# On the forward dynamical behavior of nonautonomous lattice dynamical systems
††This work is supported by NSF of China under Grants 11871368, 11801190.

Chunqiu Li,††Corresponding author. E-mail<EMAIL_ADDRESS>Jintao Wang ††
Email<EMAIL_ADDRESS>
Department of Mathematics, Wenzhou University,
Wenzhou, Zhejiang Province, 325035, P. R. China

###### Abstract

In this article, we study the forward dynamical behavior of nonautonomous
lattice systems. We first construct a family of sets
$\\{{\mathcal{A}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ in arbitrary
small neighborhood of a global attractor of the skew-product flow generated by
a general nonautonomous lattice system, which is forward invariant and
uniformly forward attracts any bounded subset of the phase space. Moreover,
under some suitable conditions, we further construct a family of sets
$\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ such that it
uniformly forward exponentially attracts bounded subsets of the phase space.
As an application, we study the discrete Gray-Scott model in detail and
illustrate how to apply our abstract results to some concrete lattice system.

Keywords: Lattice dynamical system; Pullback attractor; Forward attraction;
Exponential attraction; Nonautonomous system MSC2010: 35B40, 35B41, 35Q92,
37L50.

##  1 Introduction

It is well known that the global attractor is an appropriate object to
describe the asymptotic behavior of autonomous systems, and the corresponding
theory of attractors has been fully developed in the past decades; see [23,
34, 37, 38]. In contrast, the case for nonautonomous systems seems to be far
more difficult, though the study of the dynamics of nonautonomous systems has
been paid much attention in recent years; see e.g. [7, 11]. In fact, even the
notions of attractors are still under investigations.

Similar to autonomous systems, there are several concepts, uniform attractors,
kernel sections, pullback attractors and forward attractors which can be used
to capture the dynamics of nonautonomous dynamical systems (NDSs in short).
These attractors can be regarded as a natural generalization of global
attractors of autonomous systems. Among these attractors, pullback attractors
and forward attractors are very important and extensively studied by
researchers. On the existence of a pullback attractor, there are quite general
results for both lattice systems and continuous systems; see e.g. [6, 7, 21,
42, 43, 50], etc. On the other hand, in most cases we are more concerned with
the evolution of a nonautonomous lattice system in the future. However, some
relative works mainly concentrate on the continuous case and its pullback
asymptotic behavior; see e.g. [5, 26, 27]. In general, a pullback attractor
may not be forward attracting and provide little information on this aspect.

Lattice dynamical systems (LDSs in short) are spatiotemporal systems with
discretization in some variables, which appear in many different fields such
as biology [29], chemical reaction theory [17], electrical engineering [13]
and so on. Nowadays, the asymptotic behavior of LDSs has attracted much
attention from researchers; see e.g. [2, 4, 9, 25, 31, 33, 40, 52, 53, 54].
Particularly, Zhou and Zhao [54] gave some sufficient and necessary conditions
for the existence of a compact uniform attractor in the framework of processes
on a Hilbert space of infinite sequences. Concerning the uniform attractors,
kernel sections, pullback attractors of LDSs, one can find a vast body of
literature on their existence and structure; see e.g. [39, 47, 48, 49, 51,
55].

In this article, we are concerned with the forward dynamical behavior of
general nonautonomous LDSs. Generally speaking, the forward attractor may be
quite suitable to capture the forward dynamical behavior of a nonautonomous
LDS. Unfortunately, there are few results on the existence of such type of
attractors except in some particular cases of continuous systems such as the
periodic and the asymptotically autonomous ones, and these problems are still
undergoing investigations; see e.g. Carvalho et al. [8, p. 595], Cui et al.
[10], Cheban et al. [12], Kloeden et al. [30], Langa et al. [32], and Wang et
al. [41]. It is well known that if the pullback attraction of a pullback
attractor ${\mathcal{A}}=\\{A(\sigma)\\}_{\sigma\in\Sigma}$ of a family of
processes $\\{U_{\sigma}(t,0)\\}_{t\geqslant 0},\sigma\in\Sigma$ in the space
$E$ is uniform with respect to $\sigma\in\Sigma$, where $\Sigma$ is a compact
metric space and $E$ is a complete metric space, then it is also a forward
attractor. Using this result, the authors [12] considered the continuous
system and verified that if the section $A(\sigma)$ of ${\mathcal{A}}$ is
lower semicontinuous in $\sigma$, then ${\mathcal{A}}$ is a forward attractor.
However, it is hard to prove the lower semicontinuity of a family of sets
generally.

Motivated by these works mentioned above, we shall construct a family of sets,
which may be used to describe the forward dynamical behavior of a general LDS
in some sense. Specifically, let $\\{U_{\sigma}(t,0)\\}_{t\geqslant
0},\sigma\in\Sigma$ be a family of processes generated by the LDS in a phase
space $E$, where $\sigma\in\Sigma$ and $\Sigma$ is called a symbol space. If
the mappings $(u,\sigma)\rightarrow U_{\sigma}(t,0)u$ are continuous from
$E\times\Sigma$ to $E$, the family of continuous processes can reduce to a
autonomous semigroup $\Psi$, called a skew-product flow. It will be shown that
in any small $\varepsilon$-neighborhood of the global attractor
${\mathcal{A}}$ of $\Psi$, there exists a family of sets
$\\{{\mathcal{A}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$, which is
forward invariant for $\\{U_{\sigma}(t,0)\\}_{t\geqslant 0},\sigma\in\Sigma$,
such that it uniformly forward attracts each bounded subset of $E$.

The second result of this article is to construct a family of sets
$\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ in an arbitrary
small $\varepsilon$-neighborhood of the global attractor such that it
uniformly forward exponentially attracts each bounded subset of $E$. For this
purpose, we first construct a pullback exponential attractor for
$\\{U_{\sigma}(t,0)\\}_{t\geqslant 0},\sigma\in\Sigma$. As we all know, an
exponential attractor of an autonomous system, which was first constructed by
Eden et al. [18], attracts the trajectories at an exponential rate and is more
robust under perturbations. For nonautonomous systems, pullback exponential
attractors are extensively studied; see e.g. [55] for lattice systems and [14,
15, 19, 20, 46] for continuous systems. Particularly, the authors [20]
constructed pullback exponential attractors of nonautonomous systems in the
framework of a process on Banach spaces. Later, based on the results in [20],
Zhou and Han [55] presented a sufficient and necessary condition for the
existence of pullback exponential attractors of the process generated by a
nonautonomous LDS. The interested readers are referred to [1, 3, 16, 22, 24]
for some concrete continuous and lattice systems on the existence and
construction of exponential attractors. By the exponential attractivity of the
pullback exponential attractor, we will further show that the family of sets
$\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ uniformly
forward exponentially attracts each bounded subset of the phase space under
some suitable conditions.

As an application of our main results, we consider the nonautonomous discrete
three-component reversible Gray-Scott model driven by quasi-periodic external
forces. When there are no external forces, the Gray-Scott equations reduce to
the corresponding autonomous Gray-Scott model, which was first introduced by
H. Mahara et al. [36] to describe the isothermal, cubic autocatalytic
continuously fed and diffusive reactions of two chemicals [44]. Concerning the
initial boundary value and Cauchy problems for both discrete and continuous
Gray-Scott equations, there are many works in its autonomous and nonautonomous
cases; see [28, 44, 45] and the references therein.

The paper is organized as follows. In section 2, we make some preliminaries.
Section 3 is devoted to our main results. In section 4 we give an example to
illustrate our results. Finally, we give a remark on some extensions.

##  2 Preliminaries

In this section, we collect some basic concepts on dynamical systems. One can
also see [7, 11] etc., for details. Let $E$ be a complete metric space and
$\Sigma$ a metric space with the metrics $d$ and $\rho$, respectively. Given
two subsets $A,B$ of $E$. Define the Hausdorff-semidistance and Hausdorff
distance of $A$ and $B$, respectively, as

${\rm d}_{H}(A,B)=\sup_{x\in A}d(x,B),\hskip
11.38092pt{\rm\delta}_{H}(A,B)=\max\\{{\rm d}_{H}(A,B),{\rm d}_{H}(B,A)\\}.$

The $\varepsilon$-neighborhood of $A$ is defined by the set

$N_{\varepsilon}(A)=\\{x\in X:d(x,A)<\varepsilon\\}.$

Let $E$ be a Banach space with norm $\|\cdot\|$. A function $f(t)\in
L_{loc}^{2}(\mathbb{R},E)$ is called translation bounded in
$L_{loc}^{2}(\mathbb{R},E)$ if

$\|f\|_{L_{b}^{2}(\mathbb{R},E)}=\sup_{t\in\mathbb{R}}\int^{t+1}_{t}\|f\|_{E}^{2}{\rm
d}s<+\infty.$

Denote $L_{b}^{2}(\mathbb{R},E)$ the sets of translation bounded functions in
$L_{loc}^{2}(\mathbb{R},E)$.

A function $f(t)\in L_{loc}^{2}(\mathbb{R},E)$ is said to be translation
compact in $L_{loc}^{2}(\mathbb{R},E)$, if the set
$\\{f(\tau+\cdot):\tau\in\mathbb{R}\\}$ is precompact in
$L_{loc}^{2}(\mathbb{R},E)$. By $L_{c}^{2}(\mathbb{R},E)$ we denote the union
of translation compact functions in $L_{loc}^{2}(\mathbb{R},E)$. Then
$L_{c}^{2}(\mathbb{R},E)\subset L_{loc}^{2}(\mathbb{R},E)$ (see [35]).

Let $E$ be a Banach space. A family of mappings
$\\{U(t,\tau)\\}_{t\geqslant\tau}$ is called a process in $E$ if

  1. (i)

$U(t,s)U(s,\tau)=U(t,\tau),\hskip 11.38092ptt\geqslant
s\geqslant\tau,\tau\in\mathbb{R};$

  2. (ii)

$U(\tau,\tau)={\rm Id}_{E},\hskip 11.38092pt\tau\in\mathbb{R}.$

$\\{U_{\sigma}(t,\tau)\\}_{t\geqslant\tau},\sigma\in\Xi$ is said to be a
family of processes in $E$, if for any $\sigma\in\Xi$,
$\\{U_{\sigma}(t,\tau)\\}_{t\geqslant\tau}$ is a process in $E$, where $\Xi$
is called a symbol space.

######  Definition 2.1.

A set $B_{0}\subset E$ is said to be uniformly (w.r.t. $\sigma\in\Xi$)
absorbing for the family of processes
$\\{U_{\sigma}(t,\tau)\\}_{t\geqslant\tau},\sigma\in\Xi$, if for each
$\tau\in\mathbb{R}$ and bounded set $B\subset E$, there exists
$t_{0}=t_{0}(\tau,B)\geqslant\tau$ such that

$\bigcup_{\sigma\in\Xi}U_{\sigma}(t,\tau)B\subset B_{0},\hskip
11.38092ptt\geqslant t_{0}.$

######  Definition 2.2.

A closed set ${\mathcal{A}}_{\Xi}$ is said to be a uniformly (w.r.t.
$\sigma\in\Xi$) attractor for the family of processes
$\\{U_{\sigma}(t,\tau)\\}_{t\geqslant\tau},\sigma\in\Xi$, if

  1. (i)

$\lim\limits_{t\rightarrow+\infty}\sup\limits_{\sigma\in\Xi}{\rm
d}_{H}(U_{\sigma}(t,\tau)B,{\mathcal{A}}_{\Xi})=0,$ for each fixed
$\tau\in\mathbb{R}$ and each bounded set $B\subset E$;

  2. (ii)

${\mathcal{A}}_{\Xi}$ is the minimal set (for inclusion relation) among those
satisfying (i).

Now we always assume that the following assumptions hold true.

Assumption (I). Let $\\{\theta_{s}:s\geqslant 0\\}$ be the natural translation
semigroup (see [11]) acting on $\Xi$ and satisfy

  1. (a)

$\theta_{s}\Xi=\Xi$ for all $s\geqslant 0$;

  2. (b)

$U_{\sigma}(t+s,\tau+s)=U_{\theta_{s}\sigma}(t,\tau)$.

######  Lemma 2.3.

Let $\tau_{0}\in\mathbb{R}$ be fixed. Then for any $\tau\in\mathbb{R}$ and
$\sigma\in\Sigma$, there exists unique $\sigma_{1}\in\Sigma$ such that

$U_{\sigma}(t,\tau)=U_{\sigma_{1}}(t-\tau+\tau_{0},\tau_{0}),\hskip
11.38092ptt\geqslant\tau.$ (2.1)

###### Proof.

Let $\sigma\in\Sigma$. Since

$U_{\sigma}(t,\tau)=U_{\sigma}(t-\tau+\tau_{0}+\tau-\tau_{0},\tau-\tau_{0}+\tau_{0}),$

we deduce from Assumption (I) that (2.1) holds true with
$\sigma_{1}=\theta_{\tau-\tau_{0}}\sigma$. Note that
$\theta_{t}:\Sigma\mapsto\Sigma$ is a homeomorphism for each $t\in\mathbb{R}$.
The result of the lemma holds. ∎

######  Remark 2.4.

If $\tau_{0}=0,$ then the above lemma shows

$U_{\sigma}(t,\tau)=U_{\sigma_{1}}(t-\tau,0),\hskip 11.38092ptt\geqslant\tau$

for each $\tau\in\mathbb{R}$ and $\sigma\in\Sigma$.

