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Upper semicontinuity of attractors for small perturbations of Klein-Gordon-Schrödinger lattice system
Advances in Difference Equations volume 2014, Article number: 300 (2014)
This paper studies the existence of the random attractor for a Klein-Gordon-Schödinger system under a small ε-random perturbation on a high dimensional infinite lattice. Firstly, we prove the asymptotic compactness of the random dynamical system and obtain the random attractor. Then, by comparing to the case without random perturbation (), we show the upper semicontinuity of the attractors.
In this paper, we consider the discrete Klein-Gordon-Schrödinger system with a small random perturbation,
with initial conditions
where , , , ℤ is the integer set, α, μ, and ε are positive constants, , , and , , , the space of bounded continuous functions from ℝ into . and are two independent two-side real valued standard Wiener processes, linear operator A and space will be described in detail in the next section.
The coupled Klein-Gordon-Schrödinger (KGS) system is an important model in nonlinear science. It is encountered in several diverse branch of physics, for example in the description of the interaction of a scalar nucleon interacting with a neutral scalar meson, as in the following nonautonomous KGS equations in ℝ:
where u denotes a complex scalar nucleon field and v represents a real meson field; the complex-valued function and the real-valued function both are the time-dependent external sources. By using the Galerkin method, Fukuda and Tsutsumi  first studied the coupled KGS equations, and they obtained the existence of global strong solutions. We refer the readers to [2–7] for the existence of global solutions, asymptotic behavior, and stability for the KGS system and KGS system on infinite lattices.
Various properties of solutions for lattice dynamical systems (LDSs) have been extensively investigated. For example, the long time behavior of LDSs were studied in [6, 8–17]; the traveling wave solutions of LDSs were studied in ; the chaotic properties of solutions for LDSs were examined in . Lattice dynamical systems play an important role in their potential applications such as biology , chemical reactions , pattern recognition and image processing , electrical engineering , laser systems , etc. However, a system in reality is usually affected by external perturbations, which in many cases are of great uncertainty or have a random influence. These random effects are not only introduced to compensate for the defects in some deterministic models but also to explain the intrinsic phenomena. Therefore, there is much work concerned with stochastic lattice dynamical systems [24–27]. To the authors’ knowledge, Ruelle  first introduced a corresponding generalization of the attractor (random attractor) to the stochastic PDEs. After that, the study of random attractors gained considerable attention during the past decade; see [29, 30] for a comprehensive survey. Bates et al.  first investigated the existence of global random attractor for a kind of first order dynamical systems driven by white noise on lattice ℤ; then Lv and Sun  extended the results of Bates to the higher dimensional lattices. After that, there are several papers considering with stochastic evolution equations in an infinite lattice [26, 27, 31]. In this paper, we first extend the result  to the higher dimensional lattices, then, by comparing to the case without random perturbation  (), we see the relationship between a random attractor and a global attractor for a small ε random perturbed Klein-Gordon-Schrödinger lattice system, i.e. upper semicontinuity of the attractors for a small ε perturbed Klein-Gordon-Schrödinger lattice system. Roughly speaking, let be the attractor of the perturbed system, be the attractor of the unperturbed system; we say that those attractors have upper semicontinuity if
where denotes the Hausdorff semidistance.
This paper is organized as follows. In Section 2, we recall some basic concepts and already known results related to random dynamical systems and random attractors. In Section 3, we prove the existence of the random attractor for stochastic KGS lattice dynamical systems of (1.1) on . In Section 4, by comparing to the case without the random perturbation, i.e. case, we obtain the upper semicontinuity of the attractors.
Let be a Hilbert space, be a probability space.
Definition is called a metric dynamical systems, if is -measurable, , for all , and for all .
Definition A stochastic process is called a continuous random dynamical system (RDS) over if ϕ is -measurable, and for all
the mapping is continuous;
for all and (cocycle property).
Definition 2.1 A random bounded set is called tempered with respect to if for a.e. and all
Consider a continuous random dynamical system over and let be the collection of all tempered random set of H.
