\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2002(2002), No. 63, pp. 1--28.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2002 Southwest Texas State University.}
\vspace{1cm}}

\begin{document}

\title[\hfilneg EJDE--2002/63\hfil Robust exponential attractors]
{Robust exponential attractors for singularly perturbed phase-field
type equations}

\author[Alain Miranville \&  Sergey Zelik\hfil EJDE--2002/63\hfilneg]
{Alain Miranville \&  Sergey Zelik }

\address{ Laboratoire d'Applications des Math\'ematiques - SP2MI \hfil\break\indent
Boulevard Marie et Pierre Curie - T\'el\'eport 2\hfil\break\indent
 86962 Chasseneuil Futuroscope Cedex - France}
\email[Alain Miranville]{miranv@mathlabo.univ-poitiers.fr}
\email[Sergey Zelik]{zelik@mathlabo.univ-poitiers.fr}

\date{}
\thanks{Submitted April 18, 2002. Published July 4, 2002.}
\thanks{This research was partially supported by INTAS project 00-899.}
\subjclass{35B40, 35B45}
\keywords{Phase-field equations, exponential attractors, \hfil\break\indent
upper and lower semicontinuity}

\begin{abstract}
 In this article, we construct robust (i.e. lower and upper
 semicontinuous) exponential attractors for singularly
 perturbed phase-field type equations. Moreover, we obtain  
 estimates for the symmetric distance between these exponential
 attractors and that of the limit Cahn-Hilliard equation
 in terms of the perturbation parameter. We can note that the
 continuity is obtained without time shifts as it is the case 
 in previous results.
\end{abstract}

\maketitle

\newcommand{\dist}{\mathop{\rm dist}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\numberwithin{equation}{section}

\section*{Introduction} \label{sec0}

In this article, we are interested in the study of the asymptotic
behavior of phase-field type equations. The corresponding
equations consist of a system
of two parabolic equations involving two unknowns, namely the
temperature $u(t,x)$ at point $x$ and time $t$ of a substance
which can appear  in two different  phases (e.g. liquid-solid)
and  a phase-field function $\phi(t,x)$, also called order parameter,
which describes  the current phase at $x$ and $t$. Such models were introduced
in order
to study the evolution of interfaces in phase transitions. They have
also led to other models of phase transitions and motion of
interfaces as singular limits (e.g. the Stefan, Hele-Shaw and Cahn-Hilliard
models). We refer the interested reader  to
\cite{6,7,8,9,10,19,20,21,25,27}  and the references therein for more details.

The long time behavior of such models was extensively studied in
\cite{2,3,4,5,10,11,12,13,17}. In particular, the existence of the
global attractor and exponential attractors is obtained in \cite{3,4,5}.
 Furthermore, the upper semicontinuity of the
global attractor for a singularly perturbed phase-field model is
proved in \cite{12} (see also \cite{11} for a logarithmic nonlinearity)
for two limit equations, namely the viscous
Cahn-Hilliard and Cahn-Hilliard equations. The lower semicontinuity
of the global attractor was studied in \cite{10}, but only in one space
dimension. In that case, the authors do not need any assumption on the
hyperbolicity of the stationary solutions, as it is usually the case
to obtain the lower semicontinuity of the global attractor
 for dynamical systems which possess a global
Lyapunov function  \cite{1,22}.

In \cite{14,15}, we constructed families of robust
 (i.e. upper and lower semicontinuous) exponential attractors for
singularly perturbed viscous Cahn-Hilliard equations and damped
wave equations. We can note that these results are not based on
the study of stationary solutions and their unstable
manifolds, as it is the case for regular global attractors
\cite{1,22}; in particular, this allows us to obtain
explicit estimates on the different constants (appearing e.g. in the
estimate for the symmetric distance  between the exponential
attractors of the perturbed and unperturbed problems, see \cite{14},
\cite{15} and below).

Our aim in this article is to obtain a similar result for singularly
perturbed phase-field equations. Actually, we consider
a more general system of equations,
which does not possess a global Lyapunov function,
 by adding a nonlinear term in the
equation for the temperature $u$.

In Section \ref{sec1}, we derive uniform estimates which are necessary for the
study of the singular limit. Then, in Section \ref{sec2},
we study the asymptotic expansion of the solutions with respect to
the singular perturbation parameter $\varepsilon$ and
obtain estimates
on the difference of solutions which are essential for our
construction of exponential attractors. Finally, in Section \ref{sec3}, we
construct a family of continuous exponential attractors for our
problem and obtain in particular an explicit estimate for the symmetric
distance between the exponential attractors of
the perturbed and unperturbed equations
in terms of the perturbation parameter $\varepsilon$
(see Theorem \ref{thm3.1} below).
The case of Neumann boundary conditions is briefly addressed in
Section~\ref{sec4}.

\subsection*{Setting of the problem}

We consider the following system of singularly perturbed
reaction-diffusion equations:
\begin{equation} \begin{gathered}
\delta\partial_t \phi=\Delta_x\phi-f_1(\phi)+u+g_1,\quad \phi\big|_{\partial\Omega}=0,
\\
\varepsilon\partial_t u+\partial_t\phi=\Delta_x u-f_2(u)+g_2,\quad u\big|_{\partial\Omega}=0,\\
\phi\big|_{t=0}=\phi_0,\quad u\big|_{t=0}=u_0,
\end{gathered}\label{0.1}
\end{equation}
where  $\Omega$ is a bounded regular domain of
 $\mathbb{R}^3$,
 $(\phi(t,x),u(t,x))$ is an unknown pair of functions,
   $\Delta_x$ is the Laplacian with respect to the variable $x$,
 $g_i=g_i(x)\in L^2(\Omega)$, $i=1,2$,
 are given external forces and $\delta$ and $\varepsilon>0$
 are given constants.

We assume that the nonlinear terms $f_i$ belong
 to $C^3(\mathbb{R},\mathbb{R})$, $i=1,2$, and
satisfy
the following dissipativity conditions:
\begin{equation} \begin{gathered}
f_1(v).v\ge-C,\quad C\ge0\\
f'_1(v)\ge-K,\quad K\ge0\\
f'_2(v)\ge0,\quad f_2(0)=0. \end{gathered}
\label{0.2}
\end{equation}
Finally, we assume that the initial data $(\phi_0,u_0)$ belongs to the phase
space $\Phi$, defined by
\begin{equation}
\Phi:=\left(H^2(\Omega)\cap H^1_0(\Omega)\right)\times\left(
H^2(\Omega)\cap H^1_0(\Omega)\right).
\label{0.3}
\end{equation}

\begin{remark} \label{rm0.1} Taking $f_2\equiv0$ in \eqref{0.1}, we
recover the phase-field system considered in \cite{2,3,4,5,10,11,12,13,17}.
\end{remark}

\section{Uniform a priori estimates} \label{sec1}

In this section, we derive several uniform (with respect to $\varepsilon\ll1$)
estimates for the solutions of problem \eqref{0.1} which are
necessary for the study of the singular limit $\varepsilon\to0$. We start with
the following  lemma.

\begin{lemma} \label{lm1.1} Let the above assumptions hold and let the pair
$(\phi(t),u(t))\in C(\mathbb{R},\Phi)$ be a solution of \eqref{0.1}.
Then, the following estimate is valid:
\begin{multline}
\|\nabla_x\phi(t)\|_{L^2}^2+\varepsilon\|u(t)\|_{L^2}^2+
\left(F_1(\phi(t)),1\right)+\\+
\int_t^{t+1}\left(\|\partial_t\phi(s)\|_{L^2}^2+
\|\nabla_x u(s)\|_{L^2}^2+(f_2(u(s)),u(s))\right)\,ds\le\\\le
C\left(\|\nabla_x\phi(0)\|_{L^2}^2+\varepsilon\|u(0)\|_{L^2}^2+
\left(F_1(\phi(0)),1\right)\right)e^{-\gamma t}+\\
+C\left(\|g_1\|_{L^2}^2+\|g_2\|_{L^2}^2\right),
\label{1.1}
\end{multline}
where $F_1(v):=\int_0^vf_1(s)\,ds$, $(\cdot,\cdot)$ denotes the standard
inner product in $L^2(\Omega)$ and the positive constants $C_1$, $C_2$
and
$\gamma$ are independent of $\varepsilon$.
\end{lemma}

\paragraph{Proof} Taking the inner product in $L^2(\Omega)$ of the first
equation
 of \eqref{0.1}  by $\partial_t\phi(t)$ and of the second
equation by $u(t)$ and summing the relations that we obtain, we have
\begin{multline}
\partial_t[\delta\|\nabla_x \phi(t)\|_{L^2}^2+2(F_1(\phi(t)),1)+\varepsilon\|u(t)\|_{L^2}^2-
2(g_1,\phi(t))]+\\+
2\delta\|\partial_t\phi(t)\|^2_{L^2}+2\|\nabla_x u(t)\|^2_{L^2}
+2(f_2(u(t)),u(t))-2(g_2,u(t))
=0.
\label{1.2}
\end{multline}
Taking now the inner product in $L^2(\Omega)$ of the first equation
of \eqref{0.1} by $2\beta\phi(t)$, where $\beta$ is a
sufficiently small positive number, and summing the relation that we obtain
with equation \eqref{1.2}, we find
\begin{equation}
\partial_t E(t)+\gamma E(t)=h(t),
\label{1.3}
\end{equation}
where
\[
E(t):=
\delta\|\nabla_x\phi(t)\|_{L^2}^2+2(F_1(\phi(t)),1)+\varepsilon\|u(t)\|_{L^2}^2-
2(g_1,\phi(t))+\beta\delta\|\phi(t)\|_{L^2}^2,
\]
$0<\gamma<\beta$ is another small positive parameter which will be
fixed below and
\begin{multline}
h(t):=(\gamma \delta -2\beta )\|\nabla_x\phi(t)\|_{L^2}^2+
2\gamma\left(F_1(\phi(t))-f_1(\phi(t))\phi(t),1\right)+\\+
2(\gamma-\beta)(f_1(\phi(t)),\phi(t))-
2\delta\|\partial_t\phi(t)\|^2_{L^2}-2\|\nabla_x u(t)\|^2_{L^2}
-\\-2(f_2(u(t)),u(t))+2(g_2,u(t))+
\gamma\varepsilon\|u(t)\|_{L^2}^2+
2(\beta-\gamma)(g_1,\phi(t))+\\+\beta\delta\gamma\|\phi(t)\|_{L^2}^2+
2\beta(u(t),\phi(t)).
\label{1.4}
\end{multline}
It follows from conditions \eqref{0.2} that
\begin{equation}
f_1(v).v+K|v|^2\ge F_1(v),\quad \forall v\in\mathbb{R},
\label{1.5}
\end{equation}
(see e.g. \cite{26}). Consequently, it is possible to fix the
small positive parameters $\beta$ and $\gamma$
(which are independent of $0<\varepsilon<1$)  such  that the
following estimate holds:
\begin{equation}
h(t)\le C_1\left(1+\|g_1\|_{L^2}^2+\|g_2\|_{L^2}^2\right),
\label{1.6}
\end{equation}
where $C_1$ is independent of $\varepsilon$. Applying now  Gronwall's
inequality to relation \eqref{1.3} and using  estimate
\eqref{1.6} and  equation \eqref{1.2}, we find estimate
\eqref{1.1} and Lemma \ref{lm1.1} is proved. \hfill \qed

The next lemma gives uniform (with respect to $\varepsilon$) estimates
of $(\phi,u)$ in the space $H^2(\Omega)\times H^1(\Omega)$.

\begin{lemma} \label{lm1.2} Let the above assumptions hold. Then, the following
estimate is valid for a solution $(\phi(t),u(t))$ of  equation
\eqref{0.1}:
\begin{multline}
\|\phi(t)\|_{H^2}^2\!+\!\|\partial_t \phi(t)\|_{L^2}^2+\|u(t)\|_{H^1}^2+
\int_t^{t+1}\left(\|\partial_t\phi(s)\|_{H^1}^2\!+\!\varepsilon\|\partial_t u(s)\|^2_{L^2}\right)\,ds\le\\
\le Q(\|\phi(0)\|^2_{H^2}+\|u(0)\|^2_{H^1})e^{-\gamma t}+
Q(\|g_1\|_{L^2}^2+\|g_2\|_{L^2}^2),
\label{1.7}
\end{multline}
where the constant $\gamma>0$ and the monotonic function
$Q$ are independent of $\varepsilon>0$.
\end{lemma}

\paragraph{Proof} We set $\psi(t):=\partial_t\phi(t)$. Then, this function satisfies
\begin{equation}
\delta\partial_t\psi=\Delta_x\psi-f_1'(\phi)\psi+\partial_t u,\  \psi(0)=\delta^{-1}\left(\Delta_x\phi(0)-f_1(\phi(0))+u(0)+g_1\right).
\label{1.8}
\end{equation}
Taking the inner product in $L^2(\Omega)$ of the equation by $\psi(t)$
and of the second equation of \eqref{0.1} by $\partial_t u$ and summing
the relations that we obtain, we  have
\begin{multline}
\partial_t[\delta\|\partial_t \phi(t)\|_{L^2}^2+\|\nabla_x u(t)\|_{L^2}^2+2(F_2(u(t)),1)-
2(g_2,u(t))]+\\+
[\delta\|\partial_t \phi(t)\|_{L^2}^2+\|\nabla_x u(t)\|_{L^2}^2+2(F_2(u(t)),1)-
2(g_2,u(t))]+\\
+2\|\nabla_x\psi(t)\|_{L^2}^2+2\varepsilon\|\partial_t u(t)\|_{L^2}^2
=h_1(t),
\label{1.9}
\end{multline}
where $F_2(v):=\int_0^vf_2(s)\,ds$
 and
\begin{multline}
h_1(t):=
[\delta\|\partial_t \phi(t)\|_{L^2}^2+\|\nabla_x
u(t)\|_{L^2}^2+2(F_2(u(t)),1)-2(g_2,u(t))]+\\+
2(g_1,\partial_t\phi(t))-2(f_1'(\phi(t))\partial_t\phi(t),\partial_t\phi(t)).
\label{1.10}
\end{multline}
Analogously to \eqref{1.5}, we have
\begin{equation}
f_2(v).v\ge F_2(v).
\label{1.11}
\end{equation}
Furthermore, thanks to \eqref{0.2} and \eqref{1.11},   we find
\begin{multline}
h_1(t)\le  C_1\left(1+\|g_1\|_{L^2}^2+\|g_2\|_{L^2}^2\right)+\\+
C_2\left((f_2(u(t)),u(t))+\|\partial_t \phi(t)\|_{L^2}^2+\|\nabla_x u(t)\|_{L^2}^2\right).
\label{1.12}
\end{multline}
Applying  Gronwall's inequality to  relation \eqref{1.9} and using
estimates \eqref{1.1} and \eqref{1.12}, we obtain
\begin{multline}
\|\partial_t \phi(t)\|_{L^2}^2+\|u(t)\|_{H^1}^2+
\int_t^{t+1}\left(\|\partial_t\phi(s)\|_{H^1}^2+\varepsilon\|\partial_t u(s)\|^2_{L^2}\right)\,ds\le\\
\le Q(\|\phi(0)\|^2_{H^2}+\|u(0)\|^2_{H^1})e^{-\gamma t}+
Q(\|g_1\|_{L^2}^2+\|g_2\|_{L^2}^2),
\label{1.13}
\end{multline}
for appropriate constant $\gamma>0$ and  monotonic function $Q$
 which are independent of
$\varepsilon$. There now remains  to estimate the $H^2$-norm of $\phi(t)$.
To this end, we rewrite the first equation of \eqref{0.1} in the form
\begin{equation}
\Delta_x \phi(t)-f_1(\phi(t))=h_2(t),\quad \phi(t)\big|_{\partial\Omega}=0,
\label{1.14}
\end{equation}
where $h_2(t):=\delta\partial_t\phi(t)-u(t)-g_1$. Indeed,
according to  estimate \eqref{1.13}, we have
\begin{equation}
\|h_2(t)\|_{L^2}^2\le
 Q(\|\phi(0)\|^2_{H^2}+\|u(0)\|^2_{H^1})e^{-\gamma t}+
Q(\|g_1\|_{L^2}^2+\|g_2\|_{L^2}^2).
\label{1.15}
\end{equation}
Taking then the inner product in $L^2(\Omega)$ of equation \eqref{1.14} by
$\Delta_x \phi(t)$ and using \eqref{0.2}, we obtain
\begin{equation}
\|\Delta_x\phi(t)\|_{L^2}^2\le 2K\|\nabla_x\phi(t)\|_{L^2}^2+2\|h_2(t)\|_{L^2}^2.
\label{1.16}
\end{equation}
Inserting finally  estimates \eqref{1.15} and \eqref{1.13} into the
right-hand side of \eqref{1.16}, we derive the necessary estimate for
the $H^2$-norm of $\phi(t)$ and Lemma \ref{lm1.2} is proved.\hfill\qed

