Random Attractors for Stochastic Reaction-Diffusion Equations with Distribution Derivatives on Unbounded Domains

Abstract

In this paper, we prove the existence of random attractors for a stochastic reaction-diffusion equation with distribution derivatives on unbounded domains. The nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears spatially distributed temporal white noise. The stochastic reaction-diffusion equation is recast as a continuous random dynamical system and asymptotic compactness for this demonstrated by using uniform estimates far-field values of solutions. The results are new and appear to be optimal.

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Ahmed, E. , Abdelmajid, A. , Xu, L. and Ma, Q. (2015) Random Attractors for Stochastic Reaction-Diffusion Equations with Distribution Derivatives on Unbounded Domains. Applied Mathematics, 6, 1790-1807. doi: 10.4236/am.2015.610159.

1. Introduction

The understanding of the asymptotic behavior of dynamical system is one of the most important problems of modern mathematical physics; one way to attack the problem for dissipative deterministic dynamical systems is to consider its global attractors. This is an invariant set that attracts all the trajectories of the system. Its geometry can be very complicated and reflects the complexity of the long-time dynamical of the systems. In this paper we investigate the asymptotic behavior of solutions to the following stochastic reaction-diffusion equations with distribution derivatives and additive noise defined in the space:

(1.1)

with initial data

(1.2)

where is a positive constant; is distribution derivatives;; f is a

nonlinear function satisfying certain dissipative conditions; hj is given functions defined on; and is independent two sided real-valued wiener processes on probability space which will be specified later.

Stochastic differential equations of this type arise from many physical systems when random spatio-temporal forcing is taken into account. In order to capture the essential dynamics of random systems with wide fluctuations, the concept of pullback random attractors was introduced in [1] , being an extension to stochastic systems of the theory of attractors for deterministic equations found in [2] - [5] , for instance. The existence of such random attractors has been studied for stochastic PDE on bounded domains; see, e.g. [6] [7] , and for stochastic PDE on unbounded domains, see, e.g. [8] [9] , and the references therein. In the present paper, we prove the existence of such a random attractor for stochastic reaction-diffusion Equation (1.1) defined in which is not founded.

Notice that the unboundedness of domain introduces a major difficulty for proving the existence of an attractor because Sobolev embedding theorem is no longer compact and so the asymptotic compactness of solutions cannot be obtained by the standard method. In the case of deterministic equations, this difficulty can be overcome by the energy equation approach, introduced by Ball in [10] and then employed by several authors to prove the asymptotic compactness of deterministic equations in unbounded domains. This idea was developed in [5] to prove asymptotic compactness for the deterministic version of (1.1) on. In this paper, we provide uniform estimates on the far-field values of solutions to circumvent the difficulty caused by the unboundedness of the domains. The main contribution of this paper is to extend the method of using tail estimates of the case stochastic dissipative PDEs and prove the existence of random attractor for the stochastic reaction-diffusion equation with distribution derivatives on the unbounded domain.

The paper is organized as follows. In Section 2, we recall some preliminaries and abstract results on the existence of a pullback random attractor for random dynamical systems. In Section 3, we transform (1.1) into a continuous random dynamical system. Section 4 is devoted to obtain the uniform estimates of solution as. These estimates are necessary for proving the existence of bounded absorbing sets and the asymptotic compactness of the equation. In Section 5, we first establish the asymptotic compactness of the solution operator by giving uniform estimates on the tails of solutions, and then prove the estimates of a random attractor.

We denote by and the norm and the inner product in and use to denote the norm in. Otherwise, the norm of a general Banach space X is written as. The letters C and are generic positive constants which may change their values form line to line or even in the same line.

As mentioned in the introduction, our main purpose is to prove the existence of a random attractor. For that matter, first, we will recapitulate basic concepts related to random attractors for stochastic dynamical systems. The reader is referred to [6] [11] -[13] for more details. Let be a separable Hilbert space with Borel s-algebra B(X), and let be a probability space.

Definition 2.1. is called a metric dynamical system if: is - measurable, is the identity on, for all and for all.

Definition 2.2. A continuous random dynamical system (RDS) on X over a metric dynamical system is a mapping

which is―measurable and satisfies, for P-a.e., (1) is the iden- tity on X; (2) for all (3) is continuous for all. Hereafter, we always assume that is a continuous RDS on X over.

Definition 2.3. A random bounded set of X is called tempered with respect to if for P-a.e,

where.

