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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 143, pp. 1--23.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/143\hfil Exponential attractors]
{Exponential attractors for a Cahn-Hilliard model in bounded
domains with permeable walls}

\author[C. G. Gal\hfil EJDE-2006/143\hfilneg]
{Ciprian G. Gal}

\address{Ciprian G. Gal \newline
Department of Mathematical Sciences\\
Uinversity of Memphis\\
Memphis, TN 38152, USA}
\email{cgal@memphis.edu}

\date{}
\thanks{Submitted August 30, 2006. Published November 16, 2006.}
\subjclass[2000]{35K55, 74N20, 35B40, 35B45, 37L30}
\keywords{Phase separation; Cahn-Hilliard equations; 
 exponential attractors; \hfill\break\indent 
 global attractors; dynamic boundary conditions;
Laplace-Beltrami differential operators}

\begin{abstract}
 In a previous article \cite{g1}, we proposed a model of phase separation in a
 binary mixture confined to a bounded region which may be contained within
 porous walls. The boundary conditions were derived from a mass conservation
 law and variational methods. In the present paper, we study the problem
 further. Using a Faedo-Galerkin method, we obtain the existence and
 uniqueness of a global solution to our problem, under more general
 assumptions than those in \cite{g1}. We then study its asymptotic behavior and
 prove the existence of an exponential attractor (and thus of a global
 attractor) with finite dimension.
\end{abstract}

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


\section{Introduction}

 In this article, we are interested in the asymptotic behavior of
a Cahn-Hilliard model that was introduced in  Gal \cite{g1}. The
corresponding equations were studied as an approximate problem of a system
of two parabolic equations with dynamical and Wentzell boundary conditions
involving two unknowns, namely a temperature $u(x,t)$ at a
point $x$ and time $t$ of a substance which can appear in different phases
and an order parameter $\phi (t,x)$, which describes the
current phase at $x$ and time $t$. Such models are phase-field equations of
Caginalp type. Different versions of Cahn-Hilliard models were studied
extensively by many authors in \cite{e1,e2,k1,m1,n1} and the
references cited there in. For instance, Racke \& Zheng \cite{r1}
 show the existence and uniqueness of a global solution to the
Cahn-Hilliard equation with dynamic boundary conditions, and later  Pruss,
Racke \&  Zheng \cite{r1} study the problem of maximal
$L^{p}$-regularity and asymptotic behavior of the solution and
prove the existence
of a global attractor to the same Cahn-Hilliard system.
 Miranville \& Zelik \cite{m1} prove the existence and uniqueness of a
solution under more general assumptions on the potential function (compared
to \cite{r1,t1}) and construct a robust family of exponential
attractors to a regularized version of their problem. We would also like to
refer the reader to the papers of  Wu \&  Zheng \cite{w1} and
 Chill, Fasangova and  Pruss \cite{c1}. They study the
problem of convergence to equilibrium of solutions of a Cahn-Hilliard
equation with dynamic boundary conditions.

In \cite{g1}, we have proposed a model of phase separation in a
binary mixture confined to a bounded region $\Omega $ which may be contained
within porous walls $\Gamma =\partial \Omega $ and there we have proved the
existence and uniqueness of global solutions to this model. The course of
phase separation in \cite{g1} is completely changed from the one
described by the Cahn-Hilliard equations with boundary conditions considered
previously in the cited papers. Furthermore, the system of equations in
\cite{g1} was also compared to the models studied earlier by these
authors and in some cases, we showed that the solutions of the two different
systems resemble each other in suitable Sobolev norms.

In this paper concerns the following system of initial value
problems in a bounded domain $\Omega \subset \mathbf{R}^{N}$, $N=2,3$:
\begin{gather}
\partial _{t}\phi =\Delta \mu \quad\text{in }[0,T]\times \Omega ,
\label{e1.1} \\
\mu =-\Delta \phi +f(\phi )\quad\text{in }[0,T]\times
\Omega ,  \label{e1.2}
\end{gather}
and
\begin{gather}
\partial _{t}\phi +b\partial _{n}\mu +c\mu =0\quad\text{on }[0,T]
\times \Gamma ,  \label{e1.3}
\\
-\alpha \Delta _{\Gamma }\phi +\partial _{n}\phi +\beta \phi =\frac{\mu }{b}
\quad\text{on }[0,T]\times \Gamma .  \label{e1.4}
\end{gather}
and $\phi\big|_{t=0}=\phi _{0}$, where $\phi $ and $\mu $ are unknown
functions, $\Delta _{\Gamma }$ is the Laplace-Beltrami operator on the
boundary, $\alpha ,\beta ,c,b$ are positive constants. Moreover, $f$ is a
given nonlinear function that belongs to $C^{2}(\mathbf{R,R})$
that satisfies the following assumption:
\begin{equation}
\lim_{|s|\to \infty }\inf f'(s)>0.  \label{e1.5}
\end{equation}
Compared with the result obtained in Gal \cite{g1}, these allows
us to consider a potential $f$ with arbitrary growth (even in the case
$n=3$). The boundary condition \eqref{e1.3} is derived from mass
conservation laws that include an external mass source (energy density) on
$\Gamma $. This may be realized, for example, by an appropriate choice of the
surface material of the wall, that is, the wall $\Gamma $ may be replaced by
a penetrable permeable membrane. For a complete discussion of \eqref{e1.3}, in the context of heat and wave equations, we refer the reader
to Goldstein \cite{g4}. The condition \eqref{e1.4}
is similarly derived, since the system tends to minimize its surface free
energy, in the presence of surface interactions on $\Gamma $. On the other
hand, equation \eqref{e1.4} can be viewed as describing the chemical
potential on the walls of the region $\Omega $, and due to \eqref{e1.4},
we assume it to be directly proportional to the driving force
which equals the left hand side of \eqref{e1.4}. Therefore, the
occurrence of a nontrivial pore surface phase behavior is possible. Let us
also mention that this Cahn-Hilliard system is not conservative as discussed
in \cite{g1}. We note that if the value of $\phi (t)$
is known for some value $t=T$, then the value of the chemical potential
$\mu(T)$ can be found from the problem
\begin{gather}
\mu (T)=-\Delta \phi (T)+f(\phi (
T))\quad\text{in }\Omega ,  \label{e1.6}
\\
\frac{1}{b}\mu (T)=-\Delta _{\Gamma }\phi (T)
+\partial _{n}\phi (T)+\beta \phi (T)\quad\text{on }
\Gamma ,  \label{e1.7}
\end{gather}
Similarly, if $\partial _{t}\phi (T)$ is known, we can solve
for $\mu (T)$ from the following elliptic boundary value
problem:
\begin{gather}
-\Delta \mu (T)=-\partial _{t}\phi (T)\quad\text{in }
\Omega ,  \label{e1.6ast}
\\
b\partial _{n}\mu (T)+c\mu (T)=-\partial _{t}\phi
(T)\quad\text{on }\Gamma .  \label{e1.7ast}
\end{gather}
Thus, we only need to find the function $\phi $. It is in fact more
convenient to introduce new unknown functions
$\psi (t):=\phi (t)\big|_{\Gamma }$,
$\varpi (t):=\mu (t)\big|_{\Gamma }$ and to rewrite our system
\eqref{e1.1}--\eqref{e1.4} as follows:
\begin{gather}
\partial _{t}\phi =\Delta \mu \quad\text{in }[0,T]\times \Omega ,
\label{e1.8}
\\
\partial _{t}\psi +b\partial _{n}\mu +c\varpi =0\quad\text{on }[0,T]
\times \Gamma ,  \label{e1.9}
\end{gather}
and
\begin{gather}
\mu =-\Delta \phi +f(\phi )\quad\text{in }[0,T]\times
\Omega ,  \label{e1.10}
\\
\frac{\varpi }{b}=-\alpha \Delta _{\Gamma }\psi +\partial _{n}\phi +\beta
\psi \quad\text{on }[0,T]\times \Gamma ,  \label{e1.11}
\end{gather}
with $\phi\big|_{t=0}=\phi _{0}$ and $\psi\big|_{t=0}=\psi _{0}$. The
boundary condition \eqref{e1.4} is now interpreted as an additional
elliptic equation on the boundary $\Gamma $. We note that
\eqref{e1.10}--\eqref{e1.11} still form elliptic boundary value
problems in the sense of Agmon \& Douglis \& Nirenberg \cite{c1},
H\"{o}rmander \cite{k1}, Peetre \cite{t1} or Vi\v{s}ik \cite{w2}.

We organize our paper as follows: in Section $2$, we discuss the linear
problems associated with our equations and construct the phase spaces
necessary for the study of our Cahn-Hilliard system. In Section $3$, we
derive uniform estimates which are needed to study our problem and in
Section $4$, we discuss the existence and uniqueness of solutions. Finally,
in Section $5$, we obtain the existence of global attractors with finite
dimension.

\section{Preliminary results}

 The solvability of a similar problem to \eqref{e1.1} equipped
with the boundary condition \eqref{e1.3} was given in
\cite{f1,g1}. From now on, through out the paper, we denote
the norms on $H^{s}(\Omega )$ and $H^{s}(\Gamma )$
by $\|\cdot \|_{s}$ and $\|\cdot \|_{s,\Gamma }$, respectively.
The inner product in these spaces will be
denotes by $\langle \cdot ,\cdot \rangle _{s}$ and $\langle
\cdot ,\cdot \rangle _{s,\Gamma }$, respectively. The spaces $H^{s}$,
$s>0$ are defined the usual way found in standard textbooks. For example, we
can define $H^{s}(\Gamma )$, using the Laplace-Beltrami
operator as follows; let $H^{2m}(\Gamma )=\{f\in
L^{2}(\Gamma ):\Delta _{\Gamma }^{m}f\in L^{2}(\Gamma)\}$
and its norm defined to be the equivalent norm of
$C_{1}\|f\|_{0,\Gamma }+C_{2}\|\Delta _{\Gamma}^{m}f\|_{0,\Gamma }$.
It follows that any space $H^{s}(\Gamma )$, $s>0$ can be defined
by interpolation. Furthermore, every
product space $H^{s}(\Omega )\oplus H^{s}(\Gamma )$
($s\in \mathbf{N}$) is the completion of
$(u\big|_{\Omega },u\big|_{\Gamma })\in C^{s}(\Omega )\times C^{s}(\Gamma
)$ under the natural Sobolev norms on $H^{s}$. The dynamical boundary
condition \eqref{e1.3} is also related to a Wentzell boundary
condition, studied by other authors (see e.g.
\cite{g2,h1}). We will make this clear later in this section. Let us consider
the space $\mathcal{H}=L^{2}(\overline{\Omega },dx\oplus \frac{dS}{b})$
with norm
\begin{equation}
\|u\|_{\mathcal{H}}=\Big(\int_{\Omega
}|u(x)|^{2}dx+\int_{\Gamma
}|u(x)|^{2}\frac{dS_{x}}{b}\Big)^{1/2}.  \label{e2.1}
\end{equation}
Here we identify $\mathcal{H}$ with $L^{2}(\Omega ,dx)\oplus
L^{2}(\Gamma ,\frac{dS}{b})$. If $u\in C(\overline{\Omega
})$, we identify $u$ with the vector $U=(u\big|_{\Omega
},u\big|_{\Gamma })\in C(\Omega )\times C(\Gamma
)$. We define $\mathcal{H}$ to be the completion of $C(
\overline{\Omega })$ with respect to the norm \eqref{e2.1}.
For a complete discussion of this space, we refer the reader to
\cite{f1}. Let us also define, for $s=0,1$, the spaces
\begin{equation*}
\mathbb{V}_{s}=\overline{C^{s}(\overline{\Omega })}^{\|
\cdot \|_{\mathbb{V}_{s}}},
\end{equation*}
where the norms $\|\cdot \|_{\mathbb{V}_{s}}$ are given by
\begin{equation*}
\|(\phi ,\psi )\|_{\mathbb{V}
_{1}}=\int_{\Omega }|\nabla \phi |
^{2}dx+\int_{\Gamma }\alpha |\nabla _{\Gamma }\psi
|^{2}dS+\int_{\Gamma }\beta |\psi |^{2}dS
\end{equation*}
and
\begin{equation*}
\|(\phi ,\psi )\|_{\mathbb{V}
_{0}}=\int_{\Omega }|\phi |
^{2}dx+\int_{\Gamma }|\psi |^{2}dS,
\end{equation*}
respectively. It easy to see that we can identify $\mathbb{V}_{s}$ with
$H^{s}(\Omega )\oplus H^{s}(\Gamma )$ under these
norms, when $s=0,1$. Moreover, $\mathbb{V}_{0}=\mathcal{H}$ up to an
equivalent inner product and $\mathbb{V}_{s}$ is compactly embedded in
$\mathbb{V}_{s-1}$ for all $s\geq 1$.

