#  Regularity of pullback attractors and equilibrium for non-autonomous
stochastic FitzHugh-Nagumo system on unbounded domains

Wenqiang Zhao, Anhui Gu  Wenqiang Zhao
School Of Mathematics and Statistics, Chongqing Technology and Business
University, Chongqing 400067, China<EMAIL_ADDRESS>Anhu Gu
School of Mathematics and statistics, Southwest University, Chongqing 400715,
China<EMAIL_ADDRESS>
###### Abstract.

A theory on bi-spatial random attractors developed recently by Li _et al._ is
extended to study stochastic Fitzhugh-Nagumo system driven by a non-autonomous
term as well as a general multiplicative noise. By using the so-called notions
of uniform absorption and uniformly pullback asymptotic compactness, it is
showed that every generated random cocycle has a pullback attractor in
$L^{l}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$ with $l\in(2,p]$, and the
family of obtained attractors is upper semi-continuous at any intensity of
noise. Moreover, if some additional conditions are added, then the system
possesses a unique equilibrium and is attracted by a single point.

###### Key words and phrases:

Random dynamical systems; non-autonomous FitzHugh-Nagumo systems; upper semi-
continuity; pullback attractor; multiplicative noise;equilibria

###### 2000 Mathematics Subject Classification:

60H15, 35R60, 35B40, 35B41

##  1\. Introduction

In this paper, we consider the random dynamics of solutions of the following
non-autonomous FitzHugh-Nagumo system defined on $\mathbb{R}^{N}$ perturbed by
a $\varepsilon$-multiplicative noise:

$\displaystyle
d\tilde{u}+(\lambda\tilde{u}-\Delta\tilde{u}+\alpha\tilde{v})dt=f(x,\tilde{u})dt+g(t,x)dt+\varepsilon\tilde{u}\circ
d\omega(t),t>\tau,$ (1.1) $\displaystyle
d\tilde{v}+(\sigma\tilde{v}-\beta\tilde{u})dt=h(t,x)dt+\varepsilon\tilde{v}\circ
d\omega(t),t>\tau,$ (1.2)

with initial value numbers

$\displaystyle\tilde{u}(x,\tau)=\tilde{u}_{\tau}(x),\ \ \ \ \ \
\tilde{v}(x,\tau)=\tilde{v}_{\tau}(x),$ (1.3)

where the initial condition $(\tilde{u}_{0},\tilde{v}_{0})\in
L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$, the coefficients
$\lambda,\alpha,\beta,\sigma$ are positive constants, the non-autonomous terms
$g,h\in L^{2}_{loc}(\mathbb{R},L^{2}(\mathbb{R}^{N}))$, the nonlinearity $f$
is a smooth function satisfying some polynomial growth, $\varepsilon$ is the
intensity of noise with $\varepsilon\in[-a,a]\setminus\\{0\\},a>0$,
$\omega(t)$ is a Wiener process defined on a probability space
$(\Omega,\mathcal{F},P)$, where $\Omega=\\{\omega\in
C(\mathbb{R},\mathbb{R});\omega(0)=0\\}$, and $\mathcal{F}$ be the Borel
$\sigma$-algebra induced by the compact-open topology of $\Omega$ and ${P}$ be
the corresponding Wiener measure on $(\Omega,\mathcal{F})$.

The deterministic FitzHugh-Nagumo system is an important mathematical model to
describe the signal transmission across axons in neurobiology, see [22, 10,
15, 4] and references therein. It is well studied in the literature, see e.g.
[20, 26, 21, 23]. In the random case, when $g$ and $h$ do not depend on the
time, Wang [28] proved the existence and uniqueness of random attractors in
$L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$. For the general non-
autonomous forcings $g$ and $h$, under additive noises, Adili and Wang [2]
obtained the pullback attractors in $L^{2}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})$, and Bao [5] developed this result and obtained the
regularity in $H^{1}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$. For our
problem (1.1)-(1.3) , _i.e._ , under multiplicative noise, Adili and Wang [1]
proved the existence and upper semi-continuity of attractors in
$L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$ recently. For the
stochastic lattice FitzHugh-Nagumo system, the existences of random attractors
are widely studied in [14, 12, 13]. However, to our knowledge, there are no
literature to investigate the asymptotic high-order integrability of solutions
to the FitzHugh-Nagumo system, even for the deterministic case.

In this paper, we strengthen these results offered by [1] and devote to obtain
the asymptotic high-order integrability of solutions of problem (1.1)-(1.3).
To this end, a theory on bi-spatial random attractors developed recently by Li
_et al._ [16, 17] is extended to stochastic partial differential equations
(SPDE) with both non-autonomous terms and random noises, see Theorem 2.9 and
2.10. It is showed that the _uniform absorption_ and _uniformly pullback
asymptotic compactness_ are the appropriate notions to depict the existence
and upper semi-continuity of attractor in both the initial space and terminate
space [16]. As for the theory on the upper semi-continuity of pullback
attractors in an initial space and its applications, we may refer to [29, 27,
31, 37] and references therein.

Then we apply the obtained theorems to prove that the problem (1.1)-(1.3)
admits a unique pullback attractor in $L^{p}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})$, with the functions $f,g$ and $h$ satisfying almost the
same conditions as in [1]. Furthermore, we derive the upper semi-continuity of
pullback attractors of system in $L^{p}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})$ as the intensity $\varepsilon$ approaches any
$\varepsilon_{0}\in\mathbb{R}$. These are achieved by checking the uniform
absorption and uniformly pullback asymptotic compactness properties of random
cocycles. The uniformly pullback asymptotic compactness in
$L^{p}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$ is completely proved by
estimate of the $L^{2}$ and $L^{p}$-uniform boundedness and the
$L^{p}$-truncation of solutions, see Lemma 4.4 and 4.5. it seems that the
estimate of $L^{2}$-truncation is unnecessary, see [16, 17, 18, 34, 35, 19,
32, 36]. It is worth mention that an additional assumption on the non-
autonomous terms (see [1]) is not used in our proof, see section 3.

The third goal is to study _stochastic fixed point_ or _random equilibrium_ of
random dynamical system, see [6]. In this paper, we introduce the notion of
equilibrium for SPDE with both non-autonomous terms and white noises. It is
showed that if the parameters satisfy some additional conditions, then the
system admits a unique equilibrium and is attracted by a single point.

This paper is organized as follows. In the next section, we introduce some
concepts required for our further discussions and extend the results developed
by [16, 17] to the general SPDE with non-autonomous forcing. In section 3, we
give the assumptions on $g,h$ and $f$, and define a family of continuous
random cocycles for problem (1.1)-(1.3). In section 4, we prove the existence
and upper semi-continuity of pullback attractors in
$L^{p}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$. The final section is
concerned with the existence of equilibrium of the random cocycle.

##  2\. Preliminaries and abstract results

In this section, we give the sufficient conditions for the existence and upper
semi-continuity of pullback attractors in _the terminate space_ for random
dynamical systems over two parametric spaces, which are applicable to SPDE
with both non-autonomous deterministic and random terms, and the structure of
the pullback attractor is presented. This is an extension of the corresponding
results just established by Li _et al._ [16], which is suitable for random
systems with only one stochastic terms. The reader is also referred to [29,
30, 38] for the theory of pullback attractors and its applications in _the
initial space_ over two parametric spaces, and to [8, 25, 11, 9] for one
parametric random attractors. The reader may also refer to [7, 24] for
pullback attractor of deterministic dynamical systems.

###  2.1. Preliminaries

Let both $(X,\|.\|_{X})$ and $(Y,\|.\|_{Y})$ be separable Banach spaces, where
$X$ is called _an initial space_ which contains all initial data, and $Y$ is
called _a terminate space_ which contains all values of solutions for a SPDE
[16]. Both $X$ and $Y$ may not be embedded in any direction, but we assume
that they are _limit-uniqueness_ in the following sense:

_(H1) If $\\{x_{n}\\}_{n}\subset X\cap Y$ such that $x_{n}\rightarrow x$ in
$X$ and $x_{n}\rightarrow y$ in $Y$, respectively, then we have $x=y$._

Let $Q$ be a nonempty set and $(\Omega,\mathcal{F},{P})$ be a probability
space. We assume that there are two groups $\\{\sigma_{t}\\}_{t\in\mathbb{R}}$
and $\\{\vartheta_{t}\\}_{t\in\mathbb{R}}$ over $Q$ and $\Omega$,
respectively. Specifically, the mapping $\sigma:\mathbb{R}\times Q\mapsto Q$
satisfies that $\sigma_{0}$ is the identity on $Q$, and
$\sigma_{s+t}=\sigma_{s}\circ\sigma_{t}$ for all $s,t\in\mathbb{R}$.
Similarly, $\vartheta:\mathbb{R}\times\Omega\mapsto\Omega$ is a
$(\mathcal{B}(\mathbb{R})\times\mathcal{F},\mathcal{F})$-measurable mapping
such that $\vartheta_{0}$ is the identity on $\Omega$,
$\vartheta_{s+t}=\vartheta_{s}\circ\vartheta_{t}$ for all $s,t\in\mathbb{R}$
and $\vartheta_{t}P=P$ for all $t\in\mathbb{R}$. In particular, we call both
$(Q,\\{\sigma_{t}\\}_{t\in\mathbb{R}})$ and
$(\Omega,\mathcal{F},{P},\\{\vartheta_{t}\\}_{t\in\mathbb{R}})$ parametric
dynamical systems. Let $\mathbb{R}^{+}=\\{x\in\mathbb{R};x\geq 0\\}$ and
$2^{X}$ be the collection of all subsets of $X$.

Definition 2.1.  _A measurable mapping $\varphi:\mathbb{R}^{+}\times
Q\times\Omega\times X\rightarrow X,(t,q,\omega,x)\mapsto\varphi(t,q,\omega,x)$
is called to be a random cocycle on $X$ over
$(Q,\\{\sigma_{t}\\}_{t\in\mathbb{R}})$ and
$(\Omega,\mathcal{F},{P},\\{\vartheta_{t}\\}_{t\in\mathbb{R}})$ if for every
$q\in Q,\omega\in\Omega$ and $s,t\in\mathbb{R}^{+}$ the following statements
are satisfied:_

$\displaystyle(i)\ \varphi(0,q,\omega,.)\ \mbox{is the identity on X},\ \ \
(ii)\
\varphi(t+s,q,\omega,.)=\varphi(t,\sigma_{s}q,\vartheta_{s}\omega,.)\circ\varphi(s,q,\omega,.).$

_An random cocycle is said to be continuous in $X$ iff each operator
$\varphi(t,q,\omega,.)$ is continuous in $X$ for each $q\in Q,\omega\in\Omega$
and $t\in\mathbb{R}^{+}$.
_

In particular, it is pointed out that in this paper, we need further to assume
that the random cocycle $\varphi$ acting on $X$ takes its values in the
terminate space $Y$ for for all $t>0$ (except that $t=0$), _i.e.,_

_(H2) For every $t>0,q\in Q,$ and $\omega\in\Omega$,
$\varphi(t,q,\omega,.):X\rightarrow Y$._

In the sequel, we use $\mathcal{D}$ to denote a collection of some families of
nonempty subsets of $X$ which is parameterized by
$(q,\omega)\in(Q\times\Omega)$:

$\mathcal{D}=\\{D=\\{\emptyset\neq D(q,\omega)\in 2^{X};q\in
Q,\omega\in\Omega\\};f_{D}\ \mbox{satisfies some conditions}\\}.$

We further assume that $\mathcal{D}$ is inclusion closed, that is, for each
$D\in\mathcal{D}$,

$\\{\tilde{D}(q,\omega):\tilde{D}(q,\omega)\ \mbox{is a nonempty subset of}\
D(q,\omega),\forall\ q\in Q,\omega\in\Omega\\}\in\mathcal{D}.$

Given $D_{1},D_{2}\in\mathcal{D}$, we say that $D_{1}=D_{2}$ if and only if
$D_{1}(q,\omega)=D_{2}(q,\omega)$ for each $q\in Q$ and $\omega\in\Omega$.

Throughout this paper, all assertions about $\omega$ are assumed to hold on a
$\vartheta_{t}$-invariant set of full measure (unless some exceptional cases
needed).

Definition 2.2._A set-valued mapping $K:Q\times\Omega\rightarrow 2^{X}$ is
called measurable in $X$ with respect to $\mathcal{F}$ in $\Omega$ if the
mapping $\omega\in\Omega\mapsto\mbox{dist}_{X}(x,K(q,\omega))$ is
($\mathcal{F},\mathcal{B}(\mathbb{R}))$-measurable for every fixed $x\in X$
and $q\in Q$, where $\mbox{dist}_{X}$ is the Haustorff semi-metric in $X$,
i.e., for the two nonempty subsets $A,B\in 2^{X}$,_

$dis_{X}(A,B)=\sup_{a\in A}\inf_{b\in B}\|a-b\|_{X}.$

Definition 2.3. _Suppose $\varphi$ is a random cocycle on $X$ over
$(Q,\\{\sigma_{t}\\}_{t\in\mathbb{R}})$ and
$(\Omega,\mathcal{F},{P},\\{\vartheta_{t}\\}_{t\in\mathbb{R}})$ and takes its
value in $Y$. A set valued mappping $\mathcal{A}:Q\times\Omega\mapsto 2^{X\cap
Y}$ is called a (X,Y)-pullback attractor for $\varphi$ if_

_(i) $\mathcal{A}$ is measurable in $X$ (w.r.t the $P$-completion of
$\mathcal{F}$ in $\Omega$), and $\mathcal{A}(q,\omega)$ is compact in $Y$ for
all $q\in Q,\omega\in\Omega$,_

_(ii) $\mathcal{A}$ is invariant, that is, for every $q\in
Q,\omega\in\Omega$,_

$\varphi(t,q,\omega,\mathcal{A}(q,\omega))=\mathcal{A}(\sigma_{t}q,\vartheta_{t}\omega),\forall\
t\geq 0,$

_(iii) $\mathcal{A}$ attracts every element $D\in\mathcal{D}$ in $Y$, that is,
for every $q\in Q,\omega\in\Omega$,_

$\mathop{\lim}\limits_{t\rightarrow+\infty}\mbox{dist}_{Y}(\varphi(t,\sigma_{-t}q,\vartheta_{-t}\omega,D(\sigma_{-t}q,\vartheta_{-t}\omega)),\mathcal{A}(q,\omega))=0,$

where $dist_{Y}$ is the Haustorff semi-distance in $Y$ and the set
$\varphi(t,q,\omega,D(q,\omega))=\\{\varphi(t,q,\omega,x);x\in
D(q,\omega)\\}$.

If $X=Y$, the above concept reduces to the well known notion of a
$\mathcal{D}$-pullback attractors, which is first introduced in [30]. We also
remark that the measurability of $\mathcal{A}$ is assumed in the initial space
$X$.

Definition 2.4.(see[15].) _Let both $Z$ and $I$ be two metric spaces. A family
$\\{\mathcal{A}_{\alpha}\\}_{\alpha\in I}$ of sets in $Z$ is said to be upper
semi-continuous at $\alpha_{0}$ if_

$\displaystyle\lim_{\alpha\rightarrow\alpha_{0}}\mbox{dist}_{Z}(\mathcal{A}_{\alpha},\mathcal{A}_{\alpha_{0}})=0.$

_A family $\mathcal{A}_{\alpha}$ of set-mappings over $Q$ and $\Omega$ is
called to be upper semi-continuous if $\mathcal{A}_{\alpha}(q,\omega)$ is
upper semi-continuous for each $q\in Q$ and $\omega\in\Omega$._

In the sequel, we need to consider a family of random cocycles
$\\{\varphi_{\alpha}\\}_{\alpha\in I}$ with $I=[-a,a]\setminus\\{0\\}$, where
$a>0$ and $\varphi_{0}$ is a deterministic cocycle over the parametric space
$(Q,\\{\sigma_{t}\\}_{t\in\mathbb{R}})$.