Definition 2.4. Let D be a collection of random subsets of X and. Then is

called a random absorbing set for in D if for every and P-a.e, there exists such that

Definition 2.5. Let D be a collection of random subsets of X. Then is said to be D-pullback asymptotical-

ly compact in X if for P-a.e, has a convergent subsequence in X whenever,

and with.

Definition 2.6. Let D be a collection of random sunsets of X. Then a random set of X is called a D-random attractor (or D-pullback attractor) for if the following conditions are satisfied, for P-a.e., (1) is compact, and is measurable for every; (2) is invariant, that is,

(3) attracts every set in D, that is, for every,

where d is the Hausdorff semi-metric given by for any and. The following existence result for a random attractor for a continuous RDS can be found in [8] [13] . First, recall that a collection D of random subsets is called inclusion closed if whenever is an arbitrary random set, and is in D with for all, then must belong to D.

Definition 2.7. Let D be an inclusion-closed collection of random subsets of X and a continuous RDS on X over. Suppose that is a closed random absorbing set for in D and is D-pullback asymptotically compact in X. Then has a unique D-random attractor which is given by

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

3. The Reaction-Diffusion Equation on Rn with Distribution Derivatives and Additive Noise

(3.1)

with initial condition

(3.2)

where is a positive constant, , , for some,

are distribution derivative, are independent two-side real-valued wiener processes on a probability

space which will be specified below, and with the following assumptions:

(3.3)

(3.4)

(3.5)

for and, where, , are positive constants and.

In the sequel, we consider the probability space where

.

is the Borel s-algebra induced by the compact-open topology of, and P the corresponding wiener measure on. Then we identify with

Define the time shift by

Then is a metric dynamical system.

We now associate a continuous random dynamical system with the stochastic reaction-diffusion equation over. To this end, we need to convert the stochastic equation with a random additive term in to a deterministic equation with a random parameter. Given consider the One-dimensional Ornstein- uhlenbeck equation

(3.6)

The solution of (3.6) is given by

Note that the random variable is tempered and is P-a.e continuous, therefore, it follows form proposition 4.3.3 in [11] that there exists a tempered function such that

(3.7)

where satisfies for P-a.e

(3.8)

Then it follows form (3.7), (3.8) that, for P-a.e.

(3.9)

Putting by (3.6) we have

The existence and uniqueness of solutions to the stochastic partial differential Equation (3.1) with initial condition (3.2) which can be obtained by standard Fatou-Galerkin methods. To show that problem (3.1), (3.2) generates a random system, we let where u is a solution of problem (3.1), (3.2), then satisfies

(3.10)

By a Galerkin method, one can show that if f satisfies (3.3)-(3.5), then in the case of a bounded domain with Dirichlet boundary conditions, for P-a.e., and for all, (3.10) has a unique solution

with for every T > 0, one may take the domain to be a sequence of Balls with radius approaching to deduce the existence of a weak solution to (3.10) on, further, one may show that is unique and continuous with respect to in for all. Let

.

Then the process u is the solution of problem (3.1), (3.2), we now define a mapping by

(3.11)

Then is satisfies conditions (1)-(3) in Definition 2.2 therefore is a continuous random dynamical system associated with the stochastic reaction-diffusion equation on. In the next two sections, we establish uniform estimates for the solutions of problem (3.1), (3.2) and prove the existence of a random attractor for.

4. Uniform Estimates of Solutions

In this section, we drive uniform estimates on the solutions of (3.1), (3.2) defined on when with the purpose of proving the existence of a bounded random absorbing set and the asymptotic compactness of the random dynamical system associated with the equation. In particular, we will show that the tails of the solutions, i.e. solutions evaluated at large values of, are uniformly small when the time is sufficiently large.

We always assume that D is the collection of all tempered subsets of with respect to the next lemma shows that has a random absorbing set in D.

Lemma 4.1. Assume that gj, , and (3.3)-(3.5) hold. Then there exists such that

is a random absorbing set for in D, that is, for any and P-a.e,

there is such that

Proof. We first derive uniform estimates on from which the uniform estimates on. Multipling (3.10) by and then integrating over, we have

(4.1)

For the nonlinear term, by (3.3)-(3.5) we obtain

(4.2)

on the other hand, the next two terms on the right-hand side of (4.1) are bounded by

(4.3)

the last term on the right-hand side of (4.1) is bounded by

(4.4)

where and.