We can rewrite the equations \eqref{e1.8}, \eqref{e1.9} as
\begin{equation}
-\binom{\partial _{t}\phi }{\partial _{t}\psi }=A_{0}\binom{\mu }{\varpi },
\label{e2.2}
\end{equation}
where $A_{0}$ is defined formally as
\begin{equation}
A_{0}\binom{\mu }{\varpi }=\binom{-\Delta \mu }{b\partial _{n}\mu +c\varpi },
\label{e2.3}
\end{equation}
for functions $(\mu ,\varpi )\in H^{3/2+\delta }(\Omega
)\times L^{2}(\Gamma )$, for some $\delta >0$, with
$\varpi =\mu\big|_{\Gamma }$, such that $\Delta \mu \in L^{2}(\Omega
)$. Note that $\varpi $ and $\partial _{n}\mu $ belong (in the trace
sense) to $H^{1+\delta }(\Gamma )$ and $L^{2}(\Gamma)$, respectively.
Here, we have also identified $\mu $ with the
vector $(\mu ,\varpi )$. Next, we consider the bilinear
form:
\begin{equation}
%\begin{aligned}
\big\langle \binom{\mu }{\varpi },\binom{\Psi }{\Psi\big|_{\Gamma }}
\big\rangle _{D}
=\big\langle A_{0}\binom{\mu }{\varpi },\binom{\Psi }{
\Psi\big|_{\Gamma }}\big\rangle _{\mathcal{H}}
=\int_{\Omega }\nabla \mu \cdot \nabla \Psi dx+\int_{\Gamma
}c\varpi \Psi\big|_{\Gamma }\frac{dS}{b},
%\end{aligned}
\label{e2.4}
\end{equation}
for all $\Psi \in H^{1}(\Omega )$. Note that
$\Psi\big|_{\Gamma }$ is a well defined member of of
$H^{1/2}(\Gamma )$ in
the trace sense. Define $D$ to be the completion of
$C^{1}(\overline{\Omega })$ with respect to the inner product \eqref{e2.4}.
Notice that $D$ is densely injected and continuous in
$H^{1}(\Omega)$ and in fact, $D$ is isometrically isomorphic to
$H^{1}(\Omega )$. It is easy to see that \eqref{e2.4} defines a
closed bilinear form $a(\cdot ,\cdot )$ with domain
$D(a(\cdot ,\cdot ))=H^{1}(\Omega )\oplus L^{2}(\Gamma )$
which can be identified with $D$ up to an
equivalent inner product. The form is also densely defined ($D$ is dense in
$\mathcal{H}$) and nonnegative in $\mathcal{H}$. Then, it is well known (see
e.g. \cite{g2}) that the bilinear form given by \eqref{e2.4}
defines a strictly positive self-adjoint unbounded operator
$A:D(A)=\{(\mu ,\varpi )^{\rm tr}\in D:A(\mu ,\varpi
)^{\rm tr}\in \mathcal{H}\}\to \mathcal{H}$
such that, for all $(\mu ,\varpi )^{\rm tr}\in D(A)$ and for any
$(\Psi ,\Psi\big|_{\Gamma })^{\rm tr}\in D$, we have:
\begin{equation}
\big\langle A\binom{\mu }{\varpi },\binom{\Psi }{\Psi\big|_{\Gamma }}
\big\rangle _{\mathcal{H}}
=\big\langle \binom{\mu }{\varpi },\binom{\Psi
}{\Psi\big|_{\Gamma }}\big\rangle _{D}.  \label{e2.5}
\end{equation}
Clearly, we view the operator $A$ as the self-adjoint extension of $A_{0}$.
Here and everywhere in the paper, the superscript ``tr'' will denote
transposition. The operator $A$ is also a bijection from $D(A)$
into $\mathcal{H}$, since $c>0$ and $\mathcal{N}:=A^{-1}:\mathcal{H}
\to \mathcal{H}$ is a linear, self-adjoint and compact operator on
$\mathcal{H}$ (see \cite{g2}). In other words, we can view the
inverse operator $\mathcal{N}:D^{\ast }\to D$ by the condition
\begin{equation*}
A\Big(\mathcal{N}\binom{\mu }{\varpi }\Big)
=\binom{\mu }{\varpi },\quad\text{for all }\binom{\mu }{\varpi }\in D^{\ast },
\end{equation*}
namely, $\mathcal{N}\binom{\mu _{2}}{\varpi _{2}}$ is the solution of the
generalized problem
\begin{gather*}
-\Delta \mu =\mu _{2}\quad\text{in }\Omega ,
\\
b\partial _{n}\mu +c\mu =\varpi _{2}\quad\text{on }\Gamma ,
\end{gather*}
hence $\mathcal{N}\binom{\mu _{2}}{\varpi _{2}}\in D$ if $(\mu
_{2},\varpi _{2})\in \mathcal{H}$. If in addition, $\mu _{2}\in
D=H^{1}(\Omega )$ (thus, by regularity of trace theory, $\varpi
_{2}=\mu _{2\mid \Gamma }\in H^{1/2}(\Gamma )$), standard
elliptic theory implies that $\mathcal{N}\binom{\mu _{2}}{\varpi _{2}}\in
H^{2}(\Omega )$. Furthermore, we infer from standard spectral
theory that there exists a complete ortho-normal family of eigenvectors
$\{\eta _{j}\}_{j}$ with $\eta _{j}\in D(A)$ and a
sequence $\lambda _{j}$, $0<\lambda _{1}\leq \lambda _{2}\leq \dots \leq
\lambda _{j}\to \infty $ as $j\to \infty $ and $A\eta
_{j}=\lambda _{j}\eta _{j}$. Also, by spectral theory, $A^{-s}$, $s\in
\mathbb{N}$ is defined as an infinite series, using the standard spectral
decomposition of $A$. To this end, we define $D(A^{-s})$ to be
the completion of $\mathcal{H}$ with respect to the norm
\begin{equation}
\|\Theta \|_{-s}^{2}=\sum_{j=1}^{\infty }\frac{1}{
\lambda _{j}^{2s}}|\langle \Theta ,\eta _{j}\rangle _{
\mathcal{H}}|^{2}.  \label{e2.6}
\end{equation}
We have from \eqref{e2.5} that $D=D(A^{1/2})$.
Furthermore, it follows from \eqref{e2.5} and the definition of the
inner products in $D$ and $\mathcal{H}$, that, for any $(\mu ,\varpi
)^{\rm tr}\in D(A)$ and $(\Psi ,\Psi\big|_{\Gamma})^{\rm tr}\in D$,
\begin{equation*}
\int_{\Omega }(A\mu )\Psi dx+\int_{\Gamma }(
A\mu )\Psi\big|_{\Gamma }\frac{dS_{x}}{b}=\int_{\Omega
}\nabla \mu \cdot \nabla \Psi dx+\int_{\Gamma }c\varpi \Psi\big|_{
\Gamma }\frac{dS}{b}.
\end{equation*}
Integration by parts in the identity above yields for
$\mu(=(\mu,\varpi ))\in D$, that $A\mu =-\Delta \mu \in \mathcal{H}$ and
\begin{equation*}
\int_{\Gamma }-(\Delta \mu )\Psi\big|_{\Gamma }\frac{
dS_{x}}{b}=\int_{\Gamma }b\partial _{n}\mu \Psi\big|_{\Gamma }\frac{
dS}{b}+\int_{\Gamma }c\varpi \Psi\big|_{\Gamma }\frac{dS}{b},
\end{equation*}
holds for all $\Psi\big|_{\Gamma }\in H^{1/2}(\Gamma )$. Thus,
the following boundary condition holds in $H^{-1/2}(\Gamma )$
(that is, the dual of $H^{1/2}(\Gamma )$):
\begin{equation*}
(\Delta \mu +b\partial _{n}\mu +c\mu )\big|_{\Gamma }=0.
\end{equation*}
This is a Wentzell boundary condition for $\mu $. Such boundary conditions
have been considered in many papers (see e.g. \cite{f1}-\cite{g4}, \cite{h2}, \cite{w1}).
 Similarly, it follows that the eigenvector $\eta _{j}$ satisfies
$A\eta _{j}=\lambda _{j}\eta _{j}$, that is,
\begin{gather*}
-\Delta \eta _{j}=\lambda _{j}\eta _{j}\quad\text{in }\Omega ,
\\
(\Delta \eta _{j}+b\partial _{n}\eta _{j}+c\eta _{j})\big|_{
\Gamma }=0,
\end{gather*}
where this boundary condition may be replaced by the eigenvalue dependent
boundary condition:
\begin{equation*}
b\partial _{n}\eta _{j}+(c-\lambda _{j})\eta _{j}=0\quad\text{on }
\Gamma .
\end{equation*}
Such problems with explicit eigenparameter dependence in the boundary
condition were widely considered in the literature, therefore we will not
dwell on this issue any further (see \cite{f1} for further
references). Furthermore, let
\begin{equation*}
W=\big\{u\in H^{2}(\Omega ):(\Delta u)\big|_{
\Gamma }\in L^{2}(\Gamma ,\frac{dS}{b}),\quad (\Delta
u+b\partial _{n}u+cu)\big|_{\Gamma }=0\big\},
\end{equation*}
where the space $W$ is endowed with the natural norm
\begin{equation}
\|u\|_{W}^{2}=\|u\|_{2}^{2}+\|
(\Delta u)\big|_{\Gamma }\|_{L^{2}(\Gamma ,
\frac{dS}{b})}^{2}.  \label{e2.7}
\end{equation}
Let us also note that the embeddings $W\subset D\subset \mathcal{H}=\mathcal{
H}^{\ast }\subset D^{\ast }\subset W^{\ast }$ are dense and continuous. We
can then consider $D^{\ast }$ endowed with the norm, for $(v,\xi
)\in \mathcal{H},$
\begin{equation}
\|(v,\xi )\|_{D^{\ast }}^{2}=\|
\mathcal{N}^{1/2}\binom{v}{\xi }\|_{\mathcal{H}}^{2}=\langle
\mathcal{N}\binom{v}{\xi },\binom{v}{\xi }\rangle _{\mathcal{H}},
\label{e2.8}
\end{equation}
which can be defined in terms of the spectral decomposition of $A$ (see
\eqref{e2.6}). It follows that the following relations
\begin{gather}
\big\langle A\binom{\mu }{\varpi },\mathcal{N}\binom{\phi }{\psi }
\big\rangle _{D}
=\big\langle \binom{\mu }{\varpi },\binom{\phi }{\psi }
\big\rangle _{D},  \label{e2.9}
\\
\big\langle \binom{\mu }{\varpi },\mathcal{N}\binom{\phi }{\psi }
\big\rangle _{\mathcal{H}}
=\big\langle \binom{\mu }{\varpi },\binom{\phi
}{\psi }\big\rangle _{D^{\ast }}  \label{e2.10}
\end{gather}
hold, for all $(\mu ,\varpi )\in D$, $(\phi ,\psi ) \in D^{\ast }$.

Having established this framework, we introduce the phase space for our
problem \eqref{e1.8}--\eqref{e1.11}:
\begin{equation}
\begin{aligned}
\mathbb{Y}:=\big\{&(\phi ,\psi )\in H^{2}(\Omega )
\times H^{2}(\Gamma ):\mu \in H^{1}(\Omega ),\\
&\varpi \in L^{2}(\Gamma ,\frac{cdS}{b}),
 \phi\big|_{\Gamma }=\psi ,\;\mu\big|_{\Gamma }=\varpi
\big\},
\end{aligned} \label{e2.11}
\end{equation}
with the obvious norm
\begin{equation}
\|(\phi ,\psi )\|_{\mathbb{Y}
}^{2}:=\|\phi \|_{2}^{2}+\|\psi \|
_{2,\Gamma }^{2}+\|\nabla \mu \|_{0}^{2}+\frac{c}{b}
\|\varpi \|_{0,\Gamma }^{2}.  \label{e2.12}
\end{equation}
We recall that $(\mu ,\varpi )$ is computed from
$(\phi,\psi )$ via \eqref{e1.6}, \eqref{e1.7} or \eqref{e1.6ast}, \eqref{e1.7ast}.

\begin{definition} \label{def1}\rm
Let us consider $T>0$ be fixed, but otherwise arbitrary. By a solution of
\eqref{e1.8}--\eqref{e1.11} we mean a pair of functions
$(\phi (t),\psi (t))\in L^{\infty }([0,T],\mathbb{Y})$ with
$\partial _{t}\phi \in L^{2}([0,T],H^{1}(\Omega ))$ and
$\partial _{t}\psi \in L^{2}([0,T],H^{1}(\Gamma
))$ which satisfy the equations in the average sense of the
spaces $L^{2}([0,T],L^{2}(\Omega ))$
and $L^{2}([0,T],L^{2}(\Gamma ))$.
Moreover, since $\Omega \subset \mathbf{R}^{N}$, $N=2,3$, we have the
embedding $H^{2}\subset C$, therefore the nonlinearity $f$ in
\eqref{e1.10} is well defined and belongs to the space
$C([0,T],L^{2}(\Omega ))$. Also, by regularity theory,
since $(\partial _{t}\phi ,\partial _{t}\psi )\in L^{2}(
[0,T],H^{1}(\Omega )\times H^{1}(\Gamma))$, we get
$(\mu ,\varpi )\in L^{2}([0,T],H^{2}(\Omega )\times H^{3/2}(\Gamma ))$
and thus the boundary conditions are well defined.
\end{definition}

We close this section with the definition of the weak energy space
$\mathbb{X}:=D^{\ast }$ for our problem \eqref{e1.8}--\eqref{e1.11}
through the norm given by
\begin{equation}
\|(\phi ,\psi )\|_{\mathbb{X}}:=\|
(\phi ,\psi )\|_{D^{\ast }}  \label{e2.13}
\end{equation}
where $D^{\ast }$ is endowed with the inner product given by
\eqref{e2.8}.

\section{Uniform a priori estimates}

In this section, we derive several estimates for the solutions of the
problem \eqref{e1.8}--\eqref{e1.11} which are necessary for the
study of the asymptotic behavior. In the first step, we obtain dissipative
estimates for solutions in the spaces $\mathbb{X}$ and $\mathbb{Y}$.

