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

\AtBeginDocument{{\noindent\small 
\emph{Electronic Journal of Differential Equations}, 
Vol. 2011 (2011), No. 46, pp. 1--21.\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{8mm}}

\begin{document}
\title[\hfilneg EJDE-2011/46\hfil Cahn-Hilliard-Boussinesq system]
{Large time behavior of a Cahn-Hilliard-Boussinesq system on a
bounded domain}

\author[K. Zhao\hfil EJDE-2011/46\hfilneg]
{Kun Zhao}

\address{Kun Zhao \newline
Mathematical Biosciences Institute, Ohio State University,
Columbus, OH 43210, USA}
\email{kzhao@mbi.ohio-state.edu}

\thanks{Submitted April 11, 2010. Published April 6, 2011.}
\subjclass[2000]{35Q35, 35B40}
\keywords{Cahn-Hilliard-Boussinesq equations; classical solution;
\hfill\break\indent asymptotic behavior}

\begin{abstract}
 We study the asymptotic behavior of classical solutions to an
 initial-boundary value problem (IBVP) for a coupled
 Cahn-Hilliard-Boussinesq system on bounded domains with
 large initial data. A sufficient condition is established
 under which the solutions decay exponentially to constant
 states as time approaches infinity.
\end{abstract}

\maketitle \numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}


\section{Introduction}

As one of the fundamental modelling equations, the Cahn-Hilliard
equation \cite{C,CH} plays an important role in the mathematical
study of multi-phase flows, and has been studied intensively in
the literature both analytically and numerically (see e.g.
\cite{ABP,BF,CGS,DD,EG,KW,KKL,LT,M,O,WW}). The couplings of the
Cahn-Hilliard equation with other basic modelling equations have
been proposed in various situations to study complicated phenomena
in fluid mechanics involving phase transition. For example, the
coupled Cahn-Hilliard-Navier-Stokes (CHNS) system and its
variations, which describe the motion of an incompressible
two-phase flow under shear through an order parameter formulation,
have been used in order to understand the phenomena of phase
transition in incompressible fluid flows (c.f.
\cite{Feng,GPV,LS}). Recently, a closely related model to the CHNS
system has been developed in \cite{CFLW,CLW,GLL,GLL-1} to
understand the spinodal decomposition of binary fluid in a
Hele-Shaw cell, tumor growth, cell sorting, and two phase flows in
porous media, which is referred as the Cahn-Hilliard-Hele-Shaw
(CHHS) system. In this paper, we consider the following system of
equations:
\begin{equation} \label{e1.1}
\begin{gathered}
 \phi_t+U\cdot\nabla\phi=\Delta\mu,\quad \mathbf{x}\in\mathbb{R}^n,\;
  t>0,\\
 \mu=-\alpha\Delta\phi+F'(\phi),\\
 U_t+U\cdot\nabla U+\nabla P=\mu\nabla\phi+\theta{\textbf{e}_n},\\
 \theta_t+U\cdot\nabla\theta=\kappa\Delta\theta,\\
 \nabla\cdot U=0,
\end{gathered}
\end{equation}
which is a system of strongly coupled partial differential
equations  obtained by coupling the Cahn-Hilliard equation to the
inviscid heat-conductive Boussinesq equations. It describes the
motion of an incompressible inviscid two-phase flow subject to
convective heat transfer under the influence of gravitational
force through order parameter formulation. Here,
$U=(u_1,\dots,u_n)$, $P$ and $\theta$ denotes the velocity,
pressure and temperature respectively. $\phi$ is the order
parameter and $\mu$ is a chemical potential derived from a
coarse-grained study of the free energy of the fluid (c.f.
\cite{GSS}). The constant $\kappa>0$ and $\alpha>0$ models heat
conduction and diffusion respectively, and $\textbf{e}_n$ is the
$n$-th unit vector in $\mathbb{R}^n$. The function $F$ usually has
a physical-relevant, double-well structure, each of them
representing the two phases of the fluid. A typical example of $F$
takes the form (c.f. \cite{CV,GSS}): $F(z)=\frac{1}{4}(z^2-1)^2$.
In this paper, we consider a general scenario by imposing
appropriate growth conditions on $F$. We remark that, system
\eqref{e1.1} reduces to the CHHS model if the temperature equation
and the hydrodynamic effect are dropped. On the other hand,
\eqref{e1.1} becomes the CHNS system if the temperature equation is
removed and the viscosity of fluid is added to the velocity
equation.


In the real world, flows often move in bounded domains with
constraints from boundaries, where the initial-boundary value
problems appear. The solutions of the initial-boundary value
problems usually exhibit different behaviors and much richer
phenomena comparing with the Cauchy problem. In this paper, we
consider system \eqref{e1.1} on a bounded domain in $\mathbb{R}^n$.
The system is supplemented by the following initial and boundary
conditions:
\begin{equation} \label{e1.2}
\begin{gathered}
(\phi,\mu,U,\theta)(\mathbf{x},0)=(\phi_0,\mu_0,U_0,\theta_0)(\mathbf{x}),\\
\nabla\phi\cdot\textbf{n}|_{\partial\Omega}=0,\quad
 \nabla\mu\cdot\textbf{n}|_{\partial\Omega}=0,\quad
 U\cdot\mathbf{n}|_{\partial\Omega}=0,\quad
 \theta|_{\partial\Omega}=\bar{\theta},
\end{gathered}
\end{equation}
where $\Omega\subset\mathbb{R}^n$ is a bounded domain with smooth
boundary $\partial\Omega$, $\mathbf{n}$ is the unit outward normal
to $\partial\Omega$ and $\bar{\theta}$ is a constant.

The initial-boundary value problem \eqref{e1.1}--\eqref{e1.2} was
first studied in \cite{Z-1}, where the global existence and
uniqueness of classical solutions are established, for large
initial data with finite energy in 2D. However, the large time
asymptotic behavior of the solutions is not investigated due to
the lack of uniform-in-time estimates of the solutions. We give
definite answer to this unsolved issue in current paper for the 2D
case, based on new findings of the structure of the system.

Suggested by the conservation of total mass and the boundary
conditions, it is expected that the global attractors of $\phi$
and $\theta$ should be
$\bar{\phi}=\frac{1}{|\Omega|}\int_\Omega\phi_0(\mathbf{x})d\mathbf{x}$
and $\bar{\theta}$, respectively, due to diffusion and boundary
effects. In this paper, we provide a sufficient condition that
guarantees the decay of the solution. We will show that when the
diffusion coefficient $\alpha$ passes a threshold value determined
by $F$ and $\Omega$, the functions $\phi$ and $\theta$ will
converge exponentially in time to $\bar{\phi}$ and $\bar{\theta}$,
respectively, regardless of the magnitude of the initial
perturbation. To be precise, we shall assume that
$\alpha-F_3c_0>0$, where $F_3>0$ is a constant such that
$F''\ge-F_3$, and $c_0$ is the constant in Poincar\'e inequality
on $\Omega$. This condition is crucial in our analysis due to the
fact that it produces a positive constant multiple of
$\|\phi-\bar{\phi}\|^2_{H^2}$ which is one of the major
dissipative terms controlling the exponential decay of $\phi$. The
condition will trigger a chain reaction leading the energy
estimate performed in \cite{Z-1} to a whole new scenario. As
consequences of the convergence of $\phi$ and $\theta$, we will
show that the velocity and vorticity are uniformly bounded in
time.

