Random attractors for stochastic retarded strongly damped wave equations with additive noise on bounded domains

Xiaoyao Jia; Xiaoquan Ding

doi:10.1515/math-2019-0038

Artikel Open Access

Random attractors for stochastic retarded strongly damped wave equations with additive noise on bounded domains

Xiaoyao Jia und Xiaoquan Ding

Veröffentlicht/Copyright: 30. Mai 2019

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$Open Mathematics$

Aus der Zeitschrift Open Mathematics Band 17 Heft 1

Abstract

In this paper we study the asymptotic behavior for a class of stochastic retarded strongly damped wave equation with additive noise on a bounded smooth domain in ℝ^d. We get the existence of the random attractor for the random dynamical systems associated with the equation.

Keywords: stochastic wave function; time-delay; random attractor; random dynamical system; additive noise

MSC 2010: 35B40; 35B41; 35L05; 35L20

1 Introduction

The main aim of this paper is to investigate the asymptotic behavior of the solution to the following stochastic strongly damped wave equation with time-delay and with additive noise on a bounded set D ⊂ ℝ^d:

utt−αΔut−Δu+ut+λu=f(ut)+g(x)+∑j=1mhj(x)dWjdt, (1)

with the initial value conditions

u(t,x)=u0(t,x), ut(t,x)=∂∂tu0(t,x), for t∈[−h,0], x∈D,

and the boundary condition

u(t,x)=0, for t∈[−h,∞), x∈∂D.

Here λ and α are positive constants, W_j(j = 1, 2, ⋯, m) is a real-valued two-sided Wiener process on a probability space (Ω, ℱ, ℙ), which will be specified later, g is a given function defined on D, u is a real function, f is a nonlinear functional satisfying some conditions which will be specified later, u^t(⋅) = u (t + ⋅).

Wave equation is a kind of hyperbolic equation, which can be used to describe the wave phenomena in nature and engineering. Hence, wave equation is a very important research field. Some evolution systems in physics, chemistry and life science, depend not only on the current status, but also on the status in the past period. These systems can be described by time-delay partial differential equations.

In this paper we study the asymptotic behavior of the solution to the stochastic time-delay wave equation (1), when time tends to infinite. As we know, the asymptotic behavior of random system can be studied by its random attractor, which was first introduced by Crauel and Flandoli [9]. In recent years, the properties of random attractors have been studied by many authors, see [10, 14, 16, 17, 18, 19, 21, 24, 25, 30, 31, 32] and references therein.

The asymptotic behavior of the solution to deterministic wave equation has been investigated by many authors, see [2, 22, 23, 29, 33] and reference therein. In [33], S. Zhou obtained the uniformly boundedness of the global attractor, and the estimate of the upper bound of the Hausdorff dimension of the global attractor for strongly damped nonlinear wave equations. In [2], J.M. Ball proved the existence of a global attractor for the semi-linear wave equation on a bounded domain Ω ⊂ ℝⁿ with Dirichlet boundary conditions. The existence of the attractor for a class of strongly damped semi-linear wave equation on a bounded domain Ω ⊂ ℝ³ had been obtained by V. Pata [22]. In [29], the authors got the existence of a global attractor for the wave equation with nonlinear damping on a bounded domain. In [23], D. Prazak proved the global attractor for a class of semi-linear damped wave equation on bounded domain in ℝⁿ (n = 2, 3) has finite fractal dimension.

The random attractor for the stochastic wave equation has been studied also by many authors [12, 26, 27, 28, 34]. In [12] X. Fan got the existence and fractal dimension of random attractors for stochastic wave equations with multiplicative noise on a bounded domain Ω ⊂ ℝⁿ (n = 1, 2, 3). Z. Wang and S. Zhou investigated the random attractor for stochastic damped wave equation with multiplicative noise [26] and additive noise [27] on unbounded domain.

However, the results for the stochastic retarded wave equation are very few. Retarded wave equations are widely used in engineering, biology, physics and chemistry. Therefore, it is important for us to study the asymptotic behavior of random attractor for stochastic retarded strongly damped wave equation. To prove the existence of random attractor, we need to get some kind of compactness. Hence, we need the higher regularity for the solution. To this end, by using the method in [15, 20], we decompose the solution into two parts. One part has exponential decay with time and the other part has higher regularity. It follows from the higher regularity of the solution and Ascoli theorem that the random absorbing set are compact.

This paper is organized as follows. In Section 2, we recall a theorem for the existence of random attractor, and show that Equation (1) generates a random dynamical system. In Section 3, we prove the dynamical system has a random absorbing set, and give the uniform estimates of the solution. In Section 4, we prove the existence of random attractor.

2 Preliminaries and Random Dynamical Systems

In this section, we first recall a result for the existence of random attractor for a continuous random dynamical system (RDS), and then introduce some notations which will be used in this paper. At last, we show that the equation (1) generates a random dynamical system.

Let (X, ∥⋅∥_X) be a Banach space with Borel σ-algebra ℬ(X). Suppose that (Ω, ℱ, ℙ, (θ_t)_t∈ℝ) is a metric dynamical system on the probability space (Ω, ℱ, ℙ). Suppose that ϕ is a continuous RDS of X over (Ω, ℱ, ℙ, (θ_t)_t∈ℝ). And suppose 𝒟 be a collection of subsets of X. The reader can refer to [3] [7] for more basic knowledge about random dynamical systems.

