# Lower semicontinuity of pullback attractors for a non-autonomous coupled
system of strongly damped wave equations

Everaldo M. Bonotto† Instituto de Ciências Matemáticas e de Computação,
Universidade de São Paulo, Campus de São Carlos, Caixa Postal 668, 13560-970,
São Carlos SP, Brazil<EMAIL_ADDRESS>,  Alexandre N. Carvalho††
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo,
Campus de São Carlos, Caixa Postal 668, 13560-970, São Carlos SP, Brazil.
<EMAIL_ADDRESS>,  Marcelo J. D. Nascimento⋆ Universidade Federal de São
Carlos, Departamento de Matemática, 13565-905, São Carlos SP, Brazil.
<EMAIL_ADDRESS>and  Eric B. Santiago  Instituto de Ciências Matemáticas
e de Computação, Universidade de São Paulo, Campus de São Carlos, Caixa Postal
668, 13560-970, São Carlos SP, Brazil<EMAIL_ADDRESS>
###### Abstract.

The aim of this paper is to study the robustness of the family of pullback
attractors associated to a non-autonomous coupled system of strongly damped
wave equations given by the following evolution system

$\begin{cases}u_{tt}-\Delta
u+u+\eta(-\Delta)^{1/2}u_{t}+a_{\epsilon}(t)(-\Delta)^{1/2}v_{t}=f(u),&(x,t)\in\Omega\times(\tau,\infty),\\\
v_{tt}-\Delta
v+\eta(-\Delta)^{1/2}v_{t}-a_{\epsilon}(t)(-\Delta)^{1/2}u_{t}=0,&(x,t)\in\Omega\times(\tau,\infty),\end{cases}$

subject to boundary conditions

$u=v=0,\;(x,t)\in\partial\Omega\times(\tau,\infty),$

and initial conditions

$u(\tau,x)=u_{0}(x),\ u_{t}(\tau,x)=u_{1}(x),\ v(\tau,x)=v_{0}(x),\
v_{t}(\tau,x)=v_{1}(x),\ x\in\Omega,\ \tau\in\mathbb{R},$

where $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq 3$, with
the boundary $\partial\Omega$ assumed to be regular enough, $\eta>0$ is a
constant, $a_{\epsilon}$ is a Hölder continuous function satisfying uniform
boundedness conditions, and $f\in C^{1}(\mathbb{R})$ is a dissipative
nonlinearity with subcritical growth. This problem is a modified version of
the well known Klein-Gordon-Zakharov system. Under suitable hyperbolicity
conditions, we obtain the gradient-like structure of the limit pullback
attractor associated with this evolution system, and we prove the continuity
of the family of pullback attractors at $\epsilon=0$.

Mathematics Subject Classification 2020: Primary: 35B41, 37B55. Secondary:
35B40, 35K40.
Keywords and phrases: Non-autonomous coupled system, wave equations, pullback
attractors, exponential dichotomy, lower semicontinuity.

${\dagger}$Research partially supported by FAPESP #2020/14075-6 and CNPq #
310540/2019-4

${\dagger}{\dagger}$Research partially supported by FAPESP #2020/14075-6

⋆Research partially supported by CNPq # 302743/2022-7 and FAPESP #
2020/14075-6.

##  1\. Introduction

The continuity of attractors with respect to perturbations is a growing
subject that has called the attention of many mathematicians over the years.
In the literature, the study of the robustness of attractors is usually
divided into two main features: upper semicontinuity and lower semicontinuity.

Roughly speaking, upper semicontinuity ensures that the unperturbed attractor
remains stable when the perturbation is well-behaved. On the other hand, lower
semicontinuity means that the unperturbed attractor does not undergo a
collapse or degeneration. The first property is typically expected due to the
convergence of the perturbed dynamical system under a small perturbation.
However, obtaining a result on lower semicontinuity is considerably more
challenging, as it necessitates specific structural assumptions about the
dynamics within the unperturbed attractor. Moreover, characterizing the
structure of attractors is a crucial step towards achieving a result on lower
semicontinuity.

In the case of an autonomous dynamical system (nonlinear semigroup) subject to
an autonomous perturbation, only a structural assumption for the limit
dynamical system is necessary. This means that only the limit global attractor
needs to be gradient, and all the equilibrium points of the limit semigroup
are hyperbolic. For further details, refer to [17], where the authors
established an abstract result and also explored applications to partial
differential equations. In other words, no additional characterization was
needed for the perturbed problem.

Later, in [7], the authors explored a scenario in which the perturbed
dynamical system is non-autonomous, while the associated limit dynamical
system is autonomous. In this context, the attractor is defined as the union
of unstable manifolds around hyperbolic equilibrium points. Furthermore,
through the introduction of a non-autonomous term into an autonomous equation,
they proved how the concept of a hyperbolic equilibrium point (in the
autonomous case) turns to the concept of a hyperbolic global solution for
evolution processes (within the non-autonomous framework). Additionally, they
established the continuity of local unstable and stable manifolds in relation
to perturbations.

For non-autonomous problems under regular perturbation, the work [8] focuses
on the property of exponential dichotomy for non-autonomous linear equations.
It provides more general results that enable the treatment of cases where the
unperturbed pullback attractor is given as the closure of the union of
unstable manifolds around an infinite collection of hyperbolic time-dependent
global solutions.

Regarding applications of the abstract results in the literature, we refer to
[5], where the authors obtained the continuity of pullback attractors at
$\epsilon=0$ for a non-autonomous strongly damped wave equation, given by the
following formulation

$u_{tt}-\Delta u-\gamma(t)\Delta u_{t}+\beta_{\epsilon}(t)u_{t}=f(u),$

in a bounded domain $\Omega\subset\mathbb{R}^{N}$, where the time-dependent
terms $\gamma(t)$ and $\beta_{\epsilon}(t)$ are continuously differentiable in
$\mathbb{R}$ satisfying some uniform boundedness conditions, and the nonlinear
term $f$ has appropriate growth restrictions.

Meanwhile, in [25], the author studied the lower semicontinuity of the family
of pullback attractors for the following singularly non-autonomous plate
equation

$\begin{cases}u_{tt}+a_{\epsilon}(t,x)u_{t}+(-\Delta)u_{t}+(-\Delta)^{2}u+\lambda
u=f(u),&\mbox{in}\quad\Omega,\\\ u=\Delta
u=0,&\mbox{on}\quad\partial\Omega,\end{cases}$

with respect to the damping coefficient $a_{\epsilon}$, where $\Omega$ is a
bounded domain in $\mathbb{R}^{N}$, $\lambda>0$, and $f$ is a suitable
nonlinearity. This type of problem is related to models of vibrations in
elastic systems.

In this paper, we aim to investigate the robustness of the family of pullback
attractors associated with a non-autonomous coupled system of strongly damped
wave equations, as examined in [4]. This system is defined by the following
evolution equations:

(1.1) $\begin{cases}u_{tt}-\Delta
u+u+\eta(-\Delta)^{1/2}u_{t}+a_{\epsilon}(t)(-\Delta)^{1/2}v_{t}=f(u),&(x,t)\in\Omega\times(\tau,\infty),\\\
v_{tt}-\Delta
v+\eta(-\Delta)^{1/2}v_{t}-a_{\epsilon}(t)(-\Delta)^{1/2}u_{t}=0,&(x,t)\in\Omega\times(\tau,\infty),\\\
u=v=0,&(x,t)\in\partial\Omega\times(\tau,\infty),\end{cases}$

with initial conditions

$u(\tau,x)=u_{0}(x),\ u_{t}(\tau,x)=u_{1}(x),\ v(\tau,x)=v_{0}(x),\
v_{t}(\tau,x)=v_{1}(x),\ x\in\Omega,\ \tau\in\mathbb{R},$

where $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq 3$, with
the boundary $\partial\Omega$ assumed to be regular enough, $\eta>0$ is a
constant, $a_{\epsilon}$ is a Hölder continuous function satisfying uniform
boundedness conditions, and $f\in C^{1}(\mathbb{R})$ is a dissipative
nonlinearity with a subcritical growth restriction.

The problem (1.1) is a non-autonomous modified version of the so called Klein-
Gordon-Zakharov (KGZ) system. In the autonomous case with space dimension
$N=3$, it is well known that the KGZ system arises to describe the interaction
between a Langmuir wave and an ion acoustic wave in a plasma, see [3, 13, 23]
and the references therein.

In [4], the authors used the theory of uniform sectorial operators to
establish the local and global well-posedness of the problem (1.1) in the
product space $H_{0}^{1}(\Omega)\times L^{2}(\Omega)\times
H_{0}^{1}(\Omega)\times L^{2}(\Omega)$. By proving that the nonlinear
evolution process, associated to the global solution of (1.1), is a compact
map, it was established the existence of pullback attractors for the problem
(1.1). Using energy estimates and progressive increases of regularity, the
authors were able to improve the regularity of the pullback attractors.
Lastly, under a suitable assumption on the functional parameter
$a_{\epsilon}(t)$, they obtained the upper semicontinuity of the family of
pullback attractors at $\epsilon=0$.

In this work, we study the limit system ($\epsilon=0$) associated with (1.1)
in order to obtain a characterization for the unperturbed pullback attractor
in terms of the unstable manifolds around equilibrium points, assuming that
all of them are hyperbolic for the limit problem. With the aid of a Lyapunov
Functional, we prove that any bounded global solution in the limit pullback
attractor is forward and backward asymptotic to equilibria. We also study the
linearized system associated with (1.1) in order to prove a result on
exponential dichotomy under perturbation, see Proposition 5.3 and Corollary
5.4. Moreover, thanks to abstract general results on lower semicontinuity from
[5, 7, 8], we obtain the lower semicontinuity of pullback attractors at
$\epsilon=0$. Consequently, we achieve the continuity of the family of
pullback attractors at $\epsilon=0$, concluding in this way our study of
robustness for the problem (1.1).

It is important to mention that, in our system (1.1), the coupling terms

$a_{\epsilon}(t)(-\Delta)^{1/2}v_{t}\quad\mbox{and}\quad-
a_{\epsilon}(t)(-\Delta)^{1/2}u_{t}$

have the same order, in the sense of fractional powers, as well as the strong
damping terms $\eta(-\Delta)^{1/2}u_{t}$ and $\eta(-\Delta)^{1/2}v_{t}$. This
scenario generated a situation where we were not able to apply [18, Theorem
7.6.11] directly in order to obtain our result on exponential dichotomy for
the problem (1.1). For more details, see Section 5, in particular Lemma 5.2,
Proposition 5.3 and Corollary 5.4, where we had to carry out an analysis that
is not straightforward, and then we apply [18, Theorem 7.6.10] to conclude our
proof on exponential dichotomy. In fact, there were several difficulties that
we had to overcome, and to the best of our knowledge, the analysis presented
in this work is novel up to the present moment. The reader may compare our
results with those presented in [5] and [25], for instance. It is worth
mentioning that if we replace the coupling terms
$a_{\epsilon}(t)(-\Delta)^{1/2}v_{t}$ and
$-a_{\epsilon}(t)(-\Delta)^{1/2}u_{t}$ in system (1.1), respectively, by
$a_{\epsilon}(t)v_{t}$ and $-a_{\epsilon}(t)u_{t}$, then one may proceed
exactly as in [5] in order to obtain the continuity of the family of pullback
attractors at $\epsilon=0$.

In what follows, we describe the outline of this paper.

  1. $\bullet$

Section 2 is devoted to formulate the non-autonomous problem (1.1) in an
abstract form within the framework of nonlinear evolution processes. We
present sufficient assumptions that ensure global well-posedness and the
existence of pullback attractors for the problem (1.1) in an appropriate
energy space. Additionally, we recall some auxiliary results established in
[4].

  2. $\bullet$

Section 3 deals with the structure of the limit problem (that is, with
parameter $\epsilon=0$) associated to (1.1). In Proposition 3.7, we exhibit a
Lyapunov functional associated with system (1.1). In Theorem 3.8, we establish
the gradient-like structure of the limit pullback attractor, that is, it can
be written as the union of the unstable manifolds around equilibria.

  3. $\bullet$

Section 4 encompasses a spectral analysis of the matrix operator associated to
the linearization of problem (1.1) around an equilibrium point. Here, we prove
a continuity result for the inverse operator concerning the parameter
$\epsilon$ (see Proposition 4.2). Furthermore, we derive sectorial estimates
that are uniform in both time and parameter $\epsilon$. Using these uniform
estimates, we prove the convergence of linear semigroups, uniformly on compact
subsets, as $\epsilon$ approaches to zero (see Proposition 4.6).

  4. $\bullet$

Section 5 presents the results regarding the robustness of problem (1.1).
Through a combination of arguments and results from [10] and [18], we
establish a result on exponential dichotomy under perturbation (with respect
to parameter $\epsilon$), see Proposition 5.3 and Corollary 5.4. The lower
semicontinuity of the family of pullback attractors at $\epsilon=0$ is
obtained in Theorem 5.5.

  5. $\bullet$

Appendix A is dedicated to presenting a collection of preliminary facts that
are useful for the development of the article. We provide definitions and
results concerning the theory of nonlinear evolution processes and pullback
attractors.

  6. $\bullet$

Appendix A addresses an abstract result on lower semicontinuity of pullback
attractors which will be used in the main result.

##  2\. Abstract setting of the problem

Consider the following initial boundary value problem

(2.1) $\begin{cases}u_{tt}-\Delta
u+u+\eta(-\Delta)^{1/2}u_{t}+a_{\epsilon}(t)(-\Delta)^{1/2}v_{t}=f(u),&(x,t)\in\Omega\times(\tau,\infty),\\\
v_{tt}-\Delta
v+\eta(-\Delta)^{1/2}v_{t}-a_{\epsilon}(t)(-\Delta)^{1/2}u_{t}=0,&(x,t)\in\Omega\times(\tau,\infty),\end{cases}$

subject to boundary conditions

$u=v=0,\;(x,t)\in\partial\Omega\times(\tau,\infty),$

and initial conditions

(2.2) $u(\tau,x)=u_{0}(x),\ u_{t}(\tau,x)=u_{1}(x),\ v(\tau,x)=v_{0}(x),\
v_{t}(\tau,x)=v_{1}(x),\ x\in\Omega,\ \tau\in\mathbb{R}.$

In what follows, we recall the sufficient conditions that ensure the global
well-posedness of the non-autonomous problem $(\ref{edp01})$-$(\ref{cond01})$,
which is established in [4].

At first place, the non-autonomous problem $(\ref{edp01})$-$(\ref{cond01})$
can be formulated in a nonlinear evolution process setting by introducing the
following notations. Let $X=L^{2}(\Omega)$ and denote by $A\colon D(A)\subset
X\to X$ the negative Laplacian operator with the Dirichlet boundary
conditions, that is, $Au=(-\Delta)u$ for all $u\in D(A)$, where
$D(A)=H^{2}(\Omega)\cap H_{0}^{1}(\Omega)$. It is well-known that $A$ is a
positive self-adjoint operator in $X$ with compact resolvent and, following
Henry [18], $A$ is a sectorial operator in $X$. Thus, $-A$ generates a compact
analytic semigroup on $X$. Now, denote by $X^{\alpha}$, $\alpha>0$, the
fractional power spaces associated with the operator $A$, that is,
$X^{\alpha}=D(A^{\alpha})$ endowed with the graph norm. With this notation, we
have $X^{-\alpha}=(X^{\alpha})^{\prime}$ for all $\alpha>0$, see [1].

With this setting, the non-autonomous problem $(\ref{edp01})$-$(\ref{cond01})$
can be rewritten as an ordinary differential equation in the following
abstract form

(2.3) $\begin{cases}W_{t}+\mathcal{A}_{\epsilon}(t)W=F(W),&\ t>\tau,\\\
W(\tau)=W_{0},&\ \tau\in\mathbb{R},\end{cases}$

where $W=W(t)$, for all $t\in\mathbb{R}$, and $W_{0}=W(\tau)$ are respectively
given by

$W=\begin{bmatrix}u\\\ u_{t}\\\ v\\\ v_{t}\\\ \end{bmatrix}\ \text{and}\ \
W_{0}=\begin{bmatrix}u_{0}\\\ u_{1}\\\ v_{0}\\\ v_{1}\\\ \end{bmatrix},$

and, for each $t\in\mathbb{R}$, the unbounded linear operator
$\mathcal{A}(t)\colon D(\mathcal{A}(t))\subset Y_{0}\to Y_{0}$ is defined by

(2.4) $\begin{split}\mathcal{A}_{\epsilon}(t)\\!\\!\begin{bmatrix}u\\\ v\\\
w\\\ z\\\ \end{bmatrix}\\!\\!&=\\!\\!\begin{bmatrix}0&-I&0&0\\\ A+I&\eta
A^{1/2}&0&a_{\epsilon}(t)A^{1/2}\\\ 0&0&0&-I\\\
0&-a_{\epsilon}(t)A^{1/2}&A&\eta A^{1/2}\end{bmatrix}\\!\\!\begin{bmatrix}u\\\
v\\\ w\\\ z\\\ \end{bmatrix}\\\ \end{split}$

for each $\begin{bmatrix}u&v&w&z\end{bmatrix}^{T}$ in the domain
$D(\mathcal{A}_{\epsilon}(t))$ defined by

(2.5) $D(\mathcal{A}_{\epsilon}(t))=X^{1}\times X^{1/2}\times X^{1}\times
X^{1/2},$

where $Y_{0}=X^{1/2}\times X\times X^{1/2}\times X$ is the phase space of the
problem $(\ref{edp01})$-$(\ref{cond01})$. The nonlinearity $F$ is given by

(2.6) $F(W)=\begin{bmatrix}0\\\ f^{e}(u)\\\ 0\\\ 0\\\ \end{bmatrix},$

where $f^{e}(u)$ is the Nemitskiĭ operator associated with $f(u)$, that is,
$f^{e}(u)(x)=f(u(x)),$ for all $x\in\Omega.$

In order to ensure the global well-posedness of the problem
$(\ref{edp01})$-$(\ref{cond01})$ in $Y_{0}$, some suitable assumptions on the
functional parameter $a_{\epsilon}$ and on the nonlinearity $f$ are needed.
Assume that the function $a_{\epsilon}\colon\mathbb{R}\to(0,\infty)$ is
continuously differentiable in $\mathbb{R}$ and satisfies the following
condition:

(2.7) $0<a_{0}\leq a_{\epsilon}(t)\leq a_{1},$

for all $\epsilon\in[0,1]$ and $t\in\mathbb{R}$, with positive constants
$a_{0}$ and $a_{1}$, and we also assume that the first derivative of
$a_{\epsilon}$ is uniformly bounded in $t$ and $\epsilon$, that is, there
exists a constant $b_{0}>0$ such that

$|a_{\epsilon}^{\prime}(t)|\leq b_{0}\quad\mbox{for all}\quad t\in\mathbb{R},\
\epsilon\in[0,1].$

The functional parameter $a_{\epsilon}$ will be assumed to be
$(\beta,C)$-Hölder continuous for each $\epsilon\in[0,1]$, that is,

(2.8) $|a_{\epsilon}(t)-a_{\epsilon}(s)|\leq C|t-s|^{\beta},$

for all $t,s\in\mathbb{R}$ and $\epsilon\in[0,1],$ where the constants $C>0$
and $0<\beta\leq 1$ are both independent of the time $t\in\mathbb{R}$ and also
of the parameter $\epsilon$. Additionally, we will assume that

(2.9) $\|a_{\epsilon}-a_{0}\|_{L^{\infty}(\mathbb{R})}\rightarrow
0\quad\mbox{as}\quad\epsilon\rightarrow 0^{+}.$

Moreover, suppose that $f\in C^{1}(\mathbb{R})$ and it satisfies the
dissipativeness condition

(2.10) $\limsup\limits_{|s|\to\infty}\frac{f(s)}{s}\leq 0,$

and also satisfies the subcritical growth condition

(2.11) $|f^{\prime}(s)|\leq c(1+|s|^{\rho-1}),\quad\text{for
all}\;\;s\in\mathbb{R},$

where $1<\rho<\frac{N+2}{N-2}$, with $N\geq 3$, and $c>0$ is a constant. For a
detailed study of equations of type (2.3) in the parabolic case, we refer to
[10, 26].

