The mR scheme to the shallow water equation with horizontal density gradients in one and two dimensions

1.   Introduction

The colored noise was first introduced in ^[23,24] in order to obtain the information of velocity of randomly moving particles, which cannot be obtained from the white noise since the the Wiener process is nowhere differentiable. Moreover, for many physical systems, the stochastic fluctuations are correlated and should be modeled by the colored noise rather than the white noise, see ^[20].

This paper is concerned the asymptotic behavior of the plate equation driven by nonlinear colored noise in unbounded domains:

where $\tau\in\mathbb{R}$, $\alpha, \nu$ are positive constants, $\mu$ is the memory kernel, $f$ and $h$ are given nonlinearity, $g \in L^2_{loc}(\mathbb{R}, H^1(\mathbb{R}^{n}))$, and $\zeta_\delta$ is a colored noise with correlation time $\delta > 0$.

It is clear that (1.1) becomes a deterministic plate equation as $\mu\equiv0$ and $h\equiv0$. In this case, we can characterize the long-time behavior of solutions by virtue of the concept of global attractors under the framework of semigroup. Some authors have extensively studied the existence of global attractors for the autonomous plate equation. For instance, the attractors of deterministic plate equations have been investigated in ^{[2,8,12,14,30,32,33,34,35,44]} in bounded domains. In ^[2,30,34,35], the authors considered global attractor for the plate equation with thermal memory; Khanmamedov investigated a global attractor for the plate equation with displacement-dependent damping in ^[8]; Liu and Ma obtained the existence of time-dependent strong pullback attractors for non-autonomous plate equations in ^[12,14]; Yang and Zhong studied the uniform attractor and global attractor for non-autonomous plate equations with nonlinear damping in ^[32,33], respectively; In ^[44], the author obtained global existence and blow-up of solutions for a Kirchhoff type plate equation with damping. For the case of unbounded domains, see refereces ^{[9,10,13,31,42]}.

The existence and uniqueness of pathwise random attractors of stochastic plate equations have been studied in ^{[15,16,21,22]} in the case of bounded domains; and in ^{[36,37,38,39,40,41]} in the case of unbounded domains. In all these publications (^{[36,37,38,39,40,41]}), only the additive white noise and linear multiplicative white noise were considered. Notice that the random equation (1.1) is driven by the colored noise rather than the white noise. In general, it is very hard to study the asymptotic dynamics of differential equations driven by nonlinear white noise, including the random attractors. Indeed, only when the white noise is linear, the stochastic equations can be transformed into a deterministic equations, then one can obtain the existence of random attractors of the plate equation (1.1). However, this transformation does not apply to stochastic equations driven by nonlinear white noise, and that is why we are currently unable to prove the existence of random attractors for systems with nonlinear white noise.

For the colored noise, even it is nonlinear, we are able to show system (1.1) has a random attractor in $H^2(\mathbb{R}^n)\times L^2(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}$ (the definition of $\mathfrak{R}_{\mu, 2}$ see Section 3), which is quite different from the nonlinear white noise. The reader is referred to ^[6,7,26,27] for more details on random attractors of differential equations driven by colored noise. However, for the random plate equations driven by colored noise (1.1), we find that there is no results available to the existence of random attractors. In the present paper, we will prove that (1.1) is pathwise well-posed and generate a continuous cocycle, and the cocycle possesses a unique tempered random attractor. This is different from the corresponding stochastic system driven by white noise

where the symbol $\circ$ indicates that the equation is understood in the sense of stratonovich integration. For (1.2), one can define a random dynamical system when $h(\cdot, \cdot, u)$ is a linear function, see ^[41]. But for a general nonlinear function $h$, random dynamical system associated with (1.2) can not be defined due to the absence of appropriate transformation, hence asymptotic behavior of such stochastic equations has not been investigated until now by the random dynamical system approach. This paper indicates that the colored noise is much easier to handle than the white noise for studying pathwise dynamics of such stochastic equations.

The main purpose of the paper is establish the existence and uniqueness of measurable tempered random attractors in $H^2(\mathbb{R}^n)\times L^2(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}$ for the dynamical system associated with (1.1). The key for achieving our goal is to establish the tempered pullback asymptotic compactness of solutions of (1.1) in $H^2(\mathbb{R}^n)\times L^2(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}$. Involving to our problem (1.1), there are two essential difficulties in verifying the compactness. On the one hand, notice that system (1.1) is defined in the unbounded domain $\mathbb{R}^{n}$ where the noncompactness of Sobolev embeddings on unbounded domains gives rise to difficulty in showing the pullback asymptotic compactness of solutions, to get through of it, we use the tail-estimates method (as in^[25]) and the splitting technique (see ^[3]) to obtain the pullback asymptotic compactness. On the other hand, there is no applicable compact embedding property in the "history'' space. In this case, we solve it with the help of a useful result in ^[19]. For our purpose, we introduce a new variable and an extend Hilbert space.

The rest of this article consists of four sections. In the next section, we define some functions sets and recall some useful results. In Section 3, we first establish the existence, uniqueness and continuity of solutions in initial data of (1.1) in $H^2(\mathbb{R}^n)\times L^2(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}$, then define a non-autonomous random dynamical system based on the solution operator of problem (1.1). The last two section are devoted to derive necessary estimates of solutions of (1.1) and the existence of random attractors.

Throughout the paper, the inner product and the norm of $L^2(\mathbb{R}^n)$ will be denoted by $(\cdot, \cdot)$ and $||\cdot||$, respectively. The letters $c$ and $c_i\; (i = 1, 2, \ldots)$ are generic positive constants which may depend on some parameters in the contexts.

2.   Asymptotic compactness of cocycles

In this section, we define some functions sets and recall some useful results, see ^{[4,17,18,28,29,43]}. These results will be used to establish the asymptotic compactness of the solutions and attractor for the random plate equation defined on the entire space $\mathbb{R}^n$.