Now, we refer to [3] [13] for the following result for the existence of random attractor for continuous RDS.

Proposition 2.1

Let {K(ω)}_ω∈Ω ∈ 𝒟 be a random absorbing set for the continuous RDS ϕ in 𝒟 and ϕ is 𝒟-pullback asymptotically compact in X. Then ϕ has a unique 𝒟-random attractor {A(ω)}_ω∈Ω which is given by

A(ω)=∩τ≥0∪t≥τϕ(t,θ−tω,K(θ−tω))¯. (2)

This result will be used to prove the existence of random attractor for the RDS generating by stochastic strongly damped wave equation with time-delay (1).

The following notation will be used in the rest of the paper. We use 〈⋅, ⋅〉 and ∥⋅∥ to denote the inner product and the norm in L²(D), and use the notation ∥⋅∥_X to denote the norm of a general Banach space X. For h > 0, let u^t be the function defined on [−h, 0] by the relation u^t(s) = u(t + s), s ∈ [−h, 0]. For any continuous function ξ:[−h, 0] → L²(D), define

||ξ||S=sups∈[−h,0]||ξ(s)||. (3)

Let S be the collection of all continuous function ξ : [−h, 0] → L²(D) with ∥ξ∥_S < ∞. Then, it is easy to check that (S, ∥⋅∥_S) is a Banach space.

In the following, let (Ω, ℱ, ℙ) be a probability space, with

Ω={ω=(ω1,ω2,⋯,ωm)∈C(R,Rm): ω(0)=0}. (4)

The Borel σ-algebra ℱ on Ω is generated by the compact open topology [1] and ℙ is the corresponding Wiener measure on ℱ. (θ_t)_t∈ℝ on Ω is defined by

θtω(s)=ω(s+t)−ω(t), s,t∈R. (5)

Then, (Ω, ℱ, ℙ, (θ_t)_t∈ℝ) is an ergodic metric dynamical system.

In the rest of this section, we show that there is a RDS generated by the following stochastic strongly damped wave equation:

utt−αΔut−Δu+ut+λu=f(ut)+g(x)+∑j=1mhj(x)dWjdt, (6)

with the initial value conditions

u(t,x)=u0(t,x), ut(t,x)=∂∂tu0(t,x), for t∈[−h,0], x∈D, (7)

and the boundary condition

u(t,x)=0, for t∈[−h,∞), x∈∂D. (8)

Here α, λ and h are positive constants, g is a given function in L²(D), and hj′s (j = 1, 2, ⋯, m) are given functions in H02 (D), W(t) ≡ (W₁(t), W₂(t), ⋯, W_m(t)) is a two-sided Wiener process on the probability space (Ω, ℱ, ℙ). We identify ω(t) with W(t), i.e. W_j(t) = ω_j(t) (j = 1, 2, ⋯, m). f : S → L²(D) is a continuous functional satisfying the following conditions:

f(0) = 0;
there exists a constant L_f > 0 such that, for all ξ, η ∈ S,

||f(ξ)−f(η)||≤Lf||ξ−η||S; (9)
there exist positive constants β₀ > 0, C_f > 0 such that ∀ β ∈ (0, β₀), t > 0, u ∈ C([−h, t]; L²(D)),

∫0teβs|f(us)|2ds≤Cf2∫−hteβs|u(s)|2ds. (10)

Let σ > 0 be a fixed constant such that

λ+σ2−σ>0, 1−σ>0, 1−σα>0. (11)

For convenience, we introduce the following natations. Set

β1=λ+σ2−σ, β2=1−σ, β3=1−σα. (12)

Set H = H¹(D) × L²(D), and the norm on H is the following

∥(u,v)∥H2=β1∥u∥2+β3∥∇u∥2+∥v∥2, for (u,v)∈H. (13)

Set

H=(u,v): u∈S, |∇u|∈S, v∈S, (14)

with the norm ||(u,v)||H2=β1||u||S2+β3||∇u||S2+||v||S2.

Set η = u_t + σ u, then (6)-(8) can be rewritten as the following form:

dudt=η−σu,dηdt=−β2η+αΔη−β1u+β3Δu+f(ut)+g(x)+∑j=1mhj(x)dWjdt, (15)

with the initial value conditions

u(t,x)=u0(t,x), η(t,x)=η0(t,x)≡∂∂tu0(t,x)+σu0(t,x), for t∈[−h,0], x∈D, (16)

and the boundary condition

u(t,x)=0, for t∈[−h,∞), x∈∂D. (17)

To show that the equation (15) generates a continuous RDS, we first transform (15) into a deterministic equation with random parameters. We use famous Ornstein-Uhlenbeck process to do that. For j = 1, 2, ⋯, m, set

zj(θtωj)≡−∫−∞0es(θtωj)(s)ds, t∈R. (18)

It is an Ornstein-Uhlenbeck process, and it is the solution of the following Itô equation:

dzj+zjdt=dWj. (19)

Moreover, the random variable z_j(θ_t ω_j) is tempered, and z_j(θ_t ω_j) is ℙ-a.e. continuous [3]. Set

z(θtω)=∑j=1mhj(x)zj(θtωj),

then (19) implies that

dz+zdt=∑j=1mhjdWj. (20)