But before we derive our estimates, it is convenient to rewrite our system
of equations in a different manner, as follows:
\begin{equation}
-\binom{\Delta \phi }{b\alpha \Delta _{\Gamma }\psi +b\partial _{n}\phi
+b\beta \psi }+\binom{f(\phi )}{0}=\binom{\mu }{\varpi },
\label{e3.1}
\end{equation}
with the obvious coupling $\psi (t):=\phi (t)
\big|_{\Gamma }$, $\varpi (t):=\mu (t)\big|_{
\Gamma }$, where the vector $(\mu ,\varpi )^{\rm tr}$ is given by
\eqref{e2.2} via the operator $\mathcal{N}=A^{-1}$, that is,
\begin{equation}
\binom{\mu }{\varpi }=-\mathcal{N}\binom{\partial _{t}\phi }{\partial
_{t}\psi }.  \label{e3.2}
\end{equation}
Then \eqref{e3.1} becomes the following functional equation:
\begin{equation}
\mathcal{N}\binom{\partial _{t}\phi }{\partial _{t}\psi }-\binom{\Delta \phi
}{b\alpha \Delta _{\Gamma }\psi +b\partial _{n}\phi +b\beta \psi }+\binom{
f(\phi )}{0}=0,\quad
\psi (t)=\phi (t)\big|_{\Gamma }.  \label{e3.3}
\end{equation}
Let us define the following function $F(v)
=\int_{0}^{v}f(s)ds$. Without loss of generality, we let
$\alpha =\beta =1$ for the rest of this section. We have the following
result.

\begin{proposition} \label{prop2}
Let the nonlinearity $f$ satisfy  \eqref{e1.5} and let
$(\phi (t),\psi (t))$ be a given
solution of \eqref{e3.3}. Then
\begin{equation}  \label{e3.4}
\begin{aligned}
&\|(\phi (t),\psi (t))
\|_{\mathbb{X}}^{2}+\int_{t}^{t+1}(\|\phi
(s)\|_{1}^{2}+\|\psi (s)\|_{1,\Gamma }^{2})ds
+\int_{t}^{t+1}\|F(\phi (s))\|_{L^{1}(\Omega )}ds
\\
&\leq C_{1}\|(\phi (0),\psi (0)
)\|_{\mathbb{X}}^{2}e^{-\rho t}+C_{2},
\end{aligned}
\end{equation}
where $C_{1}$, $C_{2}$, $\rho $ are positive constants independent of $t$.
\end{proposition}

\begin{proof}
First, we take the inner product in $\mathcal{H}$ of \eqref{e3.3}
with the vector $(\phi (t),\psi (t))^{\rm tr}$, and use the fact
that $-(\mu (t),\varpi (t))^{\rm tr}=\mathcal{N}(\partial _{t}\phi (t)
,\partial _{t}\psi (t))^{\rm tr}$. Then, relation
\eqref{e2.8} and integration by parts yield the following equation:
\begin{equation}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}[\|(\phi (t),\psi (t))\|_{D^{\ast }}^{2}]
+\|\nabla \phi (t)\|_{0}^{2}+\|\nabla _{\Gamma }\psi (t)\|_{0,\Gamma }^{2}\\
&+\|\psi (t)\|_{0,\Gamma }^{2}
+\langle f(\phi (t)),\phi (t) \rangle _{0}=0.
\end{aligned}\label{e3.5}
\end{equation}
Due to assumption \eqref{e1.5}, we have
\begin{equation}
\frac{1}{2}|f(v)|(1+|
v|)\leq f(v)v+C_{f},  \label{e3.6}
\end{equation}
for each $v\in \mathbf{R}$. Here $C_{f}$ is a positive, sufficiently large
constant. Moreover, by the obvious inequality
$\|\phi \|_{1}^{2}\leq C(\|\nabla \phi \|_{0}^{2}+\|
\psi \|_{1,\Gamma }^{2})$, we obtain from \eqref{e3.5}:
\begin{equation}
\begin{aligned}
&\frac{d}{dt}\big[\|(\phi (t),\psi (
t))\|_{D^{\ast }}^{2}\big]+\rho (\|
\phi (t)\|_{1}^{2}+\|\psi (t)
\|_{1,\Gamma }^{2})+\|\nabla \phi (t)
\|_{0}^{2}
\\
&+\|\nabla _{\Gamma }\psi (t)\|_{0,\Gamma
}^{2}+\langle f(\phi (t)),\phi (t)\rangle _{0}=0.
\end{aligned} \label{e3.7}
\end{equation}
Consequently, the inequality $\|\phi \|
_{1}^{2}+\|\psi \|_{1,\Gamma }^{2}\geq C\|(
\phi ,\psi )\|_{\mathcal{H}}^{2}\geq \widetilde{C}
\|(\phi ,\psi )\|_{D^{\ast }}^{2}$, \eqref{e3.6} and \eqref{e3.7} yield
\begin{equation}
\begin{aligned}
&\|(\phi (t),\psi (t))
\|_{D^{\ast }}^{2}+\int_{t}^{t+1}(\|\phi
(s)\|_{1}^{2}+\|\psi (s)\|_{1,\Gamma }^{2})ds
+\int_{t}^{t+1}(|f(\phi (s))
|,(1+|\phi (s)|) )_{0}ds
\\
&\leq C_{1}\|(\phi (0),\psi (0)
)\|_{D^{\ast }}^{2}e^{-\rho t}+C_{2}.
\end{aligned}\label{e3.8}
\end{equation}
To deduce \eqref{e3.4} from \eqref{e3.8}, we
observe that the assumption \eqref{e1.5} also implies that
$|F(v)|-C\leq |f(v)|(1+|v|)$, for some positive
constant $C$ and all $v\in \mathbf{R.}$ The proof is complete.
\end{proof}

\begin{proposition} \label{prop3}
Let the assumptions of Proposition \ref{prop2} hold and let
$(\phi (t),\psi (t))$ be a solution of \eqref{e3.3}.
Then, the following estimate holds:
\begin{equation}
\|(\phi (t),\psi (t))
\|_{\mathbb{Y}}^{2}+\int_{0}^{t}(\|\partial
_{t}\phi (s)\|_{1}^{2}+\|\partial _{t}\psi
(s)\|_{1,\Gamma }^{2})ds
\leq Q(\|(\phi (0),\psi (0)
)\|_{\mathbb{Y}})e^{Mt},  \label{e3.9}
\end{equation}
where the positive constant $M$ and the monotonic function $Q$ are
independent of $t$.
\end{proposition}

\begin{proof}
We give a formal derivation of \eqref{e3.9}, which can be justified
by a standard regularization of the solution, that is, we can define
$\widehat{\phi }(t):=\int_{0}^{\infty }K_{r}(t-s)\phi (s)ds$,
where $K_{r}$ is smooth and
$\mathop{\rm supp} K_{r}\subset [0,r]$,
$\int_{0}^{\infty }K_{r}(s)ds=1$. Then passing to the limit $r\to 0$, we have
$\widehat{\phi }(t)\to \phi (t)$.
Therefore, without loss of generality, we can (and do) differentiate
\eqref{e3.3} and define
\begin{equation*}
(u(t),p(t),v(t),q(t)):=(\partial _{t}\phi (t),
\mu _{t}(t),\partial _{t}\psi (t),\varpi _{t}(t)).
\end{equation*}
Then, we have
\begin{equation}
\mathcal{N}\binom{u_{t}(t)}{v_{t}(t)}-\binom{
\Delta u(t)}{b\alpha \Delta _{\Gamma }v(t)
+b\partial _{n}u(t)+b\beta v(t)}+\binom{
f'(\phi (t))u(t)}{0}=0,
\label{e3.10}
\end{equation}
where $u(t)\big|_{\Gamma }=v(t)$ and $p(t)\big|_{\Gamma }=q(t)$.
The identity \eqref{e3.2} implies
$-(p(t),q(t))^{\rm tr}=\mathcal{N}(u_{t}(t),v_{t}(t))^{\rm tr}$.
Taking the inner product in $\mathcal{H}$ of \eqref{e3.10} with
$(u,v)^{\rm tr}$ and integrating by parts again (as in \eqref{e3.5}),
we deduce
\begin{equation}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\big[\|(u(t),v(t))\|_{\mathbb{X}}^{2}\big]
+\|\nabla u(t)\|_{0}^{2}+\|\nabla _{\Gamma }v(t)\|_{0,\Gamma }^{2}\\
&+\|v(t)\|_{0,\Gamma }^{2}
+\langle f'(\phi (t))u(t),u(t)\rangle _{0}=0.
\end{aligned} \label{e3.11}
\end{equation}
Due to assumption \eqref{e1.5}, we have $f'(v)\geq -M$, for some
positive constant $M$ and each $v\in \mathbf{R}$.
Consequently, applying Gronwall's inequality on \eqref{e3.11} and
then using the interpolation inequality
$\|(u,v)\|_{\mathcal{H}}^{2}\leq C\|(u,v)\|_{\mathbb{X}}\|u\|_{H^{1}(\Omega )}$, we
obtain,
\begin{equation}
\begin{aligned}
&\|(u(t),v(t))\|_{\mathbb{X}}^{2}+\int_{0}^{t}(\|\nabla u(s)
\|_{0}^{2}+\|\nabla _{\Gamma }v(s)
\|_{0,\Gamma }^{2}+\|v(s)\|_{0,\Gamma }^{2})ds
\\
&\leq C_{3}\|(u(0),v(0))
\|_{\mathbb{X}}^{2}e^{M_{2}t},
\end{aligned} \label{e3.12}
\end{equation}
where $C_{3}$ and $M_{2}$ are positive constants independent of $t$. Recall
that $(u(t),v(t))^{\rm tr}=-A(\mu(t),\varpi (t))^{\rm tr}$, since
$\ u(t)=\partial _{t}\phi (t)$ and $v(t)
=\partial _{t}\psi (t)$. Thus, using the relations \eqref{e2.4}--\eqref{e2.10},
we can rewrite \eqref{e3.12} as
\begin{equation}
\begin{aligned}
&\|\nabla \mu (t)\|_{0}^{2}+\frac{c}{b}
\|\varpi (t)\|_{0,\Gamma }^{2}
+\int_{0}^{t}(\|\partial _{t}\phi (s)
\|_{1}^{2}+\|\partial _{t}\psi (s)
\|_{1,\Gamma }^{2})ds
\\
&\leq C_{3}e^{M_{2}t}(\|\nabla \mu (0)\|
_{0}^{2}+\frac{c}{b}\|\varpi (0)\|_{0,\Gamma}^{2}).
\end{aligned} \label{e3.13}
\end{equation}
To obtain estimate \eqref{e3.9}, it remains to deduce an
estimate for $\|\phi \|_{2}^{2}$ and $\|\psi\|_{2,\Gamma }^{2}$.
The required estimates for the $H^{2}$-norms
of $\phi $ and $\psi $ can be obtained by rewriting our problem
\eqref{e1.8}--\eqref{e1.11} as a second order nonlinear elliptic
problem where the chemical potentials are considered as external forces. We
have
\begin{gather}
-\Delta \phi +f(\phi )=g_{1}(t):=\mu (
t)\quad\text{in }\Omega ,\quad \phi\big|_{\Gamma }=\psi   \label{e3.14}
\\
-\Delta _{\Gamma }\psi +\beta \psi +\partial _{n}\phi =g_{2}(t)
:=\frac{\varpi (t)}{b}\quad\text{in }\Gamma ,\quad \mu\big|_{
\Gamma }=\varpi .  \label{e3.15}
\end{gather}
We note that the estimates \eqref{e3.12}  and \eqref{e3.13} imply
\begin{equation}
\|g_{1}(t)\|_{0}^{2}+\|g_{2}(
t)\|_{0,\Gamma }^{2}\leq C_{3}\|(\phi (
0),\psi (0))\|_{\mathbb{Y}
}^{2}e^{M_{2}t}.  \label{e3.16}
\end{equation}
Applying now the maximum principle \cite[Lemma A.2]{m2} to
the problem \eqref{e3.14}, \eqref{e3.15}, we obtain
\begin{align*}
\|\phi (t)\|_{\infty }^{2}+\|\psi
(t)\|_{\infty ,\Gamma }^{2}
&\leq C_{f}+\| g_{1}(t)\|_{0}^{2}+\|g_{2}(t)
\|_{0,\Gamma }^{2}\\
&\leq C_{4}(1+\|(\phi (0),\psi (0))\|_{\mathbb{Y}}^{2})e^{M_{2}t}.
\end{align*}
Finally, applying the above estimate combined with a $H^{2}$-regularity
theorem \cite[Lemma A.1]{m2} to the elliptic boundary value
problem above, but with the nonlinearity $f$ acting as an external force, we
easily deduce that
\begin{equation*}
\|\phi (t)\|_{2}^{2}+\|\psi (
t)\|_{2,\Gamma }^{2}\leq Q(\|(\phi
(0),\psi (0))\|_{\mathbb{Y}
})e^{M_{2}t},
\end{equation*}
where $Q$ is a monotonic function independent of $t$. The proof is complete.
\end{proof}

In the second step, we state additional smoothing properties for the
solutions of \eqref{e1.8}--\eqref{e1.11}.