Now, let $\alpha\in(0,1)$ be such that

(2.12) $1<\rho<\frac{N+2(1-\alpha)}{N-2},$

with $N\geq 3$. Recall that $Y_{\alpha-1}=[Y_{-1},Y_{0}]_{\alpha}$, where

$Y_{-1}=X\times X^{-1/2}\times X\times X^{-1/2}$

is the extrapolation space of $Y_{0}$, and $[\cdot,\cdot]_{\alpha}$ denotes
the complex interpolation functor (see [27]).

Under the previous conditions, the following result on global well-posedness
holds, see [4, Theorem 2.1].

######  Theorem 2.1.

[Global Well-Posedness] Let $1<\rho<\frac{N+2}{N-2}$, $N\geq 3$,
$\alpha\in(0,1)$ satisfying condition (2.12), and let $f\in C^{1}(\mathbb{R})$
be a function satisfying (2.10)-(2.11). Assume that conditions (2.7)-(2.8)
hold and let $F\colon Y_{0}\to Y_{\alpha-1}$ be defined as in (2.6). Then for
any initial data $W_{0}\in Y_{0}$ the problem (2.3) has a unique global
solution $W(t)$ such that $W(t)\in C([\tau,\infty),Y_{0}).$ Moreover, such
solutions are continuous with respect to the initial data on $Y_{0}$.

Since the problem (2.1)-(2.2) has a global solution $W(t)$ in $Y_{0}$, we can
define an evolution process $\\{S(t,\tau)\colon t\geq\tau\in\mathbb{R}\\}$ in
$Y_{0}$ by

(2.13) $S(t,\tau)W_{0}=W(t)=(u,u_{t},v,v_{t}),\quad t\geq\tau\in\mathbb{R}.$

See [4, Theorem 5.1] and [4, Proposition 5.4] for a proof of Theorems 2.2 and
2.3 below.

######  Theorem 2.2.

There exists $R>0$ such that for any bounded subset $B\subset Y_{0}$ one can
find $t_{0}(B)>0$ satisfying $\|S(t,\tau)W_{0}\|_{Y_{0}}^{2}\leq R$ for all
$t\geq\tau+t_{0}(B)$ and $W_{0}\in B$. In particular, the evolution process
$\\{S(t,\tau)\colon t\geq\tau\in\mathbb{R}\\}$ defined in (2.13) is pullback
strongly bounded dissipative.

######  Theorem 2.3.

For each $t>\tau\in\mathbb{R}$, the evolution process $S(t,\tau)\colon
Y_{0}\to Y_{0}$ given in (2.13) is a compact map.

In [4, Theorem 2.2] it was established that, for each $\epsilon\in[0,1],$ the
nonlinear evolution process $\\{S_{\epsilon}(t,\tau)\colon
t\geq\tau\in\mathbb{R}\\}$, associated to the problem (2.1)-(2.2), has a
pullback attractor $\\{\mathbb{A}_{\epsilon}(t)\colon t\in\mathbb{R}\\}$ in
the phase space $Y_{0}$.

######  Theorem 2.4.

[Existence of Pullback Attractors] Under the conditions of Theorem 2.1, for
each $\epsilon\in[0,1],$ the problem (2.1)-(2.2) has a pullback attractor
$\\{\mathbb{A}_{\epsilon}(t)\colon t\in\mathbb{R}\\}$ in $Y_{0}$ and,
moreover, the union
$\bigcup\limits_{t\in\mathbb{R}}\mathbb{A}_{\epsilon}(t)\subset Y_{0}$ is
bounded.

The next result states the regularity of the pullback attractors obtained in
Theorem 2.4, see [4, Theorem 2.3] for a proof.

######  Theorem 2.5.

[Regularity of Pullback Attractors] Assume that
$\frac{N-1}{N-2}\leq\rho<\frac{N}{N-2}$, $N\geq 3$. For each
$\epsilon\in[0,1],$ the pullback attractor $\\{\mathbb{A}_{\epsilon}(t)\colon
t\in\mathbb{R}\\}$ for the problem (2.1)-(2.2), obtained in Theorem 2.4, lies
in a more regular space than $Y_{0}$. More precisely, we have
$\bigcup\limits_{t\in\mathbb{R}}\mathbb{A}_{\epsilon}(t)$ is a bounded subset
of $Y_{1}=X^{1}\times X^{1/2}\times X^{1}\times X^{1/2}.$

As a consequence of Theorem 2.5, it follows by the compact embedding
$Y_{1}\hookrightarrow Y_{0}$ that

(2.14)
$\overline{\bigcup\limits_{\epsilon\in[0,1]}\bigcup\limits_{t\in\mathbb{R}}\mathbb{A}_{\epsilon}(t)}\mbox{\,\,
is compact in \,}Y_{0}.$

Lemma 2.6 deals with the continuity of the evolution process
$\\{S_{\epsilon}(t,\tau):t\geq\tau\in\mathbb{R}\\}$ as $\epsilon\to 0^{+}$.
This result is proved in [4, Theorem 2.4].

######  Lemma 2.6.

Let $\\{S_{\epsilon}(t,\tau)\colon t\geq\tau\in\mathbb{R}\\}$ be the evolution
processes associated to the problem (2.1)-(2.2). Then
$S_{\epsilon}(t,\tau)x\xrightarrow{\epsilon\rightarrow 0^{+}}S_{0}(t,\tau)x$
uniformly in bounded subsets of $\\{(t,\tau)\in\mathbb{R}^{2}\colon
t\geq\tau\\}\times Y_{0}$.

In [4, Theorem 2.4] it was also proved the upper semicontinuity of the family
of pullback attractors associated to the problem (2.1)-(2.2). The next theorem
states this result.

######  Theorem 2.7.

[Upper Semicontinuity] Under condition (2.9), the family of pullback
attractors $\\{\mathbb{A}_{\epsilon}(t)\colon t\in\mathbb{R}\\}$, associated
to the problem (2.1)-(2.2), is upper semicontinuous at $\epsilon=0$, that is,
for each $t\in\mathbb{R}$,

${\rm d_{H}}(\mathbb{A}_{\epsilon}(t),\mathbb{A}_{0}(t))\rightarrow
0\quad\mbox{as}\quad\epsilon\rightarrow 0^{+}.$

##  3\. Structure of the limit problem

In this section, we are going to study the structure of the limit problem

(3.1) $\begin{cases}u_{tt}-\Delta
u+u+\eta(-\Delta)^{1/2}u_{t}+a_{0}(t)(-\Delta)^{1/2}v_{t}=f(u),&(x,t)\in\Omega\times(\tau,\infty),\\\
v_{tt}-\Delta
v+\eta(-\Delta)^{1/2}v_{t}-a_{0}(t)(-\Delta)^{1/2}u_{t}=0,&(x,t)\in\Omega\times(\tau,\infty),\\\
u=v=0,&(x,t)\in\partial\Omega\times(\tau,\infty),\\\
(u(\tau),u_{t}(\tau),v(\tau),v_{t}(\tau))=(u_{0},u_{1},v_{0},v_{1}),\end{cases}$

which is associated to the problem (2.1)-(2.2) with parameter $\epsilon=0$.
Firstly, we will prove a result on asymptotic stability for solutions of (3.1)
starting in the pullback attractor. Thereafter, this will be used to show that
the limit pullback attractor $\\{\mathbb{A}_{0}(t)\colon t\in\mathbb{R}\\}$
can be written as the union of the unstable manifolds around equilibria.

From now on, we shall assume that $\frac{N-1}{N-2}\leq\rho<\frac{N}{N-2}$.
Recall from Theorem 2.5 that this condition ensures that
$\overline{\bigcup\limits_{\epsilon\in[0,1]}\bigcup\limits_{t\in\mathbb{R}}\mathbb{A}_{\epsilon}(t)}$
is compact in $Y_{0}$.

###  3.1. Asymptotic stability of solutions

Recall that $a_{\epsilon}$ is $(\beta,C)$-Hölder continuous for each
$\epsilon\in[0,1]$, with $0<\beta\leq 1$ (see (2.8)). By Arzelà-Ascoli
Theorem, for each fixed $t\in\mathbb{R}$, given a sequence
$\\{t_{n}\\}_{n\in\mathbb{N}}\subset\mathbb{R}$, we have

(3.2) $\\{a_{n}(t)=a_{0}(t_{n}+t)\\}_{n\in\mathbb{N}}$

has a convergent subsequence (which we denote by the same notation) such that

$a_{n}(t)\xrightarrow{n\rightarrow\infty}a_{\infty}(t)$

uniformly for $t$ in compact subsets of $\mathbb{R}$. Note that the limit
function $a_{\infty}$ is also bounded as in (2.7). Thus, we may consider the
following problems

(3.3) $\begin{cases}u_{tt}-\Delta
u+u+\eta(-\Delta)^{1/2}u_{t}+a_{n}(t)(-\Delta)^{1/2}v_{t}=f(u),&(x,t)\in\Omega\times(\tau,\infty),\\\
v_{tt}-\Delta
v+\eta(-\Delta)^{1/2}v_{t}-a_{n}(t)(-\Delta)^{1/2}u_{t}=0,&(x,t)\in\Omega\times(\tau,\infty),\\\
u=v=0,&(x,t)\in\partial\Omega\times(\tau,\infty),\\\
(u(\tau),u_{t}(\tau),v(\tau),v_{t}(\tau))=(u_{0},u_{1},v_{0},v_{1})\in
Y_{0},\end{cases}$

and

(3.4) $\begin{cases}u_{tt}-\Delta
u+u+\eta(-\Delta)^{1/2}u_{t}+a_{\infty}(t)(-\Delta)^{1/2}v_{t}=f(u),&(x,t)\in\Omega\times(\tau,\infty),\\\
v_{tt}-\Delta
v+\eta(-\Delta)^{1/2}v_{t}-a_{\infty}(t)(-\Delta)^{1/2}u_{t}=0,&(x,t)\in\Omega\times(\tau,\infty),\\\
u=v=0,&(x,t)\in\partial\Omega\times(\tau,\infty),\\\
(u(\tau),u_{t}(\tau),v(\tau),v_{t}(\tau))=(u_{0},u_{1},v_{0},v_{1})\in
Y_{0},\end{cases}$

with global solutions given, respectively, by
$W^{n}(t)=(u^{n}(t),u_{t}^{n}(t),v^{n}(t),v_{t}^{n}(t))$ and
$W^{\infty}(t)=(u^{\infty}(t),u_{t}^{\infty}(t),v^{\infty}(t),v_{t}^{\infty}(t))$,
$t\geq\tau\in\mathbb{R}.$

Moreover, for $n\in\mathbb{N}$, let

$\\{\mathbb{A}_{n}(t)\colon
t\in\mathbb{R}\\}\quad\mbox{and}\quad\\{\mathbb{A}_{\infty}(t)\colon
t\in\mathbb{R}\\}$

be the pullback attractors associated with the evolution problems (3.3) and
(3.4), respectively.

######  Theorem 3.1.

If
$W^{n}(\cdot)=(u^{n}(\cdot),u_{t}^{n}(\cdot),v^{n}(\cdot),v_{t}^{n}(\cdot))$
and
$W^{\infty}(\cdot)=(u^{\infty}(\cdot),u_{t}^{\infty}(\cdot),v^{\infty}(\cdot),v_{t}^{\infty}(\cdot))$
are the global solutions associated to the problems (3.3) and (3.4),
respectively, starting in their pullback attractors, then

$\|W^{n}(t)-W^{\infty}(t)\|_{Y_{0}}^{2}\xrightarrow{n\rightarrow\infty}0$

in compact subsets of $\mathbb{R}$.

###### Proof.

If we denote $u=u^{n}-u^{\infty}$ and $v=v^{n}-v^{\infty}$, then from systems
(3.3) and (3.4), we are able to obtain the following system

$\begin{cases}u_{tt}-\Delta
u+u+\eta(-\Delta)^{1/2}u_{t}+a_{n}(t)(-\Delta)^{1/2}v_{t}^{n}-a_{\infty}(t)(-\Delta)^{1/2}v_{t}^{\infty}=f(u^{n})-f(u^{\infty}),\\\
v_{tt}-\Delta
v+\eta(-\Delta)^{1/2}v_{t}-a_{n}(t)(-\Delta)^{1/2}u_{t}^{n}+a_{\infty}(t)(-\Delta)^{1/2}u_{t}^{\infty}=0,\end{cases}$

for all $t>\tau$ and $x\in\Omega$, which can be rewritten in the following way

$\begin{cases}u_{tt}-\Delta
u+u+\eta(-\Delta)^{1/2}u_{t}+a_{n}(t)(-\Delta)^{1/2}v_{t}+[a_{n}(t)-a_{\infty}(t)](-\Delta)^{1/2}v_{t}^{\infty}=f(u^{n})-f(u^{\infty}),\\\
v_{tt}-\Delta
v+\eta(-\Delta)^{1/2}v_{t}-a_{n}(t)(-\Delta)^{1/2}u_{t}+[a_{\infty}(t)-a_{n}(t)](-\Delta)^{1/2}u_{t}^{\infty}=0,\end{cases}$

for all $t>\tau$ and $x\in\Omega$. Taking the inner product of the first
equation in the previous system with $u_{t}$ in $X$, and also the inner
product of the second equation with $v_{t}$ in $X$, we get

$\begin{split}&\frac{1}{2}\frac{d}{dt}\int_{\Omega}|u_{t}|^{2}\,dx+\frac{1}{2}\frac{d}{dt}\int_{\Omega}|\nabla
u|^{2}\,dx+\frac{1}{2}\frac{d}{dt}\int_{\Omega}|u|^{2}\,dx+\eta\|(-\Delta)^{1/4}u_{t}\|_{X}^{2}\\\
&+a_{n}(t)\langle(-\Delta)^{1/2}v_{t},u_{t}\rangle_{X}+[a_{n}(t)-a_{\infty}(t)]\langle(-\Delta)^{1/2}v_{t}^{\infty},u_{t}\rangle_{X}\\\
&=\langle f(u^{n})-f(u^{\infty}),u_{t}\rangle_{X}\end{split}$

and

$\begin{split}&\frac{1}{2}\frac{d}{dt}\int_{\Omega}|v_{t}|^{2}\,dx+\frac{1}{2}\frac{d}{dt}\int_{\Omega}|\nabla
v|^{2}\,dx+\eta\|(-\Delta)^{1/4}v_{t}\|_{X}^{2}\\\
&-a_{n}(t)\langle(-\Delta)^{1/2}u_{t},v_{t}\rangle_{X}+[a_{\infty}(t)-a_{n}(t)]\langle(-\Delta)^{1/2}u_{t}^{\infty},v_{t}\rangle_{X}=0,\end{split}$

respectively. Adding up these two last equations, it follows that

$\begin{split}&\frac{1}{2}\frac{d}{dt}\left(\int_{\Omega}|\nabla
u|^{2}\,dx+\int_{\Omega}|u|^{2}\,dx+\int_{\Omega}|u_{t}|^{2}\,dx+\int_{\Omega}|\nabla
v|^{2}\,dx+\int_{\Omega}|v_{t}|^{2}\,dx\right)\\\
&\quad+\eta\|(-\Delta)^{1/4}u_{t}\|_{X}^{2}+\eta\|(-\Delta)^{1/4}v_{t}\|_{X}^{2}+[a_{n}(t)-a_{\infty}(t)]\langle(-\Delta)^{1/2}v_{t}^{\infty},u_{t}\rangle_{X}\\\
&\quad+[a_{\infty}(t)-a_{n}(t)]\langle(-\Delta)^{1/2}u_{t}^{\infty},v_{t}\rangle_{X}\\\
&=\int_{\Omega}[f(u^{n})-f(u^{\infty})]u_{t}\,dx,\end{split}$

which leads to

$\begin{split}&\frac{d}{dt}\left(\|u\|_{X^{1/2}}^{2}+\|u\|_{X}^{2}+\|u_{t}\|_{X}^{2}+\|v\|_{X^{1/2}}^{2}+\|v_{t}\|_{X}^{2}\right)\\\
&=-2\eta\left(\|(-\Delta)^{1/4}u_{t}\|_{X}^{2}+\|(-\Delta)^{1/4}v_{t}\|_{X}^{2}\right)+2[a_{\infty}(t)-a_{n}(t)]\langle(-\Delta)^{1/2}v_{t}^{\infty},u_{t}\rangle_{X}\\\
&\quad+2[a_{n}(t)-a_{\infty}(t)]\langle(-\Delta)^{1/2}u_{t}^{\infty},v_{t}\rangle_{X}+2\int_{\Omega}[f(u^{n})-f(u^{\infty})]u_{t}\,dx\\\
&\leq
2|a_{\infty}(t)-a_{n}(t)|\|v_{t}^{\infty}\|_{X^{1/2}}\left(\|u_{t}^{n}\|_{X}+\|u_{t}^{\infty}\|_{X}\right)\\\
&\quad+2|a_{n}(t)-a_{\infty}(t)|\|u_{t}^{\infty}\|_{X^{1/2}}\left(\|v_{t}^{n}\|_{X}+\|v_{t}^{\infty}\|_{X}\right)+2\int_{\Omega}|(f(u^{n})-f(u^{\infty}))u_{t}|\,dx.\end{split}$

From the regularity theorem of the pullback attractor (see [4, Theorem 2.3]),
there exists a constant $C_{1}>0$, which is independent of both the parameter
$\epsilon$ and $t$, such that

$\|u_{t}^{\infty}\|_{X^{1/2}},\,\|v_{t}^{\infty}\|_{X^{1/2}}\leq C_{1}.$

Moreover, due to the theorem on existence of global solution, it follows that
the norms

$\|u_{t}^{n}\|_{X},\,\|u_{t}^{\infty}\|_{X},\,\|v_{t}^{n}\|_{X},\,\|v_{t}^{\infty}\|_{X}$

are all bounded and this boundedness depends only on the initial data (see the
proof of [4, Theorem 2.1]). Thus, it turns out that we have

(3.5)
$\begin{split}&\frac{d}{dt}\left(\|u\|_{X^{1/2}}^{2}+\|u\|_{X}^{2}+\|u_{t}\|_{X}^{2}+\|v\|_{X^{1/2}}^{2}+\|v_{t}\|_{X}^{2}\right)\\\
&\leq
C^{\prime}|a_{n}(t)-a_{\infty}(t)|+2\int_{\Omega}|(f(u^{n})-f(u^{\infty}))u_{t}|\,dx,\end{split}$

for some constant $C^{\prime}>0$.