Notice that h_j ∈ H02 (D) (j = 1, 2, ⋯, m). Thus, there exists a constant c > 0, such that,

||z(ω)||2≤c∑j=1m|zj(ωj)|2; ||∇z(ω)||2≤c∑j=1m|zj(ωj)|2; ||Δz(ω)||2≤c∑j=1m|zj(ωj)|2. (21)

It follows from Proposition 4.3.3 [1] that, there exists a tempered function r(ω) > 0 such that

∑j=1m|zj(ωj)|2≤r(ω), (22)

where r(ω) satisfies, for ℙ-a.e. ω ∈ Ω,

r(θtω)≤eβ2|t|r(ω), t∈R. (23)

Here β is a positive constant which will be fixed later. Then, it follows that, for ℙ-a.e. ω ∈ Ω, t ∈ ℝ

||z(θtω)||2+||∇z(θtω)||2+||Δz(θtω)||2≤ceβ2|t|r(ω). (24)

Set

v(t,x)=η(t,x)−z(θtω). (25)

Then the equation (15) can be rewritten as the following form

dudt=v−σu+z(θtω),dvdt=−β2v+αΔv−β1u+β3Δu+f(ut)+g(x)+σz(θtω)+αΔz(θtω), (26)

with the initial conditions

u(t,x)=u0(t,x), v(t,x)=v0(t,x)≡∂∂tu0(t,x)+σu0(t,x)−z(θtω), for t∈[−h,0], x∈D, (27)

and the boundary condition

u(t,x)=0, for t∈[−h,∞), x∈∂D. (28)

The equation (26) is a deterministic equation with random parameters. By [6], under the conditions (A1)-(A3), for each (u₀, v₀) ∈ 𝓗, (26) has a unique solution (u(t, ω, u₀), v(t, ω, v₀)) for a.e. ω ∈ Ω. For any T > 0, by the regularity of solutions for an analytic semigroup [4], we can get that

(u,v)∈C([−h,T];H1(D))×C([−h,T];L2(D)). (29)

The global solution can be obtained by the boundedness of solution of (26), by Lemma 3.1 below. Hence, equation (26) generates a continuousrandom dynamical system ϕ with

ϕ(t,ω,ϕ0)=(ut,vt), for t≥0, ω∈Ω, and ϕ0≡(u0,v0)∈H. (30)

Notice η(t, ω, η₀) = v(t, ω, η₀) + z(θ_t ω). Define ψ by

ψ(t,ω,ψ0)=(ut,ηt), for t≥0, ω∈Ω, and ψ0≡(u0,η0)∈H. (31)

Then, ψ also generates a continuous RDS associate with (15). It is easy to see that two random dynamical systems are equivalent. Thus, we need only to consider the random dynamical system ϕ.

3 Uniform estimates of solutions

In this section we prove some uniform estimates for the solution of the equation (26). To show that the RDS ϕ has an absorbing set, we need the following assumption:

Cf2<4σβ1β2. (32)

By condition (32), there exist two positive constants β₄, β₅, such that

Cf2=4β4β5, β4<σβ1, β5<β2. (33)

Throughout the rest of this paper we assume that 𝒟 is the collection of all tempered random subsets of 𝓗.

Lemma 3.1

Assume that conditions (A1)-(A3) and (32) hold. Then, there exists a random absorbing set {K(ω)} ∈ 𝒟 for ϕ, that is, for any {B(ω)} ∈ 𝒟, and for ℙ-a.e. ω ∈ Ω, there exists T₁(B, ω) > 0, such that

ϕ(t,θ−tω,B(θ−tω))⊂K(ω), for all t≥T1(B,ω). (34)

Proof

Taking the inner product of the first equation in (26) with u, −Δ u, and the second equation in (26) with v, respectively, we have that

12ddt||u||2=〈v,u〉−σ||u||2+〈z(θtω),u〉, (35)

12ddt||∇u||2=〈∇u,∇v〉−σ||∇u||2+〈∇z(θtω),∇u〉, (36)

12ddt||v||2=−β2||v||2−α||∇v||2−β1〈u,v〉−β3〈∇u,∇v〉+〈f(ut),v〉+ 〈g,v〉+σ〈z(θtω),v〉+α〈Δz(θtω),v〉. (37)

Then, summing up (35)×β₁, (36)×β₃ and (37), one has that

12ddt∥u,v∥H2=−σβ1||u||2−σβ3||∇u||2−β2||v||2−α||∇v||2+ 〈f(ut),v〉+〈g,v〉+β1〈z(θtω),u〉+β3〈∇z(θtω),∇u〉+σ〈z(θtω),v〉+α〈Δz(θtω),v〉. (38)

Now, we estimate the terms on the right hand side of (38). By (33), there exists a constant β₆ > 0 small enough, such that β₆ < β₂ − β₅. Using Young inequality, we obtain that

〈f(ut),v〉+〈g,v〉≤∥f(ut)∥ ∥v∥+∥g∥ ∥v∥≤14β5||f(ut)||2+β5∥v∥2+14β6||g||2+β6||v||2. (39)

Using Young inequality again, one has that

β1〈z(θtω),u〉+β3〈∇z(θtω),∇u〉+σ〈z(θtω),v〉+α〈Δz(θtω),v〉≤β1||z(θtω)|| ||u||+β3||∇z(θtω)|| ||∇u||+σ||z(θtω)|| ||v||+α||Δz(θtω)|| ||v||≤β124β7||z(θtω)||2+β7||u||2+β324β9||∇z(θtω)||2+β9||∇u||2+σ22β8||z(θtω)||2+β82||v||2+α22β8||Δz(θtω)||2+β82||v||2≤c1r(θtω)+β7||u||2+β8||v||2+β9||∇u||2. (40)

