\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 25, pp. 1--28.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/25\hfil A reaction-diffusion system]
{Global dynamics of a reaction-diffusion system}

\author[Y. You\hfil EJDE-2011/25\hfilneg]
{Yuncheng You}

\address{Yuncheng You \newline
Department of Mathematics and Statistics \\
University of South Florida \\
Tampa, FL 33620, USA}
\email{you@mail.usf.edu}

\thanks{Submitted July 28, 2010. Published February 10, 2011.}
\subjclass[2000]{37L30, 35B40, 35B41, 35K55, 35K57, 80A32, 92B05}
\keywords{Reaction-diffusion system; Brusselator; two-cell model;
\hfill\break\indent
global attractor; absorbing set; asymptotic compactness;
exponential attractor}

\begin{abstract}
 In this work the existence of a global attractor for the semiflow
 of weak solutions of a two-cell Brusselator system is proved.
 The method of grouping estimation is exploited to deal with
 the challenge in proving the absorbing property and the
 asymptotic compactness of this type of coupled reaction-diffusion
 systems with cubic autocatalytic nonlinearity and linear coupling.
 It is proved that the Hausdorff dimension and the fractal dimension
 of the global attractor are finite. Moreover, the existence of an
 exponential attractor for this solution semiflow is shown.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\newcommand{\inpt}[1]{\langle #1 \rangle}


\section{Introduction}

Consider a reaction-diffusion systems consisting of four coupled two-cell Brusselator equations associated with cubic autocatalytic kinetics \cite{GLI07, KS80, aK83, SM82},
\begin{gather}
    \frac{\partial u}{\partial t} = d_1 \Delta u + a - (b + 1)u + u^2 v + D_1 (w - u), \label{equ} \\
    \frac{\partial v}{\partial t} = d_2 \Delta v + bu - u^2 v + D_2 (z - v), \label{eqv} \\
    \frac{\partial w}{\partial t} = d_1 \Delta w + a - (b + 1)w + w^2 z + D_1 (u - w), \label{eqw} \\
    \frac{\partial z}{\partial t} = d_2 \Delta z + bw - w^2 z + D_2 (v - z),  \label{eqz}
\end{gather}
for $t > 0$, on a bounded domain $\Omega \subset \Re^{n}, n \leq 3$,
 that has a locally Lipschitz continuous boundary, with the
homogeneous Dirichlet boundary condition
\begin{equation} \label{dbc}
    u(t, x) = v(t, x) = w (t, x) = z (t, x) = 0, \quad t > 0, \; x \in \partial \Omega,
\end{equation}
and an initial condition
\begin{equation} \label{ic}
        u(0,x) = u_0 (x), \quad v(0, x) = v_0 (x), \quad
 w(0,x) = w_0 (x), \quad z(0, x) = z_0 (x),\quad x \in \Omega,
\end{equation}
where $d_1, d_2, a, b, D_1$, and $D_2$ are positive constants.
In this work, we shall study the asymptotic dynamics of the
solution semiflow generated by this problem.

The Brusselator model is originally a system of two ordinary
differential equations describing kinetics of cubic autocatalytic
chemical or biochemical reactions, proposed by the scientists
in the Brussels school led by the renowned Nobel Prize
laureate (1977), Ilya Prigogine, cf.  \cite{PL68, AN75}.
 Brusselator kinetics describes the following scheme of chemical
reactions
\begin{gather*}
    \text{A}  \longrightarrow \text{U},  \\
    \text{B} + \text{U}  \longrightarrow \text{V} + \text{D}, \\
    2 \text{U} + \text{V}  \longrightarrow 3 \text{U}, \\
     \text{U} \longrightarrow \text{E},
\end{gather*}
where \text{A}, \text{B}, \text{D}, \text{E}, \text{U},
and \text{V} are chemical reactants or products. Let $u(t, x)$
 and $v(t, x)$ be the concentrations of \text{U} and \text{V},
and assume that the concentrations of the input compounds \text{A}
and \text{B} are held constant during the reaction process,
denoted by $a$ and $b$ respectively. Then by the law of mass
action and the Fick's law one obtains a system of two nonlinear
reaction-diffusion equations called (diffusive)
\emph{Brusselator equations},
\begin{gather}
    \frac{\partial u}{\partial t} = d_1 \Delta u + u^{2}v  - (b + 1)u + a,  \label{bru}\\
    \frac{\partial v}{\partial t} = d_2 \Delta v - u^{2}v + bu, \label{brv}
\end{gather}

Several known examples of autocatalysis which can be modelled  by
the Brusselator equations, such as ferrocyanide-iodate-sulphite
reaction, chlorite-iodide-malonic acid reaction, arsenite-iodate
reaction, and some enzyme catalytic reactions, cf. \cite{AO78,
AN75, BD95}.

Numerous studies by numerical simulations or by mathematical
analysis,  especially after the seminal publications \cite{LMOS93,
jP93} in 1993, have shown that the autocatalytic
reaction-diffusion systems such as the Brusselator equations and
the Gray-Scott equations \cite{GS83, GS84} exhibit rich spatial
patterns (including but not restricted to Turing patterns) and
complex bifurcations \cite{AO78, BGMLF00, BD95, DKZ97, PPG01,
PW05, WW03} as well as interesting dynamics \cite{CQ07, ER83,
KCD97, KEW06, RR95, RPP97, Wet96} on 1D or 2D domains.

For Brusselator equations and the other cubic autocatalytic  model
equations of space dimension $n \leq 3$, however, we have not seen
substantial research results in the front of global dynamics until
recently \cite{yY07, yY09a, yY09b, yY10}.

In this paper, we shall prove the existence of a global  attractor
in the product $L^2$ phase space for the solution semiflow of the
coupled two-cell  Brusselator system \eqref{equ}--\eqref{eqz} with
homogeneous Dirichlet boundary conditions \eqref{dbc}.

This study of global dynamics of such a reaction-diffusion system
of two cells or two compartments consisting of four coupled
components is a substantial advance from the one-cell model of
two-component reaction-diffusion systems toward the biological
network dynamics \cite{GLI07, hK02}. Multi-cell or
multi-compartment models generically mean the coupled ODEs or PDEs
with large number of unknowns (interpreted as components in
chemical kinetics or species in ecology), which appear widely in
the literature of systems biology as well as cell biology. Here
understandably "cell" is a generic term that may not be narrowly
or directly interpreted as a biological cell. Coupled cells with
diffusive reaction and mutual mass exchange are often adopted as
model systems for description of processes in living cells and
tissues, or in distributed chemical reactions and transport for
compartmental reactors \cite{TCN01, SM82}.

In this regard, unfortunately, the problems with high
dimensionality  can occur and puzzle the research, when the number
of molecular species in the system turns out to be very large,
which makes the behavior simulation extremely difficult or
computationally too inefficient. Thus theoretical research results
on multi-cell dynamics can give insights to deeper exploration of
various signal transductions and spatio-temporal pattern
formations or chaos.

For most reaction-diffusion systems consisting of two or more
equations arising from the scenarios of autocatalytic chemical
reactions or biochemical activator-inhibitor reactions, such as
the Brusselator equations and the coupled two-cell Brusselator
systems here, the asymptotically dissipative sign condition in
vector version
$$
    \lim_{|s| \to \infty} F(s) \cdot s \leq C,
$$
where $C \geq 0$ is a constant, is inherently not satisfied by
the opposite-signed and coupled nonlinear terms, see \eqref{opF}
later. Besides serious challenge arises in dealing with the
coupling of the two groups of variables $u, v$ and $w, z$. The
novel mathematical feature in this paper is to overcome this
coupling obstacle and make the \emph{a priori} estimates by a
method of \emph{grouping estimation} combined with the other
techniques to show the globally dissipative and attractive
dynamics.

We start with the formulation of an evolutionary equation
associated with the two-cell Brusselator equations. Define the
product Hilbert spaces as follows,
\[
        H = [L^2 (\Omega)]^4, \quad E =  [H_{0}^{1}(\Omega)]^4, \quad
\text{and} \quad \Pi =  [(H_{0}^{1}(\Omega) \cap H^{2}(\Omega))]^4.
\]
The norm and inner-product of $H$ or the component space  $L^2
(\Omega)$ will be denoted by $\|  \cdot \|$ and $\inpt{\cdot ,
\cdot}$, respectively. The norm of $L^p (\Omega)$ will be denoted by
$\|  \cdot  \|_{L^{p}}$ if $p \ne 2$. By the Poincar\'e inequality
and the homogeneous Dirichlet boundary condition \eqref{dbc},
there is a constant $\gamma > 0$ such that
\begin{equation} \label{pcr}
    \| \nabla \varphi \|^2 \geq \gamma \| \varphi \|^2, \quad \text{for }
  \varphi \in H_{0}^{1}(\Omega)  \text{ or } E,
\end{equation}
and we shall take $\| \nabla \varphi \|$ to be the equivalent norm
$\| \varphi \|_E$ of the space $E$ and of the component space
$H_{0}^{1}(\Omega)$. We use $| \cdot |$ to denote an absolute value
or a vector norm in a Euclidean space.

It is easy to check that, by the Lumer-Phillips theorem and the
analytic semigroup generation theorem \cite{SY02}, the linear
operator
\begin{equation} \label{opA}
        A =
        \begin{pmatrix}
            d_1 \Delta     & 0    &0    &0\\
            0 & d_2 \Delta    &0   &0\\
            0 &0  &d_1 \Delta &0\\
            0 &0 &0 &d_2 \Delta
        \end{pmatrix}
        : D(A) (= \Pi) \longrightarrow H
\end{equation}
is the generator of an analytic $C_0$-semigroup on the Hilbert
space $H$, which will be denoted by $\{e^{At}, t \geq 0\}$.
It is known \cite{MK05, SY02, rT88} that $A$ in \eqref{opA} is
extended to be a bounded linear operator from $E$ to $E^*$.
By the fact that $H_{0}^{1}(\Omega) \hookrightarrow L^6(\Omega)$ is a
continuous embedding for $n \leq 3$ and using the generalized
H\"{o}lder inequality,
$$
    \| u^{2}v \| \leq \| u \|_{L^6}^2 \| v \|_{L^6}, \quad
\| w^{2}z \| \leq \| w \|_{L^6}^2 \| z \|_{L^6}, \quad
\text{for} \; u, v, w, z \in L^6 (\Omega),
$$
one can verify that the nonlinear mapping
\begin{equation} \label{opF}
    F(g) =
        \begin{pmatrix}
            a - (b+1)u + u^2 v + D_1 (w - u)  \\
            bu - u^2 v + D_2 (z - v) \\
            a - (b+1)w + w^2 z + D_1 (u - w)  \\
            bw - w^2 z + D_2 (v - z)
        \end{pmatrix}
        : E \longrightarrow H,
\end{equation}
where $g = (u, v, w, z)$, is well defined on $E$ and is
locally Lipschitz continuous. Thus the initial-boundary
value problem \eqref{equ}--\eqref{ic} is formulated into the
following initial value problem,
\begin{gather} \label{eveq}
    \frac{dg}{dt} = A g + F(g), \quad t > 0, \\
        g(0) = g_0 = \operatorname{col} (u_0, v_0, w_0, z_0). \notag
\end{gather}
where $g (t) = \operatorname{col} (u(t, \cdot), v(t, \cdot),
w(t, \cdot), z(t, \cdot))$, which is simply written as \\
$(u(t, \cdot), v(t, \cdot), w(t, \cdot), z(t, \cdot))$.
We shall also simply write $g_0 = (u_0, v_0, w_0, z_0)$.

The local existence of solution to a multi-component
reaction-diffusion system such as \eqref{eveq} with certain
regularity requirement is not a trivial issue. There are two
different approaches to get a solution. One is the mild solution
provided by the "variation-of-constant formula" in terms of the
associated linear semigroup $\{e^{At}\}_{t \geq 0}$ but the the
parabolic theory of mild solution requires that $g_0 \in E$
instead of $g_0 \in H$ assumed here. The other is the weak
solution obtained through the Galerkin approximation (the spectral
approximation) and the Lions-Magenes type of compactness approach,
cf. \cite{CV02, jL69, SY02}.

\begin{definition} \label{D:wksn} \rm
A function $g(t, x), (t, x) \in [0, \tau] \times \Omega$, is called a
weak solution to the initial value problem of the parabolic
evolutionary equation \eqref{eveq}, if the following two
 conditions are satisfied:
\begin{itemize}
\item[(i)] $\frac{d}{dt} (g, \zeta) = (Ag, \zeta) + (F(g),
\zeta)$ is satisfied for a.e. $t \in [0, \tau]$ and for any $\zeta
\in E$;

\item[(ii)] $g(t, \cdot) \in L^2 (0, \tau; E) \cap C_w ([0,
\tau]; H)$ such that $g(0) = g_0$.

\end{itemize}
Here $(\cdot , \cdot)$ stands for the dual product of $E^*$
(the dual space of $E$) and $E$, $C_w$ stands for the weakly
continuous functions valued in $H$, and \eqref{eveq} is
satisfied in the space $E^*$.
\end{definition}

\begin{proposition} \label{P:locwk}
For any given initial data $g_0 \in H$, there exists a unique,
local weak solution $g(t) = (u(t), v(t), w(t), z(t)), t \in [0,\tau]$
for some $\tau > 0$, of the Brusselator evolutionary
equation \eqref{eveq}, which becomes a strong solution on
$(0, \tau]$, namely, it satisfies
\begin{equation} \label{soln}
    g \in C([0, \tau]; H) \cap C^1 ((0, \tau); H) \cap L^2 (0, \tau; E)
\end{equation}
and \eqref{eveq} is satisfied in the space $H$ for $t \in (0, \tau]$.
\end{proposition}

The proof of Proposition \ref{P:locwk} is made by conducting
\emph{a priori} estimates on the Galerkin approximate solutions
of the initial value problem \eqref{eveq}
(these estimates are similar to what we shall present in Section 2)
and by the weak/weak$^*$ convergence argument, as well as the use
of the properties of the function space, cf. \cite{CV02, jL69},
$$
    \Phi (0, \tau) = \{\varphi (\cdot ):
 \varphi \in L^2 (0, \tau; E), \, \text{(distributional)} \,
\partial_t \varphi \in L^2 (0, \tau; E^*)\},
$$
with the norm
$$
    \| \varphi \|_\Phi = \|\varphi \|_{L^2 (0, \tau; E)} + \| \partial_t \varphi \|_{L^2 (0, \tau; E^*)}.
$$
The detail is omitted here.

