Random uniform exponential attractor for stochastic non-autonomous reaction-diffusion equation with multiplicative noise in ℝ³

1 Introduction

The concept of the exponential attractor was introduced by A. Eden et al., which is a compact positively invariant set with finite fractal dimension and attracts trajectories exponentially fast, see [1]. It can describe the asymptotic behavior of trajectories of autonomous dynamical system or the solutions to dissipative autonomous evolution equations. In contrast to a global attractor, the exponential attractor has finite fractal dimension, if it exists, the asymptotic behavior of infinite dimensional dynamical systems can be characterized by the dynamics on the finite dimensional compact set (i.e., exponential attractor). Besides, exponential attractors are stable under perturbation because of the exponential rate of convergence of trajectories to it. We should note that an exponential attractor is not necessarily unique since it is not invariant, and includes a global attractor in general.

By the notion of pullback attraction, the concept of the exponential attractor can be extended to the case of non-autonomous dynamical system, called pullback exponential attractor, see [2, 3, 4, 5, 6] and the references therein. An alternative extension to the case of non-autonomous dynamical system of the concept of the exponential attractor was based on the work [7] of Chepyzhov and Vishik (see also [8, Chapter 4]), in which they introduced an approach to study a family of non-autonomous evolution equations of the form

where u ∈ E (Banach space) and Σ is an appropriate compact symbol space. More precisely, they constructed a semigroup (skew-product semiflow) associated with (1) on extended phase space Σ × E and used the theory of semigroup to study the longtime behavior of solutions of non-autonomous evolution equations. Newly extended attractors for non-autonomous dynamical system are called uniform exponential attractors, see [9, 10, 11, 12, 13] and the references therein. By its definition, a uniform exponential attractor is time independent exponentially attracting compact set and has finite fractal dimension. Because we regard extended phase space Σ × E as a whole, symbol space Σ require to be compact and finite dimensional when we use semigroup on Σ × E to study the existence of a uniform exponential attractor of non-autonomous evolution equations, this is why we choose k-dimensional torus 𝕋^k, k ∈ ℤ₊ (corresponding to the hull of quasi-periodic functions, see [8]) as the symbol space.

Our aim in this article is to extend the uniform exponential attractor to the random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system, and give a theorem on the existence of a random uniform exponential attractor for a jointly continuous NRDS. By definition, a random uniform exponential attractor is a random compact set with finite fractal dimension, which (pullback) attracts uniformly every element of attraction universe 𝓓 with exponential rate. We emphasize that the attraction universe considered in the definition is autonomous attraction universe (it contains only autonomous random set, see [14, 15]). We also emphasize that the random uniform exponential attractor has no (positive) invariance property along the sample path. According to [16], we have a criterion for the existence of a random exponential attractor for a continuous random dynamical system on a separable Hilbert space. Other results on existence criteria of a random exponential attractor for a random dynamical system can be found in [17, 18]. With the help of the concept of skew-product cocycle (i.e., random dynamical system) introduced in [15], we consider the existence of the random exponential attractor for a continuous skew-product cocycle (generated by a jointly continuous NRDS ϕ and a base flow θ) on the extended phase space 𝕋^k × E, and project this random exponential attractor onto the phase space E. Then we obtain the random uniform exponential attractor for the jointly continuous NRDS ϕ. Consequently, we formulate Theorem 2.8 on the existence of a random uniform exponential attractor for a jointly continuous NRDS.

As an application of Theorem 2.8, we will consider the following reaction-diffusion equation with quasi-periodic external force and multiplicative noise on ℝ³,

where u = u(x, t) is real-valued functions defined on ℝ³ × [0, +∞), the coefficients λ, b are positive constants. W(t) is a two-sided real-valued Brownian motion on a probability space which will be specified later. The symbol “∘” means that the stochastic integration in system is in the Stratonovich sense. σ͠(t) = (xt + σ)mod (𝕋^k), where σ ∈ 𝕋^k, x = (x₁, …, x_k) ∈ ℝ^k is a fixed vector satisfying that x₁, …, x_k are rationally independent. The functions g, f are assumed to satisfy some conditions.

There have been many works concerning random attractors for stochastic reaction-diffusion equation, see [19, Introduction] for detailed summary. It is worth noting that Zhou [19] considered the existence of random exponential attractor for non-autonomous stochastic reaction-diffusion equation with multiplicative noise in ℝ³. In the setting of f, g in [19], we here further assume that the external force g is quasi-periodic and prove the existence of a random uniform exponential attractor.

