Finite fractal dimension of random attractor for stochastic non-autonomous strongly damped wave equation

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Finite fractal dimension of random attractor for stochastic non-autonomous strongly damped wave equation✩ Zhaojuan Wang *, Lingping Zhang School of Mathematical Science, Huaiyin Normal University, Huaian 223300, PR China

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Article history: Received 7 July 2017 Received in revised form 7 January 2018 Accepted 4 February 2018 Available online xxxx

a b s t r a c t In this paper, we first prove the existence of a random attractor for stochastic nonautonomous strongly damped wave equations with additive white noise. Then we apply a criteria to obtain an upper bound of fractal dimension of the random attractor of considered system. © 2018 Elsevier Ltd. All rights reserved.

Keywords: Stochastic strongly damped wave equation Random attractor Fractal dimension

1. Introduction Let (Ω , F , P) be a probability space, where Ω = {ω ∈ C (R, R) : ω(0) = 0}, the Borel σ -algebra F on Ω is generated by the compact open topology, and P is the corresponding Wiener measure on F . For any t ∈ R, define a mapping θt on Ω by θt ω(·) = ω(t + ·) − ω(t) for ω ∈ Ω , then (Ω , F , P, (θt )t ∈R ) is an ergodic metric dynamical system [1]. Consider the following initial boundary valued problem of non-autonomous strongly damped wave equations with additive white noise:

⎧ ˙ (t), t > τ , x ∈ U , τ ∈ R, ⎨utt + ut − ∆u − α ∆ut + f (u, x) = g(x, t) + h(x)W u(x, t)|x∈∂ U = 0, t ≥ τ , τ ∈ R, ⎩ u(x, τ ) = uτ (x), ut (x, τ ) = u1τ (x), x ∈ U , τ ∈ R,

(1.1)

where U is an open bounded set of Rn (n ≤ 3) with a smooth boundary ∂ U ; α > 0 is the strong damping coefficient; u(t) = u(x, t) is a real-valued function on U × [τ , +∞), τ ∈ R; f (·, x) ∈ C 1 (R, R); h(·) ∈ H01 (U) ∩ H 2 (U); g(·, t) ∈ Cb (R, H01 (U)), Cb (R, H01 (U)) denotes the set of continuous bounded functions from R into H01 (U); W (t) is a two-sided real-valued Wiener process on the probability space (Ω , F , P); the initial data uτ (x), u1τ (x) are assumed to be independent of ω, but u(t , τ , ω, x) and ut (t , τ , ω, x) depend on ω for t > τ . Eq. (1.1) can model a random perturbation of strongly damped wave equation. There have been a lot of profound results on the dynamics of a variety of systems related to Eq. (1.1). For example, the asymptotical behavior of solutions for deterministic strongly damped wave equation has been studied by many authors (see [2–15], etc.). For autonomous stochastic strongly damped wave equation, Wang and Zhou [16] have studied the asymptotical behavior of solutions (where g is independent of t). Eq. (1.1) is a non-autonomous stochastic system where the external term g is time-dependent. For such a system, Wang established an efficacious theory about the existence of random attractors for corresponding non-autonomous random ✩ This work is supported by the National Natural Science Foundation of China (under Grant No. 11401244). Corresponding author. E-mail address: [email protected] (Z. Wang).

*

https://doi.org/10.1016/j.camwa.2018.02.002 0898-1221/© 2018 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Z. Wang, L. Zhang, Finite fractal dimension of random attractor for stochastic non-autonomous strongly damped wave equation, Computers and Mathematics with Applications (2018), https://doi.org/10.1016/j.camwa.2018.02.002.

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dynamical systems (called cocycle in [17]), see [17,18]. For the non-autonomous stochastic strongly damped wave equation on the unbounded domain, Wang and Zhou proved the existence of a random attractor, see [19]. Jones and Wang studied the asymptotic behavior of non-autonomous stochastic nonlinear wave equations with dispersive and dissipative terms defined on an unbounded domain, see [20]. The fractal dimension estimate of random attractors is an important problem (see [21,22]). The attractor has finite fractal dimension which means that the attractor can be mapped onto a compact subset of a finite dimensional Euclidean space. Recently, Zhou and Zhao in [23,24] gave some sufficient conditions to bound the fractal dimension of a random invariant set for a cocycle and applied these conditions to get the finiteness of fractal dimension of random attractor for non-autonomous stochastic damped wave equation with multiplicative white noise and additive white noise. Zhou and Tian et al. in [25] also gave similar sufficient conditions to bound the fractal dimension of a random invariant set for a cocycle and obtained the finiteness of fractal dimension of random attractor for stochastic reaction–diffusion equations with multiplicative white noise and additive white noise. So far as we know, there were no results about the boundedness of fractal dimensions of random attractor of nonautonomous stochastic strongly damped wave equations. In this paper, motivated by the idea of [17,18,23–26], by carefully splitting the positivity of the linear operator in the corresponding random evolution equation of the first order in time, and by carefully decomposing the solutions of the system, we first prove the cocycle associated with non-autonomous stochastic strongly damped wave equations (1.1) has a random attractor in H01 (U) × L2 (U) which is bounded in [H 2 (U) ∩ H01 (U)]× H01 (U) through a recurrence method, then we apply a criteria to get an upper bound of fractal dimension of the random attractor of considered system. This paper is organized as follows. In Section 2, we present some mathematical setting for our system. We first transfer the stochastic differential equation (1.1) into an equivalent random differential equation (2.5), then we show that the solutions mapping for this random equation (2.5) generate a continuous cocycle. In Section 3, we estimate the solutions of Eq. (2.5). We first consider the concrete bounds of the solutions, then we decompose the solutions of Eq. (2.5) into two parts. In Section 4, we obtain an upper bound of fractal dimension of random attractor for our system (1.1). And in Section 5, we give some remarks. In this paper, the letters ci (i ∈ N) below are generic positive constants which do not depend on ω, τ and t. 2. Mathematical setting Let A = −∆, then A is a self-adjoint positive linear sectorial operator with eigenvalues {λi }i∈N : 0 < λ1 ≤ λ2 ≤ · · · ≤ λm ≤ · · · , λm → +∞ as m → +∞. Define the rth powers Ar of A for r ∈ R. Let D(Ar ) be a Hilbert space with inner product (u, v )2r = (Ar u, Ar v ), and 1 r E = D(Ar + 2 ) × D(Ar ) for r ∈ R. Denote the inner and norm of L2 (U) as (·, ·) and ∥ · ∥, respectively. Write E = H01 (U) × L2 (U), and introduce a new weight inner product and norm in the Hilbert space E, 1

1

1

(ϕ1 , ϕ2 )E = µ(A 2 u1 , A 2 u2 ) + (v1 , v2 ) = µ(u1 , u2 )1 + (v1 , v2 ),

∥ϕ∥E = (ϕ, ϕ )E2 ,

(2.1)

for any ϕ1 = (u1 , v1 ) , ϕ2 = (u2 , v2 ) , ϕ ∈ E, where µ is chosen as ⊤

µ=

4 + (αλ1 + 1)α +

⊤

1

λ1

4 + 2(αλ1 + 1)α +

∈

1

λ1

(1 2

) ,1 .

(2.2)

Clearly, the norm ∥ · ∥E in (2.1) is equivalent to the usual norm ∥ · ∥H 1 ×L2 of E. 0 In the following, we convert the problem∫(1.1) into a random system without noise terms. Identify ω(t) with W (t), 0 i.e., ω(t) = W (t), t ∈ R, and let z(θt ω) := − −∞ es (θt ω)(s)ds (t ∈ R) be an Ornstein–Uhlenbeck stationary process which solves the Itô equation dz + zdt = dW (t). It is known from [1,27] that for almost every ω ∈ Ω , t ↦ → z(θt ω) is continuous in t and lim e−γ t |z(θ−t ω)| = 0,

∀γ > 0;

t →+∞

E[|z(θt ω)|r ] =

Γ ( 1+2 r ) , √ π

∀r > 0 , t ∈ R ,

(2.3)

where Γ is Gamma function. In the following, we identify ‘‘a.e. ω ∈ Ω ’’ and ‘‘ω ∈ Ω ’’. Let

v (t , τ , ω) = ut (t , τ , ω) + εu(t , τ , ω) − z(θt ω)h(x),

t ≥ τ,

τ ∈ R,

(2.4)

where ε is chosen as

ε=

αλ1 + 1 4 + 2(αλ1 + 1)α +

1

.