Throughout this paper, $\|\cdot\|_{L^p}$, $\|\cdot\|_{L^\infty}$
and $\|\cdot\|_{W^{s,p}}$ denote the norms of the usual Lebesgue
measurable function spaces $L^p$ $(1\leq p<\infty)$, $L^\infty$ and
the usual Sobolev space $W^{s,p}$, respectively. For $p=2$, we
denote the norm $\|\cdot\|_{L^2}$ by $\|\cdot\|$ and
$\|\cdot\|_{W^{s,2}}$ by $\|\cdot\|_{H^s}$, respectively. The
function spaces under consideration are $C([0,T];H^r(\Omega))$ and
$L^2([0,T];H^s(\Omega))$, equipped with norms $\sup_{0\leq t\leq
T}\|\Psi(\cdot,t)\|_{H^r}$ and
$\big(\int_0^T\|\Psi(\cdot,\tau)\|_{H^s}^2d\tau\big)^{1/2}$,
respectively, where $r,s$ are positive integers. Unless specified,
$c_i$ will denote generic constants which are independent of
$\phi,\mu,U,\theta$ and $t$, but may depend on
$\alpha,\kappa,\Omega$ and initial data.


For the sake of completeness, we first state the results  obtained
in \cite{Z-1}.


\begin{theorem} \label{thm1.1}
Let $\Omega\subset \mathbb{R}^2$ be a bounded domain with
smooth boundary. Suppose
that $F(\cdot)$ satisfies the following conditions:
\begin{itemize}
\item $F(\cdot)$ is of $C^6$ class and $F(\cdot)\ge0$;
\item There exist constants $F_1,F_2>0$ such that $|F^{(n)}(\phi)|\leq F_1|\phi|^{p-n}+F_2$, $n=1,.\dots,6,\ \forall\,6\leq p<\infty$ and $\phi\in\mathbb{R}$;
\item There exists a constant $F_3\ge0$ such that $F''\ge -F_3$.
\end{itemize}
If the initial data $\phi_0(\mathbf{x})\in H^5(\Omega)$,
$\mu_0(\mathbf{x})\in H^3(\Omega)$ and
$(\theta_0(\mathbf{x}),U_0(\mathbf{x}))\in H^3(\Omega)$ are
compatible with the boundary conditions, then there exists a
unique solution $(\phi,\mu,\theta, U)$ of \eqref{e1.1}--\eqref{e1.2}
globally in time such that
\begin{gather*}
\phi\in C([0, T];H^5(\Omega))\cap L^2([0,T];H^7(\Omega)),\\
\mu\in C([0, T];H^3(\Omega))\cap L^2([0,T];H^5(\Omega)),
\end{gather*}
$U\in C([0,T];H^3(\Omega))$ and
$\theta\in C([0,T];H^3(\Omega))\cap L^2([0,T];H^4(\Omega))$ for
$0<T<\infty$.
\end{theorem}

The next theorem is the main result of this paper regarding the
large-time asymptotic behavior of the solution obtained in Theorem
\ref{thm1.1}.

\begin{theorem}\label{thm1.2}
Suppose that the assumptions in Theorem \ref{thm1.1} are in force
and assume that the constant $\alpha-F_3c_0>0$, where $c_0$ is the
constant in Poincar\'e inequality on $\Omega$. Then the solution
to \eqref{e1.1}--\eqref{e1.2} satisfies
\begin{gather*}
\|\phi(\cdot,t)-\bar{\phi}\|_{H^5}+\|\mu(\cdot,t)-F'(\bar{\phi})\|_{H^3}
+\|\theta(\cdot,t)-\bar{\theta}\|_{H^3}\leq \gamma e^{-\beta
t};\\
\|U(\cdot,t)\|_{W^{1,p}}\leq \gamma(p),\quad \forall\, 1\leq
p<\infty;\quad \|\omega(\cdot,t)\|_{L^\infty}\le\bar{\gamma},\quad
\forall\, t\ge0,
\end{gather*}
for some constants $\gamma,\beta,\gamma(p),\bar{\gamma}>0$
independent of $t$, where
$\bar{\phi}=\frac{1}{|\Omega|}\int_\Omega
\phi_0(\mathbf{x})d\mathbf{x}$ and $\omega=v_x-u_y$ is the 2D
vorticity.
\end{theorem}


\begin{remark}  \label{rmk1.1} \rm
It should be pointed out that, in the theorems obtained above, no
smallness restriction is put upon the initial data.
\end{remark}

\begin{remark}  \label{rmk1.2} \rm
We observe that, by assuming small initial perturbation around the
equilibrium state and by exploring the structure of the function
$F$, one can show the exponential decay of the solution. However,
the asymptotic result obtained in Theorem \ref{thm1.2} has an
obvious advantage over the case for small perturbation. Indeed,
Theorem \ref{thm1.2} provides a convenient criterion for
determining whether the solution collapses to a constant state as
time evolves. Based on the result, one only needs to measure the
volume of the domain, instead of measuring the ``smallness'' of
the initial perturbation which is usually laborious to perform, to
determine whether the solution decays or not when other system
parameters are fixed.
\end{remark}

\begin{remark}  \label{rmk1.3} \rm
It is well-known that the Cahn-Hilliard equation is an effective
model in the study of sharp interfaces in two-phase fluid flows.
However, based on our results, the order parameter $\phi$ tends to
a uniform constant $\bar{\phi}$ instead of $\pm1$. This suggests
that under the conditions of Theorem \ref{thm1.1} and Theorem
\ref{thm1.2}, the modelling equations \eqref{e1.1} indeed fail to
model the sharp interfacial phenomenon. Therefore, our results
exhibit some bifurcation phenomena on the effectiveness of the
modelling equations.
\end{remark}


The proof of Theorem \ref{thm1.2} involves a series of accurate
combinations of energy estimates. The estimates are delicate
mainly due to the coupling between the equations by convection,
gravitational force and boundary effects. Great efforts have been
made to simplify the proof. Current proof involves intensive
applications of Sobolev embeddings and Ladyzhenskaya type
inequalities, see Lemma \ref{lem2.1}. Roughly speaking, because of
the lack of the spatial derivatives of the solution at the
boundary, our energy framework proceeds as follows: We first apply
the standard energy estimate on the solution and the temporal
derivatives of the solution. We then apply standard results on
elliptic equations to recover estimates of the spatial
derivatives. Such a process will be repeated up to third order,
and then with the aid of the assumption on $\alpha$, the carefully
coupled estimates will be composed into a desired one leading to
the exponential decay of the solution.

The rest of the paper is organized as follows. In Section 2, we
give some basic facts that will be used in the proof of Theorem
\ref{thm1.2}. In Section 3, we prove some uniform-in-time energy
estimates of the solution based on which the combinations of
energy estimates will be performed. We then complete the proof of
Theorem \ref{thm1.2} in Section 4.

