Attractors for the Stochastic Lattice Selkov Equations with Additive Noises

In this paper, by proving the pullback asymptotic compactness of the stochastic lattice Selkov equations with the cubic nonlinearity, the existence of a random attractor of the stochastic lattice reversible Selkov equations on infinite lattice with additive noises is proved.


Introduction
In this paper, we study that the stochastic lattice Selkov system with the cubic nonlinearity and additive white noises on an infinite lattice is considered in [1] and [2]: where  denotes the integer set, The reversible Selkov model is derived from a set of the two reversible chemical reactions, which has been studied by [3] [4] and other authors: The original Selkov model corresponds to the two irreversible reactions, where the product 1 Q is an inert product. Let t u and t v are respectively the concentrations of the reactants Q and P, Equation (1.1) can be regarded as a Selkov system (see [5]) on  : For the Equation (1.1), the solution mapping defines a random dynamical system, which is a parametric dynamical system, and pullback absorbing set has been proved, see [1] and [2]. Random attractors are the appropriate objects for describing asymptotic dynamics of such a parametric dynamical system. Therefore, in this paper, we would prove the existence of a random attractor for the stochastic lattice Selkov Equation (1.1).
This paper is organized as follows. In the next section, we recall basic concepts and results related to random attractors. In Section 3, using the transformation of Ornstein-Uhlenbeck process, the stochastic Selkov equation with white noise is transformed into a noiseless determined Selkov equation with random variables as parameters. In Section 4, we prove the pullback asymptotic compactness for the random dynamical system. Then the existence of a random attractor is proved.

Preliminaries
Firstly, we introduce the relevant definitions of random attractor, which can be taken from [ , u v u ± ± , we need 6 6 , Introducing an Ornstein-Uhlenbeck process (O-U process) (see [10] given by the Wiener process: 2) the random variables with the initial value condition

Pullback Asymptotic Compactness
From Theorem 2.1, to prove the existence of a random attractor for the random dynamical system generated by (1.1), it is necessary to obtain the pullback absorbing property and the pullback asymptotic compactness. The pullback absorbing property has been obtained by [2]. For the pullback asymptotic compactness, we have the following lamma.
Proof. We choose a smooth function ρ such that 0 and there exists a positive constant 1 c , such that We first consider the random Equation (3.1). Let r be a fixed positive integer which will be specified later. Taking the inner product of the Equation Then from (4.2)-(4.5), we find that ( ) For the third term and forth term in the right-hand side of (4.6), we have 1 1 1 Mathematics and Physics   2  2  2  2  2  2  2   3  2 , For the fifth term and sixth term in the right-hand side of (4.6), we have ( ) ( ) ( ) are positive constants depending only on 1 2 1 2 , , , d d a a . For the last two terms in the right-hand side of (4.6), From (4.6)-(4.9), we have a a . By Gronwall's inequality in [11], we have that for     Thus, there exists a 4 , .
Recall that