Finally, we recall the definition of a pullback exponential attractor for the
family of processes $\\{U_{\sigma}(t,0)\\}_{t\geqslant 0},\sigma\in\Sigma.$

######  Definition 2.5.

A family of sets $\\{{\mathscr{M}}(\sigma)\\}_{\sigma\in\Sigma}$ is said to be
a pullback exponential attractor $\\{U_{\sigma}(t,0)\\}_{t\geqslant
0},\sigma\in\Sigma$, if it satisfies the following properties:

  1. (1)

${\mathscr{M}}(\sigma)$ is a compact subset of $E$ for all $\sigma\in\Sigma$,
and its fractal dimension is finite uniformly with respect to
$\sigma\in\Sigma$, i.e.
$\sup_{\sigma\in\Sigma}\dim_{f}{\mathscr{M}}(\sigma)<+\infty.$

  2. (2)

It is forward invariant:

$U_{\sigma}(t,0){\mathscr{M}}(\sigma)\subset{\mathscr{M}}(\theta_{t}\sigma),\hskip
22.76228ptt\geqslant 0,\hskip 11.38092pt\sigma\in\Sigma.$

  3. (3)

There exists $\alpha>0$ such that for each $\sigma\in\Sigma$ and any bounded
subset $B$ of $E$, there exists $T_{\sigma,B}>0$ with

${\rm d}_{H}(U_{\sigma}(0,-t)B,{\mathscr{M}}(\sigma))\leqslant C_{B}{\rm
e}^{-\alpha t},\hskip 11.38092pt\forall t\geqslant T_{\sigma,B},$

where $C_{B}$ is a positive constant depending on $B$.

##  3 Forward dynamical behavior of LDSs

In this section we consider the forward dynamical behavior of general
nonautonomous LDSs.

Set

$\ell^{2}=\\{u=(u_{m})_{m\in\mathbb{Z}}:\ u_{m}\in\mathbb{R},\hskip
11.38092pt\sum\limits_{m\in\mathbb{Z}}u_{m}^{2}<+\infty\\},$

which is equipped with the inner product and norm defined by

$(u,v)=\sum\limits_{m\in\mathbb{Z}}u_{m}v_{m},\hskip
11.38092pt\|u\|^{2}=(u,u),\hskip 22.76228pt\forall
u=(u_{m})_{m\in\mathbb{Z}},\,v=(u_{m})_{m\in\mathbb{Z}}\in\ell^{2}.$

For convenience, we use $E$ to denote $\ell^{2}$. Then $E$ is a Hilbert space.

Consider the following abstract nonautonomous LDSs in $E$:

$\displaystyle\dot{u}=f(u)+\sigma(t),\hskip 11.38092ptt>t_{0};$ (3.1)
$\displaystyle u(t_{0})=u_{0},\hskip 11.38092ptt_{0}\in\mathbb{R},$ (3.2)

where $u=(u_{m})_{m\in\mathbb{Z}},u_{0}=(u_{m,0})_{m\in\mathbb{Z}}\in E$,
$f=(f_{m})_{m\in\mathbb{Z}}$ is a mapping from $E$ to $E$ satisfying some
suitable conditions, and $\sigma(t)=(\sigma_{m}(t))_{m\in\mathbb{Z}}$ is a
function from $\mathbb{R}$ to $E$.

Let the function $\sigma_{0}(t)\in L_{c}^{2}(\mathbb{R},E)$ and take

$\Sigma:=\Sigma(\sigma_{0})=\mbox{the closure of
$\\{\sigma_{0}(s+\cdot):s\in\mathbb{R}\\}$ in $L_{loc}^{2}(\mathbb{R},E)$}$

as the symbol space. Define the translation group $\theta$ on $\Sigma$ as

$\theta_{t}\sigma=\sigma(t+\cdot),\hskip
11.38092ptt\in\mathbb{R},\,\sigma\in\Sigma.$

Assume that for each fixed $u_{0}\in E,\sigma\in\Sigma$ and
$t_{0}\in\mathbb{R}$, equations (3.1)-(3.2) has a unique continuous solution
$u(t)=u(t,t_{0};u_{0},\sigma)$ on $[t_{0},+\infty)$ in the sense that

$u:[t_{0},+\infty)\rightarrow E\,\mbox{is continuous with $u(t_{0})=u_{0}$,
and satisfies the equation \eqref{e3.1}.}$

Moreover, for $t_{0}=0$, operators of solutions

$U_{\sigma}(t,0):u_{0}\mapsto u(t,0;u_{0},\sigma)=U_{\sigma}(t,0)u_{0},\hskip
11.38092ptt\geqslant 0,\,u_{0}\in E,\,\sigma\in\Sigma.$

generate a family of continuous processes $\\{U_{\sigma}(t,0)\\}_{t\geqslant
0},\sigma\in\Sigma$ in $E$. Then one can easily check that the assumption (I)
in Section 2 holds true for $\\{U_{\sigma}(t,0)\\}_{t\geqslant
0},\sigma\in\Sigma$.

For convenience in statement, we define a semigroup
$\Psi=\\{\Psi(t)\\}_{t\geqslant 0}$, called a skew-product flow on
$E\times\Sigma$ corresponding to $\\{U_{\sigma}(t,0)\\}_{t\geqslant
0},\sigma\in\Sigma$ by

$\Psi(t)(u,\sigma)=(U_{\sigma}(t,0)u,\theta_{t}\sigma),\hskip
11.38092pt(u,\sigma)\in E\times\Sigma.$ (3.3)

In the following, we state and prove our main results in this section. We
first construct a family of sets
$\\{{\mathcal{A}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ in an arbitrary
small $\varepsilon$-neighborhood of the global attractor of $\Psi$, such that
$\\{{\mathcal{A}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ uniformly
(w.r.t. $\sigma\in\Sigma$) forward attracts each bounded subset of $E$.

######  Theorem 3.1.

Assume that the family of processes $\\{U_{\sigma}(t,0)\\}_{t\geqslant
0},\sigma\in\Sigma$ has a bounded uniformly (w.r.t. $\sigma\in\Sigma$)
absorbing set $B_{0}$. Moreover,

$\lim_{t\rightarrow+\infty}\lim_{M\rightarrow+\infty}\sup_{u\in
B_{0}}\sum_{|m|>M}|\big{(}U_{\sigma}(t,0)u\big{)}_{m}|^{2}=0,\hskip
22.76228pt\forall\sigma\in\Sigma.$

Then for any $\varepsilon>0$, there exists a family of sets
$\\{{\mathcal{A}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ with

$A(\sigma)\subset{\mathcal{A}}_{\varepsilon}(\sigma)\subset{\mathcal{N}}_{\varepsilon}(\sigma),\hskip
22.76228pt\forall\sigma\in\Sigma,$

such that it is forward invariant for $\\{U_{\sigma}(t,0)\\}_{t\geqslant
0},\sigma\in\Sigma$, where ${\mathcal{N}}_{\varepsilon}(\sigma)$ denotes the
$\sigma$-section of an $\varepsilon$-neighborhood of
$\bigcup\limits_{\sigma\in\Sigma}(A(\sigma)\times\\{\sigma\\})$. Furthermore,
$\\{{\mathcal{A}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ uniformly
(w.r.t. $\sigma\in\Sigma$) forward attracts each bounded set $B\subset E$:

$\lim_{t\rightarrow+\infty}\sup_{\sigma\in\Sigma}{\rm
d}_{H}(U_{\sigma}(t,0)B,{\mathcal{A}}_{\varepsilon}(\theta_{t}\sigma))=0.$
(3.4)

###### Proof.

Since $B_{0}$ is a bounded uniformly (w.r.t. $\sigma\in\Sigma$) absorbing set
for the family of processes $\\{U_{\sigma}(t,0)\\}_{t\geqslant
0},\sigma\in\Sigma$ in $E$ and

$\lim_{t\rightarrow+\infty}\lim_{M\rightarrow+\infty}\sup_{u\in
B_{0}}\sum_{|m|>M}|\big{(}U_{\sigma}(t,0)u\big{)}_{m}|^{2}=0,\hskip
22.76228pt\forall\sigma\in\Sigma,$

one can conclude from [54, Theorem 3.1] that
$\\{U_{\sigma}(t,0)\\}_{t\geqslant 0},\sigma\in\Sigma$ has a compact uniform
attractor $D=\omega_{\Sigma}(B_{0})$ in $E$.

Now let $\Psi$ be the semigroup given by (3.3). Then we infer from [7, 11]
that $\Psi$ has a global attractor ${\mathcal{A}}$ on $E\times\Sigma$ with

${\mathcal{A}}=\bigcup_{\sigma\in\Sigma}\big{(}A(\sigma)\times\\{\sigma\\}\big{)},$
(3.5)

where $\\{A(\sigma)\\}_{\sigma\in\Sigma}$ is the pullback attractor of
$\\{U_{\sigma}(t,0)\\}_{t\geqslant 0},\sigma\in\Sigma$ and
$D=\bigcup_{\sigma\in\Sigma}A(\sigma)$.

Let $\varepsilon>0$ be fixed. Write ${\mathcal{D}}=D\times\Sigma$. Noticing
that the global attractor ${\mathcal{A}}$ attracts ${\mathcal{D}}$, there
exists $T:=T(\varepsilon,{\mathcal{D}})>0$ independent of $\sigma\in\Sigma$
such that

$\Psi(t){\mathcal{D}}\subset N_{\varepsilon}({\mathcal{A}}),\hskip
22.76228ptt\geqslant T.$

Set ${\mathcal{A}}_{\varepsilon}=\bigcup_{s\geqslant T}\Psi(s){\mathcal{D}}$.
By the definition of semigroups, one can easily verify that the set
${\mathcal{A}}_{\varepsilon}$ is $\Psi$-positively invariant and the
$\sigma$-section ${\mathcal{A}}_{\varepsilon}(\sigma)$ of
${\mathcal{A}}_{\varepsilon}$ satisfies

${\mathcal{A}}_{\varepsilon}(\sigma)=\bigcup_{s\geqslant T}U_{\sigma}(0,-s)D.$

That is

$\begin{array}[]{l}\Psi(t)\bigcup_{\sigma\in\Sigma}\big{(}{\mathcal{A}}_{\varepsilon}(\sigma)\times\\{\sigma\\}\big{)}\subset\bigcup_{\sigma\in\Sigma}\big{(}{\mathcal{A}}_{\varepsilon}(\sigma)\times\\{\sigma\\}\big{)},\hskip
11.38092pt\forall t\geqslant 0.\end{array}$

Hence

$\begin{array}[]{l}\bigcup_{\sigma\in\Sigma}\big{(}U_{\sigma}(t,0){\mathcal{A}}_{\varepsilon}(\sigma)\times\\{\theta_{t}\sigma\\}\big{)}\subset\bigcup_{\sigma\in\Sigma}\big{(}{\mathcal{A}}_{\varepsilon}(\theta_{t}\sigma)\times\\{\theta_{t}\sigma\\}\big{)}\hskip
11.38092ptt\geqslant 0,\end{array}$

which implies the family of sets
$\\{{\mathcal{A}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ is forward
invariant for $\\{U_{\sigma}(t,0)\\}_{t\geqslant 0},\sigma\in\Sigma$. By the
invariance of the pullback attractor $\\{A(\sigma)\\}_{\sigma\in\Sigma}$, one
has $A(\sigma)\subset{\mathcal{A}}_{\varepsilon}(\sigma),\sigma\in\Sigma$. We
infer from the construction of ${\mathcal{A}}_{\varepsilon}$ that
${\mathcal{A}}_{\varepsilon}\subset N_{\varepsilon}({\mathcal{A}})$, which
shows

$\bigcup_{\sigma\in\Sigma}\big{(}{\mathcal{A}}_{\varepsilon}(\sigma)\times\\{\sigma\\}\big{)}\subset\bigcup_{\sigma\in\Sigma}\big{(}{\mathcal{N}}_{\varepsilon}(\sigma)\times\\{\sigma\\}\big{)},$

where ${\mathcal{N}}_{\varepsilon}(\sigma)$ denotes the $\sigma$-section of
$N_{\varepsilon}({\mathcal{A}})$. Thereby
${\mathcal{A}}_{\varepsilon}(\sigma)\subset{\mathcal{N}}_{\varepsilon}(\sigma)$
for $\sigma\in\Sigma$. Thus

$A(\sigma)\subset{\mathcal{A}}_{\varepsilon}(\sigma)\subset{\mathcal{N}}_{\varepsilon}(\sigma),\hskip
11.38092pt\sigma\in\Sigma.$

Write ${\mathcal{A}}_{\varepsilon}^{T}(\sigma)=U_{\sigma}(0,-T)D$. Then the
set-valued mapping $\sigma\rightarrow{\mathcal{A}}_{\varepsilon}^{T}(\sigma)$
is continuous at $\sigma\in\Sigma$. Indeed, for each $u\in D$ and the fixed
$T>0$, by the continuity of processes, we see that $U_{\cdot}(0,-T)u$ is
continuous from $\Sigma$ to $E$. Since the set $D$ is compact, one deduces
that the mapping $U_{\cdot}(0,-T)D$ is continuous from $\Sigma$ to $E$ in the
sense of Hausdorff distance. Thereby it follows from the construction of
$\mathcal{A}^{T}_{\varepsilon}(\sigma)$ that the mapping
$\sigma\rightarrow{\mathcal{A}}_{\varepsilon}^{T}(\sigma)$ is continuous at
$\sigma\in\Sigma$. In the following we shall prove that the family of sets
$\\{{\mathcal{A}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ uniformly
forward attracts each bounded set $B\subset E$. For any $\sigma\in\Sigma$ and
$t\geqslant T$, we observe that

$\displaystyle{\rm
d}_{H}(U_{\sigma}(t,0)B,{\mathcal{A}}_{\varepsilon}(\theta_{t}\sigma))$
$\displaystyle\leqslant{\rm
d}_{H}(U_{\sigma}(t,0)B,{\mathcal{A}}_{\varepsilon}^{T}(\theta_{t}\sigma))$
$\displaystyle={\rm d}_{H}(U_{\sigma}(t,0)B,U_{\sigma}(t,t-T)D)$
$\displaystyle={\rm
d}_{H}(U_{\theta_{t-T}\sigma}(T,0)U_{\sigma}(t-T,0)B,U_{\theta_{t-T}\sigma}(T,0)D).$
(3.6)

Recalling that $D$ is the uniform attractor, we have

$\sup_{\sigma\in\Sigma}{\rm d}_{H}(U_{\sigma}(t-T,0)B,D)\rightarrow 0\hskip
11.38092pt\mbox{as}\hskip 11.38092ptt\rightarrow\infty.$ (3.7)

Therefore, by the continuity of $U_{\cdot}(T,0)$ on $D$, we conclude from (3)
and (3.7) that the result holds true. ∎

######  Remark 3.2.

In the proof of the above theorem, one sees that for each $\varepsilon>0$,

$A(\sigma)\subset{\mathcal{A}}_{\varepsilon}(\sigma)\subset{\mathcal{N}}_{\varepsilon}(\sigma),\hskip
11.38092pt\sigma\in\Sigma,$

which implies

$\delta_{H}({\mathcal{A}}_{\varepsilon}(\sigma),A(\sigma))\rightarrow 0\hskip
11.38092pt\mbox{as}\hskip 11.38092pt\varepsilon\rightarrow 0.$

Moreover, the family of sets
$\\{{\mathcal{A}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ is forward
attracting. Thus the sets
$\\{{\mathcal{A}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ may be used to
describe the forward dynamical behavior of nonautonomous LDSs.

