Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains

A system of stochastic retarded reaction-diffusion equations with multiplicative noise and deterministic non-autonomous forcing on thin domains is considered. Relations between the asymptotic behavior for the stochastic retarded equations defined on thin domains in ${\mathbb R}^{n+1}$ and an equation on a domain in ${\mathbb R}^{n}$ are investigated. We first show the existence and uniqueness of tempered random attractors for these equations. Then, we analyze convergence properties of the solutions as well as the attractors.


1.
Introduction. Let Q ⊂ R n be a bounded C 2 -domain and O ε ⊂ R n+1 be the domain where g ∈ C 2 (Q, (0, +∞)) and 0 < ε ≤ 1. Since g ∈ C 2 (Q, (0, +∞)), there exist two positive constants γ 1 and γ 2 such that Denote O = Q × (0, 1) and O = Q × (0, γ 2 ) which contains O ε for 0 < ε ≤ 1. Given τ ∈ R, in this paper, we study the asymptotical behavior of the following stochastic retarded reaction-diffusion equation with multiplicative noise defined on the thin domain O ε : x,û ε (t)) + f (t, x,û ε (t − ρ 0 (t))) + G (t, x)) dt with the initial condition where ν ε is the unit outward normal vector to ∂O ε , H is a superlinear source term, f : R × O × R → R is a nonlinearity capturing the time delay, ρ 0 : R → [0, ρ] is an adequate given delay function, where ρ is a positive constant, G is a function defined on R × O, c j ∈ R for j = 1, . . . , m, w j , j = 1, 2, . . . , m, are independent two-sided real-valued Wiener processes on a probability space, and the symbol • indicates that the equation is understood in the sense of Stratonovich integration. The domain O ε is the so-called thin domain when ε is small. We will investigate the limiting behavior of (2) as ε → 0. For the deterministic reaction-diffusion equations without delay, in [26,27], Hale and Raugel first studied this problem. Some extensions of their results can be also found in [1,3,4,5,19,36]. However, systems are always subject to environmental noise. The environmental noise is an intrinsic effect in a variety of settings and spatial scales. It is worth mentioning that the ergodicity of stochastic 3D Navier-Stokes equations in a thin domain was recently investigated in [17,18], the synchronization of semilinear parabolic stochastic equations in thin bounded tubular domains was studied in [11], and the upper semicontinuity of random attractors for reaction-diffusion equations in thin domains was established in [32,34]. However, as far as the author is aware, the limiting dynamics for stochastic retarded equations on thin domains are not well studied, even for deterministic retarded equations on thin domains. In this paper, we will investigate this problem.
Let X be a Banach space. The norm of X is written as · X . For each fixed ρ > 0, we use C([−ρ, 0], X) to denote the set of all continuous functions from . We denote by (·, ·) Y the inner product in a Hilbert space Y . The letter c is a generic positive constant which may change its values from line to line.
We organize the paper as follows. In the next section, we establish the existence of a continuous cocycle in N for the stochastic retarded equation defined on the fixed domain O, which is converted from (2)-(3). We also describe the existence of a continuous cocycle in M for the stochastic retarded equation (4)- (5). Section 3 contains all necessary uniform estimates of the solutions. We then prove the existence and uniqueness of tempered random attractors for the stochastic retarded equations in section 4, and analyze convergence properties of the solutions as well as the random attractors in section 5.