λ1

Please cite this article in press as: Z. Wang, L. Zhang, Finite fractal dimension of random attractor for stochastic non-autonomous strongly damped wave equation, Computers and Mathematics with Applications (2018), https://doi.org/10.1016/j.camwa.2018.02.002.

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Then (1.1) is equivalent to the following random systems in Hilbert space E, for t > τ , x ∈ U , τ ∈ R:

⎧ du ⎪ ⎪ = v − ε u + z(θt ω)h(x), ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎨ dv = α ∆v + (1 − αε)∆u + (ε − 1)v − (ε2 − ε)u + α z(θt ω)∆h(x) dt

⎪ − f (u, x) + g(x, t) + ε z(θt ω)h(x), ⎪ ⎪ ⎪ ⎪ ⎪ u(x , t) |x∈∂ U = 0, ⎪ ⎪ ⎩ u(τ , τ , x) = uτ (x), v (τ , τ , x) = vτ (x) = u1τ (x) + ε uτ (x) − z(θτ ω)h(x),

(2.5)

which has the following vector form

{

ϕ˙ + Λϕ = G(ϕ, θt ω, t), t ≥ τ , ϕ (τ , ω) = ϕτ (ω) = (uτ , u1,τ + ε uτ − z(θτ ω)h(x))⊤ ,

τ ∈ R,

(2.6)

where

ϕ=

( ) u

v

,

Λ=

G(ϕ, θt ω, t) =

(

) εI −I , A(1 − αε ) − ε (1 − ε )I α A − (ε − 1)I

(

z(θt ω)h(x)

)

. −f (u, x) + g(x, t) + ε z(θt ω)h(x) + α z(θt ω)∆h(x) ∫u Denote by F (u, x) = 0 f (r , x)dr, for x ∈ Rn and u ∈ R. Throughout this article, we make the following assumptions on functions F (u, x) and f (u, x), for every x ∈ Rn and u ∈ R, f (0, x) = 0,

|fu′ (u, x)| ≤ c1 (1 + |u|p−1 ),

(2.7)

c2 |u|p+1 − φ1 (x) ≤ F (u, x) ≤ c3 f (u, x)u + φ2 (x),

(2.8)

where 1 ≤ p < ∞, when n = 1, 2; 1 ≤ p < 3, when n = 3; φ1 (x), φ2 (x) ∈ L (U); c1 , c2 and c3 , are positive constants. From [19] and Lemma 3.1, we can have the following theorem. 2

Theorem 2.1. If (2.7) and (2.8) hold, then for any τ ∈ R, ω ∈ Ω and ϕτ = (uτ , vτ )⊤ ∈ E, problem (2.6) admits a uniquely and globally (weak) solution ϕ (·, τ , ω, ϕτ ) ∈ C ([τ , +∞); E) with ϕ (τ , τ , ω, ϕτ ) = ϕτ . Moreover, ϕ (t , τ , ω, ϕτ ) can define a continuous cocycle over R and (Ω , F , P, (θt )t ∈R ) with state space E: for t ∈ R+ , τ ∈ R, and ω ∈ Ω ,

Φ (t , τ , ω, ϕτ (ω)) = Φ (t , τ , ω)ϕτ (ω) = ϕ (t + τ , τ , θ−τ ω, ϕτ (θ−τ ω)), where Φ (0, τ , ω)ϕτ (ω) = ϕτ (θ−τ ω), Φ (t , τ − t , θ−t ω)ϕτ −t (θ−t ω) = ϕ (τ , τ − t , θ−τ ω, ϕτ −t (θ−τ ω)). 3. Estimates of solutions 3.1. Bound of solutions In this subsection, we estimate the bound of solutions of (2.6). In the following, let D(E) be the collection of all tempered families of nonempty subsets of E with respect to (θt )t ∈R . Lemma 3.1. If (2.7) and (2.8) hold, then for any τ ∈ R, ω ∈ Ω , there exists a tempered variable M0 (ω) (independent of τ ) such that, for any set B ∈ D(E) and ϕτ −t (θ−τ ω) ∈ B(τ − t , θ−t ω), there exists T = T (τ , ω, B) ≥ 0 such that, for all t ≥ T , the solution ϕ of (2.6) satisfies

∥ϕ (τ , τ − t , θ−τ ω, ϕτ −t (θ−τ ω))∥E ≤ M0 (ω).

(3.1)

Proof. For any τ ∈ R, ω ∈ Ω , t ≥ 0, let ϕ (r) = ϕ (r , τ − t , θ−τ ω, ϕτ −t (θ−τ ω)) = (u(r), v (r))⊤ ∈ E (r ≥ τ − t) be a solution of (2.6) with ϕ (τ − t) = ϕτ −t (θ−τ ω) = (uτ −t , u1,τ −t + ε uτ −t − z(θ−t ω)h(x))⊤ ∈ E . Taking the inner product (·, ·)E of (2.6) with ϕ (r), we find that for r ≥ τ − t , 1 d ∥ϕ (r)∥2E + (Λϕ, ϕ )E 2 (dr ) ( ) ( ) = µ z(θr −τ ω)h(x), u 1 + f (u), v + g(x, r) + ε z(θr −τ ω)h(x), v

( ) + α z(θr −τ ω)∆h(x), v .

(3.2)

Please cite this article in press as: Z. Wang, L. Zhang, Finite fractal dimension of random attractor for stochastic non-autonomous strongly damped wave equation, Computers and Mathematics with Applications (2018), https://doi.org/10.1016/j.camwa.2018.02.002.

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By computation, we have

( ) σµ µ µ z(θr −τ ω)h(x), u 1 ≤ ∥z(θr −τ ω)h(x)∥21 + ∥u∥21 , σ 4 ( ) α α z(θr −τ ω)∆h(x), v ≤ α∥z(θr −τ ω)h(x)∥21 + ∥v∥21 ,

(3.3) (3.4)

4

g(x, r) + ε z(θr −τ ω)h(x), v ≤ ∥g ∥2 + ε 2 ∥z(θr −τ ω)h(x)∥2 +

(

)

1 2

∥v∥2 ,

(3.5)

where ∥g ∥2 = supr ∈R ∥g(·, r)∥2 < ∞. By [13–15], we have (Λϕ, ϕ )E ≥ σ ∥ϕ∥2E +

α 2

1

∥v∥21 + ∥v∥2 ,

(3.6)

2

where

σ =

λ1 α + 1 , √ γ + γ (γ − 4)

γ = α (αλ1 + 1) +

1

λ1

+ 4.

By the first equation of (2.5), we have

v=

du

+ ε u − z(θt ω)h(x).

dt

(3.7)

By (3.7), we have du

f (u), v = f (u),

(

)

(

=

d

+ ε u − z(θr −τ ω)h(x)

∫ dt

)

F (u(r , x), x)dx + ε f (u), u − f (u), z(θr −τ ω)h(x) .

dt

(

)

(

)

(3.8)

U

By (2.7) and (2.8), we have

ε (f (u, x), u) = ε

∫

f (u, x)udx

U

≥

∫ ε( c3

F (u(r , x), x)dx −

(3.9)

∫

U

) φ2 (x)dx , U

|f (u, x)| ≤ c1 (|u| + |u|p ).