Now, let us deal with the integral term
$\int_{\Omega}|(f(u^{n})-f(u^{\infty}))u_{t}|\,dx$. In fact, using the Mean
Value Theorem, there exists $\sigma\in(0,1)$ such that

$|f(u^{n})-f(u^{\infty})|=|f^{\prime}(\sigma
u^{n}+(1-\sigma)u^{\infty})|\,|u^{n}-u^{\infty}|=|f^{\prime}(\sigma
u^{n}+(1-\sigma)u^{\infty})|\,|u|.$

On the other hand, since $\frac{\rho-1}{2\rho}+\frac{1}{2\rho}+\frac{1}{2}=1$,
then we apply Hölder inequality to obtain

(3.6)
$\begin{split}&\int_{\Omega}|(f(u^{n})-f(u^{\infty}))u_{t}|\,dx=\int_{\Omega}|f^{\prime}(\sigma
u^{n}+(1-\sigma)u^{\infty})|\,|u|\,|u_{t}|\,dx\\\ &\leq\|f^{\prime}(\sigma
u^{n}+(1-\sigma)u^{\infty})\|_{L^{\frac{2\rho}{\rho-1}}(\Omega)}\|u\|_{L^{2\rho}(\Omega)}\|u_{t}\|_{L^{2}(\Omega)}.\end{split}$

The embedding $X^{1/2}\hookrightarrow L^{2\rho}(\Omega)$ holds as
$\rho\in\left[\frac{N-1}{N-2},\frac{N}{N-2}\right)$. Thus, we can use the
growth condition (2.11) to obtain a bound for the norm $\|f^{\prime}(\sigma
u^{n}+(1-\sigma)u^{\infty})\|_{L^{\frac{2\rho}{\rho-1}}(\Omega)}$ as follows

(3.7) $\begin{split}&\|f^{\prime}(\sigma
u^{n}+(1-\sigma)u^{\infty})\|_{L^{\frac{2\rho}{\rho-1}}(\Omega)}\leq\left(\int_{\Omega}[c(1+|\sigma
u^{n}+(1-\sigma)u^{\infty}|^{\rho-1})]^{\frac{2\rho}{\rho-1}}\,dx\right)^{\frac{\rho-1}{2\rho}}\\\
&\leq c_{1}\left(|\Omega|^{\frac{\rho-1}{2\rho}}+\|\sigma
u^{n}+(1-\sigma)u^{\infty}\|_{L^{2\rho}(\Omega)}^{\rho-1}\right)\\\ &\leq
c_{2}\left(1+\|u^{n}\|_{L^{2\rho}(\Omega)}^{\rho-1}+\|u^{\infty}\|_{L^{2\rho}(\Omega)}^{\rho-1}\right)\\\
&\leq
c_{3}\left(1+\|u^{n}\|_{X^{1/2}}^{\rho-1}+\|u^{\infty}\|_{X^{1/2}}^{\rho-1}\right)\leq
C_{0},\end{split}$

where the norms $\|u^{n}\|_{X^{1/2}}$ and $\|u^{\infty}\|_{X^{1/2}}$ are
bounded by the theorem on existence of global solution, and this boundedness
depends only on the initial data (see the proof of [4, Theorem 2.1]).

Thus, combining the estimates (3.5), (3.6) and (3.7), we obtain

$\begin{split}&\frac{d}{dt}\left(\|u\|_{X^{1/2}}^{2}+\|u\|_{X}^{2}+\|u_{t}\|_{X}^{2}+\|v\|_{X^{1/2}}^{2}+\|v_{t}\|_{X}^{2}\right)\\\
&\leq
C^{\prime}|a_{n}(t)-a_{\infty}(t)|+2C_{0}\|u\|_{L^{2\rho}(\Omega)}\|u_{t}\|_{L^{2}(\Omega)}\\\
&\leq
C^{\prime}|a_{n}(t)-a_{\infty}(t)|+2\hat{C}\|u\|_{X^{1/2}}\|u_{t}\|_{X}\\\
&\leq
C^{\prime}|a_{n}(t)-a_{\infty}(t)|+\hat{C}\left(\|u\|_{X^{1/2}}^{2}+\|u\|_{X}^{2}+\|u_{t}\|_{X}^{2}+\|v\|_{X^{1/2}}^{2}+\|v_{t}\|_{X}^{2}\right),\end{split}$

that is,

$\frac{d}{dt}G(t)\leq
C^{\prime\prime}|a_{n}(t)-a_{\infty}(t)|+C^{\prime\prime}G(t),\quad t>\tau,$

where $C^{\prime\prime}=\max\\{C^{\prime},\hat{C}\\}>0$ and

$G(t)=\|u(t)\|_{X^{1/2}}^{2}+\|u(t)\|_{X}^{2}+\|u_{t}(t)\|_{X}^{2}+\|v(t)\|_{X^{1/2}}^{2}+\|v_{t}(t)\|_{X}^{2}.$

Since

$(u^{n}(\tau),u_{t}^{n}(\tau),v^{n}(\tau),v_{t}^{n}(\tau))=(u_{0},u_{1},v_{0},v_{1})=(u^{\infty}(\tau),u_{t}^{\infty}(\tau),v^{\infty}(\tau),v_{t}^{\infty}(\tau)),$

$u=u^{n}-u^{\infty}$ and $v=v^{n}-v^{\infty}$, then $G(\tau)=0$. With this, we
can integrate the inequality

$\begin{split}\frac{d}{ds}\left(G(s)e^{-C^{\prime\prime}(s-\tau)}\right)&\leq
C^{\prime\prime}|a_{n}(s)-a_{\infty}(s)|e^{-C^{\prime\prime}(s-\tau)},\quad\tau\leq
s\leq t,\\\ \end{split}$

from $\tau$ to $t$ in order to obtain

$G(t)e^{-C^{\prime\prime}(t-\tau)}\leq-\|a_{n}-a_{\infty}\|_{L^{\infty}([\tau,t])}e^{-C^{\prime\prime}(t-\tau)}+\|a_{n}-a_{\infty}\|_{L^{\infty}([\tau,t])},$

that is,

$G(t)\leq-\|a_{n}-a_{\infty}\|_{L^{\infty}([\tau,t])}+\|a_{n}-a_{\infty}\|_{L^{\infty}([\tau,t])}e^{C^{\prime\prime}(t-\tau)}.$

Hence, as $a_{n}(s)\xrightarrow{n\rightarrow\infty}a_{\infty}(s)$ uniformly
for $s$ in compact subsets of $\mathbb{R}$, it follows that

$\begin{split}\|W^{n}(t)-W^{\infty}(t)\|_{Y_{0}}^{2}&=\|(u(t),u_{t}(t),v(t),v_{t}(t))\|_{Y_{0}}^{2}\\\
&=\|u(t)\|_{X^{1/2}}^{2}+\|u_{t}(t)\|_{X}^{2}+\|v(t)\|_{X^{1/2}}^{2}+\|v_{t}(t)\|_{X}^{2}\\\
&\leq
G(t)\leq\|a_{n}-a_{\infty}\|_{L^{\infty}([\tau,t])}e^{C^{\prime\prime}(t-\tau)}\xrightarrow{n\rightarrow\infty}0,\end{split}$

in compact subsets of $\mathbb{R}$. ∎

Next, we aim to prove that if $\xi\colon\mathbb{R}\to Y_{0}$ is a bounded
global solution of the problem (3.1) and $\xi^{n}\colon\mathbb{R}\to Y_{0}$ is
a bounded global solution of the problem (3.3), $n\in\mathbb{N}$, then
$\xi(t_{n}+t)=\xi^{n}(t)$ for all $t\in\mathbb{R}$. For that, we exhibit some
auxiliary results.

According to [4, Proposition 4.1], for each fixed $t\in\mathbb{R}$ and
$\epsilon\in[0,1]$, the operator $\mathcal{A}_{\epsilon}(t)$ defined in
(2.4)-(2.5) is maximal accretive (i.e., $-\mathcal{A}_{\epsilon}(t)$ is
dissipative and $R(I+\mathcal{A}_{\epsilon}(t))=Y_{0}$). By the Lumer-Phillips
Theorem ([22, Theorem 4.3]), we have $-\mathcal{A}_{\epsilon}(t)$ is the
infinitesimal generator of a $C_{0}$-semigroup of contractions in $Y_{0}$
given by $\\{e^{-\tau\mathcal{A}_{\epsilon}(t)}:\tau\geq 0\\}$, for each
$t\in\mathbb{R}$ and $\epsilon\in[0,1]$. Consequently, for each
$t\in\mathbb{R}$ and $\epsilon\in[0,1]$, it follows by [22, Corollary 3.6]
that $\rho(-\mathcal{A}_{\epsilon}(t))\supset\\{\lambda\in\mathbb{C}:{\rm
Re}(\lambda)>0\\}$ and

(3.8) $\|(\lambda
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\leq\dfrac{1}{{\rm
Re}(\lambda)},$

whenever $\lambda\in\mathbb{C}$ with ${\rm Re}(\lambda)>0$.

On the other hand, as proved in [4, Theorem 4.7], for each fixed
$t\in\mathbb{R}$ and $\epsilon\in[0,1]$, the operator
$\mathcal{A}_{\epsilon}(t)$ satisfies

(3.9) $|\beta|\|(i\beta
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\leq M,$

for all $\beta\in\mathbb{R}$, where
$M=1+2\left(2\eta+\frac{2(a_{1}^{2}+1)}{\eta}+4\right)+\frac{2a_{1}+2}{\eta}>0$
is independent of $t\in\mathbb{R}$ and $\epsilon\in[0,1]$.

######  Lemma 3.2.

There exists a constant $\mathcal{C}>0$, which is independent of
$t\in\mathbb{R}$ and $\epsilon\in[0,1]$, such that $\|(\lambda
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\leq\dfrac{\mathcal{C}}{1+|\lambda|}$
uniformly in $t\in\mathbb{R}$ and $\epsilon\in[0,1],$ for all $\lambda$ in a
sector $\Gamma$ whose angle is greater than $\frac{\pi}{2}$.

###### Proof.

Let $t\in\mathbb{R}$ and $\epsilon\in[0,1]$. By the proof of [4, Proposition
4.3], there exists a constant $N>0$, independent of $t$ and $\epsilon$, such
that $\|\mathcal{A}_{\epsilon}^{-1}(t)\|_{\mathcal{L}(Y_{0})}\leq N$. Thus,
$\|\lambda\mathcal{A}_{\epsilon}^{-1}(t)\|_{\mathcal{L}(Y_{0})}\leq\frac{1}{2}$
provided $|\lambda|\leq\frac{1}{2N}$, $\lambda\in\mathbb{C}$. Therefore,
$(\lambda I+\mathcal{A}_{\epsilon}(t))^{-1}$ exists for
$|\lambda|\leq\frac{1}{2N}$ and

$\|(\lambda
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\leq\|(I+\lambda\mathcal{A}_{\epsilon}^{-1}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\|\mathcal{A}_{\epsilon}^{-1}(t)\|_{\mathcal{L}(Y_{0})}\leq
2N.$

Consequently,

(3.10) $\|(\lambda
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\leq\frac{1}{|\lambda|},$

whenever $|\lambda|\leq\frac{1}{2N}$.

Now, let $\varphi\in(0,\frac{\pi}{2})$ be such that $\tan(\varphi)=4M$, where
$M>0$ comes from (3.9). Set
$\Gamma_{1}:=\\{\lambda\in\mathbb{C}\colon|\lambda|\geq\frac{1}{2N}\;\text{and}\;|arg(\lambda)|\leq\varphi\\}$.

Claim 1: $\|(\lambda
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\leq\dfrac{\frac{2}{1-\sin(\varphi)}}{|\lambda|}$
for all $\lambda\in\Gamma_{1}$.

Indeed, let $\lambda\in\Gamma_{1}$. Then

$|\lambda|\leq|{\rm Re}(\lambda)|+|{\rm Im}(\lambda)|\leq|{\rm
Re}(\lambda)|+\sin(\varphi)|\lambda|.$

Consequently, it follows by (3.8) that

$\|(\lambda
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\leq\frac{1}{{\rm
Re}(\lambda)}\leq\frac{\frac{2}{1-\sin(\varphi)}}{|\lambda|}.$

Define $\Gamma_{2}=\bigcup_{t\neq 0}\\{\mu\in\mathbb{C}\colon|\mu-
it|<\frac{|t|}{2M}\\}$.

Claim 2: If $\mu\in\Gamma_{2}$, then $\|(\mu
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\leq\dfrac{2M+1}{|\mu|}$.

In fact, let $\mu\in\Gamma_{2}$ and $t\in\mathbb{R}$, $t\neq 0$, be such that
$|\mu-it|<\frac{|t|}{2M}$. Using (3.9), we deduce

$\begin{split}\|(\mu-
it)(itI+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}&=|\mu-
it|\|(itI+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}<\frac{1}{2}\end{split}$

and, thanks to the Neumann series, we obtain

$\|[(\mu-
it)(itI+\mathcal{A}_{\epsilon}(t))^{-1}+I]^{-1}\|_{\mathcal{L}(Y_{0})}\leq 2.$

Now, using (3.9) again, we get

$\begin{split}\|(\mu
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}&=\|[(\mu-
it)(itI+\mathcal{A}_{\epsilon}(t))^{-1}+I]^{-1}(itI+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\\\
&\leq\frac{2M}{|t|}\leq\left(\frac{|\mu-
it|}{|t|}+1\right)\frac{2M}{|\mu|}<\frac{2M+1}{|\mu|}.\end{split}$

Since
$\partial\\{\lambda\in\Gamma_{1}:|arg(\lambda)|=\varphi\\}=\left\\{\left(\frac{|t|}{4M},t\right):t\in\mathbb{R}\right\\}\subset\Gamma_{2}$,
it follows by (3.10), Claim 1 and Claim 2 that we have construct a sector
$\Gamma$ whose angle is greater than $\frac{\pi}{2}$ such that

(3.11) $\|(\lambda
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\leq\frac{\tilde{C}}{|\lambda|},$

where $\tilde{C}=\max\\{2M+1,\frac{2}{1-\sin(\varphi)}\\}>0$ is independent of
$t\in\mathbb{R}$ and $\epsilon\in[0,1]$.

Lastly, for a given $\lambda\in\Gamma$, we have

(3.12) $\|\mathcal{A}_{\epsilon}(t)(\lambda
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\leq
1+|\lambda|\|(\lambda
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\leq 1+\tilde{C},$

which together with the uniform boundedness of
$\|\mathcal{A}_{\epsilon}^{-1}(t)\|_{\mathcal{L}(Y_{0})}$, we conclude that

(3.13) $\|(\lambda
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\leq\|\mathcal{A}_{\epsilon}(t)(\lambda
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\|\mathcal{A}_{\epsilon}^{-1}(t)\|_{\mathcal{L}(Y_{0})}\leq
K,$

for some constant $K>0$ independent of $t\in\mathbb{R}$ and
$\epsilon\in[0,1]$.

From (3.11) and (3.13),

$(1+|\lambda|)\|(\lambda
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\leq\tilde{C}+K:=\mathcal{C},$

for all $\lambda\in\Gamma$. ∎

######  Remark 3.3.

Since the estimates (3.12) and (3.13) do not depend on $\epsilon$, then there
exist $\delta>0$ and $K>0$ such that:

  1. $(i)$

$\|e^{-\tau\mathcal{A}_{\epsilon}(t)}\|_{\mathcal{L}(Y_{0})}\leq
Ke^{-\delta\tau}$, for all $\tau\geq 0$ and $t\in\mathbb{R}$;

  2. $(ii)$

$\|\mathcal{A}_{\epsilon}(t)e^{-\tau\mathcal{A}_{\epsilon}(t)}\|_{\mathcal{L}(Y_{0})}\leq\dfrac{Ke^{-\delta\tau}}{\tau}$,
for all $\tau>0$ and $t\in\mathbb{R}$.

######  Lemma 3.4.

For any $T>0$, there are constants $C>0$ and $0<\beta\leq 1$, both independent
of $T$ and $\epsilon$, such that

$\|[\mathcal{A}_{\epsilon}(t)-\mathcal{A}_{\epsilon}(s)]\mathcal{A}_{\epsilon}^{-1}(\tau)\|_{\mathcal{L}(Y_{0})}\leq
C|t-s|^{\beta}$

for every $t,s,\tau\in[-T,T]$. In particular,
$\|\mathcal{A}_{\epsilon}(t)\mathcal{A}_{\epsilon}^{-1}(\tau)\|_{\mathcal{L}(Y_{0})}\leq
1+2a_{1}$, for all $t,\tau\in\mathbb{R}$, where the constant $a_{1}$ comes
from (2.7).

###### Proof.

Let $T>0$ be fixed and arbitrary, and let $t,s,\tau\in[-T,T]$. Since

$\mathcal{A}_{\epsilon}(t)-\mathcal{A}_{\epsilon}(s)\,\\!\\!=\\!\\!\,[a_{\epsilon}(t)-a_{\epsilon}(s)]\begin{bmatrix}0&0&0&0\\\
0&0&0&A^{1/2}\\\ 0&0&0&0\\\ 0&-A^{1/2}&0&0\end{bmatrix}$

and

$\small\mathcal{A}_{\epsilon}^{-1}(\tau)=\begin{bmatrix}\eta
A^{1/2}(A+I)^{-1}&(A+I)^{-1}&a_{\epsilon}(\tau)A^{1/2}(A+I)^{-1}&0\\\
-I&0&0&0\\\ -a_{\epsilon}(\tau)A^{-1/2}&0&\eta A^{-1/2}&A^{-1}\\\
0&0&-I&0\end{bmatrix},$

we obtain

$\begin{split}\left\|[\mathcal{A}_{\epsilon}(t)-\mathcal{A}_{\epsilon}(s)]\mathcal{A}_{\epsilon}^{-1}(\tau)\begin{bmatrix}u\\\
v\\\ w\\\
z\end{bmatrix}\right\|_{Y_{0}}&=|a_{\epsilon}(t)-a_{\epsilon}(s)|\left(\left\|-A^{1/2}w\right\|_{X}+\left\|A^{1/2}u\right\|_{X}\right)\\\
&\leq
C|t-s|^{\beta}\left(\left\|u\right\|_{X^{1/2}}+\left\|v\right\|_{X}+\left\|w\right\|_{X^{1/2}}+\left\|z\right\|_{X}\right)\\\
&=C|t-s|^{\beta}\left\|\begin{bmatrix}u\\\ v\\\ w\\\
z\end{bmatrix}\right\|_{Y_{0}}\end{split}$

for every $[u\,\,v\,\,w\,\,z]^{T}\in Y_{0}$, where we have used (2.8). Hence,

$\|[\mathcal{A}_{\epsilon}(t)-\mathcal{A}_{\epsilon}(s)]\mathcal{A}_{\epsilon}^{-1}(\tau)\|_{\mathcal{L}(Y_{0})}\leq
C|t-s|^{\beta}$

for every $t,s,\tau\in[-T,T],$ and the result is proved. ∎

######  Proposition 3.5.

Let $\xi\colon\mathbb{R}\to Y_{0}$ be a bounded global solution of the problem
(3.1) and $\xi^{n}\colon\mathbb{R}\to Y_{0}$ be a bounded global solution of
the problem (3.3), $n\in\mathbb{N}$. Then $\xi(t_{n}+t)=\xi^{n}(t)$ for all
$t\in\mathbb{R}$.

