Automorphism group and twisted modules of the twisted Heisenberg-Virasoro vertex operator algebra

1.   Introduction

Let $\mathcal Q\subset {\mathbb R}^n$ be a bounded $C^2$-domain and $\mathcal O_\varepsilon\subset {\mathbb R}^{n+1}$ be the domain

where $g\in C^2(\overline {\mathcal Q}, (0,+\infty))$ and $0<\varepsilon\leq 1$. Since $g\in C^2(\overline {\mathcal Q}, (0,+\infty))$, there exist two positive constants $\gamma_1$ and $\gamma_2$ such that

Denote $\mathcal{O} = \mathcal Q \times (0,1)$ and $\widetilde{\mathcal O} = \mathcal Q \times (0,\gamma_2 )$ which contains $\mathcal O_\varepsilon$ for $0<\varepsilon\leq 1.$ Given $\tau\in {\mathbb R}$, we will study the limit of asymptotical behavior of the following stochastic reaction-diffusions equation with multiplicative noise defined on the thin domain $\mathcal O_{\varepsilon}$ as $\varepsilon$ tends to 0:

with the initial condition

where $\nu_\varepsilon$ is the unit outward normal vector to $\partial \mathcal O_\varepsilon$, $H$ is a superlinear source term, $G$ is a function defined on ${\mathbb R}\times \widetilde{\mathcal O}$, $c_j\in {\mathbb R}$ for $j = 1,2,\dots,m$, $w_j$, $j = 1,2,\dots,m$, are independent two-sided real-valued Wiener processes on a probability space, and the symbol $\circ$ indicates that the equation is understood in the sense of Stratonovich integration.

As $\varepsilon\rightarrow 0$, we will show in certain sense that the limiting behavior of (2) is governed by the following equation:

with the initial condition

where $\nu_0$ is the unit outward normal to $\partial \mathcal Q$.

Random attractors have been investigated in [2,5,10,19,9] in the autonomous stochastic case, and in [3,21,22,23] in the non-autonomous stochastic case. Recently, the limiting dynamical behavior of stochastic partial differential equations on thin domain was studied in [16,20,13,14,11,12,17,4]. However, in [17,13], we only investigated the limiting behavior of random attractors in $L^2(\mathcal O)$ of stochastic evolution equations on thin domain. In this paper, we will prove the existence and uniqueness of bi-spatial pullback attractor for the systems defined on fixed domain $\mathcal O$ converted from (2)-(3) when the initial space is $L^2 \left( \mathcal O \right)$ and the terminate space is $H^1 \left( \mathcal O \right)$ and establish upper semicontinuity result for the corresponding family of random attractors in $H^1 \left( \mathcal O \right)$ as $\varepsilon$ approaches 0.

Let $X$ be a Banach space. The norm of $X$ is written as $\|\cdot\|_X$. Let $\mathcal M = L^2 \left(\mathcal Q \right)$ and $\mathcal N = L^2 \left( \mathcal O \right)$. We denote by $(\cdot,\cdot)_{Y}$ the inner product in a Hilbert space $Y$. The letter $c$ and $c_i$, $i\in \mathbb N$, are generic positive constants which may change its values from line to line.

We organize the paper as follows. In the next section, we establish the existence of a continuous cocycle in $\mathcal N$ for the stochastic equation defined on the fixed domain $\mathcal O$ converted from (2)-(3). We also describe the existence of a continuous cocycle in $\mathcal M$ for the stochastic equation (4)-(5). Section 3 contains all necessary uniform estimates of the solutions. We then prove the existence and uniqueness of regular random attractors for the stochastic equations in section 4, and analyze convergence properties of the solutions as well as the random attractors in $H^1(\mathcal O)$ in section 5.

2.   Cocycles associated with non-autonomous stochastic equations

Here we show that there is a continuous cocycle generated by the reaction-diffusion equation defined on $\mathcal O_\varepsilon$ with multiplicative noise and deterministic non-autonomous forcing:

with the initial condition

where $\nu_\varepsilon$ is the unit outward normal to $\partial \mathcal O _\varepsilon$, $G: {\mathbb R}\times\widetilde{\mathcal O}\rightarrow \mathbb R$ belongs to $L^2_{loc}({\mathbb R}, {L^\infty( { {\widetilde {\mathcal O}} } )})$, $c_j\in {\mathbb R}$, $w_j$ ($j = 1,2,\dots,m$) are independent two-sided real-valued Wiener processes on a probability space which will be specified later, and $H$ is a nonlinear function satisfying the following conditions: for all $x\in \widetilde{\mathcal O}$ and $t,s\in {\mathbb{R}}$,

where $p>2$, $\lambda_1$ $\lambda_2$ and $\lambda_3$ are positive constants, $\varphi _1 \in L_{loc}^\infty({\mathbb{R}},{L^\infty( { {\widetilde {\mathcal O}} } )})$ and $\varphi _2, \psi _3 \in L_{loc}^2 ({\mathbb{R}},{L^\infty( { {\widetilde {\mathcal O}} } )}).$

Throughout this paper, we fix a positive number $\lambda \in (0, \lambda_1)$ and write

for all $x\in \widetilde{\mathcal O}$ and $t,s\in {\mathbb{R}}$. Then it follows from (8)-(11) that there exist positive numbers $\alpha_1$, $\alpha_2$, $\beta$, $b_1$ and $b_2$ such that

where $\psi_1(t,x) = \varphi_1(t,x) +b_1$ and $\psi_2(t,x) = \varphi_2(t,x) +b_2$ for $x\in \widetilde{\mathcal O}$ and $t,s\in {\mathbb{R}}$.