From now on, we assume $(\Omega, \mathcal{F}, P)$ is the canonical probability space where $\Omega = \{\omega\in C(\mathbb{R}, \mathbb{R}):\omega(0) = 0\}$ with compact-open topology, $\mathcal{F}$ is the Borel $\sigma$-algebra of $\Omega$, and $P$ is the Wiener measure on $(\Omega, \mathcal{F})$. Recall the standard group of transformations $\{\theta_t\}_{t\in\mathbb{R}}$ on $\Omega$:

Suppose $\Phi:\mathbb{R}^+\times\mathbb{R}\times\Omega\times X\rightarrow X$ is a continuous cocycle on $X$ over $(\Omega, \mathcal{F}, P, \{\theta_t\}_{t\in\mathbb{R}})$. Let $\mathcal{D}$ be a collection of some families of nonempty subset of $X$:

Suppose $\Phi$ has a $\mathcal{D}$-pullback absorbing set $K = \{K(\tau, \omega):\tau\in\mathbb{R}, \omega\in\Omega\}\in\mathcal{D}$; that is, for every $\tau\in\mathbb{R}, \ \omega\in\Omega$ and $D\in\mathcal{D}$ there exists $T = T(\tau, \omega, D) > 0$ such that for all $t\geq T$,

Assume that

where both $\Phi_1$ and $\Phi_2$ are mappings from $\mathbb{R}^+\times\mathbb{R}\times\Omega\times X$ to $X$.

Given $k\in \mathbb{N}$, denote by $\mathcal{O}_k = \{x\in\mathbb{R}^n:|x| < k\}$ and $\tilde{\mathcal{O}}_k = \{x\in\mathbb{R}^n:|x| > k\}$. Let $X$ be a Banach space with norm $\|\cdot\|_X$ which consists of some functions defined on $\mathbb{R}^n$. Given a function $u:\mathbb{R}^n\rightarrow \mathbb{R}$, the restrictions of $u$ to $\mathcal{O}_k$ and $\tilde{\mathcal{O}}_k$ are written as $u|_{\mathcal{O}_k}$ and $u|_{\tilde{\mathcal{O}}_k}$, respectively. Denote by

Suppose $X_{\mathcal{O}_k}$ and $X_{\tilde{\mathcal{O}}_k}$ are Banach spaces with norm $\|\cdot\|_{\mathcal{O}_k}$ and $\|\cdot\|_{\tilde{\mathcal{O}}_k}$, respectively, and

We further assume that for every $\delta > 0, \ \tau\in\mathbb{R}$, and $\omega\in\Omega$, there exists $t_0 = t_0(\delta, \tau, \omega, K) > 0$ and $k_0 = k_0(\delta, \tau, \omega)\geq1$ such that

and

In addition, we assume that for every $k\in\mathbb{N}, \ t\in\mathbb{R}^+, \ \tau\in\mathbb{R}$, and $\omega\in\Omega$, the set

Theorem 2.1 ^[29]. If (2.1)-(2.6) hold, then the cocycle $\Phi$ is $\mathcal{D}$-pullback asymptotically compact in X; that is, the sequence $\{ \Phi(t_n, \tau-t_n, \theta_{-t_n}\omega, x_n)\}^\infty_{n = 1}$ is precompact in $X$ for any $\tau\in\mathbb{R}, \omega\in\Omega, D\in\mathcal{D}, t_n\rightarrow \infty$ monotonically, and $x_n\in D(\tau-t_n, \theta_{-t_n}\omega)$.

Theorem 2.2 ^[29]. Let $\mathcal{D}$ be an inclusion closed collection of some families of nonempty subsets of $X$, and $\Phi$ be a continuous cocycle on $X$ over $(\Omega, \mathcal{F}, P, \{\theta_t\}_{t\in\mathbb{R} })$. Then $\Phi$ has a unique $\mathcal{D}$-pullback random attractor $\mathcal{A}$ in $\mathcal{D}$ if $\Phi$ is $\mathcal{D}$-pullback asymptotically compact in $X$ and $\Phi$ has a closed measurable $\mathcal{D}$-pullback absorbing set $K$ in $\mathcal{D}$.

3.   Cocycles of random plate equations

In this section, we first establish the existence of solution for problem (1.1), then define a non-autonomous cocycle of (1.1).

Given $\delta > 0$, let $\zeta_\delta(\theta_t\omega)$ be the unique stationary solution of the stochastic equation:

where $W$ is a two-sided real-valued Wiener process on $(\Omega, \mathcal{F}, P)$. The process $\zeta_\delta(\theta_t\omega)$ is called the one-dimensional colored noise. Recall that there exists a $\theta_t$-invariant subset of full measure (see ^[1]), which is still denoted by $\Omega$, such that for all $\omega\in\Omega$, $\zeta_\delta(\theta_t\omega)$ is continuous in $t\in\mathbb{R}$ and

Let $-\Delta$ 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

Following Dafermos ^[5], we introduce a Hilbert "history" space $\mathfrak{R}_{\mu, 2} = L^2_{\mu}(\mathbb{R}^+, V_2)$ with the inner product

and new variables

By differentiation we have

Then (1.1) can be rewritten as the equivalent system

We introduce the following hypotheses to complete the uniform estimates.

Assume that the memory kernel function $\mu\in C^1(\mathbb{R}^+)\cap L^1(\mathbb{R}^+)$, and satisfy the following conditions:

$\forall\ s\in\mathbb{R}^+ \ \text{and some} \ \varrho > 0$.

note that (3.3) implies $\varpi\stackrel{\mathrm{def}}{ = }\|\mu\|_{L^1(\mathbb{R}^+)} = \int_0^\infty\mu(s)ds > 0$.

Let $f:\mathbb{R}^n\times\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function and $F(x, r) = \int^r_0f(x, s)ds$ for all $x\in\mathbb{R}^n, r\in\mathbb{R}$ and $s, s_1, s_2\in\mathbb{R}$,

where $p > 0$ for $1\leq n\leq 4$ and $0 < p\leq\frac{4}{n-4}$ for $n\geq5$, $\alpha_1$ is a positive constant, $\varphi_1\in L^1(\mathbb{R}^n)$, and $\varphi\in L^\infty(\mathbb{R}^n)$.

