Open Access

Dynamic behaviors of a local modified stochastic Swift-Hohenberg equation with multiplicative noise

Boundary Value Problems20172017:19

https://doi.org/10.1186/s13661-017-0753-5

Received: 17 October 2016

Accepted: 20 January 2017

Published: 2 February 2017

Abstract

In this paper, we investigate a global random attractor for a stochastic local modified Swift-Hohenberg equation with multiplicative noise in Stratonovich sense. Through the Ornstein-Uhlenbeck (O-U) transformation, we obtain the random dynamical system associated with the stochastic local modified Swift-Hohenberg equation. Using the properties of the O-U process, we derive the specific uniform a priori estimates, using which we prove the existence of global random attractor for the corresponding random dynamical system.

Keywords

local modified stochastic Swift-Hohenberg equationuniform a priori estimatesrandom attractor

1 Introduction

Swift and Hohenberg [1] proposed a model for the convective instability in the Rayleigh-Bénard convection, also known as the Swift-Hohenberg (S-H) equation, which is included as an important equation in different branches of physics, such as Taylor-Couette flow [2, 3], the study of lasers [4], and so on. After that, Doelman and Standstede [5] proposed the following modified Swift-Hohenberg equation for a pattern formation system near the onset to instability:
$$ u_{t}+\Delta^{2}u+2\Delta u+\alpha u+b\vert \nabla u \vert ^{2}+u^{3}=0, $$
(1.1)
where α and b are arbitrary constants. In the case of \(b=0\), it is the usual Swift-Hohenberg equation. The additional term \(b\vert \nabla u\vert ^{2}\) arises in the study of various pattern formation phenomena involving some kind of phase turbulence or phase transition that breaks the symmetry \(u\rightarrow-u\). For more references, one can see [68] and the references therein. The dynamical properties of the S-H equation are important for the studies of pattern formation and global attractors, and the stability of stationary solution and pattern selections of the S-H equation have been extensively investigated; see [911].
It turns out that a stochastic equation can conform to physical phenomena better in some cases. These random perturbations are intrinsic effects in a variety of settings and spatial scales. In fact, when the Rayleigh number is near thermal equilibrium, the influence of small noise or molecular noise is detected in various convection experiments [12, 13]. As the effect of thermal fluctuations on the onset of convective motion into the Bénard system is considered, the stochastic local S-H equation with additive noise [1] is proposed:
$$ u_{t}=\mu u-(1+\partial_{xx})^{2} u-u^{3}+\sigma\xi. $$
(1.2)
Furthermore, a local stochastic S-H equation driven by multiplicative noise arises when the effects of small possible noise from μ is considered [14]:
$$ u_{t}=\mu u-(1+\partial_{xx})^{2} u-u^{3}+\sigma u \circ\xi, $$
(1.3)
where \(\sigma>0\), and \(\xi=\frac{dW}{dt}\) is the generalized derivative of a real-valued Brownian motion.

There have been a lot of outstanding work and important results related to the existence and uniqueness of solution and attractors for stochastic partial differential equations. For research progress on these aspects, we refer to [1519]. Until now, there are few results on the dynamics behaviors of the stochastic local modified Swift-Hohenberg equation with multiplicative noise in Stratonovich sense. This is our main purpose. After making use of the O-U transform and changing the stochastic equation into the corresponding deterministic equation with random parameter, we obtain uniform a priori estimates under some additional assumptions and prove the existence of global random attractor for the random dynamical system associated with the stochastic local modified Swift-Hohenberg equation. It allows us to overcome the computational difficulties according to the properties of the local modified Swift-Hohenberg equation. In particular, showing the existence of a random attractor needs a lot of technical skills to obtain the desired results.

In this paper, we consider the following one-dimensional stochastic local modified Swift-Hohenberg equation with multiplicative noise:
$$ du+\bigl(\Delta^{2}u+2\Delta u+\alpha u+b\vert u_{x} \vert ^{2}-u^{3}\bigr)\,dt=\sigma u \circ dW(t) $$
(1.4)
with initial condition
$$ u(x,0)=u_{0}(x) ,\quad x\in D, $$
(1.5)
and the boundary conditions
$$ u|_{\partial D}=\Delta u|_{\partial D}=0 ,\quad x\in\partial D, $$
(1.6)
where D is a bounded open interval, \(\vert b\vert \ll1\) is a constant, Δu means \(u_{xx}\), and \(\Delta^{2} u\) means \(u_{xxxx}\).

An outline of this paper is as follows. We devote Section 2 to recall some definitions and results referred to global random attractors and to present some notation. In Section 3, we not only introduce the O-U transformer, but also obtain uniform a priori estimates of the solution for the stochastic local modified Swift-Hohenberg equation. Finally, the proof of the main theorem on the existence of global random attractor is presented in Section 4.

2 Preliminaries

There are many research results on random attractors and related issues. For simplicity of the structure of the article, we only list the definitions; for the relevant theorems, we refer to [20, 21], and so on. Let \((X,\Vert \cdot \Vert _{X})\) be a completely separable Hilbert space with Borel σ-algebra \(\mathscr{B}(X)\), and let \((\Omega, \mathcal{F}, \mathbb{P}, (\theta_{t})_{t\in\mathbb{R}})\) be an ergodic metric dynamical system.

Definition 2.1

See [15, 16]

A measurable mapping
$$\varphi: \mathbb{R}^{+}\times\Omega\times X \rightarrow X, \quad (t,\omega, x)\mapsto\varphi(t,\omega, x), $$
has the cocycle property:
  1. (1)

    \(\varphi(0,\omega,x)\) is the identity mapping on X;

     
  2. (2)

    \(\varphi(t+s,\omega, x)=\varphi(t,\theta_{s}\omega, \varphi(s,\omega , x))\) for all \(s,t \geq0\), \(x\in X\), and \(\omega\in\Omega\).

     

We call φ a random dynamical system (RDS) on X over \((\Omega, \mathcal{F}, \mathbb{P}, (\theta_{t})_{t\in \mathbb{R}})\). Furthermore, the RDS φ is continuous if \(\varphi(t,\omega,\cdot)=\varphi(t,\omega):X \rightarrow X\) is continuous for all \(t\geq0\).

Definition 2.2

See [15, 16]

A random compact set \(\mathcal {A}(\omega)\) is said to be a random attractor for RDS φ if the following conditions hold:
  1. (1)
    pullback attracting property:
    $$\lim_{t\rightarrow+\infty}\,d\bigl(\varphi\bigl(t,\theta_{-t}\omega, \mathscr {D}(\theta_{-t}\omega)\bigr); \mathcal{A}(\omega)\bigr)=0, $$
     
  2. (2)

    the invariance property: \(\varphi(t,\omega,\mathcal{A}(\omega ))=\mathcal{A}(\theta_{t}\omega)\) for \(\mathbb{P}\)-a.e. \(\omega\in\Omega\) and all \(t\geq0\).

     

Remark

Let \((\Omega,\mathcal{F},\mathbb{P})\) be a probability space with Wiener measure \(\mathbb{P}\). The Wiener shift \((\theta_{t})_{t\in\mathbb{R}}\) is defined by
$$\theta_{s} \omega(t)=\omega(t+s)-\omega(s),\quad t,s\in\mathbb{R}. $$
Then \((\Omega,\mathcal{F},\mathbb{P},(\theta_{t})_{t\in\mathbb{R}})\) is an ergodic metric dynamical system; see [20].