Substituting (12) into (6) we get for $t > \tau$,

with the initial condition

We now transfer problem (17)-(18) into an initial boundary value problem on the fixed domain $\mathcal O$. To that end, we introduce a transformation $T_\varepsilon: \mathcal O_\varepsilon \rightarrow \mathcal O$ by $T_\varepsilon (x^{\ast}, x_{n+1}) = \left (x^{\ast}, {\frac {x_{n+1}}{\varepsilon g(x^{\ast})}} \right )$ for $x = (x^{\ast}, x_{n+1}) \in \mathcal O_\varepsilon$. Let $y = (y^{\ast}, y_{n+1}) = T_\varepsilon (x^{\ast}, x_{n+1})$. Then we have

It follows from [18] that the Laplace operator in the original variable $x\in {\mathcal{O}}_\varepsilon$ and in the new variable $y\in \mathcal{O}$ are related by

where we denote by $u(y) = \hat u(x)$ and $P_\varepsilon$ is the operator given by

In the sequel, we abuse the notation a little bit by writing $h(t,x, s)$ and $G(t, x)$ as $h(t, x^\ast, x_{n+1}, s)$ and $G(t, x^\ast, x_{n+1})$ for $x = (x^\ast, x_{n+1})$, respectively. With this agreement, for any function $F(t,y,s)$, we introduce

where $y = \left( {y^{\ast} ,y_{n+1} } \right) \in \mathcal O$ and $t,s\in {\mathbb{R}}$, Then problem (17)-(18) is equivalent to the following system for $t > \tau$,

with the initial condition

where $\nu$ is the unit outward normal to $\partial \mathcal O$.

Given $t\in {\mathbb R}$, define a translation $\theta_{1,t}$ on ${\mathbb R}$ by

Then $\{\theta_{1,t}\}_{t\in {\mathbb R}}$ is a group acting on ${\mathbb R}$. We now specify the probability space. Denote by

Let $\mathcal{F}$ is the Borel $\sigma$-algebra induced by the compact-open topology of $\Omega$, and $P$ the corresponding Wiener measure on $(\Omega, \mathcal{F})$. There is a classical group $\left\{ {\theta _t } \right\}_{t \in {\mathbb R}}$ acting on $(\Omega, \mathcal{F} , P)$, which is defined by

Then $(\Omega, \mathcal{F}, P, \{\theta_{t}\}_{t\in {\mathbb R}})$ is a metric dynamical system (see [1]). On the other hand, let us consider the one-dimensional stochastic differential equation

for $\alpha>0$. This equation has a random fixed point in the sense of random dynamical systems generating a stationary solution known as the stationary Ornstein-Uhlenbeck process (see [6] for more details). In fact, we have

Lemma 2.1. There exists a $\{\theta_{t}\}_{t\in {\mathbb R}}$-invariant subset $\Omega ^{'} \in \mathcal F$ of full measure such that

and, for such $\omega$, the random variable given by

is well defined. Moreover, for $\omega\in \Omega ^{'},$ the mapping

is a stationary solution of (23) with continuous trajectories. In addition, for $\omega\in \Omega ^{'}$

Denote by $z_j^ *$ the associated Ornstein-Uhlenbeck process corresponding to (23) with $\alpha = 1$ and $w$ replaced by $w_j$ for $j = 1,\ldots, m$. Then for any $j = 1,\ldots, m$, we have a stationary Ornstein-Uhlenbeck process generated by a random variable $z_j^ *(\omega)$ on $\Omega ^{'}_j$ with properties formulated in Lemma 2.1 defined on a metric dynamical system $( {\Omega _j^{'} ,{\mathcal{F}}_j ,P_j , \{\theta_t\}_{t\in\mathbb{R}} } )$. We set

Then $( {\tilde\Omega , \mathcal{F}, P, \{\theta_t\}_{t\in\mathbb{R}}} )$ is a metric dynamical system.

Denote by

and

Then for every $\omega\in \Omega ^{'}$, $\mathcal T\left( \omega \right)$ is a homeomorphism on $L^2 \left(\mathcal O \right)$, and its inverse operator is given by

It follows that $\|{\mathcal T}^{-1}(\theta_t\omega)\|$ has sub-exponential growth as $t\rightarrow \pm \infty$ for any $\omega\in \tilde \Omega$. Hence $\|{\mathcal T}^{-1}\|$ is tempered. Analogously, $\|{\mathcal T}\|$ is also tempered. Obviously, $\mathop {\sup }\limits_{s \in \left[ {s_0 - a,s_0 + a} \right]} \left\| {{\mathcal T}\left( {\theta _s \omega } \right)} \right\|$ is still tempered for every $s_0\in {\mathbb R}$ and $a\in {\mathbb R}^+$.

On the other hand, since $z_j^ * ,j = 1, \ldots ,m,$ are independent Gaussian random variables, by the ergodic theorem we still have a $\left\{ {\theta _t } \right\}_{t \in {\mathbb R}}$-invariant set $\hat \Omega\in \mathcal F$ of full measure such that

and

Remark 1. We now consider $\theta$ defined in (22) on $\tilde \Omega \cap \hat \Omega$ instead of $\Omega$. This mapping possesses the same properties as the original one if we choose $\mathcal F$ as the trace $\sigma$-algebra with respect to $\tilde \Omega \cap \hat \Omega$. The corresponding metric dynamical system is still denoted by $(\Omega, \mathcal F , P, \{\theta_t\}_{t\in\mathbb{R}})$ throughout this paper.

Next, we define a continuous cocycle for system (19)-(20) in $\mathcal N$. This can be achieved by transferring the stochastic system into a deterministic one with random parameters in a standard manner. Let $u^\varepsilon$ be a solution to (19)-(20) and denote by $v^\varepsilon(t) = {\mathcal T}^{ - 1} \left( {\theta _t \omega } \right) u^\varepsilon(t)$ and $\delta \left( \omega \right) = \sum\limits_{j = 1}^m {c_j z_j^ * \left( \omega \right)}.$ Then $v^\varepsilon$ satisfies

with the initial conditions

where $\psi^\varepsilon = ( {{ {\mathcal T}}^{ - 1} ( {\theta _\tau \omega } )} )\phi^\varepsilon$.

