\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 47, pp. 1--35.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/47\hfil Boundary conditions with memory]
{Robust exponential attractors for Coleman-Gurtin equations
with dynamic boundary conditions possessing memory}

\author[J. L. Shomberg \hfil EJDE-2016/47\hfilneg]
{Joseph L. Shomberg}

\address{Joseph L. Shomberg \newline
Department of Mathematics and Computer Science,
Providence College, Providence, RI 02918, USA}
\email{jshomber@providence.edu}


\thanks{Submitted August 15, 2015. Published February 10, 2016.}
\subjclass[2010]{35B40, 35B41, 45K05, 35Q79}
\keywords{Coleman-Gurtin equation; dynamic boundary conditions;
\hfill\break\indent  memory relaxation; exponential attractor; basin of attraction;
global attractor;
\hfill\break\indent  finite dimensional dynamics; robustness}

\begin{abstract}
 Well-posedness of generalized Coleman-Gurtin equations equipped with
 dynamic boundary conditions with memory was recently established by the
 author with C. G. Gal. In this article we report advances concerning the 
 asymptotic behavior and stability of this heat transfer model.
 For the model under consideration, we obtain a family of exponential
 attractors that is robust/H\"{o}lder continuous with respect to a perturbation
 parameter occurring in a singularly perturbed memory kernel.
 We show that the basin of attraction of these exponential attractors
 is the entire phase space. The existence of (finite dimensional) global
 attractors follows. The results are obtained by assuming the nonlinear terms
 defined on the interior of the domain and on the boundary satisfy standard
 dissipation assumptions. Also, we work under a crucial assumption that
 dictates the memory response in the interior of the domain matches that
 on the boundary.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{problem}[theorem]{Problem}
\allowdisplaybreaks

\section{Introduction to the model problem}


In the framework of \cite{GPM98}, let us only consider a thermodynamic
process based on heat conduction.
Suppose that a bounded domain $\Omega
\subset \mathbb{R}^{n}$, $n\geq 1$, is occupied by a body which may be
inhomogeneous, but has a configuration constant in time.
Thermodynamic processes taking place inside $\Omega $,
with sources also present at the
boundary $\Gamma $, give rise to the following model for the temperature
field $u$:
\begin{equation} \label{eq1m}
\begin{aligned}
& \partial_{t}u - \omega\Delta u - (1-\omega)
\int_0^{\infty} k(s) \Delta u(x,t-s) \,\mathrm{d}s + f(u)   \\
& + \alpha(1-\omega) \int_0^\infty k(s)u(x,t-s) \,\mathrm{d}s =0,
\end{aligned}
\end{equation}
in $\Omega \times ( 0,\infty )$, subject to the  boundary condition
\begin{equation} \label{eq2m}
\begin{aligned}
& \partial_{t}u - \omega\Delta _{\Gamma }u + \omega\partial_{\mathbf{n}}u
 + (1-\omega) \int_0^\infty k(s)\partial_{\mathbf{n}} u(x,t-s)\,\mathrm{d}s  \\
& + (1-\omega) \int_0^\infty k(s)(-\Delta_\Gamma+\beta)u (x,t-s)\,\mathrm{d}s
+ g(u) = 0,
\end{aligned}
\end{equation}
on $\Gamma \times (0,\infty)$, for every $\alpha\ge0$, $\beta\ge0$,
$\omega \in [ 0,1)$, and where $k:[0,\infty )\to \mathbb{R}$
is a continuous nonnegative function, smooth on $(0,\infty)$,
vanishing at infinity and satisfying the relation
\begin{equation*}
\int_0^{\infty} k(s) \,\mathrm{d}s = 1,
\end{equation*}
$\partial_{\mathbf{n}}$ represents the normal derivative and
$-\Delta_\Gamma$ is the Laplace-Beltrami operator.
The cases $\omega =0$ and $\omega >0$ in \eqref{eq1m} are usually referred
as the Gurtin-Pipkin and the Coleman-Gurtin models, respectively.
The literature contains a full treatment of equation \eqref{eq1m} only in the
case of standard boundary conditions (Dirichlet, Neumann and periodic boundary
conditions).
In light of new results and extensions for the phase field equations
(see, e.g., \cite{CGGM10, GGM08} and references therein), we must consider
more general dynamic boundary conditions.
In particular, we quote \cite{GMS2010}:
\begin{quote}
In most works, the equations are endowed with Neumann boundary conditions
for both [unknowns] $u$ and $w$ (which means that the interface is orthogonal
to the boundary and that there is no mass flux at the boundary) or with periodic
boundary conditions.
Now, recently, physicists have introduced the so-called dynamic boundary conditions,
in the sense that the kinetics, i.e., $\partial_t u$, appears explicitly in the
boundary conditions, in order to account for the interaction of the components
with the walls for a confined system.
\end{quote}
The derivation of \eqref{eq2m} in the context of \eqref{eq1m} can be derived
in a similar fashion as in \cite{Gal&Grasselli08, Gold06} exploiting first
and second laws of thermodynamics.
Let $\omega \in [ 0,1)$ be fixed. It is clear that if we (formally) choose
$k=\delta _0$ (the Dirac mass at zero), equations \eqref{eq1m}-\eqref{eq2m}
turn into the  system
\begin{gather}
\partial_{t}u - \Delta u + f(u) + \alpha(1-\omega)u = 0, \quad
\text{in } \Omega \times (0,\infty),  \label{eq3d} \\
\partial_{t}u - \Delta_{\Gamma}u + \partial_{\mathbf{n}}u + g(u)
+ \beta(1-\omega) u = 0, \quad \text{on}\ \Gamma \times ( 0,\infty ).
\label{eq4d}
\end{gather}
The latter has been investigated quite extensively recently in many contexts
(i.e., phase-field systems, heat conduction with a source at $\Gamma $,
Stefan problems, etc).

Now we define, for $\varepsilon \in (0,1]$,
\begin{equation*}
k_{\varepsilon}(s) = \frac{1}{\varepsilon}k(\frac{s}{\varepsilon}),
\end{equation*}
and we consider the same family of equations \eqref{eq1m}-\eqref{eq2m},
replacing $k$ with $k_{\varepsilon }$.
Thus, $k_{\varepsilon }\to\delta _0$ when $\varepsilon \to 0$.
Our goal is to show in what sense does the system \eqref{eq1m}-\eqref{eq2m}
converge to \eqref{eq3d}-\eqref{eq4d} as $\varepsilon \to 0$.

Such results seem to have begun with the hyperbolic relaxation of
a Chaffee-Infante reaction diffusion equation in \cite{Hale&Raugel88}.
The motivation for such a hyperbolic relaxation is similar to the motivation
for applying a memory relaxation; it alleviates the parabolic problems
from the sometimes unwanted property of ``infinite speed of propagation''.
In \cite{Hale&Raugel88} however, Hale and Raugel proved the existence of a
family of global attractors that is upper-semicontinuous in the phase space.
A global attractor is a unique compact invariant subset of the phase space
that attracts all trajectories of the associated dynamical system, even at
arbitrarily slow rates (cf. \cite{Kostin98} and \cite[Theorem 14.6]{Robinson01}).
In a sense which will become clearer below, upper-semicontinuity guarantees
the attractors to not ``blow-up'' as the perturbation parameter vanishes; i.e.,
\[
\sup_{x\in A_\varepsilon}\inf_{y\in A_0}\|x-y\|_{X_\varepsilon}\to 0 \quad\text{as } \varepsilon\to 0^+.
\]

Unlike global attractors, exponential attractors (sometimes called, inertial sets)
are compact positively invariant sets possessing finite fractal dimension that
attract bounded subsets of the phase space exponentially fast.
It can readily be seen that when both a global attractor $\mathcal{A}$ and an
exponential attractor $\mathfrak{M}$ exist, then $\mathcal{A}\subseteq \mathfrak{M}$
 provided that the basin of attraction of $\mathfrak{M}$ is the whole phase space,
and so the global attractor is also finite dimensional.
When we turn our attention to proving the existence of exponential attractors,
certain higher-order dissipative estimates are required.
In some interesting cases, it has not yet been shown how to obtain the appropriate
estimates (which would provide the existence of a compact absorbing set,
for example) {\em{independent}} of the perturbation parameter
(cf. e.g. \cite{Frigeri&ShombergXX,Gal&Shomberg15}).
It is precisely because we are able to provide a higher-order uniform bound
for the model problems here that we do not give a separate upper-semicontinuity
result for the global attractors.
An appropriate uniform higher-order bound will essentially/almost mean that
a robustness result may be found (but it is not guaranteed).

Robust families of exponential attractors (that is, both upper- and
lower-semi\-continuous with explicit control over semidistances in terms of
the perturbation parameter) of the type reported in \cite{GGMP05} have
successfully been shown to exist in many different applications,
of which we will limit ourselves to mention only \cite{GMPZ10} which contains
some applications of memory relaxation of reaction diffusion equations:
Cahn-Hilliard equations, phase-field equations, wave equations, beam equations,
and numerous others.
The main idea behind robustness is typically an estimate of the form
\begin{equation}  \label{robust-intro}
\|S_\varepsilon(t)x-\mathcal{L}S_0(t)\Pi x\|_{X_\varepsilon}\leq C\varepsilon^p,
\end{equation}
for all $t$ in some interval, where $x\in X_\varepsilon$,
$S_\varepsilon(t):X_\varepsilon\to X_\varepsilon$ and $S_0(t):X_0\to X_0$ are semigroups
generated by the solutions of the perturbed problem and the limit problem,
respectively, $\Pi$ denotes a projection from $X_\varepsilon$ onto $X_0$ and
$\mathcal{L}$ is a ``lift'' from $X_0$ into $X_\varepsilon$, and finally
$C,p>0$ are constants.
Controlling this difference in a suitable norm is crucial to obtaining our
continuity results (see (C5) in Proposition \ref{abstract2}).
The estimate \eqref{robust-intro} means we can approximate the limit problem
 with the perturbation with control explicitly written in terms of the perturbation
 parameter.
Usually such control is only exhibited on compact time intervals.
Observe, a result of this type will ensure that for every problem of
type \eqref{eq3d}-\eqref{eq4d}, there is an ``memory relaxation'' of
the form \eqref{eq1m}-\eqref{eq2m} {\em{close by}} in the sense that
the difference of corresponding trajectories satisfies \eqref{robust-intro}.

We carefully treat the following issues:

\begin{itemize}
\item[(1)] Well-posedness of the system comprising of equations
\eqref{eq1m}-\eqref{eq2m} and \eqref{eq3d}-\eqref{eq4d}.

\item[(2)] Dissipation: the existence of bounded absorbing set,
and a {\em{compact}} absorbing set, each of which is uniform with respect
to the perturbation parameter $\varepsilon$.

\item[(3)] Stability: existence of a family of exponential attractors for
each $\varepsilon\in[0,1]$ and an analysis of the continuity properties
(robustness/H\"{o}lder) with respect to $\varepsilon$.

\item[(4)] The basin of attraction for each exponential attractor is the
 entire phase space, and in demonstrating this result we see that the
semigroup of solution operators also admits a family of global attractors.
\end{itemize}

Concerning Issue 1, the well-posedness for a more general system, which includes
the one above, was given recently by \cite{Gal-Shomberg15-2}.
The relevant results from that work are cited below in Section 2.
In this article we explore Issues 2, 3, and 4 in much more depth; in particular,
the existence of an exponential attractor for each $\varepsilon\in[0,1]$,
and the continuity of these attractors with respect to $\varepsilon$.

As is now customary (cf. \cite{CDGP-2010,CPS05,CPS06,Grasselli&Pata02-2})
we introduce the so-called integrated past history of $u$, i.e., the auxiliary
variable
\begin{equation*}
\eta ^{t}(x,s) =\int_0^{s} u(x,t-y) \,\mathrm{d}y,
\end{equation*}
for $s,t>0$.
Setting
\begin{equation*}
\mu(s) = -(1-\omega)k'(s),
\end{equation*}
formal integration by parts into \eqref{eq1m}-\eqref{eq2m} yields
\begin{gather*}
(1-\omega) \int_0^{\infty} k_{\varepsilon}(s) \Delta u(x,t-s) \,\mathrm{d}s
 =\int_0^{\infty}\mu_{\varepsilon}(s) \Delta \eta^{t}(x,s) \,\mathrm{d}s,  \\
(1-\omega) \int_0^{\infty} k_{\varepsilon}(s) u(x,t-s) \,\mathrm{d}s
 = \int_0^{\infty} \mu_{\varepsilon}(s) \eta^{t}(x,s) \,\mathrm{d}s,  \\
(1-\omega) \int_0^{\infty } k_{\varepsilon}(s) \partial_{\mathbf{n}} u(x,t-s)
\,\mathrm{d}s  = \int_0^{\infty} \mu_{\varepsilon}(s)
\partial_{\mathbf{n}} \eta^{t}(x,s) \,\mathrm{d}s, \\
(1-\omega) \int_0^{\infty} k_{\varepsilon}(s)(-\Delta_{\Gamma} 
+ \beta) u(x,t-s) \,\mathrm{d}s 
= \int_0^{\infty} \mu_{\varepsilon}(s) (-\Delta_{\Gamma} 
+ \beta) \eta^t(x,s) \,\mathrm{d}s,
\end{gather*}
where
\begin{equation}  \label{mu-scaled}
\mu_{\varepsilon }(s) = \frac{1}{\varepsilon^2}\mu( \frac{s}{\varepsilon}).
\end{equation}

For each $\varepsilon\in(0,1]$, the (perturbation) problem under consideration
can now be stated.

\begin{problem} \label{Pe} \rm
Let $\alpha,\beta\ge0$, and $\omega\in(0,1)$.
Find a function $(u,\eta)$ such that
\begin{equation}
\partial_tu-\omega\Delta u-\int_0^\infty\mu_\varepsilon(s)\Delta\eta^t(s)
\,\mathrm{d}s+\alpha\int_0^\infty\mu_\varepsilon(s)\eta^t(s)\,\mathrm{d}s+f(u)=0
 \label{problemp-1}
\end{equation}
in $\Omega\times(0,\infty)$, subject to the boundary conditions
\begin{equation}
\begin{aligned}
&\partial_tu-\omega\Delta_\Gamma u+\omega\partial_{\mathbf{n}}u
+\int_0^\infty\mu_\varepsilon(s)\partial_{\mathbf{n}}\eta^t(s)\,\mathrm{d}s \\
&+\int_0^\infty\mu_\varepsilon(s)( -\Delta_\Gamma+\beta )\eta^t(s)\,\mathrm{d}s+g(u)=0
\end{aligned} \label{problemp-2}
\end{equation}
on $\Gamma\times(0,\infty)$, and
\begin{equation}
\partial_t\eta^t(s)+\partial_s\eta^t(s)=u(t)\quad \text{in } {\overline{\Omega}}\times(0,\infty),  \label{problemp-3}
\end{equation}
with
\begin{equation}
\eta^t(0)=0\quad \text{in } {\overline{\Omega}}\times(0,\infty),  \label{problemp-4}
\end{equation}
and the initial conditions
\begin{gather}
u(0)=u_0\quad \text{in } \Omega, \quad u(0)=v_0\quad \text{on } \Gamma,
 \label{problemp-5} \\
\eta^0(s)=\eta_0:=\int_0^su_0(x,-y)\,\mathrm{d}y \quad\text{in } \Omega,
\text{ for } s>0,  \label{problemp-6}
\\
\eta^0(s)=\xi_0:=\int_0^sv_0(x,-y)\,\mathrm{d}y \quad\text{on } \Gamma,
 \text{ for } s>0.  \label{problemp-7}
\end{gather}
\end{problem}

We will also discuss the problem corresponding to $\varepsilon=0$.
The results for this problem may already be found in works in parabolic
equations and the Wentzell Laplacian
(see \cite{Gal12-2,Gal12-1,Gal-15Z,Gal&Warma10}).
The singular (limit) problem is

\begin{problem} \label{P0} \rm
Let $\alpha,\beta\ge0$ and $\omega\in(0,1)$.
Find a function $u$ such that
\begin{equation}
\partial_{t}u - \Delta u + f(u) + \alpha(1-\omega)u = 0  \label{problem0-1}
\end{equation}
in $\Omega\times(0,\infty)$, subject to the boundary conditions
\begin{equation}
\partial_{t}u - \Delta_{\Gamma}u + \partial_{\mathbf{n}}u + g(u) + \beta(1-\omega) u = 0  \label{problem0-2}
\end{equation}
on $\Gamma\times(0,\infty)$, with the initial conditions
\begin{equation}
u(0)=u_0\quad \text{in } \Omega \quad\text{and}\quad u(0)=v_0\quad \text{on }\Gamma.  \label{problem0-5}
\end{equation}
\end{problem}

\begin{remark}  \label{on-traces} \rm
It need not be the case that the boundary traces of $u_0$ and $\eta_0$
be equal to $v_0$ and $\xi_0$, respectively.
Thus, we are solving a much more general problem in which equation
\eqref{problemp-1} is interpreted as an evolution equation in the bulk
$\Omega$ properly coupled with the equation \eqref{problemp-2} on the boundary
$\Gamma$. Finally, from now on both $\eta_0$ and $\xi_0$ will be regarded
as independent of the initial data $u_0$ and $v_0$.
Indeed, below we will consider a more general problem with respect to the
original one. This will require a rigorous notion of solution to
Problem \eqref{Pe} (cf. Definitions \ref{d:weak-solution},
\ref{d:strong-solution}), hence we introduce the functional setting
associated with this system.
\end{remark}

Here below is the framework used to prove Hadamard well-posedness for
Problem \eqref{Pe}.
Consider the space $\mathbb{X}^2:=L^2( \overline{\Omega }, \mathrm{d}\mu ) $,
where
\begin{equation*}
\mathrm{d}\mu = \mathrm{d}x|_{\Omega }\oplus \mathrm{d}\sigma ,
\end{equation*}
where $\,\mathrm{d}x$ denotes the Lebesgue measure on $\Omega $
and $\mathrm{d}\sigma $ denotes the natural surface measure on $\Gamma $. It is
easy to see that $\mathbb{X}^2=L^2( \Omega , \mathrm{d}x)
\oplus L^2( \Gamma , \mathrm{d}\sigma ) $ may be identified
under the natural norm
\begin{equation*}
\| u\| _{\mathbb{X}^2}^2=\int_{\Omega}| u|^2 \,\mathrm{d}x
+ \int_{\Gamma }| u|^2 \mathrm{d}\sigma.
\end{equation*}
Moreover, if we identify every $u\in C(\overline{\Omega})$ with
$U=( u|_{\Omega },u|_{\Gamma }) \in C( \Omega
) \times C( \Gamma ) $, we may also define $\mathbb{X}^2$
to be the completion of $C(\overline{\Omega})$ in the norm
$\|\cdot\|_{\mathbb{X}^2}$.
In general, any function $u\in \mathbb{X}^2$ will be of the form
$u=\binom{u_{1}}{u_{2}}$ with $u_{1}\in L^2( \Omega , \,\mathrm{d}x) $
and $u_{2}\in L^2( \Gamma , \mathrm{d}\sigma )$, and there need not be
any connection between $u_{1}$ and $u_{2}$.
From now on, the inner product in the Hilbert space $\mathbb{X}^2$ will
be denoted by $\langle \cdot,\cdot \rangle _{\mathbb{X}^2}$.
Hereafter, the spaces $L^2(\Omega , \,\mathrm{d}x) $ and
$L^2(\Gamma, \mathrm{d}\sigma )$ will simply be denoted by $L^2(\Omega) $
and $L^2( \Gamma)$.

