Abstract and Applied Analysis

Volume 2013 (2013), Article ID 981576, 16 pages

http://dx.doi.org/10.1155/2013/981576

Research Article

Random Attractors for Stochastic Retarded Reaction-Diffusion Equations on Unbounded Domains

Xiaoquan Ding¹ and Jifa Jiang²

¹School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, Henan 471023, China

²School of Mathematics and Science, Shanghai Normal University, Shanghai 200234, China

Received 10 March 2013; Accepted 28 May 2013

Academic Editor: Grzegorz Lukaszewicz

Copyright © 2013 Xiaoquan Ding and Jifa Jiang. 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.

This paper is devoted to a stochastic retarded reaction-diffusion equation on all -dimensional space with additive white noise. We first show that the stochastic retarded reaction-diffusion equation generates a random dynamical system by transforming this stochastic equation into a random one through a tempered stationary random homeomorphism. Then, we establish the existence of a random attractor for the random equation. And the existence of a random attractor for the stochastic equation follows from the conjugation relation between two random dynamical systems. The pullback asymptotic compactness is proved by uniform estimates on solutions for large space and time variables. These estimates are obtained by a cut-off technique.

1.  Introduction

The study of stochastic functional differential equations is motivated by the fact that, when one wants to model some evolution phenomena arising in physics, chemistry, biology, and other sciences, some hereditary characteristics such as aftereffect, time-lag, and time delay can appear in the variables. On the other hand, one of the most interesting problems concerning stochastic functional differential equations is to understood the asymptotic behavior of the solutions when time grows to infinite, since it can provide useful information about the future of the phenomenon described in the model.  

It is known that the asymptotic behavior of random systems can be captured by a random attractor, which was introduced in [1, 2] as an extension of the attractor theory of deterministic systems in [3–5]. For stochastic PDEs without any hereditary features, the existence of random attractors has been investigated by many authors; see, for example, [6–20] and the references therein. However, this problem is not well studied in the case of stochastic retarded PDEs.

In this paper, we investigate the asymptotic behavior of solutions to the following stochastic retarded reaction-diffusion equation with additive noise defined in the entire space :

where is a positive constant, is a given function defined on , is a nonlinear functional satisfying certain conditions, and is a two-sided infinite dimensional Wiener process on a probability space which will be specified later.

We note that the asymptotic behavior of several deterministic retarded PDEs on bounded domains was studied in [21–25], and the case of retarded Navier-Stokes equations on some unbounded domains was treated in [26]. The random attractor for retarded stochastic differential equations was considered in [27] by monotone methods. Recently, in the case of stochastic retarded lattice dynamical systems defined on the entire integer set, the existence of a random attractor was proved in [28, 29]. Here we prove the existence of a random attractor for the stochastic retarded reaction-diffusion equation defined in . It is worth mentioning that the asymptotic behavior of the nonretarded version of (1) was investigated recently in [9].

Notice that Sobolev embeddings are not compact when domains are unbounded. This introduces a major obstacle for proving existence of attractors for PDEs on unbounded domains. Under certain circumstances, the tail-estimates method can be used to deal with the problem caused by the unboundedness of domains. This approach was developed in [30, 31] for deterministic nonretarded PDEs and used in [9–11, 13, 17, 18] for stochastic systems. At the same time, the present of delays is another obstacle, which makes phase spaces not reflexive and increases the difficulty of uniform estimates. In this paper, we will develop a tail-estimates approach for stochastic retarded PDEs on unbounded domains and prove the existence of a compact random attractor for the stochastic retarded reaction-diffusion equation (1), in particular, defined on the unbounded domain . The idea is based on the observation that the solutions of the equation are uniformly small when space and time variables are sufficiently large. It is clear that our method can be used for a variety of other equations, as it was for the nonretarded case.

For convenience, hereafter we adopt the following notations. We denote by and the norm and the inner product in . Otherwise, the norm of a general Banach space is written as . For , let denote the Banach space of all continuous functions endowed with the supremum norm . For any real number , , and any continuous function , denotes the element of given by for .

The rest of the paper is organized as follows. In the next section, we introduce basic concepts concerning random dynamical systems and random attractors. In Section 3, we define a continuous random dynamical system for the stochastic retarded reaction-diffusion equation on . The existence of the random attractor is given in Section 4.

2. Preliminaries

In this section, we introduce some basic concepts related to random attractors for random dynamical systems. The reader is referred to [1, 2, 6, 32–34] for more details.

Let be a separable Banach space with Borel -algebra and a probability space.