\begin{proposition} \label{prop4}
Let the assumptions of Proposition \ref{prop2} hold and let
$(\phi (t),\psi (t))$ be a solution of \eqref{e1.8}--\eqref{e1.11}.
Then, we have the  estimate
\begin{equation}
\|(\phi (t),\psi (t))\|_{\mathbb{Y}}^{2}
\leq C_{t_{0}}Q(\|(\phi(0),\psi (0))\|_{\mathbb{X}
}^{2}),  \label{e3.17}
\end{equation}
for every $t\in [t_{0},1]$, where $t_{0}$ is a fixed value in
$(0,1)$ and the positive constant $C_{t_{0}}$ and $Q$ are
independent of $t$, but depend on $t_{0}$.
\end{proposition}

\begin{proof}
We take the inner product in $\mathcal{H}$ of \eqref{e3.3} with
$(\partial _{t}\phi (t),\partial _{t}\psi (t))^{\rm tr}$,
 then using the expressions for $\mu (t)$ and
$\varpi (t)$, and integrating by parts, we obtain
\begin{align*}
\|(\partial _{t}\phi (t),\partial _{t}\psi
(t))\|_{\mathbb{X}}^{2}
&=-\langle \binom{\mu (t)}{\varpi (t)},\binom{\partial _{t}\phi
(t)}{\partial _{t}\psi (t)}\rangle _{\mathcal{H}}
\\
&=-\frac{1}{2}\frac{d}{dt}\big[\|\phi (t)\|
_{1}^{2}+\|\psi (t)\|_{1,\Gamma
}^{2}+2\langle F(\phi (t)),1\rangle _{0}\big].
\end{align*}
It follows that
\begin{equation*}
\frac{d}{dt}\big[\frac{1}{2}\|\phi (t)\|
_{1}^{2}+\frac{1}{2}\|\psi (t)\|_{1,\Gamma
}^{2}+\langle F(\phi (t)),1\rangle _{0}\big]
+2\|(\partial _{t}\phi (t),\partial _{t}\psi
(t))\|_{\mathbb{X}}^{2}=0.
\end{equation*}
Multiplying the above identity by $t$ and integrating over $[0,t]$,
$t\in [0,1]$, we have
\begin{align*}
&t[\|\phi (t)\|_{1}^{2}+\|\psi
(t)\|_{1,\Gamma }^{2}+\langle F(\phi
(t)),1\rangle _{0}]
+2\int_{0}^{t}s\|(\partial _{t}\phi (s)
,\partial _{t}\psi (s))\|_{\mathbb{X}}^{2}ds
\\
&=\int_{0}^{t}\big[\|\phi (s)\|
_{1}^{2}+\|\psi (s)\|_{1,\Gamma
}^{2}+2\langle F(\phi (s)),1\rangle _{0}
\big]ds.
\end{align*}
We estimate the term on the right hand side of the above equation, using the
estimate \eqref{e3.4} and obtain
\begin{equation*}
\int_{0}^{t}s\|(\partial _{t}\phi (s)
,\partial _{t}\psi (s))\|_{\mathbb{X}
}^{2}ds\leq C\|(\phi (0),\psi (0)
)\|_{\mathbb{X}}^{2}+C_{\ast },
\end{equation*}
where $C,C_{\ast }$ are independent of the solution
$(\phi ,\psi)$. It is an easy consequence of the above,
 that there exists $t\in (t_{0}/2,t_{0})$ such that we have
\begin{equation}
\|(\partial _{t}\phi (t),\partial _{t}\psi
(t))\|_{\mathbb{X}}^{2}\leq \frac{C}{t_{0}}
[1+\|(\phi (0),\psi (0)
)\|_{\mathbb{X}}^{2}].  \label{e3.18}
\end{equation}
Arguing exactly as in the derivation of \eqref{e3.4}, we can verify
that \eqref{e3.18} holds for every $t\in [t_{0},1]$.
Using now the relations $(2.8)-(2.10)$ and $(
3.2)$, we can rewrite the left hand side of \eqref{e3.18} and
have
\begin{equation}
\|\mu (t)\|_{1}^{2}\leq C\big(\|
\nabla \mu (t)\|_{0}^{2}+\frac{c}{b}\|\varpi
(t)\|_{0,\Gamma }^{2}\big)
\leq \frac{C}{t_{0}}
\big[1+\|(\phi (0),\psi (0)
)\|_{\mathbb{X}}^{2}\big].  \label{e3.19}
\end{equation}
Now, we may proceed as in \eqref{e3.14}--\eqref{e3.15} to
obtain the $L^{\infty }$ bound for the solution
$(\phi (t),\psi (t))$ which together with the $H^{2}$-elliptic
estimate of \cite[Lemma A.1]{m2} yields
\begin{equation}
\|\phi (t)\|_{2}^{2}+\|\psi (
t)\|_{2,\Gamma }^{2}
\leq \frac{C}{t_{0}}Q\big(\|
(\phi (0),\psi (0))\|_{
\mathbb{X}}^{2}\big),  \label{e3.20}
\end{equation}
for a suitable monotonic function $Q$. Finally, the estimates
\eqref{e3.19} and \eqref{e3.20} yield the conclusion \eqref{e3.17}. This concludes the proof.
\end{proof}

The next theorem follows as consequence of the estimate \eqref{e3.9}
for $t\leq 1$ and estimates \eqref{e3.4}, \eqref{e3.17} for
$t\geq 1$.

\begin{theorem} \label{thm5}
Let the assumptions of Proposition \ref{prop2} hold. Then, every solution of
\eqref{e1.8}--\eqref{e1.11} satisfies the following dissipative estimate
\begin{equation}
\|(\phi (t),\psi (t))
\|_{\mathbb{Y}}^{2}\leq Q\big(\|(\phi (
0),\psi (0))\|_{\mathbb{Y}}^{2}\big)
e^{-\rho t}+C_{4},  \label{e3.21}
\end{equation}
where the positive constants $C_{4}$, $\rho $ and the monotonic function $Q$
are independent of $t$.
\end{theorem}

We close this section with a theorem that gives uniform bounds for solutions
$(\phi ,\psi )$ and $\mu $ of our problem in $H^{3}(
\Omega )\times H^{3}(\Gamma )$ and $W$ respectively.

\begin{theorem} \label{thm6}
Let the assumptions of Proposition \ref{prop2} hold and let
$\gamma \in [0,1/2)$. Then, every solution of
\eqref{e1.8}--\eqref{e1.11} satisfies the following dissipative estimates:
\begin{gather}
\|\phi (t)\|_{3+\gamma }^{2}+\|\psi
(t)\|_{3+\gamma ,\Gamma }^{2}\leq \frac{1}{t}
Q_{1}(\|(\phi (0),\psi (0)
)\|_{\mathbb{Y}}^{2})+C_{5},\quad t\geq t_{0},
\label{e3.22}
\\
\|\mu (t)\|_{W}^{2}\leq \frac{1}{t}
Q_{1}(\|(\phi (0),\psi (0)
)\|_{\mathbb{Y}}^{2}),\quad t\geq t_{0},
\label{e3.23}
\end{gather}
where $C_{5}$, $\rho >0$, $t_{0}>0$ and the monotonic function $Q_{1}$ are
independent of $t$, $\phi (t)$, $\psi (t)$ and
$\mu (t)$, $\varpi (t)$.
\end{theorem}

\begin{proof}
Taking the inner product in $\mathcal{H}$ of \eqref{e3.3} with
$(\partial _{t}\phi (t),\partial _{t}\psi (t)
)^{\rm tr}$, we obtain
\begin{equation}
\frac{d}{dt}\big[\|\phi (t)\|
_{1}^{2}+\|\psi (t)\|_{1,\Gamma
}^{2}+\langle F(\phi (t)),1\rangle _{0}
\big]+\|(\partial _{t}\phi (t),\partial
_{t}\psi (t))\|_{\mathbb{X}}^{2}=0.  \label{e3.24}
\end{equation}
Integrating over $[0,t]$, and using \eqref{e3.21}, we
deduce
\begin{equation}
\int_{0}^{t}\|(\partial _{t}\phi (s)
,\partial _{t}\psi (s))\|_{\mathbb{X}
}^{2}ds\leq Q_{1}\big(\|(\phi (0),\psi (
0))\|_{\mathbb{Y}}^{2}\big),  \label{e3.25}
\end{equation}
for a suitable monotonic function $Q_{1}$ independent of $t$ and the
solution $(\phi (t),\psi (t))$.
Furthermore, multiplying \eqref{e3.24} by $t$ and then integrating
over $[0,t]$, we deduce that
\begin{align*}
&t[\|\phi (t)\|_{1}^{2}+\|\psi
(t)\|_{1,\Gamma }^{2}+\langle F(\phi
(t)),1\rangle _{0}]+\int_{0}^{t}s
\|(\partial _{t}\phi (s),\partial _{t}\psi
(s))\|_{\mathbb{X}}^{2}ds
\\
&=\int_{0}^{t}\big[\|\phi (s)\|
_{1}^{2}+\|\psi (s)\|_{1,\Gamma
}^{2}+\langle F(\phi (s)),1\rangle _{0}
\big]ds.
\end{align*}
Using \eqref{e3.4} to estimate the right hand side term in the above
relation, we obtain
\begin{equation}
\int_{0}^{t}s\|(\partial _{t}\phi (s)
,\partial _{t}\psi (s))\|_{\mathbb{X}
}^{2}ds\leq C\|(\phi (0),\psi (0)
)\|_{\mathbb{X}}^{2}e^{-\rho t}+C,  \label{e3.26}
\end{equation}
for some positive constant $C$ and $\rho $.
Recall that by \eqref{e3.10}, we have
\begin{equation}
\mathcal{N}\binom{u_{t}(t)}{v_{t}(t)}-\binom{
\Delta u(t)}{b\alpha \Delta _{\Gamma }v(t)
+b\partial _{n}u(t)+b\beta v(t)}+\binom{
f'(\phi (t))u(t)}{0}=0,
\label{e3.27}
\end{equation}
where $u(t)\big|_{\Gamma }=v(t)$ and
$p(t)\big|_{\Gamma }=q(t)$. Taking the inner product of
\eqref{e3.27} with $(u(t),v(t))^{\rm tr}$ in $\mathcal{H}$ and
integrating by parts again, we deduce
\begin{equation}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\|(u(t),v(t)
)\|_{\mathbb{X}}^{2}+\|\nabla u(t)
\|_{0}^{2}+\|\nabla _{\Gamma }v(t)
\|_{0,\Gamma }^{2} \\
&+\|v(t)\|_{0,\Gamma }^{2}+\langle f'(\phi (t))u(
t),u(t)\rangle _{0}=0.
\end{aligned}\label{e3.28}
\end{equation}
Due to  assumption \eqref{e1.5}, we have $f'(v)\geq -N$, for some $N>0$
and $v\in \mathbf{R}$. We estimate the
last term in \eqref{e3.28} as follows:
\begin{equation}
\begin{aligned}
|\langle f'(\phi (t))u(t),u(t)\rangle _{0}|
&\leq N\|u(t)\|_{0}^{2}\leq N\|(u(t),v(t))\|_{\mathcal{H}}^{2}\\
&\leq C\|(u(t),v(t)) \|_{\mathbb{X}}\|u(t)\|_{1}\\
&\leq \frac{C}{2}\|(u(t),v(t))
\|_{\mathbb{X}}^{2}+\frac{1}{2}\|u(t)\|_{1}^{2},
\end{aligned}\label{e3.29}
\end{equation}
Here, we have used the fact that $\|u\|_{0}^{2}\leq
C\|(u,v)\|_{\mathcal{H}}^{2}\leq \widetilde{C}\|u\|_{1}\|(u,v)\|_{
\mathbb{X}}$ and $D=H^{1}(\Omega )$. Multiplying
\eqref{e3.28} by $t$, integrating \eqref{e3.28} over $[0,t]$, and using the
above estimates together with \eqref{e3.25}, \eqref{e3.26}, we obtain
\begin{equation}
\int_{0}^{t}s[\|u(s)\|
_{1}^{2}+\|v(s)\|_{1,\Gamma }^{2}]ds+t\|(u(t),v(t))\|
_{\mathbb{X}}^{2}\leq Q_{2}(\|(\phi (0)
,\psi (0))\|_{\mathbb{Y}}^{2}),
\label{e3.30}
\end{equation}
for $t\in [0,T]$, where $Q_{2}$ is independent of $t$.

Finally, taking the inner product in $\mathcal{H}$ of \eqref{e3.27}
with $(u_{t}(t),v_{t}(t))^{\rm tr}$, then
multiplying the resulting equation by $t^{2}$, we have
\begin{equation}
\begin{aligned}
& t^{2}\|(u_{t}(t),v_{t}(t))
\|_{\mathbb{X}}^{2}+\frac{1}{2}\frac{d}{dt}(t^{2}\|
u(t)\|_{1}^{2}+t^{2}\|v(t)
\|_{1,\Gamma }^{2})
+t^{2}\langle f'(\phi (t))u(t),u_{t}(t)\rangle _{0}\\
&=2t[\|u(t)\|_{1}^{2}+\|v(t)
\|_{1,\Gamma }^{2}].
\end{aligned}\label{e3.31}
\end{equation}
Estimating the last term on the right hand side of \eqref{e3.30} as
in \eqref{e3.29}, we obtain
\begin{equation}
\begin{aligned}
&t^{2}|\langle f'(\phi (t)
)u(t),u_{t}(t)\rangle _{0}|\\
&\leq Ct^{2}\|f'(\phi (t))
u(t)\|_{1}\|(u_{t}(t)
,v_{t}(t))\|_{\mathbb{X}}
\\
&\leq \frac{t^{2}}{2}\|(u_{t}(t),v_{t}(
t))\|_{\mathbb{X}}^{2}+\frac{Ct^{2}}{2}Q_{3}(
\|\phi (t)\|_{2}^{2})\|u(t)\|_{1}^{2},
\end{aligned} \label{e3.32}
\end{equation}
for a suitable function $Q_{3}$ independent of $t$. Integrating
\eqref{e3.31} now over $[0,t]$, and inserting relation \eqref{e3.32}, we obtain
\begin{equation}
\begin{aligned}
&t^{2}[\|u(t)\|_{1}^{2}+\|v(t)\|_{1,\Gamma }^{2}]
+\int_{0}^{t}s^{2}\|(u_{t}(s),v_{t}(s))\|_{\mathbb{X}}^{2}ds
\\
&\leq Ct\int_{0}^{t}Q_{3}(\|\phi (s)
\|_{2}^{2})s\|u(s)\|
_{1}^{2}ds+\int_{0}^{t}s[\|u(s)
\|_{1}^{2}+\|v(s)\|_{1,\Gamma }^{2}]ds.
\end{aligned}\label{e3.33}
\end{equation}
Furthermore, estimating the terms on the right hand side of
$(3.33)$ using \eqref{e3.17} and \eqref{e3.30}, and the
fact that $u(t)=\partial _{t}\phi (t)$,
$v(t)=\partial _{t}\psi (t)$, we deduce that
\begin{equation}
\|\partial _{t}\phi (t)\|_{1}^{2}+\|
\partial _{t}\psi (t)\|_{1,\Gamma }^{2}\leq \frac{t+1
}{t^{2}}Q_{4}(\|(\phi (0),\psi (
0))\|_{\mathbb{Y}}^{2}),  \label{e3.34}
\end{equation}
for $t>0$ and a suitable monotonic function $Q_{4}$ independent of $t$.