\vskip 0.1cm 

We are now in a position to derive a priori estimates for the solutions
of \eqref{0.1} in the phase space $\Phi$.

\begin{lemma} \label{lm1.3} Let the above assumptions hold. Then, the
following estimate holds, for every solution $(\phi(t),u(t))$
of problem \eqref{0.1}:
\begin{multline}
\|\phi(t)\|_{H^2}^2+\|u(t)\|^2_{H^2}+\varepsilon^2\|\partial_t u(t)\|^2_{L^2}
\le\\\le
Q(\|\phi(0)\|_{H^2}+\|u(0)\|_{H^2})e^{-\alpha t}+Q(\|g_1\|_{L^2}+
\|g_2\|_{L^2}),
\label{1.17}
\end{multline}
where the positive constant $\alpha$ and the monotonic function $Q$
are independent of $\varepsilon$.
\end{lemma}

\paragraph{Proof} We rewrite the second equation of  system
 \eqref{0.1} in the form
\begin{equation}
\varepsilon\partial_t u-\Delta_x u+f_2(u)=h(t):=g_2-\partial_t\phi(t).
\label{1.18}
\end{equation}
Rescaling now the time variable ($t:=\varepsilon\tau$), we  have
\begin{equation}
\partial_\tau u-\Delta_x u+f_2(u)=\tilde h(\tau):=h(\varepsilon\tau),\quad u\big|_{\tau=0}=u_0.
\label{1.19}
\end{equation}
Moreover, it follows from \eqref{1.7} that
\begin{equation}
\|\tilde h(\tau)\|_{L^2}
\le Q(\|\phi(0)\|^2_{H^2}+\|u(0)\|^2_{H^1})e^{-\gamma \varepsilon\tau}+
Q(\|g_1\|_{L^2}^2+\|g_2\|_{L^2}^2),
\label{1.20}
\end{equation}
where $Q$ and $\alpha$ are independent of $\varepsilon$.
Applying the standard maximum
principle to equation \eqref{1.19}, using the fact that $f_2(u).u\ge0$ and noting that $H^2\subset C$ (since $n=3$), we obtain the
estimate
\begin{equation}
\|u(\tau)\|_{L^\infty}\le C\|u(0)\|_{H^2}e^{-\beta \tau}+
C\sup_{s\in[0,\tau]}\left\{e^{-\beta(\tau-s)}\|\tilde h(s)\|_{L^2}\right\},
\label{1.21}
\end{equation}
for appropriate positive constants $\beta$ and $C$
(see e.g. \cite{18} for details). Inserting
estimate \eqref{1.20} into the right-hand side of \eqref{1.21}
and returning to the time variable $t$, we find
\begin{equation}
\|u(t)\|_{L^\infty}
\le Q(\|\phi(0)\|^2_{H^2}+\|u(0)\|^2_{H^2})e^{-\gamma t}+
Q(\|g_1\|_{L^2}^2+\|g_2\|_{L^2}^2),
\label{1.22}
\end{equation}
where the constant $\gamma$ and the function $Q$ are independent of $\varepsilon$.

Let us now derive a uniform estimate for the $H^2$-norm of $u(t)$.
To this end, we introduce the functions $G_i=G_i(x):=(-\Delta_x)^{-1}g_i$,
$i=1,2$,
 and
split the solution $(\phi(t),u(t))$ as follows:
\begin{equation}
\phi(t):=G_1+\phi_1(t)+\phi_2(t),\quad u(t):=G_2+u_1(t)+u_2(t)+u_3(t),
\label{1.23}
\end{equation}
where $u_1(t)$ solves
\begin{equation}
\varepsilon\partial_t u_1=\Delta_x u_1,\quad u_1\big|_{t=0}=u_0-G_2,
\label{1.24}
\end{equation}
the function $u_2(t)$ is solution of
\begin{equation}
\varepsilon\partial_t u_2=\Delta_x u_2-\partial_t\phi_1,\  u_2\big|_{t=0}=0,
\label{1.25}
\end{equation}
with
\begin{equation}
\delta\partial_t\phi_1=\Delta_x\phi_1,\quad \phi_1\big|_{t=0}=\phi_0-G_1,
\label{1.26}
\end{equation}
and the function $u_3(t)$ solves
\begin{equation}
\varepsilon\partial_t u_3-\Delta_x u_3=h_3(t):=-\partial_t\phi_2(t)-f_2(u(t)),\quad u_3\big|_{t=0}=0,
\label{1.27}
\end{equation}
with
\begin{equation}
\delta\partial_t\phi_2-\Delta_x\phi_2=h_4(t):= u(t)-f_1(\phi(t)),\quad \phi_2\big|_{t=0}=0.
\label{1.28}
\end{equation}
Obviously, $G_i\in H^2(\Omega)$ and
\begin{equation}
\|G_i\|_{H^2}\le C\|g_i\|_{L^2}, \quad i=1,2.
\label{1.29}
\end{equation}
Moreover, since $-\Delta_x$ generates an analytic semigroup in
$H^2(\Omega)$, then
\begin{equation}
\|u_1(t)\|_{H^2}\le Ce^{-\gamma t/\varepsilon}\left(\|u_0\|_{H^2}+\|g_2\|_{L^2}\right),
\label{1.30}
\end{equation}
where the constants $C$ and $\gamma$ are independent of $\varepsilon$.
 Let us then estimate  $u_2(t)$. To this end, we note that
\begin{equation}
u_2(t)=\frac{\delta}{\delta-\varepsilon}\left(\phi_1(t)-{\tilde u}_0(t)\right),
\label{1.31}
\end{equation}
for $\varepsilon\ll1$, where the function ${\tilde u}_0(t)$ solves the problem
\[
\varepsilon\partial_t {\tilde u}_0=\Delta_x {\tilde u}_0,\quad {\tilde u}_0\big|_{t=0}=\phi_0-G_1.
\]
Analogously to \eqref{1.30}, we have
\begin{equation}
\|u_2(t)\|_{H^2}\le Ce^{-\beta t}\left(\|\phi_0\|_{H^2}+\|g_1\|_{L^2}\right),
\label{1.32}
\end{equation}
where $C$ and $\beta$ are independent of $\varepsilon$. So, there only remains to
estimate $u_3(t)$. To this end, we note that,
 due to estimate \eqref{1.7} and due to the fact that
$H^2\subset C$, the function $h_4$ defined in \eqref{1.28}
satisfies
\begin{equation}
\|h_4(t)\|_{H^1}\le
  Q(\|\phi(0)\|_{H^2}+\|u(0)\|_{H^2})e^{-\gamma t}+
Q(\|g_1\|_{L^2}+\|g_2\|_{L^2}),
\label{1.33}
\end{equation}
for  appropriate $\gamma$ and $Q$ which are independent of $\varepsilon$.
Applying the parabolic regularity theorem (see e.g. \cite{18}) to equation
\eqref{1.28}, we obtain
\begin{equation}
\|\partial_t\phi_2(t)\|_{H^{1-\beta}}\le
 Q_\beta(\|\phi(0)\|_{H^2}+\|u(0)\|_{H^2})e^{-\gamma t}+
Q_\beta(\|g_1\|_{L^2}+\|g_2\|_{L^2}),
\label{1.34}
\end{equation}
where $0<\beta<1$ and $\gamma$ and $Q_\beta$ are independent of $\varepsilon$.
Consequently, according to \eqref{1.7}, \eqref{1.22} and
\eqref{1.34}, we have the following estimate for the function
$h_3(t)$ in the right-hand side of \eqref{1.17}:
\begin{equation}
\|h_3(t)\|_{H^{1-\beta}}\le
 Q_\beta(\|\phi(0)\|_{H^2}+\|u(0)\|_{H^2})e^{-\gamma t}+
Q_\beta(\|g_1\|_{L^2}+\|g_2\|_{L^2}),
\label{1.35}
\end{equation}
for appropriate $\gamma$ and $Q_\beta$ which are independent of $\varepsilon$.
Applying now the standard parabolic regularity theorem to  equation
\eqref{1.27} and rescaling the time as above  ($t:=\varepsilon\tau$)
in order to eliminate the dependence on $\varepsilon$
(analogously to \eqref{1.18}--\eqref{1.22}), we deduce from
\eqref{1.35} that
\begin{equation}
\|u_3(t)\|_{H^2}\le
Q(\|\phi(0)\|_{H^2}+\|u(0)\|_{H^2})e^{-\gamma t}+
Q(\|g_1\|_{L^2}+\|g_2\|_{L^2}),
\label{1.36}
\end{equation}
where the positive constant $\gamma$ and the monotonic function $Q$
are independent of $\varepsilon$. Combining \eqref{1.29}, \eqref{1.30},
\eqref{1.32} and \eqref{1.36}, we finally have
\begin{equation}
\|u(t)\|_{H^2}\le
Q(\|\phi(0)\|_{H^2}+\|u(0)\|_{H^2})e^{-\gamma t}+
Q(\|g_1\|_{L^2}+\|g_2\|_{L^2}),
\label{1.37}
\end{equation}
for some new positive constant $\gamma$ and monotonic function
$Q$ which are
independent of $\varepsilon$. Thus, the uniform estimate for
the $H^2$-norm of $u(t)$ is obtained. The uniform estimate
for the $L^2$-norm of $\varepsilon\partial_t u(t)$ is an immediate corollary
of \eqref{1.7}, \eqref{1.37} and of the second equation in \eqref{0.1}.
This finishes the proof of Lemma \ref{lm1.3}.\hfill\qed


\begin{lemma} \label{lm1.4} Let the above assumptions hold. Then,
for every $(\phi_0,u_0) \in\Phi$,
problem \eqref{0.1} has a unique solution $(\phi(t),u(t))\in C(\mathbb{R},\Phi)$
which satisfies  estimate \eqref{1.17}. Moreover, for any
solutions $(\phi_i(t),u_i(t))\in\Phi$, $i=1,2$, the following
inequality
holds:
\begin{multline}
\|\phi_1(t)-\phi_2(t)\|_{H^1}^2+\varepsilon\|u_1(t)-u_2(t)\|_{L^2}^2+
\\+
\int_t^{t+1}\left(\|\partial_t\phi_1(s)-\partial_t\phi_2(s)\|^2_{L^2}+
\|\nabla_x u_1(s)-\nabla_x u_2(s)\|_{L^2}^2\right)\,ds\le\\
\le
Ce^{Lt}\left(\|\phi_1(0)-\phi_2(0)\|_{H^1}^2+\varepsilon\|u_1(0)-u_2(0)\|_{L^2}^2\right),
\label{1.38}
\end{multline}
where the constants $C$ and $L$ depend on $\|\phi_i(0)\|_{H^2}$ and
on  $\|u_i(0)\|_{H^2}$, but are independent of $\varepsilon$.
\end{lemma}

\paragraph{Proof}
The existence of a solution can be proved in a
standard way, based on a priori estimate \eqref{1.17} and
on the Leray-Schauder fixed point theorem (see e.g. \cite{18}).
So, there remains to deduce estimate \eqref{1.38}. To this end,
we set
$v(t):=\phi_1(t)-\phi_2(t)$ and
$w(t):=u_1(t)-u_2(t)$.
These functions satisfy the equations
\begin{equation}
\begin{gathered}
\delta\partial_t v=\Delta_x v-l_1(t)v+w,
\quad v\big|_{t=0}=\phi_1(0)-\phi_2(0), \quad v\big|_{\partial\Omega}=0, \\
\varepsilon\partial_t w+\partial_t v=\Delta_x w-l_2(t)w,\quad
w\big|_{t=0}=u_1(0)-u_2(0),\quad w\big|_{\partial\Omega}=0,\\
\end{gathered}
\label{1.39}
\end{equation}
where
\[
l_1(t):=\int_0^1f_1'(s\phi_1(t)+(1-s)\phi_2(t))\,ds,\quad l_2(t)
:=\int_0^1f_2'(su_1(t)+(1-s)u_2(t))\,ds.
\]
It now follows  from  estimates \eqref{1.7} and \eqref{1.17}
and from the embedding $H^2\subset C$ that
\begin{multline}
\|l_1(t)\|_{H^2}+\|\partial_t l_1(t)\|_{L^2}+\|l_2(t)\|_{H^2}\le\\\le
L:=Q(\|(\phi_1(0),u_1(0))\|_\Phi+\|(\phi_2(0),u_2(0))\|_\Phi),
\label{1.40}
\end{multline}
for a  monotonic function $Q$ which is independent of $\varepsilon$.
Moreover, due to our assumptions on $f_2'$, we have
\begin{equation}
l_2(t)\ge 0.
\label{1.41}
\end{equation}
Multiplying now the first equation of \eqref{1.39} by $\partial_t v(t)$
and the second one by $w(t)$, integrating over $\Omega$ and
summing the relations that we obtain, we find, taking into  account
estimates \eqref{1.40} and  \eqref{1.41}
\begin{multline}
\partial_t[\|\nabla_x v(t)\|_{L^2}^2+\varepsilon\|w(t)\|_{L^2}^2]+2\delta\|\partial_t v(t)\|_{L^2}^2+
2\|\nabla_x w(t)\|_{L^2}^2\le\\\le
 L^2\delta^{-1}\|v(t)\|_{L^2}^2+\delta\|\partial_t v(t)\|^2_{L^2}.
\label{1.42}
\end{multline}
Applying  Gronwall's inequality to this relation, we derive  estimate
\eqref{1.38} and Lem\-ma~1.4 is proved.\hfill\qed


\begin{corollary}\label{coro1.1}
Let the above assumptions hold.
Then, for every $\varepsilon>0$,  problem \eqref{0.1} defines a semigroup
$S_t^\varepsilon$ in the phase space $\Phi$ by
\begin{equation}
S_t^\varepsilon:\Phi\to\Phi,\quad S_t^\varepsilon(\phi_0,u_0)=(\phi(t),u(t)),
\label{1.43}
\end{equation}
where the function $(\phi(t),u(t))$ solves \eqref{0.1}.
\end{corollary}

Let us now consider  the limit equation of \eqref{0.1} (i.e. $\varepsilon=0$
in
\eqref{0.1}):
\begin{equation}
\begin{gathered}
\delta\partial_t\bar\phi_0=\Delta_x\bar\phi_0-f_1(\bar \phi_0)
+\bar u_0+g_1,\quad \bar\phi_0\big|_{t=0}=\phi_0,\quad
\bar\phi_0\big|_{\partial\Omega}=0,\\
\partial_t\bar\phi_0=\Delta_x \bar u_0-f_2(\bar u_0)+g_2,\
 \ \bar u_0\big|_{\partial\Omega}=0.
\end{gathered}
\label{1.44}
\end{equation}
We note  that, in contrast to the case $\varepsilon>0$,  the values
of $(\bar \phi_0(t),\bar u_0(t))$ are not independent
in that case. Indeed, it follows from
\eqref{1.44} that
\begin{equation}
\delta\Delta_x \bar u_0(t)-\delta
f_2(\bar u_0(t))-\bar
u_0(t)=\Delta_x\bar\phi_0(t)-f_1(\bar\phi_0(t))+g_1-\delta g_2.
\label{1.45}
\end{equation}
Moroever, as shown in the following proposition, the value of $\bar u_0(t)$
is uniquely defined by \eqref{1.45}, if the value $\bar\phi_0(t)$ is
known.
\begin{proposition} \label{prop1.1}
Let the above assumptions hold. Then, the
nonlinear operator in the left-hand side of \eqref{1.45} is
invertible in $H^2(\Omega)\cap H^1_0(\Omega)$, i.e. there exists
a nonlinear $C^1$-operator
\begin{equation}
\mathcal{L}\in C^1(H^2(\Omega)\cap H^1_0(\Omega),H^2(\Omega)\cap H^1_0(\Omega)),
\label{1.46}
\end{equation}
such that \eqref{1.45} is equivalent to
\begin{equation}
\bar u_0(t)=\mathcal{L}(\bar\phi_0(t)).
\label{1.47}
\end{equation}
\end{proposition}

This proposition is an immediate corollary
of the condition $f_2'(v)\ge0$ (which provides the invertibility
of the operator in the left-hand side of \eqref{1.45}) and of
standard elliptic estimates.