We refer to \cite{jH88, SY02, rT88} and many references therein
for the concepts and basic facts in the theory of infinite
dimensional dynamical systems, including few given below for clarity.

\begin{definition} \label{D:abs} \rm
Let $\{S(t)\}_{t \geq 0}$ be a semiflow on a Banach space $X$.
A bounded subset $B_0$ of $X$ is called an \emph{absorbing set}
in $X$ if, for any bounded subset $B \subset X$, there is some
finite time $t_0 \geq 0$ depending on $B$ such that
$S(t)B \subset B_0$ for all $t > t_0$.
\end{definition}

\begin{definition} \label{D:asp} \rm
A semiflow $\{S(t)\}_{t \geq 0}$ on a Banach space $X$ is
called \emph{asymptotically compact} if for any bounded
sequences $\{x_n \}$ in $X$ and $\{t_n \} \subset (0, \infty)$
with $t_n \to \infty$, there exist subsequences $\{x_{n_k}\}$
of $\{x_n \}$ and $\{t_{n_k}\}$ of $\{t_n\}$, such that
$\lim_{k \to \infty} S(t_{n_k})x_{n_k}$ exists in $X$.
\end{definition}

\begin{definition} \label{D:atr} \rm
Let $\{S(t)\}_{t \geq 0}$ be a semiflow on a Banach space $X$.
A subset $\mathscr{A}$ of $X$ is called a \emph{global attractor}
for this semiflow, if the following conditions are satisfied:
\begin{itemize}
\item[(i)] $\mathscr{A}$ is a nonempty, compact, and invariant set in the
sense that
$$
    S(t)\mathscr{A} = \mathscr{A} \quad \text{for any }   t \geq 0.
$$

\item[(ii)] $\mathscr{A}$ attracts any bounded set $B$ of $X$ in terms of
the Hausdorff distance, i.e.
$$
    \operatorname{dist} (S(t)B, \mathscr{A})
= \sup_{x \in B} \inf_{y \in \mathscr{A}} \| S(t) x - y\|_{X} \to 0,
\quad \text{as }  t \to \infty.
$$
\end{itemize}
\end{definition}

Now we state the main result of this paper. We emphasize that
this result is established unconditionally, neither assuming initial
data or solutions are nonnegative, nor imposing any restriction
on any positive parameters involved in the equations
\eqref{equ}--\eqref{eqz}.

\begin{theorem}[Main Theorem] \label{Mthm}
For any positive parameters $d_1, d_2, a, b, D_1,D_2$,
there exists a global attractor $\mathscr{A}$ in the phase space $H$
for the solution semiflow $\{S(t)\}_{t\geq0}$ generated by the Brusselator
evolutionary equation \eqref{eveq}.
\end{theorem}

The following proposition states concisely the basic result on the
existence of a global attractor for a semiflow,
 cf. \cite{jH88, SY02, rT88}.

\begin{proposition} \label{P:kpac}
Let $\{S(t)\}_{t\geq0}$ be a semiflow on a Banach space $X$. If the following
conditions are satisfied:
\begin{itemize}
\item[(i)] $\{S(t)\}_{t\geq0}$ has a bounded absorbing set $B_0$ in $X$,
and

\item[(ii)] $\{S(t)\}_{t\geq0}$ is asymptotically compact, \\
then there exists a global attractor $\mathscr{A}$ in $X$ for this semiflow, which is given by
$$
    \mathscr{A} = \omega (B_0) \overset{\rm def}{=}
 \cap_{\tau \geq 0} \operatorname{Cl}_{X}  \cup_{t \geq \tau} (S(t)B_0).
$$
\end{itemize}
\end{proposition}

In Section 2 we shall prove the global existence of the weak
solutions of the Brusselator evolutionary equation \eqref{eveq}
and the absorbing property of this solution semiflow. In Section 3
we shall prove the asymptotic compactness of this solutions
semiflow. In Section 4 we show the existence of a global attractor
in space $H$ for this Busselator semiflow and its properties as
being the $(H, E)$ global attractor and the $\mathbb{L}^\infty$
regularity. We also prove that the global attractor has a finite
Hausdorff dimension and a finite fractal dimension. In Section 5,
the existence of an exponential attractor for this semiflow is
shown.

As a remark, with some adjustment in proof, these results are
also valid for the homogeneous Neumann boundary condition.
Furthermore, corresponding results can be shown for the coupled
two-cell Gray-Scott equations, Selkov equations, and Schnackenberg
equations.

\section{Global solutions and absorbing property}

In this article, we shall write $u(t, x)$, $v(t, x)$, $w(t, x)$, and
$z(t,x)$ simply as $u(t)$, $v(t)$, $w(t)$, and $z(t)$, or even as
$u$, $v$, $w$, and $z$, and similarly for other functions of $(t,x)$.

\begin{lemma} \label{L:glsn}
For any initial data $g_0 =(u_0, v_0, w_0, z_0) \in H$, there
exists a unique, global weak solution $g(t) = (u(t), v(t), w(t),
z(t)), \, t \in [0, \infty)$, of the Brusselator evolutionary
equation \eqref{eveq} and it becomes a strong solution on the time
interval $(0, \infty)$.
\end{lemma}

\begin{proof}
By Proposition \ref{P:locwk}, the local weak solution $g(t) =
(u(t), v(t), w(t), z(t))$ exists uniquely on $[0, T_{\rm max})$,
the maximal interval of existence. Taking the inner products
$\inpt{\eqref{eqv}, v(t)}$ and $\inpt{\eqref{eqz}, z(t)}$ and
summing up, we obtain
\begin{equation} \label{vziq}
    \begin{split}
    \frac{1}{2} &\Big(\frac{d}{dt} \| v \|^2 + \frac{d}{dt}
\| z \|^2 \Big) + d_2 \left(\| \nabla v \|^2 + \| \nabla z \|^2\right) \\
    & = \int_{\Omega} \left(-u^2 v^2 + buv - w^2 z^2
+ bwz - D_2 [v^2 - 2vz + z^2] \right) \, dx   \\
    & = \int_{\Omega} - \Big[\big(uv - \frac{b}{2}\big)^2
+ \big(wz - \frac{b}{2}\big)^2 + D_2 (v - z)^2 \Big]\, dx +
\frac{1}{2} b^2 |\Omega |\\
&\leq  \frac{1}{2} b^2 |\Omega |.
  \end{split}
\end{equation}
It follows that
\[
    \frac{d}{dt} \left(\| v \|^2 + \| z \|^2 \right)
+ 2\gamma d_2 \left(\| v \|^2 + \| z \|^2 \right) \leq b^2 |\Omega |,
\]
which yields
\begin{equation} \label{vz}
    \| v(t) \|^2 + \| z (t)\|^2 \leq e^{- 2\gamma d_2 t}
\left(\| v_0 \|^2 + \| z_0 \|^2 \right) + \frac{b^2 |\Omega |}{2\gamma
d_2}, \quad \text{for } t \in [0, T_{\rm max}).
\end{equation}

Let $y(t, x) = u(t, x) + v(t, x) + w(t,x) + z(t,x)$.
In order to treat the $u$-component and the $w$-component,
first we add up \eqref{equ}, \eqref{eqv}, \eqref{eqw} and \eqref{eqz}
altogether to get the following equation satisfied by
$y(t) = y(t, x)$,
\begin{equation} \label{eqy}
    \frac{\partial y}{\partial t} = d_1 \Delta y - y + \left[(d_2 - d_1)\Delta (v + z) + (v + z) + 2a\right].
\end{equation}
Taking the inner-product $\inpt{\eqref{eqy},y(t)}$ we obtain
\begin{align*}
    &\frac{1}{2} \frac{d}{dt} \| y \|^2
+ d_1 \| \nabla y \|^2 + \| y \|^2\\
    &= \int_{\Omega} \, \left[(d_2 - d_1)\Delta (v+z) + (v+ z)
+ 2a\right]y \, dx \\
    & \leq | d_1 - d_2 | \| \nabla (v + z) \| \| \nabla y \|
+ \| v + z \| \| y \| + 2a | \Omega |^{1/2} \| y \|\\
    & \leq \frac{d_1}{2} \| \nabla y \|^2 + \frac{|d_1
- d_2 |^2}{2d_1} \| \nabla (v + z) \|^2
+ \frac{1}{2} \| y \|^2 + \| v + z \|^2 +  4a^2 | \Omega |,
\end{align*}
so that
\begin{equation} \label{nyiq}
    \frac{d}{dt} \| y \|^2 + d_1 \| \nabla y \|^2 + \| y \|^2
\leq  \frac{| d_1 - d_2 |^2}{d_1} \| \nabla (v+z) \|^2
+ 4\left(\| v\|^2 + \|z \|^2\right) + 8a^2 | \Omega |.
\end{equation}
By substituting \eqref{vz} for $\|v \|^2 + \| z \|^2$ in the above
inequality, we obtain
\begin{equation} \label{yiq}
    \frac{d}{dt} \| y \|^2 + d_1 \| \nabla y \|^2 + \| y \|^2
\leq \frac{| d_1 - d_2 |^2}{d_1} \| \nabla (v+z) \|^2
+ C_1(v_0, z_0, t),
\end{equation}
where
\[
    C_1(v_0, z_0, t) = 4 e^{- 2\gamma d_2 t} \left(\| v_0 \|^2
+ \| z_0 \|^2\right) + \big(\frac{4b^2}{\gamma d_2} + 8a^2 \big) |\Omega |.
\]
Integrate the inequality \eqref{yiq}. Then the weak solution $y(t)$
 of \eqref{eqy} satisfies the  estimate
\begin{equation} \label{yy}
    \begin{split}
    \| y(t) \|^2 &\leq \| u_0 + v_0 + w_0 + z_0\|^2
+ \frac{| d_1 - d_2 |^2}{d_1} \int_{0}^{t} \| \nabla (v(s)+z(s)) \|^2
\, ds  \\
    &\quad + \frac{2}{\gamma d_2} \left(\| v_0 \|^2 + \| v_0 \|^2\right)
+ \big(\frac{4b^2}{\gamma d_2 } + 8a^2 \big) |\Omega |  t, \quad
t \in [0, T_{\rm max}).
   \end{split}
\end{equation}
From \eqref{vziq} we also have
\begin{align*}
    d_2 \int_0^{t} \| \nabla (v(s) + z(s)) \|^2 \, ds
&\leq 2d_2 \int_0^{t} \Big(\|\nabla v(s) \|^2 + \|\nabla z(s)
\|^2\Big)\, ds \\
&\leq \left(\| v_0 \|^2 + \|z_0 \|^2\right) + b^2 | \Omega | t.
\end{align*}
Substitute this into \eqref{yy} to obtain
\begin{equation} \label{yy1}
    \begin{split}
    \| y(t) \|^2 & \leq \| u_0 + v_0 + w_0 + z_0\|^2
+ \Big(\frac{| d_1 - d_2 |^2}{d_1 \, d_2}
+ \frac{2}{\gamma d_2} \Big) \left( \| v_0 \|^2 + \|z_0 \|^2\right)\\
    &\quad + \Big[ \Big(\frac{| d_1 - d_2 |^2}{d_1 \, d_2}
+  \frac{4}{\gamma d_2 } \Big) b^2+ 8a^2 \Big] |\Omega | \,t, \quad
t \in [0, T_{\rm max}).
  \end{split}
\end{equation}
Let $p(t) = u(t) + w(t)$. Then by \eqref{vz} and \eqref{yy1} we
have shown that
\begin{equation} \label{pp}
    \begin{split}
    &\| p(t) \|^2 = \|u(t) + w(t) \|^2 = \|y(t) - (v(t) + z(t))\|^2 \\
    & \leq 2\Big(\|u_0 +v_0+w_0+z_0\|^2
+ \Big(1 + \frac{| d_1 - d_2 |^2}{d_1 \, d_2}
+ \frac{2}{\gamma d_2} \Big)(\|v_0\|^2+\|z_0\|^2) \Big) + C_2 \, t,
       \end{split}
\end{equation}
for $t \in [0, T_{\rm max})$, where $C_2$ is a constant
independent of the initial data $g_0$.

