(L,H)-Random Attractors for Stochastic Reaction-Diffusion Equation on Unbounded Domains

and Applied Analysis 3 of all nonempty subsets of X and S X the class of all families D = {D(ω)} ω∈Ω ⊂ P(X). P(Y), and S Y can be defined in the same way. We consider the given nonempty subclasses D X ,D Y , whereD X ⊂ S X ,D Y ⊂ S Y . Definition 5. A family B = {B(ω)} ω∈Ω ∈ D Y is said to be (X, Y)-random absorbing for φ if, for everyD = {D(ω)} ω∈Ω ∈ D X , there exists T ?̂? (ω) > 0 such that, for P-a.e. ω ∈ Ω, φ (t, θ −t ω,D (θ −t ω)) ⊂ B (ω) , ∀t ≥ T ?̂? (ω) . (11) Definition 6. A family C = {C(ω)} ω∈Ω ∈ D Y is said to be (X, Y)-random attracting for φ if, for everyD = {D(ω)} ω∈Ω ∈ D X , we have, for P-a.e. ω ∈ Ω, lim t→∞ d Y (φ (t, θ −t ω,D (θ −t ω)) , C (ω)) = 0, (12) where d Y (C 1 , C 2 ) denotes the Hausdorff semi-distance between C 1 and C 2 in Y; that is, d Y (C 1 , C 2 ) = sup x∈C 1 inf y∈C 2 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩Y for C 1 , C 2 ⊂ Y. (13) Definition 7. The RDS φ is said to be (X, Y)-asymptotically compact if, for P-a.e. ω ∈ Ω, {φ(t n , θ −t n ω, x n )} ∞ n=1 has a convergent subsequence in Y whenever t n → ∞ and x n ∈ D(θ −t n ω) withD = {D(ω)} ω∈Ω ∈ D X . Definition 8. A random set A = {A(ω)} ω∈Ω ∈ S Y is said to be an (X, Y)-random attractor if the following conditions are satisfied for P-a.e. ω ∈ Ω, (i) A(ω) is closed inX and compact in Y; (ii) A = {A(ω)} ω∈Ω is invariant; that is, φ(t, ω, A(ω)) = A(θ t ω) for all t ≥ 0; (iii) A = {A(ω)} ω∈Ω attracts every random set in D X in the norm topology of Y in the sense of (12). 2.2. Abstract Results. Now, we present the main abstract results. Recall that a collectionD of random subsets is called inclusion closed if whenever {E(ω)} ω∈Ω is an arbitrary random set and {F(ω)} ω∈Ω is in D with E(ω) ⊂ F(ω) for all ω ∈ Ω, then {E(ω)} ω∈Ω ∈ D. The following theorem is an adaptation of a result of [25] to the case of RDS. The proof is similar to that of [25], and here we omit it. Theorem 9. Let φ be a continuous RDS on X and an RDS on Y over (Ω,F,P, (θ t ) t∈R), respectively, andDX andDY are inclusion closed. (i) Case 1 (X = Y) (see [1]). Assume that the family B 0 = {B 0 (ω)} ω∈Ω ∈ D X is a closed (X,X)-random absorbing set for φ and φ is (X,X)-asymptotically compact.Thenφ has a unique (X,X)-randomattractor A = {A(ω)} ω∈Ω which is given by A (ω) = ⋂ s≥0 ⋃ t≥s φ (t, θ −t ω, B 0 (θ −t ω)) X , (14) where AX denotes the closure of A with respect to the norm topology inX. (ii) Case 2 (X ̸ =Y). If the assumptions in (i) are satisfied, moreover, we assume that B 1 = {B 1 (ω)} ω∈Ω ∈ D Y is (X, Y)-random absorbing and φ is (X, Y)asymptotically compact. Then φ has an (X, Y)-random attractor A = {A󸀠(ω)} ω∈Ω which is given by A 󸀠 (ω) = ⋂ s≥0 ⋃ t≥s φ (t, θ −t ω, B 0 (θ −t ω)⋂B 1 (θ −t ω)) X = ⋂ s≥0 ⋃ t≥s φ (t, θ −t ω, B 0 (θ −t ω)⋂B 1 (θ −t ω)) Y , (15) where B 0 = {B 0 (ω)} ω∈Ω ∈ D X is the (X,X)-random absorbing set in (i). In the following of this paper we only consider X = L 2 (R), Y = H(R), and D X = D 2 , D Y = DV, where D2 andDV denote the collections of all tempered random subsets of L2(Rn) andH(R), respectively. Theorem10. Assume thatφ is an RDS onL(R) andH(R), respectively, and then φ is (L(R),H(R))-asymptotically compact if (i) for every D = {D(ω)} ω∈Ω ∈ D 2 , P-a.e. ω ∈ Ω and every ε > 0, there exist k 0 = k 0 (ω, ε) > 0 and T ?