Since (26) is a deterministic equation, by the Galerkin method, one can show that if $H$ satisfies (8)-(11), then for every $\omega\in \Omega$, $\tau\in {\mathbb R}$ and $\psi^\varepsilon\in \mathcal N$, (26)-(27) has a unique solution $v^\varepsilon\left( { t ,\tau,\omega ,\psi^\varepsilon} \right) \in C\left ( {\left[ {\tau,\tau+T} \right),L^2(\mathcal O) } \right) \cap L^2 \left( {\left( {\tau,\tau+T} \right),H^1(\mathcal O) } \right) \cap C\left( {[ {\tau+\epsilon,\tau+T} ),H^1(\mathcal O) } \right)$ with $v_\tau\left( {\cdot,\tau,\omega ,\psi^\varepsilon } \right) = \psi^\varepsilon$ for every $T>0$ and $0<\epsilon<T$. Furthermore, one may show that $v^\varepsilon(t, \tau,\omega, \psi^\varepsilon)$ is $(\mathcal{F} , \mathcal{B}(\mathcal N))$-measurable in $\omega\in \Omega$ and continuous with respect to $\psi^\varepsilon$ in $\mathcal N$ for all $t \geq \tau$. Since $u^\varepsilon\left( {t,\tau,\omega ,\phi^\varepsilon } \right) = {\mathcal T}\left( {\theta _{t} \omega } \right)v^\varepsilon\left( {t,\tau,\omega ,\psi^\varepsilon} \right)$ with $\phi^\varepsilon = ( {{ {\mathcal T}}( {\theta _\tau \omega } )} ) \psi^\varepsilon$, we find that $u^\varepsilon(t)$ is continuous in both $t\geq \tau$ and $\phi^\varepsilon \in \mathcal N$ and is $(\mathcal{F} , \mathcal{B}(\mathcal N))$-measurable in $\omega\in \Omega$. In addition, it follows from (26) that $u^\varepsilon$ is a solution of problem (19)-(20). We now define ${\Phi}_\varepsilon :{\mathbb R}^ +\times {\mathbb R} \times \Omega \times \mathcal N \to \mathcal N$ by

By the properties of $u^\varepsilon$, we find that $\Phi_\varepsilon$ is a continuous cocycle on $\mathcal N$ over $({\mathbb R}, \{\theta_{1,t}\}_{t\in {\mathbb R}})$ and $(\Omega, \mathcal{F}, P, \{\theta_{t}\}_{t\in {\mathbb R}})$, where $\{\theta_{1,t}\}_{t\in {\mathbb R}}$ and $\{\theta_{t}\}_{t\in {\mathbb R}}$ are given by (21) and (22), respectively. In this paper, we will first prove the asymptotic compactness of solutions in $H^1(\mathcal O)$ and then establish the existence and upper semicontinuity in $H^1(\mathcal O)$ of $(\mathcal N,H^1(\mathcal O))$-random attractors.

Let $R_\varepsilon :L^2 ( {\mathcal O_\varepsilon } ) \to L^2 ( \mathcal O )$ be an affine mapping of the form

Given $t\in {\mathbb{R}}^+$, $\tau\in {\mathbb{R}}$, $\omega\in \Omega$ and $\hat \phi_\varepsilon \in L^2 ( {\mathcal O_\varepsilon } )$, we can define a continuous cocycle $\hat\Phi_\varepsilon$ for problem (6)-(7) by the formula

The same change of unknown variable $v^0(t) = {\mathcal T}^{ - 1} \left( {\theta _t \omega } \right) u^0(t)$ transforms equation (4) into the following random partial differential equation on $\mathcal Q$:

with the initial conditions

where $\psi^0 = ( {{\tilde {\mathcal T}}^{ - 1} ( {\theta _\tau \omega } )} )\phi^0$.

The same argument as above allows us to prove that problem (4) and (5) generates a continuous cocycle $\Phi_0(t,\tau,\omega,\phi^0)$ in the space $\mathcal M$.

Now we want to write equation (26)-(27) as an abstract evolutionary equation. We introduce the inner product $\left( { \cdot , \cdot } \right)_{H_g(\mathcal O) }$ on $\mathcal N$ defined by

and denote by $H_g(\mathcal O)$ the space equipped with this inner product. Since $g$ is a continuous function on $\overline{\mathcal Q}$ and satisfies (1), one easily shows that $H_g(\mathcal O)$ is a Hilbert space with norm equivalent to the natural norm of $\mathcal N$.

For $0<\varepsilon\leq 1,$ we introduce a bilinear form $a_\varepsilon \left( { \cdot , \cdot } \right)$:$\,\,H^1 \left( \mathcal O \right) \times H^1 \left( \mathcal O \right) \to {\mathbb R}$, given by

where

By introducing on $H^1(\mathcal O)$ the equivalent norm, for every $0<\varepsilon\leq1$,

we see that there exist positive constants $\varepsilon_0$, $\eta_1$ and $\eta_2$ such that for all $0<\varepsilon\leq \varepsilon_0$ and $u\in H^1(\mathcal O),$

and

Denote by $A_\varepsilon$ an unbounded operator on $H_g(\mathcal O)$ with domain

as defined by

Then we have

Using $A_\varepsilon$, (26)-(27) can be written as

To reformulate system (29)-(30), we introduce the inner product $\left( { \cdot , \cdot } \right)_{H_g(\mathcal Q) }$ on $\mathcal M$ defined by

and denote by $H_g(\mathcal Q)$ the space equipped with this inner product. Let $a_0 \left( { \cdot , \cdot } \right)$:$\,\, H^1 \left( \mathcal Q\right) \times H^1 \left( \mathcal Q \right) \to \mathbb R$ be a bilinear form given by