On the other hand, let $\psi (t,x) = u(t, x) + v(t, x)
- w(t,x) - z(t,x)$, which satisfies the equation
\begin{equation} \label{eqps}
    \frac{\partial \psi}{\partial t} = d_1 \Delta \psi - (1 + 2D_1) \psi
+ \left[(d_2 - d_1)\Delta (v - z) + (1 + 2(D_1 - D_2))(v - z)\right].
\end{equation}
Taking the inner-product $\inpt{\eqref{eqps},\psi(t)}$ we obtain
\begin{align*}
    \frac{1}{2} & \frac{d}{dt} \|\psi \|^2 + d_1 \|\nabla \psi \|^2
+ \|\psi \|^2 \leq \frac{1}{2} \frac{d}{dt} \|\psi \|^2
 + d_1 \|\nabla \psi \|^2 + (1 + 2D_1) \|\psi \|^2 \\[5pt]
    & \leq (d_1 - d_2) \|\nabla (v - z)\| \|\nabla \psi \|
+ |1 + 2(D_1 - D_2)| \|v - z\| \|\psi \| \\
    & \leq \frac{d_1}{2} \|\nabla \psi\|^2
+ \frac{|d_1 - d_2|^2}{2d_1} \|\nabla (v-z)\|^2
+ \frac{1}{2} \|\psi\|^2 + \frac{1}{2} |1 + 2(D_1 - D_2)|^2 \|v - z\|^2,
\end{align*}
so that
\begin{equation} \label{psiq}
    \frac{d}{dt} \|\psi \|^2 + d_1 \|\nabla \psi \|^2 + \|\psi \|^2 \leq \frac{|d_1 - d_2|^2}{d_1} \|\nabla (v-z)\|^2 + C_3 (v_0, z_0, t),
\end{equation}
where
\[
    C_3(v_0, z_0, t) = 2 |1 + 2(D_1 - D_2)|^2
\Big( e^{- 2\gamma d_2 t} \big(\| v_0 \|^2 + \| z_0 \|^2\big)
+ \frac{b^2}{2\gamma d_2} |\Omega|\Big).
\]
Integration of \eqref{psiq} yields
\begin{equation} \label{ppsi}
    \begin{split}
    \|\psi\|^2 & \leq \|u_0+v_0 - w_0 - z_0\|^2
+ \frac{|d_1 - d_2|^2}{d_1} \int_0^t \|\nabla (v(s) - z(s) )\|^2\, ds \\
   &\quad  + |1 + 2(D_1 - D_2)|^2 \Big(\frac{1}{\gamma d_2} (\|v_0\|^2
+ \|z_0\|^2) + \frac{b^2 |\Omega|}{\gamma d_2}\, t \Big), \quad t \in
[0, T_{\rm max}).
    \end{split}
\end{equation}
Note that
\begin{align*}
    d_2 \int_0^{t} \| \nabla (v(s) - z(s)) \|^2 \, ds
& \leq 2d_2 \int_0^{t} \left(\|\nabla v(s) \|^2 + \|\nabla z(s)
\|^2\right)\, ds \\
&\leq \left(\| v_0 \|^2 + \|z_0 \|^2\right) + b^2 | \Omega | t.
\end{align*}
From \eqref{ppsi} it follows that
\begin{equation} \label{ppsi1}
    \begin{split}
    \|\psi\|^2 &\leq \| u_0 + v_0 - w_0 - z_0\|^2
+ \frac{| d_1 - d_2 |^2}{d_1 \, d_2}  \left( \| v_0 \|^2
+ \|z_0 \|^2 + b^2 |\Omega|\, t \right) \\
        &\quad +  |1 + 2(D_1 - D_2)|^2 \Big(\frac{1}{\gamma d_2}
(\|v_0\|^2 + \|z_0\|^2) + \frac{b^2 |\Omega|}{\gamma d_2}\, t \Big), \quad
t \in [0, T_{\rm max}).
    \end{split}
\end{equation}
Let $q(t) = u(t) - w(t)$. Then by \eqref{vz} and \eqref{ppsi1} we find that
\begin{equation} \label{qq}
    \begin{split}
     \| q(t) \|^2
&= \|u(t) - w(t) \|^2 = \|\psi (t) - (v(t) - z(t))\|^2 \leq 2\|u_0
+v_0-w_0-z_0\|^2 \\
&\quad + 2\Big(1 + \frac{|d_1 - d_2 |^2}{d_1 d_2} + \frac{|1 +
2(D_1 - D_2)|^2}{\gamma d_2}\Big) (\|v_0\|^2+\|z_0\|^2) + C_4 t,
    \end{split}
\end{equation}
for $ t \in [0, T_{\rm max})$, where $C_4$ is a constant
independent of the initial data $g_0$.

Finally combining \eqref{pp} and \eqref{qq} we can conclude that
for each initial data $g_0 \in H$, the components $u(t) = (1/2)
(p(t) + q(t))$ and $w(t) = (1/2)(p(t) - q(t))$  are bounded if
$T_{\rm max}$ of the maximal interval of existence of the solution
is finite. Together with \eqref{vz}, it shows that, for each $g_0
\in H$, the weak solution $g(t) = (u(t), v(t), w(t), z(t))$ of the
Brusselator evolutionary equation \eqref{eveq} will never blow up
in $H$ at any finite time and it exists globally.
\end{proof}

By the global existence and uniqueness of the weak solutions and
their continuous dependence on initial data shown in Proposition
\ref{P:locwk} and Lemma \ref{L:glsn}, the family of all the global
weak solutions $\{g(t; g_0):\, t \geq 0, g_0 \in H \}$ defines a
semiflow on $H$,
$$
    S(t): g_0 \mapsto g(t; g_0),  \quad g_0 \in H, \; t \geq 0,
$$
which is called the two-cell Brusselator semiflow, or simply  the
\emph{Brusselator semiflow}, generated by the Brusselator
evolutionary equation \eqref{eveq}.

\begin{lemma} \label{L:absb}
There exists a constant $K_1 > 0$, such that the set
\begin{equation} \label{bk}
        B_0 = \left\{ \| g \| \in H : \| g \|^2 \leq K_1 \right\}
\end{equation}
is an absorbing set in $H$ for the Brusselator semiflow $\{S(t)\}_{ t \geq 0}$.
\end{lemma}

\begin{proof}
For this two-cell Brusselator semiflow, from \eqref{vz} we obtain
\begin{equation} \label{vzsup}
        \limsup_{t \to \infty} \, (\| v(t) \|^2 + \|z(t)\|^2) < R_0
= \frac{b^2 |\Omega |}{\gamma d_2}
\end{equation}
and that for any given bounded set $B \subset H$ and $g_0 \in B$ there is a finite time $t_1 (B) \geq 0$ such that
\begin{equation} \label{t1b}
    \| v(t; g_0)\|^2 + \|z(t; g_0)\|^2 < R_0, \quad
\text{for any } t > t_1 (B).
\end{equation}
Moreover, for any $t \geq 0$, \eqref{vziq} also implies that
\begin{equation} \label{vztt}
    \begin{split}
 &\int_{t}^{t+1} (\| \nabla v(s) \|^2 + \|\nabla z(s)\|^2) \, ds\\
&\leq \frac{1}{d_2} (\| v(t) \|^2 + \|z(t)\|^2 + b^2 |\Omega |) \\
        &\leq \frac{1}{d_2} \Big( e^{- 2\gamma d_2 t} (\| v_0 \|^2
+ \|z_0\|^2) + \frac{b^2 |\Omega |}{2\gamma d_2}\Big)
        + \frac{b^2 |\Omega |}{d_2}.
    \end{split}
\end{equation}
which is for later use.

From \eqref{yiq} we can deduce that
\[
    \frac{d}{dt} \left( e^t \| y(t) \|^2 \right)
\leq \frac{| d_1 - d_2 |^2}{d_1} \, e^t \| \nabla (v(t)
+ z(t)) \|^2 + e^t C_1(v_0, z_0, t).
\]
Integrate this differential inequality to obtain
\begin{equation} \label{ey}
    \begin{split}
    \| y(t) \|^2 &\leq e^{-t} \| u_0 + v_0 +w_0+z_0 \|^2 \\
    &\quad+ \frac{| d_1 - d_2 |^2}{d_1} \, \int_{0}^{t}
e^{- (t -\tau)} \| \nabla (v(\tau) + z(\tau))\|^2 \, d\tau + C_5 (v_0, z_0, t),
        \end{split}
\end{equation}
where
\begin{align*}
    C_5 (v_0, z_0, t)
&= e^{-t}  \int_{0}^{t}  4e^{(1 - 2\gamma d_2)\tau}\, d\tau \,
(\| v_0 \|^2 + \|z_0\|^2) + \big( \frac{4b^2}{\gamma d_2} + 8a^2 \big) |\Omega |  \\
    &\leq 4\alpha (t) (\| v_0 \|^2 + \|z_0\|^2)
+ \big( \frac{4b^2}{\gamma d_2} + 8a^2 \big) |\Omega |,
\end{align*}
in which
\begin{equation} \label{apht}
    \alpha (t) = e^{-t}  \int_{0}^{t} e^{(1 - 2\gamma d_2)\tau}\, d\tau =
    \begin{cases}
        \frac{1}{| 1 - 2\gamma d_2 |} | e^{- 2\gamma d_2 t} - e^{-t} |,  & \text{if  $1 - 2\gamma d_2 \ne 0$;} \\
        t e^{-t} \leq 2e^{-1} e^{-t/2}, & \text{if $ 1 - 2\gamma d_2 = 0$.}
    \end{cases}
\end{equation}
On the other hand, multiplying \eqref{vziq} by $e^t$ and then
integrating each term of the resulting inequality, we obtain
$$
    \frac{1}{2} \int_{0}^{t} \, e^{\tau} \frac{d}{d\tau}
\left(\| v(\tau) \|^2 + \|z(\tau)\|^2 \right) \, d\tau
+ d_2 \int_{0}^{t} \, e^{\tau} (\| \nabla v(\tau) \|^2
+ \|z(\tau)\|^2) \, d\tau \leq \frac{1}{2} b^2 | \Omega | e^t,
$$
so that, by integration by parts and using \eqref{vz}, we obtain
\begin{equation} \label{evz}
    \begin{split}
&d_2 \int_{0}^{t} \, e^{\tau} (\| \nabla v(\tau) \|^2
+ \| \nabla z(\tau) \|^2 )\, d\tau  \\
&\leq \frac{1}{2} b^2 | \Omega | e^t - \frac{1}{2} \int_{0}^{t}
\, e^{\tau} \frac{d}{d\tau} \left(\| v(\tau) \|^2
+ \| \nabla z(\tau) \|^2 \right) \, d\tau   \\
    & = \frac{1}{2} b^2 | \Omega | e^t - \frac{1}{2}
\Big[ e^t (\| v(t) \|^2 + \| z(t) \|^2 ) - (\| v_0 \|^2
+ \|z_0\|^2) \\
&\quad - \int_{0}^{t} \, e^{\tau} (\| v(\tau) \|^2
+ \|z(\tau)\|^2) \, d\tau \Big]  \\
    & \leq b^2 | \Omega | e^t + (\| v_0 \|^2 + \|z_0\|^2)
+ \int_{0}^{t} \, e^{(1 - 2\gamma d_2)\tau} (\| v_0 \|^2
+ \|z_0\|^2)\, d\tau + \frac{b^2 |\Omega |}{2\gamma d_2} e^t   \\
    & \leq \big(1 + \frac{1}{2\gamma d_2} \big) b^2 |\Omega | e^t
+ \left( 1 + \alpha (t) e^t \right) (\| v_0 \|^2 + \|z_0\|^2),
\quad \text{for} \; t \geq 0.
        \end{split}
\end{equation}
Substituting \eqref{evz} into \eqref{ey}, we obtain that for $t
\geq 0$,
\begin{equation} \label{yyf}
    \begin{split}
   &\| y(t) \|^2 \\
& \leq e^{-t}\| u_0 + v_0 +w_0+z_0\|^2 + C_5 (v_0, z_0, t)   \\[5pt]
     &\quad+ \frac{2| d_1 - d_2 |^2}{d_1\, d_2} e^{-t}
\Big[ \big(1 + \frac{1}{2\gamma d_2} \big) b^2 |\Omega | e^t
+ \big( 1 + e^t \alpha (t) \big) (\| v_0 \|^2+ \|z_0\|^2)\Big]   \\
     & \leq e^{-t}\| u_0 + v_0 +w_0 +z_0\|^2
+  4\alpha (t) (\| v_0 \|^2 + \|z_0\|^2)
+ \big( \frac{4b^2}{\gamma d_2} + 8a^2 \big) |\Omega |  \\
     &\quad + \frac{2| d_1 - d_2 |^2}{d_1\, d_2} e^{-t}
\Big[ \big(1 + \frac{1}{2\gamma d_2} \big) b^2 |\Omega | e^t + \left( 1
+ e^t \alpha (t) \right) (\| v_0 \|^2 + \|z_0\|^2) \Big].
        \end{split}
\end{equation}
Note that \eqref{apht} shows $\alpha (t) \to 0$, as $t \to 0$.
From \eqref{yyf} we find that
\begin{equation} \label{ysup}
    \limsup_{t \to \infty} \|y(t) \|^2 < R_1
 = 1 + \big( \frac{4b^2}{\gamma d_2} + 8a^2 \big) |\Omega |
+  \frac{2| d_1 - d_2 |^2}{d_1\, d_2}  \big(1 + \frac{1}{2\gamma d_2} \big) b^2 |\Omega |.
\end{equation}
The combination of \eqref{vzsup} and \eqref{ysup} gives us
\begin{equation} \label{psup}
    \limsup_{t \to \infty} \| u(t) + w(t) \|^2 =  \limsup_{t \to \infty} \| y(t) - (v(t) + z(t))\|^2 < 4R_0 + 2R_1.
\end{equation}

Similarly, from the inequality \eqref{psiq} satisfied by $\psi (t)
= u(t) + v(t) - w(t) -z(t)$, we obtain
\[
    \frac{d}{dt} \left( e^t \| \psi(t) \|^2 \right)
\leq \frac{| d_1 - d_2 |^2}{d_1} \, e^t \| \nabla (v(t) - z(t)) \|^2
 + e^t C_3(v_0, z_0, t).
\]
Integrate this differential inequality to obtain
\begin{equation} \label{epsi}
    \begin{split}
    \| \psi(t) \|^2 &\leq e^{-t} \| u_0 + v_0 - w_0 - z_0 \|^2 \\
    &\quad + \frac{| d_1 - d_2 |^2}{d_1} \,
 \int_{0}^{t} e^{- (t -\tau)} \| \nabla (v(\tau)
- z(\tau))\|^2 \, d\tau + C_6 (v_0, z_0, t),
        \end{split}
\end{equation}
where
\begin{align*}
    C_6 (v_0, z_0, t)
&= 2 |1 + 2(D_1 - D_2)|^2 \Big(e^{-t}  \int_{0}^{t} e^{(1 - 2\gamma
d_2)\tau}\, d\tau \, (\| v_0 \|^2 + \|z_0\|^2)
+  \frac{b^2}{\gamma d_2} |\Omega| \Big)   \\
 &\leq 2 |1 + 2(D_1 - D_2)|^2 \Big(\alpha (t) (\| v_0 \|^2
+ \|z_0\|^2) + \frac{b^2}{\gamma d_2} |\Omega |\Big).
\end{align*}
Using \eqref{evz} to treat the integral term in \eqref{epsi},
we obtain that
\begin{equation} \label{ppsif}
    \begin{split}
&\| \psi(t) \|^2\\
& \leq e^{-t}\| u_0 + v_0 - w_0 - z_0\|^2
+ C_6 (v_0, z_0, t)   \\
     &\quad + \frac{2| d_1 - d_2 |^2}{d_1\, d_2} e^{-t}
 \int_0^t e^{\tau} (\| \nabla v(\tau) \|^2+ \|\nabla z(\tau)\|^2) \,
 d\tau  \\
& \leq e^{-t}\| u_0 + v_0 - w_0 - z_0\|^2 +  2 |1
+ 2(D_1 - D_2)|^2 \\
&\quad \times \Big(\alpha (t) (\| v_0 \|^2 + \|z_0\|^2)
 + \frac{b^2}{\gamma d_2} |\Omega |\Big) \\
&\quad + \frac{2| d_1 - d_2 |^2}{d_1\, d_2} e^{-t}
\Big[ \big(1 + \frac{1}{2\gamma d_2} \big) b^2 |\Omega | e^t + \left( 1
+ e^t \alpha (t) \right) (\| v_0 \|^2 + \|z_0\|^2) \Big],
\end{split}
\end{equation}
for $ t \geq 0$.
Therefore, since $\alpha (t) \to 0$ as $t \to 0$, from
\eqref{ppsif} we obtain
\begin{equation} \label{pssup}
    \limsup_{t \to \infty} \| \psi (t) \|^2 < R_2
= 1 + 2b^2 |\Omega| \Big[\frac{|1 + 2(D_1 - D_2)|^2}{\gamma d_2} +
\frac{| d_1 - d_2 |^2}{d_1\, d_2}  \big(1 + \frac{1}{2\gamma
d_2}\big) \Big].
\end{equation}
The combination of \eqref{vzsup} and \eqref{pssup} gives us
\begin{equation} \label{qsup}
    \limsup_{t \to \infty} \| u(t) - w(t) \|^2
=  \limsup_{t \to \infty} \| \psi (t) - (v(t) - z(t))\|^2 < 4R_0 + 2R_2.
\end{equation}