In the third step, we use (21) and (22). Here β₇, β₈, β₉ are positive constants such that β₇ < σβ₁ − β₄, β₈ < β₂ − β₅ − β₆, β₉ < σ β₃, and c₁ = cβ124β7+σ2+α22β8+β324β9 . Then, it follows from (38)-(40) that,

12ddt∥u,v∥H2≤−(σβ1−β4−β7)||u||2−(σβ3−β9)||∇u||2−(β2−β5−β6−β8)||v||2−α||∇v||2+ 14β5||f(ut)||2−β4||u||2+14β6||g||2+c1r(θtω). (41)

Set

β10=minσβ1−β4−β7β1,σβ3−β9β3,β2−β5−β6−β8. (42)

Then, one has that

ddt∥u,v∥H2≤−2β10∥u,v∥H2−2α||∇v||2+12β5||f(ut)||2−2β4||u||2+12β6||g||2+2c1r(θtω). (43)

Choose β ∈ (0, β₀) small enough, such that β < β₁₀. It follows from (43) that

ddteβt∥u,v∥H2≤−2β10−βeβt∥u,v∥H2−2αeβt||∇v||2+eβt2β5||f(ut)||2− 2β4eβt||u||2+eβt2β6||g||2+2c1eβtr(θtω). (44)

Integrate over t in both side of the last inequality,

eβt∥u(t),v(t)∥H2≤∥u(0),v(0)∥H2−2β10−β∫0teβs∥u(s),v(s)∥H2ds−2α∫0teβs||∇v(s)||2ds+12β5∫0teβs||f(us)||2ds−2β4∫0teβs||u(s)||2ds+∫0teβs2β6||g||2ds+2c1∫0teβsr(θsω)ds. (45)

By condition (A3) and (32), we find that

12β5∫0teβs||f(us)||2ds−2β4∫0teβs||u(s)||2ds≤Cf22β5∫−hteβs||u(s)||2ds−2β4∫0teβs||u(s)||2ds=2β4∫−h0eβs||u(s)||2ds≤2β4h||u0||S2. (46)

Noting that β < 2β₁₀, (45) and (46) imply that

eβt∥u(t),v(t)∥H2+2α∫0teβs||∇v(s)||2ds≤∥u(0),v(0)∥H2+2β4h||u0||S2+eβt2ββ6||g||2+2c1∫0teβsr(θsω)ds. (47)

Since ||u0||S≤1β1||(u0,v0)||H, and ∥u(0),v(0)∥H2≤∥u0,v0∥H2 , it follows from (47) that

∥u(t),v(t)∥H2+2α∫0teβ(s−t)||∇v(s)||2ds≤1+2β4hβ1e−βt∥u0,v0∥H2+12ββ6||g||2+2c1∫0teβ(s−t)r(θsω)ds. (48)

For τ ∈ [−h, 0], by (48), one has that, for t + τ > 0,

∥u(t+τ),v(t+τ)∥H2+2α∫0t+τeβ(s−t−τ)||∇v(s)||2ds≤1+2β4hβ1e−β(t+τ)∥u0,v0∥H2+12ββ6||g||2+2c1∫0t+τeβ(s−t−τ)r(θsω)ds≤1+2β4hβ1eβ(h−t)∥u0,v0∥H2+12ββ6||g||2+2c1eβh∫0teβ(s−t)r(θsω)ds, (49)

and for t + τ ≤ 0

∥u(t+τ),v(t+τ)∥H2+2α∫0t+τeβ(s−t−τ)||∇v(s)||2ds≤∥u0,v0∥H2≤1+2β4hβ1eβ(h−t)∥u0,v0∥H2+12ββ6||g||2+2c1eβh∫0teβ(s−t)r(θsω)ds. (50)

By (49) and (50), we obtain that, for t ≥ 0, τ ∈ [−h, 0]

ut,vtH2+2α∫0t+τeβ(s−t−τ)||∇v(s)||2ds≤1+2β4hβ1eβ(h−t)∥u0,v0∥H2+12ββ6||g||2+2c1eβh∫0teβ(s−t)r(θsω)ds. (51)

Replacing ω by θ_−t ω in (51), we obtain that,

ut(θ−tω,u0(θ−tω)),vt(θ−tω,v0(θ−tω))H2+2α∫0t+τeβ(s−t−τ)||∇v(s,θ−tω,v0(θ−tω))||2ds≤1+2β4hβ1eβ(h−t)∥u0(θ−tω),v0(θ−tω)∥H2+12ββ6||g||2+2c1eβh∫0teβ(s−t)r(θs−tω)ds=1+2β4hβ1eβ(h−t)∥u0(θ−tω),v0(θ−tω)∥H2+12ββ6||g||2+4c1βeβhr(ω). (52)

In the last term, we use (23). Notice that (u₀(θ_−tω), v₀(θ_−tω)) ∈ B(θ_−tω), and {B(ω)} is tempered. Then,

limt→+∞1+2β4hβ1eβ(h−t)∥u0(θ−tω),v0(θ−tω)∥H2=0. (53)