For the convenience of the following contents, we introduce some functional spaces and some notations. \(L^{q}(D)\) is the Lebesgue space with norm \(\Vert \cdot \Vert _{L^{q}}\), and \(\Vert \cdot \Vert _{L^{2}}=\Vert \cdot \Vert \). Particularly, \(\Vert u\Vert _{L^{\infty}}=\) \(\operatorname{ ess}\sup_{x\in D}\vert u(x)\vert \) for \(q=\infty\).

\(H^{\sigma}(D)\) is the Sobolev space \(\{u\in L^{2}(D), D^{k}u\in L^{2}(D), k\leq\sigma\}\) with norm \(\Vert \cdot \Vert _{H^{\sigma}}=\Vert \cdot \Vert _{\sigma}\). Especially, \(H_{0}^{2}(D)\) is the Sobolev space \(\{u\in L^{2}(D), D^{k}u\in L^{2}(D), k\leq2, \frac{\partial u}{\partial n}|_{x\in\partial D}=0\} \).

For notational simplicity, C is a generic constant that may take various values from line to line; Δv means \(v_{xx}\), and \(\Delta^{2} v\) means \(v_{xxxx}\).

3 Uniform a priori estimates of solution

In this section, we mainly show uniform a priori estimates of a solution for the stochastic local modified Swift-Hohenberg equation.

The original equation (1.4) can be rewritten as follows:
$$ u(t)=u_{0}- \int_{0}^{t}\bigl(\alpha u+2\Delta u + \Delta^{2} u+b\vert u_{x}\vert ^{2}-u^{3} \bigr)\,ds +\sigma \int_{0}^{t} u(s)\circ dW(s). $$
We now introduce an O-U process \(z(\theta_{t} \omega)\) that solves the following Itô equation:
$$ dz+z\,dt=\sigma \,dW(t). $$
By [20, 22], the random variable \(z(\theta_{t} \omega)\) is tempered, and for every \(\omega\in\tilde{\Omega}, t\mapsto z(\theta _{t} \omega)\) is continuous with respect to t. Especially, the properties \(\lim_{t\rightarrow\pm\infty}\frac{\vert z(\theta_{t} \omega)\vert }{\vert t\vert }=0\) and \(\lim_{t\rightarrow\pm\infty}\frac{\int_{0}^{t} z(\theta_{s} \omega )\,ds}{t}=0\) hold.
Let \(v=e^{-z(\theta_{t}\omega)}u(t)\). Using the Itô equation, combined with the original equation (1.4), we get
$$\begin{aligned} dv(t)&=e^{-z(\theta_{t} \omega)}\,du(t)-u(t)e^{-z(\theta_{t} \omega)}\circ d\bigl(z( \theta_{t} \omega)\bigr) \\ &=-\bigl(\Delta^{2}v+2\Delta v+\alpha v+e^{2z(\theta_{t}\omega)}v^{3}+be^{z(\theta _{t}\omega)}v_{x}^{2} \bigr)\,dt+z(\theta_{t}\omega)v(t)\,dt. \end{aligned}$$
Thus, we consider the following system:
$$\begin{aligned} & \frac{dv}{dt}+\bigl(\alpha-z(\theta_{t}\omega) \bigr)v+2\Delta v+\Delta ^{2}v+e^{2z(\theta_{t}\omega)}v^{3}+be^{z(\theta_{t}\omega)}v_{x}^{2}=0, \end{aligned}$$
(3.1)
$$\begin{aligned} &v(x,0)=e^{-z(\omega)}u_{0}, \end{aligned}$$
(3.2)
$$\begin{aligned} & v|_{\partial D}=\Delta v|_{\partial D}=0. \end{aligned}$$
(3.3)
Similarly to [8, 23], by the Galerkin method and some a priori estimates we can prove that \(v(t,\omega,v_{0})\) is unique and continuous with respect to initial value \(v_{0}\) in \(H^{2}_{0}(D)\) for \(\mathbb{P}\)-a.e. \(\omega\in\Omega\), where \(v(t,\omega,v_{0})\) is the solution of system (3.1)-(3.3). Define the continuous random dynamical system \(\{\psi(t)\}_{t\geq0}\) by
$$ \psi(t,\omega,v_{0})=v(t,\omega,v_{0}) $$
for all \(v_{0}\in H^{2}_{0}(D)\), \(t\geq0\), and \(\omega\in\Omega\). Furthermore, setting \(u(t,\omega,u_{0})=e^{z(\theta_{t}\omega)}v(t,\omega,v_{0})\), we have
$$ \phi(t,\omega,u_{0})=u(t,\omega,u_{0})=e^{z(\theta_{t}\omega)}v(t, \omega ,v_{0})=e^{z(\theta_{t}\omega)}\psi(t,\omega,v_{0}). $$
Then ϕ is a continuous random dynamical system on \(H_{0}^{2}(D)\). It is straightforward to show that the existence of a random attractor for ϕ is equivalent to the existence of a random attractor for ψ.

In the following, provided that \(\mathscr{D}\) is a collection of tempered random subsets of \(H_{0}^{2}(D)\), we will prove the existence of an absorbing set in \(H_{0}^{2}(D)\).

Lemma 3.1

Provided that \(v_{0} \in B=\{B(\omega)\}_{\omega} \subset\mathscr{D}\), there exist a random radius \(\rho_{1}(\omega)>0\) and \(T_{1B}(\omega)>0\) for \(\mathbb{P}\)-a.e. \(\omega\in\Omega\) such that
$$\begin{aligned} \bigl\Vert v\bigl(t,\theta_{-t}\omega, v_{0}(\theta_{-t}\omega)\bigr)\bigr\Vert ^{2} \leq\rho_{1}(\omega ),\quad t>T_{1B}(\omega). \end{aligned}$$
(3.4)

