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Discrete Dynamics in Nature and Society
Volume 2014, Article ID 712183, 14 pages
http://dx.doi.org/10.1155/2014/712183
Research Article

Uniform Attractor for the Fractional Nonautonomous Long-Short Wave Equations

School of Mathematics and Statistics Science, Ludong University, Yantai 264025, China

Received 14 March 2014; Accepted 3 September 2014; Published 20 October 2014

Academic Editor: Prasanta K. Panigrahi

Copyright © 2014 Huanmin Ge 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.

Abstract

We firstly proved the existence and the uniqueness of the solution for the -periodic fractional nonautonomous long-short wave equations with translation compact force by using Galerkin method and then obtained the compact uniform attractor of the system.

1. Introduction

In this paper, we consider the following fractional nonautonomous long-short wave equations with translation compact forces: with initial and periodic boundary conditions: where is an unknown and complex-valued function, is an unknown real valued function, ; , and nonautonomous terms and are time-dependant external forces and translation compact (see Definition 1).

We all know that the long-short wave resonance equations play an important role in fluid mechanics and have rich physical and mathematical properties. There are more and more resent papers treating the long-short wave resonance equations. Guo studied the global solution for one class of the system of LS nonlinear wave interaction in [1] and the periodic initial value problems and initial value problems for one class of generalized long-short type equations in [2]. The papers [35] studied the existence of a global attractor of it. Cui et al. developed the weakly compact uniform attractor for the nonautonomous long-short wave equations with translation compact forces in [6].

The Schrödinger type equation has been of great importance describing nonrelativistic quantum mechanical behavior. It is well known that Feynman and Hibbs derive the standard (nonfractional) Schrödinger type equation by applying path integrals over Brownian paths in [7]. Recently Laskin generalized the Schrödinger equation to space fractional cases using path integrals over Lévy trajectories in [8, 9]. In [10], the authors discussed the models and numerical methods of the fractional calculus. The fractional Schrödinger type equation is used to describe better physical phenomenon and has attracted more and more attention of researchers. Guo and Xu studied some applications of the Schrödinger equation in physics (see [11]). In [12], the authors obtained the approximate analytical solutions of the fractional nonlinear Schrödinger equations by using the homotopy perturbation method. Eid et al. studied the -dimensional fractional Schrödinger equation and obtained its exact solutions in [13]. Guo et al. investigated the fractional nonlinear Schrödinger equation and showed the existence and uniqueness of its global smooth solution by using energy method in [14]. Goubet [15] studied regularity of the attractor for a weakly damped nonlinear Schrödinger equation in .

The rest of this paper is arranged as follows. In Section 2, we recall some basic definitions, introduce preparation results, and analyse some fractional calculation laws which depend heavily on -period. In Section 3, we introduce some preparation lemma and give the uniform a priori estimates (uniform in initial data and symbol in the symbol space and large time). In Section 4, we show the existence and uniqueness of the solution of the system. In Section 5, we prove the existence of strong compact uniform attractor of the system.

Through the paper, we denote the norm of with the usual inner product by . We denote the norm of for all by . For simplicity and convenience, the letter represents a constant, which may be different from one to others. represents the constant expressed by the parameters appearing in the parentheses.

2. Preliminaries

In this section, we introduce notations definition and preliminary facts. We firstly recall the following known definitions (see [6, 1618]) and some main lemmas (see [16, 19, 20]).

Definition 1. Suppose is a Banach space, is a function, and is the translation operator. The hull of is defined by (i)is said to be translation bounded in if is bounded in which Then consists of all the translation bounded functions in .(ii) is called translation compact in if is compact in , where the convergence is taken in the sense of local quadratic mean convergence topology of . The collection of all the translation compact functions in is denoted by .

Let be a Banach space, and the following definitions are common.

Definition 2. Let be a parameter set. , is said to be a family of processes in , if, for each , is a process; that is, the two-parameter mapping from to satisfies(i), (ii), where is called the symbol space and is called the symbol.

A subset is said to be uniformly absorbing set for the family of processes , if, for any   and subset denoting the set of all bounded subsets of , there exists such that for all . A set is called uniformly attracting for the family of process , if, for each fixed and every , it satisfies that

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

Definition 4. , a family of processes in , is said to be -continuous, if, for any fixed   and , , projection is continuous from to .