̂? (ω, ε) > 0 such that, for all t ≥ T ?̂? (ω, ε), 󵄩󵄩󵄩󵄩󵄩 χ Q c k0 ⋅ φ (t, θ −t ω, x) 󵄩󵄩󵄩󵄩H1(Qc k0 ) ≤ ε, ∀x ∈ D (θ −t ω) , (16) (ii) χ Q k ⋅ φ is (L2(Rn),H1(Q k ))-asymptotically compact, ∀k ≥ 1, where Q k 0 = {x ∈ Rn : |x| ≤ k 0 }, Qc k 0 = Rn \ Q k 0 , Q k = {x ∈ Rn : |x| ≤ k}, and χ A is the identical function on A. Proof. It suffices to check that, for all D = {D(ω)} ω∈Ω ∈ D 2 and P-a.e. ω ∈ Ω, we can extract a Cauchy subsequence {φ(t n 󸀠 , θ −t n 󸀠 ω, x n 󸀠)} from {φ(t n , θ −t n ω, x n )}, whenever t n → ∞ and x n ∈ D (θ −t n ω). We assume that there is Ω̃ ⊂ Ω of full P-measure such that assumption (i) holds for every ω ∈ Ω̃. We now fix ω ∈ Ω̃ and ε > 0, and then by (i) there exist k 0 = k 0 (ω, ε) > 0 and T ?̂? (ω, ε) > 0 such that for all t ≥ T ?̂? (ω, ε), 󵄩󵄩󵄩󵄩󵄩 χ Q c k0 ⋅ φ (t, θ −t ω, x) 󵄩󵄩󵄩󵄩󵄩H1(Qc k0 ) ≤ ε, ∀x ∈ D (θ −t ω) . (17) On the other hand, by (ii), χ Q k ⋅ φ is (L2(Rn),H1(Q k ))asymptotically compact, for all k ≥ 1. For the above k 0 , there is a subsequence {φ (t n 󸀠 , θ −t n 󸀠 ω, x n 󸀠)} such that {χ Q k0 ⋅ φ (t n 󸀠 , θ −t n 󸀠 ω, x n 󸀠)} is convergent inH(Q k 0 ). Therefore, there 4 Abstract and Applied Analysis exists an integer N(D, ω, ε) > 0 such that for all n󸀠, m󸀠 ≥ N(D, ω, ε), we have 󵄩󵄩󵄩󵄩 φ (t n 󸀠 , θ −t n 󸀠 ω, x n 󸀠) −φ (t m 󸀠 , θ −t m 󸀠 ω, x m 󸀠) 󵄩󵄩󵄩󵄩 2 H 1 (R) = 󵄩󵄩󵄩󵄩󵄩 χ Q k0 ⋅ (φ (t n 󸀠 , θ −t n 󸀠 ω, x n 󸀠) −φ (t m 󸀠 , θ −t m 󸀠 ω, x m 󸀠)) 󵄩󵄩󵄩󵄩󵄩 2 H 1 (k0) + 󵄩󵄩󵄩󵄩󵄩 χ Q c k0 ⋅ (φ (t n 󸀠 , θ −t n 󸀠 ω, x n 󸀠) −φ (t m 󸀠 , θ −t m 󸀠 ω, x m 󸀠)) 󵄩󵄩󵄩󵄩󵄩 2 H 1 (Q c k0 ) ≤ 󵄩󵄩󵄩󵄩󵄩 χ Q k0 ⋅ φ (t n 󸀠 , θ −t n 󸀠 ω, x n 󸀠) −χ Q k0 ⋅ φ (t m 󸀠 , θ −t m 󸀠 ω, x m 󸀠) 󵄩󵄩󵄩󵄩󵄩 2 H 1 (k0) + 2 󵄩󵄩󵄩󵄩󵄩 χ Q c k0 ⋅ φ (t n 󸀠 , θ −t n 󸀠 ω, x n 󸀠) 󵄩󵄩󵄩󵄩󵄩 2

As we know, the asymptotic behavior of a random dynamical system (RDS) is characterized by random attractors, which were first introduced by Crauel and Flandoli [3] and Schmalfuss [4] and then developed in [1,2,[5][6][7][8][9][10][11][12] and among others.Recently, the existence of random attractors of the RDS associated with problem (1)-( 2) was studied by many authors.For example, in [1,2] the authors proved the existence of ( 2 (R  ),  2 (R  ))-random attractor and ( 2 (R  ),   (R  ))-random attractor, respectively, in the case of additive noise.Wang and Zhou obtained ( 2 (R  ),  2 (R  ))random attractor in [12] and Li et al. proved the existence of ( 2 (),   ())-random attractor in bounded domains in [10] in the case of multiplicative noise.A necessary and sufficient condition for the existence of random attractors for the socalled quasicontinuous RDS was established in [9], and in the most recent papers [13,14], the author employed this result to prove the existence of random attractors for some reactiondiffusion equations with additive noise and multiplicative noise on  1 , respectively, when the domain is bounded.In this paper, we study the existence of ( 2 (R  ),  1 (R  ))random attractor with additive noise for the same problem in the entire space R  .