Definition 2.5.(see[16].)  _A family of random cocycles
$\\{\varphi_{\alpha}\\}_{\alpha\in I}$ is said to be convergent at point
$\alpha=\alpha_{0}$ in $X$ if for each $q\in Q,\omega\in\Omega,$ and
$x,x_{0}\in X$,_

$\varphi_{\alpha}(t,q,\omega,x)\rightarrow\varphi_{\alpha_{0}}(t,q,\omega,x_{0})\
\mbox{in}\ X,$

_whenever $\alpha\rightarrow\alpha_{0}$ and $x\rightarrow x_{0}$. A family of
random cocycles $\\{\varphi_{\alpha}\\}_{\alpha\in I}$ is said to be
convergent in $X$ if it is convergent at any point $\alpha$. We say a family
of random cocycles $\alpha_{\varepsilon}(\alpha\in(0,a])$ converges to a
deterministic cocycle $\varphi_{0}$ in $X$ if for each $q\in Q,$ and
$x,x_{0}\in X$,_

$\varphi_{\alpha}(t,q,\omega,x)\rightarrow\varphi_{0}(t,q,x_{0})\ \mbox{in}\
X,$

_whenever $\alpha\rightarrow\alpha_{0}$ and $x\rightarrow x_{0}$._

Definition 2.6.(see[16].)  _A family of random cocycles
$\\{\varphi_{\alpha}\\}_{\alpha\in I}$ is said to be uniformly absorbing in
$X$ if each $\varphi_{\alpha}$ has a closed and measurable pullback absorbing
set $K_{\alpha}$ in $X$ such that the closure
$\overline{K}=\\{\overline{\cup_{\alpha\in I}K_{\alpha}(q,\omega)};q\in
Q,\omega\in\Omega\\}\in\mathcal{D}$ and for each $q\in Q,\omega\in\Omega$,_

$\limsup_{\alpha\rightarrow 0}\|K_{\alpha}(q,\omega)\|_{X}\leq c\ \ \mbox{for
some deterministic constant}\ c>0.$

_Here a pullback absorbing set $K_{\alpha}$ means that for every
$D\in\mathcal{D}$, there exists an absorbing time $T=T(D,q,\omega)>0$ such
that for each $q\in Q,\omega\in\Omega$,_

$\varphi_{\alpha}(t,\sigma_{-t}q,\vartheta_{-t}\omega,D(\sigma_{-t}q,\vartheta_{-t}\omega))\subseteq
K(q,\omega)\ \ \ \ \ \ \mbox{for all}\ t\geq T.$

Definition 2.7.  _A family of random cocycles
$\\{\varphi_{\alpha}\\}_{\alpha\in I}$ is said to be uniformly pullback
asymptotically compact over $I$ in $X$ if for each $q\in Q,\omega\in\Omega$,
$D\in\mathcal{D}$, the sequence_

$\displaystyle\\{\varphi_{\alpha_{n}}(t_{n},\sigma_{-t_{n}}q,\vartheta_{-t_{n}}\omega,x_{n})\\}\
\mbox{has a convergence subsequence in}\ X,$ (2.1)

_whenever $\alpha_{n}\in I,t_{n}\rightarrow\infty$, and $x_{n}\in
D(\sigma_{-t_{n}}q,\vartheta_{-t_{n}}\omega)$. A family of random cocycles
$\\{\varphi_{\alpha}\\}_{\alpha\in I}$ is uniformly pullback asymptotically
compact in $Y$ if the convergence in (2.1) holds under $Y$-norm. A single
cocycle $\varphi_{\alpha_{0}}$ is pullback asymptotically compact in $X$ if
(2.1) holds for a single point $\alpha=\alpha_{0}$._

Definition 2.8. _Let $\mathcal{D}$ be a collection of some families of
nonempty subsets of $X$, and $\varphi$ be a random cocycle on $X$ over
$(Q,\\{\sigma_{t}\\}_{t\in\mathbb{R}})$ and
$(\Omega,\mathcal{F},{P},\\{\vartheta_{t}\\}_{t\in\mathbb{R}})$. A mapping
$\psi:\mathbb{R}\times Q\times\Omega\rightarrow X$ is called _a complete
orbit_ of $\varphi$ if for each $\tau\in\mathbb{R},t\in\mathbb{R}^{+},q\in Q$
and $\omega\in\Omega$, there holds:_

$\varphi(t,\sigma_{\tau}q,\sigma_{\tau}\omega,\psi(\tau,q,\omega))=\psi(t+\tau,q,\omega).$

_If in addition, there exists $D=\\{D(q,\omega);q\in
Q,\omega\in\Omega\in\mathcal{D}\\}$ such that $\psi(\tau,q,\omega)\in
D(\sigma_{\tau}q,\sigma_{\tau}\omega)$, then $\psi$ is called a
$\mathcal{D}$-complete orbit of $\varphi$._

###  2.2. Abstract results

By slightly modifying the arguments of Theorem 3.1 in Li _et al_. [16], we can
extend the corresponding theory to the case in which the random systems with
non-autonomous term as well as random noises are considered.

Theorem 2.9. _Let $(X,Y)$ be a pair of Banach spaces satisfying hypothesis
_(H1)_ , and $\varphi$ a continuous random cocycle in $X$ over
$(Q,\\{\sigma_{t}\\}_{t\in\mathbb{R}})$ and
$(\Omega,\mathcal{F},{P},\\{\vartheta_{t}\\}_{t\in\mathbb{R}})$ such that
hypothesis _(H2)_ holds. Assume further that
_

_(i) $\varphi$ has a closed and measurable (w.r.t. the $P$-completion of
$\mathcal{F}$) pullback absorbing set $K=\\{K(q,\omega);q\in
Q,\omega\in\Omega\\}\in\mathcal{D}$ in $X$;_

_(ii) $\varphi$ is pullback asymptotically compact in $X$;_

_(iii) $\varphi$ is pullback asymptotically compact in $Y$._

_Then the random cocycle $\varphi$ admits a unique (X,Y)-pullback attractor
$\mathcal{A}\in\mathcal{D}$, which is structured by_

$\displaystyle\mathcal{A}(q,\omega)$
$\displaystyle=\cap_{\tau>0}\overline{\cup_{t\geq\tau}(t,\sigma_{-t}q,\vartheta_{-t}\omega,K(\sigma_{-t}q,\vartheta_{-t}\omega))}^{Y}$
$\displaystyle=\\{\psi(0,q,\omega);\psi\ \mbox{is a}\
\mathcal{D}\mbox{-complete orbit of the random cocycle}\ \varphi\\}.$ (2.2)

_Moreover, $\mathcal{A}=\mathcal{A}_{X}$, where $\mathcal{A}_{X}$ is the
$(X,X)$-pullback attractor._

In the following, we will consider both the semi-continuity of a family of the
bi-spatial pullback attractors and the existence problem. We give a unified
result, where the concepts of uniform absorption, uniform convergence and
uniformly pullback asymptotic compactness are used. To this end, we need to
consider a family of random cocycles $\\{\varphi_{\alpha}\\}_{\alpha\in I}$
with $I=[-a,a]\setminus\\{0\\}$, where $a>0$ and $\varphi_{0}$ is a
deterministic cocycle over the parametric space
$(Q,\\{\sigma_{t}\\}_{t\in\mathbb{R}})$.

Then analogous to the proofs of Theorem 3.1 in Li _et al._ [17], or some small
modifying the argument of Theorem 4.1 in Li _et al._ [16], we can extend their
results about random attractors with single random noises to the random
cocycle with non-autonomous term as well as random noises.

Theorem 2.10. Let $\mathcal{D}$ be a collection of some families of nonempty
subsets of $X$ and $(X,Y)$ a pair of Banach spaces satisfying hypothesis
_(H1)_. Suppose that $\\{\varphi_{\alpha}\\}_{\alpha\in I}$ is a family of
continuous random cocycles in $X$ over $(Q,\\{\sigma_{t}\\}_{t\in\mathbb{R}})$
and $(\Omega,\mathcal{F},{P},\\{\vartheta_{t}\\}_{t\in\mathbb{R}})$ such that
hypothesis _(H2)_ holds, and $\varphi_{0}$ is a continuous deterministic
cocycle in $X$ over $(Q,\\{\sigma_{t}\\}_{t\in\mathbb{R}})$ satisfying
$\varphi_{0}(t,q,.):X\rightarrow Y$ for all $t>0$ and $q\in Q$. Assume further
that

_(i) $\varphi_{\alpha}$ is convergent in $X$ at any $\alpha\in[-a,a]$;_

_(ii) $\varphi_{\alpha}(\alpha\in I)$ is uniformly absorbing in $X$ ;_

_(iii) $\varphi_{\alpha}(\alpha\in I)$ is uniformly pullback asymptotically
compact in $X$;_

_(iv) $\varphi_{\alpha}(\alpha\in I)$ is uniformly pullback asymptotically
compact in $Y$._

_Then each random cocycle $\varphi_{\alpha}$ admits a unique (X,Y)-pullback
attractor $\mathcal{A}_{\alpha}\in\mathcal{D}$, such that the family
$\\{\varphi_{\alpha}\\}_{\alpha\in I}$ is upper semi-continuous at any
$\alpha\in I$ in both $X$ and $Y$. If in addition, $\varphi_{0}$ has an
(X,Y)-attracting set $A_{0}$, then the $\\{\varphi_{\alpha}\\}_{\alpha\in I}$
is upper semi-continuous at $\alpha=0$ in both $X$ and $Y$._

##  3\.  Non-autonomous FitzHugh-Nagumo system on $\mathbb{R}^{N}$ with
multiplicative noise

For the non-autonomous FitzHugh-Nagumo system (1.1)-(1.3), the nonlinearity
$f(x,s)$ satisfy almost the same assumptions as in [1], _i.e._ , for
$x\in\mathbb{R}^{N}$ and $s\in\mathbb{R}$,

$\displaystyle f(x,s)s\leq-\alpha_{1}|s|^{p}+\psi_{1}(x),$ (3.1)
$\displaystyle|f(x,s)|\leq\alpha_{2}|s|^{p-1}+\psi_{2}(x),$ (3.2)
$\displaystyle\frac{\partial f}{\partial s}f(x,s)\leq\alpha_{3},$ (3.3)
$\displaystyle\Big{|}\frac{\partial f}{\partial
x}f(x,s)\Big{|}\leq\psi_{3}(x).$ (3.4)

where $p\geq 2$, $\alpha_{i}>0(i=1,2,3)$ are determined constants,
$\psi_{1}\in L^{1}(\mathbb{R}^{N})\cap L^{\frac{p}{2}}(\mathbb{R}^{N})$, and
$\psi_{2},\psi_{3}\in L^{2}(\mathbb{R}^{N})$. The non-autonomous terms $g\in
L^{2}_{loc}(\mathbb{R},L^{2}(\mathbb{R}^{N}))$ and $h\in
L^{2}_{loc}(\mathbb{R},L^{2}(\mathbb{R}^{N}))$ satisfy that for every
$\tau\in\mathbb{R}$ and some $0<\delta_{0}<\delta=\min\\{\lambda,\sigma\\}$,

$\displaystyle\int\limits_{-\infty}^{\tau}e^{\delta_{0}s}(\|g(s,.)\|_{L^{2}(\mathbb{R}^{N})}^{2}+\|h(s,.)\|_{L^{2}(\mathbb{R}^{N})}^{2})ds<+\infty,$
(3.5)

where $\lambda$ and $\delta$ are as in (1.1)-(1.3). The $H^{1}$-condition on
the non-autonomous term $h$ in (3.5) is required to prove the asymptotic
compactness of solutions in $L^{2}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})$, see [1]:

$\int\limits_{-\infty}^{\tau}e^{\delta_{0}s}\|h(s,.)\|_{H^{1}(\mathbb{R}^{N})}^{2}ds<+\infty.$

In order to model the random noises in system (1.1)-(1.3), we need to define a
shift operator $\vartheta$ on $\Omega$ (which is defined in the introduction)
by $\vartheta_{t}\omega(s)=\omega(s+t)-\omega(t)$ for every
$\omega\in\Omega,t,s\in\mathbb{R}.$ Then $\vartheta_{t}$ is a measure
preserving transformation group on $(\Omega,\mathcal{F},P)$, that is,
$(\Omega,\mathcal{F},P,\\{\vartheta_{t}\\}_{t\in\mathbb{R}})$ is a parametric
dynamical system. By the law of the iterated logarithm (see [8]), there exists
a $\vartheta_{t}$-invarant set $\tilde{\Omega}\subset\Omega$ of full measure
sucht that for $\omega\in\tilde{\Omega}$,

$\displaystyle\frac{\omega(t)}{t}\rightarrow 0,\ \mbox{as}\
|t|\rightarrow+\infty.$ (3.6)

Put $Q=\mathbb{R}$. Define a family of shift operator
$\\{\sigma_{t}\\}_{t\in\mathbb{R}}$ by $\sigma_{t}(\tau)=t+\tau$ for all
$t,\tau\in\mathbb{R}$. Then both
$\\{\mathbb{R},\\{\sigma_{t}\\}_{t\in\mathbb{R}}\\}$ and
$(\Omega,\mathcal{F},P,\\{\vartheta_{t}\\}_{t\in\mathbb{R}})$ are parametric
dynamical systems. We will define a continuous random cocycle for system
(1.1)-(1.3) over $(Q,\\{\sigma_{t}\\}_{t\in\mathbb{R}})$ and
$(\Omega,\mathcal{F},P,\\{\vartheta_{t}\\}_{t\in\mathbb{R}})$.

Given $\omega\in\Omega$, put
$z(t,\omega)=z_{\varepsilon}(t,\omega)=e^{-\varepsilon\omega(t)}$. Then we
have $dz+\varepsilon z\circ d\omega(t)=0.$ Let $(\tilde{u},\tilde{v})$ satisfy
problem (1.1)-(1.3) and write

$\displaystyle
u(t,\tau,\omega,u_{0})=z(t,\omega)\tilde{u}(t,\tau,\omega,\tilde{u}_{0})\ \
\mbox{and}\
v(t,\tau,\omega,v_{0})=z(t,\omega)\tilde{v}(t,\tau,\omega,\tilde{v}_{0}).$
(3.7)

Then $(u,v)$ solves the follow system

$\displaystyle\frac{du}{dt}+\lambda u-\Delta u+\alpha
v=z(t,\omega)f(x,z^{-1}(t,\omega)u)+z(t,\omega)g(t,x),$ (3.8)
$\displaystyle\frac{dv}{dt}+\sigma v-\beta u=z(t,\omega)h(t,x),$ (3.9)

with initial conditions $u_{\tau}=z(\tau,\omega)\tilde{u}_{\tau}$ and
$v_{\tau}=z(\tau,\omega)\tilde{v}_{\tau}$.