This paper is organized as follows. In the next section, we show some preliminaries and give a theorem on the existence of a random uniform exponential attractor for a jointly continuous NRDS. In section 3, we study the existence of a random uniform exponential attractor for the non-autonomous stochastic reaction-diffusion equation (2) defined on ℝ³.

2 Existence of random uniform exponential attractors

In this section, we present some notations and provide a criterion concerning the existence of a random uniform exponential attractor for a jointly continuous NRDS.

Let X be a separable Hilbert space with norm ∥⋅∥_X and Borel σ-algebra 𝓑(X), d_X denotes the metric induced from norm ∥⋅∥_X, the Hausdorff semi-distance between two nonempty subsets F₁, F₂ in X is defined by dist_X (F₁, F₂) = sup_u∈F1 inf_v∈F2 ∥u − v∥_X.

Let 𝕋^k be the k-dimensional torus:

with the identification

and the topology, metric induced from the topology, metric on ℝ^k. Thus, the norm in 𝕋^k is given by

Let x = (x₁, …, x_k) ∈ ℝ^k be a fixed vector such that x₁, …, x_k are rationally independent, i.e., if there exist integers l₁, …, l_k such that ∑j=1n l_jx_j = 0, then l_j = 0 for j = 1, …, k. For t ∈ ℝ, define

then {θ_t}_t∈ℝ is a translation group on 𝕋^k with

and

When the time symbol is quasiperiodic, we consider 𝕋^k as the symbol space (see [8]). 𝓑 (𝕋^k) denotes the Borel σ-algebra of 𝕋^k.

Let (Ω, 𝓕, ℙ) be a probability space, and let (Ω, 𝓕, ℙ, {ϑ}_t∈ℝ) be an ergodic metric dynamical system, where {ϑ}_t∈ℝ satisfies: (i) ϑ₀ is the identity on Ω; (ii) ϑ_s ∘ ϑ_t = ϑ_s+t, ∀ t, s ∈ ℝ; (iii) (t, ω) → ϑ_t ω is (𝓑(ℝ) × 𝓕, 𝓕) -measurable; (iv) ℙ-preserving: ℙ(ϑ_tF) = ℙ(F), ∀ t ≤ 0, F ∈ 𝓕; (v) if for any F ∈ 𝓕, provided P(ϑt−1F△F)=0, it holds ℙ(F) = 0 or 1, ∀ t ∈ ℝ; (vi) ϑ_t Ω = Ω, ∀ t ∈ ℝ (see [20]).

Two groups {θ_t}_t∈ℝ and {ϑ}_t∈ℝ are called base flows.

A RDS is said to be continuous if for each t ∈ ℝ⁺, ω ∈ Ω, the mapping ψ (t, ω, ⋅) is continuous.

A NRDS is said to be continuous if for each t ∈ ℝ⁺, ω ∈ Ω and σ ∈ 𝕋^k, the mapping ϕ (t, ω, σ, ⋅) is continuous. It is called jointly continuous in 𝕋^k and X if the mapping ϕ (t, ω, ⋅, ⋅) is continuous for each t ∈ ℝ⁺ and ω ∈ Ω. We obtain the general definition of NRDS by replacing torus 𝕋^k with general symbol space Σ in Definition 2.2 (see [15, Definition 2.1]).

We often write D(⋅) as D or {D(ω)}_ω∈Ω. Given two random sets D₁, D₂, we write D₁ ⊆ D₂ if D₁(ω) ⊆ D₂(ω) for a.e. ω ∈ Ω.

Hereafter, we denote by 𝓓(X) the collection of all tempered bounded random subset of X. For simplicity, we identify “ a.e. ω ∈ Ω” and “ω ∈ Ω” unless otherwise stated.

Define the extended space 𝕏 ≐ 𝕋^k × X with norm:

and Borel σ-algebra 𝓑(𝕏).

Obviously, any subset B ⊆ 𝕏 has the form 𝔹 = ∪_σ∈𝕋^k{σ} × B(σ), where B(σ) (possibly empty) is called the σ-section of 𝔹. Let P_σ 𝔹 ≐ B(σ), ∀ 𝔹 ⊆ 𝕏, and let

Then P_X is the projection from 𝕏 to X. Denote by P_𝕋^k the projection from 𝕏 to 𝕋^k.

By [15], condition (7) is equivalent to

which implies the stochastic perturbation happens only to the X-component.