###### Proof.

Consider the homogeneous problems

(3.14) $\begin{cases}u_{tt}-\Delta
u+u+\eta(-\Delta)^{1/2}u_{t}+a_{0}(t)(-\Delta)^{1/2}v_{t}=0,&(x,t)\in\Omega\times(\tau,\infty),\\\
v_{tt}-\Delta
v+\eta(-\Delta)^{1/2}v_{t}-a_{0}(t)(-\Delta)^{1/2}u_{t}=0,&(x,t)\in\Omega\times(\tau,\infty),\\\
u=v=0,&(x,t)\in\partial\Omega\times(\tau,\infty),\\\
(u(\tau),u_{t}(\tau),v(\tau),v_{t}(\tau))=(u_{0},u_{1},v_{0},v_{1}),\end{cases}$

and

(3.15) $\begin{cases}u_{tt}-\Delta
u+u+\eta(-\Delta)^{1/2}u_{t}+a_{n}(t)(-\Delta)^{1/2}v_{t}=0,&(x,t)\in\Omega\times(\tau,\infty),\\\
v_{tt}-\Delta
v+\eta(-\Delta)^{1/2}v_{t}-a_{n}(t)(-\Delta)^{1/2}u_{t}=0,&(x,t)\in\Omega\times(\tau,\infty),\\\
u=v=0,&(x,t)\in\partial\Omega\times(\tau,\infty),\\\
(u(\tau),u_{t}(\tau),v(\tau),v_{t}(\tau))=(u_{0},u_{1},v_{0},v_{1})\in
Y_{0},\end{cases}$

Let $\\{L(t,s)\colon t\geq s\in\mathbb{R}\\}$ and $\\{L_{n}(t,s)\colon t\geq
s\in\mathbb{R}\\}$ be the linear evolution processes in $Y_{0}$ associated
with (3.14) and (3.15), respectively. Since Lemmas 3.2 and 3.4 hold, it
follows by [26] that

$L(t,s)=e^{-(t-s)\mathcal{A}_{0}(s)}+\int_{s}^{t}L(t,\tau)[\mathcal{A}_{0}(s)-\mathcal{A}_{0}(\tau)]e^{-(\tau-s)\mathcal{A}_{0}(s)}d\tau,$

and

$L_{n}(t,s)=e^{-(t-s)\mathcal{A}_{n}(s)}+\int_{s}^{t}L_{n}(t,\tau)[\mathcal{A}_{n}(s)-\mathcal{A}_{n}(\tau)]e^{-(\tau-s)\mathcal{A}_{n}(s)}d\tau,$

where

$\begin{split}\mathcal{A}_{n}(t)&=\\!\\!\begin{bmatrix}0&-I&0&0\\\ A+I&\eta
A^{1/2}&0&a_{n}(t)A^{1/2}\\\ 0&0&0&-I\\\ 0&-a_{n}(t)A^{1/2}&A&\eta
A^{1/2}\end{bmatrix},\\!\\!\\\ \end{split}$

with $t\in\mathbb{R}$ and $n\in\mathbb{N}\cup\\{0\\}$. Moreover,

$\xi(t)=L(t,s)\xi(s)+\int_{s}^{t}L(t,\tau)F(\xi(\tau))d\tau,\quad
t,s\in\mathbb{R},$

and

$\xi^{n}(t)=L_{n}(t,s)\xi(s)+\int_{s}^{t}L_{n}(t,\tau)F(\xi^{n}(\tau))d\tau,\quad
t,s\in\mathbb{R},\;n\in\mathbb{N}.$

According to the proof of [4, Theorem 2.3], we conclude that

$\xi(t)=\int_{-\infty}^{t}L(t,s)F(\xi(s))ds,\quad t\in\mathbb{R},$

and

$\xi^{n}(t)=\int_{-\infty}^{t}L_{n}(t,s)F(\xi^{n}(s))ds,\quad
t\in\mathbb{R},\;n\in\mathbb{N},$

with $F$ is given by (2.6). Furthermore, note that
$\mathcal{A}_{0}(s+t_{n})=\mathcal{A}_{n}(s)$ for all $s\in\mathbb{R}$ and
$n\in\mathbb{N}$. Consequently, we deduce

$\begin{split}&L(t+t_{n},s+t_{n})=\\\
&=e^{-(t-s)\mathcal{A}_{0}(s+t_{n})}+\int_{s+t_{n}}^{t+t_{n}}L(t+t_{n},\tau)[\mathcal{A}_{0}(s+t_{n})-\mathcal{A}_{0}(\tau)]e^{-(\tau-
s-t_{n})\mathcal{A}_{0}(s+t_{n})}d\tau\\\
&=e^{-(t-s)\mathcal{A}_{n}(s)}+\int_{s}^{t}L(t+t_{n},\tau+t_{n})[\mathcal{A}_{0}(s+t_{n})-\mathcal{A}_{0}(\tau+t_{n})]e^{-(\tau-s)\mathcal{A}_{0}(s+t_{n})}d\tau\\\
&=e^{-(t-s)\mathcal{A}_{n}(s)}+\int_{s}^{t}L(t+t_{n},\tau+t_{n})[\mathcal{A}_{n}(s)-\mathcal{A}_{n}(\tau)]e^{-(\tau-s)\mathcal{A}_{n}(s)}d\tau,\end{split}$

for all $t,s\in\mathbb{R}$ and $n\in\mathbb{N}$. Using Remark 3.3, Lemma 3.4
and the Gronwall inequality, we conclude that $L(t+t_{n},s+t_{n})=L_{n}(t,s)$
for all $t\geq s\in\mathbb{R}$. In conclusion,

$\xi(t+t_{n})=\int_{-\infty}^{t+t_{n}}L(t+t_{n},s)F(\xi(s))ds=\int_{-\infty}^{t}L(t+t_{n},s+t_{n})F(\xi(s+t_{n}))ds$
$=\int_{-\infty}^{t}L_{n}(t,s)F(\xi(s+t_{n}))ds,$

for all $t\in\mathbb{R}$ and $n\in\mathbb{N}$. Since (see [4, Proposition
4.9])

$\|F(z_{1})-F(z_{2})\|_{Y_{\alpha-1}}\leq
c\|z_{1}-z_{2}\|_{Y_{0}}(1+\|z_{1}\|_{Y_{0}}^{\rho-1}+\|z_{2}\|_{Y_{0}}^{\rho-1}),\quad
z_{1},z_{2}\in Y_{0},\quad\alpha\in(0,1),$

where $Y_{\alpha-1}=[Y_{-1},Y_{0}]_{\alpha}$ (complex interpolation), and
$\|L(t,s)\|_{\mathcal{L}(Y_{\alpha-1},Y_{0})}\leq c(\alpha)(t-s)^{\alpha-1}$
(by choosing the phase space $Z_{0}=Y_{-1}$, $Z_{\alpha}=Y_{\alpha-1}$ and the
closed extension of $\mathcal{A}_{\epsilon}(t)$ to $Y_{0}$, and take into
account that [4, Proposition 4.4.] holds, we have
$\|L(t,s)\|_{\mathcal{L}(Y_{\alpha-1},Y_{0})}=\|L(t,s)\|_{\mathcal{L}(Z_{\alpha},Z_{1})}$),
it follows again by using the Gronwall inequality, that
$\xi(t_{n}+t)=\xi^{n}(t)$ for all $t\in\mathbb{R}$. ∎

######  Corollary 3.6.

If $\xi\colon\mathbb{R}\to Y_{0}$ and $\xi^{\infty}\colon\mathbb{R}\to Y_{0}$
are bounded global solutions associated to the problems (3.3) and (3.4),
respectively, starting in their pullback attractors, then

$\|\xi(t+t_{n})-\xi^{\infty}(t)\|_{Y_{0}}^{2}\xrightarrow{n\rightarrow\infty}0$

in compact subsets of $\mathbb{R}$.

###  3.2. Gradient-like structure of the limit pullback attractor

From here on, we shall assume that the problem (2.1)-(2.2) admits only a
finite number of equilibrium points (time independent solutions), that is,
there exist only finitely many solutions
$\\{(u_{1}^{\ast},0),\ldots,(u_{r}^{\ast},0)\\}$ of the elliptic problem

$\begin{cases}(-\Delta)u+u=f(u)&\mbox{in}\quad\Omega,\\\
(-\Delta)v=0&\mbox{in}\quad\Omega,\\\
u=v=0&\mbox{on}\quad\partial\Omega,\end{cases}$

and we will denote the set of equilibria as
$\mathcal{E}^{\ast}=\\{e_{1}^{\ast},\ldots,e_{r}^{\ast}\\}$, where each
$e_{j}^{\ast}$ is of the form

$e_{j}^{\ast}\,\\!\\!=\\!\\!\begin{bmatrix}u_{j}^{\ast}\\\ 0\\\ 0\\\
0\end{bmatrix},\quad 1\leq j\leq r.$

Further, we shall assume that all equilibrium points in $\mathcal{E}^{\ast}$
are hyperbolic, in the sense of Definition A.2, for the limit problem
($\epsilon=0$) associated to the system (2.1)-(2.2).

In order to pursue a result on lower semicontinuity of pullback attractors at
$\epsilon=0$, we will assume the following hyperbolicity condition:

(3.16) $0\notin\sigma(A+I-f^{\prime}(u))\quad\mbox{whenever}\quad u\in
H_{0}^{1}(\Omega)$

and the spectrum $\sigma(A+I-f^{\prime}(u))$ has no intersection with the
imaginary axis.

######  Proposition 3.7.

The evolution system (2.1)-(2.2) admits a Lyapunov functional.

###### Proof.

Let $(u(t),u_{t}(t),v(t),v_{t}(t))$ be the global solution of the problem
(2.1)-(2.2) in $Y_{0}$. Multiplying the first equation in (2.1) by $u_{t}$,
and the second by $v_{t}$, we obtain

$\begin{split}&\frac{1}{2}\frac{d}{dt}\int_{\Omega}|u_{t}|^{2}\,dx+\frac{1}{2}\frac{d}{dt}\int_{\Omega}|\nabla
u|^{2}\,dx+\frac{1}{2}\frac{d}{dt}\int_{\Omega}|u|^{2}\,dx+\eta\|(-\Delta)^{1/4}u_{t}\|_{X}^{2}\\\
&+a_{\epsilon}(t)\langle(-\Delta)^{1/2}v_{t},u_{t}\rangle_{X}=\frac{d}{dt}\int_{\Omega}\int_{0}^{u}f(s)\,dsdx\end{split}$

and

$\frac{1}{2}\frac{d}{dt}\int_{\Omega}|v_{t}|^{2}\,dx+\frac{1}{2}\frac{d}{dt}\int_{\Omega}|\nabla
v|^{2}\,dx+\eta\|(-\Delta)^{1/4}v_{t}\|_{X}^{2}-a_{\epsilon}(t)\langle(-\Delta)^{1/2}u_{t},v_{t}\rangle_{X}=0,$

for all $t>\tau$, which combined leads to

(3.17)
$\begin{split}&\frac{d}{dt}\frac{1}{2}\left(\|u\|_{X^{1/2}}^{2}+\|u\|_{X}^{2}+\|u_{t}\|_{X}^{2}+\|v\|_{X^{1/2}}^{2}+\|v_{t}\|_{X}^{2}-2\int_{\Omega}\int_{0}^{u}f(s)\,dsdx\right)\\\
&=-\eta\|(-\Delta)^{1/4}u_{t}\|_{X}^{2}-\eta\|(-\Delta)^{1/4}v_{t}\|_{X}^{2}\end{split}$

for all $t>\tau$.

Thus, we may define an energy functional in the phase space $Y_{0}$ through
the map $\mathcal{E}\colon Y_{0}\to\mathbb{R}$ given by

(3.18)
$\begin{split}\mathcal{E}(\phi,\varphi,\psi,\Phi)&=\frac{1}{2}\left(\|\phi\|_{X^{1/2}}^{2}+\|\phi\|_{X}^{2}+\|\varphi\|_{X}^{2}+\|\psi\|_{X^{1/2}}^{2}+\|\Phi\|_{X}^{2}\right)\\\
&-\int_{\Omega}\int_{0}^{\phi}f(s)\,dsdx,\end{split}$

for $(\phi,\varphi,\psi,\Phi)\in Y_{0}$. By (3.17) and (3.18), it follows that

$\frac{d}{dt}\mathcal{E}(u(t),u_{t}(t),v(t),v_{t}(t))=-\eta\|(-\Delta)^{1/4}u_{t}\|_{X}^{2}-\eta\|(-\Delta)^{1/4}v_{t}\|_{X}^{2}$

for all $t>\tau$ and, therefore, the energy functional $\mathcal{E}$ is
monotone decreasing along solutions.

Moreover, if $\mathcal{E}$ is a constant function with respect to
$t\in\mathbb{R}$, then it follows from (3.17) that
$-\eta\|(-\Delta)^{1/4}u_{t}\|_{X}^{2}-\eta\|(-\Delta)^{1/4}v_{t}\|_{X}^{2}=0$.
Consequently, $|(-\Delta)^{1/4}u_{t}|^{2}=|(-\Delta)^{1/4}v_{t}|^{2}=0\mbox{
almost everywhere on }\Omega$, i.e., $u_{t}=0\mbox{ and }v_{t}=0\mbox{ almost
everywhere on }\Omega$. Consequently, the global solutions where the energy
functional $\mathcal{E}$ is constant in time $t\in\mathbb{R}$ must belong to
the set of equilibria $\mathcal{E}^{\ast}$ (set of time independent
solutions). This proves that $\mathcal{E}\colon Y_{0}\to\mathbb{R}$, defined
in (3.18) is a Lyapunov functional for the evolution system (2.1)-(2.2). ∎

In the next result, we consider the unstable manifold around an equilibria
$e_{i}^{\ast}$, $1\leq i\leq r$, given by

$W^{u}(e_{i}^{\ast})=\\{(t,\zeta)\in\mathbb{R}\times Y_{0}:\text{ there is a
global solution }\xi\colon\mathbb{R}\to X\text{ such that }$
$\xi(t)=\zeta\text{ and
}\displaystyle\lim_{\tau\to-\infty}dist(\xi(\tau),e_{i}^{\ast})=0\\}$

and its section at time $t\in\mathbb{R}$

$W^{u}(e_{i}^{\ast})(t)=\\{\zeta\in Y_{0}\colon(t,\zeta)\in
W^{u}(e_{i}^{\ast})\\}.$

######  Theorem 3.8.

The limit pullback attractor $\\{\mathbb{A}_{0}(t)\colon t\in\mathbb{R}\\}$,
associated to the problem (2.1)-(2.2) with $\epsilon=0$, can be written as

$\mathbb{A}_{0}(t)=\bigcup_{i=1}^{r}W^{u}(e_{i}^{\ast})(t)\quad\mbox{for
all}\quad t\in\mathbb{R}.$

###### Proof.

At first place, we will prove that any bounded global solution in the pullback
attractor $\\{\mathbb{A}_{0}(t)\colon t\in\mathbb{R}\\}$ is both forwards and
backwards asymptotic to some equilibria in $\mathcal{E}^{\ast}$. Indeed, let
$\xi\colon\mathbb{R}\to Y_{0}$ be a bounded global solution of the problem
(3.1) such that $\xi(t)\in\mathbb{A}_{0}(t)$ for all $t\in\mathbb{R}$. Since
$\bigcup_{t\in\mathbb{R}}\mathbb{A}_{0}(t)$ is compact (see (2.14)) and the
energy functional $\mathcal{E}$ is monotone decreasing along solutions, there
exist $\ell_{i},\ell_{j}\in\mathbb{R}$ such that

(3.19)
$\ell_{i}\xleftarrow{t\rightarrow-\infty}\mathcal{E}(\xi(t+s))\xrightarrow{t\rightarrow+\infty}\ell_{j}$

for all $s\in\mathbb{R}$.

Let $\\{t_{n}\\}_{n\in\mathbb{N}}\subset\mathbb{R}$ be a sequence such that
$t_{n}\xrightarrow{n\rightarrow\infty}+\infty$. We may assume without loss of
generality that

$a_{0}(t_{n}+s)=a_{n}(s)\xrightarrow{n\rightarrow\infty}a_{\infty}(s)$

uniformly for $s$ in compact subsets of $\mathbb{R}$. Consider problem (3.3)
with this function $a_{n}$. By Theorem 3.1 and Proposition 3.5, if

$\xi^{n}(\cdot)=(u^{n}(\cdot),u_{t}^{n}(\cdot),v^{n}(\cdot),v_{t}^{n}(\cdot))\quad\mbox{and}\quad\xi^{\infty}(\cdot)=(u^{\infty}(\cdot),u_{t}^{\infty}(\cdot),v^{\infty}(\cdot),v_{t}^{\infty}(\cdot))$

are global solutions associated with the problems (3.3) and (3.4),
respectively, then we have

(3.20)
$\xi(t_{n}+t)=\xi^{n}(t)\xrightarrow{n\rightarrow\infty}\xi^{\infty}(t)\quad\mbox{in}\quad
Y_{0}$

in compact subsets of $\mathbb{R}$. By (3.19) and (3.20),

$\mathcal{E}(\xi^{\infty}(t))=\ell_{j}$

for all $t\in\mathbb{R}$. Since the global solutions where the energy
functional $\mathcal{E}$ is constant in $t\in\mathbb{R}$ are the equilibria in
$\mathcal{E}^{\ast}$, there exists an equilibrium point
$e_{k}^{\ast}\in\mathcal{E}^{\ast}$, for some $1\leq k\leq r$, such that
$\xi^{\infty}(\cdot)\equiv e_{k}^{\ast}$. Hence,

$\xi(t_{n}+t)\xrightarrow{n\rightarrow+\infty}e_{k}^{\ast}\in\mathcal{E}^{\ast}\quad\mbox{in}\quad
Y_{0}$

uniformly on compact subsets of $\mathbb{R}$. Similarly, we can take a
sequence $\\{t_{n}\\}_{n\in\mathbb{N}}\subset\mathbb{R}$ with
$t_{n}\xrightarrow{n\rightarrow\infty}-\infty$ and prove that there exists an
equilibrium point $e_{p}^{\ast}\in\mathcal{E}^{\ast}$, for some $1\leq p\leq
r$, such that

$\xi(t_{n}+t)\xrightarrow{n\rightarrow\infty}e_{p}^{\ast}\in\mathcal{E}^{\ast}\quad\mbox{in}\quad
Y_{0}$

uniformly on compact subsets of $\mathbb{R}$.