######  Remark 3.3.

The continuity of the mapping
$\sigma\rightarrow{\mathcal{A}}_{\varepsilon}^{T}(\sigma)$ shows some
synchronous properties of the set ${\mathcal{A}}_{\varepsilon}^{T}(\sigma)$
with $\sigma$. For example, if $\sigma\in\Sigma$ is (almost) periodic for
$t\in\mathbb{R}$, then the mapping
$t\rightarrow{\mathcal{A}}_{\varepsilon}^{T}(\theta_{t}\sigma)$ is (almost)
periodic as well.

Next, we continue to construct a family of sets
$\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ such that it
forward exponentially attracts bounded sets. For this purpose, we make the
following assumptions: (A) There is a closed neighborhood
$\\{{\mathcal{N}}(\sigma)\\}_{\sigma\in\Sigma}$ of the pullback attractor
$\\{A(\sigma)\\}_{\sigma\in\Sigma}$ such that

  1. (i)

There exists $\tau^{*}>0$ such that for any $\sigma\in\Sigma$, the operator
$U_{\sigma}(\tau^{*},0)$ is a compact perturbation of the contraction on
${\mathcal{N}}(\sigma)$:

$\|U_{\sigma}(\tau^{*},0)u-U_{\sigma}(\tau^{*},0)v\|_{E}\leqslant\delta\|u-v\|_{E}+\|K(\sigma)u-K(\sigma)v\|_{E}$

for any $u,v\in{\mathcal{N}}(\sigma)$, where $0<\delta<1/2$ and $K(\sigma)$ is
an operator from ${\mathcal{N}}(\sigma)$ to $F$, which is a Banach space
compactly embedded into $E$ and satisfies

$\|K(\sigma)u-K(\sigma)v\|_{F}\leqslant L_{1}\|u-v\|_{E},\hskip
11.38092ptu,v\in{\mathcal{N}}(\sigma)$

with $L_{1}>0$ independent of $\sigma.$

  2. (ii)

There is a positive constant $L_{2}$ (independent of $\tau$ and $\sigma$)
satisfying

$\|U_{\sigma}(\tau,0)u-U_{\sigma}(\tau,0)v\|_{E}\leqslant
L_{2}\|u-v\|_{E},\hskip 11.38092ptu,v\in{\mathcal{N}}(\sigma)$

for any $\tau\in[0,\tau^{*}]$ and $\sigma\in\Sigma$.

Then we have

######  Theorem 3.4.

Assume that the conditions in Theorem 3.1 are satisfied and let the assumption
(A) hold true. Then for each $\varepsilon>0$, there exists a family of sets
$\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ such that
$\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ is forward
invariant under the acting of $\\{U_{\sigma}(t,0)\\}_{t\geqslant
0},\sigma\in\Sigma.$ Moreover, the system has a pullback exponential attractor
$\\{{\mathscr{M}}(\sigma)\\}_{\sigma\in\Sigma}$ with

${\mathscr{M}}(\sigma)\subset{\mathcal{B}}_{\varepsilon}(\sigma),\hskip
11.38092pt\forall\sigma\in\Sigma.$

Furthermore, the family of sets
$\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ uniformly
forward exponentially attracts each bounded set $B\subset E$.

###### Proof.

First, let ${\mathcal{A}}$ be the global attractor of the semigroup $\Psi$
given by (3.3). Write ${\mathcal{B}}_{0}=B_{0}\times\Sigma$. Then for each
fixed $\varepsilon>0$, there exists $T:=T({\mathcal{B}}_{0},\varepsilon)>0$
such that

$\Psi(t){\mathcal{B}}\subset{\mathcal{N}}_{\varepsilon}({\mathcal{A}}),\hskip
22.76228ptt\geqslant T.$

Set ${\mathcal{B}}_{\varepsilon}=\bigcup_{s\geqslant
T}\Psi(s){\mathcal{B}}_{0}$. Similar to
$\\{{\mathcal{A}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$, one can also
construct a family of sets
$\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ defined by

${\mathcal{B}}_{\varepsilon}(\sigma)=\bigcup_{s\geqslant
T}U_{\sigma}(0,-s)B_{0},$

such that $\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ is
forward invariant for $\\{U_{\sigma}(t,0)\\}_{t\geqslant 0},\sigma\in\Sigma$
and
${\mathcal{B}}_{\varepsilon}(\sigma)\subset{\mathcal{N}}_{\varepsilon}(\sigma)$
for all $\sigma\in\Sigma$.

Noticing that
$\bigcup_{\sigma\in\Sigma}\left({\mathcal{N}}(\sigma)\times\\{\sigma\\}\right)$
is a closed neighborhood of the global attractor ${\mathcal{A}}$, we see that
for each bounded set $B\subset E$, there is $T_{B}>0$ independent of
$\sigma\in\Sigma$ such that

$\Psi(t){\mathcal{B}}\subset\bigcup_{\sigma\in\Sigma}\left({\mathcal{N}}(\sigma)\times\\{\sigma\\}\right)\hskip
11.38092ptt\geqslant T_{B},$

where ${\mathcal{B}}=B\times\Sigma$. It then follows from the definition of
$\Psi(t)$ that if $t\geqslant T_{B}$,

$\bigcup_{\sigma\in\Sigma}\left(U_{\sigma}(t,0)B\times\\{\theta_{t}\sigma\\}\right)\subset\bigcup_{\sigma\in\Sigma}\left({\mathcal{N}}(\theta_{t}\sigma)\times\\{\theta_{t}\sigma\\}\right),$

from which it can be deduced that the family of sets
$\\{{\mathcal{N}}(\sigma)\\}_{\sigma\in\Sigma}$ is uniformly (w.r.t.
$\sigma\in\Sigma$) absorbing for the family of processes
$\\{U_{\sigma}(t,0)\\}_{t\geqslant 0},\sigma\in\Sigma.$ Hence we may suppose
that the $T$ in the construction of
$\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ is sufficiently
large so that

${\mathcal{B}}_{\varepsilon}(\sigma)\subset{\mathcal{N}}(\sigma)\hskip
11.38092pt\mbox{for all $\sigma\in\Sigma$}.$

Since the set $B_{0}$ in the construction of
${\mathcal{B}}_{\varepsilon}(\sigma)$ is uniformly absorbing, the family of
sets $\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ is
uniformly absorbing as well. Clearly,
$\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ satisfies
conditions (i) and (ii) in assumption $\mathbf{(A)}$ and is uniformly bounded
with respect to $\sigma\in\Sigma$.

Now, we show that the family of processes $\\{U_{\sigma}(t,0)\\}_{t\geqslant
0},\sigma\in\Sigma$ has a pullback exponential attractor
$\\{{\mathscr{M}}(\sigma)\\}_{\sigma\in\Sigma}$. Note that $\theta_{t}$ is a
homeomorphism for each $t\in\mathbb{R}$. Then for any $\sigma\in\Sigma$, there
exists $s\in\mathbb{R}$ such that $\sigma=\theta_{s}\sigma_{0}$. Define

$U(t,s)u=U_{\sigma_{0}}(t,s)u,\hskip 11.38092ptt\geqslant s,\hskip
11.38092ptu\in E,$ (3.8)

where $\sigma_{0}$ is the function given in Section 3. It is easy to see that
$\\{U(t,s)\\}_{t\geqslant s}$ is a process in $E$. Set

${\mathcal{B}}_{\varepsilon}(t):={\mathcal{B}}_{\varepsilon}(\theta_{t}\sigma_{0}),\hskip
11.38092ptt\in\mathbb{R}.$ (3.9)

Then $\\{{\mathcal{B}}_{\varepsilon}(t)\\}_{t\in\mathbb{R}}$ is positively
invariant for the process $\\{U(t,s)\\}_{t\geqslant s}$.

In what follows we first check that the family of sets
$\\{{\mathcal{B}}_{\varepsilon}(s)\\}_{s\in\mathbb{R}}$ satisfies the
conditions (1)-(5) in [20, p. 656], and hence by [20, Theorem 2.1] the process
$\\{U(t,s)\\}_{t\geqslant s}$ has a pullback exponential attractor
$\\{{\mathscr{M}}(t)\\}_{t\in\mathbb{R}}$ with
${\mathscr{M}}(t)\subset{\mathcal{B}}_{\varepsilon}(t)$ for all
$t\in\mathbb{R}$. Indeed, observing that
$\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ satisfies
conditions (i) and (ii) in assumption $\mathbf{(A)}$, we conclude that for
each $\sigma\in\Sigma$ with $\sigma=\theta_{s}\sigma_{0}$ for some
$s\in\mathbb{R}$, and the $\tau^{*}>0$,

$\|U_{\sigma_{0}}(\tau^{*}+s,s)u-U_{\sigma_{0}}(\tau^{*}+s,s)v\|_{E}\leqslant\delta\|u-v\|_{E}+\|K(\theta_{s}\sigma_{0})u-K(\theta_{s}\sigma_{0})v\|_{E}$

for any $u,v\in{\mathcal{B}}_{\varepsilon}(\theta_{s}\sigma_{0})$ and

$\|K(\theta_{s}\sigma_{0})u-K(\theta_{s}\sigma_{0})v\|_{F}\leqslant
L_{1}\|u-v\|_{E},\hskip
11.38092ptu,v\in{\mathcal{B}}_{\varepsilon}(\theta_{s}\sigma_{0}).$

Moreover, for any $\tau\in[0,\tau^{*}],$

$\|U_{\sigma_{0}}(\tau+s,s)u-U_{\sigma_{0}}(\tau+s,s)v\|_{E}\leqslant
L_{2}\|u-v\|_{E},\hskip
11.38092ptu,v\in{\mathcal{B}}_{\varepsilon}(\theta_{s}\sigma_{0}).$

By (3.8) and (3.9), one finds that

$\|U(\tau^{*}+s,s)u-U(\tau^{*}+s,s)v\|_{E}\leqslant\delta\|u-v\|_{E}+\|K(s)u-K(s)v\|_{E}$

for any $u,v\in{\mathcal{B}}_{\varepsilon}(s)$,

$\|K(s)u-K(s)v\|_{F}\leqslant L_{1}\|u-v\|_{E},\hskip
11.38092ptu,v\in{\mathcal{B}}_{\varepsilon}(s)$

and for any $\tau\in[0,\tau^{*}]$,

$\|U(\tau+s,s)u-U(\tau+s,s)v\|_{E}\leqslant L_{2}\|u-v\|_{E},\hskip
11.38092ptu,v\in{\mathcal{B}}_{\varepsilon}(s),$

which shows that $\\{{\mathcal{B}}_{\varepsilon}(s)\\}_{s\in\mathbb{R}}$
satisfies the conditions (4)-(5) in [20, p. 656] for the process
$\\{U(t,s)\\}_{t\geqslant s}$ defined by (3.8). Similarly, it is easy to
verify that $\\{{\mathcal{B}}(s)\\}_{s\in\mathbb{R}}$ satisfies conditions
(1)-(3) in [20, p. 656]. Therefore one can immediately conclude from [20,
Theorem 2.1] that the process $\\{U(t,s)\\}_{t\geqslant s}$ has a pullback
exponential attractor $\\{{\mathcal{M}}(t)\\}_{t\in\mathbb{R}}$ with
${\mathcal{M}}(t)\subset{\mathcal{B}}_{\varepsilon}(t)$ for all
$t\in\mathbb{R}$, and

${\rm
d}_{H}\big{(}U(t,s){\mathcal{B}}_{\varepsilon}(s),{\mathcal{M}}(t)\big{)}\leqslant
C{\rm e}^{-\alpha(t-s)},\hskip 11.38092pts\in\mathbb{R},\,s\leqslant
t<+\infty,$ (3.10)

where $C,\alpha$ are some positive constants independent of $s\in\mathbb{R}$,
and $C$ depends on the boundedness of the family of sets
$\\{{\mathcal{B}}_{\varepsilon}(s)\\}_{s\in\mathbb{R}}$ uniformly for
$s\in\mathbb{R}$.