To deduce estimate \eqref{e3.23}, it remains to write
\eqref{e1.8}, \eqref{e1.9} as an elliptic boundary value
problem for the chemical potential $\mu $, that is, we have
\begin{equation}
\begin{gathered}
-\Delta \mu =-\partial _{t}\phi \quad\text{in }\Omega , \\
b\partial _{n}\mu +c\varpi =-\partial _{t}\psi \quad\text{on }\Gamma ,\quad
\varpi =\mu\big|_{\Gamma }.
\end{gathered}
 \label{e3.35}
\end{equation}
Thus, we have the estimate
\begin{equation}
\|\mu (t)\|_{2}^{2}\leq C\big(\|
\partial _{t}\phi (t)\|_{0}^{2}+\|\mu (
t)\|_{1}^{2}+\|\partial _{t}\psi (t)
\|_{\frac{1}{2},\Gamma }^{2}\big).  \label{e3.36}
\end{equation}
Consequently, a classical trace theorem and estimate $(3.36)$,
 imply
\begin{equation}
\|\mu (t)\|_{2}^{2}+\|\varpi (
t)\|_{\frac{3}{2},\Gamma }^{2}\leq \frac{t+1}{t^{2}}
Q_{4}(\|(\phi (0),\psi (0)
)\|_{\mathbb{Y}}^{2}).  \label{e3.37}
\end{equation}
As in the proof of Proposition \ref{prop3}, we now rewrite problem
\eqref{e1.10}, \eqref{e1.11} as an elliptic boundary-value problem:
\begin{equation}
\begin{gathered}
-\Delta \phi =g_{1}(t):=\mu (t)-f(\phi
)\quad\text{in }\Omega ,\quad \phi\big|_{\Gamma }=\psi \\
-\Delta _{\Gamma }\psi +\beta \psi +\partial _{n}\phi =g_{2}(t)
:=\frac{\varpi (t)}{b}\quad\text{in }\Gamma ,\; \mu\big|_{
\Gamma }=\varpi .
\end{gathered} \label{e3.38}
\end{equation}
Applying the $H^{s}$-elliptic estimate of \cite[Lemma A.1]{m2},
with $s\in \mathbf{R,}$ $s+1/2\notin \mathbf{N}$, but with the nonlinear
term $f$ acting as an external force, we deduce from known embedding
theorems:
\begin{equation*}
\|\phi (t)\|_{3+\gamma }^{2}+\|\psi
(t)\|_{3+\gamma ,\Gamma }^{2}\leq C(\|
\mu (t)\|_{1+\gamma }^{2}+\|\varpi (
t)\|_{1+\gamma ,\Gamma }^{2}+\|f(\phi
)\|_{1+\gamma }^{2}).
\end{equation*}
Combining the estimate $(3.37)$ and the fact that
$H^{3/2}(\Gamma )\subset H^{1+\gamma }(\Gamma )$,
$H^{2}(\Omega )\subset H^{1+\gamma }(\Omega )$, for
$\gamma \in [0,1/2)$, $H^{2}\subset C$ together with
\eqref{e2.2}, \eqref{e3.19}, we easily verify our conclusion. Thus
Theorem \ref{thm6} is proven.
\end{proof}

\section{Existence and uniqueness of solutions}

 The existence of solutions to our problem \eqref{e1.8}--\eqref{e1.11} or equivalently
\eqref{e3.3} can be proved in a standard way, based on the a priori estimates
derived in Section $3$ and on
a standard Faedo-Galerkin approximation scheme. To this end, let us consider
the operator $\mathcal{B}:\mathcal{H}\to \mathcal{H}$ given formally
by
\begin{equation*}
\mathcal{B}\binom{\phi }{\psi }=\binom{-\Delta \phi }{-b\alpha \Delta
_{\Gamma }\psi +b\partial _{n}\phi +b\beta \psi }.
\end{equation*}
Then, according to \cite{g2}, \cite{r1}, $\mathcal{B}$
defines a positive self-adjoint operator on $\mathcal{H}$ such that
$D(\mathcal{B})=\{H^{2}(\Omega )\times H^{2}(\Gamma ):
\phi\big|_{\Gamma }=\psi \}$. Thus, for $i\in \mathbf{N}$, we take
a complete system of eigenfunctions $\{\Phi_{i}=(\phi _{i},\psi _{i})\}_{i}$
of the problem $\mathcal{B}\Phi _{i}=\lambda _{i}\Phi _{i}$ in
$\mathbb{V}_{1}^{\ast }$ with
$\Phi _{i}\in D(\mathcal{B})$. According to the general
spectral theory, the eigenvalues $\lambda _{i}$ can be increasingly ordered
and counted according to their multiplicities in order to form a real
divergent sequence. Moreover, the respective eigenvectors turn out to form
an orthogonal basis both in $\mathbb{V}_{1}$ and $\mathbb{V}_{0}=\mathcal{H}$
and may be assumed to be normalized in the norm of $\mathbb{X}$. At this
point, we set the spaces
\begin{equation*}
K_{n}=\mathop{\rm span}\{\Phi _{1},\Phi _{2},\dots ,\Phi _{n}\},\quad
K_{\infty }=\cup _{n=1}^{\infty }K_{n}.
\end{equation*}
Clearly, $K_{\infty }$ is a dense subspace of both $\mathbb{V}_{1}$ and
$\mathbb{V}_{2}$. For any, $n\in \mathbf{N}$, we look for functions of the
form
\begin{equation}
\Phi =\Phi _{n}=\sum_{i=1}^{n}c_{i}(t)\Phi _{i}
\label{e4.1}
\end{equation}
solving the approximate problem that we will introduce below. Note that
$\Lambda _{n}=(\mu _{n},\varpi _{n})$ can be found in terms of
$\Phi _{n}$ from $(1.6)-(1.7)$. That is, as
mentioned previously, it is enough to solve for $\Phi _{n}$. Note that in
the definition of $\Phi _{n}$, $c_{i}(t)$ are sought to be
suitably regular real valued functions. As approximations for the initial
data $\Phi _{0}=(\phi _{0},\psi _{0})$, we take
\begin{equation*}
\Phi _{n0}\in \mathbb{Y}\quad \text{such that }\underset{n\to \infty }{
\lim }\Phi _{n0}=\Phi _{0}\text{ in }\mathbb{Y}.
\end{equation*}
The problem that we must solve is given by $(P_{n})$, for any
$n\geq 1$,
\begin{equation}
\langle \partial _{t}(\mathcal{N}\Phi _{n}),\overline{\Phi
}\rangle _{\mathcal{H}}+\langle \mathcal{B}\Phi _{n},\overline{
\Phi }\rangle _{\mathcal{H}}+\langle \mathcal{F}(\Phi
_{n}),\overline{\Phi }\rangle _{\mathcal{H}}=0,  \label{e4.2}
\end{equation}
and
\begin{equation*}
\langle \Phi _{n}(0),\overline{\Phi }\rangle _{
\mathcal{H}}=\langle \Phi _{n0},\overline{\Phi }\rangle _{
\mathcal{H}},
\end{equation*}
for all $\overline{\Phi }=(\overline{\phi },\overline{\psi })\in K_{n}$.
 Here the operator $\mathcal{F}:\mathcal{H}\to \mathcal{H}$ is given
by $\mathcal{F}(\Phi )=(f(\phi ),0)^{\rm tr}$.

We aim to apply the standard existence theorems for ODE's. For this purpose,
if $n$ is fixed, let us choose $\overline{\Phi }=\Phi _{j}$,
$1\leq j\leq n$
and substitute the expressions \eqref{e4.1} to the unknowns
$\Phi_{n} $ in \eqref{e4.2}. Performing direct calculations, we actually
derive the equation:
\begin{equation}
\sum_{i=1}^{n}\langle \Phi _{i},\Phi _{j}\rangle _{
\mathbb{X}}\frac{dc_{i}(t)}{dt}+\sum_{i=1}^{n}
\langle \mathcal{B}\Phi _{i},\Phi _{j}\rangle _{\mathcal{H}
}c_{i}(t)+\widetilde{\mathcal{F}}_{j}(c_{i}(
t))=0,  \label{e4.3}
\end{equation}
for $1\leq j\leq n$, where
\begin{equation*}
\widetilde{\mathcal{F}}_{j}(c_{i}(t))=\langle
\mathcal{F}(\sum_{i=1}^{n}c_{i}(t)\Phi
_{i}),\Phi _{j}\rangle _{\mathcal{H}}=\langle f(
\sum_{i=1}^{n}c_{i}(t)\Phi _{i}),\phi
_{j}\rangle _{0}.
\end{equation*}
Note that the matrix coefficient of $c'(t)$ in
\eqref{e4.3} is symmetric and positive-definite, hence, non-singular.
Since the bilinear form $\langle \mathcal{B}\Phi _{i},\Phi
_{j}\rangle _{\mathcal{H}}=\langle \Phi _{i},\Phi
_{j}\rangle _{\mathbb{V}_{1}}=\lambda _{i}\langle \Phi _{i},\Phi
_{j}\rangle _{\mathcal{H}}$ is $\mathbb{V}_{1}$-coercive and $f\in
C^{2}(\mathbf{R})$, applying Cauchy's theorem for ODE's, we
find a small time $t_{n}\in (0,T)$ such that \eqref{e4.3} holds
for all $t\in [0,t_{n}]$. This gives the desired local
solution $\Phi $ to our problem \eqref{e4.2}, since $\Phi _{n}$
satisfies \eqref{e4.3}. Now, based on the uniform a priori estimates
with respect to $t$, derived for the solution $\Phi $ of \eqref{e3.3},
we obtain, in particular, that any local solution is a actually a global
solution that is defined on the whole interval $[0,T]$. It
remains then to pass to the limit as $n\to \infty $.

According to the a priori estimates derived in Section 3, we have
\begin{equation*}
\|\Phi _{n}\|_{L^{\infty }([0,T];
\mathbb{V}_{2})}+\|\Phi _{n}\|_{L^{2}([
0,T];\mathbb{V}_{3})}+\|\partial _{t}\Phi
_{n}\|_{L^{2}([0,T];\mathbb{V}_{1})}\leq C,
\end{equation*}
and for $\Lambda _{n}=(\mu _{n},\varpi _{n})$, $n\in \mathbf{N}$,
\begin{equation*}
\|\Lambda _{n}\|_{L^{\infty }([0,T]
;H^{1}(\Omega )\times H^{1/2}(\Gamma ))
}+\|\Lambda _{n}\|_{L^{2}([0,T]
;W\times H^{3/2}(\Gamma ))}\leq C,
\end{equation*}
where $C$ depends on $\Omega $, $\Gamma $, $T$, $\Phi _{0}$, but is
independent of $n$ and $t$. From this point on, all convergence relations
will be intended to hold up to the extraction of suitable subsequences,
generally not labelled. Thus, we observe that weak and weak star compactness
results applied to the above sequences $\Phi _{n}$ and $\Lambda _{n}$ entail
that there exist $\Phi =(\phi ,\psi )$ and $\Lambda =(\mu
,\varpi )$ such that as $n\to \infty $, the following
properties hold:
\begin{gather*}
\Phi _{n}\to \Phi \quad\text{weakly star in }L^{\infty }([0,T
];\mathbb{V}_{2}),
\\
\Phi _{n}\to \Phi \quad\text{weakly in }L^{\infty }([0,T
];\mathbb{V}_{3}),
\\
\partial _{n}\Phi _{n}\to \partial _{t}\Phi \quad\text{weakly in }
L^{2}([0,T];\mathbb{V}_{1})
\end{gather*}
and
\begin{gather*}
\Lambda _{n}\to \Lambda \quad\text{weakly star in }L^{\infty }(
[0,T];H^{1}(\Omega )\times H^{1/2}(\Gamma
)),
\\
\Lambda _{n}\to \Lambda \quad\text{weakly in }L^{2}([0,T
];W\times H^{3/2}(\Gamma )),
\end{gather*}
where recall that $\|u\|_{W}=\|u\|_{2}+\|(1/b)(\Delta u)\big|_{\Gamma
}\|_{0,\Gamma }$. Then, standard interpolation (for instance,
$H^{2-\delta }\subset C$, for $\delta \in (0,1/2)$, since
$\Omega \subset \mathbf{R}^{N}$ with $N\leq 3$) and compact embedding results
for vector valued functions \cite[Lemma 10]{g1} ensure that
\begin{equation}
\Phi _{n}\to \Phi \quad\text{strongly in }C([0,T]
;C(\Omega )\times C(\Gamma )).  \label{e4.4}
\end{equation}
Standard arguments and \eqref{e4.4} imply that $\Phi (0)=\Phi _{0}$.
By the Lipschitz continuity of $f$, the converges above allows
us to infer that
\begin{equation*}
\mathcal{F}(\Phi _{n})\to \mathcal{F}(\Phi
)\quad\text{strongly in }C([0,T];\mathcal{H}).
\end{equation*}
Thus, passing to the limit in \eqref{e4.2} and using the above
convergence properties, we immediately have that the solution $\Phi $
satisfies \eqref{e3.3} in the sense introduced in Definition \ref{def1},
Section 2.