Recall that the Dirichlet trace map
${\mathrm{tr_{D}}}:C^{\infty }(\overline{\Omega}) \to C^{\infty}(\Gamma)$,
defined by ${\mathrm{tr_{D}}}( u) =u|_{\Gamma }$ extends to a linear continuous
operator ${\mathrm{tr_{D}}}:H^{r}( \Omega )
\to H^{r-1/2}( \Gamma ) $, for all $r>1/2$, which is onto
for $1/2<r<3/2$. This map also possesses a bounded right inverse
${\mathrm{tr_{D}}}^{-1}:H^{r-1/2}(\Gamma) \to H^{r}(\Omega)$ such that
${\mathrm{tr_{D}}}({\mathrm{tr_{D}}}^{-1}\psi) =\psi $, for any
$\psi \in H^{r-1/2}(\Gamma) $.
We can thus introduce the subspaces of $H^{r}(\Omega) \times H^{r}(\Gamma)$,
\begin{equation}
\mathbb{V}^{r}:=\{( u,\psi ) \in H^{r}( \Omega )
\times H^{r}( \Gamma ) :{\mathrm{tr_{D}}}( u) =\psi \},
\label{vvv}
\end{equation}
for every $r>1/2$, and note that we have the following dense and compact
embeddings $\mathbb{V}^{r_{1}}\hookrightarrow \mathbb{V}^{r_{2}}$,
for any $r_{1}>r_{2}>1/2$. Finally, we think of
$\mathbb{V}^{1}\simeq H^{1}(\Omega ) \oplus H^{1}( \Gamma ) $
as the completion of $C^{1}( \overline{\Omega }) $ in the norm
\begin{equation}
\| u\| _{\mathbb{V}^{1}}^2:=\int_{\Omega } (
| \nabla u| ^2 + \alpha| u| ^2 ) \,\mathrm{d}x
+ \int_{\Gamma }(| \nabla _{\Gamma }u|^2+\beta| u|^2) \mathrm{d}\sigma  \label{v1b}
\end{equation}
(or some other equivalent norm in $H^{1}( \Omega ) \times H^{1}( \Gamma ) $).
Naturally, the norm on the space $\mathbb{V}^r$ is defined as
\begin{equation}\label{Vr-norm}
\|u\|^2_{\mathbb{V}^r} := \|u\|^2_{H^r(\Omega)} + \|u\|^2_{H^r(\Gamma)}.
\end{equation}

For $U=(u,u|_{\Gamma})^{\mathrm{tr}}\in\mathbb{V}^1$, let $C_\Omega>0$
denote the best constant in which the Sobolev-Poincar\'e inequality holds
\begin{equation}  \label{Poincare}
\| u-\langle u \rangle_\Gamma \| _{L^{s}( \Omega ) }
\leq C_\Omega\| \nabla u\| _{L^{s}(\Omega)},
\end{equation}
for $s\geq 1$ (see \cite[Lemma 3.1]{RBT01}).
Here
$$
\langle u \rangle_{\Gamma}:=\frac{1}{| \Gamma | }
\int_{\Gamma}u|_{\Gamma} \mathrm{d}\sigma.
$$

Let us now introduce the spaces for the memory variable $\eta $. For a
nonnegative measurable function $\theta $ defined on $\mathbb{R}_{+}$ and a
real Hilbert space $W$ (with inner product denoted by
$\langle \cdot ,\cdot \rangle_W$), let
$L_{\theta }^2( \mathbb{R}_{+};W) $ be the Hilbert space of $W$-valued
functions on $\mathbb{R}_{+}$, endowed with the following inner
product
\begin{equation}
\langle \phi _{1},\phi _{2}\rangle _{L_{\theta}^2(
\mathbb{R}_{+};W) }:=\int_0^{\infty }\theta(s)\langle \phi
_{1}( s) ,\phi _{2}( s) \rangle _W \,\mathrm{d}s.
\end{equation}  \label{sc-2}
Consequently, for $r>1/2$ we set
\begin{equation*}
\mathcal{M}^r_{\varepsilon}:= \begin{cases}
L_{\mu_{\varepsilon}}^2( \mathbb{R}_{+};\mathbb{V}^{r}) & \text{for }\varepsilon\in(0,1], \\
\{0\} & \text{when } \varepsilon=0, \end{cases}
\end{equation*}
and when $r=0$ set
\begin{equation*}
\mathcal{M}^0_{\varepsilon}:=
\begin{cases}
L_{\mu_{\varepsilon}}^2( \mathbb{R}_{+};\mathbb{X}^2) & \text{for }
 \varepsilon\in(0,1], \\
\{0\} & \text{when } \varepsilon=0.
\end{cases}
\end{equation*}
One can see from \cite[Lemma 5.1]{GMPZ10} that for $\varepsilon_1\ge\varepsilon_2>0$ and
for fixed $r=0$ or $r>1/2$, there holds the continuous embedding
$\mathcal{M}^{r}_{\varepsilon_1}\hookrightarrow\mathcal{M}^{r}_{\varepsilon_2}$.
As a matter of convenience, the inner-product in $\mathcal{M}^1_{\varepsilon}$
is given by
\begin{equation} \label{sc}
\begin{aligned}
& \Big\langle \binom{\eta_1}{\xi_1},\binom{\eta_2}{\xi_2}
\Big\rangle_{\mathcal{M}^1_{\varepsilon}}   \\
&=\int_0^\infty \mu_{\varepsilon}(s)( \langle \nabla\eta_1(s),\nabla\eta_2(s) \rangle_{L^2(\Omega)}+\alpha\langle \eta_1(s),\eta(s) \rangle_{L^2(\Omega)} ) \,\mathrm{d}s  \\
& + \int_0^\infty \mu_{\varepsilon}(s)( \langle \nabla_\Gamma\xi_1(s),\nabla_\Gamma\xi_2(s) \rangle_{L^2(\Gamma)} + \beta\langle \xi_1(s),\xi_2(s) \rangle_{L^2(\Gamma)} ) \,\mathrm{d}s.
\end{aligned}
\end{equation}
When it is convenient, we will use the notation
\begin{align}
\mathcal{H}^0_{\varepsilon}  := \mathbb{X}^2\times\mathcal{M}^1_{\varepsilon}  \\
\mathcal{H}^1_{\varepsilon}  := \mathbb{V}^1\times\mathcal{M}^2_{\varepsilon}.
\end{align}
Each space is equipped with the corresponding ``graph norm,'' whose square
is defined by, for all $\varepsilon\in[0,1]$ and $(U,\Phi)\in\mathcal{H}^i_\varepsilon$, $i=0,1$,
\begin{equation*}
\|(U,\Phi)\|^2_{\mathcal{H}^0_{\varepsilon}}
:=\|U\|^2_{\mathbb{X}^2}+\|\Phi\|^2_{\mathcal{M}^1_{\varepsilon}}
\quad \text{and} \quad
\|(U,\Phi)\|^2_{\mathcal{H}^1_{\varepsilon}}:=\|U\|^2_{\mathbb{V}^1}
+\|\Phi\|^2_{\mathcal{M}^2_{\varepsilon}}.
\end{equation*}

For the kernel $\mu$, we take the following
assumptions (cf. e.g. \cite{CPS06,GPM98,GPM00}). Assume
\begin{gather}
 \mu\in C^1(\mathbb{R}_+)\cap L^1(\mathbb{R}_+),  \label{mu-1} \\
 \mu(s) \geq 0 \quad\text{for all}\quad s\geq 0,  \label{mu-2} \\
 \mu'(s) \leq 0 \quad\text{for all}\quad s\geq 0,  \label{mu-3} \\
 \mu'(s) + \delta\mu(s) \leq 0 \quad
\text{for all $s\geq 0$ and some } \delta>0.  \label{mu-4}
\end{gather}
The assumptions \eqref{mu-1}-\eqref{mu-3} are equivalent to assuming
$k(s)$ be a bounded, positive, nonincreasing, convex function of class
$\mathcal{C}^2$.
Moreover, assumption \eqref{mu-4} guarantees exponential decay of the
function $\mu(s)$ while allowing a singularity at $s=0$.
Assumptions \eqref{mu-1}-\eqref{mu-3} are used in the literature
(see \cite{CDGP-2010,CPS06,GPM98,Grasselli&Pata02-2} for example) to establish the
existence and uniqueness of continuous global weak solutions to a system
of equations similar to \eqref{problemp-1}, \eqref{problemp-3},
but with Dirichlet boundary conditions. In the literature,
assumption \eqref{mu-4} is used to obtain a bounded absorbing set for
the associated semigroup of solution operators.

For each $\varepsilon\in(0,1]$, define
\begin{equation}\label{memory-4}
D(\mathrm{T}_{\varepsilon})
=\{ \Phi\in\mathcal{M}^1_{\varepsilon}:\partial_s\Phi\in\mathcal{M}^1_{\varepsilon},
 \Phi(0)=0 \}
\end{equation}
where (with an abuse of notation) $\partial_s\Phi$ is the distributional
derivative of $\Phi$ and the equality $\Phi(0)=0$ is meant in the following sense
\begin{equation*}
\lim_{s\to0}\|\Phi(s)\|_{\mathbb{X}^2} = 0.
\end{equation*}
Then define the linear (unbounded) operator
$\mathrm{T}_{\varepsilon}: D(\mathrm{T}_{\varepsilon})
\to \mathcal{M}^1_{\varepsilon}$ by, for all
$\Phi\in D(\mathrm{T}_{\varepsilon})$,
\begin{equation*}
\mathrm{T}_{\varepsilon}\Phi=-\frac{\mathrm{d}}{\mathrm{d}s}\Phi.
\end{equation*}
For each $t\in[0,T]$, the equation
\begin{equation}\label{memory-2}
\partial_t\Phi^t = \mathrm{T}_{\varepsilon}\Phi^t + U(t)
\end{equation}
holds as an ODE in $\mathcal{M}^1_\varepsilon$ subject to the initial condition
\begin{equation}\label{memory-3}
\Phi^0=\Phi_0\in\mathcal{M}^1_{\varepsilon}.
\end{equation}
Concerning the solution to the IVP \eqref{memory-2}-\eqref{memory-3},
we have the following proposition.
The result is a generalization of \cite[Theorem 3.1]{Grasselli&Pata02-2}.

\begin{proposition}\label{t:generator-T}
For each $\varepsilon\in(0,1]$, the operator $\mathrm{T}_{\varepsilon}$
with domain $D(\mathrm{T}_{\varepsilon})$ is an infinitesimal generator
of a strongly continuous semigroup of contractions on $\mathcal{M}^1_{\varepsilon}$,
denoted $e^{\mathrm{T}_{\varepsilon} t}$.
\end{proposition}

We now have (cf. e.g. \cite[Corollary IV.2.2]{Pazy83}).

\begin{corollary}\label{t:memory-regularity-1}
When $U\in L^1([0,T];\mathbb{V}^1)$ for each $T>0$, then,
for every $\Phi_0\in\mathcal{M}^1_{\varepsilon}$, the Cauchy problem
\begin{equation}\label{memory-1}
\begin{gathered}
\partial_t\Phi^t=\mathrm{T}_{\varepsilon}\Phi^t+U(t), \quad \text{for } t>0, \\
\Phi^0=\Phi_0,
\end{gathered}
\end{equation}
has a unique solution $\Phi\in C([0,T];\mathcal{M}^1_{\varepsilon})$
which can be explicitly given as (cf. \cite[Section 3.2]{CPS06}
and \cite[Section 3]{Grasselli&Pata02-2})
\begin{equation}  \label{representation-formula-1}
\Phi^t(s)=\begin{cases}
\int_0^s U(t-y) \,\mathrm{d}y, & \text{for } 0<s\leq t, \\
\Phi_0(s-t) + \int_0^t U(t-y) \,\mathrm{d}y, & \text{when } s>t.
\end{cases}
\end{equation}
(The interested reader can also see \cite[Section 3]{CPS06},
\cite[pp. 346--347]{GPM98} and \cite[Section 3]{Grasselli&Pata02-2}
for more details concerning the case with static boundary conditions.)
\end{corollary}

Furthermore, we also know that, for each $\varepsilon\in(0,1]$,
$\mathrm{T_\varepsilon}$ is the infinitesimal generator of a strongly
continuous (the right-translation) semigroup of contractions on
$\mathcal{M}^1_\varepsilon$ satisfying \eqref{operator-T-1} below;
in particular, $\mathrm{Range}(\mathrm{I}-\mathrm{T}_\varepsilon)
=\mathcal{M}^1_\varepsilon$.

Following \eqref{mu-4}, there is the useful inequality.
(Also see \cite[see equation (3.4)]{CPS06} and
\cite[Section 3, proof of Theorem]{Grasselli&Pata02-2}.)

\begin{corollary} \label{t:operator-T-1}
There holds, for all $\Phi\in D(\mathrm{T}_\varepsilon)$,
\begin{equation} \label{operator-T-1}
\langle \mathrm{T}_\varepsilon\Phi,\Phi \rangle_{\mathcal{M}^1_\varepsilon}
\leq -\frac{\delta}{2\varepsilon}\|\Phi\|^2_{\mathcal{M}^1_\varepsilon}.
\end{equation}
\end{corollary}

Even though the embedding $\mathbb{V}^1\hookrightarrow\mathbb{X}^2$
is compact, it does not follow that the embedding
$\mathcal{M}^{1}_{\varepsilon}\hookrightarrow \mathcal{M}^{0}_{\varepsilon}$ is also compact.
Indeed, see \cite{Pata-Zucchi-2001} for a counterexample.
Moreover, this means the embedding
$\mathcal{H}^1_\varepsilon\hookrightarrow\mathcal{H}^0_\varepsilon$ is not compact.
Such compactness between the ``natural phase spaces'' is essential to the
construction of finite dimensional exponential attractors.
To alleviate this issue we follow \cite{CPS06,GMPZ10} and define for any
$\varepsilon\in(0,1]$ the so-called {\em{tail function}} of $\Phi\in\mathcal{M}^{0}_\varepsilon$ by,
for all $\tau\ge0$,
\begin{equation*}
\mathbb{T}_{\varepsilon}(\tau;\Phi) := \int\limits_{(0,1/\tau)\cup(\tau,\infty)}
\varepsilon\mu_\varepsilon(s) \|\Phi(s)\|^2_{\mathbb{V}^1} \,\mathrm{d}s,
\end{equation*}
With this we set, for $\varepsilon\in(0,1]$,
\begin{equation*}
\mathcal{K}^2_\varepsilon := \{ \Phi\in\mathcal{M}^2_\varepsilon :
\partial_s\Phi\in\mathcal{M}^{0}_\varepsilon,\ \Phi(0)=0,\;
\sup_{\tau\ge1} \tau\mathbb{T}_\varepsilon(\tau;\Phi)<\infty \}.
\end{equation*}
The space $\mathcal{K}^2_\varepsilon$ is Banach with the norm whose square is defined by
\begin{equation}
\|\Phi\|^2_{\mathcal{K}^2_\varepsilon} := \|\Phi\|^2_{\mathcal{M}^2_\varepsilon}
+ \varepsilon\|\partial_s\Phi\|^2_{\mathcal{M}^{0}_\varepsilon}
+ \sup_{\tau\ge1} \tau\mathbb{T}_\varepsilon(\tau;\Phi).  \label{new-norm}
\end{equation}
When $\varepsilon=0$, we set $\mathcal{K}^2_0=\{0\}$.
Importantly, for each $\varepsilon\in(0,1]$, the embedding
$\mathcal{K}^2_{\varepsilon}\hookrightarrow\mathcal{M}^1_{\varepsilon}$ is compact.
(cf. \cite[Proposition 5.4]{GMPZ10}).
Hence, let us now also define the space
\[
\mathcal{V}^1_{\varepsilon}  := \mathbb{V}^1\times\mathcal{K}^2_{\varepsilon},
\]
and the desired compact embedding $\mathcal{V}^1_\varepsilon\hookrightarrow\mathcal{H}^0_\varepsilon$
holds. Again, each space is equipped with the corresponding graph norm whose square
is defined by, for all $\varepsilon\in[0,1]$ and $(U,\Phi)\in\mathcal{V}^1_\varepsilon$,
\begin{equation*}
\|(U,\Phi)\|^2_{\mathcal{V}^1_{\varepsilon}}:=\|U\|^2_{\mathbb{V}^1}
+\|\Phi\|^2_{\mathcal{K}^2_{\varepsilon}}.
\end{equation*}

In regards to the system in Corollary \ref{t:memory-regularity-1} above,
we will also call upon the following simple generalizations of
\cite[Lemmas 3.3, 3.4, and 3.6]{CPS06}.

\begin{lemma}  \label{what-1}
Let $\varepsilon\in(0,1]$ and $\Phi_0\in D({\rm{T}}_\varepsilon)$.
Assume there is $\rho>0$ such that, for all $t\ge0$,
$\|U(t)\|_{\mathbb{V}^1}\le\rho$.
Then for all $t\ge0$,
\begin{align}
\varepsilon\|{\rm{T}}_\varepsilon\Phi^t\|^2_{\mathcal{M}^1_\varepsilon}
\le \varepsilon e^{-\delta t}\|{\rm{T}}_\varepsilon\Phi_0\|^2_{\mathcal{M}^1_\varepsilon}
+ \rho^2\|\mu\|_{L^1(\mathbb{R}_+)}.
\end{align}
\end{lemma}

\begin{remark}  \label{r:what-trans} \rm
The above result will also be needed later in the weaker space $\mathcal{M}^0_\varepsilon$
(see Step 3 in the proof of Lemma \ref{t:to-C2}).
The result for the weaker space can be obtained by suitably transforming
\eqref{memory-1}-\eqref{representation-formula-1} and applying an appropriate
bound on $U$.
\end{remark}

\begin{lemma}  \label{what-2}
Let $\varepsilon\in(0,1]$ and $\Phi_0\in D({\rm{T}}_\varepsilon)$.
Assume there is $\rho>0$ such that, for all $t\ge0$,
$\|U(t)\|_{\mathbb{V}^1}\le\rho$.
Then there is a constant $C>0$ such that, for all $t\ge0$,
\begin{align}
\sup_{\tau\ge1} \tau\mathbb{T}_\varepsilon(\tau;\Phi^t)
\le 2 ( t+2 )e^{-\delta t} \sup_{\tau\ge1} \tau\mathbb{T}_\varepsilon(\tau;\Phi_0)
+ C\rho^2.
\end{align}
\end{lemma}

Finally, we give a version of Lemma \ref{what-2} for compact intervals.

\begin{lemma}  \label{what-3}
Let $\varepsilon\in(0,1]$, $T>0$, and $\Phi_0\in D({\rm{T}}_\varepsilon)$.
Assume there is $\rho>0$ such that
\[
\int_0^T \|U(\tau)\|^2_{\mathbb{V}^1} \mathrm{d}\tau \le \rho.
\]
Then there is a positive constant $C(T)$ such that, for all $t\in[0,T]$,
\[
\sup_{\tau\ge1} \tau\mathbb{T}_\varepsilon(\tau;\Phi^t)
\le C(T) \Big( \rho + \sup_{\tau\ge1} \tau\mathbb{T}_\varepsilon(\tau;\Phi_0) \Big).
\]
\end{lemma}

We now discuss the linear operator associated with the model problem.
In our case it is given by the following (note that in \cite[Section 3.1]{CPS06}
the basic tool is the Laplacian with Dirichlet boundary conditions; in our case,
the analogue operator turns out to be the so-called ``Wentzell'' Laplace operator).

\begin{proposition}
Let $\Omega $ be a bounded open set of $\mathbb{R}^{n}$ with Lipschitz
boundary $\Gamma $. For $\alpha,\beta\ge0$, define the operator
 ${\mathrm{A_W^{\alpha,\beta}}}$ on $\mathbb{X}^2$,
by
\begin{align}
\mathrm{A_W^{\alpha,\beta}} & :=
\begin{pmatrix}
-\Delta+\alpha{\mathrm{I}} & 0 \\
\partial_{\mathbf{n}}(\cdot) & -\Delta_\Gamma + \beta{\mathrm{I}}
\end{pmatrix},  \label{A_Wentzell1}
\end{align}
with
\begin{equation}
\begin{aligned}
D( {\mathrm{A_W^{\alpha,\beta}}})
:=\Big\{ & U=(u_1,u_2)^{\mathrm{tr}}\in \mathbb{V}^{1}
:-\Delta u_{1}+\alpha u_1\in L^2( \Omega ) ,\\
&\partial_{\mathbf{n}} u_{1}-\Delta _{\Gamma }u_{2}
 +\beta u_2\in L^2( \Gamma ) \Big\} .
\end{aligned}  \label{A_Wentzell2}
\end{equation}
Then, $({\mathrm{A_W^{\alpha,\beta}}},D({\mathrm{A_W^{\alpha,\beta}}})) $
is self-adjoint and nonnegative operator on $\mathbb{X}^2$ whenever
$\alpha,\beta\ge0$, and ${\mathrm{A_W^{\alpha,\beta}}}>0$ (is strictly positive)
if either $\alpha>0$ or $\beta>0$.
Moreover, the resolvent operator
$({\mathrm{I}}+{\mathrm{A_W^{\alpha,\beta}}}) ^{-1}\in \mathcal{L}(\mathbb{X}^2) $
 is compact.
If the boundary $\Gamma $ is of class $\mathcal{C}^2$, then
 $D({\mathrm{A_W^{\alpha,\beta}}}) =\mathbb{V}^2$
(see, e.g., \cite[Theorem 2.3]{CGGM10}).
Indeed, for any $\alpha,\beta\ge 0$, the map
$\Psi:U\mapsto {\mathrm{A_W^{\alpha,\beta}}}U$, when viewed as a map
from $\mathbb{V}^2$ into $\mathbb{X}^2=L^2(\Omega)\times L^2(\Gamma)$,
is an isomorphism, and there exists a positive constant $C_*$, independent
of $U=(u,\psi)^{\mathrm{tr}}$, such that
\begin{equation}
C_*^{-1}\|U\|_{\mathbb{V}^2}\le\|\Psi(U)\|_{\mathbb{X}^2}
\le C_*\|U\|_{\mathbb{V}^2},  \label{equiv}
\end{equation}
for all $U\in\mathbb{V}^2$ (cf. Lemma \ref{t:appendix-lemma-3}).
\end{proposition}

We can refer the reader to \cite{CFGGGOR09} for an extensive survey of
recent results concerning the ``Wentzell'' Laplacian
${\mathrm{A_W^{\alpha,\beta}}}$.

For the nonlinear terms, assume $f,g \in C^{1}(\mathbb{R})$ satisfy the
growth assumptions: there exist positive constants $\ell_1$ and $\ell_2$,
and $r_1,r_2\in[1,\frac{5}{2})$ such that for all $s\in \mathbb{R}$,
\begin{gather}
|f'(s)| \leq \ell_1(1+|s|^{r_1}),  \label{assm-1} \\
|g'(s)| \leq \ell_2(1+|s|^{r_2}).  \label{assm-2}
\end{gather}
We also assume there are positive constants $M_f$ and $M_g$ so that
for all $s\in\mathbb{R}$,
\begin{gather}
f'(s)>-M_f,  \label{assm-3} \\
g'(s)>-M_g.  \label{assm-4}
\end{gather}
Consequently, \eqref{assm-3}-\eqref{assm-4} imply there are
$\kappa_i>0$, $i=1,2,3,4$, so that for all $s\in\mathbb{R}$,
\begin{gather}
f(s)s \ge -\kappa_1 s^2 - \kappa_2,  \label{cons-1} \\
g(s)s \ge -\kappa_3 s^2 - \kappa_4.  \label{cons-2}
\end{gather}

\begin{remark} \rm
Observe that here we do not allow for the critical polynomial growth exponent
 (of $5$) which appears in several works with static boundary conditions
(cf. e.g. \cite{CDGP-2010,CPS06}).
Indeed, in order for us to obtain a notion of strong solution
(see Definition \ref{d:strong-solution} below), the arguments in the proof
of Theorem \ref{t:strong-solutions} do not allow for $r_i\ge 5/2$, $i=1,2$.
\end{remark}

We can follow \cite[Section 4]{CPS06} or, more precisely \cite{GPM98,GPM00}
to deduce the existence and uniqueness of weak solutions in the
above class exploiting both semigroup methods and energy methods in the
framework of a Galerkin scheme which can be constructed for problems with
dynamic boundary conditions (see, \cite[Theorem 2.3]{CGGM10}).

Constants appearing below are independent of $\varepsilon$ and $\omega$,
 unless specified otherwise, but may depend on various structural parameters
such as $\alpha$, $\beta$, $|\Omega|$, $|\Gamma|$, $\ell_f$ and $\ell_g$,
and the constants may even change from line to line.
We denote by $Q(\cdot)$ a generic monotonically increasing function.
We will use $\|B\|_{W}:=\sup_{\Upsilon\in B}\|\Upsilon\|_W$ to denote the
``size'' of the subset $B$ in the Banach space $W$.

\section{Review of well-posedness and regularity}

Here we provide some definitions and cite the relevant global well-posedness
results concerning Problem \eqref{Pe}.
For the remainder of this article we choose to set $n=3$, which is of
course the most relevant physical dimension.