It is known (see [1]) that for every $(u_{\tau},v_{\tau})\in
L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$ the problem (1.1) possesses
a unique solution $(u,v)$ such that $u\in
C([\tau,+\infty),L^{2}(\mathbb{R}^{N}))\cap
L^{2}(\tau,T,H^{1}(\mathbb{R}^{N}))\cap L^{p}(\tau,T,L^{p}(\mathbb{R}^{N}))$
and $v\in C([0,+\infty),L^{2}(\mathbb{R}^{N}))$. In addition, the solution
$(u,v)$ is continuous in $L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$
with respect to the initial value $(u_{\tau},v_{\tau})$ . Then formally
$(\tilde{u},\tilde{v})=(z^{-1}(t,\omega)u,z^{-1}(t,\omega)v)$ is the solution
to problem (1.1)-(1.3) with the initial value
$\tilde{u}_{\tau}=z^{-1}(\tau,\omega){u}_{\tau}$ and
$\tilde{v}_{\tau}=z^{-1}(\tau,\omega){v}_{\tau}$.

We are at the position to give the continuous random cocycle $\varphi$
associated with problem (1.1)-(1.3) over
$(Q,\\{\sigma_{t}\\}_{t\in\mathbb{R}})$ and
$(\Omega,\mathcal{F},P,\\{\vartheta_{t}\\}_{t\in\mathbb{R}})$. Define

$\displaystyle\ \ \ \ \ \ \ \ \
\varphi(t,\tau,\omega,(\tilde{u}_{\tau},\tilde{v}_{\tau}))=(\tilde{u}(t+\tau,\tau,\vartheta_{-\tau}\omega,\tilde{u}_{\tau}),\tilde{v}(t+\tau,\tau,\vartheta_{-\tau}\omega,\tilde{v}_{\tau}))$
$\displaystyle=(z^{-1}(t+\tau,\vartheta_{-\tau}\omega)u(t+\tau,\tau,\vartheta_{-\tau}\omega,u_{\tau}),z^{-1}(t+\tau,\vartheta_{-\tau}\omega)v(t+\tau,\tau,\vartheta_{-\tau}\omega,v_{\tau})),$
(3.10)

where $u_{\tau}=z(\tau,\omega)\tilde{u}_{\tau}$ and
$v_{\tau}=z(\tau,\omega)\tilde{v}_{\tau}$.

Suppose that for every $\tau\in\mathbb{R}$ and $\omega\in\Omega$.

$\displaystyle\lim\limits_{t\rightarrow+\infty}e^{-\delta_{1}t}\|D(\tau-t,\vartheta_{-t}\omega)\|_{L^{2}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})}^{2}=0,$ (3.11)

where $0<\delta_{0}<\delta_{1}<\delta=\min\\{\lambda,\sigma\\}$. Denote by
$\mathcal{D}_{\delta}$ the collection of all families of nonempty subsets of
$L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$ such that (3.11) holds.
Then it is obvious that $\mathcal{D}_{\delta}$ is inclusion closed.

We emphasize that the choices of the constants $\delta_{1},\delta_{0}$ in
(3.11) and (3.5) respectively are different from the ones used in [1]. It
makes us omit the additional assumption

$\displaystyle\lim_{t\rightarrow-\infty}e^{\delta_{1}}\int\limits_{-\infty}^{0}e^{\delta
s}(\|g(s,.)\|_{L^{2}(\mathbb{R}^{N})}^{2}+\|h(s,.)\|_{L^{2}(\mathbb{R}^{N})}^{2})ds=0,$
(3.12)

which is intrinsically used in [1], see the detailed proof of Lemma 4.1 in the
following section.

Note that Adili and Wang [1] established the existence and upper semi-
continuous of pullback attractors for problem (1.1)-(1.3) in
$L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$. In this paper, we obtain
an identical result in $L^{p}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$,
but we do not increase the restrictions (except that $\psi_{1}\in
L^{p/2}(\mathbb{R}^{N})$ as in (3.1)) on the nonlinearity $f$. On the
contrary, the restrictive assumption (3.12) on the non-autonomous terms $g$
and $h$ given in [1] is omitted. Furthermore, we construct a unique random
equilibrium for this system when some additional assumptions on the physical
parameters are added.

##  4\. Existence and upper semi-continuous of attractors in $L^{p}\times
L^{2}$

From now on, we assume without loss of generality that $\varepsilon\in(0,a]$
for any $a>0$. Consider that $e^{-a|\omega(s)|}\leq
z_{\varepsilon}(s,\omega)=e^{-\varepsilon\omega(s)}\leq e^{a|\omega(s)|}$ for
$\varepsilon\in I$, and $\omega(.)$ is continuous on $[-2,0]$. Then there
exist two positive random constants $E=E(\omega)$ and $F=F(\omega)$ such that
for each $\omega\in\Omega$,

$\displaystyle E\leq z_{\varepsilon}(s,\omega)\leq F\ \ \ \ \mbox{for all}\
s\in[-2,0]\ \mbox{and}\ \varepsilon\in(0,a].$ (4.1)

Hereafter, we denote by $\|.\|$ and $\|.\|_{p}$ the norms in
$L^{2}(\mathbb{R}^{N})$ and $L^{p}(\mathbb{R}^{N})(p>2)$, respectively.
Throughout this paper, the number $c$ is a generic positive constant
independent of $\tau,\omega,D$ and $\varepsilon$ in any place, which may vary
its values everywhere.

###  4.1. Uniform absorption and uniformly asymptotic compactness in
$L^{2}\times L^{2}$

This subsection is concern with some uniform estimates of solutions on a
certain compact interval $[\tau-1,\tau]$ for $\tau\in\mathbb{R}$. The uniform
absorption of the family of random cocycles $\varphi_{\varepsilon}$ is proved.
Note that the notations $(u,v)$, $(\tilde{u},\tilde{v})$, and $\varphi$ are
the abbreviations of $(u_{\varepsilon},v_{\varepsilon})$,
$(\tilde{u}_{\varepsilon},\tilde{v}_{\varepsilon})$ and
$\varphi_{\varepsilon}$ respectively, where the later implies the dependence
of solutions on $\varepsilon$, omitting the subscript $\varepsilon$ if there
is no confusion.

Lemma 4.1. _Assume that ( 3.1)-(3.5) holds and $a>0$. Given
$\tau\in\mathbb{R},\omega\in\Omega$ and
$D=\\{D(\tau,\omega);\tau\in\mathbb{R},\omega\in\Omega\\}\in\mathcal{D}_{\delta}$,
then there exists a constant $T=T(\tau,\omega,D)\geq 2$ such that for all
$t\geq T$, $\varepsilon\in(0,a]$, and
$(\tilde{u}_{\tau-t},\tilde{v}_{\tau-t})\in D(\tau-t,\vartheta_{-t}\omega)$,
the solution $(u_{\varepsilon},v_{\varepsilon})$ of problem (3.8)-(3.9)
satisfies_

$\displaystyle\|(u_{\varepsilon}(\xi,\tau-t,\vartheta_{-\tau}\omega,{u}_{\tau-t}),v_{\varepsilon}(\xi,\tau-t,\vartheta_{-\tau}\omega,{v}_{\tau-t}))\|^{2}\leq
ce^{2\varepsilon\omega(-\tau)}(1+L_{\varepsilon}(\tau,\omega)),\xi\in[\tau-1,\tau],$
(4.2)
$\displaystyle\|(\tilde{u}_{\varepsilon}(\xi,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{\tau-t}),\tilde{v}_{\varepsilon}(\xi,\tau-t,\vartheta_{-\tau}\omega,\tilde{v}_{\tau-t}))\|^{2}\leq
c(1+L_{\varepsilon}(\tau,\omega)),\ \ \xi\in[\tau-1,\tau],$ (4.3)

_and_

$\displaystyle\int\limits_{\tau-t}^{\xi}e^{\delta(s-\tau)}\Big{(}$
$\displaystyle\|v_{\varepsilon}(\tau,\tau-t,\vartheta_{-\tau}\omega,{v}_{\tau-t})\|^{2}$
$\displaystyle+z_{\varepsilon}^{2-p}(s,\vartheta_{-\tau}\omega)\|u_{\varepsilon}(s,\tau-t,\vartheta_{-\tau}\omega,{u}_{\tau-t})\|_{p}^{p}\Big{)}ds\leq
ce^{2\varepsilon\omega(-\tau)}(1+L_{\varepsilon}(\tau,\omega)),$ (4.4)

_where
$(u_{\tau-t},v_{\tau-t})=z_{\varepsilon}(\tau-t,\vartheta_{-\tau}\omega)(\tilde{u}_{\tau-t},\tilde{v}_{\tau-t})$,
and $L_{\varepsilon}(\tau,\omega)$ is given by_

$\displaystyle
L_{\varepsilon}(\tau,\omega)=\int\limits_{-\infty}^{0}e^{\delta_{01}s+2\varepsilon|\omega(s)|}\Big{(}\|g(s+\tau,.)\|^{2}+\|h(s+\tau,.)\|^{2}+1\Big{)}ds,$
(4.5)

such that $\varepsilon\rightarrow L_{\varepsilon}(\tau,\omega)$ is an
increasing function on $(0,+\infty)$, where
$0<\delta_{0}<\delta_{01}<\delta_{1}<\delta$.

_In particular, the family of random cocycles
$\varphi_{\varepsilon}(\varepsilon\in(0,a])$ defined by (3) is uniformly
absorbing on $(0,a]$ for any $a>0$ in the sense of Definition 2.6._

_Proof_ Taking the inner products of (3.8) and (3.9) with $u$ and $v$,
respectively, by using (3.1), we have

$\displaystyle\frac{d}{dt}(\beta\|u\|^{2}+\alpha\|v\|^{2})$
$\displaystyle+\delta(\beta\|u\|^{2}+\alpha\|v\|^{2})$
$\displaystyle+2\alpha_{1}\beta z^{2-p}(t,\omega)\|{u}\|_{p}^{p}\leq
cz^{2}(t,\omega)(\|g(t,.)\|^{2}+\|h(t,.)\|^{2}+\|\psi_{1}\|_{1}).$ (4.6)

By applying the Gronwall lemma over the interval $[\tau-t,\xi]$ with
$\xi\in[\tau-1,\tau]$ and $t\geq 1$, along with $\omega$ replaced by
$\vartheta_{-\tau}\omega$, we get from (3.7) that

$\displaystyle\|u(\xi,\tau-t,\vartheta_{-\tau}\omega,v_{\tau-t})\|^{2}+\|v(\xi,\tau-t,\vartheta_{-\tau}\omega,v_{\tau-t})\|^{2}$
$\displaystyle+\int\limits_{\tau-t}^{\xi}e^{-\delta(\xi-s)}(\|v(s,\tau-t,\vartheta_{-\tau}\omega,v_{\tau-t})\|^{2}+z^{2-p}(s,\vartheta_{-\tau}\omega)\|u(s,\tau-t,\vartheta_{-\tau}\omega,v_{\tau-t})\|_{p}^{p})ds$
$\displaystyle\leq
ce^{-\delta(\xi-\tau+t)}(\|u_{\tau-t}\|^{2}+\|v_{\tau-t}\|^{2})+c\int\limits_{\tau-t}^{\xi}e^{-\delta(\xi-s)}z^{2}(s,\vartheta_{-\tau}\omega)(\|g(s,.)\|^{2}+\|h(s,.)\|^{2}+1)ds$
$\displaystyle\leq ce^{2\varepsilon\omega(-\tau)}\Big{(}e^{-\delta
t}z^{2}(-t,\omega)(\|\tilde{u}_{\tau-t}\|^{2}+\|\tilde{v}_{\tau-t}\|^{2})+\int\limits_{\tau-t}^{\tau}e^{-\delta(\tau-s)-2\varepsilon\omega(s-\tau)}(\|g(s,.)\|^{2}+\|h(s,.)\|^{2}+1)ds\Big{)}$
$\displaystyle\leq ce^{2\varepsilon\omega(-\tau)}\Big{(}e^{-\delta
t}z^{2}(-t,\omega)(\|\tilde{u}_{\tau-t}\|^{2}+\|\tilde{v}_{\tau-t}\|^{2})+\int\limits_{-t}^{0}e^{\delta_{01}s-2\varepsilon\omega(s)}(\|g(s+\tau,.)\|^{2}+\|h(s+\tau,.)\|^{2}+1)ds\Big{)},$
(4.7)

where $\delta=\min\\{\lambda,\sigma\\}$ and $\delta_{01}<\delta$. By (3.6) we
calculate that
$\lim\limits_{t\rightarrow+\infty}e^{-(\delta-\delta_{1})t}z^{2}(-t,\omega)=0$.
Then from the property of $\mathcal{D}_{\delta}$ in (3.11) it follows that

$\displaystyle\lim_{t\rightarrow+\infty}e^{-\delta
t}z^{2}(-t,\omega)(\|\tilde{u}_{\tau-t}\|^{2}+\|\tilde{v}_{\tau-t}\|^{2})=0.$
(4.8)

Thus (4.1) and (4.8) together implies that there exists a random constant
$T=T(\tau,\omega,D)\geq 1$ such that for each $\varepsilon\in(0,a]$ and all
$t\geq T$, (4.2) and (4.1) hold. By (3.7) and (4.2) it is showed that (4.3)
hold true for all $t\geq T$.

On the other hand, from (3.5) and (3.6) it follow that the integral in
$L_{\varepsilon}(\tau,\omega)$ is meaningful and thus
$L_{\varepsilon}(\tau,\omega)$ is finite. Further,

$\displaystyle L_{\varepsilon}(\tau-t,\vartheta_{-t}\omega)$
$\displaystyle=\int\limits_{-\infty}^{0}e^{\delta_{01}s+2\varepsilon|\vartheta_{-t}\omega(s)|}\Big{(}\|g(s+\tau-t,.)\|^{2}+\|h(s+\tau-t,.)\|^{2}+1\Big{)}ds$
$\displaystyle(\mbox{letting}\ s-t=s^{\prime})$ $\displaystyle\leq
e^{\delta_{01}t+2\varepsilon|\omega(-t)|}\int\limits_{-\infty}^{-t}e^{\delta_{01}s+2\varepsilon|\omega(s)|}\Big{(}\|g(s+\tau,.)\|^{2}+\|h(s+\tau,.)\|^{2}+1\Big{)}.$
(4.9)

We see from (3.6) and the relation $\delta_{0}<\delta_{01}$ that
$\lim\limits_{s\rightarrow-\infty}e^{-(\delta_{01}-\delta_{0})s+2\varepsilon|\omega(s)|}=0$,
so there exists a positive variable $a(\omega)$ such that

$0<e^{-(\delta_{01}-\delta_{0})s+2\varepsilon|\omega(s)|}\leq a(\omega),\ \
s\in(-\infty,0],$

from which it follows that

$\displaystyle L_{\varepsilon}(\tau-t,\vartheta_{-t}\omega)$
$\displaystyle\leq
a(\omega)e^{\delta_{01}t+2\varepsilon|\omega(-t)|}\int\limits_{-\infty}^{-t}e^{\delta_{0}s}\Big{(}\|g(s+\tau,.)\|^{2}+\|h(s+\tau,.)\|^{2}+1\Big{)}ds.$
(4.10)

Then from (3.11),(3.6), (3.5) and (4.10) we deduce that

$K_{\varepsilon}=\\{K_{\varepsilon}(\tau,\omega)=\\{(\tilde{u},\tilde{v})\in(L^{2}(\mathbb{R}))^{2};\
\ \|(\tilde{u},\tilde{v})\|^{2}\leq
c(1+L_{\varepsilon}(\tau,\omega))\\};\tau\in\mathbb{R},\omega\in\Omega\\}\in\mathcal{D}_{\delta},$

and further the union
$\overline{\cup_{\varepsilon\in(0,a]}K_{\varepsilon}(\tau,\omega)}\subset
K_{a}(\tau,\omega)$. Thus
$\overline{K}=\\{\overline{\cup_{\varepsilon\in(0,a]}K_{\varepsilon}(\tau,\omega)};\tau\in\mathbb{R},\omega\in\Omega\\}\in\mathcal{D}_{\delta}$.
The measurability of the absorbing set $K_{\varepsilon}(\tau,\omega)$ follows
from the measurability of the variable $L_{\varepsilon}(\tau,\omega)$. Finally
since by (4.5) and (3.5) and the relation $\delta_{0}<\delta_{01}$,

$\limsup_{\varepsilon\rightarrow 0}\|K_{\varepsilon}(\tau,\omega)\|\leq
L_{0}(\tau,\omega)=\int\limits_{-\infty}^{0}e^{\delta_{01}s}|\Big{(}\|g(s+\tau,.)\|^{2}+\|h(s+\tau,.)\|^{2}+1\Big{)}ds<+\infty,$

then we have showed the uniformly absorbing of
$\varphi_{\varepsilon}(\varepsilon\in(0,a])$ for any $a>0.$ $\ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \Box$

The uniformly asymptotic compactness in $L^{2}\times L^{2}$ has been proved by
[1].

