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Existence of a random attractor for non-autonomous stochastic plate equations with additive noise and nonlinear damping on \(\mathbb{R}^{n}\)

Abstract

Based on the abstract theory of pullback attractors of non-autonomous non-compact dynamical systems by differential equations with both dependent-time deterministic and stochastic forcing terms, introduced by Wang in (J. Differ. Equ. 253:1544–1583, 2012), we investigate the existence of pullback attractors for the non-autonomous stochastic plate equations with additive noise and nonlinear damping on \(\mathbb{R}^{n}\).

Introduction

Plate equations have been studied for many years because of their worth in certain physical areas such as vibration and elasticity theories of solid mechanics. The research of the long-time dynamical behavior of plate equations has become an important area in the field of the infinite-dimensional dynamical system.

The purpose of this paper is to investigate the following non-autonomous stochastic plate equations with additive noise and nonlinear damping defined in the entire space \(\mathbb{R}^{n}\):

$$ u_{tt}+h(u_{t})+\Delta^{2}u+\lambda u +f(x,u)=g(x,t)+\phi(x) \frac{dW}{dt}, $$
(1.1)

with the initial value conditions

$$ u(x,\tau)=u_{0}(x),\qquad u_{t}(x,\tau)=u_{1}(x), $$
(1.2)

where \(x\in\mathbb{R}^{n}\), \(t>\tau\) with \(\tau\in\mathbb{R}\), λ is a positive constant, f is a nonlinearity that satisfies certain growth and dissipative conditions, \(g(x,\cdot)\) and ϕ are given functions in \(L^{2}_{loc}(\mathbb{R}, H^{1}(\mathbb{R}^{n}))\) and \(H^{2}(\mathbb{R}^{n})\cap H^{3}(\mathbb{R}^{n})\), respectively, \(W(t)\) is a two-sided real-valued Wiener process on a probability space.

As we know, the attractor is regarded as a proper notation describing the long-time dynamics of solutions, and many classical literature works and monographs have appeared for both the deterministic and stochastic dynamical systems over the last decades, see [1, 5, 7, 8, 10, 11, 14, 19, 23, 25] and the references therein. However, in reality, a system is always affected by some random factors such as external noise. In order to scrutinize the large-time behavior and characterization of solution for the stochastic partial differential equations driven by noise, Crauel and Flandoi [7, 8], Flandoi and Schmalfuss [10], and Schmalfuss [19] introduced the concept of pullback attractors and established some abstract results for the existence of such attractors about compact dynamical system [1, 8, 10, 14, 15]. Since these methods required the compactness of a pullback absorbing set for systems, they could not be used to deal with the stochastic PDEs on unbounded domains. Therefore, in [3], Bates, Lisei, and Lu presented the concept of asymptotic compactness for random dynamical systems, which is an extension of deterministic systems. And then, using these abstract results, they proved the existence of random attractors for reaction-diffusion equations on unbounded domain in [4]. Wang in [25] further extended the concept of asymptotic compactness to the case of partial differential equations with both random and time-dependent forcing terms; moreover, he applied these criteria into the stochastic reaction-diffusion equation with additive noise on \(\mathbb{R}^{n}\) and obtained the existence of a unique pullback attractor. For most of works on stochastic PDEs, please refer to [9, 22, 2729, 32] and the references therein.

Just for problem (1.1)–(1.2) and the corresponding plate equations, in the deterministic case (i.e., \(\varepsilon=0\)), existence of global attractors has been studied by several authors, see for instance [2, 1214, 30, 31, 33, 34, 37]. As far as the stochastic case driven by additive noise goes, when the deterministic forcing term g is independent of time, that is, \(g(x, t) \equiv g(x)\), the existence of a random pullback attractor on bounded domain has been obtained in [17, 20, 21]. Recently, on the unbounded domain, the authors investigated the existence and upper semi-continuity of random attractors for stochastic plate equation with rotational inertia and Kelvin–Voigt dissipative term as well as dependent-on-time terms (see [36] for details) and asymptotic behavior for non-autonomous stochastic plate equation on unbounded domains [35]. To the best of our knowledge, it has not been considered by any predecessors for the stochastic plate equation with additive noise and nonlinear damping on unbounded domain. It is well known that nonlinear damping makes the problem more complex and interesting even to the case of bounded domain. Besides, the theory and applications of Wang in [2426] gave us the idea of solving this problem and inspired us, so we decided to study the existence of pullback attractors for problem (1.1)–(1.2).

Notice that (1.1) is a non-autonomous stochastic equation, i.e., the external term g is time-dependent. In this case, as in [25], we introduce two parametric spaces to present its dynamics: one is for the deterministic non-autonomous perturbations, while the other for the stochastic perturbations. In addition, since Sobolev embeddings are not compact on \(\mathbb{R}^{n}\), we cannot get the asymptotic compactness directly from the regularity of solutions. We conquer the difficulty by using the uniform estimates on the tails of solutions outside a bounded ball in \(\mathbb{R}^{n}\) and the splitting technique [27] and the compactness methods introduced in [16].

The organization of this paper is as follows: In Sect. 2, we present some notations and a proposition about random dynamical systems. In Sect. 3, we establish a continuous cocycle for Eq. (1.1) in \(H^{2}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})\). In Sect. 4, we obtain all necessary uniform estimates of solutions. Finally, in Sect. 5, we show the existence and uniqueness of a random attractor for (1.1)–(1.2), denoted by \(\mathbb{R}^{n}\).

Throughout the paper, the letters c and \(c_{i}\) (\(i = 1, 2, \ldots\)) are positive constants which may change their values from line to line or even in the same line.

Preliminaries

In order to state and prove our main results, we introduce some notations and a proposition related to random attractors for stochastic dynamical systems.

Let X be a separable Banach space and \((\varOmega,\mathcal{F},\mathcal{P})\) be the standard probability space, where \(\varOmega=\{\omega\in C(\mathbb{R},\mathbb{R}):\omega(0)=0\}\), \(\mathcal{F}\) is the Borel σ-algebra induced by the compact open topology of Ω, and \(\mathcal{P}\) is the Wiener measure on \((\varOmega,\mathcal{F})\). There is a classical group \(\{\theta_{t}\}_{t\in\mathbb{R}}\) acting on \((\varOmega,\mathcal{F},\mathcal{P})\) which is defined by

$$ \theta_{t}\omega(\cdot)=\omega(\cdot+t)-\omega(t) \quad \text{for all } \omega\in\varOmega, t\in\mathbb{R}. $$
(2.1)

We often say that \((\varOmega,\mathcal{F},\mathcal{P},\{\theta_{t}\}_{t\in\mathbb{R} })\) is a parametric dynamical system.

Definition 2.1

([6])

A mapping \(\varPhi:\mathbb{R}^{+}\times\mathbb{R}\times\varOmega\times X \rightarrow X\) is called a continuous cocycle on X over \(\mathbb{R}\) and \((\varOmega,\mathcal{F},\mathcal{P},\{\theta_{t}\}_{t\in\mathbb{R} })\) if, for all \(\tau\in\mathbb{R}\), \(\omega\in\varOmega\), and \(t,s\in\mathbb{R}^{+}\), the following conditions (1)–(4) are satisfied:

  1. (1)

    \(\varPhi(\cdot,\tau,\cdot,\cdot):\mathbb{R}^{+}\times\varOmega \times X\rightarrow X\) is \((\mathcal{B}(\mathbb{R}^{+})\times\mathcal{F}\times\mathcal{B}(X), \mathcal{B}(X))\)-measurable;

  2. (2)

    \(\varPhi(0,\tau,\omega,\cdot)\) is the identity on X;

  3. (3)

    \(\varPhi(t+s,\tau,\omega,\cdot)=\varPhi(t,\tau+s,\theta_{s}\omega, \cdot)\circ\varPhi(s,\tau,\omega,\cdot)\);

  4. (4)

    \(\varPhi(t,\tau,\omega,\cdot):X\rightarrow X\) is continuous.

Definition 2.2

([6])

Assume that Φ is a continuous cocycle on X over \(\mathbb{R}\) and \((\varOmega,\mathcal{F},\mathcal{P}, \{\theta_{t}\}_{t\in\mathbb{R} })\), and \(\mathcal{D}\) is the collection of all tempered families of nonempty bounded subsets of X parameterized by \(\tau\in\mathbb{R}\) and \(\omega\in\varOmega\):

$$ \mathcal{D}=\bigl\{ D=\bigl\{ D(\tau,\omega)\subseteq X:D(\tau,\omega)\neq \emptyset,\tau\in\mathbb{R},\omega\in\varOmega\bigr\} \bigr\} . $$

Definition 2.3

([6])

\(\mathcal{D}\) is said to be tempered if there exists \(x_{0}\in X\) such that, for every \(c > 0\), \(\tau\in\mathbb{R}\), and \(\omega\in\varOmega\), the following holds:

$$ \lim_{t\rightarrow-\infty}e^{ct}d\bigl(D(\tau+t, \theta_{t}\omega),x_{0}\bigr)=0. $$
(2.2)

Definition 2.4

([6])

Given \(D\in\mathcal{D}\), the family \(\varOmega(D)=\{\varOmega(D,\tau,\omega):\tau\in\mathbb{R},\omega \in\varOmega\}\) is called the Ω-limit set of D where

$$ \varOmega(D,\tau,\omega)=\bigcap_{s\geq0} \overline{ \bigcup_{t\geq s}\varPhi\bigl(t,\tau-t, \theta_{-t}\omega,D(\tau -t,\theta_{-t}\omega)\bigr)}. $$
(2.3)

Definition 2.5

([6])

The cocycle Φ is said to be \(\mathcal{D}\)-pullback asymptotically compact in X if, for all \(\tau\in\mathbb{R}\) and \(\omega\in\varOmega\), the sequence

$$ \bigl\{ \varPhi(t_{n},\tau-t_{n},\theta_{-t_{n}} \omega,x_{n})\bigr\} ^{\infty}_{n=1} \text{ has a convergent subsequence in } X $$
(2.4)

whenever \(t_{n}\rightarrow\infty\), and \(x_{n}\in D(\tau-t_{n},\theta_{-t_{n}}\omega)\) with \(\{D(\tau,\omega):\tau\in\mathbb{R},\omega\in\varOmega\}\in \mathcal{D}\).

Definition 2.6

([6])

A family \(K=\{K(\tau,\omega):\tau\in\mathbb{R},\omega\in\varOmega\}\in \mathcal{D}\) is called a \(\mathcal{D}\)-pullback absorbing set for Φ if, for all \(\tau\in\mathbb{R}\) and \(\omega\in\varOmega\) and for every \(D\in\mathcal{D}\), there exists \(T=T(D,\tau,\omega)>0\) such that

$$ \varPhi\bigl(t,\tau-t,\theta_{-t}\omega,D(\tau-t, \theta_{-t}\omega)\bigr) \subseteq K(\tau,\omega) \quad \text{for all } t \geq T. $$
(2.5)

Definition 2.7

([6])

K is called a closed measurable \(\mathcal{D}\)-pullback absorbing set for Φ if \(K(\tau,\omega)\) is closed in X and is measurable in ω with respect to \(\mathcal{F}\).

Definition 2.8

([6])

A family \(\mathcal{A}=\{\mathcal{A}(\tau,\omega):\tau\in\mathbb{R},\omega \in\varOmega\}\in\mathcal{D}\) is called a \(\mathcal{D}\)-pullback attractor for Φ if the following conditions (1)–(3) are fulfilled: for all \(t\in\mathbb{R}^{+}\), \(\tau\in\mathbb{R}\), and \(\omega\in\varOmega\),

  1. (1)

    \(\mathcal{A}(\tau,\omega)\) is compact in X and is measurable in ω with respect to \(\mathcal{F}\);

  2. (2)

    \(\mathcal{A}\) is invariant, that is,

    $$ \varPhi\bigl(t,\tau,\omega,\mathcal{A}(\tau,\omega)\bigr)=\mathcal{A}(\tau+t, \theta_{t}\omega); $$
    (2.6)
  3. (3)

    For every \(D=\{D(\tau,\omega):\tau\in\mathbb{R},\omega\in\varOmega\}\in \mathcal{D}\),

    $$ \lim_{t\rightarrow\infty}d_{H}\bigl(\varPhi\bigl(t,\tau-t, \theta_{-t}\omega,D( \tau-t,\theta_{-t}\omega)\bigr), \mathcal{A}(\tau,\omega)\bigr)=0, $$
    (2.7)

    where \(d_{H}\) is the Hausdorff semi-distance given by \(d_{H}(F,G)=\sup_{u\in F}\inf_{v\in G}\|u-v\|_{X}\) for any F, \(G\subset X\).

Definition 2.9

([6])

A mapping \(\varPsi:\mathbb{R}\times\mathbb{R}\times\varOmega\rightarrow X\) is called a random complete solution of Φ if, for every \(\tau\in\mathbb{R}^{+}\), \(s,\tau\in\mathbb{R}\), and \(\omega\in\varOmega\),

$$ \varPhi\bigl(t,\tau+s,\theta_{s}\omega,\varPsi(s,\tau,\omega) \bigr)=\varPsi(t+s, \tau,\omega). $$
(2.8)

Definition 2.10

([6])

Ψ is called a tempered random complete solution of Φ, if there exists a tempered family \(D=\{D(\tau,\omega):\tau\in\mathbb{R},\omega\in\varOmega\}\) such that \(\varPsi(t,\tau,\omega)\) belongs to \(D(\tau+t,\theta_{t}\omega)\) for every \(t\in\mathbb{R}\), \(\tau\in\mathbb{R}\) and \(\omega\in\varOmega\).

Proposition 2.1

([25])

Suppose thatΦis\(\mathcal{D}\)-pullback asymptotically compact inXand has a closed measurable\(\mathcal{D}\)-pullback absorbing setKin\(\mathcal{D}\). ThenΦhas a unique\(\mathcal{D}\)-pullback attractor\(\mathcal{A}\)in\(\mathcal{D}\)which is given by, for each\(\tau\in\mathbb{R}\)and\(\omega\in\varOmega\),

$$\begin{aligned} \mathcal{A}(\tau,\omega) =&\varOmega(K,\tau,\omega)=\bigcup _{D \in\mathcal{D}}\varOmega(D,\tau,\omega) \end{aligned}$$
(2.9)
$$\begin{aligned} =&\bigl\{ \varPsi(0,\tau,\omega):\varPsi \textit{ is a tempered random complete solution of } \varPhi\bigr\} . \end{aligned}$$
(2.10)

Cocycles for stochastic plate equation

In this section, we firstly present the precise hypotheses on problem (1.1)–(1.2), then show that it generates a continuous cocycle in \(H^{2}(\mathbb{R}^{n})\times H^{1}(\mathbb{R}^{n})\).

Let −Δ denote the Laplace operator in \(\mathbb{R}^{n}\), \(A=\Delta^{2}\) with the domain \(D(A)=H^{4}(\mathbb{R}^{n}) \). We can also define the powers \(A^{\nu}\) of A for \(\nu\in\mathbb{R}\). The space \(V_{\nu} = D(A^{\frac{\nu}{4}})\) is a Hilbert space with the following inner product and norm:

$$ (u,v)_{\nu}=\bigl(A^{\frac{\nu}{4}}u,A^{\frac{\nu}{4}}v\bigr),\qquad \Vert \cdot \Vert _{\nu}= \bigl\Vert A^{\frac{\nu}{4}}\cdot \bigr\Vert . $$

As usual, \((\cdot, \cdot)\) denotes \(L^{2}\)-inner product and \(\|\cdot\|\) denotes the \(L^{2}\)-norm.

Let \(E =H^{2 }\times L^{2}\), with the Sobolev norm

$$ \Vert y \Vert _{H^{2}\times L^{2}}=\bigl( \Vert v \Vert ^{2}+ \Vert u \Vert ^{2}+ \Vert \Delta u \Vert ^{2} \bigr)^{ \frac{1}{2}} \quad \text{for } y=(u,v)^{\top}\in E. $$
(3.1)

Let \(\xi=u_{t}+\delta u\), where δ is a small positive constant whose value will be determined later, then (1.1)–(1.2) can be reduced to the equivalent system

$$ \textstyle\begin{cases} \frac{du}{dt}+\delta u=\xi, \\ \frac{d\xi}{dt}-\delta\xi+(\lambda+\delta^{2}+A)u+h(\xi-\delta u)+f(x,u)=g(x,t)+ \phi(x)\frac{dW}{dt}, \end{cases} $$
(3.2)

with the initial value conditions

$$ u(x,\tau)=u_{0}(x),\qquad \xi(x,\tau)=\xi_{0}(x), $$
(3.3)

where \(\xi_{0}(x)=u_{1}(x)+\delta u_{0}(x)\), \(x\in\mathbb{R}^{n}\).