Below we will set $F:\mathbb{R}^2\to\mathbb{R}^2$,
\begin{equation}
F(U):=\begin{pmatrix}f(u) \\ \widetilde{g}(u)\end{pmatrix},  \label{func}
\end{equation}
where $\widetilde{g}(s):=g(s)-\omega\beta s$, for $s\in\mathbb{R}$.
(To offset $\widetilde{g}$, the term $\omega\beta u$ will be incorporated
in the operator $\mathrm{A_{W}^{0,0}}$ as $\mathrm{A_{W}^{0,\beta}}$.)

\begin{definition}  \label{d:weak-solution} \rm
Let $\varepsilon\in(0,1]$, $\omega\in(0,1)$ and $T>0$.
Given $U_0=(u_0,v_0)^{\mathrm{tr}}\in\mathbb{X}^2$ and
$\Phi_0=(\eta_0,\xi_0)^{\mathrm{tr}}\in\mathcal{M}^1_{\varepsilon}$,
the pair $U(t)=(u(t),v(t))^{{\mathrm{tr}}}$ and
$\Phi^t=(\eta^t,\xi^t)^{{\mathrm{tr}}}$ satisfying
\begin{gather}
U \in L^{\infty }([0,T];\mathbb{X}^2)\cap L^2([0,T];\mathbb{V}^{1}), \label{defn-1} \\
u \in L^{r_1}(\Omega\times[0,T]), \label{defn-2} \\
v \in L^{r_2}(\Gamma\times[0,T]), \label{defn-3} \\
\Phi \in L^{\infty }([0,T];\mathcal{M}^1_{\varepsilon}), \label{defn-4} \\
\partial_t U \in L^2([0,T];(\mathbb{V}^1)^*)
\oplus ( L^{r_1'}(\Omega\times[0,T]) \times L^{r_2'}(\Gamma\times[0,T]) ),
 \label{defn-5} \\
\partial_t \Phi \in L^2( [0,T];H^{-1}_{\mu_{\varepsilon}}
 (\mathbb{R}_+;\mathbb{V}^1) ), \label{defn-6}
\end{gather}
is said to be a weak solution to Problem \eqref{Pe} if, $v(t)=u|_{\Gamma}(t)$
and $\xi^{t}=\eta^{t}|_{\Gamma}$ for almost all $t\in[0,T]$, and
for all $\Xi =(\varsigma ,\varsigma |_{\Gamma })^{\mathrm{tr}}\in \mathbb{V}^{1}
 \cap ( L^{r_1}(\Omega) \times L^{r_2}(\Gamma) )$,
$\Pi =(\rho ,\rho |_{\Gamma })^{\mathrm{tr}}\in \mathcal{M}^1_{\varepsilon}$,
and for almost all $t\in[0,T]$, there holds,
\begin{gather}
 \langle \partial _{t}U(t),\Xi \rangle _{\mathbb{X}^2} + \omega
\langle \mathrm{{A_W^{0,\beta}}}U(t),\Xi \rangle _{\mathbb{X}^2}
+ \langle \Phi^t, \Xi \rangle_{\mathcal{M}^1_{\varepsilon}} + \langle
F( U(t)) ,\Xi \rangle _{\mathbb{X}^2}=0,  \label{weak-solution-1}
\\
\label{weak-solution-2}
\langle \partial _{t}\Phi ^{t},\Pi \rangle _{\mathcal{M}^1_{\varepsilon}}
=\langle \mathrm{T}_{\varepsilon}\Phi ^{t},\Pi
\rangle _{\mathcal{M}^1_{\varepsilon}}+\langle U(t),\Pi
\rangle _{\mathcal{M}^1_{\varepsilon}},
\end{gather}
in addition,
\begin{equation}
U(0)=U_0 \quad\text{and}\quad \Phi^{0}=\Phi_0. \label{weak-solution-3}
\end{equation}
The function $[0,T]\ni t\mapsto (U(t),\Phi^t )$ is called a global weak
solution if it is a weak solution for every $T>0$.
\end{definition}

\begin{remark}\label{r:trace-map}
When we have a weak solution to Problem \eqref{Pe}, the above restrictions
$u|_{\Gamma}(t)$ and $\eta|_{\Gamma}^t$ are well-defined by virtue of the
Dirichlet trace map, ${\mathrm{tr_D}}:H^1(\Omega)\to H^{1/2}(\Gamma)$.
However, this is not necessarily the case for $\partial_t U$.
\end{remark}

\begin{remark}  \label{r:continuity} \rm
The continuity properties $U\in C([0,T];\mathbb{X}^2)$ follow from the
classical embedding (cf. e.g. \cite[Lemma 5.51]{Tanabe79}),
\[
\{ \chi\in L^2([0,T];V), \ \partial_t\chi\in L^2([0,T];V') \}
\hookrightarrow C([0,T];H),
\]
where $H$ and $V$ are reflexive Banach spaces with continuous embeddings
$V\hookrightarrow H \hookrightarrow V'$, the injection $V\hookrightarrow H$
being compact.
\end{remark}

\begin{definition} \label{d:strong-solution} \rm
The pair $U(t)=(u(t),v(t))^{\mathrm{tr}}$ and
$\Phi^t=(\eta^t,\xi^t)^{\mathrm{tr}}$ is called a (global) strong
solution of Problem \eqref{Pe} if it is a weak solution in the sense
of Definition \ref{d:weak-solution}, and if it satisfies the following
regularity properties:
\begin{gather}
U \in L^{\infty }([0,\infty);\mathbb{V}^{1})\cap L^2([0,\infty);\mathbb{V}^2), \label{strong-defn-1} \\
\Phi \in L^{\infty }([0,\infty);\mathcal{M}^2_{\varepsilon}), \label{strong-defn-2} \\
\partial_t U \in L^\infty([0,\infty);\mathbb{X}^2 )\cap L^2([0,\infty);
 \mathbb{V}^1), \label{strong-defn-3} \\
\partial_t \Phi \in L^\infty( [0,\infty);\mathcal{M}^1_{\varepsilon} ).
 \label{strong-defn-4}
\end{gather}
Therefore, $(U(t),\Phi^t)$ satisfies the equations
\eqref{weak-solution-1}-\eqref{weak-solution-2} almost everywhere, i.e.,
is a strong solution.
\end{definition}

\begin{theorem}[Weak solutions]\label{t:weak-solutions}
Assume \eqref{mu-1}-\eqref{mu-3} and \eqref{assm-1}-\eqref{assm-4} hold.
For each $\varepsilon\in(0,1]$, $\omega\in(0,1)$ and $T>0$, and for any
$U_0=(u_0,v_0)^{ {\mathrm{tr}} }\in
\mathbb{X}^2$ and $\Phi_0=(\eta_0,\xi_0)^{ {\mathrm{tr}} }
\in\mathcal{M}^1_{\varepsilon}$, there exists a unique (global)
 weak solution to Problem \eqref{Pe} in the sense of Definition
\ref{d:weak-solution} which depends continuously on the initial data
in the following way; there exists a constant $C>0$, independent of
$U_i$, $\Phi_i$, $i=1,2$, and $T>0$ in which, for all $t\in[0,T]$, there holds
\begin{equation}
\| U_1(t)-U_2(t) \|_{\mathbb{X}^2}
+ \| \Phi^t_1-\Phi^t_2 \|_{\mathcal{M}^1_{\varepsilon}}
\le ( \| U_1(0)-U_2(0) \|_{\mathbb{X}^2}
+ \| \Phi^0_1-\Phi^0_2 \|_{\mathcal{M}^1_{\varepsilon}} ) e^{Ct}. \label{cde}
\end{equation}
\end{theorem}

\begin{proof} Cf. \cite[Theorem 3.8]{Gal-Shomberg15-2} for existence
and  \cite[Proposition 3.10]{Gal-Shomberg15-2} for \eqref{cde}.
\end{proof}

We conclude the preliminary results for Problem \eqref{Pe} with the following
result.

\begin{theorem}[Strong solutions]  \label{t:strong-solutions}
Assume  \eqref{mu-1}--\eqref{mu-3} and \eqref{assm-1}--\eqref{assm-4} hold.
For each $\varepsilon\in(0,1]$, $\omega\in(0,1)$, and $T>0$, and for
any $U_0=(u_0,v_0)^{ {\mathrm{tr}} }\in
\mathbb{V}^1$ and $\Phi_0=(\eta_0,\xi_0)^{ {\mathrm{tr}} }
\in\mathcal{M}^2_{\varepsilon}$, there exists a unique (global) strong
solution to Problem \eqref{Pe} in the sense of Definition \ref{d:strong-solution}.
\end{theorem}

For a proof of the above theorem see \cite[Theorem 3.11]{Gal-Shomberg15-2}.
Here we recall some important aspects and relevant results for Problem
\eqref{P0}.
The interested reader can also see \cite{Gal12-2,Gal12-1,Gal-15Z,Gal&Warma10}
for further details.

\begin{definition}  \label{d:weak-solution-0}
Let $\omega\in(0,1)$ and $T>0$.
Given $U_0=(u_0,v_0)^{\mathrm{tr}}\in\mathbb{X}^2$, the pair
$U(t)=(u(t),v(t))^{{\mathrm{tr}}}$ satisfying
\begin{gather}
U \in L^{\infty }([0,T];\mathbb{X}^2)\cap L^2([0,T];\mathbb{V}^{1}),
 \label{defn-1-0} \\
u \in L^{r_1}(\Omega\times[0,T]), \label{defn-2-0} \\
v \in L^{r_2}(\Gamma\times[0,T]), \label{defn-3-0} \\
\partial_t U \in L^2([0,T];(\mathbb{V}^1)^*)
 \oplus ( L^{r_1'}(\Omega\times[0,T]) \times L^{r_2'}(\Gamma\times[0,T]) ),
  \label{defn-5-0}
\end{gather}
is said to be a weak solution to Problem \eqref{P0} if, $v(t)=u|_{\Gamma}(t)$
for almost all $t\in[0,T]$, and
for all $\Xi =(\varsigma ,\varsigma |_{\Gamma })^{\mathrm{tr}}\in
\mathbb{V}^{1} \cap ( L^{r_1}(\Omega) \times L^{r_2}(\Gamma) )$,
and for almost all $t\in[0,T]$, there holds
\begin{equation}
 \langle \partial _{t}U(t),\Xi \rangle _{\mathbb{X}^2} + \omega
\langle \mathrm{{A_W^{0,\beta}}}U(t),\Xi \rangle _{\mathbb{X}^2} + \langle
F( U(t)) ,\Xi \rangle _{\mathbb{X}^2}=0,  \label{weak-solution-1-0}
\end{equation}
with
\begin{equation}
U(0)=U_0. \label{weak-solution-3-0}
\end{equation}
The function $[0,T]\ni t\mapsto U(t)$ is called a global weak solution
if it is a weak solution for every $T>0$.
\end{definition}

We remind the reader of Remark \ref{r:trace-map} on the issue of traces.
We conclude this section with the following result.

\begin{theorem}[Weak solutions]  \label{t:weak-solutions-0}
Assume \eqref{assm-1}-\eqref{assm-4} hold.
For each $\omega\in(0,1)$ and $T>0$, and for any
$U_0=(u_0,v_0)^{ {\mathrm{tr}} }\in \mathbb{X}^2$, there exists a unique
(global) weak solution to Problem \eqref{P0} in the sense of Definition
\ref{d:weak-solution-0} which depends continuously on the initial data as
follows: there exists a constant $C>0$, independent of $U_1$ and $U_2$,
and $T>0$ in which, for all $t\in[0,T]$, there holds
\begin{align}
\| U_1(t)-U_2(t) \|_{\mathbb{X}^2} \le \| U_1(0)-U_2(0) \|_{\mathbb{X}^2}
e^{Ct}. \label{cde-0}
\end{align}
\end{theorem}

For a proof of the above theorem see
 \cite[Theorem 2.2]{Gal12-1}.

\section{Asymptotic behavior and attractors}

\subsection{Preliminary estimates}

Concerning Problem \eqref{Pe} and following directly from Theorem
\ref{t:weak-solutions}, we have the first preliminary result for this section.

\begin{corollary}
Problem \eqref{Pe} defines a (nonlinear) strongly continuous semigroup
$\mathcal{S}_{\varepsilon}(t)$ on the phase space
$\mathcal{H}^0_{\varepsilon} = \mathbb{X}^2\times\mathcal{M}^1_{\varepsilon}$ by
\begin{equation*}
\mathcal{S}_{\varepsilon}(t)\Upsilon_0 := ( U(t),\Phi^t ),
\end{equation*}
where $\Upsilon_0=(U_0,\Phi_0)\in \mathcal{H}^0_{\varepsilon}$ and
$( U(t),\Phi^t )$ is the unique solution to Problem \eqref{Pe}.
The semigroup is Lipschitz continuous on $\mathcal{H}^0_{\varepsilon}$
via the continuous dependence estimate \eqref{cde}.
\end{corollary}

The next preliminary result concerns a uniform bound on the weak solutions.
This result follows from an estimate which proves the existence of a bounded
absorbing set for the semigroup of solution operators.
This result provides a basic but important first step in showing the associated
dynamical system is dissipative (cf. e.g. \cite{Babin&Vishik92,Temam88}).
It is important to note that throughout the remainder of this article,
whereby we are now concerned with the asymptotic behavior of the solutions
to Problem \eqref{Pe} and Problem \eqref{P0},

\begin{itemize}
\item[(A1)] we will assume that \eqref{mu-4} holds.
\end{itemize}

Additionally, we introduce a smallness criteria for certain parameters
relating to the linear operator $\mathrm{A_{W}^{\alpha,\beta}}$ and the
nonlinear map $F$.

\begin{itemize}
\item[(A2)] \emph{Smallness criteria:}
 Fix $\varepsilon\in(0,1]$ and $\omega\in(0,1)$.
Denote by $C_{\overline{\Omega}}$ the positive constant that arises from
the embedding $\mathbb{V}^1\hookrightarrow\mathbb{X}^2$; i.e.,
$\|U\|^2_{\mathbb{X}^2}\le C_{\overline{\Omega}}\|U\|^2_{\mathbb{V}^1}$.
The smallness criteria is that $\kappa_1,\kappa_3,\beta>0$
(cf. \eqref{A_Wentzell1} and  \eqref{cons-1}-\eqref{cons-2}) satisfy
\begin{equation}  \label{smallness-criteria}
\max\{\kappa_1,\kappa_3+\beta\} < \omega C^{-1}_{\overline{\Omega}}.
\end{equation}
\end{itemize}

As a final note, we remind the reader that all formal multiplication below
can be rigorously justified using the Galerkin procedure developed in the
proof of Theorem \ref{t:weak-solutions} in \cite{Gal-Shomberg15-2}.

\begin{lemma}  \label{weak-ball}
Let $\varepsilon\in(0,1]$ and $\omega\in(0,1)$.
In addition to the assumptions of Theorem \ref{t:weak-solutions}, assume
\eqref{mu-4} holds and that $\kappa_1,\kappa_3,\beta>0$ satisfy the smallness
criteria \eqref{smallness-criteria}.
For all $R>0$ and $\Upsilon_0=(U_0,\Phi_0)\in\mathcal{H}^0_{\varepsilon}
=\mathbb{X}^2\times\mathcal{M}^1_{\varepsilon}$ with
$\|\Upsilon_0\|_{\mathcal{H}^0_{\varepsilon}}\le R$ for all
$\varepsilon\in(0,1]$, there exist positive constants
$\nu_0=\nu_0(\omega,C_{\overline{\Omega}}, \kappa_1,\kappa_3,\beta,\delta)$
and $P_0=P_0(\kappa_2,\kappa_4,\nu_0)$, and there is a positive monotonically
increasing function $Q(\cdot)$ each independent of $\varepsilon$, in which,
for all $t\ge0$,
\begin{equation}
\| ( U(t),\Phi^t ) \|^2_{\mathcal{H}^0_{\varepsilon}}
\le Q(R)e^{-\nu_0t} + P_0.  \label{weak-decay}
\end{equation}
Moreover, the set
\begin{equation}  \label{ball-0}
\mathcal{B}^0_{\varepsilon}
:=\big\{ (U,\Phi)\in \mathcal{H}^0_{\varepsilon} :
\| (U,\Phi) \|_{\mathcal{H}^0_{\varepsilon}} \le \sqrt{P_0+1} \big\}.
\end{equation}
is absorbing and positively invariant for the semigroup
$\mathcal{S}_{\varepsilon}(t)$.
\end{lemma}

\begin{proof}
Let $\varepsilon\in(0,1]$ and $\omega\in(0,1)$.
Let $\Upsilon_0=(U_0,\Phi_0)\in\mathcal{H}^0_{\varepsilon}
=\mathbb{X}^2\times\mathcal{M}^1_{\varepsilon}$.
From the equations \eqref{weak-solution-1} and \eqref{weak-solution-2},
we take the corresponding weak solution $\Xi=U(t)$ and $\Pi(s)=\Phi^t(s)$.
We then obtain the identities
\begin{gather}
\langle \partial_t U,U \rangle_{\mathbb{X}^2}
 + \omega\langle \mathrm{A_{W}^{0,\beta}}U,U \rangle_{\mathbb{X}^2}
+ \langle \Phi^t,U \rangle_{\mathcal{M}^1_{\varepsilon}}
+ \langle F(U),U \rangle_{\mathbb{X}^2} = 0,  \label{fus-1}
\\
\langle \partial_t\Phi^t,\Phi^t \rangle_{\mathcal{M}^1_{\varepsilon}}
= \langle  \mathrm{T_{\varepsilon}}\Phi^t,\Phi^t
\rangle_{\mathcal{M}^1_{\varepsilon}}
+ \langle U,\Phi^t \rangle_{\mathcal{M}^1_{\varepsilon}}.  \label{fus-2}
\end{gather}
Observe that
\begin{gather}
 \langle \partial_t U,U \rangle_{\mathbb{X}^2}
= \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\| U \|^2_{\mathbb{X}^2},
 \label{fus-3} \\
\langle \mathrm{A_{W}^{0,\beta}}U,U \rangle_{\mathbb{X}^2}
= \|\nabla u\|^2_{L^2(\Omega)}+\|\nabla_\Gamma u\|^2_{L^2(\Gamma)}
+\beta\|u\|^2_{L^2(\Gamma)}, \label{fus-4} \\
\langle \partial_t\Phi^t,\Phi^t \rangle_{\mathcal{M}^1_{\varepsilon}}
= \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}
\| \Phi^t \|^2_{\mathcal{M}^1_{\varepsilon}}.  \label{fus-5}
\end{gather}
Combining \eqref{fus-1}-\eqref{fus-5} produces the differential identity,
 which holds for almost all $t\ge0$,
\begin{equation} \label{fus-6}
\begin{aligned}
& \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}
\big\{ \| U \|^2_{\mathbb{X}^2} + \| \Phi^t \|^2_{\mathcal{M}^1_{\varepsilon}} \big\}
 \\
& + \omega ( \|\nabla u\|^2_{L^2(\Omega)} + \|\nabla_\Gamma u\|^2_{L^2(\Gamma)}
 + \beta\|u\|^2_{L^2(\Gamma)} )   \\
& - \langle \mathrm{T}_{\varepsilon}\Phi^t,\Phi^t
 \rangle_{\mathcal{M}^1_{\varepsilon}} + \langle F(U),U \rangle_{\mathbb{X}^2} = 0.
\end{aligned}
\end{equation}
Because of assumption \eqref{mu-4},  we may directly apply \eqref{operator-T-1}
from Corollary \ref{t:operator-T-1}; i.e.,
\begin{equation}  \label{fus-7}
-\langle \mathrm{T}_{\varepsilon}\Phi^t,\Phi^t \rangle_{\mathcal{M}^1_{\varepsilon}}
\ge \frac{\delta}{2\varepsilon}\|\Phi^t\|^2_{\mathcal{M}^1_{\varepsilon}}.
\end{equation}
From \eqref{cons-1} and \eqref{cons-2}, we know that
\begin{equation} \label{fus-8}
\begin{aligned}
\langle F(U),U \rangle_{\mathbb{X}^2}
& \ge -\kappa_1\|u\|^2_{L^2(\Omega)}-(\kappa_3+\omega\beta)\|u\|^2_{L^2(\Gamma)}
 - ( \kappa_2+\kappa_4 )   \\
& \ge -\kappa_1\|u\|^2_{L^2(\Omega)}-(\kappa_3 + \beta)\|u\|^2_{L^2(\Gamma)}
 - ( \kappa_2+\kappa_4 )   \\
& = -C_F\|U\|^2_{\mathbb{X}^2} - ( \kappa_2+\kappa_4 ),
\end{aligned}
\end{equation}
where $C_F:=\max\{\kappa_1,\kappa_3+\beta\}$.
Finally, due the embedding $\mathbb{V}^1\hookrightarrow\mathbb{X}^2$, we have
\begin{equation}\label{fus-9}
C_{\overline{\Omega}}^{-1}\|U\|^2_{\mathbb{X}^2}\le \|U\|^2_{\mathbb{V}^1},
\end{equation}
for some $C_{\overline{\Omega}}>0$.
Hence, \eqref{fus-6}-\eqref{fus-9} yields the differential inequality
(minimizing the left-hand side by setting $\varepsilon=1$),
\begin{align*}
 \frac{\mathrm{d}}{\mathrm{d}t}\big\{ \| U \|^2_{\mathbb{X}^2}
 + \| \Phi^t \|^2_{\mathcal{M}^1_{\varepsilon}} \big\}
 + 2( \omega C^{-1}_{\overline{\Omega}} - C_F ) \| U \|^2_{\mathbb{X}^2}
 + \delta \|\Phi^t\|^2_{\mathcal{M}^1_{\varepsilon}}   
 \le 2(\kappa_2+\kappa_4).
\end{align*}
By the smallness criteria \eqref{smallness-criteria} there holds
\begin{equation*}
\omega C^{-1}_{\overline{\Omega}} - C_F>0.
\end{equation*}
Thus we arrive at the differential inequality, which holds for almost all $t\ge0$,
\begin{equation} \label{fus-10}
\frac{\mathrm{d}}{\mathrm{d}t}\big\{ \| U \|^2_{\mathbb{X}^2}
+ \| \Phi^t \|^2_{\mathcal{M}^1_{\varepsilon}} \big\}
 + m_0 ( \| U \|^2_{\mathbb{X}^2} + \|\Phi^t\|^2_{\mathcal{M}^1_{\varepsilon}} )
 \le C.
\end{equation}
where $m_0:=\min\{ 2( \omega C^{-1}_{\overline{\Omega}} - C_F),\delta \}>0$, and
$C>0$ depends only on $\kappa_2$ and $\kappa_4$.
(The absolute continuity of the mapping
$t\mapsto \| U(t) \|^2_{\mathbb{X}^2}+\| \Phi^t \|^2_{\mathcal{M}^1_{\varepsilon}}$
can be established as in \cite[Lemma III.1.1]{Temam88}, for example.)
After applying a suitable Gr\"{o}nwall inequality, the estimate \eqref{weak-decay}
follows with $\nu_0=m_0$ and $P_0=\frac{C}{m_0};$ indeed, \eqref{fus-10} yields,
for all $t\ge 0$,
\begin{equation} \label{fus-11}
 \| U(t) \|^2_{\mathbb{X}^2} + \| \Phi^t \|^2_{\mathcal{M}^1_{\varepsilon}}
\le e^{-m_0 t} \Big( \| U_0 \|^2_{\mathbb{X}^2}
 + \| \Phi_0 \|^2_{\mathcal{M}^1_{\varepsilon}} \Big) + P_0.
\end{equation}
Now we see \eqref{weak-decay} holds for any $R>0$ and
$\Upsilon_0=(U_0,\Phi_0)\in\mathcal{H}^0_{\varepsilon}$ such that
$\|\Upsilon_0\|_{\mathcal{H}^0_{\varepsilon}}\le R$ for all $\varepsilon\in(0,1]$.