Proof

Taking the inner product of equation (3.1) with v, we have
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt}\Vert v\Vert ^{2}+ \bigl(\alpha-z(\theta _{t}\omega)\bigr)\bigl\Vert v(t)\bigr\Vert ^{2}+(2\Delta v,v)+\Vert \Delta v\Vert ^{2}+ \bigl(e^{2z(\theta_{t}\omega )}v^{3},v\bigr) \\ &\quad{}+\bigl(be^{z(\theta_{t}\omega)}v^{2}_{x},v \bigr)=0. \end{aligned}$$
(3.5)
Noticing that \(e^{2z(\theta_{t}\omega)}(v^{3},v)=e^{2z(\theta_{t}\omega)}\Vert v\Vert ^{4}_{L^{4}}\) and applying the Hölder inequality and ϵ-Young inequality, we get
$$\begin{aligned} \bigl\vert (2\Delta v,v)\bigr\vert \leq\frac{1}{4}\Vert \Delta v \Vert ^{2}+4\Vert v\Vert ^{2}. \end{aligned}$$
Now, we deal with the last term on the left side of equation (3.5). By integration by parts we have
$$\begin{aligned} \int_{D}be^{z(\theta_{t}\omega)}v^{2}_{x}v\,dx=-be^{z(\theta_{t}\omega )} \int_{D}v(v_{x}v)_{x}\,dx=-be^{z(\theta_{t}\omega)} \int_{D}\bigl(v^{2}v_{xx}+vv^{2}_{x} \bigr)\,dx, \end{aligned}$$
and thus
$$\begin{aligned} be^{z(\theta_{t}\omega)} \int_{D}v^{2}_{x}v\,dx=- \frac{b}{2}e^{z(\theta _{t}\omega)} \int_{D}v^{2}v_{xx}\,dx. \end{aligned}$$
Applying the Hölder inequality and ϵ-Young inequality again, we get
$$\begin{aligned} \biggl\vert be^{z(\theta_{t}\omega)} \int_{D}v^{2}_{x}v\,dx\biggr\vert \leq bCe^{z(\theta _{t}\omega)}\Vert v\Vert ^{2}_{L^{4}}\Vert v_{xx}\Vert \leq\frac{1}{4}\Vert v_{xx}\Vert ^{2}+b^{2}Ce^{2z(\theta_{t}\omega)}\Vert v\Vert ^{4}_{L^{4}}. \end{aligned}$$
Taking b small enough such that \(b^{2}C\leq\frac{1}{2}\), we obtain
$$\begin{aligned} \biggl\vert be^{z(\theta_{t}\omega)} \int_{D}v^{2}_{x}v\,dx\biggr\vert \leq \frac{1}{4}\Vert v_{xx}\Vert ^{2}+ \frac{1}{2}e^{2z(\theta_{t}\omega)}\Vert v\Vert ^{4}_{L^{4}}. \end{aligned}$$
Putting all these inequalities together, we have
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt}\Vert v\Vert ^{2}+ \frac{1}{2}\Vert \Delta v\Vert ^{2}+\bigl(\beta-z( \theta_{t}\omega)\bigr)\Vert v\Vert ^{2}+(\alpha-\beta-4) \Vert v\Vert ^{2} \\ &\quad{}+\frac{1}{2}e^{2z(\theta_{t}\omega)}\Vert v\Vert ^{4}_{L^{4}}\leq0, \end{aligned}$$
(3.6)
where \(\beta>0\) is a constant such that \(\alpha-\beta-4<0\).
By the Sobolev imbedding \(L^{4}(D)\subset L^{2}(D)\) we get
$$\begin{aligned} &\frac{d}{dt}\Vert v\Vert ^{2}+2\bigl(\beta-z( \theta_{t}\omega)\bigr)\Vert v\Vert ^{2}+\Vert \Delta v \Vert ^{2}\\ &\quad \leq-e^{2z(\theta_{t}\omega)}\Vert v\Vert ^{4}_{L^{4}}-2C( \alpha-\beta-4)\Vert v\Vert ^{2}_{L^{4}}. \end{aligned}$$
We can change the right-hand side of this inequality as follows:
$$\begin{aligned} &{-}e^{2z(\theta_{t}\omega)}\Vert v\Vert ^{4}_{L^{4}}-2C(\alpha- \beta-4)\Vert v\Vert ^{2}_{L^{4}} \\ &\quad =-e^{2z(\theta_{t}\omega)}\bigl(\Vert v\Vert ^{2}_{L^{4}}+Ce^{-2z(\theta_{t}\omega)} (\alpha-\beta-4)\bigr)^{2}+C(\alpha-\beta-4)^{2}e^{-2z(\theta_{t}\omega)}. \end{aligned}$$
Then we have
$$\begin{aligned} \frac{d}{dt}\Vert v\Vert ^{2}+2\bigl(\beta-z( \theta_{t}\omega)\bigr)\Vert v\Vert ^{2}+\Vert \Delta v \Vert ^{2}\leq C(\alpha-\beta-4)^{2}e^{-2z(\theta_{t}\omega)}. \end{aligned}$$
(3.7)
By the Gronwall inequality we have
$$\begin{aligned} \bigl\Vert v\bigl(t,\omega,v_{0}(\omega)\bigr)\bigr\Vert ^{2}\leq{}& e^{-2\beta t+\int_{0}^{t}2z(\theta_{\tau}\omega)\,d\tau}\bigl\Vert v_{0}(\omega) \bigr\Vert ^{2} \\ &{}+C(\alpha-\beta-4)^{2} \int_{0}^{t}e^{-2z(\theta_{s}\omega)-2\beta(t-s)+\int _{s}^{t}2z(\theta_{\tau}\omega)\,d\tau}\,ds. \end{aligned}$$
(3.8)
Furthermore, replacing ω with \(\theta_{-t}\omega\) in (3.8), then we have
$$\begin{aligned} &\bigl\Vert v\bigl(t,\theta_{-t}\omega,v_{0}( \theta_{-t}\omega)\bigr)\bigr\Vert ^{2} \\ &\quad \leq e^{-2t(\beta-\frac{\int_{-t}^{0}2z(\theta_{\tau}\omega)\,d\tau}{t})}\bigl\Vert v_{0}(\theta_{-t}\omega) \bigr\Vert ^{2} \\ &\qquad{}+C(\alpha-\beta-4)^{2} \int_{0}^{t}e^{-2z(\theta _{s-t}\omega)-2\beta(t-s) +\int_{s-t}^{0}2z(\theta_{\tau}\omega)\,d\tau}\,ds \\ &\quad =e^{-2t(\beta-\frac{\int_{-t}^{0}2z(\theta_{\tau}\omega)\,d\tau}{t})}\bigl\Vert v_{0}(\theta_{-t}\omega) \bigr\Vert ^{2}+C(\alpha-\beta-4)^{2} \int_{-t}^{0}e^{-2z(\theta _{s}\omega)+2\beta s +\int_{s}^{0}2z(\theta_{\tau}\omega)\,d\tau}\,ds \\ &\quad\leq e^{-2t(\beta-\frac{\int_{-t}^{0}2z(\theta_{\tau}\omega)\,d\tau}{t})}\bigl\Vert v_{0}(\theta_{-t}\omega) \bigr\Vert ^{2} \\ &\qquad{}+C(\alpha-\beta-4)^{2} \int_{-\infty }^{0}e^{2s(\beta-\frac{z(\theta_{s}\omega)}{s} +\frac{\int_{s}^{0}z(\theta_{\tau}\omega)\,d\tau}{s})}\,ds. \end{aligned}$$
(3.9)
Because of the properties of \(z(\theta_{t}\omega)\), there exists \(T_{1B}(\omega)>0\) such that, for all \(t\geq T_{1B}(\omega)\),
$$\frac{\int_{-t}^{0}2z(\theta_{\tau}\omega)\,d\tau}{t}\leq\frac{\beta}{2}. $$
It follows that
$$e^{-2t(\beta-\frac{\int_{-t}^{0}2z(\theta_{\tau}\omega)\,d\tau}{t})}\bigl\Vert v_{0}(\theta_{-t}\omega)\bigr\Vert ^{2}\leq e^{-\beta t}\bigl\Vert v_{0}( \theta_{-t}\omega)\bigr\Vert ^{2}. $$
The random set \(\mathscr{D}\) is tempered, which implies the boundedness of the first term on the right-hand side of (3.9). The second term on the right-hand side of (3.9) is convergent.
Thus, there exist \(T_{1B}(\omega)>0\) and a random variable \(\rho _{1}(\omega)\) such that, for \(\mathbb{P}\)-a.e. \(\omega\in\Omega\) and all \(t>T_{1B}(\omega)\),
$$\begin{aligned} \bigl\Vert v\bigl(t,\theta_{-t}\omega,v_{0}( \theta_{-t}\omega)\bigr)\bigr\Vert ^{2}\leq \rho_{1}(\omega). \end{aligned}$$
 □