Let $h:\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}\rightarrow \times\mathbb{R}$ be continuous such that for all $t, s, s_1, s_2\in\mathbb{R}$ and $x\in\mathbb{R}^n$,

where $\alpha_2$ and $\alpha_3$ are positive constants, and $\varphi_2\in L^2_{loc}(\mathbb{R}, L^2(\mathbb{R}^n))$.

By (3.3), the space $\mathfrak{R}_{\mu, r} = L^2_{\mu}(\mathbb{R}^+, V_r)(r\in\mathbb{R})$ is a Hilbert space of $V_r$-valued functions on $\mathbb{R}^+$ with the inner product and norm

and on $\mathfrak{R}_{\mu, r}$, the linear operator $-\partial_s$ has domain

Definition 3.1. Given $\tau\in\mathbb{R}, \omega\in\Omega, \ T > 0, u_0\in H^2(\mathbb{R}^n)$, $u_{1, 0}\in L^2(\mathbb{R}^n)$, and $\eta^0\in\mathfrak{R}_{\mu, 2}$, a function $z(t) = (u, u_t, \eta^t)$ is called a (weak) solution of (3.2) if the following conditions are fulfilled:

(i) $u(\cdot, \tau, \omega, u_0, u_{1, 0})\in L^\infty(\tau, \tau+T; H^2(\mathbb{R}^n))\cap C([\tau, \tau+T], L^2(\mathbb{R}^n))$ with $u(\tau, \tau, \omega, u_0, u_{1, 0}) = u_0, u_t(\cdot, \tau, \omega, u_0, u_{1, 0})\in L^\infty(\tau, \tau+T; L^2(\mathbb{R}^n))\cap C([\tau, \tau+T], L^2(\mathbb{R}^n))$ with $u_t(\tau, \tau, \omega, u_0, u_{1, 0}) = u_{1, 0}$ and $\eta^t(\cdot, \tau, \omega, \eta^0, s)\in L^\infty(\tau, \tau+T; \mathfrak{R}_{\mu, 2})\cap C([\tau, \tau+T], L^2(\mathbb{R}^n))$ with $\eta^t(\tau, \tau, \omega, \eta^0, s) = \eta^0$.

(ii) $u(t, \tau, \cdot, u_0, u_{1, 0}):\Omega\rightarrow H^2(\mathbb{R}^n)$ is $(\mathcal{F}, \mathcal{B}(H^2(\mathbb{R}^n))$-measurable, $u_t(t, \tau, \cdot, u_0, u_{1, 0}):\Omega\rightarrow L^2(\mathbb{R}^n)$ is $(\mathcal{F}, \mathcal{B}(L^2(\mathbb{R}^n))$-measurable, and $\eta^t(t, \tau, \cdot, \eta^0, s):\Omega\rightarrow \mathfrak{R}_{\mu, 2}$ is $(\mathcal{F}, \mathcal{B}(\mathfrak{R}_{\mu, 2})$-measurable.

(iii) For all $\xi\in C^\infty_0((\tau, \tau+T)\times\mathbb{R}^n)$,

In order to investigate the long-time dynamics, we are now ready to prove the existence and uniqueness of solutions of (3.2). We first recall the following well-known existence and uniqueness of solutions for the corresponding linear plate equations of (1.1)(see ^[34,35]).

Lemma 3.1. Let $u_0\in H^2(\mathbb{R}^n), u_{1, 0}\in L^2(\mathbb{R}^n)$ and $g\in L^1(\tau, \tau+T; L^2(\mathbb{R}^n))$ with $\tau\in\mathbb{R}$ and $T > 0$. Then the linear plate equation

with the initial conditions

possesses a unique solution $(u, u_t, \eta^t)$ in the sense of Definition 3.1. In addition,

and there exists a positive number $C$ depending only on $\nu$ (but independent of $\tau, T, u_0, u_{1, 0}$ and $g$) such that for all $t \in[\tau, \tau+ T]$,

Furthermore, the solution $(u, u_t, \eta^t)$ satisfies the energy equation

and

for almost all $t \in[\tau, \tau+ T]$.

Theorem 3.1. Let $\tau\in\mathbb{R}, u_0\in H^2(\mathbb{R}^n), u_{1, 0}\in L^2(\mathbb{R}^n)$ and $\eta^0\in\mathfrak{R}_{\mu, 2}$. Suppose (3.3)-(3.8) hold, then:

(a) Problem (3.2) possesses a solution $z(t) = (u, u_t, \eta^t)$ in the sense of Definition 3.1;

(b) The solution $z(t) = (u, u_t, \eta^t)$ to problem (3.2) is unique, continuous in initial data in $H^2(\mathbb{R}^n)\times L^2(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}$, and

Moreover, the solution $z(t) = (u, u_t, \eta^t)$ to problem (3.2) satisfies the energy equation:

for almost all $t\in[\tau, \tau+T]$.

Proof. The proof will be divided into four steps. We first construct a sequence of approximate solutions, and then derive uniform estimates, in the last two steps we take the limit of those approximate solutions to prove the uniqueness of solutions.