Now, we argue that this assertion does not depend on the particular choices of
subsequences, and then we can conclude our result. Indeed, suppose that there
are sequences $t_{n},s_{n}\xrightarrow{n\rightarrow\infty}\infty$ such that

$\xi(t_{n})\xrightarrow{n\rightarrow\infty}e_{i}^{\ast}\neq
e_{j}^{\ast}\xleftarrow{n\rightarrow\infty}\xi(s_{n}).$

Reindexing these sequences, if necessary, we can assume without loss of
generality that $t_{n+1}>s_{n}>t_{n}$ for all $n\in\mathbb{N}$. Let
$\delta=\|e_{i}^{\ast}-e_{j}^{\ast}\|_{Y_{0}}>0$. There is
$n_{0}\in\mathbb{N}$ such that $\xi(t_{n})\in
B(e_{i}^{\ast},\frac{\delta}{3})$ and $\xi(s_{n})\in
B(e_{j}^{\ast},\frac{\delta}{3})$ for all $n\geq n_{0}$. Let
$\tau_{n}\in(t_{n},s_{n})$ be such that $\xi(\tau_{n})\notin
B(e_{j}^{\ast},\frac{\delta}{3})\cup B(e_{i}^{\ast},\frac{\delta}{3})$, for
$n\geq n_{0}$. Thus, $\tau_{n}\xrightarrow{n\rightarrow\infty}\infty$ and,
taking subsequences if necessary, we also have

$a_{0}(\tau_{n}+s)=a_{n}(s)\xrightarrow{n\rightarrow\infty}\tilde{a}(s)$

uniformly for $s$ in compact subsets of $\mathbb{R}$. Repeating the previous
argument, we obtain

$\xi(\tau_{n}+t)\xrightarrow{n\rightarrow\infty}\tilde{\xi}(t)$

in compact subsets of $\mathbb{R}$, where $\tilde{\xi}$ is the global solution
of the problem (3.4) with $\bar{a}$ replaced by $\tilde{a}$, with

$\tilde{\xi}(\cdot)\equiv e_{m}^{\ast}\in\mathcal{E}^{\ast}.$

Since $\xi(\tau_{n})\notin B(e_{j}^{\ast},\frac{\delta}{3})\cup
B(e_{i}^{\ast},\frac{\delta}{3})$, for $n\geq n_{0}$, we conclude that

$e_{m}^{\ast}\in\mathcal{E}^{\ast}\backslash\\{e_{i}^{\ast},e_{j}^{\ast}\\}.$

Continuing with the previous argument, we obtain a contradiction due to the
fact that there are only finitely many equilibria in $\mathcal{E}^{\ast}$.
Hence, we can write the pullback attractor $\\{\mathbb{A}_{0}(t)\colon
t\in\mathbb{R}\\}$ as

$\mathbb{A}_{0}(t)=\bigcup_{i=1}^{r}W^{u}(e_{i}^{\ast})(t)\quad\mbox{for
all}\quad t\in\mathbb{R},$

and the proof is complete. ∎

##  4\. The linearized system

Consider the linearization of the problem (2.1)-(2.2) around an equilibrium
point $e_{j}^{\ast}\in\mathcal{E}^{\ast}$, $1\leq j\leq r$, which is given by
the system

$\begin{cases}\tilde{u}_{tt}+(-\Delta)\tilde{u}+\tilde{u}+\eta(-\Delta)^{1/2}\tilde{u}_{t}+a_{\epsilon}(t)(-\Delta)^{1/2}\tilde{v}_{t}-f^{\prime}(u_{j}^{\ast})\tilde{u}=h(\tilde{u}),&\mbox{in}\quad\Omega,\\\
\tilde{v}_{tt}+(-\Delta)\tilde{v}+\eta(-\Delta)^{1/2}\tilde{v}_{t}-a_{\epsilon}(t)(-\Delta)^{1/2}\tilde{u}_{t}=0,&\mbox{in}\quad\Omega,\\\
\tilde{u}=\tilde{v}=0,&\mbox{on}\quad\partial\Omega,\\\ \end{cases}$

where

$h(\tilde{u})=f(\tilde{u}+u_{j}^{\ast})-f(u_{j}^{\ast})-f^{\prime}(u_{j}^{\ast})\tilde{u}.$

This system can be rewritten in the following abstract form

(4.1)
$\tilde{W}_{t}+\tilde{\mathcal{A}}_{\epsilon}(t)\tilde{W}=\tilde{F}(\tilde{W}),\quad
t>\tau\in\mathbb{R},$

where

(4.2)
$\tilde{\mathcal{A}}_{\epsilon}(t)\,\\!\\!=\\!\\!\begin{bmatrix}0&-I&0&0\\\
A+I-f^{\prime}(u_{j}^{\ast})&\eta A^{1/2}&0&a_{\epsilon}(t)A^{1/2}\\\
0&0&0&-I\\\ 0&-a_{\epsilon}(t)A^{1/2}&A&\eta A^{1/2}\end{bmatrix}$

with

$\tilde{W}=\begin{bmatrix}\tilde{u}\\\ \tilde{u}_{t}\\\ \tilde{v}\\\
\tilde{v}_{t}\\\
\end{bmatrix}\quad\mbox{and}\quad\tilde{F}(\tilde{W})=\begin{bmatrix}0\\\
h(\tilde{u})\\\ 0\\\ 0\\\ \end{bmatrix}.$

We may write $\tilde{\mathcal{A}}_{\epsilon}(t)=\mathcal{A}_{\epsilon}(t)+B$,
where $\mathcal{A}_{\epsilon}(t)$ is defined by (2.4)-(2.5) and $B$ is given
by

(4.3) $B\,\\!\\!=\\!\\!\begin{bmatrix}0&0&0&0\\\
-f^{\prime}(u_{j}^{\ast})&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{bmatrix},$

with $D(B)=D(\mathcal{A}_{\epsilon}(t))=Y_{1}$.

######  Proposition 4.1.

The operator $B$, defined in (4.3), satisfies

$\|B\|_{\mathcal{L}(Y_{0})}\leq c\left(1+\max\limits_{1\leq i\leq
r}\|e_{i}^{\ast}\|_{Y_{0}}^{\rho-1}\right),$

where $c>0$ is a constant.

###### Proof.

For a given $\vartheta=\begin{bmatrix}u&v&w&z\end{bmatrix}^{T}\in Y_{0}$ and
recalling that
$e_{j}^{\ast}=\begin{bmatrix}u_{j}^{\ast}&0&0&0\end{bmatrix}^{T}\in\mathcal{E}^{\ast},$
we have

(4.4)
$\begin{split}&\|B\vartheta\|_{Y_{0}}\,\\!\\!=\\!\\!\,\left\|\begin{bmatrix}0\\\
-f^{\prime}(u_{j}^{\ast})u\\\ 0\\\
0\end{bmatrix}\right\|_{Y_{0}}=\|f^{\prime}(u_{j}^{\ast})u\|_{L^{2}(\Omega)}\\\
&\leq\|f^{\prime}(u_{j}^{\ast})\|_{L^{\frac{2\rho}{\rho-1}}(\Omega)}\|u\|_{L^{2\rho}(\Omega)}\leq\tilde{c}\|f^{\prime}(u_{j}^{\ast})\|_{L^{\frac{2\rho}{\rho-1}}(\Omega)}\|u\|_{X^{1/2}}\\\
&\leq\tilde{c}\|f^{\prime}(u_{j}^{\ast})\|_{L^{\frac{2\rho}{\rho-1}}(\Omega)}\|\vartheta\|_{Y_{0}},\end{split}$

where we have used the Hölder inequality with
$\frac{\rho-1}{2\rho}+\frac{1}{2\rho}=\frac{1}{2}$, and the embedding
$X^{1/2}\hookrightarrow L^{2\rho}(\Omega)$ (as condition
$\frac{N-1}{N-2}\leq\rho<\frac{N}{N-2}$ holds). On the other hand, we can use
the growth condition (2.11) to obtain

(4.5)
$\begin{split}&\|f^{\prime}(u_{j}^{\ast})\|_{L^{\frac{2\rho}{\rho-1}}(\Omega)}\leq\left(\int_{\Omega}[c(1+|u_{j}^{\ast}|^{\rho-1})]^{\frac{2\rho}{\rho-1}}\,dx\right)^{\frac{\rho-1}{2\rho}}\\\
&\leq
c_{1}\left(|\Omega|+\int_{\Omega}|u_{j}^{\ast}|^{2\rho}\,dx\right)^{\frac{\rho-1}{2\rho}}\\\
&\leq
c_{2}\left(|\Omega|^{\frac{\rho-1}{2\rho}}+\|u_{j}^{\ast}\|_{L^{2\rho}(\Omega)}^{\rho-1}\right)\leq
c_{3}\left(1+\|u_{j}^{\ast}\|_{L^{2\rho}(\Omega)}^{\rho-1}\right)\\\ &\leq
c_{4}\left(1+\|u_{j}^{\ast}\|_{X^{1/2}}^{\rho-1}\right)=c_{4}\left(1+\|e_{j}^{\ast}\|_{Y_{0}}^{\rho-1}\right).\end{split}$

By (4.4) and (4.5), we obtain

$\|B\vartheta\|_{Y_{0}}\leq
c_{5}\left(1+\|e_{j}^{\ast}\|_{Y_{0}}^{\rho-1}\right)\|\vartheta\|_{Y_{0}}\leq
c_{5}\left(1+\max\limits_{1\leq i\leq
r}\|e_{i}^{\ast}\|_{Y_{0}}^{\rho-1}\right)\|\vartheta\|_{Y_{0}},$

which yields $\|B\|_{\mathcal{L}(Y_{0})}\leq c_{5}\left(1+\max\limits_{1\leq
i\leq r}\|e_{i}^{\ast}\|_{Y_{0}}^{\rho-1}\right),$ with $c_{5}>0$ being a
constant, which proves the result. ∎

######  Proposition 4.2.

For each fixed $t\in\mathbb{R}$, there holds
$\|\tilde{\mathcal{A}}_{\epsilon}^{-1}(t)-\tilde{\mathcal{A}}_{0}^{-1}(t)\|_{\mathcal{L}(Y_{0})}\rightarrow
0$ as $\epsilon\rightarrow 0^{+}$.

###### Proof.

Note that

(4.6) $\small\tilde{\mathcal{A}}_{\epsilon}^{-1}(t)=\begin{bmatrix}\eta
A^{1/2}(A+I-f^{\prime}(u_{j}^{\ast}))^{-1}&(A+I-f^{\prime}(u_{j}^{\ast}))^{-1}&a_{\epsilon}(t)A^{1/2}(A+I-f^{\prime}(u_{j}^{\ast}))^{-1}&0\\\
-I&0&0&0\\\ -a_{\epsilon}(t)A^{-1/2}&0&\eta A^{-1/2}&A^{-1}\\\
0&0&-I&0\end{bmatrix}.$

Given $\vartheta=\begin{bmatrix}u&v&w&z\end{bmatrix}^{T}\in Y_{0}$, we have

(4.7)
$\begin{split}&\|(\tilde{\mathcal{A}}_{\epsilon}^{-1}(t)-\tilde{\mathcal{A}}_{0}^{-1}(t))\vartheta\|_{Y_{0}}\\\
&=\left\|\begin{bmatrix}[a_{\epsilon}(t)-a_{0}(t)]A^{1/2}(A+I-f^{\prime}(u_{j}^{\ast}))^{-1}w\\\
0\\\ [a_{0}(t)-a_{\epsilon}(t)]A^{-1/2}u\\\ 0\end{bmatrix}\right\|_{Y_{0}}\\\
&=\|[a_{\epsilon}(t)-a_{0}(t)]A^{1/2}(A+I-f^{\prime}(u_{j}^{\ast}))^{-1}w\|_{X^{1/2}}+\|[a_{0}(t)-a_{\epsilon}(t)]A^{-1/2}u\|_{X^{1/2}}\\\
&=|a_{\epsilon}(t)-a_{0}(t)|\left(\|A(A+I-f^{\prime}(u_{j}^{\ast}))^{-1}w\|_{X}+\|u\|_{X}\right).\end{split}$

By condition (3.16), there exists a constant $\kappa_{0}>0$ such that

$\|(A+I-f^{\prime}(u_{j}^{\ast}))^{-1}\|_{\mathcal{L}(X)}\leq\kappa_{0}.$

Following the proof of Proposition 4.1 (see (4.5)), there exists a constant
$c>0$ such that

$\|f^{\prime}(u_{j}^{\ast})\|_{L^{\frac{2\rho}{\rho-1}}(\Omega)}\leq
c\left(1+\|e_{j}^{\ast}\|_{Y_{0}}^{\rho-1}\right)\leq
c\left(1+\max\limits_{1\leq i\leq r}\|e_{i}^{\ast}\|_{Y_{0}}^{\rho-1}\right).$

Now, we use the embedding $X^{1/2}\hookrightarrow L^{2\rho}(\Omega)$ (since
$\frac{N-1}{N-2}\leq\rho<\frac{N}{N-2}$) and Hölder inequality to obtain

(4.8) $\begin{split}&\|A(A+I-f^{\prime}(u_{j}^{\ast}))^{-1}w\|_{X}\\\
&=\left\|I-(I-f^{\prime}(u_{j}^{\ast}))\left(A+I-f^{\prime}(u_{j}^{\ast}))^{-1}\right)w\right\|_{X}\\\
&\leq\|w\|_{X}+\|(I-f^{\prime}(u_{j}^{\ast}))(A+I-f^{\prime}(u_{j}^{\ast}))^{-1}w\|_{X}\\\
&\leq\lambda_{1}^{-1/2}\|w\|_{X^{1/2}}+\|(A+I-f^{\prime}(u_{j}^{\ast}))^{-1}\|_{\mathcal{L}(X)}\|(I-f^{\prime}(u_{j}^{\ast}))w\|_{X}\\\
&\leq\lambda_{1}^{-1/2}\|w\|_{X^{1/2}}+\kappa_{0}\left(\|w\|_{X}+\|f^{\prime}(u_{j}^{\ast})w\|_{X}\right)\\\
&\leq\lambda_{1}^{-1/2}\|w\|_{X^{1/2}}+\kappa_{0}\lambda_{1}^{-1/2}\|w\|_{X^{1/2}}+\kappa_{0}\|f^{\prime}(u_{j}^{\ast})w\|_{L^{2}(\Omega)}\\\
&\leq(1+\kappa_{0})\lambda_{1}^{-1/2}\|w\|_{X^{1/2}}+\kappa_{0}\|f^{\prime}(u_{j}^{\ast})\|_{L^{\frac{2\rho}{\rho-1}}(\Omega)}\|w\|_{L^{2\rho}(\Omega)}\\\
&\leq(1+\kappa_{0})\lambda_{1}^{-1/2}\|w\|_{X^{1/2}}+\kappa_{0}c\left(1+\max\limits_{1\leq
i\leq r}\|e_{i}^{\ast}\|_{Y_{0}}^{\rho-1}\right)\tilde{c}\|w\|_{X^{1/2}}\\\
&=\tilde{\kappa}\|w\|_{X^{1/2}},\end{split}$

where
$\tilde{\kappa}=(1+\kappa_{0})\lambda_{1}^{-1/2}+\kappa_{0}c\tilde{c}\left(1+\max\limits_{1\leq
i\leq r}\|e_{i}^{\ast}\|_{Y_{0}}^{\rho-1}\right),$ with $\tilde{c}>0$ being
the embedding constant of $X^{1/2}\hookrightarrow L^{2\rho}(\Omega)$, and
$\lambda_{1}>0$ denotes the first eigenvalue of the negative Laplacian
operator with homogeneous Dirichlet boundary condition.

Combining (4.7) and (4.8), we get

$\begin{split}&\|(\tilde{\mathcal{A}}_{\epsilon}^{-1}(t)-\tilde{\mathcal{A}}_{0}^{-1}(t))\vartheta\|_{Y_{0}}\leq|a_{\epsilon}(t)-a_{0}(t)|\left(\tilde{\kappa}\|w\|_{X^{1/2}}+\lambda_{1}^{-1/2}\|u\|_{X^{1/2}}\right)\\\
&\leq|a_{\epsilon}(t)-a_{0}(t)|\max\\{\tilde{\kappa},\lambda_{1}^{-1/2}\\}\left(\|u\|_{X^{1/2}}+\|w\|_{X^{1/2}}\right)\\\
&\leq|a_{\epsilon}(t)-a_{0}(t)|\max\\{\tilde{\kappa},\lambda_{1}^{-1/2}\\}\left(\|u\|_{X^{1/2}}+\|v\|_{X}+\|w\|_{X^{1/2}}+\|z\|_{X}\right)\\\
&=|a_{\epsilon}(t)-a_{0}(t)|\tilde{\tilde{\kappa}}\|\vartheta\|_{Y_{0}},\end{split}$

where the constant
$\tilde{\tilde{\kappa}}=\max\\{\tilde{\kappa},\lambda_{1}^{-1/2}\\}>0$ does
not depend on $t\in\mathbb{R}$ neither on the parameter $\epsilon$. Therefore,

$\|\tilde{\mathcal{A}}_{\epsilon}^{-1}(t)-\tilde{\mathcal{A}}_{0}^{-1}(t)\|_{\mathcal{L}(Y_{0})}\leq\tilde{\tilde{\kappa}}\|a_{\epsilon}-a_{0}\|_{L^{\infty}(\mathbb{R})}\rightarrow
0\quad\text{as}\quad\epsilon\rightarrow 0^{+},$

which proves our result. ∎

######  Proposition 4.3.

For each fixed $t,\tau\in\mathbb{R}$, there holds

$\|[\tilde{\mathcal{A}}_{\epsilon}(t)-\tilde{\mathcal{A}}_{0}(t)]\tilde{\mathcal{A}}^{-1}_{\epsilon}(\tau)\|_{\mathcal{L}(Y_{0})}\leqslant\|a_{\epsilon}-a_{0}\|_{L^{\infty}(\mathbb{R})}.$

###### Proof.

Given $\vartheta=\begin{bmatrix}u&v&w&z\end{bmatrix}^{T}\in Y_{0}$ and using
(4.6), we have

$\begin{split}\|[\tilde{\mathcal{A}}_{\epsilon}(t)-\tilde{\mathcal{A}}_{0}(t)]\tilde{\mathcal{A}}^{-1}_{\epsilon}(\tau)\vartheta\|_{Y_{0}}&=\left\|\begin{bmatrix}0\\\
-[a_{\epsilon}(t)-a_{0}(t)]A^{\frac{1}{2}}w\\\ 0\\\
[a_{\epsilon}(t)-a_{0}(t)]A^{\frac{1}{2}}u\end{bmatrix}\right\|_{Y_{0}}\\\
&=\|[a_{\epsilon}(t)-a_{0}(t)]A^{\frac{1}{2}}w\|_{X}+\|-[a_{\epsilon}(t)-a_{0}(t)]A^{\frac{1}{2}}u\|_{X}\\\
&\leqslant\|a_{\epsilon}-a_{0}\|_{L^{\infty}(\mathbb{R})}\|\vartheta\|_{Y_{0}}.\end{split}$

∎

###  4.1. Uniform sectorial estimates and convergence of the linear semigroups

Consider the sector $\Gamma$, whose angle is greater than $\frac{\pi}{2}$,
given in Lemma 3.2.

######  Lemma 4.4.

There exists $R_{0}>0$ sufficiently large such that, if $\mu\in\Gamma$
satisfies $|\mu|\geq R_{0}$, then the operator
$\tilde{\mathcal{A}}_{\epsilon}(t)=\mathcal{A}_{\epsilon}(t)+B$, where $B$ is
given by (4.3), satisfies the estimate

$\|(\mu
I+\tilde{\mathcal{A}}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\leq\frac{\overline{C}}{|\mu|},$

where the constant $\overline{C}>0$ is independent of both $t\in\mathbb{R}$
and $\epsilon\in[0,1]$.