Assume that the functions h, f satisfy the following conditions:

(1) Let \(F(x,u)=\int_{0}^{u}f(x,s)\,ds\) for \(x\in\mathbb{R}^{n}\) and \(u\in\mathbb{R}\), there exist positive constants \(c_{i}\) (\(i=1,2,3,4\)) such that

$$\begin{aligned} & \bigl\vert f(x,u) \bigr\vert \leq c_{1} \vert u \vert ^{p}+\eta_{1}(x), \quad \eta_{1} \in L^{2}\bigl( \mathbb{R}^{n}\bigr), \end{aligned}$$
(3.4)
$$\begin{aligned} &f(x,u)u-c_{2}F(x,u)\geq\eta_{2}(x),\quad \eta_{2} \in L^{1}\bigl( \mathbb{R}^{n}\bigr), \end{aligned}$$
(3.5)
$$\begin{aligned} &F(x,u)\geq c_{3} \vert u \vert ^{p+1}- \eta_{3}(x),\quad \eta_{3} \in L^{1}\bigl( \mathbb{R}^{n}\bigr), \end{aligned}$$
(3.6)
$$\begin{aligned} & \biggl\vert \frac{\partial f}{\partial u} (x,u) \biggr\vert \leq\beta,\qquad \biggl\vert \frac{\partial f}{\partial x} (x,u) \biggr\vert \leq\eta_{4}(x),\quad \eta_{4} \in L^{2}\bigl(\mathbb{R}^{n}\bigr), \end{aligned}$$
(3.7)

where \(\beta>0\), \(1\leq p\leq\frac{n+4}{n-4}\). Note that (3.4) and (3.5) imply

$$ F(x,u)\leq c\bigl( \vert u \vert ^{2}+ \vert u \vert ^{p+1}+\eta_{1}^{2}+\eta_{2} \bigr). $$
(3.8)

(2) There exist two constants \(\beta_{1}\), \(\beta_{2}\) such that

$$ h(0)=0,\quad 0< \beta_{1}\leq h'(v)\leq \beta_{2}< \infty. $$
(3.9)

We identify \(\omega(t)\) with \(W(t)\), i.e., \(\omega(t)=W(t)=W(t,x)\), \(t\in\mathbb{R}\). To study the dynamical behavior of problem (3.2)–(3.3), we need to convert the stochastic system into a deterministic one with a random parameter. To this end, we set \(v(t)=\xi(t)-\phi\omega(t)\), we obtain the equivalent system of (3.2)–(3.3):

$$ \textstyle\begin{cases} \frac{du}{dt}+\delta u=v+\phi\omega(t), \\ \frac{dv}{dt}-\delta v+(\lambda+\delta^{2}+A)u+f(x,u)=g(x,t)-h(v+ \phi\omega(t)-\delta u)+ \delta\phi\omega(t), \end{cases} $$
(3.10)

with the initial value conditions

$$ u(x,\tau)=u_{0}(x),\qquad v(x,\tau)=v_{0}(x), $$
(3.11)

where \(v_{0}(x)=\xi_{0}(x)-\phi\omega(t)\), \(x\in\mathbb{R}^{n}\).

By a standard method as in [5, 18, 23, 36], one may show the following lemma under conditions (3.4)–(3.9).

Lemma 3.1

Put\(\varphi(t+\tau,\tau,\theta_{-\tau}\omega,\varphi_{0})=(u(t+ \tau,\tau,\theta_{-\tau}\omega,u_{0}),v(t+\tau,\tau,\theta_{- \tau}\omega,v_{0}))^{\top}\), where\(\varphi_{0}=(u_{0},v_{0})^{\top}\), and let (3.4)(3.9) hold. Then, for every\(\omega\in\varOmega\), \(\tau\in\mathbb{R}\), and\(\varphi_{0}\in E(\mathbb{R}^{n})\), problem (3.10)(3.11) has a unique\((\mathcal{F}, \mathcal{B}(H^{2}(\mathbb{R}^{n}))\times\mathcal{B}(L^{2}( \mathbb{R}^{n})))\)-measurable solution\(\varphi(\cdot,\tau,\omega,\varphi_{0})\in C([\tau,\infty),E( \mathbb{R}^{n}))\)with\(\varphi(\tau,\tau,\omega,\varphi_{0})=\varphi_{0}\), \(\varphi(t, \tau,\omega,\varphi_{0})\in E(\mathbb{R}^{n})\)being continuous in\(\varphi_{0}\)with respect to the usual norm of\(E(\mathbb{R}^{n})\)for each\(t>\tau\). Moreover, for every\((t,\tau,\omega,\varphi_{0})\in\mathbb{R}^{+}\times\mathbb{R} \times\varOmega\times E(\mathbb{R}^{n})\), the mapping

$$ \varPhi(t,\tau,\omega,\varphi_{0})=\varphi(t+\tau,\tau, \theta_{- \tau}\omega,\varphi_{0}) $$
(3.12)

generates a continuous cocycle from\(\mathbb{R}^{+}\times\mathbb{R}\times\varOmega\times E(\mathbb{R}^{n})\)to\(E(\mathbb{R}^{n})\)over\(\mathbb{R}\)and\((\varOmega,\mathcal{F},\mathcal{P}, \{\theta_{t}\}_{t\in\mathbb{R} })\).

Introducing the homeomorphism \(P(\theta_{t}\omega)(u,v)^{\top}=(u,v+z(\theta_{t}\omega))^{\top}\), \((u,v)^{\top}\in E(\mathbb{R}^{n})\) whose inverse homeomorphism \(P^{-1}(\theta_{t}\omega)(u,v)^{\top}=(u,v-z(\theta_{t}\omega ))^{\top}\). Then the transformation

$$ \widetilde{\varPhi}\bigl(t,\tau,\omega,(u_{0},\xi_{0}) \bigr)=P(\theta_{t} \omega)\varPhi\bigl(t,\tau,\omega,(u_{0},v_{0}) \bigr)P^{-1}(\theta_{t}\omega) $$
(3.13)

also generates a continuous cocycle with (3.2)–(3.3) over \(\mathbb{R}\) and \((\varOmega,\mathcal{F},\mathcal{P},\{\theta_{t}\}_{t\in\mathbb{R} })\).

Note that these two continuous cocycles are equivalent. By (3.13), it is easy to check that Φ̃ has a random attractor provided Φ possesses a random attractor. Then we only need to consider the continuous cocycle Φ.

Next we make another assumption:

Assume that σ, δ, and g satisfy the following conditions:

$$\begin{aligned}& \sigma=\min\biggl\{ \delta,\frac{\delta c_{2}}{2}\biggr\} , \end{aligned}$$
(3.14)
$$\begin{aligned}& \lambda+\delta^{2}-\beta_{2}\delta>0 \quad \text{and} \quad \beta _{1}> 4\delta+ \frac{3\beta^{2}}{\delta(\lambda+\delta^{2}-\beta_{2}\delta)}. \end{aligned}$$
(3.15)

Moreover,

$$ \int_{-\infty}^{0} e^{\sigma s} \bigl\Vert g( \cdot, \tau+s) \bigr\Vert _{1}^{2} \,ds< \infty,\quad \forall \tau\in\mathbb{R}, $$
(3.16)

and

$$ \lim_{k\rightarrow\infty} \int_{-\infty}^{0}e^{\sigma s} \int_{ \vert x \vert \geq k} \bigl\vert g(x,\tau+s) \bigr\vert ^{2}\,dx\,ds=0,\quad \forall \tau\in\mathbb{R}, $$
(3.17)

where \(|\cdot|\) denotes the absolute value of a real number in \(\mathbb{R}\).

Given a bounded nonempty subset B of E, we write \(\|B\|=\sup_{\phi\in B}\|\phi\|_{E}\). Let \(D=\{D(\tau,\omega):\tau\in\mathbb{R}, \omega\in\varOmega\}\) be a family of bounded nonempty subsets of E such that, for every \(\tau\in\mathbb{R}\), \(\omega\in\varOmega\),

$$ \lim_{s\rightarrow-\infty}e^{\sigma s} \bigl\Vert D(\tau+s, \theta_{s} \omega) \bigr\Vert ^{2}_{E}=0. $$
(3.18)

Let \(\mathcal{D}\) be the collection of all such families, that is,

$$ \mathcal{D}=\bigl\{ D=\bigl\{ D(\tau,\omega):\tau\in\mathbb{R},\omega\in \varOmega \bigr\} : D \text{ satisfies (3.18)}\bigr\} . $$
(3.19)

Uniform estimates of solutions

In this section, we derive uniform estimates on the solutions of the stochastic plate equations (3.2)–(3.3) defined on \(\mathbb{R}^{n}\).

We define a new norm \(\|\cdot\|_{E}\) by

$$ \Vert Y \Vert _{E}=\bigl( \Vert v \Vert ^{2}+ \bigl(\lambda+\delta^{2}-\beta_{2}\delta\bigr) \Vert u \Vert ^{2}+ \Vert \Delta u \Vert ^{2} \bigr)^{\frac{1}{2}}\quad \text{for } Y=(u,v)\in E. $$
(4.1)

It is easy to check that \(\|\cdot\|_{E}\) is equivalent to the usual norm \(\|\cdot\|_{H^{2}\times L^{2}}\) in (3.1).

The next lemma shows that the cocycle Φ has a pullback \(\mathcal{D}\)-absorbing set in \(\mathcal{D}\).

Lemma 4.1

Under (3.4)(3.9) and (3.14)(3.17), for every\(\tau\in\mathbb{R}\), \(\omega\in\varOmega\), \(D=\{D(\tau,\omega):\tau\in\mathbb{R},\omega\in\varOmega\}\in \mathcal{D}\), there exists\(T=T(\tau,\omega,D)>0\)such that, for all\(t\geq T\), the solution of problem (3.10)(3.11) satisfies

$$\begin{aligned}& \bigl\Vert Y\bigl(\tau,\tau-t,\theta_{-\tau}\omega,D(\tau-t, \theta_{-t} \omega)\bigr) \bigr\Vert ^{2}_{E} \leq R_{1}(\tau,\omega), \\& e^{-\sigma t} \int^{t}_{\tau-t}e^{\sigma s} \bigl\Vert Y \bigl(s,\tau-t,\theta_{- \tau}\omega,D(\tau-t,\theta_{-t} \omega)\bigr) \bigr\Vert ^{2}_{E}\,ds\leq R_{1}( \tau,\omega), \end{aligned}$$

and\(R_{1}(\tau,\omega)\)is given by

$$ \begin{aligned}R_{1}(\tau,\omega)=M+M \int^{\tau}_{-\infty }e^{\sigma(s- \tau)} \bigl\Vert g(x, s) \bigr\Vert ^{2}\,ds+c \int^{0}_{ -\infty}e^{\sigma s }\bigl( \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\vert \omega(s) \bigr\vert ^{p+1}\bigr)\,ds, \end{aligned} $$
(4.2)

whereMis a positive constant independent ofτ, ω, D.

Proof

Taking the inner product of the second equation of (3.10) with v in \(L^{2}(\mathbb{R}^{n})\), we find that

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \Vert v \Vert ^{2}- \delta \Vert v \Vert ^{2}+\bigl(\lambda+\delta^{2} \bigr) (u,v)+ (Au,v)+\bigl(f(x,u),v\bigr) \\ &\quad =\bigl(g(x,t),v\bigr)-\bigl(h\bigl(v+ \phi\omega(t)-\delta u\bigr),v \bigr)+ \delta(\phi,v) \omega(t). \end{aligned}$$
(4.3)

By the first equation of (3.10), we have

$$ v=u_{t}- \phi\omega(t)+\delta u. $$
(4.4)

By Lagrange’s mean value theorem and (3.9), we get

$$\begin{aligned}& -\bigl(h\bigl(v+ \phi\omega(t)-\delta u\bigr),v\bigr) \\& \quad = -\bigl(h\bigl(v+ \phi\omega(t)-\delta u\bigr)-h(0),v\bigr) \\& \quad = -\bigl(h'(\vartheta) \bigl(v+ \phi\omega(t)-\delta u \bigr),v\bigr) \\& \quad \leq -\beta_{1} \Vert v \Vert ^{2}- \bigl(h'(\vartheta) \bigl( \phi\omega(t)-\delta u\bigr),v\bigr) \\& \quad \leq -\beta_{1} \Vert v \Vert ^{2}+ \beta_{2} \bigl\vert \omega(t) \bigr\vert \Vert \phi \Vert \Vert v \Vert +h'( \vartheta)\delta(u,v) \\& \quad \leq -\beta_{1} \Vert v \Vert ^{2}+ \frac{\beta_{1}-\delta}{6} \Vert v \Vert ^{2}+ \frac{3\beta_{2}^{2}}{2(\beta_{1}-\delta)} \bigl\vert \omega(t) \bigr\vert ^{2} \Vert \phi \Vert ^{2}+h'(\vartheta)\delta(u,v), \end{aligned}$$
(4.5)

where ϑ is between 0 and \(v+ \phi\omega(t)-\delta u\).

By (3.9) and (4.4), we get

$$\begin{aligned}& h'(\vartheta)\delta(u,v) \\& \quad = h'(\vartheta)\delta\bigl(u,u_{t}-\phi \omega(t)+\delta u\bigr) \\& \quad \leq \beta_{2}\delta\cdot\frac{1}{2} \frac{d}{dt} \Vert u \Vert ^{2}+\beta_{2} \delta^{2} \Vert u \Vert ^{2}+ \beta_{2} \delta \bigl\vert \omega(t) \bigr\vert \Vert \phi \Vert \Vert u \Vert \\& \quad \leq \beta_{2}\delta\cdot\frac{1}{2} \frac{d}{dt} \Vert u \Vert ^{2}+\beta_{2} \delta^{2} \Vert u \Vert ^{2}+\frac{1}{4} \delta\bigl(\lambda+\delta^{2}-\beta_{2} \delta\bigr) \Vert u \Vert ^{2} +c \bigl\vert \omega(t) \bigr\vert ^{2} \Vert \phi \Vert ^{2}. \end{aligned}$$
(4.6)

Then substituting the v in (4.4) into the third and fourth terms on the left-hand side of (4.3), we find that

$$\begin{aligned}& \bigl(\lambda+\delta^{2}\bigr) (u,v) \\& \quad =\bigl(\lambda+\delta^{2}\bigr) \bigl(u,u_{t}- \phi\omega(t)+\delta u\bigr) \\& \quad \geq\frac{1}{2}\bigl(\lambda+\delta^{2}\bigr) \frac{d}{dt} \Vert u \Vert ^{2}+\delta\bigl( \lambda+ \delta^{2}\bigr) \Vert u \Vert ^{2}-\bigl(\lambda+ \delta^{2}\bigr) \bigl\vert \omega(t) \bigr\vert \Vert \phi \Vert \Vert u \Vert \\& \quad \geq\frac{1}{2}\bigl(\lambda+\delta^{2}\bigr) \frac{d}{dt} \Vert u \Vert ^{2}+\delta\bigl( \lambda+ \delta^{2}\bigr) \Vert u \Vert ^{2}- \frac{1}{4}\delta\bigl(\lambda+\delta^{2}- \beta_{2}\delta\bigr) \Vert u \Vert ^{2} - c \bigl\vert \omega(t) \bigr\vert ^{2} \Vert \phi \Vert ^{2}, \end{aligned}$$
(4.7)
$$\begin{aligned}& \begin{aligned}[b] (Au,v) &=(\Delta u,\Delta v)=\bigl(\Delta u,\Delta u_{t}-\omega(t)\Delta \phi+\delta\Delta u\bigr) \\ &\geq\frac{1}{2}\frac{d}{dt} \Vert \Delta u \Vert ^{2}+\delta \Vert \Delta u \Vert ^{2}- \bigl\vert \omega(t) \bigr\vert \Vert \Delta\phi \Vert \Vert \Delta u \Vert \\ &\geq\frac{1}{2}\frac{d}{dt} \Vert \Delta u \Vert ^{2}+\frac{\delta}{2} \Vert \Delta u \Vert ^{2}- \frac{1}{2\delta} \bigl\vert \omega(t) \bigr\vert ^{2} \Vert \Delta\phi \Vert ^{2}. \end{aligned} \end{aligned}$$
(4.8)