Thus, the solution $(\bar\phi_0(t),\bar u_0(t))$ of problem \eqref{1.44}
exists only for initial data $(\phi_0,u_0)$ that belong to the infinite dimensional
submanifold $\mathbb{L}$ of the phase space $\Phi$ defined by
\begin{equation}
\mathbb{L}:=\{(\phi_0,u_0)\in\Phi,\quad u_0=\mathcal{L}(\phi_0)\}\subset\Phi.
\label{1.48}
\end{equation}

\begin{lemma} \label{lm1.5} Let the above assumptions hold. Then, for every
$(\phi_0,u_0)\in\mathbb{L}$,  problem \eqref{1.44} has a unique
solution $(\bar\phi_0(t),\bar u_0(t))\in\mathbb{L}$, for $t\ge0$, which satisfies
the estimate
\begin{multline}
\|\bar\phi_0(t)\|_{H^2}^2+\|\partial_t\bar\phi_0(t)\|_{L^2}^2+\|\bar u_0(t)\|_{H^2}^2+
\int_t^{t+1}\|\partial_t\bar\phi_0(s)\|_{H^1}^2\,ds\le\\\le
Q(\|\bar\phi_0(0)\|_{H^2}^2)e^{-\gamma t}+Q(\|g_1\|_{L^2}^2+\|g_2\|_{L^2}^2),
\label{1.49}
\end{multline}
for a  positive constant $\gamma$ and a monotonic function  $Q$.
 Consequently,
equation \eqref{1.44} defines a semigroup $S^0_t$ on the manifold
$\mathbb{L}$ by
\begin{equation}
 S^0_t:\mathbb{L}\to\mathbb{L},\quad S^0_t(\phi_0,u_0):=(\bar\phi_0(t),\bar u_0(t)),
\label{1.50}
\end{equation}
where the function $(\bar\phi_0(t),\bar u_0(t))$ solves \eqref{1.44}.
\end{lemma}

\paragraph{Proof}
Since  estimates \eqref{1.7} and \eqref{1.17} are
uniform with respect to $\varepsilon$, then, passing to the limit $\varepsilon\to0$
in  equations \eqref{0.1}, we obtain a solution
$(\bar \phi_0(t),\bar u_0(t))$ for  problem \eqref{1.44} which satisfies
\eqref{1.49}. The uniqueness of this solution can be proved exactly
as in Lemma \ref{lm1.4}.\hfill\qed

In the sequel, we will also need  the estimates for $\partial_t \bar u_0$
and $\partial_t^2 \bar u_0$ that are given in the following lemma.

\begin{lemma} \label{lm1.6} Let the above assumptions hold. Then, the
following estimate is valid for the solution $(\bar\phi_0(t),\bar u_0(t))$
of  problem \eqref{1.44}:
\begin{multline}
\|\partial_t \bar u_0(t)\|_{L^2}^2+\int_t^{t+1}\left(\|\partial_t \bar u_0(s)\|_{H^1}^2+
\|\partial_t^2\bar u_0(s)\|_{H^{-1}}^2\right)\,ds
\le\\\le
Q(\|\bar\phi_0(0)\|_{H^2}^2)e^{-\gamma t}+Q(\|g_1\|_{L^2}^2+\|g_2\|_{L^2}^2),
\label{1.51}
\end{multline}
for a  positive constant $\gamma$ and a monotonic function $Q$.
\end{lemma}

\paragraph{Proof} Let us first derive  estimate \eqref{1.51}
for the first derivative $\partial_t \bar u_0(t)$. To this end, we differentiate
relation \eqref{1.45} with respect to $t$ and split $\partial_t \bar u_0$
 as follows:
\begin{equation}
\partial_t \bar u_0(t)=\delta^{-1}\partial_t\bar\phi_0(t)+\psi_0(t).
\label{1.52}
\end{equation}
After straightforward substitutions, we find
\begin{multline}
\delta\Delta_x\psi_0(t)-\delta f'_2(\bar u_0(t))\psi_0(t)-\psi_0(t)=\\=
(f_2'(\bar u_0(t))-f_1'(\bar\phi_0(t))+\delta^{-1})\partial_t\bar\phi_0(t):=\Psi(t).
\label{1.53}
\end{multline}
It then follows from \eqref{1.49} that
\[
\|\Psi(t)\|_{L^2}\le
Q(\|\bar\phi_0(0)\|_{H^2})e^{-\gamma t}+Q(\|g_1\|_{L^2}+\|g_2\|_{L^2}),
\]
and, consequently, due to the assumption $f_2'\ge0$, it follows from
\eqref{1.53} (using standard elliptic estimates) that
\begin{equation}
\|\psi_0(t)\|_{H^2}\le
 Q(\|\bar \phi_0(0)\|_{H^2})e^{-\gamma t}+Q(\|g_1\|_{L^2}^2+\|g_2\|_{L^2}^2).
\label{1.54}
\end{equation}
Estimates \eqref{1.49} and \eqref{1.54} imply the part of
\eqref{1.51} for $\partial_t \bar u_0$. So, there remains to estimate $\partial_t^2\bar
u_0$
only. To this end, we differentiate the first equation of \eqref{1.44}
with respect to $t$:
\begin{equation}
\delta\partial_t^2\bar\phi_0(t)=\Delta_x\partial_t\bar\phi_0(t)-f_1'(\bar\phi_0(t))\partial_t\bar\phi_0(t)+
\delta^{-1}\partial_t\bar\phi_0(t)+\psi_0(t),
\label{1.55}
\end{equation}
and obtain, using \eqref{1.49} and \eqref{1.54}
\begin{equation}
\int_t^{t+1}\|\partial_t^2\bar\phi_0(s)\|_{H^{-1}}^2\,ds\le
Q(\|\bar\phi_0(0)\|_{H^2}^2)e^{-\gamma t}+Q(\|g_1\|_{L^2}^2+\|g_2\|_{L^2}^2).
\label{1.56}
\end{equation}
Differentiating now equation \eqref{1.53} with respect to
 $t$ and setting
$\theta_0(t):=\partial_t \psi_0(t)$, we have
\begin{multline}
\delta\Delta_x\theta_0-\delta f_2'(\bar u_0)\theta_0-\theta_0=
\bigg[(f_2'(\bar u_0)-f_1'(\bar\phi_0)+\delta^{-1})\partial_t^2\bar\phi_0\bigg]+\\+
\bigg[(\delta^{-1}f_2''(\bar u_0)-f_1''(\bar\phi_0))(\partial_t\bar\phi_0)^2\bigg]+
\bigg[f''_2(\bar u_0)(\delta\psi_0+2\partial_t\bar\phi_0)\psi_0\bigg]:=\\:=
I_1(t)+I_2(t)+I_3(t).
\label{1.57}
\end{multline}
Multiplying \eqref{1.57} by $\theta_0(t)$, integrating over
$\Omega$ and noting that $f_2'\ge0$, we obtain
the inequality
\begin{multline}
\delta\|\nabla_x\theta_0(t)\|^2_{L^2}+\|\theta_0(t)\|^2_{L^2}\le\\\le
|\left(I_1(t),\theta_0(t)\right)|+|\left(I_2(t),\theta_0(t)\right)|+
|\left(I_3(t),\theta_0(t)\right)|.
\label{1.58}
\end{multline}
Let us estimate each term in the right-hand side of \eqref{1.58}.
Using  Schwarz' inequality
and the embeddings  $H^2\subset C$ and $H^1\subset L^6$,
we have
\begin{multline}
|\left(I_1(t),\theta_0(t)\right)|\le\\\le C\|\partial_t^2\bar\phi_0(t)\|_{H^{-1}}
\|\nabla_x[(f_2'(\bar u_0(t))-f_1'(\bar\phi_0(t))+
\delta^{-1})\theta_0(t)]\|_{L^2}\le\\ \le
Q(\|\bar \phi_0(t)\|_{H^2})\|\partial_t^2\bar\phi_0(t)\|_{H^{-1}}\|\nabla_x\theta_0(t)\|_{L^2}\le\\\le
\frac\delta4\|\nabla_x\theta_0(t)\|_{L^2}^2+
Q_1(\|\bar \phi_0(t)\|_{H^2})\|\partial_t^2\bar\phi_0(t)\|_{H^{-1}}^2,
\label{1.59}
\end{multline}
where $Q$ and $Q_1$ are appropriate monotonic functions
(here, we implicitly used formula \eqref{1.47} in order to estimate
$\|\bar u_0(t)\|_{H^2}$ through $\|\bar\phi_0 (t)\|_{H^2}$).
Thanks to
H\"older's inequality, we  can estimate the second term:
\begin{multline}
|\left(I_2(t),\theta_0(t)\right)|\le Q(\|\bar\phi_0(t)\|_{H^2})
\|\partial_t\bar\phi_0(t)\|_{L^2}\|\partial_t\bar\phi_0(t)\|_{L^3}\|\theta_0(t)\|_{L^6}\le\\\le
Q_1(\|\bar\phi_0(t)\|_{H^2})\|\partial_t\bar\phi_0(t)\|_{L^2}^2
\|\partial_t\bar\phi_0(t)\|_{H^1}^2+
\frac\delta4\|\nabla_x\theta_0(t)\|^2_{L^2}.
\label{1.60}
\end{multline}
Finally, using estimates \eqref{1.49} and \eqref{1.54}, we have
\begin{equation}
|\left(I_3(t),\theta_0(t)\right)|\le \|\theta_0(t)\|_{L^2}^2+
Q(\|\bar\phi_0(0)\|_{H^2})e^{-\gamma t}+Q(\|g_1\|_{L^2}^2+\|g_2\|_{L^2}^2).
\label{1.61}
\end{equation}
Inserting estimates \eqref{1.59}-\eqref{1.61} into
\eqref{1.58}, integrating the inequality that we obtain over $[t,t+1]$
and using estimates \eqref{1.49} and \eqref{1.54} again,
we find
\begin{equation}
\int_t^{t+1}\|\theta_0(s)\|_{H^1}^2\,ds\le
Q(\|\bar\phi_0(0)\|_{H^2}^2)e^{-\gamma t}+Q(\|g_1\|_{L^2}^2+\|g_2\|_{L^2}^2),
\label{1.62}
\end{equation}
for a positive constant $\gamma$ and a  monotonic function
$Q$. There now remains to recall that
$\partial_t^2\bar u_0:=\delta^{-1}\partial_t^2\bar\phi_0+\theta_0$
and that the appropriate estimate for $\partial_t^2\bar\phi_0$ is given by
\eqref{1.56} to finish the proof of the lemma.\hfill\qed




\section{Estimates on the difference of solutions} \label{sec2}

In this section, we derive several estimates on the difference
of two solutions of problem \eqref{0.1} which are of fundamental
significance for our study of exponential attractors.

We start with computing the first terms of the asymptotic expansions
of the solution $(\phi(t),u(t))$ of  problem \eqref{0.1} as
$\varepsilon\to0$. To this end, following the general procedure (see
e.g. \cite{24}), we introduce the fast variable $\tau:=\frac t\varepsilon$
and expand the solution as follows:
\begin{equation}
\phi(t)=\phi_0(t,\tau)+\varepsilon\phi_1(t,\tau)+\cdots,\quad  u(t)=u_0(t,\tau)+\varepsilon u_1(t,\tau)+\cdots,
\label{2.1}
\end{equation}
where the functions $u_i(t,\tau)$ are of the form
\begin{equation}
u_i(t,\tau):=\bar u_i(t)+\tilde u_i(\tau),\quad \phi_i(t,\tau):=\bar\phi_i(t)+\tilde\phi_i(\tau),
\label{2.2}
\end{equation}
and satisfy the additional conditions
\begin{equation}
\lim_{\tau\to\infty}\tilde
u_i(\tau)=\lim_{\tau\to\infty}\tilde\phi(\tau)=0.
\label{2.3}
\end{equation}
Inserting these expansions into  system \eqref{0.1} and assuming
that the $u_i(t,\tau)$ are independent of $\varepsilon$,
we can obtain the recurrent equations for $u_i(t,\tau)$ and
$\phi_i(t,\tau)$. Indeed, at order $\varepsilon^{-1}$, it follows
from the first equation of \eqref{0.1} that
\[
\partial_\tau\tilde\phi_0(\tau)=0\  \text{ and, consequently, }
\tilde\phi_0(\tau )\equiv0.
\]
At order $\varepsilon^0$, we obtain
\[
\delta\partial_\tau\tilde\phi_1(\tau)=\tilde u_0(\tau), \quad
\delta\partial_t \bar\phi_0(t)=\Delta_x\bar\phi_0(t)-f_1(\bar\phi_0)
+\bar u_0+g_1.
\]
Analogously, we deduce from the second equation of \eqref{0.1}
that
\[
\partial_t\bar\phi_0(t)=\Delta_x\bar u_0(t)-f_2(\bar u_0(t))+g_2,
\]
and
\[
\partial_\tau\tilde u_0(\tau)=\Delta_x\tilde u_0(\tau)-
[f_2(\bar u_0(0)+\tilde u_0(\tau))-f_2(\bar u_0(0))]-
\partial_\tau\tilde\phi_1(\tau).
\]
Expanding now the initial data for $(\phi(t),u(t))$, we have
\[
\bar \phi_0(0)=\phi(0),\quad  \bar\phi_1(0)+\tilde\phi_1(0)=0,
\quad \tilde u(0)=u(0)-\bar u_0(0).
\]
Thus, the function $(\bar \phi_0(t),\bar u_0(t))$ solves  equation
\eqref{1.44} with  initial data  $\bar\phi_0(0)=\phi(0)$,
i.e.
\begin{equation}
(\bar\phi_0(t),\bar u_0(t))=S_t^0(\phi(0),\mathcal{L}(\phi(0))),
\label{2.4}
\end{equation}
and the first boundary layer term $\tilde u_0(\tau)$ can
be found as a solution of the following problem:
\begin{multline}
\partial_\tau\tilde u_0(\tau)=\Delta_x\tilde u_0(\tau)-
[f_2(\bar u_0(0)+\tilde u_0(\tau))-f_2(\bar
u_0(0))]-\delta^{-1}\tilde u_0(\tau),\\
\tilde u_0(0)=u(0)-\mathcal{L}(\phi(0)),\quad \tilde u_0\big|_{\partial\Omega}=0.
\label{2.5}
\end{multline}
Then, the boundary layer term $\tilde\phi_1(\tau)$
is given by
\begin{equation}
\tilde\phi_1(\tau)=\delta^{-1}\int_\tau^\infty\tilde u_0(s)\,ds.
\label{2.6}
\end{equation}
We restrict ourselves to  the first boundary layer
term in the asymptotic expansions \eqref{2.1} only  and  estimate
the rest (which is in  fact sufficient for our purposes).
To be more precise, we seek for a  solution of  equations
\eqref{0.1} of the form
\begin{equation}
\phi(t):=\bar\phi_0(t)+\varepsilon\tilde\phi(t/\varepsilon)+\varepsilon\widehat\phi(t),\quad  u(t):=\bar u_0(t)+\tilde u(t/\varepsilon)+\varepsilon \widehat u(t),
\label{2.7}
\end{equation}
where $(\bar\phi_0(t),\bar u_0(t))$ solves the limit problem
\eqref{1.44}, the boundary layer term $\tilde u(\tau)$
solves
\begin{multline}
\partial_\tau\tilde u(\tau)=\Delta_x\tilde u(\tau)-
[f_2(\bar u_0(\varepsilon\tau)+\tilde u(\tau))-f_2(\bar
u_0(\varepsilon\tau))]-\delta^{-1}\tilde u(\tau),\\
\tilde u(0)=u(0)-\mathcal{L}(\phi(0)),\quad \tilde u\big|_{\partial\Omega}=0,
\label{2.8}
\end{multline}
and the boundary layer term $\tilde\phi(\tau)$ is defined by
\begin{equation}
\tilde\phi(\tau)=\delta^{-1}\int_\tau^\infty\tilde u(s)\,ds.
\label{2.9}
\end{equation}
Equation \eqref{2.8} on $\tilde u(\tau)$ differs
slightly from equation \eqref{2.5} for the function
$\tilde u_0(\tau)$ (the term $\bar u_0(0)$ is replaced
by $\bar u_0(t):=\bar u_0(\varepsilon\tau)$). We note however that the difference
$\tilde u(\tau)-\tilde u_0(\tau)$ is of  order $\varepsilon^1$
and, consequently, can be interpreted as a part of the rest
in the asymptotic expansions \eqref{2.1}.