For any $s\in\mathbb{R}$, define

${\mathscr{M}}(\sigma)={\mathcal{M}}(s),\hskip
11.38092pt\sigma=\theta_{s}\sigma_{0}.$ (3.11)

Clearly,
${\mathscr{M}}(\sigma)\subset{\mathcal{B}}_{\varepsilon}(\sigma),\sigma\in\Sigma.$
We shall show that $\\{{\mathscr{M}}(\sigma)\\}_{\sigma\in\Sigma}$ is a
pullback exponential attractor of $\\{U_{\sigma}(t,0)\\}_{t\geqslant
0},\sigma\in\Sigma$. First, since $\theta_{t}:\Sigma\mapsto\Sigma$ is a
homeomorphism for $t\in\mathbb{R}$, for any $\sigma\in\Sigma$, there exists
$s\in\mathbb{R}$ such that $\sigma=\theta_{s}\sigma_{0}$. Thus
${\mathscr{M}}(\sigma)$ is compact, as the set ${\mathcal{M}}(s)$ is compact.
For the fractal dimension of ${\mathscr{M}}(\sigma)$, noticing that the
fractal dimension of ${\mathcal{M}}(t)$ is finite and uniformly bounded for
$t\in\mathbb{R}$, one can conclude that the fractal dimension of
${\mathcal{M}}(\sigma)$ is finite and uniformly bounded with respect to
$\sigma\in\Sigma$ as well. Now we verify that the family of sets
$\\{{\mathscr{M}}(\sigma)\\}_{\sigma\in\Sigma}$ is forward invariant for
$\\{U_{\sigma}(t,0)\\}_{t\geqslant 0},\sigma\in\Sigma$. Let
$\sigma=\theta_{s}\sigma_{0}$ for some $s\in\mathbb{R}$ and $\tau\geqslant 0$.
We infer from (3.8), (3.11) and the positive invariance of
$\\{{\mathcal{M}}(t)\\}_{t\in\mathbb{R}}$ for $\\{U(t,s)\\}_{t\geqslant s}$
that

$\displaystyle
U_{\sigma}(\tau,0){\mathscr{M}}(\sigma)=U_{\sigma_{0}}(\tau+s,s){\mathscr{M}}(\theta_{s}\sigma_{0})$
$\displaystyle=U(\tau+s,s){\mathcal{M}}(s)$
$\displaystyle\subset{\mathcal{M}}(\tau+s)={\mathscr{M}}(\theta_{\tau+s}\sigma_{0})={\mathscr{M}}(\theta_{\tau}\sigma).$

Finally, it remains to prove the exponential attractivity of
$\\{{\mathscr{M}}(\sigma)\\}_{\sigma\in\Sigma}.$ For each $B\subset E$, we
conclude from the definition of the pullback exponential attractor
$\\{{\mathcal{M}}(t)\\}_{t\in\mathbb{R}}$ for $\\{U(t,s)\\}_{t\geqslant s}$
that if $\tau\geqslant 0$ and $\sigma=\theta_{s}\sigma_{0}$, there exist
constants $\alpha>0$, $C_{B}$ and $T_{\sigma,B}$ depending on $T_{s,B}$ such
that

$\displaystyle{\rm
d}_{H}\left(U_{\sigma}(0,-\tau)B,{\mathscr{M}}(\sigma)\right)$
$\displaystyle={\rm
d}_{H}(U_{\sigma_{0}}(s,s-\tau)B,{\mathscr{M}}(\theta_{s}\sigma_{0}))$
$\displaystyle={\rm d}_{H}(U(s,s-\tau)B,{\mathcal{M}}(s))\leqslant C_{B}{\rm
e}^{-\alpha\tau},\hskip 11.38092pt\tau\geqslant T_{\sigma,B}.$

Thus, the family of sets $\\{{\mathscr{M}}(\sigma)\\}_{\sigma\in\Sigma}$ is a
pullback exponential attractor for the family of processes
$\\{U_{\sigma}(t,0)\\}_{t\geqslant 0},\sigma\in\Sigma$ with
${\mathscr{M}}(\sigma)\subset{\mathcal{B}}_{\varepsilon}(\sigma)$ for all
$\sigma\in\Sigma$.

To complete the proof of the theorem, it remains to prove the forward
exponential attractivity of
$\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$. By the
construction of a pullback exponential attractor (see (3.10)) and (3.8),
(3.9), (3.11) that

${\rm
d}_{H}\left(U_{\sigma}(0,-t){\mathcal{B}}_{\varepsilon}(\theta_{-t}\sigma),{\mathscr{M}}(\sigma)\right)\leqslant
C{\rm e}^{-\alpha t},\hskip 22.76228pt\forall\sigma\in\Sigma,\,\,\forall
t\geqslant 0.$ (3.12)

Let $B\subset E$ be a bounded set. Since
$\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ is uniformly
absorbing, it follows from (3.12) that there exists $T_{B}>0$ (independent of
$\sigma$) such that

${\rm d}_{H}\left(U_{\sigma}(0,-t)B,{\mathscr{M}}(\sigma)\right)\leqslant
C_{1}{\rm e}^{-\alpha t},\hskip 11.38092pt\forall t\geqslant T_{B},$

where $C_{1}$ is a positive constant depending on $C$ and $T_{B}$. Recalling
that ${\mathscr{M}}(\sigma)\subset{\mathcal{B}}_{\varepsilon}(\sigma)$ for all
$\sigma\in\Sigma$, one can deduce that

${\rm
d}_{H}\left(U_{\sigma}(0,-t)B,{\mathcal{B}}_{\varepsilon}(\sigma)\right)\leqslant{\rm
d}_{H}\left(U_{\sigma}(0,-t)B,{\mathscr{M}}(\sigma)\right).$

Hence we conclude that if $t\geqslant T_{B}$, then

${\rm
d}_{H}\left(U_{\sigma}(0,-t)B,{\mathcal{B}}_{\varepsilon}(\sigma)\right)\leqslant
C_{1}{\rm e}^{-\alpha t},\hskip 11.38092pt\forall\sigma\in\Sigma,$

from which it can be seen that

$\sup_{\sigma\in\Sigma}{\rm
d}_{H}\left(U_{\sigma}(0,-t)B,{\mathcal{B}}_{\varepsilon}(\sigma)\right)\leqslant
C_{1}{\rm e}^{-\alpha t},\hskip 11.38092pt\forall t\geqslant T_{B}.$

Therefore

$\sup_{\sigma\in\Sigma}{\rm
d}_{H}\left(U_{\sigma}(t,0)B,{\mathcal{B}}_{\varepsilon}(\theta_{t}\sigma)\right)\leqslant
C_{1}{\rm e}^{-\alpha t},\hskip 11.38092pt\forall t\geqslant T_{B}.$

The proof of the theorem is complete. ∎

######  Remark 3.5.

In the proof of the above theorem, one can see that

$A(\sigma)\subset{\mathscr{M}}(\sigma)\subset{\mathcal{B}}_{\varepsilon}(\sigma)\subset{\mathcal{N}}_{\varepsilon}(\sigma),\hskip
11.38092pt\forall\sigma\in\Sigma,$

which implies

$\mathrm{\delta}_{H}({\mathscr{M}}(\sigma),A(\sigma))\rightarrow 0,\hskip
11.38092pt\delta_{H}({\mathcal{B}}_{\varepsilon}(\sigma),{\mathscr{M}}(\sigma))\rightarrow
0\hskip 11.38092pt\mbox{as}\hskip 11.38092pt\varepsilon\rightarrow 0.$

We infer from the proof of Theorem 3.4 that the existence of a pullback
exponential attractor plays an important role in proving the exponential
attraction of $\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$.
Therefore by using the results on the existence of pullback exponential
attractors in a Hilbert space, see [55, Theorem 2], we can obtain the
following main results, which seem to be more convenient in applications.

######  Theorem 3.6.

Let $\\{U_{\sigma}(t,0)\\}_{t\geqslant 0},\sigma\in\Sigma$ be the family of
processes generated by equations (3.1)-(3.2) in $H$, where $H$ is a Hilbert
space. Assume that $B_{0}\subset H$ is a bounded uniformly (w.r.t.
$\sigma\in\Sigma$) absorbing set for $\\{U_{\sigma}(t,0)\\}_{t\geqslant
0},\sigma\in\Sigma$ and satisfies

$\lim_{t\rightarrow+\infty}\lim_{M\rightarrow+\infty}\sup_{u\in
B_{0}}\sum_{|m|>M}|\big{(}U_{\sigma}(t,0)u\big{)}_{m}|^{2}=0,\hskip
22.76228pt\forall\sigma\in\Sigma,$

and for some $T_{0}:=T(B_{0})>0$ such that

$U_{\sigma}(t,0)B_{0}\subset B_{0},\hskip 11.38092pt\forall t\geqslant
T_{0},\,\,\forall\sigma\in\Sigma.$

In addition, if

  1. (A1)

there exist positive constants $T^{*}>T_{0}$ and $L=L_{T^{*}}$ such that for
each $\sigma\in\Sigma$,

$\|U_{\sigma}(t,0)u-U_{\sigma}(t,0)v\|_{H}\leqslant L\|u-v\|_{H},\hskip
11.38092ptu,v\in B_{0},\,\,t\in[T_{0},T^{*}];$

  2. (A2)

there are $0\leqslant\beta<1/2$ and a finite dimensional subspace $H_{m}$ of
$H$ such that for any $\sigma\in\Sigma$,

$\|(I-P_{m})(U_{\sigma}(T^{*},0)u-U_{\sigma}(T^{*},0)v)\|_{H}\leqslant\beta\|u-v\|_{H},\hskip
11.38092ptu,v\in B_{0},$

where $P_{m}:H\mapsto H_{m}$ is a projection, $\beta$ and $m\in\mathbb{N}$
(depending on $T^{*}$) are independent of $\sigma\in\Sigma$,

then for each $\varepsilon>0$, there exists a family of sets
$\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ such that
$\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ is forward
invariant for $\\{U_{\sigma}(t,0)\\}_{t\geqslant 0},\sigma\in\Sigma$ and
$\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ uniformly
forward exponentially attracts each bounded subset of $H$.

###### Proof.

For each fixed $\varepsilon>0$, let
$\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ be the family of
sets given in the proof of Theorem 3.4, where

${\mathcal{B}}_{\varepsilon}(\sigma)=\bigcup_{s\geqslant
T}U_{\sigma}(0,-s)B_{0}.$

Since the set $B_{0}$ is uniformly absorbing, it may be assumed that the $T$
in the construction of ${\mathcal{B}}_{\varepsilon}(\sigma)$ is sufficiently
large so that

$U_{\sigma}(0,-s)B\subset B_{0},\hskip 11.38092pts\geqslant T$

uniformly for $\sigma\in\Sigma$. Thus
$\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ satisfies the
conditions (A1)-(A2).

By (3.8), we infer from conditions (A1)-(A2) that for
$\sigma=\theta_{s}\sigma_{0}$,

$\|U(t+s,s)u-U(t+s,s)v\|_{H}\leqslant L\|u-v\|_{H},\hskip 11.38092ptu,v\in
B_{0},\,\,t\in[T_{0},T^{*}]$

and

$\|(I-P_{m})(U(T^{*}+s,s)u-U(T^{*}+s,s)v)\|_{H}\leqslant\beta\|u-v\|_{H},\hskip
11.38092ptu,v\in B_{0}.$

Hence by [55, Theorem 2], one can conclude that the process
$\\{U(t,s)\\}_{t\geqslant s}$ has a pullback exponential attractor
$\\{{\mathcal{M}}(t)\\}_{t\in\mathbb{R}}$. Note that the family of sets
$\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ is forward
invariant and uniformly absorbing for $\\{U_{\sigma}(t,0)\\}_{t\geqslant
0},\sigma\in\Sigma$. Repeating the same argument below (3.10), it can be shown
that $\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\Sigma}$ uniformly
forward exponentially attracts each bounded subset of $H$. ∎

##  4 Applications

In this section, we study the nonautonomous discrete Gray-Scott equations
driven by quasi-periodic external forces, illustrating how to apply our
abstract results to some concrete equations.

Consider the following nonautonomous LDS:

$\displaystyle\dot{u}_{m}$
$\displaystyle=-d_{1}(Au)_{m}-(\lambda+k)u_{m}+u_{m}^{2}v_{m}-\alpha
u_{m}^{3}+\beta z_{m}+b_{1m}f_{1m}(\sigma(t)),$ (4.1)
$\displaystyle\dot{v}_{m}$
$\displaystyle=-d_{2}(Av)_{m}+\lambda(a_{m}-v_{m})-u_{m}^{2}v_{m}+\alpha
u_{m}^{3}+b_{2m}f_{2m}(\sigma(t)),$ (4.2) $\displaystyle\dot{z}_{m}$
$\displaystyle=-d_{3}(Az)_{m}+ku_{m}-(\lambda+\beta)z_{m}+b_{3m}f_{3m}(\sigma(t))$
(4.3)

for $m\in\mathbb{Z},t>t_{0},t_{0}\in\mathbb{R}$, associated with the initial
conditions

$\displaystyle u_{m}(t_{0})=u_{m,0},\hskip
11.38092ptv_{m}(t_{0})=v_{m,0},\hskip 11.38092ptz_{m}(t_{0})=z_{m,0},\hskip
22.76228ptm\in\mathbb{Z},$ (4.4)

where $d_{1},d_{2},d_{3},k,\alpha,\beta,\lambda$ are positive numbers,
$a_{m}\in\mathbb{R},b_{im}\in\mathbb{R},f_{im}\in\mathbb{R},i=1,2,3,\sigma\in\mathbb{T}^{\kappa}(\kappa$-dimensional
torus), and $A$ is a linear operator defined by

$(Au)_{m}=2u_{m}-u_{m+1}-u_{m-1},\hskip 22.76228pt\forall
u=(u_{m})_{m\in\mathbb{Z}}.$ (4.5)

Equations (4.1)-(4.3) can be regarded as a discrete analogue of the following
nonautonomous three-component reversible Gray-Scott model on $\mathbb{R}$:

$\displaystyle\frac{\partial u}{\partial t}=$ $\displaystyle
d_{1}u_{xx}-(\lambda+k)u+u^{2}v-\alpha u^{3}+\beta z+b_{1}f_{1}(x,\sigma),$
(4.6) $\displaystyle\frac{\partial v}{\partial t}=$ $\displaystyle
d_{2}v_{xx}-\lambda v-u^{2}v+\alpha u^{3}+\lambda a+b_{2}f_{2}(x,\sigma),$
(4.7) $\displaystyle\frac{\partial z}{\partial t}=$ $\displaystyle
d_{3}z_{xx}+ku-(\lambda+\beta)z+b_{3}f_{3}(x,\sigma),$ (4.8)

where the positive numbers $d_{1},d_{2},d_{3},k,\alpha,\beta,\lambda$ are
defined by nondimensionalization, see [44] for details.