(3.10)

By (2.8) and (3.10), we get

( ) f (u, x), z(θr −τ ω)h(x) ∫ ( ) ≤ c1 |u| + |u|p · |z(θr −τ ω)h(x)|dx U ∫ ≤ c1 ∥u∥ · ∥z(θr −τ ω)h(x)∥ + c1 |u|p · |z(θr −τ ω)h(x)|dx U

c1

≤ √ ∥u∥1 · ∥z(θr −τ ω)h(x)∥ + c1 λ1

(∫

c1

c1

p+1

|u|

) p+p 1 dx

∥z(θr −τ ω)h(x)∥Lp+1

U

(∫

) p+p 1

≤ √ ∥u∥1 · ∥z(θr −τ ω)h(x)∥ + F (u(r , x), x) + φ1 (x) dx c2 λ1 U ∫ σµ ε ≤ ∥u∥21 + c5 ∥z(θr −τ ω)h(x)∥2 + F (u(r , x), x)dx 4 2c3 U ∫ ε + φ1 (x)dx + c5 ∥z(θr −τ ω)h(x)∥p1+1 . 2c3

(

)

∥z(θr −τ ω)h(x)∥Lp+1

(3.11)

U

By putting (3.6)–(3.9) and (3.11) into (3.2), we have that for r ≥ τ − t , 1 d 2 dr

[∥ϕ (r)∥2E + 2

∫

F (u(r , x), x)dx] + U

σ 2

∥ϕ (r)∥2E +

α 4

∥v∥21 +

≤ c6 ∥z(θr −τ ω)h(x)∥21 + ∥g ∥2 + c6 ∥z(θr −τ ω)h(x)∥2 ∫ ∫ ε ε + φ2 (x)dx + φ1 (x)dx + c5 ∥z(θr −τ ω)h(x)∥p1+1 . c3

U

2c3

ε 2c3

∫

F (u(r , x), x)dx

U

(3.12)

U

Please cite this article in press as: Z. Wang, L. Zhang, Finite fractal dimension of random attractor for stochastic non-autonomous strongly damped wave equation, Computers and Mathematics with Applications (2018), https://doi.org/10.1016/j.camwa.2018.02.002.

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{

Let ρ = min σ ,

ε

}

2c3

)

–

5

and

y(r) = ∥ϕ (r)∥2E + 2

∫

F (u(r , x), x)dx ≥ ∥ϕ (r)∥2E − 2

∫

φ1 (x)dx,

(3.13)

U

U

q(θr −τ ω) = 2c6 ∥z(θr −τ ω)h(x)∥21 + 2∥g ∥2 + 2c6 ∥z(θr −τ ω)h(x)∥2 ∫ ∫ 2ε ε + φ2 (x)dx + φ1 (x)dx + 2c5 ∥z(θr −τ ω)h(x)∥p1+1 c3 U c3 U

= c7 + c7 |z(θr −τ ω)|p+1 ,

(3.14)

then by (3.12), we have d dr

y(r) + ρ y(r) +

α 2

∥v∥21 ≤ q(θr −τ ω).

(3.15)

By Gronwall’s inequality to (3.15) on [τ − t , r ] (r ≥ τ − t), we have that for r ≥ τ − t , y(r , τ − t , θ−τ ω, ϕτ −t (θ−τ ω))

α

∫

r

∥v (r , τ − t , θ−τ ω, ϕτ −t (θ−τ ω))∥21 e−ρ (r −s) ds ∫ r ≤ y(τ − t , τ − t , θ−τ ω, ϕτ −t (θ−τ ω))e−ρ (r +t −τ ) + q(θs−τ ω)e−ρ (r −s) ds, +

2

τ −t

(3.16)

τ −t

where y(τ − t , τ − t , θ−τ ω, ϕτ −t (θ−τ ω))

= ∥ϕτ −t (θ−τ ω)∥ + 2 2 E

∫

F (u(τ − t , x), x)dx U

≤ ∥ϕτ −t (θ−τ ω)∥ + c1 c3 ∥uτ −t ∥ + 2 E

2

p+1 c1 c3 uτ −t Lp+1

∥

∥

( ) ≤ ∥ϕτ −t (θ−τ ω)∥2E + c8 ∥uτ −t ∥2 + ∥uτ −t ∥p1+1 +

∫

φ2 (x)dx,

+ U

∫

φ2 (x)dx, U

and r

∫

τ −t

q(θs−τ ω)e

−ρ (r −s)

∫ ds = c9 + c7

r

τ −t

|z(θs−τ ω)|p+1 e−ρ (r −s) ds.

Thus, by (3.13) and (3.16), we have for r ≥ τ − t

∥ϕ (r , τ − t , θ−τ ω, ϕτ −t (θ−τ ω))∥2E ∫ ≤ y(r) + 2 φ1 (x)dx U ( ) ( ) ∫ ≤ ∥ϕτ −t (θ−τ ω)∥2E + c8 ∥uτ −t ∥2 + ∥uτ −t ∥p1+1 + φ2 (x)dx e−ρ (r +t −τ ) U ∫ r ∫ + q(θs−τ ω)e−ρ (r −s) ds + 2 φ1 (x)dx. τ −t

U

Therefore,

∥ϕ (τ , τ − t , θ−τ ω, ϕτ −t (θ−τ ω))∥2E ( ) ≤ ∥ϕτ −t (θ−τ ω)∥2E + c8 ∥uτ −t ∥2 + c8 ∥uτ −t ∥p1+1 e−ρ t ∫ 0 + c10 + c7 |z(θs ω)|p+1 eρ s ds.

(3.17)

−∞

For any set B(τ , ω) ∈ B ∈ D(E),

ϕτ −t (θ−τ ω) = (uτ −t , u1,τ −t + ε uτ −t − z(θ−t ω)h(x))⊤ ∈ B(τ − t , θ−t ω) ∈ D(E), we have lim sup t →+∞

(

) ∥ϕτ −t (θ−τ ω)∥2E + c8 ∥uτ −t ∥2 + c8 ∥uτ −t ∥p1+1 e−ρ t = 0.

(3.18)

Please cite this article in press as: Z. Wang, L. Zhang, Finite fractal dimension of random attractor for stochastic non-autonomous strongly damped wave equation, Computers and Mathematics with Applications (2018), https://doi.org/10.1016/j.camwa.2018.02.002.

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Taking

∫

M02 (ω) = 2c10 + 2c7

0

|z(θs ω)|p+1 eρ s ds < ∞,

(3.19)

−∞

which is a tempered random variable, then by (3.17)–(3.19), we have that (3.1) holds. The proof is completed. Let B0 (ω) = {ϕ ∈ E : ∥ϕ∥E ≤ M0 (ω)}.

(3.20)

By (3.1) and (3.20), there exist T (τ , ω, B0 ) ≥ 0 such that

ϕ (r , τ − t , θ−τ ω, B0 (θ−t ω)) ∈ B0 (θr −τ ω), ∀t ≥ T (τ , ω, B0 ),

τ − t ≤ r ≤ τ.

(3.21)

By (2.3) and (3.19), for any τ ∈ R,

(

Γ ( 2+p ) θ ω)) = 2 c10 + c7 √2 ρ π

) ,

E(M02 ( τ

(3.22)

and for k > 1,

(

k E(M02k (θτ ω)) ≤ 22k c10 + c7k E

(

0

∫

)k ) |z(θs+τ ω)|p+1 eρ s ds

−∞ 2k

≤2

(

k c10

+

c7k

(∫

0

k

ρs

e 2(k−1) ds

)k−1

·E

−∞

(

k = 22k c10 + c7k

( 2(k − 1) )k−1 kρ

(

∫

0

k

e 2 ρ s |z(θs+τ ω)|(p+1)k ds

))

−∞ 1+(p+1)k ) ) 2 Γ( · < ∞. √2 kρ π

(3.23)

3.2. Decomposition of solutions In this subsection, for proving the existence of random attracting set for the cocycle Φ , we have to decompose the solutions of system (2.6) with different initial data into two parts. For any τ ∈ R and ω ∈ Ω , set B1 (τ , ω) =

⋃

ϕ (τ , τ − t , θ−τ ω, B0 (θ−t ω)) ⊆ B0 (ω).

t ≥T (τ ,ω,B0 )

Let ϕ (r) = ϕ (r , τ − t , θ−τ ω, ϕτ −t (θ−τ ω)) (r ≥ τ − t, t ≥ 0) be a solution of system (2.6) with ϕτ −t (θ−τ ω) ∈ B1 (τ − t , θ−t ω) ⊆ B0 (θ−t ω), then it follows from (3.21) that ϕ (r) ∈ B0 (θr −τ ω) for all r ≥ τ − t,

∥ϕ (r , τ − t , θ−τ ω, ϕτ −t (θ−τ ω))∥E ≤ M0 (θr −τ ω).