The next lemma shows that the function $\tilde u(\tau)$, solution
of equation \eqref{2.8}, is indeed a boundary
layer term.

\begin{lemma} \label{lm2.1} Let the above assumptions hold. Then,
the solution $\tilde u(\tau)$ of problem \eqref{2.8} satisfies
the estimate
\begin{equation}
\|\tilde u(\tau)\|_{H^2}+\|\partial_\tau\tilde u(\tau)\|_{L^2}\le
 Q\bigg(\|(\phi(0),u(0))\|_\Phi\bigg)\|\tilde u(0)\|_{H^2}e^{-\gamma\tau},
\label{2.10}
\end{equation}
where $\gamma>0$ is a positive constant
and $Q$ is a monotonic function that are
both independent of $\varepsilon$.
\end{lemma}

\paragraph{Proof} We set $\tilde v(\tau):=\tilde u(\tau)^2$. Then,
due to the assumption $f_2'\ge0$, this function satisfies the inequation
\[
\partial_\tau \tilde v(\tau)-\Delta_x \tilde v(\tau)-2\delta^{-1}\tilde
v(\tau)\le0,
\quad \tilde v(0)=\tilde u(0)^2,
\]
and, consequently, due to the comparison principle, we have
\begin{equation}
\|\tilde u(\tau)\|_{L^\infty}\le C\|\tilde u(0)\|_{L^\infty}
e^{-\gamma t}.
\label{2.11}
\end{equation}
Having estimate \eqref{2.11} for the $L^\infty$-norm of
$\tilde u(\tau)$ and  estimates \eqref{1.49}
and \eqref{1.51} for $\bar u_0(t)$, we deduce \eqref{2.10} by
applying standard parabolic regularity arguments
to equation \eqref{2.8} and Lemma \ref{lm2.1} is proved.\hfill\qed

We are now in a position to estimate the rest $(\widehat\phi(t),\widehat
u(t))$ in expansions~\eqref{2.7}.

\begin{lemma} \label{lm2.2} Let the above assumptions hold. Then, the
rest $(\widehat\phi(t),\widehat u(t))$ in the asymptotic expansions
\eqref{2.7} enjoys the following estimate:
\begin{equation}
\|\widehat\phi(t)\|_{H^2}+\|\widehat u(t)\|_{H^2}+
\|\partial_t\widehat\phi(t)\|_{L^2}+\varepsilon\|\partial_t\widehat u(t)\|_{L^2}\le
Ce^{Lt},
\label{2.12}
\end{equation}
where the constants $C$ and $L$ depend on $\|(\phi(0),u(0))\|_{\Phi}$,
but are independent of $\varepsilon$.
\end{lemma}

\paragraph{Proof} The functions $\widehat\phi(t)$ and $\widehat u(t)$
 satisfy the equations
\begin{equation}
\begin{gathered}
\delta\partial_t\widehat\phi=\Delta_x\widehat\phi-
\frac1\varepsilon\bigg[f_1(\bar \phi_0+\varepsilon\tilde\phi+\varepsilon\widehat\phi)-
f_1(\bar\phi_0)\bigg]+\widehat u+\Delta_x\tilde\phi,\\
\varepsilon\partial_t\widehat u=\Delta_x\widehat u-\frac1\varepsilon\bigg[f_2(\bar u_0+\tilde u+\varepsilon\widehat u)-
f_2(\bar u_0+\tilde u)\bigg]-\partial_t\widehat\phi-\partial_t\bar u_0,\\
\widehat\phi\big|_{t=0}=-\tilde\phi(0),\quad \widehat u\big|_{t=0}=0.
\end{gathered}
\label{2.13}
\end{equation}
We first note that, according to \eqref{2.9} and \eqref{2.10},
we have
\begin{equation}
\|\tilde\phi(\tau)\|_{H^2}\le Q\bigg(\|(\phi(0),u(0))\|_\Phi\bigg)
\|\tilde u(0)\|_{H^2}e^{-\gamma\tau},
\label{2.14}
\end{equation}
where $Q$ is independent of $\epsilon $, and, consequently, the initial data in \eqref{2.13} is uniformly
bounded in $H^2(\Omega)$ as $\varepsilon\to0$.

Multiplying the first equation of \eqref{2.13} by $\widehat\phi(t)$
and integrating over $\Omega$, we have, noting that $f'_1\ge-K$
\begin{multline}
\delta\partial_t\|\widehat\phi(t)\|_{L^2}^2+\frac32\|\nabla_x\widehat\phi(t)\|_{L^2}^2\le
2K\|\widehat\phi(t)\|_{L^2}^2+\\+C\left(\|\widehat u(t)\|_{L^2}^2+
\|\nabla_x\tilde\phi(\frac t\varepsilon)\|_{L^2}^2\right).
\label{2.15}
\end{multline}
We now differentiate the first equation of \eqref{2.13} with respect
to $t$, multiply the relation that we obtain by $\partial_t \widehat\phi(t)$ and integrate over
 $\Omega$ to find
\begin{multline}
\delta\partial_t\|\partial_t\widehat\phi(t)\|_{L^2}^2+2\|\nabla_x\partial_t\widehat\phi(t)\|_{L^2}^2-
2(\partial_t\widehat\phi(t),\partial_t\widehat u(t))\le
2K\|\partial_t\widehat\phi(t)\|_{L^2}^2-\\-
\frac2\varepsilon\left([f_1'(\bar
\phi_0+\varepsilon\tilde\phi+\varepsilon\widehat\phi)-f_1'(\bar\phi_0)]\partial_t\bar\phi_0,\partial_t\widehat\phi\right)-
2(f_1'(\bar\phi_0+\varepsilon\tilde\phi+\varepsilon\widehat\phi)\partial_t\tilde\phi,\partial_t\widehat\phi)+\\+
\|\partial_t\Delta_x\tilde\phi\|_{L^2}\left(1+\|\partial_t\widehat\phi(t)\|_{L^2}^2\right).
\label{2.16}
\end{multline}
Since the functions $\tilde\phi$ and $\varepsilon\widehat\phi$
are uniformly bounded
(with respect to $\varepsilon$)
in $H^2(\Omega)$ and $\partial_t\bar\phi_0$ is bounded in
$L^2(\Omega)$ (see \eqref{1.17}, \eqref{1.51} and \eqref{2.14}),
it follows that
\begin{multline}
\frac2\varepsilon\left([
f_1'(\bar\phi_0)-f_1'(\bar
\phi_0+\varepsilon\tilde\phi+\varepsilon\widehat\phi)]\partial_t\bar\phi_0,\partial_t\widehat\phi\right)
\le C\left((1+|\widehat\phi|)|\partial_t\bar\phi_0|,|\partial_t\widehat\phi|\right)
\le\\\le
C\left(1+\|\partial_t\widehat \phi(t)\|_{L^2}^2+\|\widehat\phi(t)\|^2_{L^2}\right)+
\frac12\|\nabla_x\widehat\phi(t)\|_{L^2}^2+\|\nabla_x\partial_t\widehat\phi(t)\|_{L^2}^2,
\label{2.17}
\end{multline}
where the constant $C$ depends on $\|(\phi_0,u_0)\|_\Phi$,
but is independent of $\varepsilon$. Analogously, we have
\begin{equation}
2|(f_1'(\bar\phi_0+\varepsilon\tilde\phi+\varepsilon\widehat\phi)\partial_t\tilde\phi,\partial_t\widehat\phi)|\le
C\|\partial_t\tilde\phi\|_{H^2}(1+\|\partial_t\widehat\phi(t)\|_{L^2}^2),
\label{2.18}
\end{equation}
where $C$ is independent of $\varepsilon$.
Inserting estimates \eqref{2.17} and \eqref{2.18} into
estimate \eqref{2.16} and  summing the relation that we obtain with inequality
\eqref{2.15},
we find
\begin{multline}
\delta\partial_t\left(\|\widehat\phi(t)\|_{L^2}^2+\|\partial_t\widehat\phi(t)\|_{L^2}^2+1\right)+
\|\nabla_x\partial_t\widehat\phi(t)\|_{L^2}^2+\|\nabla_x\widehat\phi(t)\|_{L^2}^2-\\-
2\left(\partial_t\widehat\phi(t),\partial_t\widehat u(t)\right)\le\\\le
C\left(1+\|\partial_t\tilde\phi\|_{H^2}\right)
\left(1+\|\widehat\phi(t)\|_{L^2}^2+\|\partial_t\widehat\phi(t)\|_{L^2}^2+
\|\widehat u(t)\|_{L^2}^2\right),
\label{2.19}
\end{multline}
where the constant $C$ depends on $\|(\phi_0,u_0)\|_\Phi$,
but is independent of $\varepsilon$.

Multiplying now the second equation of \eqref{2.13} by $\partial_t\widehat u(t)$
and integrating over $\Omega$, we  have
\begin{multline}
\partial_t\left(\|\nabla_x u(t)\|_{L^2}^2-2(\partial_t\bar u_0(t),\widehat u(t))\right)+
2(\partial_t\widehat\phi(t),\partial_t\widehat u(t))+\varepsilon\|\partial_t\widehat u(t)\|_{L^2}^2\le\\\le-
\frac2\varepsilon\left([f_2(\bar u_0+\tilde u+\varepsilon\widehat u)-
f_2(\bar u_0+\tilde u)],\partial_t\widehat u(t)\right)-\\-
\|\partial_t^2\bar u_0\|_{H^{-1}}(1+\|\widehat u(t)\|_{H^1}^2).
\label{2.20}
\end{multline}
In order to transform \eqref{2.20}, we use the following identity:
\begin{multline}
\frac1\varepsilon\left([f_2(\bar u_0+\tilde u+\varepsilon\widehat u)-
f_2(\bar u_0+\tilde u)],\partial_t\widehat u(t)\right)=\\=
\partial_t\bigg[\frac1{\varepsilon^2}\left(F_2(\bar u_0+\tilde u+\varepsilon\widehat u)-F_2(\bar
u_0+\tilde u)-\varepsilon f_2(\bar u_0+\tilde u)\widehat u,1\right)\bigg]-\\-
\bigg[\frac1{\varepsilon^2}\left(f_2(\bar u_0+\tilde u+\varepsilon\widehat u)-
f_2(\bar u_0+\tilde u_0)-\varepsilon f_2'(\bar u_0+\tilde u)\widehat u,\partial_t\bar
u_0+\partial_t\tilde u\right)\bigg]:=\\:=\partial_t\Theta_\varepsilon(t)-\theta_\varepsilon(t),
\label{2.21}
\end{multline}
where $F_2(v):=\int_0^vf_2(s)\,ds$.
We now note  that, due to the assumption $f_2'(v)\ge0$ and due
to the condition $\widehat u(0)=0$, we have
\begin{equation}
\Theta_\varepsilon(t)\ge0,\quad \Theta_\varepsilon(0)=0.
\label{2.22}
\end{equation}
Moreover, arguing in a standard way, we can obtain the following
estimate for $\theta_\varepsilon(t)$:
\begin{equation}
|\theta_\varepsilon(t)|\le C\left(|\widehat u(t)|^2,|\partial_t\tilde u|+|\partial_t\bar u_0|\right)\le
C_1(\|\partial_t \tilde u\|_{L^2}+1)\|\widehat u(t)\|_{H^1}^2,
\label{2.23}
\end{equation}
where the constants $C$ and $C_1$ depend on $\|(\phi_0,u_0)\|_\Phi$,
but are independent of $\varepsilon$. Inserting identity \eqref{2.21}
and inequality \eqref{2.23} into relation \eqref{2.20}
and summing the relation that we obtain with  inequality \eqref{2.19},
we finally find
\begin{multline*}
\partial_t\bigg[
\delta\|\widehat\phi(t)\|_{L^2}^2+\delta\|\partial_t\widehat\phi(t)\|_{L^2}^2+
\|\widehat u(t)\|_{H^1}^2-2(\partial_t \bar u_0(t),\widehat
u(t))+2\Theta_\varepsilon(t)+C_2\bigg]\le\\
\le C_3\left(1+\|\partial_t\tilde\phi(t/\varepsilon)\|_{H^2}+\|\partial_t\tilde u(t/\varepsilon)\|_{L^2}+
\|\partial_t^2\bar u_0(t)\|_{H^{-1}}^2\right)\times\\\times\bigg[
\delta\|\widehat\phi(t)\|_{L^2}^2+\delta\|\partial_t\widehat\phi(t)\|_{L^2}^2+
\|\widehat u(t)\|_{H^1}^2-2(\partial_t \bar u_0(t),\widehat
u(t))+2\Theta_\varepsilon(t)+C_2\bigg],
\end{multline*}
where the constants $C_2$ and $C_3$ depend on $\|(\phi_0,u_0)\|_\Phi$,
but are independent of $\varepsilon$.  Moreover, the constant $C_2$
can be chosen such that the expression in square brackets in the right-hand
side of the above inequality is positive
(it is possible to do so thanks to  estimates \eqref{1.51} and
\eqref{2.22}). Applying Gronwall's inequality to this relation
and noting that \eqref{1.51} and
\eqref{2.10} yield the estimate
\begin{equation}
\int_t^{t+1}\left(
\|\partial_t\tilde\phi(s/\varepsilon)\|_{H^2}+\|\partial_t\tilde u(s/\varepsilon)\|_{L^2}+
\|\partial_t^2\bar u_0(s)\|_{H^{-1}}^2\right)\,ds\le C_4,
\label{2.24}
\end{equation}
where $C_4$ is independent of
$\varepsilon$, we find the estimate
\begin{equation}
\|\widehat\phi(t)\|_{L^2}^2+\|\partial_t\widehat\phi(t)\|_{L^2}^2+
\|\widehat u(t)\|_{H^1}^2\le C_5e^{L_1 t},
\label{2.25}
\end{equation}
where the constants $C_5$ and $L_1$ depend on $\|(\phi_0,u_0)\|_\Phi$,
but are independent of $\varepsilon$.

Estimate \eqref{2.12} can be deduced from  estimate
\eqref{2.25}, based on  standard parabolic regularity arguments,
exactly as in the proof of Lemma \ref{lm1.3}, which finishes the proof of
Lemma \ref{lm2.2}.\hfill\qed

\vskip 0.1cm

Let us now formulate several useful corollaries of
 estimate \eqref{2.12}.
\begin{corollary} \label{coro2.1}
Let the above assumptions hold. We also assume that
$(\phi(t),u(t))$ is  solution of  equation \eqref{0.1} and
$(\bar\phi_0(t), \bar u_0(t))$ is  solution of the limit problem
 \eqref{1.44},
with  $\bar\phi_0(0)=\phi(0)$. Then, the following estimate is valid:
\begin{multline}
\|\phi(t)-\bar\phi_0(t)\|_{H^2}+\|u(t)-\bar u_0(t)\|_{H^2}+\|\partial_t\phi(t)-\partial_t\bar\phi_0(t)\|_{L^2}+\\+
\varepsilon\|\partial_t u(t)-\partial_t \bar u_0(t)\|_{L^2}\le
 C\left(\|u(0)-\mathcal{L}(\phi(0))\|_{H^2}e^{-\gamma\frac t\varepsilon}+
\varepsilon e^{L t}\right),
\label{2.26}
\end{multline}
where $\gamma>0$ is a  positive constant depending only on
$\Omega$ and the constants $C$ and $L$ depend on
$\|(\phi(0),u(0))\|_\Phi$, but are independent of $\varepsilon$.
\end{corollary}

Indeed, estimate \eqref{2.26} is an immediate corollary
of the asymptotic expansions \eqref{2.7} and of  estimates
\eqref{2.10}, \eqref{2.12} and \eqref{2.14}.

\begin{corollary} \label{coro2.2}
 Let the above assumptions hold and let
$(\phi(t),u(t))$ be solution of problem \eqref{0.1}. Then, the
following estimates hold:
\begin{equation}
\|\partial_t u(t)\|_{L^2}\le
Q\left(\|(\phi(0),u(0))\|_\Phi\right)\left(1+
\frac1\varepsilon\|u(0)-\mathcal{L}(\phi(0))\|_{H^2}e^{-\gamma\frac t\varepsilon}\right),
\label{2.27}
\end{equation}
and
\begin{equation}
\|u(t)-\mathcal{L}(\phi(t))\|_{H^2}\le
Q\left(\|(\phi(0),u(0))\|_\Phi\right)\left(\varepsilon+
\|u(0)-\mathcal{L}(\phi(0))\|_{H^2}e^{-\gamma\frac t\varepsilon}\right),
\label{2.28}
\end{equation}
where the constant $\gamma>0$ and the function $Q$ are independent of $\varepsilon$.
\end{corollary}

\paragraph{Proof} Without loss of generality, we can derive  estimates
\eqref{2.27} and \eqref{2.28} for $t\le1$ only. Now,  estimate
\eqref{2.27} is an immediate corollary of \eqref{2.26} and
\eqref{1.51}. So, there only  remains to deduce  estimate \eqref{2.28}.
To this end, we recall that, by definition of the operator $\mathcal{L}$, we have
$\bar u_0(t)=\mathcal{L}(\bar \phi_0(t))$ and, consequently
\begin{equation}
\|u(t)-\mathcal{L}(\phi(t))\|_{H^2}\le \|u(t)-\bar u_0(t)\|_{H^2}+
\|\mathcal{L}(\phi(t))-\mathcal{L}(\bar\phi_0(t))\|_{H^2}.
\label{2.29}
\end{equation}
Estimate \eqref{2.28} is now  a corollary of \eqref{2.26},
\eqref{2.29} and  of Proposition \ref{prop1.1}.\hfill\qed


\begin{remark} \label{rm2.1} Let the function $\tilde U(\tau)$ be
solution
of the problem
\begin{equation}
\partial_\tau \tilde U=\Delta_x\tilde U-\delta^{-1}\tilde U,\quad \tilde
U\big|_{t=0}=u(0)-\mathcal{L}(\phi(0)),
\label{2.30}
\end{equation}
i.e. $\tilde U(\tau):=e^{-(-\Delta_x+\delta^{-1}I)\tau}(u(0)-\mathcal{L}(\phi(0)))$. Then, it is not difficult to verify that the quantity
$\tilde u(t/\varepsilon)-\tilde U(t/\varepsilon)$ is of order $\varepsilon^1$ as $\varepsilon\to0$
and, consequently, the boundary layer term in  expansions
\eqref{2.7} can be simplified as follows:
\begin{equation}
u(t)=\bar u_0(t)+e^{-(-\Delta_x+\delta^{-1}I)\frac t\varepsilon}[u(0)-\bar u_0(0)]+
O(\varepsilon).
\label{ 2.31}
\end{equation}
\end{remark}