In order to express equations (4.1)-(4.4) as an abstract first-order ODE, we
set
$u=(u_{m})_{m\in\mathbb{Z}},v=(v_{m})_{m\in\mathbb{Z}},z=(z_{m})_{m\in\mathbb{Z}},u^{2}v=(u_{m}^{2}v_{m})_{m\in\mathbb{Z}},u^{3}=(u_{m}^{3})_{m\in\mathbb{Z}},$
$b_{i}f_{i}(\sigma(t))=(b_{im}f_{im}(\sigma(t)))_{m\in\mathbb{Z}},i=1,2,3,a=(a_{m})_{m\in\mathbb{Z}},\sigma(t)=({\bf
x}t+\sigma_{0})\,{\rm mod}(\mathbb{T}^{\kappa}).$ Then

$\displaystyle\dot{u}$ $\displaystyle=-d_{1}Au-(\lambda+k)u+u^{2}v-\alpha
u^{3}+\beta z+b_{1}f_{1}(\sigma(t)),$ (4.9) $\displaystyle\dot{v}$
$\displaystyle=-d_{2}Av-\lambda v-u^{2}v+\alpha u^{3}+\lambda
a+b_{2}f_{2}(\sigma(t)),$ (4.10) $\displaystyle\dot{z}$
$\displaystyle=-d_{3}Az+ku-(\lambda+\beta)z+b_{3}f_{3}(\sigma(t)),$ (4.11)

associated with initial conditions

$u(t_{0})=(u_{m,0})_{m\in\mathbb{Z}},\,v(t_{0})=(v_{m,0})_{m\in\mathbb{Z}},\,z(t_{0})=(z_{m,0})_{m\in\mathbb{Z}},\hskip
11.38092ptt_{0}\in\mathbb{R},$ (4.12)

where $\sigma_{0}\in\mathbb{T}^{\kappa}$ (x and $\mathbb{T}^{\kappa}$ will be
introduced below), and $A$ is a linear operator from $\ell^{2}$ to $\ell^{2}$
defined by (4.5).

###  4.1 Mathematical setting

We first introduce some operators and spaces.

Let $\ell^{2}$ be the space given in Section 3. Define the linear operators
$B$ and $B^{*}$ on $\ell^{2}$ as

$(Bu)_{m}=u_{m+1}-u_{m},\,\,(B^{*}u)_{m}=u_{m-1}-u_{m},\hskip
11.38092ptm\in\mathbb{Z},\,\forall u=(u_{m})_{m\in\mathbb{Z}}\in\ell^{2}.$

Then it is easy to see that $B^{*}$ is the adjoint operator of $B$. Moreover,
for any $u,v\in\ell^{2}$,

$\displaystyle(Au,v)$ $\displaystyle=(B^{*}Bu,v)=(Bu,Bv),\hskip
11.38092pt(Bu,v)=(u,B^{*}v),$ $\displaystyle\|Bu\|^{2}$
$\displaystyle=\|B^{*}u\|^{2}\leqslant 4\|u\|^{2},\hskip
11.38092pt\|Au\|^{2}\leqslant 16\|u\|^{2}.$

Let $\mathbb{T}^{\kappa}$ be a $\kappa$-dimensional torus

$\mathbb{T}^{\kappa}=\\{\sigma=(\sigma_{1},\cdot\cdot\cdot,\sigma_{\kappa}):\sigma_{i}\in[-\pi,\pi],\hskip
11.38092pt\forall i=1\cdot\cdot\cdot\kappa\\}$

with the following identification

$(\sigma_{1},\cdot\cdot\cdot,\sigma_{i},-\pi,\sigma_{i+1},\cdot\cdot\cdot\sigma_{\kappa})\sim(\sigma_{1},\cdot\cdot\cdot,\sigma_{i},\pi,\sigma_{i+1},\cdot\cdot\cdot\sigma_{\kappa}),\hskip
22.76228pt\forall i=1,\cdot\cdot\cdot,\kappa,$

and the topology and metric induced from the topology and metric on
$\mathbb{R}^{\kappa}$. Thus we consider the norm on $\mathbb{T}^{\kappa}$
given by

$\|\sigma\|_{\mathbb{T}^{\kappa}}=(\sum\limits_{i=1}^{\kappa}\sigma_{i}^{2})^{1/2},\hskip
22.76228pt\forall\sigma=(\sigma_{1},\cdot\cdot\cdot,\sigma_{\kappa})\in\mathbb{T}^{\kappa}.$

Let $\mathbf{x}=(x_{1},\cdot\cdot\cdot,x_{\kappa})\in\mathbb{R}^{\kappa}$ be a
fixed vector and $x_{1},\cdot\cdot\cdot,x_{\kappa}$ be rationally independent.
For $s\in\mathbb{R}$, define

$\theta_{s}\sigma_{0}=(\mathbf{x}s+\sigma_{0}){\rm
mod}(\mathbb{T}^{\kappa}),\hskip 22.76228pt\sigma_{0}\in\mathbb{T}^{\kappa}.$

Then one can easily check that

$\theta_{s}\mathbb{T}^{\kappa}=\mathbb{T}^{\kappa},\hskip 22.76228pt\forall
s\in\mathbb{R}.$

Denote by $E$ the product space $\ell^{2}\times\ell^{2}\times\ell^{2}.$ Then
$E$ is a Hilbert space as well. We also use the notations $(\cdot,\cdot)$ and
$||\cdot||$ to denote the inner product and norm, respectively. Define the
extended space $\tilde{E}=E\times\mathbb{T}^{\kappa}$ with the norm defined by

$\|\widetilde{\varphi}\|=(\|\varphi\|^{2}+\|\sigma\|^{2}_{\mathbb{T}^{\kappa}})^{1/2},\hskip
22.76228pt\forall\widetilde{\varphi}=(\varphi,\sigma)\in\tilde{E}.$

Now we rewrite equations (4.9)-(4.12) as an abstract first-order ODE:

$\displaystyle\dot{\varphi}+\Theta\varphi=G(\varphi,\sigma,t),\hskip
22.76228ptt>t_{0},$ (4.13)
$\displaystyle\varphi(t_{0})=\varphi_{0}=(u_{0},v_{0},z_{0})^{T},\hskip
22.76228ptt_{0}\in\mathbb{R},$ (4.14)

where $\varphi=(u,v,z)^{T}$, and

$G(\varphi,\sigma,t)=(u^{2}v-\alpha u^{3}+b_{1}f_{1}(\sigma(t)),\lambda
a-u^{2}v+\alpha u^{3}+b_{2}f_{2}(\sigma(t)),b_{3}f_{3}(\sigma(t)))^{T},$
$\begin{array}[]{ccc}\Theta=\left(\begin{array}[]{ccc}d_{1}A+(\lambda+k)I&0&-\beta
I\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr 0&d_{2}A+\lambda I&0\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt\cr-
kI&0&d_{3}A+(\lambda+\beta)I\end{array}\right).\end{array}$

For the parameters $a,b_{i}$ and functions $f_{i}$ for $i=1,2,3$ in equations
(4.9)-(4.11), we make the following assumptions: (H1) Assume
$a=(a_{m})_{m\in\mathbb{Z}}\in\ell^{2},b_{i}=(b_{i,m})_{m\in\mathbb{Z}}\in\ell^{2},i=1,2,3$,
and that

$2\beta<k<\min\big{\\{}\frac{3\lambda}{2\mu-1},\beta+\lambda\\},$

where $\mu=k/\beta$.

(H2) $f_{im}(0_{\mathbb{T}^{\kappa}})=0$ for all $m\in\mathbb{Z},i=1,2,3$, and
there exist $c_{1},c_{2},c_{3}>0$ such that

$|f_{im}(\sigma_{01})-f_{im}(\sigma_{02})|\leqslant
c_{i}\|\sigma_{01}-\sigma_{02}\|_{\mathbb{T}^{\kappa}},\hskip
22.76228pt\forall
m\in\mathbb{Z},\,\sigma_{01},\sigma_{02}\in\mathbb{T}^{\kappa},\hskip
11.38092pti=1,2,3.$

###  4.2 Unique existence and boundedness of solutions

We prove the existence and uniqueness of solutions and then verify the
existence of bounded absorbing set for skew-product flows. For simplicity, we
let

$Z(t)=\frac{\beta}{k}z(t).$

Then equations (4.9)-(4.11) can be transformed into the following equivalent
form

$\displaystyle\dot{u}$ $\displaystyle=-d_{1}Au-(\lambda+k)u+u^{2}v-\alpha
u^{3}+kZ+b_{1}f_{1}(\sigma(t)),$ (4.15) $\displaystyle\dot{v}$
$\displaystyle=-d_{2}Av-\lambda v-u^{2}v+\alpha u^{3}+\lambda
a+b_{2}f_{2}(\sigma(t)),$ (4.16) $\displaystyle\mu\dot{Z}$
$\displaystyle=-d_{3}\mu AZ+ku-(\mu\lambda+k)Z+b_{3}f_{3}(\sigma(t)).$ (4.17)

######  Theorem 4.1.

Let the assumptions $(\mathbf{H}1)$-$(\mathbf{H}2)$ hold. Then for any initial
value $\varphi_{0}=(u_{0},v_{0},z_{0})\in E,t_{0}\in\mathbb{R}$ and
$\sigma\in\mathbb{T}^{\kappa}$, equations (4.13)-(4.14) has a unique solution
$\varphi(t)=\varphi(t,t_{0};\varphi_{0},\sigma)$ with

$\varphi(t)=(u(t),v(t),z(t))\in{\mathcal{C}}([t_{0},+\infty),E)\cap{\mathcal{C}}^{1}((t_{0},+\infty),E),\,\,t\geqslant
t_{0}.$

###### Proof.

Let

$\tilde{G}(\varphi,\sigma,t)=G(\varphi,\sigma,t)-\Theta\varphi,\hskip
22.76228pt\sigma\in\mathbb{T}^{\kappa}.$

Then under the assumptions, one can check that
$\tilde{G}:\tilde{E}\times\mathbb{R}\mapsto E$ is locally Lipschitz continuous
respect to $\varphi$ for any $\sigma\in\mathbb{T}^{\kappa}$ and $t$ in any
compact interval of $\mathbb{R}$. By the classical theory of ODEs, we can
obtain the existence and uniqueness of a local solution of equations
(4.13)-(4.14) and

$\varphi(t)=(u(t),v(t),z(t))\in{\mathcal{C}}([t_{0},T),E)\cap{\mathcal{C}}^{1}((t_{0},T),E)$

for some $T>t_{0}$. Moreover, if $T<+\infty,$ then $\lim\limits_{t\rightarrow
T^{-}}\|\varphi(t)\|=+\infty.$

In what follows, we prove that $T=+\infty.$ Taking the inner product
$(\cdot,\cdot)$ of the three equations of (4.15)-(4.17) with $\alpha
u(t),v(t),\mu\alpha Z(t)$, respectively, and summing up, we deduce that

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}(\alpha\|u\|^{2}+\|v\|^{2}+\mu^{2}\alpha\|Z\|^{2})+\alpha(\lambda+k)\|u\|^{2}+\lambda\|v\|^{2}+\mu\alpha(\mu\lambda+k)\|Z\|^{2}$
$\displaystyle\leqslant$
$\displaystyle\,k\alpha(1+\mu)\sum\limits_{m\in\mathbb{Z}}u_{m}Z_{m}+\lambda\sum\limits_{m\in\mathbb{Z}}a_{m}v_{m}+\alpha\sum\limits_{m\in\mathbb{Z}}b_{1m}f_{1m}(\sigma)u_{m}+\sum\limits_{m\in\mathbb{Z}}b_{2m}f_{2m}(\sigma)v_{m}$
$\displaystyle+\mu\alpha\sum\limits_{m\in\mathbb{Z}}b_{3m}f_{3m}(\sigma)Z_{m}.$
(4.18)

Since

$\displaystyle\lambda\sum\limits_{m\in\mathbb{Z}}a_{m}v_{m}$
$\displaystyle\leqslant\frac{\lambda}{2}(\|a\|^{2}+\|v\|^{2}),$ (4.19)
$\displaystyle k\alpha(1+\mu)\sum\limits_{m\in\mathbb{Z}}u_{m}Z_{m}$
$\displaystyle\leqslant\frac{k\alpha(1+\mu)}{2}(\|u\|^{2}+\|Z\|^{2}),$ (4.20)

and by $(\mathbf{H}1)$-$(\mathbf{H}2)$, we have

$\displaystyle\alpha\sum\limits_{m\in\mathbb{Z}}b_{1m}f_{1m}(\sigma)u_{m}$
$\displaystyle\leqslant\frac{\alpha(\lambda+k)}{4}\|u\|^{2}+\frac{\alpha\kappa
c_{1}^{2}\pi^{2}}{\lambda+k}\|b_{1}\|^{2},$ (4.21)
$\displaystyle\sum\limits_{m\in\mathbb{Z}}b_{2m}f_{2m}(\sigma)v_{m}$
$\displaystyle\leqslant\frac{\lambda}{4}\|v\|^{2}+\frac{\kappa
c_{2}^{2}\pi^{2}}{\lambda}\|b_{2}\|^{2},$ (4.22)
$\displaystyle\mu\alpha\sum\limits_{m\in\mathbb{Z}}b_{3m}f_{3m}(\sigma)Z_{m}$
$\displaystyle\leqslant\frac{\mu\alpha(\mu\lambda+k)}{4}\|Z\|^{2}+\frac{\mu\alpha\kappa
c_{3}^{2}\pi^{2}}{\mu\lambda+k}\|b_{3}\|^{2}.$ (4.23)