(3.24)

We decompose ϕ (r) into ϕ (r) = ˜ ϕ (r) + ˆ ϕ (r), where ˜ ϕ (r) = (˜ u, ˜ v ) and ˆ ϕ (r) = (ˆ u, ˆ v ) satisfy ⊤

{ ˜ ϕ˙ + Λ˜ ϕ = 0,

⊤

r > τ − t,

˜ ϕ (τ − t , τ − t , θ−τ ω, ϕτ −t (θ−τ ω)) = ϕτ −t (θ−τ ω),

(3.25)

and

{ ˆ ϕ˙ + Λˆ ϕ = G(ϕ, θr −τ ω, r),

r > τ − t,

ˆ ϕ (τ − t , τ − t , θ−τ ω, ϕτ −t (θ−τ ω)) = (0, 0)⊤ ,

(3.26)

respectively. First, let us estimate the component ˜ ϕ which decays exponentially. Lemma 3.2. If (2.7) and (2.8) hold, then for any τ ∈ R, ω ∈ Ω , t ≥ 0, r ≥ τ − t and ϕτ −t (θ−τ ω) ∈ B0 (θ−t ω),

∥˜ ϕ (r , τ − t , θ−τ ω, ϕτ −t (θ−τ ω))∥E ≤ M0 (θ−t ω)e−σ (t +r −τ ) ,

(3.27)

where M0 (ω) is as in (3.19). Proof. Taking the inner product (·, ·)E of (3.25) with ˜ ϕ (r), we find that for r ≥ τ − t , 1 d 2 dr

∥˜ ϕ (r)∥2E + (Λ˜ ϕ, ˜ ϕ )E = 0.

(3.28)

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Similar to (3.6), we have (Λ˜ ϕ, ˜ ϕ )E ≥ σ ∥˜ ϕ ∥2E +

α

2 By (3.28) and (3.29), we have

1

∥˜ v∥21 + ∥˜ v∥2 .

(3.29)

2

d ϕ ∥2E ≤ 0. ∥˜ ϕ (r)∥2E + 2σ ∥˜ dr By Gronwall’s inequality to (3.30) on [τ − t , r ] (r ≥ τ − t), we have that for r ≥ τ − t ,

∥˜ ϕ (r , τ − t , θ−τ ω, ϕτ −t (θ−τ ω))∥2E ≤ ∥ϕτ −t (θ−τ ω)∥2E e−2σ (r +t −τ ) .

(3.30)

(3.31)

Note that ϕτ −t (θ−τ ω) ∈ B0 (θ−t ω). Then by (3.24) and (3.31), we get that (3.27) holds. Thus we complete the proof. For the component ˆ ϕ which is ultimately pullback bounded in a ‘‘higher regular’’ space, we have the following estimate. Lemma 3.3. If (2.7) and (2.8) hold, then for any τ ∈ R, ω ∈ Ω and t ≥ 0, there exist a positive constant

( { }] 1 3−p ν ∈ 0, min , 4

(3.32)

4

and positive-value random variable M2ν (ω) > 0 such that the solution ˆ ϕ (r) = (ˆ u(r), ˆ v (r))⊤ of (3.26) satisfies that for t ≥ 0 and ϕτ −t (θ−τ ω) ∈ B0 (θ−t ω),

∥Aνˆ ϕ (τ , τ − t , θ−τ ω, ϕτ −t (θ−τ ω))∥2E ≤ M22ν (ω).

(3.33)

Proof. By (3.26), we have

ˆ v=

dˆ u dt

+ εˆ u − z(θt ω)h(x).

(3.34)

Taking the inner product of (3.26) in E with A2νˆ ϕ = (A2νˆ u, A2νˆ v )⊤ , and using (3.34) we have for r ≥ τ − t, 1 d 2 dr

(

∥Aνˆ ϕ ∥2E + 2

∫

)

f (u, x) · A2νˆ udx

∫U ∫ ( ) 2ν 2ν + Λˆ ϕ, A ˆ ϕ E + ε f (u, x) · A ˆ udx − fu′ (u, x)ut · A2νˆ udx U U ( ) ( ) = µ z(θr −τ ω)h(x), A2νˆ u 1 + α z(θr −τ ω)∆h(x), A2νˆ v ( ) ( ) + f (u, x), z(θr −τ ω)A2ν h(x) + g(x, r) + εz(θr −τ ω)h(x), A2νˆ v .

(3.35)

Similar to (3.6), we get

( ) α 1 1 Λˆ ϕ , A2νˆ ϕ E ≥ σ ∥Aνˆ ϕ ∥2E + ∥Aν+ 2 ˆ v∥2 + ∥Aνˆ v∥2 .

2 By computation, we have that for r ≥ τ − t ,

2

( ) α z(θr −τ ω)∆h(x), A2νˆ v α α 1 1 ≤ ∥z(θr −τ ω)Aν+ 2 h(x)∥2 + ∥Aν+ 2 ˆ v∥2 , 2( 2 ) µ z(θr −τ ω)h(x), A2νˆ u 1 ≤

µ2 σ 1 1 ∥z(θr −τ ω)Aν+ 2 h(x)∥2 + ∥Aν+ 2 ˆ u∥ 2 , σ 4 ( ) g(x, r) + ε z(θr −τ ω)h(x), A2νˆ v

1 [∥Aν g ∥2 + ε 2 ∥z(θr −τ ω)Aν h∥2 ] + ∥Aνˆ v∥2 . 2 2 By (3.10) and (3.24), we have that for r ≥ τ − t ,

≤

1

f (u, x), z(θr −τ ω)A2ν h(x)

(

)

≤ ∥f (u, x)∥ · ∥z(θr −τ ω)A2ν h(x)∥ ≤ ∥c1 (|u| + |u|p )∥ · ∥z(θr −τ ω)A2ν h(x)∥ ) c1 ( ≤ ∥|u| + |u|p ∥2 + ∥z(θr −τ ω)A2ν h(x)∥2 2 ( ) ≤ c11 M02p (θr −τ ω) + z 2 (θr −τ ω) , where c11 depends on ∥h∥1 . Please cite this article in press as: Z. Wang, L. Zhang, Finite fractal dimension of random attractor for stochastic non-autonomous strongly damped wave equation, Computers and Mathematics with Applications (2018), https://doi.org/10.1016/j.camwa.2018.02.002.

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ν

By (3.32), we have H0ν (U) = D(A 2 ) = H ν (U); 1

H a1 (U) ⊂ H a2 (U), if a1 > a2 ; and H ν (U) ⊂ Lq (U), where

q

=

1 2

−

ν 3

.

(3.36)

Then by Hölder’s inequality, (2.7), (3.7), (3.24) and (3.36), we have that for r ≥ τ − t ,

∫ U

fu′ (u, x)ut · A2νˆ udx

∫

(

≤ c1 U

(∫

u|dx 1 + |u|p−1 · |ut | · |A2νˆ

)

≤ c1

6

p−1 2−4ν

(

)

1 + | u|

) 2−64ν (∫

) 21 (∫

2

6 1+4ν

2ν

) 1+64ν

dx u| |ut | dx |A ˆ dx U U ) ( ) 1 ≤ c12 1 + ∥u∥p1−1 · ∥v∥ + ∥ε u∥ + ∥z(θr −τ ω)h(x)∥ · ∥Aν+ 2 ˆ u∥ σ 1 u∥ 2 , ≤ c13 M02p (θr −τ ω) + ∥Aν+ 2 ˆ

(U

4 where c13 depends on ∥h∥. By putting above inequalities into (3.35), we have 1 d 2 dr

(

∥Aνˆ ϕ ∥2E + 2

)

∫ U

f (u, x) · A2νˆ udx

+

σ 2

∥Aνˆ ϕ ∥2E + ε

∫ U

f (u, x) · A2νˆ udx

) ( ≤ c14 z 2 (θr −τ ω) + M02p (θr −τ ω) .

(3.37)

where c14 depends on ∥h∥1 and ∥g ∥1 = supr ∈R ∥g(·, r)∥1 < ∞. Let

ρ1 = min{σ , ε},

(3.38)

y2 (r) = ∥Aνˆ ϕ ∥E + 2 2

∫ U

udx, f (u, x) · A2νˆ

(3.39)

and 2p

q1 (θr −τ ω) = 2c14 z 2 (θr −τ ω) + M0 (θr −τ ω) ,

(

)

it then follows from (3.37) that d dr Note that

y2 (r) + ρ1 y2 (r) ≤ q1 (θr −τ ω),

∀ r ≥ τ − t.