The existence of the bounded set $\mathcal{B}^0_{\varepsilon}$ in
$\mathcal{H}^0_{\varepsilon}$ that is absorbing and positively invariant for
$\mathcal{S}_{\varepsilon}(t)$ follows from \eqref{fus-11}
(cf. e.g. \cite[Proposition 2.64]{Milani&Koksch05}).
Given any nonempty bounded subset $B$ in
$\mathcal{H}^0_{\varepsilon}\setminus\mathcal{B}^0_{\varepsilon}$,
then we have that $\mathcal{S}_{\varepsilon}(t)B\subseteq\mathcal{B}^0$,
in $\mathcal{H}^0_{\varepsilon}$, for all $t\ge t_0$ where
\begin{equation}\label{time0}
t_0\ge\frac{1}{m_0}\ln\big( \|B\|^2_{\mathcal{H}^0_{\varepsilon}} \big).
\end{equation}
(Observe that  $t_0>0$ because $\|B\|_{\mathcal{H}^0_{\varepsilon}}>1$.)
This completes the proof.
\end{proof}

\begin{corollary}
From \eqref{weak-decay} it follows that for each $\varepsilon\in(0,1]$
and $\omega\in(0,1)$, any weak solution $(U(t),\Phi^t)$ to Problem \eqref{Pe},
according to Definition \ref{d:weak-solution}, is bounded uniformly in $t$.
Indeed, for all $\Upsilon_0\in\mathcal{H}^0_\varepsilon$,
\begin{equation}\label{weak-bound}
\limsup_{t\to+\infty} \|\mathcal{S}_\varepsilon(t)
\Upsilon_0\|_{\mathcal{H}^0_{\varepsilon}} \le \widetilde{P}_0,
\end{equation}
where $\widetilde{P}_0$ depends on $P_0$ and the initial datum.
\end{corollary}

\begin{corollary}
Problem \eqref{Pe} defines a (nonlinear) strongly continuous semigroup
$\mathcal{S}_{\varepsilon}(t)$ on the phase space
$\mathcal{H}^0_{\varepsilon} = \mathbb{X}^2\times\mathcal{M}^1_{\varepsilon}$ by
\begin{equation*}
\mathcal{S}_{\varepsilon}(t)\Upsilon_0 := ( U(t),\Phi^t ),
\end{equation*}
where $\Upsilon_0=(U_0,\Phi_0)\in \mathcal{H}^0_{\varepsilon}$ and
$( U(t),\Phi^t )$ is the unique solution to Problem \eqref{Pe}.
The semigroup is Lipschitz continuous on $\mathcal{H}^0_{\varepsilon}$
via the continuous dependence estimate \eqref{cde}.
\end{corollary}

\begin{remark} \rm
Thanks to the uniformity of the above estimates with respect to the
perturbation parameter $\varepsilon$, it is easy to see that there exists a
bounded absorbing set $\mathcal{B}^0_0$ for the semigroup
$\mathcal{S}_0:\mathcal{H}^0_0=\mathbb{X}^2\to\mathbb{X}^2$ generated
by the weak solutions of Problem \eqref{P0}.
Moreover, we also easily see that Problem \eqref{P0} defines a
semigroup $\mathcal{S}_0(t):\mathcal{H}^0_0=\mathbb{X}^2\to\mathbb{X}^2$
by $\mathcal{S}_0(t)U_0:=U(t)$.
(See the references mentioned above for further details.)
\end{remark}

\subsection{Exponential attractors}

Exponential attractors (sometimes called inertial sets) are positively
invariant sets possessing finite fractal dimension that attract bounded
subsets of their basin of attraction exponentially fast.
This section will focus on the existence of exponential attractors.
The existence of an exponential attractor depends on certain properties
of the semigroup; namely, the smoothing property for the difference of
any two trajectories and the existence of a more regular bounded absorbing
set in the phase space (see e.g. \cite{EFNT95,EMZ00,GGMP05} and in
particular \cite{CPS06,GMPZ10}).
The basin of attraction will be discussed in the next section.

The main result of this section is the following.

\begin{theorem}  \label{t:exponential-attractors}
For each $\varepsilon \in [0,1]$ and $\omega\in(0,1)$, the dynamical
system $( \mathcal{S}_{\varepsilon},\mathcal{H}^0_{\varepsilon }) $
associated with Problem \eqref{Pe} admits an exponential
attractor $\mathfrak{M}_{\varepsilon }$ compact in $\mathcal{H}^0_{\varepsilon}$,
and bounded in $\mathcal{V}^1_{\varepsilon }$.
Moreover,
\begin{itemize}
\item[(i)] For each $t\geq 0$, $\mathcal{S}_{\varepsilon }
(t)\mathfrak{M}_{\varepsilon}\subseteq \mathfrak{M}_{\varepsilon }$.

\item[(ii)] The fractal dimension of $\mathfrak{M}_{\varepsilon }$ with respect to
the metric $\mathcal{H}_{\varepsilon }^0$ is finite, uniformly in $\varepsilon$;
namely,
\begin{equation*}
\dim _{\rm{F}}( \mathfrak{M}_{\varepsilon },\mathcal{H}^0_{\varepsilon })
\leq C<\infty ,
\end{equation*}
for some positive constant $C$ independent of $\varepsilon$.

\item[(iii)] There exist $\varrho >0$ and a positive nondecreasing function $Q$
such that, for all $t\geq 0$,
\begin{equation*}
\operatorname{dist}_{\mathcal{H}^0_{\varepsilon }}(\mathcal{S}_{\varepsilon }(t)B,
\mathfrak{M}_{\varepsilon })\leq Q(\Vert B\Vert _{\mathcal{H}^0_{\varepsilon }})
e^{-\varrho t},
\end{equation*}
for every nonempty bounded subset $B$ of $\mathcal{H}^0_{\varepsilon }$.
\end{itemize}
\end{theorem}

\begin{remark} \rm
Above, the fractal dimension of $\mathfrak{M}_{\varepsilon}$ in
$\mathcal{H}^0_{\varepsilon}$ is given by
\begin{equation*}
\dim _{\mathrm{F}}(\mathfrak{M}_{\varepsilon },\mathcal{H}^0_{\varepsilon})
:=\limsup_{r\to 0}\frac{\ln \mu _{\mathcal{H}^0_{\varepsilon }}
(\mathfrak{M}_{\varepsilon },r)}{-\ln r}<\infty
\end{equation*}
where $\mu _{\mathcal{H}^0_{\varepsilon }}(\mathcal{X},r)$ denotes the
minimum number of $r$-balls from $\mathcal{H}^0_{\varepsilon }$ required to
cover $\mathcal{X}$.
\end{remark}

The proof of Theorem \ref{t:exponential-attractors} follows from the application
 of an abstract result reported here for our problem
(see e.g. \cite{CPS06,GMPZ10}; cf. also Remark \ref{rem_att} below).

\begin{proposition} \label{abstract1}
Let $( \mathcal{S}_{\varepsilon },\mathcal{H}^0_{\varepsilon
}) $ be a dynamical system for each $\varepsilon\in[0,1]$. Assume the
following hypotheses hold:
\begin{itemize}
\item[(C1)] There exists a bounded absorbing set $\mathcal{B}_{\varepsilon
}^{1}\subset \mathcal{V}^1_{\varepsilon }$ which is positively invariant
for $\mathcal{S}_{\varepsilon }(t)$.
More precisely, there exists a time $t_{1}>0$, uniform in $\varepsilon$, such that
\begin{equation*}
\mathcal{S}_{\varepsilon }(t)\mathcal{B}_{\varepsilon }^{1}\subset \mathcal{B}_{\varepsilon }^{1}
\end{equation*}
for all $t\geq t_{1}$ where $\mathcal{B}_{\varepsilon }^{1}$ is endowed with
the topology of $\mathcal{H}^0_{\varepsilon }$.

\item[(C2)] There is $t^{\ast }\geq t_{1}$ such that the map
$\mathcal{S}_{\varepsilon}(t^{\ast })$ admits the decomposition,
for each $\varepsilon \in (0,1]$ and for all
$\Upsilon _0,\Xi_0\in \mathcal{B}_{\varepsilon }^{1}$,
\begin{equation*}
\mathcal{S}_{\varepsilon }(t^{\ast })\Upsilon _0
- \mathcal{S}_{\varepsilon }(t^{\ast })\Xi_0 = L_{\varepsilon }
(\Upsilon_0,\Xi_0) + R_{\varepsilon }(\Upsilon_0,\Xi_0)
\end{equation*}
where, for some constants $\alpha ^{\ast }\in (0,\frac{1}{2})$ and $\Lambda
^{\ast }=\Lambda ^{\ast }(\Omega ,t^{\ast },\omega)\geq 0$, the following hold:
\begin{gather}
\| L_{\varepsilon }(\Upsilon_0,\Xi_0) \|_{\mathcal{H}^0_{\varepsilon }}
\leq \alpha ^{\ast } \| \Upsilon _0-\Xi_0 \|_{\mathcal{H}^0_{\varepsilon}} ,
\label{difference-decomposition-L} \\
\| R_{\varepsilon}(\Upsilon_0,\Xi_0) \|_{\mathcal{V}^1_{\varepsilon }}
\leq \Lambda ^{\ast } \| \Upsilon_0-\Xi_0 \|_{\mathcal{H}^0_{\varepsilon }}.
 \label{difference-decomposition-K}
\end{gather}

\item[(C3)] The map
\begin{equation*}
(t,\Upsilon) \mapsto \mathcal{S}_{\varepsilon }(t)\Upsilon:[t^{\ast },
2t^{\ast }]\times \mathcal{B}_{\varepsilon }^{1}\to \mathcal{B}_{\varepsilon }^{1}
\end{equation*}
is Lipschitz continuous on $\mathcal{B}_{\varepsilon }^{1}$ in the topology
of $\mathcal{H}^0_{\varepsilon }$.
\end{itemize}
Then $(\mathcal{S}_{\varepsilon },\mathcal{H}^0_{\varepsilon }) $
possesses an exponential attractor $\mathfrak{M}_{\varepsilon }$
in $\mathcal{B}_{\varepsilon }^{1}$.
\end{proposition}

We now prove the hypotheses of Proposition \ref{abstract1} and
we again remind the reader that for the remainder of the article,
we assume that the smallness criteria \eqref{smallness-criteria} holds,
in addition to the assumption \eqref{mu-4}.
We begin with the perturbation Problem \eqref{Pe}.
The results for the singular Problem \eqref{P0} will follow.

\begin{lemma}  \label{t:to-C1}
Condition {\rm (C1)} holds for each $\varepsilon\in(0,1]$ and $\omega\in(0,1)$.
Moreover, for all $R>0$ and
 $\Upsilon_0=(U_0,\Phi_0)\in\mathcal{V}^1_{\varepsilon}
=\mathbb{V}^1\times\mathcal{K}^2_{\varepsilon}$ with
$\|\Upsilon_0\|_{\mathcal{V}^1_\varepsilon}\le R$ for all $\varepsilon\in(0,1]$,
there exists a positive constant $P_1=P_1(\nu_1,\widetilde{P}_0)$ and a
positive monotonically increasing function $Q(\cdot)$, each independent of $\varepsilon$,
such that, for all $t\ge0$,
\begin{align}
\| ( U(t),\Phi^t ) \|^2_{\mathcal{V}^1_{\varepsilon}}
\le Q(R) e^{-\min\{\delta,1\}t} ( t+1 ) + 2P_1.  \label{strong-decay}
\end{align}
\end{lemma}

\begin{proof}
Let $\varepsilon\in(0,1]$, $\omega\in(0,1)$ and
$\Upsilon_0=(U_0,\Phi_0)\in\mathcal{V}^1_{\varepsilon}
=\mathbb{V}^1\times\mathcal{K}^2_{\varepsilon}$.
For all $s,t\ge0$, let $Z(t)={\rm{A_W^{\alpha,\beta}}}U(t)$ and
$\Theta^t(s)={\rm{A_W^{\alpha,\beta}}}\Phi^t(s)$.
In equations \eqref{weak-solution-1}-\eqref{weak-solution-2}, take
$\Xi=Z(t)$ and $\Pi=\Theta^t(s)$.
Proceeding as in \cite[proof of Theorem 3.11]{Gal-Shomberg15-2}
(however, this time we are able to enjoy the uniform bounds \eqref{strong-defn-1}),
we obtain the identities
\begin{gather}
\langle \partial _{t}U,{Z}\rangle _{\mathbb{X}^2}
+\omega\langle \mathrm{A_{W}^{0,\beta}}U,{Z}\rangle _{\mathbb{X}^2}
+\langle \Phi^{t}, Z\rangle _{\mathcal{M}^{1}_{\varepsilon}}
+\langle F(U),{Z}\rangle _{\mathbb{X}^2}=0,  \label{qest3}
\\
\langle \partial _{t}\Phi^{t},\Theta^{t}\rangle _{\mathcal{M}^{1}_{\varepsilon}}
=\langle \mathrm{T}_{\varepsilon}\Phi^{t},\Theta^{t}
\rangle _{\mathcal{M}^{1}_{\varepsilon}}+\langle U,\Theta^{t}
\rangle _{\mathcal{M}^{1}_{\varepsilon}}.  \label{qest4}
\end{gather}
These two identities may be combined together after we observe that,
from the definition of the product given in \eqref{sc},
\begin{equation} \label{qest2}
\begin{aligned}
\langle \Phi^{t}, Z\rangle _{\mathcal{M}^{1}_{\varepsilon}}
& =\int_0^{\infty }\mu_{\varepsilon}(s)\langle \Phi^{t}( s),
 Z\rangle _{\mathbb{V}^{1}}\,\mathrm{d}s   \\
& =\int_0^{\infty }\mu_{\varepsilon}(s)\langle
\mathrm{A_{W}^{\alpha,\beta}}\Phi^{t}( s)
,Z\rangle _{\mathbb{X}^2}\,\mathrm{d}s   \\
& =\int_0^{\infty }\mu_{\varepsilon}(s)
 \langle \mathrm{A_{W}^{\alpha,\beta}}\Phi^{t}( s) ,
 \mathrm{A_{W}^{\alpha,\beta}}U\rangle _{\mathbb{X}^2}\,\mathrm{d}s   \\
& =\int_0^{\infty }\mu_{\varepsilon}(s)\langle \Theta^{t}( s),
 \mathrm{A_{W}^{\alpha,\beta}}U\rangle _{\mathbb{X}^2}\,\mathrm{d}s   \\
& =\int_0^{\infty }\mu_{\varepsilon}(s)\langle \Theta^{t}( s) ,
 U\rangle _{\mathbb{V}^{1}}\,\mathrm{d}s   \\
& =\langle U,\Theta^{t}\rangle _{\mathcal{M}^{1}_{\varepsilon}}.
\end{aligned}
\end{equation}
Now inserting \eqref{qest2} into \eqref{qest3} and adding the result
to \eqref{qest4}, we obtain the identity
\begin{equation} \label{qest5}
\begin{aligned}
&\langle \partial _{t}U,{Z}\rangle _{\mathbb{X}^2}
  + \omega\langle \mathrm{A_{W}^{0,\beta}}U,{Z}\rangle _{\mathbb{X}^2}
  + \langle \partial _{t}\Phi^{t},\Theta^{t}
 \rangle _{\mathcal{M}^{1}_{\varepsilon}} \\
& - \langle \mathrm{T}_{\varepsilon}\Phi^{t},\Theta^{t}
 \rangle _{\mathcal{M}^{1}_{\varepsilon}}
 + \langle F(U),{Z}\rangle _{\mathbb{X}^2} = 0.
\end{aligned}
\end{equation}
Next we write
\begin{equation} \label{qest10}
\langle \partial _{t}U,{Z}\rangle _{\mathbb{X}^2}
=\langle \partial _{t}U,{\mathrm{A^{\alpha,\beta}_{W}}}{U}
 \rangle _{\mathbb{X}^2}  \\
=\langle \partial _{t}U,{U}\rangle _{\mathbb{V}^{1}}   \\
=\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \| U \|^2_{\mathbb{V}^1},
\end{equation}
\begin{equation} \label{qest11}
\begin{aligned}
\omega\langle {\mathrm{A_{W}^{0,\beta}}}U,{Z}\rangle _{\mathbb{X}^2}
& = \omega( \langle \nabla u,\nabla z \rangle_{L^2(\Omega)}
  + \langle \nabla_\Gamma u,\nabla_\Gamma z \rangle_{L^2(\Gamma)}
  + \beta\langle v,z \rangle_{L^2(\Gamma)} )    \\
& = \omega\langle \mathrm{A_{W}^{\alpha,\beta}}U,Z \rangle_{\mathbb{X}^2}
 - \omega\alpha\langle u,z\rangle_{L^2(\Omega)}   \\
& = \omega\| Z \|^2_{\mathbb{X}^2} - \omega\alpha\langle u,z\rangle_{L^2(\Omega)}  ,
\end{aligned}
\end{equation}
and
\begin{equation} \label{qest12}
\begin{aligned}
\langle \partial _{t}\Phi^{t},\Theta^{t}\rangle _{\mathcal{M}^{1}_{\varepsilon}}
& = \int_0^\infty \mu_{\varepsilon}(s) \langle \partial_t\Phi^t(s),\Theta^{t}(s)
 \rangle_{\mathbb{V}^1}\,\mathrm{d}s  \\
& = \int_0^\infty \mu_{\varepsilon}(s)
 \langle \partial_t{\mathrm{A^{\alpha,\beta}_{W}}}\Phi^t(s),
 \Theta^t(s) \rangle_{\mathbb{X}^2}\,\mathrm{d}s   \\
& = \int_0^\infty \mu_{\varepsilon}(s) \langle \partial_t\Theta^t(s),
 \Theta^t(s) \rangle_{\mathbb{X}^2}\,\mathrm{d}s   \\
& = \langle \partial_t\Theta^t,\Theta^t \rangle_{\mathcal{M}^0_\varepsilon}   \\
& = \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|
 \Theta^t \|^2_{\mathcal{M}^0_\varepsilon}.
\end{aligned}
\end{equation}
Combining \eqref{qest5}-\eqref{qest12} brings us to the differential identity,
which holds for almost all $t\ge0$,
\begin{equation}  \label{qest13}
\begin{aligned}
&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}
 \big\{ \| U \|^2_{\mathbb{V}^1} + \| \Theta^t \|^2_{\mathcal{M}^0_\varepsilon} \big\}
 + \omega\| Z \|^2_{\mathbb{X}^2} - \langle \mathrm{T}_{\varepsilon}\Phi^{t},
 \Theta^{t}\rangle _{\mathcal{M}^{1}_{\varepsilon}}
 + \langle F(U),{Z}\rangle _{\mathbb{X}^2}   \\
& = \omega\alpha\langle u,z \rangle_{L^2(\Omega)}.
\end{aligned}
\end{equation}
With assumption \eqref{mu-4} we are able to obtain
\begin{equation} \label{qest14}
\begin{aligned}
\langle {\mathrm{T}_{\varepsilon}}\Phi^t,\Theta^t \rangle_{\mathcal{M}^1_\varepsilon}
 & = \int_0^\infty\mu_{\varepsilon}(s)\langle {\mathrm{T}_{\varepsilon}}\Phi^t(s),
 \Theta^t(s) \rangle_{\mathbb{V}^1}\,\mathrm{d}s   \\
& = \int_0^\infty\mu_{\varepsilon}(s)
 \langle {\mathrm{T}_{\varepsilon}}{\mathrm{A^{\alpha,\beta}_{W}}}\Phi^t(s),
 \Theta^t(s) \rangle_{\mathbb{X}^2}\,\mathrm{d}s   \\
& = \int_0^\infty\mu_{\varepsilon}(s)\langle {\mathrm{T}_{\varepsilon}}\Theta^t(s),
 \Theta^t(s) \rangle_{\mathbb{X}^2}\,\mathrm{d}s   \\
& = -\int_0^\infty\mu_{\varepsilon}(s)\langle \frac{\mathrm{d}}{\mathrm{d}s}
\Theta^t(s),\Theta^t(s) \rangle_{\mathbb{X}^2}\,\mathrm{d}s   \\
& = -\frac{1}{2}\int_0^\infty\mu_{\varepsilon}(s)\frac{\mathrm{d}}{\mathrm{d}s}
\|\Theta^t(s)\|^2_{\mathbb{X}^2}\,\mathrm{d}s   \\
& = \underbrace{-\frac{1}{2}\int_0^\infty\frac{\mathrm{d}}{\mathrm{d}s}
 (\mu_{\varepsilon}(s)\|\Theta^t(s)\|^2_{\mathbb{X}^2})\,\mathrm{d}s}_{=0}
 + \frac{1}{2}\int_0^\infty\mu'_{\varepsilon}(s)\|\Theta^t(s)\|^2_{\mathbb{X}^2}
 \,\mathrm{d}s   \\
& \le -\frac{\delta}{2}\int_0^\infty\mu_{\varepsilon}(s)\|
 \Theta^t(s)\|^2_{\mathbb{X}^2}\,\mathrm{d}s   \\
& = -\frac{\delta}{2}\|\Theta^t\|^2_{\mathcal{M}^0_\varepsilon}.
\end{aligned}
\end{equation}
Multiplying the nonlinear term by $Z$ in $\mathbb{X}^2$ produces, with
an application of integration by parts,
\begin{equation} \label{qest5bis}
\begin{aligned}
&\langle F(U),Z\rangle _{\mathbb{X}^2}\\
& =\int_{\Omega }f( u) ( -\Delta u + \alpha u )\,\mathrm{d}x
+\int_{\Gamma }\widetilde{g}( u) ( -\Delta
_{\Gamma }u+\partial_{\mathbf{n}}u+\beta u) \mathrm{d}\sigma
 \\
& =\int_{\Omega }f'( u) | \nabla
u| ^2\,\mathrm{d}x+\int_{\Gamma }\widetilde{g}'(
u) | \nabla _{\Gamma }u| ^2\mathrm{d}\sigma\\
&\quad +\alpha\int_\Omega f(u)u \,\mathrm{d}x+\beta\int_{\Gamma }\widetilde{g}( u)
  u \mathrm{d}\sigma
 +\int_{\Gamma }( \widetilde{g}( u) -f(
u) ) \partial_{\mathbf{n}}u \mathrm{d}\sigma.
\end{aligned}
\end{equation}
Directly from \eqref{assm-3} and \eqref{assm-4}, we see that
\begin{equation}
\int_{\Omega }f'( u) | \nabla
u| ^2\,\mathrm{d}x+\int_{\Gamma }\widetilde{g}'(
u) | \nabla _{\Gamma }u| ^2\mathrm{d}\sigma \ge -M_f\|\nabla u\|^2_{L^2(\Omega)}
-M_g\|\nabla_\Gamma u\|^2_{L^2(\Gamma)},  \label{qest6}
\end{equation}
and from \eqref{cons-1}-\eqref{cons-2}, we obtain
\begin{equation} \label{qest7}
\begin{aligned}
& \alpha\int_\Omega f(u)u \,\mathrm{d}x + \beta\int_{\Gamma }\widetilde{g}( u)
 u \mathrm{d}\sigma   \\
& = \alpha\int_\Omega f(u)u \,\mathrm{d}s + \beta\int_{\Gamma}g(u)u \mathrm{d}\sigma
  - \int_\Gamma \omega\beta^2 u^2 \mathrm{d}\sigma    \\
& \ge -\alpha\kappa_1\|u\|^2_{L^2(\Omega)} - \alpha\kappa_2
 -\beta\kappa_3\|u\|^2_{L^2(\Gamma)}-\beta\kappa_4-\omega\beta^2\|u\|^2_{L^2(\Gamma)}   \\
& \ge -C( \| U \|^2_{\mathbb{X}^2}+1 ),
\end{aligned}
\end{equation}
for some constant $C>0$, independent of $t$.
For the last term in \eqref{qest5bis} we recall
\cite[Proof of Theorem 3.11]{Gal-Shomberg15-2}.
Due to the assumptions \eqref{assm-1}-\eqref{assm-2} it suffices
to bound boundary integrals of the form, for some $r<\frac{5}{2}$,
\begin{equation*}
I:=\int_\Gamma u^{r+1}\partial_{\mathbf{n}} u \mathrm{d}\sigma.
\end{equation*}
Indeed, thanks to the trace and regularity embeddings, for
all $\omega\in(0,1)$ and for some $C_\omega\sim\frac{C}{\omega}>0$,
\begin{equation}
I  \le \|\partial_{\mathbf{n}}u\|_{H^{1/2}(\Gamma)}
\|u^{r+1}\|_{H^{-1/2}(\Gamma)}
 \le \frac{\omega}{4}\|u\|^2_{H^2(\Omega)}
 + C_\omega\|u^{r+1}\|^2_{H^{-1/2}(\Gamma)}.  \label{qest8}
\end{equation}
To bound the last term in \eqref{qest8} we will employ the
Sobolev embeddings (recall $\Gamma$ is two-dimensional)
$H^{1/2}(\Gamma) \hookrightarrow L^{4}(\Gamma)$ and
$H^{1}(\Gamma) \hookrightarrow L^{s}(\Gamma)$, for any $s\in (\frac{4}{3},\infty )$.
Then, by employing some basic H\"{o}lder inequalities
\begin{align}
\| u^{r+1}\| _{H^{-1/2}( \Gamma ) }^2
& = \sup_{\psi \in H^{1/2}( \Gamma ) :\| \psi \|
_{H^{1/2}( \Gamma ) }=1}| \langle u^{r+1},\psi
\rangle_{L^2(\Gamma)} | ^2   \\
& \le \sup_{\psi \in H^{1/2}( \Gamma ) :\| \psi \|
_{H^{1/2}( \Gamma ) }=1}\|u^{r+1}\psi \|_{L^1(\Gamma)}^2   \\
& \leq \| u\| _{L^{s}( \Gamma )
}^2\| u\| _{L^{\overline{s}r}( \Gamma )}^{2r}   \\
& \leq C\| u\| _{H^{1}( \Gamma )
}^2\| u\| _{L^{\overline{s}r}( \Gamma )
}^{2r},  \label{qest5qqq}
\end{align}
for some positive constant $C$ and for sufficiently
large $s\in (\frac{4}{3},\infty )$, where
$\overline{s}:=4s/(3s-4) >4/3$.
Next we exploit the interpolation inequality
\begin{equation*}
\| u\| _{L^{\overline{s}r}( \Gamma ) }
\leq C\| u\| _{H^2( \Gamma ) }^{1/2r} \| u\| _{L^2( \Gamma ) }^{1-1/2r},
\end{equation*}
provided that $r=1+2/\overline{s}<5/2$, where we further infer from
\eqref{qest5qqq} that
\begin{equation} \label{qest5qq}
\begin{aligned}
\| u^{r+1}\| _{H^{-1/2}( \Gamma ) }^2 &
\leq C\| u\| _{H^{1}( \Gamma )
}^2\| u \| _{H^2( \Gamma ) }\|
u \| _{L^2( \Gamma ) }^{2r-1}   \\
& \leq \eta \| u \| _{H^2( \Gamma )
}^2+C_{\eta }\| u \| _{H^{1}( \Gamma )
}^2\Big( \| u \| _{H^{1}( \Gamma )
}^2\| u \| _{L^2( \Gamma ) }^{2(
2r-1) }\Big),
\end{aligned}
\end{equation}
for any $\eta \in (0,1]$.
Inserting \eqref{qest5qq} into \eqref{qest8} and choosing a sufficiently
small $\eta =\omega /C_{\omega}$, by virtue of \eqref{equiv}, we easily deduce
\begin{equation}
I \leq \frac{\omega}{4} \| Z\| _{\mathbb{X}^2}^2
+ C_{\omega}\| u \| _{H^{1}( \Gamma )
}^2\Big( \| u \| _{H^{1}( \Gamma )
}^2\| u \| _{L^2( \Gamma ) }^{2(
2r-1) }\Big).  \label{qest5last}
\end{equation}
Together, \eqref{qest6}-\eqref{qest5last} provide the following bound
 on \eqref{qest5bis} for all $\omega>0$, and for some positive constants
$C$ and $C_{\omega}\sim\frac{C}{\omega}$,
\begin{equation} \label{qest9}
\begin{aligned}
\langle F(U),Z\rangle _{\mathbb{X}^2}
&\ge  - C(\|U\|^2_{\mathbb{X}^2}+1) - \frac{\omega}{4}\|Z\|^2_{\mathbb{X}^2}   \\
& \quad - C_{\omega}\| u \| _{H^{1}( \Gamma )
}^2\Big( \| u \| _{H^{1}( \Gamma )
}^2\| u \| _{L^2( \Gamma ) }^{2(
2r-1) }\Big).
\end{aligned}
\end{equation}
Also with Young's inequality,
\begin{equation} \label{q2-1}
\begin{aligned}
\omega\alpha\langle u,z \rangle_{L^2(\Omega)}
& \le \omega\alpha^2\|u\|^2_{L^2(\Omega)} + \frac{\omega}{4}\|z\|^2_{L^2(\Omega)} \\
& \le \omega\alpha^2\|u\|^2_{L^2(\Omega)} + \frac{\omega}{4}\|Z\|^2_{\mathbb{X}^2}.
\end{aligned}
\end{equation}
Applying the estimates \eqref{qest14}, \eqref{qest9} and \eqref{q2-1}
to \eqref{qest13}, we arrive at the differential inequality, which holds
for almost all $t\ge0$, and for $0<r<5/2$,
\begin{equation}  \label{cabsball-1}
\begin{aligned}
&\frac{\mathrm{d}}{\mathrm{d}t}\big\{ \| U \|^2_{\mathbb{V}^1}
 +\| \Theta^t \|^2_{\mathcal{M}^0_\varepsilon} \big\}
+ \omega\|Z\|^2_{\mathbb{X}^2} + \delta\|\Theta^t \|^2_{\mathcal{M}^0_\varepsilon}  \\
&\le C( \| U \|^2_{\mathbb{X}^2}+1)
+ C_\omega \|u\|^2_{H^1(\Gamma)}( \| u \|^2_{H^1(\Gamma)}
 \| u \|^{2(2r-1)}_{L^2(\Gamma)} ).
\end{aligned}
\end{equation}
On the left-hand side, we estimate the term $\omega\|Z\|^2_{\mathbb{X}^2}$ using
\begin{equation}
\|U\|^2_{\mathbb{V}^1}
=\langle U,{\rm{A_W^{\alpha,\beta}}}U \rangle_{\mathbb{X}^2}
=\langle U,Z\rangle_{\mathbb{X}^2}
 \le C_\omega\|U\|^2_{\mathbb{X}^2}+\omega\|Z\|^2_{\mathbb{X}^2}.  \label{cabsball-5}
\end{equation}
Finally, with \eqref{cabsball-5} and the uniform bounds \eqref{weak-bound},
we now obtain from \eqref{cabsball-1}, with $m_1:=\min\{1,\delta\}>0$,
\begin{equation} \label{cabsball-2}
\begin{aligned}
&\frac{\mathrm{d}}{\mathrm{d}t}\big\{ \| U \|^2_{\mathbb{V}^1}
 +\| \Theta^t \|^2_{\mathcal{M}^0_\varepsilon} \big\}
 + m_1\big( \|U\|^2_{\mathbb{V}^1} + \|\Theta^t \|^2_{\mathcal{M}^0_\varepsilon} \big)   \\
& \le C_\omega ( 1+\|u\|^2_{H^1(\Gamma)} ) ( \|U\|^2_{\mathbb{V}^1}
 + \|\Theta^t\|^2_{\mathcal{M}^0_\varepsilon} ) + C,
\end{aligned}
\end{equation}
where $C_\omega>0$ depends on $\widetilde{P}_0$ from \eqref{weak-bound}.
Now from \eqref{fus-6}, we immediately find the following dissipation integral
\begin{equation}
\omega\int_t^{t+1} \|U(\tau)\|^2_{\mathbb{V}^1} \mathrm{d}\tau \le C,
\end{equation}
and we may apply a Gr\"{o}nwall-type inequality
(see e.g. Proposition \ref{GL} below) to \eqref{cabsball-2}.
We also recall \eqref{equiv} yields, for some $C_*>0$,
\begin{equation}
C_*^{-1}\| \Phi^t \|^2_{\mathbb{V}^2}
\le \| \mathrm{A_{W}^{\alpha,\beta}}\Phi^t \|^2_{\mathcal{M}^0_\varepsilon}
= \|\Theta^t\|^2_{\mathcal{M}^0_\varepsilon} \le C_*\| \Phi^t \|^2_{\mathbb{V}^2}.
\label{eqivi2}
\end{equation}
Hence, there are constants $M_1\ge 1$ and $P_1>0$, both uniform in $t$,
such that for all $t\ge0$, \eqref{cabsball-2} produces, for all $t\ge0$,
\begin{equation} \label{cabsball-9}
\begin{aligned}
 \| U(t) \|^2_{\mathbb{V}^1} + \| \Phi^t \|^2_{\mathcal{M}^2_{\varepsilon}}
& \le M_1 e^{-m_1t}( \| U_0 \|^2_{\mathbb{V}^1}
 + \| \Phi_0 \|^2_{\mathcal{M}^2_{\varepsilon}} ) + P_1   \\
& \le M_1 R e^{-m_1t} + P_1,
\end{aligned}
\end{equation}
where the last inequality follows because
$\|\Phi_0\|_{\mathcal{M}^2_{\varepsilon}}
\le\|\Phi_0\|_{\mathcal{K}^2_{\varepsilon}}\le R$.