We are now able to verify the  uniform (with respect to $\varepsilon$)
Lipschitz continuity of the semigroups $S_t^\varepsilon$ associated with
problem \eqref{0.1} in the phase space $\Phi$.

\begin{lemma} \label{lm2.3} Let the assumptions of Lemma \ref{lm1.1} hold and let
$(\phi_1(t),u_1(t))$ and $(\phi_2(t),u_2(t))$ be two solutions
of problem \eqref{0.1} with  initial data in $\Phi$.
Then, the following estimate is valid:
\begin{multline}
\|\phi_1(t)-\phi_2(t)\|_{H^2}^2+\|u_1(t)-u_2(t)\|_{H^2}^2+
\|\partial_t \phi_1(t)-\partial_t\phi_2(t)\|_{L^2}^2+\\+
\varepsilon^2\|\partial_t u_1(t)-\partial_t u_2(t)\|_{L^2}^2\le\\
\le
Ce^{Lt}\left(\|\phi_1(0)-\phi_2(0)\|_{H^2}^2+\|u_1(0)-u_2(0)\|_{H^2}^2\right),
\label{2.32}
\end{multline}
where the constants $C$ and $L$ depend on $\|\phi_i(0)\|_{H^2}$ and
on  $\|u_i(0)\|_{H^2}$, but are independent of $\varepsilon$.
\end{lemma}

\paragraph{Proof} We set $v(t):=\phi_1(t)-\phi_2(t)$ and
$w(t):=u_1(t)-u_2(t)$.
These functions  satisfy  equation \eqref{1.39}.
Moreover, due to estimate \eqref{2.27}
as well as \eqref{1.40} and \eqref{1.41}, we  also have  the
uniform estimate
\begin{equation}
\int_t^{t+1}\|\partial_t l_2(s)\|_{L^2}\,ds \le L.
\label{2.33}
\end{equation}
Differentiating now the first equation of \eqref{1.39} with respect
 to $t$,
multiplying by $\partial_t v(t)$, summing the relation that we obtain with
the second equation of \eqref{1.39} multiplied by $\partial_t w(t)$
and integrating over $\Omega$, we obtain
\begin{multline}
\partial_t[\delta\|\partial_t v(t)\|_{L^2}^2+\|\nabla_x
w(t)\|_{L^2}^2+(l_2(t)w(t),w(t))]+
2\|\nabla_x\partial_t v(t)\|_{L^2}^2
\le\\\le
 -2(l_1(t)\partial_t v(t),\partial_t v(t))-
2(\partial_t l_1(t)v(t),\partial_t v(t))+(\partial_t l_2(t)w(t),w(t)),
\label{2.34}
\end{multline}
where
\[
l_1(t):=\int_0^1f_1'(s\phi_1(t)+(1-s)\phi_2(t))\,ds,\quad
l_2(t):=\int_0^1f_2'(su_1(t)+(1-s)u_2(t))\,ds.
\]
Estimates \eqref{2.34}, \eqref{1.40} and \eqref{1.41} imply
that
\begin{multline}
\partial_t[\delta\|\partial_t v(t)\|_{L^2}^2+\|\nabla_x
w(t)\|_{L^2}^2+(l_2(t)w(t),w(t))]+
2\|\nabla_x\partial_t v(t)\|_{L^2}^2
\le\\\le C\left(1+\|\partial_t l_2(t)\|_{L^2}\right)[
\delta\|\partial_t v(t)\|_{L^2}^2+\|\nabla_x
w(t)\|_{L^2}^2+(l_2(t)w(t),w(t))+\\+
2\|\nabla_x\partial_t v(t)\|_{L^2}^2]+C\|v(t)\|_{L^2}^2,
\label{2.35}
\end{multline}
where the constant $C$ depends on $\|(\phi_i(0),u_i(0))\|_{\Phi}$,
but is independent of $\varepsilon$.
Applying Gronwall's inequality to  relation \eqref{2.35}
and taking into account  inequalities \eqref{2.33} and
\eqref{1.38}, we find
\begin{equation}
\|\partial_t v(t)\|_{L^2}^2+\|w(t)\|^2_{H^1}\le
Ce^{Lt}\left(\|v(0)\|_{H^2}^2+\|w(0)\|^2_{H^1}\right),
\label{2.36}
\end{equation}
where the constants $C$ and $L$ depend on
 $\|(\phi_i(0),u_i(0))\|_{\Phi}$,
but are independent of~$\varepsilon$.

Estimate \eqref{2.32} is a  corollary of \eqref{2.36}
and of standard parabolic regularity arguments (see the proof of
Lemma \ref{lm1.3}).  This finishes the proof of Lemma
\ref{lm2.3}.\hfill\qed

\vskip 0.1cm

To conclude this section, we finally derive   standard smoothing estimates
for the difference of solutions of \eqref{0.1} which are necessary
for our construction of exponential attractors.


\begin{lemma} \label{lm2.4} Let the assumptions of Lemma \ref{lm1.1} hold and let
$(\phi_1(t),u_1(t))$ and $(\phi_2(t),u_2(t))$ be two solutions
of problem \eqref{0.1} with  initial data in $\Phi$.
Then, the following estimate is valid:
\begin{multline}
\|\phi_1(t)-\phi_2(t)\|_{H^3}^2+\|u_1(t)-u_2(t)\|_{H^3}^2\le\\
\le
Ce^{Lt}\frac{t+1}t\left(\|\phi_1(0)-\phi_2(0)\|_{H^2}^2+
\|u_1(0)-u_2(0)\|_{H^2}^2\right),\quad t>0,
\label{2.37}
\end{multline}
where the constants $C$ and $L$ depend on $\|\phi_i(0)\|_{H^2}$ and
on  $\|u_i(0)\|_{H^2}$, but are independent of $\varepsilon$.
\end{lemma}

\paragraph{Proof} We split the solution $(v(t),w(t))$
of problem \eqref{1.39} into a sum of two functions
\begin{equation}
v(t):=v_1(t)+v_2(t),\quad w(t):=w_1(t)+w_2(t),
\label{2.38}
\end{equation}
where the function $(v_1(t),w_1(t))$ solves
\begin{equation}
\begin{gathered}
\delta\partial_t v_1-\Delta_x v_1=H_1(t):=w(t)-l_1(t)v(t),\quad v_1\big|_{t=0}=0,\\
\varepsilon\partial_t w_1+\partial_t v_1-\Delta_x w_1=H_2(t):=-l_2(t)w(t),\quad w_1\big|_{t=0}=0,
\end{gathered}
\label{2.39}
\end{equation}
with
\[
l_1(t):=\int_0^1f_1'(s\phi_1(t)+(1-s)\phi_2(t))\,ds,\quad
l_2(t):=\int_0^1f_2'(su_1(t)+(1-s)u_2(t))\,ds,
\]
and the function $(v_2(t),w_2(t))$ solves
\begin{equation}
\begin{gathered}
\delta\partial_t v_2-\Delta_x v_2=0,\quad v_2\big|_{t=0}=v(0),\\
\varepsilon\partial_t w_2+\partial_t v_2-\Delta_x w_2=0,\quad w_2\big|_{t=0}
=w(0).
\end{gathered}
\label{2.40}
\end{equation}
It follows from estimates \eqref{2.32} and from the assumption
$f_i\in C^3$, $i=1,2$, that
\begin{equation}
\|H_1(t)\|_{H^2}+\|H_2(t)\|_{H^2}\le
 Ce^{L t}\left(\|v(0)\|_{H^2}+\|w(0)\|_{H^2}\right),
\label{2.41}
\end{equation}
and, consequently, due to  standard parabolic regularity arguments
(see the proof of Lemma \ref{lm1.3}), we have
\begin{equation}
\|v_1(t)\|_{H^3}+\|w_1(t)\|_{H^3}\le
 Ce^{L t}\left(\|v(0)\|_{H^2}+\|w(0)\|_{H^2}\right),
\label{2.42}
\end{equation}
where the constants $C$ and $L$ depend on
$\|(\phi_i(0),u_i(0))\|_{\Phi}$,
but are independent of~$\varepsilon$.

 The solution $(v_2(t),w_2(t))$ of  problem \eqref{2.40} can
be easily found by using the analytic semigroups theory (see
\cite{16} and \cite{23}).
 More precisely, we have
\begin{equation}
v_2(t)=e^{-A\frac t\delta}v(0),\quad w_2(t)=e^{-A\frac t\varepsilon}w(0)+
\frac\delta{\delta-\varepsilon}\left(e^{-A\frac t\delta}-e^{-A\frac t\varepsilon}v(0)\right),
\label{2.43}
\end{equation}
where $A:=-\Delta_x$, associated with Dirichlet boundary conditions. A standard smoothing estimate for  analytic
semigroups (see e.g. \cite{16}), applied to \eqref{2.43}, implies that
\begin{equation}
\|v_2(t)\|_{H^3}^2+\|w_2(t)\|_{H^3}^2\le Ct^{-1}e^{-\gamma t}
\left(\|v(0)\|_{H^2}^2+\|w(0)\|_{H^2}^2\right),\quad  t>0,
\label{2.44}
\end{equation}
where $C$ and $\gamma>0$ are independent of $\varepsilon$. Combining
estimates \eqref{2.42} and \eqref{2.44}, we derive \eqref{2.37}
and Lemma \ref{lm2.4} is proved.\hfill\qed


\begin{remark} \label{rm2.2} We recall that  estimates \eqref{2.32} and
\eqref{2.37} hold uniformly with respect to $\varepsilon>0$. Consequently,
passing to the limit $\varepsilon\to0$ in these estimates, we see that
the same estimates remain valid for the difference of
solutions of the limit problem \eqref{1.39}.
\end{remark}


\section{Robust exponential attractors} \label{sec3}

In this section, we construct a uniform family
of exponential attractors $\mathcal{M}_\varepsilon$ in $\Phi$ for  problem
\eqref{0.1}
which
converges as $\varepsilon\to0$ to the limit exponential attractor $\mathcal{M}_0$
of problem \eqref{1.44}. To be more precise, the main result
of this section is the following theorem.

\begin{theorem} \label{thm3.1} Let assumptions (0.2) hold. Then,
there exists a family of compact sets $\mathcal{M}_\varepsilon\subset\Phi$, $\varepsilon\in[0,1]$,
such that

1. These sets are semi-invariant with respect to the flows $S_t^\varepsilon$
associated with problem \eqref{0.1}, i.e.
\begin{equation}
S_t^\varepsilon\mathcal{M}_\varepsilon\subset\mathcal{M}_\varepsilon. \label{3.1}
\end{equation}

2. The fractal dimension of the sets $\mathcal{M}_\varepsilon$ is finite and
uniformly bounded with respect to $\varepsilon$:
\begin{equation}
\dim_F(\mathcal{M}_\varepsilon,\Phi)\le C<\infty,
\label{3.2}
\end{equation}
where $C$ is independent of $\varepsilon$.

3. These sets attract exponentially the bounded subsets of $\Phi$, i.e.
there exists a positive constant $\alpha>0$ and a monotonic function $Q$
which are independent of $\varepsilon$ such that, for every bounded
subset $B$ in the phase space $\Phi$, we have
\begin{equation}
\dist\nolimits_\Phi(S_t^\varepsilon B,\mathcal{M}_\varepsilon)\le Q(\|B\|_\Phi)e^{-\alpha t},\quad  \varepsilon\in[0,1],
\label{3.3}
\end{equation}
where $\dist_\Phi$ denotes the nonsymmetric Hausdorff distance between
sets in $\Phi$ (for  $\varepsilon=0$, we should take $B\subset\mathbb{L}$).

4. The symmetric Hausdorff distance between the limit attractor
 $\mathcal{M}_0$ and the attractors $\mathcal{M}_\varepsilon$ enjoys the following estimate:
\begin{equation}
\dist\nolimits_{sym,\Phi}
(\mathcal{M}_\varepsilon,\mathcal{M}_0)\le C\varepsilon^\kappa,
\label{3.4}
\end{equation}
where the constants $C>0$ and $0<\kappa<1$ are independent of $\varepsilon$
and can be computed explicitly.
\end{theorem}

The proof of this result is based on the following abstract result
for exponential attractors of singularly perturbed discrete maps.

\begin{proposition} \label{prop3.1}
Let $B_\varepsilon\subset\Phi$, $\varepsilon\in[0,1]$,
be a family of closed and bounded subsets of a Banach space $\Phi$ and let
$S^\varepsilon: B_\varepsilon\to B_\varepsilon$ be a family of maps which satisfies the
following properties:

1. There exists another Banach space $\Phi_1$, which is compactly embedded
into $\Phi$, such that, for every $b_\varepsilon^1,b_\varepsilon^2\in B_\varepsilon$, the following
estimate holds:
\begin{equation}
\|S^\varepsilon b_\varepsilon^1-S^\varepsilon b_\varepsilon^2\|_{\Phi_1}\le
K\|b_\varepsilon^1-b^2_\varepsilon\|_{\Phi},
\label{3.5}
\end{equation}
where the constant $K$ is independent of $\varepsilon$.

2. There exist  nonlinear 'projectors' $\Pi_\varepsilon: B_\varepsilon\to B_0$
such that, for every $b_\varepsilon\in B_\varepsilon$
\begin{equation}
\|S^\varepsilon_{(k)}b_\varepsilon-S^0_{(k)}\Pi_\varepsilon
b_\varepsilon\|_{\Phi}\le \varepsilon L^k,\quad k\in\mathbb{N},
\label{3.6}
\end{equation}
where $S_{(k)}$ denotes the $k$th iteration of  $S$ and the
constant $L$ is independent of $\varepsilon$.

 Then, the maps $S^\varepsilon$ possess a  family of exponential
attractors $\mathcal{M}_\varepsilon^d$ which satisfies \eqref{3.1}, \eqref{3.2},
\eqref{3.4} uniformly with respect to $\varepsilon$ and such that
\begin{equation}
\dist\nolimits_\Phi(S^\varepsilon_{(k)}B_\varepsilon,\mathcal{M}_\varepsilon^d)\le Ce^{-\gamma k},
\label{3.7}
\end{equation}
where $C$ and $\gamma>0$ are also independent of $\varepsilon$ and can be
computed explicitly.
\end{proposition}

The proof of this proposition is given in
\cite{15} in a more general setting.

\vskip 0.1cm

\paragraph{Proof of Theorem \ref{thm3.1}} We  apply the abstract result
of Proposition \ref{prop3.1} to our situation. To this end, we define
the sets $B_\varepsilon\subset\Phi$ for $\varepsilon\ne0$ by
\begin{equation}
B_\varepsilon=B:=\{(\phi_0,u_0)\in\Phi,\,
\|(\phi_0,u_0)\|_{\Phi}^2\le 2Q(\|g_1\|^2_{L^2}+\|g_2\|_{L^2}^2)\},
\label{3.8}
\end{equation}
where the function $Q$ is defined in \eqref{1.17}, and, for $\varepsilon=0$,
we set
\begin{equation}
B_0=\{(\phi_0,u_0)\in\Phi,\
\|\phi_0\|_{H^2}^2\le 2Q(\|g_1\|_{L^2}^2+\|g_2\|_{L^2}^2),\ u_0=\mathcal{L}(\phi_0)\}.
\label{3.9}
\end{equation}
Then, it follows from the uniform estimate \eqref{1.17} that there exists
a time $T>0$ which is independent of $\varepsilon$ such that
\begin{equation}
S_T^\varepsilon B_\varepsilon\subset B_\varepsilon,\quad \varepsilon\in[0,1].
\label{3.10}
\end{equation}
We now set
\begin{equation}
S^\varepsilon:=S_T^\varepsilon,\quad \varepsilon\in[0,1],
\label{3.11}
\end{equation}
and verify that the operators \eqref{3.11} satisfy all the assumptions
of Proposition \ref{prop3.1}. Indeed, according to \eqref{3.10}, the maps
$S^\varepsilon$ are well defined on $B_\varepsilon$. Estimate \eqref{3.5},
with $\Phi_1:=H^3(\Omega)\times H^3(\Omega)$, is an immediate
corollary of Lemma \ref{lm2.4}. So, there remains to verify \eqref{3.5}. To this
end, we define the nonlinear projector $\Pi_\varepsilon$ by
\begin{equation}
 \Pi_\varepsilon: B_\varepsilon\to B_0,\quad \Pi_\varepsilon(\phi_0,u_0)
:=(\phi_0,\mathcal{L}(\phi_0)).
\label{3.12}
\end{equation}
Then,  estimate \eqref{3.6} is an immediate corollary of
\eqref{2.26} (in which the boundary layer term disappears since
$t=T>0$ and $T$ is independent of $\varepsilon$).
 Thus, all the assumptions of Proposition \ref{prop3.1} are satisfied
for the family of maps \eqref{3.11} and, consequently, these maps
possess a family of discrete exponential attractors $\mathcal{M}_\varepsilon^d$
which satisfies \eqref{3.1}, \eqref{3.2}, \eqref{3.4} and \eqref{3.7}.