Finally, putting together \eqref{psup} and \eqref{qsup}, we assert
that
\begin{equation} \label{uwsup}
    \limsup_{t \to \infty} (\|u(t)\|^2 + \| w(t) \|^2 ) < 8R_0 + 2(R_1 + R_2).
\end{equation}
Moreover, from \eqref{vz}, \eqref{yyf} and \eqref{ppsif} we see that for any given bounded set $B \subset H$ and $g_0 \in B$ there is a finite time $t_2 (B) \geq 0$ such that
\begin{equation} \label{T2B}
    \| u(t; g_0)\|^2 + \|w(t; g_0)\|^2 < 8R_0 + 2(R_1 + R_2), \quad \text{for any } t > t_2 (B).
\end{equation}
Then assembling \eqref{vzsup} and \eqref{uwsup}, we end up with
\[
    \limsup_{t \to \infty} \|g(t)\|^2
= \limsup_{t \to \infty} (\|u(t)\|^2 + \|v(t)\|^2 + \| w(t) \|^2
+ \|z(t)\|^2 ) < 9R_0 + 2(R_1 + R_2).
\]
Moreover, \eqref{t1b} and \eqref{T2B} show that for any given bounded
set $B \subset H$ and $g_0 \in B$ the solution $g(t; g_0)$ satisfies
\[
    \|g(t; g_0)\|^2 < 9R_0 + 2(R_1 + R_2), \quad \text{for any } t > \max \{t_1 (B), t_2 (B)\}.
\]
Thus this lemma is proved with $K_1 =  9R_0 + 2(R_1 + R_2)$
in \eqref{bk}. And $K_1$ is a universal positive constant
independent of initial data.
\end{proof}

Next we show the absorbing properties of the $(v, z)$ components
of this Brusselator semiflow in the product Banach spaces
$[L^{2p} (\Omega)]^2$, for any integer $1 \leq p \leq 3$.

\begin{lemma} \label{L:absbp}
    For any given integer $1 \leq p \leq 3$, there exists a positive
constant $K_p$ such that the absorbing inequality
\begin{equation} \label{lsupp}
    \limsup_{t \to \infty} \, \|(v(t), z(t))\|_{L^{2p}}^{2p} < K_p
\end{equation}
is satisfied by the $(v, z)$ components of the Brusselator semiflow
$\{S(t)\}_{t \geq 0}$ for any initial data $g_0 \in H$.
\end{lemma}

\begin{proof}
The case $p = 1$ has been shown in Lemma \ref{L:absb}.
According to the solution property \eqref{soln} satisfied by all
the global weak solutions on $[0, \infty)$, we know that for any
given initial status $g_0 \in H$ there exists a time $t_0 \in (0, 1)$
such that
\begin{equation} \label{tog}
    S(t_0) g_0  \in E = [H_0^1 (\Omega)]^6 \hookrightarrow \mathbb{L}^6 (\Omega) \hookrightarrow \mathbb{L}^4 (\Omega).
\end{equation}
Then the weak solution $g(t) = S(t)g_0$ becomes a strong solution
on $[t_0, \infty)$ and satisfies
\begin{equation} \label{esf}
    S(\cdot) g_0 \in C([t_0, \infty); E) \cap L^2 (t_0, \infty; \Pi)
\subset C([t_0, \infty); \mathbb{L}^6 (\Omega)) \subset C([t_0, \infty);
 \mathbb{L}^4 (\Omega)),
\end{equation}
for $n \leq 3$. Based on this observation, without loss of generality,
we can simply \emph{assume} that $g_0 \in \mathbb{L}^6 (\Omega)$ for
the purpose of studying the long-time dynamics. Thus parabolic
regularity \eqref{esf} of strong solutions implies the
$S(t)g_0 \in E \subset \mathbb{L}^6 (\Omega), t \geq 0$.
Then by the bootstrap argument, again without loss of generality,
one can \emph{assume} that $g_0 \in \Pi \subset \mathbb{L}^8 (\Omega)$
so that $S(t)g_0 \in \Pi \subset \mathbb{L}^8 (\Omega), t \geq 0$.

Take the $L^2$ inner-product $\inpt{\eqref{eqv},v^5}$ and
$\inpt{\eqref{eqz},z^5}$ and sum up to obtain
\begin{equation} \label{vz6}
    \begin{split}
   & \frac{1}{6} \frac{d}{dt} \left(\|v(t)\|_{L^6}^6 + \|z(t)\|_{L^6}^6\right) + 5d_2 \left(\|v(t)^2 \nabla v(t)\|^2 + \|z(t)^2 \nabla z(t)\|^2\right) \\
    &= \int_{\Omega} \left(bu(t,x)v^5 (t,x) - u^2(t,x)v^6(t,x) + bw(t,x)z^5(t,x) - w^2(t, x) z^6(t,x)\right) dx\\
    & + D_2 \int_{\Omega} \left[(z(t,x) - v(t,x))v^5(t,x) + (v(t,x) - z(t,x))z^5(t,x)\right] dx.
    \end{split}
\end{equation}
By Young's inequality, we have
\begin{align*}
&\int_{\Omega} \left[\left(buv^5 - u^2v^6\right)
+ \left(bwz^5 - w^2z^6\right)\right] dx\\
&\leq \frac{1}{2}
\Big(\int_{\Omega} b^2 (v^4 + z^4)\, dx
- \int_{\Omega} (u^2 v^6 + w^2 z^6)\, dx \Big),
\end{align*}
and
\[
     \int_{\Omega} \left[(z - v)v^5 + (v - z)z^5\right] dx
\leq  \int_{\Omega}  \Big[- v^6 + \big(\frac{1}{6} z^6
+ \frac{5}{6} v^6 \big) + \big(\frac{1}{6} v^6
+ \frac{5}{6} z^6 \big) - z^6 \Big] dx = 0.
\]
Substitute the above two inequalities into \eqref{vz6} and use
Poincar\'{e} inequality, we obtain the following inequality
relating $\|(v, z)\|_{L^6}^6$ to $\|(v, z)\|_{L^4}^4$,
\begin{align*}
    &\frac{d}{dt} \left(\|v(t)\|_{L^6}^6 + \|z(t)\|_{L^6}^6\right)
 + 10\gamma d_2 \left(\|v(t)\|_{L^6}^6 + \|z(t)\|_{L^6}^6 \right) \\
    & \leq \frac{d}{dt} \left(\|v(t)\|_{L^6}^6 + \|z(t)\|_{L^6}^6\right)
+ 10 d_2 \left(\|\nabla v^3 (t)\|^2 + \| \nabla z^3(t)\|^2\right)\\
& \leq 3b^2 (\|(v(t))\|_{L^4}^4 + \|(z(t))\|_{L^4}^4).
\end{align*}
Similarly we can get the corresponding inequality relating $\|(v, z)\|_{L^4}^4$ to $\|(v, z)\|^2$,
\begin{align*}
    &\frac{d}{dt} \left(\|(v(t))\|_{L^4}^4 + \|(z(t))\|_{L^4}^4\right)
+ 6\gamma d_2 \left(\|(v(t))\|_{L^4}^4 + \|(z(t))\|_{L^4}^4 \right) \\
    & \leq \frac{d}{dt} \left(\|(v(t))\|_{L^4}^4 + \|(z(t))\|_{L^4}^4\right)
+ 6 d_2 \left(\|\nabla v^2 (t)\|^2 + \| \nabla z^2(t)\|^2\right)\\
& \leq 2b^2 (\|v(t)\|^2 + \|z(t)\|^2).
\end{align*}
Applying Gronwall inequality to the above two inequalities and
using \eqref{vz}, we obtain
\begin{align*}
&\|(v(t))\|_{L^4}^4 + \|(z(t))\|_{L^4}^4\\
& \leq e^{-6\gamma d_2 t} \left(\|v_0\|_{L^4}^4 + \|z_0\|_{L^4}^4\right) \\
    &\quad + \int_0^t e^{-6\gamma d_2 (t-\tau)} 2b^2 (\|v(\tau)\|^2
+ \|z(\tau)\|^2)d\tau \\
    &\leq e^{-6\gamma d_2 t} \left(\|v_0\|_{L^4}^4 + \|z_0\|_{L^4}^4\right)
+ \int_{\Omega} e^{-6\gamma d_2(t-\tau)-2\gamma d_2 \tau}2b^2(\|v_0\|^2 + \|z_0\|^2)\, d\tau + \frac{b^4 |\Omega |}{6\gamma^2 d_2^2} \\
    & \leq e^{-2\gamma d_2 t} C_7 \left(\|v_0\|_{L^6}^6 + \|z_0\|_{L^6}^6\right)
 + \frac{b^4 |\Omega |}{6\gamma^2 d_2^2}, \quad t \geq 0,
\end{align*}
where $C_7$ is a uniform positive constant, and then
\begin{align*}
&\|v(t)\|_{L^6}^6 + \|z(t)\|_{L^6}^6\\
&\leq e^{- 10\gamma d_2 t} \left(\|v_0\|_{L^6}^6 + \|z_0\|_{L^6}^6\right)
 + \int_0^t e^{- 10\gamma d_2 (t-\tau)} 3b^2 (\|v(\tau)\|_{L^4}^4
+ \|z(\tau)\|_{L^4}^4)d\tau \\
&\leq e^{- 10\gamma d_2 t} \left(\|v_0\|_{L^6}^6 + \|z_0\|_{L^6}^6\right)
+ \int_{\Omega} e^{- 10\gamma d_2(t-\tau)-2\gamma d_2 \tau}3b^2 C_7(\|v_0\|_{L^6}^6
+ \|z_0\|_{L^6}^6)\, d\tau\\
&\quad + \frac{b^6 |\Omega |}{20\gamma^3 d_2^3} \\
&\leq e^{-2\gamma d_2 t} \Big(1 + \frac{3b^2 C_7}{8\gamma d_2} \Big)
\left(\|v_0\|_{L^6}^6 + \|z_0\|_{L^6}^6\right) + \frac{b^6 |\Omega |}{20\gamma^3 d_2^3},
\quad t \geq 0.
\end{align*}
It follows that
\begin{gather}
    \limsup_{t \to \infty}  \left(\|v(t)\|_{L^4}^4 + \|z(t)\|_{L^4}^4\right)
 < K_2 = 1 + \frac{b^4 |\Omega |}{6\gamma^2 d_2^2},  \label{vz4sup} \\
    \limsup_{t \to \infty}  \left(\|v(t)\|_{L^6}^6 + \|z(t)\|_{L^6}^6\right)
< K_3 = 1 + \frac{b^6 |\Omega |}{20\gamma^3 d_2^3}. \label{vz6sup}
\end{gather}
Thus \eqref{lsupp} is proved.
\end{proof}

\section{Asymptotic compactness}

The lack of inherent dissipation and the appearance of cross-cell
coupling make the attempt of showing the asymptotic compactness of
the two-cell Brusselator semiflow also challenging. In this
section we shall prove this asymptotic compactness through the
following two lemmas.

Since $H_0^1 (\Omega) \hookrightarrow L^4 (\Omega)$ and  $H_0^1 (\Omega)
\hookrightarrow L^6 (\Omega)$ are continuous embeddings, there are
constants $\delta > 0$ and $\eta > 0$ such that $\| \cdot \|_{L^4}^2
\leq \delta \|\nabla (\cdot)\|^2$ and $\| \cdot \|_{L^6}^2 \leq \eta
\|\nabla (\cdot)\|^2$. We shall use the notation $\|(y_1, y_2)\|^2
= \|y_1\|^2 + \|y_2\|^2$ and $\|\nabla (y_1, y_2)\|^2 = \|\nabla
y_1\|^2 + \|\nabla y_2\|^2$ for conciseness. The following
proposition is about the uniform Gronwall inequality, which is an
instrumental tool in the analysis of asymptotic compactness, cf.
\cite{MK05, SY02, rT88}.