Denote by $A_0$ an unbounded operator on $H_g(\mathcal Q)$ with domain

as defined by

Then we have

Using $A_0$, (29)-(30) can be written as

Hereafter, we set $X_0 = \mathcal M$, $X_\varepsilon = L^2(\mathcal O_\varepsilon)$ and $X_1 = \mathcal N$. For every $i = \varepsilon, 0$ or $1$, a family $B_i = \{B_i\left( {\tau , \omega} \right): \tau \in \mathbb{R},\omega \in \Omega \}$ of nonempty subsets of $X_i$ is called tempered if for every $c > 0$, we have:

where $\|B_i\|_{X_i} = \sup_{ x\in B_i} \|x\|_{X_i}$. The collection of all families of tempered nonempty subsets of $X_i$ is denoted by $\mathcal{D}_i$, i.e.,

Our main purpose of the paper is to prove that the cocycle $\hat\Phi_\varepsilon$ and $\Phi_0$ possess a unique $(L^2(\mathcal O_\varepsilon),H^1(\mathcal O_\varepsilon))$-random attractor $\hat{\mathcal A}_\varepsilon$ and $(\mathcal M,\mathcal J)$-random attractor ${\mathcal A}_0$, respectively. Furthermore $\hat{\mathcal A}_\varepsilon$ is upper-semicontinuous at $\varepsilon = 0$, that is, for every $\tau\in {\mathbb{R}}$ and $\omega\in \Omega$,

To prove (38), we only need to show that the cocycle $\Phi_\varepsilon$ has a unique $(\mathcal N,\mathcal H)$-random attractor ${\mathcal A}_\varepsilon$ and it is upper-semicontinuous at $\varepsilon = 0$ in the sense that for every $\tau\in {\mathbb{R}}$ and $\omega\in \Omega$,

which will be established in the last section of the paper.

Furthermore, we suppose that there exists $\lambda_0>0$ such that

Let us consider the mapping

By the ergodic theory and (39) we have

The following condition will be needed when deriving uniform estimates of solutions:

When constructing tempered pullback attractors, we will assume

Since $\psi_1 = \varphi_1 + b_1$ for some positive constant $b_1$, it is evident that (42) and (43) imply

and

for any $\sigma>0$.

3.   Uniform estimates of solutions

In this section, we recall and generalize some results in [17] and derive some new uniform estimates of solutions of problem (36) or (19)-(20) which are needed for proving the existence of $\mathcal{D}_1$-pullback absorbing sets and the $\mathcal{D}_1$-pullback asymptotic compactness in $H^1(\mathcal O)$ of the cocycle $\Phi_\varepsilon$.

Lemma 3.1. Assume that (8)-(11), (39) and (42) hold. Then for every $0<\varepsilon\leq \varepsilon_0$, $\tau\in \mathbb R$, $\omega\in \Omega$, and $D_1 = \{ D_1\left( {\tau ,\omega} \right):\tau \in \mathbb R,$ $\omega \in \Omega \} \in \mathcal{D}_1$, there exists $T = T(\tau,\omega,D_1)\geq 2$, independent of $\varepsilon$, such that for all $t\geq T$, $\lambda_1> \lambda_0$ and $\psi^\varepsilon \in D_1 \left( {\tau - t,\theta _{- t} \omega } \right)$, the solution $v^\varepsilon$ of (36) with $\omega$ replaced by $\theta_{-\tau}\omega$ satisfies

where $R_2(\tau,\omega)$ is determined by

where $R_1(\tau,\omega)$ is determined by

and $r_1(\omega)$ is a tempered function, and $c$ is independent of $\varepsilon$.

Proof. The proof is similar as that of Lemma 3.4 in [17], so we only sketch the proof here. Taking the inner product of (36) with $v^\varepsilon$ in $H_g(\mathcal O)$, we find that

By (13), we have

Then, we have for any $\sigma\geq\tau$,

where $\gamma \left( {\theta _t \omega } \right) = - \lambda_0+\delta(\theta_t\omega)$.

Thus by the similar arguments as Lemma 3.1 in [17] we get for every $\tau\in \mathbb R$, $\omega\in \Omega$, and $D_1\in {\mathcal D}_1$, there exists $T = T(\tau, \omega, D_1) > 0$ such that for all $t\geq T$,

Moreover, taking the inner product of (36) with $A_\varepsilon v^\varepsilon$ in $H_g(\mathcal O)$, we find that

By (15)-(16) we have

The left proof is similar of that Lemma 3.4 in [17], so we omit it here.

We are now in a position to establish the uniform estimates for the solution $u^\varepsilon$ of the stochastic equation (19)-(20) by using those estimates for the solution $v^\varepsilon$ of (36) and the relation between $v^\varepsilon$ and $u^\varepsilon$.

Lemma 3.2. Assume that (8)-(11), (39) and (42) hold. Then for every $0<\varepsilon\leq \varepsilon_0$, $\tau\in \mathbb R$, $\omega\in \Omega$, and $D_1 = \left\{ {D_1\left( {\tau ,\omega } \right):\tau \in \mathbb R,\omega \in \Omega } \right\}\in \mathcal{D}_1$, there exists $T = T(\tau,\omega,D_1)\geq 2$, independent of $\varepsilon$, such that for all $t\geq T$, $\lambda_1> \lambda_0$ and $\phi^\varepsilon \in D_1 \left( {\tau - t,\theta _{ - t} \omega } \right)$, the solution $u^\varepsilon$ of (19)-(20) with $\omega$ replaced by $\theta_{-\tau}\omega$ satisfies

where $r_2(\omega)$ is a tempered function and $R_2(\tau,\omega)$ is given by (47).