To show \eqref{strong-decay} holds we need to control the last two terms
of the norm \eqref{new-norm}.
First, it is easy to see from \eqref{cabsball-9} that for all $t\ge0$
\begin{align}
\| U(t) \|^2_{\mathbb{V}^1} \le \| U(t) \|^2_{\mathbb{V}^1}
 + \| \Phi^t \|^2_{\mathcal{M}^2_{\varepsilon}} \le M_1R+P_1.
\end{align}
Then the conclusions of Lemmas \ref{what-1} and \ref{what-2} given above
now take the form
\begin{equation}
\begin{aligned}
& \varepsilon\|{\rm{T}}_\varepsilon\Phi^t\|^2_{\mathcal{M}^0_\varepsilon}
 + \sup_{\tau\ge0} \tau \mathbb{T}_\varepsilon(\tau;\Phi^t)   \\
& \le e^{-\delta t} \Big( \varepsilon \|{\rm{T}}_\varepsilon\Phi_0\|^2_{\mathcal{M}^0_\varepsilon}
 + 2\sup_{\tau\ge1} \tau\mathbb{T}_\varepsilon(\tau;\Phi_0) ( t+2 ) \Big)
 + M_1Re^{-m_1t}+P_1   \\
& \le e^{-m_1t} ( R(M_1+1) + Q(R) ( t+1 ) ) + P_1   \\
& \le Q(R) e^{-m_1t} ( t+1 ) + P_1.  \label{strong-bound-1}
\end{aligned}
\end{equation}
Together, the estimates \eqref{cabsball-9} and \eqref{strong-bound-1}
show that \eqref{strong-decay} holds.

The existence of a bounded set $\mathcal{B}^1_\varepsilon$ in $\mathcal{V}^1_{\varepsilon}$
that is absorbing and positively invariant for $\mathcal{S}_\varepsilon(t)$ follows
from \eqref{strong-decay}.
Indeed, define
\begin{equation*}
\mathcal{B}^1_{\varepsilon}:=\big\{ (U,\Phi)\in \mathcal{V}^1_{\varepsilon}
: \| (U,\Phi) \|_{\mathcal{V}^1_{\varepsilon}}\le \sqrt{2P_1+1} \big\}.
\end{equation*}
Then, given any nonempty bounded subset $B$ in
$\mathcal{H}^0_{\varepsilon}\setminus\mathcal{B}^1_{\varepsilon}$,
and after possibly enlarging the radius of $\mathcal{B}^1_{\varepsilon}$
in $\mathcal{H}^0_{\varepsilon}$ due to the embedding
$\mathcal{V}^1_{\varepsilon}\hookrightarrow\mathcal{H}^0_{\varepsilon}$,
we have that $\mathcal{S}_{\varepsilon}(t)B\subseteq\mathcal{B}^1_{\varepsilon}$,
in $\mathcal{H}^0_{\varepsilon}$, for all $t\ge t_1$ where $t_1=t_1(R)\ge0$
is such that there holds
\begin{equation}
e^{-\min\{\delta,1\}t_1}( t_1+1 ) \le \frac{1}{Q(R)}.  \label{time1}
\end{equation}
This establishes (C1) and completes the proof when $\varepsilon\in(0,1]$.
\end{proof}

The following result refers to the strong solutions developed
in \cite[Theorem 3.11]{Gal-Shomberg15-2} (see Theorem \ref{t:strong-solutions}
above) whose initial data is now taken in
$\mathcal{V}^1_\varepsilon\subset\mathcal{H}^0_\varepsilon$.

\begin{corollary}
For all $\Upsilon=(U_0,\Phi_0)\in\mathcal{H}^1_{\varepsilon}
=\mathbb{V}^1\times\mathcal{M}^2_\varepsilon$, it follows that any strong
solution $(U(t),\Phi^t)$ to Problem \eqref{Pe} is bounded, uniformly in
$t$ and $\varepsilon$; indeed, thanks to \eqref{strong-decay} there is a
constant $\widetilde{P}_1>0$, depending on the bound $P_1$ and the initial datum,
but independent of $t$ and $\varepsilon$, in which,
\begin{equation} \label{strong-bound}
\limsup_{t\to+\infty} \|\mathcal{S}_\varepsilon(t)(U_0,\Phi_0)
\|_{\mathcal{V}^1_{\varepsilon}} \le \widetilde{P}_1.
\end{equation}
\end{corollary}

We can now give a decay estimate for $\Phi^t$ in $\mathcal{M}^1_\varepsilon$.

\begin{lemma}\label{t:Phi-decay-1}
There holds, for all $\varepsilon\in(0,1]$, $\omega\in(0,1)$,
$\Upsilon_0=(U_0,\Phi_0)\in\mathcal{V}^1_\varepsilon$, and for all $t\geq 0$,
\begin{equation}
\label{Phi-decay-1}
\| \Phi^t \|^2_{\mathcal{M}^1_\varepsilon}
\leq \| \Phi_0 \|^2_{\mathcal{M}^1_\varepsilon} e^{-\delta t/2\varepsilon}
+ C(\widetilde{P}_0)\varepsilon.
\end{equation}
\end{lemma}

\begin{proof}
Let $\varepsilon\in(0,1]$, $\omega\in(0,1)$ and
$\Upsilon_0=(U_0,\Phi_0)\in\mathcal{H}^0_\varepsilon$.
As in the proof of Lemma \ref{weak-ball}, take $\Pi=\Phi^t(s)$
in equation \eqref{weak-solution-2} to obtain
\begin{align*}
&\int_0^\infty  \mu_\varepsilon(s) \langle \partial_t \Phi^t(s),
 {\rm{A_W^{\alpha,\beta}}} \Phi^t(s) \rangle_{\mathbb{X}^2} \,\mathrm{d}s \\
& = \int_0^\infty \mu_\varepsilon(s) \langle {\rm{T}}_\varepsilon
\Phi^t(s),{\rm{A_W^{\alpha,\beta}}} \Phi^t(s) \rangle_{\mathbb{X}^2}
\,\mathrm{d}s + \int_0^\infty \mu_\varepsilon(s)
\langle U,{\rm{A_W^{\alpha,\beta}}} \Phi^t(s) \rangle_{\mathbb{X}^2} \,\mathrm{d}s.
\end{align*}
Combining \eqref{operator-T-1}, \eqref{fus-2}, \eqref{fus-5}, and \eqref{fus-7},
we obtain
\begin{equation}
\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \| \Phi^t \|^2_{\mathcal{M}^1_\varepsilon}
+ \frac{\delta}{2\varepsilon}\|\Phi^t\|^2_{\mathcal{M}^1_\varepsilon}
\le \langle U,\Phi^t \rangle_{\mathcal{M}^1_\varepsilon}.  \label{Phi-decay-3}
\end{equation}
Estimating the product on the right-hand side with Young's inequality,
\begin{equation} \label{Phi-decay-5}
\begin{aligned}
\langle U,\Phi^t \rangle_{\mathcal{M}^1_\varepsilon}
& = \int_0^\infty \mu_\varepsilon(s) \langle U,
 \Phi^t \rangle_{\mathbb{V}^1} \,\mathrm{d}s \\
& \le \int_0^\infty \mu_\varepsilon(s) \| U \|_{\mathbb{V}^1}
 \|\Phi^t \|_{\mathbb{V}^1} \,\mathrm{d}s \\
& \leq \|U\|_{\mathbb{V}^1}\|\Phi^t\|_{\mathcal{M}^1_\varepsilon} \\
& \leq \frac{1}{\delta}\|U\|^2_{\mathbb{V}^1}
 + \frac{\delta}{4\varepsilon}\| \Phi^t \|^2_{\mathcal{M}^1_\varepsilon},
\end{aligned}
\end{equation}
we combine \eqref{Phi-decay-3} and \eqref{Phi-decay-5} to find that,
for almost all $t\geq 0$,
\begin{equation} \label{Phi-decay-6}
\frac{\mathrm{d}}{\mathrm{d}t}\| \Phi^t \|^2_{\mathcal{M}^1_\varepsilon}
+ \frac{\delta}{4\varepsilon} \| \Phi^t \|^2_{\mathcal{M}^1_\varepsilon}
\leq \frac{1}{\delta}\|U\|^2_{\mathbb{V}^1}.
\end{equation}
Thus, applying a Gr\"{o}nwall type inequality whereby integrating
\eqref{Phi-decay-6} over the interval $(0,t)$, recalling the uniform
bound \eqref{strong-bound}, produces \eqref{Phi-decay-1}.
\end{proof}

\begin{corollary}
From Lemma \ref{t:Phi-decay-1} we obtain the limit, for each $t>0$ fixed,
\begin{equation}
\label{Phi-decay-2}
\lim_{\varepsilon\to 0}\| \Phi^t \|_{\mathcal{M}^1_\varepsilon} = 0.
\end{equation}
In addition, since $e^{-\delta t/2\varepsilon} < e^{-\delta t/2}
\varepsilon^{\delta t/2} < \varepsilon^{\delta T/2}$ for all
$\varepsilon\in(0,1]$ and for all $t$ in the compact interval $[0,T]$,
for some $T>0$, then inequality \eqref{Phi-decay-1} is estimated by,
\[
\| \Phi^t \|^2_{\mathcal{M}^1_\varepsilon}
\leq \max\left\{ \| \Phi_0 \|^2_{\mathcal{M}^1_\varepsilon},
C(\widetilde{P}_0) \right\} ( e^{\delta T/2}+\varepsilon ).
\]
Define the constants
$\Lambda_0 = \max\big\{ \| \Phi_0 \|^2_{\mathcal{M}^1_\varepsilon}
e^{-\delta T/2},C(\widetilde{P}_0) \big\}^{1/2}$ and
$p_0=\min\{ \frac{\delta T}{4},\frac{1}{2}\}$.
Then, for all $\varepsilon\in(0,1]$ and for all $t\in[0,T]$, there holds
\[
\| \Phi^t \|_{\mathcal{M}^1_\varepsilon} \leq \Lambda_0 \varepsilon^{p_0}.
\]
\end{corollary}

We now go on to establish the next condition of Proposition \ref{abstract1}.

\begin{lemma}  \label{t:to-C2}
Condition {\rm (C2)} holds for each $\varepsilon\in(0,1]$ and $\omega\in(0,1)$.
The constants $t^*$ and $\ell^*$ depend on $\omega, \delta$ and
the constant due to the embedding $\mathbb{V}^1\hookrightarrow \mathbb{X}^2$.
\end{lemma}

\begin{proof}
Let $\varepsilon\in(0,1]$ and $\omega\in(0,1)$.
Let $\Upsilon_0=(U_0,\Phi_0), \Xi_0=(V_0,\Psi_0)\in \mathcal{B}^{1}_{\varepsilon}$.
Define the pair of trajectories, for $t\geq 0$,
$\Upsilon(t)=\mathcal{S}_{\varepsilon}(t)\Upsilon _0=(U(t),\Phi^t)$ and
$\Xi(t)=\mathcal{S}_{\varepsilon}(t)\Xi _0=(V(t),\Psi^t)$.
For each $t\geq 0$, decompose the difference
$\overline{\Delta}(t):=\Upsilon (t)-\Xi(t)$ with
$\overline{\Delta}_0:=\Upsilon_0-\Xi _0$ as follows:
\begin{equation*}
\overline{\Delta}(t)=\widehat{\Upsilon}(t)+\widehat{\Xi}(t)
\end{equation*}
where $\widehat{\Upsilon}(t)=(\widehat{V}(t),\widehat{\Psi}^t)$
and $\widehat{\Xi}(t)=(\widehat{W}(t),\widehat{\Theta}^t)$ are solutions
of the problems:
\begin{equation} \label{diff-decomp-u}
 \begin{gathered}
\partial_t \widehat{V}(t) + \omega {\mathrm{A^{0,\beta}_{W}}}\widehat{V}(t)
+ \int_0^\infty \mu_{\varepsilon}(s) {\mathrm{A^{\alpha,\beta}_{W}}}
\widehat{\Psi}^t(s) \,\mathrm{d}s = 0, \\
\partial_t\widehat{\Psi}^t(s) = {\rm{T}_{\varepsilon}} \widehat{\Psi}^t(s)
+ \widehat{V}(t), \\
\widehat{\Upsilon}(0) = \Upsilon_0-\Xi_0,
\end{gathered}
\end{equation}
and
\begin{equation} \label{diff-decomp-v}
\begin{gathered}
\partial_t \widehat{W}(t) + \omega {\mathrm{A^{0,\beta}_{W}}}\widehat{W}(t)
 + \int_0^\infty \mu_{\varepsilon}(s) {\mathrm{A^{\alpha,\beta}_{W}}}
\widehat{\Theta}^t(s) \,\mathrm{d}s + F(U(t)) - F(V(t)) = 0, \\
\partial_t \widehat{\Theta}^t(s) = {\rm{T}_{\varepsilon}} \widehat{\Theta}^t(s)
+ \widehat{W}(t), \\
\widehat{\Xi}(0) = {\bf{0}}.
\end{gathered}
\end{equation}

