#  Regularity of pullback attractors for non-autonomous stochastic FitzHugh-
Nagumo systems with additive noises on unbounded domains

Wenqiang Zhao  Wenqiang Zhao
School Of Mathematics and Statistics, Chongqing Technology and Business
University, Chongqing 400067, China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove the existences of pullback attractors in
$L^{p}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$ for stochastic Fitzhugh-
Nagumo system driven by both additive noises and deterministic non-autonomous
forcings. The nonlinearity is polynomial like growth with exponent $p-1$. The
asymptotic compactness for the cocycle in $L^{p}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})$ is proved by using asymptotic a priori method, where
the plus and minus signs of the nonlinearity at large value are not required.

###### Key words and phrases:

Random dynamical systems; non-autonomous FitzHugh-Nagumo systems; upper semi-
continuity; $\mathcal{D}$-pullback attractor; multiplicative noise

###### 2000 Mathematics Subject Classification:

60H15, 35R60, 35B40, 35B41

##  1\. Introduction

In this paper, we consider the regularity of pullback attractors for the
following non-autonomous FitzHugh-Nagumo system defined on $\mathbb{R}^{N}$
perturbed by additive noises:

$\begin{cases}d\tilde{u}+(\lambda\tilde{u}-\Delta\tilde{u}+\alpha\tilde{v})dt=f(x,\tilde{u})dt+g(t,x)dt+h_{1}d\omega_{1}(t),\\\
d\tilde{v}+(\sigma\tilde{v}-\beta\tilde{u})dt=h(t,x)dt+h_{2}d\omega_{2}(t),\\\
\tilde{u}(x,\tau)=\tilde{u}_{0}(x),\\\
\tilde{v}(x,\tau)=\tilde{v}_{0}(x),\end{cases}$ (1.1)

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, $h_{1}$ and $h_{2}$ are
given functions on $\mathbb{R}^{N}$ satisfying some regular conditions, 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
with exponent $p-1,p>2$ and $\omega(t)=(\omega_{1}(t),\omega_{2}(t))$ is a
Wiener process defined on a probability space $(\Omega,\mathcal{F},{P})$,
which will be specified later.

The deterministic FitzHugh-Nagumo system is an important mathematical model to
describe the signal transmission across axons in neurobiology, which is well
studied in the literature, see, e.g., [20, 8, 13, 18, 22, 19]. In the
stochastic case, Wang [23] proved the existence and uniqueness of random
attractors in the initial space $L^{2}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})$ for the autonomous FitzHugh-Nagumo system. For the
general non-autonomous forcings $g$ and $h$, Adili and Wang [1, 2] obtained
the pullback attractors in the initial space $L^{2}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})$, and Bao [4] strengthened this result and proved the
regularity of pullback attractors in the non-initial space
$H^{1}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$. Recently, [12, 10, 11]
studied the existences of random attractors for stochastic lattice FitzHugh-
Nagumo system

However, to our knowledge, there are no literature to investigate the high-
order integrability of solutions of problem (1.1), even for the deterministic
case.

In this paper, we prove that the problem (1.1) admits a unique pullback
attractor in the non-initial space $L^{\varpi}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})$ for every $\varpi\in(2,p]$, with the functions $f,g$
and $h$ satisfying almost the same conditions as in [2]. Actually, we show
that the obtained pullback attractors in a family of spaces
$L^{\varpi}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})(\varpi\in(2,p])$ are
the same object as in the initial space $L^{2}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})$. Furthermore, it is compact and attracts every nonempty
bounded subset of $L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$ under
the topology of $L^{\varpi}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})(\varpi\in(2,p])$. The asymptotic compactness in
$L^{p}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$ for the cocycle is proved
by using asymptotic a priori method which is developed by [29]. Because of the
interactions of the variables in the system (1.1), proving that the unbounded
part of solutions vanishes in $L^{p}$ norm is a more technical task. We prove
this in a relatively simple way just using asymptotic a priori method, in
which the plus and minus signs of the nonlinearity at large value are not
required, comparing with some similar work as in [14, 28, 17, 25].

This paper is organized as follows. In the next section, we recall some
preliminaries required for our further discussions and then establish an
abstract result regarding the existence of pullback attractors in some non-
initial spaces, which can be applicable to stochastic partial differential
equations (SPDEs) with both random noise and a deterministic non-autonomous
term. In Section 3, we give the assumptions on $g,h$ and $f$ and define a
continuous cocycle for problem (1.1). Finally we prove the existences of
pullback attractors in $L^{\varpi}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})$ in Section 4.

##  2\. Preliminaries and abstract results

Let $(X,\|.\|_{X})$ and $(Y,\|.\|_{Y})$ be two complete separable Banach
spaces with Borel sigma-algebra $\mathcal{B}(X)$ and $\mathcal{B}(Y)$,
respectively. For convenience, we call $X$ _the initial space_ (contains all
initial data of a SPDE) and $Y$ _the associated non-initial space_ (usually
the regular solutions (of a SPDE) located space).

In this section, we give a sufficient standard for the existence of pullback
attractors in the non-initial space $Y$ for the random dynamical system (RDS)
over two parametric spaces. For the existence of random attractors in the non-
initial space $Y$ for the RDS over one parametric space, the readers may refer
to [26, 27, 28, 17, 14, 15, 16, 25].

We also mention that regarding the existence of random attractors in the
initial space $X$ for the RDS over one parametric space, the good references
are [3, 5, 6, 21, 9, 7]. However, here we recall from [24] some basic notions
regarding RDS over two parametric spaces. Let $\Omega_{1}$ be a nonempty set
on which there is a mapping $\vartheta_{1,t}:\Omega_{1}\rightarrow\Omega_{1}$
such that $\vartheta_{1,0}$ is the identity on $\Omega_{1}$ and
$\vartheta_{1,s+t}=\vartheta_{1,t}\circ\vartheta_{1,s}$ for all
$s,t\in\mathbb{R}$. Let $(\Omega_{2},\mathcal{F}_{2},{P})$ be a probability
space on which there is a measure preserving transformation $\vartheta_{2,t}$
such that $\vartheta_{2,0}$ is the identity on $\Omega_{2}$ and
$\vartheta_{2,s+t}=\vartheta_{2,t}\circ\vartheta_{2,s}$ for all
$s,t\in\mathbb{R}$. We often use
$(\Omega_{1},\\{\vartheta_{1,t}\\}_{t\in\mathbb{R}})$ and
$(\Omega_{2},\mathcal{F}_{2},{P},\\{\vartheta_{2,t}\\}_{t\in\mathbb{R}})$ to
denote the two parametric spaces respectively.

A mapping $\varphi:\mathbb{R}^{+}\times\Omega_{1}\times\Omega_{2}\times
X\mapsto X$ is called a cocycle on $X$ over
$(\Omega_{1},\\{\vartheta_{1,t}\\}_{t\in\mathbb{R}})$ and
$(\Omega_{2},\mathcal{F}_{2},{P},\\{\vartheta_{2,t}\\}_{t\in\mathbb{R}})$ if
for every $\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}$ and
$t,s\in\mathbb{R}^{+}$, the following conditions are satisfied:

(i) $\varphi(.,\omega_{1},.,.):\mathbb{R}^{+}\times\Omega_{2}\times X\mapsto
X$ is
$(\mathcal{B}(\mathbb{R}^{+})\times\mathcal{F}_{2}\times\mathcal{B}(X),\mathcal{B}(X))$-measurable,

(ii) $\varphi(0,\omega_{1},\omega_{2},.)$ is the identity on X,

(iii)
$\varphi(t+s,\omega_{1},\omega_{2},.)=\varphi(t,\vartheta_{1,s}\omega_{1},\vartheta_{2,s}\omega_{2},.)\circ\varphi(s,\omega_{1},\omega_{2},.).$
In addition, if $\varphi(t,\omega_{1},\omega_{2},.):X\rightarrow X$ is
continuous for every
$t\in\mathbb{R}^{+},\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}$, then
$\varphi$ is called a continuous cocycle on $X$ over
$(\Omega_{1},\\{\vartheta_{1,t}\\}_{t\in\mathbb{R}})$ and
$(\Omega_{2},\mathcal{F}_{2},{P},\\{\vartheta_{2,t}\\}_{t\in\mathbb{R}})$.

A set-valued mapping $K:\Omega_{1}\times\Omega_{2}\mapsto 2^{X}$ is called
measurable in $X$ with respect to $\mathcal{F}_{2}$ in $\Omega_{2}$ if the
mapping
$\omega_{2}\in\Omega_{2}\mapsto\mbox{dist}_{X}(x,K(\omega_{1},\omega_{2}))$ is
($\mathcal{F}_{2},\mathcal{B}(\mathbb{R}))$-measurable for every fixed $x\in
X$ and $\omega_{1}\in\Omega_{1}$, where $\mbox{dist}_{X}$ is the Haustorff
semi-metric in $X$. And furthermore if the value $K(\omega_{1},\omega_{2})$ is
a closed nonempty subset of $X$ for every $\omega_{1}\in\Omega_{1}$ and
$\omega_{2}\in\Omega_{2}$ then
$\\{K(\omega_{1},\omega_{2});\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}\\}$
is called a closed measurable subset of $X$ with respect to $\mathcal{F}_{2}$
in $\Omega_{2}$.

In the sequel, we assume that the cocycle $\varphi$ together with $X$ and $Y$
satisfy the following

_(H1) For every fixed $t>0,\omega_{1}\in\Omega_{1}$ and
$\omega_{2}\in\Omega_{2}$, $\varphi(t,\omega_{1},\omega_{2},.):X\mapsto Y$. _

We always assume that $\varphi$ is a continuous cocycle on the initial space
$X$ over $(\Omega_{1},\\{\vartheta_{1,t}\\}_{t\in\mathbb{R}})$ and
$(\Omega_{2},\mathcal{F}_{2},{P},\\{\vartheta_{2,t}\\}_{t\in\mathbb{R}})$, and
$\mathcal{D}$ a collection of some families of nonempty subsets of $X$
parametrized by $\omega_{1}\in\Omega_{1}$ and $\omega_{2}\in\Omega_{2}$,
_i.e.,_

$\mathcal{D}=\\{D=\\{D(\omega_{1},\omega_{2})\in
2^{X};D(\omega_{1},\omega_{2})\neq\emptyset,\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}\\};f_{D}\
\mbox{satisfies some conditions}\\}.$

Let $D_{1}$ and $D_{2}$ belong to $\mathcal{D}$. We say $D_{1}=D_{2}$ if and
only if $D_{1}(\omega_{1},\omega_{2})=D_{2}(\omega_{1},\omega_{2})$ for all
$\omega_{1}\in\Omega_{1}$ and $\omega_{2}\in\Omega_{2}$. Note that for our
problem the continuity of the cocycle $\varphi$ in the non-initial space $Y$
is not necessary.