(3.40)

y2 (τ − t , τ − t , θ−τ ω, ϕτ −t (θ−τ ω)) = 0, then by applying the Gronwall’s inequality to (3.40) on [τ − t , r ] (r ≥ τ − t), we have y2 (r , τ − t , θ−τ ω, ϕτ −t (θ−τ ω)) ≤

r

∫

τ −t

q1 (θξ −τ ω)e−ρ1 (r −ξ ) dξ ,

∀ r ≥ τ − t.

(3.41)

By (3.10), we have

≤ ≤ ≤ ≤

⏐∫ ⏐ ⏐ ⏐ udx⏐ ⏐ f (u, x) · A2νˆ ∫U c1 (|u| + |u|p ) · |A2νˆ u|dx U ∫ ( ( ) 5−64ν (∫ ) 1+64ν 6 ) 6 c1 |u| + |u|p 5−4ν dx |A2νˆ u| 1+4ν dx U U ( ) 1 p c15 ∥u∥1 + ∥u∥1 · ∥Aν+ 2 ˆ u∥ µ 1 2p c16 M0 (θr −τ ω) + ∥Aν+ 2 ˆ u∥ 2 .

4 It then follows from (3.39) and (3.42) that, y2 (r) ≥

1 2

∥Aνˆ ϕ ∥2E − 2c16 M02p (θr −τ ω).

(3.42)

(3.43)

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Therefore, by (3.41) and (3.43), we have that for t ≥ 0,

∥Aνˆ ϕ (τ , τ − t , θ−τ ω, ϕτ −t (θ−τ ω))∥2E

≤ 2y2 (τ , τ − t , θ−τ ω, ϕτ −t (θ−τ ω)) + 4c16 M02p (ω) ∫ τ ( 2 ) 2p 2p ≤ 2c14 z (θξ −τ ω) + M0 (θξ −τ ω) e−ρ1 (τ −ξ ) dξ + 4c16 M0 (ω).

(3.44)

τ −t

Taking M22ν (ω) = 2c14

0

∫

(

z 2 (θξ ω) + M0 (θξ ω) eρ1 ξ dξ + 4c16 M0 (ω), 2p

2p

)

(3.45)

−∞

then (3.44) and (3.45) imply (3.33). The proof is completed. Lemma 3.4. For any τ ∈ R, ω ∈ Ω and t ≥ 0, assume that Bν (τ , ω) ⊆ B1 (τ , ω) ⊆ B0 (ω) and Bν (τ , ω) ∈ D(E ν ), where ν is as in (3.32). If (2.7) and (2.8) hold, then there exist a random variable tν (ω) > 0 and a tempered random variable Mν (ω) > 0 (independent of τ and t) such that for any t ≥ tν (ω), ϕτ −t (θ−τ ω) ∈ Bν (τ − t , θ−t ω) ⊆ B0 (θ−t ω) ∩ D(E ν ), the solution ϕ of (2.6) satisfies

∥ϕ (τ , τ − t , θ−τ ω, ϕτ −t (θ−τ ω))∥2E ν = ∥Aν ϕ (τ , τ − t , θ−τ ω, ϕτ −t (θ−τ ω))∥2E ≤ Mν2 (ω).

(3.46)

Proof. Taking the inner product of (2.6) in E with A2ν ϕ = (A2ν u, A2ν v )⊤ , we have that for r ≥ τ − t, 1 d

(

2 dr

ν

∥A ϕ∥ + 2 2 E

∫

2ν

)

f (u, x)A udx

∫U

∫ ( ) + Λϕ, A2ν ϕ E + ε f (u, x) · A2ν udx − fu′ (u, x)ut · A2ν udx U U ( ) ( ) = µ z(θr −τ ω)h(x), A2ν u 1 + α z(θr −τ ω)∆h(x), A2ν v ( ) ( ) + f (u, x), z(θr −τ ω)A2ν h(x) + g(x, r) + εz(θr −τ ω)h(x), A2ν v .

(3.47)

Similar to (3.42), we have that for r ≥ τ − t,

⏐∫ ⏐ µ 1 ⏐ ⏐ 2p ⏐ f (u, x) · A2ν udx⏐ ≤ c16 M0 (θr −τ ω) + ∥Aν+ 2 u∥2 .

(3.48)

4

U

By (3.47), (3.48) and similar to (3.40), we have that for r ≥ τ − t, d dr

y3 (r) + ρ1 y3 (r) ≤ q2 (θr −τ ω),

(3.49)

where ρ1 is as in (3.38), y3 (r) = ∥Aν ϕ (r)∥E + 2 2

∫

f (u, x) · A2ν u(r)dx U

≥

1 2

∥Aν ϕ (r)∥2E − 2c16 M02p (θr −τ ω),

(3.50)

and

(

)

2p

q2 (θr −τ ω) = c17 z 2 (θr −τ ω) + M0 (θr −τ ω) .

(3.51)

By applying the Gronwall’s inequality to (3.49) on [τ − t , r ] (r ≥ τ − t), we have y3 (r) ≤ y3 (τ − t)e−ρ1 (r +t −τ ) +

∫

r

τ −t

q2 (θξ −τ ω)e−ρ1 (r −ξ ) dξ .

(3.52)

By (3.50)–(3.52), we have

∥Aν ϕ (τ , τ − t , θ−τ ω, ϕτ −t (θ−τ ω))∥2E ≤ 2y3 (τ , τ − t , θ−τ ω, ϕτ −t (θ−τ ω)) + 4c16 M02p (ω) ∫ τ 2p −ρ1 t ≤ 2y3 (τ − t)e +2 q2 (θξ −τ ω)e−ρ1 (τ −ξ ) dξ + 4c16 M0 (ω) τ −t

≤ 2y3 (τ − t)e−ρ1 t + 2c17

∫

0

(

)

z 2 (θξ ω) + M0 (θξ ω) eρ1 ξ dξ + 4c16 M0 (ω). 2p

2p

(3.53)

−t

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By (3.48), (3.50) and ϕτ −t (θ−τ ω) ∈ B0 (θ−t ω) ∩ D(E ν ), we have

(

y3 (τ − t)e−ρ1 t ≤

3 2

) ∥Aν ϕτ −t ∥2E + 2c16 M02p (θ−t ω) e−ρ1 t → 0 as t → +∞.

Taking Mν2 (ω) = 4c17

∫

0

(

)

z 2 (θξ ω) + M0 (θξ ω) eρ1 ξ dξ + 8c16 M0 (ω), 4p

2p

(3.54)

−∞

then the proof is completed. Basing on Lemma 3.2–3.4 and recursion, we have the following Lemma. Lemma 3.5. For any τ ∈ R, ω ∈ Ω , t ≥ 0, let Bκ (τ , ω) ⊆ B1 (τ , ω), Bκ (τ , ω) ∈ D(E κ ) and ϕτ −t (θ−τ ω) ∈ Bκ (τ − t , θ−t ω) . If ˜ κ (ω) > 0, Mκ (ω) > 0 (independent of τ and (2.7) and (2.8) hold, then there exist Tκ (ω) ≥ 0 and tempered random variables M t) such that for t ≥ Tκ (ω), the solution ϕ of (2.6) satisfies (i) for ν ≤ κ ≤ 1,

∥ϕ (τ , τ − t , θ−τ ω, ϕτ −t (θ−τ ω))∥2E κ 1

= ∥Aκ+ 2 u(τ , τ − t , θ−τ ω, ϕτ −t (θ−τ ω))∥2 + ∥Aκ v (τ , τ − t , θ−τ ω, ϕτ −t (θ−τ ω))∥2 ˜ κ2 (ω); ≤M

(3.55)

(ii) for ν ≤ κ ≤ 1 − ν ,

∥ˆ ϕ (τ , τ − t , θ−τ ω, ϕτ −t (θ−τ ω))∥2E κ+ν 1

u(τ , τ − t , θ−τ ω, ϕτ −t (θ−τ ω))∥2 + ∥Aκ+νˆ v (τ , τ − t , θ−τ ω, ϕτ −t (θ−τ ω))∥2 = ∥Aκ+ν+ 2 ˆ

