Open Access

Existence of random attractors for the floating beam equation with strong damping and white noise

Boundary Value Problems20152015:126

https://doi.org/10.1186/s13661-015-0391-8

Received: 13 April 2015

Accepted: 14 July 2015

Published: 31 July 2015

Abstract

In this paper, we investigate the existence of a compact random attractor for the random dynamical system generated by a model for nonlinear oscillations in a floating beam equation with strong damping and white noise.

Keywords

floating beam equationrandom dynamical systemrandom attractors

MSC

35Q3535B4035B41

1 Introduction

In this paper, we consider the following stochastic floating beam equations:
$$ \left \{ \textstyle\begin{array}{l} u_{tt}+\triangle^{2}u+\triangle^{2} u_{t}+bu^{+}+f(u)=q(x)\dot{W},\quad \text{in } \Omega\times[\tau,+\infty),\tau\in\mathbb{R} , \\ \triangle u(x,t)=\nabla\triangle u(x,t)=0, \quad x\in\partial\Omega , t\geq\tau, \\ u(x,\tau)= u_{0}(x),\qquad u_{t}(x,\tau)=u_{1}(x), \end{array}\displaystyle \right . $$
(1.1)
where \(b>0\) is a measure of the cross section of the floating beam, Ω is an open bounded subset of \(\mathbb{R}^{2}\) with sufficiently smooth boundary Ω. \(u=u(x,t)\) represents the depth of the bottom of the floating beam as it floats, \(u^{+}=u\) for \(u\geq0\) and \(u^{+}=0\) for \(u<0\). \(q(x)\in H^{3}(\Omega)\) is not identically equal to zero, f is a nonlinear function satisfying certain conditions. \(\dot{W}\) is the derivative of a one-dimensional two-sided real-valued Wiener process \(W(t)\) and \(q(x)\dot{W}\) formally describes white noise.
We assume that the nonlinear function \(f\in C^{2}(\mathbb{R},\mathbb {R})\) with \(f(0)=0\), which satisfies the following assumptions.
  1. (a)
    Growth conditions:
    $$ \bigl\vert f(s)\bigr\vert \leq C_{0}\bigl(1+|s|^{p} \bigr),\quad p\geq1, \forall s\in\mathbb{R}, $$
    (1.2)
    where \(C_{0}\) is a positive constant. For example, obviously, \(f(s)=|s|^{p-1}s\) satisfies (1.2).
     
  2. (b)
    Dissipation conditions:
    $$ F(s):=\int_{0}^{s}f(r)\, dr\geq C_{1} \bigl(|s|^{p+1}-1\bigr), \quad p\geq1, \forall s\in \mathbb{R} $$
    (1.3)
    and
    $$ sf(s)\geq C_{2}\bigl(F(s)-1\bigr),\quad \forall s\in \mathbb{R}, $$
    (1.4)
    where \(C_{1}\), \(C_{2}\) are positive constants.
     

When \(f(u)\equiv0\) and \(q(x)\equiv0\), equation (1.1) is regarded as a model of naval structures, which is originally in [1] introduced by Lazer and McKenna. In the actual problems, it can be presented as ships, submarines, hovercraft, gliders etc. To the best of our knowledge, the author investigated the existence of a global attractor for the deterministic floating beam in [2], that is, the ‘noise’ is absent in (1.1). Until now, we find that no one else has studied the long-time behavior of the solutions about these problems, it is just our interest in this paper. As far as the other related problems are concerned, we refer the reader to [310] and the references therein.

It is well known that Crauel and Flandoli originally introduced the random attractor for the infinite-dimensional RDS [11, 12]. A random attractor of RDS is a measurable and compact invariant random set attracting all orbits. It is the appropriate generalization of the now classical attractor from the deterministic dynamical systems to RDS. The reason is that if such a random attractor exists, it is the smallest attracting compact set and the largest invariant set [13]. These abstract results have been successfully applied to many stochastic dissipative partial differential equations. For instance, Fan [14] proved the existence of a random attractor for a damped Sine-Gordon equation with white noise. The existence of random attractors for the wave equations has been investigated by several authors [1517]. Yang et al. [18] studied random attractors for stochastic semi-linear degenerate parabolic equations. Ma and Ma [19] investigated attractors for stochastic strongly damped plate equations with additive noise. In this article, we study the existence of random attractors for the floating beam equation with white noise by means of the methods established in [1113].

The outline of this paper is as follows. Background material on RDS and random attractors is iterated in Section 2. We present the existence and uniqueness of the solution corresponding to system (1.1) which determines RDS in Section 3. Finally, the existence of random attractors for RDS is shown in the last section.

2 Random dynamical system

In this section, we recall some basic concepts related to RDS and a random attractor for RDS in [1113], which are important for getting our main results.

Let \((X, \|\cdot\|_{X})\) be a separable Hilbert space with Borel σ-algebra \(\mathcal{B}(X)\), and let \((\Omega,\mathcal{F},P)\) be a probability space. \({\theta_{t}:\Omega\rightarrow\Omega, t\in\mathbb {R}}\) is a family of measure preserving transformations such that \((t,\omega)\mapsto\theta_{t}\omega\) is measurable, \(\theta_{0}=\mathrm{id}\) and \(\theta_{t+s}=\theta_{t}\theta_{s}\) for all \(t,s\in\mathbb{R}\). The flow \(\theta_{t}\) together with the probability space \((\Omega,\mathcal {F},P,(\theta_{t})_{t\in\mathbb{R}})\) is called a metric dynamical system.

Definition 2.1

Let \((\Omega,\mathcal{F},P,(\theta _{t})_{t\in\mathbb{R}})\) be a metric dynamical system. Suppose that the mapping \(\phi:\mathbb{R}^{+}\times\Omega\times X\rightarrow X\) is \((\mathcal{B}(\mathbb{R}^{+})\times\mathcal{F}\times\mathcal {B}(X),\mathcal{B}(X))\)-measurable and satisfies the following properties:
  1. (i)

    \(\phi(0,\omega)x=x\), \(x\in X\) and \(\omega\in\Omega\);

     
  2. (ii)

    \(\phi(t+s,\omega)=\phi(t,\theta_{s}\omega)\circ\phi(s,\omega )\) for all \(t,s\in\mathbb{R^{+}}\), \(x\in X\) and \(\omega\in\Omega\).

     
Then ϕ is called a random dynamical system (RDS). Moreover, ϕ is called a continuous RDS if ϕ is continuous with respect to x for \(t\geq0\) and \(\omega\in\Omega\).

Definition 2.2

A set-valued map \(D:\Omega\rightarrow 2^{X}\) is said to be a closed (compact) random set if \(D(\omega)\) is closed (compact) for P-a.s. \(\omega\in\Omega\), and \(\omega\mapsto \mathrm{d}(x,D(\omega))\) is P-a.s. measurable for all \(x\in X\).