Definition 2.1. _Let $\mathcal{D}$ be a collection of some families of
nonempty subsets of $X$.
$\\{K(\omega_{1},\omega_{2});\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}\\}\in\mathcal{D}$
is called a $\mathcal{D}$-pullback absorbing set for a cocycle $\varphi$ in
$X$ if for every $\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}$ and
$D\in\mathcal{D}$, there exists a absorbing time
$T=T(\omega_{1},\omega_{2},D)>0$ such that_

$\varphi(t,\vartheta_{1,-t}\omega_{1},\vartheta_{2,-t}\omega_{2},D(\vartheta_{1,-t}\omega_{1},\vartheta_{2,-t}\omega_{2}))\subseteq
K(\omega_{2},\omega_{2})\ \ \ \ \ \ \mbox{for all}\ t\geq T.$

Definition 2.2. _Let $\mathcal{D}$ be a collection of some families of
nonempty subsets of $X$. A cocycle $\varphi$ is said to be
$\mathcal{D}$-pullback asymptotically compact in $X\ (\mbox{resp. in}\ Y)$ if
for every $\omega_{1}\in\Omega_{1}$ and $\omega_{2}\in\Omega_{2}$,_

$\\{\varphi(t_{n},\vartheta_{1,-t_{n}}\omega_{1},\vartheta_{2,-t_{n}}\omega_{2},x_{n})\\}\
\mbox{has a convergent subsequence in}\ X(\mbox{resp. in}\ Y)$

_whenever $t_{n}\rightarrow\infty$ and $x_{n}\in
D(\vartheta_{1,-t}\omega_{1},\vartheta_{2,-t}\omega_{2})$ with
$D=\\{D(\omega_{1},\omega_{2});\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}\\}\in\mathcal{D}$._

Definition 2.3. _Let $\mathcal{D}$ be a collection of some families of
nonempty subsets of $X$.
$\\{\mathcal{A}(\omega_{1},\omega_{2});\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}\\}\in\mathcal{D}$
is called a $\mathcal{D}$-pullback attractor for a cocycle $\varphi$ in $X\
(\mbox{resp. in}\ Y)$ over
$(\Omega_{1},\\{\vartheta_{1,t}\\}_{t\in\mathbb{R}})$ and
$(\Omega_{2},\mathcal{F}_{2},{P},\\{\vartheta_{2,t}\\}_{t\in\mathbb{R}})$ if
the following statements are satisfied:_

_(i) $\mathcal{A}$ is measurable in $X$ with respect to the $P$-completion of
$\mathcal{F}_{2}$ in $\Omega_{2}$ and $\mathcal{A}(\omega_{1},\omega_{2})$ is
compact in $X\ (\mbox{resp. in}\ Y)$ for all
$\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}$,_

_(ii) $\mathcal{A}$ is invariant, that is, for every
$\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}$,
$\varphi(t,\omega_{1},\omega_{2},\mathcal{A}(\omega_{1},\omega_{2}))=\mathcal{A}(\vartheta_{1,t}\omega_{1},\vartheta_{2,t}\omega_{2})$
for all $t\geq 0,$_

_(iii) $\mathcal{A}$ attracts every element
$D=\\{D(\omega_{1},\omega_{2});\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}\\}\in\mathcal{D}$
in $X\ (\mbox{resp. in}\ Y)$, that is, for every
$\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}$,_

$\mathop{\lim}\limits_{t\rightarrow+\infty}\mbox{dist}_{X}(\varphi(t,\vartheta_{1,-t}\omega_{1},\vartheta_{2,-t}\omega_{2},D(\vartheta_{1,-t}\omega_{1},\vartheta_{2,-t}\omega_{2})),\mathcal{A}(\omega_{1},\omega_{2}))=0$
$(\mbox{resp.}\mathop{\lim}\limits_{t\rightarrow+\infty}\mbox{dist}_{Y}(\varphi(t,\vartheta_{1,-t}\omega_{1},\vartheta_{2,-t}\omega_{2},D(\vartheta_{1,-t}\omega_{1},\vartheta_{2,-t}\omega_{2})),\mathcal{A}(\omega_{1},\omega_{2}))=0).$

The following Theorem 2.6 is a useful tool to prove the existence of pullback
attractors for SPDEs with both non-autonomous deterministic term and random
term in a associated non-initial space. We need to assume that the spaces $X$
and $Y$ satisfy sequence limits uniqueness property:

_(H2) 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 $x=y$._

Theorem 2.6. _Let $\mathcal{D}$ be a collection of some families of nonempty
subsets of $X$ which is inclusion closed. Suppose that $\varphi$ is a
continuous cocycle on the initial space $X$ over
$(\Omega_{1},\\{\vartheta_{1,t}\\}_{t\in\mathbb{R}})$ and
$(\Omega_{2},\mathcal{F}_{2},{P},\\{\vartheta_{2,t}\\}_{t\in\mathbb{R}})$.
Assume that_

_(i) $\varphi$ has a closed and measurable (w.r.t. the $P$-completion of
$\mathcal{F}_{2}$ in $\Omega_{2}$) $\mathcal{D}$-pullback absorbing set
$K=\\{K(\omega_{1},\omega_{2});\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}\\}\in\mathcal{D}$
in $X$;_

_(ii) $\varphi$ is $\mathcal{D}$-pullback asymptotically compac_t in $X$.
_Then the cocycle $\varphi$ has a unique $\mathcal{D}$-pullback attractor
$\mathcal{A}_{X}=\\{\mathcal{A}_{X}(\omega_{1},\omega_{2});\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}\\}$
in $X$, which is given by_

$\displaystyle\mathcal{A}_{X}(\omega_{1},\omega_{2})=\cap_{s>0}\overline{\cup_{t\geq
s}\varphi(t,\vartheta_{1,-t}\omega_{1},\vartheta_{2,-t}\omega_{2},K(\vartheta_{1,-t}\omega_{1},\vartheta_{2,-t}\omega_{2}))}^{X},\
\ \omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2},$ (2.1)

_where the closure is taken over in $X$._

_If further (H1)-(H2) hold and $\varphi$ is $\mathcal{D}$-pullback
asymptotically compact in $Y$,_ _then the cocycle $\varphi$ has a unique
$\mathcal{D}$-pullback attractor
$\mathcal{A}_{Y}=\\{\mathcal{A}_{Y}(\omega_{1},\omega_{2});\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}\\}\in\mathcal{D}$
in $Y$, which is given by_

$\displaystyle\mathcal{A}_{Y}(\omega_{1},\omega_{2})=\cap_{s>0}\overline{\cup_{t\geq
s}\varphi(t,\vartheta_{1,-t}\omega_{1},\vartheta_{2,-t}\omega_{2},K(\vartheta_{1,-t}\omega_{1},\vartheta_{2,-t}\omega_{2}))}^{Y},\
\ \omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2},$ (2.2)

_where the closure is taken over in $Y$._ _In addition, we have
$\mathcal{A}_{Y}=\mathcal{A}_{X}$, i.e., for every
$\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}$,
$\mathcal{A}_{Y}(\omega_{1},\omega_{2})=\mathcal{A}_{X}(\omega_{1},\omega_{2})$_.

Proof The first result is stated in [24]. We only prove the second result.
Indeed, (2.2) makes sense by using _(H1)_ , and$\mathcal{A}_{Y}\neq\emptyset$
by the asymptotic compactness of the cocycle $\varphi$ in $Y$. In the
following, we show that $\mathcal{A}_{Y}$ is a unique $\mathcal{D}$-pullback
attractor in $Y$.

_Step 1._ Since $\mathcal{A}_{X}$ is measurable in $X$ (with respect to the
$P$-completion of $\mathcal{F}_{2}$ in $\Omega_{2}$) and
$\mathcal{A}_{X}\in\mathcal{D}$ is invariant (the measurability of
$\mathcal{A}_{X}$ is proved by Theorem 2.14 in [24]), then the measurability
and invariance of $\mathcal{A}_{Y}$ can be proved by showing that
$\mathcal{A}_{Y}=\mathcal{A}_{X}$. We have to prove that for every
$\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}$,
$\mathcal{A}_{Y}(\omega_{1},\omega_{2})=\mathcal{A}_{X}(\omega_{1},\omega_{2})$,
where $\mathcal{A}_{X}(\omega_{1},\omega_{2})$ and
$\mathcal{A}_{Y}(\omega_{1},\omega_{2})$ defined in (2.1) and (2.2)
respectively. Indeed, taking $x\in\mathcal{A}_{X}(\omega_{1},\omega_{2})$, by
(2.1), there are $t_{n}\rightarrow+\infty$ and $x_{n}\in
K(\vartheta_{1,-t_{n}}\omega_{1},\vartheta_{2,-t_{n}}\omega_{2})$ such that

$\displaystyle\varphi(t_{n},\vartheta_{1,-t_{n}}\omega_{1},\vartheta_{2,-t_{n}}\omega_{2},x_{n})\xrightarrow[n\rightarrow\infty]{\|.\|_{X}}x.$
(2.3)

Since $\varphi$ is $\mathcal{D}$-asymptotically compact in $Y$, then there is
a $y\in Y$ such that up to a subsequence (we still denote this by itself),

$\displaystyle\varphi(t_{n},\vartheta_{1,-t_{n}}\omega_{1},\vartheta_{2,-t_{n}}\omega_{2},x_{n})\xrightarrow[n\rightarrow\infty]{\|.\|_{Y}}y.$
(2.4)

Then it follows from (2.2) that $y\in\mathcal{A}_{Y}(\omega_{1},\omega_{2})$.
Then by _(H2)_ and (2.3) and (2.4) we have
$x=y\in\mathcal{A}_{X}(\omega_{1},\omega_{2})$ and thus
$\mathcal{A}_{X}(\omega_{1},\omega_{2})\subseteq\mathcal{A}_{Y}(\omega_{1},\omega_{2})$.
The inverse inclusion can be proved similarly. Thus
$\mathcal{A}_{X}=\mathcal{A}_{Y}$ as required.