Using the Cauchy–Schwarz inequality and Young’s inequality, we have

$$ \begin{aligned}\delta\bigl(\phi\omega(t),v\bigr)\leq\delta \bigl\vert \omega(t) \bigr\vert \Vert \phi \Vert \Vert v \Vert \leq \frac{3 \delta^{2}}{2(\beta_{1}-\delta)} \Vert \phi \Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2}+\frac{\beta_{1}-\delta}{6} \Vert v \Vert ^{2} \end{aligned} $$
(4.9)

and

$$ (g,v)\leq \Vert g \Vert \Vert v \Vert \leq\frac{3}{2(\beta_{1}-\delta)} \Vert g \Vert ^{2}+ \frac{\beta_{1}-\delta}{6} \Vert v \Vert ^{2}. $$
(4.10)

Let \(\widetilde{F}(x,u)=\int_{\mathbb{R}^{n}}F(x,u)\,dx\). Then, for the last term on the left-hand side of (4.3), we have

$$ \begin{aligned}\bigl(f(x,u),v\bigr)&=\bigl(f(x,u),u_{t}- \phi\omega(t) +\delta u\bigr) \\ &=\frac{d}{dt}\widetilde{F}(x,u)+\delta\bigl(f(x,u),u\bigr)- \bigl(f(x,u),\phi \omega(t)\bigr). \end{aligned} $$
(4.11)

By condition (3.5) we get

$$ \bigl(f(x,u),u\bigr)\geq c_{2}\widetilde{F}(x,u)+ \int_{\mathbb{R}^{n}}\eta_{2}(x)\,dx. $$
(4.12)

Using condition (3.4) and (3.6), we obtain

$$\begin{aligned}& \bigl(f(x,u),\phi\omega(t)\bigr) \\& \quad \leq \int_{\mathbb{R}^{n}}\bigl(c_{1} \vert u \vert ^{p}+\eta_{1}(x)\bigr) \bigl\vert \phi\omega(t) \bigr\vert \,dx \\& \quad \leq \bigl\Vert \eta_{1}(x) \bigr\Vert \Vert \phi \Vert \bigl\vert \omega(t) \bigr\vert +c_{1}\biggl( \int_{\mathbb {R}^{n}} \vert u \vert ^{p+1}\,dx \biggr)^{ \frac{p}{p+1}} \Vert \phi \Vert _{p+1} \bigl\vert \omega(t) \bigr\vert \\& \quad \leq \bigl\Vert \eta_{1}(x) \bigr\Vert \Vert \phi \Vert \bigl\vert \omega(t) \bigr\vert +c_{1}\biggl( \int_{\mathbb{R}^{n}}\bigl(F(x,u)+ \eta_{3}(x)\bigr)\,dx \biggr)^{\frac{p}{p+1}} \Vert \phi \Vert _{p+1} \bigl\vert \omega(t) \bigr\vert \\& \quad \leq \frac{1}{2} \bigl\Vert \eta_{1}(x) \bigr\Vert ^{2}+\frac{1}{2} \Vert \phi \Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2}+ \frac{\delta c_{2}}{2}\widetilde{F}(x,u) \\& \qquad {}+ \frac{\delta c_{2}}{2} \int_{\mathbb{R}^{n}}\eta_{3}(x)\,dx+c \Vert \phi \Vert ^{p+1}_{H^{2}} \bigl\vert \omega(t) \bigr\vert ^{p+1}. \end{aligned}$$
(4.13)

By (4.11)–(4.13), we get

$$\begin{aligned}& \delta\bigl(f(x,u),u\bigr)-\bigl(f(x,u),\phi\omega(t) \bigr) \\& \quad \geq \frac{\delta c_{2}}{2}\widetilde{F}(x,u)+\delta \int_{ \mathbb{R}^{n}}\eta_{2}(x)\,dx-\frac{1}{2} \bigl\Vert \eta_{1}(x) \bigr\Vert ^{2}- \frac{1}{2} \Vert \phi \Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2} \\& \qquad {} -\frac{\delta c_{2}}{2} \int_{\mathbb{R}^{n}}\eta_{3}(x)\,dx-c \Vert \phi \Vert ^{p+1}_{H^{2}} \bigl\vert \omega(t) \bigr\vert ^{p+1}. \end{aligned}$$
(4.14)

Substitute (4.5)–(4.14) into (4.3) to obtain

$$\begin{aligned}& \frac{1}{2}\frac{d}{dt}\bigl( \Vert v \Vert ^{2}+\bigl(\lambda+\delta^{2}-\beta_{2} \delta\bigr) \Vert u \Vert ^{2}+ \Vert \Delta u \Vert ^{2}+2\widetilde{F}(x,u)\bigr) \\& \qquad {} +\delta\bigl( \Vert v \Vert ^{2}+\bigl(\lambda+ \delta^{2}-\beta_{2}\delta\bigr) \Vert u \Vert ^{2}+ \Vert \Delta u \Vert ^{2}\bigr)+ \frac{\delta c_{2}}{2}\widetilde{F}(x,u) \\& \quad \leq \frac{\delta}{2} \Vert v \Vert ^{2}+ \frac{2\delta-\beta_{1} }{2} \Vert v \Vert ^{2}+ \frac{\delta}{2} \bigl(\lambda+\delta^{2}-\beta_{2}\delta\bigr) \Vert u \Vert ^{2} + \frac{\delta}{2} \Vert \Delta u \Vert ^{2} \\& \qquad {} +c\bigl(1+ \bigl\vert \omega(t) \bigr\vert ^{2}+ \bigl\vert \omega(t) \bigr\vert ^{p+1}\bigr)+ \frac{3}{2(\beta_{1}-\delta)} \Vert g \Vert ^{2}. \end{aligned}$$
(4.15)

Let \(\sigma=\min\{\delta,\frac{\delta c_{2}}{2}\}\), then

$$\begin{aligned}& \frac{d}{dt}\bigl( \Vert v \Vert ^{2}+\bigl(\lambda+ \delta^{2}-\beta_{2}\delta\bigr) \Vert u \Vert ^{2}+ \Vert \Delta u \Vert ^{2}+2\widetilde{F}(x,u) \bigr) \\& \qquad {} +\sigma\bigl( \Vert v \Vert ^{2}+\bigl(\lambda+ \delta^{2}-\beta_{2}\delta\bigr) \Vert u \Vert ^{2}+ \Vert \Delta u \Vert ^{2}+2\widetilde{F}(x,u) \bigr) \\& \quad \leq \frac{3}{ (\beta_{1}-\delta)} \Vert g \Vert ^{2}+c\bigl(1+ \bigl\vert \omega(t) \bigr\vert ^{2}+ \bigl\vert \omega(t) \bigr\vert ^{p+1}\bigr). \end{aligned}$$
(4.16)

Multiplying (4.16) by \(e^{\sigma t}\) and then integrating over \((\tau-t,\tau)\), we have

$$\begin{aligned}& e^{\sigma\tau} \bigl( \bigl\Vert v(\tau,\tau-t,\omega,v_{0}) \bigr\Vert ^{2}+\bigl(\lambda+ \delta^{2}- \beta_{2}\delta\bigr) \bigl\Vert u(\tau,\tau-t,\omega,u_{0}) \bigr\Vert ^{2} \\& \qquad {} + \bigl\Vert \Delta u(\tau,\tau-t,\omega,u_{0}) \bigr\Vert ^{2}+2\widetilde{F}\bigl(x,( \tau,\tau-t,\omega,u_{0}) \bigr)\bigr) \\& \quad \leq e^{\sigma(\tau-t)}\bigl( \Vert v_{0} \Vert ^{2}+\bigl(\lambda+\delta^{2}-\beta_{2} \delta\bigr) \Vert u_{0} \Vert ^{2}+ \Vert \Delta u_{0} \Vert ^{2}+2\widetilde{F}(x,u_{0}) \bigr) \\& \qquad {} +\frac{3}{ (\beta_{1}-\delta)} \int^{\tau}_{\tau-t}e^{\sigma s} \bigl\Vert g(x,s) \bigr\Vert ^{2}\,ds+c \int^{\tau}_{\tau-t}e^{\sigma s}\bigl(1+ \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\vert \omega(s) \bigr\vert ^{p+1}\bigr)\,ds. \end{aligned}$$

Replacing ω by \(\theta_{-\tau}\omega\) in the above, we obtain, for every \(t\in\mathbb{R}^{+}\), \(\tau\in\mathbb{R}\), and \(\omega\in \varOmega\),

$$\begin{aligned}& \bigl\Vert v(\tau,\tau-t,\theta_{-\tau}\omega,v_{0}) \bigr\Vert ^{2}+\bigl(\lambda+ \delta^{2}- \beta_{2}\delta\bigr) \bigl\Vert u(\tau,\tau-t,\theta_{-\tau} \omega,u_{0}) \bigr\Vert ^{2} \\& \qquad {} + \bigl\Vert \Delta u(\tau,\tau-t,\omega,u_{0}) \bigr\Vert ^{2}+2\widetilde{F}\bigl(x,u( \tau,\tau-t, \theta_{-\tau}\omega,u_{0})\bigr) \\& \quad \leq e^{-\sigma t}\bigl( \Vert v_{0} \Vert ^{2}+\bigl(\lambda+\delta^{2}-\beta_{2} \delta\bigr) \Vert u_{0} \Vert ^{2}+ \Vert \Delta u_{0} \Vert ^{2}+2\widetilde{F}(x,u_{0}) \bigr) \\& \qquad {}+ \frac{3}{ (\beta_{1}-\delta)} \int^{\tau}_{\tau-t}e^{\sigma(s- \tau)} \bigl\Vert g(x,s) \bigr\Vert ^{2}\,ds \\& \qquad {}+c \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)}\bigl(1+ \bigl\vert \theta_{-\tau}\omega(s) \bigr\vert ^{2}+ \bigl\vert \theta_{-\tau}\omega(s) \bigr\vert ^{p+1}\bigr)\,ds. \end{aligned}$$
(4.17)

Again, by (3.9), we get

$$ \widetilde{F}(x,u_{0})\leq c\bigl(1+ \Vert u_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{p+1} \bigr). $$

Thus, for the first term on the right-hand side of (4.17), we have

$$\begin{aligned}& e^{-\sigma t}\bigl( \Vert v_{0} \Vert ^{2}+ \bigl(\lambda+\delta^{2}-\beta_{2}\delta\bigr) \Vert u_{0} \Vert ^{2}+ \Vert \Delta u_{0} \Vert ^{2}+2\widetilde{F}(x,u_{0})\bigr) \\& \quad \leq ce^{- \sigma t}\bigl(1+ \Vert v_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}_{H^{2}}+ \Vert u_{0} \Vert ^{p+1}_{H^{2}}\bigr). \end{aligned}$$

Since \((u_{0},v_{0})^{\top}\in D(\tau-t,\theta_{-t}\omega)\) and \(D\in\mathcal{D}\), then we find

$$ \lim_{t\rightarrow+\infty}e^{-\sigma t}\bigl( \Vert v_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}_{H^{2}}+ \Vert u_{0} \Vert ^{p+1}_{H^{2}}\bigr)=0. $$

Therefore, there exists \(T=T(\tau,\omega,D)>0\) such that, for all \(t\geq T\),

$$ e^{-\sigma t}\bigl(1+ \Vert v_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}_{H^{2}}+ \Vert u_{0} \Vert ^{p+1}_{H^{2}}\bigr) \leq1. $$
(4.18)

For the last term on the right-hand side of (4.17), we find

$$\begin{aligned}& c \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)}\bigl(1+ \bigl\vert \theta_{-\tau} \omega(s) \bigr\vert ^{2}+ \bigl\vert \theta_{-\tau}\omega(s) \bigr\vert ^{p+1}\bigr)\,ds \\& \quad \leq c \int^{0}_{ -t}e^{\sigma s }\bigl(1+ \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\vert \omega(s) \bigr\vert ^{p+1}\bigr)\,ds \\& \quad \leq c \int^{0}_{ -\infty}e^{\sigma s }\bigl(1+ \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\vert \omega(s) \bigr\vert ^{p+1}\bigr)\,ds \\& \quad \leq \frac{c}{\sigma}+c \int^{0}_{ -\infty}e^{\sigma s }\bigl( \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\vert \omega(s) \bigr\vert ^{p+1}\bigr)\,ds. \end{aligned}$$
(4.19)

Notice that \(\omega(s)\) has at most linear growth at \(|s|\rightarrow\infty\), which combines (3.19), we can have

$$ c \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)}\bigl(1+ \bigl\vert \theta_{-\tau}\omega (s) \bigr\vert ^{2}+ \bigl\vert \theta_{-\tau}\omega(s) \bigr\vert ^{p+1}\bigr)\,ds \rightarrow\frac{c}{\sigma}\quad (t\rightarrow\infty). $$
(4.20)

Finally, we estimate the fourth term on the left-hand side of (4.17). Thanks to (3.6), we obtain that, for all \(t\geq0\),

$$ -2\widetilde{F}\bigl(x,u(\tau,\tau-t,\theta_{-\tau} \omega,u_{0})\bigr) \leq2 \int_{\mathbb{R}^{n}}\eta_{3}\,dx. $$
(4.21)

It follows from (4.18)–(4.21) that

$$\begin{aligned}& \bigl\Vert v(\tau,\tau-t,\theta_{-\tau}\omega,v_{0}) \bigr\Vert ^{2} +\bigl(\lambda+ \delta^{2}- \beta_{2}\delta\bigr) \bigl\Vert u(\tau,\tau-t,\theta_{-\tau} \omega,u_{0}) \bigr\Vert ^{2} + \bigl\Vert \Delta u( \tau,\tau-t,\omega,u_{0}) \bigr\Vert ^{2} \\& \quad \leq c+c \int^{\tau}_{-\infty}e^{\sigma(s-\tau)} \bigl\Vert g(x, s) \bigr\Vert ^{2}\,ds+c \int^{0}_{ -\infty}e^{\sigma s }\bigl( \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\vert \omega(s) \bigr\vert ^{p+1}\bigr)\,ds. \end{aligned}$$
(4.22)

Thus the proof is completed. □

Lemma 4.2

Under (3.4)(3.9) and (3.14)(3.17), for every\(\tau\in\mathbb{R}\), \(\omega\in\varOmega\), \(D=\{D(\tau,\omega):\tau\in\mathbb{R},\omega\in\varOmega\}\in \mathcal{D}\), there exists\(T=T(\tau,\omega,D)>0\)such that, for all\(t\geq T\), the solution of problem (3.10)(3.11) satisfies

$$ \bigl\Vert A^{\frac{1}{4}}Y\bigl(\tau,\tau-t,\theta_{-\tau} \omega,D(\tau-t, \theta_{-t}\omega)\bigr) \bigr\Vert ^{2}_{E}\leq R_{2}(\tau,\omega), $$

and\(R_{2}(\tau,\omega)\)is given by

$$\begin{aligned} R_{2}(\tau,\omega) =& ce^{-\sigma t}\bigl( \bigl\Vert A^{\frac{1}{4}}v_{0} \bigr\Vert ^{2}+ \bigl\Vert A^{ \frac{1}{4}}u_{0} \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}} u_{0} \bigr\Vert ^{2} \bigr) \\ &{}+ c \int^{\tau}_{-\infty}e^{\sigma(s-\tau)} \bigl\Vert g(x,s) \bigr\Vert _{1}^{2}\,ds+c \int^{0}_{-\infty}e^{\sigma s }\bigl(1+ \bigl\vert \omega(s) \bigr\vert ^{2} \bigr)\,ds. \end{aligned}$$
(4.23)

Proof

Taking the inner product of the second equation of (3.10) with \(A^{\frac{1}{2}}v\) in \(L^{2}(\mathbb{R}^{n})\), we find that

