Asymptotic behavior for non-autonomous stochastic plate equation on unbounded domains

1 Introduction

Consider the following non-autonomous stochastic plate equation with additive noise defined in the entire space ℝⁿ:

with the initial value conditions

where x ∈ ℝⁿ, t > τ with τ ∈ ℝ, α > 0, λ > 0 and β are constants, f is a nonlinearity satisfying certain growth and dissipative conditions, g(x, ⋅) and h are given functions in Lloc2 (ℝ, L²(ℝⁿ)) and H²(ℝⁿ), respectively, W(t) is a two-sided real-valued Wiener process on a probability space.

Plate equations have been investigated for many years due to their importance in some physical areas such as vibration and elasticity theories of solid mechanics. The study of the long-time dynamics of plate equations has become an outstanding area in the field of the infinite-dimensional dynamical system. While the attractors are regarded as a proper notation to describe the long-time dynamics of solutions. To the best of our knowledge, there have been many works on the investigation of the attractors for the plate equations over the last few years. For instance, if the random term is vanished and g(x, t) = g(x), then (1.1)-(1.2) change into a deterministic autonomous plate equation. The existence and uniqueness of the global attractor of the corresponding dynamical system was studied in [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]; besides, the uniform attractor of the dynamical system generated by the non-autonomous plate equation was established in [11].

For the stochastic plate equations, if the forcing term g(x, t) = g(x), then the existence of a random attractor of (1.1)-(1.2) on bounded domain have been proved in [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. However, it is not yet considered by any predecessors to the stochastic plate equation on unbounded domain. In recent years, the existence of random attractors for stochastic dynamical system on unbounded domains have been investigated by several authors, such as Reaction-diffusion equations with additive noise [15], Reaction-diffusion equations with multiplicative noise [16], FitzHugh-Nagumo equations with additive noise [17, 18], Navier-Stokes equations with additive noise [19], wave equations with additive noise [20, 21, 22], wave equations with multiplicative noise [23].

Motivated by above literatures, the goal of the present paper is to study random attractors and its upper semicontinuity of non-autonomous stochastic equation (1.1). By applying the abstract results in [24, 25, 26], we will first prove the stochastic plate equation (1.1) has tempered random attractors in H²(ℝⁿ) × L²(ℝⁿ), then establish the upper semicontinuity of the random attractors.

In general, the existence of global random attractor depends on some kind compactness (see, e.g., [27, 28, 29, 30]). To prove the existence of random attractors for (1.1) in H²(ℝⁿ) × L²(ℝⁿ), we must establish the pullback asymptotic compactness of solutions. Since Sobolev embeddings are not compact on unbounded domain, we cannot get the desired asymptotic compactness directly from the regularity of solutions. We here overcome the difficulty by using the uniform estimates on the tails of solutions outside a bounded ball in ℝⁿ and the splitting technique, see [20, 31] for details.

The framework of this paper is as follows. In the next Section, we recall a sufficient and necessary criterion for existence of pullback attractors for cocycle or nonautonomous random dynamical systems. In Section 3, we define a continuous cocycle for Eq. (1.1) in H²(ℝⁿ) × L²(ℝⁿ). Then we derive all necessary uniform estimates of solutions in Section 4. In Section 5, we prove the existence and uniqueness of tempered random attractor for the non-autonomous stochastic plate equation. Finally, in Section 6, we prove the upper semicontinuity of random attractors as β to zero.

Throughout the paper, the letters c and c_i (i = 1, 2, …) are generic positive constants which may change their values from line to line or even in the same line.

2 Preliminaries

In this section, we recall some basic concepts related to random attractors for stochastic dynamical systems.

Let X be a separable Banach space and (Ω, 𝓕, 𝓟) be the standard probability space, where Ω = {ω ∈ C(ℝ, ℝ) : ω(0) = 0}, 𝓕 is the Borel σ-algebra induced by the compact open topology of Ω, and 𝓟 is the Wiener measure on (Ω, 𝓕). There is a classical group {θ_t}_t∈ℝ acting on (Ω, 𝓕, 𝓟) which is defined by

We often say that (Ω, 𝓕, 𝓟, {θ_t}_t∈ℝ) is a parametric dynamical system.

The following four definitions and one proposition are from [24].