Lemma 4.2. _Assume that ( 3.1)-(3.5) hold. Then the family of random cocycles
$\varphi_{\varepsilon}$ defined by (3) is uniformly pullback asymptotically
compact over $\varepsilon\in(0,a]$ in $L^{2}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})$. _

###  4.2. Uniformly asymptotic compactness in $L^{p}\times L^{2}$

In this subsection, we prove that the family of random cocycles
$\varphi_{\varepsilon}(\varepsilon(0,a])$ is uniformly asymptotically compact
in $L^{p}\times L^{2}$. We need to prove the $L^{p}$-uniform boundedness of
the first component of solution $u_{\varepsilon}$ as well as the uniform
smallness of truncation of $u_{\varepsilon}$ in $L^{p}$ norm.

Lemma 4.3. _Assume that ( 3.1)-(3.5) hold. Given
$\tau\in\mathbb{R},\omega\in\Omega$ and
$D=\\{D(\tau,\omega);\tau\in\mathbb{R},\omega\in\Omega\\}\in\mathcal{D}_{\delta}$,
then there exist some random constants $C=C(\tau,\omega)$ and
$T=T(\tau,\omega,D)\geq 2$ such that for all
$(\tilde{u}_{\tau-t},\tilde{v}_{\tau-t})\in D(\tau-t,\vartheta_{-t}\omega)$,
the solution $(u_{\varepsilon},v_{\varepsilon})$ of problem (3.8)-(3.9)
satisfies_

$\displaystyle\sup_{t\geq
T}\sup_{\xi\in[\tau-1,\tau]}\sup_{\varepsilon\in(0,a]}\|{u}_{\varepsilon}(\xi,\tau-t,\vartheta_{-\tau}\omega,{u}_{\tau-t})\|_{p}^{p}\leq
C(\tau,\omega),$ (4.11)

where $C(\tau,\omega)$ is independent $\varepsilon$.

_Proof_ Multiplying (3.8) by $|u|^{p-2}u$ and then integrating over
$\mathbb{R}^{N}$, we have

$\displaystyle\ \ \ \ \ \ \ \
\frac{1}{p}\frac{d}{dt}\|u\|_{p}^{p}+\lambda\|u\|_{p}^{p}\leq\alpha\int\limits_{\mathbb{R}^{N}}v|u|^{p-1}dx$
$\displaystyle+z(t,\omega)\int\limits_{\mathbb{R}^{N}}f(x,z^{-1}(t,\omega)u)|u|^{p-2}udx+z(t,\omega)\int\limits_{\mathbb{R}^{N}}g(t,x)|u|^{p-2}udx.$
(4.12)

From (3.1) and $\psi\in L^{p/2}$, applying Young inequality, we obtain that

$\displaystyle
z(t,\omega)\int\limits_{\mathbb{R}^{N}}f(x,z^{-1}(t,\omega)u)|u|^{p-2}udx\leq-\alpha_{1}z^{2-p}(t,\omega)\|u\|_{2p-2}^{2p-2}+\frac{\lambda}{4}\|u\|_{p}^{p}+cz^{p}(t,\omega)\|\psi_{1}\|^{p/2}_{p/2}.$
(4.13)

On the other hand,

$\displaystyle\alpha\int\limits_{\mathbb{R}^{N}}v|u|^{p-1}dx\leq\frac{1}{4}\alpha_{1}z^{2-p}(t,\omega)\|u\|_{2p-2}^{2p-2}+cz^{p-2}(t,\omega)\|v\|^{2},$
(4.14)

and

$\displaystyle
z(t,\omega)\int\limits_{\mathbb{R}^{N}}g(t,x)|u|^{p-2}udx\leq\frac{1}{4}\alpha_{1}z^{2-p}(t,\omega)\|u\|_{2p-2}^{2p-2}+cz^{p}(t,\omega)\|g(t,.)\|^{2}.$
(4.15)

Then combination (4.2)-(4.15), it give that

$\displaystyle\frac{d}{dt}\|u\|_{p}^{p}+\delta\|u\|_{p}^{p}\leq
cz^{p-2}(t,\omega)\|v\|^{2}+cz^{p}(t,\omega)(\|g(t,.)\|^{2}+\|\psi_{1}\|^{p/2}_{p/2}),$
(4.16)

where $\delta=\min\\{\lambda,\sigma\\}$. Note that $\frac{1}{\xi-\tau+2}\leq
1$ for $\xi\in[\tau-1,\tau]$. Applying Gronwall lemma (see also Lemma 5.1 in
[33]) over the interval $[\tau-2,\xi]$, along with $\omega$ replaced by
$\vartheta_{-\tau}\omega$, we deduce that

$\displaystyle\|u(\xi,\tau-t,\vartheta_{-\tau}\omega,{u}_{\tau-t})\|_{p}^{p}$
$\displaystyle\leq
c\int\limits_{\tau-2}^{\tau}e^{\delta(s-\tau)}\|{u}(s,\tau-t,\vartheta_{-\tau}\omega,{u}_{\tau-t})\|_{p}^{p}ds$
$\displaystyle+c\int\limits_{\tau-2}^{\tau}e^{\delta(s-\tau)}z^{p-2}(s,\vartheta_{-\tau})\|v(s,\tau-t,\vartheta_{-\tau}\omega,{v}_{\tau-t})\|^{2}ds$
$\displaystyle+c\int\limits_{\tau-2}^{\tau}e^{\delta(s-\tau)}z^{p}(s,\vartheta_{-\tau})(\|g(s,.)\|^{2}+1)ds.$
(4.17)

We now estimate every term on the right hand side of (4.2). First from (4.1)
it follows that for all $s\in[\tau-2,\tau]$ and $\varepsilon\in(0,a]$,
$z^{2-p}(s,\vartheta_{-\tau}\omega)=e^{\varepsilon(2-p)\omega(-\tau)}z^{2-p}(s-\tau,\omega)\geq
e^{\varepsilon(2-p)\omega(-\tau)}F^{2-p}$. Then from (4.1), there exists
$T=T(\tau,\omega,D)\geq 2$, such that for all $\varepsilon\in(0,a]$ and $t\geq
T$,

$\displaystyle\int\limits_{\tau-2}^{\tau}e^{\delta(s-\tau)}\|u(s,\tau-t,\vartheta_{-\tau}\omega,{u}_{\tau-t})\|_{p}^{p}ds\leq
cF^{p-2}e^{ap|\omega(-\tau)|}(1+L_{a}(\tau,\omega)).$ (4.18)

Noticing that $z^{p-2}(s,\vartheta_{-\tau})\leq
e^{(p-2)\varepsilon\omega(-\tau)}F^{p-2}$ for $s\in[\tau-2,\tau]$, then from
(4.1) again we see that

$\displaystyle\int\limits_{\tau-2}^{\tau}e^{\delta(s-\tau)}z^{p-2}(s,\vartheta_{-\tau})\|v(s,\tau-t,\vartheta_{-\tau}\omega,{v}_{\tau-t})\|^{2}ds\leq
cF^{p-2}e^{ap|\omega(-\tau)|}(1+L_{a}(\tau,\omega)).$ (4.19)

On the other hand, by (3.5),

$\displaystyle\int\limits_{\tau-2}^{\tau}e^{\delta(s-\tau)}z^{p}(s,\vartheta_{-\tau})(\|g(s,.)\|^{2}+1)ds\leq
F^{p}e^{ap|\omega(-\tau)|}\int\limits_{-2}^{0}e^{\delta
s}(\|g(s+\tau,.)\|^{2}+1)ds<+\infty.$ (4.20)

Hence (4.2)-(4.20) implies the desired. $\ \ \ \ \ \ \Box$

Let $M=M(\tau,\omega)>0$. Denote by $(u-M)_{+}$ the positive part of
$u-M$,_i.e._ ,

$(u-M)_{+}=\left\\{\begin{array}[]{ll}u-M,\ \ \mbox{if}\ u>M;\\\ 0,\ \ \ \ \ \
\ \ \ \mbox{if}\ u\leq M.\end{array}\right.$

The next lemma will show that the unbounded part of the absolute value $|u|$
approaches zero in $L^{p}$-norm on the state domain
$\mathbb{R}^{N}(|u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{\tau-t})|\geq M)$
for $M$ large enough, where

$\displaystyle\mathbb{R}^{N}(|u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{\tau-t})|\geq
M)=\\{x\in\mathbb{R}^{N};|u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{\tau-t})|\geq
M|\\}.$

Note that we need not to prove some auxiliary lemmas except Lemma 4.1 and
Lemma 4.3, see [16, 17, 18, 19, 34, 35].

Lemma 4.4. Assume that (3.1)-(3.5) hold. Given
$\tau\in\mathbb{R},\omega\in\Omega$ and
$D=\\{D(\tau,\omega);\tau\in\mathbb{R},\omega\in\Omega\\}\in\mathcal{D}_{\delta}$,
then for any $\eta>0$, there exist random constants
$M=M(\tau,\omega,\eta,D)>1$ and $T=T(\tau,\omega,D)\geq 2$ such that for all
$(\tilde{u}_{\tau-t},\tilde{v}_{\tau-t})\in D(\tau-t,\vartheta_{-t}\omega)$,
the first component $\tilde{u}_{\varepsilon}$ of solutions
$(\tilde{u}_{\varepsilon},\tilde{v}_{\varepsilon})$ of problem (1.1)-(1.3)
satisfies

$\displaystyle\sup_{t\geq
T}\sup_{\varepsilon\in(0,a]}\int\limits_{\mathbb{R}^{N}(|\tilde{u}_{\varepsilon}|\geq
M)}|\tilde{u}_{\varepsilon}(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{\tau-t})|^{p}dx\leq\eta,$

where $\mathbb{R}^{N}(|\tilde{u}_{\varepsilon}|\geq
M)=\mathbb{R}^{N}(|\tilde{u}_{\varepsilon}(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{\tau-t})|\geq
M)$, and $M,T$ are independent of $\varepsilon$.

_Proof_ Let $s\in[\tau-1,\tau]$ and $t\geq T\geq 2$, where $T$ is determined
by Lemma 4.1 and Lemma 4.2. Replacing $\omega$ by $\vartheta_{-\tau}\omega$ in
(3.8)-(3.9), we see that
$u=u(s,\tau-t,\vartheta_{-\tau}\omega,u_{\tau-t}),v=v(s,\tau-t,\vartheta_{-\tau}\omega,v_{\tau-t})$,
is a solution of the following system

$\displaystyle\frac{du}{ds}+\lambda u-\Delta u+\alpha
v=\frac{z(s-\tau,\omega)}{z(-\tau,\omega)}f(x,\tilde{u})+\frac{z(s-\tau,\omega)}{z(-\tau,\omega)}{g}(s,x),$
(4.21) $\displaystyle\frac{dv}{ds}+\sigma v-\beta
u=\frac{z(s-\tau,\omega)}{z(-\tau,\omega)}{h}(s,x).$ (4.22)

For fixed $\tau\in\mathbb{R}$ and $\omega\in\Omega$, we assume that
$M=M(\tau,\omega)>1$. We multiply (4.21) by $(u-M)_{+}^{p-1}$ and integrate
over $\mathbb{R}^{N}$ to yield

$\displaystyle\frac{1}{p}\frac{d}{ds}\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{p}dx$
$\displaystyle+\lambda\int\limits_{\mathbb{R}^{N}}u(u-M)_{+}^{p-1}dx-\int\limits_{\mathbb{R}^{N}}\Delta
u(u-M)_{+}^{p-1}dx$
$\displaystyle=-\alpha\int\limits_{\mathbb{R}^{N}}v(u-M)_{+}^{p-1}dx+\frac{z(s-\tau,\omega)}{z(-\tau,\omega)}\int\limits_{\mathbb{R}^{N}}f(x,\tilde{u})(u-M)_{+}^{p-1}dx$
$\displaystyle\ \ \ \ \ \ \
+\frac{z(s-\tau,\omega)}{z(-\tau,\omega)}\int\limits_{\mathbb{R}^{N}}{g}(s,x)(u-M)_{+}^{p-1}dx.$
(4.23)

We now have to estimate every term in (4.2). First, it is obvious that

$\displaystyle-\int\limits_{\mathbb{R}^{N}}\Delta
u(u-M)_{+}^{p-1}dx=(p-1)\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{p-2}|\nabla
u|^{2}dx\geq 0,$ (4.24)
$\displaystyle\lambda\int\limits_{\mathbb{R}^{N}}u(u-M)_{+}^{p-1}dx\geq\lambda\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{p}dx.$
(4.25)

The most involved work is to calculate the nonlinearity in (4.2). Consider
that for $u(s)>M$ for $\in[\tau-1,\tau]$, we have
$\tilde{u}(s)=z^{-1}(s,\vartheta_{-\tau}\omega)u(s)=\frac{z(-\tau,\omega)}{z(s-\tau,\omega)}u(s)>0$,
and thus by (3.1) and (4.1), we find that for every $s\in[\tau-1,\tau]$,

$\displaystyle f(x,\tilde{u})$
$\displaystyle\leq-\alpha_{1}|\tilde{u}|^{p-1}+\frac{1}{\tilde{u}}\psi_{1}(x)$
$\displaystyle=-\alpha_{1}\Big{(}\frac{z(s-\tau,\omega)}{z(-\tau,\omega)}\Big{)}^{1-p}|u|^{p-1}+\frac{z(s-\tau,\omega)}{z(-\tau,\omega)}\frac{\psi_{1}(x)}{u}$
$\displaystyle\leq-\frac{\alpha_{1}z^{p-1}(-\tau,\omega)}{2F^{p-1}}M^{p-2}(u-M)-\frac{\alpha_{1}z^{p-1}(-\tau,\omega)}{2F^{p-1}}(u-M)^{p-1}$
$\displaystyle+\frac{F}{z(-\tau,\omega)}|\psi_{1}(x)|(u-M)^{-1},$

from which and (4.1) again it follows that

$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\frac{z(s-\tau,\omega)}{z(-\tau,\omega)}\int\limits_{\mathbb{R}^{N}}f(x,\tilde{u})(u-M)_{+}^{p-1}dx$
$\displaystyle\leq-\frac{\alpha_{1}Ez^{p-1}(-\tau,\omega)}{2F^{p-1}}M^{p-2}\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{p}dx-\frac{\alpha_{1}Ez^{p-1}(-\tau,\omega)}{2F^{p-1}}\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{2p-2}dx$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\frac{F^{2}}{z^{2}(-\tau,\omega)}\int\limits_{\mathbb{R}^{N}}|\psi_{1}(x)|(u-M)_{+}^{p-2}dx$
$\displaystyle\leq-\frac{\alpha_{1}Ez^{p-1}(-\tau,\omega)}{2F^{p-1}}M^{p-2}\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{p}dx-\frac{\alpha_{1}Ez^{p-1}(-\tau,\omega)}{2F^{p-1}}\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{2p-2}dx$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\frac{1}{2}\lambda\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{p}dx+\frac{cF^{p}}{z^{p}(-\tau,\omega)}\int\limits_{\mathbb{R}^{N}(u\geq
M)}|\psi_{1}(x)|^{p/2}dx,$ (4.26)

in which we have used the Young inequality in the last term, and here
$c=(\frac{2}{\lambda})^{p-2/2}$. On the other hand by using the Young
inequality again, we get that for $s\in[\tau-1,\tau]$,