_Step 2._ We prove the attraction of $\mathcal{A}_{Y}$ in $Y$ by a
contradiction argument. Indeed, if for fixed $\omega_{1}\in\Omega_{1}$ and
$\omega_{2}\in\Omega_{2}$, there exist $\delta>0$, $x_{n}\in
D(\vartheta_{1,-t_{n}}\omega_{1},\vartheta_{2,-t_{n}}\omega_{2})$ with
$D=\\{D(\omega_{1},\omega_{2});\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}\\}\in\mathcal{D}$
and $t_{n}\rightarrow+\infty$ such that

$\displaystyle\mbox{dist}_{Y}\Big{(}\varphi(t_{n},\vartheta_{1,-t_{n}}\omega_{1},\vartheta_{2,-t_{n}}\omega_{2},x_{n}),\mathcal{A}_{Y}(\tau,\omega)\Big{)}\geq\delta.$
(2.5)

By the asymptotic compactness of $\varphi$ in $Y$, there exists $y_{0}\in Y$
such that up to a subsequence,

$\displaystyle\varphi(t_{n},\vartheta_{1,-t_{n}}\omega_{1},\vartheta_{2,-t_{n}}\omega_{2},x_{n})\xrightarrow[n\rightarrow\infty]{\|.\|_{Y}}y_{0}.$
(2.6)

On the other hand, by our assumption (i), there exist a large time $T>0$ such
that

$\displaystyle
y_{n}=\varphi(T,\vartheta_{1,-t_{n}}\omega_{1},\vartheta_{2,-t_{n}}\omega_{2},x_{n})$
$\displaystyle=\varphi(T,\vartheta_{1,-T}\vartheta_{1,-t_{n}+T}\omega_{1},\vartheta_{2,-T}\vartheta_{2,-t_{n}+T}\omega_{2},x_{n})$
$\displaystyle\in
K(\vartheta_{1,-t_{n}+T}\omega_{1},\vartheta_{2,-t_{n}+T}\omega_{2}).$ (2.7)

Then by the cocycle property of $\varphi$, along with (2.6) and (2), we infer
that as $n\rightarrow\infty$,

$\displaystyle\varphi(t_{n},\vartheta_{1,-t_{n}}\omega_{1},\vartheta_{2,-t_{n}}\omega_{2},x_{n})$
$\displaystyle=\varphi(t_{n}-T,\vartheta_{1,-t_{n}+T}\omega_{1},\vartheta_{2,-t_{n}+T}\omega_{2},y_{n})\rightarrow
y_{0}\ \ \ \mbox{in}\ Y.$

Therefore by (2.2), $y_{0}\in\mathcal{A}_{Y}(\omega_{1},\omega_{2})$. This
implies that

$\displaystyle\mbox{dist}_{Y}\Big{(}\varphi(t_{n},\vartheta_{1,-t_{n}}\omega_{1},\vartheta_{2,-t_{n}}\omega_{2},x_{n}),\mathcal{A}_{Y}(\omega_{1},\omega_{2})\Big{)}\rightarrow
0$

as $n\rightarrow\infty$, which is a contradiction to (2.5).

_Step 3._ It remains to prove the compactness of $A_{Y}$ in $Y$. Let
$\\{y_{n}\\}_{n=1}^{\infty}$ be a sequence in $A_{Y}(\omega_{1},\omega_{2})$.
By the invariance of $A_{Y}(\omega_{1},\omega_{2})$ which is proved in Step 1,
we have

$\varphi(t,\vartheta_{1,-t}\omega_{1},\vartheta_{2,-t}\omega_{2},\mathcal{A}_{Y}(\vartheta_{1,-t}\omega_{1},\vartheta_{2,-t}\omega_{2}))=\mathcal{A}(\omega_{1},\omega_{2}).$

Then it follows that there is a sequence
$\\{z_{n}\\}_{n=1}^{\infty}\in\mathcal{A}_{Y}(\vartheta_{1,-t}\omega_{1},\vartheta_{2,-t}\omega_{2})$
such that for every $n\in\mathbb{N}$,

$\displaystyle
y_{n}=\varphi(t_{n},\vartheta_{1,-t_{n}}\omega_{1},\vartheta_{2,-t_{n}}\omega_{2},z_{n}).$

Note that $A_{Y}\in\mathcal{D}$. Then by the asymptotic compactness of
$\varphi$ in $Y$, $\\{y_{n}\\}$ has a convergence subsequence in $Y$, _i.e._ ,
there is a $y_{0}\in Y$ such that

$\lim_{n\rightarrow\infty}y_{n}=y_{0}\ \ \mbox{in}\ Y.$

But $A_{Y}(\omega_{1},\omega_{2})$ is closed in $Y$, so $y_{0}\in
A_{Y}(\omega_{1},\omega_{2})$.

The uniqueness is easily followed by the attraction property of $\varphi$ and
$A_{Y}\in\mathcal{D}$. This completes the total proofs.$\ \ \ \ \ \ \Box$

Remark 2.7. (i) We emphasize that the assumption _(H1)_ is necessary to
guarantee that the closure of the set
$\varphi(t,\vartheta_{1,-t}\omega_{1},\vartheta_{2,-t}\omega_{2},K(\vartheta_{1,-t}\omega_{1},\vartheta_{2,-t}\omega_{2}))$
in $Y$ makes sense for all $t>0$, as in (2.2).

(ii) We emphasize that the attractor $A_{Y}$ is measurable in $X$ and
completely structured by a closed and measurable $\mathcal{D}$-pullback
absorbing set in the initial space $X$. Thus the existence of absorbing set in
the non-initial space $Y$ is unnecessary. This is different from the
construction in [4].

(iii ) The above Theorem 2.6 generalizes Theorem 3.1 in [16] regarding the bi-
spatial random attractor which is suitable for the random dynamical system
over one parametric space. Our result here is applicable to the SPDE with
random noises and deterministic non-autonomous forcing.

In particular, if the initial space $X=L^{2}(\mathbb{R}^{N})$ and the
associated non-initial space $Y=L^{r}(\mathbb{R}^{N})$, then we have an easy
condition on the asymptotic compactness of the cocycle $\varphi$ in the non-
initial space $Y=L^{r}(\mathbb{R}^{N}),r>2$. First we let $\mathcal{D}$ be the
collection of some family of nonempty subsets of $L^{2}(\mathbb{R}^{N})$. Then
we have

Proposition 2.8. _Assume that the cocycle $\varphi$ is asymptotically compact
in $L^{2}(\mathbb{R}^{N})$. Given
$\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}$ and
$D=\\{D(\omega_{1},\omega_{2});\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}\\}\in\mathcal{D}$,
suppose that for every $\eta>0$, there are constants
$M=M(\omega_{1},\omega_{2},D,\eta)>0$ and $T=T(\omega_{1},\omega_{2},D)>0$
such that_

$\displaystyle\sup\limits_{t\geq
T}\sup\limits_{u_{0}\in{D}(\vartheta_{1,-t}\omega_{1},\vartheta_{2,-t}\omega_{2}))}\int\limits_{\mathbb{R}^{N}(|\varphi_{t}|\geq
M)}|\varphi_{t}|^{r}dx\leq\eta,$

_where
$\varphi_{t}=\varphi(t,\vartheta_{1,-t}\omega_{1},\vartheta_{2,-t}\omega_{2},u_{0})$.
Then the cocycle $\varphi$ is asymptotically compact in
$L^{r}(\mathbb{R}^{N})$, that is, for every $\omega_{1}\in\Omega_{1}$ and
$\omega_{2}\in\Omega_{2}$, the sequence
$\\{\varphi(t,\vartheta_{1,-t_{n}}\omega_{1},\vartheta_{2,-t_{n}}\omega_{2},{u}_{0,n})\\}$
has a convergent subsequence in $L^{r}(\mathbb{R}^{N})$ whenever
$t_{n}\rightarrow+\infty$ and ${u}_{0,n}\in
D(\vartheta_{1,-t_{n}}\omega_{1},\vartheta_{2,-t_{n}}\omega_{2})\in\mathcal{D}$._

_Proof_ The proof is very similar to the one parameter case, in which we can
find the proof in [27, 14, 15] and so we omit it here. $\ \ \ \ \ \ \ \ \
\Box$

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

In this section, we give the conditions on the functions $f,g$ and $h$. We
also give the parametric dynamical systems
$(\Omega_{1},\\{\vartheta_{1,t}\\}_{t\in\mathbb{R}}$ and
$(\Omega_{2},\mathcal{F}_{2},P,\\{\vartheta_{2,t}\\}_{t\in\mathbb{R}})$
respectively, and show the existence of continuous cocycle in the initial
space $L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$. For the non-
autonomous FitzHugh-Nagumo system (1.1), the nonlinearity $f(x,s)$ satisfies
almost the same assumptions as in [2], _i.e._ , for all $x\in\mathbb{R}^{N}$
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 $p>2$, $\alpha_{i}>0(i=1,2,3)$ are determined constants, $\psi_{1}\in
L^{1}(\mathbb{R}^{N})\cap L^{\frac{p}{2}}(\mathbb{R}^{N})$, $\psi_{2}$ and
$\psi_{3}\in L^{2}(\mathbb{R}^{N})$. 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
$\tau\in\mathbb{R}$,

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

where $\delta=\min\\{\lambda,\sigma\\}$, $\lambda$ and $\sigma$ are as in
(1.1). The functions $h_{1}\in H^{2}(\mathbb{R}^{N})\cap
W^{2,p}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})$ and $h_{2}\in
H^{1}(\mathbb{R}^{N})$. We remark that the assumption $\psi_{1}$ is
essentially bounded is not required and then the plus and minus signs of the
nonlinearity $f(x,s)$ at large value are unknown by condition (3.1).