We now define the desired family $\mathcal{M}_\varepsilon$ of exponential attractors
by the standard expression:
\begin{equation}
\mathcal{M}_\varepsilon:=\cup_{t\in[1,T+1]}S_t^\varepsilon\mathcal{M}_\varepsilon^d.
\label{3.13}
\end{equation}
The semi-invariance \eqref{3.1} is then an immediate corollary
of the semi-invariance of $\mathcal{M}_\varepsilon^d$ and of  definition
\eqref{3.13}. The exponential attraction \eqref{3.3} follows from
the fact that the $B_\varepsilon$ are uniform (with respect to $\varepsilon$) absorbing sets
for $S_t^\varepsilon$ (due to \eqref{1.17}) and from the uniform Lipschitz
continuity \eqref{2.32}. Estimate \eqref{3.4} for the symmetric
distance is also a  corollary of an  analogous result
for the discrete exponential attractors and of  estimates \eqref{2.26}
and \eqref{2.32}.
We note  that the boundary layer term in \eqref{2.26} also
disappears, due to estimate \eqref{2.28}, since
\begin{equation}
\mathcal{M}_\varepsilon\subset S_1^\varepsilon B_\varepsilon.
\label{3.14}
\end{equation}
 Thus, there only remains to verify
estimate \eqref{3.3} for the fractal dimension. To this end, we
need the following lemma.

\begin{lemma} \label{lm3.1} Let assumptions (0.2) hold. Then, the
solution $(\phi(t),u(t))$ of equation \eqref{0.1} is H\"older
continuous with respect to $t$, with  H\"older exponent $1/3$ if
$t\ge1$, i.e., for every $t\ge1$ and $0\le s\le 1$, we have
\begin{equation}
\|\phi(t+s)-\phi(t)\|_{H^2}+\|u(t+s)-u(t)\|_{H^2}\le
Q(\|(\phi_0,u_0)\|_{\Phi})s^{1/3},
\label{3.15}
\end{equation}
for an appropriate monotonic function $Q$ which is independent of
$\varepsilon$.
\end{lemma}

\paragraph{Proof} According to  estimates \eqref{2.27} and
\eqref{1.17} and since $t\ge1$, we  have
\begin{equation}
\|\phi(t+s)-\phi(t)\|_{L^2}+\|u(t+s)-u(t)\|_{L^2}\le
Q(\|(\phi_0,u_0)\|_{\Phi})s^{1},
\label{3.16}
\end{equation}
for an appropriate monotonic function $Q$ which is independent of
$\varepsilon$. In order to derive \eqref{3.15} from \eqref{3.16}, we note
that, due
to estimate \eqref{2.37} and due to a standard interpolation
inequality
\begin{multline*}
\|\phi(t+s)-\phi(t)\|_{H^2}+\|u(t+s)-u(t)\|_{H^2}\le
\|\phi(t+s)-\phi(t)\|_{L^2}^{1/3}\|\phi(t+s)-\phi(t)\|_{H^3}^{2/3}+\\+
\|u(t+s)-u(t)\|_{L^2}^{1/3}\|u(t+s)-u(t)\|_{H^3}^{2/3}\le
Q_1(\|(\phi_0,u_0)\|_{\Phi})s^{1/3},
\end{multline*}
which completes the proof of Lemma \ref{lm3.1}.\hfill\qed


The Lipschitz continuity \eqref{2.32} of $S^\varepsilon_t$ with respect to
the initial data $(\phi_0,u_0)$, together with the H\"older continuity
\eqref{3.15}, imply that
\begin{equation}
\dim_F(\mathcal{M}_\varepsilon,\Phi)\le \dim_F(\mathcal{M}_\varepsilon^d,\Phi)+3,
\label{3.17}
\end{equation}
and Theorem \ref{thm3.1} is proved.\hfill\qed


\section{The case of Neumann boundary conditions} \label{sec4}

In this concluding section, we briefly consider system \eqref{0.1} of
phase-field equations associated with the Neumann boundary conditions
\begin{equation}
\partial_n\phi\big|_{\partial\Omega}=\partial_n
u\big|_{\partial\Omega}=0.
\label{4.1}
\end{equation}
We first note that, if the nonlinear function $f_2$ is strictly
monotone, i.e.
\begin{equation}
f_2'(v)\ge \alpha,\quad \forall v\in\mathbb{R},
\label{4.2}
\end{equation}
for some strictly positive constant $\alpha$ (which improves the nonstrict
monotonicity assumption \eqref{0.2} 3.), then, repeating word by word
the proofs given above for the case of Dirichlet boundary conditions,
we easily extend all the results of Sections \ref{sec1}-\ref{sec3}
 to the case of Neumann boundary conditions.
That is the reason why we consider only the case where assumption
\eqref{4.2} is violated below. More precisely, we assume that $f_2\equiv0$ and
$g_2\equiv0$. Then, system \eqref{0.1} reads
\begin{equation}
\begin{gathered}
\delta\partial_t\phi=\Delta_x\phi-f(\phi)+u+g,\quad \partial_n\phi\big|_{\partial\Omega}=0,\\
\varepsilon\partial_t u+\partial_t\phi=\Delta_x u,\quad \partial_n u\big|_{\partial\Omega}=0,\\
\phi\big|_{t=0}=\phi_0,\quad u\big|_{t=0}=u_0,
\end{gathered}
\label{4.3}
\end{equation}
which corresponds to the standard phase-field system.
We assume that $g\in L^2(\Omega)$ and that the nonlinear
function $f\in C^3(\mathbb{R},\mathbb{R})$ satisfies the assumptions
\begin{equation}
1. \  f(v).v\ge\mu{{|v|}^2}-{\mu ^ \prime },\quad 2. \ f'(v)\ge -K,\ \text{ for every $v\in\mathbb{R}$},
\label{4.4}
\end{equation}
with $\mu>0$ and ${\mu^ \prime }$, $K\ge 0$.

The main difference between system \eqref{4.3} and system \eqref{0.1} with
Dirichlet boundary conditions is the existence of a conservation law.
Indeed, integrating the second equation of \eqref{4.3} over $\Omega$,
we have
\begin{equation}
\varepsilon\partial_t\langle u(t)\rangle +\partial_t\langle \phi(t)\rangle
=0,\quad \text{where}\quad
\langle v\rangle :=\frac1{|\Omega|}\int_\Omega v(x)\,dx.
\label{4.5}
\end{equation}
Integrating then \eqref{4.5} with respect to $t$, we obtain the conservation law
mentioned above:
\begin{equation}
\varepsilon\langle u(t)\rangle +\langle \phi(t)\rangle =\varepsilon\langle u(0)\rangle +\langle \phi(0)\rangle :=I_0(u_0,\phi_0),\quad t\in\mathbb{R}_+.
\label{4.6}
\end{equation}
Therefore, we cannot expect the existence of the global
dissipative estimate \eqref{1.17} for the solutions of \eqref{4.3} in
the phase space $\Phi$. Nevertheless, we will show in this section that all the results obtained above for
Dirichlet boundary conditions remain valid (after minor changes)
for Neumann boundary conditions. To this end, we need to
modify the phase space $\Phi$ for problem \eqref{4.3} by fixing
explicitly the bounds for the possible values of
 the conserved integral $I_0$,
namely, for every $M>0$, we define the phase space $\Phi_M$ for
problem \eqref{4.3} as follows:
\begin{multline}
\Phi_M:=\{(\phi_0,u_0)\in H^2(\Omega)\times H^2(\Omega),\\
\partial_n\phi_0\big|_{\partial\Omega}=
\partial_nu_0\big|_{\partial\Omega}=0,\quad |I_0(u_0,\phi_0)|\le M\}.
\label{4.7}
\end{multline}
The following theorem gives a dissipative estimate for the solutions
of \eqref{4.3} in the phase space $\Phi_M$, similar to that given in
Lemma \ref{lm1.3}.

\begin{theorem} \label{thm4.1}
Let assumption \eqref{4.4} hold. Then, for
every $M>0$ and every $(\phi_0,u_0)\in\Phi_M$, problem \eqref{4.3}
possesses a unique solution $(\phi(t),u(t))$ which satisfies the
following estimate:
\begin{multline}
\|\phi(t)\|_{H^2}^2+\|u(t)\|_{L^2}^2+\varepsilon^2\|\partial_t u(t)\|_{L^2}^2\le\\
\le Q_M(\|\phi(0)\|_{H^2}^2+\|u(0)\|_{H^2}^2)e^{-\alpha t}+Q_M(\|g\|_{L^2}),
\label{4.8}
\end{multline}
where $\alpha>0$ and the monotonic function $Q_M$ depend on $M$, but
are independent of $\varepsilon$.
\end{theorem}

\paragraph{Proof} Let us first derive the analogue of estimate
\eqref{1.1} for equation \eqref{4.3}. Taking the scalar
product of the first equation of \eqref{4.3} by
$\partial_t\phi(t)+\beta\phi(t)$, of the second equation by $u(t)$ and
summing the relations that we obtain, we have (analogously to \eqref{1.3})
\begin{equation}
\partial_t E(t)+\gamma E(t)=h(t),
\label{4.9}
\end{equation}
where $\beta$ and $\gamma$ are small positive numbers such that $\beta>\gamma$,
\begin{multline}
E(t):=\delta\|\nabla_x\phi(t)\|_{L^2}^2+2(F(\phi(t)),1)+
\varepsilon\|u(t)\|_{L^2}^2-\\-2(g,\phi(t))+\beta\delta\|\phi(t)\|_{L^2}^2,
\label{4.10}
\end{multline}
$F(v)={\int _0^ v}f(s)\,ds$ and the function $h(t)$ is defined by
\begin{multline}
h(t):=(\gamma \delta -2\beta )\|\nabla_x\phi(t)\|_{L^2}^2+
2\gamma(F(\phi(t))-f(\phi(t))\phi(t),1)-\\-2(\beta-\gamma)(f(\phi(t)),\phi(t))
-2\delta\|\partial_t\phi(t)\|_{L^2}^2-2\|\nabla_x
u(t)\|_{L^2}^2+\gamma\varepsilon\|u(t)\|_{L^2}^2+\\+
2(\beta-\gamma)(g,\phi(t))+
\beta\gamma\delta\|\phi(t)\|_{L^2}^2+2\beta(u(t),\phi(t)).
\label{4.11}
\end{multline}
We now transform the last term in the right-hand side of \eqref{4.11}
as follows,
using the conservation law \eqref{4.6}:
\begin{multline}
2\beta(u,\phi)=2\beta(u-|\Omega|\langle u\rangle ,\phi)+2\beta|\Omega|^2\langle \phi\rangle \langle u\rangle =\\=
2\beta(u-|\Omega|\langle u\rangle ,\phi)-2\beta|\Omega|^2\varepsilon\langle u\rangle ^2+2\beta
I_0(u_0,\phi_0)|\Omega|^2\langle u\rangle .
\label{4.12}
\end{multline}
Inserting this relation into the right-side of \eqref{4.11} and using
inequalities \eqref{1.6}, \eqref{4.4} and the following analogue
of Friedrichs' inequality:
\begin{equation}
\|u-|\Omega|\langle u\rangle \|_{L^2}^2\le C_\Omega\|\nabla_x u\|_{L^2}^2,
\label{4.13}
\end{equation}
for sufficiently small (but independent of $\varepsilon$) constants
$\gamma$ and $\beta$,
we obtain the estimate
\begin{multline}
h(t)\le -{1\over 2}(2\beta -\gamma \delta )\|\phi(t)\|_{H^1}^2-
(\beta-\gamma)(f(\phi(t)),\phi(t))-\delta\|\partial_t\phi(t)\|_{L^2}^2-\\-
\|\nabla_x u(t)\|_{L^2}^2-2(\beta-\gamma)\varepsilon|\Omega|^2\langle u(t)
\rangle ^2+ C\left(1+\|g\|_{L^2}^2\right)+
2\beta|\Omega|^2\langle u(t)\rangle ,
\label{4.14}
\end{multline}
for some constant $C$ that is independent of $\varepsilon$ (in contrast to
the case of Dirichlet boundary conditions, we now need assumption
\eqref{4.4} 1. {\it with strictly positive} constant $\mu$
because, in the case of Neumann boundary conditions,
 the term $\|\nabla_x\phi(t)\|_{L^2}^2$ does not bound the
$L^2$-norm of $\phi$ and we obtain the estimate for this norm
from the third term in the right-hand side of \eqref{4.11}).