Lemma 3.3. Assume that (8)-(11), (39) and (42) hold. Then for every $0<\varepsilon\leq \varepsilon_0$, $\tau\in \mathbb R$, $\omega\in \Omega$, and $D_1 = \{ D_1\left( {\tau ,\omega} \right):\tau \in \mathbb R,$ $\omega \in \Omega \} \in \mathcal{D}_1$, there exists $T = T(\tau,\omega,D_1)\geq 2$, independent of $\varepsilon$, such that for all $t\geq T$, $\lambda_1 > \lambda_0$ and $\psi^\varepsilon \in D_1 \left( {\tau - t,\theta _{ - t} \omega } \right)$, the solution $v^\varepsilon$ of (36) with $\omega$ replaced by $\theta_{-\tau}\omega$ satisfies

where $R_3(\tau,\omega)<\infty$ for every $\tau\in \mathbb R$ and $\omega\in \Omega$.

Proof. The proof is similar as that of Lemma 3.6 in [14], so we omit it here.

Lemma 3.4. Assume that (8)-(11), (39) and (42) hold. Then for every $\eta>0$, $\tau\in \mathbb R$, $\omega\in \Omega$, and $D_1 = \left\{ {D_1\left( {\tau ,\omega } \right):\tau \in \mathbb R,\omega \in \Omega } \right\}\in \mathcal{D}_1$, there exist $T = T(\tau,\omega,D_1)\geq 2$, $\gamma = \gamma(\omega)>0$, a large $M = M(\tau,\omega,\eta)>0$ and $0<\varepsilon_1<\varepsilon_0$ such that for all $t\geq T$, $\lambda_1>\lambda_0$, $0<\varepsilon<\varepsilon_1$ and $\psi^\varepsilon \in D_1 \left( {\tau - t,\theta _{ - t} \omega } \right)$, the solution $v^\varepsilon$ of (36) with $\omega$ replaced by $\theta_{-\tau}\omega$ satisfies

Proof. Let $M$ be a positive number to be specified later. Taking the scalar product of (36) with $(v^\varepsilon-M)_+^{p-1}$, where $(v^\varepsilon-M)_+ = \max\{v^\varepsilon-M,0\}$, we have

For the first term on the right side of (59) we have

For the second term on the right-hand side of (59), by (8), we obtain, for $v^\varepsilon>M$,

which implies

The last term in (59) is bounded by

All above estimates yield

Multiplying (63) by $e^{-\int_0^t( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr }$, then integrating on $(\tau -1, \tau)$ we obtain

where $|\mathcal O|$ stands for the Lebesgue measure of $\mathcal O$. Replacing $\omega$ by $\theta_{-\tau} \omega$ in (64) we get

Since $\omega$ is continuous on $[-1,0]$, there exist $c_1 = c_1(\omega, p, \alpha_1)>0$ and $c_2 = c_2(\omega, p, \alpha_1)>0$ such that

By (66) we obtain

For the left-hand side of (65), by (67) we find that there exists $c_3 = c_3(\omega)>0$ such that

For the first term on the right-hand side of (65), by (67) we obtain

Similarly, for the second terms on the right-hand side of (65), we have from (67) there exists $c_4 = c_4(\omega)>0$ such that

Since $\varphi_1 \in L^\infty_{loc}( \mathbb{R}, {L^\infty(\widetilde {\mathcal O})})$ and $G\in L^2_{loc} ( \mathbb{R}, {L^\infty(\widetilde {\mathcal O})})$, for the two three terms on the right-hand side of (65), by (67) we obtain there exists $c_5 = c_5(\tau,\omega)>0$ such that

By (68)-(71) we get from (65) that

which together with Lemma 3.2 and Lemma 3.3 implies that there exist $c_6 = c_6(\tau, \omega)>0$ and $T = T(\tau, \omega, D_1) \ge 2$ such that for all $t \ge T$,

Since $p>2$, we find that for every $\eta>0$, there exists $M_0 = M_0(\tau, \omega, \eta)>0$ such that for all $M\ge M_0$ and $t\ge T$,

Note that $|v| \le 2 (v-M)_+$ for $v\ge2 M$, which together with (73) yields that for all $M \ge M_0$ and $t\ge T$,

Similarly, one can verify that there exist $M_1 = M_1(\tau, \omega, \eta)>0$ and $T_1 = (\tau, \omega, D) \ge 2$ such that for all $M\ge M_1$ and $t\ge T_1$,

Then Lemma 3.4 follows from (3) and (75) immediately.

Note that $A_\varepsilon$ is a family of linear operators in $Y_\varepsilon$, for $0\leq\varepsilon\leq \varepsilon_0$, where $Y_\varepsilon = H_g(\mathcal O)$, for $0<\varepsilon\leq \varepsilon_0$, and $Y_0 = H_g(\mathcal Q)$, is self-adjoint and has a compact resolvent. Then, $\sigma(A_\varepsilon)$ consists of only eigenvalues $\{\lambda_n^\varepsilon\}_{n = 1}^\infty$ with finite multiplicity:

and their associated eigenfunctions $\{\varpi_n^\varepsilon\}_{n = 1}^{\infty}$ form an orthonormal basis of $Y_\varepsilon$.

It follows from Corollary 9.7 in [8] that the eigenvalues and the eigenfunctions of $A_\varepsilon$ are convergent with respect to $\varepsilon$.

Next, we introduce the spectral projections. We use $P^\varepsilon_m$ to denote the projection from $Y_\varepsilon$ onto the eigenspace $\text{span}\{\varpi_i^\varepsilon\}_{i = 1}^m$ given by

We use $Q^\varepsilon_m$ to denote its orthogonal complement projection, i.e., $P^\varepsilon_m+Q^\varepsilon_m = I_\varepsilon$, where $I_\varepsilon$ is the identity operators on $Y_\varepsilon$. It is clear that

and

Let $u^\varepsilon = u_1^\varepsilon + u_2^\varepsilon$ and $v^\varepsilon = v_1^\varepsilon + v_2^\varepsilon$, where $u_1^\varepsilon = P_m^\varepsilon u^\varepsilon$, $u_2^\varepsilon = Q_m^\varepsilon u^\varepsilon$, $v_1^\varepsilon = P_m^\varepsilon v^\varepsilon$, and $v_2^\varepsilon = Q_m^\varepsilon v^\varepsilon$, respectively.