For the probability space $(\Omega,\mathcal{F},P)$, we write
$\Omega=\\{\omega\in C(\mathbb{R},\mathbb{R}^{2});\omega(0)=0\\}$. Let
$\mathcal{F}$ be the Borel $\sigma$-algebra induced by the compact-open
topology of $\Omega$ and ${P}$ be the corresponding Wiener measure on
$(\Omega,\mathcal{F})$. We define a shift operator $\vartheta$ on $\Omega$ by

$\vartheta_{t}\omega(s)=\omega(s+t)-\omega(t),\ \mbox{for every}\
\omega\in\Omega,t,s\in\mathbb{R}.$

It is easy to show that $\vartheta_{t}$ is a measure preserving transformation
on the given probability space $(\Omega,\mathcal{F},P)$. Let
$\\{\vartheta_{1,t}\\}_{t\in\mathbb{R}}$ be a group on $\mathbb{R}$ given by
$\vartheta_{1,t}(\tau)=\tau+t$ for all $\tau,t\in\mathbb{R}$. Then
$(\Omega_{1},\\{\vartheta_{1,t}\\}_{t\in\mathbb{R}}$ and
$(\Omega,\mathcal{F},P,\\{\vartheta_{t}\\}_{t\in\mathbb{R}})$ are the
requested parametric dynamical systems.

Given $t\in\mathbb{R},\omega=(\omega_{1},\omega_{2})\in\Omega$, we often write
$\theta_{t}\omega=(\theta_{t}\omega_{1},\theta_{t}\omega_{2})$. Put

$z_{1}(\vartheta_{t}\omega_{1})=-\lambda\int\limits_{-\infty}^{0}e^{\lambda
s}(\vartheta_{t}\omega_{1})(s)ds;\ \ \
z_{2}(\vartheta_{t}\omega_{2})=-\sigma\int\limits_{-\infty}^{0}e^{\sigma
s}(\vartheta_{t}\omega_{2})(s)ds.$

Then we have

$dz_{1}(\vartheta_{t}\omega_{1})+\lambda
z_{1}(\vartheta_{t}\omega_{1})dt=d\omega_{1}(t),\ \
dz_{2}(\vartheta_{t}\omega_{1})+\sigma
z_{2}(\vartheta_{t}\omega_{1})dt=d\omega_{2}(t).$

By [3] we know that both $z_{1}(\vartheta_{t}\omega_{1})$ and
$z_{2}(\vartheta_{t}\omega_{2})$ are continuous in $t$ on $\mathbb{R}$, and
further both $z_{1}(\omega_{1})$ and $z_{2}(\omega_{1})$ are tempered.

Let $(\tilde{u},\tilde{v})$ satisfy problem (1.1) and write

$\displaystyle
u(t,\tau,\omega,u_{0})=\tilde{u}(t,\tau,\omega,\tilde{u}_{0})-h_{1}z_{1}(\vartheta_{t}\omega_{1})\
\ \mbox{and}\
v(t,\tau,\omega,v_{0})=\tilde{v}(t,\tau,\omega,\tilde{v}_{0})-h_{2}z_{2}(\vartheta_{t}\omega_{2}).$
(3.6)

Then $(u,v)$ solves the follow non-autonomous FitzHugh-Nagumo system

$\displaystyle\frac{du}{dt}+\lambda u-\Delta u+\alpha
v=f(x,u+h_{1}z_{1}(\vartheta_{t}\omega_{1})+g(t,x)+\Delta
z_{1}(\vartheta_{t}\omega_{1})-\alpha h_{2}z_{2}(\vartheta_{t}\omega_{2}),$
(3.7) $\displaystyle\frac{dv}{dt}+\sigma v-\beta u=h(t,x)+\beta
h_{1}z_{1}(\vartheta_{t}\omega_{1}),$ (3.8)

with the initial conditions
$u_{0}=\tilde{u}_{0}-h_{1}z_{1}(\vartheta_{\tau}\omega_{1})$ and
$v_{0}=\tilde{v}_{0}-h_{2}z_{2}(\vartheta_{\tau}\omega_{2})$.

It is known (see [2]) that for every $(u_{0},v_{0})\in
L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$ the problem (3.7)-(3.8)
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 $v\in C([\tau,+\infty),L^{2}(\mathbb{R}^{N}))$. 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 $(u_{0},v_{0})$ . Then formally
$(\tilde{u},\tilde{v})=(u+h_{1}z_{1}(\vartheta_{t}\omega_{1}),v+h_{2}z_{2}(\vartheta_{t}\omega_{2}))$
is the solution to problem (1.1) with the initial value
$\tilde{u}_{0}={u}_{0}+h_{1}z_{1}(\vartheta_{\tau}\omega_{1})$ and
$v\tilde{}_{0}={v}_{0}+h_{2}z_{2}(\vartheta_{\tau}\omega_{2})$.

The continuous cocycle $\varphi$ over $\mathbb{R}$ and
$(\Omega,\mathcal{F},P,\\{\vartheta_{t}\\}_{t\in\mathbb{R}})$ is defined by

$\displaystyle\ \ \ \ \ \ \ \ \
\varphi(t,\tau,\omega,(\tilde{u}_{0},\tilde{v}_{0}))=(\tilde{u}(t+\tau,\tau,\vartheta_{-\tau}\omega,\tilde{u}_{0}),\tilde{v}(t+\tau,\tau,\vartheta_{-\tau}\omega,\tilde{v}_{0}))$
$\displaystyle=(u(t+\tau,\tau,\vartheta_{-\tau}\omega,u_{0})+h_{1}z_{1}(\vartheta_{t}\omega_{1}),v(t+\tau,\tau,\vartheta_{-\tau}\omega,v_{0})+h_{2}z_{2}(\vartheta_{t}\omega_{2})),$
(3.9)

where $u_{0}=\tilde{u}_{0}-h_{1}z_{1}(\omega_{1})$ and
$v_{0}=\tilde{v}_{0}-h_{2}z_{2}(\omega_{2})$.

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

$\displaystyle\lim\limits_{t\rightarrow+\infty}e^{-\delta
t}\|D(\tau-t,\vartheta_{-t}\omega)\|_{L^{2}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})}^{2}=0,$ (3.10)

where $\delta$ is as in (3.5). Denote by $\mathcal{D}_{\delta}$ the collection
of all families of bounded nonempty subsets of $L^{2}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})$ which satisfies (3.10). Then it is obvious that
$\mathcal{D}_{\delta}$ is inclusion closed.

In fact, the existence of $\mathcal{D}_{\delta}$-pullback attractors for the
cocycle $\varphi$ defined by (3) in the initial space
$L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$ was proved in [2].

Theorem 3.1([2])._Assume that ( 3.1)-(3.5) hold. Then the cocycle $\varphi$
has a unique $\mathcal{D}_{\delta}$-pullback attractor
$\mathcal{A}=\\{\mathcal{A}(\tau,\omega),\tau\in\mathbb{R},\omega\in\Omega\\}$
in $L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$. Here for every fixed
$\tau\in\mathbb{R}$ and $\omega\in\Omega$, $\mathcal{A}(\tau,\omega)$ is given
by_

$\displaystyle\mathcal{A}(\tau,\omega)=\bigcap_{s>0}\overline{\bigcup_{t\geq
s}\varphi(t,\tau-t,\vartheta_{-t}\omega,K(\tau-t,\vartheta_{-t}\omega))}^{L^{2}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})},$ (3.11)

_where $K$ is a closed and measurable $\mathcal{D}_{\delta}$-pullback
absorbing set of $\varphi$ in $L^{2}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})$ which is stated by_

$\displaystyle K(\tau,\omega)=\\{(\tilde{u},\tilde{v})\in
L^{2}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N});\|\tilde{u}\|^{2}+\|\tilde{v}\|^{2}\leq\rho(\tau,\omega)\\},$
(3.12)

_where $\rho(\tau,\omega)$ is the random constant:_

$\displaystyle\rho(\tau,\omega)=c(1+|z_{1}(\omega_{1})|^{2}+|z_{2}(\omega_{2})|^{2})+ce^{-\delta\tau}\int\limits_{-\infty}^{\tau}e^{\delta
s}(\|g(s,.)\|^{2}+\|h(s,.)\|^{2})ds$ $\displaystyle\ \ \ \ \ \ \ \ \ \
+c\int\limits_{-\infty}^{0}e^{\delta
s}(|z_{1}(\vartheta_{s}\omega_{1})|^{p}+|z_{2}(\vartheta_{s}\omega_{2})|^{2})ds.$
(3.13)

_Here $c$ is a deterministic positive constant._

In the sequel, we will prove that the set $\mathcal{A}$ defined by (3.11) is
also a $\mathcal{D}_{\delta}$-pullback attractor in
$L^{\varpi}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$ for every
$\varpi\in(2,p]$ with $p>2$.

##  4\. Existence of pullback random attractors in
$L^{\varpi}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$

In this section, we will establish the existence of
$\mathcal{D}_{\delta}$-pullback attractors in
$L^{\varpi}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})(\varpi\in(2,p])$ for
the cocycle $\varphi$ defined by (3). Actually, we will show that the family
of obtained pullback attractors are the same one.

Note that we do not increase any restrictions on the nonlinearity $f(x,s)$ and
non-autonomous terms $g$ and $h$ given in [2], except that $\psi_{1}\in
L^{p/2}(\mathbb{R}^{N})$ as in (3.1). For our problem, the key point is prove
that the unbounded part of the first component $\tilde{u}$ of solutions of
(1.1) vanishes in $L^{p}$, where $p>2$.

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 $\tau,\omega,D,\eta$ in any place.

Note that [2] has proved the following energy inequality (see the proof of
Lemma 4.1 there):

$\displaystyle\frac{d}{dt}(\alpha\|v\|^{2}$
$\displaystyle+\beta\|u\|^{2})+\delta(\alpha\|v\|^{2}+\beta\|u\|^{2})+\frac{1}{2}\delta\alpha\|v\|^{2}+\alpha_{1}\beta\|{u}\|_{p}^{p}$
$\displaystyle\leq\frac{4\beta}{\lambda}\|g(t,.)\|^{2}+\frac{4\alpha}{\sigma}\|h(t,.)\|^{2}+c(|z_{1}(\vartheta_{t}\omega_{1})|^{p}+|z_{2}(\vartheta_{t}\omega_{2})|^{2}+1).$
(4.1)

###  4.1. Uniform estimates of solutions

First we present a lemma which is sated in [2].