Definition 5. The space denotes all measurable functions with the norm for , and

Lemma 6. Let be a compact metric space and suppose is a family of operators defined on , satisfying(i)(ii)translation identity: where is an arbitrary process in compact metric space . Note that 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 7. Assumption (11) holds if the system has a unique solution.

Lemma 8. Let be a uniform convex Banach space (particularly, a Hilbert space), and let be a sequence in . If and , then .

Lemma 9. Let be a sequence in space . If , then
Since the solution , if it exists, is a -periodic function, we have the Fourier expansion: where . Therefore, and is defined by Since the following definitions make sense. Let and let be a complete space of the set under the norm: Then we can easily check that is a Banach space and that, for , is space-periodic with the period and its order derivatives are in . And for , when . is a Hilbert space with the inner product
For brevity, we introduce and . We denote the space of functions by with norm Similarly, we denote the space of by with norm

Assumption 10. Suppose that the symbol belongs to 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 assume .

Remark 11. Due to the conception of translation compact/boundedness, we remark that(i), ;(ii), where is a translation operator.

3. A Uniform A Priori Estimates

In this section, we will get some uniform a priori estimates which hold uniformly independently of initial data, time, and symbols in symbol space (). In the following, we denote that and , which will not cause any confusions.

We first recall the Gagliardo-Nirenberg and the Young inequalities (see [21]).

Lemma 12. Let . Then for , there exists a constant such that where .

Lemma 13. Let . Then for each satisfying , it holds that

Lemma 14. Assume that(i) satisfy Assumption 10;(ii) and solves problem (1)–(4).Then there exist positive constants and such that

Proof. Taking the inner product of (1) with , we have Taking the imaginary part of (27), we get By (28) and Remark 11, we have By Gronwall inequality we get the lemma.

Lemma 15. Assume that(i), and satisfy Assumption 10;(ii) and solves problem (1)–(4).Then there exist positive constants and such that

Proof. Taking the inner product of (1) with , we have Taking the real part of (31), we have where So we have Taking the inner product of (1) with , we have Taking the real part of (35), we have By (34) and (36), we have By Lemmas 12 and 13 and the condition , we have So we reduce that Similarly, we also derive that Taking the inner product of (2) with , we have By (1), we have where By (41)(43), we get By Lemmas 1214 and the condition , we have So we obtain that Similarly, we can also get that Set So by (39) and (46), we get By (40) and (47), we also get Setting , , then we deduce that By Gronwall inequality, we have Obviously for any , we have So by (52)~(54), there exists a such that for any .
By (48),(53), and (55), we get Then setting , we get By using Lemma 14, we conclude the lemma.

Lemma 16. Assume that(i), and satisfy Assumption 10;(ii) and solves problem (1)–(4).Then there exist positive constants and such that

Proof. Taking the inner product of (1) with , we have Taking the real part of (59), we have where, by (1) and (2), we have By Lemmas 12, 14, and 15 and , we see that By (60)~(62), Lemmas 12~ 15, and Hölder inequality, we can see that Taking the inner product of (2) with , we have where, by Lemmas 12~ 15 and , we can see that So by (64) and (65), we get Set and Then by (63) and (66), we can deduce that which has the same form with (51) in the proof of Lemma 15. Similar to the study of (51), there exist positive constants and such that which conclude the proof of Lemma 16.

4. Unique Existence of the Solution

In this section, we show the unique existence theorem of the solutions. Since uniform a priori estimates have been established in the above section, one can readily get the existence of the solution by ’s method (see [20, 2224]). We show the theorem and prove it briefly for readers’ convenience.

Theorem 17. Set , and satisfy Assumption 10; for each , then system (1)–(4) has a unique global solution , .

Proof. We prove this theorem by two steps.
Step  1. The existence of solution.
By ’s method, we construct the approximate solution of the periodic initial value problem (1)(4). We apply the following approximate solution: to approach , the solution of the problem (1)–(4). And for satisfies We see that system (71) is an initial boundary value problem of ordinary differential equations (ODE). By the standard existence theory for ODE and uniform a priori estimates in Section 3, for any , there exists a unique solution of (71), such that There is a subsequence of and such that Due to the above proof and the continuous extension theorem, is the solution Of (1)(4).
Step  2. The uniqueness of solution.
Suppose are two solutions of problem (1)–(4). Let , and then satisfies Obviously, is uniformly bounded. Note that .
Taking the inner product of (74) with and taking the imaginary part, we can get Taking the inner product of (75) with , we can obtain Taking the inner product of (74) with and taking the real part, we can get