Lemma 3.5. Assume that (8)-(11), (39) and (42) hold. Then for every $\tau\in \mathbb R$, $\omega\in \Omega$, $\eta>0$ and $D_1 = \left\{ {D_1\left( {\tau ,\omega } \right):\tau \in \mathbb R,\omega \in \Omega } \right\}\in \mathcal{D}_1$, there exists $T = T(\tau,\omega,D_1,\eta)\geq 2$, $m = m(\tau,\omega,D,\eta)\in \mathbb N$ and $0<\varepsilon_1 = \varepsilon_1(n)<\varepsilon_0$ such that for all $t\geq T$, $0<\varepsilon<\varepsilon_1$ and $\phi^\varepsilon \in D_1 \left( {\tau - t,\theta _{ - t} \omega } \right)$, the solution $u^\varepsilon$ of (19)-(20) with $\omega$ replaced by $\theta_{-\tau}\omega$ satisfies

Proof. Taking the inner product (36) with $A_\varepsilon v_{2}^\varepsilon$ in $H_g \left( \mathcal O \right)$, we get

For the first term on the right-hand side of (78), we have

For the superlinear term, we have from (9) that

For the last term on the right-hand side of (78), we have

Noting that $\left\| {A_\varepsilon v_2^\varepsilon } \right\|^2 \ge \lambda _{n + 1}^\varepsilon a_\varepsilon(v_2^\varepsilon,v_2^\varepsilon)$, we obtain from all above estimates that

Taking $\xi\in (\tau-1,\tau)$, multiplying (82) by $e^{ \lambda _{n + 1}^\varepsilon}t$, first integrating with respect to $t$ on $(\xi, \tau)$, integrating with respect to $\xi$ on $(\tau-1,\tau)$, and then replacing $\omega$ by $\theta_{-\tau}\omega$, we get

Since $\varphi_2,G \in L^2_{loc}( \mathbb{R}, {L^\infty(\widetilde {\mathcal O})})$, $\|\mathcal T(\theta_t\omega)\|$ is continuous on $[-1,0]$ and $\lambda^\varepsilon_{n+1}$ approaches $\lambda^0_{n+1}$ as $\varepsilon\rightarrow 0$, we find that there exists $c = c(\omega) >0$ and $0<\varepsilon^{*}<\varepsilon_0$ such that for $0<\varepsilon<\varepsilon^{*}$,

Given $\eta>0$, let $T = T (\tau, \omega, D_1) \geq 2, \gamma = \gamma(\omega) > 0$, $M = M(\tau, \omega, \eta) \geq 1$ and $0<\varepsilon_1<\varepsilon^{*}$ be the constants in Lemma 3.4. Choose $N_1 = N_1(\tau, \omega, \eta) \geq1$ large enough such that $\lambda_{n+1}^0-1\geq \gamma M^{p-2}$ for all $n \geq N_1$. Then, by Lemma 3.4, we obtain, for all $t \geq T$, $n \geq N_1$ and $0<\varepsilon<\varepsilon_1$,

For the last three terms on the right-hand side of (84), by Lemma 3.1, we find that there exist $c_1 = c_1(\tau, \omega)> 0$ and $T_1 = T_1(\tau, \omega, D_1) \geq T$ such that for all $t \geq T_1$,

Since $\lambda_{n+ 1}^0\rightarrow \infty$ as $n\rightarrow \infty$, we obtain from (84)-(86) that there exists $N_2 = N_2(\tau, \omega, \eta) \geq N_1$ such that for all $n \geq N_2$, $t \geq T_1$ and $0<\varepsilon<\varepsilon_1$,

which together $v^\varepsilon(t) = {\mathcal T}^{ - 1} \left( {\theta _t \omega } \right) u^\varepsilon(t)$ and (77) completes the proof.

4.   Existence of pullback attractor

In this subsection, we establish the existence of $\mathcal D_1$-pullback attractor for the cocycle $\Phi_\varepsilon$ associated with the stochastic problem (19)-(20). We first show that problem (19)-(20) has a tempered pullback absorbing set as stated below.

Lemma 4.1. Suppose (8)-(11), (39) and (43) hold. Then the cocycle $\Phi_\varepsilon$ associated with problem (19)-(20) has a closed measurable $\mathcal D_1$-pullback absorbing set $K = \{K \left( {\tau ,\omega } \right):\tau\in \mathbb R,\omega\in \Omega\}\in \mathcal D_1$.

Proof. We first notice that, by Lemma 3.2, $\Phi_\varepsilon$ has a closed $\mathcal D_1$-pullback absorbing set $K$ in $H^1(\mathcal O).$ More precisely, given $\tau\in \mathbb R$ and $\omega\in \Omega$, let

where $L\left( {\tau ,\omega } \right)$ is the constant given by the right-hand side of (55). It is evident that, for each $\tau\in \mathbb R,$ $L(\tau,\cdot ) : \Omega \rightarrow \mathbb R$ is $( \mathcal{F}, \mathcal{B} ( \mathbb R ))$-measurable. In addition, for every $\tau\in \mathbb R$, $\omega\in \Omega$, and $D\in \mathcal{D}_1$, there exists $T = T(\tau,\omega,D)\geq 2$ such that for all $t\geq T$,

Thus we find that $K = \left\{ {K \left( {\tau ,\omega } \right):\tau \in \mathbb R,\omega \in \Omega } \right\}$ is a closed measurable set and pullback-attracts all elements in $\mathcal D_1$. By the similar argument as in [15] we can obtain easily from (43) that $K = \{K (\tau, \omega) : \tau\in \mathbb R, \omega \in \Omega\}$ is tempered. Consequently, $K$ is a closed measurable $D_1$-pullback absorbing set for $\Phi_\varepsilon$ in $D_1$.