Step (i): Approximate solutions. Given $k\in\mathbb{N}$, define a function $\eta_k:\mathbb{R}\rightarrow \mathbb{R}$ by

Then for every fixed $k\in\mathbb{N}$, the function $\eta_k$ as defined by (3.14) is bounded and Lipschitz continuous; more precisely, for all $s, s_1, s_2\in\mathbb{R}$

For all $x\in\mathbb{R}^n$ and $t, s\in\mathbb{R}$, denote

By (3.4) we know that there exists $k_0\in\mathbb{N}$ such that for all $|s|\geq k_0$ and $x\in\mathbb{R}^n$,

thus, for all $k\geq k_0$ and $x\in\mathbb{R}^n$,

By (3.5)-(3.6), (3.15)-(3.16) and (3.18) we know that for all $s, s_1, s_2\in\mathbb{R}$ and $x\in\mathbb{R}^n$,

and

By (3.19) we get that for all $s\in\mathbb{N}$ and $x\in\mathbb{R}^n$,

By (3.7)-(3.8) and (3.15)-(3.16) we obtain that for all $k\geq1, t, s, s_1, s_2\in\mathbb{R}$ and $x\in\mathbb{R}^n$,

By (3.3) and (3.15)-(3.16), we find that for all $k\in\mathbb{N}, s, s_1, s_2\in\mathbb{N}$ and $x\in\mathbb{R}^n$,

For every $k\in\mathbb{N}$, consider the following approximate system for $u_k, \eta^t_k$:

From (3.23)-(3.24), $\varphi\in L^\infty(\mathbb{R}^n)$ and the standard method (see, e.g., ^[11]), it follows that for each $\tau\in\mathbb{R}, \omega\in\Omega, u_0\in H^2(\mathbb{R}^n), u_{1, 0}\in L^2(\mathbb{R}^n)$ and $\eta^0\in\mathfrak{R}_{\mu, 2}$, problem (3.26) has a unique global solution $(u_k, \partial_t u_k, \eta^t_k)$ defined on $[\tau, \tau+T]$ for every $T > 0$ in the sense of Definition 3.1. In particular, $u_k(\cdot, \tau, \omega, u_0)\in C([\tau, \tau+T], H^2(\mathbb{R}^n))$ and $u_k(t, \tau, \omega, u_0)$ is measurable with respect to $\omega\in\Omega$ in $H^2(\mathbb{R}^n)$ for every $t\in[\tau, \tau+T]$; $\partial_tu_k(\cdot, \tau, \omega, u_0)\in C([\tau, \tau+T], L^2(\mathbb{R}^n))$ and $\partial_tu_k(t, \tau, \omega, u_0)$ is measurable with respect to $\omega\in\Omega$ in $L^2(\mathbb{R}^n)$ for every $t\in[\tau, \tau+T]$; $\eta^t_k(\cdot, \tau, \omega, \eta_0, s)\in C([\tau, \tau+T], \mathfrak{R}_{\mu, 2})$ and $\eta^t_k(t, \tau, \omega, \eta_0, s)$ is measurable with respect to $\omega\in\Omega$ in $\mathfrak{R}_{\mu, 2}$ for every $t\in[\tau, \tau+T]$ Furthermore, the solution $u_k$ satisfies the energy equation:

for almost all $t\in[\tau, \tau+T]$. Next, we use the energy equation (3.25) to derive uniform estimate on the sequence $\{u_k, \partial_tu_k, \eta^t_k\}^\infty_{k = 1}$.

Step (ii): Uniform estimates.

For the last term on the right-hand side of (3.25), by (3.21) we have

By Young's inequality, we get

By (3.27)–(3.29) together with (3.3), it follows that for almost all $t\in[\tau, \tau+T]$,

where $c_1 > 0$ depends only on $\alpha_2$, but independent of $k$.

By (3.20) and (3.30) we obtain

where $c_2 > 0$ depends only on $\nu$ and $\alpha_2$, but independent of $k$.

Multiplying (3.31) with $e^{-c_2\int^t_0(1+|\zeta_\delta(\theta_r\omega)|)dr}$, and then integrating the inequality on $(\tau, t)$, we have

By (3.21) we get, for all $k\geq1$,

By (3.32)-(3.33) imply that there exists a positive constant $c_3 = c_3(\tau, T, \varphi, \varphi_1, \varphi_2, g, \omega, \delta, \alpha_1, \nu)$ (but independent of $k, u_0, u_{1, 0}$) such that for all $t\in[\tau, \tau+T]$ and $k\geq1$,

which along with (3.20) show that for all $t\in[\tau, \tau+T]$ and $k\geq k_0$,

thus,

By (3.19), there exists a positive constant $c_4 = c_4(p, n, \alpha_1)$ such that

which along with the embedding $H^{2}(\mathbb{R}^n)\hookrightarrow L^{2(p+1)}(\mathbb{R}^n)$ and the assumption $\varphi\in L^{\infty}(\mathbb{R}^n)$ implies that there exists $c_5 = c_5(p, n, \alpha_1, \varphi) > 0$ (independent of $k$) such that

By (3.35) and (3.38) we see that

By (3.22) we get

which together with (3.35) shows that

By (3.35)–(3.37) and (3.39)-(3.40), it follows that there exists $u\in L^\infty(\tau, \tau+T; H^{2}(\mathbb{R}^n))$ with $\partial_tu\in L^\infty(\tau, \tau+T; L^{2}(\mathbb{R}^n)), \kappa_1\in L^{2}(\tau, \tau+T; L^{2}(\mathbb{R}^n)), \kappa_2\in L^2(\tau, \tau+T; L^{2}(\mathbb{R}^n)), v^{\tau+T}\in H^{2}(\mathbb{R}^n)$ and $v^{\tau+T}_1\in L^{2}(\mathbb{R}^n)$ such that

It follows from (3.41)-(3.42) that there exists a subsequence which is still denoted $u_k$, such that

By (3.15) and (3.48) we get that for almost all $(t, x)\in[\tau, \tau+T]\times\mathbb{R}^n$,

By (3.49), we have

It follows from (3.44)-(3.45), (3.50)-(3.51) that

Step (iii): Existence of solutions.

Choosing an arbitrary $\xi\in C^\infty_0((\tau, \tau+T)\times\mathbb{R}^n)$. By (3.26) we get

Letting $k\rightarrow \infty$ in (3.54), it follows from (3.41)-(3.43) and (3.52)-(3.53) that for any $\xi\in C^\infty_0((\tau, \tau+T)\times\mathbb{R}^n)$,

Notice that

By (3.56) we obtain

We claim that

In fact, by (3.5) we obtain that there exists some $c_6 = c_6(p, n, \alpha_1, \varphi) > 0$ such that

which along with (3.56) to obtain (3.58).

By (3.54)–(3.58), we can get

where $H^{-2}(\mathbb{R}^n)$ is the dual space of $H^{2}(\mathbb{R}^n)$.