Definition 2.3

If K and B are random sets such that for P-a.s. ω there exists a time \(t_{B}(\omega)\) such that for all \(t\geq t_{B}(\omega)\),
$$\phi(t,\theta_{-t}\omega)B(\theta_{-t}\omega)\subset K( \omega), $$
then K is said to absorb B, and \(t_{B}(\omega)\) is called the absorption time.

Definition 2.4

A random set \(\mathcal{A}=\{A(\omega)\} _{\omega\in\Omega}\subset X\) is called a random attractor associated to the RDS ϕ if P-a.s.:
  1. (i)

    \(\mathcal{A}\) is a random compact set, i.e., \(A(\omega )\) is compact for P-a.s. \(\omega\in\Omega\), and the map \(\omega \mapsto\mathrm{d}(x,A(\omega))\) is measurable for every \(x\in X\);

     
  2. (ii)

    \(\mathcal{A}\) is ϕ-invariant, i.e., \(\phi (t,\omega)A(\omega)=A(\theta_{t}\omega)\) for all \(t\geq0\) and P-a.s. \(\omega\in\Omega\);

     
  3. (iii)
    \(\mathcal{A}\) attracts every set B in X, i.e., for all bounded (and non-random) \(B\subset X\),
    $$\lim_{t\rightarrow\infty}\mathrm{d}\bigl(\phi(t,\theta _{-t} \omega)B(\theta_{-t}\omega), A(\omega)\bigr)=0, $$
    where \(\mathrm{d}(\cdot,\cdot)\) denotes the Hausdorff semi-distance:
    $$\mathrm{d}(A,B)=\sup_{x\in A} \inf_{y\in B} \mathrm{d}(x,y), \quad A, B\in X. $$
     

Note that \(\phi(t,\theta_{-t}\omega)x\) can be interpreted as the position of the trajectory which was in x at time −t. Thus, the attraction property holds from \(t=-\infty\).

Theorem 2.5

(Existence of a random attractor [11])

Let ϕ be a continuous random dynamical system on X over \((\Omega ,\mathcal{F},P,(\theta_{t})_{t\in\mathbb{R}})\). Suppose that there exists a random compact set \(K(\omega)\) absorbing every bounded non-random set \(B\subset X\). Then the set
$$\mathcal{A}=\bigl\{ A(\omega)\bigr\} _{\omega\in\Omega}=\overline{\bigcup _{B\subset X}\Lambda_{B}(\omega)} $$
is a global random attractor for ϕ, where the union is taken over all bounded \(B\subset X\), and \(\Lambda_{B}(\omega)\) is the ω-limits set of B given by
$$\Lambda_{B}(\omega)=\bigcap_{s\geq0} \overline{\bigcup_{t\geq s}\phi(t,\theta_{-t} \omega)B(\theta_{-t}\omega)}. $$

3 Existence and uniqueness of solutions

With the usual notation, we denote
$$\begin{aligned}& H=L^{2}(\Omega), \qquad V=H^{2}(\Omega), \\& D(A)=H^{2}(\Omega)\cap H_{0}^{1}(\Omega), \\& D\bigl(A^{2}\bigr)= \bigl\{ u\in H^{4}(\Omega) : A^{2}u\in L^{2}(\Omega) \bigr\} , \end{aligned}$$
where \(A=-\triangle\), \(A^{2}=\triangle^{2}\). We denote H, V with the following inner products and norms, respectively:
$$\begin{aligned}& (u,v)=\int_{\Omega}uv\, dx, \qquad \|u\|^{2}=(u,u), \quad \forall u, v\in H, \\& \bigl((u,v)\bigr)=\int_{\Omega}\triangle u \triangle v \, dx, \qquad \|u\| ^{2}_{2}=\bigl((u,u)\bigr), \quad \forall u, v\in V. \end{aligned}$$
And we introduce the space \(E=D(A)\times H\) which is used throughout the paper and endow the space E with the following usual scalar product and norm:
$$(y_{1},y_{2})_{E}=\bigl((u_{1},u_{2}) \bigr)+(v_{1},v_{2}), \qquad \|y\|_{E}^{2}=(y,y)_{E} $$
for all \(y_{i}=(u_{i},v_{i})^{T}, y=(u,v)^{T}\in E\), here T denotes the transposition. Moreover, the norm of \(L^{p}(\Omega)\) is written as \(\| \cdot\|_{p}\).
Let \(\lambda>0\) be the eigenvalue of \(A^{2}v=\lambda v\), \(\triangle v(x,t)=\nabla\triangle v(x,t)=0\), \(x\in\partial\Omega\), by the Poincaré inequality, we have
$$\|u\|_{2}^{2}\geq\lambda\|u\|^{2}, \quad \forall u \in D(A). $$
It is convenient to reduce (1.1) to an evolution equation of the first order in time
$$ \left \{ \textstyle\begin{array}{l} u_{t}=v, \\ v_{t}=-A^{2}u-A^{2}v-bu^{+}-f(u)+q(x)\dot{W}, \\ u(x,\tau)=u_{0}(x), \qquad u_{t}(x,\tau)=u_{1}(x),\quad x\in\Omega, \end{array}\displaystyle \right . $$
(3.1)
whose equivalent Itó equation is
$$ \left \{ \textstyle\begin{array}{l} du=v\, dt, \\ dv=-A^{2}u\, dt-A^{2}v\, dt-bu^{+}\, dt-f(u)\, dt+q(x)\, dW, \\ u(x,\tau)=u_{0}(x), \qquad u_{t}(x,\tau)=u_{1}(x), \quad x\in\Omega, \end{array}\displaystyle \right . $$
(3.2)
where \(W(t)\) is a one-dimensional two-sided real-valued Wiener process on \((\Omega,\mathcal{F},P, (\theta_{t})_{t\in\mathbb{R}})\). We can assume without loss of generality that
$$\Omega= \bigl\{ \omega(t)=W(t)\in C(\mathbb{R},\mathbb{R}) : \omega (0)=0 \bigr\} , $$
that P is a Wiener measure. We can define a family of measure preserving and ergodic transformations (flow) \(\{\theta_{t}\}_{t\in \mathbb{R}}\) by
$$\theta_{t}\omega(\cdot)=\omega(\cdot+t)-\omega(t), \quad t\in\mathbb {R}, \omega\in\Omega. $$
Let \(z=v-q(x)W\), then \(v=z+q(x)W\). We consider the random partial differential equation equivalent to (3.2)
$$ \left \{ \textstyle\begin{array}{l} \frac{du}{dt}=z+q(x)W, \\ \frac{dz}{dt}=-A^{2}u-A^{2}z-bu^{+}-f(u)-A^{2}q(x)W, \\ u(x,\tau)=u_{0}(x), \qquad z(\tau,\omega)=z(x,\tau,\omega )=u_{1}(x)-q(x)W(\tau), \quad x\in\Omega. \end{array}\displaystyle \right . $$
(3.3)
It apparently contrasts to the stochastic differential equation (3.2), no stochastic differential appears here. Let
$$\varphi= \left (\textstyle\begin{array}{@{}c@{}} u \\ z \end{array}\displaystyle \right ),\qquad L= \left (\textstyle\begin{array}{@{}c@{\quad}c@{}} 0 & I \\ -A^{2}& -A^{2} \end{array}\displaystyle \right ) $$
and
$$F(\varphi,\omega)= \left (\textstyle\begin{array}{@{}c@{}} q(x)W \\ -bu^{+}-f(u)-A^{2}q(x)W \end{array}\displaystyle \right ), $$
then (3.3) can be written as
$$ \dot{\varphi}=L\varphi+F(\varphi,\omega), \qquad \varphi(\tau , \omega)=\bigl(u_{0},z(\tau,\omega)\bigr)^{T}. $$
(3.4)
We know from [20] that L is the infinitesimal generators of \(C_{0}\)-semigroup \(e^{Lt}\) on E. It is not difficult to check that the function \(F(\cdot,\omega ): E\mapsto E\) is locally Lipschitz continuous with respect to φ and bounded for every \(\omega\in\Omega\). By the classical semigroup theory of existence and uniqueness of solutions of evolution differential equations [20], the random partial differential equation (3.4) has a unique solution in the mild sense
$$\varphi(t,\omega)=e^{L(t-\tau)}\varphi(\tau,\omega)+\int_{\tau}^{t}e^{L(t-s)}F\bigl(\varphi(s),\omega\bigr)\, ds $$
for any \(\varphi(\tau,\omega)\in E\). We can prove that for P-a.s. every \(\omega\in\Omega\) the following statements hold for all \(T>0\):
  1. (i)