Lemma 3.2

Provided that \(v_{0} \in B=\{B(\omega)\}_{\omega} \subset\mathscr{D}\), there exist a random radius \(\rho_{2}(\omega)>0\) and \(T_{2B}(\omega)>0\) for \(\mathbb{P}\)-a.e. \(\omega\in\Omega\) such that
$$\begin{aligned} \int_{t}^{t+1}\bigl\Vert \Delta v\bigl(s, \theta_{-t-1}\omega, v_{0}(\theta_{-t-1}\omega )\bigr) \bigr\Vert ^{2} \,ds \leq\rho_{2}(\omega),\quad t>T_{2B}(\omega). \end{aligned}$$
(3.10)

Proof

Multiplying equation (3.7) by \(e^{2\beta t-\int_{0}^{t}2z(\theta_{\tau }\omega)\,d\tau}\), we get
$$\begin{aligned} &\frac{d}{dt}\bigl[e^{2\beta t-\int_{0}^{t}2z(\theta_{\tau }\omega)\,d\tau} \Vert v\Vert ^{2}\bigr]+e^{2\beta t-\int_{0}^{t}2z(\theta_{\tau}\omega)\,d\tau }\Vert \Delta v\Vert ^{2} \\ &\quad \leq C(\alpha-\beta-4)^{2}e^{-2z(\theta_{t}\omega)+2\beta t-\int _{0}^{t}2z(\theta_{\tau}\omega)\,d\tau}. \end{aligned}$$
(3.11)
Set \(\widehat{T}\leq t\leq t+1\). Integrating from to t, we have
$$\begin{aligned} &e^{2\beta t-\int_{0}^{t}2z(\theta_{\tau}\omega)\,d\tau}\bigl\Vert v(t)\bigr\Vert ^{2}+ \int_{\widehat{T}}^{t}e^{2\beta s-\int_{0}^{s}2z(\theta_{\tau}\omega)\,d\tau }\bigl\Vert \Delta v(s) \bigr\Vert ^{2}\,ds \\ &\quad \leq C(\alpha-\beta-4)^{2} \int_{\widehat{T}}^{t}e^{-2z(\theta_{s}\omega )+2\beta s-\int_{0}^{s}2z(\theta_{\tau}\omega)\,d\tau}\,ds+e^{2\beta\widehat {T}-\int_{0}^{\widehat{T}}2z(\theta_{\tau}\omega)\,d\tau}\bigl\Vert v\bigl(\widehat {T},\omega,v_{0}(\omega)\bigr)\bigr\Vert ^{2}. \end{aligned}$$
Multiplying this inequality by \(e^{-2\beta t+\int_{0}^{t}2z(\theta_{\tau }\omega)\,d\tau}\) and getting rid of the first term, we obtain
$$\begin{aligned} &\int_{\widehat{T}}^{t}e^{2\beta(s-t)+\int _{s}^{t}2z(\theta_{\tau}\omega)\,d\tau}\bigl\Vert \Delta v(s) \bigr\Vert ^{2}\,ds \\ &\quad\leq e^{2\beta (\widehat{T}-t)+\int_{\widehat{T}}^{t}2z(\theta_{\tau}\omega)\,d\tau}\bigl\Vert v\bigl( \widehat{T},\omega,v_{0}(\omega)\bigr)\bigr\Vert ^{2} \\ &\qquad{}+C(\alpha-\beta-4)^{2} \int_{\widehat{T}}^{t}e^{-2z(\theta_{s}\omega)+2\beta (s-t)+\int_{s}^{t}2z(\theta_{\tau}\omega)\,d\tau}\,ds. \end{aligned}$$
(3.12)
Now, substituting t for in (3.8), we obtain
$$\begin{aligned} \bigl\Vert v\bigl(\widehat{T},\omega,v_{0}(\omega) \bigr)\bigr\Vert ^{2}\leq{}& e^{-2\beta\widehat{T}+\int_{0}^{\widehat{T}}2z(\theta_{\tau}\omega)\,d\tau}\bigl\Vert v_{0}(\omega)\bigr\Vert ^{2} \\ &{}+C(\alpha-\beta-4)^{2} \int_{0}^{\widehat{T}}e^{-2z(\theta_{s}\omega)-2\beta (\widehat{T}-s)+\int_{s}^{\widehat{T}}2z(\theta_{\tau}\omega)\,d\tau}\,ds. \end{aligned}$$
(3.13)
If we plug (3.13) back into (3.12), we have
$$\begin{aligned} & \int_{\widehat{T}}^{t}e^{2\beta(s-t)+\int _{s}^{t}2z(\theta_{\tau}\omega)\,d\tau}\bigl\Vert \Delta v \bigl(s,\omega,v_{0}(\omega)\bigr)\bigr\Vert ^{2}\,ds \\ &\quad \leq e^{-2\beta t+\int_{0}^{t}2z(\theta_{\tau}\omega)\,d\tau}\bigl\Vert v_{0}(\omega)\bigr\Vert ^{2} \\ &\qquad{}+C(\alpha-\beta-4)^{2} \int_{0}^{t}e^{-2z(\theta_{s}\omega) -2\beta(t-s)+\int_{s}^{t}2z(\theta_{\tau}\omega)\,d\tau}\,ds. \end{aligned}$$
(3.14)
Replacing ω with \(\theta_{-t}\omega\) in (3.14), we get
$$\begin{aligned} & \int_{\widehat{T}}^{t}e^{2\beta(s-t)+\int_{s}^{t}2z(\theta_{\tau-t}\omega )\,d\tau}\bigl\Vert \Delta v \bigl(s,\theta_{-t}\omega,v_{0}(\theta_{-t}\omega) \bigr)\bigr\Vert ^{2}\,ds \\ &\quad\leq C(\alpha-\beta-4)^{2} \int_{0}^{t}e^{-2z(\theta_{s-t}\omega) -2\beta(t-s)+\int_{s}^{t}2z(\theta_{\tau-t}\omega)\,d\tau}\,ds \\ &\qquad{}+e^{-2\beta t+\int_{0}^{t}2z(\theta_{\tau-t}\omega)\,d\tau}\bigl\Vert v_{0}(\theta _{-t}\omega) \bigr\Vert ^{2}. \end{aligned}$$
(3.15)
In order to obtain the result, we need to substitute for t and t for \(t+1\) in (3.15) as follows:
$$\begin{aligned} & \int_{t}^{t+1}e^{2\beta(s-t-1)+\int_{s}^{t+1}2z(\theta_{\tau -t-1}\omega)\,d\tau}\bigl\Vert \Delta v \bigl(s,\theta_{-t-1}\omega,v_{0}(\theta _{-t-1} \omega)\bigr)\bigr\Vert ^{2}\,ds \\ &\quad\leq e^{-2\beta(t+1)+\int_{0}^{t+1}2z(\theta_{\tau-t-1}\omega)\,d\tau}\bigl\Vert v_{0}(\theta_{-t-1}\omega) \bigr\Vert ^{2} \\ &\qquad{}+C(\alpha-\beta-4)^{2} \int_{0}^{t+1}e^{-2z(\theta_{s-t-1}\omega) -2\beta(t+1-s)+\int_{s}^{t+1}2z(\theta_{\tau-t-1}\omega)\,d\tau}\,ds, \end{aligned}$$
that is,
$$\begin{aligned} & \int_{t}^{t+1}e^{2\beta(s-t-1)+\int _{s-t-1}^{0}2z(\theta_{\tau}\omega)\,d\tau}\bigl\Vert \Delta v \bigl(s,\theta_{-t-1}\omega ,v_{0}(\theta_{-t-1} \omega)\bigr)\bigr\Vert ^{2}\,ds \\ &\quad \leq e^{-2(t+1)(\beta-\frac{\int_{-t-1}^{0}2z(\theta_{\tau}\omega)\,d\tau }{2(t+1)})}\bigl\Vert v_{0}(\theta_{-t-1}\omega) \bigr\Vert ^{2} \\ &\qquad{}+C(\alpha-\beta-4)^{2} \int _{-t-1}^{0}e^{-2z(\theta_{s}\omega)+2\beta s+\int_{s}^{0}2z(\theta_{\tau}\omega )\,d\tau}\,ds \\ &\quad \leq e^{-2(t+1)(\beta-\frac{\int_{-t-1}^{0}2z(\theta_{\tau}\omega)\,d\tau }{2(t+1)})}\bigl\Vert v_{0}(\theta_{-t-1}\omega) \bigr\Vert ^{2} \\ &\qquad{}+C(\alpha-\beta-4)^{2} \int_{-\infty}^{0}e^{-2z(\theta_{s}\omega)+2\beta s+\int_{s}^{0}2z(\theta_{\tau}\omega)\,d\tau}\,ds. \end{aligned}$$
(3.16)
According to the properties of \(z(\theta_{t}\omega)\), when \(-1\leq s-t-1\leq0\), we can deduce
$$\begin{aligned} & \int_{t}^{t+1}e^{2\beta(s-t-1)+\int _{s-t-1}^{0}2z(\theta_{\tau}\omega)\,d\tau}\bigl\Vert \Delta v \bigl(s,\theta_{-t-1}\omega ,v_{0}(\theta_{-t-1} \omega)\bigr)\bigr\Vert ^{2}\,ds \\ & \quad \geq e^{-2\beta-2\max_{-1\leq\tau\leq0}\vert z(\theta_{\tau}\omega)\vert } \int _{t}^{t+1}\bigl\Vert \Delta v\bigl(s, \theta_{-t-1}\omega,v_{0}(\theta_{-t-1}\omega)\bigr) \bigr\Vert ^{2}\,ds. \end{aligned}$$
(3.17)
Combining estimates (3.16) and (3.17), we prove that there exist a random variable \(\rho_{2}(\omega)\) and \(T_{2B}(\omega)>0\) such that, for \(\mathbb{P}\)-a.e. \(\omega\in\Omega\) and all \(t>T_{2B}(\omega)\),
$$\begin{aligned} \int_{t}^{t+1}\bigl\Vert \Delta v\bigl(s, \theta_{-t-1}\omega,v_{0}(\theta _{-t-1}\omega)\bigr) \bigr\Vert ^{2}\,ds\leq\rho_{2}(\omega). \end{aligned}$$
The proof of the lemma is now complete. □