\begin{proposition} \label{P:uGw}
    Let $\beta, \zeta$, and $h$ be nonnnegative functions in $L_{loc}^1 ([0, \infty); \mathbb{R})$. Assume that $\beta$ is absolutely continuous on $(0, \infty)$ and the following differential inequality is satisfied,
$$
    \frac{d\beta}{dt} \leq \zeta \beta + h, \quad \text{for }  t > 0.
$$
If there is a finite time $t_1 > 0$ and some $r > 0$ such that
$$
    \int_t^{t+r} \zeta (\tau)\, d\tau \leq A, \quad
\int_t^{t+r} \beta (\tau)\, d\tau \leq B, \quad
 \int_t^{t+r} h(\tau)\, d\tau \leq C,
$$
for any $t > t_1$, where $A, B$, and $C$ are some positive constants,
then
$$
    \beta (t) \leq \big(\frac{B}{r} + C\big) e^{A},
\quad \text{for any } t > t_1 + r.
$$
\end{proposition}

\begin{lemma} \label{L:uwc}
    For any given initial data $g_0 \in B_0$, the $(u, w)$ components
of the solution trajectories $g(t) = S(t)g_0$ of the IVP \eqref{eveq}
satisfy
\begin{equation} \label{uwq}
    \|\nabla (u(t), w(t)) \|^2 \leq M_1, \quad \text{for }  t > T_1,
\end{equation}
where $M_1 > 0$ is a uniform constant depending on $K_1$ and
$|\Omega |$ but independent of initial data, and $T_1 > 0$ is finite
and only depends on the absorbing ball $B_0$.
\end{lemma}

\begin{proof}
    Take the inner-products $\inpt{\eqref{equ}, -\Delta u(t)}$ and
$\inpt{\eqref{eqw}, -\Delta w(t)}$ and then sum up the two equalities
to obtain
\begin{align*}
    &\frac{1}{2} \frac{d}{dt} \|\nabla(u, w)\|^2 + d_1 \|\Delta (u,w)\|^2 + (b+ 1)\|\nabla(u, w)\|^2 \\
    &= - \int_\Omega a (\Delta u +\Delta w)\, dx - \int_\Omega (u^2 v \Delta u + w^2 z \Delta w)\, dx  \\
    &\quad - D_1 \int_\Omega  (|\nabla u |^2 - 2 \nabla u \cdot \nabla w + |\nabla w |^2 )\, dx \\
    &\leq \big(\frac{d_1}{4} + \frac{d_1}{4} + \frac{d_1}{2}\big) \|\Delta (u, w)\|^2 + \frac{a^2}{d_1} |\Omega |  + \frac{1}{2d_1} \int_\Omega \left(u^4 v^2 + w^4 v^2 \right) dx.
\end{align*}
It follows that
\begin{equation} \label{nuw}
    \begin{split}
&\frac{d}{dt} \|\nabla(u, w)\|^2 + 2(b+ 1)\|\nabla(u, w)\|^2\\
&\leq \frac{2 a^2 }{d_1} |\Omega | + \frac{1}{d_1}
\left(\|u^2\|^2 \| v\|^2 + \|w^2\|^2 \| z\|^2 \right) \\
    & \leq \frac{2 a^2 }{d_1} |\Omega | + \frac{\delta^2}{d_1}
 \left(\|v\|^2 \|\nabla u \|^4 + \|z\|^2 \|\nabla w \|^4 \right).
    \end{split}
\end{equation}
By the absorbing property shown in Lemma \ref{L:absb}, there is
a finite time $T_0 = T_0 (B_0) \geq 0$ such that
$S(t) B_0 \subset B_0$ for all $t > T_0$. Therefore, for any
$g_0 \in B_0$, by \eqref{bk} we have
\begin{equation} \label{vzvp}
    \| (u(t), w(t))\|^2 + \| (v(t), z(t))\|^2 \leq K_1, \quad \text{for } t > T_0.
\end{equation}
Substitute \eqref{vzvp} into \eqref{nuw} to obtain
\begin{equation} \label{nuwiq}
\begin{split}
\frac{d}{dt} \|\nabla (u, w)\|^2
&\leq \frac{d}{dt} \|\nabla (u, w)\|^2 + 2(b+ 1)\|\nabla(u, w)\|^2\\
&\leq \frac{\delta^2 K_1}{d_1}\|\nabla (u, w)\|^4
+  \frac{2 a^2}{d_1} |\Omega |,
\end{split}
\end{equation}
which can be written as the inequality
\begin{equation} \label{ugi}
    \frac{d \rho}{dt} \leq \beta \rho + \frac{2 a^2}{d_1} |\Omega |,
\end{equation}
where
$$
    \rho (t) = \|\nabla (u(t), w(t))\|^2 \quad \text{and} \quad
\beta (t) = \frac{\delta^2 K_1}{d_1} \rho (t).
$$
In view of the inequality \eqref{nyiq}, \eqref{vztt} and \eqref{vzvp},
we have
\begin{equation} \label{ytt}
    \begin{split}
    &\int_t^{t+1} \|\nabla y(\tau)\|^2 \, d\tau \\
&\leq \frac{2|d_1 - d_2|^2}{d_1^2} \int_t^{t+1}
\|\nabla (v + z)\|^2 d\tau  \\
    &\quad  + \frac{1}{d_1} \Big(\|y(t)\|^2 + \int_t^{t+1}
\frac{8}{\gamma} (\|v(\tau)\|^2 +\|z(\tau)\|^2 + 2a^2 |\Omega |)\, d\tau \Big)
 \leq C_{8},
    \end{split}
\end{equation}
for  $t> T_0$, where
$$
    C_{8} = \frac{4|d_1 - d_2|^2}{d_1^2 d_2}
\big[K_1 + \big(1 + \frac{1}{2\gamma d_2}\big)b^2 |\Omega |\big]
+ \frac{1}{d_1} \Big(K_1 + \frac{8}{\gamma}(K_1 + 2a^2 |\Omega|)\Big).
$$
From the inequality \eqref{psiq}, \eqref{vztt} and \eqref{vzvp}
and with a similar estimation, there exists a uniform constant
$C_{9} > 0$ such that
\begin{equation} \label{ptt}
    \int_t^{t+1} \|\nabla \psi(\tau)\|^2 \,d\tau \leq C_{9}, \quad \text{for } t > T_0.
\end{equation}
Then we can put together \eqref{vztt}, \eqref{ytt} and \eqref{ptt}
to get
\begin{equation} \label{rtt}
    \begin{split}
&\int_t^{t+1} \rho (\tau) \, d\tau\\
&= \int_t^{t+1} (\|\nabla u(\tau)\|^2 + \|\nabla w(\tau)\|^2)\, d\tau \\
    &\leq \frac{1}{2} \int_t^{t+1} \left(\|\nabla (y(\tau) - (v(\tau) + z(\tau))\|^2 + \|\nabla (\psi(\tau) - (v(\tau) - z(\tau))\|^2 \right) d\tau \\
    &\leq \int_t^{t+1} \left(\|\nabla y(\tau)\|^2 + \|\nabla \psi(\tau)\|^2 + \|\nabla (v + z)\|^2 + \|\nabla (v - z)\|^2\right) d\tau \\
    &\leq C_{8} + C_{9} + \frac{4}{d_2}
\big[K_1 + \big(1 + \frac{1}{2\gamma d_2}\big)b^2 |\Omega|\big]
\overset{\rm def}{=} C_{10}, \quad \text{for } t > T_0.
    \end{split}
\end{equation}
Now we can apply the uniform Gronwall inequality in
Proposition \ref{P:uGw} where $r =1$ to \eqref{ugi} and
use \eqref{rtt} to reach the conclusion \eqref{uwq} with
$$
    M_1 = \Big(C_{10} +  \frac{2 a^2}{d_1} |\Omega | \Big)
e^{\delta^2 K_1C_{10}/d_1}
$$
and $T_1 = T_0 (B_0) + 1$. The proof is completed.
\end{proof}

\begin{lemma} \label{L:vzc}
    For any given initial data $g_0 \in B_0$, the $(v, z)$
components of the trajectory $g(t) = S(t)g_0$ of the IVP \eqref{eveq}
satisfy
\begin{equation} \label{q2}
    \|\nabla (v(t), z(t)) \|^2 \leq M_2, \quad \text{for }  t > T_2,
\end{equation}
where $M_2 > 0$ is a uniform constants depending on $K_1$
and $|\Omega |$ but independent of initial data, and $T_2\, (> T_1 > 0)$ is finite and only depends on the absorbing ball $B_0$.
\end{lemma}

\begin{proof}
    Take the inner-products $\inpt{\eqref{eqv}, -\Delta v(t)}$ and
$\inpt{\eqref{eqz}, -\Delta z(t)}$ and sum up the two equalities
to obtain
\begin{align*}
    &\frac{1}{2} \frac{d}{dt} \|\nabla (v, z)\|^2
+ d_2 \|\Delta (v, z)\|^2 \\
&= -\int_\Omega b(u\Delta v + w \Delta z)\, dx \\
    &\quad + \int_\Omega (u^2 v \Delta v + w^2 z \Delta z )\, dx
- D_2 \int_\Omega [(z - v)\Delta v + (v - z) \Delta z ]\, dx \\
    & \leq \frac{d_2}{2} \|\Delta (v, z)\|^2
+ \frac{b^2}{d_2} \|(u, w)\|^2 + \frac{1}{d_2}
\int_\Omega (u^4 v^2 + w^4 z^2) \, dx \\
    &\quad - D_2 \int_\Omega  (|\nabla v |^2 - 2 \nabla v \cdot \nabla z
+ |\nabla z |^2 )\, dx \\
    &\leq \frac{d_2}{2} \|\Delta (v, z)\|^2 + \frac{b^2}{d_2} \|(u, w)\|^2
+ \frac{1}{d_2} \int_\Omega (u^4 v^2 + w^4 z^2) \, dx, \quad t > T_0.
\end{align*}
Since
\begin{align*}
&\|\nabla (v(t), z(t))\|^2 \\
&= - (\inpt{v, \Delta v}
 + \inpt{z, \Delta z}) \leq \frac{1}{2} \Big(\|v(t)\|^2 +\|z(t)|^2 \|
+ \|\Delta v(t)\|^2 + \|\Delta z(t)\|^2 \Big),
\end{align*}
by using H\"{o}lder inequality and the embedding inequality
mentioned in the beginning of this section and by Lemma
\ref{L:uwc}, from the above inequality we obtain
\begin{equation} \label{nbvz}
    \begin{split}
    &\frac{d}{dt}\|\nabla (v, z)\|^2 + d_2 \|\nabla (v, z)\|^2  \\
    &\leq d_2 \|(v, z)\|^2+ \frac{2b^2}{d_2} \|(u, w)\|^2 + \frac{2}{d_2} (\|u\|_{L^6}^4 \|v\|_{L^6}^2 + \|w\|_{L^6}^4 \|z\|_{L^6}^2 ) \\
     &\leq \big(d_2 + \frac{2b^2}{d_2}\big)K_1
+ \frac{2\eta^6}{d_2} (\|\nabla u \|^4 + \|\nabla w \|^4)\|\nabla (v, z)\|^2 \\
     & \leq K_1 \big(d_2 + \frac{2b^2}{d_2} \big)
+ \frac{2\eta^6 M_1^2}{d_2} \|\nabla (v, z)\|^2, \, t > T_1.
    \end{split}
\end{equation}
Applying the uniform Gronwall inequality in Proposition \ref{P:uGw}
to \eqref{nbvz} and using \eqref{vztt}, we can assert that
\begin{equation} \label{nvziq}
    \|\nabla (v(t), z(t))\|^2 \leq M_2, \quad \text{for } t > T_1 + 1,
\end{equation}
where
$$
    M_2 = \Big(\frac{1}{d_2} \big[K_1
+ \big(1 + \frac{1}{2\gamma d_2}\big) b^2 |\Omega |\big] +  K_1
\big[d_2 + \frac{2b^2}{d_2} \big]\Big) e^{2 \eta^6 M_1^2/d_2} .
$$
Thus \eqref{q2} is proved with this $M_2$ and $T_2 = T_1 + 1$.
\end{proof}

\section{{The existence of a global attractor and its properties}}

In this section we finally prove Theorem \ref{Mthm} on the
existence of a global attractor, which will be denoted by $\mathscr{A}$,
for the Brusselator semiflow $\{S(t)\}_{t\geq0}$ and we shall investigate the
properties of $\mathscr{A}$, including its finite fractal dimensionality.

\begin{proof}[Proof of Theorem \ref{Mthm}]
In Lemma \ref{L:absb}, we have shown that the Brusselator semiflow
$\{S(t)\}_{t\geq0}$ has a bounded absorbing set $B_0$ in $H$.
Combining Lemma \ref{L:uwc} and Lemma \ref{L:vzc} we proved that
$$
    \| S(t) g_0 \|_E^2 \leq M_1 + M_2, \quad \text{for }  t > T_2
\text{ and for }  g_0 \in B_0,
$$
which implies that $\{S(t) B_0: t > T_2\}$ is a bounded set in
space $E$ and consequently a precompact set in space $H$.
Therefore, the Brusselator semiflow $\{S(t)\}_{t\geq0}$ is asymptotically
compact in $H$. Finally we apply Proposition \ref{P:kpac}
to reach the conclusion that there exists a global attractor
$\mathscr{A}$ in $H$ for this Brusselator semiflow $\{S(t)\}_{t\geq0}$.
\end{proof}

Now we show that the global attractor $\mathscr{A}$ of
the Brusselator semiflow is an $(H, E)$ global attractor with
the regularity $\mathscr{A} \subset \mathbb{L}^\infty (\Omega)$.
The concept of $(H, E)$ global attractor was introduced
in \cite{BV83}.

\begin{definition} \label{D:hea} \rm
    Let $\{\Sigma (t)\}_{t\geq 0}$ be a semiflow on a Banach
space $X$ and let $Y$ be a compactly imbedded subspace of $X$.
A subset $\mathcal{A}$ of $Y$ is called an $(X, Y)$ global
attractor for this semiflow if $\mathcal{A}$ has the
following properties,
\begin{itemize}
\item[(i)] $\mathcal{A}$ is a nonempty, compact, and invariant
set in $Y$.