\noindent\textbf{Step 1.} (Proof of \eqref{difference-decomposition-L}.)
By estimating along the usual lines, after multiplying \eqref{diff-decomp-u}$_1$
 by $\widehat{V}$ in $\mathbb{X}^2$ and multiplying equation
\eqref{diff-decomp-u}$_2$ by ${\mathrm{A^{\alpha,\beta}_{W}}}\widehat{\Psi}^t$
in $\mathcal{M}^0_\varepsilon=L^2_{\mu_{\varepsilon}}(\mathbb{R}_+;\mathbb{X}^2)$,
we easily obtain the differential inequality,
\begin{equation}
\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}
 \big\{ \| \widehat{V} \|^2_{\mathbb{X}^2}
+ \| \widehat{\Psi}^t \|^2_{\mathcal{M}^1_{\varepsilon}} \big\}
+ C^{-1}_{\overline{\Omega}}\omega\| \widehat{V} \|^2_{\mathbb{X}^2}
+ \frac{\delta}{2}\| \widehat{\Psi}^t \|^2_{\mathcal{M}^1_{\varepsilon}}
\le 0,   \label{diff-decomp-1}
\end{equation}
where the constant $C_{\overline{\Omega}}>0$ is due to the embedding
$\mathbb{V}^1\hookrightarrow \mathbb{X}^2$; i.e.,
$\|\widehat{V}\|^2_{\mathbb{X}^2}
\leq C_{\overline{\Omega}}\|\widehat{V}\|^2_{\mathbb{V}^1}$.
Set $m_2:=\min\{ 2C^{-1}_{\overline{\Omega}}\omega,\delta \} > 0$.
Thus, \eqref{diff-decomp-1} becomes, for almost all $t\geq 0$,
\begin{equation*}
\frac{\mathrm{d}}{\mathrm{d}t}\big\{ \| \widehat{V} \|^2_{\mathbb{X}^2}
+ \| \widehat{\Psi}^t \|^2_{\mathcal{M}^1_{\varepsilon}} \big\}
+ m_2 ( \| \widehat{V} \|^2_{\mathbb{X}^2}
+ \| \widehat{\Psi}^t \|^2_{\mathcal{M}^1_{\varepsilon}} ) \leq 0.
\end{equation*}
After applying a Gr\"{o}nwall inequality, we have that for all $t\geq 0$,
\begin{equation}  \label{to-C2-L}
\| ( \widehat{V}(t),\widehat{\Psi}^t ) \|_{\mathcal{H}^0_{\varepsilon}}
\leq \|\overline{\Delta}_0 \|_{\mathcal{H}^0_{\varepsilon}} e^{-m_2 t/2}.
\end{equation}
Set $t^{\ast }:=\max \{t_{1},\frac{2}{m_2}\ln 4 \}$ (recall $t_1$
was defined in \eqref{time1} in the proof of Lemma \ref{t:to-C1}).
Then, for all $t\geq t^{\ast }$, \eqref{difference-decomposition-L} holds
with $L=\widehat{\Upsilon}(t^\ast)=(\widehat{V}(t^{\ast}),\widehat{\Phi}^{t^{\ast}})$,
 and
\begin{equation*}
\ell^{\ast } = e^{-m_2 t^{\ast }/2} < \frac{1}{2}.
\end{equation*}

Before we show that \eqref{difference-decomposition-K} holds, we need
to establish a crucial bound.
\smallskip

\noindent\textbf{Step 2.} (A preliminary bound for $\widehat{W}$ and
$\widehat{\Theta}^t$.)
We claim, for each $0<T<\infty$, there holds
\begin{gather}
 \widehat{W}\in L^\infty([0,T];\mathbb{X}^2)\cap L^2([0,T];\mathbb{V}^1),  \label{hat-claim-1} \\
 \widehat{\Theta}^t\in L^\infty([0,T];\mathcal{M}^1_{\varepsilon}).  \label{hat-claim-2}
\end{gather}

To show this, we multiply equation \eqref{diff-decomp-v}$_1$ by
$\widehat{W}$ in $\mathbb{X}^2$ and multiply equation \eqref{diff-decomp-v}$_2$
by $\mathrm{A_{W}^{\alpha,\beta}}\widehat{\Theta}^t$ in $\mathcal{M}^0_\varepsilon$.
Summing the resulting two identities produces,
\begin{equation} \label{ba-vid-4}
\begin{aligned}
& \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}
\big\{ \| \widehat{W} \|^2_{\mathbb{X}^2}
 + \| \widehat{\Theta}^t \|^2_{\mathcal{M}^1_{\varepsilon}} \big\}
 + \omega\langle \mathrm{A_{W}^{0,\beta}}\widehat{W},\widehat{W}
 \rangle_{\mathbb{X}^2} - \langle \mathrm{T}_{\varepsilon}\widehat{\Theta}^t,
 \widehat{\Theta}^t \rangle_{\mathcal{M}^1_{\varepsilon}}   \\
& + \langle F(U)-F(V),\widehat{W} \rangle_{\mathbb{X}^2}   \\
& = 0.
\end{aligned}
\end{equation}
The first of the three products above can be re-written, using the definition
of the $\mathbb{V}^1$ norm (see \eqref{v1b}), as
\begin{equation}  \label{ba-pop-2}
\begin{aligned}
\omega\langle \mathrm{A_{W}^{0,\beta}}\widehat{W},\widehat{W}
\rangle_{\mathbb{X}^2}
& = \omega\langle \mathrm{A_{W}^{\alpha,\beta}}\widehat{W},\widehat{W}
\rangle_{\mathbb{X}^2} - \omega\alpha\langle \widehat{w},\widehat{w}
\rangle_{L^2(\Omega)}  \\
& = \omega\|\widehat{W}\|^2_{\mathbb{V}^1} - \omega\alpha \| \widehat{w} \|^2_{L^2(\Omega)}   \\
& = \omega \Big( \| \nabla \widehat{w} \|^2_{L^2(\Omega)}
+ \| \nabla_\Gamma \widehat{w} \|^2_{L^2(\Gamma)}
+ \beta\| \widehat{w} \|^2_{L^2(\Gamma)} \Big).
\end{aligned}
\end{equation}
As with the above estimate \eqref{qest14}, we have
\begin{align}
\langle  \mathrm{T}_{\varepsilon}\widehat{\Theta}^t,\widehat{\Theta}^t
\rangle_{\mathcal{M}^1_{\varepsilon}}
 \leq -\frac{\delta}{2}\| \widehat{\Theta}^t \|^2_{\mathcal{M}^1_{\varepsilon}}.
 \label{ba-pop-4}
\end{align}
Using assumptions \eqref{assm-1} and \eqref{assm-2} with data in the bounded
set $\mathcal{B}^1_\varepsilon$ and the uniform bound \eqref{weak-bound},
we now estimate the nonlinear terms as follows
\begin{equation} \label{func-1}
\begin{aligned}
\langle f(u)-f(v),\widehat{w}\rangle_{L^2(\Omega)}
& \le \|(f(u)-f(v))\widehat{w}\|_{L^1(\Omega)}   \\
& \le \|f(u)-f(v)\|_{L^{6/5}(\Omega)}\|\widehat{w}\|_{L^6(\Omega)}   \\
& \le \ell_1\|(u-v)(1+|u-v|^{r_1})\|_{L^{6/5}(\Omega)}\|\widehat{w}\|_{L^6(\Omega)}
  \\
& \le \ell_1\|u-v\|_{L^{6}(\Omega)}(1+\|u-v\|^{r_1}_{L^{3r_1/2}(\Omega)})
 \|\widehat{w}\|_{L^6(\Omega)}   \\
& \le C\|\widehat{w}\|_{H^1(\Omega)},
\end{aligned}
\end{equation}
where $C=C(\ell_1,\Omega,\widetilde{P}_0,r_1)>0$ and the last inequality
follows from the fact that $H^1(\Omega)\hookrightarrow L^6(\Omega)$ and
$H^1(\Omega)\hookrightarrow L^{3r_1/2}(\Omega)$ because $1\le r_1<\frac{5}{2}$.
Similarly for $\widetilde{g}$ (here the estimate is easier because
$H^1(\Gamma)\hookrightarrow L^p(\Gamma)$ for $1\le p<\infty$ as $\Gamma$
is two dimensional),
\begin{equation}
\langle \widetilde{g}(u)-\widetilde{g}(v),\widehat{w}\rangle_{L^2(\Gamma)}
\le C\|\widehat{w}\|_{H^1(\Gamma)}.  \label{func-2}
\end{equation}
Thus, \eqref{func-1} and \eqref{func-2} show that
\begin{equation}
\big|\langle F(U)-F(V),\widehat{W} \rangle_{\mathbb{X}^2}\big|
 \le C_\omega\| {\overline{\Delta}}_0 \|^2_{\mathcal{H}^0_{\varepsilon}}
+ \frac{\omega}{2}\| \widehat{W} \|^2_{\mathbb{V}^1},  \label{ba-pop-5}
\end{equation}
where $C_\omega\sim\frac{C}{\omega}$.
Together \eqref{ba-vid-4}-\eqref{ba-pop-5} yields the differential inequality,
 which holds for almost all $t\ge0$,
\begin{equation} \label{ba-pop-9}
\begin{aligned}
& \frac{\mathrm{d}}{\mathrm{d}t}
\big\{ \| \widehat{W} \|^2_{\mathbb{X}^2}
 + \| \widehat{\Theta}^t \|^2_{\mathcal{M}^1_{\varepsilon}} \big\}    \\
& + \omega ( \| \nabla \widehat{w} \|^2_{L^2(\Omega)}
 + \| \nabla_\Gamma \widehat{w} \|^2_{L^2(\Gamma)}
 + \beta\| \widehat{w} \|^2_{L^2(\Gamma)} )
 + \delta\| \widehat{\Theta}^t \|^2_{\mathcal{M}^1_{\varepsilon}}   \\
& \le C_{\omega}\| {\overline{\Delta}}_0 \|^2_{\mathcal{H}^0_{\varepsilon}}.
\end{aligned}
\end{equation}
Now integrating \eqref{ba-pop-9} with respect to $t$ in $[0,T]$, for some
fixed $0<T<\infty$, we obtain
\begin{equation} \label{ba-pop-10}
\begin{aligned}
& \| \widehat{W}(t) \|^2_{\mathbb{X}^2} + \| \widehat{\Theta}^t
 \|^2_{\mathcal{M}^1_{\varepsilon}}
+ \int_0^t \Big( \omega \Big( \| \nabla \widehat{w}(\tau) \|^2_{L^2(\Omega)} \\
&+ \| \nabla_\Gamma \widehat{w}(\tau) \|^2_{L^2(\Gamma)}
+ \beta\| \widehat{w}(\tau) \|^2_{L^2(\Gamma)} \Big)
+ \delta\| \widehat{\Theta}^\tau \|^2_{\mathcal{M}^1_{\varepsilon}} \Big)
\mathrm{d}\tau   \\
& \le C_{\omega}\| {\overline{\Delta}}_0 \|^2_{\mathcal{H}^0_{\varepsilon}}T.
\end{aligned}
\end{equation}
Using \eqref{ba-pop-10}, we easily deduce the claim
\eqref{hat-claim-1}-\eqref{hat-claim-2}.
\smallskip

\noindent\textbf{Step 3.} (Proof of \eqref{difference-decomposition-K})
We begin by multiplying equation \eqref{diff-decomp-v}$_1$ by
$K=\mathrm{A_{W}^{\alpha,\beta}}\widehat{W}$ in $\mathbb{X}^2$,
then, after applying $\mathrm{A_{W}^{\alpha,\beta}}$ to equation
\eqref{diff-decomp-v}$_2$, we multiply the result by
$\Lambda^t=\mathrm{A_{W}^{\alpha,\beta}}\widehat{\Theta}^t$
in $\mathcal{M}^0_\varepsilon=L^2_{\mu_{\varepsilon}}(\mathbb{R}_+;\mathbb{X}^2)$.
This leaves us with the two identities,
\begin{equation} \label{vid-1}
\begin{aligned}
&  \langle \partial_t\widehat{W},K \rangle_{\mathbb{X}^2}
 + \omega\langle \mathrm{A_{W}^{0,\beta}}\widehat{W},K \rangle_{\mathbb{X}^2}
 + \langle \mathrm{A_{W}^{\alpha,\beta}}\widehat{\Theta}^t,K
 \rangle_{\mathcal{M}^0_\varepsilon}   \\
& + \langle F(U)-F(V),K \rangle_{\mathbb{X}^2}   = 0.
\end{aligned}
\end{equation}
and
\begin{equation}
\langle \partial_t\mathrm{A_{W}^{\alpha,\beta}}\widehat{\Theta}^t,
\Lambda^t \rangle_{\mathcal{M}^0_\varepsilon}
= \langle \mathrm{A_{W}^{\alpha,\beta}} \mathrm{T}_{\varepsilon}
\widehat{\Theta}^t,\Lambda^t \rangle_{\mathcal{M}^0_\varepsilon}
+ \langle \mathrm{A_{W}^{\alpha,\beta}}\widehat{W},\Lambda^t
\rangle_{\mathcal{M}^0_\varepsilon}.  \label{vid-2}
\end{equation}
Observe that
\begin{equation}
\langle \mathrm{A_{W}^{\alpha,\beta}}\widehat{\Theta}^t,
K \rangle_{\mathcal{M}^0_\varepsilon}  = \langle \Lambda^t,
\mathrm{A_{W}^{\alpha,\beta}}\widehat{W} \rangle_{\mathcal{M}^0_\varepsilon}.
 \label{vid-22}
\end{equation}
Hence, combining \eqref{vid-1} and \eqref{vid-2} through \eqref{vid-22},
\begin{equation} \label{vid-3}
\begin{aligned}
& \langle \partial_t\widehat{W},K \rangle_{\mathbb{X}^2}
  + \omega\langle \mathrm{A_{W}^{0,\beta}}\widehat{W},K \rangle_{\mathbb{X}^2}
  + \langle \partial_t\mathrm{A_{W}^{\alpha,\beta}}\widehat{\Theta}^t,
 \Lambda^t \rangle_{\mathcal{M}^0_\varepsilon}\\
& - \langle \mathrm{A_{W}^{\alpha,\beta}} \mathrm{T}_{\varepsilon}
 \widehat{\Theta}^t,\Lambda^t \rangle_{\mathcal{M}^0_\varepsilon}
 + \langle F(U)-F(V),K \rangle_{\mathbb{X}^2}
 = 0.
\end{aligned}
\end{equation}
The first three products can be re-written as follows,
\begin{gather} \label{pop-1}
\langle \partial_t\widehat{W},K \rangle_{\mathbb{X}^2}
 = \langle \partial_t\widehat{W},\mathrm{A_{W}^{\alpha,\beta}}\widehat{W}
\rangle_{\mathbb{X}^2}
 = \langle \partial_t\widehat{W},\widehat{W} \rangle_{\mathbb{V}^1}
 = \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\| \widehat{W} \|^2_{\mathbb{V}^1},\\
\label{pop-2}
\begin{aligned}
\omega\langle \mathrm{A_{W}^{0,\beta}}\widehat{W},K \rangle_{\mathbb{X}^2}
& = \omega\langle \mathrm{A_{W}^{\alpha,\beta}}\widehat{W},K
 \rangle_{\mathbb{X}^2} - \omega\alpha\langle \widehat{w},
 k \rangle_{L^2(\Omega)}  \\
& = \omega\|K\|^2_{\mathbb{X}^2} - \omega\alpha\langle \widehat{w},k
\rangle_{L^2(\Omega)},
\end{aligned} \\
 \label{pop-3}
\langle \partial_t\mathrm{A_{W}^{\alpha,\beta}}\widehat{\Theta}^t,
\Lambda^t \rangle_{\mathcal{M}^0_\varepsilon}
 = \langle \partial_t\Lambda^t, \Lambda^t \rangle_{\mathcal{M}^0_\varepsilon}
 = \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\| \Lambda^t \|^2_{\mathcal{M}^0_\varepsilon}.
\end{gather}
Inserting \eqref{pop-1}-\eqref{pop-3} into \eqref{vid-3} gives us the
 differential identity
\begin{equation} \label{vid-4}
\begin{aligned}
&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}
\big\{ \| \widehat{W} \|^2_{\mathbb{V}^1} + \| \Lambda^t \|^2_{\mathcal{M}^0_\varepsilon}
\big\} + \omega \| K \|^2_{\mathbb{X}^2}
 - \langle \mathrm{T}_{\varepsilon}\widehat{\Theta}^t,\Lambda^t
 \rangle_{\mathcal{M}^0_\varepsilon}
+ \langle F(U)-F(V),K \rangle_{\mathbb{X}^2}\\
&= \omega\alpha\langle \widehat{w},k \rangle_{L^2(\Omega)}.
\end{aligned}
\end{equation}
Similar to \eqref{qest14}, we have
\begin{equation}
\langle \mathrm{A_{W}^{\alpha,\beta}} \mathrm{T}_{\varepsilon}\widehat{\Theta}^t,\Lambda^t
\rangle_{\mathcal{M}^0_\varepsilon} \leq -\frac{\delta}{2}
\| \Lambda^t \|^2_{\mathcal{M}^0_\varepsilon},  \label{pop-4}
\end{equation}
and in a similar fashion to \eqref{ba-pop-5}, we find
\begin{equation}
|\langle F(U)-F(V),K \rangle_{\mathbb{X}^2}|
 \le C_\omega\| {\overline{\Delta}}_0 \|^2_{\mathcal{H}^0_{\varepsilon}}
+ \frac{\omega}{2}\|K\|^2_{\mathbb{V}^1},  \label{pop-5}
\end{equation}
where $C_\omega\sim\frac{C}{\omega}$.
We also estimate
\begin{equation}
\omega\alpha\langle \widehat{w},k \rangle_{L^2(\Omega)}
\le \omega\alpha^2\| \widehat{W} \|^2_{\mathbb{X}^2}
+ \frac{\omega}{4}\| K \|^2_{\mathbb{X}^2}  \label{pop-6}
\end{equation}
Together \eqref{vid-4}-\eqref{pop-6} yields the differential inequality,
 which holds for almost all $t\ge0$,
\begin{equation} \label{pop-9}
\begin{aligned}
& \frac{\mathrm{d}}{\mathrm{d}t}
\big\{ \| \widehat{W} \|^2_{\mathbb{V}^1}
 + \| \Lambda^t \|^2_{\mathcal{M}^0_\varepsilon} \big\}
+ \omega \| \mathrm{A_{W}^{\alpha,\beta}}\widehat{W} \|^2_{\mathbb{X}^2}
+ \delta\| \Lambda^t \|^2_{\mathcal{M}^0_\varepsilon}   \\
& \le \alpha\| \widehat{W} \|^2_{\mathbb{X}^2}
+ C_{\omega}\| {\overline{\Delta}}_0 \|^2_{\mathcal{H}^0_{\varepsilon}}.
\end{aligned}
\end{equation}
Now integrating \eqref{pop-9} with respect to $t$ in $[0,T]$,
for some fixed $0<T<\infty$, we obtain
\begin{align}
& \| \widehat{W}(t) \|^2_{\mathbb{V}^1} + \| \Lambda^t \|^2_{\mathcal{M}^0_\varepsilon} + \int_0^t ( \omega \| \mathrm{A_{W}^{0,\beta}}\widehat{W}(\tau) \|^2_{\mathbb{X}^2} + \delta\| \Lambda^\tau \|^2_{\mathcal{M}^0_\varepsilon} ) \mathrm{d}\tau   \\
& \le \int_0^t \alpha\| \widehat{W}(\tau) \|^2_{\mathbb{X}^2} \mathrm{d}\tau + C_{\omega}\| {\overline{\Delta}}_0 \|^2_{\mathcal{H}^0_{\varepsilon}}T,  \label{pop-10}
\end{align}
where the right-hand side of the inequality makes sense thanks to \eqref{hat-claim-1}.
Now omitting the second and third terms from the left-hand side of \eqref{pop-10}, the following bound follows easily with Gr\"{o}nwall's inequality
\begin{equation}
\| \widehat{W}(t) \|^2_{\mathbb{V}^1}
\le C_{\omega}\| {\overline{\Delta}}_0 \|^2_{\mathcal{H}^0_{\varepsilon}}
T e^{\alpha T},  \label{david-1}
\end{equation}
and with this
\begin{equation}
\|\Lambda^t\|^2_{\mathcal{M}^0_\varepsilon}
 \le C_{\omega}\| {\overline{\Delta}}_0 \|^2_{\mathcal{H}^0_{\varepsilon}}
T e^{\alpha T},  \label{david-2}
\end{equation}
also follows.