4 The existence of random attractor

Lemma 4.1

Provided that \(v_{0} \in B=\{B(\omega)\}_{\omega } \subset\mathscr{D}\), there exist a random radius \(\rho_{3}(\omega)>0\) and \(T_{3B}(\omega)>0\) for \(\mathbb{P}\)-a.e. \(\omega\in\Omega\) such that
$$\begin{aligned} \bigl\Vert \Delta v\bigl(t,\theta_{-t}\omega, v_{0}(\theta_{-t}\omega)\bigr)\bigr\Vert ^{2} \leq\rho _{3}(\omega),\quad t>T_{3B}(\omega). \end{aligned}$$
(4.1)

Proof

Taking the inner product of equation (3.1) with \(\Delta^{2}v\), we have
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt}\Vert \Delta v\Vert ^{2}+\bigl\Vert \Delta^{2}v\bigr\Vert ^{2}+ \bigl(2\Delta v,\Delta^{2}v\bigr)+\bigl(\alpha-z(\theta_{t} \omega)\bigr)\Vert \Delta v\Vert ^{2} \\ &\quad{}+\bigl(e^{2z(\theta_{t}\omega)}v^{3},\Delta^{2}v\bigr)+ \bigl(be^{z(\theta_{t}\omega )}(v_{x})^{2},\Delta^{2}v \bigr)=0. \end{aligned}$$
(4.2)
Applying the Hölder inequality and ϵ-Young inequality, we get
$$\bigl\vert \bigl(2\Delta v,\Delta^{2}v\bigr)\bigr\vert \leq \frac{1}{4}\bigl\Vert \Delta^{2}v\bigr\Vert ^{2}+4 \Vert \Delta v\Vert ^{2}. $$
By the Gagliardo-Nirenberg inequality \(\Vert v\Vert _{L^{6}}\leq C\Vert v\Vert ^{\frac {11}{12}}\Vert \Delta^{2}v\Vert ^{\frac{1}{12}}\) (see [8]) and the ϵ-Young inequality we have
$$\begin{aligned} \bigl\vert \bigl(e^{2z(\theta_{t}\omega)}v^{3},\Delta^{2}v\bigr) \bigr\vert &\leq Ce^{2z(\theta _{t}\omega)}\Vert v\Vert ^{3}_{L^{6}} \bigl\Vert \Delta^{2}v\bigr\Vert \leq Ce^{2z(\theta_{t}\omega)}\bigl\Vert \Delta^{2}v\bigr\Vert \cdot \Vert v\Vert ^{\frac{11}{4}}\bigl\Vert \Delta^{2}v\bigr\Vert ^{\frac{1}{4}} \\ &\leq\eta_{1}\bigl\Vert \Delta^{2}v\bigr\Vert ^{2}+C(\eta_{1})e^{\frac{16z(\theta_{t}\omega)}{3}}\Vert v\Vert ^{\frac{22}{3}}. \end{aligned}$$
Similarly, by the Gagliardo-Nirenberg inequality \(\Vert v_{x}\Vert _{L^{4}}\leq C\Vert v\Vert ^{\frac{11}{16}}\Vert \Delta^{2}v\Vert ^{\frac{5}{16}}\) and the ϵ-Young inequality again, we obtain
$$\begin{aligned} \bigl\vert \bigl(be^{z(\theta_{t}\omega)}v^{2}_{x}, \Delta^{2}v\bigr)\bigr\vert &\leq \vert b\vert Ce^{z(\theta_{t}\omega)} \bigl\Vert \Delta^{2}v\bigr\Vert \Vert v_{x}\Vert ^{2}_{L^{4}}\leq \vert b\vert Ce^{z(\theta_{t}\omega)}\bigl\Vert \Delta^{2}v\bigr\Vert ^{\frac{13}{8}}\Vert v\Vert ^{\frac {11}{8}} \\ &\leq\eta_{2}\bigl\Vert \Delta^{2}v\bigr\Vert ^{2}+C(\eta_{2})e^{\frac{16}{3}z(\theta_{t}\omega)}\Vert v\Vert ^{\frac{22}{3}}. \end{aligned}$$
Taking \(\eta_{1}=\eta_{2}=\frac{1}{4}\), we deduce
$$\begin{aligned} &\frac{d}{dt}\Vert \Delta v\Vert ^{2}+2\bigl( \beta-z(\theta _{t}\omega)\bigr)\Vert \Delta v\Vert ^{2}+ \frac{1}{2}\bigl\Vert \Delta^{2}v\bigr\Vert ^{2} \\ &\quad\leq Ce^{\frac {16}{3}z(\theta_{t}\omega)}\Vert v\Vert ^{\frac{22}{3}}+2(4-\alpha+\beta)\Vert \Delta v\Vert ^{2}. \end{aligned}$$
(4.3)
Integrating (4.3) from s to \(t+1\) with respect to t, we have
$$\begin{aligned} &\bigl\Vert \Delta v\bigl(t+1,\omega,v_{0}(\omega)\bigr)\bigr\Vert ^{2}\\ &\quad \leq\bigl\Vert \Delta v\bigl(s,\omega,v_{0}(\omega) \bigr)\bigr\Vert ^{2}+2 \int_{s}^{t+1}\bigl(z(\theta_{\tau}\omega)- \beta\bigr)\bigl\Vert \Delta v\bigl(\tau,\omega,v_{0}(\omega)\bigr) \bigr\Vert ^{2}\,d\tau \\ &\qquad{}+ \int_{s}^{t+1}\bigl[Ce^{\frac{16}{3}z(\theta_{\tau}\omega)}\bigl\Vert v \bigl(\tau,\omega ,v_{0}(\omega)\bigr)\bigr\Vert ^{\frac{22}{3}}+2(4- \alpha+\beta)\bigl\Vert \Delta v\bigl(\tau,\omega ,v_{0}(\omega)\bigr) \bigr\Vert ^{2}\bigr]\,d\tau. \end{aligned}$$
Integrating from t to \(t+1\) with respect to s again, we get
$$\begin{aligned} &\bigl\Vert \Delta v\bigl(t+1,\omega,v_{0}(\omega) \bigr)\bigr\Vert ^{2} \\ &\quad \leq \int _{t}^{t+1}\bigl\Vert \Delta v\bigl(s, \omega,v_{0}(\omega)\bigr)\bigr\Vert ^{2}\,ds+2 \int_{t}^{t+1}\bigl\vert \bigl(z(\theta _{\tau}\omega)-\beta\bigr)\bigr\vert \bigl\Vert \Delta v\bigl(\tau, \omega,v_{0}(\omega)\bigr)\bigr\Vert ^{2}\,d\tau \\ &\qquad{}+ \int_{t}^{t+1}Ce^{\frac{16}{3}z(\theta_{\tau}\omega)}\bigl\Vert v\bigl( \tau,\omega ,v_{0}(\omega)\bigr)\bigr\Vert ^{\frac{22}{3}}\,d\tau+2(4- \alpha+\beta) \int_{t}^{t+1}\bigl\Vert \Delta v\bigl(\tau, \omega,v_{0}(\omega)\bigr)\bigr\Vert ^{2}\,d\tau \\ &\quad\leq(9-2\alpha+4\beta) \int_{t}^{t+1}\bigl\Vert \Delta v\bigl(s, \omega,v_{0}(\omega)\bigr)\bigr\Vert ^{2}\,ds+2 \int_{t}^{t+1}\bigl\vert z(\theta_{\tau}\omega)\bigr\vert \bigl\Vert \Delta v\bigl(\tau,\omega ,v_{0}( \omega)\bigr)\bigr\Vert ^{2}\,d\tau \\ &\qquad{}+ \int_{t}^{t+1}Ce^{\frac{16}{3}z(\theta_{\tau}\omega)}\bigl\Vert v\bigl( \tau,\omega ,v_{0}(\omega)\bigr)\bigr\Vert ^{\frac{22}{3}}\,d\tau. \end{aligned}$$