Hereafter, we assume Φ is a continuous cocycle on X over ℝ and (Ω, 𝓕, 𝓟, {θ_t}_t∈ℝ), and 𝓓 is the collection of all tempered families of nonempty bounded subsets of X parameterized by τ ∈ ℝ and ω ∈ Ω:

𝓓 is said to be tempered if there exists x₀ ∈ X such that for every c > 0, τ ∈ ℝ and ω ∈ Ω, the following holds:

Given D ∈ 𝓓, the family Ω(D) = {Ω(D, τ, ω) : τ ∈ ℝ, ω ∈ Ω} is called the Ω-limit set of D where

The cocycle Φ is said to be 𝓓-pullback asymptotically compact in X if for all τ ∈ ℝ and ω ∈ Ω, the sequence

whenever t_n → ∞, and x_n ∈ D(τ − t_n, θ_−tn ω) with {D(τ, ω) : τ ∈ ℝ, ω ∈ Ω} ∈ 𝓓.

If, in addition, K(τ, ω) is closed in X and is measurable in ω with respect to 𝓕, then K is called a closed measurable 𝓓-pullback absorbing set for Φ.

As in the deterministic case, random complete solutions can be used to characterized the structure of a 𝓓-pullback attractor. The definition of such solutions are given below.

If, in addition, there exists a tempered family D = {D(τ, ω) : τ ∈ ℝ, ω ∈ Ω} such that Ψ(t, τ, ω) belongs to D(τ + t, θ_t ω) for every t ∈ ℝ, τ ∈ ℝ and ω ∈ Ω, then Ψ is called a tempered random complete solution of Φ.

3 Cocycles for stochastic plate equation

In this section, we outline some basic settings about (1.1)-(1.2) and show that it generates a continuous cocycle in H²(ℝⁿ) × L²(ℝⁿ).

Let −Δ denote the Laplace operator in ℝⁿ, A = Δ² with the domain D(A) = H⁴(ℝⁿ). We can define the powers A^ν of A for ν ∈ ℝ. The space Vν=D(Aν4) is a Hilbert space with the following inner product and norm

For brevity, the notation (⋅, ⋅) for L²-inner product will also be used for the notation of duality pairing between dual spaces.

We denote 𝓗 = H²(ℝⁿ) × L²(ℝⁿ).

We define a new norm ∥⋅∥_𝓗 by

for Y = (u, v)^⊤ ∈ 𝓗, where ⊤ stands for the transposition.

Let ξ = u_t + δ u, where δ is a small positive constant whose value will be determined later, then (1.1)-(1.2) can be rewritten as the equivalent system

with the initial value conditions

where ξ₀(x) = u₁(x) + δ u₀(x), x ∈ ℝⁿ.

Let F(x,u)=∫0uf(x,s)ds for x ∈ ℝⁿ and u ∈ ℝ. The function f will be assumed to satisfy the following conditions:

1. |f(x, u)| ≤ c₁|u|^k + η₁(x),

2. f(x, u)u − c₂F(x, u) ≥ η₂(x),

3. F(x, u) ≥ c₃|u|^k+1 − η₃(x),

4. |f_u(x, u)| ≤ ϖ,

where ϖ > 0, 1 ≤ k ≤ n+4n−4 , η₁ ∈ L²(ℝⁿ), η₂ ∈ L¹(ℝⁿ), η₃ ∈ L¹(ℝⁿ).

Note that (F1) and (F2) imply

We also need the following condition on g: there exists a positive constants σ such that

which implies that

where |⋅| denotes the absolute value of real number in ℝ.

For our purpose, it is convenient to convert the problem (1.1)-(1.2) (or (3.2)-(3.3)) into a deterministic system with a random parameter, and then show that it generates a cocycle over ℝ and (Ω, 𝓕, 𝓟, {θ_t}_t∈ℝ).

