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.