Thus it follows from (4.2)-(4.23) that for $t\geqslant t_{0}$,

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}(\alpha\|u\|^{2}+\|v\|^{2}+\mu^{2}\alpha\|Z\|^{2})+\theta_{1}(\alpha\|u\|^{2}+\|v\|^{2}+\mu^{2}\alpha\|Z\|^{2})\leqslant
R_{0}^{2},$ (4.24)

where

$\displaystyle\theta_{1}=\min\\{\frac{3}{2}\lambda-(\mu-\frac{1}{2})k,\frac{\lambda}{2},\frac{3}{2}\lambda+\frac{(\frac{\mu}{2}-1)k}{\mu^{2}}\\},$
$\displaystyle R_{0}^{2}=\lambda\|a\|^{2}+\frac{2\alpha\kappa
c_{1}^{2}\pi^{2}}{\lambda+k}\|b_{1}\|^{2}+\frac{2\kappa
c_{2}^{2}\pi^{2}}{\lambda}\|b_{2}\|^{2}+\frac{2\mu\alpha\kappa
c_{3}^{2}\pi^{2}}{\mu\lambda+k}\|b_{3}\|^{2}.$

Applying Gronwall inequality to (4.24) on $[t_{0},t]$ with $t\geqslant t_{0}$,
we obtain

$\displaystyle\alpha\|u\|^{2}\\!+\\!\|v\|^{2}\\!+\\!\alpha\|z\|^{2}\leqslant\mathrm{e}^{-\theta_{1}(t-t_{0})}(\alpha\|u(t_{0})\|^{2}\\!+\\!\|v(t_{0})\|^{2}\\!+\\!\alpha\|z(t_{0})\|^{2})\\!+\\!\frac{R_{0}^{2}}{\theta_{1}}.$
(4.25)

Let $\delta_{1}=\min\\{1,\alpha\\},\delta_{2}=\max\\{1,\alpha\\}$. Then (4.25)
shows that

$\|\varphi(t)\|^{2}\leqslant\frac{\delta_{2}}{\delta_{1}}\mathrm{e}^{-\theta_{1}(t-t_{0})}\|\varphi(t_{0})\|^{2}+\frac{R_{0}^{2}}{\theta_{1}\delta_{1}}.\hskip
22.76228ptt\geqslant t_{0},$

which implies that $T=+\infty$. The proof is complete. ∎

Set

$U_{\sigma}(t,0)\varphi_{0}=\varphi(t,0;\varphi_{0},\sigma),\hskip
11.38092ptt\geqslant 0,\,\,\sigma\in\mathbb{T}^{\kappa}.$

Then one can easily verify that $\\{U_{\sigma}(t,0)\\}_{t\geqslant
0},\sigma\in\mathbb{T}^{\kappa}$ is a family of continuous processes in $E$.
Moreover, $\Psi$ is the corresponding semigroup. As a direct consequence of
Theorem 4.1, we can obtain the following result.

######  Lemma 4.2.

Suppose the assumptions $(\mathbf{H}1)$-$(\mathbf{H}2)$ hold. Then the family
of processes $\\{U_{\sigma}(t,0)\\}_{t\geqslant
0},\sigma\in\mathbb{T}^{\kappa}$ generated by (4.13)-(4.14) possesses a
bounded uniformly $(w.r.t.\,\sigma\in\mathbb{T}^{\kappa})$ absorbing set
$B_{0}:=B(0,R)$ in $E$ centered at $0$ with radius $\displaystyle
R=2R_{0}/\sqrt{\theta_{1}\delta_{1}}$. Specifically, for any bounded set $B$
of $E$, there exists $t(B)>0$ (independent of $\sigma\in\mathbb{T}^{\kappa}$)
such that

$U_{\sigma}(t,0)B\subset B_{0},\hskip 22.76228pt\forall t\geqslant t(B),\hskip
11.38092pt\forall\sigma\in\mathbb{T}^{\kappa}.$

Particularly, there exists $T_{0}=t(B_{0})>0$ such that

$U_{\sigma}(t,0)B_{0}\subset B_{0},\hskip 22.76228pt\forall t\geqslant
T_{0},\hskip 11.38092pt\forall\sigma\in\mathbb{T}^{\kappa}.$

Next we prove that the solutions of (4.13)-(4.14) have uniformly asymptotic
end tails, from which one can also obtain the existence of a uniform
attractor.

######  Lemma 4.3.

Suppose the assumptions $(\mathbf{H}1)$-$(\mathbf{H}2)$ hold. Then for any
$\varepsilon>0$, there exist $t_{1}(\varepsilon,B_{0})>0$ and
$N_{1}(\varepsilon,B_{0})\in\mathbb{N}$, such that the solution
$\varphi(t)=U_{\sigma}(t,0)\varphi_{0}=(u_{m}(t),v_{m}(t),z_{m}(t))_{m\in\mathbb{Z}}$
$\in E$ of equations (4.13)-(4.14) with initial data $\varphi_{0}\in B_{0}$
satisfies

$\sup\limits_{\varphi_{0}\in B_{0}}\sum\limits_{|m|\geqslant
N_{1}(\varepsilon,B_{0})}|\varphi_{m}(t)|^{2}_{E}\leqslant\varepsilon^{2},\quad\forall
t\geqslant t_{1}(\varepsilon,B_{0}),\hskip
11.38092pt\forall\sigma\in\mathbb{T}^{\kappa},$

where $|\varphi_{m}(t)|^{2}_{E}=u_{m}^{2}(t)+v_{m}^{2}(t)+z_{m}^{2}(t)$.

###### Proof.

Define a function $\chi(x)\in{\mathcal{C}}^{1}({\bf\mathbb{R}}_{+},[0,1])$
such that

$\displaystyle\quad\chi(x)=\left\\{\begin{array}[]{ll}0,\enskip 0\leqslant
x\leqslant 1;\\\ 1,\enskip x\geqslant 2,\end{array}\right.{\rm
and}\,|\chi^{\prime}(x)|\leqslant\chi_{0}\,({\rm positive\enskip
constant}),\,\forall x\geqslant 0.$ (4.28)

Set

$\displaystyle p=(p_{m})_{m\in\mathbb{Z}},\hskip
11.38092ptq=(q_{m})_{m\in\mathbb{Z}},\hskip
11.38092ptW=(W_{m})_{m\in\mathbb{Z}},$

with

$p_{m}=\chi(\frac{|m|}{M})u_{m},\hskip
11.38092ptq_{m}=\chi(\frac{|m|}{M})v_{m},\hskip
11.38092ptW_{m}=\chi(\frac{|m|}{M})Z_{m},$

where $M$ is a positive integer. Taking the inner product $(\cdot,\cdot)$ of
these three equations of (4.15)-(4.17) with $\alpha p,q$ and $\mu\alpha W,$
respectively, and adding these equations, we obtain

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})(\alpha
u_{m}^{2}\\!+\\!v_{m}^{2}\\!+\\!\mu^{2}\alpha
Z_{m}^{2})\\!+\\!d_{1}\alpha(Bu,Bp)\\!+\\!d_{2}(Bv,Bq)\\!+\\!\mu^{2}\alpha
d_{3}(BZ,BW)$
$\displaystyle+\alpha(\lambda+k)\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})u_{m}^{2}+\lambda\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})v_{m}^{2}+\mu\alpha(\mu\lambda+k)\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})Z_{m}^{2}$
$\displaystyle\leqslant$
$\displaystyle\,k\alpha(1+\mu)\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})u_{m}Z_{m}+\lambda\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})a_{m}v_{m}+\alpha\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})u_{m}b_{1m}f_{1m}(\sigma)$
$\displaystyle+\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})v_{m}b_{2m}f_{2m}(\sigma)+\mu\alpha\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})Z_{m}b_{3m}f_{3m}(\sigma).$
(4.29)

Elementary computations show that

$\displaystyle(Bu,Bp)$
$\displaystyle=\sum\limits_{m\in\mathbb{Z}}(Bu)_{m}(Bp)_{m}=\sum\limits_{m\in\mathbb{Z}}(Bu)_{m}[\chi(\frac{|m+1|}{M})u_{m+1}-\chi(\frac{|m|}{M})u_{m}]$
$\displaystyle=\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})|(Bu)_{m}|^{2}+\sum\limits_{m\in\mathbb{Z}}\chi^{\prime}(\frac{\tilde{m}}{M})\frac{1}{M}(u_{m+1}-u_{m})u_{m+1}$
$\displaystyle\geqslant\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})|(Bu)_{m}|^{2}-\frac{2\chi_{0}R^{2}}{M},\hskip
22.76228ptt\geqslant T_{0},$ (4.30)

where $\tilde{m}$ locates between $|m|$ and $|m+1|$. Likewise,

$\displaystyle(Bv,Bq)$
$\displaystyle\geqslant\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})|(Bv)_{m}|^{2}-\frac{2\chi_{0}R^{2}}{M},\hskip
22.76228ptt\geqslant T_{0},$ (4.31) $\displaystyle(BZ,BW)$
$\displaystyle\geqslant\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})|(BZ)_{m}|^{2}-\frac{2\chi_{0}R^{2}}{M\mu^{2}},\hskip
22.76228ptt\geqslant T_{0}.$ (4.32)

Now

$\displaystyle\lambda\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})a_{m}v_{m}$
$\displaystyle\leqslant\frac{\lambda}{2}\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})v_{m}^{2}+\frac{\lambda}{2}\sum\limits_{m\in\mathbb{Z}}\chi(\frac{m}{M})a_{m}^{2},$
(4.33) $\displaystyle
k\alpha(1+\mu)\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})u_{m}Z_{m}$
$\displaystyle\leqslant\frac{k\alpha(1+\mu)}{2}\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})(u_{m}^{2}+Z_{m}^{2}),$
(4.34)

and by (H1)-(H2), we have

$\displaystyle\alpha\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})u_{m}b_{1m}f_{1m}(\sigma)\\!$
$\displaystyle\leqslant\\!\frac{\alpha(\lambda+k)}{4}\\!\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})u_{m}^{2}\\!+\\!\frac{\alpha\kappa
c_{1}^{2}\pi^{2}}{\lambda+k}\\!\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})b_{1m}^{2},$
(4.35)
$\displaystyle\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})v_{m}b_{2m}f_{2m}(\sigma)$
$\displaystyle\leqslant\frac{\lambda}{4}\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})v_{m}^{2}+\frac{\kappa
c_{2}^{2}\pi^{2}}{\lambda}\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})b_{2m}^{2},$
(4.36)
$\displaystyle\mu\alpha\sum\limits_{m\in\mathbb{Z}}\\!\chi(\frac{|m|}{M})Z_{m}b_{3m}f_{3m}(\sigma)$
$\displaystyle\leqslant\\!\frac{\mu\alpha(\mu\lambda+k)}{4}\\!\sum\limits_{m\in\mathbb{Z}}\\!\chi(\frac{|m|}{M})Z_{m}^{2}\\!+\\!\frac{\mu\alpha\kappa
c_{3}^{2}\pi^{2}}{\mu\lambda+k}\\!\sum\limits_{m\in\mathbb{Z}}\\!\chi(\frac{|m|}{M})b_{3m}^{2}.$
(4.37)

Thus we conclude from (4.2)-(4.37) that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})(\alpha
u_{m}^{2}+v_{m}^{2}+\mu^{2}\alpha
Z_{m}^{2})+[\frac{3}{2}\lambda-(\mu-\frac{1}{2})k]\alpha\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})u_{m}^{2}$
$\displaystyle+\frac{\lambda}{2}\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})v_{m}^{2}+[\frac{3\lambda}{2}+\frac{(\frac{\mu}{2}-1)k}{\mu^{2}}]\mu^{2}\alpha\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})Z_{m}^{2}$
$\displaystyle\leqslant$ $\displaystyle\,\lambda\sum\limits_{|m|\geqslant
M}a_{m}^{2}+(\frac{2\alpha c_{1}^{2}}{\lambda+k}\sum\limits_{|m|\geqslant
M}b_{1m}^{2}+\frac{2c_{2}^{2}}{\lambda}\sum_{|m|\geqslant
M}b_{2m}^{2}+\frac{2\mu\alpha c_{3}^{2}}{\mu\lambda+k}\sum_{|m|\geqslant
M}b_{3m}^{2})\kappa\pi^{2}$
$\displaystyle+\frac{4\chi_{0}R^{2}}{M}(d_{1}\alpha+d_{2}+d_{3}\alpha),\hskip
22.76228ptt\geqslant T_{0}.$

Then

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})(\alpha
u_{m}^{2}+v_{m}^{2}+\mu^{2}\alpha
Z_{m}^{2})+\theta_{1}\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})(\alpha
u_{m}^{2}+v_{m}^{2}+\mu^{2}\alpha Z_{m}^{2})$ $\displaystyle\leqslant$
$\displaystyle\,C(M,a,b_{1},b_{2},b_{3}),\hskip 22.76228ptt\geqslant T_{0},$
(4.38)

where

$\displaystyle C(M,a,b_{1},b_{2},b_{3})$ $\displaystyle=(\frac{2\alpha
c_{1}^{2}}{\lambda+k}\sum\limits_{|m|\geqslant
M}b_{1m}^{2}+\frac{2c_{2}^{2}}{\lambda}\sum_{|m|\geqslant
M}b_{2m}^{2}+\frac{2\mu\alpha c_{3}^{2}}{\mu\lambda+k}\sum_{|m|\geqslant
M}b_{3m}^{2})\kappa\pi^{2}$ $\displaystyle+\lambda\sum\limits_{|m|\geqslant
M}a_{m}^{2}+\frac{4\chi_{0}R^{2}}{M}(d_{1}\alpha+d_{2}+d_{3}\alpha).$

Applying Gronwall inequality to (4.2), we obtain for any $t\geqslant T_{0},$

$\displaystyle\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})(\alpha
u_{m}^{2}+v_{m}^{2}+\mu^{2}\alpha
Z_{m}^{2})=\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})(\alpha
u_{m}^{2}+v_{m}^{2}+\alpha z_{m}^{2})$ $\displaystyle\leqslant$
$\displaystyle\,\mathrm{e}^{-\theta_{1}(t-t_{0})}\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})(\alpha
u_{m,0}^{2}+v_{m,0}^{2}+\alpha
z_{m,0}^{2})+\frac{C(M,a,b_{1},b_{2},b_{3})}{\theta_{1}}.$