2.
Cocycles associated with stochastic retarded equations. Here we show that there is a continuous cocycle generated by the delay reaction-diffusion equation defined on O ε with multiplicative noise and deterministic non-autonomous forcing: with the initial condition where ν ε is the unit outward normal to ∂O ε , G : R × O → R belongs to L 2 loc (R, L ∞ ( O)), c j ∈ R, w j (j = 1, 2, . . . , m) are independent two-sided real-valued Wiener processes on a probability space which will be specified later. f : R × O × R → R is a continuous function and has the property that there exist a positive constant K and a function ψ and there exists L f > 0 such that for all t ∈ R, x ∈ O and s 1 , s 2 ∈ R. ρ 0 : R → [0, ρ] is a continuously differentiable function with |ρ 0 (t)| ≤ ρ * < 1 for all t ∈ R. H is a nonlinear function satisfying the following conditions: for all x ∈ O and t, s ∈ R, where p ≥ 2, λ 1 , λ 2 and λ 3 are positive constants, ϕ 2 ∈ L ∞ loc (R, L ∞ ( O)) and ϕ 3 , ψ 4 ∈ L 2 loc (R, L ∞ ( O)). Throughout this paper, we fix a positive number λ ∈ (0, λ 1 ) and write for all x ∈ O and t, s ∈ R. Then it follows from (10)-(13) that there exist positive numbers α 1 , α 2 , β, b 1 and b 2 such that where ψ 2 (t, x) = ϕ 2 (t, x) + b 1 and ψ 3 (t, x) = ϕ 3 (t, x) + b 2 for x ∈ O and t, s ∈ R. Substituting (14) into (6) we get for t > τ , with the initial condition We now transfer problem (19)- (20) into an initial boundary value problem on the fixed domain O. To that end, we introduce a transformation T ε : After some calculations, we find that the Jacobian matrix of T ε is given by The determinant of J is |J| = 1 εg(y * ) . Let J * be the transport of J. Then we have It follows from [35] (see also [26]) that the gradient operator and the Laplace operator in the original variable x ∈ O ε and in the new variable y ∈ O are related by ∇ xû (x) = J * ∇ y u(y) and ∆ xû (x) = |J|div y (|J| −1 JJ * ∇ y u(y)) = 1 g div y (P ε u(y)), where we denote by u(y) =û(x), ∇ x and ∆ x are the gradient operator and the Laplace operator in x ∈ O ε respectively, div y and ∇ y are the divergence operator and the gradient operator in y ∈ O respectively, and P ε is the operator given by In the sequel, we abuse the notation a little bit by writing h(t, x, s), f (t, x, s) and G(t, x) as h(t, x * , x n+1 , s), f (t, x * , x n+1 , s) and G(t, x * , x n+1 ) for x = (x * , x n+1 ), respectively. With this agreement, for any function F (t, y, s), we introduce F ε (t, y * , y n+1 , s) = F (t, y * , εg (y * ) y n+1 , s) , F 0 (t, y * , s) = F (t, y * , 0, s) , where y = (y * , y n+1 ) ∈ O and t, s ∈ R. Then problem (19)- (20) is equivalent to the following system for t > τ , with the initial condition where ν is the unit outward normal to ∂O. Note that the boundary condition in (21) follows from the original boundary condition in (19) (see [34] for the details). Given t ∈ R, define a translation θ 1,t on R by Then {θ 1,t } t∈R is a group acting on R. We now specify the probability space. Denote by Ω = {ω ∈ C (R, R) : ω (0) = 0} . Let F be the Borel σ-algebra induced by the compact-open topology of Ω, and P be the corresponding Wiener measure on (Ω, F). There is a classical group {θ t } t∈R acting on (Ω, F, P ), which is defined by