Lemma 4.1([2]). _Assume that ( 3.1)-(3.5) hold. Let
$\tau\in\mathbb{R},\omega\in\Omega$ and
$D=\\{D(\tau,\omega);\tau\in\mathbb{R},\omega\in\Omega\\}\in\mathcal{D}_{\delta}$.
Then there exists a constant $T=T(\tau,\omega,D)>0$ such that for all $t\geq
T$, the solution $(u,v)$ of problem (3.7)-(3.8) with $\omega$ replaced by
$\vartheta_{-\tau}\omega$ satisfies_

$\displaystyle\|u(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{0}-h_{1}z_{1}(\vartheta_{-t}\omega_{1})\|^{2}+\|v(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{v}_{0}-h_{2}z_{2}(\vartheta_{-t}\omega_{2})\|^{2}\leq
R(\tau,\omega),$ (4.2)
$\displaystyle\int\limits_{\tau-t}^{\tau}e^{\delta(s-\tau)}\Big{(}\|v(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{v}_{0}-h_{2}z_{2}(\vartheta_{-t}\omega_{2})\|^{2}+\|\tilde{u}(s,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{0})\|_{p}^{p}\Big{)}ds\leq
R(\tau,\omega),$ (4.3)

_where $(\tilde{u}_{0},\tilde{v}_{0})\in
D_{\delta}(\tau-t,\vartheta_{-t}\omega)$ and_

$\displaystyle R(\tau,\omega)$
$\displaystyle=c+ce^{-\delta\tau}\int\limits_{-\infty}^{\tau}e^{\delta
s}(\|g(s,.)\|^{2}+\|h(s,.)\|^{2})ds$
$\displaystyle+c\int\limits_{-\infty}^{0}e^{\delta
s}(|z_{1}(\vartheta_{s}\omega_{1})|^{2p-2}+|z_{1}(\vartheta_{s}\omega_{1})|^{p}+|z_{1}(\vartheta_{s}\omega_{1})|^{2}+|z_{2}(\vartheta_{s}\omega_{2})|^{2})ds.$
(4.4)

In fact, for our problem we need further to show that (4.2) holds in the
compact interval $[\tau-1.\tau]$. In particular, we have

Lemma 4.2. _Assume that ( 3.1)-(3.5) hold. Let
$\tau\in\mathbb{R},\omega\in\Omega$ 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
$t\geq T$, the first component of the solution of problem (3.7)-(3.8)
satisfies that for $\xi\in[\tau-1,\tau]$,_

$\displaystyle\|u(\xi,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{0}-h_{1}z_{1}(\vartheta_{-t}\omega_{1})\|^{2}+\|{v}(\xi,\tau-t,\vartheta_{-\tau}\omega,\tilde{v}_{0}-h_{2}z_{2}(\vartheta_{-t}\omega_{2})\|^{2}\leq
R(\tau,\omega),$

_where $(\tilde{u}_{0},\tilde{v}_{0})\in
D_{\delta}(\tau-t,\vartheta_{-t}\omega)$ and $R(\tau,\omega)$ is as in (4.1)_
only with a difference in the constant $c$.

_Proof_ We first let $t\geq 2$ and $\xi\in[\tau-1,\tau]$. Replacing $t$ by $s$
in (4) and multiplying by $e^{\delta(s-\tau)}$ and then integrating with
respect to $s$ over $(\tau-t,\xi)$ we deduce that, along with $\omega$
replaced by $\vartheta_{-\tau}\omega$,

$\displaystyle
e^{-\delta}(\beta\|u(\xi,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{0}-h_{1}z_{1}(\vartheta_{-t}\omega_{1})\|^{2}+\alpha\|v(\xi,\tau-t,\vartheta_{-\tau}\omega,\tilde{v}_{0}-h_{2}z_{2}(\vartheta_{-t}\omega_{2})\|^{2})$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
\leq\int\limits_{\tau-t}^{\tau}e^{\delta(s-\tau)}(\frac{4\beta}{\lambda}\|g(s,.)\|^{2}+\frac{4\alpha}{\sigma}\|h(s,.)\|^{2})ds$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
+c\int\limits_{\tau-t}^{\tau}e^{\delta(s-\tau)}(|z_{1}(\vartheta_{s-\tau}\omega_{1})|^{p}+|z_{2}(\vartheta_{s-\tau}\omega_{2})|^{2}+1)ds+e^{-\delta
t}(\beta\|u_{0}\|^{2}+\alpha\|v_{0}\|^{2})$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\ \ \ \leq ce^{-\delta\tau}\int\limits_{-\infty}^{\tau}e^{\delta
s}(\|g(s,.)\|^{2}+\|h(s,.)\|^{2})ds$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
\ +c\int\limits_{-\infty}^{0}e^{\delta
s}(|z_{1}(\vartheta_{s}\omega_{1})|^{p}+|z_{2}(\vartheta_{s}\omega_{2})|^{2}+1)ds+ce^{-\delta
t}(\|\tilde{u}_{0}\|^{2}+\|\tilde{v}_{0}\|^{2})$ $\displaystyle\ \ \ \ \ \ \ \
\ \ \ \ \ \ \ +ce^{-\delta
t}(\|z_{1}(\vartheta_{-t}\omega_{1})\|^{2}+\|z_{2}(\vartheta_{-t}\omega_{2})\|^{2}).$
(4.5)

Note that $z_{1}(\omega_{1})$ and $z_{2}(\omega_{2})$ are tempered and
$(\tilde{u}_{0},\tilde{v}_{0})\in D(\tau-t,\vartheta_{-t}\omega)$. Then (4.1)
implies the desired. $\ \ \ \ \ \ \Box$

A result on the uniform $L^{p}$-estimate of the first component of solution of
problem (3.7)-(3.8) on the compact interval $[\tau-1.\tau]$ is also needed to
show the uniform small of the unbounded part of solution in $L^{p}$ topology.

Lemma 4.3. _Assume that ( 3.1)-(3.5) hold. Let
$\tau\in\mathbb{R},\omega\in\Omega$ 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
$t\geq T$, the first component of the solution of problem (3.7)-(3.8)
satisfies that for $\xi\in[\tau-1,\tau]$,_

$\displaystyle\|{u}(\xi,\tau-t,\vartheta_{-\tau}\omega,{u}_{0})\|_{p}^{p}\leq
R(\tau,\omega),$

_where $(\tilde{u}_{0},\tilde{v}_{0})\in
D_{\delta}(\tau-t,\vartheta_{-t}\omega)$ and $R(\tau,\omega)$ is as in (4.1)_
only with a difference in the constant $c$.

_Proof_ Multiplying (3.7) by $|u|^{p-2}u$ and then integrating over
$\mathbb{R}^{N}$, we have

$\displaystyle\frac{1}{p}\frac{d}{dt}\|u\|_{p}^{p}+\lambda\|u\|_{p}^{p}$
$\displaystyle\leq\int\limits_{\mathbb{R}^{N}}f(x,\tilde{u})|u|^{p-2}udx-\alpha\int\limits_{\mathbb{R}^{N}}v|u|^{p-1}udx$
$\displaystyle+\int\limits_{\mathbb{R}^{N}}g(t,x)|u|^{p-2}udx+\int\limits_{\mathbb{R}^{N}}(\Delta
h_{1}z_{1}(\vartheta_{t}\omega_{1})-\alpha
h_{2}z_{2}(\vartheta_{t}\omega_{2}))|u|^{p-2}udx.$ (4.6)

By using (3.1) and (3.2) it infer us that

$\displaystyle
f(x,\tilde{u})u\leq-\frac{\alpha_{1}}{2^{p}}|{u}|^{p}+c|h_{1}z_{1}(\vartheta_{t}\omega_{1})|^{p}+\psi_{1}(x)+\psi_{2}(x)|h_{1}z_{1}(\vartheta_{t}\omega_{1})|,$

from which and by using Young inequality we deduce that

$\displaystyle
f(x,\tilde{u})u^{p-2}u\leq-\frac{\alpha_{1}}{2^{p+1}}|{u}|^{2p-2}+\frac{\lambda}{2}u^{p}+c|h_{1}z_{1}(\vartheta_{t}\omega_{1})|^{2p-2}+c\psi_{1}(x)^{p/2}+c|\psi_{2}(x)|^{2}.$

Then by $\psi\in L^{p/2}(\mathbb{R}^{N}$) and $\psi_{2}\in
L^{2}(\mathbb{R}^{N})$, we get the estimate of the nonlinearity in (4.1),

$\displaystyle\int\limits_{\mathbb{R}^{N}}f(x,\tilde{u})u^{p-2}udx$
$\displaystyle\leq-\frac{\alpha_{1}}{2^{p+1}}\|u\|_{2p-2}^{2p-2}+\frac{\lambda}{2}\|u\|_{p}^{p}+c|z_{1}(\vartheta_{t}\omega_{1})|^{2p-2}+c.$
(4.7)

On the other hand, by using Young inequality again we get that

$\displaystyle\alpha\int\limits_{\mathbb{R}^{N}}v|u|^{p-1}udx+\int\limits_{\mathbb{R}^{N}}g(t,x)|u|^{p-2}udx$
$\displaystyle+\int\limits_{\mathbb{R}^{N}}(\Delta
h_{1}z_{1}(\vartheta_{t}\omega_{1})-\alpha
h_{2}z_{2}(\vartheta_{t}\omega_{2}))|u|^{p-2}udx$
$\displaystyle\leq\frac{\alpha_{1}}{2^{p+1}}\|{u}\|_{2p-2}^{2p-2}+c(\|v\|^{2}+\|g(t,.)\|^{2}+|z_{1}(\vartheta_{t}\omega_{1})|^{2}+|z_{2}(\vartheta_{t}\omega_{2})|^{2}).$
(4.8)

Then combination (4.1)-(4.1), we find that

$\displaystyle\frac{d}{dt}\|u\|_{p}^{p}+\delta\|u\|_{p}^{p}\leq
c(\|v\|^{2}+\|g(t,.)\|^{2}+|z_{1}(\vartheta_{t}\omega_{1})|^{2p-2}+|z_{1}(\vartheta_{t}\omega_{1})|^{2}+|z_{2}(\vartheta_{t}\omega_{2})|^{2}+1),$
(4.9)

where $\delta=\min\\{\lambda,\sigma\\}$. We first replace $t$ by $s$ in (4.9).
Then by applying Gronwwall lemma (see also Lemma 5.1 in [26]) over the
interval $[\tau-t,\xi]$ for $t\geq 2$ and $\xi\in[\tau-1,\tau]$, we find that,
along with $\omega$ replaced by $\vartheta_{-\tau}\omega$,