\section{Preliminaries}

In this section, we shall collect several facts which will be used
in the proof of Theorem \ref{thm1.2}. First, we recall some
inequalities of Sobolev and Ladyzhenskaya type (c.f.
\cite{Adam,LSU}).



\begin{lemma}  \label{lem2.1}
Let $\Omega\subset \mathbb{R}^2$ be
any bounded domain with smooth boundary $\partial\Omega$. Then
\begin{itemize}
\item[(i)] $ \|f\|_{L^{\infty}}\leq c_1\|f\|_{H^2}$;

\item[(ii)] $\|f\|_{L^{\infty}}\leq c_2\|f\|_{W^{1,p}}$ for all $p>2$;

\item[(iii)] $\|f\|_{L^p}\leq c_3\|f\|_{H^1}$ for all $1\leq p<\infty$;

\item[(iv)] $ \|f\|^2_{L^4}\leq c_4\big(\|f\|\|\nabla
f\|+\|f\|^2\big)$;

\item[(v)] $\|f\|^2_{L^8}\leq c_5\big(\|f\|\|\nabla f\|_{L^4}
+\|f\|^2\big)$,
\end{itemize}
for some constants $c_i=c_i(p,\Omega)$, $i=1,\dots,5$.
\end{lemma}

Next, we recall some classical results on elliptic equations (c.f.
\cite{A-D-N,L-M-1,L-M-2}), which are useful in the estimation of
$\theta$.


\begin{lemma} \label{lem2.2}
 Let $\Omega\subset \mathbb{R}^2$ be
any bounded domain with smooth boundary $\partial\Omega$. Consider
the Dirichlet problem
\begin{gather*}
\kappa\Delta \Theta=f\quad\text{in } \Omega,\\
\Theta=0\quad\text{on } \partial\Omega.
 \end{gather*}
If $f\in W^{m,p}$, then $\Theta\in W^{m+2,p}$ and there exists a
constant $c_6=c_6(p,\kappa,m,\Omega)$ such that
\begin{equation*}
\|\Theta\|_{W^{m+2,p}}\leq c_6\|f\|_{W^{m,p}}
\end{equation*}
for any $p\in (1, \infty)$  and the integer $m\ge -1$.
\end{lemma}

The next lemma is useful for the estimation of the velocity field
(see \cite{B-B}).

\begin{lemma}  \label{lem2.3}
Let $\Omega\subset \mathbb{R}^2$ be
any bounded domain with smooth boundary $\partial\Omega$, and let
$F\in W^{s,p}(\Omega)$ be a vector-valued function satisfying
$F\cdot \mathbf{n}|_{\partial\Omega}=0$, where $\mathbf{n}$ is the
unit outward normal to $\partial\Omega$. Then there exists a
constant $c_7=c_7(s,p,\Omega)$ such that
\[
\|F\|_{W^{s,p}}\leq c_7(\|\nabla\times F\|_{W^{s-1,p}}+\|\nabla\cdot
F\|_{W^{s-1,p}}+\|F\|_{L^p})
\]
for any $s\ge 1$ and $p\in(1,\infty)$.
\end{lemma}


Finally, we recall some Poincar\'e type inequalities, which will
be used in the estimation of $\phi$, whose proof is
straightforward.

\begin{lemma} \label{lem2.4}
Let $\Omega\subset \mathbb{R}^n$ be any bounded domain with smooth
boundary $\partial\Omega$. Then, for any function $H^s(\Omega)\ni
f:\Omega\to\mathbb{R}$, there exists a constant
$c_8=c_8(s,\Omega)>0$ such that
\begin{itemize}
\item[(1)] $\|f-\bar{f}\|_{H^{2s}}\leq c_8\|\Delta^s f\|$
and $\|f-\bar{f}\|_{H^{2s+1}}\leq c_8\|\nabla\Delta^s f\|$,
$s\ge1$, if $ \nabla f\cdot\mathbf{n}|_{\partial\Omega}=0$;

\item[(2)] $\|f\|_{H^{2s}}\leq c_8\|\Delta^s f\|$ and
$\|f\|_{H^{2s+1}}\leq c_8\|\nabla\Delta^s f\|$, $s\ge1$, if
$f|_{\partial\Omega}=0$,
\end{itemize}
where $\bar{f}=\frac{1}{|\Omega|}\int_\Omega fd\mathbf{x}$.
\end{lemma}

\section{Uniform energy estimates}

In this section, we establish some uniform-in-time energy
estimates of the solution under the condition $\alpha-F_3c_0>0$,
based on  which the exponential decay rate of the solution will be
proved. The results are stated as a sequence of lemmas and the
proofs are carried out by carefully exploring the condition
$\alpha-F_3c_0>0$ and delicate applications of Cauchy-Schwarz and
Gronwall inequalities.

To study the asymptotic behavior, we first reformulate the
original problem to get the one for the perturbations. For this
purpose, let $\Phi=\phi-\bar{\phi}$ and
$\Theta=\theta-\bar{\theta}$. After plugging $\Phi$ and $\Theta$
into \eqref{e1.1} we obtain
\begin{equation}  \label{e3.1}
\begin{gathered}
 {\Phi}_t+U\cdot\nabla{\Phi}=\Delta\mu,\\
 \mu=-\alpha\Delta{\Phi}+F'(\phi),\\
 U_t+U\cdot\nabla U+\nabla \tilde{P}=\mu\nabla{\Phi}+\Theta{\textbf{e}_2},\\
 \Theta_t+U\cdot\nabla\Theta=\kappa\Delta\Theta,\\
 \nabla\cdot U=0,
\end{gathered}
\end{equation}
which is equivalent to \eqref{e1.1} for sufficiently smooth
solutions, where $\tilde{P}=P-\bar{\theta}y$, and the initial and
boundary conditions become
\begin{equation}  \label{e3.2}
\begin{gathered}
(\Phi,\mu,U,\Theta)(\mathbf{x},0)=(\Phi_0,\mu_0,U_0,\Theta_0)(\mathbf{x})\equiv (\phi_0-\bar{\phi},\mu_0,U_0,\theta_0-\bar{\theta})(\mathbf{x}),\\
\nabla\Phi\cdot\mathbf{n}|_{\partial\Omega}=0,\quad
\nabla\mu\cdot\mathbf{n}|_{\partial\Omega}=0,\quad
U\cdot\mathbf{n}|_{\partial\Omega}=0,\quad
\Theta|_{\partial\Omega}=0.
\end{gathered}
\end{equation}

We begin with the uniform estimate of $\|\Phi\|_{L^2}$.

\begin{lemma} \label{lem3.1}
 Under the assumptions of Theorem \ref{thm1.2}, it holds that
 \begin{equation}  \label{e3.3}
\|\Phi(\cdot,t)\|^2+\int_0^t\|\Phi(\cdot,\tau)\|_{H^2}^2d\tau\leq
c_{9},\quad \forall\,t\ge0.
\end{equation}
\end{lemma}

\begin{proof}
Taking $L^2$ inner product of \eqref{e3.1}$_1$ with $\Phi$, after
integrating by parts we have
\begin{equation}  \label{e3.4}
\frac{1}{2}\frac{d}{dt}\|\Phi\|^2 =\int_{\Omega}\Phi\Delta\mu \mathbf{x}
=-\int_{\Omega}\nabla\mu\cdot\nabla\Phi d\mathbf{x}
=-\alpha\|\Delta\Phi\|^2-\int_{\Omega}F''(\phi)|\nabla\Phi|^2d\mathbf{x},
\end{equation}
which gives
\begin{equation}  \label{e3.5}
\frac{1}{2}\frac{d}{dt}\|\Phi\|^2+\alpha\|\Delta\Phi\|^2
\leq F_3\|\nabla\Phi\|^2,
\end{equation}
where we have used the condition on $F$. Since
$\Phi=\phi-\bar{\phi}$ and $\bar{\phi}$ is a constant, using the
boundary conditions, Cauchy-Schwarz and Poincar\'e inequalities we
have
\begin{equation}  \label{e3.6}
\|\nabla\Phi\|^2=-\int_\Omega \Phi\Delta\Phi d\mathbf{x}
\leq \frac{1}{2c_0}\|\Phi\|^2+\frac{c_0}{2}\|\Delta \Phi\|^2
\leq \frac{1}{2}\|\nabla\Phi\|^2+\frac{c_0}{2}\|\Delta \Phi\|^2,
\end{equation}
which implies
\begin{equation}  \label{e3.7}
\|\nabla\Phi\|^2\leq c_0\|\Delta \Phi\|^2,
\end{equation}
where $c_0$ is the constant in Poincar\'e inequality on $\Omega$.
Let $\alpha_1\equiv \alpha-F_3c_0>0$.
 Substituting \eqref{e3.7} in \eqref{e3.5} we
have
\begin{equation} \label{e3.8}
\frac{1}{2}\frac{d}{dt}\|\Phi\|^2+\alpha_1\|\Delta\Phi\|^2\le0.
\end{equation}
Upon integrating \eqref{e3.8} in time over $[0,t]$ and using
Lemma \ref{lem2.4} we obtain \eqref{e3.3}. This completes the proof.
\end{proof}