Hence, for any B ∈ 𝒟, there exists a random variable T₁(B, ω) > 0 such that, for all t > T₁(B, ω),

ut(θ−tω,u0(θ−tω)),vt(θ−tω,v0(θ−tω))H2+2α∫0t+τeβ(s−t−τ)||∇v(s,θ−tω,v0(θ−tω))||2ds≤1+12ββ6||g||2+4c1βeβhr(ω)≡r1(ω). (54)

It is easy to see that r₁(ω) is tempered. This ends the proof. □

To show the existence of random attractor, we need to get some estimates for Δ u and Δ v. To this end, we decompose the solution of (26) into two parts, and then obtain some priori estimates for the solutions. Set

u=u~+u^; v=v~+v^. (55)

Here (ũ, ṽ) and (û, v̂) are the functions which satisfy the following equations respectively

du~dt=v~−σu~,dv~dt=−β2v~+αΔv~−β1u~+β3Δu~,u~(t,x)=u0(t,x), v~(t,x)=v0(t,x), t∈[−h,0], x∈D,u~(t,x)=0, t∈[−h,+∞), x∈∂D. (56)

and

du^dt=v^−σu^+z(θtω),dv^dt=−β2v^+αΔv^−β1u^+β3Δu^+f(ut)+g(x)+σz(θtω)+αΔz(θtω),u^(t,x)=0, v^(t,x)=0, t∈[−h,0], x∈D,u^(t,x)=0, t∈[−h,+∞), x∈∂D. (57)

First, we estimate (ũ, ṽ). By the proof of Lemma 3.1, it follows from (52) that, for t > 0, τ ∈ [−h, 0]

u~t(θ−tω,u0(θ−tω)),v~t(θ−tω,v0(θ−tω))H2+2α∫0t+τeβ(s−t−τ)||∇v~(s,θ−tω,v0(θ−tω))||2ds ≤1+2β4hβ1eβ(h−t)∥u0(θ−tω),v0(θ−tω)∥H2. (58)

It follows from (52) and (58) that

u^t(θ−tω,0),v^t(θ−tω,0)H2+2α∫0t+τeβ(s−t−τ)||∇v^(s,θ−tω,0))||2ds≤2ut(θ−tω,u0(θ−tω)),vt(θ−tω,v0(θ−tω))H2+2u~t(θ−tω,u0(θ−tω)),v~t(θ−tω,v0(θ−tω))H2+ +4α∫0t+τeβ(s−t−τ)∇v(s,θ−tω,v0(θ−tω)))2+∇v^(s,θ−tω,v0(θ−tω)))2ds≤41+2β4hβ1eβ(h−t)∥u0(θ−tω),v0(θ−tω)∥H2+1ββ6||g||2+8c1βeβhr(ω). (59)

Thus, (54) and (59) imply that, for t > T₁(B, ω), τ ∈ [−h, 0],

u^(t,θ−tω,0),v^(t,θ−tω,0)H2+2α∫0t+τeβ(s−t−τ)||∇v^(s,θ−tω,0)||2ds≤2r1(ω)+2. (60)

Here T₁(B, ω) and r₁(ω) are random variables defined in Lemma 3.1. Next, we give a priori estimate for (Δ û, Δ û).

Lemma 3.2

Assume that conditions (A1)-(A3) and (32) hold. Let B ∈ 𝒟 and (u₀(ω), v₀(ω)) ∈ B(ω). Then there exist two random variables r₂(ω) and T₂(B, ω) > 0, such that, the solution (û(t, ω, 0), v̂(t, ω, 0)) of (57) satisfies, for t > T₂(B, ω),

β3||Δu^(t,θ−tω,0)||2+||∇v^(t,θ−tω,0)||2+α∫0teβ(s−t)||Δv^(s,θ−tω,0)||2ds≤r2(ω). (61)

Proof

Using the first equation in (57), one has that

ddtΔu^=Δv^−σΔu^+Δz(θtω). (62)

Taking the inner product of (62) and the second equation in (57) with Δ û and −Δ v̂ respectively, we obtain

12ddt||Δu^||2=〈Δv^,Δu^〉−σ||Δu^||2+〈Δz(θtω),Δu^〉, (63)

12ddt||∇v^||2=−β2||∇v^||2−α||Δv^||2−β3〈Δu^,Δv^〉+β1〈u^,Δv^〉−〈f(ut)+g+σz(θtω)+αΔz(θtω),Δv^〉. (64)

Adding up (63)×β₃ and (64), we get

12ddtβ3||Δu^||2+||∇v^||2=−σβ3||Δu^||2−β2||∇v^||2−α||Δv^||2+β1〈u^,Δv^〉+ β3〈Δz(θtω),Δu^〉−〈f(ut)+g+σz(θtω)+αΔz(θtω),Δv^〉. (65)

It follows from (21), (22) and Young inequality, that

β3〈Δz(θtω),Δu^〉≤σβ32||Δu^||2+β32σ||Δz||2≤σβ32||Δu^||2+cβ32σr(θtω), (66)

and

β1〈u^,Δv^〉−〈f(ut)+g+σz(θtω)+αΔz(θtω),Δv^〉≤α2||Δv^||2+5β122α||u^||2+52α||f(ut)||2+52α||g||2+5c(σ2+α2)2αr(θtω). (67)