So, there remains to estimate the last term in the right-hand side
 of \eqref{4.14}. To this end, we integrate the first equation of
\eqref{4.3} over $\Omega$ and express $\langle \phi(t)\rangle $ through
 $\langle u(t)\rangle $ by using the conservation law to obtain
\begin{equation}
\varepsilon\partial_t\langle u(t)\rangle +\langle u(t)\rangle
=\langle f(\phi(t))\rangle -\langle g\rangle .
\label{4.15}
\end{equation}
We also note that, due to \eqref{4.4} 1. and the continuity of the
function $f$, we have
\begin{equation}
|\langle f(\phi(t))\rangle |\le\langle |f(\phi(t))|\rangle
\le \nu (f(\phi(t)),\phi(t))+C_\nu,
\label{4.16}
\end{equation}
where the positive constant $\nu$ can be arbitrarily small. We
now multiply \eqref{4.15} by
 $\kappa:=\frac{2\beta|\Omega|^2I_0(u_0,\phi_0)}{1-\gamma\varepsilon}$
and sum the relation that we obtain with \eqref{4.9}.
Then, according to \eqref{4.11}-\eqref{4.12}
and \eqref{4.14}-\eqref{4.16}, we have
\begin{multline}
\partial_t[\kappa\varepsilon\langle u(t)\rangle +E(t)]
+\gamma[\kappa\varepsilon\langle u(t)\rangle +E(t)]+\\+
\gamma'\left(\|\phi(t)\|_{H^1}^2+\|\nabla_x u(t)\|_{L^2}^2+\|\partial_t
\phi(t)\|_{L^2}^2\right)\le C(1+\|g\|_{L^2}^2),
\label{4.17}
\end{multline}
where all the constants are positive and are independent of $\varepsilon$. We
also recall that, due to  \eqref{4.10}
\begin{multline}
C_M^{-1}\left(\varepsilon\|u(t)\|_{L^2}^2+(F(\phi(t)),1)+\delta\|\phi(t)\|_{H^1}^2
-1-\|g\|_{L^2}^2
\right)\le\\\le \kappa\varepsilon\langle u(t)\rangle +E(t)\le\\\le
C_M\left(\varepsilon\|u(t)\|_{L^2}^2+(F(\phi(t)),1)+\delta\|\phi(t)\|_{H^1}^2
+1+\|g\|_{L^2}^2\right),
\label{4.18}
\end{multline}
for every $(\phi(t),u(t))\in\Phi_M$. Here, the constant $C_M$ depends on
$M$, but is independent of $\varepsilon$. Applying Gronwall's inequality to
\eqref{4.17} and using \eqref{4.18}, we have the following
estimate (which is similar to that obtained in Lemma \ref{lm1.1}):
\begin{multline}
\varepsilon\|u(t)\|_{L^2}^2+(F(\phi(t)),1)+\delta\|\phi(t)\|_{H^1}^2+\\+
\int_t^{t+1}(\|\partial_t \phi(s)\|_{L^2}^2+\|\nabla_x u(s)\|_{L^2}^2)\,ds\le
\\\le C_M
\left(\varepsilon\|u(t)\|_{L^2}^2+(F(\phi(t)),1)+
\delta\|\phi(t)\|_{H^1}^2\right)e^{-\alpha t}+C_M\left(1+\|g\|_{L^2}^2\right),
\label{4.19}
\end{multline}
where the constant $C_M$ depends on $M$, but is independent of $\varepsilon$.
Our aim is now to derive the analogue of estimate \eqref{1.7}. Arguing as in
the proof of Lemma \ref{lm1.2}, we have
\begin{multline*}
\partial_t[\delta\|\partial_t\phi(t)\|_{L^2}^2+\|\nabla_x u(t)\|_{L^2}^2]
+[\delta\|\partial_t\phi(t)\|_{L^2}^2+\|\nabla_x u(t)\|_{L^2}^2]+\\+
2\|\partial_t\nabla_x\phi(t)\|_{L^2}^2
+2\varepsilon\|\partial_t u(t)\|_{L^2}^2\le
[\delta\|\partial_t\phi(t)\|_{L^2}^2+\|\nabla_x u(t)\|_{L^2}^2]+\\+2(g,\partial_t\phi(t))-
2(f'(\phi(t))\partial_t\phi(t),\partial_t\phi(t)).
\end{multline*}
Applying Gronwall's inequality to this relation and using \eqref{4.19},
we obtain the estimate
\begin{multline}
\delta\|\partial_t\phi(t)\|_{L^2}^2+\|\nabla_x u(t)\|_{L^2}^2+
\int_t^{t+1}(\|\partial_t\phi(s)\|_{H^1}^2+\varepsilon\|\partial_t u(s)\|_{L^2}^2)\,ds\le\\\le
Q(\|\phi(0)\|_{H^2}^2+\|u(0)\|_{H^2}^2)e^{-\alpha t}+Q(\|g\|_{L^2}),
\label{4.20}
\end{multline}
where the function $Q$ depends on $M$, but is independent of $\varepsilon$.
Multiplying then the first equation of \eqref{4.3} by $\Delta_x \phi(t)$,
integrating by parts in $(u(t),\Delta_x \phi(t))$ and using estimate
\eqref{4.20}, we have, analogously to \eqref{1.14}-\eqref{1.16}
\begin{equation}
\|\phi(t)\|_{H^2}^2\le
Q_1(\|\phi(0)\|_{H^2}^2+\|u(0)\|_{H^2}^2)e^{-\alpha t}+Q_1(\|g\|_{L^2}),
\label{4.21}
\end{equation}
for some function $Q_1$ which is independent of $\varepsilon$. Since
$H^2\subset C$, \eqref{4.21} implies the estimate
\begin{equation}
\|f(\phi(t))\|_{L^2}^2\le
Q_2(\|\phi(0)\|_{H^2}^2+\|u(0)\|_{H^2}^2)e^{-\alpha t}+Q_2(\|g\|_{L^2}),
\label{4.22}
\end{equation}
for an appropriate function $Q_2$ which depends on $M$, but is
independent of $\varepsilon$. Returning now to equation \eqref{4.15} and using
\eqref{4.22}, we find
\begin{equation}
\langle u(t)\rangle \le
 Q_3(\|\phi(0)\|_{H^2}^2+\|u(0)\|_{H^2}^2)e^{-\alpha t}+Q_3(\|g\|_{L^2})
\label{4.23}
\end{equation}
(see \eqref{1.19}-\eqref{1.22}). Finally, estimates \eqref{4.20},
\eqref{4.21} and \eqref{4.23} imply the analogue of estimate
\eqref{1.7} for the case of Neumann boundary conditions:
\begin{multline}
\delta\|\partial_t\phi(t)\|_{L^2}^2+\|u(t)\|_{H^1}^2+\|\phi(t)\|_{H^2}^2+\\+
\int_t^{t+1}(\|\partial_t\phi(s)\|_{H^1}^2+\varepsilon\|\partial_t u(s)\|_{L^2}^2)\,ds\le\\\le
Q(\|\phi(0)\|_{H^2}^2+\|u(0)\|_{H^2}^2)e^{-\alpha t}+Q(\|g\|_{L^2}),
\label{4.24}
\end{multline}
where the monotonic
function $Q$ depends on $M$, but is independent of $\varepsilon$. Estimate
\eqref{4.8} follows from \eqref{4.24}, exactly as in Lemma \ref{lm1.3}, and
Theorem \ref{thm4.1} is proved.\hfill\qed

\vskip 0.1cm


We now formulate
the analogues of Lemma \ref{lm2.3} and Lemma \ref{lm2.4}
 for the difference of two solutions of \eqref{4.3}
 (since $f_2\equiv0$, we do not need
estimate \eqref{2.27} in order to prove this result).

\begin{theorem} \label{thm4.2} Let the above assumptions hold and let
 $(\phi_1,u_1)$ and $(\phi_2,u_2)$ be two solutions of \eqref{4.3}
belonging to $\Phi_M$. Then, the following estimate is valid:
\begin{multline}
\|\phi_1(t)-\phi_2(t)\|_{H^2}^2+\|u_1(t)-u_2(t)\|_{H^2}^2+\\+
\varepsilon^2\|\partial_t u_1(t)-\partial_t u_2(t)\|_{L^2}^2\le\\ \le
Ce^{Lt}\left(\|\phi_1(0)-\phi_2(0)\|_{H^2}^2+\|u_1(0)-u_2(0)\|_{H^2}^2\right),
\label{4.25}
\end{multline}
where the constants $C$ and $L$ depend on $M$, $\|\phi_i(0)\|_{H^2}$
and $\|u_i(0)\|_{H^2}$, but are independent of $\varepsilon$. Moreover, the
following smoothing estimate holds:
\begin{multline}
\|\phi_1(t)-\phi_2(t)\|_{H^3}^2+\|u_1(t)-u_2(t)\|_{H^3}^2
\le\\\le
Ce^{Lt}\frac{t+1}t
\left(\|\phi_1(0)-\phi_2(0)\|_{H^2}^2+\|u_1(0)-u_2(0)\|_{H^2}^2\right),
\quad t>0.
\label{4.26}
\end{multline}
\end{theorem}

\paragraph{Proof} We set $v(t):=\phi_1(t)-\phi_2(t)$ and
$w(t):=u_1(t)-u_2(t)$. These functions satisfy
\begin{equation}
\partial_t v=\Delta_x v+w+G(t),\quad \varepsilon\partial_t w+\partial_t v=\Delta_x u,\quad \partial_n
v\big|_{\partial\Omega}=\partial_n w\big|_{\partial\Omega}=0,
\label{4.27}
\end{equation}
where $G(t):=\int_0^1f'(s\phi_1(t)+(1-s)\phi_2(t))\,ds\cdot v(t)$.
We note that system \eqref{4.27} also possesses a conservation law:
\begin{equation}
I_0(v(t),w(t)):=I_0(\phi_1(t),u_1(t))-I_0(\phi_2(t),u_2(t))\equiv
const.
\label{4.28}
\end{equation}
Moreover, obviously
\begin{equation}
|I_0(v(t),w(t))|^2\le C\left(\|v(0)\|_{H^2}^2+\|w(0)\|_{H^2}^2\right)
\label{4.29}
\end{equation}
and, due to estimate \eqref{4.8} and the embedding $H^2\subset C$
\begin{equation}
\|G(t)\|_{L^2}^2+\|\partial_t G(t)\|_{L^2}^2\le C\left(\|\partial_t v(t)\|_{L^2}^2+
\|v(t)\|_{L^2}^2\right),
\label{4.30}
\end{equation}
where $C$ depends on $M$ and on the $H^2$-norm of the initial data, but
is independent of~$\varepsilon$. Interpreting now the function $G(t)$ in
\eqref{4.27} as a nonautonomous external force, repeating word by
word the proof of Theorem \ref{thm4.1} and using estimates \eqref{4.29} and
\eqref{4.30}, we find estimate \eqref{4.25}.
Having estimate \eqref{4.25}, we can prove the smoothing property
\eqref{4.26} exactly as in Lemma \ref{lm2.4} and Theorem \ref{thm4.2}
is proved.\hfill\qed

\vskip 0.1cm

As in Section \ref{sec1}, we now study the limit problem \eqref{4.3} with
$\varepsilon=0$:
\begin{equation}
\begin{gathered}
\delta \partial_t \bar\phi_0=\Delta_x\bar\phi_0-f(\bar\phi_0)+\bar u_0+g,\quad \partial_t\bar\phi_0=\Delta_x\bar u_0,\\
\bar\phi_0\big|_{t=0}=\phi_0,\quad \partial_n\bar\phi_0\big|_{\partial\Omega}=
 \partial_n\bar u_0\big|_{\partial\Omega}=0.
\end{gathered}
\label{4.31}
\end{equation}
Again, the variables
$(\bar \phi_0,\bar u_0)$ are not independent, but satisfy the relation
\begin{equation}
\delta\Delta_x\bar u_0(t)-\bar u_0(t)=\Delta_x\bar\phi_0(t)-f(\bar\phi_0)+g, \quad t\in\mathbb{R}_+
\label{4.32}
\end{equation}
(compare with \eqref{1.45}) and, consequently, there exists a nonlinear
 operator
\begin{equation}
\mathcal{L}\in C^1(H^2(\Omega),\{v\in H^2(\Omega),\
\partial_n v\big|_{\partial\Omega}=0\}),
\label{4.33}
\end{equation}
such that
\begin{equation}
\bar u_0(t)=\mathcal{L}(\bar\phi_0(t)),\quad t\in\mathbb{R}_+,
\label{4.34}
\end{equation}
for every solution $(\bar\phi_0(t),\bar u_0(t))$ of problem
\eqref{4.31}. Thus, problem \eqref{4.31} defines a semigroup
in the infinite dimensional submanifold of $\Phi$ defined by
\begin{equation}
\mathbb{L}_M:=\{(\phi_0,u_0)\in H^2(\Omega ),\quad u_0=\mathcal{L}(\phi_0),\quad \partial_n\phi_0\big|_{\partial\Omega}=0,\quad |\langle \phi_0\rangle |\le M\}.
\label{4.35}
\end{equation}
The next theorem gives the analogue of Lemmas 1.5 and 1.6 for equation \eqref{4.31}.

\begin{theorem} \label{thm4.3} Let the above assumptions hold. Then, for every
$(\phi_0,u_0)\in\mathbb{L}_M$,  problem \eqref{1.44} has a unique
solution $(\bar\phi_0(t),\bar u_0(t))\in\mathbb{L}_M$, $t\ge0$, which satisfies
the estimate
\begin{multline}
\|\bar\phi_0(t)\|_{H^2}^2+\|\partial_t\bar\phi_0(t)\|_{L^2}^2+\|\bar u_0(t)\|_{H^2}^2+
\int_t^{t+1}\|\partial_t\bar\phi_0(s)\|_{H^1}^2\,ds\le\\\le
Q(\|\bar\phi_0(0)\|_{H^2}^2)e^{-\gamma t}+Q(\|g\|_{L^2}),
\label{4.36}
\end{multline}
for a positive constant $\gamma$ and a monotonic function  $Q$ which
depend on $M$. Moreover, estimates \eqref{4.25} and \eqref{4.26}
remain valid for the difference of solutions of the limit problem
\eqref{4.31} and the following analogue of estimate \eqref{1.51} holds:
\begin{multline}
\|\partial_t \bar u_0(t)\|_{L^2}^2+\int_t^{t+1}\left(\|\partial_t \bar u_0(s)\|_{H^1}^2+
\|\partial_t^2\bar u_0(s)\|_{H^{-1}}^2\right)\,ds
\le\\\le
Q(\|\bar\phi_0(0)\|_{H^2}^2)e^{-\gamma t}+Q(\|g\|_{L^2}),
\label{4.37}
\end{multline}
where ${H^{-1}}(\Omega )$ denotes here the
dual of $H^1(\Omega)$.
\end{theorem}

\paragraph{Proof}
Since the constant $\alpha$ and the monotonic function $Q_M$
in \eqref{4.8} are independent of $\varepsilon$, then, passing to the limit
$\varepsilon\to0$, we have estimate \eqref{4.36}. The estimates for the
difference of solutions can be obtained similarly. Finally, estimate
\eqref{4.37} can be verified exactly as in Lemma \ref{lm1.6} and
Theorem \ref{thm4.3} is proved.\hfill\qed

\vskip 0.1cm

We now extend the asymptotic expansions for $(\phi(t),u(t))$
as $\varepsilon\to0$ (obtained in Section \ref{sec2} for the case of Dirichlet boundary
conditions) to the case of Neumann boundary conditions. We note that
the formulae for the first boundary layer term are simpler
now, since $f_2\equiv0$. Indeed,
analogously to \eqref{2.8} and \eqref{2.9},
we obtain the following system for $\tilde \phi(\tau)$ and
$\tilde u(\tau)$, $\tau:=\frac t\varepsilon$:
\begin{equation}
\begin{gathered}
\delta\partial_\tau\tilde\phi(\tau)=u(\tau),\quad \partial_\tau\tilde u(\tau) =
(\Delta_x-\delta^{-1})\tilde u(\tau),\quad \partial_n\tilde u\big|_{\partial\Omega}=0,\\
\tilde u(0)=u(0)-\mathcal{L}(\phi(0)),\quad \lim\nolimits_{\tau\to\infty}\tilde\phi(\tau)=0.
\end{gathered}
\label{4.38}
\end{equation}
The solution $(\tilde \phi(\tau),\tilde u(\tau))$ can be expressed
explicitly, using the analytic semigroups theory:
\begin{equation}
\begin{gathered}
\tilde\phi(\tau)=(I-\delta\Delta_x)^{-1}
e^{(\Delta_x-\delta^{-1}I)\tau}(u(0)-\mathcal{L}(\phi(0)),
\\
\tilde u(\tau):=
e^{(\Delta_x-\delta^{-1}I)\tau}(u(0)-\mathcal{L}(\phi(0)),
\end{gathered}
\label{4.39}
\end{equation}
where $\Delta_x$ is associated with Neumann boundary
conditions. As in Section \ref{sec2},  we seek for asymptotic
expansions for $(\phi(t),u(t))$ near $t=0$ of the form
\begin{equation}
\phi(t):=\bar\phi_0(t)+\varepsilon\tilde\phi(\frac t\varepsilon)+\varepsilon\widehat\phi(t),\quad u(t):=\bar u_0(t)+\tilde u(\frac t\varepsilon)+\varepsilon\widehat u(t),
\label{4.40}
\end{equation}
where $(\bar \phi_0(t),\bar u_0(t))$ is solution of \eqref{4.31}
with $\bar\phi_0(0):=\phi(0)$ and $(\tilde \phi, \tilde u)$ is defined
by \eqref{4.39}. The following theorem is an analogue of Lemma \ref{lm2.2}.

\begin{theorem} \label{thm4.4}
Let the above assumptions hold. Then, the rest
$(\widehat\phi(t),\widehat u(t))$ in the asymptotic expansions \eqref{4.40}
enjoys the following estimate:
\begin{equation}
\|\widehat \phi(t)\|_{H^2}+\|\widehat u(t)\|_{H^2}+\|\partial_t \widehat\phi(t)\|_{L^2}+
\varepsilon\|\widehat u(t)\|_{L^2}\le Ce^{Lt},
\label{4.41}
\end{equation}
where the constants $C$ and $L$ depend on $\|\phi(0)\|_{H^2}$ and
$\|u(0)\|_{H^2}$, but are independent of $\varepsilon$.
\end{theorem}

\paragraph{Proof} The functions $\tilde u(t)$ and $\tilde \phi(t)$ satisfy
the equations
\begin{equation}
\begin{gathered}
\delta\partial_t\widehat\phi=\Delta_x\widehat\phi-
\frac1\varepsilon\bigg[f(\bar\phi_0+\varepsilon\tilde\phi+\varepsilon\widehat\phi)-f(\bar\phi_0)\bigg]+
\widehat u+\Delta_x\tilde\phi,\\
\varepsilon\partial_t \widehat u=\Delta_x\widehat u-\partial_t\widehat\phi-\partial_t\bar u_0,\quad \phi\big|_{t=0}=-\tilde\phi(0),\quad  \widehat u\big|_{t=0}=0,\\
\partial_n\widehat\phi\big|_{\partial\Omega}=
\partial_n \widehat u\big|_{\partial\Omega}=0.
\end{gathered}
\label{4.42}
\end{equation}
Arguing as in \eqref{2.15}-\eqref{2.23}
(with $f_2\equiv0$),
we obtain the estimate
\begin{multline}
\partial_t\bigg[
\delta\|\widehat\phi(t)\|_{L^2}^2+\delta\|\partial_t\widehat\phi(t)\|_{L^2}^2+
\|\nabla_x\widehat u(t)\|_{L^2}^2-2(\partial_t \bar u_0(t),\widehat
u(t))+C_1\bigg]\le\\
\le C_2\left(1+\|\partial_t\tilde\phi(t/\varepsilon)\|_{H^2}+\|\partial_t\tilde u(t/\varepsilon)\|_{L^2}+
\|\partial_t^2\bar u_0(t)\|_{H^{-1}}^2\right)\times\\\times\bigg[
\delta\|\widehat\phi(t)\|_{L^2}^2+\delta\|\partial_t\widehat\phi(t)\|_{L^2}^2+
\|\nabla_x \widehat u(t)\|_{L^2}^2-2(\partial_t \bar u_0(t),\widehat
u(t))+C_1+\langle u(t)\rangle ^2\bigg],
\label{4.43}
\end{multline}
where we have the additional term $\langle u(t)\rangle ^2$ in the right-hand side
(which appears because of Friedrichs' inequality \eqref{4.13}
for Neumann boundary conditions) and
the constants $C_1$ and $C_2$ are independent of $\varepsilon$. In order to
estimate this term, we integrate the first equation of \eqref{4.42}
over $\Omega$:
\begin{equation}
\langle \widehat u(t)\rangle =\delta\langle \partial_t\widehat \phi(t)\rangle +
\langle \frac1\varepsilon\bigg[f(\bar\phi_0+\varepsilon\tilde\phi+\varepsilon\widehat\phi)-
f(\bar\phi_0)\bigg ]\rangle .
\label{4.44}
\end{equation}
 Since, due to Theorems \ref{thm4.1} and \ref{thm4.3}, the $L^\infty$-norms of
$\phi(t):=\bar\phi_0(t)+\varepsilon\tilde\phi(t/\varepsilon)+\varepsilon\widehat\phi(t)$
 and $\bar\phi_0(t)$ are uniformly (with respect to $\varepsilon$) bounded,
it follows from \eqref{4.44} that
\begin{equation}
\langle u(t)\rangle ^2\le C\left(1+\|\partial_t\widehat \phi(t)\|_{L^2}^2+\|\widehat \phi(t)\|_{L^2}^2\right),
\label{4.45}
\end{equation}
where the constant $C$ is independent of $\varepsilon$. Applying now
Gronwall's inequality to \eqref{4.43} and using \eqref{2.24} and
\eqref{4.45}, we have
\begin{equation}
\|\widehat\phi(t)\|_{L^2}^2+\|\partial_t\widehat\phi(t)\|_{L^2}^2+\|\widehat
u(t)\|_{H^1}^2\le Ce^{Lt},
\label{4.46}
\end{equation}
where the constants $C$ and $L$ are independent of $\varepsilon$. Estimate
\eqref{4.41} can be deduced from \eqref{4.46} exactly as in Lemma
\ref{lm1.3}. This finishes the proof of Theorem \ref{thm4.4}.\hfill\qed


\begin{corollary} \label{coro4.1}
Under the assumptions of Theorem \ref{thm4.4},
estimates \eqref{2.26}, \eqref{2.27} and \eqref{2.28} remain valid
(for the case of Neumann boundary conditions).
\end{corollary}

Indeed, these estimates can be deduced from \eqref{4.46} exactly
as in Corollaries \ref{coro2.1} and \ref{coro2.2}.