\begin{remark}   \label{rmk3.1} \rm
The estimate \eqref{e3.8} already implies the decay of $\|\Phi\|^2$.
 However, our ultimate goal is to show the decay rate of the
higher order derivatives of the solution. Hence, for the sake of
completeness, we leave the proof of the decay rate in the next section.
\end{remark}


Next, we prove uniform estimates of $\Theta$, which will be used to settle down the
uniform bound of $\|U\|_{H^1}$.


\begin{lemma} \label{lem3.2}
 Under the assumptions of Theorem \ref{thm1.2}, there exists a constant
$\beta_0$ independent of $t$ such that for any $t\ge0$, it holds that
\begin{equation} \label{e3.9}
 \|\Theta(\cdot,t)\|^2\leq \|\Theta_0\|^2e^{-2\beta_0 t},\quad
\int_0^t\|\nabla\Theta(\cdot,\tau)\|^2e^{\beta_0 \tau}d\tau\leq
\frac{1}{\kappa}\|\Theta_0\|^2.
\end{equation}
\end{lemma}

\begin{proof}
Taking the $L^2$ inner product of \eqref{e3.1}$_4$ with $\Theta$ we have
\begin{equation} \label{e3.10}
 \frac{1}{2}\frac{d}{dt}\|\Theta\|^2+\kappa\|\nabla\Theta\|^2=0.
\end{equation}
Since $\Theta|_{\partial\Omega}=0$, Poincar\'e's inequality
implies
\begin{equation} \label{e3.11}
 \frac{d}{dt}\|\Theta\|^2+\frac{2\kappa}{c_0}\|\Theta\|^2\le0,
\end{equation}
which yields immediately
\begin{equation} \label{e3.12}
 \|\Theta(\cdot,t)\|^2\leq \|\Theta_0\|^2e^{-2\beta_0 t},
\end{equation}
where $\beta_0=\kappa/c_0$. This proves the first part
of \eqref{e3.9}.

Next, we multiply \eqref{e3.10} by $e^{\beta_0 t}$ and
use \eqref{e3.12} to obtain
\begin{equation} \label{e3.13}
\frac{d}{dt}\big(e^{\beta_0 t}\|\Theta\|^2\big)+2\kappa e^{\beta_0
t}\|\nabla\Theta\|^2\leq \beta_0e^{-\beta_0 t}\|\Theta_0\|^2.
\end{equation}
For any $t>0$, upon integrating \eqref{e3.13} in time we obtain
\begin{equation} \label{e3.14}
e^{\beta_0
t}\|\Theta(\cdot,t)\|^2-\|\Theta_0\|^2+2\kappa\int_0^te^{\beta_0
\tau}\|\nabla\Theta(\cdot,\tau)\|^2d\tau \leq \big(1-e^{-\beta_0
t}\big)\|\Theta_0\|^2,
\end{equation}
which implies the second part of \eqref{e3.9} immediately.
This completes the proof.
\end{proof}

With the help of Lemma \ref{lem3.2} we now prove the uniform estimates
of $\|U\|_{H^1}$ and $\|\Phi\|_{H^1}$.

\begin{lemma} \label{lem3.3}
 Under the assumptions of Theorem \ref{thm1.2}, for all $t\ge0$,
 \begin{equation} \label{e3.15}
\|U(\cdot,t)\|^2_{H^1}+\|\Phi(\cdot,t)\|^2_{H^1}
+\int_0^t\big(\|\nabla\mu(\cdot,\tau)\|^2
+\|\Phi(\cdot,\tau)\|^2_{H^3}\big)d\tau\leq c_{10}.
\end{equation}
\end{lemma}

\begin{proof}
{\bf Step 1.} Note that due to Lemma \ref{lem2.3}
and the boundary condition on $U$,
it suffices to estimate $\|U\|^2$ and $\|\omega\|^2$, in order
estimate $\|U(\cdot,t)\|^2_{H^1}$.
Taking the $L^2$ inner product of \eqref{e3.1}$_3$ with $U$ we have
\begin{equation} \label{e3.16}
 \frac{1}{2}\frac{d}{dt}\|U\|^2=\int_{\Omega}\mu(\nabla\Phi\cdot
U)d\textbf{x}+\int_{\Omega}\Theta \mathbf{e}_2\cdot Ud\textbf{x}.
\end{equation}
Taking $L^2$ inner product of $\eqref{e3.1}_1$ with $\mu$ we have
\begin{equation} \label{e3.17}
 \frac{d}{dt}\Big(\frac{\alpha}{2}\|\nabla\Phi\|^2
+\int_{\Omega}F(\phi)d\textbf{x}\Big)+\|\nabla\mu\|^2
=-\int_{\Omega}\mu(\nabla\Phi\cdot U)d\textbf{x}.
\end{equation}
Adding \eqref{e3.16} and \eqref{e3.17}, we obtain
\begin{equation} \label{e3.18}
 \frac{d}{dt}\Big(\frac{1}{2}\|U\|^2+\frac{\alpha}{2}\|\nabla\Phi\|^2
+\int_{\Omega}F(\phi)d\textbf{x}\Big)
+\|\nabla\mu\|^2=\int_{\Omega}\Theta \mathbf{e}_2\cdot U d\mathbf{x}.
\end{equation}
Applying Cauchy-Schwarz inequality to the right-hand side
of \eqref{e3.18} and using \eqref{e3.12}, we obtain
\begin{equation} \label{e3.19}
 \frac{d}{dt}\Big(\frac{1}{2}\|U\|^2+\frac{\alpha}{2}\|\nabla\Phi\|^2+\int_{\Omega}F(\phi)d\textbf{x}\Big)
+\|\nabla\mu\|^2\leq e^{-\beta_0t}\|U\|^2+
e^{-\beta_0t}\|\Theta_0\|^2.
\end{equation}
After dropping $\|\nabla\mu\|^2$ from the left hand side (LHS) of
\eqref{e3.19}, we have
\begin{equation} \label{e3.20}
\frac{d}{dt}\Big(\frac{1}{2}\|U\|^2+\frac{\alpha}{2}\|\nabla\Phi\|^2+\int_{\Omega}F(\phi)d\textbf{x}\Big)
\leq e^{-\beta_0t}\|U\|^2+
e^{-\beta_0t}\|\Theta_0\|^2.
\end{equation}
Since $F\ge0$, Gronwall's inequality then gives
\begin{equation} \label{e3.21}
 \frac{1}{2}\|U\|^2+\frac{\alpha}{2}\|\nabla\Phi\|^2
+\int_{\Omega}F(\phi)d\textbf{x}\leq c_{11},\quad \forall\,
t\ge0.
\end{equation}
Applying \eqref{e3.21} to \eqref{e3.19} and integrating with
respect to $t$ we obtain
\begin{equation} \label{e3.22}
\|U\|^2+\|\Phi\|^2_{H^1}+\int_{\Omega}F(\phi)d\textbf{x}
+\int_0^t\|\nabla\mu(\cdot,\tau)\|^2d\tau\leq c_{12},\quad \forall\,
t\ge0.
\end{equation}