$\displaystyle\|$ $\displaystyle
u(\xi,\tau-t,\vartheta_{-\tau}\omega,{u}_{0})\|_{p}^{p}$
$\displaystyle\leq\frac{e^{\delta}}{\xi-\tau+t}\int\limits_{\tau-t}^{\xi}e^{\delta(s-\tau)}\|u(s,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{0}-h_{1}z_{1}(\vartheta_{-t}\omega_{1})\|_{p}^{p}ds$
$\displaystyle+c\int\limits_{\tau-t}^{\xi}e^{\delta(s-\tau)}\|v(s,\tau-t,\vartheta_{-\tau}\omega,\tilde{v}_{0}-h_{2}z_{2}(\vartheta_{-t}\omega_{2}))\|^{2}ds+c\int\limits_{\tau-t}^{\xi}e^{\delta(s-\tau)}\|g(s,.)\|^{2}ds$
$\displaystyle+c\int\limits_{\tau-t}^{\xi}e^{\delta(s-\tau)}(|z_{1}(\vartheta_{s-\tau}\omega_{1})|^{2p-2}+|z_{1}(\vartheta_{s-\tau}\omega_{1})|^{2}+|z_{2}(\vartheta_{s-\tau}\omega_{2})|^{2}+1)ds$
$\displaystyle\leq
c\int\limits_{\tau-t}^{\tau}e^{\delta(s-\tau)}\|\tilde{u}(s,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{0})\|_{p}^{p}ds$
$\displaystyle+c\int\limits_{\tau-t}^{\tau}e^{\delta(s-\tau)}\|v(s,\tau-t,\vartheta_{-\tau}\omega,\tilde{v}_{0}-h_{2}z_{2}(\vartheta_{-t}\omega_{2}))\|^{2}ds+ce^{-\delta\tau}\int\limits_{-\infty}^{\tau}e^{\delta
s}\|g(s,.)\|^{2}ds$ $\displaystyle+c\int\limits_{-\infty}^{0}e^{\delta
s}(|z_{1}(\vartheta_{s}\omega_{1})|^{2p-2}+|z_{1}(\vartheta_{s}\omega_{1})|^{p}+|z_{1}(\vartheta_{s}\omega_{1})|^{2}+|z_{2}(\vartheta_{s}\omega_{2})|^{2})ds+c,$
(4.10)

where we have used that $\frac{1}{\xi-\tau+t}\leq 1$ for $t\geq 2$. Then by
applying (4.3) to (4.1), we find that we have got the desired. $\ \ \ \ \ \
\Box$

_Remark:_ A similar calculation gives us that $u\in L^{p}(\mathbb{R}^{N})$,
which implies that for every $t>0,\tau\in\mathbb{R},\omega\in\Omega$ and
$t>\tau$, the mapping $\varphi(t,\tau,\omega,.):L^{2}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})\mapsto L^{p}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})$. This demonstrate that the cocycle $\varphi$ defined by
(3) satisfies condition _(H1)_.

Let $u$ be the first component of solutions of problem (3.7)-(3.8). Given
$\tau\in\mathbb{R},\omega\in\Omega$, let $M=M(\tau,\omega)>0$ and

$\displaystyle\mathbb{R}^{N}(|u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{0})|\geq
M)=\\{x\in\mathbb{R}^{N};|u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{0})|\geq
M|\\}.$

Then we have

Lemma 4.4. _Assume that ( 3.1)-(3.5) hold. Let
$\tau\in\mathbb{R},\omega\in\Omega$ and
$D=\\{D(\tau,\omega);\tau\in\mathbb{R},\omega\in\Omega\\}\in\mathcal{D}_{\delta}$.
Then for any $\eta>0$, there exist constants $M=M(\tau,\omega,\eta)>0$ and
$T=T(\tau,\omega,D)\geq 2$ such that for all $t\geq T$, the first component
$u$ of solution $(u,v)$ of problem (3.7)-(3.8) satisfies_

$\displaystyle
meas(\mathbb{R}^{N}(|u(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{0}-h_{1}z_{1}(\omega_{1})|\geq
M)\leq\eta,$

_where meas(.) is the measure of a set and $(\tilde{u}_{0},\tilde{v}_{0})\in
D(\tau-t,\vartheta_{-t}\omega)$._

_Proof_ By (4.2), there exist positive constants $T=T(\tau,\omega,D)>0$ and
$R=R(\tau,\omega)$ such that for all $t\geq T$,

$\displaystyle\|u(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{0}-h_{1}z_{1}(\vartheta_{-t}\omega_{1})\|^{2}\leq
R(\tau,\omega),$ (4.11)

where $(\tilde{u}_{0},\tilde{v}_{0})\in D(\tau-t,\vartheta_{-t}\omega)$. Then
for any $M>0$ and
$|u(\tau)|=|u(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{0}-h_{1}z_{1}(\omega_{1})|\geq
M$, by using (4.11) we have

$\displaystyle
M^{2}meas(\mathbb{R}^{N}(|u(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{0}-h_{1}z_{1}(\omega_{1})|\geq
M)$ $\displaystyle\leq\int\limits_{\mathbb{R}^{N}(|u(\tau)|\geq
M)}|u(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{0}-h_{1}z_{1}(\vartheta_{-t}\omega_{1})|^{2}dx\leq
R(\tau,\omega).$ (4.12)

By choosing $M=M(\tau,\omega,\eta)>(\frac{R(\tau,\omega)}{\eta})^{1/2}$, we
get the result from (4.1). This ends the proof. $\ \ \ \ \ \ \Box$

To prove the following lemma, we need to introduce some notations. We will use
a truncated version of the solution, $u-M$. Define

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

The next lemma will show that the unbounded part of the absolute value $|u|$
vanishes in $L^{p}$-norm on the state domain
$\mathbb{R}^{N}(|u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{0})|\geq M)$ for $M$
large enough.

Lemma 4.5. Assume that (3.1)-(3.5) hold. Let
$\tau\in\mathbb{R},\omega\in\Omega$ and
$D=\\{D(\tau,\omega);\tau\in\mathbb{R},\omega\in\Omega\\}\in\mathcal{D}_{\delta}$.
Then for any $\eta>0$, there exist constants $M=M(\tau,\omega,\eta,D)>1$ and
$T=T(\tau,\omega,D)\geq 2$ such that the first component $\tilde{u}$ of
solutions $(\tilde{u},\tilde{v})$ of problem (1.1) satisfies

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

where $(\tilde{u}_{0},\tilde{v}_{0})\in D(\tau-t,\vartheta_{-t}\omega)$ and
$\mathbb{R}^{N}(|\tilde{u}(\tau)|\geq
M)=\mathbb{R}^{N}(|\tilde{u}(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{0})|\geq
M)$.

_Proof_ Given $s\in[\tau-1,\tau]$, by replacing $\omega$ by
$\vartheta_{-\tau}\omega$ in (3.7) we get that

$u=u(s)=:u(s,\tau-t,\vartheta_{-\tau}\omega,u_{0}),v=v(s)=:v(s,\tau-t,\vartheta_{-\tau}\omega,v_{0})\
\ s\in[\tau-1,\tau],$

satisfy the following differential equations:

$\displaystyle\frac{du}{ds}+\lambda u-\Delta u+\alpha
v=f(x,\tilde{u})+{g}(s,x)+\Delta
h_{1}z_{1}(\vartheta_{s-\tau}\omega_{1})-\alpha
h_{2}z_{2}(\vartheta_{s-\tau}\omega_{2}),$ (4.13)

where
$(u_{0},v_{0})=(\tilde{u}_{0}-h_{1}z_{1}(\omega_{1}),\tilde{v}_{0}-h_{2}z_{2}(\omega_{2}))$
with $(\tilde{u}_{0},\tilde{v}_{0})\in D(\tau-t,\vartheta_{-t}\omega)$.

For fixed $\tau\in\mathbb{R}$ and $\omega\in\Omega$, we write

$\displaystyle
Z=Z(\tau,\omega)=\max\limits_{s\in[\tau-1,\tau]}\\{\|h_{1}z_{1}(\vartheta_{s-\tau}\omega_{1})\|_{L^{\infty}(\mathbb{R}^{n})}\\},\
\ \ M=M(\tau,\omega)>\max\\{1,Z\\}.$ (4.14)

We multiply (4.13) by $(u-M)_{+}^{p-1}$ and integrate over $\mathbb{R}^{N}$ to
get that for every $s\in[\tau-1,\tau]$,

$\displaystyle\frac{1}{p}\frac{d}{ds}\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{p}dx+\lambda\int\limits_{\mathbb{R}^{N}}u(u-M)_{+}^{p-1}dx-\int\limits_{\mathbb{R}^{N}}\Delta
u(u-M)_{+}^{p-1}dx$ $\displaystyle\ \ \
=-\alpha\int\limits_{\mathbb{R}^{N}}v(u-M)_{+}^{p-1}dx+\int\limits_{\mathbb{R}^{N}}f(x,\tilde{u})(u-M)_{+}^{p-1}dx$
$\displaystyle\ \ \
+\int\limits_{\mathbb{R}^{N}}{g}(s,x)(u-M)_{+}^{p-1}dx+\int\limits_{\mathbb{R}^{N}}(\Delta
h_{1}z_{1}(\vartheta_{s-\tau}\omega_{1})-\alpha
h_{2}z_{2}(\vartheta_{s-\tau}\omega_{2}))(u-M)_{+}^{p-1}dx.$ (4.15)

We now have to estimate every term in (4.1). First, it is obvious that

$\displaystyle-\int\limits_{\mathbb{R}^{N}}\Delta
u(u-M)_{+}^{p-1}dx=(p-1)\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{p-2}|\nabla
u|^{2}dx\geq 0,$ (4.16)
$\displaystyle\lambda\int\limits_{\mathbb{R}^{N}}u(u-M)_{+}^{p-1}dx\geq\lambda\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{p}dx.$
(4.17)

The most involved calculations are to estimate the nonlinearity in (4.1). Note
that

$\displaystyle|\tilde{u}(s)|^{p-1}\geq
2^{2-p}|u(s)|^{p-1}-|h_{1}z_{1}(\vartheta_{s-\tau}\omega_{1})|^{p-1},$ (4.18)

and by (4.14), if $u(s)>M$ then we have for all $s\in[\tau-1,\tau]$,

$\displaystyle\tilde{u}(s)=u(s)+h_{1}z_{1}(\vartheta_{s-\tau}\omega_{1})\geq
u(s)-|h_{1}z_{1}(\vartheta_{s-\tau}\omega_{1})|\geq u(s)-M>0.$ (4.19)

Then by (3.1) and along with (4.18)-(4.19), we find that for
$s\in[\tau-1,\tau]$ and $u(s)>M$,