For our problem, there are two difficulties when we consider the existence of ( 2 ,  1 )-random attractor.The first is the lack of compactness of Sobolev embeddings when the domain is unbounded.It is worth mentioning that in deterministic case differential equations of this type were extensively studied in both autonomous and nonautonomous cases and in both bounded domains and unbounded domains [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29].In the case of unbounded domains the difficulty of noncompact embeddings can be overcome by the energy equation approach introduced by Ball in [30,31] and other methods.We are interested in the method used in [22] for the deterministic version of the initial problem ( 1)-( 2) on R  .In [22] the author approached R  by a bounded ball and found that the approximation error of the norm of solutions is arbitrary small uniformly for large time, and thus they proved asymptotic compactness by passing the limit of the energy equation.More recently, the idea of the tail estimate was used in [1] to prove the existence of random attractor in  2 (R  ) for the SRDE (1)- (2).In this paper, we use an extended version of the tail estimate described in [22] to overcome the difficulty of noncompact embeddings.
Another difficulty is that one can not differentiate the stochastic equation with respect to time  in usual sense.In the case of deterministic equation, by differentiating the reaction-diffusion equation with respect to , one can prove the existence of ( 2 (R  ),  1 (R  )) or ( 2 (),  1 0 ()) ( ⊂ R  is bounded) attractors; see [24,27,29,32] for autonomous equations and [23,26,33] for non-autonomous equations.But in stochastic case this idea breaks down, since, as we know, neither the Winner process nor the Ornstein-Uhlenbeck process is differentiable with respect to  in usual sense.However, this is only a matter of method or estimate.In [25], the author used a result for compactness in  1 (R  ) introduced in [17] to establish the asymptotic compactness in  1 (R  ) without differentiating the equation.Unfortunately, the growth order  is restricted in that case.In this paper, we overcome this drawback by using an appropriate estimate motivated by the works in [19] and the estimate is accurate enough so that we needn't differentiate the equation as usual.
This paper is organized as follows.In Section 2, we recall some basic notions of bispaces random attractors for RDS.In Section 3, we transform the problem (1)-( 2) into a parameterized evolution equation and obtain the corresponding RDS.In Section 4, we give some uniform estimates of the solutions as  → ∞.In Section 5, we prove the asymptotic compactness and the existence of an ( 2 (R  ),  1 (R  ))random attractor.
Throughout this paper, we denote by ‖ ⋅ ‖  the norm of Banach space  and by (⋅, ⋅)  the inner product in Hilbert space .The inner product and norm of  2 (R  ) are written as (⋅, ⋅) and ‖⋅‖, respectively.We also use ‖‖  to denote the norm of  ∈   (R  ) ( ≥ 1,  ̸ = 2) and || to denote the modular of .The letter  denotes any positive constant which may be different from line to line or even in the same line (sometimes for special case, we also denote the different positive constants by   ( = 1, 2, . ..)).