    If \(\varphi(\tau,\omega)\in E\), then \(\varphi(t,\omega)\in C([\tau,\tau+T); D(A))\times C([\tau,\tau +T); H)\).

     
  2. (ii)

    \(\varphi(t,\varphi(\tau,\omega))\) is continuous in t and \(\varphi(\tau,\omega)\).

     
  3. (iii)

    The solution mapping of (3.4) satisfies the properties of RDS.

     
Equation (3.4) has a unique solution for every \(\omega\in\Omega\). Hence the solution mapping
$$ \bar{S}(t,\omega): \varphi(\tau,\omega)\mapsto\varphi(t,\omega ) $$
(3.5)
generates a random dynamical system. So the transformation
$$ S(t,\omega):\varphi(\tau,\omega)+\bigl(0, q(x)W(\tau)\bigr)^{T} \mapsto\varphi(t,\omega)+\bigl(0, q(x)W(t)\bigr)^{T} $$
(3.6)
also determines a random dynamical system corresponding to equation (3.1).

4 Existence of a random attractor

In this section, we prove the existence of a random attractor for RDS (3.6) in the space E.

Let \(\bar{z}=z+\varepsilon u\), \(\psi=(u,\bar{z})^{T}\), where
$$ \varepsilon=\frac{\lambda^{2}}{4\lambda^{2}+3\lambda+4}. $$
(4.1)
Hence equation (3.3) can be written as
$$ \dot{\psi}+Q\psi=\bar{F}(\psi,\omega), \qquad \psi(\tau,\omega )= \bigl(u_{0},z(\tau,\omega)+\varepsilon u_{0} \bigr)^{T}, \quad t\geqslant\tau , $$
(4.2)
where
$$\begin{aligned}& Q= \left (\textstyle\begin{array}{@{}c@{\quad}c@{}} \varepsilon I & -I \\ (1-\varepsilon)A^{2}+\varepsilon^{2}I & A^{2}-\varepsilon I \end{array}\displaystyle \right ), \\& \bar{F}(\psi,\omega)= \left (\textstyle\begin{array}{@{}c@{}} q(x)W \\ -bu^{+}-f(u)-A^{2}q(x)W+\varepsilon q(x)W \end{array}\displaystyle \right ). \end{aligned}$$
The mapping
$$\bar{S}_{\varepsilon}(t,\omega):\bigl(u_{0}, z (\tau,\omega)+ \varepsilon u_{0}\bigr)^{T}\mapsto \bigl(u(t), z(t)+ \varepsilon u(t)\bigr)^{T}, \qquad E\rightarrow E,\quad t\geq\tau $$
is defined by (4.2).
To show the conjugation of the solution of the stochastic partial differential equation (1.1) and the random partial differential equation (4.2), we introduce the homeomorphism
$$R_{\varepsilon}: (u, z)^{T} \mapsto(u, z+\varepsilon u)^{T} $$
with the inverse homeomorphism \(R_{-\varepsilon}\). Then the transformation
$$ \bar{S}_{\varepsilon}(t,\omega)=R_{\varepsilon}S(t,\omega )R_{-\varepsilon} $$
(4.3)
also determines RDS corresponding to equation (1.1). Therefore, for RDS (3.6) we only need consider the equivalent random dynamical system \(S_{\varepsilon}(t,\omega)=R_{\varepsilon}S(t,\omega)R_{-\varepsilon }\), where \(S_{\varepsilon}(t,\omega)\) is decided by
$$ \eta_{t}+Q\eta=G(\eta,\omega), \qquad \eta(\tau,\omega )=(u_{0},u_{1}+\varepsilon u_{0})^{T}, \quad t\geq\tau, $$
(4.4)
where \(\eta(t)=(u(t), u_{t}(t)+\varepsilon u(t))^{T}\) and
$$G(\eta,\omega)= \left (\textstyle\begin{array}{@{}c@{}} 0\\ -bu^{+}-f(u)+q(x)\dot{W} \end{array}\displaystyle \right ). $$

Next, we prove a positivity property of the operator Q in E that plays a vital role throughout the paper.