\item[(ii)] $\mathcal{A}$ attracts any bounded set $B \subset X$
with respect to the $Y$-norm; namely, there is a
$\tau = \tau (B)$ such that $\Sigma (t)B \subset Y$ for $t > \tau$
and $\operatorname{dist}_Y (\Sigma (t)B, \mathcal{A}) \to 0$, as $t \to \infty$.
\end{itemize}
\end{definition}

\begin{lemma} \label{L:ste}
    Let $\{g_m\}$ be a sequence in $E$ such that $\{g_m\}$ converges
to $g_0 \in E$ weakly in $E$ and $\{g_m\}$ converges to $g_0$
strongly in $H$, as $m \to \infty$. Then
    $$
        \lim_{m \to \infty} S(t) g_m = S(t) g_0 \quad
\text{strongly in } E,
    $$
where the convergence is uniform with respect to $t$ in any given
compact interval $[t_0, t_1] \subset (0, \infty)$.
\end{lemma}

The proof of this lemma is found in \cite[Lemma 10]{yY10}.

\begin{theorem} \label{Thm2}
    The global attractor $\mathscr{A}$ in $H$ for the Brusselator semiflow
$\{S(t)\}_{t\geq0}$ is indeed an $(H, E)$ global attractor and $\mathscr{A}$
is a bounded subset in $\mathbb{L}^\infty (\Omega)$.
\end{theorem}

\begin{proof}
    By Lemmas \ref{L:absb},  \ref{L:uwc} and  \ref{L:vzc},
we can assert that for the Brusselator semiflow $\{S(t)\}_{t\geq0}$ defined
on $H$ there exists a bounded absorbing set $B_1 \subset E$
and this absorbing is in the $E$-norm. Indeed,
$$
    B_1 = \{g \in E: \| g \|_E^2 = \|\nabla g \|^2 \leq M_1 + M_2\}.
$$

 Now we show that the Brusselator semiflow $\{S(t)\}_{t\geq0}$ is asymptotically
compact with respect to the strong topology in $E$. For any time
sequence $\{t_n \}, t_n \to \infty$, and any bounded sequence
$\{g_n \} \subset E$, there exists a finite time $t_0 \geq 0$
such that $S(t) \{g_n\} \subset B_0$, for any $t > t_0$.
Then for an arbitrarily given $T > t_0 + T_2$, where $T_2$
is the time specified in Lemma \ref{L:vzc}, there is an
integer $n_0 \geq 1$ such that $t_n > 2T$ for all $n > n_0$.
By Lemma \ref{L:uwc} and Lemma \ref{L:vzc},
$$
    \{S(t_n - T) g_n\}_{n > n_0} \; \text{is a bounded set in}\; E.
$$
Since $E$ is a Hilbert space, there is an increasing sequence
of integers $\{n_j\}_{j=1}^\infty$ with $n_1 > n_0$, such that
$$
      \lim_{j \to \infty} S(t_{n_j} - T) g_{n_j} = g^* \quad
 \text{weakly in } E.
$$
By the compact imbedding $E \hookrightarrow H$, there is a further
subsequence of $\{n_j\}$, but relabeled as the same as $\{n_j\}$,
 such that
$$
    \lim_{j \to \infty} S(t_{n_j} - T) g_{n_j} = g^* \quad
 \text{strongly in }  H.
$$
Then by Lemma \ref{L:ste}, we have the following convergence
with respect to the $E$-norm,
$$
    \lim_{j \to \infty} S(t_{n_j}) g_{n_j}
= \lim_{j \to \infty} S(T) S(t_{n_j} - T) g_{n_j}
 = S(T) g^* \quad \text{strongly in }  E.
$$
This proves that $\{S(t)\}_{t\geq0}$ is asymptotically compact in $E$.

Therefore, by Proposition \ref{P:kpac}, there exists a global
 attractor $\mathscr{A}_E$ for the extended Brusselator semiflow
$\{S(t)\}_{t\geq0}$ in the space $E$. According to Definition \ref{D:hea}
and the fact that $B_1$ attracts $B_0$ in the $E$-norm due
to the combination of Lemma \ref{L:uwc} and Lemma \ref{L:vzc},
 we see that this global attractor $\mathscr{A}_E$ is an $(H, E)$
global attractor. Moreover, the invariance and the boundedness
of $\mathscr{A}$ in $H$ and the invariance and boundedness of
$\mathscr{A}_E$ in $E$ imply that
\begin{quote}
    $\mathscr{A}_E$  attracts  $\mathscr{A}$  in $E$,  so that
 $\mathscr{A} \subset \mathscr{A}_E$, and \\
$\mathscr{A}$   attracts   $\mathscr{A}_E$   in $H$,
  so that  $\mathscr{A}_E \subset \mathscr{A}$.
\end{quote}
Therefore, $\mathscr{A} = \mathscr{A}_E$ and we proved that the global
attractor $\mathscr{A}$ in $H$ is itself an $(H, E)$ global attractor
for this Brusselator semiflow.

Next we show that $\mathscr{A}$ is a bounded subset in
$\mathbb{L}^\infty (\Omega)$. By the $(L^p, L^\infty)$ regularity of
the analytic $C_0$-semigroup $\{e^{At}\}_{t\geq 0}$ stated in
\cite[Theorem 38.10]{SY02}, one has $e^{At}: \mathbb{L}^p (\Omega)
\to \mathbb{L}^\infty (\Omega)$ for $t > 0$, and there is a constant
$C(p) > 0$ such that
\begin{equation} \label{cp}
    \| e^{At} \|_{\mathcal{L} (\mathbb{L}^p, \mathbb{L}^\infty)}
\leq C(p) \, t^{- \frac{n}{2p}}, \;\; t > 0, \quad \text{where } n
= \dim  \Omega.
\end{equation}
By the variation-of-constant formula satisfied by the mild
solutions (of course by strong solutions), for any
$g \in \mathscr{A} \, (\subset E)$, we have
\begin{equation} \label{mld}
    \begin{split}
    \|S(t) g\|_{L^\infty}
& \leq \|e^{At} \|_{\mathcal{L} (L^2,L^\infty)} \|g\|
 + \int_0^t \|e^{A(t- \sigma)}\|_{\mathcal{L} (L^2, L^\infty)}
 \| f(S(\sigma)g) \| \, d\sigma \\
&\leq  C(2) t^{-3/4} \|g\| + \int_0^t C(2)
(t-\sigma)^{-3/4} L(M_1, M_2) \|S(\sigma) g\|_E \, d\sigma,
    \end{split}
\end{equation}
$t \geq 0$, where $C(2)$ is in the sense of \eqref{cp} with $p = 2$,
 and $L(M_1, M_2)$ is the Lipschitz constant of the nonlinear
map $f$ on the closed, bounded ball with radius $M_1 + M_2$ in $E$.
By the invariance of the global attractor $\mathscr{A}$, surely we have
$$
\{S(t) \mathscr{A}: t \geq 0\} \subset B_0 \, (\subset H)
\quad\text{and} \quad
\{S(t) \mathscr{A}: t \geq 0\} \subset B_1 \, (\subset E).
$$
Then from \eqref{mld} we obtain
\begin{equation} \label{bdift}
    \begin{split}
    \|S(t) g\|_{L^\infty} &\leq C(2) K_1 t^{-3/4}
+ \int_0^t C(2) L(M_1, M_2) (M_1 + M_2) (t- \sigma)^{-3/4} \, d\sigma \\
    &= C(2) [K_1 t^{-3/4} +  4 L(M_1, M_2) (M_1 + M_2)
t^{1/4}], \quad \text{for } t > 0.
    \end{split}
\end{equation}
Specifically one can take $t = 1$ in \eqref{bdift} and use the
invariance of $\mathscr{A}$ to obtain
$$
    \| g \|_{L^\infty} \leq C(2)( K_1 +  4 L(M_1, M_2) (M_1 + M_2)),
\quad \text{for any } g \in \mathscr{A}.
$$
Thus the global attractor $\mathscr{A}$ is a bounded subset in
$\mathbb{L}^\infty (\Omega)$.
\end{proof}

Now consider the Hausdorff dimension and fractal dimension of the
 global attractor of the Brusselator semiflow $\{S(t)\}_{t\geq0}$ in $H$.
Let $q_{m} = \limsup_{t \to \infty} \, q_{m} (t)$, where
\begin{equation} \label{trq}
    q_{m} (t) = \sup_{g_0 \in \mathscr{A}} \;
 \sup_{\substack{g_{i} \in H, \|g_{i} \| = 1\\ i = 1, \dots, m}}
 \Big( \frac{1}{t} \int_{0}^{t} \operatorname{Tr}  \left( A + F'
(S(\tau) g_0 ) \right) \circ Q_{m} (\tau) \, d\tau \Big),
\end{equation}
in which $\operatorname{Tr} \, (A + F' (S(\tau)g_0))$ is the
trace of the linear operator $A + F' (S(\tau)g_0)$, with $F(g)$
being the nonlinear map in \eqref{eveq}, and $Q_{m} (t)$ stands
for the orthogonal projection of space $H$ on the subspace
spanned by $G_1 (t), \dots, G_{m} (t)$, with
\begin{equation} \label{Frder}
    G_{i} (t) = L(S(t), g_0)g_{i},   \quad i = 1, \dots, m.
\end{equation}
Here $F'(S(\tau)g_0)$ is the Fr\'{e}chet derivative of the map
$F$ at  $S(\tau)g_0$, and $L(S(t), g_0)$ is the Fr\'{e}chet
derivative of the map $S(t)$ at $g_0$, with $t$ being fixed.

Next we study Hausdorff and fractal dimensions of the global
attractor $\mathscr{A}$. The following proposition is seen in
\cite[Chapter 5]{rT88}.

\begin{proposition} \label{P:HFd}
If there is an integer $m$ such that $q_{m} < 0$, then
the Hausdorff dimension $d_{H} (\mathscr{A})$ and the fractal
dimension $d_{F} (\mathscr{A})$ of $\mathscr{A}$ satisfy
\begin{equation}  \label{hfd}
    d_{H} (\mathscr{A}) \leq m,  \quad \text{and} \quad
d_{F} (\mathscr{A}) \leq m \max_{1 \leq j \leq m - 1}
\Big( 1 + \frac{(q_{j})_{+}}{| q_{m} |} \Big) \leq 2m.
\end{equation}
\end{proposition}

It is standard to show that for any given $t > 0$, $S(t)$
on $H$ is Fr\'{e}chet differentiable and its Fr\'{e}chet
derivative at $g_0$ is the bounded linear operator $L(S(t), g_0)$
given by
$$
    L(S(t), g_0)G_0 \overset{\rm def}{=} G(t)
= (U(t), V(t), W(t), Z(t)),
$$
for any $G_0 = (U_0, V_0, W_0, Z_0) \in H$, where
$(U(t), V(t), W(t), Z(t))$ is the strong solution of the
following initial-boundary value problem of the variational equations
\begin{equation} \label{vareq}
    \begin{gathered}
    \frac{\partial U}{\partial t}  = d_1 \Delta U + 2u(t)v(t) U + u^2 (t) V - (b + 1) U + D_1 (W - U),  \\
    \frac{\partial V}{\partial t}  = d_2 \Delta V - 2u(t)v(t) U - u^2 (t) V + b U + D_2 (Z - V), \\
    \frac{\partial W}{\partial t}  = d_1 \Delta W + 2w(t)z(t) W + w^2 (t) Z - (b + 1) W + D_1 (U - W),  \\
    \frac{\partial Z}{\partial t}  = d_2 \Delta Z - 2w(t)z(t) W - w^2 (t) Z + b W + D_2 (V - Z),  \\
     U\mid_{\partial \Omega} =  V\mid_{\partial \Omega} = W\mid_{\partial \Omega} = Z\mid_{\partial \Omega} = 0, \quad t > 0, \\
     U(0) = U_0, \quad V(0) = V_0, \quad W(0) = W_0, \quad Z(0) = Z_0.
    \end{gathered}
\end{equation}
Here $g(t) = (u(t), v(t), w(t), z(t)) = S(t)g_0$
is the solution of \eqref{eveq} with the initial condition
$g(0) = g_0$.  The initial-boundary value problem \eqref{vareq}
can be written as
\begin{equation} \label{vareveq}
    \frac{dG}{dt} = (A + F' (S(t)g_0))G,   \quad G(0) = G_0.
\end{equation}
From Lemma \ref{L:uwc}, Lemma \ref{L:vzc} and the invariance
of $\mathscr{A}$ it follows that
\begin{equation} \label{mbd}
    \sup_{g_0 \in \mathscr{A}}  \| S(t)g_0 \|_E^2 \leq M_1 + M_2.
\end{equation}

\begin{theorem} \label{Dmn}
The global attractors $\mathscr{A}$ for the Brusselator semiflow $\{S(t)\}_{t\geq0}$ has a finite Hausdorff dimension and a finite fractal dimension.
\end{theorem}