To obtain the desired bound from \eqref{david-1} and \eqref{david-2},
first recall that there is $C_*>0$ (cf. \eqref{equiv}) such that
\begin{equation*}
\|\Lambda^t\|^2_{\mathcal{M}^0_\varepsilon}
= \|\mathrm{A_{W}^{\alpha,\beta}}\widehat{\Theta}^t\|^2_{\mathcal{M}^0_\varepsilon}
\ge C_*^{-1} \|\widehat{\Theta}^t\|^2_{\mathcal{M}^2_{\varepsilon}}.
\end{equation*}
Thus, letting $T=t^*$ (from Step 1), we obtain, for some positive monotonically
increasing function $M_2(\cdot)$,
\begin{equation}  \label{top-4}
\| ( \widehat{W}(t^*),\widehat{\Theta}^{t^*} ) \| _{\mathcal{H}^1_{\varepsilon}}
 \le M_2(t^*) \| {\overline{\Delta}}_0 \|_{\mathcal{H}^0_{\varepsilon}}.
\end{equation}
Now it suffices to show that for some positive constant $C(T)$,
there holds for all $t\in[0,T]$,
\begin{equation}
\|\widehat{\Theta}^t\|^2_{\mathcal{K}^2_{\varepsilon}}
 \le C(T)\| {\overline{\Delta}}_0 \|^2_{\mathcal{H}^0_{\varepsilon}}.  \label{david-3}
\end{equation}
First, we see that with an application of Lemma \ref{what-3} with
\eqref{david-1} and \ref{david-2} there holds, for all $t\in[0,T]$,
\begin{equation}
\sup_{\tau\ge1} \tau\mathbb{T}_\varepsilon(\tau;\widehat{\Theta}^t)
\le C(T)\| {\overline{\Delta}}_0 \|^2_{\mathcal{H}^0_{\varepsilon}},  \label{david-4}
\end{equation}
and secondly, by applying the weak form of Lemma \ref{what-1}
(see Remark \ref{r:what-trans}), we find that for all $t\in[0,T]$,
\begin{equation}
\varepsilon\|{\rm{T}}_\varepsilon\Phi^t\|^2_{\mathcal{M}^0_\varepsilon}
\le C(T)\| {\overline{\Delta}}_0 \|^2_{\mathcal{H}^0_{\varepsilon}}.  \label{david-5}
\end{equation}
Together \eqref{david-4}-\eqref{david-5} establish \eqref{david-3}.
Therefore, inequality \eqref{difference-decomposition-K} now follows
with $R= \Xi(t^{\ast})=(\widehat{W}(t^*),\widehat{\Theta}^{t^*})$ and
$\wp^{\ast}=M_2(t^{\ast})\geq 0$ (for a suitably updated function $M_2$).
This completes the proof of (C2).
\end{proof}

\begin{lemma} \label{t:to-C3}
Condition (C3) holds for each $\varepsilon\in(0,1]$ and $\omega\in(0,1)$.
\end{lemma}

\begin{proof}
Let $\varepsilon\in(0,1]$ and $\omega\in(0,1)$.
Let $R>0$ and $\Upsilon_0=(U_0,\Phi_0)\in\mathcal{V}^1_{\varepsilon}$
where $\|\Upsilon_0\|_{\mathcal{V}^1_{\varepsilon}}\le R$.
Directly from \eqref{strong-bound}, there holds,
\begin{equation*}
\| \mathcal{S}_{\varepsilon}(t)\Upsilon_0 \|_{\mathcal{V}^1_{\varepsilon}}
\leq \widetilde{P}_1,
\end{equation*}
but where now the size of the initial data, $R$, depends on the size of
$\mathcal{B}^{1}_{\varepsilon}$.
Hence, on the compact interval $[t^{\ast },2t^{\ast }]$, the map
$t\mapsto S(t)\Upsilon _0$ is Lipschitz continuous for each fixed $\Upsilon _0\in
\mathcal{B}^{1}_{\varepsilon}$.
This means there is a constant $L=L(t^{\ast })>0$ such that
\begin{equation*}
\Vert \mathcal{S}_\varepsilon(t_1)\Upsilon _0 - \mathcal{S}_\varepsilon(t_2)
\Upsilon_0\Vert _{\mathcal{H}^0_{\varepsilon}} \leq L|t_1-t_2|.
\end{equation*}
Together with the continuous dependence estimate \eqref{cde}, (C3) follows.
\end{proof}

\begin{remark} \rm
According to Proposition \ref{abstract1}, for each $\varepsilon\in(0,1]$,
the semigroup
$\mathcal{S}_{\varepsilon}(t):\mathcal{H}^0_{\varepsilon}
\to \mathcal{H}^0_{\varepsilon}$ possesses an exponential attractor,
$\mathfrak{M}_{\varepsilon}\subset \mathcal{B}^{1}_{\varepsilon}$,
which attracts bounded subsets of $\mathcal{B}^{1}_{\varepsilon}$ exponentially
fast (in the topology of $\mathcal{H}^0_{\varepsilon}$).
Moreover, in light of the results in this section -- which are {\em{uniform}}
in the perturbation parameter $\varepsilon$ -- we now simply accept the corresponding
results for the simpler limit Problem \eqref{P0}.
In this setting we use the notation for the compact absorbing set
$\mathcal{B}^1_0$ and the exponential attractor $\mathfrak{M}_0$
admitted by the semigroup
$\mathcal{S}_0(t):\mathcal{H}^0_0=\mathbb{X}^2\to\mathbb{X}^2$.
\end{remark}

\begin{remark}  \label{rem_att} \rm
To show that the attraction property (iii) in Theorem \ref{t:exponential-attractors}
also holds -- that is,  to show that the basin of attraction of
$\mathfrak{M}_\varepsilon$ is all of $\mathcal{H}^0_\varepsilon$ -- we appeal to the
transitivity of the exponential attraction in Proposition \ref{t:exp-attr}
and Theorem \ref{t:trans-out} below.
\end{remark}


\subsection{Basin of attraction (and global attractors)}


The main result in this section has two purposes:
primary, per the above remark, it will help us show that the exponential
attractors we seek attract every bounded subset in $\mathcal{H}^0_\varepsilon$
(not just $\mathcal{B}^1_\varepsilon$).
This property is sometimes not obvious because of the difficulties using
spaces involving memory (we refer the reader to Section 1 of this article
and to the rate of attraction of $\mathcal{B}^1_\varepsilon$ as found in
Lemma \ref{t:to-C1}).
However, we overcome this problem, partly, by proving a condition on the
solution semigroup $\mathcal{S}_\varepsilon$ that is also essential for the existence
of global attractors (also called a universal attractors); we refer to the
asymptotic compactness/smoothing of $\mathcal{S}_\varepsilon$, which happens
to occur in our case with an exponential rate.
Together, the asymptotic compactness os $\mathcal{S}_\varepsilon$
(Theorem \ref{t:trans-out} below) and the existence of an absorbing sets
in $\mathcal{H}^0_\varepsilon$ (Lemma \ref{weak-ball}) will guarantee the existence
of a global attractor that is compact in $\mathcal{H}^0_\varepsilon$ and bounded
in $\mathcal{V}^1_\varepsilon$.

\begin{theorem}  \label{t:trans-out}
For each $\varepsilon\in[0,1]$, there is a positive constant $\varrho_1$
and a monotonically increasing function $Q(\cdot)$ in which for every
nonempty bounded subset $B$ of $\mathcal{H}^0_\varepsilon$ there holds, for all $t\ge0$,
\begin{equation*}
\operatorname{dist}_{\mathcal{H}^0_{\varepsilon }}(\mathcal{S}_{\varepsilon }(t)B,
\mathcal{B}^1_{\varepsilon })
\leq Q(\Vert B\Vert _{\mathcal{H}^0_{\varepsilon }})e^{-\varrho_1 t}.
\end{equation*}
\end{theorem}

\begin{proof}
Because of the smoothing properties of the associated with the Wentzell
parabolic Problem \eqref{P0} (cf. \cite{Gal12-1}), we limit ourselves
to the case when $\varepsilon\in(0,1]$.

Let $\varepsilon\in(0,1]$ and $B$ be a nonempty bounded subset of $\mathcal{H}^0_\varepsilon$.
By recalling Lemma \ref{weak-ball}, we already know that there is a bounded
absorbing set that is exponentially attracting in $\mathcal{H}^0_\varepsilon$, i.e.,
for all $t\ge0$ there holds
\begin{equation*}
\operatorname{dist}_{\mathcal{H}^0_{\varepsilon }}(\mathcal{S}_{\varepsilon }
(t)B,\mathcal{B}^0_{\varepsilon })
\leq Q(\Vert B\Vert _{\mathcal{H}^0_{\varepsilon }})e^{-\nu_0 t},
\end{equation*}
so owing once again to the transitivity of exponential attraction
(cf. Proposition \ref{t:exp-attr} below) it suffices to show that, for all $t\ge0$,
\begin{equation}
\operatorname{dist}_{\mathcal{H}^0_{\varepsilon }}(\mathcal{S}_{\varepsilon }
(t)\mathcal{B}^0_\varepsilon,\mathcal{B}^1_{\varepsilon })\leq Q(P_0)e^{-\varrho_0 t},
 \label{smooth-1}
\end{equation}
for some positive constant $\varrho_0$ and for some positive monotonically
increasing function $Q(\cdot)$, each independent of $\varepsilon$.
(Recall from \eqref{ball-0} that $\sqrt{P_0+1}$ is the radius of
$\mathcal{B}^0_\varepsilon$.)

To prove \eqref{smooth-1}, the idea is to show that for each $\varepsilon\in(0,1]$
and for each $\Upsilon_0\in\mathcal{H}^0_\varepsilon$ we can decompose the semigroup
\[
\mathcal{S}_\varepsilon(t)\Upsilon_0 = \mathcal{Z}_\varepsilon(t)\Upsilon_0
 + \mathcal{K}_\varepsilon(t)\Upsilon_0
\]
where the operators $\mathcal{Z}_\varepsilon$ are uniformly (exponentially)
decaying to zero and $\mathcal{K}_\varepsilon$ are uniformly compact
(bounded in $\mathcal{V}^1_\varepsilon$) for large $t$.
This is done in the following lemmas.
\end{proof}

The following decomposition and subsequently more general lemmas,
as we will allow the datum to belong to {\em{any}} bounded subset of the
phase space $\mathcal{H}^0_\varepsilon$, can be seen to follow
\cite[Theorem 6.10--Lemma 6.12]{CPS06} with obvious changes to account
for the dynamic boundary conditions with memory.
Hence, we will limit the proofs to sketches of the most important details.

First, choose a constant $M_F>0$, based on \eqref{assm-3}, \eqref{assm-4},
and \eqref{func}, so that the map defined by, for all $s\in\mathbb{R}$,
\begin{equation*}
F_0(s):=F(s)+M_Fs,
\end{equation*}
satisfies, for every $s\in\mathbb{R}$,
\begin{equation}
F'_0(s)\ge \binom{0}{0}.  \label{up}
\end{equation}
Next, let $\Upsilon_0=(U_0,\Phi_0)\in \mathcal{H}^0_{\varepsilon }$.
Then rewrite Problem \eqref{Pe} into the system of equations in $(V,\Psi)$
and $(W,\Theta)$, where $(V,\Psi)+(W,\Theta)=(U,\Phi)$,
\begin{equation}  \label{decomp-v}
\begin{gathered}
\partial_t V(t) + \omega {\mathrm{A^{0,\beta}_{W}}}V(t)
+ \int_0^\infty \mu_{\varepsilon}(s) {\mathrm{A^{\alpha,\beta}_{W}}} \Psi^t(s)
\,\mathrm{d}s + F_0(U(t)) - F_0(W(t)) = 0, \\
\partial_t\Psi^t(s) = {\rm{T}_{\varepsilon}} \Psi^t(s) + V(t), \\
(V(0),\Psi^0)=\Upsilon_0,
\end{gathered}
\end{equation}
and
\begin{gather}
\partial_t W(t) + \omega {\mathrm{A^{0,\beta}_{W}}}W(t)
+ \int_0^\infty \mu_{\varepsilon}(s) {\mathrm{A^{\alpha,\beta}_{W}}}
 \Theta^t(s) \mathrm{d}s + F_0(W(t)) - M_F U(t) = 0, \nonumber \\
\partial_t \Theta^t(s) = {\rm{T}_{\varepsilon}} \Theta^t(s) + W(t), 
 \label{decomp-w} \\
(W(0),\Theta^0)={\mathbf{0}}. \nonumber
\end{gather}

In view of Lemmas \ref{t:decay} and \ref{t:compact} below,
we define the one-parameter family of maps,
$\mathcal{K}_{\varepsilon }(t):\mathcal{H}^0_{\varepsilon }
\to \mathcal{H}^0_{\varepsilon }$, by
\begin{equation*}
\mathcal{K}_{\varepsilon }(t)\Upsilon_0:=( W(t),\Theta^t),
\end{equation*}
where $(W,\Theta)$ is a solution of \eqref{decomp-w}.
With such $(W,\Theta)$, we may define a second function $(V,\Psi)$
as the solution of \eqref{decomp-v}.
Through the dependence of $(V,\Psi)$ on $(W,\Theta)$ and $(U(0),\Phi^0)=\Upsilon_0$,
the solution of \eqref{decomp-v} defines a
one-parameter family of maps,
$\mathcal{Z}_{\varepsilon}(t):\mathcal{H}^0_{\varepsilon}
\to \mathcal{H}^0_{\varepsilon }$, defined by
\begin{equation*}
\mathcal{Z}_{\varepsilon}(t)\Upsilon_0:=(V(t),\Psi^t).
\end{equation*}
Notice that if $(V,\Psi)$ and $(W,\Theta)$ are solutions to \eqref{decomp-v}
and \eqref{decomp-w}, respectively, then the function
$(U(t),\Phi^t):=(V(t),\Psi^t)+(W(t),\Theta^t)$ is a solution to Problem
\eqref{Pe}.

The next result shows that the operators $\mathcal{Z}_{\varepsilon }$
are uniformly decaying to zero in $\mathcal{H}_{\varepsilon}$.

\begin{lemma}  \label{t:decay}
For each $\varepsilon \in (0,1]$ and
$\Upsilon_0=(U_0,\Phi_0)\in \mathcal{H}^0_{\varepsilon }$,
there exists a unique global weak solution
$(V,\Psi)\in C([0,\infty);\mathcal{H}^0_{\varepsilon })$
to problem \eqref{decomp-v}.
Moreover, given $R>0$, then for all $\Upsilon_0\in\mathcal{H}^0_\varepsilon$
with $\|\Upsilon_0\|_{\mathcal{H}^0_{\varepsilon}}\leq R$ for all
$\varepsilon \in (0,1]$, there exists $\nu_0'>0 $, independent of $\varepsilon $,
such that, for all $t\geq 0$,
\begin{equation}
\|\mathcal{Z}_{\varepsilon}(t)\Upsilon_0\|_{\mathcal{H}^0_{\varepsilon}}\leq
Q(R)e^{-\nu_0' t}.  \label{eq:uniform-decay}
\end{equation}
\end{lemma}

\begin{proof}
The existence of a global weak solution to \eqref{decomp-v} follows as
the proof of \cite[Theorem \ref{t:weak-solutions}]{Gal&Shomberg15}.
It remains to show that \eqref{eq:uniform-decay} holds.

The proof is very similar to the proof of Lemma \ref{weak-ball} save that
the assumptions \eqref{assm-3}-\eqref{assm-4} become crucial.
Indeed, the constant $C$ on the right-hand side of \eqref{fus-10} vanishes
because nonlinear terms now satisfy the bound
\[
\langle F_0(U)-F_0(W),V \rangle_{\mathbb{X}^2} \ge 0
\]
as here $V=U-W$ and \eqref{up} holds.
\end{proof}

The following lemma establishes the uniform compactness of the operators
$\mathcal{K}_{\varepsilon }$.

\begin{lemma}  \label{t:compact}
For each $\varepsilon \in (0,1]$ and
$\Upsilon_0=(U_0,\Phi_0)\in \mathcal{H}^0_{\varepsilon}$,
there exists a unique global weak solution
$(W,\Theta)\in C([0,\infty);\mathcal{H}^0_{\varepsilon})$
to problem \eqref{decomp-w}.
Moreover, given $R>0$, then for all $\Upsilon_0\in \mathcal{H}^0_{\varepsilon }$
with $\|\Upsilon_0\|_{\mathcal{H}^0_{\varepsilon}} \leq R$ for all
$\varepsilon \in (0,1]$, there holds for all $t\geq 0$,
\begin{equation*}
\|\mathcal{K}_{\varepsilon}(t)\Upsilon_0\|_{\mathcal{V}^1_{\varepsilon }}\leq Q(R),
\end{equation*}
Furthermore, the operators $K_{\varepsilon }$ are uniformly compact
in $\mathcal{H}^0_{\varepsilon }$.
\end{lemma}

\begin{proof}
Again, in light of \cite[Theorem \ref{t:weak-solutions}]{Gal&Shomberg15},
it remains to show that the operators $\mathcal{K}_{\varepsilon }$ are
uniformly compact in $\mathcal{H}^0_{\varepsilon }$.

This time we appeal to Lemma \ref{t:to-C1} whereby only trivial changes
are required in the proof in order to show Lemma \ref{t:compact} holds.
\end{proof}

\begin{remark} \rm
These results -- with datum contained to the absorbing set $\mathcal{B}^0_\varepsilon$ --
complete the proof of Theorem \ref{t:trans-out}.
Consequently, the existence of a (finite dimensional) global attractor
 $\mathcal{A}_\varepsilon$, $\varepsilon\in(0,1]$, for $\mathcal{S}_\varepsilon$ follows.
\end{remark}

\begin{theorem} \label{t:global-attractors}
For each $\varepsilon \in (0,1]$, the semigroup $\mathcal{S}_\varepsilon$ admits
a unique global attractor
\begin{equation*}
\mathcal{A}_{\varepsilon } = \omega (\mathcal{B}^0_{\varepsilon})
:= \cap_{s\geq 0}{\overline{\cup_{t\geq s}\mathcal{S}_{\varepsilon }(t)
\mathcal{B}^0_{\varepsilon}}}^{\mathcal{H}^0_{\varepsilon }}
\end{equation*}
in $\mathcal{H}^0_{\varepsilon }$.
Moreover, the following statements hold:
\begin{itemize}

\item[(1)] For each $t\geq 0$,
$\mathcal{S}_{\varepsilon }(t)\mathcal{A}_{\varepsilon }
= \mathcal{A}_{\varepsilon }$, and

\item[(2)] For every nonempty bounded subset $B$ of $\mathcal{H}^0_{\varepsilon }$,
\begin{equation}
\lim_{t\to \infty }\operatorname{dist}_{\mathcal{H}^0_{\varepsilon
}}(\mathcal{S}_{\varepsilon }(t)B,\mathcal{A}_{\varepsilon })=0.
\label{gl_attraction}
\end{equation}

\item[(3)] The global attractor $\mathcal{A}_{\varepsilon }$ is bounded in
$\mathcal{V}^1_{\varepsilon }$ (hence, compact in $\mathcal{H}^0_\varepsilon$)
and trajectories on $\mathcal{A}_{\varepsilon }$ are strong solutions
(in the sense of Definitions \ref{d:strong-solution}).

\item[(4)] The fractal dimension is bounded, uniformly in $\varepsilon$, i.e.,
\[
\dim_{\mathrm{F}}(\mathcal{A}_\varepsilon,\mathcal{H}^0_\varepsilon)
\le \dim_{\mathrm{F}}(\mathfrak{M}_\varepsilon,\mathcal{H}^0_\varepsilon) \le C < \infty,
\]
for some constant $C>0$ independent of $\varepsilon$.
\end{itemize}
\end{theorem}

\begin{proof}
The existence and boundedness of the global attractor for Problem \eqref{P0}
can be found in \cite[Theorem 2.3]{Gal12-2} and the references therein.
Thus, it suffices to show the result for the perturbation Problem \eqref{Pe},
with $\varepsilon\in(0,1]$.
By referring to the standard literature (cf. e.g. \cite{Babin&Vishik92,Temam88})
and Lemma \ref{weak-ball}, Lemma \ref{t:to-C1}, and Theorem \ref{t:trans-out},
the proof is complete.
\end{proof}


\subsection{Robustness and H\"{o}lder continuity of the exponential attractors}

What remains in this section is to show that the family of exponential attractors
is robust, or H\"{o}lder continuous with respect to the perturbation parameter $\varepsilon$.
As a preliminary step, we follow, for example \cite[see p. 177]{CPS05} among others,
and define the so-called {\em{canonical extension}} map,
$\mathcal{E}:\mathbb{X}^2 \to \mathfrak{M}_\varepsilon$, by
\begin{equation}
\label{canonical-extension-map-2}
\mathcal{E}(U) = 0.
\end{equation}
With this, define the {\em{lift}} mapping, $\mathcal{L}\mathbb{X}^2 \to \mathcal{H}^0_\varepsilon$, by
\begin{equation}
\label{lift}
\mathcal{L}(U) = (U,\mathcal{E}(U)) = (U,0).
\end{equation}

\begin{theorem} \label{cont_att}
Let the assumptions of Theorem \ref{t:exponential-attractors} be satisfied.
For each $\varepsilon$, the semigroup of solution operators,
$\mathcal{S}_\varepsilon(t)$ admits an exponential attractor
$\mathfrak{M}_\varepsilon$ in which the family of compact sets
$(\mathbb{M}_{\varepsilon }) _{\varepsilon \in [ 0,1] }$ defined by
\begin{equation}\label{global-attractors-family}
\mathbb{M}_\varepsilon :=  \begin{cases}
\mathcal{L}\mathfrak{M}_0 & \text{for }\varepsilon=0 \\
\mathfrak{M}_\varepsilon & \text{for }\varepsilon\in(0,1]
\end{cases}
\end{equation}
is H\"{o}lder continuous for every $\varepsilon \in [ 0,1] $, i.e.,
there exist constants $\Lambda >0$, $\tau \in (0,1/2]$ independent of
$\varepsilon $, such that, for every $0\leq \varepsilon _{2}<\varepsilon _{1}\leq 1$,
the symmetric Hausdorff distance satisfies
\begin{equation}  \label{symm-diff}
\operatorname{dist}_{\mathcal{H}^0_{\varepsilon _{1}}}^{\mathrm{sym}}
(\mathfrak{M}_{\varepsilon _{1}},\mathfrak{M}_{\varepsilon _{2}})
\leq \Lambda (\varepsilon _{1}-\varepsilon _{2})^{\tau }.
\end{equation}
\end{theorem}

\begin{remark} \rm
The {\em{symmetric Hausdorff distance}} between two subsets $A,B$ of a
Banach space $X$ is defined as
\begin{align}
\operatorname{dist}_{X}^{\mathrm{sym}}(A,B)
:= \max\{ \operatorname{dist}_{X}(A,B),\operatorname{dist}_{X}(B,A) \}.
\end{align}
More precisely, the condition given in \eqref{symm-diff} implies the
family of attractors is both upper- and lower-semicontinuous (thus, continuous)
at each value of the perturbation parameter $\varepsilon\in[0,1)$.
\end{remark}

To prove Theorem \ref{cont_att} we develop the main assumptions of the abstract
results found in the seminal works \cite{GGMP05,GMPZ10}.
As in Proposition \ref{abstract1} above, the assumptions suited specifically
for our needs appear in \cite[(H2) and (H3) of Theorem A.2]{CPS06}.

As above, the number $L>0$ shown below is used to denote the
(local) Lipschitz constant of the mapping $F:\mathbb{V}^1\to\mathbb{X}^2$.

\begin{proposition}  \label{abstract2}
Let the assumptions of Proposition \ref{abstract1} be
satisfied. In addition, assume the following:

\begin{itemize}
\item[(C4)] The canonical extension map $\mathcal{E}|_{\mathcal{B}_0^{1}}:
\mathbb{X}^2 \to \mathcal{H}^0_\varepsilon$ given by
\eqref{canonical-extension-map-2} is Lipschitz continuous.