Next, we prove ($u, u_t, \eta^t)$ satisfy the initial conditions $(3.2)_2$.

By (3.26), we get that for any $v\in C^\infty_0(\mathbb{R}^n)$ and $\psi\in C^2([\tau, \tau+T])$,

Letting $k\rightarrow \infty$ in (3.60), by (3.41)-(3.43), (3.46)-(3.47) and (3.52)-(3.53) we obtain, for any $v\in C^\infty_0(\mathbb{R}^n)$ and $\psi\in C^2([\tau, \tau+T])$,

By (3.55) we get that for any $v\in C^\infty_0(\mathbb{R}^n)$,

By (3.62) we find that for any $v\in C^\infty_0(\mathbb{R}^n)$ and $\psi\in C^2([\tau, \tau+T])$,

together with (3.61) to obtain, for $v\in C^\infty_0(\mathbb{R}^n)$ and $\psi\in C^2([\tau, \tau+T])$,

Let $\psi\in C^2([\tau, \tau+T])$ such that $\psi(\tau+T) = \psi'(\tau+T) = \psi'(\tau) = 0$ and $\psi(\tau) = 1$, by (3.64), we have

Let $\psi\in C^2([\tau, \tau+T])$ such that $\psi(\tau+T) = \psi'(\tau+T) = \psi(\tau) = 0$ and $\psi'(\tau) = 1$, by (3.64), we have

which together with (3.65) that $(u, u_t, \eta^t)$ satisfies the initial conditions $(3.2)_2$.

Through choosing proper $\psi\in C^2([\tau, \tau+T])$, we can also obtain from (3.64) that

which along with (3.46)-(3.47) implies that

thereby,

Similar to (3.67)-(3.69), one can verify that for any $t\in[\tau, \tau+T]$,

By (3.70)–(3.72), we get the that $(u, u_t, \eta^t)$ is a solution of (3.2) in the sense of Definition 3.1.

Step (iv): Uniqueness of solutions.

Let $(u_1, (u_1)_t, \eta^t_1)$ and $(u_2, (u_2)_t, \eta^t_2)$ be solutions to (3.2), denote $v = u_1-u_2, \bar{\eta}^t = \eta^t_1-\eta^t_2$. Then we have

by (3.10), we get

Since $H^2(\mathbb{R}^n)\hookrightarrow L^{2(p+1)}(\mathbb{R}^n)$ for $0 < p\leq\frac{4}{n-4}$, by (3.5), we get

and hence

By (3.8) we get

It follows from (3.74)–(3.76) that

where $c_{7} > 0$ depends on $\tau$ and $T$. Since $u_1, u_2\in L^\infty(\tau, \tau+T; H^{2}(\mathbb{R}^n))$, then applying the Gronwall's lemma on $[\tau, \tau+T]$, we can obtain that the uniqueness of solution as well as the continuous dependence property of solution with initial data.

We now define a mapping $\Phi:\mathbb{R}^+\times\mathbb{R}\times\Omega\times H^{2}(\mathbb{R}^n)\times L^{2}(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}\rightarrow H^{2}(\mathbb{R}^n)\times L^{2}(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}$ such that for all $t\in\mathbb{R}^+, \tau\in\mathbb{R}, \omega\in\Omega$ and $(u_0, u_{1, 0}, \eta^0)\in H^{2}(\mathbb{R}^n)\times L^{2}(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}$,

where $(u, u_t, \eta^t)$ is the solution of (3.2). Then $\Phi$ is a continuous cocycle on $H^{2}(\mathbb{R}^n)\times L^{2}(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}$ over $(\Omega, \mathcal{F}, P, \{\theta_t\}_{t\in\mathbb{R}})$.

4.   Uniform estimates of solutions

In this section, we derive necessary estimates of solutions of (3.2) under stronger conditions than (3.4)-(3.8) on the nonlinear functions $f$ and $h$. These estimates are useful for proving the asymptotic compactness of the solutions and the existence of pullback random attractors.

From now on, we assume $f$ satisfies: for all $x\in\mathbb{R}^n$ and $s\in\mathbb{R}$,

where $p > 0$ for $1\leq n\leq 4$ and $0 < p\leq\frac{4}{n-4}$ for $n\geq5$, $\gamma\in(0, 1]$, $\alpha_4, \varsigma$ are positive constants, $\varphi_3\in L^1(\mathbb{R}^n)$, and $\varphi_4\in L^2(\mathbb{R}^n)\cap L^\infty(\mathbb{R}^n), \iota > 0$ will be denoted later.

By (3.5) and (4.1) we get that for all $x\in\mathbb{R}^n$ and $s\in\mathbb{R}$,

Assume the nonlinearity $h$ satisfies: for all $x\in\mathbb{R}^n$ and $t, s\in\mathbb{R}$,

where $\varphi_5\in L^\infty(\mathbb{R}^n)\cap L^{2+\frac{4}{p}}(\mathbb{R}^n)$, $\varphi_6\in L^2(\mathbb{R}^n)$, and $\varphi_7\in L^2(\mathbb{R}^n)\cap L^\infty(\mathbb{R}^n)$.

Let $\mathcal{D}$ be the set of all tempered families of nonempty bounded subsets of $H^{2}(\mathbb{R}^n)\times L^{2}(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}$. $D = \{D(\tau, \Omega):\tau\in\mathbb{R}, \omega\in\Omega\}$ is called tempered if for any $c > 0$,

where $\|D\|_{H^{2}(\mathbb{R}^n)\times L^{2}(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}} = \sup\limits_{\xi\in D}\|\xi\|_{H^{2}(\mathbb{R}^n)\times L^{2}(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}}$.