$\displaystyle f(x,\tilde{u})$
$\displaystyle\leq-\alpha_{1}|\tilde{u}|^{p-1}+\frac{1}{\tilde{u}}\psi_{1}(x)\leq-\alpha_{1}2^{2-p}|u|^{p-1}+\alpha_{1}|h_{1}z_{1}(\vartheta_{s-\tau}\omega_{1})|^{p-1}+\frac{\psi_{1}(x)}{\tilde{u}}.$
(4.20)

Then by (4.20) and along with (4.19) we get that for $u(s)>M$,

$\displaystyle f(x,\tilde{u})$
$\displaystyle\leq-\alpha_{1}2^{2-p}|u|^{p-1}+\alpha_{1}|h_{1}z_{1}(\vartheta_{s-\tau}\omega_{1})|^{p-1}+|\psi_{1}(x)|(u-M)^{-1}$
$\displaystyle\leq-\alpha_{1}2^{1-p}M^{p-2}(u-M)-\alpha_{1}2^{1-p}(u-M)^{p-1}$
$\displaystyle+\alpha_{1}|h_{1}z_{1}(\vartheta_{s-\tau}\omega_{1})|^{p-1}+|\psi_{1}(x)|(u-M)^{-1},$

from which it follows that

$\displaystyle\ \ \
\int\limits_{\mathbb{R}^{N}}f(x,\tilde{u})(u-M)_{+}^{p-1}dx$
$\displaystyle\leq-\alpha_{1}2^{1-p}M^{p-2}\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{p}dx-\alpha_{1}2^{1-p}\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{2p-2}dx$
$\displaystyle\ \ \
+\alpha_{1}\int\limits_{\mathbb{R}^{N}}|h_{1}z_{1}(\vartheta_{s-\tau}\omega_{1})|^{p-1}(u-M)_{+}^{p-1}dx+\int\limits_{\mathbb{R}^{N}}|\psi_{1}(x)|(u-M)_{+}^{p-2}dx.$
(4.21)

Since $h_{1}\in L^{2}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})$, by the
Sobolev interpolation, $h_{1}\in L^{2p-2}(\mathbb{R}^{N})$. Then by using
Young inequality we have

$\displaystyle\alpha_{1}\int\limits_{\mathbb{R}^{N}}|h_{1}z_{1}(\vartheta_{s-\tau}\omega_{1})|^{p-1}(u-M)_{+}^{p-1}dx\leq\frac{1}{2}\alpha_{1}2^{1-p}\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{2p-2}dx+c|z_{1}(\vartheta_{s-\tau}\omega_{1})|^{2p-2},$
(4.22)

and by $\psi_{1}\in L^{p/2}(\mathbb{R}^{N})$,

$\displaystyle\int\limits_{\mathbb{R}^{N}}|\psi_{1}(x)|(u-M)_{+}^{p-2}dx\leq\frac{1}{2}\lambda\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{p}dx+c\|\psi_{1}\|^{p/2}_{p/2}.$
(4.23)

Combination (4.1)-(4.23) we find

$\displaystyle\ \ \ \ \
\int\limits_{\mathbb{R}^{N}}f(x,\tilde{u})(u-M)_{+}^{p-1}dx$
$\displaystyle\leq-\alpha_{1}2^{1-p}M^{p-2}\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{p}dx-\alpha_{1}2^{-p}\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{2p-2}dx$
$\displaystyle\ \ \ \ \
+c|z_{1}(\vartheta_{s-\tau}\omega_{1})|^{2p-2}+\frac{1}{2}\lambda\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{p}dx+c\|\psi_{1}\|^{p/2}_{p/2}.$
(4.24)

On the other hand, by using Young inequality, we also get that

$\displaystyle-\alpha\int\limits_{\mathbb{R}^{N}}v(u-M)_{+}^{p-1}dx\leq\frac{1}{4}\alpha_{1}2^{-p}\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{2p-2}dx+c\int\limits_{\mathbb{R}^{N}(u(s)\geq
M)}v^{2}dx,$ (4.25)
$\displaystyle\int\limits_{\mathbb{R}^{N}}{g}(s,x)(u-M)_{+}^{p-1}dx\leq\frac{1}{8}\alpha_{1}2^{-p}\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{2p-2}dx+c\int\limits_{\mathbb{R}^{N}(u\geq
M)}{g}^{2}(s,x)dx,$ (4.26)

and by $h_{1}\in H^{2}(\mathbb{R}^{N})$ and $h_{2}\in L^{2}(\mathbb{R}^{N})$,

$\displaystyle\int\limits_{\mathbb{R}^{N}}(\Delta
h_{1}z_{1}(\vartheta_{s-\tau}\omega_{1})-\alpha
h_{2}z_{2}(\vartheta_{s-\tau}\omega_{2}))(u-M)_{+}^{p-1}dx$
$\displaystyle\leq\frac{1}{8}\alpha_{1}2^{-p}\int\limits_{\mathbb{R}^{N}}(u-M)_{+}^{2p-2}dx+c(|z_{1}(\vartheta_{s-\tau}\omega_{1})|^{2}+|z_{2}(\vartheta_{s-\tau}\omega_{2})|^{2}).$
(4.27)

Then Combination (4.1)-(4.17) and (4.1)-(4.1), we obtain that

$\displaystyle\frac{d}{ds}$
$\displaystyle\int\limits_{\mathbb{R}^{N}}(u(s)-M)_{+}^{p}dx+\alpha_{1}2^{1-p}M^{p-2}\int\limits_{\mathbb{R}^{N}}(u(s)-M)_{+}^{p}dx\leq
c(\|{g}(s,.)\|^{2}+\|v(s)\|^{2})$ $\displaystyle\ \ \ \
+c(\|\psi_{1}\|_{p/2}^{p/2}+|z_{1}(\vartheta_{s-\tau}\omega_{1})|^{2p-2}+|z_{1}(\vartheta_{s-\tau}\omega_{1})|^{2}+|z_{2}(\vartheta_{s-\tau}\omega_{2})|^{2}),$
(4.28)

where the positive constant $c$ is independent of $\tau,\omega$ and $M$. For
convenience, we put

$k=k(\tau,\omega,M)=2^{1-p}M^{p-2},$

and

$\chi(\vartheta_{s}\omega)=c(\|\psi_{1}\|_{p/2}^{p/2}+|z_{1}(\vartheta_{s}\omega_{1})|^{2p-2}+|z_{1}(\vartheta_{s}\omega_{1})|^{2}+|z_{2}(\vartheta_{s}\omega_{2})|^{2}).$

Then we write (4.1) as

$\displaystyle\frac{d}{ds}\int\limits_{\mathbb{R}^{N}}(u(s)-M)_{+}^{p}dx$
$\displaystyle+k\int\limits_{\mathbb{R}^{N}}(u(s)-M)_{+}^{p}dx\leq
c(\|{g}(s,.)\|^{2}+\|v(s)\|^{2})+\chi(\vartheta_{s-\tau}\omega),$ (4.29)

where $s\in[\tau-1,\tau]$. Multiplying (4.29) by $e^{k(s-\tau)}$ and
integrating first with respect to $s$ over $[\sigma,\tau]$ for
$\sigma\leq\tau$ and then with respect to $\sigma$ over $[\tau-1,\tau]$, we
find that

$\displaystyle\int\limits_{\mathbb{R}^{N}}\Big{(}u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{0})-M\Big{)}_{+}^{p}dx$
$\displaystyle\leq\int\limits_{\tau-1}^{\tau}e^{k(s-\tau)}\int\limits_{\mathbb{R}^{N}}\Big{(}u(s,\tau-t,\vartheta_{-\tau}\omega,u_{0})-M\Big{)}_{+}^{p}dxds$
$\displaystyle+c\int\limits_{\tau-1}^{\tau}e^{k(s-\tau)}(\|g(s,.)\|^{2}+\|v(s,\tau-t,\vartheta_{-\tau}\omega,v_{0})\|^{2})ds+\int\limits_{\tau-1}^{\tau}e^{k(s-\tau)}\chi(\vartheta_{s-\tau}\omega)ds$
$\displaystyle\leq\int\limits_{\tau-1}^{\tau}e^{k(s-\tau)}(\|u(s,\tau-t,\vartheta_{-\tau}\omega,u_{0})\|_{p}^{p}+\|v(s,\tau-t,\vartheta_{-\tau}\omega,v_{0})\|^{2})ds$
$\displaystyle+c\int\limits_{\tau-1}^{\tau}e^{k(s-\tau)}\|g(s,.)\|^{2}ds+\int\limits_{\tau-1}^{\tau}e^{k(s-\tau)}\chi(\vartheta_{s-\tau}\omega)ds$
$\displaystyle=I_{1}+I_{2}+I_{3},$ (4.30)

where

$I_{1}=\int\limits_{\tau-1}^{\tau}e^{k(s-\tau)}(\|u(s,\tau-t,\vartheta_{-\tau}\omega,u_{0})\|_{p}^{p}+\|v(s,\tau-t,\vartheta_{-\tau}\omega,v_{0})\|^{2})ds,$
$I_{2}=c\int\limits_{\tau-1}^{\tau}e^{k(s-\tau)}\|g(s,.)\|^{2}ds,$

and

$I_{3}=\int\limits_{\tau-1}^{\tau}e^{k(s-\tau)}\chi(\vartheta_{s-\tau}\omega)ds.$

We now calculate that $I_{1},I_{2}$ and $I_{3}$ are not larger than $\eta/3$
as $M\rightarrow+\infty$, respectively. By Lemma 4.2 and 4.3, for every fixed
$\tau\in\mathbb{R}$ and $\omega\in\Omega$ there exists
$T_{1}=T_{1}(\tau,\omega,D)\geq 2$ such that for all $t\geq T_{1}$,

$\displaystyle I_{1}\leq
2R(\tau,\omega)\int\limits_{\tau-1}^{\tau}e^{k(s-\tau)}ds\leq\frac{2R(\tau,\omega)}{k}\leq\eta/3,$
(4.31)

as $M$ large enough, since $k\rightarrow+\infty$ as $M\rightarrow+\infty$. In
order to estimate $I_{2}$, choosing $k>\delta$ and taking $\varsigma\in(0,1)$,
we have

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

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 is small. Then when $M\rightarrow+\infty$, we
have

$\displaystyle I_{2}\leq\eta/3.$ (4.32)

