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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.

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

with initial data

where

nonlinear function satisfying certain dissipative conditions; h_{j} is given functions defined on

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 [

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 [

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

We denote by

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 [

Definition 2.1.

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

which is

Definition 2.3. A random bounded set

where

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

called a random absorbing set for

Definition 2.5. Let D be a collection of random subsets of X. Then

ly compact in X if for P-a.e

and

Definition 2.6. Let D be a collection of random sunsets of X. Then a random set

(3)

where d is the Hausdorff semi-metric given by

Definition 2.7. Let D be an inclusion-closed collection of random subsets of X and

In this paper, we will take D as the collection of all tempered random subsets of

with initial condition

where

are distribution derivative,

space which will be specified below, and

for

In the sequel, we consider the probability space

Define the time shift by

Then

We now associate a continuous random dynamical system with the stochastic reaction-diffusion equation over

The solution of (3.6) is given by

Note that the random variable

where

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

Putting

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

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.

with

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

Then

In this section, we drive uniform estimates on the solutions of (3.1), (3.2) defined on

We always assume that D is the collection of all tempered subsets of

Lemma 4.1. Assume that g^{j},

there is

Proof. We first derive uniform estimates on

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

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

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

where

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

Note that

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

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

which implies that for all

Let

By replacing

Note that

So from (4.11) we get that, for all

By assumption

therefore, if

which along with (4.12) shows that, for all

Given

Then

Which completes the Proof. ,

We next drive uniform estimates for u in

Lemma 4.2. Assume that

Then for every

where C is a positive deterministic constant independent of

Proof. First, replacing t by

Multiply the above by

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

From (4.16), (4.17) it follows that

By (4.8) we find that, for

Dropping the first term on the left-hand side of (4.19) and replacing

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

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

This completes the proof. ,

Lemma 4.3. Assume that g^{j},

Then for P-a.e

where C is a positive deterministic constant and

Proof. First replacing t by t + 1 and then replacing

Note that

Since

which along with (4.23) shows that, for all

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

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 g^{j},

Then for P-a.e

where C is a positive deterministic constant and

Proof. Let

By (3.9) we obtain

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

Then the lemma follows from (4.28). ,

Lemma 4.5. Assume that g^{j},

where C is a positive deterministic constant and

Proof. Taking the inner product of (3.10) with

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

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

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

By (4.29)-(4.32) we get that

Let

Since

which along with (3.9) shows that

By (4.33), (4.34) we have

Let

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

Replacing

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

Then by 4.38 and 3.9, we have, for all

which completes the proof. ,

Lemma 4.6. Assume that g^{j},

Then for every

Proof. Choose a smooth function

Then there exists a constant C such that

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

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

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

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

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

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

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

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

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

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

Note that (4.49) implies that

By lemma 4.1 and 4.5, there is

Now integrating (4.50) over

Replacing

In what follows, we estimate the terms in (4.53). First replacing t by

where we have used (4.7). By (4.54), we find that, given

By lemma 4.2, there is

And hence, there is

First replacing t by s and then replacing

This implies that there exist

Then the second term on the right-hand side of (4.53), there exist

Note that

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

Note that

that for all

where

Let

which shows that for all

This completes the proof. ,

Lemma 4.7. Assume that g^{j},

Then for every

Proof. Let

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

which completes the proof. ,

In this section, we prove the existence of a D-random attractor for the random dynamical system

Lemma 5.1. Assume that g^{j},

pullback asymptotically compact in

has a convergent subsequence in

Proof. Let

Hence, there is

Next, we prove the weak convergence of (5.1) is actually strong convergence. Given

Since

On the other hand, by lemma 4.1 and 4.5, there

Let _{n} ≥ T_{2} for n ≥ N_{2}. then by (5.4) we have that, for all n ≥ N_{2},

Denote by

which shows that for the given

Note that

let

which shows that

as wanted. ,

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

Theorem 5.2. Assume that

Proof. Notice that

asymptotically compact in

This work was supported by the NSFC (11101334).

Eshag MohamedAhmed,Ali DafallahAbdelmajid,LingXu,QiaozhenMa, (2015) Random Attractors for Stochastic Reaction-Diffusion Equations with Distribution Derivatives on Unbounded Domains. Applied Mathematics,06,1790-1807. doi: 10.4236/am.2015.610159