Under $\alpha > 0, \nu > 0, \varrho > 0, \varpi > 0$ and $\gamma\in (0, 1]$, we can choose a sufficiently small positive constant $\varepsilon$ such that

We also assume

Lemma 4.1. Let (3.3)–(3.5), (3.8), (4.1)-(4.2) and (4.5)–(4.8) hold. Then for any $\tau\in\mathbb{R}, \omega\in\Omega$ and $D\in\mathcal{D}$, there exists $T = T(\tau, \omega, D) > 0$ such that for all $t\geq T$, the solution of (3.2) satisfies

where $(u_0, u_{1, 0}, \eta^0) \in D(\tau-t, \theta_{-t}\omega)$ and $M_1$ is a positive constant independent of $\tau, \omega$ and $D$.

Proof. By (3.11), (3.13), (4.1) and (4.10) we obtain, for almost all $t\in[\tau, \tau+T]$,

By (3.3), (4.2) and (4.5) we have

where $c_4 > 0$ depends on $\alpha, \nu, \gamma, \varepsilon$.

It follows from (4.10)-(4.13) and rewrite the result obtained, we have

where $c_5 > 0$ depends on $\alpha, \nu, \gamma, \varepsilon$.

For the second term on the right-hand side of (4.14) we get

By (4.14)-(4.15) we get

Multiplying (4.14) by $e^{\frac{1}{4}\varepsilon\gamma t}$, and then integrating the inequality $[\tau-t, \tau]$, after replacing $\omega$ by $\theta_{-\tau}\omega$, we get

For the first term on the right-hand side of (4.17), by (4.4) we get

By (4.7) we get

which along with (4.2) and (4.18) that for all $t\geq T$,

Then the proof is completed.

Based on Lemma 4.1, we can easily obtain the following Lemma that implies the existence of tempered random absorbing sets of $\Phi$.

Lemma 4.2. If (3.3)-(3.5), (3.8), (4.1)-(4.2) and (4.5)-(4.9) hold, then the cocycle $\Phi$ possesses a closed measurable $\mathcal{D}$-pullback absorbing set $B = \{B(\tau, \omega):\tau\in\mathbb{R}, \omega\in\Omega\}\in\mathcal{D}$, which is given by

where

In order to derive the uniform tail-estimates of the solutions of (3.2) for large space variables when times is large enough, we need to derive the regularity of the solutions in a space higher than $H^2(\mathbb{R}^n)$.

Lemma 4.3. Let (3.3)–(3.5), (3.8), (4.1)-(4.2) and (4.5)–(4.8) hold. Then for any $\tau\in\mathbb{R}, \omega\in\Omega$ and $D\in\mathcal{D}$, there exists $T = T(\tau, \omega, D) > 0$ such that for all $t\geq T$, the solution of (3.2) satisfies

where $(u_0, u_{1, 0}, \eta^0) \in D(\tau-t, \theta_{-\tau}\omega)$ and $M_2$ is a positive number independent of $\tau, \omega$ and $D$.

Proof. Taking the inner product of $(3.2)_1$ with $A^{\frac{1}{2}} u$ in $L^2(\mathbb{R}^n)$, we have

Taking the inner product of $(1.1)_1$ with $A^{\frac{1}{2}} u_t$ in $L^2(\mathbb{R}^n)$, we find that

By (4.20) and (4.21), we get

By (3.3), (4.5), (4.6) and Lemma 4.1, we have

From (4.3) and Lemma 4.1 yields

where the definition of $L$ see Lemma 4.2, and $C$ is the positive constant satisfying

Choosing

thus, we get

By (4.22)–(4.26), we get

which can be rewritten as

For the last term on the right-hand side of (4.27) we have

which together with (4.27), we get

Similar to the remainder of Lemma 4.1, we can obtain the desired result.

Lemma 4.4. Let (3.3)–(3.5), (3.8), (4.1)-(4.2) and (4.5)–(4.8) hold. Then for every $\eta > 0, \tau\in\mathbb{R}, \omega\in\Omega$ and $D\in\mathcal{D}$, there exists $T_0 = T_0(\eta, \tau, \omega, D) > 0$ and $m_0 = m_0(\eta, \tau, \omega)\geq1$ such that for all $t\geq T_0$, $m\geq m_0$ and $(u_0, u_{1, 0}, \eta^0) \in D(\tau-t, \theta_{-\tau}\omega)$, the solution of (3.2) satisfies

Proof. Let $\rho:\mathbb{R}^n\rightarrow \mathbb{R}$ be a smooth function such that $0\leq\rho(x)\leq1$ for all $x\in\mathbb{R}^n$, and

For every $m\in\mathbb{N}$, let

Then there exist positive constants $c_{11}$ and $c_{12}$ independent of $m$ such that $|\nabla\rho_m(x)|\leq\frac{1}{m}c_{11}$, $|\Delta\rho_m(x)|\leq\frac{1}{m}c_{12}$ for all $x\in\mathbb{R}^n$ and $m\in\mathbb{N}$.

Similar to the energy equation (3.11), we have

Taking the inner product of $(3.2)_1$ with $\rho_m(x)u$ in $L^2(\mathbb{R}^n)$, we have

By (4.28)-(4.29) and (4.1), we get

Similar to the arguments of (4.11)-(4.13), we have the following estimates:

where $c_{13}$ depends only on $\alpha, \nu, \gamma$ and $\varepsilon$.

By (4.30)–(4.33) we get

where $c_{14} > 0$ depends only on $\alpha, \nu, \gamma$ and $\varepsilon$, but not on $m$.