We identify ω(t) with W(t), i.e., ω(t) = W(t) = W(t, x), t ∈ ℝ. Consider Ornstein-Uhlenbeck equation dy + ydt = dW(t), and Ornstein-Uhlenbeck process

From [32], it is known that the random variable |y(ω)| is tempered, and there is a θ_t-invariant set Ω͠ ⊂ Ω of full 𝓟 measure such that y(θ_t ω) is continuous in t for every ω ∈ Ω͠. Put

which solves

Now, let v(t, τ, ω) = ξ(t, τ, ω) − β z(θ_t ω), we obtain the equivalent system of (3.2)-(3.3),

with the initial value conditions

where v₀(x) = ξ₀(x) − z(θ_τ ω), x ∈ ℝⁿ. We will consider (3.9)-(3.10) for ω ∈ Ω͠ and write Ω͠ as Ω from now on.

The well-posedness of the deterministic problem (3.9)-(3.10) in H²(ℝⁿ) × L²(ℝⁿ) can be established by standard methods as in [34, 35]. If (F1)-(F4) are fulfilled, let φ^(β)(t + τ, τ, θ_−τ ω, φ0(β) ) = (u(t + τ, τ, θ_−τ ω, u₀), v(t + τ, τ, θ_−τ ω, v₀))^⊤, where φ0(β) = (u₀, v₀)^⊤. Then for every ω ∈ Ω, τ ∈ ℝ and φ0(β) ∈ 𝓗(ℝⁿ), problem (3.9)-(3.10) has a unique (𝓕, 𝓑(H²(ℝⁿ)) × 𝓑(L²(ℝⁿ)))-measurable solution φ^(β)(⋅, τ, ω, φ0(β) ) ∈ C([τ, ∞), 𝓗(ℝⁿ)) with φ^(β)(τ, τ, ω, φ0(β) ) = φ0(β) , φ^(β)(t, τ, ω, φ0(β) ) ∈ 𝓗₀(ℝⁿ) being continuous in φ0(β) for each t > τ. Moreover, for every (t, τ, ω, φ₀) ∈ ℝ⁺ × ℝ × Ω × 𝓗(ℝⁿ), the mapping

generates a continuous cocycle from ℝ⁺ × ℝ × Ω × 𝓗(ℝⁿ) to 𝓗(ℝⁿ) over ℝ and (Ω, 𝓕, 𝓟, {θ_t}_t∈ℝ).

Introducing the homeomorphism P(θ_t ω)(u, v)^⊤ = (u, v + z(θ_t ω))^⊤, (u, v)^⊤ ∈ 𝓗(ℝⁿ) with an inverse homeomorphism P⁻¹(θ_t ω)(u, v)^⊤ = (u, v − z(θ_t ω))^⊤. Then, the transformation

generates a continuous cocycle with (3.2)-(3.3) over over ℝ and (Ω, 𝓕, 𝓟, {θ_t}_t∈ℝ).

Note that these two continuous cocycles are equivalent. By (3.12), it is easy to check that Φ͠_β has a random attractor provided Φ_β possesses a random attractor. Then, we only need to consider the continuous cocycle Φ_β.

One can prove Φ_β is measurable by using the same method as in the following paper: H. Cui, J.A. Langa, Y. Li, Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Differ. Equ. 30 (2018) 1873-1898.

4 Uniform estimates of solutions

In this subsection, we derive uniform estimates on the solutions of the stochastic plate equations (3.9)-(3.10) defined on ℝⁿ when t → ∞. These estimates are necessary for proving the existence of bounded absorbing sets and the asymptotic compactness of the random dynamical system associated with the equations. In particular, we will show that the tails of the solutions for large space variables are uniformly small when time is sufficiently large.

Let δ ∈ (0, 1) be small enough such that

and define σ appearing in (3.5) by

where c₂ is the positive constant in (F2).

Choose a smooth function ρ, such that 0 ≤ ρ ≤ 1 for s ∈ ℝ, and

and there exist constants μ₁, μ₂, μ₃, μ₄ such that |ρ′(s)| ≤ μ₁, |ρ″(s)| ≤ μ₂, |ρ″′(s)| ≤ μ₃, |ρ″″(s)| ≤ μ₄ for s ∈ ℝ.

Given r ≥ 1, denote ℍ_r = {x ∈ ℝⁿ : |x| < r} and ℝⁿ ∖ ℍ_r the complement of ℍ_r. To prove asymptotic compactness of solution on ℝⁿ, we prove the following lemma.

Substituting v in (4.4) into the second, third and last terms on the right-hand side of (4.23), using Young inequality and the Sobolev interpolation inequality

we conclude that