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

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.


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 n  : where λ is a positive constant; j j D x ; f is a nonlinear function satisfying certain dissipative conditions; h j is given functions defined on n  ; and { } 1 m j j w = 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 n  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 n  .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 n  .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 t → ∞ .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

Preliminaries and Abstract Results
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.

{ }
A ω ω ∈Ω attracts every set in D, that is, for every where d is the Hausdorff semi-metric given by ( ) for any Y X ⊆ and Z X ⊆ .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 ( ) E ω ω ∈Ω is an arbitrary random set, and ( ) ω ω ⊂ for all ω ∈ Ω , then ( )

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

The Reaction-Diffusion Equation on  n with Distribution Derivatives and Additive
with initial condition where λ is a positive constant, ( ) are independent two-side real-valued wiener processes on a probability space which will be specified below, and ( ) 2k k λ > + .
In the sequel, we consider the probability space ( ) , , , , : 0 0 F is the Borel σ-algebra induced by the compact-open topology of Ω , and P the corresponding wiener measure on ( ) υ ω υ υ = 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 n  , further, one may show that ( ) 0 , , t υ ω υ is unique and continuous with respect to 0 υ in ( ) Then the process u is the solution of problem (3.1), (3.2), we now define a mapping : , .
Then φ is satisfies conditions ( 1)-( 3) in Definition 2.2 therefore φ is a continuous random dynamical sys- tem associated with the stochastic reaction-diffusion equation on n  .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 φ .

Uniform Estimates of Solutions
In this section, we drive uniform estimates on the solutions of (3.1), (3.2) defined on n  when t → ∞ 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 x , are uniformly small when the time is sufficiently large.
We always assume that D is the collection of all tempered subsets of ( ) the next lemma shows that φ has a random absorbing set in D.
there is , , for all .
Proof.We first derive uniform estimates on ( ) ( ) ( ) from which the uniform estimates on ( ) u t .Multipling (3.10) by υ and then integrating over n  , we have 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 ( )  , therefore, the right-hand side of (4.5) is bounded as following By (3.9), we find that for P-a.e, ω ∈ Ω ( ) ( ) it follows from (4.5), (4.6) that, all 0 t ≥ , ( ) ( ) which implies that for all 0 t ≥ , ( ) ( ) 2k k λ λ = − − .Applying Gronwall's lemma, we find that, for all 0 t ≥ , ( ) ( ) By assumption is tempered.On the other hand, by definition, ( ) which along with (4.12) shows that, for all ( ) ( ) .
Then for every 1 0 T ≥ and P-a.e ω ∈ Ω, the solutions where C is a positive deterministic constant independent of 1 T and ( ) r ω is the tempered function in (3.7).Proof.First, replacing t by 1 T and then replacing ω by )  By (4.7), the second term on the right-hand side of (4.16) satisfies From (4.16), (4.17) it follows that T t t t By (4.8) we find that, for ( ) ( ) where C is a positive deterministic constant and ( ) r ω is the tempered function in (3.7).Proof.First replacing t by t + 1 and then replacing 1 T by t in (4.14), we find that ( ) 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 satisfies, for all ( ) where C is a positive deterministic constant and ( ) r ω is the tempered function in (3.9).Proof.Let ( )

B
T ω be the positive constant in lemma 4.
, , 1 where C is a positive deterministic constant and ( ) r ω is the tempered function in (3.9).Proof.Taking the inner product of (3.10) with υ ∆ in ( ) 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 ( ) ( )  ,  , ,  d  d   , ,  , ,  , ,  d .
) ) 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 0 >  and P-a.e ω ∈ Ω , there exists ( ) ( ) Then there exists a constant C such that ( ) L  , and integrating over n  we find 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 ( ) Note that (4.49) implies that By lemma 4.1 and 4.5, there is , 0 Replacing ω by t θ ω − , we obtain from (4.52) that, for all ( ) where we have used (4.7).By (4.54), we find that, given 0 >  , there is , , By lemma 4.2, there is ( ) such that the fourth term on the right-hand side of (4.53) satisfies And hence, there is ( ) This implies that there exist ( ) Then the second term on the right-hand side of (4.53), there exist ( ) ( ) where ( ) r ω is the tempered function in (3.7) and C  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):

L
 .Otherwise, the norm of a general Banach space X is written as X ⋅ .The letters C and  are generic positive constants which may change their values form line to line or even in the same line.

ω ω ∈Ω must belong to D. Definition 2 . 7 .
Let D be an inclusion-closed collection of random subsets of X and φ a continuous RDS on closed random absorbing set for φ in D and φ is D-pullback asymptotically compact in X.Then φ has a unique D-random attractor

First replacing t by s and then replacing ω by t θ ω − in ( 4 .
10), we find that the third term on the right-hand side of (4.53) satisfies

 4 . 7 .
This completes the proof. Lemma Assume that g j , P-a.e ω ∈ Ω , there exists ( ) Let T  and R  be the constant in lemma 4.6 By (4.60) and (3.7) we have, for all t T ≥  and k 4.62) and lemma 4.6, we get that, for all t T ≥  and k R ≥  proof. Let D be a collection of random subsets of X.Then φ is said to be D-pullback asymptotical- Let D be a collection of random sunsets of X.Then a random set ∈Ω of X is called tempered with respect to ( ) . .
For the five term on the right-hand side of (4.53), we have