By (4.34) we get

By Young's inequality we get

By (4.35)-(4.36) we get

By (4.7) and (4.37) we have

Multiplying (4.38) by $e^{\frac{1}{4}\varepsilon\gamma t}$, and then integrating the inequality $[\tau-t, \tau]$, after replacing $\omega$ by $\theta_{-\tau}\omega$, we get

Next, we estimate the right-hand side of (4.39). By (4.18), we know that there exists $T_1(\eta, \tau, \omega, D) > 0$ such that for all $t\geq T_1$,

For the second and the third terms on the right-hand side of (4.39) we get

By (4.8) and the conditions of $\varphi_i(x)(i = 1, 3, 5, 6)$ satisfy, we know that there exists $m_1 = m_1(\eta, \tau, \omega)\geq1$ such that for all $m\geq m_1$, the right-hand of side of (4.39) is bounded by $\eta$, i.e.,

For the last term in (4.39), by Lemma 4.1 and Lemma 4.3, we know that there exists $T_2(\eta, \tau, \omega, D)\geq T_1$ such that for all $t\geq T_2$,

where $c_{15} > 0$ depends only on $\alpha, \nu, \gamma, \varepsilon, \tau$ and $\omega$, but not on $m$. Thus, there exists $m_2 = m_2(\eta, \tau, \omega)\geq m_1$ such that for all $m\geq m_2$ and $t\geq T_2$,

By (4.39), (4.40), (4.42) and (4.43) we see that for all $m\geq m_2$ and $t\geq T_2$,

By (4.7) we have

which together with (4.2) and (4.44) yields that for all $m\geq m_2$ and $t\geq T_2$,

Since $\varphi_1\in L^1(\mathbb{R}^n)$, there exists $m_3 = m_3(\eta, \tau, \omega)\geq m_2$ such that for all $m\geq m_3$,

From (4.45)-(4.46) we obtain, for all $m\geq m_3$ and $t\geq T_2$,

5.   Existence of random attractors

In this section, we present the existence and uniqueness of $\mathcal{D}$-pullback random attractors of (3.2).

Let $z = (u, u_t, \eta^t)$ be the solution of (3.2). Denote $u = \tilde{v}+v, \eta^t = \tilde{\eta}^t+\eta$ where $(\tilde{v}, \tilde{\eta}^t)$ and $(v, \eta^t)$ are the solutions of the following equations, respectively,

and

Lemma 5.1. Suppose (3.3), (4.7)-(4.8) hold. Then for every $\tau\in\mathbb{R}, \omega\in\Omega$ and $D\in\mathcal{D}$, there exists $T = T(\tau, \omega, D) > 0$ such that for all $t\geq T$ and $r\in[-t, 0]$, the solution $\tilde{v}$ of (5.1) satisfies

where $(u_0, u_{1, 0})\in D(\tau-t, \theta_{-t}\omega)$ and $M_2$ is a positive number independent of $\tau, \omega$ and $D$.

Proof. From (3.10)-(3.11) and (5.1) we see that

In addition, we get

By (4.11)-(4.12) and (5.3)-(5.4) we have

which can be rewritten as

It follows from (4.7) and (5.5) that

Applying Gronwall's lemma to (5.6), we get for all $\tau\in\mathbb{R}, t\geq0, r\in[-t, 0]$ and $\omega\in\Omega$,

By (4.7) we have

By (5.7)-(5.8) we see that for all $\tau\in\mathbb{R}, t\geq0, r\in[-t, 0]$ and $\omega\in\Omega$,

Similar to (4.16), one can verify that

which along with (5.9) yields the desire result.

Based on Lemma 5.1, we infer that system (5.1) has a tempered pullback random absorbing set.

Lemma 5.2. Suppose (3.3), (4.8)-(4.9) hold, then (5.1) possesses a closed measurable $\mathcal{D}$-pullback absorbing set $B_1 = \{B_1(\tau, \omega):\tau\in\mathbb{R}, \omega\in\Omega\}\in\mathcal{D}$, which is given by

where

Lemma 5.3. Suppose (4.8)-(4.9) hold, then the sequence of the solutions to (5.1)

converges in $H^2(\mathbb{R}^n)\times L^2(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}$ for any $\tau\in\mathbb{R}, \omega\in\Omega, D\in\mathcal{D}, t_n\rightarrow \infty$ monotonically, and $(u^{(n)}_0, u^{(n)}_{1, 0}, \eta^{(0n)})\in D(\tau-t_n, \theta_{-t_n}\omega)$.

Proof. Let $m > n$ and

for $t\geq\tau-t_n$.

by (5.1) we get

Similar to (5.9) with $r = 0, t = t_n$ and $g = 0$, we obtain

which together with $(5.12)_2$, we get

By (5.9) with $r = -t_n$, and $t = t_m$, we obtain

It follows from (5.14)-(5.15) that for $m > n\rightarrow \infty$,

together with (5.11) implies $\{\tilde{v}(\tau, \tau-t_n, \theta_{-\tau}\omega, u^{(n)}_0), \tilde{v}_t(\tau, \tau-t_n, \theta_{-\tau}\omega, u^{(n)}_{1, 0}), \tilde{\eta}^t(\tau, \tau-t_n, \theta_{-\tau}\omega, \eta^{(0n)})\}^\infty_{n = 1}$ is a Cauchy sequence in $H^2(\mathbb{R}^n)\times L^2(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}$. This complete the proof.

Lemma 5.4. Suppose (3.3), (4.8)-(4.9) hold, then (5.1) has a unique $\mathcal{D}$-pullback random attractor $\mathcal{A}_1 = \{\mathcal{A}_1(\tau, \omega):\tau\in\mathbb{R}, \omega\in\Omega\}\in\mathcal{D}$ in $H^2(\mathbb{R}^n)\times L^2(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}$, which is actually a singleton; that is, $\mathcal{A}_1(\tau, \omega)$ consisting of a single point for all $\tau\in\mathbb{R}, \omega\in\Omega$.

Proof. From Lemmas 5.2 and 5.3 by applying the abstract results in ^[29], we can get the existence and uniqueness of the $\mathcal{D}$-pullback random attractor $\mathcal{A}_1\in\mathcal{D}$ of (5.1) in $H^2(\mathbb{R}^n)\times L^2(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}$ immediately.

Next, we prove $\mathcal{A}_1$ is a singleton. Suppose $\{t_n\}^\infty_{ n = 1}$ 1 be a sequence of numbers such that $t_n\rightarrow \infty$ as $n\rightarrow \infty$. Given $\tau\in\mathbb{R}, \omega\in\Omega$, let $(z^{(n)}_0, z^{(n)}_{1, 0}, \eta^{(0n)}), (y^{(n)}_0, y^{(n)}_{1, 0}, y^{(0n)})\in\mathcal{A}_1(\tau-t_n, \theta_{-t_n}\omega)$.