$\displaystyle\Big{|}\frac{z(s-\tau,\omega)}{z(-\tau,\omega)}$
$\displaystyle\int\limits_{\mathbb{R}^{N}}{g}(s,x)(u-M)_{+}^{p-1}dx\Big{|}\leq\frac{F}{z(-\tau,\omega)}\Big{|}\int\limits_{\mathbb{R}^{N}}{g}(s,x)(u(s)-M)_{+}^{p-1}dx\Big{|}$
$\displaystyle\leq\frac{\alpha_{1}Ez^{p-1}(-\tau,\omega)}{4F^{p-1}}\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{2p-2}dx+\frac{F^{p+1}}{\alpha_{1}Ez^{p+1}(-\tau,\omega)}\int\limits_{\mathbb{R}^{N}(u(s)\geq
M)}{g}^{2}(s,x)dx,$ (4.27)

and

$\displaystyle\Big{|}-\alpha\int\limits_{\mathbb{R}^{N}}v(u-M)_{+}^{p-1}dx\Big{|}$
$\displaystyle\leq\frac{\alpha_{1}Ez^{p-1}(-\tau,\omega)}{4F^{p-1}}\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{2p-2}dx$
$\displaystyle+\frac{\alpha^{2}F^{p-1}}{\alpha_{1}Ez^{p-1}(-\tau,\omega)}\int\limits_{\mathbb{R}^{N}(u(s)\geq
M)}v^{2}dx.$ (4.28)

For convenience of calculations, we introduce the following notations:

$\displaystyle
k=k(\tau,\omega,M)=\frac{\alpha_{1}Ee^{-(p-1)a|\omega(-\tau)|}}{2F^{p-1}}M^{p-2};$
(4.29)

which is increasing to infinite in $M$ for $p>2$, and

$\displaystyle
G(\tau,\omega)=\max\Big{\\{}\frac{F^{p+1}e^{a(p+1)|\omega(-\tau)|}}{\alpha_{1}E};\frac{\alpha^{2}F^{p-1}e^{a(p-1)|\omega(-\tau)|}}{\alpha_{1}E};(\frac{2}{\lambda})^{\frac{p-2}{2}}F^{p}z^{ap|\omega(-\tau)|}\Big{\\}},$
(4.30)

which is a nonnegative random constant depending only on $\tau,\omega$.
Combination (4.2)-(4.2) and using the notations (4.29)-(4.30), we obtain that

$\displaystyle\frac{d}{ds}\int\limits_{\mathbb{R}^{N}}(u(s)-M)_{+}^{p}dx$
$\displaystyle+k\int\limits_{\mathbb{R}^{N}}(u(s)-M)_{+}^{p}dx\leq
G(\tau,\omega)(\|{g}(s,.)\|^{2}+\|v\|^{2}+1),$ (4.31)

where $s\in[\tau-1,\tau]$. Applying Gronwall lemm (also see Lemma 5.1 in [33]
) over $[\tau-1,\tau]$, by Lemma 4.1 and Lemma 4.3, we find that for all
$t\geq T$ and $\varepsilon\in(0,a]$,

$\displaystyle\int\limits_{\mathbb{R}^{N}}\Big{(}u(\tau,\tau-t,$
$\displaystyle\vartheta_{-\tau}\omega,u_{\tau-t})-M\Big{)}_{+}^{p}dx\leq\int\limits_{\tau-1}^{\tau}e^{k(s-\tau)}\|u(s,\tau-t,\vartheta_{-\tau}\omega,u_{0})\|_{p}^{p}ds$
$\displaystyle+G(\tau,\omega)\Big{(}\int\limits_{\tau-1}^{\tau}e^{k(s-\tau)}\|v(s,\tau-t,\vartheta_{-\tau}\omega,v_{\tau-t})\|^{2}ds+\int\limits_{\tau-1}^{\tau}e^{k(s-\tau)}(\|g(s,.)\|^{2}+1)ds\Big{)}$
$\displaystyle\leq\frac{C(\tau,\omega)+cG(\tau,\omega)e^{2a|\omega(-\tau)|}(1+L_{a}(\tau,\omega))+G(\tau,\omega)}{k}$
$\displaystyle+G(\tau,\omega)\int\limits_{\tau-1}^{\tau}e^{k(s-\tau)}\|g(s,.)\|^{2}ds.$
(4.32)

For fixed $\tau\in\mathbb{R}$ and $\omega\in\Omega$, the first term in the
last inequality of (4.2) varies only with the number $k$, but $k$ is a large
number as $M\rightarrow+\infty$. Therefore, we get that this term converges to
zero when $M$ goes to infinite. It remains to prove that the second term
vanishes for $M$ large enough. First, choosing a large $M$ such that
$k=k(\tau,\omega,M)>\delta_{0}$ ($\delta_{0}$ is as in (3.5)) and taking
$\varsigma\in(0,1)$, we have

$\displaystyle\int\limits_{\tau-1}^{\tau}e^{k(s-\tau)}\|g(s,.)\|^{2}ds$
$\displaystyle=\int\limits_{\tau-1}^{\tau-\varsigma}e^{k(s-\tau)}\|g(s,.)\|^{2}ds+\int\limits_{\tau-\varsigma}^{\tau}e^{k(s-\tau)}\|g(s,.)\|^{2}ds$
$\displaystyle=e^{-k\tau}\int\limits_{\tau-1}^{\tau-\varsigma}e^{(k-\delta_{0})s}e^{\delta
s}\|g(s,.)\|^{2}ds+e^{-k\tau}\int\limits_{\tau-\varsigma}^{\tau}e^{ks}\|g(s,.)\|^{2}ds$
$\displaystyle\leq
e^{-k\varsigma}e^{\delta_{0}(\varsigma-\tau)}\int\limits_{-\infty}^{\tau}e^{\delta_{0}s}\|g(s,.)\|^{2}ds+\int\limits_{\tau-\varsigma}^{\tau}\|g(s,.)\|^{2}ds.$
(4.33)

By (3.5), the first term above vanishes as $k\rightarrow+\infty$, and by $g\in
L^{2}_{loc}(\mathbb{R},L^{2}(\mathbb{R}^{N}))$ we can choose $\varsigma$ small
enough such that the second term in (4.2) is small. In terms of these
arguments, from (4.2)and (4.2) we have proved that

$\displaystyle\sup_{t\geq
T}\sup_{\varepsilon\in(0,a]}\int\limits_{\mathbb{R}^{N}}\Big{(}u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{\tau-t})-M\Big{)}_{+}^{p}dx\rightarrow
0,$ (4.34)

when $M\rightarrow+\infty$. Therefore, for any $\eta>0$, there exists
$M_{1}=M_{1}(\tau,\omega,\eta,D)>1$ large enough such that

$\displaystyle\sup_{t\geq
T}\sup_{\varepsilon\in(0,a]}\int\limits_{\mathbb{R}^{N}}\Big{(}u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{\tau-t})-M_{1}\Big{)}_{+}^{p}dx$
$\displaystyle\leq e^{-ap|\omega(-\tau)|}\frac{\eta}{2^{p+1}}.$ (4.35)

If $u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{\tau-t})\geq 2M_{1}$, then
$u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{\tau-t})-M_{1}\geq\frac{u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{\tau-t})}{2},$
so by (4.35) it infer us that

$\displaystyle\ \ \ \sup_{t\geq
T}\sup_{\varepsilon\in(0,a]}\int\limits_{\mathbb{R}^{N}(u(\tau)\geq
2M_{1})}|{u}(\tau,\tau-t,\vartheta_{-\tau}\omega,{u}_{\tau-t})|^{p}dx\leq
e^{-ap|\omega(-\tau)|}\frac{\eta}{2}.$ (4.36)

We see from (3.7) that
$\tilde{u}(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{\tau-t})=z(-\tau,\omega)u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{\tau-t})$.
Then in terms of the fact that $e^{-a|\omega(-\tau)|}\leq
z(-\tau,\omega)=e^{-\varepsilon\omega(-\tau)}\leq e^{a|\omega(-\tau)|}$ for
all $\varepsilon\in(0,a]$, it induce that
$\mathbb{R}^{N}(\tilde{u}(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{\tau-t})\geq
2M_{1}e^{a|\omega(-\tau)|})\subseteq\mathbb{R}^{N}(u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{\tau-t})\geq
2M_{1}).$ This along with (4.36) implies that,

$\displaystyle\sup_{t\geq
T}\sup_{\varepsilon\in(0,a]}\int\limits_{\mathbb{R}^{N}(\tilde{u}(\tau)\geq
2M_{1}e^{a|\omega(-\tau)|})}|\tilde{u}(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{\tau-t})|^{p}dx$
$\displaystyle\ \ \ \ \ \ \leq\sup_{t\geq
T}\sup_{\varepsilon\in(0,a]}e^{ap|\omega(-\tau)|}\int\limits_{\mathbb{R}^{N}(u(\tau)\geq
2M_{1})}|u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{\tau-t})|^{p}dx\leq\frac{\eta}{2}.$
(4.37)

Similarly, we can deduce that there exists $M_{2}=M_{2}(\tau,\omega,\eta,D)>0$
large enough such that

$\displaystyle\sup_{t\geq
T}\sup_{\varepsilon\in(0,1]}\int\limits_{\mathbb{R}^{N}(\tilde{u}(\tau)\leq-2M_{2}e^{a|\omega(-\tau)|})}$
$\displaystyle|\tilde{u}(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{\tau-t})|^{p}dx\leq\frac{\eta}{2}.$
(4.38)

Put $M=\max\\{M_{1},M_{2}\\}\times e^{a|\omega(-\tau)|}$. Then (4.2) and
(4.38) together imply the desired. $\ \ \ \ \ \ \ \ \ \Box$

Lemma 4.5. _Assume that ( 3.1)-(3.5) hold. Then for every
$\tau\in\mathbb{R},\omega\in\Omega$,
$\\{\tilde{u}_{\varepsilon_{n}}(\tau,\tau-
t_{n},\vartheta_{-t_{n}}\omega,\tilde{u}_{0,n})\\}$ has a convergent
subsequence in $L^{p}(\mathbb{R}^{N})$ whenever $\varepsilon_{n}\in(0,a]$,
$t_{n}\rightarrow+\infty$ and $(\tilde{u}_{0,n},\tilde{v}_{0,n})\in D(\tau-
t_{n},\vartheta_{-t_{n}}\omega)\in\mathcal{D}_{\delta}$._

_Proof._ Denote by $\tilde{u}_{n}(\tau)=\tilde{u}_{\varepsilon_{n}}(\tau,\tau-
t_{n},\vartheta_{-\tau}\omega,\tilde{u}_{0,n})$. From Lemma 4.3, for any
$\eta>0$, there exist random constants $M=M(\tau,\omega,\eta,D)>1$ and
$\mathcal{Z}_{1}=\mathcal{Z}_{1}(\tau,\omega,D)\in\mathbb{Z}^{+}$ such that
the solution $\tilde{u}_{n}(\tau)$ satisfies that for all
$n\geq\mathcal{Z}_{1}$,

$\displaystyle\sup_{\varepsilon_{n}\in(0,a]}\int\limits_{\mathbb{R}^{N}(|\tilde{u}_{n}(\tau)|\geq
M)}|\tilde{u}_{n}(\tau)|^{p}dx\leq\frac{\eta^{p}}{2^{p+2}}.$ (4.39)

On the other hand, Lemma 4.2 also implies that there exists a
$\mathcal{Z}_{2}=\mathcal{Z}_{1}(\tau,\omega,B)\in\mathbb{Z}^{+}$ such that
for all $n,m\geq\mathcal{Z}_{2}$,

$\displaystyle\int\limits_{\mathbb{R}^{N}}|\tilde{u}_{n}(\tau)-\tilde{u}_{m}(\tau)|^{2}dx\leq\frac{1}{(2M)^{p-2}}\frac{\eta^{p}}{4},$
(4.40)

whenever $\varepsilon_{n},\varepsilon_{m}\in(0,a]$. Here $M$ is as in (4.39).
We then decompose the entire space $\mathbb{R}^{N}$ by
$\mathbb{R}^{N}=\mathcal{O}_{1}\cup\mathcal{O}_{2}\cup\mathcal{O}_{3}\cup\mathcal{O}_{4}$,
where

$\displaystyle\mathcal{O}_{1}=\mathbb{R}^{N}(|\tilde{u}_{n}(\tau)|\leq
M)\cap\mathcal{O}^{N}(|\tilde{u}_{m}(\tau)|\leq M);\ \
\mathcal{O}_{2}=\mathbb{R}^{N}(|u_{n}(\tau)|\geq
M)\cap\mathbb{R}^{N}(|\tilde{u}_{m}(\tau)|\leq M);$
$\displaystyle\mathcal{O}_{3}=\mathbb{R}^{N}(|\tilde{u}_{n}(\tau)|\leq
M)\cap\mathbb{R}^{N}(|\tilde{u}_{m}(\tau)|\geq M);\ \
\mathcal{O}_{4}=\mathbb{R}^{N}(|\tilde{u}_{n}(\tau)|\geq
M)\cap\mathbb{R}^{N}(|\tilde{u}_{m}(\tau)|\geq M).$

We now put $\mathcal{Z}=\max\\{\mathcal{Z}_{1},\mathcal{Z}_{2}\\}$. Then for
all $n,m\geq\mathcal{Z}$, (4.39) and (4.40) hold true. By (4.40), we have

$\displaystyle\int\limits_{\mathcal{O}_{1}}|\tilde{u}_{n}(\tau)-\tilde{u}_{m}(\tau)|^{p}dx$
$\displaystyle\leq\int\limits_{\mathbb{R}^{N}(|\tilde{u}_{n}(\tau)-\tilde{u}_{m}(\tau)|\leq
2M)}|\tilde{u}_{n}(\tau)-\tilde{u}_{m}(\tau)|^{p}dx$
$\displaystyle\leq(2M)^{p-2}\|\tilde{u}_{n}(\tau)-\tilde{u}_{m}(\tau)\|^{2}\leq(2M)^{p-2}.(2M)^{2-p}(\frac{\eta^{p}}{4})=\frac{\eta^{p}}{4}.$
(4.41)