Lemma 4.1

For any \(\varphi=(u, z)^{T}\in E\), there holds
$$(Q\varphi, \varphi)_{E}\geq\frac{\varepsilon}{2}\|\varphi\| _{E}^{2}+\frac{\varepsilon}{4}\|u\|_{2}^{2} +\frac{\lambda}{2}\|z\|^{2}. $$

Proof

Since \(Q\varphi=(\varepsilon u-z, (1-\varepsilon)A^{2}u+\varepsilon ^{2}u+A^{2}z-\varepsilon z)^{T}\), using the Poincaré inequality and the Young inequality, we get
$$\begin{aligned} (Q\varphi, \varphi)_{E} =&\varepsilon\|u\|_{2}^{2}- \varepsilon(Au, Az)+\varepsilon^{2}(u,z)+\|Az\| ^{2}- \varepsilon\|z\|^{2} \\ \geq&\varepsilon\|u\|_{2}^{2}-\frac{\varepsilon}{8}\|u\| _{2}^{2}-2\varepsilon\|Az\|^{2}- \frac{\varepsilon}{8} \|u\|_{2}^{2} -\frac{2\varepsilon^{3}}{\lambda}\|z\|^{2}+\|Az \|^{2}-\varepsilon\|z\|^{2} \\ \geq&\frac{\varepsilon}{2}\|\varphi\|_{E}^{2}+ \frac{\varepsilon}{4}\| u\|_{2}^{2}+(1-2\varepsilon)\lambda\|z \|^{2} - \biggl(\frac{2\varepsilon}{\lambda}+\frac{3\varepsilon}{2} \biggr)\|z \|^{2}, \\ =&\frac{\varepsilon}{2}\|\varphi\|_{E}^{2}+\frac{\varepsilon}{4} \|u\| _{2}^{2}+\frac{\lambda}{2}\|z\|^{2}, \end{aligned}$$
noting that we used the fact \(\varepsilon=\frac{\lambda^{2}}{4\lambda ^{2}+3\lambda+4}\) in the last inequality. □

Lemma 4.2

Let (1.2)-(1.4) hold. There exist a random variable \(r_{1}(\omega)>0\) and a bounded ball \(B_{0}\) of E centered at 0 with random radius \(r_{0}(\omega)>0\) such that for any bounded non-random set B of E, there exists a deterministic \(T(B)\leq-1\) such that the solution \(\psi(t,\omega;\psi(\tau ,\omega))=(u(t,\omega),\bar{z}(t,\omega))^{T}\) of (4.2) with initial value \((u_{0}, u_{1}+\varepsilon u_{0})^{T}\in B\) satisfies, for P-a.s. \(\omega\in\Omega\),
$$\bigl\Vert \psi\bigl(-1,\omega;\psi(\tau,\omega)\bigr)\bigr\Vert _{E}\leq r_{0}(\omega),\quad \tau\leq T(B), $$
and for all \(\tau\leq t\leq0\),
$$\begin{aligned}& \bigl\Vert \psi\bigl(t,\omega;\psi(\tau,\omega)\bigr)\bigr\Vert _{E}^{2} \\& \quad \leq 2e^{-\varepsilon _{1}(t-\tau)}\biggl(\|u_{0} \|_{2}^{2}+\|u_{1}+\varepsilon u_{0} \|^{2}+\|q\|^{2}\bigl\vert W(\tau )\bigr\vert ^{2}+\int_{\Omega}F(u_{0})\, dx \biggr)+r_{1}^{2}(\omega), \end{aligned}$$
(4.5)
where \(\bar{z}(t,\omega)=u_{t}(t)+\varepsilon u(t)-q(x)W(t)\).
Besides, it is easy to deduce a similar absorption result for
$$\eta(-1)=(\eta_{1},\eta_{2})=\bigl(u(-1),u_{t}(-1)+ \varepsilon u(-1)\bigr)^{T} $$
instead of \(\psi(-1)\).