Similar to (5.13) we have

Due to $\mathcal{A}_1\in\mathcal{D}$, we see that the right-hand side of (5.16) tends to zero as $n\rightarrow \infty$, and thus we get

which together with the invariance of $\mathcal{A}_1$, we know that the $\mathcal{D}$-pullback random attractor $\mathcal{A}_1$ is a singleton. This complete the proof.

To obtain the asymptotic compactness of the solutions of (5.2), we need the following Lemma.

Lemma 5.5. Let $u_0\in H^2(\mathbb{R}^n)$, $u_{1, 0}\in L^2(\mathbb{R}^n), \eta^0\in\mathfrak{R}_{\mu, 2}, \tau\in\mathbb{R}, \omega\in\Omega$ and $T > 0$. If (3.3)-(3.5), (3.8), (4.1)-(4.2) and (4.5)-(4.8) hold, then the solution of (5.2) satisfies, for all $t\in[\tau, \tau+T]$,

where $C$ is a positive number depending on $\tau, \omega, T$ and $R$ when $\|(u_0, u_{1, 0}, \eta^0)\|_{H^2(\mathbb{R}^n)\times L^2(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}}\leq R$.

Proof. This is an immediate consequence of Lemma 4.3.

Lemma 5.6. Let (3.3)–(3.5), (3.6), (4.1)–(4.3) and (4.5)–(4.9) hold. Then the cocycle $\Phi$ is $\mathcal{D}$-pullback asymptotically compact in $H^2(\mathbb{R}^n)\times L^2(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}$; that is, the sequence $\{\Phi(t_n, \tau-t_n, \theta_{-t_n}\omega, (u^{(n)}_0, u^{(n)}_{1, 0}), \eta^{(0n)}\}^\infty_{n = 1}$ has a convergent subsequence in $H^2(\mathbb{R}^n)\times L^2(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}$ for any $\tau\in\mathbb{R}, \omega\in\Omega, D\in\mathcal{D}, t_n\rightarrow \infty$ and $(u^{(n)}_0, u^{(n)}_{1, 0}, \eta^{(0n)})\in D(\tau-t_n, \theta_{-t_n}\omega)$.

Proof. Given $t\in\mathbb{R}^+, \tau\in\mathbb{R}, \omega\in\Omega$ and $(u_0, u_{1, 0}, \eta^0)\in H^2(\mathbb{R}^n)\times L^2(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}$, define

where $(\tilde{v}, \tilde{\eta}^t)$ and $(v, \eta^t)$ are the solutions of (5.1) and (5.2), respectively.

By(3.78) we have

Let $B\in\mathcal{D}$ be the $\in\mathcal{D}$-pullback absorbing set of $\Phi$ given by (4.19). From Lemmas 4.2, 4.4 and 5.4 we see that for every $\delta > 0$ there exists $t_0 = t_0(\delta, \tau, \omega, B) > 0$ and $k_0 = k_0(\delta, \tau, \omega)\geq1$ such that for all $(u_0, u_{1, 0}, \eta^0)\in B(\tau-t_0, \theta_{-t_0}\omega)$,

with $\tilde{\mathcal{O}}_{k_0} = \{x\in\mathbb{R}^n:|x| > k_0\}$, and

in $H^2(\mathbb{R}^n)\times L^2(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}$.

In addition, by Lemma 5.5 we know that for every $t\in\mathbb{R}^+, \tau\in\mathbb{R}, \omega\in\Omega$ and $k\in\mathbb{N}$,

and thus for each $k\in\mathbb{N}$,

with $\mathcal{O}_{k} = \{x\in\mathbb{R}^n:|x| < k\}$.

It follows from (5.17)–(5.20) we get that all conditions of Theorem 2.1 are satisfied, so $\Phi$ is $\mathcal{D}$-pullback asymptotically compact in $H^{2}(\mathbb{R}^n)\times L^{2}(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}$.

Since Lemma 4.2 implies a closed measurable $\mathcal{D}$-pullback absorbing set for $\Phi$, and $\Phi$ is $\mathcal{D}$-pullback asymptotically compact in $H^{2}(\mathbb{R}^n)\times L^{2}(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}$ from Lemma 5.6, we immediately get the following existence theorem by Theorem 2.2.

Theorem 5.1. Let (3.3)–(3.5), (3.6), (4.1)–(4.3) and (4.5)–(4.9) hold. Then the cocycle $\Phi$ has a unique $\mathcal{D}$-pullback random attractor in $H^2(\mathbb{R}^n)\times L^2(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}$.

6.   The discussion of the proposed method's theoretical analysis

In this paper, we use the uniform estimates on the tails of solutions and the splitting technique to obtained the existence and uniqueness of $\mathcal{D}$-pullback attractor for the problem (1.1). The method used in this paper is proposed by P. W. Bates et al ^[3], they applied the method to deal with the asymptotic behavior of the non-automatous random system on unbounded domains. More precisely, one first need to show that the tails of the solutions of (1.1) are uniformly small outside a bounded domain for large time, and then derive the asymptotic compactness of solutions in bounded domains by splitting the solutions as two parts: one part has trivial dynamics in the sense that it possesses a unique tempered attracting random solution; and the other part has regularity higher than $H^2(\mathbb{R}^n)\times L^2(\mathbb{R}^n)\times\mathfrak{R}_{\mu, 2}$ based on the estimates of solutions (see Lemma 4.3).

7.   Conclusions

Using the uniform estimates on the tails of solutions and the splitting technique, we obtained the existence and uniqueness of $\mathcal{D}$-pullback attractor for the problem (1.1). It is well-known that the pullback random attractors are employed to describe long-time behavior for an non-autonomous dynamical system with random term, while the $\mathcal{D}$-pullback attractor that we obtained can characterize the asymptotic behavior of the equation like (1.1), which is featured with both stochastic term and non-autonomous term.

Acknowledgments

The author X. Yao was supported by the Natural Science Foundation of China (No. 12161071, 11961059).

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this article.