It follows from (65)-(67) that

ddtβ3||Δu^||2+||∇v^||2≤−β11β3||Δu^||2+||∇v^||2−α||Δv^||2+ 5β12α||u^||2+5α||f(ut)||2+5α||g||2+β12r(θtω), (68)

with β₁₁ = min {σ, 2β₂} and β12=5c(σ2+α2)α+cβ3σ . Let β > 0 be the constant defined in Lemma 3.1. Noting that β₁₀ < β₁₁, we have β < β₁₁. Thus, multiplying e^βt in both side of (68), we have

ddtβ3eβt||Δu^||2+eβt||∇v^||2≤−(β11−β)β3eβt||Δu^||2+eβt||∇v^||2−αeβt||Δv^||2+ 5β12αeβt||u^||2+5αeβt||f(ut)||2+5αeβt||g||2+β12eβtr(θtω), (69)

Noticing (û(0), v̂(0)) = (0, 0) and β < β₁₁, integrate with respect to t in both side of (69),

β3||Δu^(t)||2+||∇v^(t)||2+α∫0teβ(s−t)||Δv^(s)||2ds≤5β12α∫0teβ(s−t)||u^(s)||2ds+5α∫0teβ(s−t)||f(us)||2ds+5αβ||g||2+β12∫0teβ(s−t)r(θsω)ds. (70)

Replace ω by θ_−t ω in last inequality,

β3||Δu^(t,θ−tω,0)||2+||∇v^(t,θ−tω,0)||2+α∫0teβ(s−t)||Δv^(s,θ−tω,0)||2ds≤5β12α∫0teβ(s−t)||u^(s,θ−tω,0)||2ds+5α∫0teβ(s−t)||f(us(θ−tω,u0(θ−tω)))||2ds+5αβ||g||2+β12∫0teβ(s−t)r(θs−tω)ds≤5β12α∫0teβ(s−t)||u^(s,θ−tω,0)||2ds+5α∫0teβ(s−t)||f(us(θ−tω,u0(θ−tω)))||2ds+5αβ||g||2+2β12βr(ω). (71)

In the second step, we use (23). Next, we estimate the first two terms on the right hand side of (71). For the first term, using (23) and (59), we have

5β12α∫0teβ(s−t)||u^(s,θ−tω,0)||2ds≤20β1(β1+2β4h)α∫0teβ(h−t)∥u0(θ−tω),v0(θ−tω)∥H2ds+5β12αβ2β6||g||2+40c1β12αβeβh∫0teβ(s−t)r(θs−tω)ds≤20β1(β1+2β4h)αteβ(h−t)∥u0(θ−tω),v0(θ−tω)∥H2+5β12αβ2β6||g||2+80c1β12αβ2eβhr(ω). (72)

By Condition (A3), one has that

5α∫0teβ(s−t)||f(us(θ−tω,u0(θ−tω)))||2ds≤5Cf2α∫−hteβ(s−t)||u(s,θ−tω,u0(θ−tω))||2ds=5Cf2α∫−h0eβ(s−t)||u(s,θ−tω,u0(θ−tω))||2ds+5Cf2α∫0teβ(s−t)||u(s,θ−tω,u0(θ−tω))||2ds≤5Cf2hαe−βt||u0(θ−tω))||S2+5Cf2α∫0teβ(s−t)||u(s,θ−tω,u0(θ−tω))||2ds. (73)

Similarly as in (72), by (52), one has that

5Cf2α∫0teβ(s−t)||u(s,θ−tω,u0(θ−tω))||2ds≤5Cf2(β1+2β4h)αβ1teβ(h−t)∥u0(θ−tω),v0(θ−tω)∥H2+5Cf22αβ2β6||g||2+20c1Cf2αβ2eβhr(ω). (74)

Notice (u₀(ω), v₀(ω)) ∈ B(ω). It follows that

limt→+∞teβ(h−t)∥u0(θ−tω),v0(θ−tω)∥H2+e−βt||u0(θ−tω))||S2=0. (75)

Therefore, it follows from (71)-(75) that, for any (u₀(ω), v₀(ω)) ∈ B(ω), there exists T₂(B, ω) > 0, such that, for all t > T₂(B, ω),

β3||Δu^(t,θ−tω,0)||2+||∇v^(t,θ−tω,0)||2+α∫0teβ(s−t)||Δv^(s,θ−tω,0)||2ds≤1+5αβ+5β12αβ2β6+5Cf22αβ2β6||g||2+2β12β+80c1β12αβ2eβh+20c1Cf2αβ2eβhr(ω)≡r2(ω). (76)

This completes the proof. □

Lemma 3.3

Assume that conditions (A1)-(A3) and (32) hold. Let B(ω) ∈ 𝒟 and (u₀( ω), v₀(ω)) ∈ B(ω). Then there exist two random variables r₃(ω) and T₃(B, ω), such that, the solution (û(t, ω, 0), v̂(t, ω, 0)) of (57) satisfies, for t > T₃(B, ω) and −h ≤ σ₁ < σ₂ ≤ 0,

u^t(σ1,θ−tω,0),v^t(σ1,θ−tω,0)−u^t(σ2,θ−tω,0),v^t(σ2,θ−tω,0)H≤r3(ω)|σ1−σ2|1/2. (77)