\noindent {\bf Step 2.} By the definition of $\Phi$,
Lemma \ref{lem2.4} and
\eqref{e3.1}$_2$, we observe that
\begin{equation} \label{e3.23}
\|\Phi\|^2_{H^3}\leq c_8\|\nabla(\Delta\Phi)\|^2
\leq c_{13}\big(\|\nabla\mu\|^2+\|F''(\phi)\nabla\Phi\|^2\big).
\end{equation}
Using the condition on $F$, H\"older inequality, Lemma
\ref{lem2.1} (iii) and \eqref{e3.22} we have
\begin{equation} \label{e3.24}
\begin{aligned}
\|F''(\phi)\nabla\Phi\|^2
&\leq c_{14}\big(\|\phi\|^{2(p-2)}_{L^{4(p-2)}}
\|\nabla\Phi\|_{L^4}^2+\|\nabla\Phi\|^2\big)\\
&\leq c_{15}\big((\|\Phi\|^{2(p-2)}_{H^1}+\|\bar{\phi}\|^{2(p-2)}_{H^1})
\|\nabla\Phi\|_{H^1}^2+\|\nabla\Phi\|^2\big)\\
&\leq c_{16}\|\Phi\|^2_{H^2}.
\end{aligned}
\end{equation}
substituting \eqref{e3.24} in \eqref{e3.23} we have
\begin{equation} \label{e3.25}
\begin{aligned}
\|\Phi\|^2_{H^3}\leq c_{17}\big(\|\nabla\mu\|^2+\|\Phi\|^2_{H^2}\big),
\end{aligned}
\end{equation}
which, together with \eqref{e3.3} and \eqref{e3.22}, implies
\begin{equation} \label{e3.26}
\int_0^t\|\Phi(\cdot,\tau)\|^2_{H^3}d\tau\leq c_{18},\quad \forall\,
t\ge0.
\end{equation}

\noindent {\bf Step 3.} By taking the curl of the velocity equation, we
obtain
\begin{equation} \label{e3.27}
 \omega_t+U\cdot\nabla\omega=\mu_x\Phi_y-\mu_y\Phi_x+\Theta_x,
\end{equation}
where $\omega=v_x-u_y$ is the 2D vorticity. Taking $L^2$ inner
product of \eqref{e3.27} with $\omega$ and applying H\"older inequality we have
\begin{equation} \label{e3.28}
\frac{d}{dt}\|\omega\|\leq
2\|\nabla\mu\|\|\nabla\Phi\|_{L^\infty}+\|\nabla\Theta\|.
\end{equation}
Upon integrating \eqref{e3.28} in time using H\"older
and Sobolev inequalities we have
\begin{equation} \label{e3.29}
\begin{aligned}
\|\omega(\cdot,t)\|
&\le\int_0^t \big(2\|\nabla\mu\|\|\nabla\Phi\|_{L^\infty}+\|\nabla\Theta\|\big)d\tau+\|\omega_0\|\\
&\leq c_{19}\Big(\int_0^t
\|\nabla\mu\|^2d\tau\Big)^{1/2}\Big(\int_0^t
\|\Phi\|^2_{H^3}d\tau\Big)^{1/2}\\
&\quad + \Big(\int_0^t e^{\beta_0
\tau/2}\|\nabla\Theta\|^2d\tau\Big)^{1/2}\Big(\int_0^t
e^{-\beta_0 \tau/2}d\tau\Big)^{1/2}+\|\omega_0\|.
\end{aligned}
\end{equation}
Since the right hand side of \eqref{e3.29} is uniformly bounded
in time by virtue of previous estimates, we have
\begin{equation} \label{e3.30}
\|\omega(\cdot,t)\|\leq c_{20},\quad \forall\,t\ge0.
\end{equation}
Thus, \eqref{e3.15} follows from \eqref{e3.22}, \eqref{e3.26}
and \eqref{e3.30}. This completes the proof.
\end{proof}

With the aid of Lemma \ref{lem3.3} we are now able to improve the estimates of
$\Phi$ and $\mu$. Due to the lack of spatial derivatives of
the solution on $\partial\Omega$, we shall alternatively work
on the temporal derivatives and use an iteration program
to recover the spatial derivatives.

\begin{lemma} \label{lem3.4}
 Under the assumptions of Theorem \ref{thm1.2}, it holds that
\begin{equation} \label{e3.31}
\|\Phi(\cdot,t)\|_{H^2}^2+\|\mu(\cdot,t)\|^2
+\int_0^t\big(\|\Phi_t\|^2+\|\nabla\mu\|^2_{H^1}\big)d\tau\leq c_{21},
\quad \forall\,t\ge0.
\end{equation}
\end{lemma}

\begin{proof}
{\bf Step 1.} By taking $L^2$ inner product of \eqref{e3.1}$_1$
with $\Phi_t$ we have
\begin{equation} \label{e3.32}
\|\Phi_t\|^2+\int_{\Omega}\Phi_t(U\cdot\nabla\Phi)d\mathbf{x}
=\int_{\Omega}\Phi_t\Delta\mu
d\mathbf{x}.
\end{equation}
Using the boundary conditions we calculate the RHS of \eqref{e3.32} as:
\begin{equation} \label{e3.33}
\begin{aligned}
 \int_{\Omega}\Phi_t\Delta\mu d\mathbf{x}
=-\frac{d}{dt}\Big(\frac{\alpha}{2}\|\Delta\Phi\|^2
+\frac{1}{2}\int_{\Omega}F''(\phi)|\nabla\Phi|^2 d\mathbf{x}\Big)
+\frac{1}{2}\int_{\Omega}F'''(\phi)\Phi_t|\nabla\Phi|^2d\mathbf{x}.
\end{aligned}
\end{equation}
Substituting \eqref{e3.33} in \eqref{e3.32} we obtain
\begin{equation} \label{e3.34}
\begin{aligned}
&\frac{d}{dt}\Big(\frac{\alpha}{2}\|\Delta\Phi\|^2
 +\frac{1}{2}\int_{\Omega}F''(\phi)|\nabla\Phi|^2
d\mathbf{x}\Big)+\|\Phi_t\|^2\\
&=\frac{1}{2}\int_{\Omega}F'''(\phi)\Phi_t|\nabla\Phi|^2d\mathbf{x}
-\int_{\Omega}\Phi_t(U\cdot\nabla\Phi)d\mathbf{x}.
\end{aligned}
\end{equation}

