1. Introduction

In this paper, we consider the asymptotic behavior of solutions to the following stochastic reaction-diffusion equation (SRDE) with multiplicative noise:

(1.1)

with the initial-boundary value conditions

(1.2)

where and is a bounded open set with regular boundary and is a two-sided real-valued Wiener process on a probability space which will be specified later.

The nonlinearity satisfies the following conditions:

(1.3)

(1.4)

for some and for all

The asymptotic behavior of a random dynamical system (RDS) is captured by random attractors, which were first introduced in ^{[5, 11]}. They are compact invariant random sets attracting all the orbits, but the attraction to it may be arbitrary. This drawback can be overcome by creating the notion of exponential attractor, which is a compact, positively invariant set of finite dimension and exponentially attract each orbit at an exponential rate. The existence of exponential attractors for deterministic case has been extensively studied since 1994, ^[7] (^{[3, 6, 8, 9, 10]}). The concept of random exponential attractors was first introduced by A. Shirikyan and S. Zelik in ^[12]. They construct a random exponential attractor for abstract RDS and study its dependence on a parameter. In this paper, we devote to construct an exponential attractor for RDS and discuss the exponential attractive property of a random attractor. Firstly, we extend the deterministic result in ^[9] to stochastic case. Since we mainly concentrate on the exponential attractive property, we don't intend to discuss the time regularity of exponential attractors and its dependence on a parameter as in ^[12]. We then prove that a random attractor is actually an exponential attractor when the RDS satisfies Lipschitz continuity with small coefficient. Finally, we apply the abstract results to Eq.(1.1) to show that the corresponding RDS possesses an exponential attractors. When the derivative of the nonlinearity satisfies some restrictive condition, the random attractor become a point, i.e., the random equilibrium, and it attracts every orbit exponentially.

We organize this paper as follows. In section 2, we recall some basic notions of random attractors for RDS. In section 3, we present our main results and give the proofs. In section 4, we show our application to Eq.(1.1).

Throughout this paper, we denote by the norm of Banach space The norm of is written as denotes the random attractor for RDS in a Banach space

2. Preliminaries and Main Results

Let be a probability space, and be the Borel -algebra of In this paper, the term -a.s. (the abbreviation for almost surely) denote that an event happens with probability one. In other words, the set of possible exceptions may be non-empty, but it has probability zero. Moreover, we need the following definitions, see ^{[2, 4, 5, 13]} for more details.

Definition 2.1. is called a (discrete or continuous) metric dynamical system (MDS) if is -measurable, is the identity on or all and for all

Definition 2.2. The RDS on X over an is a mapping which is -measurable and satisfies for -

(i) on

(ii) (cocycle property) on X for all

An RDS is said to be continuous on if is continuous for all and -

Definition 2.3. A random bounded set of is called tempered with respect to if for -

where

Definition 2.4. (1) A random set is said to be a random absorbing set for if for every there exists such that for -

(2) A random set is said to be -pullback attracting if for any we have for -

where denotes the Hausdorff semi-distance between and in given by

(3) A random set is said to be a random attractor if the following conditions are satisfied for -,

(i) is compact, and is measurable for every

(ii) is invariant, that is for all

(iii) attracts every random set in

(4) A random set is said to be a random exponential attractor if the following conditions are satisfied for -

(i) is compact;

(ii) is positively-invariant, that is, for all

(iii) attracts every random set in exponentially, that is, there is such that for

(iv) has finite fractal dimension, that is, there exists a number such that

Our main results read as:

Theorem 2.1. Let and be Banach spaces such that is compactly embedded in Assume that is positively invariant under a nonlinear map and, for can be decomposed into a sum of two maps

(2.1)

and, for and any there exist and such that

(2.2)

and

(2.3)

Then, the discrete possesses a random exponential attractor.

In particular, when we have

Theorem 2.2. Let and be Banach spaces such that is compactly embedded in Let be a bounded random set positively invariant under Assume that, for

(2.4)

Then the discrete possesses a random exponential attractor.

Theorem 2.3. Let and be Banach spaces such that is compactly embedded in Let be a bounded random set positively invariant under Assume that, for

(2.5)

and

(2.6)

where Then the random exponential attractor is identical with the random attractor i.e., attracts every obit exponentially.

Assume that is an RDS on over an we define

Then, using the cocycle property of we have

This implies that is a discrete RDS over the on where

In the following, we still use instead of

Once the existence of exponential attractors for discrete case is proved the result for the continuous case follows in a standard manner (e.g., see ^[7]).

Theorem 2.4. Suppose that there is a such that satisfies of theorem 2.1, and the map is Hölder continuous from into for any Then has a random exponential attractor.

Next, we construct based on the random attractor

Lemma 2.5. For any fixed there exists an integer m0 such that for any

Furthermore,

Proof. Since is the random attractor, is a random absorbing set for any Thus, the first assertion follows from the definition of random absorbing set.

By the continuity of on and the cocycle property, we get

The proof is complete.

Set and for any fixed then we have

Lemma 2.6. for any Furthermore, is a random absorbing set for

Proof. On one hand, from lemma 2.5, we have

By replacing by we get

this implies

On the other hand, since we get

Thus, the first assertion hold. For any since is absorbing for there exists such that for all

Therefore

The proof is complete.

