Advances in Mathematical Physics
Volume 2019 (2019), Article ID 2763245, 15 pages
https://doi.org/10.1155/2019/2763245

Research Article

Upper Semicontinuity of Random Attractors for Nonautonomous Stochastic Reversible Selkov System with Multiplicative Noise

Chunxiao Guo,¹ Yanfeng Guo,2,3 and Xiaohan Li¹

¹Department of Mathematics, China University of Mining and Technology, Beijing, Beijing 100083, China
²School of Science, Guangxi University of Science and Technology, Liuzhou, Guangxi 545006, China
³School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China

Correspondence should be addressed to Yanfeng Guo; guoyan_feng@163.com

Received 23 April 2019; Accepted 12 June 2019; Published 27 June 2019

Academic Editor: Ciprian G. Gal

Copyright © 2019 Chunxiao Guo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

Selkov equations [1] 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 [4]. Li proved the existence of random attractor of the stochastic reversible Selkov system on infinite lattice with additive noise in [5]. In this paper, we consider the upper semicontinuity of random attractors for nonautonomous stochastic reversible Selkov system with multiplicative noise

where is a bounded smooth domain. There are boundary conditions

and initial conditions

All 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 [23] and the existence and upper semicontinuity of random attractor for stochastic three-component reversible Gray-Scott system was proved in [24]. In [25], 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.

2. Preliminaries

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 spaces

The 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 , where

the -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 process

which 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. Let

where is the Ornstein-Uhlenbeck process. Then

So we can consider the following equations with random coefficients, but without white noise:

for and , with the homogeneous Dirichlet boundary conditions

and the initial value conditions

For every , , the initial-boundary value problem (19)–(22) is formulated into the following reversible Selkov equation:

where , and

and

for 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 and

This is similar to the autonomous case studied in [4]. 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 that

Sometimes, 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 .

Lemma 4. Suppose that . Then, for every , and , there exist a number and a tempered random variable

such that for all , the solution of (19)–(22) with satisfies

Proof. Multiplying (19) by and (20) by , we get

and

Adding up the above two equations, we obtain

By the -Young inequality, we have

It follows that

Let , . It is easy to see that

By Gronwall’s lemma on the interval for above inequality and substituting by , we can deduce

Note that

By (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 have

Thus, we get

where the integral is convergent due to (46). Similarly, we also obtain

Thus, we yield

By (49), we get that

According to the properties of the Ornstein-Uhlenbeck process,

Noticing that is tempered, then for any , we have

Then, is tempered. By (49), the proof is completed.

Lemma 5. Suppose that . Then, for every , and , there exist a number and tempered random variables ,

such that for all , the solution of (19)–(22) with satisfies

Proof. Substituting for in (41), for , we yield

Applying Gronwall’s lemma to (40) over and by (55), we obtain

It is easy to show that

and

Now let . According to (56), (57), and (58), there exists a such that for all , we can get

Similarly, substituting for in (41), applying Gronwall’s lemma to (40) over , and combining the estimate (49), there exists a such that for all , we get

It 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

and

such 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 , it follows that

where and are positive constants.
Let , . Then, the transformed system (19)-(20) becomes a rescaled system

Multiplying (65) by , we obtain

Multiplying (66) by , we get

Summing up the resulting equalities, we have

Using the Young inequality, we have

and

Recalling that , it follows that

where . By Gronwall’s lemma on the interval , , for the above inequality, and replacing by , we get

Recalling that , , we have

Similarly, multiplying (65) by and (66) by , we get

Put . Applying Gronwall’s inequality over and replacing by , we have

Recalling that , , we get

The proof is completed.

Lemma 7. Suppose that . Then, for every , and , there exist a number and a tempered random variable

such that for all , the solution of (19)–(22) with satisfies

where .

Proof. Multiplying (19) by and (20) by , we have

Then summing up the resulting equalities, we have

Due to Hölder’s inequality and -Young inequality, we obtain

For the third term on the right-hand side of (82), we have

For the last term in (82), we get

Recalling that and , it follows that

For and , we integrate (86) over to get

Now integrating the above with respect to over , we find that for , it follows that

By the Mean Value Theorem in (60), there is a time such that

Then by Lemma 6, for , there is a random variable such that

By Lemma 5, it follows from (77) that, for all ,

It is easy to see that is tempered. This completes the proof.