≤ Mκ2 (ω),

(3.56)

where ν is as in (3.32). 4. Random attractor In this section, we will consider the existence and boundedness of fractal dimension of random attractor for the cocycle

Φ in E. 4.1. Existence of random attractor In this subsection, we prove the existence of a random attractor for the cocycle Φ basing on Lemma 4.2 and Lemma 3.5. We first present the concept of random attractor and an existence criteria of random attractor for the cocycle Φ . Definition 4.1. (1) A family K = {K (τ , ω) : τ ∈ R, ω ∈ Ω } ∈ D(E) of nonempty subsets of E is called a measurable D(E)-pullback attracting set for the cocycle Φ if (i) K is measurable with respect to F in Ω ; (ii) for all τ ∈ R, ω ∈ Ω and for every B ∈ D(E), limt →+∞ dE (Φ (t , τ − t , θ−t ω, B(τ − t , θ−t ω)), K (τ , ω)) = 0, where ‘‘dE (·, ·)’’ denotes the Hausdorff semi-distance between two subsets of E. (2) A family A = {A(τ , ω) : τ ∈ R, ω ∈ Ω } ∈ D(E) is called a D(E)-pullback random attractor for the cocycle Φ if (i) A(τ , ω) is measurable in ω and compact in E for any τ ∈ R, ω ∈ Ω ; (ii) A is invariant, i.e., for any τ ∈ R, ω ∈ Ω , t ≥ 0, Φ (t , τ , ω, A(τ , ω)) = A(t + τ , θt ω); (iii) A is an attracting set in D(E). Lemma 4.2. If the cocycle Φ has a compact measurable (w.r.t. F ) D(E) -pullback attracting set K in D(E), then the cocycle Φ has a unique D(E)-pullback random attractor A in D(E) given by: for each τ ∈ R and ω ∈ Ω , A(τ , ω) =

⋂⋃

Φ (t , τ − t , θ−t ω, K (τ − t , θ−t ω)).

r ≥0 t ≥r

Lemma 4.3. If (2.7) and (2.8) hold, then for any τ ∈ R, ω ∈ Ω , there exist a Tν (ω) ≥ 0, a random bounded ball ˆ B1 (ω) of E 1 , a positive number ˆ ρ and a tempered random variable ˆ Q (ω) > 0 such that, for t ≥ Tν (ω) and ϕτ −t (θ−τ ω) ∈ B1 (τ − t , θ−t ω), the solution ϕ of (2.6) satisfies ρt dE (ϕ (τ , τ − t , θ−τ ω, B1 (τ − t , θ−t ω)), ˆ B1 (ω)) ≤ ˆ Q (θ−t ω)e−ˆ .

(4.1)

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By Lemma 7.6 in [28], Lemmas 3.2 and 3.5, we can prove Lemma 4.3. Since the proof of Lemma 4.3 is similar to that of Lemma 3.8 of [23], we omit it here. Combining Lemmas 4.2 and 4.3, we can prove the existence of a random attractor for the cocycle Φ . Theorem 4.4. If (2.7) and (2.8) hold, then the cocycle Φ associated with (2.6) possesses a D(E)-pullback random attractor A ∈ D(E) such that for any τ ∈ R, ω ∈ Ω , A(τ , ω) ⊆ ˆ B1 (ω) ∩ B0 (ω), where B0 (ω) and ˆ B1 (ω) are as in (3.20) and Lemma 4.3, respectively. Proof. For any τ ∈ R and ω ∈ Ω , by Lemma 4.3 and the compactness of embedding E 1 ↪→ E, ˆ B1 (ω) is a compact measurable D(E)-pullback attracting ball in E. By Lemma 4.2, the cocycle Φ possesses a D(E)-pullback random attractor A ∈ D(E) such that for any τ ∈ R, ω ∈ Ω , A(τ , ω) ⊆ ˆ B1 (ω) ∩ B0 (ω). The proof is completed. 4.2. Fractal dimension of random attractor In this subsection, we estimate the upper bound of fractal dimension of A(τ , ω) for the cocycle Φ in E basing on the following result, which can be found in Theorem 2.2 (II) of [23]. Lemma 4.5. Assume that the random attractor A = {A(τ , ω)}τ ∈R,ω∈Ω of the cocycle Φ satisfying the following conditions: for any τ ∈ R and ω ∈ Ω , (H1) there exists a tempered random variable Rω (independent of τ ) such that supτ ∈R supu∈A(τ ,ω) ∥u∥X ≤ Rω < ∞, and Rθt ω is continuous in t for all t ∈ R; ˜ δ , t0 , random variable C0 (ω) ≥ 0 and m -dimensional projector Pm : H → Pm H (H2) there exist positive numbers λ, (dim(Pm H) = m) such that for any τ ∈ R, ω ∈ Ω and any u, v ∈ A(τ , ω), ∫ t0

∥Pm Φ (t0 , τ , ω)u − Pm Φ (t0 , τ , ω)v∥E ≤ e

0

C0 (θs ω)ds

∥u − v∥E

(4.2)

and ∫ t0

∥(I − Pm )Φ (t0 , τ , ω)u − (I − Pm )Φ (t0 , τ , ω)v∥E ≤ (e−λt0 + δ e ˜

0

C0 (θs ω)ds

)∥u − v∥E ,

(4.3)

˜ δ, t0 , m are independent of τ and ω; where λ, ˜ t0 , δ satisfy: (H3) C0 (ω), λ,

⎧ ⎪ ⎪ ⎪ E[C02 (ω)] < ∞, ⎪ ⎪ ⎪ ⎨ ln 4 , t0 ≥ 1 + ⎪ λ˜ ⎪ ⎪ ⎪ − 4 t 2 E[C 2 (ω)] ⎪ ⎪ ⎩0 < δ ≤ 1 e ln 32 0 0 ;

(4.4)

8

then for any τ ∈ R, ω ∈ Ω , the fractal dimension of A(τ , ω): dimf A(τ , ω) ≤

2m ln

(√

m

δ

ln

4 3

) +1

.

(4.5)

Now let us check the random attractor A = {A(τ , ω)}τ ∈R,ω∈Ω satisfies conditions (H1)–(H3) in Lemma 4.5. For any τ ∈ R,

ω ∈ Ω , t ≥ 0 and

ϕj,τ −t (θ−τ ω) = (uj,τ −t (θ−τ ω), vj,τ −t (θ−τ ω))⊤ ∈ A(τ − t , θ−t ω),

j = 1, 2,

then by the invariance of A(τ , ω) and Theorem 4.4, the solutions

ϕj (r) = ϕj (r , τ − t , θ−τ ω, ϕj,τ −t (θ−τ ω)) = (uj (r), vj (r))⊤ of system (2.6) satisfy that for r ≥ τ − t ,

ϕ1 (r), ϕ2 (r) ∈ A(r , θr −τ ω) ⊆ ˆ B1 (θr −τ ω) ⊂ E 1 and

∥ϕ1 (r)∥E 1 ≤ b1 (θr −τ ω),

∥ϕ2 (r)∥E 1 ≤ b1 (θr −τ ω),

(4.6)

which implies that for r ≥ τ − t ,

|uj (r)| ≤ c19 (Ω )∥Auj (r)∥ ≤ c20 b1 (θr −τ ω),

j = 1, 2,

(4.7)

Please cite this article in press as: Z. Wang, L. Zhang, Finite fractal dimension of random attractor for stochastic non-autonomous strongly damped wave equation, Computers and Mathematics with Applications (2018), https://doi.org/10.1016/j.camwa.2018.02.002.

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where b1 (ω) is the radius of the bounded ball ˆ B1 (ω) ⊂ E 1 . From (3.54) and Lemma 3.5, the radius b1 (ω) can be expressed in the following form

(

4p([ ν1 ]+2)

)

2p

b21 (ω) = c21 + c22 M02 (θ−t ω) + M0 (θ−t ω) e−ρ1 t + c23 M0

(∫

0

+ c24

(

)

z 2 (θξ ω) + M0 (θξ ω) eρ1 ξ dξ 4p

)2([ ν1 ]+2)

(ω )

.