Proof

By the definition of ∥⋅∥_H, one has that

(u^t(σ1,θ−tω,0),v^t(σ1,θ−tω,0))−(u^t(σ2,θ−tω,0),v^t(σ2,θ−tω,0))H≤β1u^(t+σ1,θ−tω,0)−u^(t+σ2,θ−tω,0)+β3∇u^(t+σ1,θ−tω,0)−∇u^(t+σ2,θ−tω,0) +v^(t+σ1,θ−tω,0)−v^(t+σ2,θ−tω,0)≤β1∫t+σ1t+σ2u^′(s,θ−tω,0)ds+β3∫t+σ1t+σ2∇u^′(s,θ−tω,0)ds+∫t+σ1t+σ2v^′(s,θ−tω,0)ds. (78)

Now, we estimate the terms on the right hand side of (78). For the first term, using (57), (60) and (24), we obtain that, for t > T₁(B, ω),

∫t+σ1t+σ2u^′(s,θ−tω,0)ds≤∫t+σ1t+σ2v^(s,θ−tω,0)ds+σ∫t+σ1t+σ2u^(s,θ−tω,0)ds+∫t+σ1t+σ2z(θs−tω)ds≤∫t+σ1t+σ2v^(s,θ−tω,0)2ds1/2+σ∫t+σ1t+σ2u^(s,θ−tω,0)2ds1/2+∫t+σ1t+σ2z(θs−tω)2ds1/2|σ1−σ2|1/2≤1+σβ1∫t+σ1t+σ22r1(θs−tω)+2ds1/2+∫t+σ1t+σ2z(θs−tω)2ds1/2|σ1−σ2|1/2≤1+σβ1h1/22supτ∈[−h,0]r1(θτω)+21/2+c1/2h1/2eβh/4r1/2(ω)|σ1−σ2|1/2. (79)

In the third step, we use (60), and in the last step, we use (24). Notice that

∫0t+σ2eβ(s−t−σ2)||∇v^(s,θ−tω,0)||2ds≥∫t+σ1t+σ2eβ(s−t−σ2)||∇v^(s,θ−tω,0)||2ds≥e−βh∫t+σ1t+σ2||∇v^(s,θ−tω,0)||2ds.

Hence, it follows from (60) that, for t > T₁(B, ω),

∫t+σ1t+σ2||∇v^(s,θ−tω,0)||2ds≤eβhαr1(ω)+1. (80)

Using similar computation as (79), we can get the following estimate: for t > T₁(B, ω),

∫t+σ1t+σ2∇u^′(s,θ−tω,0)ds≤∫t+σ1t+σ2∇v^(s,θ−tω,0)2ds1/2+σ∫t+σ1t+σ2∇u^(s,θ−tω,0)2ds1/2+∫t+σ1t+σ2∇z(θs−tω)2ds1/2|σ1−σ2|1/2≤eβh/2α(r1(ω)+1)1/2+σβ3h1/22supτ∈[−h,0]r1(θτω)+21/2+c1/2h1/2eβh/4r1/2(ω)|σ1−σ2|1/2. (81)

Next, we estimate the third term on the right hand side of (78). By the second equation in (58), we get that

∫t+σ1t+σ2v^′(s,θ−tω,0)ds≤∫t+σ1t+σ2β2v^(s,θ−tω,0)+αΔv^(s,θ−tω,0)+β1u^(s,θ−tω,0)+ β3Δu^(s,θ−tω,0)+f(us(θ−tω,u0(θ−tω)))+g+σz(θs−tω)+αΔz(θs−tω)ds. (82)

By (60), we obtain that, for t > T₁(B, ω),

∫t+σ1t+σ2β2v^(s,θ−tω,0)+β1u^(s,θ−tω,0)ds≤(1+β2)∫t+σ1t+σ2(u^(s,θ−tω,0),v^(s,θ−tω,0))H2ds1/2|σ1−σ2|1/2≤(1+β2)h1/2supτ∈[−h,0]2r1(θτω)+21/2|σ1−σ2|1/2. (83)

By Lemma 3.2, for t > T₂(B, ω),

∫t+σ1t+σ2β3Δu^(s,θ−tω,0)+αΔv^(s,θ−tω,0)ds≤β3∫t+σ1t+σ2Δu^(s,θ−tω,0)2ds1/2+α∫t+σ1t+σ2Δv^(s,θ−tω,0)2ds1/2|σ1−σ2|1/2≤β3hsupτ∈[−h,0]r2(θτω)1/2+αr21/2(ω)|σ1−σ2|1/2. (84)

By Condition (A1), (A2) and Lemma 3.1, for t > T₁(B, ω)

∫t+σ1t+σ2f(us(θ−tω,u0(θ−tω)))ds≤Lf∫t+σ1t+σ2us(θ−tω,u0(θ−tω)))Hds≤Lfh1/2supτ∈[−h,0]r11/2(θτω)|σ1−σ2|1/2. (85)

By (24), one has that

∫t+σ1t+σ2g+σz(θs−tω)+αΔz(θs−tω)ds≤||g||+(σ+α)c1/2eβh/4r1/2(ω)h1/2|σ1−σ2|1/2. (86)

Set

T3(B,ω)=maxT1(B,ω),T2(B,ω);r3(ω)≡β11+σβ1h1/22supτ∈[−h,0]r1(θτω)+21/2+c1/2h1/2eβh/4r1/2(ω)+ β3eβh/2α(r1(ω)+1)1/2+σβ3h1/22supτ∈[−h,0]r1(θτω)+21/2+c1/2h1/2eβh/4r1/2(ω)+ (1+β2)h1/2supτ∈[−h,0]2r1(θτω)+21/2+β3hsupτ∈[−h,0]r2(θτω)1/2+αr21/2(ω)+ Lfh1/2supτ∈[−h,0]r11/2(θτω)+||g||+(σ+α)c1/2eβh/4r1/2(ω)h1/2. (87)