On the other hand, according to (4.39),

$\displaystyle\int\limits_{\mathcal{O}_{2}}|\tilde{u}_{n}(\tau)-\tilde{u}_{m}(\tau)|^{p}dx$
$\displaystyle\leq 2^{p}\int\limits_{\mathbb{R}^{N}(|\tilde{u}_{n}(\tau)|\geq
M)}|\tilde{u}_{n}(\tau)|^{p}dx\leq\frac{\eta^{p}}{4},$ (4.42)
$\displaystyle\int\limits_{\mathcal{O}_{3}}|\tilde{u}_{n}(\tau)-\tilde{u}_{m}(\tau)|^{p}dx$
$\displaystyle\leq 2^{p}\int\limits_{\mathbb{R}^{N}(|\tilde{u}_{m}(\tau)|\geq
M)}|\tilde{u}_{m}(\tau)|^{p}dx\leq\frac{\eta^{p}}{4},$ (4.43)
$\displaystyle\int\limits_{\mathcal{O}_{4}}|\tilde{u}_{n}(\tau)-\tilde{u}_{m}(\tau)|^{p}dx$
$\displaystyle\leq 2^{p-1}\Big{(}\int\limits_{\mathbb{R}^{N}(|u(\tau)|\geq
M)}|\tilde{u}_{n}(\tau)|^{p}dx+\int\limits_{\mathbb{R}^{N}(|\tilde{u}_{m}(\tau)|\geq
M)}|\tilde{u}_{m}(\tau)|^{p}dx\Big{)}\leq\frac{\eta^{p}}{4}.$ (4.44)

It follows form (4.2)-(4.44) that

$\|\tilde{u}_{n}(\tau)-\tilde{u}_{m}(\tau)\|_{p}\leq\eta\ \ \ \mbox{for all}\
n,m\geq\mathcal{Z},$

whenever $\varepsilon_{n},\varepsilon_{m}\in(0,a]$, which shows that
$\\{\tilde{u}_{n}(\tau)\\}$ also has a convergent subsequence in
$L^{p}(\mathbb{R}^{N})$. Then the proof is concluded. $\ \ \ \ \ \ \ \ \ \Box$

By Lemma 4.2 and Lemma 4.5 we immediately have

Lemma 4.6. _Assume that ( 3.1)-(3.5) hold. Then the family of random cocycles
$\varphi_{\varepsilon}$ defined by (3) is uniformly pullback asymptotically
compact over $\varepsilon\in(0,a]$ in $L^{p}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})$. In particular, each $\varphi_{\varepsilon}$ has a
unique $(L^{2}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N}),L^{p}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N}))$-pullback attractor $\mathcal{A}_{\varepsilon}$_

_Proof_ The uniformly pullback asymptotic compactness is followed from Lemma
4.2 and Lemma 4.4, and the existence and uniqueness of bi-spatial pullback
attractor are from Theorem 2.9, or Theorem 2.10. $\ \ \ \ \ \ \ \ \ \Box$

###  4.3. Convergence of the family $\varphi_{\varepsilon}$ on $(0,a]$ in
$L^{2}\times L^{2}$

This subsection deal with the convergence of solutions at any intension
$\varepsilon$ of noise. The convergence at zero has been shown by [1]. Here we
need to prove it also converges at any $\varepsilon>0$. To this end, the
following assumption on the nonlinearity $f$ as in [2] is also required. That
is, for all $x\in\mathbb{R}^{N}$ and $s\in\mathbb{R}$,

$\displaystyle\Big{|}\frac{\partial}{\partial
s}(x,s)\Big{|}\leq\alpha_{4}|s|^{p-2}+\psi_{4}(x),$ (4.45)

where $\alpha_{4}>0$, $\psi_{4}\in L^{\infty}(\mathbb{R}^{N})$ if $p=2$ and
$\psi_{4}\in L^{\frac{p}{p-2}}(\mathbb{R}^{N})$ if $p>2$. We need further to
assume that $\psi_{2}\in L^{q}(\mathbb{R}^{N})$, where $\psi_{2}$ is as in
(3.2) and $q=\frac{p}{p-1}$ is conjugation of $p$.

To begin with, from (4.1), it is very easy to derive the following inequality.

Lemma 4.7. _Assume that ( 3.1)-(3.5) hold. Then for each $\tau\in\mathbb{R}$,
$\omega\in\Omega$ and
$(\tilde{u}_{\varepsilon}(\tau),\tilde{v}_{\varepsilon}(\tau))\in
L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$, the solution $({u},{v})$
of problem (3.8) satisfies for all $t\geq\tau$,_

$\displaystyle\|u_{\varepsilon}(t,\tau,\omega,u(\tau))\|^{2}+\|v_{\varepsilon}(t,\tau,\omega,v(\tau))\|^{2}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \
+\int\limits_{\tau}^{t}e^{\delta(s-t)}\Big{(}\|v_{\varepsilon}(s,\tau,\omega,v_{\varepsilon}(\tau))\|^{2}+z_{\varepsilon}^{2}(s,\omega)\|\tilde{u}_{\varepsilon}(s,\tau,\omega,\tilde{u}_{\varepsilon}(\tau))\|^{p}_{p}\Big{)}ds$
$\displaystyle\leq
z_{\varepsilon}^{2}(\tau,\omega)(\|\tilde{u}_{\varepsilon}(\tau)\|^{2}+\|\tilde{v}_{\varepsilon}(\tau)\|^{2})+c\int\limits_{\tau}^{t}z_{\varepsilon}^{2}(s,\omega)(\|g(s,.)\|^{2}+\|h(s,.)\|^{2}+1)ds.$

Then by applying Lemma 4.7 we have

Lemma 4.8. _Assume that ( 3.1)-(3.5) and (4.45) hold. Let
$(\tilde{u}_{\varepsilon},\tilde{v}_{\varepsilon})$ be the solution of problem
(3.8)-(3.9) with initial data
$(\tilde{u}_{\varepsilon,\tau},\tilde{v}_{\varepsilon,\tau})$. Assume that
$\varepsilon\rightarrow\varepsilon_{0}$ and
$\|(\tilde{u}_{\varepsilon,\tau},\tilde{v}_{\varepsilon,\tau})-(\tilde{u}_{\varepsilon_{0},\tau},\tilde{v}_{\varepsilon_{0},\tau})\|\rightarrow
0$ for $\varepsilon,\varepsilon_{0}\in(0,a]$. Then for each
$\tau\in\mathbb{R}$, $\omega\in\Omega,T>0$ and every $t\in[\tau,\tau+T]$,_

$\displaystyle\lim\limits_{\varepsilon\rightarrow\varepsilon_{0}}\|(\tilde{u}_{\varepsilon}(t,\tau,\omega,\tilde{u}_{\varepsilon,\tau}),\tilde{v}_{\varepsilon}(t,\tau,\omega,\tilde{v}_{\varepsilon,\tau}))-(\tilde{u}_{\varepsilon_{0}}(t,\tau,\omega,\tilde{u}_{\varepsilon_{0},\tau}),\tilde{v}_{\varepsilon_{0}}(t,\tau,\omega,\tilde{v}_{\varepsilon_{0},\tau}))\|=0.$
(4.46)

_In particular, let $(\tilde{u},\tilde{v})$ be the solution of problem
(3.8)-(3.9) for $\varepsilon=0$ with initial data
$(\tilde{u}_{\tau},\tilde{v}_{\tau})$. Assume that $\varepsilon\rightarrow{0}$
and
$\|(\tilde{u}_{\varepsilon,\tau},\tilde{v}_{\varepsilon,\tau}))-(\tilde{u}_{\tau},\tilde{v}_{\tau})\|\rightarrow
0$. Then for each $\tau\in\mathbb{R}$, $\omega\in\Omega,T>0$
_

$\displaystyle\lim\limits_{\varepsilon\rightarrow{0}}\|(\tilde{u}_{\varepsilon}(t,\tau,\omega,\tilde{u}_{\varepsilon,\tau}),\tilde{v}_{\varepsilon}(t,\tau,\omega,\tilde{v}_{\varepsilon,\tau}))-(\tilde{u}(t,\tau,\tilde{u}_{\tau}),\tilde{v}(t,\tau,\tilde{v}_{\tau}))\|=0.$
(4.47)

_Proof_ Put
$U=U(t)=u_{\varepsilon}(t,\tau,\omega,u_{\varepsilon,\tau})-u_{\varepsilon_{0}}(t,\tau,\omega,u_{\varepsilon_{0},\tau})$
and
$V=V(t)=v_{\varepsilon}(t,\tau,\omega,v_{\varepsilon,\tau})-v_{\varepsilon_{0}}(t,\tau,\omega,v_{\varepsilon_{0},\tau})$.
Then we get the following system:

$\begin{cases}\frac{dU}{dt}+\lambda U-\Delta U+\alpha
V=e^{-\varepsilon\omega(t)}f(x,e^{\varepsilon\omega(t)}u_{\varepsilon})-e^{-\varepsilon_{0}\omega(t)}f(x,e^{\varepsilon_{0}\omega(t)}u_{\varepsilon_{0}})\\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+(e^{-\varepsilon\omega(t)}-e^{-\varepsilon_{0}\omega(t)})g(t,x),\\\
\frac{dV}{dt}+\sigma V-\beta
U=(e^{-\varepsilon\omega(t)}-e^{-\varepsilon_{0}\omega(t)})h(t,x),\end{cases}$
(4.48)

where
$u_{\varepsilon}=u_{\varepsilon}(t)=u_{\varepsilon}(t,\tau,\omega,u_{\varepsilon,\tau})$.
Let $\eta$ be a small positive number. Since $\omega$ is continuous on
$\mathbb{R}$, then there exists a $\chi=\chi(\tau,\omega,\eta,T)>0$ such that
for every
$\varepsilon\in(\varepsilon_{0}-\chi,\varepsilon_{0}+\chi)\subset(0,a]$ and
$t\in[\tau,\tau+T]$,

$\displaystyle|e^{\varepsilon\omega(t)}-e^{\varepsilon_{0}\omega(t)}|+|e^{-\varepsilon\omega(t)}-e^{-\varepsilon_{0}\omega(t)}|\leq\eta.$
(4.49)

By (4.48), we deduce that

$\displaystyle\frac{1}{2}\frac{d}{dt}(\beta\|U\|^{2}+\alpha\|V\|^{2})+\lambda\beta\|U\|^{2}+\sigma\alpha\|V\|^{2}$
$\displaystyle\leq\int\limits_{\mathbb{R}^{N}}\Big{(}e^{-\varepsilon\omega(t)}f(x,e^{\varepsilon\omega(t)}u_{\varepsilon})-e^{-\varepsilon_{0}\omega(t)}f(x,e^{\varepsilon_{0}\omega(t)}u_{\varepsilon_{0}})\Big{)}Udx$
$\displaystyle+(e^{-\varepsilon\omega(t)}-e^{-\varepsilon_{0}\omega(t)})\int\limits_{\mathbb{R}^{N}}g(t,x)Udx+(e^{-\varepsilon\omega(t)}-e^{-\varepsilon_{0}\omega(t)})\int\limits_{\mathbb{R}^{N}}h(t,x)Vdx.$
(4.50)

The first term on the right hand side of (4.3) is rewritten as

$\displaystyle\int\limits_{\mathbb{R}^{N}}\Big{(}e^{-\varepsilon\omega(t)}f(x,e^{\varepsilon\omega(t)}u_{\varepsilon})-e^{-\varepsilon_{0}\omega(t)}f(x,e^{\varepsilon_{0}\omega(t)}u_{\varepsilon_{0}})\Big{)}Udx$
$\displaystyle=e^{-\varepsilon\omega(t)}\int\limits_{\mathbb{R}^{N}}\Big{(}f(x,e^{\varepsilon\omega(t)}u_{\varepsilon})-f(x,e^{\varepsilon_{0}\omega(t)}u_{\varepsilon_{0}})\Big{)}Udx+(e^{-\varepsilon\omega(t)}-e^{-\varepsilon_{0}\omega(t)})\int\limits_{\mathbb{R}^{N}}f(x,e^{\varepsilon_{0}\omega(t)}u_{\varepsilon_{0}})Udx$
$\displaystyle\ \ \ \ \ \
=e^{-\varepsilon\omega(t)}\int\limits_{\mathbb{R}^{N}}\frac{\partial}{\partial
s}f(x,s)(e^{\varepsilon\omega(t)}u_{\varepsilon}-e^{\varepsilon_{0}\omega(t)}u_{\varepsilon_{0}})Udx+(e^{-\varepsilon\omega(t)}-e^{-\varepsilon_{0}\omega(t)})\int\limits_{\mathbb{R}^{N}}f(x,e^{\varepsilon_{0}\omega(t)}u_{\varepsilon_{0}})Udx$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
=\int\limits_{\mathbb{R}^{N}}\frac{\partial}{\partial
s}f(x,s)U^{2}dx+(e^{-\varepsilon_{0}\omega(t)}-e^{-\varepsilon\omega(t)})\int\limits_{\mathbb{R}^{N}}\frac{\partial}{\partial
s}f(x,s)\tilde{u}_{\varepsilon_{0}}Udx$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+(e^{-\varepsilon\omega(t)}-e^{-\varepsilon_{0}\omega(t)})\int\limits_{\mathbb{R}^{N}}f(x,e^{\varepsilon_{0}\omega(t)}u_{\varepsilon_{0}})Udx.$
(4.51)

By (4.45) and (4.49), the second term on the right hand side of (4.3) is
bounded by

$\displaystyle(e^{-\varepsilon_{0}\omega(t)}-e^{-\varepsilon\omega(t)})\int\limits_{\mathbb{R}^{N}}\frac{\partial}{\partial
s}f(x,s)\tilde{u}_{\varepsilon_{0}}Udx$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\leq|e^{-\varepsilon_{0}\omega(t)}-e^{-\varepsilon\omega(t)}|\int\limits_{\mathbb{R}^{N}}\Big{(}\alpha_{4}|\tilde{u}_{\varepsilon}+\tilde{u}_{\varepsilon_{0}}|^{p-2}|\tilde{u}_{\varepsilon_{0}}||U|+|\tilde{u}_{\varepsilon_{0}}||U|\psi_{4}|\Big{)}dx$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \leq
c\eta\Big{(}\|\tilde{u}_{\varepsilon}\|_{p}^{p}+\|\tilde{u}_{\varepsilon_{0}}\|_{p}^{p}+\|U\|^{p}_{p}+\|\psi_{4}\|_{\frac{p}{p-2}}^{\frac{p}{p-2}}\Big{)}.$
(4.52)

By (3.2) and (4.49),connection with $\psi_{2}\in L^{q}$, the third term on the
right hand side of (4.3) is bounded by

$\displaystyle(e^{-\varepsilon\omega(t)}-e^{-\varepsilon_{0}\omega(t)})\int\limits_{\mathbb{R}^{N}}f(x,\tilde{u}_{\varepsilon_{0}})Udx$
$\displaystyle\leq|e^{-\varepsilon\omega(t)}-e^{-\varepsilon_{0}\omega(t)}|\int\limits_{\mathbb{R}^{N}}(\varepsilon_{2}|\tilde{u}_{\varepsilon_{0}}|^{p-1}+\psi_{2})|U|dx$
$\displaystyle\leq
c\eta\Big{(}\|\tilde{u}_{\varepsilon_{0}}\|_{p}^{p}+\|U\|_{p}^{p}+\|\psi_{2}\|_{q}^{q}\Big{)},$
(4.53)

where $q=\frac{p}{p-1}$. Then combination (4.3)-(4.3), we find that for every
$\varepsilon\in(\varepsilon_{0}-\chi,\varepsilon_{0}+\chi)$ and
$t\in[\tau,\tau+T]$,