Proof of Theorem 2.1. We choose such that Since is bounded for there exists a ball of radius centered in which contains Setting It follows from (2.3) that the -ball covers Since the embedding is compact, we can cover the -ball by a finite number of balls in with centers Moreover, the finite number of ball in this covering has the following estimate

This implies that

(2.7)

It follows from (2.2) we get

(2.8)

Combining (2.7) and (2.8), we conclude that

(2.9)

where

Now, we enlarge the radius twice so that

(2.10)

and

We set

(2.11)

(2.12)

Applying the above covering process to every ball in the right-hand side in (2.10), we can generate the kth generation of centers in such that

(2.13)

(2.14)

Therefore, for any we find sets enjoy the following properties:

(2.15)

(2.16)

(2.17)

(2.18)

Now, we can construct the random exponential attractor for as follows:

(2.19)

Considering as deterministic sets with parameter then we can show that satisfies the conditions in definition 2.4 (4) (see ^[8] for deterministic case). Thus, is a random exponential attractor for The proof is completed.

Proof of Theorem 2.3. For any from (2.6) and the invariant of , we get

where Therefore

(2.20)

Combine (2.15) and (2.20), we can choose sets satisfying (2.15)-(2.18) with (2.18) replaced by

for some Therefore, the exponential attractor constructed in the proof of Theorem 2.1 is identical with the random attractor. The proof is complete.

3. Applications

In this section, we apply the above results to show that the RDS generated by Eq. (1.1) possess a random exponential attractor. To this end, we need to convert the stochastic equation into a deterministic equation with a random parameter. We consider the probability space where

is the Borel algebra induced by the compact-open topology of and is the corresponding Wienner measure on . Then we identify with i.e.,

Define the time shift by

Then is an ergodic MDS. We introduce an Otnstein-Uhlenbeck process

and it solves the Itȏ equation

From ^[14], it is known that the random variable is tempered, and there is a -invariant set of full measure such that for every is continuous in and

(3.1)

Moreover, there exists a tempered random variable such that

(3.2)

Setting from (3.1) one can easily show that and are temperate, and is continuous in t for Therefore, by using Proposition 4.3.3 in ^[1], for any there exists -slowly varying random variable such that

and satisfies, for

Therefore, for

(3.3)

If we set we can get from (3.2) and (3.3) that

(3.4)

(3.5)

for all and and is also tempered.

Let and we can consider the following evolution equation with random coefficients but without white noise:

(3.6)

with Dirichlet boundary condition

(3.7)

and initial condition

(3.8)

From ^[15] we see that for and for all the parameterized evolution equation (3.6)-(3.8) with conditions (1.3)-(1.4) has a unique solution

(3.9)

Furthermore, is continuous with respect to in for all and

Then is a solution of (1.1)-(1.2) with We now define a mapping by

(3.10)

Then is a continuous RDS on and an RDS on respectively associated with the SRDE (1.1)-(1.2) on

Theorem 3.1. (^[15]) Assume that (1.3)-(1.4) hold. Then the RDS generated by (1.1)-(1.2) has a unique random attractor in

Theorem 3.2. Assume that (1.3)-(1.4) hold. Let Then for there exists such that the solution of (1.1)-(1.2) satisfies that for all

(3.11)

where is a random variable and is the first eigenvalue of

Proof. Let , and Then satisfies the following equation

(3.12)

where and by (1.4).

Multiply the above equation with w to get

(3.13)

Thus,

(3.14)

We multiply both side with to get

(3.15)

Integrating the above inequality in we obtian

(3.16)

Next, we take inner product of (3.12) with in and use to get

(3.17)

That is

(3.18)

Integrating (3.13) from t to t + 1 and using (3.16), it yields

(3.19)

Combining (3.18) and (3.19) using Uniform Gronwall's Lemma (note that the Uniform Gronwall's inequality also hold when the right-hand side of (3.19) dependent on t), it yields

(3.20)

Let then replace by to get

(3.21)

We have used (3.5) in (3.21). By (3.1), there is for

(3.22)

Finally, by the relationship between u and v we obtain

(3.23)

The proof is complete.

Theorem 3.3. Assume that (1.3)-(1.4) hold. Then the RDS generated by Eq (1.1) has a random exponential attractor in

Proof. From theorem 2.2 and theorem 3.2, we see that, for some fixed the discrete possesses a random exponential attractor in where Moreover, by an elementary process, one can easily show that is Hölder continuous from into then by theorem 2.4 we obtain that the has a random exponential attractor in The proof is complete.

When the constant in (1.4) satisfies the random attractor reduces to a single point, i.e., a random equilibrium (see ^[15]). Moreover, we have

Theorem 3.4. Assume that (1.3)-(1.4) hold and Then the unique random equilibrium attracts every obit exponentially.

Proof. From (3.11) we get that for all

(3.24)

Using poincaré inequality, we obtain

(3.25)

Since we choose small enough and such that

(3.26)

for Then from (3.24) and (3.25), the conditions in theorem 2.3 are satisfied. From theorem 2.3 we arrive at our conclusion. The proof is complete.

4. Conclusion

In this paper, we have constructed exponential attractors for abstract RDS and discussed the exponential attractive property of a random attractor. Moreover, we have applied our abstract results to a stochastic reaction-diffusion equation. The abstract results presented in this paper have widely applications in RDS generated by many other stochastic partial differential equations, and these results will be applied in our future study.

Acknowledgments

The authors are grateful to the anonymous referees for helpful comments and suggestions that greatly improved the presentation of this paper.

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