An RDS is said to be continuous on  if (, ) :  →  is continuous for all  ∈ R + and P-a.e.  ∈ Ω. where (2) A random variable () ≥ 0 is called tempered with respect to (  ) ∈R if, for P-a.e.  ∈ Ω, Next, we introduce some notions about the bi-spaces random attractors which are motivated by the works in [2,20,25,35].We assume that  is an RDS on  and  over an MDS (Ω, F, P, (  ) ∈R ), respectively.Let P() denote the family of all nonempty subsets of  and S  the class of all families D = {()} ∈Ω ⊂ P().P(), and S  can be defined in the same way.We consider the given nonempty subclasses D  , D  , where D  ⊂ S  , D  ⊂ S  .

Abstract Results
. Now, we present the main abstract results.Recall that a collection D of random subsets is called inclusion closed if whenever {()} ∈Ω is an arbitrary random set and {( The following theorem is an adaptation of a result of [25] to the case of RDS.The proof is similar to that of [25], and here we omit it.Theorem 9. Let  be a continuous RDS on  and an RDS on  over (Ω, F, P, (  ) ∈R ), respectively, and D  and D  are inclusion closed.
In the following of this paper we only consider  =  2 (R  ),  =  1 (R  ), and D  = D 2 , D  = D V , where D 2 and D V denote the collections of all tempered random subsets of  2 (R  ) and  1 (R  ), respectively.Theorem 10.Assume that  is an RDS on  2 (R  ) and  1 (R  ), respectively, and then  is and every  > 0, there exist  0 =  0 (, ) > 0 and  D(, ) > 0 such that, for all  ≥  D(, ), and   is the identical function on .
Remark 11.If we replace  1 (R  ) by other Banach spaces in Theorem 10, such as  2 (R  ),   (R  ) and  2 (R  ), the corresponding results also hold true.In particular, in the deterministic case, it is the exact method used in [22] when  1 (R  ) is replaced by  2 (R  ).

The Reaction-Diffusion Equation on R 𝑛 with Additive Noise
We consider the probability space (Ω, F, P) where F is the Borel -algebra induced by the compact-open topology of Ω, and P the corresponding Wiener measure on (Ω, F).Then we will identify  with Define the time shift by and then (Ω, F, P, (  ) ∈R ) is an MDS.We now translate the stochastic equation ( 1)-(2) into a deterministic equation with a random parameter.
To this end, we consider the one-dimensional Ornstein-Uhlenbeck process given by where () satisfies that for P-a.e.  ∈ Ω, Therefore, for P-a.e.  ∈ Ω, Putting then by (23) we have Remark 12. From ( 24) and ( 27), we can easily show that the sum is bounded by () with a deterministic positive constant  0 .
In the following of this paper, we use the symbols () and () to denote the random variables in (24).

Uniform Estimates of Solutions
where  is a constant independent of , , and  0 .
Lemma 17. Assume that  ∈  2 (R  ) and (3)-( 6) hold.Let D = {()} ∈Ω ∈ D 2 and  0 () ∈ ().Then, for P-a.e.  ∈ Ω and for every  > 0, there exist  * =  * D(, ) > 0 and  * =  * (, ), such that the solution V (, , V 0 ()) of ( 30)-( 31) Proof.By Lemma 4.6 in [1], it suffices to prove that Let  be a smooth function defined on R + such that 0 ≤ () ≤ 1, for all  ∈ R + , and Then there is a positive constant  such that |  ()| + |  ()| ≤  for all  ∈ R + .Multiplying (30) by −(|| 2 / 2 )ΔV and integrating with respect to  over R  , we get The second term of the left-hand side is bounded by For the last term of the right-hand side of (68), we have We next consider the nonlinear term in (68).Since We now estimate each term in the right-hand side of (72).Using (4), the property of , and Cauchy's inequality, we see that the first term of the right-hand side of ( 72 For the third term of the right-hand side of (72), by using (5), we have For the last term of the right-hand side of (72), by using ( 4) and Young's inequality, we find Putting ( 73)-(76) together into (72), it yields that Then by ( 68)-( 71) and (77), we get In particular, Integrating the aforementioned inequality with respect to  over (,  + 1), and replacing  by  −−1 , we obtain In the sequel, we estimate each term in the right-hand side of (81) to show that they are arbitrary small when  and  are large enough.