Therefore,

$\displaystyle\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{M})|\varphi_{m}(t)|^{2}$
$\displaystyle\leqslant\frac{\delta_{2}}{\delta_{1}}\mathrm{e}^{-\theta_{1}(t-t_{0})}\|\varphi_{0}\|^{2}+\frac{C(M,a,b_{1},b_{2},b_{3})}{\delta_{1}\theta_{1}}$
$\displaystyle\leqslant\frac{\delta_{2}}{\delta_{1}}R^{2}\mathrm{e}^{-\theta_{1}(t-t_{0})}+\frac{C(M,a,b_{1},b_{2},b_{3})}{\delta_{1}\theta_{1}},\hskip
22.76228ptt\geqslant T_{0}.$ (4.39)

By the assumption (H1), there exists $N(\varepsilon,B_{0})\in\mathbb{N}$ so
that

$\frac{C(M,d_{1},d_{2},d_{3})}{\delta_{1}\theta_{1}}\leqslant\frac{\varepsilon^{2}}{2},\hskip
22.76228pt\forall M\geqslant N(\varepsilon,B_{0}),$ (4.40)

and there is $t(\varepsilon,B_{0})>T_{0}$ such that

$\frac{\delta_{2}}{\delta_{1}}R^{2}\mathrm{e}^{-\theta_{1}(t-t_{0})}\leqslant\frac{\varepsilon^{2}}{2},\hskip
22.76228pt\forall t\geqslant t(\varepsilon,B_{0}).$ (4.41)

Thus let $N_{1}(\varepsilon,B_{0})=2N(\varepsilon,B_{0})$ and
$t_{1}(\varepsilon,B_{0})=t(\varepsilon,B_{0})$. We deduce from (4.2)-(4.41)
that

$\sum\limits_{|m|\geqslant
N_{1}(\varepsilon,B_{0})}|\varphi_{m}(t)|^{2}\leqslant\sum\limits_{m\in\mathbb{Z}}\chi(\frac{|m|}{N(\varepsilon,B_{0})})|\varphi_{m}(t)|^{2}\leqslant\varepsilon^{2},\hskip
22.76228pt\forall t\geqslant t_{1}(\varepsilon,B_{0}).$

The proof is complete. ∎

By Theorems 3.1, 4.1 and Lemmas 4.2, 4.3, we have the following result.

######  Theorem 4.4.

Let the assumptions $(\mathbf{H}1)$-$(\mathbf{H}2)$ hold. Then for any
$\varepsilon>0$, there exists a family of sets
$\\{{\mathcal{A}}_{\varepsilon}(\sigma)\\}_{\sigma\in\mathbb{T}^{\kappa}}$
with

$A(\sigma)\subset{\mathcal{A}}_{\varepsilon}(\sigma)\subset{\mathcal{N}}_{\varepsilon}(\sigma),\hskip
22.76228pt\forall\sigma\in\mathbb{T}^{\kappa},$

such that it is forward invariant for the family of processes
$\\{U_{\sigma}(t,0)\\}_{t\geqslant 0},\sigma\in\mathbb{T}^{\kappa}$. Moreover,
$\\{{\mathcal{A}}_{\varepsilon}(\sigma)\\}_{\sigma\in\mathbb{T}^{\kappa}}$
uniformly (w.r.t. $\sigma\in\mathbb{T}^{\kappa}$) forward attracts each
bounded set $B\subset E$:

$\lim_{t\rightarrow+\infty}\sup_{\sigma\in\mathbb{T}^{\kappa}}{\rm
d}_{H}(U_{\sigma}(t,0)B,{\mathcal{A}}_{\varepsilon}(\theta_{t}\sigma))=0.$

###  4.3 Existence of exponentially attracting set

In this subsection, we show that the assumptions (A1)-(A2) in Theorem 3.6 are
satisfied.

######  Lemma 4.5.

Assume the assumptions $(\mathbf{H}1)$-$(\mathbf{H}2)$ hold. Then

  1. (1)

for any $T>T_{0}$, there exists $L_{T}>0$ such that for any
$\sigma\in\mathbb{T}^{\kappa}$ and $t\in[T_{0},T]$,

$\|U_{\sigma}(t,0)\varphi^{(1)}_{0}-U_{\sigma}(t,0)\varphi^{(2)}_{0}\|\leqslant
L_{T}\|\varphi^{(1)}_{0}-\varphi^{(2)}_{0}\|,\hskip
11.38092pt\varphi^{(1)}_{0},\varphi^{(2)}_{0}\in B_{0};$

  2. (2)

there exist $T^{*}>T_{0},0\leqslant\beta<1/2$ and a finite dimensional
projection $P_{N^{*}}:E\mapsto E_{N^{*}}(N^{*}\in\mathbb{N})$ such that for
any $\sigma\in\mathbb{T}^{\kappa}$,

$\|(I-P_{N^{*}})(U_{\sigma}(T^{*},0)\varphi^{(1)}_{0}-U_{\sigma}(T^{*},0)\varphi^{(2)}_{0})\|\leqslant\beta\|\varphi^{(1)}_{0}-\varphi^{(2)}_{0}\|,\hskip
11.38092pt\varphi^{(1)}_{0},\varphi^{(2)}_{0}\in B_{0}.$

###### Proof.

(i) For each $\sigma\in\mathbb{T}^{\kappa}$, let

$\varphi^{(i)}(t)=U_{\sigma}(t,0)\varphi^{(i)}_{0}=(u^{(i)}(t),v^{(i)}(t),z^{(i)}(t)),\hskip
11.38092pt\forall t\geqslant 0$

be two solutions of equations (4.13)-(4.14) with initial values
$\varphi^{(i)}_{0}\in B_{0}$ for $i=1,2$. Then, if $t\geqslant T_{0}$,
$\varphi^{(1)}(t),\varphi^{(2)}(t)\in B_{0}$ . Write

$\displaystyle u_{d}(t)=$ $\displaystyle u^{(1)}(t)-u^{(2)}(t),\hskip
11.38092ptv_{d}(t)=v^{(1)}(t)-v^{(2)}(t),$ $\displaystyle z_{d}(t)=$
$\displaystyle z^{(1)}(t)-z^{(2)}(t),\hskip
11.38092pt\varphi_{d}(t)=\varphi^{(1)}(t)-\varphi^{(2)}(t).$

By equations (4.13)-(4.14), we have

$\displaystyle\dot{\varphi}_{d}+\Theta\varphi_{d}=G(\varphi^{(1)},\sigma,t)-G(\varphi^{(2)},\sigma,t),$
(4.42) $\displaystyle\varphi_{d}(0)=\varphi^{(1)}_{0}-\varphi^{(2)}_{0}.$

Taking the inner product of (4.42) with $\varphi_{d}$ in $E$, one obtains

$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|\varphi_{d}\|^{2}+\big{(}\Theta\varphi_{d}-G(\varphi^{(1)},\sigma,t)+G(\varphi^{(2)},\sigma,t),\varphi_{d}\big{)}=0.$
(4.43)

Since $\Theta:E\mapsto E$ is a bounded linear operator,
$G:\widetilde{E}\times\mathbb{R}\mapsto E$ is locally Lipschitz continuous
with respect to $\varphi$ for each $\sigma\in\mathbb{T}^{\kappa}$ and
$t\in\mathbb{R}$ and the set $B_{0}$ is bounded, we conclude that there exist
positive constants $C_{0}$ and $L_{0}$ such that

$\displaystyle\hskip
11.38092pt\big{(}\Theta\varphi_{d}-G(\varphi^{(1)},\sigma,t)+G(\varphi^{(2)},\sigma,t),\varphi_{d}\big{)}$
$\displaystyle\leqslant\big{(}C_{0}\|\varphi_{d}\|+\|G(\varphi^{(2)},\sigma,t)-G(\varphi^{(1)},\sigma,t)\|\big{)}\|\varphi_{d}\|$
$\displaystyle\leqslant(C_{0}+L_{0})\|\varphi_{d}\|^{2}.$ (4.44)

Thus it follows from (4.43)-(4.44) that

$\frac{\mathrm{d}}{\mathrm{d}t}\|\varphi_{d}\|^{2}\leqslant
C_{1}\|\varphi_{d}\|^{2},$ (4.45)

where $C_{1}=2(C_{0}+L_{0})$. Applying Gronwall inequality to (4.45) on
$[0,t]$ with $t\in[T_{0},T]$, we have

$\|\varphi_{d}(t)\|^{2}\leqslant{\rm e}^{C_{1}t}\|\varphi_{d}(0)\|^{2},$
(4.46)

which implies that

$\displaystyle\|\varphi^{(1)}(t)-\varphi^{(2)}(t)\|$
$\displaystyle=\|U_{\sigma}(t,0)\varphi_{0}^{(1)}-U_{\sigma}(t,0)\varphi_{0}^{(2)}\|$
$\displaystyle\leqslant L_{T}\|\varphi_{0}^{(1)}-\varphi_{0}^{(2)}\|,\hskip
22.76228ptt\in[T_{0},T],$

where $L_{T}=\sqrt{{\rm e}^{C_{1}T}}.$ (ii) Set

$\displaystyle p_{d}$ $\displaystyle=(p_{dm})_{m\in\mathbb{Z}},\
q_{d}=(q_{dm})_{m\in\mathbb{Z}},\ w_{d}=(w_{dm})_{m\in\mathbb{Z}},$
$\displaystyle p_{dm}$ $\displaystyle=\chi(\frac{|m|}{M})u_{dm},\
q_{dm}=\chi(\frac{|m|}{M})v_{dm},\,w_{dm}=\chi(\frac{|m|}{M})z_{dm},$

where $M$ is a positive integer which will be decided later and $\chi(x)$ is
defined by (4.28). By equation (4.9), we have

$\displaystyle\dot{u}_{d}=$ $\displaystyle-
d_{1}Au_{d}-(\lambda+k)u_{d}+(u^{(1)})^{2}v^{(1)}-(u^{(2)})^{2}v^{(2)}-\alpha\left((u^{(1)})^{3}-(u^{(2)})^{3}\right)$
$\displaystyle+\beta z_{d}.$ (4.47)

Taking the inner product of (4.3) with $p_{d}$ in $\ell^{2}$ gives

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})u_{dm}^{2}$
$\displaystyle=\\!-d_{1}(Au_{d},p_{d})\\!-\\!(\lambda+k)(u_{d},p_{d})\\!+\\!\left((u^{(1)})^{2}v^{(1)}-(u^{(2)})^{2}v^{(2)},p_{d}\right)$
$\displaystyle-\alpha\left((u^{(1)})^{3}-(u^{(2)})^{3},p_{d}\right)+\beta(z_{d},p_{d}).$
(4.48)

By some computations, we have

$\displaystyle(Au_{d},p_{d})$
$\displaystyle=(Bu_{d},Bp_{d})=\sum\limits_{m\in{\bf\mathbb{Z}}}(Bu_{d})_{m}(\chi(\frac{|m+1|}{M})u_{dm+1}-\chi(\frac{|m|}{M})u_{dm})$
$\displaystyle=\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})(Bu_{d})_{m}^{2}+\sum\limits_{m\in{\bf\mathbb{Z}}}\chi^{\prime}(\frac{\tilde{m}}{M})\frac{1}{M}(u_{dm+1}-u_{dm})u_{dm+1}$
$\displaystyle\geqslant\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})(Bu_{d})_{m}^{2}-\frac{2\chi_{0}}{M}\|\varphi_{d}\|^{2}_{E},$
(4.49)

where $\tilde{m}$ is a constant locating between $|m|$ and $|m+1|$. Thanks to
Lemma 4.3, there exist $t_{1}:=t_{1}(\alpha,\lambda,B_{0})>T_{0}$ and
$N_{1}:=N_{1}(\alpha,\lambda,B_{0})\in\mathbb{N}$ such that if $t\geqslant
t_{1}$ and $M\geqslant N_{1}$,

$\displaystyle\left((u^{(1)})^{2}v^{(1)}-(u^{(2)})^{2}v^{(2)},p_{d}\right)$
$\displaystyle=$
$\displaystyle\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})((u^{(1)}_{m})^{2}v^{(1)}_{m}-(u^{(2)}_{m})^{2}v^{(2)}_{m})u_{dm}$
$\displaystyle=$
$\displaystyle\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})(u^{(1)}_{m})^{2}v_{dm}u_{dm}+\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})(u_{m}^{(1)}+u_{m}^{(2)})v_{m}^{(2)}u_{dm}^{2}$
$\displaystyle\leqslant$
$\displaystyle\frac{\lambda}{10}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})|v_{dm}u_{dm}|+\frac{\lambda}{10}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})u_{dm}^{2}$
$\displaystyle\leqslant$
$\displaystyle\frac{\lambda}{5}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})u_{dm}^{2}+\frac{\lambda}{40}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})v_{dm}^{2}$
(4.50)

and

$\displaystyle\left((u^{(1)})^{3}-(u^{(2)})^{3},p_{d}\right)$
$\displaystyle=\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})((u^{(1)}_{m})^{3}-(u^{(2)}_{m})^{3})u_{dm}$
$\displaystyle\leqslant\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})(|u_{m}^{(1)}|+|u_{m}^{(2)}|)^{2}u_{dm}^{2}$
$\displaystyle\leqslant\frac{\lambda}{10\alpha}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})u_{dm}^{2}.$
(4.51)

Note that

$\displaystyle\beta(z_{d},p_{d})=\beta\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})z_{dm}u_{dm}\leqslant\frac{\beta}{2}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})z_{dm}^{2}+\frac{\beta}{2}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})u_{dm}^{2}.$
(4.52)

Thus taking (4.3)-(4.52) into account, we obtain

$\displaystyle\hskip
11.38092pt\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})u_{dm}^{2}+(\lambda+k)\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})u_{dm}^{2}$
$\displaystyle\leqslant(\frac{3\lambda}{10}+\frac{\beta}{2})\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})u_{dm}^{2}+\frac{\lambda}{40}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})v_{dm}^{2}+\frac{\beta}{2}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})z_{dm}^{2}$
$\displaystyle\quad+\frac{2d_{1}\chi_{0}}{M}\|\varphi_{d}\|^{2}_{E},\hskip
22.76228ptt\geqslant t_{1},\,\,M\geqslant N_{1}.$ (4.53)