$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}- \delta \bigl\Vert A^{ \frac{1}{4}}v \bigr\Vert ^{2}+\bigl(\lambda+\delta^{2}\bigr) \bigl(u,A^{\frac{1}{2}}v\bigr)+ \bigl(Au,A^{ \frac{1}{2}}v\bigr)+ \bigl(f(x,u),A^{\frac{1}{2}}v\bigr) \\& \quad =\bigl(g(x,t),A^{\frac{1}{2}}v\bigr)-\bigl(h\bigl(v+ \phi\omega(t)- \delta u\bigr),A^{ \frac{1}{2}}v\bigr)+\delta\bigl(\phi,A^{\frac{1}{2}}v \bigr)\omega(t). \end{aligned}$$
(4.24)

Similar to the proof of Lemma 4.1, we have the following estimates:

$$\begin{aligned}& -\bigl(h\bigl(v+ \phi\omega(t)-\delta u\bigr),A^{\frac{1}{2}}v\bigr) \\& \quad = -\bigl(h\bigl(v+ \phi\omega(t)-\delta u\bigr)-h(0),A^{\frac{1}{2}}v \bigr) \\& \quad = -\bigl(h'(\vartheta) \bigl(v+ \phi\omega(t)-\delta u \bigr),A^{\frac{1}{2}}v\bigr) \\& \quad \leq -\beta_{1} \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}-\bigl(h'(\vartheta) \bigl( \phi \omega(t)-\delta u\bigr),A^{\frac{1}{2}}v\bigr) \\& \quad \leq -\beta_{1} \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}+\beta_{2} \bigl\vert \omega(t) \bigr\vert \bigl\Vert A^{ \frac{1}{4}}\phi \bigr\Vert \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert +h'(\vartheta)\delta\bigl(u,A^{ \frac{1}{2}}v\bigr) \\& \quad \leq -\beta_{1} \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}+ \frac{\beta_{1}-\delta}{6} \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}+c \bigl\vert \omega(t) \bigr\vert ^{2} \bigl\Vert A^{\frac{1}{4}}\phi \bigr\Vert ^{2}+h'( \vartheta)\delta\bigl(u,A^{\frac{1}{2}}v\bigr), \end{aligned}$$
(4.25)
$$\begin{aligned}& h'(\vartheta)\delta\bigl(u,A^{\frac{1}{2}}v\bigr) \\& \quad = h'(\vartheta)\delta\bigl(u,A^{\frac{1}{2}}u_{t}- \omega(t)A^{ \frac{1}{2}}\phi+\delta A^{\frac{1}{2}}u\bigr) \\& \quad \leq \beta_{2}\delta\cdot\frac{1}{2} \frac{d}{dt} \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+\beta_{2}\delta^{2} \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+ \beta_{2} \delta \bigl\vert \omega(t) \bigr\vert \bigl\Vert A^{\frac{1}{4}}\phi \bigr\Vert \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert \\& \quad \leq \beta_{2}\delta\cdot\frac{1}{2} \frac{d}{dt} \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+\beta_{2}\delta^{2} \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2} \\& \qquad {}+\frac{1}{6} \delta \bigl(\lambda+\delta^{2}-\beta_{2}\delta\bigr) \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+c \bigl\vert \omega(t) \bigr\vert ^{2} \bigl\Vert A^{\frac{1}{4}}\phi \bigr\Vert ^{2}, \end{aligned}$$
(4.26)
$$\begin{aligned}& \bigl(\lambda+\delta^{2}\bigr) \bigl(u,A^{\frac{1}{2}}v\bigr) \\& \quad = \bigl(\lambda+\delta^{2}\bigr) \bigl(u,A^{\frac{1}{2}}u_{t}- \omega(t)A^{ \frac{1}{2}}\phi+\delta A^{\frac{1}{2}}u\bigr) \\& \quad \geq \frac{1}{2}\bigl(\lambda+\delta^{2}\bigr) \frac{d}{dt} \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+\delta\bigl(\lambda+\delta^{2}\bigr) \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}-\bigl( \lambda+ \delta^{2}\bigr) \bigl\vert \omega(t) \bigr\vert \bigl\Vert A^{\frac{1}{4}}\phi \bigr\Vert \bigl\Vert A^{ \frac{1}{4}}u \bigr\Vert \\& \quad \geq \frac{1}{2}\bigl(\lambda+\delta^{2}\bigr) \frac{d}{dt} \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+\delta\bigl(\lambda+\delta^{2}\bigr) \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2} \\& \qquad {}- \frac{1}{6}\delta \bigl(\lambda+\delta^{2}-\beta_{2}\delta\bigr) \bigl\Vert A^{ \frac{1}{4}}u \bigr\Vert ^{2}-c \bigl\vert \omega(t) \bigr\vert ^{2} \bigl\Vert A^{\frac{1}{4}}\phi \bigr\Vert ^{2}, \end{aligned}$$
(4.27)
$$\begin{aligned}& \begin{aligned}[b] \bigl(Au,A^{\frac{1}{2}}v\bigr) &=\bigl(A u,A^{\frac{1}{2}} u_{t}-\omega(t)A^{ \frac{1}{2}}\phi+\delta A^{\frac{1}{2}} u \bigr) \\ &\geq \frac{1}{2}\frac{d}{dt} \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2}+\delta \bigl\Vert A^{ \frac{3}{4}} u \bigr\Vert ^{2}- \bigl\vert \omega(t) \bigr\vert \bigl\Vert A^{\frac{3}{4}} \phi \bigr\Vert \bigl\Vert A^{ \frac{3}{4}} u \bigr\Vert \\ &\geq \frac{1}{2}\frac{d}{dt} \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2}+ \frac{\delta}{2} \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2}-\frac{1}{2\delta} \bigl\vert \omega(t) \bigr\vert ^{2} \bigl\Vert A^{\frac{3}{4}}\phi \bigr\Vert ^{2}, \end{aligned} \end{aligned}$$
(4.28)
$$\begin{aligned}& \delta\bigl(\phi\omega(t),A^{\frac{1}{2}}v\bigr)\leq\delta \bigl\vert \omega(t) \bigr\vert \bigl\Vert A^{ \frac{1}{4}}\phi \bigr\Vert \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert \leq c \bigl\Vert A^{\frac{1}{4}} \phi \bigr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2}+\frac{\beta_{1}-\delta}{6} \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}, \end{aligned}$$
(4.29)
$$\begin{aligned}& \bigl(g,A^{\frac{1}{2}}v\bigr)\leq \Vert g \Vert _{1} \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert \leq \frac{3}{2(\beta_{1}-\delta)} \Vert g \Vert ^{2}_{1}+ \frac{\beta_{1}-\delta}{6} \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}. \end{aligned}$$
(4.30)

For the last term on the left-hand side of (4.24), thanks to (3.7), we have

$$\begin{aligned} - \bigl(f(x,u),A^{\frac{1}{2}}v \bigr) \leq& \bigl\vert \bigl(f(x,u),A^{\frac{1}{2}}v \bigr) \bigr\vert \\ =& \biggl\vert \int_{\mathbb{R}^{n}} \frac{\partial}{\partial x}f(x,u)\cdot A^{ \frac{1}{4}}v \,dx+ \int_{\mathbb{R}^{n}} \frac{\partial}{\partial u}f(x,u) \cdot A^{\frac{1}{4}}u\cdot A^{\frac{1}{4}}v \,dx \biggr\vert \\ \leq& \int_{\mathbb{R}^{n}} \biggl\vert \frac{\partial}{\partial x}f(x,u) \biggr\vert \cdot \bigl\vert A^{\frac{1}{4}}v \bigr\vert \,dx+ \int_{\mathbb{R}^{n}} \biggl\vert \frac{\partial}{\partial u}f(x,u) \biggr\vert \cdot \bigl\vert A^{\frac{1}{4}}u \bigr\vert \cdot \bigl\vert A^{ \frac{1}{4}}v \bigr\vert \,dx \\ \leq& \int_{\mathbb{R}^{n}} \vert \eta_{4} \vert \cdot \bigl\vert A^{\frac{1}{4}}v \bigr\vert \,dx+ \beta \int_{\mathbb{R}^{n}} \bigl\vert A^{\frac{1}{4}}u \bigr\vert \cdot \bigl\vert A^{\frac{1}{4}}v \bigr\vert \,dx \\ \leq& \Vert \eta_{4} \Vert \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert +\beta \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert \bigl\Vert A^{ \frac{1}{4}}v \bigr\Vert \\ \leq& c + \biggl(\delta+ \frac{3\beta^{2}}{2\delta(\lambda+\delta^{2}-\beta_{2}\delta)} \biggr) \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2} \\ &{}+\frac{1}{6}\delta \bigl(\lambda+\delta^{2}- \beta_{2}\delta\bigr) \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}. \end{aligned}$$
(4.31)

Plugging (4.25)–(4.31) into (4.24) and together with (3.15), we obtain

$$\begin{aligned}& \frac{1}{2}\frac{d}{dt}\bigl( \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}+\bigl(\lambda+ \delta^{2}- \beta_{2}\delta\bigr) \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2} \bigr) \\& \qquad {} +\delta\bigl( \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}+\bigl(\lambda+\delta^{2}-\beta_{2} \delta\bigr) \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}}u \bigr\Vert ^{2}\bigr) \\& \quad \leq \frac{\delta}{2} \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}+\frac{\delta}{2}\bigl( \lambda+\delta^{2}- \beta_{2}\delta\bigr) \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2} + \frac{\delta}{2} \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2} +c\bigl(1+ \bigl\vert \omega(t) \bigr\vert ^{2}+ \Vert g \Vert ^{2}_{1}\bigr), \end{aligned}$$
(4.32)

then

$$\begin{aligned}& \frac{d}{dt}\bigl( \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}+\bigl(\lambda+\delta^{2}-\beta_{2} \delta\bigr) \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2} \bigr) \\& \qquad {} +\sigma\bigl( \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}+\bigl(\lambda+\delta^{2}-\beta_{2} \delta\bigr) \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2}\bigr) \\& \quad \leq c\bigl(1+ \bigl\vert \omega(t) \bigr\vert ^{2}+ \Vert g \Vert ^{2}_{1} \bigr). \end{aligned}$$
(4.33)

Multiplying (4.33) by \(e^{\sigma t}\) and then integrating over \((\tau-t,\tau)\), we have

$$\begin{aligned}& e^{\sigma\tau} \bigl( \bigl\Vert A^{\frac{1}{4}}v(\tau,\tau-t, \omega,v_{0}) \bigr\Vert ^{2}+\bigl( \lambda+ \delta^{2}-\beta_{2}\delta\bigr) \bigl\Vert A^{\frac{1}{4}} u(\tau, \tau-t,\omega,u_{0}) \bigr\Vert ^{2} \\& \qquad {}+ \bigl\Vert A^{\frac{3}{4}} u(\tau,\tau-t, \omega,u_{0}) \bigr\Vert ^{2} \bigr) \\& \quad \leq e^{\sigma(\tau-t)}\bigl( \bigl\Vert A^{\frac{1}{4}}v_{0} \bigr\Vert ^{2}+\bigl(\lambda+ \delta^{2}- \beta_{2}\delta\bigr) \bigl\Vert A^{\frac{1}{4}} u_{0} \bigr\Vert ^{2}+ \bigl\Vert A^{ \frac{3}{4}} u_{0} \bigr\Vert ^{2} \bigr) \\& \qquad {}+c \int^{\tau}_{\tau-t}e^{\sigma s}\bigl(1+ \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\Vert g(x,s) \bigr\Vert _{1}^{2} \bigr)\,ds. \end{aligned}$$

Replacing ω by \(\theta_{-\tau}\omega\) in the above, we obtain, for every \(t\in\mathbb{R}^{+}\), \(\tau\in\mathbb{R}\), and \(\omega\in \varOmega\),

$$\begin{aligned}& \bigl\Vert A^{\frac{1}{4}}v(\tau,\tau-t,\theta_{-\tau} \omega,v_{0}) \bigr\Vert ^{2}+\bigl( \lambda+ \delta^{2}-\beta_{2}\delta\bigr) \bigl\Vert A^{\frac{1}{4}}u(\tau,\tau-t, \theta_{-\tau}\omega,u_{0}) \bigr\Vert ^{2} \\& \qquad {} + \bigl\Vert A^{\frac{3}{4}} u(\tau,\tau-t,\theta_{-\tau} \omega,u_{0}) \bigr\Vert ^{2} \\& \quad \leq e^{-\sigma t}\bigl( \bigl\Vert A^{\frac{1}{4}}v_{0} \bigr\Vert ^{2}+\bigl(\lambda+\delta^{2}- \beta_{2}\delta\bigr) \bigl\Vert A^{\frac{1}{4}}u_{0} \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}} u_{0} \bigr\Vert ^{2} \bigr) \\& \qquad {}+ \frac{3}{ (\beta_{1}-\delta)} \int^{\tau}_{\tau-t}e^{\sigma(s- \tau)} \bigl\Vert g(x,s) \bigr\Vert _{1}^{2}\,ds+c \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)}\bigl(1+ \bigl\vert \theta_{-\tau}\omega(s) \bigr\vert ^{2} \bigr)\,ds \\& \quad \leq ce^{-\sigma t}\bigl( \bigl\Vert A^{\frac{1}{4}}v_{0} \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{1}{4}}u_{0} \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}} u_{0} \bigr\Vert ^{2} \bigr) \\& \qquad {}+ c \int^{\tau}_{-\infty}e^{\sigma(s-\tau)} \bigl\Vert g(x,s) \bigr\Vert _{1}^{2}\,ds+c \int^{0}_{-\infty}e^{\sigma s }\bigl(1+ \bigl\vert \omega(s) \bigr\vert ^{2} \bigr)\,ds. \end{aligned}$$
(4.34)

Thus the proof is completed. □

Lemma 4.3

Under (3.4)(3.9) and (3.14)(3.17), for every\(\eta>0\), \(\tau\in\mathbb{R}\), \(\omega\in\varOmega\), \(D=\{D(\tau,\omega):\tau\in\mathbb{R},\omega\in\varOmega\}\in \mathcal{D}\), there exist\(T=T(\tau,\omega,D,\eta)>0\), \(K=K(\tau,\omega,\eta)\geq1\)such that, for all\(t\geq T\), \(k\geq K\), the solution of problem (3.10)(3.11) satisfies

$$ \bigl\Vert Y\bigl(\tau,\tau-t,\theta_{-\tau}\omega,D(\tau-t, \theta_{-t} \omega)\bigr) \bigr\Vert ^{2}_{E(\mathbb{R}^{n}\setminus\mathbb{B}_{k})} \leq\eta, $$
(4.35)

where for\(k\geq1\), \(\mathbb{B}_{k}=\{x\in\mathbb{R}^{n}:|x| \leq k\}\)and\(\mathbb{R}^{n}\setminus\mathbb{B}_{k}\)is the complement of\(\mathbb{B}_{k}\).