Using Cauchy-Schwarz inequality and Lemma \ref{lem3.3} we
estimate the first term on the RHS of \eqref{e3.34} as
\begin{equation} \label{e3.35}
\begin{aligned}
\Big|\frac{1}{2}\int_{\Omega}F'''(\phi)\Phi_t|\nabla\Phi|^2d\mathbf{x}
\Big|
 &\leq \frac{1}{4}\|\Phi_t\|^2+\frac{1}{4}\int_{\Omega}|F'''(\phi)|^2|\nabla\Phi|^4d\mathbf{x}\\
 &\leq \frac{1}{4}\|\Phi_t\|^2+c_{22}\|\phi\|^{2(p-3)}_{L^{4(p-3)}}\|\nabla\Phi\|^4_{L^8}+c_{22}\|\nabla\Phi\|^4_{L^4}\\
 &\leq \frac{1}{4}\|\Phi_t\|^2+c_{23}\big(\|\nabla\Phi\|^4_{L^8}+\|\nabla\Phi\|^4_{L^4}\big).
\end{aligned}
\end{equation}
Lemma \ref{lem2.1} (iii)--(v) and Lemma \ref{lem3.3} then give
\begin{equation} \label{e3.36}
\begin{aligned}
 \|\nabla\Phi\|^4_{L^4}+\|\nabla\Phi\|^4_{L^8}&\leq c_{24}\big(\|\nabla\Phi\|^2\|\nabla^2\Phi\|^2+\|\nabla\Phi\|^4+\|\nabla\Phi\|^2\|\nabla^2\Phi\|_{L^4}^2+\|\nabla\Phi\|^4\big)\\
&\leq c_{25}\big(\|\nabla^2\Phi\|^2+\|\nabla\Phi\|^2+\|\nabla^2\Phi\|^2_{H^1}\big)\\
&\leq c_{26}\|\Phi\|^2_{H^3}.
\end{aligned}
\end{equation}
So we update \eqref{e3.35} as
\begin{equation} \label{e3.37}
\begin{aligned}
\Big|\frac{1}{2}\int_{\Omega}F'''(\phi)\Phi_t|\nabla\Phi|^2d\mathbf{x}
\Big|\leq \frac{1}{4}\|\Phi_t\|^2+c_{27}\|\Phi\|^2_{H^3}.
\end{aligned}
\end{equation}

The second term on the RHS of \eqref{e3.34} is estimated as
\begin{equation} \label{e3.38}
\begin{aligned}
\Big|-\int_{\Omega}\Phi_t(U\cdot\nabla\Phi)d\mathbf{x}\Big|
&\leq \frac{1}{4}\|\Phi_t\|^2+c_{28}\|U\|_{H^1}^2\|\nabla\Phi\|^2_{H^1}\\
&\leq \frac{1}{4}\|\Phi_t\|^2+c_{29}\|\Phi\|_{H^2}^2,
\end{aligned}
\end{equation}
where we have used Lemma \ref{lem3.3}. Combining \eqref{e3.34}, \eqref{e3.37}
and \eqref{e3.38} we have
\begin{equation} \label{e3.39}
\frac{d}{dt}\Big(\frac{\alpha}{2}\|\Delta\Phi\|^2
 +\frac{1}{2}\int_{\Omega}F''(\phi)|\nabla\Phi|^2
d\mathbf{x}\Big)+\frac{1}{2}\|\Phi_t\|^2\leq c_{30}\|\Phi\|_{H^3}^2.
\end{equation}
Upon integrating \eqref{e3.39} in time and using \eqref{e3.26} we have
\begin{equation} \label{e3.40}
\frac{\alpha}{2}\|\Delta\Phi\|^2
+\frac{1}{2}\int_{\Omega}F''({\phi})|\nabla\Phi|^2
d\textbf{x}+\frac{1}{2}\int_0^t\|\Phi_t\|^2d\tau\leq c_{31}.
\end{equation}
Since $F''\ge -F_3$, we have
\begin{equation} \label{e3.41}
\int_{\Omega}F''({\phi})|\nabla\Phi|^2
d\textbf{x}\ge -F_3\|\nabla\Phi\|^2\ge -F_3c_0\|\Delta\Phi\|^2,
\end{equation}
where we have used \eqref{e3.6}. Substituting \eqref{e3.41} in
 \eqref{e3.40} we have
\begin{equation*}
\frac{{\alpha_1}}{2}\|\Delta\Phi\|^2
+\frac{1}{2}\int_0^t\|\Phi_t\|^2d\tau\leq
c_{31},
\end{equation*}
which, together with Lemma \ref{lem2.4}, implies
\begin{equation} \label{e3.42}
\|\Phi\|^2_{H^2}+\int_0^t\|\Phi_t\|^2d\tau\leq c_{32}.
\end{equation}

{\bf Step 2.} We derive some consequences of \eqref{e3.42}. From
\eqref{e3.1}$_2$ and Lemma \ref{lem2.1} (i) we see that
\begin{equation} \label{e3.43}
\begin{aligned}
\|\mu(\cdot,t)\|^2&\leq c_{33} \big(\|\Delta\Phi\|^2
+\|F'(\phi)\|^2\big)\\
&\leq c_{34} \big(\|\Delta\Phi\|^2+\|\phi\|^{2(p-1)}_{L^\infty}+1\big)\\
&\leq c_{35} \big(\|\Delta\Phi\|^2+\|\Phi\|^{2(p-1)}_{H^2}
+\|\bar{\phi}\|^{2(p-1)}_{L^\infty}+1\big).
\end{aligned}
\end{equation}
Therefore, using \eqref{e3.42} we have
\begin{equation} \label{e3.44}
\|\mu(\cdot,t)\|^2\leq c_{36},\quad \forall\,t\ge0.
\end{equation}

Since $\nabla\mu\cdot\mathbf{n}|_{\partial\Omega}=0$,
by Lemma \ref{lem2.4} and Lemma \ref{lem3.3} we have
\begin{equation} \label{e3.45}
\begin{aligned}
\|\nabla\mu\|_{H^1}^2\leq c_{37}\|\Delta\mu\|^2
&\leq c_{38}\big(\|\Phi_t\|^2+\|U\cdot\nabla\Phi\|^2\big)\\
&\leq c_{39}\big(\|\Phi_t\|^2+\|U\|_{H^1}^2\|\nabla\Phi\|_{H^1}^2\big)\\
&\leq c_{40}\big(\|\Phi_t\|^2+\|\Phi\|_{H^2}^2\big),
\end{aligned}
\end{equation}
which, together with \eqref{e3.15} and \eqref{e3.42}, implies that
\begin{equation} \label{e3.46}
\int_0^t\|\nabla\mu\|_{H^1}^2d\tau \leq c_{41}.
\end{equation}
Therefore, \eqref{e3.31} follows from \eqref{e3.42}, \eqref{e3.44} and \eqref{e3.46}. This completes
the proof.
\end{proof}

The next lemma gives the uniform estimate of $\|U_t(\cdot,t)\|^2$
whose proof requires more careful examination of the energy
estimate for the temperature.

\begin{lemma} \label{lem3.5}
Under the assumptions of Theorem \ref{thm1.2}, there exists a
constant $\beta_1>0$ independent of $t$ such that
\begin{equation} \label{e3.47}
e^{\beta_1t}\|\Theta(\cdot,t)\|^2_{H^1}+\int_0^t
e^{\beta_1\tau/2}\|\Theta(\cdot,\tau)\|_{H^2}^2d\tau
\leq c_{42},\quad \forall\,t\ge0.
\end{equation}
\end{lemma}

\begin{proof}
{\bf Step 1.} Taking $L^2$ inner product of \eqref{e3.1}$_4$ with
$\Theta_t$ we have
\begin{equation} \label{e3.48}
\frac{\kappa}{2}\frac{d}{dt}\|\nabla \Theta\|^2+\|\Theta_t\|^2
\leq \|U\cdot\nabla \Theta\|^2+\frac{1}{4}\|\Theta_t\|^2.
\end{equation}
Using Lemma \ref{lem3.3} we have
\begin{equation} \label{e3.49}
\|U\cdot\nabla \Theta\|^2\leq c_{43}\|U\|_{H^1}^2
\|\nabla \Theta\|_{L^4}^2\leq c_{44}\|\nabla \Theta\|_{L^4}^2.
\end{equation}
So we update \eqref{e3.48} as
\begin{equation} \label{e3.50}
\frac{\kappa}{2}\frac{d}{dt}\|\nabla
\Theta\|^2+\frac{3}{4}\|\Theta_t\|^2 \leq c_{44}\|\nabla
\Theta\|_{L^4}^2.
\end{equation}