Definition 1. is called a metric dynamical system if is -measurable, and is the identity on , for all and for all .

Definition 2. A set is called invariant with respect to , if, for all , it holds that

Definition 3. A continuous random dynamical system on over a metric dynamical system is a mapping

which is -measurable, and, for all ,

(i) is continuous for all ; (ii) is the identity on ; (iii) for all .

Definition 4. A random set is a multivalued mapping such that, for every , the mapping is measurable, where is the distance between the element and the set . It is said that the random set is bounded (resp., closed or compact) if is bounded (resp., closed or compact) for a.e. .

Definition 5. A random variable is called tempered with respect to , if for a.e.

A random set is called tempered if is contained in a ball with center zero and tempered radius for all .

Remark 6. If is tempered, then

(1) for any , is tempered; (2)for any and a.e.

 and is tempered.

Moreover, if, for a.e. , is continuous in , then is continuous in for such , and, for any , is tempered.

Hereafter, we always assume that is a continuous random dynamical system over , and is a collection of random subsets of .

Definition 7. A random set is called a random absorbing set in if, for every and a.e. , there exists such that

Definition 8. A random set is called a -random attractor (-pullback attractor) for if the following hold:

(i) is a random compact set; (ii) is strictly invariant; that is, for a.e. and all ,

(iii) attracts all sets in ; that is, for all and a.e. ,

 where is the Hausdorff semimetric given by for any and .

We remark that if , then this attractor is unique [33].

Definition 9. is said to be -pullback asymptotically compact in if, for all and a.e. , has a convergent subsequence in whenever , and .

The following existence result on a random attractor for a continuous random dynamical system can be found in [2, 34]. First, recall that a collection of random subsets of is called inclusion closed if whenever is an arbitrary random set and is in with for a.e. , then must belong to .

Proposition 10. Let be an inclusion-closed collection of random subsets of and a continuous random dynamical system on over . Suppose that is a closed random absorbing set for in and is -pullback asymptotically compact in . Then has a unique -random attractor which is given by

In this paper, we will take as the collection of all tempered random subsets of and prove the stochastic retarded reaction-diffusion equation on has a -random attractor.

3. Stochastic Retarded Reaction-Diffusion Equations on with Additive Noise

In this section, we show that there is a continuous random dynamical system generated by the stochastic retarded reaction-diffusion equation on with additive white noise:

with the initial condition

Here is a positive constant, is a given function in , is an -valued two-sided Wiener process with a symmetric nonnegative finite trace covariance operator defined on a probability space which will be specified below, and is a continuous mapping satisfying the following conditions:

; there exists a positive continuous function with

 for some positive integer such that, for all , with and ,

there exist positive constants and such that, for all ,  ,  , and ,

.

In the sequel, we consider the probability space where

is the Borel -algebra induced by the compact-open topology of , and the corresponding Wiener measure on with respect to the covariance operator . Let

Then is an ergodic metric dynamical system. Since the above probability space is canonical, we have

Similar to Proposition A.1 in [34], we can find that there exists a full -measure -invariant set such that for each

Let be the -completion of , and let

with

where is the smallest -algebra generated by the random variable for all , such that and is the collection of -null sets of .

Note that

so is a filtered metric dynamical system (see [32, pages 72 and 91] for more details). In addition, it is important to note that the measurability of is not true if we replace by its completion; see [32, page 547] for details.

In this paper, the solution of problem (10)-(11) is interpreted in a mild sense:

a.s. for any , where is the analytic semigroup on generated by . By the theory in [35], we deal with (22) on the complete probability space .

We now associate a continuous random dynamical system with the stochastic retarded reaction-diffusion equation (10)-(11) over . To this end, we introduce an auxiliary Ornstein-Uhlenbeck process on and transform the stochastic retarded reaction-diffusion equation into a random one. Let

Then by (18), (23) is well defined. The process , , is a stationary, Gaussian process. By Lemma 5.13 in [35], we can see that it is a mild solution of the linear equation

That is, for all and a.s.

Moreover, the random variable is tempered, and, for each , the mapping is continuous.

Setting for in (22), then by (25), we obtain a deterministic equation, a.s. in ,

which is the mild form of the evolution equation

with the initial condition

Here , .

Problem (27)-(28) is a deterministic partial functional differential equation with random coefficients, which can be solved pathwise. We now establish the following result for problem (27)-(28).

Theorem 11. Let and be fixed. Then the following properties hold:

(1) for each , problem (27)-(28) has a unique mild solution that belongs to

  for any , and for a.e.