Proof

Taking the inner product in E of (4.2) with \(\psi =(u,\bar{z})^{T}\), in which \(\bar{z}=u_{t}+\varepsilon u-q(x)W\), we find that
$$ \frac{1}{2}\frac{d}{dt}\|\psi\|_{E}^{2}+(Q \psi,\psi)_{E}=\bigl(\bar {F}(\psi,\omega),\psi\bigr)_{E}, \quad \forall t\geq\tau, $$
(4.6)
where
$$ \bigl(\bar{F}(\psi,\omega),\psi\bigr)_{E}=\bigl(\bigl(u, q(x)W\bigr) \bigr)-b\bigl(u^{+}, \bar {z}\bigr)-\bigl(f(u), \bar{z}\bigr)-\bigl(A^{2}q(x)W, \bar{z}\bigr)+\varepsilon\bigl(q(x)W, \bar {z}\bigr). $$
(4.7)
We deal with the terms in (4.7) one by one as follows:
$$\begin{aligned}& \bigl(\bigl(u, q(x)W\bigr)\bigr)\leq\frac{\varepsilon}{4}\|u\|_{2}^{2}+ \frac{\|q\| _{2}^{2}}{\varepsilon}\bigl\vert W(t)\bigr\vert ^{2}; \end{aligned}$$
(4.8)
$$\begin{aligned}& -b\bigl(u^{+},\bar{z}\bigr)=-b\bigl(u^{+},u_{t}+\varepsilon u-q(x)W\bigr) \\& \hphantom{-b\bigl(u^{+},\bar{z}\bigr)}=-\frac{1}{2}\frac{d}{dt}b\bigl\Vert u^{+}\bigr\Vert ^{2}-\varepsilon b\bigl\Vert u^{+}\bigr\Vert ^{2}+b\bigl(u^{+},q(x)W\bigr) \\& \hphantom{-b\bigl(u^{+},\bar{z}\bigr)}\leq-\frac{1}{2}\frac{d}{dt}b\bigl\Vert u^{+}\bigr\Vert ^{2}-\frac{\varepsilon b}{2}\bigl\| u^{+} \bigr\| ^{2}+\frac{b\|q\|^{2}}{2\varepsilon}\bigl\vert W(t)\bigr\vert ^{2}; \end{aligned}$$
(4.9)
$$\begin{aligned}& \bigl\vert -\bigl(A^{2}q(x)W, \bar{z}\bigr)\bigr\vert \leq \frac{\|q\|_{4}^{2}}{\lambda}\bigl\vert W(t)\bigr\vert ^{2}+\frac {\lambda}{4} \|\bar{z}\|^{2}; \end{aligned}$$
(4.10)
$$\begin{aligned}& \varepsilon\bigl(q(x)W, \bar{z}\bigr)\leq\frac{\varepsilon^{2}\|q\|^{2}}{\lambda }\bigl\vert W(t) \bigr\vert ^{2}+ \frac{\lambda}{4}\|\bar{z}\|^{2}. \end{aligned}$$
(4.11)
Using (1.2)-(1.3) and the Hölder inequality, we conclude that
$$\begin{aligned}& \bigl(f(u),q(x)W(t)\bigr) \\& \quad \leq C_{0}\int_{\Omega}\bigl(1+|u|^{p} \bigr)q(x)W(t)\, dx \\& \quad \leq C_{0}\|q\|\bigl\vert W(t)\bigr\vert +C_{0} \biggl(\int_{\Omega}|u|^{p+1}\, dx \biggr)^{\frac {p}{p+1}} \|q\|_{p+1}\bigl\vert W(t)\bigr\vert \\& \quad \leq C_{0}\|q\|\bigl\vert W(t)\bigr\vert +C_{0}C_{1}^{-\frac{p}{p+1}} \biggl(\int_{\Omega}\bigl(F(u)+C_{1}\bigr)\, dx \biggr)^{\frac{p}{p+1}}\|q\|_{p+1} \bigl\vert W(t)\bigr\vert \\& \quad \leq C_{0}\|q\|\bigl\vert W(t)\bigr\vert + \frac{\varepsilon C_{0} C_{1}^{-1}}{2}\int_{\Omega}F(u)\, dx+\frac{C_{0}}{2\varepsilon}\|q \|_{p+1}^{p+1}\bigl\vert W(t)\bigr\vert ^{p+1}+ \frac {\varepsilon C_{0}|\Omega|}{2}. \end{aligned}$$
(4.12)
Inequality (1.4) together with (4.12) yields
$$\begin{aligned}& -\bigl(f(u),\bar{z}\bigr) \\& \quad = -\bigl(f(u),u_{t}+\varepsilon u-qW(t)\bigr) \\& \quad \leq -\frac{d}{dt}\int_{\Omega}F(u)\, dx- \varepsilon C_{2}\int_{\Omega}F(u)\, dx+\varepsilon C_{2}|\Omega|+\bigl(f(u),qW(t)\bigr) \\& \quad \leq -\frac{d}{dt}\int_{\Omega}F(u)\, dx- \frac{\varepsilon(2C_{2}-C_{0} C_{1}^{-1})}{2}\int_{\Omega}F(u)\, dx+ C_{0}\|q\| \bigl\vert W(t)\bigr\vert \\& \qquad{} +\frac{C_{0}}{2\varepsilon}\|q\| _{p+1}^{p+1}\bigl\vert W(t)\bigr\vert ^{p+1}+\frac{\varepsilon(C_{0}+2C_{2})}{2}|\Omega |. \end{aligned}$$
(4.13)
Therefore, collecting with (4.6)-(4.13) and Lemma 4.1, we get that
$$\begin{aligned}& \frac{d}{dt}\biggl(\|\psi\|_{E}^{2}+b\bigl\Vert u^{+}\bigr\Vert ^{2}+2\int_{\Omega}F(u)\, dx+2C_{1}|\Omega|\biggr) \\& \qquad {}+\varepsilon\|\psi\|_{E}^{2}+\varepsilon b\bigl\Vert u^{+}\bigr\Vert ^{2}+\varepsilon \bigl(2C_{2}-C_{0} C_{1}^{-1}\bigr)\int _{\Omega}F(u)\, dx+2\varepsilon C_{1}|\Omega| \\& \quad \leq M\bigl(1+\bigl\vert W(t)\bigr\vert +\bigl\vert W(t)\bigr\vert ^{2}+\bigl\vert W(t)\bigr\vert ^{p+1}\bigr), \end{aligned}$$
where \(M=\max \{\varepsilon(C_{0}+2C_{2}+2C_{1})|\Omega|, 2C_{0}\|q\|, \frac{2\varepsilon\|q\|_{4}^{2}+2\lambda\|q\|_{2}^{2}+(2\varepsilon ^{3}+\lambda b) \|q\|^{2}}{\varepsilon\lambda}, \frac{C_{0}}{\varepsilon }\|q\|_{p+1}^{p+1} \}\). Using (1.3), we have the fact \(2\int_{\Omega}F(u)\, dx+2C_{1}|\Omega|\geq0\). Choosing \(\varepsilon_{1}=\min \{\varepsilon,\frac {\varepsilon(2C_{2}-C_{0} C_{1}^{-1})}{2} \}\), and \(C_{2}>\frac {C_{0}C_{1}^{-1}}{2}\), by the Gronwall lemma, we conclude that
$$\begin{aligned}& \bigl\Vert \psi\bigl(t,\omega;\psi(\tau,\omega)\bigr)\bigr\Vert _{E}^{2} \\& \quad \leq e^{-\varepsilon_{1}(t-\tau)}\biggl(\bigl\Vert \psi(\tau, \omega)\bigr\Vert ^{2}_{E}+b\bigl\Vert u_{0}^{+}\bigr\Vert ^{2}+2\int_{\Omega}F(u_{0})\, dx+2C_{1}|\Omega|\biggr) \\& \qquad {} +M\int_{\tau}^{t}e^{-\varepsilon _{1}(t-s)}\bigl(1+ \bigl\vert W(s)\bigr\vert +\bigl\vert W(s)\bigr\vert ^{2}+\bigl\vert W(s)\bigr\vert ^{p+1}\bigr)\, ds \\& \quad \leq 2e^{-\varepsilon_{1}(t-\tau)}\biggl(\|u_{0}\|_{2}^{2}+ \|u_{1}+\varepsilon u_{0}\| ^{2}+\|q \|^{2}\bigl\vert W(\tau)\bigr\vert ^{2}+b\bigl\Vert u_{0}^{+}\bigr\Vert ^{2}+\int _{\Omega}F(u_{0})\, dx +C_{1}|\Omega|\biggr) \\& \qquad {}+M\int_{\tau}^{t}e^{-\varepsilon _{1}(t-s)}\bigl(1+ \bigl\vert W(s)\bigr\vert +\bigl\vert W(s)\bigr\vert ^{2}+\bigl\vert W(s)\bigr\vert ^{p+1}\bigr)\, ds. \end{aligned}$$
(4.14)
Take
$$\begin{aligned} r_{0}^{2}(\omega) =&2\Bigl(1+\sup_{\tau\leq-1}e^{\varepsilon_{1} \tau } \|q\|^{2}\bigl\vert W(\tau)\bigr\vert ^{2}\Bigr)+ \frac{M}{\varepsilon_{1}} \\ &{}+M\int_{-\infty}^{-1} e^{-\varepsilon_{1}(-1-s)} \bigl(\bigl\vert W(s)\bigr\vert +\bigl\vert W(s)\bigr\vert ^{2}+ \bigl\vert W(s)\bigr\vert ^{p+1}\bigr)\, ds \end{aligned}$$
and
$$r_{1}^{2}(\omega)=\frac{M}{\varepsilon_{1}}+M\int _{-\infty }^{0}e^{\varepsilon_{1}s}\bigl(\bigl\vert W(s) \bigr\vert +\bigl\vert W(s)\bigr\vert ^{2}+\bigl\vert W(s)\bigr\vert ^{p+1}\bigr)\, ds. $$
Since \(\lim_{t\rightarrow\infty}\frac{W(t)}{t}=0\), \(r_{0}^{2}(\omega)\) and \(r_{1}^{2}(\omega)\) are finite P-a.s., given a bounded set B of E, choose \(T(B)\leq-1\) such that
$$ e^{-\varepsilon_{1}(-1-\tau)} \biggl(\|u_{0}\|_{2}^{2}+ \|u_{1}+\varepsilon u_{0}\| ^{2}+b\bigl\Vert u_{0}^{+}\bigr\Vert ^{2}+\int _{\Omega}F(u_{0})\, dx+C_{1}|\Omega| \biggr) \leq1 $$
(4.15)
for all \((u_{0},u_{1}+\varepsilon u_{0})^{T}\in B\), and
$$ e^{\varepsilon_{1}\tau} \biggl(\|u_{0}\|_{2}^{2}+ \|u_{1}+\varepsilon u_{0}\|^{2}+b\bigl\Vert u_{0}^{+}\bigr\Vert ^{2}+\int _{\Omega}F(u_{0})\, dx+C_{1}|\Omega| \biggr) \leq1 $$
(4.16)
for all \((u_{0},u_{1}+\varepsilon u_{0})^{T}\in B\), and for all \(\tau \leq T(B)\).