###### Proof.

Note that it follows from (3.12) that

$\begin{split}\|\mu(\mu
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}&=\|I-\mathcal{A}_{\epsilon}(t)(\mu
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\\\ &\leq
1+\|\mathcal{A}_{\epsilon}(t)(\mu
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\leq
2+\tilde{C}=C_{1},\end{split}$

that is,

(4.9) $\|\mu(\mu I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\leq
C_{1},\quad\mu\in\Gamma.$

Moreover, since

$\begin{split}\mu I+\tilde{\mathcal{A}}_{\epsilon}(t)&=(\mu
I+\mathcal{A}_{\epsilon}(t))+B\\\ &=(\mu I+\mathcal{A}_{\epsilon}(t))+B(\mu
I+\mathcal{A}_{\epsilon}(t))^{-1}(\mu I+\mathcal{A}_{\epsilon}(t))\\\
&=[I+B(\mu I+\mathcal{A}_{\epsilon}(t))^{-1}](\mu
I+\mathcal{A}_{\epsilon}(t)),\end{split}$

we have

(4.10) $(\mu I+\tilde{\mathcal{A}}_{\epsilon}(t))^{-1}=(\mu
I+\mathcal{A}_{\epsilon}(t))^{-1}[I+B(\mu
I+\mathcal{A}_{\epsilon}(t))^{-1}]^{-1},\quad\mu\in\Gamma.$

On the other hand, by Proposition 4.1, there is a constant $c>0$ such that

$\|B\|_{\mathcal{L}(Y_{0})}\leq c\left(1+\max\limits_{1\leq i\leq
r}\|e_{i}^{\ast}\|_{Y_{0}}^{\rho-1}\right),$

that is, $B$ is a bounded linear operator from $Y_{0}$ into itself and,
therefore, there exists a constant $k_{0}>0$ such that $\|-Bx\|_{Y_{0}}\leq
k_{0}\|x\|_{Y_{0}}$ for all $x\in D(B)=D(\mathcal{A}_{\epsilon}(t))=Y_{1}$.

Now, given $\alpha\in(0,1),$ take $R_{0}>0$ such that
$1-\frac{k_{0}C_{1}}{R_{0}}\geq\alpha$ and let $\xi>0$ be such that
$\xi(1+\tilde{C})<\alpha$. In this way, we have

$1-\xi(1+\tilde{C})-\frac{k_{0}C_{1}}{R_{0}}\geq\alpha-\xi(1+\tilde{C})>0.$

Furthermore, we also have

$\|-Bx\|_{Y_{0}}\leq
k_{0}\|x\|_{Y_{0}}+\xi\|\mathcal{A}_{\epsilon}(t)x\|_{Y_{0}}$

for all $x\in Y_{1}$. Thus, using (3.12) and (4.9), we obtain

$\begin{split}\|-B(\mu
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}&\leq\xi\|\mathcal{A}_{\epsilon}(t)(\mu
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}+k_{0}\|(\mu
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\\\
&\leq\xi(1+\tilde{C})+\frac{k_{0}C_{1}}{|\mu|}\end{split}$

and, consequently,

$1-\|-B(\mu I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\geq
1-\xi(1+\tilde{C})-\frac{k_{0}C_{1}}{|\mu|},\quad\mu\in\Gamma,\mu\neq 0.$

An application of the Neumann series leads to

(4.11) $\begin{split}\|[I+B(\mu
I+\mathcal{A}_{\epsilon}(t))^{-1}]^{-1}\|_{\mathcal{L}(Y_{0})}&\leq\frac{1}{1-\|-B(\mu
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}}\\\
&\leq\left(1-\xi(1+\tilde{C})-\frac{k_{0}C_{1}}{|\mu|}\right)^{-1}.\end{split}$

Now, if $\mu\in\Gamma$ with $|\mu|\geq R_{0}$, taking
$C_{2}=1-\xi(1+\tilde{C})-\frac{k_{0}C_{1}}{R_{0}}>0$, we derive

(4.12)
$\left(1-\xi(1+\tilde{C})-\frac{k_{0}C_{1}}{|\mu|}\right)^{-1}\leq\frac{1}{C_{2}}$

and, combining (4.9), (4.10), (4.11) and (4.12), it follows that

(4.13) $\begin{split}\|(\mu
I+\tilde{\mathcal{A}}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}&\leq\|(\mu
I+\mathcal{A}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\|[I+B(\mu
I+\mathcal{A}_{\epsilon}(t))^{-1}]^{-1}\|_{\mathcal{L}(Y_{0})}\\\
&\leq\frac{C_{1}}{|\mu|}\frac{1}{C_{2}}=\frac{C_{3}}{|\mu|},\end{split}$

where the constant $C_{3}=\frac{C_{1}}{C_{2}}>0$ is independent of
$t\in\mathbb{R}$ and $\epsilon\in[0,1]$. This concludes the proof. ∎

######  Remark 4.5.

The estimate obtained in (4.13) holds outside a ball of radius $R_{0}>0$. Now,
let $a>R_{0}$, $\phi_{0}\in(\pi/2,\pi)$ be the angle of the sector $\Gamma$
and consider the sector

$\Sigma_{a}=\\{\lambda\in\mathbb{C}\colon|\arg(\lambda-a)|\leq\phi_{0},\,\lambda\neq
a\\}.$

If $\mu\in\Sigma_{a}$, then by (4.13), we obtain

$\begin{split}\|(\mu
I+\tilde{\mathcal{A}}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}&\leq\frac{C_{3}}{|\mu|}\leq\frac{C_{3}}{|\mu-a|}\left(1+\frac{|a|}{|\mu|}\right)\leq\frac{C_{3}}{|\mu-a|}\left(1+\frac{|a|}{R_{0}}\right)\end{split}$

provided $|\mu|\geq R_{0}$, that is,

$\|(\mu
I+\tilde{\mathcal{A}}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\leq\frac{C_{4}}{|\mu-a|}\quad\mbox{for
all}\quad\mu\in\Sigma_{a},$

where the constant $C_{4}=C_{3}\left(1+\frac{|a|}{R_{0}}\right)>0$ is
independent of $t\in\mathbb{R}$ and $\epsilon\in[0,1]$.

$x$$y$$R_{0}$$a$$\Sigma_{a}$$\Gamma^{\prime}$$\Gamma$

For each $\epsilon\in[0,1],$ let
$\\{e^{-s\tilde{\mathcal{A}}_{\epsilon}(t)}\colon s\geq
0\\}\subset\mathcal{L}(Y_{0})$ be the analytic semigroup generated by the
operator $-\tilde{\mathcal{A}}_{\epsilon}(t)$, for each $t\in\mathbb{R}$
fixed. In Proposition 4.6 below we present a result on convergence for this
family of semigroups as $\epsilon\rightarrow 0^{+}$.

######  Proposition 4.6.

For $\epsilon\in[0,1],$ the family of semigroups
$\\{e^{-s\tilde{\mathcal{A}}_{\epsilon}(t)}\colon s\geq 0\\}$ satisfies

$\|e^{-s\tilde{\mathcal{A}}_{\epsilon}(t)}-e^{-s\tilde{\mathcal{A}}_{0}(t)}\|_{\mathcal{L}(Y_{0})}\rightarrow
0\quad\mbox{as}\quad\epsilon\rightarrow 0^{+}$

uniformly for $s$ on compact subsets of $\mathbb{R}_{+}$.

###### Proof.

Let $t\in\mathbb{R}$ be fixed and arbitrary. According to [18, Theorem 1.3.4],
for each $\epsilon\in[0,1],$ the semigroup
$\\{e^{-s\tilde{\mathcal{A}}_{\epsilon}(t)}\colon s\geq 0\\}$ can be written
as

(4.14) $e^{-s\tilde{\mathcal{A}}_{\epsilon}(t)}=\frac{1}{2\pi
i}\int_{\Gamma^{\prime}}e^{\mu s}(\mu
I+\tilde{\mathcal{A}}_{\epsilon}(t))^{-1}\,d\mu,\quad s\geq 0,$

where the curve $\Gamma^{\prime}$ is a contour in
$\rho(-\tilde{\mathcal{A}}_{\epsilon}(t))$, with $\arg\mu\rightarrow\pm\theta$
as $|\mu|\rightarrow\infty$, for some $\theta\in(\pi/2,\pi)$. In addition to
the previous conditions, let us assume, without loss of generality,
$\Gamma^{\prime}$ is contained in the sector $\Sigma_{a}$ given by

$\Sigma_{a}=\\{\lambda\in\mathbb{C}\colon|\arg(\lambda-a)|\leq\theta,\,\lambda\neq
a\\},$

where $a>R_{0}$ and $R_{0}>0$ is given as in Lemma 4.4 (see also Remark 4.5).
At first place, we have

(4.15)
$\begin{split}&\|e^{-s\tilde{\mathcal{A}}_{\epsilon}(t)}-e^{-s\tilde{\mathcal{A}}_{0}(t)}\|_{\mathcal{L}(Y_{0})}\\\
&=\left\|\frac{1}{2\pi i}\int_{\Gamma^{\prime}}e^{\mu s}(\mu
I+\tilde{\mathcal{A}}_{\epsilon}(t))^{-1}\,d\mu-\frac{1}{2\pi
i}\int_{\Gamma^{\prime}}e^{\mu s}(\mu
I+\tilde{\mathcal{A}}_{0}(t))^{-1}\,d\mu\right\|_{\mathcal{L}(Y_{0})}\\\
&\leq\frac{1}{2\pi}\int_{\Gamma^{\prime}}|e^{\mu s}|\|(\mu
I+\tilde{\mathcal{A}}_{\epsilon}(t))^{-1}-(\mu
I+\tilde{\mathcal{A}}_{0}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\,d\mu.\end{split}$

Now, in order to deal with the resolvent operators in (4.15), we first write

(4.16) $\begin{split}&\|(\mu I+\tilde{\mathcal{A}}_{\epsilon}(t))^{-1}-(\mu
I+\tilde{\mathcal{A}}_{0}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\\\ &=\|(\mu
I+\tilde{\mathcal{A}}_{\epsilon}(t))^{-1}[(\mu
I+\tilde{\mathcal{A}}_{0}(t))-(\mu I+\tilde{\mathcal{A}}_{\epsilon}(t))](\mu
I+\tilde{\mathcal{A}}_{0}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\\\ &=\|(\mu
I+\tilde{\mathcal{A}}_{\epsilon}(t))^{-1}\tilde{\mathcal{A}}_{\epsilon}(t)[\tilde{\mathcal{A}}_{\epsilon}^{-1}(t)-\tilde{\mathcal{A}}_{0}^{-1}(t)]\tilde{\mathcal{A}}_{0}(t)(\mu
I+\tilde{\mathcal{A}}_{0}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\\\ &\leq\|(\mu
I+\tilde{\mathcal{A}}_{\epsilon}(t))^{-1}\tilde{\mathcal{A}}_{\epsilon}(t)\|_{\mathcal{L}(Y_{0})}\|\tilde{\mathcal{A}}_{\epsilon}^{-1}(t)-\tilde{\mathcal{A}}_{0}^{-1}(t)\|_{\mathcal{L}(Y_{0})}\|\tilde{\mathcal{A}}_{0}(t)(\mu
I+\tilde{\mathcal{A}}_{0}(t))^{-1}\|_{\mathcal{L}(Y_{0})}.\end{split}$

On the other hand, using Lemma 4.4, we have

(4.17) $\begin{split}&\|(\mu
I+\tilde{\mathcal{A}}_{\epsilon}(t))^{-1}\tilde{\mathcal{A}}_{\epsilon}(t)\|_{\mathcal{L}(Y_{0})}\leq
1+|\mu|\|(\mu
I+\tilde{\mathcal{A}}_{\epsilon}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\leq
1+\overline{C},\end{split}$

where $\hat{C}:=1+\overline{C}>0$ is independent of $t\in\mathbb{R}$ and
$\epsilon\in[0,1]$. Analogously, we also have

(4.18) $\|\tilde{\mathcal{A}}_{0}(t)(\mu
I+\tilde{\mathcal{A}}_{0}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\leq\hat{C}.$

Thus, combining (4.16), (4.17) and (4.18), together with Proposition 4.2, we
get

$\|(\mu I+\tilde{\mathcal{A}}_{\epsilon}(t))^{-1}-(\mu
I+\tilde{\mathcal{A}}_{0}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\rightarrow
0\quad\mbox{as}\quad\epsilon\rightarrow 0^{+}$

and, consequently, it follows by (4.15) and the Dominated Convergence Theorem
that

$\begin{split}&\|e^{-s\tilde{\mathcal{A}}_{\epsilon}(t)}-e^{-s\tilde{\mathcal{A}}_{0}(t)}\|_{\mathcal{L}(Y_{0})}\leq\frac{1}{2\pi}\int_{\Gamma^{\prime}}|e^{\mu
s}|\|(\mu I+\tilde{\mathcal{A}}_{\epsilon}(t))^{-1}-(\mu
I+\tilde{\mathcal{A}}_{0}(t))^{-1}\|_{\mathcal{L}(Y_{0})}\,d\mu\xrightarrow{\epsilon\rightarrow
0^{+}}0\end{split}$

uniformly for $s\geq 0$ on compact subsets of $\mathbb{R}_{+}$, which yields
the desired result. ∎

##  5\. Exponential dichotomy and lower semicontinuity of pullback attractors

######  Proposition 5.1.

The map $\mathbb{R}\ni t\mapsto\tilde{\mathcal{A}}_{\epsilon}(t)$ is uniformly
Hölder continuous in $Y_{0},$ for each $\epsilon\in[0,1],$ that is, for any
$T>0$

$\|[\tilde{\mathcal{A}}_{\epsilon}(t)-\tilde{\mathcal{A}}_{\epsilon}(s)]\tilde{\mathcal{A}}_{\epsilon}^{-1}(\tau)\|_{\mathcal{L}(Y_{0})}\leq
C|t-s|^{\beta}$

for every $t,s,\tau\in[-T,T]$, where $C>0$ and $0<\beta\leq 1$ $($both
independent of $T$ and $\epsilon$$)$ come from condition 2.8.

###### Proof.

Let $T>0$ be fixed and arbitrary, and let $t,s,\tau\in[-T,T]$. Since

$\tilde{\mathcal{A}}_{\epsilon}(t)-\tilde{\mathcal{A}}_{\epsilon}(s)\,\\!\\!=\\!\\!\,[a_{\epsilon}(t)-a_{\epsilon}(s)]\begin{bmatrix}0&0&0&0\\\
0&0&0&A^{1/2}\\\ 0&0&0&0\\\ 0&-A^{1/2}&0&0\end{bmatrix}$

and

$\small\tilde{\mathcal{A}}_{\epsilon}^{-1}(\tau)=\begin{bmatrix}\eta
A^{1/2}(A+I-f^{\prime}(u_{j}^{\ast}))^{-1}&(A+I-f^{\prime}(u_{j}^{\ast}))^{-1}&a_{\epsilon}(\tau)A^{1/2}(A+I-f^{\prime}(u_{j}^{\ast}))^{-1}&0\\\
-I&0&0&0\\\ -a_{\epsilon}(\tau)A^{-1/2}&0&\eta A^{-1/2}&A^{-1}\\\
0&0&-I&0\end{bmatrix},$

we obtain

$\begin{split}\left\|[\tilde{\mathcal{A}}_{\epsilon}(t)-\tilde{\mathcal{A}}_{\epsilon}(s)]\tilde{\mathcal{A}}_{\epsilon}^{-1}(\tau)\begin{bmatrix}u\\\
v\\\ w\\\
z\end{bmatrix}\right\|_{Y_{0}}&=\left\|[a_{\epsilon}(t)-a_{\epsilon}(s)]\begin{bmatrix}0&0&0&0\\\
0&0&-A^{1/2}&0\\\ 0&0&0&0\\\ A^{1/2}&0&0&0\end{bmatrix}\begin{bmatrix}u\\\
v\\\ w\\\ z\end{bmatrix}\right\|_{Y_{0}}\\\
&=|a_{\epsilon}(t)-a_{\epsilon}(s)|\left\|\begin{bmatrix}0\\\ -A^{1/2}w\\\
0\\\ A^{1/2}u\end{bmatrix}\right\|_{Y_{0}}\\\
&=|a_{\epsilon}(t)-a_{\epsilon}(s)|\left(\left\|-A^{1/2}w\right\|_{X}+\left\|A^{1/2}u\right\|_{X}\right)\\\
&\leq
C|t-s|^{\beta}\left(\left\|u\right\|_{X^{1/2}}+\left\|v\right\|_{X}+\left\|w\right\|_{X^{1/2}}+\left\|z\right\|_{X}\right)\\\
&=C|t-s|^{\beta}\left\|\begin{bmatrix}u\\\ v\\\ w\\\
z\end{bmatrix}\right\|_{Y_{0}}\end{split}$

for every $[u\,\,v\,\,w\,\,z]^{T}\in Y_{0}$, where we have used the Hölder
continuity condition of the function $a_{\epsilon}$ uniformly in $\epsilon$.
Thus, it follows that

$\|[\tilde{\mathcal{A}}_{\epsilon}(t)-\tilde{\mathcal{A}}_{\epsilon}(s)]\tilde{\mathcal{A}}_{\epsilon}^{-1}(\tau)\|_{\mathcal{L}(Y_{0})}\leq
C|t-s|^{\beta}$

for every $t,s,\tau\in[-T,T],$ and the result is proved. ∎

The next result uses Proposition 4.6 to obtain the convergence of the family
of linear semigroups in the norm topology of $\mathcal{L}(Y_{1})$, which will
be helpful in the proof of Proposition 5.3.