(4.4)
Replacing ω with \(\theta_{-t-1}\omega\) in (4.4), we have
$$\begin{aligned} &\bigl\Vert \Delta v\bigl(t+1,\theta_{-t-1} \omega,v_{0}(\theta _{-t-1}\omega)\bigr)\bigr\Vert ^{2} \\ &\quad \leq(9-2\alpha+4\beta) \int_{t}^{t+1}\bigl\Vert \Delta v\bigl(s, \theta_{-t-1}\omega ,v_{0}(\theta_{-t-1}\omega)\bigr) \bigr\Vert ^{2}\,ds \\ &\qquad{}+2 \int_{t}^{t+1}\bigl\vert z(\theta_{\tau-t-1} \omega)\bigr\vert \bigl\Vert \Delta v\bigl(\tau,\theta _{-t-1} \omega,v_{0}(\theta_{-t-1}\omega)\bigr)\bigr\Vert ^{2}\,d\tau \\ &\qquad{}+ \int_{t}^{t+1}Ce^{\frac{16}{3}z(\theta_{\tau-t-1}\omega)}\bigl\Vert v\bigl( \tau,\theta _{-t-1}\omega,v_{0}(\theta_{-t-1}\omega) \bigr)\bigr\Vert ^{\frac{22}{3}}\,d\tau. \end{aligned}$$
(4.5)
According to Lemma 3.2, the first term on the right-hand side of the above inequality is bounded:
$$\begin{aligned} (9-2\alpha+4\beta) \int_{t}^{t+1}\bigl\Vert \Delta v\bigl(s, \theta_{-t-1}\omega ,v_{0}(\theta_{-t-1}\omega)\bigr) \bigr\Vert ^{2}\,ds\leq \vert 9-2\alpha+4\beta \vert \rho_{2}(\omega). \end{aligned}$$
For the second term, taking into account the properties of \(z(\theta _{t}\omega)\), we have \(\vert z(\theta_{t}\omega)\vert \leq e^{\vert t\vert }r(\omega)\). Then
$$\begin{aligned} &2 \int_{t}^{t+1}\bigl\vert z(\theta_{\tau-t-1} \omega)\bigr\vert \bigl\Vert \Delta v\bigl(\tau ,\theta_{-t-1} \omega,v_{0}(\theta_{-t-1}\omega)\bigr)\bigr\Vert ^{2}\,d\tau \\ &\quad \leq 2 \int_{t}^{t+1}e^{\vert \tau-t-1\vert }r(\omega)\bigl\Vert \Delta v\bigl(\tau,\theta _{-t-1}\omega,v_{0}( \theta_{-t-1}\omega)\bigr)\bigr\Vert ^{2}\,d\tau. \end{aligned}$$
Noticing that \(\vert \tau-t-1\vert <1\) and applying Lemma 3.2, we obtain
$$\begin{aligned} 2 \int_{t}^{t+1}\bigl\vert z(\theta_{\tau-t-1} \omega)\bigr\vert \bigl\Vert \Delta v\bigl(\tau ,\theta_{-t-1} \omega,v_{0}(\theta_{-t-1}\omega)\bigr)\bigr\Vert ^{2}\,d\tau\leq2er(\omega )\rho_{2}(\omega). \end{aligned}$$
Now, we estimate the last term. Replacing t with τ in (3.8), we get
$$\begin{aligned} \bigl\Vert v\bigl(\tau,\omega,v_{0}(\omega)\bigr)\bigr\Vert ^{2}\leq{}& e^{-2\beta \tau+\int_{0}^{\tau}2z(\theta_{r}\omega)\,dr}\bigl\Vert v_{0}(\omega) \bigr\Vert ^{2} \\ &{}+C(\alpha-\beta-4)^{2} \int_{0}^{\tau}e^{-2z(\theta_{s}\omega)-2\beta(\tau -s)+\int_{s}^{\tau}2z(\theta_{r}\omega)\,dr}\,ds. \end{aligned}$$
(4.6)
Substituting ω for \(\theta_{-t-1}\omega\) in (4.6), we obtain
$$\begin{aligned} &\bigl\Vert v\bigl(\tau,\theta_{-t-1} \omega,v_{0}(\theta _{-t-1}\omega)\bigr)\bigr\Vert ^{2} \\ &\quad\leq e^{-2\beta\tau+\int_{0}^{\tau}2z(\theta _{r-t-1}\omega)\,dr}\bigl\Vert v_{0}( \theta_{-t-1}\omega)\bigr\Vert ^{2} \\ &\qquad{}+C(\alpha-\beta-4)^{2} \int_{0}^{\tau}e^{-2z(\theta_{s-t-1}\omega)-2\beta (\tau-s)+\int_{s}^{\tau}2z(\theta_{r-t-1}\omega)\,dr}\,ds. \end{aligned}$$
(4.7)
If we plug (4.7) back into (4.5), we obtain
$$\begin{aligned} & \int_{t}^{t+1}Ce^{\frac{16}{3}z(\theta_{\tau -t-1}\omega)}\biggl[e^{-2\beta\tau+\int_{0}^{\tau}2z(\theta_{r-t-1}\omega)\,dr} \bigl\Vert v_{0}(\theta_{-t-1}\omega)\bigr\Vert ^{2} \\ &\qquad{}+C(\alpha-\beta-4)^{2} \int_{0}^{\tau}e^{-2z(\theta_{s-t-1}\omega)-2\beta (\tau-s)+\int_{s}^{\tau}2z(\theta_{r-t-1}\omega)\,dr}\,ds \biggr]^{\frac{11}{3}}\,d\tau \\ &\quad \leq \int_{t}^{t+1}Ce^{\frac{16}{3}z(\theta_{\tau-t-1}\omega )}\biggl[ \bigl(e^{-2\beta\tau+\int_{0}^{\tau}2z(\theta_{r-t-1}\omega)\,dr}\bigl\Vert v_{0}(\theta _{-t-1}\omega) \bigr\Vert ^{2}\bigr)^{\frac{11}{3}} \\ &\qquad{}+C(\alpha-\beta-4)^{\frac{22}{3}}\biggl( \int_{0}^{\tau}e^{-2z(\theta _{s-t-1}\omega)-2\beta(\tau-s)+\int_{s}^{\tau}2z(\theta_{r-t-1}\omega )\,dr}\,ds \biggr)^{\frac{11}{3}}\biggr]\,d\tau \\ &\quad = \int_{t}^{t+1}Ce^{\frac{16}{3}z(\theta_{\tau-t-1}\omega)}\bigl(e^{-2\beta\tau +\int_{0}^{\tau}2z(\theta_{r-t-1}\omega)\,dr} \bigl\Vert v_{0}(\theta_{-t-1}\omega)\bigr\Vert ^{2}\bigr)^{\frac{11}{3}}\,d\tau \\ &\qquad{}+C(\alpha-\beta-4)^{\frac{22}{3}} \\ &\qquad{}\times \int_{t}^{t+1}e^{\frac{16}{3}z(\theta _{\tau-t-1}\omega)}\biggl( \int_{0}^{\tau}e^{-2z(\theta_{s-t-1}\omega)-2\beta(\tau -s)+\int_{s}^{\tau}2z(\theta_{r-t-1}\omega)\,dr}\,ds \biggr)^{\frac{11}{3}}\,d\tau. \end{aligned}$$
(4.8)
Next, we estimate each term on the right-hand side of the last inequality. For the first term, we have