(2) Let and be the mild solutions of problem (27)-(28) for the initial data and , respectively. Then there exists a constant such that for all

Proof. (1) By -, following the same lines of Theorem in [36], one can show that, for each , there exists a such that (26) has a unique solution on . Moreover, if , then

We prove now that this local solution is a global one. For fixed , by regularity of mild solutions for an analytic semigroup [37, page 145], we inform that

for any , and (27) holds for a.e. . Then, taking the inner product of (27) with in , we get that

By (), we can choose small enough such that . Using the Young inequality, we find that

Then it follows from (34) and (35) that

Choose small enough such that . Then by (36), we obtain

Now, we can also choose such that . Integrating (37) over    leads to

Using the Young inequality and , we find that

Then by (38) and (39), we obtain

where . Consequently,

Hence, for fixed , we get that, for ,

and, for ,

In view of (42) and (43), we find that, for all ,

Therefore, for all ,

which, together with (32), implies that . This proves property (1).

(2) By (44), there exists a constant such that

Then from (A₂) and (26), we have that for

Hence, for fixed , we get that, for ,

and, for ,

In view of (48) and (49), we find that, for all ,

The Gronwall inequality implies that, for all ,

This prove property (2). The proof is complete.

Conversely, if, for each , is a mild solution of problem (27)-(28) with , then by (25) the continuous process

is a mild solution of problem (10)-(11).

Theorem 12. Problem (27)-(28) generates a continuous random dynamical system over , where

Moreover, if one defines by

then is another continuous random dynamical system associated with problem (10)-(11).

Proof. By a classical successive approximation argument, one can easily show that, for fixed , is an -adapted continuous process. Hence, for fixed , is also an -adapted continuous process. On the other hand, from property (2) of Theorem 11, it follows that, for fixed and , is continuous. Consequently, is -measurable.

By (26) we have that, for , and ,

Then again by (26) we get

For each consider

Then for we have

It follows from (56) that

for all . By the uniqueness of the solution of (26) we find that

while (58) implies

Therefore, is a continuous random dynamical system.

As for , noticing that

we get from (61) that, for ,

Therefore, is also a continuous random dynamical system. Furthermore, and are conjugated random dynamical systems; that is,

where, for every , is a homeomorphism of . The proof is complete.

4. Existence of Random Attractors

In this section, we prove the existence of a -random attractor for the random dynamical system associated with the stochastic retarded reaction-diffusion equation (10)-(11) on . We first establish the existence of a -random attractor for its conjugated random dynamical system , then the existence of a -random attractor for follows from the conjugation relation between and . To this end, we will derive uniform estimates on the mild solutions of problem (27)-(28) when with the purpose of proving the existence of a bounded random absorbing set and the asymptotic compactness for . In particular, we will show that the tails of the solutions, that is, solutions evaluated at large value of , are uniformly small when time is sufficiently large.

From now on, we always assume that is the collection of all tempered subsets of with respect to . The next lemma shows that has a random absorbing set in .

Lemma 13. There exists such that is a random absorbing set for in ; that is, for any and a.e. , there exists such that

Proof. Replacing with in (41) and (44), we get that, for all ,

By assumption, is tempered. On the other hand, by Remark 6, is also tempered. Therefore, if , then there exists such that, for all ,

where

is tempered by Remark 6. Then it follows from (66), (67), and (68) that for all ,

Given , we define

where

is tempered. Then . Further, (71) indicates that is a random absorbing set for in , which completes the proof.

We next derive uniform estimates for in . From property (1) of Theorem 11 and the fact that [37, page 165]

for any , we get that .

Lemma 14. Let and . Then for a.e. , the solution of problem (27)-(28) satisfies, for all ,

where and are tempered and is the random function in Lemma 13.

Proof. Replacing with in (44), we get that, for all and ,

Integrating (76) over the interval leads to

which, together with (68), implies that, for all ,

Obviously, is tempered. Integrating (37) over the interval leads to

Using the Young inequality and , we obtain that

It follows from (79) and (80) that

Thus,

Replacing with in (82), we get from (71) that, for a.e. and all ,

Therefore, we have that, for all ,

By Remark 6, is tempered. Then the lemma follows from (78) and (84).

Lemma 15. Let and . Then for a.e. , the solution of problem (27)-(28) satisfies, for all and ,

where and are tempered and is the random function in Lemma 13.

Proof. Taking the inner product of (27) with in , we get that

Using the Young inequality, we obtain that

It follows from (86) and (87) that

Thus,

Let be the positive constant in Lemma 13, and take and . Integrating (89) over the interval leads to

Integrating the above with respect to over the interval , we get that

Replacing with and by - and Lemmas 13 and 14, we find that, for all ,

where

is tempered by Remark 6. Then we have that, for all ,

By , one can easily see that is also tempered for any tempered random variable . Hence, is tempered.