By Definition 2.4, a random set 𝔹 in 𝕏 is tempered if it satisfies e−βt∥B(ϑ−tω)∥X ⟶t→∞  0,∀β>0. Let

and

then 𝓓_1,𝕏 ⊂ 𝓓_𝕏 ⊂ 𝓓_0,𝕏 and for any element 𝔹 ∈ 𝓓_𝕏, there exist an element 𝔹₁ ∈ 𝓓_1,𝕏 such that 𝔹 ⊆ 𝔹₁.

For K-dimensional subspace X_K of X (K ∈ ℕ), we define the bounded projections ℙ_k+K : 𝕏 → 𝕏_K = 𝕋^k × X_K and Qk+K:X→XK⊥=Tk×XK⊥ as

where P_K : X → X_K is K-dimensional orthogonal projection from X into X_K, Q_K = I_X − P_K and ℚ_k+K = I_𝕏 − ℙ_k+K, where I_𝕏 is the identity operator on 𝕏.

To study the existence of a random uniform exponential attractor for a NRDS on a separable Hilbert space, we need to introduce a skew-product cocycle on extended space 𝕏 (see [10, Section 4.1]). Given an NRDS ϕ, define a mapping π : ℝ⁺ × Ω × 𝕏 → 𝕏 by

Then the mapping π is a RDS, namely, satisfying

1. π is (𝓑(ℝ⁺) × 𝓕 × 𝓑(𝕏), 𝓑(𝕏))-measurable;

2. π (0, ω, 𝓧) = 𝓧, ∀ ω ∈ Ω, 𝓧 ∈ 𝕏;

3. the cocycle property π(t + s, ω, 𝓧) = π(t, ϑ_sω, π (s, ω, 𝓧)), ∀ t, s ≥ 0, ω ∈ Ω, 𝓧 ∈ 𝕏.

The RDS π is called the skew-product cocycle generated by ϕ and θ. Note that π is continuous, that is, the mapping 𝓧 → π(⋅, ⋅, 𝓧) is continuous in 𝕏, if and only if ϕ is jointly continuous in 𝕋^k and X.

Now we define the random uniform exponential attractor for continuous NRDS {ϕ(t, ω, σ)}_{t≥0,ω∈Ω,σ∈𝕋^k} on a separable Hilbert space X.

We make the following assumptions on the continuous skew-product {π(t, ω)}_{t≥0,ω∈Ω} on extended space 𝕏 defined in (10):

1. there exists a tempered closed random set {χ(ω)}_ω∈Ω of 𝕏 such that for any ω ∈ Ω,

   1. the diameter ∥χ(ω)∥_𝕏 of χ(ω) is bounded by a tempered random variable R_ω, i.e.,

      supX,Y∈χ(ω)||X−Y||X≤Rω<∞,

      where R_ϑtω is continuous in t for all t ∈ ℝ;

   2. χ(ω) is positively invariant with respect to {ϑ_t}_t∈ℝ in the sense that π(t, ϑ_−t ω)χ(ϑ_−tω) ⊆ χ(ω), for all t ≥ 0;

   3. χ(ω) is pullback absorbing in the sense that for any family of set 𝔹 ∈ 𝓓_𝕏, there exist T_𝔹 = T_𝔹(ω) ≥ 0 such that π(t, ϑ_−tω)𝔹(ϑ_−tω) ⊆ χ(ω) for t ≥ T_𝔹;

2. there exist positive numbers λ̄, δ̄, random variables C̄₀(ω), C̄₁(ω) ≥ 0 and (k + K)-dimensional projector ℙ_k+K : 𝕏 → ℙ_k+K𝕏 (dim (ℙ_k+K𝕏) = (k + K) ∈ ℕ) such that for every ω ∈ Ω and any 𝓧, 𝓨 ∈ χ(ω),

   ∥π(t,ω)X−π(t,ω)Y∥X≤e∫08ln⁡2λ¯C¯0(θsω)ds∥X−Y∥X,∀t∈[0,8ln⁡2λ¯] (12)

   and

   (IX−Pk+K)π(8ln⁡2λ¯,ω)X−π(8ln⁡2λ¯,ω)YX≤(e−8ln⁡2+∫08ln⁡2λ¯C¯1(θsω)ds+δ¯2e∫0t¯0C¯0(θsω)ds)∥X−Y∥X, (13)

   where λ̄, K are independent of ω;

3. C̄₀(ω), C̄₁(ω), λ̄, δ̄ satisfy:

   0≤E[C¯02(ω)]<∞,0≤E[C¯1(ω)]≤λ¯16,0<δ¯≤min116,e−2ln⁡32128(ln⁡2)2λ¯2E[C¯02(ω)]+64(ln⁡2)2λ¯E[C¯0(ω)], (14)

   where “E”denotes the expectation.