######  Lemma 5.2.

For $s\geq\tau\in\mathbb{R}$, there holds

(5.1)
$\|e^{-(s-\tau)\tilde{\mathcal{A}}_{\epsilon}(\tau)}-e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\|_{\mathcal{L}(Y_{0},Y_{1})}\rightarrow
0\quad\mbox{as}\quad\epsilon\rightarrow 0^{+}.$

###### Proof.

Note that

$\displaystyle\tilde{\mathcal{A}}_{\epsilon}(\tau)\Big{[}e^{-(s-\tau)\tilde{\mathcal{A}}_{\epsilon}(\tau)}-e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\Big{]}$
$\displaystyle=\tilde{\mathcal{A}}_{\epsilon}(\tau)e^{-\frac{(s-\tau)}{2}\tilde{\mathcal{A}}_{\epsilon}(\tau)}\Big{[}e^{-\frac{(s-\tau)}{2}\tilde{\mathcal{A}}_{\epsilon}(\tau)}-e^{-\frac{(s-\tau)}{2}\tilde{\mathcal{A}}_{0}(\tau)}\Big{]}$
$\displaystyle\quad+e^{-\frac{(s-\tau)}{2}\tilde{\mathcal{A}}_{\epsilon}(\tau)}\Big{[}\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{0}(\tau)\Big{]}e^{-\frac{(s-\tau)}{2}\tilde{\mathcal{A}}_{0}(\tau)}$
$\displaystyle\quad+\Big{[}e^{-\frac{(s-\tau)}{2}\tilde{\mathcal{A}}_{\epsilon}(\tau)}-e^{-\frac{(s-\tau)}{2}\tilde{\mathcal{A}}_{0}(\tau)}\Big{]}\tilde{\mathcal{A}}_{0}(\tau)e^{-\frac{(s-\tau)}{2}\tilde{\mathcal{A}}_{0}(\tau)}$
$\displaystyle\quad+\Big{[}\tilde{\mathcal{A}}_{0}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(\tau)\Big{]}e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}.$

Due to Lemma 4.4 and (4.14), we have the following estimate

$\begin{split}\|e^{-(s-\tau)\tilde{\mathcal{A}}_{\epsilon}(\tau)}x\|_{Y_{1}}&=\|\tilde{\mathcal{A}}_{\epsilon}(\tau)e^{-(s-\tau)\tilde{\mathcal{A}}_{\epsilon}(\tau)}x\|_{Y_{0}}\leq
M_{0}(s-\tau)^{-1}e^{-\omega(s-\tau)}\|x\|_{Y_{0}}\end{split}$

for every $x\in Y_{0}$, where the constants $M_{0},\omega>0$ are both
independent of the parameters $t$ and $\epsilon$, that is,

(5.2)
$\|e^{-(s-\tau)\tilde{\mathcal{A}}_{\epsilon}(\tau)}\|_{\mathcal{L}(Y_{0},Y_{1})}\leq
M_{0}(s-\tau)^{-1}e^{-\omega(s-\tau)}.$

Using Propositions 4.3 and 4.6, condition (2.9) and (5.2) the result is
complete. ∎

######  Proposition 5.3.

For $\epsilon\in[0,1],$ let $\\{\tilde{L}_{\epsilon}(t,\tau)\colon
t\geq\tau\in\mathbb{R}\\}$ be the linear evolution process in $Y_{0}$
corresponding to the linearized problem (4.1). Then, for each
$t\geq\tau\in\mathbb{R}$,

$\|\tilde{L}_{\epsilon}(t,\tau)-\tilde{L}_{0}(t,\tau)\|_{\mathcal{L}(Y_{0})}\rightarrow
0\quad\mbox{as}\quad\epsilon\rightarrow 0^{+}.$

###### Proof.

According to [10], the process $\\{\tilde{L}_{\epsilon}(t,\tau)\colon
t\geq\tau\in\mathbb{R}\\}$ can be written as

$\tilde{L}_{\epsilon}(t,\tau)=e^{-(t-\tau)\tilde{\mathcal{A}}_{\epsilon}(\tau)}+\int_{\tau}^{t}\tilde{L}_{\epsilon}(t,s)[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)]e^{-(s-\tau)\tilde{\mathcal{A}}_{\epsilon}(\tau)}\,ds,\quad
t\geq\tau\in\mathbb{R},$

for each $\epsilon\in[0,1],$ where $\tilde{\mathcal{A}}_{\epsilon}(\tau)$ is
given as in (4.2). At first place, we have

(5.3)
$\begin{split}\|\tilde{L}_{\epsilon}(t,\tau)-\tilde{L}_{0}(t,\tau)\|_{\mathcal{L}(Y_{0})}&\leq\|e^{-(t-\tau)\tilde{\mathcal{A}}_{\epsilon}(\tau)}-e^{-(t-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\|_{\mathcal{L}(Y_{0})}\\\
&+\Big{\|}\int_{\tau}^{t}\tilde{L}_{\epsilon}(t,s)[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)]e^{-(s-\tau)\tilde{\mathcal{A}}_{\epsilon}(\tau)}\,ds\\\
&\quad\quad-\int_{\tau}^{t}\tilde{L}_{0}(t,s)[\tilde{\mathcal{A}}_{0}(\tau)-\tilde{\mathcal{A}}_{0}(s)]e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds\Big{\|}_{\mathcal{L}(Y_{0})}.\end{split}$

By Proposition 4.6, we have

(5.4)
$\|e^{-(t-\tau)\tilde{\mathcal{A}}_{\epsilon}(\tau)}-e^{-(t-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\|_{\mathcal{L}(Y_{0})}\rightarrow
0\quad\mbox{as}\quad\epsilon\rightarrow 0^{+}.$

Note that the difference between the integrals in (5.3) can be written in the
following way

(5.5)
$\small\begin{split}&\int_{\tau}^{t}\tilde{L}_{\epsilon}(t,s)[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)]e^{-(s-\tau)\tilde{\mathcal{A}}_{\epsilon}(\tau)}\,ds-\int_{\tau}^{t}\tilde{L}_{0}(t,s)[\tilde{\mathcal{A}}_{0}(\tau)-\tilde{\mathcal{A}}_{0}(s)]e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds\\\
&=\int_{\tau}^{t}\tilde{L}_{\epsilon}(t,s)[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)](e^{-(s-\tau)\tilde{\mathcal{A}}_{\epsilon}(\tau)}-e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)})\,ds\\\
&\quad+\int_{\tau}^{t}(\tilde{L}_{\epsilon}(t,s)-\tilde{L}_{0}(t,s))[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)]e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds\\\
&\quad+\int_{\tau}^{t}\tilde{L}_{0}(t,s)[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)]e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds\\\
&\quad-\int_{\tau}^{t}\tilde{L}_{0}(t,s)[\tilde{\mathcal{A}}_{0}(\tau)-\tilde{\mathcal{A}}_{0}(s)]e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds.\end{split}$

Now, we are going to analyze each one of the integrals that appear in (5.5),
following the order listed below

(A)
$\displaystyle\int_{\tau}^{t}\tilde{L}_{0}(t,s)[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)]e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds-\int_{\tau}^{t}\tilde{L}_{0}(t,s)[\tilde{\mathcal{A}}_{0}(\tau)-\tilde{\mathcal{A}}_{0}(s)]e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds$;

(B)
$\displaystyle\int_{\tau}^{t}(\tilde{L}_{\epsilon}(t,s)-\tilde{L}_{0}(t,s))[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)]e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds$;

(C)
$\displaystyle\int_{\tau}^{t}\tilde{L}_{\epsilon}(t,s)[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)](e^{-(s-\tau)\tilde{\mathcal{A}}_{\epsilon}(\tau)}-e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)})\,ds.$

In fact, for item (A), let us denote
$\hat{A}\,\\!\\!=\\!\\!\begin{bmatrix}0&0&0&0\\\ 0&0&0&A^{1/2}\\\ 0&0&0&0\\\
0&-A^{1/2}&0&0\end{bmatrix}$ and then note that

$\begin{split}&\int_{\tau}^{t}\tilde{L}_{0}(t,s)[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)]e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds-\int_{\tau}^{t}\tilde{L}_{0}(t,s)[\tilde{\mathcal{A}}_{0}(\tau)-\tilde{\mathcal{A}}_{0}(s)]e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds\\\
&=\int_{\tau}^{t}\tilde{L}_{0}(t,s)[a_{\epsilon}(\tau)-a_{\epsilon}(s)]\hat{A}\,e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds-\int_{\tau}^{t}\tilde{L}_{0}(t,s)[a_{0}(\tau)-a_{0}(s)]\hat{A}\,e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds\\\
&=[a_{\epsilon}(\tau)-a_{0}(\tau)]\int_{\tau}^{t}\tilde{L}_{0}(t,s)\hat{A}\,e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds+\int_{\tau}^{t}[a_{0}(s)-a_{\epsilon}(s)]\tilde{L}_{0}(t,s)\hat{A}\,e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds\end{split}$

which leads to

$\begin{split}&\left\|\int_{\tau}^{t}\tilde{L}_{0}(t,s)[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)]e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds\right.\\\
&\hskip
142.26378pt\left.-\int_{\tau}^{t}\tilde{L}_{0}(t,s)[\tilde{\mathcal{A}}_{0}(\tau)-\tilde{\mathcal{A}}_{0}(s)]e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds\right\|_{\mathcal{L}(Y_{0})}\\\
&\leq\left\|[a_{\epsilon}(\tau)-a_{0}(\tau)]\int_{\tau}^{t}\tilde{L}_{0}(t,s)\hat{A}\,e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds\right\|_{\mathcal{L}(Y_{0})}\\\
&\quad+\left\|\int_{\tau}^{t}[a_{0}(s)-a_{\epsilon}(s)]\tilde{L}_{0}(t,s)\hat{A}\,e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds\right\|_{\mathcal{L}(Y_{0})}\\\
\end{split}$

$\begin{split}&\leq|a_{\epsilon}(\tau)-a_{0}(\tau)|\left\|\int_{\tau}^{t}\tilde{L}_{0}(t,s)\hat{A}\,e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds\right\|_{\mathcal{L}(Y_{0})}\\\
&\quad+\int_{\tau}^{t}|a_{0}(s)-a_{\epsilon}(s)|\|\tilde{L}_{0}(t,s)\hat{A}\,e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\|_{\mathcal{L}(Y_{0})}\,ds\\\
&\leq\|a_{\epsilon}-a_{0}\|_{L^{\infty}(\mathbb{R})}\int_{\tau}^{t}\|\tilde{L}_{0}(t,s)\hat{A}\,e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\|_{\mathcal{L}(Y_{0})}\,ds\\\
&\quad+\int_{\tau}^{t}\|a_{0}-a_{\epsilon}\|_{L^{\infty}(\mathbb{R})}\|\tilde{L}_{0}(t,s)\hat{A}\,e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\|_{\mathcal{L}(Y_{0})}\,ds\\\
&=2\|a_{\epsilon}-a_{0}\|_{L^{\infty}(\mathbb{R})}\int_{\tau}^{t}\|\tilde{L}_{0}(t,s)\hat{A}\,e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\|_{\mathcal{L}(Y_{0})}\,ds\xrightarrow{\epsilon\rightarrow
0^{+}}0,\end{split}$

which proves that

(5.6)
$\begin{split}&\left\|\int_{\tau}^{t}\tilde{L}_{0}(t,s)[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)]e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds\right.\\\
&\hskip
85.35826pt\left.-\int_{\tau}^{t}\tilde{L}_{0}(t,s)[\tilde{\mathcal{A}}_{0}(\tau)-\tilde{\mathcal{A}}_{0}(s)]e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds\right\|_{\mathcal{L}(Y_{0})}\xrightarrow{\epsilon\rightarrow
0^{+}}0.\end{split}$

Next, in order to deal with item (B), we write

$\begin{split}&\|(\tilde{L}_{\epsilon}(t,s)-\tilde{L}_{0}(t,s))[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)]e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\|_{\mathcal{L}(Y_{0})}\\\
&\leq\|\tilde{L}_{\epsilon}(t,s)-\tilde{L}_{0}(t,s)\|_{\mathcal{L}(Y_{0})}\|[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)]\tilde{\mathcal{A}}_{\epsilon}^{-1}(\tau)\|_{\mathcal{L}(Y_{0})}\\\
&\hskip
142.26378pt\times\|\tilde{\mathcal{A}}_{\epsilon}(\tau)\tilde{\mathcal{A}}_{0}^{-1}(\tau)\|_{\mathcal{L}(Y_{0})}\|\tilde{\mathcal{A}}_{0}(\tau)e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\|_{\mathcal{L}(Y_{0})}\end{split}$

and then using the estimates

$\|\tilde{\mathcal{A}}_{0}(\tau)e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\|_{\mathcal{L}(Y_{0})}\leq\tilde{C}(s-\tau)^{-1},\,s-\tau>0,\,\tau\in\mathbb{R},$

for some constant $\tilde{C}>0,$

(5.7)
$\|\tilde{\mathcal{A}}_{\epsilon}(\tau)\tilde{\mathcal{A}}_{0}^{-1}(\tau)\|_{\mathcal{L}(Y_{0})}\leq
1+\|a_{\epsilon}-a_{0}\|_{L^{\infty}(\mathbb{R})},\,\tau\in\mathbb{R},$

and Proposition 5.1, it follows that

$\displaystyle\left\|\int_{\tau}^{t}(\tilde{L}_{\epsilon}(t,s)-\tilde{L}_{0}(t,s))[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)]e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds\right\|_{\mathcal{L}(Y_{0})}$
$\displaystyle\leq\int_{\tau}^{t}\|(\tilde{L}_{\epsilon}(t,s)-\tilde{L}_{0}(t,s))[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)]e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\|_{\mathcal{L}(Y_{0})}\,ds$
$\displaystyle\leq\int_{\tau}^{t}\|\tilde{L}_{\epsilon}(t,s)-\tilde{L}_{0}(t,s)\|_{\mathcal{L}(Y_{0})}\|[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)]\tilde{\mathcal{A}}_{\epsilon}^{-1}(\tau)\|_{\mathcal{L}(Y_{0})}$
$\displaystyle\hskip
142.26378pt\times\|\tilde{\mathcal{A}}_{\epsilon}(\tau)\tilde{\mathcal{A}}_{0}^{-1}(\tau)\|_{\mathcal{L}(Y_{0})}\|\tilde{\mathcal{A}}_{0}(\tau)e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\|_{\mathcal{L}(Y_{0})}\,ds$
$\displaystyle\leq\int_{\tau}^{t}\|\tilde{L}_{\epsilon}(t,s)-\tilde{L}_{0}(t,s)\|_{\mathcal{L}(Y_{0})}C|\tau-s|^{\beta}\left(1+\|a_{\epsilon}-a_{0}\|_{L^{\infty}(\mathbb{R})}\right)\tilde{C}(s-\tau)^{-1}\,ds$

(5.8)
$\displaystyle=\tilde{\tilde{C}}\left(1+\|a_{\epsilon}-a_{0}\|_{L^{\infty}(\mathbb{R})}\right)\int_{\tau}^{t}\|\tilde{L}_{\epsilon}(t,s)-\tilde{L}_{0}(t,s)\|_{\mathcal{L}(Y_{0})}|\tau-s|^{\beta}(s-\tau)^{-1}\,ds$
$\displaystyle=\tilde{\tilde{C}}\left(1+\|a_{\epsilon}-a_{0}\|_{L^{\infty}(\mathbb{R})}\right)\int_{\tau}^{t}|\tau-s|^{\beta-1}\|\tilde{L}_{\epsilon}(t,s)-\tilde{L}_{0}(t,s)\|_{\mathcal{L}(Y_{0})}\,ds.$

Now, for the integral on item (C), we write

(5.9)
$\begin{split}&\left\|\int_{\tau}^{t}\tilde{L}_{\epsilon}(t,s)[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)](e^{-(s-\tau)\tilde{\mathcal{A}}_{\epsilon}(\tau)}-e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)})\,ds\right\|_{\mathcal{L}(Y_{0})}\\\
&\leq\int_{\tau}^{t}\|\tilde{L}_{\epsilon}(t,s)[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)](e^{-(s-\tau)\tilde{\mathcal{A}}_{\epsilon}(\tau)}-e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)})\|_{\mathcal{L}(Y_{0})}\,ds\\\
&\leq\int_{\tau}^{t}\|\tilde{L}_{\epsilon}(t,s)\|_{\mathcal{L}(Y_{0})}\|\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)\|_{\mathcal{L}(Y_{1},Y_{0})}\|e^{-(s-\tau)\tilde{\mathcal{A}}_{\epsilon}(\tau)}-e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\|_{\mathcal{L}(Y_{0},Y_{1})}\,ds\end{split}$

Now, according to [10, Theorem 2.2], the process
$\\{\tilde{L}_{\epsilon}(t,\tau)\colon t\geq\tau\in\mathbb{R}\\}$ satisfies

$\|\tilde{L}_{\epsilon}(t,\tau)\|_{\mathcal{L}(Y_{0})}\leq M,\quad
t\geq\tau\in\mathbb{R},$

with constant $M>0$ being independent of the time $t\in\mathbb{R}$ and also of
the parameter $\epsilon$. With this, we can combine (5.9), Lemma 5.2,
Proposition 5.1, and the Dominated Convergence Theorem in order to obtain

$\begin{split}&\left\|\int_{\tau}^{t}\tilde{L}_{\epsilon}(t,s)[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)](e^{-(s-\tau)\tilde{\mathcal{A}}_{\epsilon}(\tau)}-e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)})\,ds\right\|_{\mathcal{L}(Y_{0})}\\\
&\leq\tilde{M}\int_{\tau}^{t}|\tau-s|^{\beta}\|e^{-(s-\tau)\tilde{\mathcal{A}}_{\epsilon}(\tau)}-e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\|_{\mathcal{L}(Y_{0},Y_{1})}\,ds\xrightarrow{\epsilon\rightarrow
0^{+}}0,\end{split}$

which yields

(5.10)
$\left\|\int_{\tau}^{t}\tilde{L}_{\epsilon}(t,s)[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)](e^{-(s-\tau)\tilde{\mathcal{A}}_{\epsilon}(\tau)}-e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)})\,ds\right\|_{\mathcal{L}(Y_{0})}\xrightarrow{\epsilon\rightarrow
0^{+}}0.$

Finally, combining (5.3), (5.5) and (5.8), we can write

$\small\begin{split}&\|\tilde{L}_{\epsilon}(t,\tau)-\tilde{L}_{0}(t,\tau)\|_{\mathcal{L}(Y_{0})}\leq\|e^{-(t-\tau)\tilde{\mathcal{A}}_{\epsilon}(\tau)}-e^{-(t-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\|_{\mathcal{L}(Y_{0})}\\\
&+\left\|\int_{\tau}^{t}\tilde{L}_{0}(t,s)[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)]e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds-\int_{\tau}^{t}\tilde{L}_{0}(t,s)[\tilde{\mathcal{A}}_{0}(\tau)-\tilde{\mathcal{A}}_{0}(s)]e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)}\,ds\right\|_{\mathcal{L}(Y_{0})}\\\
&+\tilde{\tilde{C}}\left(1+\|a_{\epsilon}-a_{0}\|_{L^{\infty}(\mathbb{R})}\right)\int_{\tau}^{t}|\tau-s|^{\beta-1}\|\tilde{L}_{\epsilon}(t,s)-\tilde{L}_{0}(t,s)\|_{\mathcal{L}(Y_{0})}\,ds\\\
&+\left\|\int_{\tau}^{t}\tilde{L}_{\epsilon}(t,s)[\tilde{\mathcal{A}}_{\epsilon}(\tau)-\tilde{\mathcal{A}}_{\epsilon}(s)](e^{-(s-\tau)\tilde{\mathcal{A}}_{\epsilon}(\tau)}-e^{-(s-\tau)\tilde{\mathcal{A}}_{0}(\tau)})\,ds\right\|_{\mathcal{L}(Y_{0})}\end{split}$

and then, using (5.4), (5.6), (5.10) and Generalized Gronwall inequality (see
[28, Theorem 1.26]) we have

$\|\tilde{L}_{\epsilon}(t,\tau)-\tilde{L}_{0}(t,\tau)\|_{\mathcal{L}(Y_{0})}\rightarrow
0,\quad\text{as }\epsilon\rightarrow 0^{+},$

which completes the proof. ∎

######  Corollary 5.4.

All the equilibrium points in $\mathcal{E}^{\ast}$ are hyperbolic for the
family of processes $\\{\tilde{L}_{\epsilon}(t,\tau)\colon
t\geq\tau\in\mathbb{R}\\},$ for every $\epsilon\in[0,\epsilon_{0}],$ for some
$\epsilon_{0}>0$ sufficiently small.