\begin{proof}
By Proposition \ref{P:HFd}, we shall estimate $\operatorname{Tr}
 (A + F' (S(\tau)g_0 )) \circ Q_{m}(\tau)$.
At any given time $\tau > 0$, let
$\{\varphi_{j} (\tau): j = 1, \dots , m\}$ be an $H$-orthonormal basis
for the subspace
$$
    Q_m(\tau) H = \operatorname{span}  \{G_1(\tau), \dots ,
 G_,(\tau) \},
$$
where $G_1 (t), \dots , G_{m} (t)$ satisfy \eqref{vareveq}
with the respective initial values $G_{1,0},\dots, G_{m,0}$ and,
without loss of generality, assuming that $G_{1,0}, \dots, G_{m,0}$
are linearly independent in $H$. By Gram-Schmidt orthogonalization
scheme,
$$
\varphi_{j}(\tau) = (\varphi_{j}^1 (\tau), \varphi_{j}^2 (\tau),
\varphi_{j}^3 (\tau), \varphi_{j}^4 (\tau) ) \in E, \quad
j = 1, \dots , m,
$$
 and $\varphi_{j} (\tau)$ are strongly measurable in $\tau$.
Let $d_0 = \min \{d_1, d_2 \}$. Then
\begin{equation} \label{Trace}
        \begin{split}
& \operatorname{Tr}  (A + F' (S(\tau)g_0 )\circ Q_m(\tau)\\
& = \sum_{j=1}^{m} \left( \langle A \varphi_{j}(\tau), \varphi_{j}(\tau) \rangle + \langle  F' (S(\tau)g_0 ) \varphi_{j}(\tau), \varphi_{j}(\tau) \rangle\right)  \\
& \leq - d_0 \sum_{j=1}^{m} \, \| \nabla \varphi_{j}(\tau) \|^2
+ J_1 + J_2 + J_3,
        \end{split}
\end{equation}
where
\begin{align*}
    J_1 & = \sum_{j=1}^{m} \int_{\Omega} 2 u(\tau) v(\tau)
\left( |\varphi_{j}^1 (\tau) |^2 -  \varphi_{j}^1 (\tau) \varphi_{j}^2 (\tau)  \right) dx \\
    &\quad+ \sum_{j=1}^{m} \, \int_{\Omega} 2 w(\tau) z(\tau)
\left( |\varphi_{j}^3 (\tau) |^2 -  \varphi_{j}^3 (\tau) \varphi_{j}^4
(\tau)  \right) dx,
\end{align*}
\begin{align*}
    J_2 &= \sum_{j=1}^{m} \int_{\Omega} \left( u^2(\tau)
\left( \varphi_{j}^1 (\tau) \varphi_{j}^2 (\tau) - | \varphi_{j}^2 (\tau) |^2 \right) + w^2(\tau) \left( \varphi_{j}^3 (\tau) \varphi_{j}^4 (\tau) - | \varphi_{j}^4 (\tau) |^2 \right)\right) dx  \\
    &\leq  \sum_{j=1}^{m} \int_{\Omega} \left( u^2(\tau)
| \varphi_{j}^1 (\tau) | |\varphi_{j}^2 (\tau)| + w^2(\tau) | \varphi_{j}^3 (\tau) | |\varphi_{j}^4 (\tau)| \right) dx,
\end{align*}
and
\begin{align*}
    J_3 & =  \sum_{j=1}^{m} \int_{\Omega} \left( - (b + 1)
(|\varphi_{j}^1 (\tau) |^2 + |\varphi_{j}^3 (\tau) |^2)
+ b (\varphi_{j}^1 (\tau) \varphi_{j}^2 (\tau) + \varphi_{j}^3
(\tau) \varphi_{j}^4 (\tau)) \right) dx \\
    &\quad - \sum_{j=1}^{m} \int_{\Omega} \left(D_1
\left(\varphi_j^1 (\tau) - \varphi_j^3 (\tau)\right)^2
+ D_2 \left(\varphi_j^3 (\tau) - \varphi_j^4 (\tau)\right)^2 \right) dx \\
    & \leq \sum_{j=1}^{m} \int_{\Omega}  b \left(\varphi_{j}^1 (\tau)
\varphi_{j}^2 (\tau) + \varphi_{j}^3 (\tau) \varphi_{j}^4 (\tau) \right) dx.
\end{align*}
By the generalized H\"{o}lder inequality and the Sobolev embedding
$H_0^1 (\Omega) \hookrightarrow  L^4 (\Omega)$ for $n \leq 3$, and using
\eqref{mbd}, we obtain
\begin{equation} \label{J1eq}
        \begin{split}
    J_1& \leq 2 \sum_{j=1}^{m} \| u(\tau) \|_{L^4} \| v(\tau) \|_{L^4}
 \left( \| \varphi_{j}^1 (\tau) \|_{L^4}^2
+ \| \varphi_{j}^1(\tau) \|_{L^4}  \| \varphi_{j}^2 (\tau) \|_{L^4}
\right) \\
    &\quad + 2 \sum_{j=1}^{m} \| w(\tau) \|_{L^4} \| z(\tau) \|_{L^4}
 \left( \| \varphi_{j}^3 (\tau) \|_{L^4}^2 + \| \varphi_{j}^3(\tau)
 \|_{L^4}  \| \varphi_{j}^4 (\tau) \|_{L^4} \right) \\
    & \leq 4 \sum_{j=1}^{m} \| S(\tau)g_0 \|_{L^4}^2
\| \varphi_{j} (\tau) \|_{L^4}^2  \leq 4 \delta (M_1 + M_2)
\sum_{j=1}^{m} \| \varphi_{j} (\tau) \|_{L^4}^2,
        \end{split}
\end{equation}
where $\delta$ is the Sobolev embedding coefficient given at
the beginning of Section 3. Now we apply the Garliardo-Nirenberg
interpolation inequality, cf. \cite[Theorem B.3]{SY02},
\begin{equation} \label{GNineq}
    \| \varphi \|_{W^{k,p}} \leq C \| \varphi \|_{W^{m,q}}^{\theta}
\| \varphi \|_{L^{r}}^{1 - \theta}, \quad \text{for }
 \varphi \in W^{m,q}(\Omega),
\end{equation}
provided that $p, q, r \geq 1, 0 < \theta \leq 1$, and
$$
    k - \frac{n}{p} \leq \theta \big( m - \frac{n}{q} \big)
 - (1 - \theta ) \frac{n}{r},   \quad \text{where }  n = \dim  \Omega.
$$
Here let $W^{k, p}(\Omega) = L^4(\Omega), W^{m, q}(\Omega) = H_{0}^{1}(\Omega),
L^{r}(\Omega) = L^2(\Omega)$, and $\theta = n/4 \leq 3/4$.
It follows from \eqref{GNineq} that
\begin{equation} \label{inter}
    \| \varphi_{j} (\tau) \|_{L^4} \leq C \| \nabla \varphi_{j}
(\tau) \|^{n/4} \| \varphi_{j} (\tau) \|^{1 - \frac{n}{4}}
= C \| \nabla \varphi_{j} (\tau) \|^{n/4}, \quad
j = 1, \dots , m,
\end{equation}
since $\| \varphi_{j}(\tau) \| = 1$, where $C$ is a uniform constant.
Substitute \eqref{inter} into \eqref{J1eq} to obtain
\begin{equation} \label{J1est}
    J_1 \leq 4\delta (M_1 + M_2) C^2 \sum_{j=1}^{m}
 \| \nabla \varphi_{j} (\tau) \|^{n/2}.
\end{equation}
Similarly, by the generalized H\"{o}lder inequality, we can get
\begin{equation} \label{J2est}
    J_2 \leq \delta (M_1 + M_2) \sum_{j=1}^{m} \| \varphi_{j}
(\tau) \|_{L^4}^2 \leq \delta (M_1 + M_2) C^2 \sum_{j=1}^{m}
 \| \nabla \varphi_{j} (\tau) \|^{n/2}.
\end{equation}
Moreover, we have
\begin{equation} \label{J3est}
    J_3 \leq \sum_{j=1}^{m}  b \| \varphi_{j} (\tau) \|^2 = b m.
\end{equation}
Substituting \eqref{J1est}, \eqref{J2est} and \eqref{J3est}
into \eqref{Trace}, we obtain
\begin{equation} \label{Trest}
\begin{split}
&\operatorname{Tr}  (A + F' (S(\tau)g_0 )\circ Q_m(\tau)\\
& \leq - d_0 \sum_{j=1}^{m} \| \nabla \varphi_{j}(\tau) \|^2
+ 5\delta (M_1 + M_2) C^2 \sum_{j=1}^{m}
\| \nabla \varphi_{j}(\tau) \|^{n/2} + b m.
\end{split}
\end{equation}
By Young's inequality, for $n \leq 3$, we have
$$
    5 \delta (M_1 + M_2) C^2 \sum_{j=1}^{m}
 \| \nabla \varphi_{j}(\tau) \|^{n/2}  \leq \frac{d_0}{2}
\sum_{j=1}^{m} \|\nabla \varphi_{j}(\tau) \|^2 + \Gamma (n) m,
$$
where $\Gamma (n)$ is a uniform constant depending only on
$n = \dim (\Omega)$ and the involved constants $\delta, C, d_0, M_1$
and $M_2$. Hence,
\[
    \operatorname{Tr}  (A + F' (S(\tau)g_0 )\circ Q_m(\tau)
 \leq - \frac{d_0}{2} \sum_{j=1}^{m} \| \nabla \varphi_{j}(\tau) \|^2
+ \left(\Gamma (n) + b \right) m, \quad \tau > 0, \, g_0 \in \mathscr{A}.
\]
According to the generalized Sobolev-Lieb-Thirring
inequality \cite[Appendix, Cor. 4.1]{rT88},
 since $\{ \varphi_1 (\tau), \dots , \varphi_{m} (\tau) \}$ is
an orthonormal set in $H$, so there exists a constant $\Psi > 0$
only depending on the shape and dimension of $\Omega$, such that
\begin{equation} \label{SLTineq}
    \sum_{j=1}^{m} \| \nabla \varphi_{j}(\tau) \|^2
\geq  \frac{\Psi \, m^{1 + \frac{2}{n}}}{|\Omega |^{2/n}}.
\end{equation}
Therefore, we end up with
\begin{equation} \label{finest}
    \operatorname{Tr} (A + F' (S(\tau)g_0 )\circ Q_m(\tau)
 \leq - \frac{d_0  \Psi}{2 |\Omega |^{2/n}} m^{1 + \frac{2}{n}}
 + \left(\Gamma (n) + b \right) m,
\end{equation}
for  $\tau > 0$ and $g_0 \in \mathscr{A}$.
Then we can conclude that
\begin{equation} \label{qmt}
        \begin{split}
    q_{m}(t) & =  \sup_{g_0 \in \mathscr{A}} \;
 \sup_{\substack{g_{i} \in H, \|g_{i} \| = 1\\ i = 1, \dots, m}}
 \Big( \frac{1}{t} \int_{0}^{t} \operatorname{Tr}
\left( A + F' (S(\tau) g_0 ) \right) \circ Q_{m} (\tau) \, d\tau \Big)  \\
    & \leq -  \frac{d_0 \Psi}{2 |\Omega |^{2/n}} m^{1 + \frac{2}{n}}
+ \left(\Gamma (n) + b\right) m, \quad \text{for any} \, \; t > 0,
        \end{split}
\end{equation}
so that
\begin{equation}  \label{qm}
    q_m = \limsup_{t \to \infty} \, q_m (t) \leq  -  \frac{d_0 \Psi}{2 |\Omega |^{2/n}} m^{1 + \frac{2}{n}}  + \left(\Gamma (n) + b\right) m < 0,
\end{equation}
if the integer $m$ satisfies the  condition
\begin{equation} \label{dimc}
    m - 1 \leq \Big( \frac{2(\Gamma (n) + b)}{d_0 \Psi} \Big)^{n/2}
| \Omega | < m.
\end{equation}
According to Proposition \ref{P:HFd}, we have shown that
the Hausdorff dimension and the fractal dimension of the global
attractor $\mathscr{A}$ are finite and their upper bounds are given by
$$
    d_{H} (\mathscr{A}) \leq m \quad \text{and} \quad d_{F} (\mathscr{A})
 \leq 2m,
$$
respectively, where $m$ satisfies \eqref{dimc}.
\end{proof}

\section{Existence of an exponential attractor}

In this final section, we shall prove the existence of an
exponential attractor for the Brusselator semiflow
$\{S(t)\}_{t\ge 0}$.

\begin{definition}\label{D:exatr} \rm
Let $X$ be a real Banach space and $\{\Sigma (t)\}_{t\ge 0}$
 be a semiflow on $X$. A set $\mathscr{E}\subset X$ is an
exponential attractor for the semiflow $\{\Sigma (t)\}_{t\ge 0}$
in $X$, if the following conditions are satisfied:
\begin{itemize}
\item[(i)]  $\mathscr{E}$ is a nonempty, compact, positively
invariant set in $X$,

\item[(ii)]  $\mathscr{E}$ has a finite fractal dimension, and

\item[(iii)]  $\mathscr{E}$ attracts every bounded set
$B\subset X$ exponentially: there exist positive constants
$\mu$ and $C(B)$ which depends on $B$, such that
$$
    \operatorname{dist}_X(\Sigma (t)B,\mathscr{E})
\leq C(B)e^{-\mu t},\quad\text{for } t\ge 0.
$$
\end{itemize}
\end{definition}

The basic theory and construction of exponential attractors
were established in \cite{EFNT94} for discrete and
continuous semiflows on Hilbert spaces. The existence theory
was generalized to semiflows on Banach spaces in \cite{DN01}
and extended to some nonlinear reaction-diffusion equations
on unbounded domains and other equations including chemotaxis
equations and some quasilinear parabolic equations.

Global attractors, exponential attractors, and inertial manifolds
 are the three major research topics in the area of infinite
dimensional dynamical systems. For a continuous semiflow on a
Hilbert space, if all the three objects (a global attractor
$\mathscr{A}$, an exponential attractor $\mathscr{E}$, and an
inertial manifold $\mathscr{M}$ of the same exponential
attraction rate) exist, then the following inclusion
relationship holds,
$$
    \mathscr{A}\subset\mathscr{E}\subset\mathscr{M}.
$$

Here we shall tackle the existence of exponential attractor for
the Brusselator semiflow by the argument of squeezing  property
\cite{EFNT94,MK05}.

\begin{definition}\label{D:sqpy} \rm
For a spectral (orthogonal) projection $P_N$ relative to a
nonnegative, self-adjoint, linear operator
$\Lambda:D(\Lambda)\to\mathcal{H}$ with a  compact resolvent,
which maps the Hilbert space $\mathcal{H}$ onto the $N$-dimensional
subspace $\mathcal{H}_N$ spanned by a set of the first
$N$ eigenvectors of the operator $\Lambda$, we defined a cone
$$
    \mathscr{C}_{P_N}
=\{y\in X:\|(I-P_N)(y)\|_{\mathcal{H}}\leq\|P_N(y)\|_{\mathcal{H}}\}.
$$
A continuous mapping $S_*$ satisfies the
\emph{discrete squeezing property} relative to a set
$B\subset\mathcal{H}$ if there exist a constant $\kappa \in(0,1/2)$
and a spectral projection $P_{N}$ on $\mathcal{H}$ such that for
any pair of points $y_0,z_0\in B$, if
$$
    S_*(y_0)-S_*(z_0)\notin\mathscr{C}_{P_N},
$$
then
$$
  \|S_*(y_0)-S_*(z_0)\|_{\mathcal{H}}\leq\kappa \|y_0-z_0\|_{\mathcal{H}}.
$$
\end{definition}

We first present the following lemma, which is a modified version
of the basic result \cite[Theorem 4.5]{MK05} on the sufficient
conditions for the existence of an exponential attractor
of a semiflow on a Hilbert space. In some sense, this lemma
provides a more accessible way to check these sufficient
conditions if we are sure there exists an $(X, Y)$ global attractor,
such as the $(H, E)$ global attractor for the Brusselator semiflow here. The following lemma was proved in \cite[Lemma 6.1]{yY10a}.

\begin{lemma}\label{L:EXatr}
Let $X$ be a real Banach space and $Y$ be a compactly embedded
subspace of $X$. Consider a semilinear evolutionary equation
\begin{equation} \label{eveqig}
    \frac{d\varphi}{dt}+\Lambda \varphi = g(\varphi),\quad t>0,
\end{equation}
where $\Lambda:D(\Lambda)\to X$ is a nonnegative, self-adjoint,
linear operator with compact resolvent, and
$g:Y=D(\Lambda^{1/2})\to W$ is a locally Lipschitz continuous mapping. Suppose that the weak solution of \eqref{eveqig} for each initial point $w(0)=w_0\in W$ uniquely exists for all $t\ge 0$, which turn out to be a strong solution for $t > 0$ and altogether form a semiflow denoted by $\{\Sigma (t)\}_{t\ge 0}$. Assume that the following four conditions are satisfied:
\begin{itemize}
\item[(i)]  There exist a compact, positively invariant,
absorbing set $\mathcal{B}_c$ in $X$.