Thus, we have the following result on the solvability of our problem
\eqref{e1.8}--\eqref{e1.11}. Let $T>0$ be fixed, but otherwise
arbitrary.

\begin{theorem} \label{thm7}
Let $(\phi _{0},\psi _{0})\in \mathbb{X}$ and suppose that the
nonlinearity $f$ satisfies assumption \eqref{e1.5}. Then, the
problem \eqref{e1.8}--\eqref{e1.11} has a unique solution in
the sense of the Definition \ref{def1} in Section $2$. Moreover, the solution
$(\phi (t),\psi (t))$ belongs to the
space $C([0,T],\mathbb{X})\cap L_{loc}^{\infty}((0,T],\mathbb{Y})$.
\end{theorem}

\begin{proof}
It remains to verify only the uniqueness. Suppose that
$(\phi _{1}(t),\psi _{1}(t))$ and
$(\phi _{1}(t),\psi _{1}(t))$ are two solutions of
\eqref{e3.3} with same initial data. We set
\begin{gather*}
(u(t),p(t)):=(\phi _{1}(
t)-\phi _{2}(t),\mu _{1}(t)-\mu _{2}(t))
\\
(v(t),q(t)):=(\psi _{1}(
t)-\psi _{2}(t),\varpi _{1}(t)-\varpi
_{2}(t))
\end{gather*}
These functions satisfy the  equation
\begin{equation}
\mathcal{N}\binom{u_{t}(t)}{v_{t}(t)}-\binom{
\Delta u(t)}{b\alpha \Delta _{\Gamma }v(t)
+b\partial _{n}u(t)+b\beta v(t)}+\binom{f(
\phi _{1}(t))-f(\phi _{2}(t))}{
0}=0,  \label{e4.5}
\end{equation}
where $\psi (t)=\phi (t)\big|_{\Gamma }$. Taking
the inner product in $\mathcal{H}$ of \eqref{e4.5} with
$(u(t),v(t))^{\rm tr}$ and using the relations
\eqref{e2.2}--\eqref{e2.10}, we deduce
\begin{equation}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}[\|(u(t),v(
t))\|_{\mathbb{X}}^{2}]+\|\nabla
u(t)\|_{0}^{2}+\|\nabla _{\Gamma }v(
t)\|_{0,\Gamma }^{2} \\
&+\|v(t)\|_{0,\Gamma }^{2}
+\langle f(\phi _{1}(t))-f(\phi
_{2}(t)),u(t)\rangle _{0}=0.
\end{aligned}\label{e4.6}
\end{equation}
Due to assumption \eqref{e1.5}, we have $f'(v)\geq -M$, for some positive
constant $M$. Consequently, \eqref{e4.6} implies
\begin{equation}
\begin{aligned}
&\frac{d}{dt}[\|(u(t),v(t)
)\|_{\mathbb{X}}^{2}]+2\|\nabla u(
t)\|_{0}^{2}+2\|\nabla _{\Gamma }v(t)
\|_{0,\Gamma }^{2}\\
&+2\|v(t)\|_{0,\Gamma }^{2}
\leq 2M\|u(t)\|_{0}^{2}.
\end{aligned} \label{e4.7}
\end{equation}
Using now the interpolation inequality $\|(u,v)
\|_{\mathcal{H}}^{2}\leq C\|(u,v)\|
_{\mathbb{X}}\|(u,v)\|_{D}$, the obvious
inequality $\|u\|_{0}\leq C\|(u,v)
\|_{\mathcal{H}}$ and the fact $D=H^{1}(\Omega )$ up
to an equivalent norm $\|u\|_{1}$, in order to estimate
the term on the right hand side of \eqref{e4.7}, and applying
Gronwall's inequality, we obtain
\begin{equation}
\|(u(t),v(t))\|_{
\mathbb{X}}^{2}\leq C\|(u(0),v(0)
)\|_{\mathbb{X}}^{2}e^{M_{3}t},  \label{e4.8}
\end{equation}
where the positive constants $C$, $M_{3}$ are independent of $t$ and the
norm of initial data. This finishes the proof of the uniqueness.

As we mentioned already, the existence of solutions in the phase space
$\mathbb{Y}$ (in the sense of Definition \ref{def1}) can be verified in a standard
way, whenever $(\phi (0),\psi (0))\in
\mathbb{Y}$. Thus, problem \eqref{e1.8}--\eqref{e1.11}
generates a semiflow
\begin{equation*}
S(t):\mathbb{Y}\to \mathbb{Y},
\end{equation*}
such that
\begin{equation*}
S(t)(\phi (0),\psi (0))
=(\phi (t),\psi (t)),
\end{equation*}
where $(\phi (t),\psi (t))$ is the
unique solution of \eqref{e1.8}--\eqref{e1.11} with initial
data in $\mathbb{Y}$. Moreover, by estimate \eqref{e4.8}, we have
the Lipschitz continuity (with respect to the initial data) in the
$\mathbb{X}$-norm:
\begin{equation}
\|S(t)(\phi _{1},\psi _{1})-S(
t)(\phi _{2},\psi _{2})\|_{\mathbb{X}}\leq
Ce^{Mt}\|(\phi _{1}-\phi _{2},,\psi _{1}-\psi _{2})
\|_{\mathbb{X}},  \label{e4.9}
\end{equation}
for all $(\phi _{i},\psi _{i})\in \mathbb{Y}$, $i=1,2$. In what
follows, we extend this semigroup, which we still denote by $S(
t)$, in a unique way by continuity such that it maps $\mathbb{X}$
into $\mathbb{Y}$. For this purpose, let $(\phi (0),\psi
(0))\in \mathbb{X}$. By the obvious dense injection
$\mathbb{Y}\hookrightarrow \mathbb{X}$, we can construct a sequence $(
\phi _{n},\psi _{n})\in \mathbb{Y}$ such that $(\phi _{n},\psi
_{n})\to (\phi (0),\psi (0)
)$ in the norm of $\mathbb{X}$. Therefore, we extend $S(
t)$ to the semigroup
\begin{equation}
S(t)(\phi (0),\psi (0))
:=\lim_{n\to \infty }S(t)(\phi _{n},\psi
_{n}),  \label{e4.10}
\end{equation}
where the convergence takes place in $\mathbb{X}$. Since the solutions
$(\phi _{n},\psi _{n})\in C([0,T],\mathbb{Y}
)$, (and clearly, $\mathbb{Y}\subset \mathbb{X}$), we get that the
limit solution $(\phi (t),\psi (t))=S(t)(\phi (0),\psi (0))$
belongs to the space $C([0,T],\mathbb{X})$, and it
satisfies the same estimates as in Section $3$. Thus, passing to the limit
as $n\to \infty $, for each $t>0$, we have $S(t):
\mathbb{X}\to \mathbb{Y}$ and $(\phi (t),\psi
(t))$ satisfies the equations \eqref{e1.8}--\eqref{e1.11} in the sense defined
in Section $2$. The proof is complete.
\end{proof}

\section{Exponential attractors}

\qquad In this section, we shall prove the existence of global attractors
for the semiflow $T$ in $\mathbb{Y}$ and moreover, the existence of a
semiflow and of a global attractor in $\mathbb{X}$ will also be obtained as
a consequence. The existence of a global attractor to our problem $(
1.8)-(1.11)$ can be deduced as a result of the uniform
and dissipative estimates of our semiflow obtained in Section $3$.

Recall that the compact set $\mathcal{A}\subset V$ is called the global
attractor for the semigroup $S(t)$ on $V$ if it is invariant by
$S(t)$, that is,
\begin{equation}
S(t)\mathcal{A=A},\quad\text{for }t\geq 0  \label{e5.1}
\end{equation}
and it attracts the bounded subsets of $V$ as $t\to \infty $, that
is, for every bounded $B\subset V,$
\begin{equation*}
\lim_{t\to \infty }\mathop{\rm dist}{}_{V}(S(t)B,\mathcal{A}
)=0,
\end{equation*}
where dist$_{V}$ is the Hausdorff semi-distance in $V$.

According to the abstract attractor existence theorem in \cite{b1},
\cite{t1}, it suffices to verify that the operators $S(
t):\mathbb{Y}\to \mathbb{Y}$ are continuous for each $t\geq 0$
and that the semigroup $S(t)$ possesses an attracting set
$\mathbb{B}$ in $\mathbb{Y}$ and that the orbits are pre-compact in $\mathbb{Y
}$.

To this end, we introduce the following ball $\mathbb{B}$ with sufficiently
large radius $R$ in the space $H^{3}(\Omega )\times H^{3}(
\Gamma )$:
\begin{equation}
\mathbb{B}_{R}=\{(\phi ,\psi )\in H^{3}(\Omega
)\times H^{3}(\Gamma ):\|(\phi ,\psi
)\|_{H^{3}(\Omega )\times H^{3}(\Gamma
)}\leq R\}.  \label{e5.2}
\end{equation}
Then obviously, $\mathbb{B}_{R}\subset \mathbb{Y}$ by \eqref{e3.22}, \eqref{e3.23}.
Moreover, due to the dissipative estimates \eqref{e3.21}, \eqref{e3.22} and the
smoothing property \eqref{e3.17}, there exist sufficiently large
$\overline{R}$ ($\geq C_{6}$)
and $T_{0}=T_{0}(\rho ,\overline{R})$ such that
$\mathbb{B}:=\mathbb{B}_{\overline{R}}$ is an absorbing set for
the semigroup $S(t)$ acting on $\mathbb{Y}$ and
$S(t)(\mathbb{B})\subset \mathbb{B}$ for $t\geq T_{0}$.
The continuity of the semigroup $S(t)$ was actually verified in
Theorem \ref{thm8}. It remains to prove the relative compactness of
orbits $(\phi ,\psi)$ in $\mathbb{Y}$. This property follows thanks
to the estimates of
Theorem \ref{thm6} and the compact embedding $H^{3}\subset H^{2}$. Thus, the
semigroup $S(t)$ possesses a compact global attractor
$\mathcal{A}\subset \mathbb{B}\subset \mathbb{Y}$. Due to the
parabolic nature of the
problem \eqref{e1.8}--\eqref{e1.11}, we have the standard
smoothing property for its solutions as given by Proposition \ref{prop2},
Theorem \ref{thm5}, \ref{thm6} and \ref{thm8} and the concrete
choice of the space
$\mathbb{Y}$ is not
essential and can be replaced by the weak energy space $\mathbb{X}$. In
fact, in Section $4$, we have extended the unique solution
$(\phi,\psi )(t)=S(t)(\phi _{0},\psi_{0})$, for every
$(\phi _{0},\psi _{0})=(\phi(0),\psi (0))\in \mathbb{Y}$ by continuity,
to the semigroup $S(t):\mathbb{X\to X}$ which possesses
the smoothing property $S(t):\mathbb{X\to Y}$ for each
fixed $t>0$. Consequently, we have proved the following result.

\begin{theorem} \label{thm8}
The semigroup $S(t)$ defined by \eqref{e4.10} possesses
a compact global attractor $\mathcal{A}\subset \mathbb{Y}$ which has the
following structure
\begin{equation*}
\mathcal{A=I}_{0}\mathcal{K},
\end{equation*}
where $\mathcal{K}$ denotes the set of all complete bounded trajectories of
the semigroup $S(t)$, that is,
\begin{align*}
\mathcal{K}=\big\{&(\phi ,\psi )\in C_{b}(\mathbb{R},
\mathbb{X}):S(t)(\phi ,\psi )=(\phi
(t+h),\psi (t+h))\\
&\text{for }t\in \mathbb{R},\; h\geq 0,\;
 \|(\phi ,\psi )\|_{\mathbb{X}}\leq C_{\phi ,\psi }\big\},
\end{align*}
and $\mathcal{I}_{0}(\phi ,\psi )\equiv (\phi (0),\psi (0))$.
\end{theorem}

\begin{remark} \label{rmk9} \rm
Recall that $\Omega \subset \mathbf{R}^{n}$, $n=2,3$. Then
Theorem \ref{thm8} and the continuous embedding of
$\mathbb{V}_{2}\subset C(\overline{\Omega })$ imply that for
each $t>0$, we have the following regularity:
\begin{equation*}
S(t):\mathbb{X\to }C(\overline{\Omega })
\quad\text{and}\quad \mathcal{A}\subset C(\overline{\Omega }).
\end{equation*}
\end{remark}

The fact that this global attractor $\mathcal{A}$ has finite fractal
dimension in the topology of $H^{2}(\Omega )\times H^{2}(\Gamma )$
will be a consequence of the existence of the exponential
attractor below (see Proposition \ref{prop10}). But first, let us recall that a
compact set $\mathcal{M}\subset V$ is called an exponential attractor for
the semigroup on $V$, if it is semi-invariant, that is,
\begin{equation}
S(t)\mathcal{M}\subset \mathcal{M},\quad\text{for }t\geq 0
\label{e5.3}
\end{equation}
and it attracts exponentially all bounded subsets of $V$, that is, there is
a constant $\eta >0$, such that for every bounded $B\subset V$, we have
\begin{equation*}
\lim_{t\to \infty }\mathop|{\rm dist}{}_{V}(S(t)B,\mathcal{M}
)\leq Q(\|B\|_{V})e^{-\eta t},
\end{equation*}
and it has finite fractal dimension in $V$, that is,
$d(\mathcal{M},V)<\infty $.