The estimate of the RHS of \eqref{e3.50} is tricky.
First, applying Lemma \ref{lem2.1} (iv) to $\nabla\Theta$ to obtain
\begin{equation} \label{e3.51}
c_{44}\|\nabla \Theta\|_{L^4}^2
\leq c_{45}\big(\|\nabla \Theta\|\|D^2\Theta\|+\|\nabla \Theta\|^2\big)
\leq c_{46}(\delta)\|\nabla \Theta\|^2+\delta\|D^2\Theta\|^2,
\end{equation}
where $\delta$ is a number to be determined. Since $\Theta|_{\partial\Omega}=0$,
by the elliptic estimate (c.f. Lemma \ref{lem2.2}), we have
\begin{equation} \label{e3.52}
\|\Theta\|^2_{H^2}\leq
c_{47}\big(\|\Theta_t\|^2+\|U\cdot\nabla\Theta\|^2\big).
\end{equation}
For the second term on the RHS of \eqref{e3.52}, we use
\eqref{e3.49} and \eqref{e3.51} to get
\begin{equation} \label{e3.53}
\|U\cdot\nabla\Theta\|^2
\leq c_{48}\big(\|\nabla\Theta\|\|D^2\Theta\|+\|\nabla\Theta\|^2\big).
\end{equation}
Then, using Cauchy-Schwarz inequality we update \eqref{e3.52} as
\begin{equation} \label{e3.54}
\begin{aligned}
\|\Theta\|^2_{H^2}
&\leq c_{49}\big(\|\Theta_t\|^2+\|\nabla\Theta\|\|D^2\Theta\|
 +\|\nabla\Theta\|^2\big)\\
&\leq c_{50}\big(\|\Theta_t\|^2+\|\nabla\Theta\|^2\big)
 +\frac{1}{2}\|\Theta\|_{H^2}^2,
\end{aligned}
\end{equation}
which implies
\begin{equation} \label{e3.55}
\|\Theta\|^2_{H^2}\leq
c_{51}\big(\|\Theta_t\|^2+\|\nabla\Theta\|^2\big).
\end{equation}
By choosing $\delta=1/(4c_{51})$ in \eqref{e3.51}, and coupling the
result with \eqref{e3.55} we have
\begin{equation} \label{e3.56}
c_{44}\|\nabla \Theta\|_{L^4}^2\leq
c_{52}\|\nabla\Theta\|^2+\frac{1}{4}\|\Theta_t\|^2.
\end{equation}
Combining \eqref{e3.50} and \eqref{e3.56} we obtain
\begin{equation} \label{e3.57}
\frac{{\kappa}}{2}\frac{d}{dt}\|\nabla
\Theta\|^2+\frac{1}{2}\|\Theta_t\|^2 \leq c_{52}\|\nabla \Theta\|^2.
\end{equation}

{\bf Step 2.} We multiply \eqref{e3.10} by $2c_{52}/\kappa$ and
add the result to \eqref{e3.57} to obtain
\begin{equation} \label{e3.58}
\frac{d}{dt}\Big(\frac{c_{52}}{\kappa}\|\Theta\|^2
+\frac{\kappa}{2}\|\nabla\Theta\|^2\Big)
+c_{52}\|\nabla\Theta\|^2+\frac{1}{2}\|\Theta_t\|^2\leq 0.
\end{equation}
It is clear that, by Poincar\'e inequality, there exists a
constant $c_{53}>0$ such that
\begin{equation} \label{e3.59}
c_{53}\Big(\frac{c_{52}}{\kappa}\|\Theta\|^2
+\frac{\kappa}{2}\|\nabla\Theta\|^2\Big)\leq
c_{52}\|\nabla\Theta\|^2.
\end{equation}
Substituting \eqref{e3.59} in \eqref{e3.58} we have
\begin{equation} \label{e3.60}
\frac{d}{dt}\Big(\frac{c_{52}}{\kappa}\|\Theta\|^2
+\frac{\kappa}{2}\|\nabla\Theta\|^2\Big)
+c_{53}\Big(\frac{c_{52}}{\kappa}\|\Theta\|^2
+\frac{\kappa}{2}\|\nabla\Theta\|^2\Big)+\frac{1}{2}\|\Theta_t\|^2\leq
0,
\end{equation}
which implies (by dropping $\frac{1}{2}\|\Theta_t\|^2$ from the LHS) that
\begin{equation} \label{e3.61}
\Big(\frac{c_{52}}{\kappa}\|\Theta(\cdot,t)\|^2+\frac{\kappa}{2}\|\nabla\Theta(\cdot,t)\|^2\Big)\leq
\Big(\frac{c_{52}}{\kappa}\|\Theta_0\|^2+\frac{\kappa}{2}\|\nabla\Theta_0\|^2\Big)e^{-c_{53}t}.
\end{equation}
Therefore, for $t\ge0$,
\begin{equation} \label{e3.62}
\|\Theta(\cdot,t)\|^2_{H^1}\leq \Big(\min\{c_{52}/\kappa,\kappa/2\}
\Big)^{-1}\Big(\frac{c_{52}}{\kappa}\|\Theta_0\|^2
+\frac{\kappa}{2}\|\nabla\Theta_0\|^2\Big)e^{-c_{53}t}.
\end{equation}

Using \eqref{e3.58} and \eqref{e3.61}, by repeating the same procedure
as in Lemma \ref{lem3.2}, we have
\begin{equation} \label{e3.63}
\begin{aligned}
&e^{c_{53}t/2}\Big(\frac{c_{52}}{\kappa}\|\Theta(\cdot,t)\|^2
 +\frac{\kappa}{2}\|\nabla\Theta(\cdot,t)\|^2\Big)\\
&+\int_0^te^{c_{53}\tau/2}\Big(c_{52}\|\nabla\Theta(\cdot,\tau)\|^2+\frac{1}{2}\|\Theta_t(\cdot,\tau)\|^2\Big)d\tau\\
&\leq 2\Big(\frac{c_{52}}{\kappa}\|\Theta_0\|^2
 +\frac{\kappa}{2}\|\nabla\Theta_0\|^2\Big),
\end{aligned}
\end{equation}
which yields
\begin{equation} \label{e3.64}
\int_0^te^{c_{53}\tau/2}\Big(c_{52}\|\nabla\Theta(\cdot,\tau)\|^2
+\frac{1}{2}\|\Theta_t(\cdot,\tau)\|^2\Big)d\tau
\leq 2\Big(\frac{c_{52}}{\kappa}\|\Theta_0\|^2
+\frac{\kappa}{2}\|\nabla\Theta_0\|^2\Big).
\end{equation}
In view of \eqref{e3.55} we see that
\begin{equation} \label{e3.65}
\int_0^te^{c_{53}\tau/2}\|\Theta(\cdot,\tau)\|^2_{H^2}d\tau
\leq c_{54}.
\end{equation}
Therefore, \eqref{e3.47} follows from \eqref{e3.62}
 and \eqref{e3.65}. This completes the proof.
\end{proof}


With the help of Lemma \ref{lem3.5}, we have the following result.