$$\begin{aligned} &C \int_{t}^{t+1}\bigl(e^{-2\beta\tau+\int_{0}^{\tau }2z(\theta_{r-t-1}\omega)\,dr}\bigl\Vert v_{0}(\theta_{-t-1}\omega)\bigr\Vert ^{2} \bigr)^{\frac{11}{3}} e^{\frac{16}{3}z(\theta_{\tau-t-1}\omega)}\,d\tau \\ &\quad =C \int_{t}^{t+1}\bigl(e^{-2\beta\tau+\int_{-t-1}^{\tau-t-1}2z(\theta_{r}\omega )\,dr}\bigl\Vert v_{0}(\theta_{-t-1}\omega)\bigr\Vert ^{2} \bigr)^{\frac{11}{3}} e^{\frac{16}{3}z(\theta_{\tau-t-1}\omega)}\,d\tau \\ &\quad =C \int_{-1}^{0}\bigl(e^{-2\beta(\tau+1+t)+\int_{-t-1}^{\tau}2z(\theta _{r}\omega)\,dr}\bigl\Vert v_{0}(\theta_{-t-1}\omega)\bigr\Vert ^{2} \bigr)^{\frac{11}{3}} e^{\frac{16}{3}z(\theta_{\tau}\omega)}\,d\tau \\ &\quad\leq Ce^{\frac{16}{3}\max_{-1\leq\tau\leq0}\vert z(\theta_{\tau}\omega)\vert } \int _{-1}^{0}\bigl(e^{-2\beta(\tau+t+1)+\int_{-t-1}^{0}2\vert z(\theta_{r}\omega )\vert \, dr} \bigr)^{\frac{11}{3}}\,d\tau \\ &\quad = Ce^{\frac{16}{3}\max_{-1\leq\tau\leq0}\vert z(\theta_{\tau}\omega)\vert } \int _{-1}^{0}\bigl(e^{-2\beta(t+1)+\int_{-t-1}^{0}2\vert z(\theta_{r}\omega)\vert \,dr} \bigr)^{\frac {11}{3}}\cdot e^{-\frac{22}{3}\beta\tau}\,d\tau \\ &\quad =Ce^{\frac{16}{3}\max_{-1\leq\tau\leq0}\vert z(\theta_{\tau}\omega )\vert }\bigl(e^{-2\beta(t+1)+\int_{-t-1}^{0}2\vert z(\theta_{r}\omega)\vert \,dr}\bigr)^{\frac {11}{3}}\cdot \int_{-1}^{0} e^{-\frac{22}{3}\beta\tau}\,d\tau \\ &\quad \leq Ce^{\frac{16}{3}\max_{-1\leq\tau\leq0}\vert z(\theta_{\tau}\omega )\vert }\bigl(e^{-2\beta(t+1)+\int_{-t-1}^{0}2\vert z(\theta_{r}\omega)\vert \,dr}\bigr)^{\frac{11}{3}}, \end{aligned}$$
(4.9)
where the first inequality is due to the properties of \(z(\theta _{t}\omega)\).
For the second term, similarly to the above method, as \(t<\tau<t+1\), we have
$$\begin{aligned} &C(\alpha-\beta-4)^{\frac{22}{3}} \int _{t}^{t+1}e^{\frac{16}{3}z(\theta_{\tau-t-1}\omega)}\biggl( \int_{0}^{\tau}e^{-2z(\theta_{s-t-1}\omega)-2\beta(\tau-s)+\int_{s}^{\tau}2z(\theta _{r-t-1}\omega)\,dr}\,ds \biggr)^{\frac{11}{3}}\,d\tau \\ &\quad \leq C(\alpha-\beta-4)^{\frac{22}{3}} \int_{t}^{t+1}e^{\frac{16}{3}z(\theta _{\tau-t-1}\omega)}\biggl( \int_{0}^{\tau}e^{-2z(\theta_{s-t-1}\omega)+\int _{s-t-1}^{0}2 \vert z(\theta_{r}\omega)\vert \,dr-2\beta(\tau-s)}\,ds \biggr)^{\frac {11}{3}}\,d\tau \\ &\quad \leq C(\alpha-\beta-4)^{\frac{22}{3}} \int_{t}^{t+1}e^{\frac {16}{3}z(\theta_{\tau-t-1}\omega)}\biggl( \int_{-t-1}^{\tau-t-1} e^{-2z(\theta _{s}\omega)+\int_{s}^{0}2 \vert z(\theta_{r}\omega)\vert \,dr-2\beta(\tau -t-1-s)}\,ds \biggr)^{\frac{11}{3}}\,d\tau \\ &\quad \leq C(\alpha-\beta-4)^{\frac{22}{3}} \\ &\qquad{}\times\int_{t}^{t+1}e^{\frac{16}{3}z(\theta _{\tau-t-1}\omega)}\biggl( \int_{-t-1}^{0} e^{-2z(\theta_{s}\omega)+2\beta s+\int_{s}^{0}2 \vert z(\theta_{r}\omega)\vert \,dr+2\beta}\,ds \biggr)^{\frac{11}{3}}\,d\tau, \end{aligned}$$
(4.10)
where by the condition \(t<\tau<t+1\), \(-2\beta(\tau-t)<0\), we get rid of the term \(e^{-2\beta(\tau-t)}\). Then the last inequality can be estimated as follows:
$$\begin{aligned} &C(\alpha-\beta-4)^{\frac{22}{3}} \int_{t}^{t+1}e^{\frac {16}{3}z(\theta_{\tau-t-1}\omega)}\biggl( \int_{0}^{\tau}e^{-2z(\theta _{s-t-1}\omega)-2\beta(\tau-s)+\int_{s}^{\tau}2z(\theta_{r-t-1}\omega )\,dr}\,ds \biggr)^{\frac{11}{3}}\,d\tau \\ &\quad \leq C(\alpha-\beta-4)^{\frac{22}{3}}e^{\frac{22}{3}\beta} \int _{t}^{t+1}e^{\frac{16}{3}z(\theta_{\tau-t-1}\omega)}\biggl( \int_{-\infty}^{0} e^{-2z(\theta_{s}\omega)+2\beta s+\int_{s}^{0}2 \vert z(\theta_{r}\omega )\vert \,dr}\,ds \biggr)^{\frac{11}{3}}\,d\tau \\ &\quad \leq C(\alpha-\beta-4)^{\frac{22}{3}}e^{\frac{22}{3}\beta} \int _{-1}^{0}e^{\frac{16}{3}z(\theta_{\tau}\omega)}\biggl( \int_{-\infty}^{0} e^{-2z(\theta_{s}\omega)+2\beta s+\int_{s}^{0}2 \vert z(\theta_{r}\omega )\vert \,dr}\,ds \biggr)^{\frac{11}{3}}\,d\tau \\ &\quad \leq C(\alpha-\beta-4)^{\frac{22}{3}}e^{\frac{22}{3}\beta}e^{\frac {16}{3}\max_{-1\leq\tau\leq0}\vert z(\theta_{\tau}\omega)\vert }\biggl( \int_{-\infty}^{0} e^{-2z(\theta_{s}\omega)+2\beta s+\int_{s}^{0}2 \vert z(\theta_{r}\omega )\vert \,dr}\,ds \biggr)^{\frac{11}{3}}. \end{aligned}$$
(4.11)
In summary, from estimates of (4.9) and (4.10) we obtain that the terms on the right-hand of inequality (4.8) are bounded. Therefore, we prove that there exist a random variable \(\rho_{3}(\omega)\) and \(T_{3B}(\omega)>0\) such that for \(\mathbb{P}\)-a.e. \(\omega\in\Omega\) and all \(t>T_{3B}(\omega)\),
$$\begin{aligned} \bigl\Vert \Delta v\bigl(t+1,\theta_{-t-1}\omega,v_{0}( \theta_{-t-1}\omega )\bigr)\bigr\Vert ^{2}\leq \rho_{3}(\omega). \end{aligned}$$
This completes the proof. □