Asymptotic Compactness in Bounded Balls.
In what follows, we prove the asymptotic compactness in any bounded ball, which together with Lemma 18 and Theorem 10 is a necessary condition for verifying the ( 2 (R  ),  1 (R  ))asymptotic compactness.For this purpose, we set  = 1 − , where  is the function described in Lemma 17.
For fixed  ≥ 1, define Then Ṽ(, , V 0 ()) ∈  1 0 ( 2 ) and where  is a positive constant, independent of , , and .Then we have Consider the following eigenvalue problem: and then problem (110) has a family of eigenfunctions forms an orthogonal basis in both  2 ( 2 ) and  1 0 ( 2 ) and Given , let   = span{ 1 , . . .,   } and   :  2 ( 2 ) →   be the projection operator.For any Ṽ ∈  1 0 ( 2 ), we write In order to prove the asymptotic compactness we need the following lemma, which can be found in [19].
From ( 118)-(119), we get Applying Cauchy's inequality, we obtain that is, where ; we first substitute  for  in the aforementioned inequality and then we replace  by  −  to get For the first term of the right-hand side of (125), we use Lemma 15: Then there exists  1 =  1 (, ) > 0 such that, for all  ≥  D() and all  ≥  1 , For the second term of the right-hand side of (125), ∀ ∈ [ − 1, ], by (26) we have This implies that there exists  2 =  2 (, ) > 0 such that ∀ ≥  2 we have For the last term of the right-hand side of (125), we can easily see that there exists  3 =  3 () > 0 such that, for all  ≥  3 , Let  * = max{ * 1 ,  * 2 }; from (138) and ( 139) we can obtain our results.The proof is complete.
Remark 21.The idea of the proof of the above lemma comes from [19] (this idea can be further traced back to Marion [16] and Robinson [18]).We see from (132) that the constant  in Lemma 20 is independent of , which is different from the  in the Lemma 3.4 in [19].This is crucial in the following estimates.As we know, the time  will vary to infinite when we consider the asymptotic behavior of an RDS.It means that if  is not a fixed constant with respect to , the following estimate will be invalid.In other words, if the function  in (1) is dependent on , our method will fail.Lemma 22. Assume that  ∈  2 (R  ) and (3)-( 6) hold.Let D = {()} ∈Ω ∈ D 2 and  0 () ∈ ().Then, for P-a.e.  ∈ Ω and for every  > 0 and all  ≥ 1, there exist  D() > 0 and (, , ) > 0 such that the solution V(, , V 0 ()) of (30)- (31) satisfies that, ∀ ≥  D(), ∀ ≥  (, , ), where Ṽ2 = ( −   )Ṽ. Proof for simplicity, hereafter, we write V( * ) = V(,  −−1 , V 0 ( −−1 )).
In the sequel, we estimate each term in the right-hand side of (150).For the first term, we use Lemma 15 and (107) to obtain that, for  ≥  D(), Thus, there exists  1 =  1 (, ), for all  ≥  1 and all  ≥  D(), we get For the second term in the right-hand side of (150), we can estimate it as follows:

Abstract and Applied Analysis
The first term in the right-hand side of (154) is bounded by when we choose appropriate  and  by Lemma 20.

Asymptotic Compactness and Random Attractors
In this section, we prove our main result, that is, the existence of an ( 2 (R  ),  1 (R  ))-random attractor for the RDS  associated with the initial value problem of SRDE (1)-(2).To this end, we should show the ( 2 (R  ),  1 (R  ))-asymptotic compactness of .From Theorem 10 and Lemmas 18 and 23, we can immediately obtain the asymptotic compactness of .
Then the RDS  generated by (1)-( 2) is ( 2 (R  ),  1 (R  )) asymptotically compact.Now, we are in a position to present our main result.
Proof.The result can be obtained by Theorems 9, and 13, Lemmas 14, and 24 immediately.