(4.8)

−∞

Let

ψ (r) = ϕ1 (r) − ϕ2 (r) = (ξ (r), η(r))⊤ ,

(4.9)

then, ψ (r) satisfies

{ ψ˙ + Λψ = G(ϕ1 , θr ω, r) − G(ϕ2 , θr ω, r), r ≥ τ, ψτ (ω) = (ξτ , ητ )⊤ = (ξτ , u1,1,τ − u2,1,τ + εξτ )⊤ = (u1τ − u2τ , v1τ − v2τ )⊤ .

(4.10)

Lemma 4.6. If (2.7) and (2.8) hold, then there exists a tempered random variable C1 (ω) > 0 such that for any τ ∈ R, t ≥ 0, ω ∈ Ω , it holds

∥ϕ1 (τ , τ − t , θ−τ ω, ϕ1,τ −t (θ−τ ω)) − ϕ2 (τ , τ − t , θ−τ ω, ϕ2,τ −t (θ−τ ω))∥E ∫t

≤e

0 C1 (θs ω)ds

∥ϕ1,τ −t − ϕ2,τ −t ∥E .

(4.11)

Proof. Taking the inner product (·, ·)E of (4.10) with ψ (r) , we find that for r ≥ τ , d dr

1

∥ψ (r)∥2E + 2σ ∥ψ (r)∥2E ≤

αλ1 + 1

∥f (u2 (r), x) − f (u1 (r), x)∥2 .

(4.12)

By (3.10) and (4.7), 1

αλ1∫ + 1 (

∥f (u2 (r), x) − f (u1 (r), x)∥2

1 + |u1 (r)|2(p−1) + |u2 (r)|2(p−1) |ξ (r)|2 dx

≤ c25

)

(U

) 2(p−1) 2(p−1) ≤ c25 1 + 2c20 b1 (θr −τ ω) ∥ξ (r)∥2 ( ) ≤ c26 µ 1 + b12(p−1) (θr −τ ω) ∥ξ (r)∥21 .

(4.13)

Thus putting (4.13) into (4.12), we have that for r ≥ τ − t, d

) ( ( ) −1) (θr −τ ω) − 2σ ∥ψ (r)∥2E . ∥ψ (r)∥2E ≤ c26 1 + b2(p 1

dr Then by applying Gronwall inequality to (4.14), it follows that for r ≥ τ − t,

(4.14)

∥ϕ1 (r) − ϕ2 (r)∥2E ∫r

≤ ∥ϕ1,τ −t − ϕ2,τ −t ∥2E e ≤ ∥ϕ1,τ −t − ϕ2,τ −t ∥2E e

(

(

2(p−1)

τ −t c26 1+b1

∫ r −τ ( −t

(

2(p−1)

c26 1+b1

)

)

(θs−τ ω) −2σ ds

)

)

(θs ω) −2σ ds

.

(4.15)

By r → τ , it follows that (4.11) holds with ) c26 ( 2(p−1) C1 (θs ω) = 1 + b1 (θ s ω ) − σ . 2 By (3.23) and (4.8), we have that for s ∈ R and k ≥ 1

(4.16)

E[b2k 1 (θs ω )] = c27 < ∞,

(4.17)

and

where c27 and c28

c26

2(p−1)

(1 + E[b1 (θs ω)]) − σ = c28 < ∞, 2 are independent of s and ω. The proof is completed.

0 < E(C1 (θs ω)) =

Let {ej }j∈N ⊂ D(A) be the eigenvectors of operator A corresponding to the eigenvalues {λj }j∈N with Aej = λj ej for j ∈ N, then {ej }j∈N form an orthonormal base of L2 (U). Let L2n (U) = span{e1 , e2 , . . . , en },

[L2n (U)]⊥ = span{en+1 , en+2 , . . .},

n ∈ N,

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13

then L2n (U) × L2n (U) is a 2n-dimensional subspace of E . Let Pn : E → L2n (U) × L2n (U),

Qn = I − Pn : E → [L2n (U)]⊥ × [L2n (U)]⊥

be the orthonormal projectors. For ϕ = (u, v )⊤ ∈ E, let ϕnq = Qn ϕ = (unq , vnq )⊤ ∈ [L2n (U)]⊥ × [L2n (U)]⊥ , then

λn+1 ∥unq ∥2 ≤ ∥unq ∥21 .

(4.18)

Lemma 4.7. If (2.7) and (2.8) hold, then for any τ ∈ R , ω ∈ Ω , t ≥ 0, there exist a random variable C2 (ω) > 0 and a 2n-dimensional finite dimensional projector Pn : E → Pn E = L2n (U) × L2n (U) (independent of τ and ω) such that for any ϕj,τ −t (θ−τ ω) ∈ A(τ − t , θ−t ω) , j = 1, 2, it holds that

∥(I − Pn )Φ (t , τ − t , θ−τ ω)ϕ1,τ −t (θ−τ ω) − (I − Pn )Φ (t , τ − t , θ−τ ω)ϕ2,τ −t (θ−τ ω)∥E ) ( ∫0 c29 e −t C2 (θs ω)ds ∥ϕ1,τ −t (θ−t ω) − ϕ2,τ −t (θ−t ω)∥E ≤ e−σ t + √ λn+1

(4.19)

and

∥Pn Φ (t , τ − t , θ−τ ω)ϕ1,τ −t (θ−τ ω) − Pn Φ (t , τ − t , θ−τ ω)ϕ2,τ −t (θ−τ ω)∥E ≤e

∫0

−t C2 (θs ω)ds

∥ϕ1,τ −t (θ−t ω) − ϕ2,τ −t (θ−t ω)∥E ,

(4.20)

where E[C22 (ω)] < ∞.

(4.21)

Proof. Taking the inner product of (4.10) with ψnq = Qn ψ in E, we have for r ≥ τ − t, d ∥ψnq (r)∥2E + 2σ ∥ψnq (r)∥2E + α∥ηnq (r)∥21 + ∥ηnq (r)∥2 ≤ 2(f (u2 , x) − f (u1 , x), ηnq ). dr By (4.18), we have

λn+1 ∥ηnq ∥2 ≤ ∥ηnq ∥21 .

(4.22)

(4.23)

By (3.10), (4.7) and Lemma 4.6, we get 2 f (u2 (r), x) − f (u1 (r), x), ηnq

( ∫

(

=2

)

f (u2 (r)) − f (u1 (r)) · ηnq dx

)

U

≤ 2∥f (u2 (r)) − f (u1 (r))∥ · ∥ηnq ∥ 1

∥f (u2 (r)) − f (u1 (r))∥2 + (αλn+1 + 1)∥ηnq (r)∥2 αλn+1 + 1 ) c30 ( 2(p−1) ≤ 1 + b1 (θr −τ ω) ∥ξ (r)∥2 + (αλn+1 + 1) ∥ηnq (r)∥2 λn+1 ≤

≤

c30

C1 (θr −τ ω)e

∫r

τ −t 2C1 (θs−τ ω)ds

λn+1 + (αλn+1 + 1) ∥ηnq (r)∥2 .

∥ϕ1,τ −t (θ−τ ω) − ϕ2,τ −t (θ−τ ω)∥2E (4.24)

Putting (4.24) into (4.22), for r ≥ τ − t, we have d

≤

∥ψnq (r , τ − t , θ−τ ω, ψ1,τ −t (θ−τ ω))∥2E + 2σ ∥ψnq (r , τ , θ−τ ω, ϕ1τ (θ−τ ω))∥2E

dr c30

λn+1

∫r

C1 (θr −τ ω)e

τ −t 2C1 (θs−τ ω)ds

∥ϕ1,τ −t (θ−τ ω) − ϕ2,τ −t (θ−τ ω)∥2E .

(4.25)

By the Gronwall inequality to (4.25) on [τ − t , τ ] (t ≥ 0), we have

∥ψnq (τ , τ − t , θ−τ ω, ψτ −t (θ−τ ω))∥2E

≤ ∥ψnq (τ − t , τ − t , θ−τ ω, ψτ −t (θ−τ ω))∥2E · e−2σ t ∫ τ ∫ r −τ c30 2 + ∥ϕ1,τ −t (θ−τ ω) − ϕ2,τ −t (θ−τ ω)∥E C1 (θr −τ ω)e −t 2C1 (θs ω)ds−2σ (τ −r) dr λn+1 τ −t ( ) ∫ c30 ∫ 0 2C1 (θs ω)ds 0 −2σ t 2 σ ≤ e + e −t C1 (θr ω)e r dr λn+1 −t × ∥ϕ1,τ −t (θ−t ω) − ϕ2,τ −t (θ−t ω)∥2E .