$\displaystyle\ \ \ \ \ \
\int\limits_{\mathbb{R}^{N}}\Big{(}e^{-\varepsilon\omega(t)}f(x,\tilde{u}_{\varepsilon})-e^{-\varepsilon_{0}\omega(t)}f(x,\tilde{u}_{\varepsilon_{0}})\Big{)}Udx$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\leq\alpha_{3}\|U\|^{2}+c\eta+c\eta\Big{(}\|\tilde{u}_{\varepsilon}\|_{p}^{p}+\|\tilde{u}_{\varepsilon_{0}}\|_{p}^{p}+\|U\|^{p}_{p}\Big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\leq\alpha_{3}\|U\|^{2}+c_{0}\eta\Big{(}\|\tilde{u}_{\varepsilon}\|_{p}^{p}+\|\tilde{u}_{\varepsilon_{0}}\|_{p}^{p}+\|U\|^{p}_{p}\Big{)}.$
(4.54)

For the last two terms on the right hand side of (4.3), by (4.49), we have for
every $\varepsilon\in(\varepsilon_{0}-\chi,\varepsilon_{0}+\chi)$ and
$t\in[\tau,\tau+T]$,

$\displaystyle(e^{-\varepsilon\omega(t)}-e^{-\varepsilon_{0}\omega(t)})\int\limits_{\mathbb{R}^{N}}g(t,x)Udx\leq\eta\|U\|^{2}+\eta\|g(t,.)\|^{2},$
(4.55)
$\displaystyle(e^{-\varepsilon\omega(t)}-e^{-\varepsilon_{0}\omega(t)})\int\limits_{\mathbb{R}^{N}}h(t,x)Vdx\leq\eta\|V\|^{2}+\eta\|h(t,.)\|^{2}.$
(4.56)

Then by (4.3) and (4.3)-(4.56), we get that for every
$\varepsilon\in(\varepsilon_{0}-\chi,\varepsilon_{0}+\chi)$ and
$t\in[\tau,\tau+T]$,

$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\frac{d}{dt}(\beta\|U\|^{2}+\alpha\|V\|^{2})\leq
c_{1}(\beta\|U\|^{2}+\alpha\|V\|^{2})$
$\displaystyle+c_{2}\eta\Big{(}\|\tilde{u}_{\varepsilon}\|_{p}^{p}+\|\tilde{u}_{\varepsilon_{0}}\|_{p}^{p}+\|u_{\varepsilon_{0}}\|_{p}^{p}+\||u_{\varepsilon}\|^{p}_{p}+\|g(t,.)\|^{2}+\|h(t,.)\|^{2}\Big{)},$
(4.57)

where $c_{1}$ and $c_{2}$ are positive constants independent of $\tau,\omega$
and $\varepsilon$. By (4.3) we immediately have for every
$\varepsilon\in(\varepsilon_{0}-\chi,\varepsilon_{0}+\chi)$ and
$t\in[\tau,\tau+T]$,

$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\|U(t)\|^{2}+\|V(t)\|^{2}\leq
c_{3}e^{c_{1}(t-\tau)}(\|U(\tau)\|^{2}+\|V(\tau)\|^{2})$
$\displaystyle+c_{4}\eta
e^{c_{1}(t-\tau)}\int\limits_{\tau}^{t}\Big{(}\|\tilde{u}_{\varepsilon}(s)\|_{p}^{p}+\|\tilde{u}_{\varepsilon_{0}}(s)\|_{p}^{p}+\||u_{\varepsilon}(s)\|^{p}_{p}+\|u_{\varepsilon_{0}}(s)\|_{p}^{p}+\|g(s,.)\|^{2}+\|h(s,.)\|^{2}\Big{)}ds.$
(4.58)

Since $e^{-\varepsilon\omega(s)}$ is continuous on $\mathbb{R}$, then for
every fixed $\tau\in\mathbb{R}$ and $\omega\in\Omega$, and
$s\in[\tau,\tau+T]$, there exist $\mu=\mu(\tau,\omega,T)$ and
$\nu=\nu(\tau,\omega,T)$ such that for all $\varepsilon\in(0,a]$, $\mu\leq
z_{\varepsilon}(s,\omega)\leq\nu$ for all $s\in[\tau,\tau+T]$. Therefor by
Lemma 4.7, it follows that for all $t\in[\tau,\tau+T]$,

$\displaystyle\int\limits_{\tau}^{t}\|\tilde{u}_{\varepsilon}(s,\tau,\omega,u_{\varepsilon}(\tau))\|^{p}_{p}ds$
$\displaystyle\leq\mu^{-2}e^{\delta(t-\tau)}\int\limits_{\tau}^{t}e^{\delta(s-t)}z_{\varepsilon}^{2}(s,\omega)\|\tilde{u}_{\varepsilon}(s,\tau,\omega,u_{\varepsilon}(\tau))\|^{p}_{p}ds$
$\displaystyle\leq\mu^{-2}e^{\delta(t-\tau)}\Big{(}z_{\varepsilon}^{2}(\tau,\omega)(\|\tilde{u}_{\varepsilon}(\tau)\|^{2}+\|\tilde{v}_{\varepsilon}(\tau)\|^{2})+c\nu^{2}\int\limits_{\tau}^{t}(\|g(s,.)\|^{2}+\|h(s,.)\|^{2}+1)ds\Big{)}$
$\displaystyle\leq\mu^{-2}e^{\delta(t-\tau)}\Big{(}e^{2a|\omega(\tau)|}(\|\tilde{u}_{\varepsilon}(\tau)\|^{2}+\|\tilde{v}_{\varepsilon}(\tau)\|^{2})+c\nu^{2}\int\limits_{\tau}^{t}(\|g(s,.)\|^{2}+\|h(s,.)\|^{2}+1)ds\Big{)}.$
(4.59)

By a similar technique we can calculate that for all $t\in[\tau,\tau+T]$,

$\displaystyle\int\limits_{\tau}^{t}\|u_{\varepsilon}(s,\tau,\omega,u_{\varepsilon}(\tau))\|^{p}_{p}ds$
$\displaystyle\leq\mu^{p-2}e^{\delta(t-\tau)}\Big{(}e^{2a|\omega(\tau)|}(\|\tilde{u}_{\varepsilon}(\tau)\|^{2}+\|\tilde{v}_{\varepsilon}(\tau)\|^{2})+c\nu^{2}\int\limits_{\tau}^{t}(\|g(s,.)\|^{2}+\|h(s,.)\|^{2}+1)ds\Big{)}.$
(4.60)

Then by (4.3-(4.3) it gives that

$\displaystyle\ \ \ \ \ \ \ \ \ \ \|U(t)\|^{2}+\|V(t)\|^{2}\leq
c_{3}e^{c_{1}(t-\tau)}(\|U(\tau)\|^{2}+\|V(\tau)\|^{2})$
$\displaystyle+c_{5}\eta
e^{c_{6}(t-\tau)}\Big{(}e^{2a|\omega(\tau)|}(\mu^{-2}+\mu^{p-2})(\|\tilde{u}_{\varepsilon}(\tau)\|^{2}+\|\tilde{u}_{\varepsilon_{0},\tau}\|^{2}+\|\tilde{v}_{\varepsilon}(\tau)\|^{2}+\|\tilde{v}_{\varepsilon_{0},\tau}\|^{2})$
$\displaystyle\ \ \ \ \ \ \ \
+\nu^{2}(\mu^{-2}+\mu^{p-2})\int\limits_{\tau}^{t}(\|g(s,.)\|^{2}+\|h(s,.)\|^{2}+1)ds+\int\limits_{\tau}^{t}(\|g(s,.)\|^{2}+\|h(s,.)\|^{2})ds\Big{)}.$
(4.61)

On the other hand, by (4.49) it follows that for every
$\varepsilon\in(\varepsilon_{0}-\chi,\varepsilon_{0}+\chi)\subset(0,a]$,

$\displaystyle\|U(\tau)\|^{2}$
$\displaystyle=\|e^{-\varepsilon\omega(\tau)}\tilde{u}_{\varepsilon}(\tau)-e^{-\varepsilon_{0}\omega(\tau)}\tilde{u}_{\varepsilon_{0},\tau}\|^{2}$
$\displaystyle\leq
2e^{-2\varepsilon\omega(\tau)}\|\tilde{u}_{\varepsilon}(\tau)-\tilde{u}_{\varepsilon_{0},\tau}\|^{2}+2|e^{-\varepsilon\omega(\tau)}-e^{-\varepsilon_{0}\omega(\tau)}|^{2}\|\tilde{u}_{\varepsilon_{0},\tau}\|^{2}$
$\displaystyle\leq
2e^{2a|\omega(\tau)|}\|\tilde{u}_{\varepsilon}(\tau)-\tilde{u}_{\varepsilon_{0},\tau}\|^{2}+2\eta^{2}\|\tilde{u}_{\varepsilon_{0},\tau}\|^{2}.$
(4.62)

Similarly,

$\displaystyle\|V(\tau)\|^{2}\leq
2e^{2a|\omega(\tau)|}\|\tilde{v}_{\varepsilon}(\tau)-\tilde{v}_{\varepsilon_{0},\tau}\|^{2}+2\eta^{2}\|\tilde{v}_{\varepsilon_{0},\tau}\|^{2}.$
(4.63)

We now let $\varepsilon\rightarrow\varepsilon_{0}$ and
$\|u_{\varepsilon,\tau}-u_{\varepsilon_{0},\tau}\|\rightarrow 0$. Then by
(4.3)-(4.63) we obtain that for all $t\in[\tau,\tau+T]$,

$\displaystyle\|U(t)\|^{2}+\|V(t)\|^{2}$
$\displaystyle=\|u_{\varepsilon}(t,\tau,\omega,u_{\varepsilon,\tau})-u_{\varepsilon_{0}}(t,\tau,\omega,u_{\varepsilon_{0},\tau})\|^{2}$
$\displaystyle+\|v_{\varepsilon}(t,\tau,\omega,v_{\varepsilon,\tau})-v_{\varepsilon_{0}}(t,\tau,\omega,v_{\varepsilon_{0},\tau})\|^{2}\rightarrow
0.$ (4.64)

Notice that by (4.49) we also have for every
$\varepsilon\in(\varepsilon_{0}-\chi,\varepsilon_{0}+\chi)$ and
$t\in[\tau,\tau+T]$,

$\displaystyle\|\tilde{u}_{\varepsilon}(t,\tau,\omega,\tilde{u}_{\varepsilon}(\tau))-\tilde{u}_{\varepsilon_{0}}(t,\tau,\omega,\tilde{u}_{\varepsilon_{0},\tau})\|^{2}$
$\displaystyle\leq
2e^{2a|\omega(t)|}\|u_{\varepsilon}(t)-u_{\varepsilon_{0}}(t)\|^{2}+2\eta^{2}\|u_{\varepsilon_{0}}(t)\|^{2},$
(4.65)
$\displaystyle\|\tilde{v}_{\varepsilon}(t,\tau,\omega,\tilde{v}_{\varepsilon}(\tau))-\tilde{v}_{\varepsilon_{0}}(t,\tau,\omega,\tilde{v}_{\varepsilon_{0},\tau})\|^{2}$
$\displaystyle\leq
2e^{2a|\omega(t)|}\|v_{\varepsilon}(t)-v_{\varepsilon_{0}}(t)\|^{2}+2\eta^{2}\|v_{\varepsilon_{0}}(t)\|^{2}.$
(4.66)

Then by (4.3)-(4.66) we get (4.46). Repeating the same arguments we can derive
(4.47). $\ \ \ \ \ \ \ \ \ \Box$

###  4.4. Main results

We are now at the point to present the main results in this paper.

Theorem 4.9. _Suppose $\varepsilon\in\mathbb{R}$ and (3.1)-(3.5) hold true.
Then_

_(i) Each random cocycle $\varphi_{\varepsilon}$ generated by (1.1)-(1.3) has
a unique pullback attractor $\mathcal{A}_{\varepsilon}$ and the corresponding
deterministic cocycle $\varphi_{0}$ has a unique pullback attractor
$\mathcal{A}_{0}$ in $L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$, both
$\mathcal{A}_{\varepsilon}$ and $\mathcal{A}_{0}$ are
$(L^{2}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N}),L^{l}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N}))$-pullback attractors._

_(ii) If further ( 4.45) holds true, then the family
$\mathcal{A}_{\varepsilon}$ is upper semi-continuous under the Hausdorff semi-
distance of $L^{l}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$ at any
$\varepsilon_{0}\in\mathbb{R}$. Here $l\in(2,p],p>2$._

_Proof_ Let $X=L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$ and
$L^{l}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$. Then it is known that the
hypothesises (_H1_)) and (_H2_)) (see Lemma 2.7 in [34]) hold true. By the
Sobolev interpolation and association with Lemma 4.2 and Lemma 4.6, we
immediately obtain the uniformly pullback asymptotic compactness in
$L^{l}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$ for any $2<l<p$. Then
along with uniform absorption (see Lemma 4.1) and convergence property (see
Lemma 4.8), all conditions of Theorem 2.10 are satisfied. $\ \ \ \ \ \ \ \ \
\Box$

##  5\. Existence of random equilibria for the generated random cocycle

Random equilibrium is a special case of omega-limit sets. We can refer to [3,
6] for the definitions and applications to monotone random dynamical system.
The problem of the construction of equilibria for a general random dynamical
system is rather complicate [6]. Recently, Gu [13] proved that the stochastic
FitzHugh-Nagumo lattice equations driven by fractional Brownian motions
possess a unique equilibrium. However, we here introduce the random
equilibrium in the case of non-autonomous stochastic dynamical system.
Specifically, we have

Definition 5.1. _Let $(Q,\\{\sigma_{t}\\}_{\mathbb{R}})$ and
$(\Omega,\mathcal{F},{P},\\{\vartheta_{t}\\}_{t\in\mathbb{R}})$ be parametric
dynamical systems. A random variable $u^{*}:Q\times\Omega\mapsto X$ is said to
be an equilibrium (or fixed point, or stationary solution) of random cocycle
$\varphi$ if it is invariant under $\varphi$, i.e., if_

$\varphi(t,q,\omega,u^{*}(q,\omega))=u^{*}(\sigma_{t}q,\vartheta_{t}\omega)\ \
for\ all\ t\geq 0,\ q\in Q,\ \omega\in\Omega.$

In this section, the parametric dynamical systems
$(Q,\\{\sigma_{t}\\}_{\mathbb{R}})$ and
$(\Omega,\mathcal{F},{P},\\{\vartheta_{t}\\}_{t\in\mathbb{R}})$ are same as in
section 3. We will prove the existence of equilibrium for problem (1.1)-(1.3)
on the whole space $\mathbb{R}^{N}$. To this end, we need to assume that

$\displaystyle\delta=\min\\{\lambda,\sigma\\}>\alpha_{3},\ \ \ \beta\geq 1,$
(5.1)

where $\alpha_{3}$ is as in (3.3) and $\lambda,\sigma,\beta$ are as in the
FitzHugh-Nagumo system (1.1)-(1.2).