Lemma 8. Suppose that . Then for each and , the continuous cocycle associated with problem (19)–(22) has a closed measurable -pullback absorbing set , which is given by

where is defined as in (32).

Theorem 9. Suppose that . Then the continuous cocycle associated with problem (19)–(22) has a unique -pullback attractor in .

Proof. By Lemma 8, the continuous cocycle has a -pullback absorbing set . By the compact embedding , the continuous cocycle is pullback asymptotically compact in . According to Theorem 1, there exists a unique random attractor in for the continuous cocycle .

4. Upper Semicontinuity of Random Attractors

In this section, we prove the upper semicontinuity of random pullback attractors for the nonautonomous stochastic reversible Selkov system when the intensity of noise . Given , we write the cocycle of problem (19)–(22) as to indicate its dependence on .

The deterministic system corresponding to (19)–(22) is given by

with the homogeneous Dirichlet boundary conditions

and the initial conditions

Similarly, we can prove that problem (93)–(96) generates a continuous cocycle in . Moreover, has a unique pullback attractor in and has a pullback absorbing set , where is given by

and

The next lemma shows the convergence of solutions of (19)–(22) as .

Lemma 10. Suppose that . Let and be the solution of (19)–(22) and (93)–(96) with initial conditions and , respectively. If , then, for and ,

Proof. Let and be the solution of (19)–(22), and , , and . Then from (19)-(20) and (93)-(94), we get

which implies

Summing up the two equalities together, it yields that

Note that

Given that , , , and , since is continuous on , we obtain that there exists a such that for all and ,

Thus, we obtain

Lemma 6 is true even if . By Hölder’s inequality and Gagliardo-Nirenberg inequality, we have

where is a constant depending on , and the Gagliardo-Nirenberg inequality constant. It follows from (105) and (106) that there exists a such that for all and ,

where , . For all and , due to Gronwall’s lemma, we get

From (108), recalling that , we find that

which finishes the proof.

Theorem 11. Provided that . Then for every and ,

Proof. To achieve the result, it suffices to verify conditions (1), (2), and (3) in Proposition 3. For condition (1), by taking a sequence and , from Lemma 10 we have

By Lemma 8 and Theorem 9, has a tempered pullback absorbing set and a unique -pullback random attractor . Here,

where is defined as in (32). It follows from (97) and (112) that for all and , we have

which deduces condition (2) immediately.
Now consider condition (3). By Lemma 7, we know that is also a closed and tempered absorbing set for in , where

Let

and

We find that for all and ,

By the compactness of embedding , we know that is compact in . By (117), it follows that is precompact in . Hence, condition (3) is clear and the proof is finished.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors would like to thank the reviewer for their valuable comments and suggestions which improve the manuscript. This paper was partially supported by the National Natural Science Foundation of China Nos. 11771444, 11861013, the Yue Qi Young Scholar Project, China University of Mining and Technology (Beijing), China Scholarship Council (CSC), Guangxi Natural Science Foundation No. 2017GXNSFAA198221, the Fundamental Research Funds for the Central Universities, and China University of Geosciences (Wuhan) No. 2018061.

References

Y. You, “Asymptotical dynamics of Selkov equations,” Discrete and Continuous Dynamical Systems - Series S, vol. 2, no. 1, pp. 193–219, 2009.

Ed. L, Mathematical Models in Molecular and Cellular Biology, L. A. Segel, Ed., Cambridge University Press, 1980.

E. E. Selkov, “Self-oscillations in glycolysis:a simple kinetic model,” European Journal of Biochemistry, vol. 4, pp. 79–86, 1968.

Y. You, “Global dynamics and robustness of reversible autocatalytic reaction–diffusion systems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 75, no. 6, pp. 3049–3071, 2012.

H. Li, “Random attractor of the stochastic lattice reversible selkov equations with additive noises,” in Proceedings of the IEEE 13th International Conference on e-Business Engineering (ICEBE), pp. 176–181, 2016.

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, Germany, 1998.

P. W. Bates, H. Lisei, and K. Lu, “Attractors for stochastic lattice dynamical systems,” Stochastics and Dynamics, vol. 6, no. 1, pp. 626–637, 2006.