Definition 2.2 A random set is called an absorbing set in if for all and a.e. there exists such that
Definition 2.3 A random set is a random -attractor for RDS ϕ if
is a random compact set, i.e., is measurable for every and is compact for a.e. ;
is strictly invariant, i.e., , and for a.e. ;
attracts all sets in , i.e., for all and a.e. we have
where , .
The collection is called the domain of attraction of .
Definition 2.4 Let ϕ be a RDS on Hilbert space H. ϕ is called asymptotically compact if for any bounded sequence and , the set is precompact in H, for any .
From , we have the following result.
Proposition 2.1 Let be an absorbing set for an asymptotically compact continuous RDS ϕ. Then ϕ has a unique global random -attractor
which is compact in H.
Throughout this paper, let be a fixed positive integer. We set
For brevity, we use H to denote the Hilbert space or , and we equip H with the inner product and norm as
where denotes the conjugate of .
In our paper, we introduce the transformation with a small positive constant. Then system (1.1) becomes
with initial conditions
where , , , and the linear operator A is defined by
In fact, the linear operator A has the following decomposition:
and there exist bounded linear operators defined by
where , such that
where denotes the norm of operator in the set of linear operators from ℍ into itself and is the adjoint operator of , , that is,
Let and , where . Here denotes the standard complete orthonormal system in , which means that the j th component of is 1 and all other elements are 0. Then and are -valued Q-Wiener processes. It is obvious that and . For details we refer to .
We abstract (2.1) as a stochastic ordinary differential equations with respect to time t in . Let , , , , and . Then the equations in (2.1) can be written as the following integral equations:
For our purpose we introduce the probability space as
endowed with the compact open topology . ℙ is the corresponding Wiener measure and is the ℙ-completion of the Borel σ-algebra on Ω.
Let , . Then is a metric dynamical system with the filtration , , where is the smallest σ-algebra generated by the random variable for all , such that ; see  for more details.
3 The existence of a random attractor
In this section, we study the dynamics of solutions of Klein-Gordon-Schrödinger lattice system under the ε-random perturbation (1.1). Then we apply Proposition 2.1 to prove the existence of a global random attractor for (1.1). In order to show the existence of a global solutions of system (2.2), we first change (2.2) into deterministic equations. Due to the special linear multiplicative noise, the first equation in system (2.1) can be reduced to an equation with random coefficients by a suitable change of variable. Consider the process , which satisfies the stochastic differential equation
The process obeys the random differential equation
Lemma 3.1 Assume . Then the solution of the first equation in (3.1) satisfies
We denote , then system (2.2) can be changed into the following system:
For each fixed , system (3.3) is a deterministic equation, and we have the following result.
Theorem 3.1 For any , system (2.2) is well-posed and admits a unique solution . Moreover, the solution of (2.2) depends continuously on the initial data .
Proof By the standard existence theorem for ODEs, it follows that system (3.3) possesses a local solution , where is the maximal interval of existence of the solution of (3.3). Now, we prove that this local solution is a global solution. Let , from (3.3) it follows that
By the definition of the linear operator A, we have
By the Young inequality, direct computation shows that
Combining the above inequalities with Lemma 3.1, we obtain
where , are constants depending on α, σ, μ, ε, and . By the Gaussian property of and , (3.4) implied that system (3.3) admits the global solution . The proof is completed. □
From the definition , we know
and combining the above theorem we have the following result.
Theorem 3.2 System (2.2) generates a continuous random dynamical system over .
The proof is similar to that of Theorem 3.2 in , so we omit it.
Now, we prove the existence of a random attractor for system (2.2). By Proposition 2.1, we first prove that RDS ϕ possesses a bounded absorbing set . We introduce an Ornstein-Uhlenbeck process in on the metric dynamical system given by a Wiener process:
where ν and λ are positive. The above integral exists in the sense that for any path ω with a subexponential growth , solve the following Itô equations:
In fact, the mapping , , are the Ornstein-Uhlenbeck process. Furthermore, there exists a invariant set of full ℙ measure such that:
the mappings , , are continuous for each ;
the random variables , , are tempered.