For convenience, here we write $\varepsilon=1$. First, we have

Lemma 5.2. _Suppose that $g$ and $h$ satisfy (3.5) and $f$ satisfies (3.1) and
(3.3) such that (5.1) holds. Then there exists a positive constant
$0<b_{0}<\delta-\alpha_{3}$ such that the solutions of problem (1.1)-(1.3)
with initial values $(\tilde{u}_{\tau-t_{i}},\tilde{v}_{\tau-
t_{i}})(i=1,2),t_{1}<t_{2}$ satisfy the following decay property:_

$\displaystyle\|\tilde{u}(\tau,\tau-
t_{1},\vartheta_{-\tau}\omega,\tilde{u}_{\tau-t_{1}})-\tilde{u}(\tau,\tau-
t_{2},\vartheta_{-\tau}\omega,\tilde{u}_{\tau-t_{2}})\|^{2}$
$\displaystyle+\|\tilde{v}(\tau,\tau-
t_{1},\vartheta_{-\tau}\omega,\tilde{v}_{\tau-t_{1}})-\tilde{v}(\tau,\tau-
t_{2},\vartheta_{-\tau}\omega,\tilde{v}_{\tau-t_{2}})\|^{2}$
$\displaystyle\leq
c\Big{(}e^{-b_{0}t_{1}}z^{2}(-t_{1},\omega)(\|\tilde{u}_{\tau-
t_{1}}\|^{2}+\|\tilde{v}_{\tau-
t_{1}}\|^{2})+e^{-b_{0}t_{2}}z^{2}(-t_{2},\omega)(\|\tilde{u}_{\tau-
t_{2}}\|^{2}+\|\tilde{v}_{\tau-t_{2}}\|^{2})\Big{)}$
$\displaystyle+ce^{(b_{0}-b)t_{1}}\int_{-\infty}^{0}e^{b_{0}s}z^{2}(s,\omega)(\|g(s+\tau,.)\|^{2}+\|h(s+\tau,.)\|^{2}+1)ds,$

_where $c$ is a deterministic non-random constant._

_Proof_ Put $\bar{u}=u(t,\tau-t_{1},\vartheta_{-\tau}\omega,u_{\tau-
t_{1}})-u(t,\tau-t_{2},\vartheta_{-\tau}\omega,u_{\tau-t_{2}})$ and

$\bar{v}=v(t,\tau-t_{1},\vartheta_{-\tau}\omega,v_{\tau-t_{1}})-v(t,\tau-
t_{2},\vartheta_{-\tau}\omega,v_{\tau-t_{2}}).$

Then from (3.8)-(3.9), along with (5.1), we have

$\displaystyle\frac{d}{dt}(\beta\|\bar{u}\|^{2}+\alpha\|\bar{v}\|^{2})+b(\beta\|\bar{u}\|^{2}+\alpha\|\bar{v}\|^{2})\leq
0,$ (5.2)

where $b=\delta-\alpha_{3}$. By applying Gronwall lemma to (5.2) over the
interval $[\tau-t_{1},\tau]$, we immediately get

$\displaystyle\|\bar{u}(\tau)\|^{2}+\|\bar{v}(\tau)\|^{2}$ $\displaystyle\leq
ce^{-bt_{1}}(\|u(\tau-t_{1},\tau-t_{2},\vartheta_{-\tau}\omega,u_{\tau-
t_{2}})-u_{\tau-t_{1}}\|^{2}$ $\displaystyle+\|v(\tau-t_{1},\tau-
t_{2},\vartheta_{-\tau}\omega,v_{\tau-t_{2}})-v_{\tau-t_{1}}\|^{2})$
$\displaystyle\leq ce^{-bt_{1}}(\|u(\tau-t_{1},\tau-
t_{2},\vartheta_{-\tau}\omega,u_{\tau-t_{2}})\|^{2}$ $\displaystyle+\|v(\tau-
t_{1},\tau-t_{2},\vartheta_{-\tau}\omega,v_{\tau-
t_{2}})\|^{2})+ce^{-bt_{1}}(\|u_{\tau-t_{1}}\|^{2}+\|v_{\tau-t_{1}}\|^{2}),$
(5.3)

where $c=c(\alpha,\beta)$ is a positive deterministic constant. Write

$\displaystyle 0<\delta_{0}<b_{0}<b=\delta-\alpha_{3},$ (5.4)

where $\delta_{0}$ is as in (3.5). From (4.1) and using (5.4), we have

$\displaystyle\frac{d}{dt}(\beta\|u\|^{2}+\alpha\|v\|^{2})$
$\displaystyle+b_{0}(\beta\|u\|^{2}+\alpha\|v\|^{2})\leq
cz^{2}(t,\omega)(\|g(t,.)\|^{2}+\|h(t,.)\|^{2}+1).$ (5.5)

Then by Gronwall lemma again over the interval $[\tau-t_{2},\tau-t_{1}]$, we
find that

$\displaystyle\|u(\tau-t_{1},\tau-t_{2},\vartheta_{-\tau}\omega,u_{\tau-
t_{2}})\|^{2}+\|v(\tau-t_{1},\tau-t_{2},\vartheta_{-\tau}\omega,v_{\tau-
t_{2}})\|^{2}$ $\displaystyle\ \ \ \ \ \leq ce^{b_{0}(t_{1}-t_{2})}(\|u_{\tau-
t_{2}}\|^{2}+\|v_{\tau-t_{2}}\|^{2})$ $\displaystyle\ \ \ \ \ \ \
+c\int\limits_{\tau-t_{2}}^{\tau-t_{1}}e^{-b_{0}(\tau-
t_{1}-s)}z^{2}(s,\vartheta_{-\tau}\omega)(\|g(s,.)\|^{2}+\|h(s,.)\|^{2}+1)ds$
$\displaystyle\ \ \ \ \ \ \ \ \ \leq ce^{b_{0}(t_{1}-t_{2})}(\|u_{\tau-
t_{2}}\|^{2}+\|v_{\tau-t_{2}}\|^{2})$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
+ce^{b_{0}t_{1}}e^{2\omega(-\tau)}\int\limits_{-\infty}^{0}e^{b_{0}s}z^{2}(s,\omega)(\|g(s+\tau,.)\|^{2}+\|h(s+\tau,.)\|^{2}+1)ds,$
(5.6)

where $c=c(\alpha,\beta)$ is a positive deterministic constant. Then
combination (5) and (5) we have

$\displaystyle\|\bar{u}(\tau)\|^{2}+\|\bar{v}(\tau)\|^{2}$ $\displaystyle\leq
ce^{-bt_{1}}(\|u_{\tau-t_{1}}\|^{2}+\|v_{\tau-
t_{1}}\|^{2})+ce^{(b_{0}-b)t_{1}}e^{-b_{0}t_{2}}(\|u_{\tau-
t_{2}}\|^{2}+\|v_{\tau-t_{2}}\|^{2})$
$\displaystyle+ce^{(b_{0}-b)t_{1}}e^{2\omega(-\tau)}\int_{-\infty}^{0}e^{b_{0}s}z^{2}(s,\omega)(\|g(s+\tau,.)\|^{2}+\|h(s+\tau,.)\|^{2}+1)ds$
$\displaystyle\leq ce^{-b_{0}t_{1}}(\|u_{\tau-t_{1}}\|^{2}+\|v_{\tau-
t_{1}}\|^{2})+e^{-b_{0}t_{2}}(\|u_{\tau-t_{2}}\|^{2}+\|v_{\tau-t_{2}}\|^{2})$
$\displaystyle+ce^{(b_{0}-b)t_{1}}e^{2\omega(-\tau)}\int_{-\infty}^{0}e^{b_{0}s}z^{2}(s,\omega)(\|g(s+\tau,.)\|^{2}+\|h(s+\tau,.)\|^{2}+1)ds,$
(5.7)

where we have used $e^{(b_{0}-b)t_{1}}\leq 1$ for $b_{0}<b$. In terms of the
relation (3.7), we get

$\displaystyle\|\bar{\tilde{u}}(\tau)\|^{2}+\|\bar{\tilde{v}}(\tau)\|^{2}$
$\displaystyle\leq
e^{-2\omega(-\tau)}\|\bar{u}(\tau)\|^{2}+\|\bar{v}(\tau)\|^{2}$
$\displaystyle\leq ce^{-2\omega(-\tau)}\Big{(}e^{-b_{0}t_{1}}(\|u_{\tau-
t_{1}}\|^{2}+\|v_{\tau-t_{1}}\|^{2})+e^{-b_{0}t_{2}}(\|u_{\tau-
t_{2}}\|^{2}+\|v_{\tau-t_{2}}\|^{2})\Big{)}$
$\displaystyle+ce^{(b_{0}-b)t_{1}}\int_{-\infty}^{0}e^{b_{0}s}z^{2}(s,\omega)(\|g(s+\tau,.)\|^{2}+\|h(s+\tau,.)\|^{2}+1)ds$
$\displaystyle=ce^{-2\omega(-\tau)}\Big{(}e^{-b_{0}t_{1}}z^{2}(\tau-
t_{1},\vartheta_{-\tau}\omega)(\|\tilde{u}_{\tau-
t_{1}}\|^{2}+\|\tilde{v}_{\tau-t_{1}}\|^{2})$
$\displaystyle+e^{-b_{0}t_{2}}z^{2}(\tau-
t_{2},\vartheta_{-\tau}\omega)(\|\tilde{u}_{\tau-
t_{2}}\|^{2}+\|\tilde{v}_{\tau-t_{2}}\|^{2})\Big{)}$
$\displaystyle+ce^{(b_{0}-b)t_{1}}\int_{-\infty}^{0}e^{b_{0}s}z^{2}(s,\omega)(\|g(s+\tau,.)\|^{2}+\|h(s+\tau,.)\|^{2}+1)ds$
$\displaystyle=c\Big{(}e^{-b_{0}t_{1}}z^{2}(-t_{1},\omega)(\|\tilde{u}_{\tau-
t_{1}}\|^{2}+\|\tilde{v}_{\tau-
t_{1}}\|^{2})+e^{-b_{0}t_{2}}z^{2}(-t_{2},\omega)(\|\tilde{u}_{\tau-
t_{2}}\|^{2}+\|\tilde{v}_{\tau-t_{2}}\|^{2})\Big{)}$
$\displaystyle+ce^{(b_{0}-b)t_{1}}\int_{-\infty}^{0}e^{b_{0}s}z^{2}(s,\omega)(\|g(s+\tau,.)\|^{2}+\|h(s+\tau,.)\|^{2}+1)ds,$

which finishes the proof.$\ \ \ \ \ \ \ \ \ \Box$

Lemma 5.3. _Suppose that $g$ and $h$ satisfy (3.5), $f$ satisfies (3.1) and
(3.3) such that (5.1) holds. Let
$D=\\{D(\tau,\omega);\tau\in\mathbb{R},\omega\in\Omega\\}\in\mathcal{D}_{\delta}$
where $\mathcal{D}_{\delta}$ is defined as in (3.11). Then for
$\tau\in\mathbb{R},\omega\in\Omega$, there exists a unique element
$u^{*}=u^{*}(\tau,\omega)\in L^{2}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})$ such that_

$\displaystyle\lim\limits_{t\rightarrow+\infty}(\tilde{u}(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{\tau-t}),\tilde{u}(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{\tau-t}))=u^{*}(\tau,\omega),\
\ \mbox{in}\ L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N}),$

_where $(\tilde{u}_{\tau-t},\tilde{v}_{\tau-t})\in
D(\tau-t,\vartheta_{-t}\omega)$. Furthermore, the convergence is uniform
(w.r.t. $(\tilde{u}_{\tau-t},\tilde{v}_{\tau-t})\in
D(\tau-t,\vartheta_{-t}\omega)$)._

_Proof_ We choose the constant $b_{0}$ in Lemma 5.2 satisfying
$b>b_{0}>\delta_{1}$, where $\delta_{1}$ is as in (3.11). If
$(\tilde{u}_{\tau-t_{i}},\tilde{v}_{\tau-t_{i}})\in D(\tau-
t_{i},\vartheta_{-t_{i}}\omega)$, $i=1,2$, then similar to (4.8) we have

$\lim\limits_{t_{i}\rightarrow+\infty}e^{-b_{0}t_{i}}z^{2}(-t_{i},\omega)\|(\tilde{u}_{\tau-
t_{i}},\tilde{v}_{\tau-t_{i}})\|^{2}=0,i=1,2.$

Thus the result is derived directly from Lemma 5.2. $\ \ \ \ \ \ \ \ \ \Box$

Lemma 5.4. _Suppose that $g$ and $h$ satisfies (3.5), $f$ satisfies (3.1) and
(3.3) such that (5.1) holds. Then for $\tau\in\mathbb{R},\omega\in\Omega$, the
element $u^{*}=u^{*}(\tau,\omega)$ defined in Lemma 5.3 is a unique random
equilibrium for the cocycle $\varphi$ defined by (3) in
$L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$, i.e.,_

$\displaystyle\varphi(t,\tau,\omega,u^{*}(\tau,\omega))=u^{*}(\tau+t,\vartheta_{t}\omega),\
\ \mbox{for every }\ t\geq 0,\ \tau\in\mathbb{R},\ \omega\in\Omega.$

_Furthermore, the random equilibrium
$\\{u^{*}(\tau,\omega),\tau\in\mathbb{R},\ \omega\in\Omega\\}$ is the unique
element of the pullback attractor
$\mathcal{A}=\\{\mathcal{A}(\tau,\omega);\tau\in\mathbb{R},\
\omega\in\Omega\\}$ for the random cocycle $\varphi$, i.e., for every
$\tau\in\mathbb{R},\omega\in\Omega$,
$\mathcal{A}(\tau,\omega)=\\{u^{*}(\tau,\omega)\\}.$_

_Proof_ From the definition of random cocycle,

$\varphi(t,\tau-t,\vartheta_{-t}\omega,(\tilde{u}_{\tau-t},\tilde{v}_{\tau-t}))=(\tilde{u}(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{\tau-t}),\tilde{v\emph{}}(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{v}_{\tau-t})),$

then for for every $\tau\in\mathbb{R},\omega\in\Omega$, we see from Lemma 5.3
that

$\displaystyle
u^{*}(\tau,\omega)=\lim\limits_{t\rightarrow+\infty}\varphi(t,\tau-t,\vartheta_{-t}\omega,(\tilde{u}_{\tau-t},\tilde{v}_{\tau-t})),$
(5.8)

where $(\tilde{u}_{\tau-t},\tilde{v}_{\tau-t})\in
D(\tau-t,\vartheta_{-t}\omega)$. Thus by the continuity and the cocycle
property of $\varphi$ and (5.8), we find that for every $t\geq
0,\tau\in\mathbb{R},\omega\in\Omega$,

$\displaystyle\varphi(t,\tau,\omega,u^{*}(\tau,\omega))$
$\displaystyle=\varphi(t,\tau,\omega,.)\circ\lim\limits_{s\rightarrow+\infty}\varphi(s,\tau-s,\vartheta_{-s}\omega,(\tilde{u}_{\tau-t},\tilde{v}_{\tau-t}))$
$\displaystyle=\lim\limits_{s\rightarrow+\infty}\varphi(t,\tau,\omega,.)\circ\varphi(s,\tau-s,\vartheta_{-s}\omega,(\tilde{u}_{\tau-t},\tilde{v}_{\tau-t}))$
$\displaystyle=\lim\limits_{s\rightarrow+\infty}\varphi(t+s,\tau-s,\vartheta_{-s}\omega,(\tilde{u}_{\tau-t},\tilde{v}_{\tau-t}))$
$\displaystyle=\lim\limits_{s\rightarrow+\infty}\varphi(t+s,(\tau+t)-t-s,\vartheta_{-s-t}\vartheta_{t}\omega,(\tilde{u}_{\tau-t},\tilde{v}_{\tau-t}))=u^{*}(\tau+t,\vartheta_{t}\omega),$

which also implies the invariance of $\mathcal{A}$, that is,
$\varphi(t,\tau,\omega,\mathcal{A}(\tau,\omega))=\mathcal{A}(\tau+t,\vartheta_{t}\omega)$.
The compactness of $\mathcal{A}(\tau,\omega)$ is obvious and the attracting
property follows from (5.8). $\ \ \ \ \ \ \ \ \ \Box$

Acknowledgments:

This work was supported by Chongqing Basis and Frontier Research Project of
China (No. cstc2014jcyjA00035), National Natural Science Foundation of China
(No. 11271388), China Postdoctoral Science Foundation Grant (No.2014M550452).

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