Let , . Integrating (89) over the interval leads to

Replacing with in (95) and by -, Lemma 13, and (94), we find that, for all and ,

where

is tempered by Remark 6. Similar to , is also tempered. Then the lemma follows from (94) and (96).

Lemma 16. Let and . Then for every and a.e. , there exist and such that the solution of problem (27)-(28) satisfies, for all ,

Proof. Let be a smooth function defined on such that for all , and

Then there exists a positive deterministic constant such that for all . Taking the inner product of (27) with in , we get that

We now estimate the terms in (100). First, we have that

Note that the second term on the right-hand side of (101) is bounded by

By (101) and (102), we find that

For the right-hand side of (100), applying the Young inequality, we obtain that

Then it follows from (100), (103), and (104) that

Consequently,

Take . For all , integrating (106) over the interval leads to

Using the Young inequality and , we obtain that

By (107) and (108), we find that

Then we have, for all ,

If we take , then by (110) we find that, for all ,

Then we have, for all ,

Replacing with , we find that

We now estimate the terms in (113) as follows. First, replacing with and then replacing with in (67), we find that

Thus,

Since and are tempered functions, is tempered set and ; we find from (115) that, for every , there exists such that, for all ,

Next, note that

By the Lebesgue theorem of dominated convergence, there exists such that, for all ,

Then it follows from (118) that, for all and ,

For the third term on the right-side of (113), we get from (70) and (94) that, for all ,

Then by (120), we obtain that

Since is tempered, the last integral in (121) exists. Therefore, there exists such that, for all ,

Finally, since , there exists such that, for all ,

Taking , it follows from (116), (119), (122), and (123) that, for all and ,

which completes the lemma.

Lemma 17. Let and . Then the solution of problem (27)-(28) satisfies, for all and ,

where and are tempered random functions.

Proof. By (120), we find that, for all and ,

Next, by - and Lemma 14, we find that, for all and (assuming for simplicity),

By Lemma 15, we obtain that

Then it follows from (127), (128), and (129) that

where

It is not difficult to see that and are tempered. The lemma follows from (127) and (131).

Lemma 18. The random dynamical system is -pullback asymptotically compact in ; that is, for a.e. , the sequence has a convergent subsequence in provided , , and .

Proof. Denote by the set for each . Since , there exists such that for all . Then by Lemma 17, we find that, for all and ,

By the compactness of embedding , it follows from (132) that for each the sequence is relatively compact in . On the other hand, by Lemma 17, we also find that, for all and ,

Hence, the sequence is equicontinuous. By the Ascoli-Arzelà theorem, for each the sequence is relatively compact in . Then, by a diagonal procedure, we can extract a subsequence such that for each , the sequence converges to in .

Obviously, for fixed and , coincides with on . Therefore, one can define unambiguously a measurable function by stipulating that it is equal to on . By Lemma 13, we obtain that, for all , , and ,

Thus,

which implies

Hence, .

For every and , by Lemma 16, there exist and such that, for all and ,

Since , there exists such that for all . Then we get from (137) that, for all ,

Take a fixed integer . Then we find from (138) that, for all integer and ,

which implies

As the sequence converges to in , there exists such that, for all ,

Let . Then it follows from (138), (140), and (141) that, for all ,

which shows that the sequence converges to in . This completes the proof.

We are now in a position to present our main result about the existence of a -random attractor for in .

Theorem 19. The random dynamical system has a unique -random attractor in .

Proof. Notice that has a closed absorbing set in by Lemma 13 and is -pullback asymptotically compact in by Lemma 18. Hence, the existence of a unique -random attractor for follows from Proposition 10 immediately.

Since and are conjugated by the random homeomorphism and is tempered, then, by Proposition in [33], has a unique -random attractor in which is given by

The proof is complete.

Acknowledgments

The authors are very grateful to the anonymous reviewers for their careful reading of the paper and valuable comments, which improved the presentation of this paper. This work is partially supported by the National Natural Science Foundation of China under Grants 11071166 and 11271110, the Key Programs for Science and Technology of the Education Department of Henan Province under Grant 12A110007, and the Scientific Research Start-up Funds of Henan University of Science and Technology.

References

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

F. Flandoli and B. Schmalfuss, “Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise,” Stochastics and Stochastics Reports, vol. 59, no. 1-2, pp. 21–45, 1996.