From equation (4.10), we have

$\displaystyle\dot{v}_{d}=-d_{2}Av_{d}-\lambda
v_{d}-\left((u^{(1)})^{2}v^{(1)}-(u^{(2)})^{2}v^{(2)}\right)+\alpha\left((u^{(1)})^{3}-(u^{(2)})^{3}\right).$
(4.54)

Taking the inner product of (4.54) with $q_{d}$ in $\ell^{2}$ gives

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})v_{dm}^{2}$
$\displaystyle=-d_{2}(Av_{d},q_{d})-\lambda(v_{d},q_{d})-\left((u^{(1)})^{2}v^{(1)}-(u^{(2)})^{2}v^{(2)},q_{d}\right)$
$\displaystyle\hskip
11.38092pt+\alpha\left((u^{(1)})^{3}-(u^{(2)})^{3},q_{d}\right).$ (4.55)

Similar to (4.3), we have

$\displaystyle(Av_{d},q_{d})\geqslant\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})(Bv_{d})_{m}^{2}-\frac{2\chi_{0}}{M}\|\varphi_{d}\|^{2}_{E}.$
(4.56)

By virtue of Lemma 4.3, there exist $t_{2}:=t_{2}(\alpha,\lambda,B_{0})$ and
$N_{2}:=N_{2}(\alpha,\lambda,B_{0})\in\mathbb{N}$ with $t_{2}>t_{1}$ and
$N_{2}>N_{1}$ such that if $t\geqslant t_{2}$ and $M\geqslant N_{2}$,

$\displaystyle\left((u^{(1)})^{2}v^{(1)}-(u^{(2)})^{2}v^{(2)},q_{d}\right)$
$\displaystyle=$
$\displaystyle\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})((u^{(1)}_{m})^{2}v^{(1)}_{m}-(u^{(2)}_{m})^{2}v^{(2)}_{m})v_{dm}$
$\displaystyle=$
$\displaystyle\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})(u^{(1)}_{m})^{2}v_{dm}^{2}+\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})(u_{m}^{(1)}+u_{m}^{(2)})v_{m}^{(2)}u_{dm}v_{dm}$
$\displaystyle\leqslant$
$\displaystyle\frac{\lambda}{10}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})u_{dm}^{2}+\frac{\lambda}{20}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})v_{dm}^{2}$
(4.57)

and

$\displaystyle\left((u^{(1)})^{3}-(u^{(2)})^{3},q_{d}\right)$
$\displaystyle=\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})((u^{(1)}_{m})^{3}-(u^{(2)}_{m})^{3})v_{dm}$
$\displaystyle\leqslant\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})(|u_{m}^{(1)}|+|u_{m}^{(2)}|)^{2}|u_{dm}v_{dm}|$
$\displaystyle\leqslant\frac{\lambda}{10\alpha}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})u_{dm}^{2}+\frac{\lambda}{40\alpha}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})v_{dm}^{2}.$
(4.58)

Combining (4.3)-(4.3), we see that for any $t\geqslant t_{2}$ and $M\geqslant
N_{2}$,

$\displaystyle\quad\,\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})v_{dm}^{2}+\frac{37\lambda}{40}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})v_{dm}^{2}$
$\displaystyle\leqslant\frac{\lambda}{5}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})u_{dm}^{2}+\frac{2d_{2}\chi_{0}}{M}\|\varphi_{d}\|^{2}_{E}.$
(4.59)

Finally, by (4.11), we have

$\displaystyle\dot{z}_{d}=-d_{3}Az_{d}+ku_{d}-(\lambda+\beta)z_{d}.$ (4.60)

Taking the inner product of (4.60) with $w_{d}$ in $\ell^{2}$ gives

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})z_{dm}^{2}=$
$\displaystyle-
d_{3}(Az_{d},w_{d})+k(u_{d},w_{d})-(\lambda+\beta)(z_{d},w_{d}).$ (4.61)

Similar to the derivation of (4.3), we deduce that

$\displaystyle\quad\,\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})z_{dm}^{2}+(\lambda+\beta-\frac{k}{2})\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})z_{dm}^{2}$
$\displaystyle\leqslant\frac{k}{2}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})u_{dm}^{2}+\frac{2d_{3}\chi_{0}}{M}\|\varphi_{d}\|^{2}_{E}.$
(4.62)

Thus, we conclude from (4.3), (4.3) and (4.3) that if $t\geqslant t_{2}$ and
$M\geqslant N_{2}$,

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})(u_{dm}^{2}+v_{dm}^{2}+z_{dm}^{2})+\frac{\lambda+k-\beta}{2}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})u_{dm}^{2}+\frac{9\lambda}{10}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})v_{dm}^{2}$
$\displaystyle+\frac{\lambda+\beta-k}{2}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})z_{dm}^{2}\leqslant\frac{2\chi_{0}(d_{1}+d_{2}+d_{3})}{M}\|\varphi_{d}\|^{2}_{E},$

which implies that for any $t\geqslant t_{2}$ and $M\geqslant N_{2}$,

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})|\varphi_{dm}|_{E}^{2}+\theta_{2}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})|\varphi_{dm}|_{E}^{2}\leqslant\frac{4\chi_{0}(d_{1}+d_{2}+d_{3})}{M}\|\varphi_{d}\|^{2}_{E},$
(4.63)

where
$\theta_{2}=\min\\{\lambda+k-\beta,\frac{9\lambda}{5},\lambda+\beta-k\\}$.
Applying Gronwall inequality to (4.63) from $t_{2}$ to $t$ with $t\geqslant
t_{2}$, we have

$\displaystyle\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})|\varphi_{dm}(t)|_{E}^{2}\leqslant$
$\displaystyle\,\mathrm{e}^{-\theta_{2}(t-t_{2})}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})|\varphi_{dm}(t_{2})|_{E}^{2}$
$\displaystyle+\frac{4\chi_{0}(d_{1}+d_{2}+d_{3})}{M}\int_{t_{2}}^{t}\|\varphi_{d}(s)\|_{E}^{2}\mathrm{e}^{-\theta_{2}(t-s)}\mathrm{d}s,$
(4.64)

provided $M\geqslant N_{2}.$ By (4.46), one has

$\displaystyle\mathrm{e}^{-\theta_{2}(t-t_{2})}\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})|\varphi_{dm}(t_{2})|_{E}^{2}\leqslant\mathrm{e}^{-\theta_{2}(t-t_{2})}\|\varphi_{d}(t_{2})\|_{E}^{2}\leqslant\mathrm{e}^{-\theta_{2}(t-t_{2})+C_{1}t_{2}}\|\varphi_{d}(0)\|_{E}^{2}$
(4.65)

and

$\displaystyle\frac{4\chi_{0}(d_{1}+d_{2}+d_{3})}{M}\int_{t_{2}}^{t}\|\varphi_{d}(s)\|_{E}^{2}\mathrm{e}^{-\theta_{2}(t-s)}\mathrm{d}s$
$\displaystyle\leqslant$
$\displaystyle\frac{4\chi_{0}(d_{1}+d_{2}+d_{3})}{M}\|\varphi_{d}(0)\|_{E}^{2}\int_{t_{2}}^{t}\mathrm{e}^{-\theta_{2}(t-s)+C_{1}s}\mathrm{d}s$
$\displaystyle\leqslant$
$\displaystyle\frac{4\chi_{0}(d_{1}+d_{2}+d_{3})}{M(\theta_{2}+C_{1})}\|\varphi_{d}(0)\|_{E}^{2}\mathrm{e}^{C_{1}t}.$
(4.66)

Thus it follows from (4.3)-(4.3) that if $t\geqslant t_{2}$ and $M\geqslant
N_{2}$,

$\displaystyle\hskip
11.38092pt\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{M})|\varphi_{dm}(t)|_{E}^{2}\leqslant$
$\displaystyle\,\mathrm{e}^{-\theta_{2}(t-t_{2})+C_{1}t_{2}}\|\varphi_{d}(0)\|_{E}^{2}$
$\displaystyle\,+\frac{4\chi_{0}(d_{1}+d_{2}+d_{3})}{M(\theta_{2}+C_{1})}\|\varphi_{d}(0)\|_{E}^{2}\mathrm{e}^{C_{1}t}.$
(4.67)

Now pick two constants $T^{*}\geqslant t_{2}$ and $N_{3}\geqslant N_{2}$ such
that

$\mathrm{e}^{-\theta_{2}(T^{*}-t_{2})+C_{1}t_{2}}+\frac{4\chi_{0}(d_{1}+d_{2}+d_{3})}{N_{3}(\theta_{2}+C_{1})}\mathrm{e}^{C_{1}T^{*}}:=\beta^{2}<\frac{1}{4}.$

Therefore if $N^{*}\geqslant 2N_{3}$, we deduce from (4.3) that

$\sum\limits_{|m|\geqslant
N^{*}}|\varphi_{dm}(T^{*})|_{E}^{2}\leqslant\sum\limits_{m\in{\bf\mathbb{Z}}}\chi(\frac{|m|}{N_{3}})|\varphi_{dm}(T^{*})|_{E}^{2}\leqslant\beta^{2}\|\varphi_{d}(0)\|_{E}^{2},$

which implies

$\|(I-P_{N^{*}})(U_{\sigma}(T^{*},0)\varphi_{0}^{(1)}-U_{\sigma}(T^{*},0)\varphi_{0}^{(2)})\|\leqslant\beta\|\varphi_{0}^{(1)}-\varphi_{0}^{(2)}\|,$

where $\beta<1/2.$ This completes the proof of the lemma. ∎

According to Lemmas 4.2, 4.5 and Theorem 3.6, we have

######  Theorem 4.6.

Assume the assumptions $(\mathbf{H}1)$-$(\mathbf{H}2)$ hold. Then for each
$\varepsilon>0$, there exists a family of sets
$\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\mathbb{T}^{\kappa}}$
which is forward invariant for the family of processes
$\\{U_{\sigma}(t,0)\\}_{t\geqslant 0},\sigma\in\mathbb{T}^{\kappa}$ generated
by equations (4.13)-(4.14) such that
$\\{{\mathcal{B}}_{\varepsilon}(\sigma)\\}_{\sigma\in\mathbb{T}^{\kappa}}$
uniformly forward exponentially attracts each bounded subset of $E$.

##  5 Remarks on some extensions

We end the article with two remarks on some possible extensions of our main
results.

######  Remark 5.1.

Our main results could be examined for the general lattice system (3.1)-(3.2)
considered in $\mathbb{Z}^{n}$ for some positive integer $n\geqslant 2$,

$\displaystyle\dot{u}=f(u)+\sigma(t),\hskip
11.38092ptu=(u_{m})_{m\in\mathbb{Z}^{n}},\,\,t>t_{0};$ (5.1) $\displaystyle
u(t_{0})=u_{0},\hskip 11.38092ptu_{0}=(u_{m,0})_{m\in\mathbb{Z}}\in l^{2},$
(5.2)

where

$l^{2}=\\{u=(u_{m})_{m\in\mathbb{Z}^{n}}:\ u_{m}\in\mathbb{R},\hskip
11.38092pt\sum\limits_{m\in\mathbb{Z}^{n}}u_{m}^{2}<+\infty\\}.$

Moreover, the corresponding results of Theorem 4.4 and Theorem 4.6 could be
verified for the following LDS with quasi-periodic external forces:

$\displaystyle\dot{u}_{m}$
$\displaystyle=-d_{1}(Au)_{m}-(\lambda+k)u_{m}+u_{m}^{2}v_{m}-\alpha
u_{m}^{3}+\beta z_{m}+b_{1m}f_{1m}(\sigma(t)),$ (5.3)
$\displaystyle\dot{v}_{m}$
$\displaystyle=-d_{2}(Av)_{m}+\lambda(a_{m}-v_{m})-u_{m}^{2}v_{m}+\alpha
u_{m}^{3}+b_{2m}f_{2m}(\sigma(t)),$ (5.4) $\displaystyle\dot{z}_{m}$
$\displaystyle=-d_{3}(Az)_{m}+ku_{m}-(\lambda+\beta)z_{m}+b_{3m}f_{3m}(\sigma(t))$
(5.5)

for $m=(m_{1},m_{2},\cdot\cdot\cdot,m_{n})\in\mathbb{Z}^{n},$ where the
operator $A$ is defined as

$\displaystyle(Au)_{m}$
$\displaystyle=(Au)_{(m_{1},m_{2},\cdot\cdot\cdot,m_{n})}$
$\displaystyle=2ku_{m}-u_{(m_{1}+1,m_{2},\cdot\cdot\cdot,m_{n})}-u_{(m_{1},m_{2}+1,\cdot\cdot\cdot,m_{n})}-\cdot\cdot\cdot-
u_{(m_{1},m_{2},\cdot\cdot\cdot,m_{n}+1)}$ $\displaystyle\hskip 11.38092pt-
u_{(m_{1}-1,m_{2},\cdot\cdot\cdot,m_{n})}-u_{(m_{1},m_{2}-1,\cdot\cdot\cdot,m_{n})}-\cdot\cdot\cdot-
u_{(m_{1},m_{2},\cdot\cdot\cdot,m_{n}-1)}.$

Thus the similar results of Theorem 4.4 and Theorem 4.6 for equations
(5.3)-(5.5) are still valid. In such case, equations (5.3)-(5.5) can be
regarded as a discrete analogue of nonautonomous Gray-Scott equations
(4.6)-(4.8) in $\mathbb{R}^{n}$.

######  Remark 5.2.

In this article, we consider the forward dynamical behavior of nonautonomous
LDSs of the form $\dot{u}=f(u)+\sigma(t)$. In fact, for a nonautonomous LDS:
$\dot{u}=f(u,t),$ if the symbol space is compact and the corresponding
conditions are satisfied, one can also obtain the similar results. We will
investigate this issue in another paper.

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