Lemma 4.2. Assume that (8)-(11), (39) and (43) hold. Then, the cocycle $\Phi_\varepsilon$ is ${\mathcal{D}}_1$-pullback asymptotically compact in $H^1(\mathcal O)$; that is, for all $\tau\in \mathbb R$ and $\omega\in \Omega$, $\left\{ {\Phi_\varepsilon \left( {{t_n},\tau-t_n ,{\theta _{ - {t_n}}}\omega ,{x_n}} \right)} \right\}_{n = 1}^\infty$ has a convergent subsequence in $H^1(\mathcal O)$ whenever $t_n\rightarrow \infty$ and ${x_n} \in D_1\left( {\tau-t_n,{\theta _{- {t_n}}}\omega } \right)$ with $\{ D_1\left( {\tau ,\omega } \right):\tau \in \mathbb R ,$ $\omega \in \Omega \}\in {\mathcal{D}}_1$.

Proof. We will show that for every $\eta>0$, the sequence $\{u^\varepsilon\left( {\tau,\tau-t_n,\theta _{ - \tau} \omega , \phi^\varepsilon} \right) \}_{n = 1}^\infty$ has a finite open cover of balls with radii less than $\eta$. By Lemma 3.5, we infer that there exists $N_1 = N_1(\tau,\omega,D_1,\eta)\geq 1$, $m_0 = m_0(\tau,\omega,D_1,\eta)\in \mathbb N$ and $0<\varepsilon_1 = \varepsilon_1(m_0)<\varepsilon_0$ such that for all $n \ge N_1$ and $0<\varepsilon<\varepsilon_1$,

On the other hand, by Lemma 3.2 we find that the sequence $\{P_{m_0} u^\varepsilon\left( {\tau,\tau-t_n,\theta _{ - \tau} \omega , \phi^\varepsilon} \right) \}_{n = 1}^\infty$ is bounded in the finite-dimensional space $P_{m_0} H^1 (\mathcal O)$ and hence is precompact, which together with (88) shows that the sequence $u^\varepsilon\left( {\tau,\tau-t_n,\theta _{ - \tau} \omega , \phi^\varepsilon} \right)$ has a finite open cover of balls with radii less than $\eta$ in $H^1(\mathcal O)$, as desired.

Theorem 4.3. Assume that (8)-(11), (39) and (43) hold. Then, the cocycle $\Phi_\varepsilon$ has a unique $\mathcal{D}_1$-pullback $(\mathcal N, H^1(\mathcal O))$-attractor $\mathcal A_\varepsilon = \{\mathcal A_\varepsilon(\tau,\omega):\tau\in \mathbb R,\omega\in\Omega\}$.

Proof. First, we know from Lemma 4.1 that $\Phi_\varepsilon$ has a a closed measurable $\mathcal D_1$-pullback absorbing set ${K \left( {\tau ,\omega } \right)}$. Second, it follows from Lemma 4.2 that $\Phi_\varepsilon$ is $\mathcal{D}_1$-pullback asymptotically compact from $\mathcal N$ to $H^1(\mathcal O)$. Hence, the existence of a unique $\mathcal{D}_1$-pullback $(\mathcal N, H^1(\mathcal O))$-attractor for the cocycle $\Phi_\varepsilon$ follows from Proposition 2.5 in [7].

Analogous results also hold for the solution of (4)-(5). In particular, we have:

Theorem 4.4. Assume that (8)-(11), (39) and (43) hold. Then, the cocycle $\Phi_0$ has a unique $\mathcal{D}_0$-pullback ($\mathcal M$, $H^1(\mathcal Q)$)-attractor $\mathcal A_0 = \{\mathcal A_0(\tau,\omega):\tau\in \mathbb R,\omega\in\Omega\}$.

5.   Upper semicontinuity of attractors

The following estimates are needed when we derive the convergence of pullback attractors. By the similar proof of that of Theorem 5.1 in [14], we get the following lemma.

Lemma 5.1. Assume that (8)-(11) and (39) hold. Then for every $0<\varepsilon\leq \varepsilon_0$, $\tau\in \mathbb R$, $\omega\in \Omega$, $T>0$, and $\lambda_1> \lambda_0$, the solution $v^\varepsilon$ of (36) satisfies, for all $t\in [\tau, \tau+T]$,

where $c$ is a positive constant depending on $\tau$, $\omega$, $\lambda_0$ and $T$, but independent of $\varepsilon$.

Similarly, one can prove

Lemma 5.2. Assume that (8)-(11) and (39) hold. Then for every $\tau\in \mathbb R$, $\omega\in \Omega$, $T>0$, and $\lambda_1>\lambda_0$, the solution $v^0$ of (37) satisfies, for all $t\in [\tau, \tau+T]$,

where $c$ is a positive constant depending on $\tau$, $\omega$, $\lambda_0$ and $T$, but independent of $\varepsilon$.

In the sequel, we further assume the functions $G$ and $H$ satisfy that for all $t,s\in {\mathbb R}$,

and

where $\kappa_1(t), \kappa_2(t)\in L^2_{loc}({\mathbb R})$.

By (12) and (90) we have, for all $x\in \widetilde{\mathcal O}$ and $t,s\in {\mathbb R}$,

Since $\mathcal M$ can be embedded naturally into $\mathcal N$ as the subspace of functions independent of $y_{n+1}$, we can consider the cocycle $\Phi_0$ as a mapping from $\mathcal M$ into $\mathcal N$. Therefore we can compare $\Phi_0$ with $\Phi_\varepsilon$.