(4.27)

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14

Z. Wang, L. Zhang / Computers and Mathematics with Applications (

Since

)

–

√

x ≤ ex , x ≥ 0, it follows that 0

∫

C1 (θr ω)e2σ r dr

−t

≤

0

(∫

C12 (θr ω)dr −t

≤

1

√ e

0

) 12 (∫

e4σ r dr

) 21

−t

∫0

2 −t C1 (θr ω)dr

2 σ

.

(4.28)

From (4.27) and (4.28), we have

∥ψnq (τ , τ − t , θ−τ ω, ψτ −t (θ−τ ω))∥2E ( ) ) ∫0 ( 2 c30 −2σ t −t 2C1 (θs ω)+C1 (θs ω) ds ∥ϕ1,τ −t (θ−t ω) − ϕ2,τ −t (θ−t ω)∥2E ≤ e + √ e 2 σ λn+1 )2 ( ∫0 c29 C2 (θs ω)ds −σ t − t +√ ≤ e ∥ϕ1,τ −t (θ−t ω) − ϕ2,τ −t (θ−t ω)∥2E , e λn+1

(4.29)

where 1

C 2 (θ s ω ) = C 1 (θ s ω ) +

2

4(p−1)

C12 (θs ω) = c31 1 + b1

(

(ω ) .

)

(4.30)

By (4.17) and (4.30), we have 4(p−1)

E[C22 (θr ω)] = c31 1 + E[b1

(

(θr ω)] = c32 < ∞.

)

(4.31)

By Lemma 4.6 and (4.30), we have (4.20) holds. We complete the proof. As a consequence of Lemmas 4.5 and 4.7, we have the main result of this section. Theorem 4.8. Suppose (2.7) and (2.8) hold. Then there exists a finite integer number n0 ∈ N such that for any τ ∈ R, ω ∈ Ω , the fractal dimension of A(τ , ω) has a finite upper bound:

(√

)

2n0 λn +1 0 c29

4n0 ln dimf A(τ , ω) ≤

ln

+1 < ∞.

4 3

(4.32)

Proof. Since U ⊂ R3 , then by Section 5 in [29], we have 2

λn ≥ c33 n 3 → +∞ as n → +∞,

(4.33)

where c33 > 0 is a constant depending on the volume of U and area of ∂ U only. Note that (4.33) implies that 1 0 < √

λn

≤ √

Letting t = t0 = 1 + satisfying

1 c33

2 ln 2

σ

12

n0 ≥

3 3 − 2 ln 23 512c29 c33 e

→ 0 as n → +∞. √ 3

(4.34)

n

in (4.19)–(4.20) , by (4.34) and 0 ≤ E[C22 (ω)] < ∞, there exists a finite integer number n0 ∈ N (

2 1+ 2 ln σ

)2

E[C22 (ω)]

− 1,

which implying that 0 < √

c29

λn0 +1

≤ √

c29

c33

√ 3

≤

n0 + 1

1 8

−

e

4 ln 3 2

(

2 1+ 4 ln ε

)2 (

E[C22 (ω)]

)

.

It then follows from Lemmas 4.5 and 4.7 that for any τ ∈ R, ω ∈ Ω ,

(√ 4n0 ln dimf A(τ , ω) ≤

2n0 λn +1 0 c29

ln

4 3

) +1 < ∞.

This completes the proof.

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5. Remark In this paper, we obtain an upper bound of fractal dimension of the random attractor for stochastic non-autonomous strongly damped wave equations. The required condition g(·, t) ∈ Cb (R, H01 (U)), is stronger than the tempered integrability of g in [17,18,30], this is because that this strong condition can prove the finiteness of fractal dimension. References [1] L. Arnold, Random Dynamical Systems, Springer, New York, 1998. [2] F. Chen, B. Guo, P. Wang, Long time behavior of strongly damped nonlinear wave equations, J. Differential Equations 147 (1998) 339–352. [3] J.M. Ghidaglia, A. Marzocchi, Longtime behavior of strongly damped nonlinear wave equations, global attractors and their dimension, SIAM. J. Math. Anal. 22 (1991) 879–895. [4] V. Kalantarov, S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations 48 (2009) 1120– 1155. [5] H. Li, S. Zhou, One-dimensional global attractor for strongly damped wave equations, Commun. Nonlinear Sci. Numer. Simul. 12 (2007) 784–793. [6] H. Li, S. Zhou, On non-autonomous strongly damped wave equations with a uniform attractor and some averaging, J. Math. Anal. Appl. 341 (2008) 791–802. [7] P. Massatt, Limiting behavior for strongly damped nonlinear wave equations, J. Differential Equations 48 (1983) 334–349. [8] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. [9] M. Yang, C. Sun, Attractors for strongly damped wave equations, Nonlinear Anal. RWA 10 (2009) 1097–1100. [10] M. Yang, C. Sun, Dynamics of strongly damped wave equations in locally uniform spaces: attractors and asymptotic regularity, Trans. Amer. Math. Soc. 361 (2009) 1069–1101. [11] M. Yang, C. Sun, Exponential attractors for the strongly damped wave equations, Nonlinear Anal. RWA 11 (2010) 913–919. [12] S. Zhou, Dimension of the global attractor for strongly damped nonlinear wave equation, J. Math. Anal. Appl. 233 (1999) 102–115. [13] S. Zhou, Global attractor for strongly damped nonlinear wave equation, Funct. Differ. Equ. 6 (1999) 451–470. [14] S. Zhou, X. Fan, Kernel sections for non-autonomous strongly damped wave equations, J. Math. Anal. Appl. 275 (2002) 850–869. [15] S. Zhou, Attractors for strongly damped wave equations with critical exponent, Appl. Math. Lett. 16 (2003) 1307–1314. [16] Z. Wang, S. Zhou, A. Gu, Random attractor of the stochastic strongly damped wave equation, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 1649– 1658. [17] B. Wang, Suffcient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations 253 (2012) 1544–1583. [18] B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst. 34 (2014) 269–300. [19] Z. Wang, S. Zhou, Random attractors for non-autonomous stochastic strongly damped wave equation on unbounded domains, J. Appl. Anal. Comput. 5 (2015) 363–387. [20] R. Jones, B. Wang, Asymptotic behavior of a class of stochastic nonlinear wave equations with dispersive and dissipative terms, Nonlinear Anal. RWA 14 (2013) 1308–1322. [21] J.A. Langa, J.C. Robinson, Fractal dimension of a random invariant set, J. Math. Pures Appl. 85 (2006) 269–294. [22] G. Wang, Y. Tang, Fractal dimension of a random invariant set and applications, J. Appl. Math. (2013) Article ID 415764, 5 pages. [23] S. Zhou, M. Zhao, Fractal dimension of random invariant sets for nonautonomous random dynamical systems and random attractor for stochastic damped wave equation, Nonlinear Anal. 133 (2016) 292–318. [24] S. Zhou, M. Zhao, Fractal dimension of random attractor for stochastic damped wave equation with multiplicative noise, Discrete Contin. Dyn. Syst. 36 (2016) 2887–2914. [25] S. Zhou, Y. Tian, Z. Wang, Fractal dimension of random attractors for stochastic non-autonomous reaction–diffusion equations, Appl. Math. Comput. 276 (2016) 80–95. [26] Z. Wang, S. Zhou, Random attractor for stochastic non-autonomous damped wave equation with critical exponent, Discrete Contin. Dyn. Syst. 37 (2017) 545–573. [27] X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise, Inter. J. Math. 19 (2008) 421–437. [28] M. Yang, J. Duan, P. Kloeden, Asymptotic behavior of solutions for random wave equations with nonlinear damping and white noise, Nonlinear Anal. RWA 12 (2011) 464–478. [29] P. Li, S.T. Yau, Estimate of eigenvalues of a compact Riemann manifold, Proc. Symp. Pure Math. 36 (1980) 205–239. [30] J. Yin, Y. Li, A. Gu, Backwards compact attractors and periodic attractors for non-autonomous damped wave equations on an unbounded domain, Comput. Math. Appl. 74 (2017) 744–758.

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