Since, we lose the invariance for the semigroup, that is, the assumption
\eqref{e5.3} instead of \eqref{e5.1}, then the exponential
attractor is not necessarily unique. However, we always have $\mathcal{A}
\subset \mathcal{M}$. The following proposition gives sufficient conditions
for the existence of an exponential attractor in Banach spaces
(see \cite{e1}).

\begin{proposition} \label{prop10}
Let $H$ and $H_{1}$ be two Banach spaces and $H_{1}$ compactly embedded in
$H$. Let $E$ be a bounded subset of $H$. We consider a nonlinear map
$L:E\to E$ such that $L$ can be decomposed in the sum of two maps
\begin{equation*}
L=L_{0}+L_{1}\quad\text{with}\quad L_{i}:E\to H\quad(i=0,1),
\end{equation*}
where $L_{0}$ is a contraction, that is,
\begin{equation}
\|L_{0}(x_{1})-L_{0}(x_{2})\|
_{H}\leq k\|x_{1}-x_{2}\|_{H},  \label{e5.4}
\end{equation}
for any $x_{1},x_{2}\in E$ with $k\leq 1/2$ and $L_{1}$ satisfies the
condition
\begin{equation}
\|L_{1}(x_{1})-L_{1}(x_{2})\|
_{H_{1}}\leq C\|x_{1}-x_{2}\|_{H},  \label{e5.5}
\end{equation}
for all $x_{1},x_{2}\in E$. Then the map $L$ possesses an exponential
attractor $\mathcal{M}^{\ast }$.
\end{proposition}

To verify the conditions (that is the assumption \eqref{e5.4} and
\eqref{e5.5}) of Proposition \ref{prop10}, we need to decompose the solution
in a sum of two components. To this end, we decompose the vector function
$(\phi (t),\psi (t))=(\widehat{
\phi }(t),\widehat{\psi }(t))+(
\widetilde{\phi }(t),\widetilde{\psi }(t))$
into the sum of an exponentially decaying and a smoothing part, where the
vector functions $(\widehat{\phi }(t),\widehat{\psi }
(t))$ and $(\widetilde{\phi }(t),
\widetilde{\psi }(t))$ satisfy the equations:
\begin{equation}
\begin{gathered}
\mathcal{N}\binom{\widehat{\phi }_{t}(t)}{\widehat{\psi }
_{t}(t)}-\binom{\Delta \widehat{\phi }(t)}{b\alpha
\Delta _{\Gamma }\widehat{\psi }(t)+b\partial _{n}\widehat{\phi
}(t)+b\beta \widehat{\psi }(t)}=0,\\
\big(\widehat{\phi }(0),\widehat{\psi }(0)\big)
=\big(\phi (0),\psi (0)\big),
\end{gathered} \label{e5.6}
\end{equation}
and
\begin{equation}
\mathcal{N}\binom{\widetilde{\phi }_{t}(t)}{\widetilde{\psi }
_{t}(t)}-\binom{\Delta \widetilde{\phi }(t)}{
b\alpha \Delta _{\Gamma }\widetilde{\psi }(t)+b\partial _{n}
\widetilde{\phi }(t)+b\beta \widetilde{\psi }(t)}+
\binom{f(\widehat{\phi }(t)+\widetilde{\phi }(
t))}{0}=0,  \label{e5.7}
\end{equation}
with $(\widetilde{\phi }(0),\widetilde{\psi }(0))=(0,0)$, respectively.

We prove our required estimates in the next lemma. Recall that $S(
t)(\mathbb{B})\subset \mathbb{B}$ for $t\geq T_{0}$,
where $\mathbb{B}$ was introduced in \eqref{e5.2}.

\begin{lemma} \label{lem11}
Let $(\phi _{1}(0),\psi _{1}(0))$ and
$(\phi _{2}(0),\psi _{2}(0))$ belong
to $\mathbb{B}$. The corresponding solutions of equation \eqref{e5.6}
and \eqref{e5.7} satisfy the following two estimates:
\begin{equation} \label{e5.8}
\begin{aligned}
&\|\widehat{\phi }_{1}(t)-\widehat{\phi }_{2}(
t)\|_{1}^{2}+\|\widehat{\psi }_{1}(t)-
\widehat{\psi }_{2}(t)\|_{1,\Gamma }^{2}
\\
&\leq C_{1}\frac{e^{-\alpha t}}{t}\|\phi _{1}(0)-\phi
_{2}(0),\psi _{1}(0)-\psi _{2}(0)
\|_{\mathbb{X}}^{2},
\end{aligned}
\end{equation}
and
\begin{equation}
\begin{aligned}
&\|\widetilde{\phi }_{1}(t)-\widetilde{\phi }_{2}(
t)\|_{1}^{2}+\|\widetilde{\psi }_{1}( t)-\widetilde{\psi
}_{2}(t)\|_{1,\Gamma }^{2}
\\
&\leq C_{2}\frac{e^{Kt}}{t}\|\phi _{1}(0)-\phi
_{2}(0),\psi _{1}(0)-\psi _{2}(0)
\|_{\mathbb{X}}^{2},
\end{aligned}\label{e5.9}
\end{equation}
for $t>0$, where $C_{1},C_{2}>0$ are independent of $b$, $t$.
\end{lemma}

\begin{proof}
We first note that, due to the estimates \eqref{e3.20} and \eqref{e5.2}, we have
\begin{equation}
\|\phi _{i}(t)\|_{2}+\|\psi_{i}(t)\|_{2,\Gamma }\leq C,  \label{e5.10}
\end{equation}
for $t\geq 0$, $i=0,1$, where the $C$ is independent of $t$ and depends at
most on $\overline{R}$. Here, $\phi _{0}:=\widehat{\phi }$, $\phi _{1}:=
\widetilde{\phi }$ and $\psi _{0}:=\widehat{\psi }$, $\psi _{1}:=\widetilde{
\psi }$ respectively. Due to the continuous embedding $H^{2}\subset C$, we
have analogous estimates for $L^{\infty }$ norms of the solution
$(\phi _{i},\psi _{i})$, which are necessary in order to handle the
nonlinear term $f$. We now set
 $(\Phi (t),\Psi (t))^{\rm tr}:=(\widehat{\phi }_{1}(t)-\widehat{
\phi }_{2}(t),\widehat{\psi }_{1}(t)-\widehat{\psi
}_{2}(t))^{\rm tr}$. Then this vector-valued function
$(\Phi (t),\Psi (t))^{\rm tr}$ satisfies
\eqref{e5.6}. Taking the inner product of \eqref{e5.6} with
the vector $(\Phi (t),\Psi (t))^{\rm tr}$
\ in $\mathcal{H}$ and integrating by parts, we deduce
\begin{equation}
\frac{1}{2}\frac{d}{dt}[\|\Phi (t),\Psi (
t)\|_{\mathbb{X}}^{2}]+\|\Phi (
t)\|_{1}^{2}+\|\Psi (t)\|
_{1,\Gamma }^{2}=0.  \label{e5.11}
\end{equation}
Consequently, the inequality
$\|\Phi \|_{1}^{2}+\|\Psi \|_{1,\Gamma }^{2}\geq C\|(
\Phi ,\Psi )\|_{\mathcal{H}}^{2}\geq \widetilde{C}
\|(\Phi ,\Psi )\|_{\mathbb{X}}^{2}$ and
relation \eqref{e5.11} yield
\begin{equation}
\|\Phi (t),\Psi (t)\|_{\mathbb{X
}}^{2}+\int_{0}^{t}\big(\|\Phi (s)\|
_{1}^{2}+\|\Psi (s)\|_{1,\Gamma }^{2}\big)
ds\leq Ce^{-\rho t}\|\Phi (0),\Psi (0)
\|_{\mathbb{X}}^{2},  \label{e5.12}
\end{equation}
for some positive constants $\rho $, $C$ independent of $t$. Similarly, we
take the inner product in $\mathcal{H}$ of \eqref{e5.6} with
$t(\Phi _{t}(t),\Psi _{t}(t))^{\rm tr}$
and integrate by parts. Consequently, we obtain
\begin{equation}
t\|\Phi _{t}(t),\Psi _{t}(t)\|_{
\mathbb{X}}^{2}+\frac{1}{2}\frac{d}{dt}[t(\|\Phi (
t)\|_{1}^{2}+\|\Psi (t)\|
_{1,\Gamma }^{2})]
=\frac{1}{2}(\|\Phi (t)\|
_{1}^{2}+\|\Psi (t)\|_{1,\Gamma }^{2}).
  \label{e5.13}
\end{equation}
Integrating now \eqref{e5.13} over $[0,t]$ and using
estimate \eqref{e5.12}, we deduce
\begin{equation*}
t(\|\Phi (t)\|_{1}^{2}+\|\Psi
(t)\|_{1,\Gamma }^{2})
+\int_{0}^{t}s\|\Phi _{t}(s),\Psi _{t}(
s)\|_{\mathbb{X}}^{2}ds
\leq Ce^{-\rho t}\|\Phi (0),\Psi (0)
\|_{\mathbb{X}}^{2}.
\end{equation*}
This last estimate yields our conclusion \eqref{e5.8}. In order to
verify estimate \eqref{e5.9}, we will use \eqref{e5.10}. We
define
\begin{equation*}
(\overline{\Phi }(t),\overline{\Psi }(t)
)^{\rm tr}:=(\widetilde{\phi }_{1}(t)-\widetilde{\phi }
_{2}(t),\widetilde{\psi }_{1}(t)-\widetilde{\psi }
_{2}(t))^{\rm tr}.
\end{equation*}
This vector-valued function satisfies \eqref{e5.7}. Arguing as in
the derivation of the estimate \eqref{e5.12}, we similarly deduce
that
\begin{equation}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}[\|\overline{\Phi }(t),
\overline{\Psi }(t)\|_{\mathbb{X}}^{2}]
+\|\overline{\Phi }(t)\|_{1}^{2}+\|
\overline{\Psi }(t)\|_{1,\Gamma }^{2}
\\
&=-\langle f(\widehat{\phi }_{1}(t)+\widetilde{\phi }
_{1}(t))-f(\widehat{\phi }_{2}(t)+
\widetilde{\phi }_{2}(t)),\overline{\Phi }(
t)\rangle _{0}.
\end{aligned}\label{e5.14}
\end{equation}
In contrast to the proof of Proposition \ref{prop2}, we can not estimate the
nonlinear term $f$ using assumption \eqref{e1.5}, so then instead we
use the uniform estimate \eqref{e5.10} and the analogous $L^{\infty
}-$estimates. We have
\begin{equation}
-\langle f(\widehat{\phi }_{1}(t)+\widetilde{\phi }
_{1}(t))-f(\widehat{\phi }_{2}(t)+
\widetilde{\phi }_{2}(t)),\overline{\Phi }(
t)\rangle _{0}\leq C\|\overline{\Phi }(t)
\|_{0}^{2}.  \label{e5.15}
\end{equation}
Using now the obvious interpolation inequality
\begin{equation}
\|\overline{\Phi }(t)\|_{0}^{2}\leq
C\|(\overline{\Phi }(t),\overline{\Psi }(
t))\|_{\mathcal{H}}^{2}\leq \widehat{C}\|
(\overline{\Phi }(t),\overline{\Psi }(t)
)\|_{D}\|(\overline{\Phi }(t),
\overline{\Psi }(t))\|_{\mathbb{X}}  \label{e5.16}
\end{equation}
and the fact $D=H^{1}(\Omega )$ (up to an equivalent norm), we
deduce from $(5.14),(5.15)$ and Gronwall's
inequality that
\begin{equation} \label{e5.17}
\begin{aligned}
&\|(\overline{\Phi }(t),\overline{\Psi }(
t))\|_{\mathbb{X}}^{2}+\int_{0}^{t}(
\|\overline{\Phi }(s)\|_{1}^{2}+\|
\overline{\Psi }(s)\|_{1,\Gamma }^{2})ds
\\
&\leq e^{Kt}\|(\phi _{1}(0)-\phi _{2}(
0),\psi _{1}(0)-\psi _{2}(0))\|_{\mathbb{X}}^{2}.
\end{aligned}
\end{equation}
Arguing as in \eqref{e5.13}, we deduce
\begin{equation}
\begin{aligned}
&t\|(\overline{\Phi }_{t}(t),\overline{\Psi }
_{t}(t))\|_{\mathbb{X}}^{2}+\frac{1}{2}\frac{d
}{dt}[t(\|\overline{\Phi }(t)\|
_{1}^{2}+\|\overline{\Psi }(t)\|_{1,\Gamma
}^{2})]
\\
&=-\langle f(\widehat{\phi }_{1}(t)+\widetilde{\phi }
_{1}(t))-f(\widehat{\phi }_{2}(t)+
\widetilde{\phi }_{2}(t)),t\overline{\Phi }_{t}(
t)\rangle _{0}\\
&\quad +\frac{1}{2}(\|\Phi (t)\|_{1}^{2}+\|\Psi (t)\|
_{1,\Gamma }^{2}).
\end{aligned}\label{e5.18}
\end{equation}
We estimate the first term on the right hand side of \eqref{e5.18},
using \eqref{e5.10}, as follows:
\begin{equation}
\begin{aligned}
&-\langle f(\widehat{\phi }_{1}(t)+\widetilde{\phi }
_{1}(t))-f(\widehat{\phi }_{2}(t)+
\widetilde{\phi }_{2}(t)),t\overline{\Phi }_{t}(
t)\rangle _{0}\\
&\leq Ct\|\overline{\Phi }(t)
\|_{1}^{2}+\frac{1}{2}t\|(\overline{\Phi }
_{t}(t),\overline{\Psi }_{t}(t))\|
_{\mathbb{X}}^{2},
\end{aligned}\label{e5.19}
\end{equation}
where we have used again the interpolation inequality \eqref{e5.16}
and the $L^{\infty }$-norm estimates for $\overline{\Phi }$ and $\Phi $.
Gronwall's lemma and the estimates \eqref{e5.17}--\eqref{e5.19}
then yield the final conclusion \eqref{e5.9}, which finishes the
proof of the lemma.
\end{proof}

Let us now fix $t^{\ast }\geq T_{0}$. It is left to observe that for every
$(\phi _{1}(0),\psi _{1}(0))$ and
$(\phi _{2}(0),\psi _{2}(0))$
belonging to $\mathbb{B}$, the functions $\overline{\Phi }$ and
$\overline{\Psi }$ (defined in the proof of Lemma \ref{lem11} satisfy
the following smoothing estimate:
\begin{equation}
\|\overline{\Phi }(t^{\ast })\|_{1}^{2}
+\|\overline{\Psi }(t^{\ast })\|_{1,\Gamma }^{2}\leq C_{T}\|\phi _{1}(0)-\phi
_{2}(0),\psi _{1}(0)-\psi _{2}(0)
\|_{\mathbb{X}}^{2},  \label{e5.20}
\end{equation}
where $C_{T}$ depends on $T$.\ Obviously, $\mathbb{V}_{1}=H^{1}(\Omega
)\times H^{1}(\Gamma )$ is compactly embedded in
$\mathbb{X}$, since $\mathbb{V}_{1}$ is compactly injected in $\mathcal{H}$
and the definition of the norm of $\mathbb{X}$ is based on that of
$\mathcal{H}$ (see \eqref{e2.6}, \eqref{e2.7}). It follows that the
assumption \eqref{e5.5} of Proposition \ref{prop10} is verified. Moreover,
due to the decaying estimate \eqref{e5.8}, the functions $\Phi $ and
$\Psi $ satisfy
\begin{equation}
\|(\Phi (t^{\ast }),\Psi (t^{\ast })
)\|_{\mathbb{X}}^{2}\leq k\|\phi _{1}(
0)-\phi _{2}(0),\psi _{1}(0)-\psi
_{2}(0)\|_{\mathbb{X}}^{2},  \label{e5.21}
\end{equation}
where $k<1/2$ (note that it is possible to do this, thanks to the estimate
\eqref{e5.8} and obvious inequality
$\widetilde{C}\|(\cdot ,\cdot )\|_{\mathbb{X}}
\leq \|(\cdot,\cdot )\|_{\mathcal{H}}\leq C\|(\cdot,\cdot )\|_{H^{1}
(\Omega )\times H^{1}(\Gamma )}$). Thus, assumption \eqref{e5.4}
is also verified.