\item[(C5)] There is a constant $\Lambda_1=\Lambda_1(L,\Omega ,t^{\ast })>0$
such that, for all $t\in [ t^{\ast },2t^{\ast }]$ and for all
$\Upsilon_0=(U_0,\Phi_0)\in\mathcal{B} _{\varepsilon }^{1}$,
\begin{equation}
\| \mathcal{S}_{\varepsilon }(t)\Upsilon_0 - \mathcal{LS}_0(t)\mathbb{P}
\Upsilon_0 \| _{\mathcal{H}^0_{\varepsilon }} \leq \Lambda_1\sqrt{\varepsilon }.
\label{robust-part}
\end{equation}
Here, $\mathbb{P} :\mathcal{H}^0_{\varepsilon }\to \mathcal{H}^0_0$
denotes the projection defined by, for all
$\Upsilon = (U,\Phi)\in \mathcal{H}^0_{\varepsilon }$,
\begin{equation*}
\mathbb{P} \Upsilon = U.
\end{equation*}

\item[(C6)] There is a constant $\Lambda_{2}=\Lambda_{2}(L,\Omega ,t^{\ast })>0$
such that, for all $t\in [ t^{\ast },2t^{\ast }]$,
$\Upsilon_0 = (U_0,\Phi_0)\in \mathcal{B}_{\varepsilon _{1}}^{1}
\subset \mathcal{H}^1_{\varepsilon_1}$
and, for all $0<\varepsilon_2<\varepsilon_1\leq 1$,
\begin{equation}
\| \mathcal{S}_{\varepsilon _{1}}(t)\Upsilon_0
- \mathcal{S}_{\varepsilon _{2}}(t)\Upsilon_0 \| _{\mathcal{H}^0_{\varepsilon_1}}
\leq \Lambda_{2}(\varepsilon_1-\varepsilon_2)^{1/2}.  \label{Holder-part}
\end{equation}
\end{itemize}

Then, the family of exponential attractors
$(\mathbb{M}_{\varepsilon})_{\varepsilon \in \left[ 0,1\right] }$
is H\"{o}lder continuous for every $\varepsilon \in \left[ 0,1\right] $
in the sense of Theorem \ref{cont_att}.
\end{proposition}

\begin{remark} \rm
Condition (C6) below does not appear in \cite{CPS06}, but rather we now
borrow \cite[(H7) of Theorem 4.4]{GMPZ10}, cf. also
\cite[(P4) of Theorem 2.1]{MPZ07}.
\end{remark}

\begin{lemma}
Condition {\rm (C4)} holds.
\end{lemma}

\begin{proof}
Based on the definition of $\mathcal{E}$ given in \eqref{canonical-extension-map-2},
the result is vacuously true.
\end{proof}

The following lemma proves condition (C5) of Proposition \ref{abstract2}.
It shows that the difference between the semigroups
$\mathcal{S}_{\varepsilon}(t)$ and the lifted limit semigroup
$\mathcal{LS}_0(t)$ in $\mathcal{H}^0_{\varepsilon }$, on finite time
intervals, is of order $\varepsilon^{1/2}$.

\begin{lemma}
Let $T>0$. For all $\varepsilon\in(0,1]$, $\omega\in(0,1)$ and
$\Upsilon_0=(U_0,\Phi_0)\in\mathcal{H}^0_\varepsilon$ such that
$\|\Upsilon_0\|_{\mathcal{H}^0_\varepsilon} \leq R$ for all
$\varepsilon\in(0,1]$, there exists a positive constant $C(T)$,
independent of $\varepsilon$, but depending on $\omega$ and $T$,
in which, for all $t\geq [0,T]$,
\begin{equation}
\label{C5-key-1}
\| \mathcal{S}_\varepsilon(t)\Upsilon_0
- \mathcal{LS}_0(t) \mathbb{P} \Upsilon_0 \|_{\mathcal{H}^0_\varepsilon}
\leq C(T)\varepsilon^{1/2}.
\end{equation}
\end{lemma}

\begin{proof}
Let $\widehat{\Upsilon}(t)=(\widehat{U}(t),\widehat{\Phi}^t)$ denote the
solution of Problem P$_\varepsilon$ corresponding to the initial data
$\Upsilon_0=(U_0,\Phi_0)\in \mathcal{B}_{\varepsilon }^{1}$ and let $U(t)$
denote the solution of Problem P$_0$ corresponding to the initial data
$\mathbb{P} \Upsilon_0 = U_0\in \mathcal{B}_0^{1}$.
With the solution $U(t)$, define the function $\Phi^t$ by the solution
to the Cauchy problem
\begin{equation}\label{C5-key-2}
\begin{gathered}
\partial_t\Phi^t = \mathrm{T}_\varepsilon\Phi^t + U(t) \\
\Phi^0 = \mathbb{Q}\Upsilon_0=\Phi_0 \in \mathcal{M}^0_\varepsilon.
\end{gathered}
\end{equation}
With the (unique) solution to \eqref{C5-key-2}
(cf. Corollary \ref{t:memory-regularity-1}), define
$\Upsilon(t) := (U(t),\Phi^t)$ for all $t\geq 0$.
Let
\begin{align*}
\widehat{\Delta}(t) = (Z(t),\Theta^t)
:&= \widehat{\Upsilon}(t) - \Upsilon(t) \\
& = (\widehat{U}(t),\widehat{\Phi}^t) - (U(t),\Phi^t) \\
& = (\widehat{U}(t) - U(t), \widehat{\Phi}^t - \Phi^t);
\end{align*}
hence, $\widehat{\Delta}(t)=(Z(t),\Theta^t)$ satisfies the system
\begin{equation} \label{C5-key-3}
 \begin{gathered}
\begin{aligned}
&\partial_t Z(t) + \omega \mathrm{A_{W}^{0,\beta}}Z(t)
+ \int_0^\infty \mu_\varepsilon(s) \mathrm{A_{W}^{\alpha,\beta}}
\Theta^t(s) \,\mathrm{d}s + F(\widehat{U}(t)) - F(U(t)) \\
&= -\int_0^\infty \mu_\varepsilon(s) \mathrm{A_{W}^{\alpha,\beta}}
\Phi^t(s) \,\mathrm{d}s,
\end{aligned}\\
\partial_t\Theta^t(s) = {\rm{T_\varepsilon}} \Theta^t(s) + Z(t), \\
( Z(0),\Theta^0 ) = {\bf{0}}.
\end{gathered}
\end{equation}
Multiply \eqref{C5-key-3}$_1$ by $Z$ in $\mathbb{X}^2$ and
\eqref{C5-key-3}$_2$ by $\mathrm{A_{W}^{\alpha,\beta}}\Theta^t$ in
$L^2_{\mu_\varepsilon}(\mathbb{R}_+;\mathbb{X}^2)$, summing the resulting
identities and estimating as in the above arguments, it is not hard to
 see that there holds, for almost all $t\geq 0$,
\begin{equation} \label{C5-key-4}
\begin{aligned}
& \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}
\big\{ \| Z \|^2_{\mathbb{X}^2} + \| \Theta^t \|^2_{\mathcal{M}^1_\varepsilon} \big\}
 + \omega\| Z \|^2_{\mathbb{V}^1}
 + \frac{\delta}{2\varepsilon}\| \Theta^t \|^2_{\mathcal{M}^1_\varepsilon} \\
& \leq -\langle F(\widehat{U}) - F(U),Z \rangle_{\mathbb{X}^2}
 + \omega\alpha\|z\|^2_{L^2(\Omega)}
 - \int_0^\infty \mu_\varepsilon(s) \langle \mathrm{A_{W}^{\alpha,\beta}}\Phi^t(s),Z
 \rangle_{\mathbb{X}^2} \,\mathrm{d}s.
\end{aligned}\end{equation}
Recall that with \eqref{assm-3} we obtain
\begin{equation} \label{C5-key-5}
-\langle F(\widehat{U})-F(U),Z \rangle_{\mathbb{X}^2}
\leq M_F \| Z \|^2_{\mathbb{X}^2}.
\end{equation}
For the remaining term on the right-hand side, we apply the definition of
the norm and Young's inequality to find
\begin{align*}
- \int_0^\infty \mu_\varepsilon(s)
\langle \mathrm{A_{W}^{\alpha,\beta}} \Phi^t(s),Z
\rangle_{\mathbb{X}^2} \,\mathrm{d}s
& = -\int_0^\infty \mu_\varepsilon(s) \langle \Phi^t(s),
 Z \rangle_{\mathbb{V}^1} \,\mathrm{d}s \\
& \leq \|Z\|_{\mathbb{V}^1} \| \Phi^t \|_{\mathcal{M}^1_\varepsilon} \\
& \leq \omega\| Z \|^2_{\mathbb{V}^1}
+ \frac{1}{4\omega} \| \Phi^t \|^2_{\mathcal{M}^1_\varepsilon}.
\end{align*}
Recall that, by \eqref{Phi-decay-1}, for all $t\geq 0$,
\begin{equation} \label{C5-key-14}
\| \Phi^t \|^2_{\mathcal{M}^1_\varepsilon}
\leq \| \Phi_0 \|^2_{\mathcal{M}^1_\varepsilon}
e^{-\delta t/2\varepsilon} + C\varepsilon,
\end{equation}
where $C>0$ depends on the bound $\widetilde{P}_0$, but is uniform in
$\varepsilon$ and $t$.
Collecting \eqref{C5-key-4}-\eqref{C5-key-14} yields,
\begin{equation}\begin{aligned}
\label{C5-key-8}
&\frac{\mathrm{d}}{\mathrm{d}t}  \big\{ \| Z \|^2_{\mathbb{X}^2}
 + \| \Theta^t \|^2_{\mathcal{M}^1_\varepsilon} \big\}
+ \delta\| \Theta^t \|^2_{\mathcal{M}^1_\varepsilon} \\
& \leq 2( \omega\alpha + M_F ) \| Z \|^2_{\mathbb{X}^2}
+ C \varepsilon \| Z \|^2_{\mathbb{X}^2}
+ C_\omega \big( \| \Phi_0 \|^2_{\mathcal{M}^1_\varepsilon}
e^{-\delta t/2\varepsilon}  + \varepsilon \big).
\end{aligned}
\end{equation}
Integrating \eqref{C5-key-8} with respect to $t$ on the interval $[0,T]$,
for $T>0$, and then applying the initial conditions \eqref{C5-key-4}$_3$,
as well as the uniform bound \eqref{weak-bound}, we have
\begin{equation}
\label{C5-key-9}
\| Z(t) \|^2_{\mathbb{X}^2} + \| \Theta^t \|^2_{\mathcal{M}^1_\varepsilon}
\leq \int_0^t C \|Z(\tau) \|^2_{\mathbb{X}^2} \mathrm{d}\tau + C(T)\varepsilon.
\end{equation}

Next we seek an appropriate bound on the term with $Z$.
It follows from \eqref{C5-key-9} and Gronwall's inequality that there holds,
for all $t\geq 0$ and for all $\varepsilon\in(0,1]$,
\begin{equation}\label{C5-key-17}
\|Z(t)\|^2_{\mathbb{X}^2} \leq C(T)\varepsilon,
\end{equation}
where $C>0$ depends on $\omega$, $\delta$, and of course $T$,
 but not $\varepsilon$.

Returning to \eqref{C5-key-9}, we now see that there holds,
for all $t\in[\sqrt{\varepsilon},T]$ and for all $\varepsilon\in(0,1]$,
\begin{equation} \label{C5-key-12}
\| ( Z(t), \Theta^t ) \|^2_{\mathcal{H}^0_\varepsilon} \leq C(T)\varepsilon.
\end{equation}
Therefore \eqref{C5-key-1} follows. This finishes the proof.
\end{proof}

We will establish the H\"{o}lder continuity with the following lemma.
With regard to \cite{GMPZ10}, in particular, hypothesis (H7) of
Theorem 4.4 there, we {\em{do not}} perform an $\varepsilon$-scaling
of the memory variable.

\begin{lemma}
Condition {\rm (C6)} holds.
\end{lemma}

\begin{proof}
Assume $0<\varepsilon _{2}<\varepsilon _{1}\leq 1$.
Let $\Upsilon_0 = (U_0,\Phi^0) \in \mathcal{B}_{1}^{1}$.
Let $\widetilde{\Upsilon}(t)=(\widetilde{U}(t),\widetilde{\Phi}^t)$
denote the solution of Problem P$_{\varepsilon_1}$ corresponding to the
initial datum $\Upsilon_0$ and let
$\widetilde{\Xi}(t)=(\widetilde{V}(t),\widetilde{\Psi}^t)$
denote the solution Problem P$_{\varepsilon_2}$ corresponding to the
same initial datum $\Upsilon_0$. Let
\begin{align*}
\widetilde{\Delta}(t) = ( \widetilde{Z}(t),\widetilde{\Theta}^t )
:&= \widetilde{\Upsilon}(t) - \widetilde{\Xi}(t) \\
& = (\widetilde{U}(t),\widetilde{\Phi}^t) - (\widetilde{V}(t),\widetilde{\Psi}^t) \\
& = ( \widetilde{U}(t) - \widetilde{V}(t),\widetilde{\Phi}^t - \widetilde{\Psi}^t ).
\end{align*}
\begin{equation} \label{tilde-z-difference}
\begin{gathered}
\begin{aligned}
&\partial_t\widetilde{Z} + \omega \mathrm{A_{W}^{0,\beta}} \widetilde{Z}
+ \int_0^\infty \mu_{\varepsilon_1}(s) \mathrm{A_{W}^{\alpha,\beta}}
\widetilde{\Theta}^t(s)\,\mathrm{d}s + F(\widetilde{U}) - F(\widetilde{V})\\
& = \int_0^\infty ( \mu_{\varepsilon_2}(s)
 - \mu_{\varepsilon_1}(s) ) \mathrm{A_{W}^{\alpha,\beta}}
 \widetilde{\Psi}^t(s)\,\mathrm{d}s
\end{aligned} \\
\partial_t\widetilde{\Theta}^t(s)
= {\rm{T}}_{\varepsilon_1}\widetilde{\Theta}^t(s) + \widetilde{Z}(t) \\
\widetilde{Z}(0) = {\bf{0}}, \quad \widetilde{\Theta}^0 = {\bf{0}}.
\end{gathered}
\end{equation}
Observe, by the definition of ${\rm{T}}_\varepsilon$,
$( {\rm{T_{\varepsilon_1}}} - {\rm{T}}_{\varepsilon_2} )\widetilde{\Psi}^t(s)=0$.
We proceed in the usual fashion by multiplying \eqref{tilde-z-difference}$_1$
by $\widetilde{Z}$ in $\mathbb{X}^2$, and multiplying equation
\eqref{tilde-z-difference}$_2$ by
$\mathrm{A_{W}^{\alpha,\beta}} \widetilde{\Theta}^t$ in
$L^2_{\mu_{\varepsilon_1}}(\mathbb{R}_+;\mathbb{X}^2)$,
summing the results, we arrive at the identity
\begin{equation} \label{C6-1}
\begin{aligned}
&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}
 \big\{ \| \widetilde{Z} \|^2_{\mathbb{X}^2}
+ \| \widetilde{\Theta}^t \|^2_{\mathcal{M}^0_{\varepsilon_1}} \big\}
+ \omega\|\widetilde{Z} \|^2_{\mathbb{V}^1}
 - \int_0^\infty \mu_{\varepsilon_1}(s)\langle {\rm{T_{\varepsilon_1}}}
 \widetilde{\Theta}^t(s), \mathrm{A_{W}^{\alpha,\beta}}
 \widetilde{\Theta}^t(s) \rangle_{\mathbb{X}^2} \,\mathrm{d}s \\
& = \int_0^\infty ( \mu_{\varepsilon_2}(s)
 - \mu_{\varepsilon_1}(s) ) \langle \widetilde{\Psi}^t(s),
 \widetilde{Z}(t) \rangle_{\mathbb{X}^2} \,\mathrm{d}s  \\
&\quad - \langle F(\widetilde{U}) - F(\widetilde{V}),
\widetilde{Z} \rangle_{\mathbb{X}^2}
 + \omega\alpha \|\tilde{z}\|^2_{L^2(\Omega)}.
\end{aligned}
\end{equation}
We estimate from here along the usual lines to obtain, for almost all $t\geq0$,
\begin{equation} \label{C6-2}
- \int_0^\infty \mu_{\varepsilon_1}(s)\langle
{\rm{T_{\varepsilon_1}}}\widetilde{\Theta}^t(s), \mathrm{A_{W}^{\alpha,\beta}} \widetilde{\Theta}^t(s) \rangle_{\mathbb{X}^2} \,\mathrm{d}s \le \frac{\delta}{2\varepsilon_1}\| \widetilde{\Theta}^t \|^2_{\mathcal{M}^1_{\varepsilon_1}}.
\end{equation}
We know there is a constant $M_F>0$ in which
\begin{equation} \label{C6-4}
- \langle F(\widetilde{U}) - F(\widetilde{V}), \widetilde{Z}
\rangle_{\mathbb{X}^2} \leq M_2 \|\widetilde{Z} \|^2_{\mathbb{X}^2},
\end{equation}
and finally, with the fact that $\widetilde{\Psi}^t$ is uniformly
bounded in $\mathcal{B}^1_{\varepsilon_1}$,
\begin{equation} \label{C6-5}
\begin{aligned}
&\int_0^\infty ( \mu_{\varepsilon_2}(s)
- \mu_{\varepsilon_1}(s) ) \langle \widetilde{\Psi}^t(s),
\widetilde{Z}(t) \rangle_{\mathbb{X}^2} \,\mathrm{d}s  \\
& = \frac{\varepsilon_1 - \varepsilon_2}{\varepsilon_1\varepsilon_2}
 \int_0^\infty \mu_{\varepsilon}(s) \langle \widetilde{\Psi}^t(s),
 \widetilde{Z}(t) \rangle_{\mathbb{X}^2} \,\mathrm{d}s \\
& \leq C\frac{\varepsilon_1 - \varepsilon_2}{\varepsilon_2}
 \| \widetilde{Z} \|_{\mathbb{X}^2} \| \widetilde{\Psi}^t
  \|_{\mathcal{M}^1_{\varepsilon_1}} \\
& \leq \frac{\varepsilon_1 - \varepsilon_2}{\varepsilon_2}Q(R_1)
 + \frac{1}{2}\| \widetilde{Z} \|^2_{\mathbb{X}^2},
\end{aligned}
\end{equation}
where $R_1>0$ is the radius of the absorbing set $\mathcal{B}^1_{\varepsilon_1}$.
After applying \eqref{C6-2}-\eqref{C6-5}, we obtain the differential inequality,
\begin{equation} \label{C6-6}
\begin{aligned}
&\frac{\mathrm{d}}{\mathrm{d}t}
\big\{ \| \widetilde{Z} \|^2_{\mathbb{X}^2}
+ \| \widetilde{\Theta}^t \|^2_{\mathcal{M}^1_{\varepsilon_1}} \big\} \\
&\leq 2( M_2+\omega+1 ) \|\widetilde{Z} \|^2_{\mathbb{X}^2}
+ C \| \widetilde{\Theta}^t \|^2_{\mathcal{M}^1_{\varepsilon_1}}
+ \frac{\varepsilon_1 - \varepsilon_2}{\varepsilon_2}Q(R_1),
\end{aligned}
\end{equation}
where $M_3:=\max\{2( M_2+\omega+1 ),C\}>0$.
We now integrate \eqref{C6-6} with respect to $t$ over $[0,T]$
which in turn yields the Gronwall-type estimate, for all $t\in[0,T]$
\[
\| (\widetilde{Z}(t),\widetilde{\Theta}^t ) \|_{\mathcal{H}^0_{\varepsilon_1}}
\leq \sqrt{\frac{\varepsilon_1 - \varepsilon_2}{\varepsilon_2}}\frac{Q(R_1)
( e^{M_3 T} - 1 )}{M_3}.
\]
Therefore, \eqref{Holder-part} follows.
\end{proof}

\begin{remark} \rm
In conclusion, by Theorem \ref{cont_att} the semigroup
$\mathcal{S}_\varepsilon$ generated by the solutions of Problem \eqref{Pe}
 admits a robust family of exponential attractors
$(\mathbb{M}_\varepsilon)_{\varepsilon\in[0,1]}$ in
$\mathcal{H}^0_\varepsilon$, H\"{o}lder continuous at each $\varepsilon\in[0,1]$.
\end{remark}

\section{Appendix}

For the reader's convenience we report some important results that are
needed in this article.
The following lemma is from \cite[Lemma 2.2]{Gal&Grasselli08}.
It is in the spirit of the $H^s$-elliptic regularity estimate that
can be found in \cite[Theorem II.5.1]{Lions&Magenes72}.

\begin{lemma}  \label{t:appendix-lemma-3}
Consider the linear boundary value problem
\begin{equation} \label{appendix-BVP}
\begin{gathered}
-\Delta u+\alpha u  = \psi_1 \quad\text{in }\Omega, \\
-\Delta_{\Gamma}u + \partial_{\mathbf{n}} u + \beta u
 = \psi_2 \quad\text{on }\Gamma.
\end{gathered}
\end{equation}
If $(\psi_1,\psi_2)^{\mathrm{tr}}\in H^s(\Omega)\times H^s(\Gamma)$,
for $s\geq 0$ and $s+\frac{1}{2} \not\in\mathbb{N}$, then the following
estimate holds for some constant $C>0$,
\begin{equation}\label{H2-regularity-estimate}
\|u\|_{H^{s+2}(\Omega)} + \|u\|_{H^{s+2}(\Gamma)}
\leq C( \|\psi_1\|_{H^s(\Omega)} + \|\psi_2\|_{H^s(\Gamma)} ).
\end{equation}
\end{lemma}

The following result is the so-called transitivity property
of exponential attraction from \cite[Theorem 5.1]{FGMZ04}).

\begin{proposition}  \label{t:exp-attr}
Let $(\mathcal{X},d)$ be a metric space and let $S_t$ be a semigroup
acting on this space such that
\[
d(S_t x_1,S_t x_2) \leq C e^{Kt} d(x_1,x_2),
\]
for appropriate constants $C$ and $K$.
Assume  there exists three subsets $U_1$,$U_2$,$U_3\subset\mathcal{X}$
such that
\[
\operatorname{dist}_\mathcal{X}(S_t U_1,U_2)
\leq C_1 e^{-\alpha_1 t}, \quad
\operatorname{dist}_\mathcal{X}(S_t U_2,U_3) \leq C_2 e^{-\alpha_2 t}.
\]
Then
\[
\operatorname{dist}_\mathcal{X}(S_t U_1,U_3) \leq C' e^{-\alpha' t},
\]
where $C'=CC_1+C_2$ and $\alpha'=\frac{\alpha_1\alpha_1}{K+\alpha_1+\alpha_2}$.
\end{proposition}

The following statement refers to a frequently used
Gr\"{o}nwall-type inequality that is useful when working with
dissipation arguments.
We also refer the reader to \cite[Lemma 2.1]{Conti-Pata-2005},
\cite[Lemma 2.2]{Grasselli&Pata02} and \cite[Lemma 5]{Pata&Zelik06}.

\begin{proposition}  \label{GL}
Let $\Lambda :\mathbb{R}_{+}\to \mathbb{R}_{+}$ be an absolutely continuous
function satisfying
\[
\frac{d}{dt}\Lambda (t)+2\eta \Lambda (t)\leq h(t)\Lambda(t)+k,
\]
where $\eta >0$, $k\geq 0$ and $\int_{s}^{t}h(\tau )d\tau \leq \eta(t-s)+m$,
for all $t\geq s\geq 0$ and some $m\geq 0$. Then, for all $t\geq 0$,
\[
\Lambda (t)\leq \Lambda (0)e^{m}e^{-\eta t}+\frac{ke^{m}}{\eta }.
\]
\end{proposition}


\subsection*{Acknowledgments}

The author is indebted to the anonymous referees for their careful
reading of the manuscript and for their helpful comments and suggestions.


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\end{document}