\vskip 0.1cm

We are now ready to construct a robust family of exponential attractors
for problem \eqref{4.3} with Neumann boundary conditions. Since we
have the dissipativity of system \eqref{4.3} in the phase spaces
$\Phi_M$ only (for every fixed $M$; for Dirichlet
boundary conditions, this property was valid in the whole space $\Phi$),
it is natural to construct the exponential attractors $\mathcal{M}_\varepsilon^M$ for
the semigroups
\begin{equation}
S^{\varepsilon,M}_t:\Phi_M\to\Phi_M,\quad  S^{\varepsilon,M}_t(\phi_0,u_0):=(\phi(t),u(t))
\label{4.47}
\end{equation}
(where $(\phi(t),u(t))$ is the corresponding solution of
\eqref{4.3}) acting in the spaces $\Phi_M$. In that case, the
exponential attractors $\mathcal{M}_\varepsilon^M$ depend obviously on $M$. We consider
the following limit semigroup $S^{0,M}_t$ for \eqref{4.47}:
\begin{equation}
S^{0,M}_t:\mathbb{L}_M\to\mathbb{L}_M,\quad S^{0,M}_t(\phi_0,u_0):=
(\bar \phi_0(t),\bar u_0(t)),
\label{4.48}
\end{equation}
associated with the limit problem \eqref{4.31} on the manifold $\mathbb{L}_M$ defined by \eqref{4.35}.

The main result of this section is the following analogue of Theorem
\ref{thm3.1} for the case of Neumann boundary conditions.

\begin{theorem} \label{thm4.5}
Let the assumptions of Theorem \ref{thm4.1} hold. Then, for every $M>0$,
there exists a family of compact sets $\mathcal{M}_\varepsilon^{M}\subset\Phi_M$,
 $\varepsilon\in[0,1]$,
such that

1. These sets are semi-invariant with respect to the flows $S_t^{\varepsilon,M}$
associated with problem \eqref{4.3}, i.e.
\begin{equation}
S_t^{\varepsilon,M}\mathcal{M}_\varepsilon^M\subset\mathcal{M}_\varepsilon^M.\quad \label{4.49}
\end{equation}

2. The fractal dimension of the sets $\mathcal{M}_\varepsilon^M$ is finite and
uniformly bounded with respect to $\varepsilon$:
\begin{equation}
\dim_F(\mathcal{M}_\varepsilon^M,\Phi_M)\le C<\infty,
\label{4.50}
\end{equation}
where $C=C(M)$ is independent of $\varepsilon$.

3. These sets attract exponentially the bounded subsets of $\Phi_M$, i.e.
there exist a positive constant $\alpha=\alpha(M)>0$
and a monotonic function $Q=Q_M$
which are independent of $\varepsilon$ such that, for every bounded
subset $B$ in the phase space $\Phi_M$, we have
\begin{equation}
\dist\nolimits_{\Phi_M}(S_t^{\varepsilon,M} B,\mathcal{M}_\varepsilon^M)\le
 Q(\|B\|_{\Phi_M})e^{-\alpha t},\quad  \varepsilon\in[0,1]
\label{4.51}
\end{equation}
(for  $\varepsilon=0$, we should take $B\subset\mathbb{L}_M$).

4. The symmetric Hausdorff distance between the limit attractor
 $\mathcal{M}_0^M$ and the attractors $\mathcal{M}_\varepsilon^M$ enjoys the following estimate:
\begin{equation}
\dist\nolimits_{sym,{\Phi_M}}(\mathcal{M}_\varepsilon^M,\mathcal{M}_0^M)\le C\varepsilon^\kappa,
\label{4.52}
\end{equation}
where the constants $C=C(M)>0$ and $0<\kappa=\kappa(M)<1$
 are independent of $\varepsilon$
and can be computed explicitly.
\end{theorem}

\paragraph{Proof} As in the case of Dirichlet boundary conditions, the
 proof of this theorem is based on the abstract result given in
Proposition \ref{prop3.1} and coincides, up to minor changes, with that of
Theorem \ref{thm3.1}. That is the reason why we only indicate these
changes below and
leave the details to the reader.

Instead of the absorbing sets $B_\varepsilon$ and $B_0$ defined by
\eqref{3.8} and \eqref{3.9} respectively, we now consider, for
every $M>0$, the sets
\begin{gather}
B_\varepsilon^M:=\{(\phi_0,u_0)\in\Phi_M,\quad \|(\phi_0, u_0)
\|_{\Phi}^2\le 2Q_M(\|g\|_{L^2})\}, \label{4.53} \\
B_0^M:=\{(\phi_0,u_0)\in\mathbb{L}_M,\quad \|\phi_0\|_{H^2}^2\le
2Q_M(\|g\|_{L^2})\}, \label{4.54}
\end{gather}
where the function $Q_M$ is the same as in \eqref{4.8}. We note that,
in contrast to the case of Dirichlet boundary conditions, the sets
$B_\varepsilon$ now {\it depend} on $\varepsilon$, since the conserved integral
\eqref{4.5} depends explicitly on $\varepsilon$.

Then, these sets are indeed uniform (with respect to $\varepsilon$)
absorbing sets for semigroups \eqref{4.47} and \eqref{4.48} (due to
estimates \eqref{4.8} and \eqref{4.36} (thus, the analogue of
\eqref{3.10} is also satisfied)).
 Moreover, condition \eqref{3.5} of Proposition \ref{prop3.1} is satisfied
for these semigroups, due to Theorem \ref{thm4.2}.

Let us now verify condition \eqref{3.6} of this proposition. To this
end, we modify slightly the construction of the nonlinear projectors
$\Pi_\varepsilon$ as follows:
\begin{equation}
\Pi_\varepsilon:B_\varepsilon^M\to B_0^M,\quad \Pi_\varepsilon(\phi_0,u_0):=
(\phi_0+\varepsilon\langle u_0\rangle ,\mathcal{L}(\phi_0+\varepsilon\langle u_0\rangle ).
\label{4.55}
\end{equation}
Since
\[
|\langle \phi_0+\varepsilon\langle u_0\rangle \rangle |=|I_0(\phi_0,u_0)|\le M,
\]
projectors \eqref{4.55} are indeed well defined. Moreover,
the analogue of condition \eqref{3.6} for our case now follows from
estimate \eqref{2.26} (see Corollary \ref{coro4.1}),
estimate \eqref{4.25} for the limit problem \eqref{4.31} and from
the obvious estimate
\begin{equation}
\|\Pi_\varepsilon(\phi_0,u_0)-(\phi_0,\mathcal{L}(\phi_0))\|_{\Phi}\le \varepsilon C_M,
\label{4.56}
\end{equation}
for every $(\phi_0,u_0)\in B_\varepsilon$.

Thus, we can apply Proposition \ref{prop3.1} to our
situation and we obtain the desired family of exponential
attractors $\mathcal{M}_\varepsilon^{M,d}$ for the discrete semigroups $S_{nT}^{\varepsilon,M}$
acting on the absorbing sets \eqref{4.53} and \eqref{4.54}. The
existence of the exponential attractors for the continuous semigroups then follows exactly as in the proof
of Theorem \ref{thm3.1} and Theorem \ref{thm4.5} is proved.\hfill\qed

\begin{remark} \label{rm4.1} We recall that the exponential attractors
$\mathcal{M}_\varepsilon^M$ constructed in Theorem \ref{thm4.5} depend on $M$. Moreover,
all the constants in estimates \eqref{4.50}-\eqref{4.52} also
depend a priori on $M$. It is possible to prove, however, that,
under natural assumptions on $f$,
there exists a positive
constant $M_0\gg1$ that is independent of $\varepsilon$ such that every solution
of equation \eqref{4.3} with initial data satisfying
\begin{equation}
|I_0(\phi_0,u_0)|=M_0,
\label{4.57}
\end{equation}
uniformly with respect to $\varepsilon$, stabilizes
exponentially to the
corresponding
 equilibrium $(\bar \phi, \bar u)\in\mathbb{R}^2$, which is the unique solution
of
\begin{equation}
\bar u=f(\bar\phi),\quad \varepsilon\bar u+\bar\phi=I_0(\phi_0,u_0).
\label{4.58}
\end{equation}
This suggests that it is possible to construct the family of exponential attractors
$\mathcal{M}_\varepsilon^M$ for \eqref{4.3} such that
all the constants in estimates \eqref{4.50}-\eqref{4.52} are
 independent of $M$. We will come back to this problem
 in a forthcoming article.
\end{remark}


\begin{thebibliography}{00}

\bibitem{1} A. V. Babin and M. I. Vishik,
\textit{Attractors of evolutionary equations},
 Nauka, Moscow, 1989.

\bibitem{2} P. W. Bates and Z. Cheng,
\textit{Inertial Manifolds and Inertial Sets for the Phase-Field
equations}, J. Dyn. Diff. Eqns., Vol. 4 (1992),  375--398.

\bibitem{3} D. Brochet, X. Chen and D. Hilhorst,
 \textit{Finite Dimensional Exponential Attractor for the Phase Field
Model},  Appl. Anal., Vol. 49 (1993), 197--212.

\bibitem{4} D. Brochet and D. Hilhorst,
\textit{Universal Attractor and Inertial Sets for the Phase Field Model},
Appl. Math. Lett., Vol. 4 (1991), 59--62.

\bibitem{5} D. Brochet, D. Hilhorst and A. Novick-Cohen,
\textit{Maximal Attractor and Inertial Sets for a Con\-ser\-ved Phase
Field Model}, Adv. Diff. Eqns., Vol. 1 (1996), 547--568.
%
\bibitem{6} G. Caginalp,
\textit{An Analysis of a Phase Field Model of a Free Boundary},
 Arch. Rational Mech. Anal., Vol. 92 (1986), 205--245.

 \bibitem{7} G. Caginalp,
\textit{The Dynamics of a Conserved Phase Field System: Stefan-like,
Hele-Shaw and Cahn-Hilliard Models as Asymptotic Limits},
 IMA J. Appl. Math., Vol. 44 (1990), 77--94.

 \bibitem{8} G. Caginalp,
\textit{Phase Field Models and Sharp Interface Limits: Some Differences
in Subtle Situations},
 Rocky Mountain J. Math., Vol. 21 (1991), 602--617.

\bibitem{9} G. Caginalp and P. C. Fife,
\textit{Dynamics of Layered Interfaces Arising from Phase Boundaries},
 SIAM J. Appl. Math., Vol. 48  (1988), 506--518.

\bibitem{10} C. Dupaix, D. Hilhorst and I. N. Kostin,
\textit{The Viscous Cahn-Hilliard Equation as a Limit of the Phase
Field Model: Lower Semicontinuity of the Attractor},
 J. Dyn. Diff. Eqns., Vol. 11  (1999), 333--353.

\bibitem{11} C. Dupaix,
\textit{A Singularly Perturbed Phase Field Model with a Logarithmic
Nonlinearity: Upper Semicontinuity of the Attractor},
 Nonlinear Anal., Vol. 41 (2000), 725--744.

\bibitem{12} C. Dupaix, D. Hilhorst and Ph. Lauren\c cot,
\textit{Upper-Semicontinuity of the Attractor for a Singularly
Perturbed
Phase Field Model},
 Adv. Math. Sci. Appl., Vol. 1 (1998), 115--143.

\bibitem{13} C. Dupaix,
\textit{A Singularly Perturbed Phase-Field Model with a Logarithmic
Nonlinearity}, Preprint.

\bibitem{14}
M. Efendiev, A. Miranville and  S. Zelik,
\textit{Exponential Attractors for a Singularly Perturbed
 Cahn-Hilliard System},
Universit\'e de Poitiers, Dept. Math., Preprint, Vol. 146 (2000).

\bibitem{15}
P. Fabrie, C. Galusinski, A. Miranville and S. Zelik,
\textit{Uniform exponential attractors for a singularly perturbed
damped wave equation},
 Submitted.

\bibitem{16} D. Henry,
\textit{Geometric Theory of Semilinear Parabolic Equations},
Springer-Verlag, Berlin, 1981.

\bibitem{17} A. Jimenez-Casas,
\textit{Dinamica en dimension infinita: modelos de campos de fase y un
termosifon cerrado},
 Ph. D. Thesis, U. Complutense Madrid, Spain, 1996.

\bibitem{18} O. A. Ladyzenskaya, O. A. Solonnikov and N. N. Uraltseva,
\textit{Linear and Quasilinear Equations of Parabolic Type},
 Translated from Russian, AMS, Providence, RI, 1967.

\bibitem{19} O. Penrose and P. Fife,
\textit{Thermodynamically Consistent Models of Phase-Field Type for
Kinetics of Phase Transitions},
 Physica D, Vol. 43 (1990), 44--62.

\bibitem{20} H. Soner,
\textit{Convergence of the Phase-Field Equations to the Mullins-Sekera
Problem with Kinetic Undercooling},
 Arch. Rational Mech. Anal., Vol. 131 (1995), 139--197.

\bibitem{21} B. Stoth,
\textit{The Cahn-Hilliard Equation as Degenerate Limit of the Phase
Field Equations},
 Quart. Appl. Math., Vol. 53  (1995), 695--700.

\bibitem{22} R. Temam,
\textit{Infinite Dimensional Dynamical Systems in Mechanics and
 Physics, 2d ed.},
 Sprin\-ger-Verlag, New-York, 1997.

\bibitem{23} H. Triebel,
\textit{Interpolation theory, function spaces,
differential operators},
  North-Hol\-land, Amsterdam,  1978.

\bibitem{24} M. I. Vishik and L. A. Lyusternik,
\textit{Regular Degeneration and Boundary Layer for Linear Differential
Equations with Small Parameter},
 Uspehi Mat. Nauk., Vol. 12
No. 5~(77) (1957), 3--122.

\bibitem{25} W. Xie,
\textit{The Classical Stefan Model as the Singular Limit of Phase Field
Equations},
 Disc. Cont. Dyn. Sys., Added Vol. II (1988), 288-302.

\bibitem{26}  S. Zelik,
\textit{The attractor for a nonlinear reaction-diffusion system with
 a supercritical nonlinearity and its dimension},
Rend.  Accad. Naz. Sci. XL Mem. Mat. Appl., Vol. 24  (2000), 1--25.

\bibitem{27} S. Zheng,
\textit{Global Existence for a Thermodynamically Consistent Model of
Phase Field Type},
 Diff. Int. Eqns., Vol. 5 (1992), 241--253.

\end{thebibliography}


\enddocument