Upper semicontinuity of attractors for small perturbations of Klein-Gordon-Schrödinger lattice system
© Li and Sun; licensee Springer. 2014
Received: 4 October 2014
Accepted: 17 November 2014
Published: 1 December 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.
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.
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.
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).
Consider a continuous random dynamical system over and let be the collection of all tempered random set of H.
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.
which is compact in H.
where denotes the conjugate of .
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 .
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
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 .
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. □
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.
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 .
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. □
where for .
and (a positive constant).
This completes the proof. □
Lemma 3.4 The random dynamical system is asymptotically compact.
Next, we show that the above weak convergence is actually strong convergence in E.
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.
where is a positive constant, and .
where , are constants depending on α, σ, μ, n, C, and .
This completes the proof. □
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).
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