Upper Semicontinuity of Random Attractors for Nonautonomous Stochastic Reversible Selkov System with Multiplicative Noise
In this paper, the existence of random attractors for nonautonomous stochastic reversible Selkov system with multiplicative noise has been proved through Ornstein-Uhlenbeck transformation. Furthermore, the upper semicontinuity of random attractors is discussed when the intensity of noise approaches zero. The main difficulty is to prove the asymptotic compactness for establishing the existence of tempered pullback random attractor.
Selkov equations  were a system of ODEs originally proposed by Selkov [2, 3] as a simplified model of a biochemical process called glycolysis, through which living cells get energy from breaking down sugar. The existence and robustness of global attractors of the reversible Selkov system have been investigated in the deterministic case by You in . Li proved the existence of random attractor of the stochastic reversible Selkov system on infinite lattice with additive noise in . In this paper, we consider the upper semicontinuity of random attractors for nonautonomous stochastic reversible Selkov system with multiplicative noisewhere is a bounded smooth domain. There are boundary conditionsand initial conditionsAll the coefficients , and are arbitrarily given positive constants; denote the time-dependent external forces. is a one-dimensional two-sided standard Wiener process on a probability space. The terms , indicate that (1)-(2) are interpreted in the sense of the Stratonovich integration.
In the past decades, great progress has been made in various aspects of random dynamical systems; see [6–22]. The long-time behavior of solutions for nonautonomous three-component reversible Gray-Scott system was discussed in  and the existence and upper semicontinuity of random attractor for stochastic three-component reversible Gray-Scott system was proved in . In , Tu and You introduced the pullback asymptotic compactness of stochastic Brusselator system with multiplicative noise. Notice that (1) and (2) are nonautonomous stochastic equations with time-dependent external terms and . For such a nonautonomous stochastic system, Wang established an efficacious theory about the existence and upper semicontinuity of the random attractor by introducing two parametric spaces [26–29].
In this paper, we first obtain the existence of random attractors for the stochastic perturbed reversible Selkov system (1)–(4) defined on . When , we will consider the limit behaviors of random attractors and prove the upper semicontinuity of these perturbed random attractors. As usual, we must prove the pullback asymptotic compactness of the solution operators in . The main difficulty for proving such compactness lines is the uniform estimates of the solutions in when we need to obtain some estimates in . We obtain the important -norm estimates by using the Mean Value Theorem.
The outline of the paper is as follows. In Section 2, some basic concepts related to the nonautonomous random dynamical system and upper semicontinuity of random attractors are introduced. Section 3 is devoted to the pullback asymptotic compactness and the existence of random attractors. In Section 4, we establish the upper semicontinuity of random attractors when the coefficient approaches zero.
In this section, we use some concepts of nonautonomous random dynamical system and random attractor. Let be a real separable Banach space with Borel -algebra and be an ergodic metric dynamical system. Some concepts and definitions of continuous random dynamical system have been seen in [6, 8, 17, 30] for more details. The following theorem is used to obtain the existence of random attractors for the continuous random dynamical system.
Theorem 1 (see [6, 8]). Let be a continuous random dynamical system on X over . If there exists a closed random tempered absorbing set of and is asymptotically compact in X, then is a random attractor of , where Moreover, is the random attractor of .
Now, we define the product Hilbert spacesThe norm and inner-product in or will be denoted by and , respectively. The norm in or will be denoted by , if . The norm in will be denoted by .
Let be the standard one-dimensional, two-sided Wiener process in the complete probability space , wherethe -algebra is generated by the compact-open topology on , and is the corresponding Wiener measure on ; see [6, 8]. The Wiener shift is defined by Consider the Ornstein-Uhlenbeck processwhich solves the Itô equation
Proposition 2 (see [6, 8]). Let the metric dynamical system and the Ornstein-Uhlenbeck process be defined as above. There is a -invariant set of full -measure such that for every , the following statements hold:
(1) The Ornstein-Uhlenbeck process has the asymptotically sublinear growth property; i.e.,(2) is continuous in and, for any fixed ,In the sequel, we consider only and will always write for .
Proposition 3 ((upper semicontinuity of random attractors) (see [8, 28])). Suppose the following conditions are satisfied:
(1) Given , we suppose that is a random dynamical system over a metric system . For P-a.e. , , , , and with , there holds(2) Assume that has a random attractor and a random absorbing set such that for some deterministic positive constant and for P-a.e. where
(3) There exists a such that for P-a.e. Then, for P-a.e. ,
To investigate the random dynamics of stochastic PDEs, we convert the stochastic evolutionary system to a system of pathwise PDEs with the random parameter. Letwhere is the Ornstein-Uhlenbeck process. ThenSo we can consider the following equations with random coefficients, but without white noise:for and , with the homogeneous Dirichlet boundary conditionsand the initial value conditionsFor every , , the initial-boundary value problem (19)–(22) is formulated into the following reversible Selkov equation:where , and andfor any with initial data
By the Galerkin method, we can prove the existence and uniqueness of the weak solution, which is continuously depending on the initial data andThis is similar to the autonomous case studied in . Define the continuous random dynamical system associated with the problem (23)-(24) as follows:for all . The continuous dynamical system and a continuous random dynamical system associated with the solutions of (1)–(4) are equivalent. Thus, we only discuss the dynamical system induced by (19)–(22).
Assume thatSometimes, we also assume that and are tempered in the following sense. For every ,
3. Existence of Random Attractors
In this section, we derive uniform estimates of solution for the stochastic reversible Selkov equations; these estimates are necessary for proving the existence of bounded absorbing sets and the asymptotic compactness of the random dynamical system. We will use to denote the collection of all tempered random subsets of from now on.
For brevity, we write as or simply as ; similarly, we write weak solution as or .
Proof. Multiplying (19) by and (20) by , we get andAdding up the above two equations, we obtain By the -Young inequality, we haveIt follows thatLet , . It is easy to see thatBy Gronwall’s lemma on the interval for above inequality and substituting by , we can deduceNote thatBy (11) and (12), we find that there exists a such that for , we have We get for all ,For the term on the right-hand side of (45), since , we haveThus, we getwhere the integral is convergent due to (46). Similarly, we also obtainThus, we yieldBy (49), we get thatAccording to the properties of the Ornstein-Uhlenbeck process,Noticing that is tempered, then for any , we haveThen, is tempered. By (49), the proof is completed.
Proof. Substituting for in (41), for , we yieldApplying Gronwall’s lemma to (40) over and by (55), we obtainIt is easy to show thatandNow let . According to (56), (57), and (58), there exists a such that for all , we can getSimilarly, substituting for in (41), applying Gronwall’s lemma to (40) over , and combining the estimate (49), there exists a such that for all , we getIt is easy to check that and are tempered. Let . This completes the proof.
Lemma 6. Let , . If there exists a such that with , where is any given random variable. Then for every , and , there exist a number and two tempered random variables and andsuch that for all , the solution of (19)–(22) with satisfies
Proof. Here we invoked Sobolev’s embedding theorem to assert that is a continuous embedding for . For with <