DINGSHI LI, KENING LU, BIXIANG WANG AND XIAOHU WANG
Then (Ω, F, P, {θ t } t∈R ) is a metric dynamical system (see [2]). On the other hand, let us consider the one-dimensional stochastic differential equation for α > 0. This equation has a random fixed point in the sense of random dynamical systems generating a stationary solution known as the stationary Ornstein-Uhlenbeck process (see [12,23] for more details). In fact, we have and, for such ω, the random variable given by is well defined. Moreover, for ω ∈ Ω , the mapping is a stationary solution of (25) with continuous trajectories. In addition, for ω ∈ Ω Denote by z * j the associated Ornstein-Uhlenbeck process corresponding to (25) with α = 1 and w replaced by w j for j = 1, . . . , m. Then for any j = 1, . . . , m, we have a stationary Ornstein-Uhlenbeck process generated by a random variable z * j (ω) on Ω j with properties formulated in Lemma 2.1 defined on a metric dynamical system (Ω j , F j , P j , {θ t } t∈R ). We set Then for every ω ∈Ω, T (ω) is a homeomorphism on L 2 (O), and its inverse operator is given by These operators can be easily extended to linear homeomorphismsT (ω) andT −1 (ω) on N . Indeed, for any ξ ∈ N , let us define It follows that T −1 (θ t ω) has sub-exponential growth as t → ±∞ for any ω ∈ Ω. Hence T −1 is tempered. Analogously, T is also tempered. Obviously, sup T (θ s ω) is still tempered for every s 0 ∈ R and a ∈ R + .
On the other hand, since z * j , j = 1, . . . , m, are independent Gaussian random variables, by the ergodic theorem we still have a {θ t } t∈R -invariant setΩ ∈ F of full measure such that Remark 1. We now consider θ defined in (24) onΩ∩Ω instead of Ω. This mapping possesses the same properties as the original one if we choose F as the trace σalgebra with respect toΩ ∩Ω. The corresponding metric dynamical system is still denoted by (Ω, F, P, {θ t } t∈R ) throughout this paper.
Next, we define a continuous cocycle for system (21)- (22) in N . This can be achieved by transferring the stochastic system into a deterministic one with random parameters in a standard manner. Let u ε be a solution to (21)- (22) and denote by where ψ ε = (T −1 (θ τ ω))φ ε . Since (28) is a deterministic equation, by the Galerkin method, one can show that if f satisfies (8)- (9) and H satisfies (10)-(13), then for every ω ∈ Ω, τ ∈ R and ψ ε ∈ N , Then we find that u ε t is continuous in both t ≥ τ and φ ε ∈ N and is (F, B(N ))-measurable in ω ∈ Ω. In addition, it follows from (28) that u ε is a solution of problem (21)- (22). We now define By the properties of u ε , we find that Φ ε is a continuous cocycle on N over (R, {θ 1,t } t∈R ) and (Ω, F, P, {θ t } t∈R ), where {θ 1,t } t∈R and {θ t } t∈R are given by (23) and (24), respectively.
The same change of unknown variable v 0 (t) = T −1 (θ t ω) u 0 (t) transforms equation (4) into the following random partial differential equation on Q: The same argument as above allows us to prove that problem (4) and (5) generates a continuous cocycle Φ 0 (t, τ, ω, φ 0 ) in the space M.
Now we want to write equation (28)-(29) as an abstract evolutionary equation. We introduce the inner product (·, ·) Hg(O) on L 2 (O) defined by and denote by H g (O) the space equipped with this inner product. Since g is a continuous function on Q and satisfies (1), one easily shows that H g (O) is a Hilbert space with norm equivalent to the natural norm of L 2 (O).
For 0 < ε ≤ 1, we introduce a bilinear form a ε (·, ·): where By introducing on H 1 (O) the equivalent norm, for every 0 < ε ≤ 1, we see that there exist positive constants ε 0 , η 1 and η 2 such that for all 0 < ε ≤ ε 0 and u ∈ H 1 (O), Denote by A ε an unbounded operator on Then we have Using A ε , (28)- (29) can be written as To reformulate system (31)-(32), we introduce the inner product (·, ·) Hg(Q) on and denote by H g (Q) the space equipped with this inner product. Let a 0 (·, ·): Then we have Using A 0 , (31)-(32) can be written as Hereafter, we set The collection of all families of tempered nonempty subsets of X i is denoted by D i , i.e., Our main purpose of the paper is to prove that the continuous cocyclesΦ ε and Φ 0 possess a unique D ε -pullback attractorÂ ε in X ε and a unique D 0 -pullback attractor A 0 in the space M, respectively. FurthermoreÂ ε is upper-semicontinuous at ε = 0, that is, for every τ ∈ R and ω ∈ Ω, To prove (39), we only need to show that the cocycle Φ ε has a unique D 1 -pullback attractor A ε in N and it is upper-semicontinuous at ε = 0 in the sense that for every τ ∈ R and ω ∈ Ω, lim ε→0 dist N (A ε (τ, ω) , A 0 (τ, ω)) = 0, which will be established in the last section of the paper.
Furthermore, we suppose that there exists λ 0 > 0 such that Let us consider the mapping By the ergodic theory and (40) we have The following condition will be needed when deriving uniform estimates of solutions: When constructing tempered pullback attractors, we will assume Notice that condition (43) does not require that G(s, ·) be bounded in L ∞ ( O), when s → ±∞. Since ψ 2 = ϕ 2 + b 1 for some positive constant b 1 , it is evident that (43) and (44) imply for any σ > 0.
3. Uniform estimates of solutions. In this section, we derive uniform estimates of solutions of problem (37) which are needed for proving the existence of D 1pullback absorbing sets and the D 1 -pullback asymptotic compactness of the continuous cocycle Φ ε . The estimates of solutions of problem (37) in C([−ρ, 0], H g (O)) are provided below.
Proof. Since λ 1 > λ 0 , we can take the λ in (14) satisfying λ ≥ λ 0 . Taking the inner product of (37) with v ε in H g (O), we find that For the third term on the right-hand side of (49), by (15), we have where |O| stands for the Lebesgue measure of O. For the last term on the right-hand side of (49), we have Consequently, it follows from (49)-(51) that Hg(O)

DINGSHI LI, KENING LU, BIXIANG WANG AND XIAOHU WANG
It follows from (52) that Then, we have for any σ ≥ τ , Now we deal with the third term on the right-hand side of (54) by means of (8).
Proof. Taking the inner product of (37) with A ε v ε in H g (O), we find that We first estimate the second term on the right-hand side of (68), by Lemma 3.3, we have We now estimate the nonlinear delay term in (68) for which, by (8) we have On the other hand, the last term on the right-hand side of (68) is bounded by By (68)-(71) we get that ). (73) Given τ ∈ R, ω ∈ Ω, and ∈ (τ + s − 1, τ + s), where s ∈ [−ρ, 0], integrating (73) on ( , τ + s), we get Now integrating the above with respect to over (τ + s − 1, τ + s) and replacing ω by θ −τ ω, we find that Let T = T (τ, ω, D 1 ) ≥ 2ρ + 1 be the positive number found in Lemma 3.2. Then it follows from the above inequality and Lemma 3.2 that, for all t ≥ T and for all ω ∈ Ω, The last term on the right-hand side of (74) satisfies )dr, which is bounded as in Lemma 3.1. Thus, Lemma 3.4 follows from Lemma 3.1.
Replacing ω with θ −τ ω, we find that For the last term on the right hand side of (77), we have which together (67), (77), Lemma 3.2 and Lemma 3.4 completes the proof.
We are now in a position to establish the uniform estimates for the solution u ε of the stochastic equation (21) and (22) by using those estimates for the solution v ε of (37).