###### Proof.

Indeed, by our assumptions, all equilibrium points in $\mathcal{E}^{\ast}$ are
hyperbolic, in the sense of Definition A.2, for the limit problem
($\epsilon=0$) associated to the system (2.1)-(2.2), which means that the
process $\\{\tilde{L}_{0}(t,\tau)\colon t\geq\tau\in\mathbb{R}\\}$ has an
exponential dichotomy in $Y_{0}$, in the sense of Definition A.1. On the other
hand, according to [10, Theorem 2.2], we have

$\|\tilde{L}_{0}(t,\tau)\|_{\mathcal{L}(Y_{0})}\leq M,\quad
t\geq\tau\in\mathbb{R},$

with $M>0$ being a constant. In particular, it follows that

$\sup\\{\|\tilde{L}_{0}(t,\tau)\|_{\mathcal{L}(Y_{0})}\colon 0\leq t-\tau\leq
1\\}<\infty.$

Now, with Proposition 5.3 in hands, and applying [18, Theorem 7.6.10], it
follows that, for some $\epsilon_{0}>0$ sufficiently small, the process
$\\{\tilde{L}_{\epsilon}(t,\tau)\colon t\geq\tau\in\mathbb{R}\\}$ has an
exponential dichotomy in $Y_{0}$, for every $\epsilon\in[0,\epsilon_{0}]$.
Consequently, every point in the set of equilibria $\mathcal{E}^{\ast}$ is
also hyperbolic for the process $\\{\tilde{L}_{\epsilon}(t,\tau)\colon
t\geq\tau\in\mathbb{R}\\},$ for every $\epsilon\in[0,\epsilon_{0}]$. ∎

######  Theorem 5.5.

The family of pullback attractors $\\{\mathbb{A}_{\epsilon}(t)\colon
t\in\mathbb{R}\\}$ associated to the problem (2.1)-(2.2) is lower
semicontinuous at $\epsilon=0$, that is, for each $t\in\mathbb{R}$,

${\rm d_{H}}(\mathbb{A}_{0}(t),\mathbb{A}_{\epsilon}(t))\rightarrow
0\quad\mbox{as}\quad\epsilon\rightarrow 0^{+}.$

###### Proof.

Applying Lemma 2.6 we have the uniform continuity, with respect to the
parameter $\epsilon$, of the nonlinear evolution processes
$\\{S_{\epsilon}(t,\tau)\colon t\geq\tau\in\mathbb{R}\\}$ associated to the
problem (2.1)-(2.2). The compactness of the closure of the union of pullback
attractors is a direct consequence of the regularity result stated in Theorem
2.5. Moreover, in Theorem 3.8, we obtained the gradient-like structure of the
limit pullback attractor $\\{\mathbb{A}_{0}(t)\colon t\in\mathbb{R}\\}$
associated to the problem (2.1)-(2.2) with $\epsilon=0$. It remains to verify
the two conditions in Theorem A.3. But, as noted in [5], they are just
consequences of the stability of the hyperbolic equilibria under perturbation,
see also [7, Theorem 6.1] and [8, Theorem 2.3]. Now, Corollary 5.4 ensures
that every equilibrium point in $\mathcal{E}^{\ast}$ is also hyperbolic with
respect to the perturbed problem (2.1)-(2.2), for every
$\epsilon\in[0,\epsilon_{0}],$ for some $\epsilon_{0}>0$ sufficiently small.
Hence, we can apply Theorem A.3, which leads to the lower semicontinuity at
$\epsilon=0$ of the family of pullback attractors
$\\{\mathbb{A}_{\epsilon}(t)\colon t\in\mathbb{R}\\}$ associated to the
problem (2.1)-(2.2). ∎

######  Corollary 5.6.

The family of pullback attractors $\\{\mathbb{A}_{\epsilon}(t)\colon
t\in\mathbb{R}\\},$ associated to the non-autonomous problem (2.1)-(2.2), is
continuous at $\epsilon=0$.

###### Proof.

This result is a direct consequence of Theorems 2.7 and 5.5. ∎

##  Appendix A Appendix

###  A.1. Evolution processes and pullback attractors

Let $(Z,d)$ be a metric space and $\mathbb{N}=\\{1,2,\ldots\\}$ be the set of
all natural numbers. An evolution process acting in $Z$ is a two-parameter
family $\\{S(t,\tau):t\geq\tau\in\mathbb{R}\\}$ of maps from $Z$ into itself
such that:

  1. $\bullet$

$S(t,t)=I$ for all $t\in\mathbb{R}$ ($I$ is the identity operator in $Z$),

  2. $\bullet$

$S(t,\tau)=S(t,s)S(s,\tau)$ for all $t\geq s\geq\tau$, and

  3. $\bullet$

the map $\\{(t,\tau)\in\mathbb{R}^{2}:t\geq\tau\\}\times Z\ni(t,\tau,x)\mapsto
S(t,\tau)x\in Z$ is continuous.

For a given set $B\subset Z$, $S(t,\tau)B=\\{S(t,\tau)x\colon x\in B\\}$ is
the image of $B$ under $\\{S(t,\tau)\colon t\geq\tau\in\mathbb{R}\\}$. If $Z$
is a Banach space and $\\{S(t,\tau)\colon
t\geq\tau\in\mathbb{R}\\}\subset\mathcal{L}(Z)$, we refer to this process as a
linear evolution process.

The Hausdorff semidistance between two nonempty subsets $A$ and $B$ of $Z$ is
given by

${\rm d_{H}}(A,B)=\sup\limits_{a\in A}\inf\limits_{b\in B}d(a,b)$

and the symmetric Hausdorff distance is given by

${\rm dist_{H}}(A,B)=\max\\{{\rm d_{H}}(A,B),{\rm d_{H}}(B,A)\\}.$

######  Definition A.1.

A family $\\{\mathbb{A}(t)\colon t\in\mathbb{R}\\}$ of compact subsets of $Z$
is a pullback attractor for an evolution process $\\{S(t,\tau)\colon
t\geq\tau\in\mathbb{R}\\}$ if the following conditions hold:

  1. $(i)$

$\\{\mathbb{A}(t)\colon t\in\mathbb{R}\\}$ is invariant, that is,
$S(t,\tau)\mathbb{A}(\tau)=\mathbb{A}(t)$ for all $t\geq\tau$,

  2. $(ii)$

$\\{\mathbb{A}(t)\colon t\in\mathbb{R}\\}$ pullback attracts bounded subsets
of $Z$, that is,

$\lim\limits_{\tau\to-\infty}{\rm d_{H}}(S(t,\tau)B,\mathbb{A}(t))=0$

for every $t\in\mathbb{R}$ and every bounded subset $B$ of $Z$, and

  3. $(iii)$

$\\{\mathbb{A}(t)\colon t\in\mathbb{R}\\}$ is the minimal family of closed
sets satisfying property $(ii)$.

A general result concerning the existence of pullback attractors states that,
if an evolution process is pullback strongly bounded dissipative and pullback
asymptotically compact, then it has a pullback attractor (see [9, Theorem
2.23]). The theory of pullback attractors for nonlinear evolution processes is
covered with details in [6], [9] and [11].

######  Definition A.2.

A global solution for an evolution process $\\{S(t,\tau)\colon
t\geq\tau\in\mathbb{R}\\}$ in $Z$ is a function $\xi\colon\mathbb{R}\to Z$
such that $S(t,\tau)\xi(\tau)=\xi(t)$ for all $t\geq\tau$.

######  Definition A.3.

An equilibrium point $\xi^{\ast}\in Z$ for an evolution process
$\\{S(t,\tau)\colon t\geq\tau\in\mathbb{R}\\}$ in $Z$ is a time independent
global solution, that is, $S(t,\tau)\xi^{\ast}=\xi^{\ast}$ for all
$t\geq\tau$.

######  Definition A.4.

Let $\\{A_{\lambda}\\}_{\lambda\in\Lambda}$ be a family of subsets of $Z$
indexed on a metric space $\Lambda$. The family
$\\{A_{\lambda}\\}_{\lambda\in\Lambda}$ is said to be:

  1. $(i)$

upper semicontinuous as $\lambda\rightarrow\lambda_{0}$ (or at $\lambda_{0}$)
if $\lim\limits_{\lambda\rightarrow\lambda_{0}}{\rm
d_{H}}(A_{\lambda},A_{\lambda_{0}})=0$;

  2. $(ii)$

lower semicontinuous as $\lambda\rightarrow\lambda_{0}$ (or at $\lambda_{0}$)
if $\lim\limits_{\lambda\rightarrow\lambda_{0}}{\rm
d_{H}}(A_{\lambda_{0}},A_{\lambda})=0$;

  3. $(iii)$

continuous as $\lambda\rightarrow\lambda_{0}$ (or at $\lambda_{0}$) if it is
both upper and lower semicontinuous as $\lambda\rightarrow\lambda_{0}$.

In the next two definitions, we consider a linear evolution process
$\\{L(t,\tau)\colon t\geq\tau\in\mathbb{R}\\}$ acting on a Banach space $X$.

######  Definition A.5.

A linear evolution process $\\{L(t,\tau)\colon t\geq\tau\in\mathbb{R}\\}$ in a
Banach space $X$ admits an exponential dichotomy with projection
$\\{P(t)\colon t\in\mathbb{R}\\}\subset\mathcal{L}(X)$, exponent $\omega>0$
and constant $M\geq 1$, if:

  1. $(i)$

$P(t)L(t,\tau)=L(t,\tau)P(\tau)$ for all $t\geq\tau$;

  2. $(ii)$

the restriction of $L(t,\tau)$ to $R(P(\tau))$ is an isomorphism from
$R(P(\tau))$ onto $R(P(t))$, where $R(P(t))$ denotes the range of the operator
$P(t)$; and

  3. $(iii)$

$\|L(t,\tau)(I-P(\tau))\|_{\mathcal{L}(X)}\leq Me^{-\omega(t-\tau)},\quad
t\geq\tau,$

$\|L(t,\tau)P(\tau)\|_{\mathcal{L}(X)}\leq Me^{\omega(t-\tau)},\quad
t\leq\tau$.

######  Remark A.6.

In Definition A.5, item $(ii)$, we denote the inverse of $L(t,\tau)$ by
$L(\tau,t)$.

######  Definition A.7.

A global solution $\xi\colon\mathbb{R}\to X$ for a linear evolution process
$\\{L(t,\tau)\colon t\geq\tau\in\mathbb{R}\\}$ in a Banach space $X$ is said
to be hyperbolic, if $\\{L(t,\tau)\colon t\geq\tau\in\mathbb{R}\\}$ has an
exponential dichotomy in the sense of Definition A.5.

##  Appendix A Appendix

###  A.1. An abstract result on lower semicontinuity of pullback attractors

Let $X$ be a Banach space, $\mathcal{B}\colon D(\mathcal{B})\subset X\to X$ be
the generator of a strongly continuous semigroup $\\{e^{\mathcal{B}t}\colon
t\geq 0\\}\subset\mathcal{L}(X),$ $y_{0}\in X$ and consider the following
abstract semilinear problem

(A.1) $\begin{cases}y_{t}=\mathcal{B}y+f_{0}(t,y),&t>\tau,\\\
y(\tau)=y_{0},&\tau\in\mathbb{R},\end{cases}$

and its associated regular perturbation, given by

(A.2)
$\begin{cases}y_{t}=\mathcal{B}y+f_{\epsilon}(t,y),&t>\tau,\quad\epsilon\in[0,1],\\\
y(\tau)=y_{0},&\tau\in\mathbb{R}.\end{cases}$

If we assume that, for each $\epsilon\in[0,1],$ the nonlinearity
$f_{\epsilon}\colon\mathbb{R}\times X\to X$ is continuous and Lipschitz
continuous in the second variable, uniformly on bounded subsets of $X$, then
the problems (A.1) and (A.2) are locally well-posed. Assuming that, for each
$\tau\in\mathbb{R}$ and $y_{0}\in X$, the solution $t\mapsto
S_{\epsilon}(t,\tau)y_{0}$ of (A.2) is defined in $[\tau,\infty),$ we write

$S_{\epsilon}(t,\tau)y_{0}=e^{\mathcal{B}(t-\tau)}y_{0}+\int_{\tau}^{t}e^{\mathcal{B}(t-s)}f_{\epsilon}(s,S_{\epsilon}(s,\tau)y_{0})\,ds.$

The family $\\{S_{\epsilon}(t,\tau)\colon
t\geq\tau\in\mathbb{R}\\},\epsilon\in[0,1],$ consists of nonlinear evolution
processes.

If $\xi_{\epsilon}^{\ast}\colon\mathbb{R}\to X,\epsilon\in[0,1],$ is a global
solution of (A.2), then we consider the linearization of (A.2) around
$\xi_{\epsilon}^{\ast}$ given by

(A.3)
$\begin{cases}z_{t}=\mathcal{B}z+(f_{\epsilon})_{z}(t,\xi_{\epsilon}^{\ast}(t))z,\\\
z(\tau)=z_{0},\end{cases}$

and the evolution process $\\{U_{\epsilon}(t,\tau)\colon
t\geq\tau\in\mathbb{R}\\},\epsilon\in[0,1],$ given by

$U_{\epsilon}(t,\tau)z_{0}=e^{\mathcal{B}(t-\tau)}z_{0}+\int_{\tau}^{t}e^{\mathcal{B}(t-s)}(f_{\epsilon})_{z}(s,\xi_{\epsilon}^{\ast}(s))U_{\epsilon}(s,\tau)z_{0}\,ds.$

######  Definition A.1.

The linearized problem (A.3) admits an exponential dichotomy with exponent
$\omega>0$ and constant $M\geq 1$, if there exists a family of projections
$\\{\mathcal{Q}_{\epsilon}(t)\colon t\in\mathbb{R}\\}\subset\mathcal{L}(X)$
such that:

  1. $(i)$

$\mathcal{Q}_{\epsilon}(t)U_{\epsilon}(t,\tau)=U_{\epsilon}(t,\tau)\mathcal{Q}_{\epsilon}(\tau)$
for all $t\geq\tau$;

  2. $(ii)$

$U_{\epsilon}(t,\tau)\colon R(\mathcal{Q}_{\epsilon}(\tau))\to
R(\mathcal{Q}_{\epsilon}(t))$ is an isomorphism, with inverse

$U_{\epsilon}(\tau,t)\colon R(\mathcal{Q}_{\epsilon}(t))\to
R(\mathcal{Q}_{\epsilon}(\tau))\quad\mbox{for all}\quad t\geq\tau;$

  3. $(iii)$

$\|U_{\epsilon}(t,\tau)(I-\mathcal{Q}_{\epsilon}(\tau))\|_{\mathcal{L}(X)}\leq
Me^{-\omega(t-\tau)},\quad\mbox{for}\quad t\geq\tau,$

$\|U_{\epsilon}(t,\tau)\mathcal{Q}_{\epsilon}(\tau)\|_{\mathcal{L}(X)}\leq
Me^{\omega(t-\tau)},\quad\mbox{for}\quad t\leq\tau.$

######  Definition A.2.

A global solution $\xi_{\epsilon}^{\ast}\colon\mathbb{R}\to
X,\epsilon\in[0,1],$ of (A.2) is said to be hyperbolic if the evolution
process $\\{U_{\epsilon}(t,\tau)\colon t\geq\tau\in\mathbb{R}\\}$ associated
to (A.3) has an exponential dichotomy in the sense of Definition A.1.

The following result concerns the lower semicontinuity of a family of pullback
attractors and it was stated in [5, Theorem 6.2]. We also refer to [8, Theorem
3.1] for a more general result.

######  Theorem A.3.

Let $\\{S_{\epsilon}(t,\tau)\colon
t\geq\tau\in\mathbb{R}\\},\epsilon\in[0,1],$ be the family of nonlinear
evolution processes associated to (A.2) in $X$ and assume that, for
$[\tau,t]\subset\mathbb{R},$

$\|S_{\epsilon}(t,\tau)x-S_{0}(t,\tau)x\|_{X}\rightarrow
0\quad\mbox{as}\quad\epsilon\rightarrow 0^{+},$

for any $x$ in a compact subset of $X$. Suppose that, for each
$\epsilon\in[0,1],$ the process $\\{S_{\epsilon}(t,\tau)\colon
t\geq\tau\in\mathbb{R}\\}$ has a pullback attractor
$\\{\mathbb{A}_{\epsilon}(t)\colon t\in\mathbb{R}\\}$ in $X$ such that, for
$\epsilon_{0}>0$ sufficiently small,

$\overline{\bigcup\limits_{t\in\mathbb{R}}\bigcup\limits_{\epsilon\in[0,\epsilon_{0}]}\mathbb{A}_{\epsilon}(t)}\mbox{
is compact in }X.$

Moreover, assume that there exists a finite collection
$\mathcal{E}^{\ast}=\\{e_{1}^{\ast},\ldots,e_{r}^{\ast}\\}$ of equilibrium
points, all of them hyperbolic for the limit problem (A.1), such that the
limit pullback attractor $\\{\mathbb{A}_{0}(t)\colon t\in\mathbb{R}\\}$ is the
union of their unstable manifolds, that is,

$\mathbb{A}_{0}(t)=\bigcup_{i=1}^{r}W^{u}(e_{i}^{\ast})(t)\quad\mbox{for
all}\quad t\in\mathbb{R},$

where $W^{u}(e_{i}^{\ast})(t)=\\{\zeta\in X:\text{ there is a global solution
}\xi\colon\mathbb{R}\to X\text{ such that }\xi(t)=\zeta\text{ and
}\displaystyle\lim_{\tau\to-\infty}dist(\xi(\tau),e_{i}^{\ast})=0\\}$.
Further, assume that for each $e_{j}^{\ast}\in\mathcal{E}^{\ast}$:

  1. $\bullet$

given $\delta>0$, there exists $\epsilon_{j,\delta}>0$ such that, for all
$0<\epsilon<\epsilon_{j,\delta}$ there is a global hyperbolic solution
$\xi_{j,\epsilon}$ of (A.2) that satisfies

$\sup\limits_{t\in\mathbb{R}}\|\xi_{j,\epsilon}(t)-e_{j}^{\ast}\|_{X}<\delta,$

  2. $\bullet$

the local unstable manifold of $\xi_{j,\epsilon}$ behaves continuously as
$\epsilon\rightarrow 0^{+}$, that is,

${\rm dist_{H}}\left(W_{0,{\rm loc}}^{u}(e_{j}^{\ast}),W_{\epsilon,{\rm
loc}}^{u}(\xi_{j,\epsilon})\right)\rightarrow
0\quad\mbox{as}\quad\epsilon\rightarrow 0^{+},$

where ${\rm dist_{H}}(\cdot,\cdot)$ is the symmetric Hausdorff distance and
$W_{{\rm loc}}^{u}(\cdot)=W^{u}(\cdot)\cap B_{X}(\cdot,\rho),$ with $\rho>0$.

Then the family $\\{\mathbb{A}_{\epsilon}(t)\colon
t\in\mathbb{R},\epsilon\in[0,\epsilon_{0}]\\}$ is lower semicontinuous at
$\epsilon=0$, that is,

${\rm d_{H}}(\mathbb{A}_{0}(t),\mathbb{A}_{\epsilon}(t))\rightarrow
0\quad\mbox{as}\quad\epsilon\rightarrow 0^{+}.$

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