Proof

Choose a smooth function ρ such that \(0\leq\rho\leq1\) for \(s\in\mathbb{R}\), and

$$ \rho(s)= \textstyle\begin{cases} 0, & \text{if } 0\leq \vert s \vert \leq1, \\ 1, & \text{if } \vert s \vert \geq2, \end{cases} $$
(4.36)

and there exist constants \(\mu_{1}\), \(\mu_{2}\), \(\mu_{3}\), \(\mu _{4}\) such that \(|\rho'(s)|\leq\mu_{1}\), \(|\rho''(s)|\leq\mu_{2}\), \(|\rho'''(s)| \leq\mu_{3}\), \(|\rho''''(s)|\leq\mu_{4}\) for \(s\in\mathbb{R}\). Taking the inner product of the second equation of (3.10) with \(\rho(\frac{|x|^{2}}{k^{2}})v\) in \(L^{2}(\mathbb{R}^{n})\), we obtain

$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert v \vert ^{2}\,dx- \delta \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert v \vert ^{2}\,dx \\& \qquad {} +\bigl(\lambda+\delta^{2}\bigr) \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)uv\,dx+ \int_{\mathbb{R}^{n}}(Au)\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)v\,dx+ \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u)v\,dx \\& \quad =\delta \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \omega(t) v\,dx - \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) h\bigl(v+ \phi\omega(t)-\delta u\bigr)v\,dx \\& \qquad {}+ \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)g(x,t)v\,dx. \end{aligned}$$
(4.37)

Similar to (4.5), we have

$$\begin{aligned}& - \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) h\bigl(v+ \phi\omega(t)- \delta u\bigr)v\,dx \\& \quad =- \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl(h \bigl(v+\phi \omega(t)-\delta u\bigr)-h(0)\bigr)v\,dx \\& \quad \leq-\beta_{1} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert v \vert ^{2}\,dx+h'( \vartheta)\delta \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)uv\,dx \\& \qquad {}+ \beta_{2} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \phi \vert \bigl\vert \omega(t) \bigr\vert \vert v \vert \,dx. \end{aligned}$$
(4.38)

Taking (4.38) into (4.37), we have

$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert v \vert ^{2}\,dx-(\delta-\beta_{1}) \int_{\mathbb{R}^{n}} \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert v \vert ^{2}\,dx+ \int_{\mathbb{R}^{n}}(Au) \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)v\,dx \\& \qquad {} +\bigl(\lambda+\delta^{2}-h'(\vartheta) \delta\bigr) \int_{\mathbb{R}^{n}} \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)uv\,dx + \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u)v\,dx \\& \quad \leq (1+\delta+\beta_{2}) \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \phi \vert \bigl\vert \omega(t) \bigr\vert \vert v \vert \,dx+ \int_{\mathbb{R}^{n}} \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)g(x,t)v\,dx \\& \quad \leq \frac{\beta_{1}-\delta}{3} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert v \vert ^{2}\,dx+c \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \phi \vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2} \,dx \\& \qquad {}+ \int_{\mathbb{R}^{n}} \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)g(x,t)v\,dx. \end{aligned}$$
(4.39)

For the fourth term on the left-hand side of (4.39), we have

$$\begin{aligned}& \bigl(\lambda+\delta^{2}-h'(\vartheta)\delta\bigr) \int_{\mathbb{R}^{n}} \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)uv\,dx \\& \quad = \bigl(\lambda+\delta^{2}-h'(\vartheta)\delta \bigr) \int_{\mathbb{R}^{n}} \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)u\biggl( \frac{du}{dt}+\delta u-\phi\omega(t)\biggr)\,dx \\& \quad = \bigl(\lambda+\delta^{2}-h'(\vartheta)\delta \bigr) \int_{\mathbb{R}^{n}} \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \biggl( \frac{1}{2}\frac{d}{dt} u^{2}+ \delta u^{2}- \phi\omega(t)u\biggr)\,dx \\& \quad \geq \bigl(\lambda+\delta^{2}-\beta_{2}\delta \bigr) \biggl(\frac{1}{2} \frac{d}{dt} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert u \vert ^{2} \,dx+ \delta \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert u \vert ^{2}\,dx\biggr) \\& \qquad {} -\bigl(\lambda+\delta^{2}-\beta_{1}\delta \bigr) \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \phi \vert \bigl\vert \omega(t) \bigr\vert \vert u \vert \,dx \\& \quad \geq \bigl(\lambda+\delta^{2}-\beta_{2}\delta \bigr) \biggl(\frac{1}{2} \frac{d}{dt} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert u \vert ^{2} \,dx+ \delta \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert u \vert ^{2}\,dx\biggr) \\& \qquad {} -\frac{\delta}{2}\bigl(\lambda+\delta^{2}- \beta_{2}\delta\bigr) \int_{ \mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert u \vert ^{2}\,dx-c \int_{ \mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \phi \vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2} \,dx. \end{aligned}$$
(4.40)

For the third term on the left-hand side of (4.39), we have

$$\begin{aligned}& \int_{\mathbb{R}^{n}}(Au)\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)v\,dx \\& \quad = \int_{\mathbb{R}^{n}}(Au)\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \biggl( \frac{du}{dt}+\delta u-\phi\omega(t)\biggr)\,dx \\& \quad = \int_{\mathbb{R}^{n}}\bigl(\Delta^{2}u\bigr)\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \biggl( \frac{du}{dt}+\delta u-\phi\omega(t) \biggr)\,dx \\& \quad = \int_{\mathbb{R}^{n}}(\Delta u)\Delta\biggl(\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \biggl( \frac{du}{dt}+\delta u-\phi\omega(t) \biggr)\biggr)\,dx \\& \quad = \int_{\mathbb{R}^{n}}(\Delta u) \biggl(\biggl(\frac{2}{k^{2}} \rho'\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)+\frac{4x^{2}}{k^{4}} \rho''\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\biggr) \biggl(\frac{du}{dt}+\delta u-\phi\omega(t)\biggr) \\& \qquad {} +2\cdot\frac{2 \vert x \vert }{k^{2}}\rho'\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\nabla\biggl( \frac{du}{dt}+\delta u-\phi \omega(t)\biggr)+\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \Delta\biggl( \frac{du}{dt}+\delta u-\phi\omega(t)\biggr)\biggr)\,dx \\& \quad \geq - \int_{k< x< \sqrt{2}k}\biggl(\frac{2\mu_{1}}{k^{2}}+ \frac{4\mu_{2}x^{2}}{k^{4}} \biggr) \bigl\vert (\Delta u)v \bigr\vert \,dx- \int_{k< x< \sqrt{2}k} \frac{4\mu_{1}x}{k^{2}} \bigl\vert (\Delta u) ( \nabla v) \bigr\vert \,dx \\& \qquad {} +\frac{1}{2}\frac{d}{dt} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert ^{2}\,dx+\delta \int_{\mathbb{R}^{n}} \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert ^{2}\,dx \\& \qquad {}- \int_{\mathbb{R}^{n}} \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert \vert \Delta\phi \vert \bigl\vert \omega(t) \bigr\vert \,dx \\& \quad \geq - \int_{\mathbb{R}^{n}}\biggl(\frac{2\mu_{1}+8\mu_{2}}{k^{2}}\biggr) \bigl\vert ( \Delta u)v \bigr\vert \,dx- \int_{\mathbb{R}^{n}}\frac{4\sqrt{2}\mu_{1}}{k} \bigl\vert ( \Delta u) ( \nabla v) \bigr\vert \,dx \\& \qquad {}+\frac{1}{2}\frac{d}{dt} \int_{\mathbb{R}^{n}} \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert ^{2}\,dx +\delta \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert ^{2}\,dx \\& \qquad {}- \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert \vert \Delta \phi \vert \bigl\vert \omega(t) \bigr\vert \,dx \\& \quad \geq -\frac{\mu_{1}+4\mu_{2}}{k^{2}}\bigl( \Vert \Delta u \Vert ^{2}+ \Vert v \Vert ^{2}\bigr)- \frac{4\sqrt{2}\mu_{1}}{k} \Vert \Delta u \Vert \Vert \nabla v \Vert +\frac{1}{2} \frac{d}{dt} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert ^{2}\,dx \\& \qquad {} +\delta \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert ^{2}\,dx- \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert \vert \Delta \phi \vert \bigl\vert \omega(t) \bigr\vert \,dx \\& \quad \geq -\frac{\mu_{1}+4\mu_{2}}{k^{2}}\bigl( \Vert \Delta u \Vert ^{2}+ \Vert v \Vert ^{2}\bigr)- \frac{2\sqrt{2}\mu_{1}}{k} \bigl( \Vert \Delta u \Vert ^{2}+ \Vert \nabla v \Vert ^{2}\bigr) \\& \qquad {} + \frac{1}{2}\frac{d}{dt} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert ^{2}\,dx +\delta \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert ^{2}\,dx \\& \qquad {}- \frac{\delta}{2} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert ^{2}\,dx-c \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta\phi \vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2}\,dx, \end{aligned}$$
(4.41)
$$\begin{aligned}& \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u)v\,dx \\& \quad = \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u) \biggl( \frac{du}{dt}+\delta u-\phi\omega(t)\biggr)\,dx \\& \quad = \frac{d}{dt} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)F(x,u)\,dx+ \delta \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u)u\,dx \\& \qquad {}- \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u) \phi \omega(t) \,dx. \end{aligned}$$
(4.42)

Similar to (4.12) and (4.13) in Lemma 4.1, we have

$$\begin{aligned}& \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u)u\,dx \geq c_{2} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)F(x,u)\,dx + \int_{ \mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \eta_{2}\,dx, \end{aligned}$$
(4.43)
$$\begin{aligned}& \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u) \phi \omega(t)\,dx \\& \quad \leq \frac{1}{2} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \eta_{1} \vert ^{2}\,dx+ \frac{1}{2} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \phi \vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2} \,dx \\& \qquad {}+\frac{\delta c_{2}}{2} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl(F(x,u)+\eta_{3}\bigr)\,dx+c \int_{\mathbb{R}^{n}} \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \phi \vert ^{p+1} \bigl\vert \omega(t) \bigr\vert ^{p+1}\,dx, \end{aligned}$$
(4.44)
$$\begin{aligned}& \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)g(x,t)v\,dx \\& \quad \leq \frac{3}{2(\beta_{1}-\delta)} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl\vert g(x,t) \bigr\vert ^{2}\,dx +\frac{\beta_{1}-\delta}{6} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert v \vert ^{2}\,dx. \end{aligned}$$
(4.45)

By (4.38)–(4.45), we have

$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert v \vert ^{2}+\bigl(\lambda+\delta^{2}- \beta_{2} \delta\bigr) \vert u \vert ^{2}+ \vert \Delta u \vert ^{2}+2F(x,u)\bigr)\,dx \\& \qquad {} +\delta \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert v \vert ^{2}+\bigl( \lambda+\delta^{2}- \beta_{2}\delta\bigr) \vert u \vert ^{2}+ \vert \Delta u \vert ^{2}\bigr)\,dx \\& \qquad {}+ \frac{\delta c_{2}}{2} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)F(x,u)\,dx \\& \quad \leq \frac{\delta}{2} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert v \vert ^{2}\,dx+\frac{2\delta-\beta_{1}}{2} \int_{ \mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert v \vert ^{2}\,dx \\& \qquad {}+ \frac{\delta}{2}\bigl(\lambda+ \delta^{2}-\beta_{2}\delta\bigr) \int_{ \mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert u \vert ^{2}\,dx \\& \qquad {} +\frac{\delta}{2} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert ^{2}\,dx+ \frac{\mu_{1}+4\mu_{2}}{k^{2}}\bigl( \Vert \Delta u \Vert ^{2}+ \Vert v \Vert ^{2}\bigr) \\& \qquad {}+ \frac{2\sqrt{2}\mu_{1}}{k}\bigl( \Vert \Delta u \Vert ^{2}+ \Vert \nabla v \Vert ^{2}\bigr) \\& \qquad {} +c \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert \eta_{1} \vert ^{2}+ \vert \eta_{2} \vert + \vert \eta_{3} \vert + \vert g \vert ^{2}+ \bigl\vert \omega(t) \bigr\vert ^{p+1} \vert \phi \vert ^{p+1}\bigr)\,dx \\& \qquad {} +c \bigl\vert \omega(t) \bigr\vert ^{2} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert \phi \vert ^{2}+ \vert \Delta\phi \vert ^{2} \bigr)\,dx. \end{aligned}$$
(4.46)

Since \(\phi\in H^{2}(\mathbb{R}^{n})\), \(\eta_{1} \in L^{2}( \mathbb{R}^{n})\), \(\eta_{2} \in L^{1}(\mathbb{R}^{n})\), \(\eta_{3} \in L^{1}(\mathbb{R}^{n})\), we obtain that there exists \(K_{1}=K_{1}(\tau,\eta)\geq1\) such that, for all \(k\geq K_{1}\),

$$\begin{aligned}& c\int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert \eta_{1} \vert ^{2}+ \vert \eta_{2} \vert + \vert \eta_{3} \vert + \bigl\vert \omega(t) \bigr\vert ^{p+1} \vert \phi \vert ^{p+1} \bigr)\,dx \\& \qquad {}+c \bigl\vert \omega(t) \bigr\vert ^{2} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert \phi \vert ^{2}+ \vert \Delta\phi \vert ^{2} \bigr)\,dx \\& \quad = c \int_{ \vert x \vert \geq k}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert \eta_{1} \vert ^{2}+ \vert \eta_{2} \vert + \vert \eta_{3} \vert + \bigl\vert \omega(t) \bigr\vert ^{p+1} \vert \phi \vert ^{p+1} \bigr)\,dx \\& \qquad {}+c \bigl\vert \omega(t) \bigr\vert ^{2} \int_{ \vert x \vert \geq k}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert \phi \vert ^{2}+ \vert \Delta \phi \vert ^{2}\bigr)\,dx \\& \quad \leq c \int_{ \vert x \vert \geq k} \bigl( \vert \eta_{1} \vert ^{2}+ \vert \eta_{2} \vert + \vert \eta_{3} \vert + \bigl\vert \omega(t) \bigr\vert ^{p+1} \vert \phi \vert ^{p+1}\bigr)\,dx \\& \qquad {}+c \bigl\vert \omega(t) \bigr\vert ^{2} \int_{ \vert x \vert \geq k} \bigl( \vert \phi \vert ^{2}+ \vert \Delta\phi \vert ^{2}\bigr)\,dx \\& \quad \leq c\eta\bigl(1+ \bigl\vert \omega(t) \bigr\vert ^{2}+ \bigl\vert \omega(t) \bigr\vert ^{p+1}\bigr), \end{aligned}$$
(4.47)

along with

$$ c \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}} \biggr)g^{2}(x,t)\,dx\leq c \int_{ \vert x \vert \geq k}g^{2}(x,t)\,dx, $$
(4.48)

we have that, for all \(k\geq K_{1}\),

$$\begin{aligned}& \frac{d}{dt} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert v \vert ^{2}+\bigl( \lambda+\delta^{2}- \beta_{2}\delta\bigr) \vert u \vert ^{2}+ \vert \Delta u \vert ^{2}+2F(x,u)\bigr)\,dx \\& \qquad {} +\sigma \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert v \vert ^{2}+\bigl( \lambda+\delta^{2}- \beta_{2}\delta\bigr) \vert u \vert ^{2}+ \vert \Delta u \vert ^{2}+2F(x,u)\bigr)\,dx \\& \quad \leq \frac{2\mu_{1}+8\mu_{2}}{k^{2}}\bigl( \Vert \Delta u \Vert ^{2}+ \Vert v \Vert ^{2}\bigr) + \frac{4\sqrt{2}\mu_{1}}{k}\bigl( \Vert \Delta u \Vert ^{2}+ \Vert \nabla v \Vert ^{2}\bigr) \\& \qquad {} +c\eta\bigl(1+ \bigl\vert \omega(t) \bigr\vert ^{2}+ \bigl\vert \omega(t) \bigr\vert ^{p+1}\bigr)+c \int_{ \vert x \vert \geq k}g^{2}(x,t)\,dx. \end{aligned}$$
(4.49)

Multiplying (4.49) by \(e^{\sigma t}\) and then integrating over \((\tau-t,\tau)\), we find

$$\begin{aligned}& \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \bigl\vert v(\tau,\tau-t, \omega,v_{0}) \bigr\vert ^{2}+ \bigl(\lambda+\delta^{2}-\beta_{2}\delta\bigr) \bigl\vert u(\tau, \tau-t,\omega,u_{0}) \bigr\vert ^{2} \\& \qquad {} + \bigl\vert \Delta u(\tau,\tau-t,\omega,u_{0}) \bigr\vert ^{2}+2F\bigl(x,u( \tau,\tau-t,\omega,u_{0})\bigr) \bigr)\,dx \\& \quad \leq e^{-\sigma t} \int_{\mathbb{R}^{n}}\rho\biggl(\frac { \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert v_{0} \vert ^{2}+\bigl( \lambda+ \delta^{2}-\beta_{2}\delta\bigr) \vert u_{0} \vert ^{2} + \vert \Delta u_{0} \vert ^{2}+2F(x,u_{0})\bigr)\,dx \\& \qquad {} + \frac{2\mu_{1}+8\mu_{2}}{k^{2}} \int^{\tau}_{\tau-t}e^{\sigma(s- \tau)}\bigl( \bigl\vert \Delta u(s,\tau-t,\omega,u_{0}) \bigr\vert ^{2}+ \bigl\vert v(s,\tau-t,\omega ,v_{0}) \bigr\vert ^{2}\bigr)\,ds \\& \qquad {} +\frac{4\sqrt{2}\mu_{1}}{k} \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)}\bigl( \bigl\vert \Delta u(s,\tau-t,\omega,u_{0}) \bigr\vert ^{2}+ \bigl\vert \nabla v(s,\tau-t,\omega ,v_{0}) \bigr\vert ^{2}\bigr)\,ds \\& \qquad {} +c\frac{\eta}{\sigma}+c\eta \int^{\tau}_{\tau-t}e^{\sigma(s- \tau)}\bigl( \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\vert \omega(s) \bigr\vert ^{p+1}\bigr)\,ds \\& \qquad {}+c \int^{\tau}_{\tau-t} \int_{ \vert x \vert \geq k}e^{\sigma(s-\tau)}g^{2}(x,s)\,dx\,ds. \end{aligned}$$
(4.50)