Theorem 5.3. Suppose (8)-(11), (39), and (89)-(90) hold. Given $\tau\in {\mathbb{R}}$, $\omega\in \Omega$, $\varepsilon_n\rightarrow 0$ and a positive number $L(\tau, \omega)$, if $\phi{^{\varepsilon_n}}\in H^1_{\varepsilon_n}(\mathcal O))$ such that $\left\| {\phi{^{\varepsilon_n}} } \right\|_{H^1_{\varepsilon_n}(\mathcal O))} \le L(\tau, \omega)$, then there exists $\phi^0\in \mathcal M$ such that, up to a subsequence, for $t\geq 0$,

Proof. Since $\phi^{\varepsilon_n}\in H^1_{\varepsilon_n}(\mathcal O)$, there exists $\phi^0\in \mathcal M$ such that $\phi^{\varepsilon_n} \rightarrow \phi^0$ in $\mathcal N$. By the similar proof of that of Theorem 5.4 in [14], for any $T>0$, we have for $t\in[\tau,\tau+T]$

where $\xi(\theta_t\omega) = \beta+|\delta(\theta_t\omega)|.$ By Lemma 5.1 and Lemma 5.2 we find that there exists a positive constant $\varrho = \varrho(\tau,\omega,T)$, independent of ${\varepsilon_n}$, such that for all $t\in[\tau,\tau+T]$,

Notice that, for all $t\in[\tau,\tau+T]$,

which together with (93) implies the desired results.

The next result is concerned with uniform compactness of attractors with respect to $\varepsilon$.

Lemma 5.4. Assume that (8)-(11), (39) and (43) hold. If $\varepsilon_n\rightarrow 0$ and $u^{\varepsilon_n} \in \mathcal A_{\varepsilon _n } \left( {\tau ,\omega } \right)$, then there exist a subsequence of $({u^{\varepsilon_n} })_{n\in \mathbb{N}}$, again denoted by $(u^{\varepsilon_n} )_{n\in \mathbb{N}}$, and $u\in H^1(\mathcal Q)$ such that

Proof. Take a sequence $t_n \to \infty$. By the invariance of $\mathcal A_{\varepsilon _n }$ there exists $\phi^{\varepsilon_n} \in \mathcal A_{\varepsilon_n } \left( {\tau - t_n ,\theta _{ - t_n } \omega } \right)$ such that

By Lemma 4.1, we have $\phi^{\varepsilon_n} \in K \left( {\tau - t_n ,\theta _{ - t_n } \omega } \right)\in D_1.$ Since $\varepsilon_n\rightarrow 0$ and $t_n \to \infty$, By Lemma 3.5, for any $\eta>0$, there exists a large enough $N_1\in \mathbb N$ such that for all $n\geq N_1$,

By Lemma 3.2, we have

It follows from (95) and (96) that $(u^{\varepsilon_n}\left( {\tau,\tau-t_n,\theta _{ - \tau} \omega , \phi^{\varepsilon_n}} \right))_{n\in \mathbb{N}}$ is precompact in $H^1(\mathcal O)$. Since the estimate (55) holds, there exists $u$ in $H^1(\mathcal Q)$ and a subsequence of $({u^{\varepsilon_n}})_{n\in \mathbb{N}}$, again denoted by $(u^{\varepsilon_n})_{n\in \mathbb{N}}$, such that

This completes the proof.

Now we are in a position to prove the main result of this paper.

Theorem 5.5. Assume that (8)-(11), (39), (43), and (89)-(90) hold. The attractors $\mathcal{A}_\varepsilon$ are upper-semicontinuous at $\varepsilon = 0$, that is, for every $\tau\in \mathbb R$ and $\omega\in \Omega$,

Proof. Given $\tau\in {\mathbb{R}}$ and $\omega \in \Omega$, by the invariance of $\mathcal{A}_\varepsilon$ and (55) we find that there exists $\varepsilon_0>0$ such that

where $L(\tau, \omega)$ is the positive constant given by the right-hand side of (55) which is independent of $\varepsilon$. If the theorem is not true, there exist $\delta>0$, a sequence $(\varepsilon_n)_{n\in \mathbb{N}}$ of positives constants, $\varepsilon_n\rightarrow 0$, and a sequence $(z_n)_{n\in \mathbb{N}}$, $z_n\in \mathcal{A}_{\varepsilon_n} (\tau,\omega)$ for all ${n\in \mathbb{N}}$, such that

By Lemma 5.4 there exists $z_{\ast}$ in $H^1(\mathcal Q)$ and a subsequence of $({z_n})_{n\in \mathbb{N}}$, again denoted by $(z_n)_{n\in \mathbb{N}}$, such that

By the invariance property of the attractor $\mathcal{A}_{\varepsilon_n} (\tau,\omega)$, for every $t>0$ there exists $y_n^t \in \mathcal{A}_{\varepsilon_n} \left( \tau-t,{\theta _{ - t} \omega } \right)$ such that

By Lemma 5.4 again there exists $y_{\ast}^t$ in $H^1(\mathcal Q)$ and a subsequence of $({y_n^t})_{n\in \mathbb{N}}$, again denoted by $(y_n^t)_{n\in \mathbb{N}}$, such that

It follows from Theorem 5.3 that for every $t>0$,

By (100), (101), (103) and uniqueness of limits we obtain

Notice that $\mathcal{A}_{\varepsilon_n} (\tau-t,\theta_{-t}\omega)\subseteq K(\tau-t,\theta_{-t}\omega)$ and $y^t_n\in \mathcal{A}_{\varepsilon_n} (\tau-t,\theta_{-t}\omega)$ for all $n\in \mathbb{N}.$ Thus by (98) we have

By (102) and (105) we get, for every $t>0$,

By $K_0\in \mathcal D_0$ and the attraction property of $\mathcal A_0$ in $\mathcal D_0$, we obtain from (104) and (106) that

This implies that $z_{\ast}\in \mathcal{A}_0(\tau,\omega)$ since $\mathcal{A}_0(\tau,\omega)$ is compact. Therefore, we have

a contradiction with (99). This completes the proof.

Acknowledgments

The authors would like to thank the anonymous referee for the useful suggestions and comments.