This completes the proof of Lemma 4.2. □

Let \(u(t)\) be a solution of problem (1.1) with initial value \((u_{0},u_{1}+\varepsilon u_{0})^{T}\in B\). We make the decomposition \(u(t)=y_{1}(t)+y_{2}(t)\), where \(y_{1}(t)\) and \(y_{2}(t)\) satisfy
$$ \left \{ \textstyle\begin{array}{l} y_{1tt}+\triangle^{2}y_{1}+\triangle^{2}y_{1t}=0, \quad \text{in } \Omega \times[\tau, +\infty), \tau\in\mathbb{R}, \\ \triangle y_{1}(x,t)=\nabla\triangle y_{1}(x,t)=0, \quad x\in\partial \Omega, t\geq\tau, \\ y_{1}(x,\tau)=u_{0}(x), \qquad y_{1t}(x,\tau)=u_{1}(x),\quad x\in\Omega \end{array}\displaystyle \right . $$
(4.17)
and
$$ \left \{ \textstyle\begin{array}{l} y_{2tt}+\triangle^{2}y_{2}+\triangle^{2} y_{2t}+bu^{+}+f(u)=q(x)\dot {W}, \quad \text{in } \Omega\times[\tau,+\infty),\tau\in\mathbb {R} , \\ \triangle y_{2}(x,t)=\nabla\triangle y_{2}(x,t)=0,\quad x\in\partial \Omega, t\geq\tau, \\ y_{2}(x,\tau)=0, \qquad y_{2t}(x,\tau)=0, \quad x\in\Omega. \end{array}\displaystyle \right . $$
(4.18)

Lemma 4.3

Let B be a bounded non-random subset of E. For any \((u_{0},u_{1}+\varepsilon u_{0})^{T}\in B\),
$$ \bigl\Vert Y_{1}(0)\bigr\Vert _{E}^{2}=\bigl\Vert y_{1}(0)\bigr\Vert _{2}^{2}+\bigl\Vert y_{1t}(0)+\varepsilon y_{1}(0)\bigr\Vert ^{2} \leq\bigl(\Vert u_{0}\Vert _{2}^{2}+\Vert u_{1}+\varepsilon u_{0}\Vert ^{2}\bigr) \frac {e^{\varepsilon\tau}}{1-\varepsilon}, $$
(4.19)
where \(Y_{1}=(y_{1},y_{1t}+\varepsilon y_{1})^{T}\) satisfies (4.17).

Proof

Taking the scalar product in \(L^{2}(\Omega)\) of (4.17) with \(v=y_{1t}+\varepsilon y_{1}\), we conclude that
$$ \frac{1}{2}\frac{d}{dt}\bigl((1-\varepsilon)\|y_{1} \|_{2}^{2}+\|v\| ^{2}\bigr)+\varepsilon(1- \varepsilon)\|y_{1}\|_{2}^{2}+\|v \|_{2}^{2}-\varepsilon\|v\| ^{2}+ \varepsilon^{2}(y_{1}, v)=0. $$
(4.20)
Due to (4.1), using the Hölder inequality and the Young inequality, we get that
$$\begin{aligned}& \varepsilon(1-\varepsilon)\|y_{1}\|_{2}^{2}+\|v \|_{2}^{2}-\varepsilon\|v\| ^{2}+ \varepsilon^{2}(y_{1}, v) \\& \quad \geq \varepsilon(1-\varepsilon)\|y_{1}\|_{2}^{2}+ \|v\|_{2}^{2}-\varepsilon\| v\|^{2}- \frac{\varepsilon(1-\varepsilon)}{2}\|y_{1}\|_{2}^{2} - \frac{\varepsilon^{3}}{2(1-\varepsilon)\lambda}\|v\|^{2} \\& \quad \geq \frac{\varepsilon(1-\varepsilon)}{2}\|y_{1}\|_{2}^{2}+ \biggl(\lambda -\frac{\varepsilon}{2\lambda}-\varepsilon\biggr)\|v\|^{2} \\& \quad \geq\frac{\varepsilon(1-\varepsilon)}{2}\|y_{1}\| _{2}^{2}+ \frac{\varepsilon}{2}\|v\|^{2}. \end{aligned}$$
(4.21)
Associating (4.20) with (4.21), we have that
$$\frac{d}{dt}\bigl((1-\varepsilon)\|y_{1}\|_{2}^{2}+ \|v\|^{2}\bigr)+\varepsilon \bigl((1-\varepsilon)\|y_{1} \|_{2}^{2}+\|v\|^{2}\bigr)\leq0. $$
The Gronwall lemma leads to (4.19). □

Lemma 4.4

Assume that (1.2) holds, there exists a random radius \(r_{2}(\omega)\) such that for P-a.s. \(\omega\in\Omega\),
$$ \bigl\Vert A^{\frac{1}{2}}Y_{2}\bigl(0,\omega;Y_{2}(\tau, \omega)\bigr)\bigr\Vert _{E}^{2}\leq r_{2}^{2}( \omega), $$
(4.22)
where \(Y_{2}=(y_{2},y_{2t}+\varepsilon y_{2}-q(x)W)^{T}\) satisfies (4.18).