Replacing ω by \(\theta_{-\tau}\omega\), it then follows from (4.50) that

$$\begin{aligned}& \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \bigl\vert v(\tau,\tau-t, \theta_{-\tau}\omega,v_{0}) \bigr\vert ^{2}+\bigl(\lambda+\delta^{2}- \beta_{2} \delta\bigr) \bigl\vert u(\tau,\tau-t, \theta_{-\tau}\omega,u_{0}) \bigr\vert ^{2} \\& \qquad {} + \bigl\vert \Delta u(\tau,\tau-t,\theta_{-\tau} \omega,u_{0}) \bigr\vert ^{2}+2F\bigl(x,u( \tau,\tau-t, \theta_{-\tau}\omega,u_{0})\bigr)\bigr)\,dx \\& \quad \leq c\eta+ e^{-\sigma t} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert v_{0} \vert ^{2}+\bigl(\lambda+ \delta^{2}-\beta_{2} \delta\bigr) \vert u_{0} \vert ^{2} + \vert \Delta u_{0} \vert ^{2}+2F(x,u_{0})\bigr)\,dx \\& \qquad {} +\frac{2\mu_{1}+8\mu_{2}}{k^{2}} \int^{\tau}_{\tau-t}e^{\sigma(s- \tau)}\bigl( \bigl\vert \Delta u(s,\tau-t,\theta_{-\tau}\omega,u_{0}) \bigr\vert ^{2}+ \bigl\vert v(s, \tau-t,\theta_{-\tau} \omega,v_{0}) \bigr\vert ^{2}\bigr)\,ds \\& \qquad {} +\frac{4\sqrt{2}\mu_{1}}{k} \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)}\bigl( \bigl\vert \Delta u(s,\tau-t,\theta_{-\tau}\omega,u_{0}) \bigr\vert ^{2}+ \bigl\vert \nabla v(s, \tau-t,\theta_{-\tau} \omega,v_{0}) \bigr\vert ^{2}\bigr)\,ds \\& \qquad {} +c\eta \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)}\bigl( \bigl\vert \theta_{-\tau} \omega(s) \bigr\vert ^{2}+ \bigl\vert \theta_{-\tau}\omega(s) \bigr\vert ^{p+1}\bigr) \,ds+c \int^{\tau}_{ \tau-t} \int_{ \vert x \vert \geq k}e^{\sigma(s-\tau)}g^{2}(x,s)\,dx\,ds \\& \quad \leq c\eta+ e^{-\sigma t} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert v_{0} \vert ^{2}+\bigl(\lambda+ \delta^{2}-\beta_{2} \delta\bigr) \vert u_{0} \vert ^{2} + \vert \Delta u_{0} \vert ^{2}+2F(x,u_{0})\bigr)\,dx \\& \qquad {} +\frac{2\mu_{1}+8\mu_{2}}{k^{2}} \int^{\tau}_{\tau-t}e^{\sigma(s- \tau)}\bigl( \bigl\vert \Delta u(s,\tau-t,\theta_{-\tau}\omega,u_{0}) \bigr\vert ^{2}+ \bigl\vert v(s, \tau-t,\theta_{-\tau} \omega,v_{0}) \bigr\vert ^{2}\bigr)\,ds \\& \qquad {} +\frac{4\sqrt{2}\mu_{1}}{k} \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)}\bigl( \bigl\vert \Delta u(s,\tau-t,\theta_{-\tau}\omega,u_{0}) \bigr\vert ^{2}+ \bigl\vert \nabla v(s, \tau-t,\theta_{-\tau} \omega,v_{0}) \bigr\vert ^{2}\bigr)\,ds \\& \qquad {} +c\eta \int^{0}_{-\infty}e^{\sigma s}\bigl( \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\vert \omega (s) \bigr\vert ^{p+1}\bigr)\,ds+c \int^{\tau}_{-\infty} \int_{ \vert x \vert \geq k}e^{\sigma(s-\tau)}g^{2}(x,s)\,dx\,ds. \end{aligned}$$
(4.51)

By (3.17), we see that there exists \(K_{2}=K_{2}(\tau,\eta)\geq K_{1}\) such that, for all \(k\geq K_{2}\),

$$ c \int^{\tau}_{-\infty} \int_{|x|\geq k}e^{\sigma(s-\tau)}g^{2}(x,s)\,dx\,ds \leq\eta. $$
(4.52)

It follows from (4.51)–(4.52), Lemma 4.1, and Lemma 4.2 that there exists \(T_{1}=T_{1}(\tau,\omega, D,\eta)>0\) such that, for all \(t\geq T_{1}\), \(k\geq K_{2}\),

$$\begin{aligned}& \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \bigl\vert v(\tau,\tau-t, \theta_{-\tau}\omega,v_{0}) \bigr\vert ^{2}+\bigl(\lambda+\delta^{2}- \beta_{2} \delta\bigr) \bigl\vert u(\tau,\tau-t, \theta_{-\tau}\omega,u_{0}) \bigr\vert ^{2} \\& \qquad {} + \bigl\vert \Delta u(\tau,\tau-t,\theta_{-\tau} \omega,u_{0}) \bigr\vert ^{2}+2F\bigl(x,u( \tau,\tau-t, \theta_{-\tau}\omega,u_{0})\bigr)\bigr)\,dx \\& \quad \leq c\eta\biggl(1+ \int^{0}_{-\infty}e^{\sigma s}\bigl( \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\vert \omega(s) \bigr\vert ^{p+1}\bigr)\,ds+ \int^{\tau}_{-\infty} \int_{ \vert x \vert \geq k}e^{ \sigma(s-\tau)}g^{2}(x,s)\,dx\,ds \biggr), \end{aligned}$$
(4.53)

where \((u_{0},v_{0})^{\top}\in D(\tau-t,\theta_{-t}\omega)\).

Note that (3.6) holds, then we find

$$ -2 \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)F(x,u)\,dx \leq2 \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \eta_{3}\,dx\leq2 \int_{ \vert x \vert \geq k}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \eta_{3}\,dx, $$

from which along with \(\eta_{3}\in L^{1}(\mathbb{R}^{n})\), we see that there exists \(K_{3}=K_{3}(\tau,\eta)\geq K_{2}\) such that, for all \(k\geq K_{3}\),

$$ -2 \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)F(x,u)\,dx \leq \eta. $$
(4.54)

Then from (4.53)–(4.54) we know that there exists \(T_{2}=T_{2}(\tau,\omega,D,\eta)>T_{1}\) such that, for all \(t\geq T_{2}\) and \(k\geq K_{3}\),

$$\begin{aligned}& \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \bigl\vert v(\tau,\tau-t, \theta_{-\tau}\omega,v_{0}) \bigr\vert ^{2}+\bigl(\lambda+\delta^{2}- \beta_{2} \delta\bigr) \bigl\vert u(\tau,\tau-t, \theta_{-\tau}\omega,u_{0}) \bigr\vert ^{2} \\& \qquad {} + \bigl\vert \Delta u(\tau,\tau-t,\theta_{-\tau} \omega,u_{0}) \bigr\vert ^{2} \bigr)\,dx \\& \quad \leq c\eta\biggl(1+ \int^{0}_{-\infty}e^{\sigma s}\bigl( \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\vert \omega(s) \bigr\vert ^{p+1}\bigr)\,ds+ \int^{\tau}_{-\infty} \int_{ \vert x \vert \geq k}e^{ \sigma(s-\tau)}g^{2}(x,s)\,dx\,ds \biggr), \end{aligned}$$
(4.55)

which completes the proof. □

Let \(\widehat{\rho}=1-\rho\) with ρ given by (4.36). Fix \(k\geq1\), and set

$$ \textstyle\begin{cases} \widehat{u}(t,\tau,\omega,\widehat{u_{0}})=\widehat{\rho}( \frac{ \vert x \vert ^{2}}{k^{2}})u(t,\tau,\omega,u_{0}), \\ \widehat{v}(t,\tau,\omega,\widehat{v_{0}})=\widehat{\rho}( \frac{ \vert x \vert ^{2}}{k^{2}})v(t,\tau,\omega,v_{0}). \end{cases} $$
(4.56)

By (3.10)–(3.11) we find that û and satisfy the following system in \(\mathbb{B}_{2k}=\{x\in\mathbb{R}^{n}:|x|<2k\}\):

$$\begin{aligned}& \frac{d\widehat{u}}{dt}=\widehat{v}+ \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}} \biggr)\phi\omega(t)-\delta\widehat{u}, \end{aligned}$$
(4.57)
$$\begin{aligned}& \frac{d\widehat{v}}{dt}-\delta\widehat{v}+\bigl(\delta^{2}+\lambda+A \bigr) \widehat{u}+\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u) \\& \quad = \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)g(x,t)-\widehat{\rho} \biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)h\bigl(v+ \phi\omega(t) -\delta u\bigr)+(1+ \delta) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\omega(t) \\& \qquad {} +4\Delta\nabla\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \nabla u+6 \Delta\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \Delta u \\& \qquad {}+4\nabla \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \Delta\nabla u+u \Delta^{2} \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr), \end{aligned}$$
(4.58)

with boundary conditions

$$ \widehat{u}=\widehat{v}=0 \quad \text{for } \vert x \vert =2k. $$
(4.59)

Let \(\{e_{n}\}^{\infty}_{n=1}\) be an orthonormal basis of \(L^{2}(\mathbb{B}_{2k})\) such that \(Ae_{n}=\lambda_{n}e_{n}\) with zero boundary condition in \(\mathbb{B}_{2k}\). Given n, let \(X_{n}=\operatorname{span}\{e_{1},\ldots,e_{n}\}\) and \(P_{n}:L^{2}(\mathbb{B}_{2k})\rightarrow X_{n}\) be the projection operator.

Lemma 4.4

Under (3.4)(3.9) and (3.14)(3.17), for every\(\eta>0\), \(\tau\in\mathbb{R}\), \(\omega\in\varOmega\), \(D=\{D(\tau,\omega):\tau\in\mathbb{R},\omega\in\varOmega\}\in \mathcal{D}\), there exist\(T=T(\tau,\omega,D,\eta)>0\), \(K=K(\tau,\omega,\eta)\geq1\), and\(N=N(\tau,\omega,\eta)\geq1\)such that, for all\(t\geq T\), \(k\geq K\), and\(n\geq N\), the solution of problem (4.57)(4.59) satisfies

$$ \bigl\Vert (I-P_{n})\widehat{Y}\bigl(\tau,\tau-t, \theta_{-\tau}\omega,D(\tau-t, \theta_{-\tau}\omega)\bigr) \bigr\Vert ^{2}_{E (\mathbb{B}_{2k})}\leq\eta. $$

Proof

Let \(\widehat{u}_{n,1}=P_{n}\widehat{u}\), \(\widehat{u}_{n,2}=(I-P_{n}) \widehat{u}\), \(\widehat{v}_{n,1}=P_{n}\widehat{v}\), \(\widehat{v}_{n,2}=(I-P_{n}) \widehat{v}\). Applying \(I-P_{n}\) to (4.57), we obtain

$$ \widehat{v}_{n,2}=\frac{d\widehat{u}_{n,2}}{dt}+\delta\widehat {u}_{n,2}-(I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\omega(t). $$
(4.60)

Applying \(I-P_{n}\) to (4.58) and taking the inner product with \(\widehat{v}_{n,2}\) in \(L^{2}(\mathbb{B}_{2k})\), we have

$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \Vert \widehat{v}_{n,2} \Vert ^{2}- \delta \Vert \widehat{v}_{n,2} \Vert ^{2} +\bigl(\lambda+\delta^{2}+A\bigr) ( \widehat{u}_{n,2}, \widehat{v}_{n,2}) +\biggl( \widehat{ \rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u), \widehat{v}_{n,2} \biggr) \\& \quad = \biggl( (I-P_{n}) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}} \biggr)g(x,t), \widehat{v}_{n,2}\biggr)+(1+\delta) \biggl(\widehat{ \rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \phi\omega(t),\widehat{v}_{n,2} \biggr) \\& \qquad {} - (I-P_{n})\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}} \biggr) \bigl(h\bigl(v+\phi\omega(t)- \delta u\bigr),\widehat{v}_{n,2} \bigr) \\& \qquad {} + \biggl(4\Delta\nabla\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}} \biggr) \nabla u+6 \Delta\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \Delta u \\& \qquad {}+4\nabla \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \Delta\nabla u+u \Delta^{2} \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr), \widehat{v}_{n,2}\biggr). \end{aligned}$$
(4.61)

Substituting \(\widehat{v}_{n,2}\) in (4.60) into the third term on the left-hand side of (4.61), we have

$$\begin{aligned} \bigl(\lambda+\delta^{2}\bigr) (\widehat{u}_{n,2}, \widehat{v}_{n,2}) =&\biggl( \widehat{u}_{n,2}, \frac{d\widehat{u}_{n,2}}{dt}+\delta\widehat {u}_{n,2}-(I-P_{n}) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\omega(t)\biggr) \\ \geq&\frac{1}{2}\bigl(\lambda+\delta^{2}\bigr) \frac{d}{dt} \Vert \widehat{u}_{n,2} \Vert ^{2}+\delta\bigl(\lambda+\delta^{2}\bigr) \Vert \widehat{u}_{n,2} \Vert ^{2} -\frac{1}{4}\delta\bigl(\lambda+\delta^{2}- \beta_{2}\delta\bigr) \Vert \widehat{u}_{n,2} \Vert ^{2} \\ &{}-c \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2}, \end{aligned}$$
(4.62)

and then

$$\begin{aligned} (A\widehat{u}_{n,2},\widehat{v}_{n,2}) =&\bigg(\Delta \widehat{u}_{n,2}, \Delta\biggl(\frac{d\widehat{u}_{n,2}}{dt}+\delta \widehat{u}_{n,2} -(I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\omega(t)\biggr)\bigg) \\ \geq&\frac{1}{2}\frac{d}{dt} \Vert \Delta \widehat{u}_{n,2} \Vert ^{2}+ \frac{3\delta}{4} \Vert \Delta\widehat{u}_{n,2} \Vert ^{2} \\ &{}-c \biggl\Vert (I-P_{n}) \Delta\biggl( \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\biggr) \biggr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2}. \end{aligned}$$
(4.63)

For the fourth term on the left-hand side of (4.61), we have

$$\begin{aligned}& \biggl( \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u), \widehat{v}_{n,2}\biggr) \\& \quad = \biggl( \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u), \frac{d\widehat{u}_{n,2}}{dt}+\delta\widehat{u}_{n,2}-(I-P_{n}) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\omega(t)\biggr) \\& \quad = \frac{d}{dt}\biggl( \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}} \biggr)f(x,u), \widehat{u}_{n,2}\biggr)- \biggl( \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f'_{u}(x,u)u_{t}, \widehat{u}_{n,2}\biggr) \\& \qquad {} + \delta\biggl(\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}} \biggr)f(x,u),\widehat{u}_{n,2}\biggr)-\biggl( \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u),(I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\omega(t)\biggr). \end{aligned}$$
(4.64)