Based on the above arguments, it is easy to deduce that there exists a random absorbing set for the random dynamical system generated by system (3.1)-(3.3) in \(H_{0}^{2}(D)\).

Theorem 4.1

Assume that \(v_{0} \in B=\{B(\omega)\}_{\omega} \subset\mathscr{D}\). There exists a random absorbing set \(B^{*}(\omega)\) for the random dynamical system associated with system (3.1)-(3.3) in \(H_{0}^{2}(D)\).

Proof

We can take \(T=\max\{T_{1B},T_{2B},T_{3B}\}\) and \(\rho(\omega)=\max\{ \rho_{1}(\omega),\rho_{2}(\omega),\rho_{3}(\omega)\}\). Then, for all \(t\geq T\) and \(\mathbb{P}\)-a.e. \(\omega\in\Omega\), there exists a random absorbing set \(B^{*}(\omega)\) for the random dynamical system associated with system (3.1)-(3.3) in \(H_{0}^{2}(D)\). □

Based on the above results, by Lemma 2.8 in [10] we claim that ψ is asymptotically compact. Therefore, the existence of a random attractor for ψ follows immediately from Theorem 2.2 in [15].

Theorem 4.2

Provided that \(v_{0} \in B=\{B(\omega)\}_{\omega} \subset\mathscr{D}\), there exists a global random attractor in \(H_{0}^{2}(D)\) for the random dynamical system associated with system (3.1)-(3.3).

Declarations

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11371183) and Guangxi Provincial Natural Science Foundation (No. FAA118016).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, China University of Mining and Technology Beijing, Beijing, China
(2)
Department of Information and Computation Science, Guangxi University of Science and of Technology, Guangxi, China

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