Proof

Provided that \(Y_{2}=(y_{2}, y_{2t}+\varepsilon y_{2}-q(x)W)^{T}\), then equation (4.18) can be reduced to
$$ Y_{2t}+QY_{2}=H(Y_{2},\omega), \qquad Y_{2}(\tau)=\bigl(0, -q(x)W(\tau )\bigr)^{T}, \quad t \geqslant\tau, $$
(4.23)
where
$$H(Y_{2},\omega)= \left (\textstyle\begin{array}{@{}c@{}} q(x)W(t) \\ -bu^{+}-f(u)-A^{2}q(x)W(t)+\varepsilon q(x)W(t) \end{array}\displaystyle \right ). $$
Taking the inner product in E of (4.23) with \(AY_{2}\), we have that
$$ \frac{1}{2}\frac{d}{dt}\bigl\Vert A^{\frac{1}{2}}Y_{2} \bigr\Vert _{E}^{2}+(QY_{2},AY_{2})_{E}= \bigl(H(Y_{2},\omega),AY_{2}\bigr)_{E}, $$
(4.24)
where
$$\begin{aligned}& \bigl(H(Y_{2},\omega),AY_{2}\bigr)_{E} \\& \quad =\bigl( \bigl(Ay_{2}, q(x)W\bigr)\bigr) \\& \qquad {}-\bigl(bu^{+}+f(u)+A^{2}q(x)W(t)- \varepsilon q(x)W(t), A\bigl(y_{2t}+\varepsilon y_{2}-q(x)W \bigr)\bigr). \end{aligned}$$
(4.25)
According to Lemma 4.1, we have that
$$ (QY_{2},AY_{2})_{E}\geq\frac{\varepsilon}{2}\bigl\Vert A^{\frac{1}{2}}Y_{2}\bigr\Vert _{E}^{2}+ \frac{\varepsilon}{4}\bigl\Vert A^{\frac{1}{2}}y_{2}\bigr\Vert _{2}^{2} +\frac{\lambda}{2}\bigl\Vert A^{\frac{1}{2}} \bigl(y_{2t}+\varepsilon y_{2}-q(x)W\bigr)\bigr\Vert ^{2}. $$
(4.26)
Thanks to the Young inequality, we obtain that
$$\begin{aligned}& \bigl(\bigl(Ay_{2}, q(x)W\bigr)\bigr)\leq\frac{\varepsilon}{4}\bigl\Vert A^{\frac {1}{2}}y_{2}\bigr\Vert _{2}^{2}+ \frac{1}{\varepsilon}\bigl\Vert A^{\frac{1}{2}}q\bigr\Vert _{2}^{2}\bigl\vert W(t)\bigr\vert ^{2}; \end{aligned}$$
(4.27)
$$\begin{aligned}& \bigl\vert -\bigl(bu^{+}, A\bigl(y_{2t}+\varepsilon y_{2}-q(x)W\bigr)\bigr)\bigr\vert \\& \quad \leq\frac {2b^{2}}{\lambda}\bigl\Vert A^{\frac{1}{2}}u^{+}\bigr\Vert ^{2}+\frac{\lambda}{8}\bigl\Vert A^{\frac {1}{2}}\bigl(y_{2t}+\varepsilon y_{2}-q(x)W \bigr)\bigr\Vert ^{2}; \end{aligned}$$
(4.28)
$$\begin{aligned}& \bigl\vert -\bigl(A^{2}q(x)W(t), A\bigl(y_{2t}+\varepsilon y_{2}-q(x)W\bigr)\bigr)\bigr\vert \\& \quad \leq\frac {2}{\lambda}\bigl\Vert A^{\frac{5}{2}}q\bigr\Vert ^{2}\bigl\vert W(t)\bigr\vert ^{2}+\frac{\lambda}{8}\bigl\Vert A^{\frac{1}{2}} \bigl(y_{2t}+\varepsilon y_{2}-q(x)W\bigr)\bigr\Vert ^{2}; \end{aligned}$$
(4.29)
$$\begin{aligned}& \bigl\vert \bigl(\varepsilon q(x)W(t), A\bigl(y_{2t}+\varepsilon y_{2}-q(x)W\bigr)\bigr)\bigr\vert \\& \quad \leq\frac {2\varepsilon^{2}}{\lambda}\bigl\Vert A^{\frac{1}{2}}q\bigr\Vert ^{2}\bigl\vert W(t)\bigr\vert ^{2}+\frac{\lambda }{8}\bigl\Vert A^{\frac{1}{2}} \bigl(y_{2t}+\varepsilon y_{2}-q(x)W\bigr)\bigr\Vert ^{2}. \end{aligned}$$
(4.30)
By (1.2), (4.5) and the Sobolev embedding theorem, we show that \(f(s)\) is uniformly bounded in \(L^{\infty}\), that is, there exists a constant \(M>0\) such that
$$ \bigl\vert f'(s)\bigr\vert _{L^{\infty}}\leq M. $$
(4.31)
Combining with (4.31), the Young inequality and the Sobolev embedding theorem, it follows that
$$\begin{aligned}& \bigl\vert -\bigl(f(u), A\bigl(y_{2t}+\varepsilon y_{2}-q(x)W\bigr)\bigr)\bigr\vert \\& \quad \leq \bigl\Vert A^{\frac{1}{2}}f(u)\bigr\Vert \bigl\Vert A^{\frac{1}{2}}\bigl(y_{2t}+\varepsilon y_{2}-q(x)W\bigr) \bigr\Vert \\& \quad \leq\frac{2\mu M^{2}}{\lambda}\|u\|_{2}^{2}+\frac{\lambda}{8} \bigl\Vert A^{\frac {1}{2}}\bigl(y_{2t}+\varepsilon y_{2}-q(x)W\bigr)\bigr\Vert ^{2}, \end{aligned}$$
(4.32)
where μ is a positive constant. Thus, collecting all (4.25)-(4.32) and (4.5), from (4.24) we have, for \(\tau\leq T(B)\),
$$\begin{aligned}& \frac{d}{dt}\bigl\Vert A^{\frac{1}{2}}Y_{2}\bigr\Vert _{E}^{2}+\bigl\Vert A^{\frac{1}{2}}Y_{2}\bigr\Vert _{E}^{2} \\& \quad \leq \frac{4\mu(b^{2}+M^{2})}{\lambda} \Vert u\Vert _{2}^{2}+ \biggl(\frac{2}{\varepsilon}\bigl\Vert A^{\frac{1}{2}}q\bigr\Vert _{2}^{2}+\frac{4}{\lambda }\bigl\Vert A^{\frac{5}{2}}q\bigr\Vert ^{2}+ \frac{4\varepsilon^{2}}{\lambda}\bigl\Vert A^{\frac{1}{2}}q\bigr\Vert ^{2} \biggr)\bigl\vert W(t)\bigr\vert ^{2} \\& \quad \leq \frac{4\mu(b^{2}+M^{2})}{\lambda} \biggl(2e^{-\varepsilon_{1}(t-\tau)}\biggl(\Vert u_{0}\Vert _{2}^{2}+\Vert u_{1}+ \varepsilon u_{0}\Vert ^{2}+\Vert q\Vert ^{2} \bigl\vert W(\tau)\bigr\vert ^{2} \\& \qquad {}+\int_{\Omega}F(u_{0})\, dx\biggr)+r_{1}^{2}(\omega)\biggr) \\& \qquad {} + \biggl(\frac{2}{\varepsilon}\bigl\Vert A^{\frac{1}{2}}q\bigr\Vert _{2}^{2}+\frac {4}{\lambda}\bigl\Vert A^{\frac{5}{2}}q\bigr\Vert ^{2}+ \frac{4\varepsilon^{2}}{\lambda}\bigl\Vert A^{\frac{1}{2}}q\bigr\Vert ^{2} \biggr)\bigl\vert W(t)\bigr\vert ^{2},\quad \tau\leq t\leq0. \end{aligned}$$
Applying the Gronwall lemma, we obtain that
$$\begin{aligned}& \bigl\Vert A^{\frac{1}{2}}Y_{2}\bigl(0,\omega;Y_{2}(\tau, \omega)\bigr)\bigr\Vert _{E}^{2} \\& \quad \leq e^{\varepsilon\tau}\bigl\Vert A^{\frac{1}{2}}q\bigr\Vert ^{2}\bigl\vert W(\tau)\bigr\vert ^{2} +\frac{4\mu(b^{2}+M^{2})}{\lambda} \biggl(\frac{2 e^{\varepsilon_{1}\tau }}{\varepsilon-\varepsilon_{1}}\biggl(\|u_{0}\|_{2}^{2}+ \|u_{1}+\varepsilon u_{0}\|^{2} \\& \qquad {}+\| q \|^{2}\bigl\vert W(\tau)\bigr\vert ^{2}+\int_{\Omega}F(u_{0})\, dx\biggr) + \frac{r_{1}^{2}(\omega)}{\varepsilon} \biggr) \\& \qquad {}+ \biggl(\frac{2}{\varepsilon}\bigl\Vert A^{\frac{1}{2}}q\bigr\Vert _{2}^{2}+ \frac {4}{\lambda}\bigl\Vert A^{\frac{5}{2}}q\bigr\Vert ^{2}+ \frac{4\varepsilon^{2}}{\lambda}\bigl\Vert A^{\frac{1}{2}}q\bigr\Vert ^{2} \biggr)\int_{\tau}^{0}e^{\varepsilon s} \bigl\vert W(s)\bigr\vert ^{2}\, ds. \end{aligned}$$
(4.33)
Set
$$\begin{aligned} r_{2}^{2}(\omega) =&\bigl\Vert A^{\frac{1}{2}}q\bigr\Vert ^{2}\sup_{\tau\leq 0}e^{\varepsilon\tau}\bigl\vert W( \tau)\bigr\vert ^{2} \\ &{}+\frac{4\mu(b^{2}+M^{2})}{\lambda(\varepsilon-\varepsilon_{1})}\Bigl(2+2\|q\| ^{2} \sup_{\tau\leq0}e^{\varepsilon_{1} \tau}\bigl\vert W(\tau )\bigr\vert ^{2}+r_{1}^{2}(\omega)\Bigr) \\ &{}+ \biggl(\frac{2}{\varepsilon}\bigl\Vert A^{\frac{1}{2}}q\bigr\Vert _{2}^{2}+\frac {4}{\lambda}\bigl\Vert A^{\frac{5}{2}}q\bigr\Vert ^{2}+ \frac{4\varepsilon^{2}}{\lambda}\bigl\Vert A^{\frac{1}{2}}q\bigr\Vert ^{2} \biggr)\int _{-\infty}^{0}e^{\varepsilon s}\bigl\vert W(s)\bigr\vert ^{2}\, ds. \end{aligned}$$
Since \(\lim_{t\rightarrow\infty}\frac{W(t)}{t}=0\), \(r_{2}^{2}(\omega)\) is finite P-a.s., together with (4.16) and (4.33), we have that
$$\bigl\Vert A^{\frac{1}{2}}Y_{2}\bigl(0,\omega;Y_{2}(\tau, \omega)\bigr)\bigr\Vert _{E}^{2}\leq r_{2}^{2}( \omega) $$
for all \((u_{0},u_{1}+\varepsilon u_{0})^{T}\in B\) and all \(\tau\leq T(B)\). □