Then it follows from (4.1)-(4.4) that

(4.5)

Note that and, therefore, the right-hand side of (4.5) is bounded as following

(4.6)

By (3.9), we find that for P-a.e,

(4.7)

it follows from (4.5), (4.6) that, all,

(4.8)

which implies that for all,

(4.9)

Let. Applying Gronwall’s lemma, we find that, for all,

(4.10)

By replacing by, we get from (4.10) and (4.7) that for all

(4.11)

Note that.

So from (4.11) we get that, for all,

(4.12)

By assumption is tempered. On the other hand, by definition, is also tempered,

therefore, if. Then there is such that for all

which along with (4.12) shows that, for all,

(4.13)

Given

Then, further, (4.13) indicates that is a random absorbing set for in D.

Which completes the Proof. ,

We next drive uniform estimates for u in and for u in.

Lemma 4.2. Assume that and (3.3)-(3.5) hold, let and.

Then for every and P-a.e, the solutions of problem (3.1), (3.2) and of (3.11) with satisfy, for all.

(4.14)

(4.15)

where C is a positive deterministic constant independent of and is the tempered function in (3.7).

Proof. First, replacing t by and then replacing by in (4.10) we find that

Multiply the above by and then simplify to get.

(4.16)

By (4.7), the second term on the right-hand side of (4.16) satisfies

(4.17)

From (4.16), (4.17) it follows that

(4.18)

By (4.8) we find that, for

(4.19)

Dropping the first term on the left-hand side of (4.19) and replacing by, we obtain that, for all

(4.20)

By (4.7), the second term on the right-hand side of (4.20) satisfies, for all

(4.21)

Then, using (4.20) and (4.21), it follows from (4.20) that

This completes the proof. ,

Lemma 4.3. Assume that gj, and (3.3)-(3.5) hold, Let and.

Then for P-a.e, there exists such that the solutions of problem (3.1), (3.2) and of (3.11) with satisfy, for all.

where C is a positive deterministic constant and is the tempered function in (3.7).

Proof. First replacing t by t + 1 and then replacing by t in (4.14), we find that

(4.22)

Note that for, hence, form (4.22) we have

(4.23)

Since and are tempered there is such that for all

which along with (4.23) shows that, for all,

(4.24)

Then from (4.10) using the same steps of last process applying on (4.15), we get that

(4.25)

The above uniform estimates is a special case lemma 4.2, then the lemma follows from (4.24)-(4.25). ,.

Lemma 4.4. Assume that gj, and (3.3)-(3.5) hold, let and.

Then for P-a.e, there exists such that the solution of problem (3.1), (3.2) satisfies, for all.

where C is a positive deterministic constant and is the tempered function in (3.9).

Proof. Let be the positive constant in lemma 4.3, take and, by (3.11) we find that

(4.26)

By (3.9) we obtain

(4.27)

Now integration (4.26) with respect to s over (t, t + 1), by lemma 4.3 and inequality (4.27), we have

(4.28)

Then the lemma follows from (4.28). ,

Lemma 4.5. Assume that gj, and (3.3)-(3.5) hold, let and. Then for P-a.e, there exists such that for all.

where C is a positive deterministic constant and is the tempered function in (3.9).

Proof. Taking the inner product of (3.10) with in, we get that

(4.29)

We estimates the first term in the right-hand side of (4.29) by (3.3), (3.4) we have

(4.30)

On the other hand, the second term on the right-hand side of (4.29) is bounded by

(4.31)

The last term on the right-hand side of (4.29) is bounded by

(4.32)

By (4.29)-(4.32) we get that

(4.33)

Let

(4.34)

Since and, there are positive constants and such that

which along with (3.9) shows that

(4.35)

By (4.33), (4.34) we have

(4.36)

Let be the positive constant in lemma 4.3 take and. Then integrate 4.36 over (s, t + 1) to get that

Now integrating the above equation with respect to s over (t, t + 1), we find that

Replacing by we obtain that

(4.37)

By lemma 4.3 and 4.4, it follows from (4.37) and (4.35) that, for all

(4.38)

Then by 4.38 and 3.9, we have, for all

which completes the proof. ,

Lemma 4.6. Assume that gj, and (3.3)-(3.5) hold, let and.