Thus, according to the conclusion of Proposition \ref{prop10}, the operator
$L=S(t^{\ast })$ possesses an exponential attractor
$\mathcal{M}^{\ast }$ on $\mathbb{B}$. Since, $\mathbb{B}$
is an absorbing set for this
semigroup on $\mathbb{X}$, then these attractors attract exponentially all
the bounded subsets of $\mathbb{X}$ with respect to the metric of
$\mathbb{X}$. Moreover, the fractal dimension of $\mathcal{M}^{\ast }$
is finite, that is,
\begin{equation*}
\dim _{F}(\mathcal{M}^{\ast },\mathbb{X})<\infty .
\end{equation*}
Next, using a standard formula, we define the desired exponential attractor
by
\begin{equation}
\mathcal{M=}\cup _{t\in [0,T]}S(t)\mathcal{M}
^{\ast }.  \label{e5.22}
\end{equation}
The semi-invariance \eqref{e5.3} is then an immediate consequence of
the semi-invariance of $\mathcal{M}^{\ast }$ and the definition
\eqref{e5.22}. The exponential attraction
\begin{equation*}
\lim_{t\to \infty }\mathop{\rm dist}{}_{\mathbb{X}}(S(t)B,
\mathcal{M})\leq Q(\|B\|_{\mathbb{X}})
e^{-\eta t},\,\text{\ }t\geq 0,
\end{equation*}
(for every bounded subset $B\subset \mathbb{X}$) follows from the fact that
$\mathbb{B}$ defined by \eqref{e5.2} is an absorbing set for $S(
t)$ (due to \eqref{e3.12}) and from the uniform Lipschitz
continuity (with respect to the initial data) due to \eqref{e4.9}.
Thus, there only remains to verify that $\mathcal{M}$ has finite fractal
dimension. Nevertheless, since the map $S(t)$ is uniformly
H\"{o}lder continuous on $[0,T]\times \mathbb{B}$ in the norm of
$\mathbb{X}$, it follows that $\mathcal{M}$ is still a compact set with
finite fractal dimension that will be exponentially attracting for the
semiflow $S(t)$ on $\mathbb{B}$. The Lipschitz continuity with
respect to the initial data was verified in \eqref{e4.9} and the
H\"{o}lder continuity with respect to $t$ follows from the fact that
$(\partial _{t}\phi (t),\partial _{t}\psi (t))$ is in
$\mathbb{X}$ if $(\phi (t),\psi (t))$ is in $\mathbb{Y}$.
This result is given below in the following
corollary.

\begin{corollary} \label{coro12}
The semigroup $S(t)$ defined in \eqref{e4.10} is Holder
continuous on $[0,T]\times \mathbb{B}$ in the topology of
$\mathbb{X}$, ($\mathbb{B}$ is defined in \eqref{e5.2}), that is,
\begin{align*}
&\|S(t_{2})(\phi _{0}^{1},\psi _{0}^{1})
-S(t_{1})(\phi _{0}^{2},\psi _{0}^{2})\|
_{\mathbb{X}}\\
&\leq C(R,T)[\|(\phi _{0}^{1}-\phi
_{0}^{2},\psi _{0}^{1}-\psi _{0}^{2})\|_{\mathbb{X}
}+|t_{2}-t_{1}|^{1/2}],
\end{align*}
for $(\phi _{0}^{i},\psi _{0}^{i})\in \mathbb{B}$ and
$t_{i}\in [0,T]$. Moreover, the constant $C(R,T)$ is
independent of $t$.
\end{corollary}

\begin{proof}
The uniform Lipschitz continuity of $S$ with respect to the initial data in
the metric of $\mathbb{X}$ was actually verified in \eqref{e4.9}. It
is left to verify the H\"{o}lder continuity with respect to $t$. Arguing as
in the proof of Theorem \ref{thm8}, \eqref{e3.25}, we obtain
\begin{equation}
\int_{T}^{T+1}\|(\partial _{t}\phi (s)
,\partial _{t}\psi (s))\|_{\mathbb{X}
}^{2}ds\leq Q(\|(\phi (0),\psi (
0))\|_{\mathbb{Y}}^{2})\leq C(R),
\label{e5.23}
\end{equation}
(for a suitable monotonic function $Q$) if
$(\phi _{0},\psi_{0})\in \mathbb{B}$. Moreover, for every
$t_{1}$, $t_{2}\in [0,T]$,
\begin{align*}
\|(\phi ,\psi )(t_{1})-(\phi ,\psi)(t_{2})\|_{\mathbb{X}}
&=\|\int_{t_{1}}^{t_{2}}\partial _{t}(\phi (s),\psi (s))ds\|_{\mathbb{X}}\\
&\leq \int_{t_{1}}^{t_{2}}\|(\partial _{t}\phi (s),\partial _{t}\psi (s))
\|_{\mathbb{X}}ds \\
&\leq \Big(\int_{t_{1}}^{t_{2}}\|(\partial _{t}\phi
(s),\partial _{t}\psi (s))\|_{
\mathbb{X}}^{2}ds\Big)^{1/2}|t_{2}-t_{1}|^{1/2}\\
&\leq \sqrt{C(R)}|t_{2}-t_{1}|^{1/2}.
\end{align*}
\end{proof}

Thus, we have constructed the exponential attractor $\mathcal{M}$ for the
metric of $\mathbb{X}$ such that we have
\begin{equation*}
\dim _{F}(\mathcal{M},\mathbb{X})\leq \dim _{F}(\mathcal{M }^{\ast
},\mathbb{X})+2.
\end{equation*}
To obtain the attractor $\mathcal{M}$ with the respect of the
metric in $H^{2}(\Omega )\times H^{2}(\Gamma )$,
it is enough to note that the semigroup $S(t)$ possesses
smoothing properties according to Theorem \ref{thm5} and \ref{thm6}
 and use the following
interpolation inequality:
\begin{equation}
\|(\phi ,\psi )\|_{H^{2}(\Omega
)\times H^{2}(\Gamma )}\leq C\|(\phi
,\psi )\|_{\mathbb{X}}^{1/6}\|(\phi ,\psi
)\|_{\mathbb{B}}^{5/6},  \label{e5.24}
\end{equation}
for some positive constant $C$ independent of the solution. Thus, arguing as
in \cite{e2}, \cite{m1}, \cite{m2}, we can
extend the above result to the case when the norm of $\mathbb{X}$ is
replaced by $H^{2}(\Omega )\times H^{2}(\Gamma )$.
We summarize this final result of this section in the following theorem on
the existence of exponential attractors for problem
\eqref{e1.8}--\eqref{e1.11}.

\begin{theorem} \label{thm13}
There exists a compact set $\mathcal{M}\subset \mathbb{Y}$ which satisfies
the following properties:
\begin{enumerate}
\item  The fractal dimension of $\mathcal{M}$ is finite, that is,
$\dim _{F}(\mathcal{M},H^{2}(\Omega )\times H^{2}(\Gamma ))<\infty$.

\item Semi-invariance: $S(t)(\mathcal{M} )\subset \mathcal{M}$, $t\geq 0$.

\item  There exists a positive constant $\rho $ and a monotonic
function $Q$, such that for every bounded subset $B$ of ${\mathbb{X}}$, we
have:
\begin{equation*}
\lim_{t\to \infty }\mathop{\rm dist}{}_{H^{2}(\Omega )\times
H^{2}(\Gamma )}(S(t)B,\mathcal{M})
\leq Q(\|B\|_{\mathbb{X}})e^{-\rho t},\quad\text{
}t\geq 0.
\end{equation*}
\end{enumerate}
\end{theorem}

\section{Final remarks and open questions}

 In this section, let us consider a conserved version of the
Cahn-Hilliard equation \eqref{e1.1}--\eqref{e1.4}, that is,
\begin{gather}
\partial _{t}\phi =\Delta \mu \quad\text{in }[0,T]\times \Omega ,
\label{e6.1} \\
\mu =-\Delta \phi +f(\phi )\quad\text{in }[0,T]\times
\Omega ,  \label{e6.2}
\end{gather}
and
\begin{gather}
\partial _{t}\psi +b\partial _{n}\mu =0\quad\text{on }[0,T]\times
\Gamma ,  \label{e6.3}\\
-\alpha \Delta _{\Gamma }\psi +\partial _{n}\phi +\beta \psi =\frac{\varpi }{
b}\quad\text{on }[0,T]\times \Gamma ,  \label{e6.4}
\end{gather}
with $\psi =\phi\big|_{\Gamma }$, $\varpi =\mu\big|_{\Gamma }$ and initial
conditions $\phi (0,x)=\phi _{0}(x)$, $\psi (0,x)=\psi _{0}(x)$.
We note that due to the divergence
theorem and equations \eqref{e6.1}, \eqref{e6.3}, we have
\begin{equation*}
\frac{d}{dt}\Big(\int_{\Omega }\phi (t,x)
dx+\int_{\Gamma }\psi (t,x)\frac{dS}{b}\Big)=0,
\end{equation*}
hence the total mass
\begin{equation}
\int_{\Omega }\phi (t,x)dx+\int_{\Gamma }\psi
(t,x)\frac{dS}{b}=\int_{\Omega }\phi _{0}(x)
dx+\int_{\Gamma }\psi _{0}(x)\frac{dS}{b}  \label{e6.5}
\end{equation}
is conserved for all time $t>0$. We also observe that the boundary condition
\eqref{e6.3} is the same as \eqref{e1.3}, if $c=0$. This
case becomes more difficult to treat since the operator defined by
\eqref{e2.4}--\eqref{e2.5} may not be invertible on $\mathcal{H}$.
Nevertheless, we may construct such an invertible operator $\mathcal{N}$, if
we restrict the vectors $(\phi (t),\psi (t))$ to a subspace of
$\mathcal{H}$ such that $(\phi (t),\psi (t))$ satisfy the conservation
property \eqref{e6.5}. Moreover, it is apparent from the results of Section 3,
that the estimates for the solution $(\phi ,\psi )$ satisfying
\eqref{e6.1}--\eqref{e6.4} are uniform with respect to the
parameter $b$. We could then regard the coefficient $b$ in the boundary
conditions \eqref{e6.3} and \eqref{e6.4} as a parameter and
consider the parameter dependent solution of this system, and construct a
nonlinear semigroup
$\mathcal{S}_{b}(t)=S(t,b):\mathbb{Y}(b)\to \mathbb{Y}(b)$, $t>0$,
for each $b\in [b_{0},+\infty )$ with $b_{0}>0$. Here the
space $\mathbb{Y}(b)$ will depend explicitly on the parameter
$b $ (also, see \eqref{e2.11}). When $b\to +\infty $, the
boundary conditions \eqref{e6.3} and \eqref{e6.4} become
formally a homogeneous Neumann condition for the chemical potential $\mu $
and a homogeneous boundary condition of second order for $\phi $,
respectively. Hence, we formally obtain the limiting system of equations:
\begin{gather}
\partial _{t}\phi =\Delta \mu \quad\text{in }[0,T]\times \Omega ,
\label{e6.6} \\
\mu =-\Delta \phi +f(\phi )\quad\text{in }[0,T]\times
\Omega ,  \label{e6.7}
\end{gather}
and
\begin{gather}
\partial _{n}\mu =0\quad\text{on }[0,T]\times \Gamma ,  \label{e6.8}
\\
-\alpha \Delta _{\Gamma }\psi +\partial _{n}\phi +\beta \psi =0\quad\text{on }
[0,T]\times \Gamma .  \label{e6.9}
\end{gather}
Such a system was considered and studied in \cite[Section 7]{r1}. We note
that \eqref{e6.6}, \eqref{e6.7} imply that the mass
\begin{equation}
\int_{\Omega }\phi (t,x)dx=\int_{\Omega }\phi
_{0}(x)dx  \label{e6.10}
\end{equation}
is conserved for all time $t>0$. Notice that \eqref{e6.5} becomes
$(6.10)$ when $b=+\infty $.

The fundamental question that arises from these initial observations is
whether we may be able to construct a family of robust exponential
attractors $\mathcal{M}_{b}$ for the semi-flow
$\mathcal{S}_{b}(t)$ associated with this problem \eqref{e6.1}--\eqref{e6.4},
whenever $b\in [b_{0},+\infty )$. Thus, if the
uniform and decaying estimates in Theorem \ref{thm6} and its supporting lemmas hold,
it is natural to ask whether the sets $\mathcal{M}_{b}$ tend to the limit
set $\mathcal{M}_{\infty }$ as $b\to +\infty $ in the following
sense:
\begin{equation*}
\mathop{\rm dist}{}_{X}(\mathcal{M}_{b};\mathcal{M}_{\infty })\to 0,
\end{equation*}
where $\mathop{\rm dist}_{X}$ denotes the Hausdorff distance in the topology of a
suitable metric space $X$. We will investigate such a problem in a
forthcoming article.

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\end{document}