Lemma 3.2 There exists a invariant set of full ℙ measure and an absorbing random set , , for the random dynamical system .
Proof We use the estimates in Theorem 3.1. By (3.4), we have
By Gronwall’s lemma, it follows that
Replace ω by in the above inequality to construct the radius of the absorbing set and define
Then is a tempered ball by the property of , , and, for any , . Here, denotes the collection of all tempered random set of Hilbert space H. This completes the proof. □
Lemma 3.3 Let , the absorbing set given in Lemma 3.2. Then for every and ℙ-a.e. , there exist and such that the solution of system (2.2) satisfies
where for .
Proof Let be a cut-off function satisfying
and (a positive constant).
Let M be a suitable large integer. Taking the inner product of (3.3) with , , and , we get
We also use the estimates in Theorem 3.1. Similar to (3.4), it follows that, for fixed constant ,
Replace ω by in (3.5). Then we estimate each of the terms on the right-hand of (3.5), and it follows that
Since , , are tempered and , , are continuous in t, there is a tempered function , such that
Combining (3.6) with (3.7), there is a constant , such that
Next, we estimate
Let and be fixed positive constants. Then, for and , we have
Since and , there exists such that for ,
Then, for and , we obtain
Direct computation shows that
Therefore, we obtain
This completes the proof. □
Lemma 3.4 The random dynamical system is asymptotically compact.
Proof We use the method of . Let . Consider a sequence with as . Since is a bounded absorbing set, for large n, , where . Then there exist and a sequence, denoted by , such that
Next, we show that the above weak convergence is actually strong convergence in E.
From Lemma 3.3, for any , there exist positive constants and such that for ,
Since , there exists such that
Let , then, from (3.11), there exists such that for ,
By (3.12)-(3.14), we find that, for ,
This completes the proof. □
Now, combining Lemma 3.2, Lemma 3.4, and Proposition 2.1, we can easily obtain the following result.
Theorem 3.3 The random dynamical system possesses a global random attractor in E.
4 Upper semicontinuity of attractors
This section studies the upper semicontinuity of random attractors for the stochastic Klein-Gordon-Schrödinger lattice system. The existence of global attractors for the Klein-Gordon-Schrödinger lattice system has been obtained by  for one dimension, and  for the high dimensional case. We assume is an attractor corresponding to the Klein-Gordon-Schrödinger lattice system, i.e., the case of system (1.1). By Theorem 2 of , we only need to prove the following lemma for upper semicontinuity of attractors. In what follows, we take the vector form for brevity.
Lemma 4.1 Assume that and are the solutions of the perturbed lattice system (1.1) and the unperturbed lattice system (the case of in (1.1)), respectively. Then, for ℙ-almost every and the absorbing set , we have
Proof Let , . Then satisfies
Now, we use the change of variables
where is a positive constant, and .
Then we change (4.1) into
Taking the imaginary part of the inner product of the first equation in (4.2) with , we have
Taking the inner product of the second and third equation in (4.2) with δ and , we have
Summing up (4.3)-(4.5), we get
In what follows, we will estimate (4.6) by term by term. By the definition of the linear operator A, we have
Using the Young inequality, we have
Therefore, by the above inequalities and Lemma 3.1, we obtain
where , are constants depending on α, σ, μ, n, C, and .
Applying the Gronwall lemma to (4.7), we get
Note that , , and , so, by (4.8), we have
which implies that
This completes the proof. □
Theorem 4.1 Assume that are attractors for system (1.1). Then, for ℙ-almost every ,
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The authors express their sincere thanks to the anonymous referees for a very careful reading and for providing many valuable comments and suggestions, which led to an improvement of this paper. The first author is supported by the Fundamental Research Foundation for PHD of NCWU (No. 40291).
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.