Theorem 4.5

Let (1.2)-(1.4) hold, \(q(x)\in H^{3}(\Omega)\). Then the random dynamical system \(S_{\varepsilon}(t,\omega)\) possesses a nonempty compact random attractor \(\mathcal{A}\).

Proof

Let \(B_{1}(\omega)\) be the ball of \(H^{3}(\Omega )\times(H^{2}(\Omega)\cap H_{0}^{1}(\Omega))\) of radius \(r_{2}(\omega)\). From the compact embedding \(H^{3}(\Omega)\), it follows that \(B_{1}(\omega)\) is compact in E. For every bounded non-random set B of E and any \(\psi(0)\in\bar{S}_{\varepsilon}(t,\theta_{-t}\omega)B\), from Lemma 4.4, we know that \(Y_{2}(0)=\psi(0)-Y_{1}(0)\in B_{1}(\omega)\), where \(Y_{2}(t,\omega)\) is given by (4.18). Therefore, for \(\tau\leq0\),
$$\inf_{l(0)\in B_{1}(\omega)}\bigl\Vert \psi(0)-l(0)\bigr\Vert _{E}^{2} \leq\bigl\Vert Y_{1}(0)\bigr\Vert _{E}^{2}\leq\bigl(\|u_{0}\|_{2}^{2}+ \|u_{1}+\varepsilon u_{0}\| ^{2}\bigr) \frac{e^{\varepsilon\tau}}{1-\varepsilon}. $$
Furthermore, for all \(t\geq0\),
$$\mathrm{d}\bigl(\bar{S}_{\varepsilon}(t,\theta_{-t}\omega)B, B_{1}(\omega)\bigr) \leq\bigl(\|u_{0}\|_{2}^{2}+ \|u_{1}+\varepsilon u_{0}\|^{2}\bigr) \frac {{e^{-\varepsilon t}}}{1-\varepsilon}. $$
Finally, from relation (4.3) between \(S_{\varepsilon}(t,\omega)\) and \(\bar{S}_{\varepsilon}(t,\omega)\), one can easily obtain that for any non-random bounded \(B\subset E\) P-a.s.,
$$\mathrm{d}\bigl(S_{\varepsilon}(t,\theta_{-t}\omega)B, B_{1}(\omega )\bigr)\rightarrow0 \quad \text{as } t\rightarrow+\infty. $$
Hence, the RDS \(S_{\varepsilon}(t,\omega)\) associated with (3.6) possesses a uniformly attracting compact set \(B_{1}(\omega)\subset E\). Then applying Theorem 2.5 we complete the proof. □

Declarations

Acknowledgements

This work was supported by the NSFC (11361053, 11101334) and the NSF of Gansu Province (145RJZA112, 145RJZA232).

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)
College of Mathematics and Statistics, Northwest Normal University

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