For the third term on the right-hand side of (4.61), we have

$$\begin{aligned}& -(I-P_{n})\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl(h \bigl(v+\phi\omega(t)- \delta u\bigr),\widehat{v}_{n,2}\bigr) \\& \quad = -(I-P_{n})\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl(h\bigl(v+\phi\omega(t)- \delta u\bigr)-h(0),\widehat{v}_{n,2} \bigr) \\& \quad \leq -\beta_{1} \Vert \widehat{v}_{n,2} \Vert ^{2}+h'(\vartheta)\delta( \widehat{u}_{n,2}, \widehat{v}_{n,2})+\frac{\beta_{1}-\delta}{6} \Vert \widehat{v}_{n,2} \Vert ^{2}+c \biggl\Vert (I-P_{n}) \widehat{\rho} \biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2} \\& \quad \leq -\beta_{1} \Vert \widehat{v}_{n,2} \Vert ^{2}+h'(\vartheta)\delta\biggl( \widehat{u}_{n,2}, \frac{d\widehat{u}_{n,2}}{dt}+\delta\widehat {u}_{n,2}-(I-P_{n}) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\omega(t)\biggr) \\& \qquad {} + \frac{\beta_{1}-\delta}{6} \Vert \widehat{v}_{n,2} \Vert ^{2}+c \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2} \\& \quad \leq -\beta_{1} \Vert \widehat{v}_{n,2} \Vert ^{2}+\beta_{2}\delta\cdot \frac{1}{2} \frac{d}{dt} \Vert \widehat{u}_{n,2} \Vert ^{2}+ \beta_{2}\delta^{2} \Vert \widehat{u}_{n,2} \Vert ^{2} \\& \qquad {}+\beta_{2}\delta \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}} \biggr)\phi \biggr\Vert \bigl\vert \omega(t) \bigr\vert \Vert \widehat{u}_{n,2} \Vert \\& \qquad {} +\frac{\beta_{1}-\delta}{6} \Vert \widehat{v}_{n,2} \Vert ^{2}+c \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2} \\& \quad \leq -\beta_{1} \Vert \widehat{v}_{n,2} \Vert ^{2}+\beta_{2}\delta\cdot \frac{1}{2} \frac{d}{dt} \Vert \widehat{u}_{n,2} \Vert ^{2}+ \beta_{2}\delta^{2} \Vert \widehat{u}_{n,2} \Vert ^{2}+\frac{1}{4}\delta \bigl(\lambda+\delta^{2}- \beta_{2}\delta\bigr) \Vert \widehat{u}_{n,2} \Vert ^{2} \\& \qquad {} +\frac{\beta_{1}-\delta}{6} \Vert \widehat{v}_{n,2} \Vert ^{2}+c \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2}. \end{aligned}$$
(4.65)

Using the Cauchy–Schwarz inequality and Young’s inequality, we get

$$\begin{aligned}& \delta\biggl(\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\omega(t), \widehat{v}_{n,2}\biggr) \\& \quad \leq \frac{\beta_{1}-\delta}{6} \Vert \widehat{v}_{n,2} \Vert ^{2}+c \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2}, \end{aligned}$$
(4.66)
$$\begin{aligned}& \biggl((I-P_{n})\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}} \biggr)g(x,t),\widehat{v}_{n,2}\biggr) \\& \quad \leq \frac{\beta_{1}-\delta}{6} \Vert \widehat{v}_{n,2} \Vert ^{2}+c \biggl\Vert (I-P_{n}) \biggl( \widehat{\rho} \biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)g(x,t)\biggr) \biggr\Vert ^{2}. \end{aligned}$$
(4.67)

Now, we estimate the last term in (4.61)

$$\begin{aligned}& \biggl(4\Delta\nabla\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\cdot\nabla u+6 \Delta\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\cdot\Delta u+4\nabla \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\cdot\Delta\nabla u+u \Delta^{2} \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr), \widehat{v}_{n,2}\biggr) \\& \quad = \biggl(4\nabla u\cdot\biggl(\frac{12 \vert x \vert }{k^{4}}\widehat{\rho} ''\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)+ \frac{8 \vert x \vert ^{3}}{k^{6}} \widehat{\rho} ''' \biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\biggr) \\& \qquad {}+ 6\Delta u\cdot\biggl( \frac{2}{k^{2}} \widehat{\rho} '\biggl(\frac{ \vert x \vert ^{2}}{r^{2}} \biggr)+\frac{4x^{2}}{k^{4}} \widehat{\rho} '' \biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\biggr)+\frac{8 \vert x \vert }{k^{2}}\Delta\nabla u\cdot\widehat{\rho} '\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \\& \qquad {}+u\biggl( \frac{12}{k^{4}}\widehat{\rho} ''\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)+\frac{48x^{2}}{k^{6}}\widehat{\rho} ''' \biggl( \frac{ \vert x \vert ^{2}}{k^{2}} \biggr) +\frac{16x^{4}}{k^{8}}\widehat{\rho} '''' \biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\biggr),\widehat{v}_{n,2}\biggr) \\& \quad \leq \frac{16\sqrt{2}(3\mu_{2}+4\mu_{3})}{k^{3}} \Vert \nabla u \Vert \cdot \Vert \widehat{v}_{n,2} \Vert +\frac{12(\mu_{1}+4\mu_{2})}{k^{2}} \Vert \Delta u \Vert \cdot \Vert \widehat{v}_{n,2} \Vert \\& \qquad {} +\frac{8\sqrt{2}\mu_{1}}{k} \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert \cdot \Vert \widehat{v}_{n,2} \Vert +\frac{4(3\mu_{2}+24\mu_{3}+16\mu_{4})}{k^{4}} \Vert u \Vert \cdot \Vert \widehat{v}_{n,2} \Vert \\& \quad \leq \frac{8(48\mu_{2}+64\mu_{3})^{2}}{(\beta_{1}-\delta) k^{6}} \Vert \nabla u \Vert ^{2}+ \frac{4(12\mu_{1}+48\mu_{2})^{2}}{(\beta_{1}-\delta)k^{4}} \Vert \Delta u \Vert ^{2} + \frac{512\mu^{2}_{1}}{(\beta_{1}-\delta)k^{2}} \bigl\Vert A^{ \frac{3}{4}} u \bigr\Vert ^{2} \\& \qquad {} + \frac{4(12\mu_{2}+96\mu_{3}+64\mu_{4})^{2}}{(\beta_{1}-\delta)k^{8}} \Vert u \Vert ^{2} + \frac{\beta_{1}-\delta}{4} \Vert \widehat{v}_{n,2} \Vert ^{2}. \end{aligned}$$
(4.68)

Assemble together (4.61)–(4.68) to obtain

$$\begin{aligned}& \frac{1}{2}\frac{d}{dt}\biggl[ \Vert \widehat{v}_{n,2} \Vert ^{2}+\bigl(\lambda+\delta^{2}- \beta_{2}\delta\bigr) \Vert \widehat{u}_{n,2} \Vert ^{2}+ \Vert \Delta\widehat{u}_{n,2} \Vert ^{2}+2\biggl( \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}} \biggr)f(x,u),\widehat{u}_{n,2}\biggr)\biggr] \\& \qquad {} +\delta\bigl[ \Vert \widehat{v}_{n,2} \Vert ^{2}+\bigl(\lambda+\delta^{2}-\beta_{2} \delta\bigr) \Vert \widehat{u}_{n,2} \Vert ^{2}+ \Vert \Delta\widehat{u}_{n,2} \Vert ^{2}\bigr]+ \delta \biggl( \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u), \widehat{u}_{n,2}\biggr) \\& \quad \leq \frac{\delta}{2} \Vert \widehat{v}_{n,2} \Vert ^{2}+ \frac{3\delta-\beta_{1}}{4} \Vert \widehat{v}_{n,2} \Vert ^{2}+ \frac{\delta}{2} \bigl(\lambda+ \delta^{2}-\beta_{2}\delta\bigr) \Vert \widehat{u}_{n,2} \Vert ^{2}+\frac{\delta}{4} \Vert \Delta\widehat{u}_{n,2} \Vert ^{2} \\& \qquad {} +\frac{2}{\beta_{1}-\delta}\biggl( \frac{4(48\mu_{2}+64\mu_{3})^{2}}{ k^{6}} \Vert \nabla u \Vert ^{2}+ \frac{2(12\mu_{1}+48\mu_{2})^{2}}{k^{4}} \Vert \Delta u \Vert ^{2}+ \frac{256\mu^{2}_{1}}{k^{2}} \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2} \\& \qquad {} +\frac{2(12\mu_{2}+96\mu_{3}+64\mu_{4})^{2}}{k^{8}} \Vert u \Vert ^{2}\biggr)+c \biggl\Vert (I-P_{n}) \biggl(\widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)g(x,t)\biggr) \biggr\Vert ^{2} \\& \qquad {} +c \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2}+ c \biggl\Vert (I-P_{n}) \Delta\biggl( \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\biggr) \biggr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2} \\& \qquad {} +\biggl( \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}} \biggr)f'_{u}(x,u)u_{t}, \widehat{u}_{n,2}\biggr)+\biggl( \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u),(I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\omega(t)\biggr). \end{aligned}$$
(4.69)

For the nonlinear terms in (4.69), by (3.7), using Cauchy’s inequality and Young’s inequality, we obtain

$$ \biggl( \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f'_{u}(x,u)u_{t}, \widehat{u}_{n,2} \biggr) \leq\frac{\delta}{4} \Vert \Delta \widehat{u}_{n,2} \Vert ^{2}+c\lambda^{-1}_{n+1} \Vert u_{t} \Vert ^{2}. $$
(4.70)

By (3.4), we know

$$\begin{aligned}& \biggl( \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u),(I-P_{n}) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\omega(t)\biggr) \\& \quad \leq c \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert \bigl\vert \omega(t) \bigr\vert +c \Vert u \Vert ^{ p }_{H^{2}(\mathbb{R}^{n})} \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert \bigl\vert \omega(t) \bigr\vert . \end{aligned}$$
(4.71)

Since \(1\leq p\leq\frac{n+4}{n-4}\) and \(\lambda_{n}\rightarrow\infty\), by Lemmas 4.1 and 4.2, there are \(N_{1}=N(\eta)\), \(K_{1}=K(\eta)\) such that, for all \(n\geq N_{1}\), \(k\geq K_{1}\),

$$\begin{aligned}& \frac{2}{\beta_{1}-\delta}\biggl( \frac{4(48\mu_{2}+64\mu_{3})^{2}}{ k^{6}} \Vert \nabla u \Vert ^{2}+ \frac{2(12\mu_{1}+48\mu_{2})^{2}}{k^{4}} \Vert \Delta u \Vert ^{2}+ \frac{256\mu^{2}_{1}}{k^{2}} \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2} \\& \qquad {} +\frac{2(12\mu_{2}+96\mu_{3}+64\mu_{4})^{2}}{k^{8}} \Vert u \Vert ^{2}\biggr)+c \lambda^{- \frac{1}{2}}_{n+1} \Vert u_{t} \Vert ^{2} \\& \qquad {} +c \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert \bigl\vert \omega(t) \bigr\vert +c \Vert u \Vert ^{ p }_{H^{2}(\mathbb{R}^{n})} \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert \bigl\vert \omega(t) \bigr\vert \\& \qquad {} +c \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2} +c \biggl\Vert (I-P_{n}) \Delta\biggl( \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\biggr) \biggr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2} \\& \quad \leq c\eta\bigl(1+ \bigl\vert \omega(t) \bigr\vert ^{2}+ \Vert u_{t} \Vert ^{2}+ \Vert u \Vert ^{18}_{H^{2}( \mathbb{R}^{n})}\bigr). \end{aligned}$$
(4.72)

Then, by (4.69)–(4.72), we obtain

$$\begin{aligned}& \frac{d}{dt}\biggl[ \Vert \widehat{v}_{n,2} \Vert ^{2}+\bigl(\lambda+\delta^{2}-\beta_{2} \delta\bigr) \Vert \widehat{u}_{n,2} \Vert ^{2}+ \Vert \Delta\widehat{u}_{n,2} \Vert ^{2}+2\biggl( \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u),\widehat{u}_{n,2} \biggr)\biggr] \\& \qquad {} +\sigma\biggl[ \Vert \widehat{v}_{n,2} \Vert ^{2}+\bigl(\lambda+\delta^{2}-\beta_{2} \delta\bigr) \Vert \widehat{u}_{n,2} \Vert ^{2}+ \Vert \Delta\widehat{u}_{n,2} \Vert ^{2}+2\biggl( \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u),\widehat{u}_{n,2} \biggr)\biggr] \\& \quad \leq c\eta\bigl(1+ \bigl\vert \omega(t) \bigr\vert ^{2}+ \Vert u_{t} \Vert ^{2}+ \Vert u \Vert ^{18}_{H^{2}( \mathbb{R}^{n})}\bigr)+c \biggl\Vert (I-P_{n}) \biggl(\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)g(x,t)\biggr) \biggr\Vert ^{2}. \end{aligned}$$
(4.73)

Multiplying (4.73) by \(e^{\sigma t}\) and then integrating over \((\tau-t,\tau)\), we have for all \(n>N_{1}\) and \(k>K_{1}\)

$$\begin{aligned}& \bigl\Vert \widehat{v}_{n,2}(\tau,\tau-t,\omega) \bigr\Vert ^{2}+\bigl(\lambda+\delta^{2}- \beta_{2} \delta\bigr) \bigl\Vert \widehat{u}_{n,2}(\tau,\tau-t,\omega) \bigr\Vert ^{2} + \bigl\Vert \Delta\widehat{u}_{n,2}(\tau, \tau-t,\omega) \bigr\Vert ^{2} \\& \qquad {}+2\biggl(\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u), \widehat{u}_{n,2}( \tau,\tau-t,\omega)\biggr) \\& \quad \leq e^{-\sigma t} \bigg( \Vert v_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}+\bigl( \lambda+\delta^{2}- \beta_{2}\delta\bigr) \Vert \Delta u_{0} \Vert ^{2}+2\biggl(\widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u),u_{0}\biggr)\bigg) \\& \qquad {}+c\eta \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)}\bigl(1+ \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\Vert u_{t}(s, \tau-t,\omega,u_{0}) \bigr\Vert ^{2} + \bigl\Vert u(s,\tau-t,\omega,u_{0}) \bigr\Vert ^{18}_{H^{2}(\mathbb{R}^{n})} \bigr)\,ds \\& \qquad {}+c \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)} \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{r^{2}}\biggr)g(x,s) \biggr\Vert ^{2}\,ds. \end{aligned}$$
(4.74)

Replacing ω by \(\theta_{-\tau}\omega\), by a similar process as in Lemma 4.1, we get

$$\begin{aligned}& \bigl\Vert \widehat{v}_{n,2}(\tau,\tau-t,\theta_{-\tau} \omega) \bigr\Vert ^{2}+\bigl( \lambda+\delta^{2}- \beta_{2}\delta\bigr) \bigl\Vert \widehat{u}_{n,2}(\tau, \tau-t,\theta_{-\tau}\omega) \bigr\Vert ^{2} \\& \qquad {} + \bigl\Vert \Delta\widehat{u}_{n,2}(\tau,\tau-t, \theta_{-\tau}\omega) \bigr\Vert ^{2} +2\biggl(\widehat{ \rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u),\widehat{u}_{n,2}( \tau, \tau-t,\theta_{-\tau}\omega)\biggr) \\& \quad \leq ce^{-\sigma t} \bigl(1+ \Vert v_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}+\bigl( \lambda+ \delta^{2}-\beta_{2}\delta\bigr) \Vert \Delta u_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}_{H^{2}( \mathbb{R}^{n})}+ \Vert u_{0} \Vert ^{p+1}_{H^{2}(\mathbb{R}^{n})}\bigr) \\& \qquad {} +c\eta \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)}\bigl(1+ \bigl\vert \theta_{-\tau} \omega(s) \bigr\vert ^{2}+ \bigl\Vert u_{t}(s,\tau-t,\theta_{-\tau}\omega,u_{0}) \bigr\Vert ^{2} \\& \qquad {} + \bigl\Vert u(s,\tau-t,\theta_{-\tau}\omega,u_{0}) \bigr\Vert ^{18}_{H^{2}( \mathbb{R}^{n})}\bigr)\,ds+c \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)} \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{r^{2}}\biggr)g(x,s) \biggr\Vert ^{2}\,ds \\& \quad \leq c e^{-\sigma t} \bigl(1+ \Vert v_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}+\bigl( \lambda+ \delta^{2}-\beta_{2}\delta\bigr) \Vert \Delta u_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}_{H^{2}( \mathbb{R}^{n})}+ \Vert u_{0} \Vert ^{p+1}_{H^{2}(\mathbb{R}^{n})}\bigr) \\& \qquad {} +c\eta \int^{0}_{-\infty}e^{\sigma s }\bigl(1+ \bigl\vert \omega(s) \bigr\vert ^{2}\bigr)\,ds+c \eta \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)}\bigl( \bigl\Vert u_{t}(s,\tau-t, \theta_{-\tau}\omega,u_{0}) \bigr\Vert ^{2} \\& \qquad {} + \bigl\Vert u(s,\tau-t,\theta_{-\tau}\omega,u_{0}) \bigr\Vert ^{18}_{H^{2}( \mathbb{R}^{n})}\bigr)\,ds \\& \qquad {}+c \int^{0}_{-\infty}e^{\sigma s } \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{r^{2}}\biggr)g(x,s+ \tau) \biggr\Vert ^{2}\,ds. \end{aligned}$$
(4.75)