\item[(ii)]  There is a positive integer $N$ such that the norm quotient $Q(t)$ defined by
\begin{equation}
\label{qn}
    Q(t)=\frac{\|\Lambda^{1/2}\left(\varphi_1(t)-\varphi_2(t)\right)\|_X^2}{\|\varphi_1(t)-\varphi_2(t)\|_X^2}
\end{equation}
for any two trajectories $\varphi_1(\cdot)$ and $\varphi_2(\cdot)$ of \eqref{eveqig} starting from the set $\mathcal{B}_c\backslash\mathscr{C}_{P_N}$ satisfies a differential inequality
$$
    \frac{dQ}{dt}\le\rho\left(\mathcal{B}_c\right)Q(t),\quad t>0,
$$
where $\rho\left(\mathcal{B}_c\right)$ is a positive constant
only depending on $\mathcal{B}_c$.

\item[(iii)]  For any given finite $T>0$ and any given
$\varphi \in\mathcal{B}_c$,
$\Sigma (\cdot)\varphi:[0,T]\to\mathcal{B}_c$ is H\"{o}lder continuous
with exponent $\theta=1/2$ and the coefficient of H\"{o}lder continuity, $K(\varphi):\mathcal{B}_c\to(0,\infty)$, is a bounded function.

\item[(iv)]  For any $t\in[0,T]$, where $T>0$ is arbitrarily given,
$\Sigma (t)(\cdot):\mathcal{B}_c\to\mathcal{B}_c$ is a Lipschitz
continuous mapping and the Lipschitz constant
$L(t):[0,T]\to(0,\infty)$ is a bounded function.
\end{itemize}
Then there exists an exponential attractor $\mathscr{E}$ in $X$
for this semiflow $\{\Sigma (t)\}_{t\ge 0}$.
\end{lemma}

The next theorem is another main result in this paper and
it shows the existence of an exponential attractor for the
 solution semiflow $\{S(t)\}_{t\ge 0}$ in $H$ by the
approach through the existence of an $(H, E)$ global attractor
and Lemma \ref{L:EXatr}.

\begin{theorem}\label{T:exstexp}
For any positive parameters $d_1, d_2, a, b, D_1$ and $D_2$,
there exists an exponential attractor $\mathscr{E}$ in $H$ for
the solution semiflow $\{S(t)\}_{t\ge 0}$ generated by the
Brusselator evolutionary equation \eqref{eveq}.
\end{theorem}

\begin{proof}
By Theorem \ref{Thm2}, there exists an $(H,E)$ global attractor
$\mathscr{A}$, which is exactly the global attractor of
the Brusselator semiflow $\{S(t)\}_{t\ge 0}$ in $H$. Consequently,
 by \cite[Corollary 5.7]{yY10a}, there exists a compact,
positively invariant, absorbing set $\mathcal{B}_E$ in $H$,
which is a bounded set in $E$, for this semiflow.

Next we prove that the second condition in Lemma \ref{L:EXatr}
is satisfied by this Brusselator semiflow. Consider any two
points $g_1(0), g_2(0)\in\mathcal{B}_E$ and let
$g_i(t)=\left(u_i(t),v_i(t),w_i (t), z_i (t)\right)$, $i=1,2$,
be the corresponding solutions, respectively.
Let $y(t)=g_1(t)-g_2(t)$, $t\geq 0$. The associated norm quotient
of the difference  $g_1 - g_2$ of two trajectories,
where $g_1(0)\ne g_2(0)$, is given by
$$
    Q(t)=\frac{\|(-A)^{1/2}y(t)\|^2}{\|y(t)\|^2},\quad t\geq 0.
$$
Directly we can calculate
\begin{equation}
\label{qnest}
\begin{split}
&\frac12 \frac{d}{dt}Q(t)\\
& =\frac1{\|y(t)\|^4}\left[\inpt{(-A)^{1/2}y(t),(-A)^{1/2}y_t}\|y(t)\|^2
    -\|(-A)^{1/2}y(t)\|^2 \inpt{y(t),y_t}\right] \\
    &=\frac1{\|y(t)\|^2}\left[\inpt{(-A)y(t),y_t}-Q(t)
\inpt{y(t),y_t}\right] \\
    &=\frac1{\|y(t)\|^2}\inpt{(-A)y(t)-Q(t)y(t),Ay(t)
+F\left(g_1(t)\right)-F\left(g_2(t)\right)} \\
    &=\frac1{\|y(t)\|^2}\inpt{(-A)y(t)-Q(t)y(t),Ay(t)
+Q(t)y(t)+F\left(g_1(t)\right)-F \left(g_2(t)\right)} \\
    &=\frac1{\|y(t)\|^2}\left[-\|Ay(t)+Q(t)y(t)\|^2
-\inpt{Ay(t)+Q(t)y(t),F \left(g_1(t)\right)-F \left(g_2(t)\right)}\right] \\
    &\leq\frac1{\|y(t)\|^2}
\Big(-\frac12\|Ay(t)+Q(t)y(t)\|^2+\frac12\|F
\left(g_1(t)\right)-F \left(g_2(t)\right)\|\Big)
\end{split}
\end{equation}
where we used the identity $- \inpt{Ay(t)+Q(t)y(t),Q(t)y(t)}=0$.
 Note that $\mathcal{B}_E$ is a bounded set in $E$ and that
$E\hookrightarrow [L^6(\Omega)]^4$ is a continuous imbedding so that there is a uniform constant $R>0$ only depending on $\mathcal{B}_E$ such that
\begin{equation}
\label{bR6}
    \|(u,v, w, z)\|_{L^6(\Omega)}^2\leq R,\quad\text{for any}\;(u,v, w, z)\in\mathcal{B}_E.
\end{equation}
It is seen that
\begin{equation} \label{fw}
\begin{split}
    &\|F\left(g_1(t)\right)-F\left(g_2(t)\right)\| \\
    &\leq \|-(b+1)\left(u_1-u_2\right)
    +\left(u_1^2v_1-u_2^2v_2\right)-D_1\left((u_1-u_2) - (w_1 - w_2)\right)\| \\
    &\quad +\|b\left(u_1-u_2\right)-\left(u_1^2v_1-u_2^2v_2\right)
    -D_2\left((v_1 - v_2) - (z_1 - z_2)\right)\| \\
    &\quad + \|-(b+1)\left(w_1-w_2\right)
    +\left(w_1^2z_1-w_2^2z_2\right)+D_1\left((u_1-u_2) - (w_1 - w_2)\right)\| \\
    &\quad +\|b\left(w_1-w_2\right)-\left(w_1^2z_1-w_2^2z_2\right)+D_2\left((v_1 - v_2) - (z_1 - z_2)\right)\|.
\end{split}
\end{equation}
Using the H\"{o}lder inequality, the imbedding inequality
$\|\cdot \|_{L^6}^2 \leq \eta \|\cdot \|_E^2$, and Poincar\'{e}
inequality orderly, we have
\begin{align*}
\|u_1-u_2\|^2
&\leq |\Omega|^{2/3}\|u_1-u_2\|_{L^6(\Omega)}^2\\
&\leq|\Omega|^{2/3}\eta\|\nabla(u_1-u_2)\|^2\\
&=c_1\|(-A)^{1/2}\left(g_1-g_2\right)\|^2,
\end{align*}
where $c_1=|\Omega|^{2/3} \eta\,d_1$. Similarly, we have
$$
    \|w_1-w_2\|^2\leq c_1\|(-A)^{1/2}\left(g_1-g_2\right)\|^2,
$$
and
$$
\|v_1-v_2\|^2\leq c_2\|(-A)^{1/2}\left(g_1-g_2\right)\|^2,
 \quad
\|z_1-z_2\|^2\leq c_2\|(-A)^{1/2}\left(g_1-g_2\right)\|^2,
$$
where $c_2=|\Omega|^{2/3} \eta\,d_2$. By the generalized H\"{o}lder
inequality and \eqref{bR6}, we have
\begin{align*}
&\|u_1^2v_1-u_2^2v_2\|^2\\
&\leq 2\|u_1-u_2\|_{L^6 (\Omega)}^2
    \|u_1+u_2\|_{L^6(\Omega)}^2\|v_2\|_{L^6(\Omega)}^2 +2\|v_1-v_2\|_{L^6(\Omega)}^2\|u_1\|_{L^6(\Omega)}^4 \\
&\leq 8R^2\|u_1-u_2\|_{L^6(\Omega)}^2+2R^2
  \|v_1-v_2\|_{L^6(\Omega)}^2 \\
& \leq c_3(R)\|(-A)^{1/2}\left(g_1-g_2\right)\|^2,
\end{align*}
and similarly,
$$
    \|w_1^2 z_1-w_2^2 z_2\|^2
\leq c_3(R)\|(-A)^{1/2}\left(g_1-g_2\right)\|^2,
$$
where  $c_3(R)=2 \eta \left(4d_1+d_2\right)R^2$.

Substituting these inequalities into \eqref{fw}, we obtain
\begin{equation}\label{Fw}
\begin{split}
&\|F\left(g_1(t)\right)-F\left(g_2(t)\right)\|\\
&\leq 2\left(\sqrt{c_1 (b + 1 + D_1)} + \sqrt{c_2\,D_2}
+ \sqrt{c_3(R)}\right)\|(-A)^{1/2}y(t)\|.
\end{split}
\end{equation}
Then substitution of \eqref{Fw} into \eqref{qnest} yields
\begin{equation}
\label{QN}
    \frac{d}{dt}\,Q(t)\leq \frac1{\|y(t)\|^2}\|F
\left(g_1(t)\right)-F\left(g_2(t)\right)\|
    \leq\rho(\mathcal{B}_E) Q(t),\quad t>0,
\end{equation}
where
$$
    \rho(\mathcal{B}_E)= 2\left(\sqrt{c_1 (b + 1 + D_1)}
+ \sqrt{c_2\,D_2} + \sqrt{c_3(R)}\right)
$$
is a positive constant only depending on $R$ which depends on
$\mathcal{B}_E$. Thus the second condition in Lemma \ref{L:EXatr}
is satisfied.

Now check the H\"{o}lder continuity of
$S(\cdot)g:[0,T]\to\mathcal{B}_E$ for any given
$g\in\mathcal{B}_E$ and any given compact interval $[0,T]$. By the
mild solution formula, for any $0\le t_1<t_2\le T$ we obtain
\begin{equation}
\label{vocs}
\begin{split}
        \|S(t_2)g-S\left(t_1\right)g\|
&\leq\|\big(e^{A(t_2-t_1)}-I\big)e^{At_1}g\|
        +\int_{t_1}^{t_2}\|e^{A\left(t_2-\sigma\right)}F(S(\sigma)g)\| d\sigma \\
 &\quad+\int_0^{t_1}\|\big(e^{A(t_2-t_1)}-I\big)
e^{A\left(t_1-\sigma\right)}F(S(\sigma)g)\| d\sigma.
\end{split}
\end{equation}
Since $\mathcal{B}_E$ is positively invariant with respect
to the Brusselator semiflow $\{S(t)\}_{t \geq 0}$ and
$\mathcal{B}_E$ is bounded in $E$, there exists a constant
$K_{\mathcal{B}_E}>0$ such that for any $g\in\mathcal{B}_E$, we have
$$
    \| S(t)g \|_E\leq K_{\mathcal{B}_E},\quad t\geq 0.
$$
Since $F:E\to H$ is locally Lipschitz continuous,
 there is a Lipschitz constant $L_{\mathcal{B}_E}>0$ of $F$
relative to this positively invariant set $\mathcal{B}_E$.
Moreover, by \cite[Theorem 37.5]{SY02}, for the analytic,
contracting, linear semigroup $\{e^{At}\}_{t\ge 0}$, there
exist positive constants $N_0$ and $N_1$ such that
$$
\|e^{At}g-g\|_H\leq N_0 \, t^{1/2}\|g\|_E,\quad
\text{for }t\geq 0,\; w\in E,
$$
and
$$
    \|e^{At}\|_{\mathcal{L}(H,E)}\leq N_1
  t^{-1/2},\quad\text{for } t>0.
$$
It follows that
$$
  \|\big(e^{A(t_2-t_1)}-I\big)e^{At_1}g\|
\le N_0 (t_2-t_1)^{1/2}K_{\mathcal{B}_E}
$$
and
$$
    \int_{t_1}^{t_2}\|e^{A\left(t_2-\sigma\right)}F(S(\sigma)g)\|d\sigma
    \le\int_{t_1}^{t_2}\frac{N_1 L_{\mathcal{B}_E}K_{\mathcal{B}_E}}{\sqrt{t_2-\sigma}}\, d\sigma
    =2K_{\mathcal{B}_E}L_{\mathcal{B}_E}N_1 (t_2-t_1)^{1/2}.
$$
Moreover,
\begin{align*}
    \int_0^{t_1}\|\big(e^{A(t_2-t_1)}-I\big)e^{A\left(t_1-\sigma\right)} F(S(\sigma)g)\| d\sigma
    &\leq N_0 (t_2-t_1)^{1/2}\int_0^{t_1}
\frac{N_1 L_{\mathcal{B}_E}K_{\mathcal{B}_E}}{\sqrt{t_1-\sigma}}\, d\sigma \\
    &=2K_{\mathcal{B}_E}L_{\mathcal{B}_E}N_0 N_1 \sqrt{T}(t_2-t_1)^{1/2}.
\end{align*}
Substituting the above three inequalities into \eqref{vocs}, we obtain
\begin{equation}
\label{holc}
    \|S(t_2)g-S\left(t_1\right)g\|
    \le K_{\mathcal{B}_E}\left(N_0 + 2L_{\mathcal{B}_E}N_1 (1+N_0 \sqrt{T})\right)
    (t_2-t_1)^{1/2},
\end{equation}
for $0\le t_1<t_2\le T$. Thus the third condition in Lemma \ref{L:EXatr} is satisfied. Namely, for any given $T>0$, the mapping $S(\cdot)g:[0,T]\to\mathcal{B}_E$ is H\"{o}lder continuous with the exponent $1/2$ and with a uniformly bounded coefficient independent of $g\in\mathcal{B}_E$.

We can use Theorem 47.8 (specifically (47.20) therein) in \cite{SY02}
to confirm the Lipschitz continuity of the mapping
$S(t)(\cdot):\mathcal{B}_E\to\mathcal{B}_E$ for any $t\in[0,T]$
where $T>0$ is arbitrarily given. Thus the fourth condition
in Lemma \ref{L:EXatr} is also satisfied. Finally, we apply
Lemma \ref{L:EXatr} to reach the conclusion of this theorem.
\end{proof}


\subsection*{Acknowledgments}
The author is partly supported by the National Science Foundation
under the grant NSF-DMS-1010998.

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\end{document}