To estimate $I_{3}$, we see that

$\displaystyle
I_{3}=\int\limits_{-1}^{0}e^{ks}\chi(\vartheta_{s}\omega)ds=\max_{0\leq s\leq
1}\chi(\vartheta_{s}\omega)\frac{1}{k}\leq\eta/3,$ (4.33)

as $k$ large enough. Then from (4.1)-(4.33), there exists
$M_{1}=M_{1}(\tau,\omega,\eta,D)>0$ large enough such that

$\displaystyle\sup_{t\geq
T_{1}}\int\limits_{\mathbb{R}^{N}}\Big{(}u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{0})-M_{1}\Big{)}_{+}^{p}dx$
$\displaystyle\leq\eta,$ (4.34)

where $T_{1}$ is as in (4.31). If
$u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{0})\geq 2M_{1}$, then
$u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{0})-M_{1}\geq\frac{u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{0})}{2},$
and therefore by (4.34) it infer us that

$\displaystyle\sup_{t\geq T_{1}}\int\limits_{\mathbb{R}^{N}(u(\tau)\geq
2M_{1})}|{u}(\tau,\tau-t,\vartheta_{-\tau}\omega,{u}_{0})|^{p}dx$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \leq\sup_{t\geq
T_{1}}2^{p}\int\limits_{\mathbb{R}^{N}}\Big{(}u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{0})-M_{1}\Big{)}_{+}^{p}dx\leq
c\eta.$ (4.35)

On the other hand, since by Sobolev interpolation we know that $h_{1}\in
L^{p}(\mathbb{R}^{N})$, then there is a small constant $\epsilon>0$ such that
for any subset $E\subset\mathbb{R}^{N}$ with $meas(E)<\epsilon$,

$\displaystyle\int\limits_{E}|h_{1}(x)|^{p}ds\leq\frac{\eta}{|z_{1}(\omega_{1})|^{p}}.$
(4.36)

But by Lemma 4.4, there exist positive constants
$M_{2}=M_{2}(\tau,\omega,\eta)>M_{1}$ and $T_{2}=T_{2}(\tau,\omega,D)>T_{1}$
such that

$\displaystyle\sup_{t\geq
T_{2}}meas(\mathbb{R}^{N}(|u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{0})|\geq
2M_{2}))\leq\min\\{\eta,\epsilon\\}.$ (4.37)

Then it follows from (4.36)-(4.37) that

$\displaystyle\sup_{t\geq T_{2}}\int\limits_{\mathbb{R}^{N}(u(\tau)\geq
2M_{2})}|h_{1}z_{1}(\omega_{1})|^{p}dx\leq\eta.$ (4.38)

On the other hand, by (4.1) we get that

$\displaystyle\sup_{t\geq T_{2}}\int\limits_{\mathbb{R}^{N}(u(\tau)\geq
2M_{2})}|{u}(\tau,\tau-t,\vartheta_{-\tau}\omega,{u}_{0})|^{p}dx\leq c\eta.$
(4.39)

By taking advantage of the relation
$\tilde{u}(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{0})=u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{0})+h_{1}z(\omega_{1})$
and association with (4.14) we can deduce that

$\displaystyle\mathbb{R}^{N}(\tilde{u}(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{0})\geq
2M_{2}+Z)\subseteq\mathbb{R}^{N}(u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{0})\geq
2M_{2}),$

from which it implies that

$\displaystyle\sup_{t\geq
T_{2}}\int\limits_{\mathbb{R}^{N}(\tilde{u}(\tau)\geq
2M_{2}+Z)}|\tilde{u}(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{0})|^{p}dx$
$\displaystyle\ \ \ \ \ \ \ \leq\sup_{t\geq
T_{2}}2^{p-1}\int\limits_{\mathbb{R}^{N}(u(\tau)\geq
2M_{2})}|u(\tau,\tau-t,\vartheta_{-\tau}\omega,u_{0})|^{p}dx$ $\displaystyle\
\ \ \ \ \ \ \ \ \ \ \ \ +\sup_{t\geq
T_{2}}2^{p-1}\int\limits_{\mathbb{R}^{N}(u(\tau)\geq
2M_{2})}|h_{1}z(\omega_{1})|^{p}dx.$ (4.40)

Therefore combination (4.38)-(4.1), we find that

$\displaystyle\sup_{t\geq
T_{2}}\int\limits_{\mathbb{R}^{N}(\tilde{u}(\tau)\geq
2M_{2}+Z)}|\tilde{u}(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{0})|^{p}dx\leq
c\eta.$ (4.41)

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

$\displaystyle\sup_{t\geq\tilde{T}_{2}}\int\limits_{\mathbb{R}^{N}(\tilde{u}(\tau)\leq-2\tilde{M}_{2}-Z)}$
$\displaystyle|\tilde{u}(\tau,\tau-t,\vartheta_{-\tau}\omega,\tilde{u}_{0})|^{p}dx\leq
c\eta.$ (4.42)

Then (4.41) and (4.42) together imply the desired. $\ \ \ \ \ \ \ \ \ \Box$

###  4.2. Existence of pullback random attractors in
$L^{\varpi}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$

Note that the existence of pullback attractor in $L^{2}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})$ is prove in [2], which is stated by Theorem 3.1 in
Section 3. In this subsection, we will prove that this also happens in
$L^{\varpi}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$ for every
$\varpi\in(2,p]$. By Theorem 2.6, it remains to prove the asymptotic
compactness of solutions of problem (1.1) in $L^{\varpi}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})$ for every $\varpi\in(2,p]$, in which case we need to
assume that $p>2$.

First the following lemma is borrowed from [2].

Lemma 4.6. _Assume that ( 3.1)-(3.5) hold. Then the cocycle $\varphi$
associated with problem (1.1) is $\mathcal{D}_{\delta}$-asymptotically compact
in $L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$; that is, for every
$\tau\in\mathbb{R},\omega\in\Omega$, the sequence $\\{\varphi(t_{n},\tau-
t_{n},\vartheta_{-t_{n}}\omega,(\tilde{u}_{0,n},\tilde{v}_{0,n}))\\}$ has a
convergent subsequence in $L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$
whenever $t_{n}\rightarrow+\infty$ and $(\tilde{u}_{0,n},\tilde{v}_{0,n})\in
D(\tau-t_{n},\vartheta_{-t_{n}}\omega)\in\mathcal{D}_{\delta}$._

By Proposition 2.8 and Lemma 4.5 and 4.6, we deduce the following, which reads

Lemma 4.7. _Assume that ( 3.1)-(3.5) hold. Then the cocycle $\varphi$
associated with problem (1.1) is $\mathcal{D}_{\delta}$-asymptotically compact
in $L^{p}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$; that is, for every
$\tau\in\mathbb{R},\omega\in\Omega$, the sequence $\\{\varphi(t_{n},\tau-
t_{n},\vartheta_{-t_{n}}\omega,(\tilde{u}_{0,n},\tilde{v}_{0,n}))\\}$ has a
convergent subsequence in $L^{p}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$
whenever $t_{n}\rightarrow+\infty$ and $(\tilde{u}_{0,n},\tilde{v}_{0,n})\in
D(\tau-t_{n},\vartheta_{-t_{n}}\omega)\in\mathcal{D}_{\delta}$._

Then by Sobolev interpolation and Lemma 4.6 and Lemma 4.7, we immediately have

Lemma 4.8. _Assume that ( 3.1)-(3.5) hold. Then the cocycle $\varphi$
associated with problem (1.1) is $\mathcal{D}_{\delta}$-pullback
asymptotically compact in $L^{\varpi}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})$ for every $\varpi\in(2,p]$; that is, for every
$\tau\in\mathbb{R},\omega\in\Omega$, and $\varpi\in(2,p]$, the sequence
$\\{\varphi(t_{n},\tau-
t_{n},\vartheta_{-t_{n}}\omega,(\tilde{u}_{0,n},\tilde{v}_{0,n}))\\}$ has a
convergent subsequence in $L^{\varpi}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})$ whenever $t_{n}\rightarrow+\infty$ and
$(\tilde{u}_{0,n},\tilde{v}_{0,n})\in D(\tau-
t_{n},\vartheta_{-t_{n}}\omega)\in\mathcal{D}_{\delta}$._

By Lemma 4.1, Lemma 4.6, Lemma 4.8, and Theorem 2.6, we get readily the
following result, which implies that the obtained
$\mathcal{D}_{\delta}$-pullback random attractor $\mathcal{A}$ in
$L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$ is compact and attracting
in $L^{\varpi}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$ for every
$\varpi\in(2,p]$ with $p>2$.

Theorem 4.9. Assume that (3.1)-(3.5) hold and $\mathcal{D}_{\sigma}$ is
defined by (3.10). Then the cocycle $\varphi$ associated with problem (1.1)
possesses a unique $\mathcal{D}_{\delta}$-pullback random attractor
$\mathcal{A}_{\varpi}=\\{\mathcal{A}_{\varpi}(\tau,\omega);\tau\in\mathbb{R},\omega\in\Omega\\}\in\mathcal{D}_{\delta}$
in $L^{\varpi}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$ for every
$\varpi\in(2,p]$. Furthermore, for every $\tau\in\mathbb{R},\omega\in\Omega$,

$\displaystyle\mathcal{A}_{\varpi}(\tau,\omega)$
$\displaystyle=\cap_{s>0}\overline{\cup_{t\geq
s}\varphi(t,\tau-t,\vartheta_{-t}\omega,K(\tau-t,\vartheta_{-t}\omega))}^{L^{\varpi}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})}$ (4.43) $\displaystyle=\cap_{s>0}\overline{\cup_{t\geq
s}\varphi(t,\tau-t,\vartheta_{-t}\omega,K(\tau-t,\vartheta_{-t}\omega))}^{L^{2}(\mathbb{R}^{N})\times
L^{2}(\mathbb{R}^{N})}=\mathcal{A}(\tau,\omega),$

where $K=\\{K(\tau,\omega);\tau\in\mathbb{R},\omega\in\Omega\\}$ and
$\mathcal{A}=\\{\mathcal{A}(\tau,\omega);\tau\in\mathbb{R},\omega\in\Omega\\}$
are the $\mathcal{D}_{\delta}$-pullback attractor and pullback absorbing set
of $\varphi$ in $L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$ defined as
in (3.11) and (3.12), respectively.

Acknowledgments:

We are appreciated to Yangrong Li for his warm discussions and useful
suggestions

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

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