Using the first equation of (3.10) as well as the Minkowski inequality, we can obtain

$$\begin{aligned}& \bigl\Vert u_{t}(s,\tau-t,\theta_{-\tau} \omega,u_{0}) \bigr\Vert ^{2} \\& \quad = \bigl\Vert -\delta u(s,\tau-t,\theta_{-\tau} \omega,u_{0})+v(s,\tau-t, \theta_{-\tau} \omega,v_{0}) +\phi\theta_{-\tau}\omega \bigr\Vert ^{2} \\& \quad \leq c\bigl( \bigl\Vert u(s,\tau-t,\theta_{-\tau} \omega,u_{0}) \bigr\Vert ^{2}+ \bigl\Vert v(s, \tau-t,\theta_{-\tau}\omega,v_{0}) \bigr\Vert ^{2}+ \vert \theta_{-\tau}\omega \vert ^{2} \bigr) \\& \quad \leq cR_{1}(\tau,\omega)+c \vert \theta_{-\tau} \omega \vert ^{2} \end{aligned}$$
(4.76)

and

$$ \bigl\Vert u(s,\tau-t,\theta_{-\tau}\omega,u_{0}) \bigr\Vert ^{18}_{H^{2}(\mathbb{R}^{n})} \leq cR^{9}_{1}( \tau,\omega), $$
(4.77)

where \(c=\max\{\delta,\|\phi\|^{2},1\}\) and \(R_{1}(\tau,\omega)\) is given in Lemma 4.1. Hence, it follows from (4.75)–(4.77) that

$$\begin{aligned}& \bigl\Vert \widehat{v}_{n,2}(\tau,\tau-t,\theta_{-\tau} \omega) \bigr\Vert ^{2}+\bigl( \lambda+\delta^{2}- \beta_{2}\delta\bigr) \bigl\Vert \widehat{u}_{n,2}(\tau, \tau-t,\theta_{-\tau}\omega) \bigr\Vert ^{2} \\& \qquad {} + \bigl\Vert \Delta\widehat{u}_{n,2}(\tau,\tau-t, \theta_{-\tau}\omega) \bigr\Vert ^{2} +2\biggl(\widehat{ \rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u),\widehat{u}_{n,2}( \tau, \tau-t,\theta_{-\tau}\omega)\biggr) \\& \quad \leq e^{-\sigma t} \bigl(1+ \Vert v_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}+\bigl( \lambda+\delta^{2}- \beta_{2}\delta\bigr) \Vert \Delta u_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}_{H^{2}( \mathbb{R}^{n})}+ \Vert u_{0} \Vert ^{p+1}_{H^{2}(\mathbb{R}^{n})}\bigr) \\& \qquad {} +c\eta R^{9}_{1}(\tau,\omega)+c\eta \int^{0}_{-\infty}e^{\sigma s }\bigl(1+ \bigl\vert \omega(s) \bigr\vert ^{2} \bigr)\,ds \\& \qquad {}+c \int^{0}_{-\infty}e^{\sigma s } \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{r^{2}}\biggr)g(x,s+ \tau) \biggr\Vert ^{2}\,ds. \end{aligned}$$
(4.78)

Since \((u_{0},v_{0})^{\top}\in D(\tau-t,\theta_{-t}\omega)\) and \(D\in\mathcal{D}\), then

$$\begin{aligned}& e^{-\sigma t} \bigl(1+ \Vert v_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}+\bigl(\lambda+ \delta^{2}- \beta_{2}\delta\bigr) \Vert \Delta u_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}_{H^{2}( \mathbb{R}^{n})}+ \Vert u_{0} \Vert ^{p+1}_{H^{2}(\mathbb{R}^{n})}\bigr) \\& \quad \rightarrow0,\quad t\rightarrow\infty. \end{aligned}$$
(4.79)

For the last term on the right-hand side of (4.78), by (3.16), there exists \(N_{2}=N_{2}(\tau,\omega,\eta)\geq N_{1}\) such that, for all \(n\geq N_{2}\),

$$ \int^{0}_{-\infty} e^{\sigma s} \biggl\Vert (I-P_{n}) \biggl(\widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}} \biggr)g(x,s+\tau)\biggr) \biggr\Vert ^{2}\,ds < \eta. $$
(4.80)

The proof is completed by (3.4), (4.79)–(4.80), and Lemma 4.1. □

Random attractors

In this section, we prove the existence of a \(\mathcal{D}\)-attractor for the stochastic system (3.10)–(3.11). We firstly apply the lemmas shown in Sect. 4 to derive the asymptotic compactness of solutions of (3.10)–(3.11).

Lemma 5.1

Under (3.4)(3.9) and (3.14)(3.17), for every\(\tau\in\mathbb{R}\), \(\omega\in\varOmega\), the sequence of weak solutions of (3.10)(3.11) \(\{Y(\tau,\tau-t_{m},\theta_{-\tau}\omega,Y_{0}(\theta_{-t_{m}} \omega))\}^{\infty}_{m=1}\)has a convergent subsequence inEwhenever\(t_{m}\rightarrow\infty\)and\(Y_{0}(\theta_{-t_{m}}\omega)\in D(\tau-t_{m},\theta_{-t_{m}} \omega)\)with\(D\in\mathcal{D}\).

Proof

Let \(t_{m}\rightarrow\infty\) and \(Y_{0}(\theta_{-t_{m}}\omega)\in D(\tau-t_{m},\theta_{-t_{m}} \omega)\) with \(D\in\mathcal{D}\). By Lemma 4.1, there exists \(m_{1}=m_{1}(\tau,\omega,D)>0\) such that, for all \(m\geq m_{1}\), we have

$$ \bigl\Vert Y\bigl(\tau,\tau-t_{m},\theta_{-\tau} \omega,Y_{0}(\theta_{-t_{m}} \omega)\bigr) \bigr\Vert ^{2}_{E}\leq R_{1}(\tau,\omega). $$
(5.1)

By Lemma 4.3, for every \(\eta>0\), there exist \(k_{0}=r_{0}(\tau,\omega,\eta)\geq1\) and \(m_{2}=m_{2}(\tau,\omega,D,\eta)\geq m_{1}\) such that, for all \(m\geq m_{2}\),

$$ \bigl\Vert Y\bigl(\tau,\tau-t,\theta_{-\tau}\omega,D(\tau-t, \theta_{-t} \omega)\bigr) \bigr\Vert ^{2}_{E(\mathbb{R}^{n}\setminus\mathbb{B}_{k_{0}})} \leq \eta. $$
(5.2)

By Lemma 4.4, there exist \(k_{1}=k_{1}(\tau,\omega,\eta)\geq k_{0}\) and \(m_{3}=m_{3}(\tau,\omega,D,\eta)\geq m_{2}\) and \(n_{1}=n_{1}(\tau,\omega,\eta)\geq0\) such that, for all \(m\geq m_{3}\),

$$ \bigl\Vert (I-P_{n})\widehat{Y}\bigl(\tau,\tau-t, \theta_{-\tau}\omega,D(\tau-t, \theta_{-\tau}\omega)\bigr) \bigr\Vert ^{2}_{E (\mathbb{B}_{2k_{1}})}\leq\eta. $$
(5.3)

Using (4.56) and (5.1), we get

$$ \bigl\Vert P_{n}\widehat{Y}\bigl(\tau,\tau-t,\theta_{-\tau} \omega,D(\tau-t, \theta_{-\tau}\omega)\bigr) \bigr\Vert ^{2}_{P_{n}E (\mathbb{B}_{2k_{1}})}\leq c R_{1}( \tau,\omega), $$
(5.4)

which together with (5.3) implies that \(\{Y(\tau,\tau-t_{m},\theta_{-\tau}\omega,Y_{0}(\theta_{-t_{m}} \omega))\}\) is precompact in \(E(\mathbb{B}_{2k_{1}})\). Note that \(\widehat{\rho}(\frac{|x|^{2}}{k^{2}_{1}})=1\) for \(|x|\leq k_{1}\). Therefore, \(\{Y(\tau,\tau-t_{m},\theta_{-\tau}\omega,Y_{0}(\theta_{-t_{m}} \omega))\}\) is precompact in \(E(\mathbb{B}_{k_{1}})\), which along with (5.2) shows the precompactness of this sequence in E. □

Theorem 5.1

Under (3.4)(3.9) and (3.14)(3.17), the random dynamical systemΦgenerated by the stochastic plate equation (3.10)(3.11) has a unique pullback\(\mathcal{D}\)-attractor\(\mathcal{A} =\{\mathcal{A}(\tau, \omega) :\tau\in\mathbb{R}, \omega\in\varOmega\}\in\mathcal{D}\)in the spaceE.

Proof

Notice that Φ is pullback \(\mathcal{D}\)-asymptotically compact in E by Lemma 5.1 and has a pullback \(\mathcal{D}\)-absorbing set by Lemma 4.1. Thus the existence of a unique \(\mathcal{D}\)-attractor follows from Proposition 2.1 immediately. □

References

  1. 1.

    Arnold, L.: Random Dynamical Systems. Springer, New York (1998)

  2. 2.

    Barbosaa, A.R.A., Ma, T.F.: Long-time dynamics of an extensible plate equation with thermal memory. J. Math. Anal. Appl. 416, 143–165 (2014)

  3. 3.

    Bates, P.W., Lisei, H., Lu, K.: Attractors for stochastic lattic dynamical systems. Stoch. Dyn. 6, 1–21 (2006)

  4. 4.

    Bates, P.W., Lu, K., Wang, B.: Random attractors for stochastic reaction-diffusion equations on unbounded domains. J. Differ. Equ. 246, 845–869 (2009)

  5. 5.

    Chepyzhov, V.V., Vishik, M.I.: Attractors for Equations of Mathematical Physics. Am. Math. Soc., Providence (2002)

  6. 6.

    Crauel, H.: Random Probability Measure on Polish Spaces. Taylor & Francis, London (2002)

  7. 7.

    Crauel, H., Debussche, A., Flandoli, F.: Random attractors. J. Dyn. Differ. Equ. 9, 307–341 (1997)

  8. 8.

    Crauel, H., Flandoli, F.: Attractors for random dynamical systems. Probab. Theory Relat. Fields 100, 365–393 (1994)

  9. 9.

    Cui, H., Langa, J.A.: Uniform attractors for non-autonomous random dynamical systems. J. Differ. Equ. 263, 1225–1268 (2017)

  10. 10.

    Flandoli, F., Schmalfuss, B.: Random attractors for the 3D stochastic Navier–Stokes equation with multiplicative noise. Stoch. Stoch. Rep. 59, 21–45 (1996)

  11. 11.

    Huang, J., Shen, W.: Pullback attractors for non-autonomous and random parabolic equations on non-smooth domains. Discrete Contin. Dyn. Syst. 24, 855–882 (2009)

  12. 12.

    Khanmamedov, A.K.: Existence of global attractor for the plate equation with the critical exponent in an unbounded domain. Appl. Math. Lett. 18, 827–832 (2005)

  13. 13.

    Khanmamedov, A.K.: Global attractors for the plate equation with a localized damping and a critical exponent in an unbounded domain. J. Differ. Equ. 225, 528–548 (2006)

  14. 14.

    Khanmamedov, A.K.: A global attractor for the plate equation with displacement-dependent damping. Nonlinear Anal. 74, 1607–1615 (2011)

  15. 15.

    Kloden, P.E., Langa, J.A.: Flattening, squeezing and the existence of random attractors. Proc. R. Soc. Lond. Ser. A 463, 163–181 (2007)

  16. 16.

    Ma, Q., Wang, S., Zhong, C.: Necessary and sufficient conditions for the existence of global attractors for semigroups and applications. Indiana Univ. Math. J. 51, 1541–1559 (2002)

  17. 17.

    Ma, W., Ma, Q.: Attractors for the stochastic strongly damped plate equations with additive noise. Electron. J. Differ. Equ. 2013, 111 (2013)

  18. 18.

    Pazy, A.: Semigroup of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)

  19. 19.

    Schmalfuss, B.: Backward cocycles and attractors of stochastic differential equations. In: International Seminar on Applied Mathematics—Nonlinear Dynamics: Attractor Approximation and Global Behavior, Dresden, pp. 185–192 (1992)

  20. 20.

    Shen, X., Ma, Q.: The existence of random attractors for plate equations with memory and additive white noise. Korean J. Math. 24, 447–467 (2016)

  21. 21.

    Shen, X., Ma, Q.: Existence of random attractors for weakly dissipative plate equations with memory and additive white noise. Comput. Math. Appl. 73, 2258–2271 (2017)

  22. 22.

    Shen, Z., Zhou, S., Shen, W.: One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation. J. Differ. Equ. 248, 1432–1457 (2010)

  23. 23.

    Temam, R.: Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1998)

  24. 24.

    Wang, B.: Asymptotic behavior of stochastic wave equations with critical exponents on \(\mathbb{R}^{3}\). Trans. Am. Math. Soc. 363, 3639–3663 (2011)

  25. 25.

    Wang, B.: Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems. J. Differ. Equ. 253, 1544–1583 (2012)

  26. 26.

    Wang, B.: Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms. Stoch. Dyn. 14, 1–31 (2014)

  27. 27.

    Wang, B., Gao, X.: Random attractors for wave equations on unbounded domains. Discrete Contin. Dyn. Syst. Spec. 2009, 800–809 (2009)

  28. 28.

    Wang, Z., Zhou, S.: Random attractor for non-autonomous stochastic strongly damped wave equation on unbounded domains. J. Math. Anal. Appl. 5, 363–387 (2015)

  29. 29.

    Wang, Z., Zhou, S., Gu, A.: Random attractor for a stochastic damped wave equation with multiplicative noise on unbounded domains. Nonlinear Anal., Real World Appl. 12, 3468–3482 (2011)

  30. 30.

    Yang, L., Zhong, C.: Uniform attractor for non-autonomous plate equations with a localized damping and a critical nonlinearity. J. Math. Anal. Appl. 338, 1243–1254 (2008)

  31. 31.

    Yang, L., Zhong, C.: Global attractor for plate equation with nonlinear damping. Nonlinear Anal. 69, 3802–3810 (2008)

  32. 32.

    Yang, M., Duan, J., Kloden, P.E.: Asymptotic behavior of solutions for random wave equations with nonlinear damping and white noise. Nonlinear Anal., Real World Appl. 12, 464–478 (2011)

  33. 33.

    Yao, B.X., Ma, Q.Z.: Global attractors for a Kirchhoff type plate equation with memory. Kodai Math. J. 40, 63–78 (2017)

  34. 34.

    Yao, B.X., Ma, Q.Z.: Global attractors of the extensible plate equations with nonlinear damping and memory. J. Funct. Spaces 2017, Article ID 4896161 (2017)

  35. 35.

    Yao, X., Liu, X.: Asymptotic behavior for non-autonomous stochastic plate equation on unbounded domains. Open Math. 17, 1281–1302 (2019)

  36. 36.

    Yao, X., Ma, Q., Liu, T.: Asymptotic behavior for stochastic plate equations with rotational inertia and Kelvin–Voigt dissipative term on unbounded domains. Discrete Contin. Dyn. Syst., Ser. B 24, 1889–1917 (2019)

  37. 37.

    Yue, G., Zhong, C.: Global attractors for plate equations with critical exponent in locally uniform spaces. Nonlinear Anal. 71, 4105–4114 (2009)

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This work was supported by the High-level Talent Program of Qinghai Nationalities University (No. 2020XJG10).

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This paper completed by XY deals with the existence of pullback attractors for the non-autonomous stochastic plate equations with additive noise and nonlinear damping on unbounded domains. The author read and approved the final manuscript.

Correspondence to Xiaobin Yao.

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Yao, X. Existence of a random attractor for non-autonomous stochastic plate equations with additive noise and nonlinear damping on \(\mathbb{R}^{n}\). Bound Value Probl 2020, 49 (2020). https://doi.org/10.1186/s13661-020-01346-z

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MSC

  • 35B40
  • 35B41

Keywords

  • Pullback attractors
  • Pate equation
  • Unbounded domains
  • The splitting technique
  • Additive noise