Then for every and P-a.e, there exists and such that the solution of (3.10) with satisfies, for all

Proof. Choose a smooth function defined on such that for all and

Then there exists a constant C such that for any, multiplying (3.10) by in

, and integrating over we find that

(4.39)

We now estimate the terms in (4.39) as follows, first we have

(4.40)

Note that the second term on the right-hand side of (4.40) is bounded by

(4.41)

By (4.40), (4.41), we find that

(4.42)

From (4.39) the first term on the right-hand side, we have

(4.43)

By (3.3), the first term on the right-hand side of (4.43) is bounded by

(4.44)

By (3.4), the second term on the right-hand side of (4.43) is bounded by

(4.45)

Then it follows from (4.43)-(4.45) we have that

(4.46)

For the second term on the right-hand side of (4.39) we have

(4.47)

For the last term on the right-hand side of (4.39), we have that

(4.48)

Finally, by (4.39), (4.42) and (4.47) (4.48), we have that

(4.49)

Note that (4.49) implies that

(4.50)

By lemma 4.1 and 4.5, there is such that for all,

(4.51)

Now integrating (4.50) over we get that, for all

(4.52)

Replacing by, we obtain from (4.52) that, for all,

(4.53)

In what follows, we estimate the terms in (4.53). First replacing t by and then replacing by in (4.10), we have the following bounds for the first term on the right-hand side of (4.53)

(4.54)

where we have used (4.7). By (4.54), we find that, given, there is such that for all

(4.55)

By lemma 4.2, there is such that the fourth term on the right-hand side of (4.53) satisfies

And hence, there is such that for all and,

(4.56)

First replacing t by s and then replacing by in (4.10), we find that the third term on the right-hand side of (4.53) satisfies

This implies that there exist and such that for all and,

(4.57)

Then the second term on the right-hand side of (4.53), there exist and such that for all and we have that

(4.58)

Note that,. therefore, there is such that for all,

For the five term on the right-hand side of (4.53), we have

(4.59)

Note that and Hence there is such

that for all and.

(4.60)

where is the tempered function in (3.7) and is the positive constant in the last term on the right-hand side of (4.60), By (4.60) and (3.7), (3.8), we have the following bounds for the last term on the right-hand side of (4.53):

(4.61)

Let and then it follows from (4.53), (4.55)-(4.61) that, for all and, we have

which shows that for all and

This completes the proof. ,

Lemma 4.7. Assume that gj, and (3.3)-(3.5) hold. Let and.

Then for every and P-a.e, there exists and such that, for all

Proof. Let and be the constant in lemma 4.6 By (4.60) and (3.7) we have, for all and

(4.62)

then by (4.62) and lemma 4.6, we get that, for all and

which completes the proof. ,

5. Random Attractors

In this section, we prove the existence of a D-random attractor for the random dynamical system associated with the stochastic reaction-diffusion Equations (3.1), (3.2) on. It follows from lemma 4.1 that has a closed random absorbing set in D, which along with the D-pullback asymptotic compactness will imply the existence of a unique D-random attractor. The D-pullback asymptotic compactness of is given below and will be proved by using the uniform estimates on the tails of solutions.

Lemma 5.1. Assume that gj, and (3.3)-(3.5) hold. Then the random dynamical system ϕ is D-

pullback asymptotically compact in; that is, for P-a.e, the sequence

has a convergent subsequence in provided, and

Proof. Let and Then by lemma 4.1 for P-a.e, we have that

Hence, there is such that, up to a subsequence,

(5.1)

Next, we prove the weak convergence of (5.1) is actually strong convergence. Given, by lemma 4.7, there is and such that for all,

(5.2)

Since, there is such that for every. Hence, it follows from (5.2) that for all,

(5.3)

On the other hand, by lemma 4.1 and 4.5, there such that for all,

(5.4)

Let be large enough such that tn ≥ T2 for n ≥ N2. then by (5.4) we have that, for all n ≥ N2,

(5.5)

Denote by the set. By the compactness of embedding, it fol- lows from (5.5) that, up to a subsequence,

which shows that for the given, there exists such that for all,

(5.6)

Note that. Therefore there exists such that

(5.7)

let and By (5.3), (5.6), and (6.7), we find that for all,

which shows that

as wanted. ,

Now we are in a position to present our main result: the existence of a D-random attractor for in

Theorem 5.2. Assume that, and (3.3)-(3.5) hold. Then the random dynamical system has a unique D-random attractor in.

Proof. Notice that has a closed random absorbing set in D by lemma 4.1, and is D-pullback

asymptotically compact in by lemma 5.1. Hence the existence of a unique D-random attractor for follows from proposition 2.7 immediately. ,

Foundation Term

This work was supported by the NSFC (11101334).

NOTES

*Corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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