P. W. Bates, K. Lu, and B. Wang, “Random attractors for stochastic reaction-diffusion equations on unbounded domains,” Journal of Differential Equations, vol. 246, no. 2, pp. 845–869, 2009.

H. Crauel and F. Flandoli, “Attractors for random dynamical systems,” Probability Theory and Related Fields, vol. 100, no. 3, pp. 365–393, 1994.

T. Caraballo, M. J. Garrido-Atienza, and T. Taniguchi, “The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 11, pp. 3671–3684, 2011.

T. Caraballo and J. A. Langa, “On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems,” Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, vol. 10, no. 4, pp. 491–513, 2003.

H. Cui and Y. Li, “Existence and upper semicontinuity of random attractors for stochastic degenerate parabolic equations with multiplicative noises,” Applied Mathematics and Computation, vol. 271, pp. 777–789, 2015.

I. Chueshov, Monotone Random Systems Theory and Applications, vol. 1779 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2002.

I. Chueshov, J. Real, and T. Caraballo, “Pullback attractors for stochastic heat equations in materials with memory,” Discrete and Continuous Dynamical Systems - Series B, vol. 9, no. 3/4, May, pp. 525–539, 2008.

I. Chueshov and M. Scheutzow, “On the structure of attractors and invariant measures for a class of monotone random systems,” Dynamical Systems, vol. 19, no. 2, pp. 127–144, 2004.

F. Flandoli and B. Schmalfuss, “Random attractors for the D stochastic Navier-Stokes equation with multiplicative white noise,” Stochastics and Stochastics Reports, vol. 59, no. 1-2, pp. 21–45, 1996.

M. J. Garrido-Atienza, A. Ogrowsky, and B. Schmalfuss, “Random differential equations with random delays,” Stochastics and Dynamics, vol. 11, pp. 369–388, 2011.

M. J. Garrido-Atienza and B. Schmalfuss, “Ergodicity of the infinite dimensional fractional Brownian motion,” Journal of Dynamics and Differential Equations, vol. 23, no. 3, pp. 671–681, 2011.

W. Shen and J. Huang, “Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains,” Discrete and Continuous Dynamical Systems - Series A, vol. 24, no. 3, pp. 855–882, 2009.

P. E. Kloeden and J. A. Langa, “Flattening, squeezing and the existence of random attractors,” Proceedings of the Royal Society A Mathematical, Physical and Engineering Sciences, vol. 463, no. 2077, pp. 163–181, 2007.

Y. Li, H. Cui, and J. Li, “Upper semi-continuity and regularity of random attractors on p-times integrable spaces and applications,” Journal of Nonlinear Analysis, vol. 109, pp. 33–44, 2014.

Y. Lv and W. Wang, “Limiting dynamics for stochastic wave equations,” Journal of Differential Equations, vol. 244, no. 1, pp. 1–23, 2008.

A. Gu, S. Zhou, and Z. Wang, “Uniform attractor of non-autonomous three-component reversible Gray–Scott system,” Applied Mathematics and Computation, vol. 219, no. 16, pp. 8718–8729, 2013.

A. Gu and H. Xiang, “Upper semicontinuity of random attractors for stochastic three-component reversible Gray-Scott system,” Applied Mathematics and Computation, vol. 225, pp. 387–400, 2013.

J. Tu and Y. You, “Random attractor of stochastic brusselator system with multiplicative noise,” Discrete and Continuous Dynamical Systems - Series A, vol. 36, no. 5, pp. 2757–2779, 2016.

B. Wang, “Asymptotic behavior of stochastic wave equations with critical exponents on ,” Transactions of the American Mathematical Society, vol. 363, no. 7, pp. 3639–3663, 2011.

B. Wang, “Random attractors for non-autonomous stochastic wave equations with multiplicative noise,” Discrete and Continuous Dynamical Systems - Series A, vol. 34, no. 1, pp. 269–300, 2014.

B. Wang, “Upper semicontinuity of random attractors for non-compact random dynamical systems,” Electronic Journal of Differential Equations, vol. 2009, pp. 1–18, 2009.

B. Wang, “Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems,” Journal of Differential Equations, vol. 253, no. 5, pp. 1544–1583, 2012.

A. Goldbeter, Models for Oscillation and Excitability in Biochemical Systems, E. Segel, Ed., Mathematical Models in Molecular and Cell Biology, Cambridge University Press, Cambridge, UK, 1980.