Then, it follows from (78)-(86) that, for t > T₃(B, ω),

(u^t(σ1,θ−tω,0),v^t(σ1,θ−tω,0)−(u^t(σ2,θ−tω,0),v^t(σ2,θ−tω,0)H2≤r3(ω)|σ1−σ2|1/2. (88)

This ends the proof. □

4 Random attractor

In this section we prove the existence of Random attractor for the RDS generated by (26). First, we will use the prior estimates in Section 3 to show that RDS ϕ is asymptotically compact.

Lemma 4.1

Assume that conditions (A1)-(A3) and (32) hold. The RDS ϕ is 𝒟-pullback asymptotically compact in 𝓗, that is, for ℙ-a.e. ω ∈ Ω, the sequence {ϕ(tn,θ−tnω,ϕ0,n(θ−tnω))}n=1∞ has convergent subsequence in 𝓗, as t_n → ∞, for any B ∈ 𝒟 and ϕ_0,n(θ_{−t_n} ω) ∈ B(θ_{−t_n} ω).

Proof

Since t_n → +∞, there exists N₁ = N₁(B, ω) large enough, such that, for n ≥ N₁, t_n ≥ T₃(B, ω) + h. Thus, it follows from (60) and Lemma 3.2 that, for n ≥ N₁ and σ ∈ [−h, 0],

u^(σ,θ−tnω,0)H2(D)2≤1β1+1β32r1(ω)+2+1β3r2(ω), v^(σ,θ−tnω,0)H1(D)2≤2r1(ω)+2+r2(ω). (89)

By the compact embedding theorem, we obtain that H¹(D) ↪ L²(D) and H²(D) ↪ H¹(D) are compact. Hence, for σ ∈ [−h, 0] and n ≥ N₁, {(û (σ, θ_{−t_n} ω, 0), v̂ (σ, θ_{−t_n} ω, 0)} is relatively compact in H. By Lemma 3.3, for n ≥ N₁, {(û (⋅, θ_{−t_n} ω, 0), v̂ (⋅, θ_{−t_n} ω, 0)} is equi-continuous in C([−h, 0], H). Hence, it follows from Ascoli theorem that, {(û (σ, θ_{−t_n} ω, 0), v̂ (σ, θ_{−t_n} ω, 0)} is relatively compact in C([−h, 0], H). Therefore, we can find a subsequence {t_{n_i}} such that,

u^tn(⋅,θ−tnω,0),v^tn(⋅,θ−tnω,0)→ζ(⋅,ω),ξ(⋅,ω)) in C([−h,0],H). (90)

For any (u₀, v₀) ∈ B, B ∈ 𝒟 and ϵ > 0, there exists N₂ = N₂(B, ω, ϵ), such that for n > N₂(B, ω, ϵ), and σ ∈ [−h, 0],

(u^tn(⋅,θ−tnω,0)−ζ(⋅,ω),v^tn(⋅,θ−tnω,0)−ξ(⋅,ω))H<ϵ. (91)

By (58), there exists there exists N₃ = N₃(B, ω, ϵ), such that for n > N₃(B, ω, ϵ),

(u~tn(⋅,θ−tnω,u0(θ−tnω)),v~tn(⋅,θ−tnω,v0(θ−tnω)))H<ϵ. (92)

By (91) and (92), we have that, for n > max {N₁, N₂, N₃}

(utn(⋅,θ−tnω,u0(θ−tnω))−ζ(⋅,ω),vtn(⋅,θ−tnω,v0(θ−tnω))−ξ(⋅,ω))H≤2(u^tn(⋅,θ−tnω,0)−ζ(⋅,ω),v^tn(⋅,θ−tnω,0)−ξ(⋅,ω))H+2(u~tn(⋅,θ−tnω,u0(θ−tnω)),v^tn(⋅,θ−tnω,v0(θ−tnω)))H<4ϵ. (93)

This completes the proof. □

Theorem 4.2

Assume that conditions (A1)-(A3) and (32) hold. Then, the random dynamical system ϕ has a unique 𝒟-random attractor in 𝓗.

Proof

By Lemma 3.1, ϕ has a closed absorbing set K(ω) in 𝒟, and by Lemma 4.1, ϕ is 𝒟-pullback asymptotically compact in 𝓗. Hence, by Proposition 2.1, ϕ has a unique 𝒟-random attractor. This ends the proof. □

Competing interests: The authors declare that there is no conflict of interest regarding the publication of this paper.

Acknowledgement

This work is partially supported by the National Natural Science Foundation of China under Grants 11301153 and 11271110, the Key Programs for Science and Technology of the Education Department of Henan Province under Grand 12A110007, and the Scientific Research Funds of Henan University of Science and Technology.

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Received: 2018-03-13

Accepted: 2019-03-06

Published Online: 2019-05-30

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Artikel in diesem Heft

https://doi.org/10.1515/math-2019-0038

Schlagwörter für diesen Artikel

stochastic wave function; time-delay; random attractor; random dynamical system; additive noise

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BY 4.0