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, USA, 1988.

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, The Netherlands, 1992.

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, NY, USA, 2nd edition, 1997.

H. Crauel, A. Debussche, and F. Flandoli, “Random attractors,” Journal of Dynamics and Differential Equations, vol. 9, no. 2, pp. 307–341, 1997.

X. Fan, “Random attractors for damped stochastic wave equations with multiplicative noise,” International Journal of Mathematics, vol. 19, no. 4, pp. 421–437, 2008.

Y. Li and B. Guo, “Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations,” Journal of Differential Equations, vol. 245, no. 7, pp. 1775–1800, 2008.

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.

B. Wang, “Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains,” Journal of Differential Equations, vol. 246, no. 6, pp. 2506–2537, 2009.

B. Wang, “Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2811–2828, 2009.

Z. Shen, S. Zhou, and W. Shen, “One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation,” Journal of Differential Equations, vol. 248, no. 6, pp. 1432–1457, 2010.

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. Gess, “Random attractors for degenerate stochastic partial differential equations,” Journal of Dynamics and Differential Equations, vol. 25, no. 1, pp. 121–157, 2013.

B. Gess, W. Liu, and M. Röckner, “Random attractors for a class of stochastic partial differential equations driven by general additive noise,” Journal of Differential Equations, vol. 251, no. 4-5, pp. 1225–1253, 2011.

W.-J. Beyn, B. Gess, P. Lescot, and M. Röckner, “The global random attractor for a class of stochastic porous media equations,” Communications in Partial Differential Equations, vol. 36, no. 3, pp. 446–469, 2011.

Z. Wang, S. Zhou, and A. Gu, “Random attractor for a stochastic damped wave equation with multiplicative noise on unbounded domains,” Nonlinear Analysis: Real World Applications, vol. 12, no. 6, pp. 3468–3482, 2011.

Z. Wang and S. Zhou, “Random attractor for stochastic reaction-diffusion equation with multiplicative noise on unbounded domains,” Journal of Mathematical Analysis and Applications, vol. 384, no. 1, pp. 160–172, 2011.

W. Zhao and Y. Li, “(L ²,L p)-random attractors for stochastic reaction-diffusion equation on unbounded domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 2, pp. 485–502, 2012.

A. Gu, “Random attractors of stochastic three-component reversible Gray-Scott system on unbounded domains,” Abstract and Applied Analysis, vol. 2012, Article ID 419472, 22 pages, 2012.

L. Wang and D. Xu, “Asymptotic behavior of a class of reaction-diffusion equations with delays,” Journal of Mathematical Analysis and Applications, vol. 281, no. 2, pp. 439–453, 2003.

T. Caraballo and J. Real, “Attractors for 2D-Navier-Stokes models with delays,” Journal of Differential Equations, vol. 205, no. 2, pp. 271–297, 2004.

J. Li and J. Huang, “Uniform attractors for non-autonomous parabolic equations with delays,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 5-6, pp. 2194–2209, 2009.

P. Marín-Rubio, J. Real, and J. Valero, “Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 5, pp. 2012–2030, 2011.

M. Conti and M. Coti Zelati, “Attractors for the Cahn-Hilliard equation with memory in 2D,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1668–1682, 2010.

P. Marín-Rubio and J. Real, “Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 10, pp. 2784–2799, 2007.

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.

X. Ding and J. Jiang, “Random attractors for stochastic retarded lattice dynamical systems,” Abstract and Applied Analysis, vol. 2012, Article ID 409282, 27 pages, 2012.

W. Yan, Y. Li, and S. Ji, “Random attractors for first order stochastic retarded lattice dynamical systems,” Journal of Mathematical Physics, vol. 51, no. 3, Article ID 032702, 17 pages, 2010.

B. Wang, “Attractors for reaction-diffusion equations in unbounded domains,” Physica D, vol. 128, no. 1, pp. 41–52, 1999.

A. Rodriguez-Bernal and B. Wang, “Attractors for partly dissipative reaction diffusion systems in ,” Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 790–803, 2000.

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

I. Chueshov, Monotone Random Systems Theory and Applications, Springer, Berlin, Germany, 2002.

P. W. Bates, H. Lisei, and K. Lu, “Attractors for stochastic lattice dynamical systems,” Stochastics and Dynamics, vol. 6, no. 1, pp. 1–21, 2006.

G. da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1992.

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, NY, USA, 1983.

A. Bensoussan, G. da Prato, M. C. Delfour, and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Birkhäauser, Boston, Mass, USA, 2nd edition, 2007.