uniform attractor for systems (1)–(4) by employing the energy equations and the energy method presented by Ball (see [16, 17]). The energy method can be concisely understood as the following two steps (e.g., in autonomous cases): construct a weakly compact attractor and prove the strong compactness of the weak attractor, that is, verify that the weak attractor is actually the strong one. To accomplish Step 1, one can construct a bounded (weakly compact) absorbing set and the weak continuity of the system. Step 2 is usually deduced by applying proper energy inequalities together with Lemmas 11 and 12. Obviously, this method is good at solving problems which are not continuous and those that lack Sobolev embeddings (such as systems defined in unbounded domains).
Besides, approximate inertial manifolds (AIM) for the system is studied afterwards. This manifold is a finite-dimensional smooth surface in a phase space, whose small vicinity attracts all the trajectories at a much higher speed than global attractors. To investigate AIM, by employing the extending phase space method we transfer the nonautonomous system to an autonomous one , and we get the AIM for by constructing the AIM for .
Main Theorem. Assume that (i), are quasiperiodic forces satisfying Assumption 1;(ii)generalized and are quasilinear functions satisfying (9) and (10).Then systems (1)–(4) generate a family of processes in . Moreover, the family of processes admits a compact uniform attractor and an AIM in .
We would like to remark that the existence of the compact uniform attractor for the system does not depend heavily on the quasiperiodicity of the forces. It still holds when the forces are just translation bounded (see Remark 14), that is, it strengthens the result in .
This paper is organized as follows. In Section 2, we show the LS equations in details and we deeply introduce the quasiperiodicity conditions. In Section 3, we get the uniform a prior estimates for the solutions. In Section 4, we study the unique existence of the solution. In Section 5 the existence of the uniform attractor for (5)–(8) is obtained by applying weak convergence method. In Section 6, AIM for (5)–(8) is constructed by extending and splitting the phase space and making use of projection operators.
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 any Banach space . Besides, notations “”, “”, and “” denote weakly converges to, weakly star converges to and strongly converges to, respectively. And we denote different constants by the same letter , and represents that the constant relies only on the parameters that appear in the brackets.
We show the nonautonomous dissipative generalized long-short wave equations with quasiperiodic forces in details as follows: with initial value conditions and periodic boundary value conditions where is an unknown complex valued vector, is an unknown real valued function, are positive constants and nonautonomous terms , and are time-depended external forces satisfying quasiperiodicity conditions (see Assumption 1); non-linear terms and are given real-valued functions, satisfying where are given positive constants for .
Let be a topological space, and is a function. The set is called the hull of in , denoted by . is translation compact (resp., translation bounded) if is compact (resp., bounded) in .
We denote all the translation compact functions in by ; is a Banach space. Apparently, implies that is translation bounded as follows:
Assumption 1. For , we suppose and it satisfies quasiperiodicity conditions; namely, and for all , where , , and are rational and independent; is differentiable to each position and
If satisfy Assumption 1, we can consider the symbol space as
Since there is a continuous mapping , from  we know that the symbol space can be replaced by . And, for each , the translation operator acting on can be defined as Therefore, is translation compact.
Proposition 2. Under Assumption 1, we can deduce the following properties:(i) is translation bounded in ; that is,where , , (ii)for all , ,(iii), which can be seen directly from (15) and the fact thatMoreover, is translation bounded in . Similarly to (18), by the continuity of , we can find a constant , which is independent of , such that
3. Uniform a Prior Estimates of the Solutions
In this section, we derive uniform a priori estimates of the solutions both in time and in symbols which come from the symbol space . First we recall the following interpolation inequality.
Lemma 3. Let , , such that , , . Then we have for , where and , .
Proof. Taking the inner product of (5) with in we get that Taking the imaginary part of (26), we obtain that By Young inequality and Proposition 2 we have Then by Gronwall lemma we can complete the proof.
In the following, we denote by , which will not cause no confusions.
Proof. Taking the inner product of (5) with in and taking the real part, we have
By (6) we know that
which shows that
where is introduced by
Taking the inner product of (5) with in and taking the real part, we get that
Multiply (34) by and add the resulting identity to (32) to get
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. For all , when is sufficiently large, by (9), Lemmas 3 and 4 we have
By (9) we deduce that
By (36)–(41) we deduce that
Similarly we can also deduce that Taking the inner product of (6) with in , we have By (5) we get that It comes from (44)–(46) that Dealing with the right hand side of inequality (47), by Lemmas 3 and 4, we get
Analogously, we can also deduce that
Set , and Then by (42) and (53) and (43) and (54) we can respectively deduce that which shows that if we set , we can deduce that where . By Gronwall lemma we have that Similarly to (39), (40), (51), and (48), for we have And then where when . Then by (58) we infer that where . By (55), (59) and (61) we infer that Choosing , we have which concludes the proof by using Lemma 4.
Proof. Taking the real part of the inner product of (5) with in , we have
By (5) and (6), we have
we know that
Multiply (5) by and take the real part, we find that
Now we deal with (70) to get (78). Due to equalities
we deduce that
We take care of terms in (72) as follows
It follows from (72)–(75) that
From (70) and (77) we have
By (65), (66), (68), and (78) we conclude that where .
For later purpose, we let
Then from (79) we have or By Lemma 5 and Agmon inequality we have In the following, we denote . By Lemma 3 and (84) we estimate the size of to get Taking the inner product of (6) with in , we see that Since by (86) we can deduce that From (5) we know that Taking the real part of the inner product to (89) with in , we have Because of it holds that By (92) and (88), we find that That is, For later use, we let Then identity (94) as being equivalent to Similarly to (85) we estimate each term in (94) and then we get Let , and By (85) and (98) we deduce that which has the same form with (57) in the proof of Lemma 5. Similarly to the study of (57) we can derive that where and when . By (80) we deduce that and then by (99), (101), and (102), we deduce that which concludes the proof by Lemma 5.
To study the AIM for the system, we construct the following higher order estimate.