Random attractor for second-order stochastic delay lattice sine-Gordon equation

*Correspondence: xintaolimath@126.com 1School of Mathematical Sciences, Xiamen University, Xiamen 361005, P.R. China Full list of author information is available at the end of the article Abstract In this paper, we prove the existence of randomD-attractor for the second-order stochastic delay sine-Gordon equation on infinite lattice with certain dissipative conditions, and then establish the upper bound of Kolmogorov ε-entropy for the randomD-attractor.


Second-order stochastic delay lattice sine-Gordon equation
Denote l p (p ≥ 1) defined by with the norm In particular, l 2 is a Hilbert space with the inner product (·, ·) and norm · given by for any u = (u i ) i∈Z k , v = (v i ) i∈Z k ∈ l 2 . Define a linear operator in the following way: (Au i ) i∈Z k = 2ku (i 1 ,...,i j ,...,i k )u (i 1 -1,...,i j ,...,i k ) -· · ·u (i 1 ,...,i j -1,...,i k ) -· · ·u (i 1 ,...,i j ,...,i k -1) u (i 1 +1,...,i j ,...,i k ) -· · ·u (i 1 ,...,i j +1,...,i k ) -· · ·u (i 1 ,...,i j ,...,i k +1) , Then B * j is the adjoint operator of B j , and By using the above equalities, we have For any u = (u i ) i∈Z k , v = (v i ) i∈Z k ∈ l 2 , we define a new inner product and norm on l 2 by It is obvious that Denote Then the norms · and · λ are equivalent. Let H = l 2 λ × l 2 be endowed with the inner product and norm In the following, we consider the probability space ( , F, P), where = ω ∈ C R, l 2 : ω(0) = 0 , F is the Borel σ -algebra induced by the compact-open topology of , and P the corresponding Wiener measure on ( , F). We will identify ω with Define the time shift by Then ( , F, P, (θ t ) t∈R ) is a metric dynamical system with the filtration For convenience, we rewrite Eq. (1.1) as where δ is a positive constant and satisfies then Eq. (3.2) can be rewritten as Also, we make the following assumptions:

By (H 1 ) and (H 3 ), we know that
where d 1 = max{2δ 2 + 54k 2 λ + 3λ + 6δ 2 (δ-α) 2 λ , 6(δα) 2 + 2(4k + λ)}. Thus, F and D map the bounded sets into bounded sets. In this way, by the standard theory of differential equations, we find that there exists a unique local solution. Then calculations in blow shows that this local solution is actually global. Indeed, suppose the solutions ϕ (1) Applying the Gronwall inequality, we have The proof is complete. Proof The proof is similar to that of Theorem 3.2 in [9], so here it is omitted.
For convenience, we denote ψ 2 where i ∈ Z k and i = max 1≤j≤k |i j |.
The proof is complete.

An upper bound of the Kolmogorov ε-entropy
In this section, we study the upper bound of the Kolmogorov ε-entropy of the global random D-attractor A(ω) given by Theorem 4.1.