\begin{lemma} \label{lem3.6}
Under the assumptions of Theorem \ref{thm1.2}, for all $t\ge0$,
\begin{equation} \label{e3.66}
\|\mu(\cdot,t)\|^2_{H^1}+\|\Phi_t(\cdot,t)\|^2
+\|U(\cdot,t)\|^2_{W^{1,4}}+\|U(\cdot,t)\|^2_{L^\infty}
+\|U_t(\cdot,t)\|^2\leq c_{55}.
\end{equation}
\end{lemma}

\begin{proof}
{\bf Step 1.} Taking $L^2$ inner product of \eqref{e3.27} with
$|\omega|^{2}\omega$ and using H\"older inequality we have
\begin{equation} \label{e3.67}
\frac{d}{dt}\|\omega\|_{L^4}
\leq 2\|\nabla\mu\|_{L^{8}}\|\nabla\Phi\|_{L^{8}}
 +\|\nabla\Theta\|_{L^{4}}
\leq c_{56}(\|\nabla\mu\|_{H^1}\|\Phi\|_{H^2}+\|\Theta\|_{H^2}).
\end{equation}
Integrating \eqref{e3.67} in time and using the previous
lemmas, we have
\begin{equation} \label{e3.68}
\begin{aligned}
\|\omega(\cdot,t)\|_{L^4}
&\leq c_{56}\Big(\int_0^t\|\nabla\mu\|_{H^1}^2d\tau\Big)^{1/2}
 \Big(\int_0^t\|\Phi\|_{H^2}^2d\tau\Big)^{1/2}\\
&\quad+ c_{56}\Big(\int_0^te^{-\beta_1\tau/2}d\tau\Big)^{1/2}
\Big(\int_0^te^{\beta_1\tau/2}\|\Theta\|_{H^2}^2d\tau\Big)^{1/2}
+\|\omega_0\|_{L^4}\\
&\leq c_{57},
\end{aligned}
\end{equation}
which, together with Lemma \ref{lem2.3} and Sobolev embedding, implies
\begin{equation} \label{e3.69}
\|U(\cdot,t)\|_{W^{1,4}}+\|U(\cdot,t)\|_{L^\infty}
\leq c_{58},\quad \forall\,t\ge0.
\end{equation}

For the estimate of $\|U_t\|^2$, taking $L^2$ inner product of
\eqref{e3.1}$_3$ with $U_t$ we have
\begin{equation} \label{e3.70}
\|U_t\|^2\leq c_{59}\big(\|U\|^2_{L^\infty}\|\nabla U\|^2
+\|\mu\|^2_{H^1}\|\Phi\|^2_{H^2}+\|\Theta_0\|^2\big)
\leq c_{60}\big(\|\nabla\mu\|^2+1\big),
\end{equation}
where we have used \eqref{e3.42}, \eqref{e3.44} and \eqref{e3.69}.

{\bf Step 2.} We now deal with $\mu$ and $\Phi_t$. Taking $L^2$
inner product of $\eqref{e3.1}_1$ with $\mu_t$ we have
\begin{equation} \label{e3.71}
 \frac{1}{2}\frac{d}{dt}\|\nabla\mu\|^2+\alpha\|\nabla\Phi_t\|^2
 =-\int_{\Omega}\big[F''({\phi})\Phi_t^2+\mu_t(U\cdot\nabla\Phi)\big]d\textbf{x}.
\end{equation}
From \eqref{e3.42} and Sobolev embedding we know that
\begin{equation} \label{e3.72}
\|\Phi\|_{L^\infty}\leq c_{61},
\end{equation}
which  according to the condition on $F$ implies
\begin{equation} \label{e3.73}
\|F^{(p-n)}(\phi)\|_{L^\infty}\leq
F_1c(p,n)\big(\|\Phi\|_{L^\infty}^{p-n}
+\|\bar{\phi}\|_{L^\infty}^{p-n}\big)+F_2\leq
c_{62},\quad n=1,.\dots,6.
\end{equation}

Using \eqref{e3.73} we estimate the RHS of \eqref{e3.71} as follows:
\begin{align*}
& \Big|-\int_{\Omega}\big[F''({\phi})\Phi_t^2
+\mu_t(U\cdot\nabla\Phi)\big]d\textbf{x}\Big|\\
 &\leq \|F''(\phi)\|_{L^\infty}\|\Phi_t\|^2+2{\alpha}\|U\|_{L^\infty}^2\|\nabla\Phi\|^2+\frac{1}{8\alpha}\|\mu_t\|^2\\
 &\leq c_{62}\|\Phi_t\|^2+c_{63}\|\nabla\Phi\|^2
+\frac{1}{8\alpha}\big(2\alpha^2\|\Delta\Phi_t\|^2
+2\|F''(\phi)\|_{L^\infty}^2\|\Phi_t\|^2\big)\\
 &\leq c_{64}\|\Phi_t\|^2+c_{65}\|\Delta\Phi\|^2
+\frac{\alpha}{4}\|\Delta\Phi_t\|^2,
\end{align*}
where we have used \eqref{e3.69} and Lemma \ref{lem2.4}.
We update \eqref{e3.71} as
\begin{equation} \label{e3.74}
 \frac{1}{2}\frac{d}{dt}\|\nabla\mu\|^2+\alpha\|\nabla\Phi_t\|^2
\leq c_{64}\|\Phi_t\|^2+c_{65}\|\Delta\Phi\|^2
+\frac{\alpha}{4}\|\Delta\Phi_t\|^2.
\end{equation}

Differentiating \eqref{e3.1}$_1$ with respect to $t$ we have
\begin{equation} \label{e3.75}
 \Phi_{tt}+U_t\cdot\nabla\Phi+U\cdot\nabla\Phi_t=\Delta\mu_t.
\end{equation}
Taking $L^2$ inner product of \eqref{e3.75} with $\Phi_t$ we have
\begin{equation} \label{e3.76}
 \frac{1}{2}\frac{d}{dt}\|\Phi_t\|^2+\alpha\|\Delta\Phi_t\|^2
=\int_{\Omega}F''({\phi})\Phi_t\Delta\Phi_t d\textbf{x}
+\int_{\Omega}\Phi(U_t\cdot\nabla\Phi_t)d\textbf{x}.
\end{equation}

Using \eqref{e3.73} and \eqref{e3.70} we estimate the RHS
of \eqref{e3.76} as follows:
\begin{equation} \label{e3.77}
\begin{aligned}
&\Big|\int_{\Omega}F''({\phi})\Phi_t\Delta\Phi_t d\textbf{x}
 +\int_{\Omega}\Phi(U_t\cdot\nabla\Phi_t)d\textbf{x}\Big|\\
&\leq \frac{\alpha}{4}\|\Delta\Phi_t\|^2
 +\frac{1}{\alpha}\|F''({\phi})\|_{L^\infty}^2\|\Phi_t\|^2
 +\frac{1}{2\alpha}\|\Phi\|_{L^\infty}^2\|U_t\|^2
 +\frac{\alpha}{2}\|\nabla\Phi_t\|^2\\
&\leq \frac{\alpha}{4}\|\Delta\Phi_t\|^2+c_{66}\|\Phi_t\|^2
 +c_{67}\|\Phi\|^2_{H^2}(\|\nabla\mu\|^2+1)
 +\frac{\alpha}{2}\|\nabla\Phi_t\|^2\\
&\leq \frac{\alpha}{4}\|\Delta\Phi_t\|^2+c_{66}\|\Phi_t\|^2
 +c_{68}\big(\|\nabla\mu\|^2+\|\Delta\Phi\|^2\big)
 +\frac{\alpha}{2}\|\nabla\Phi_t\|^2.
\end{aligned}
\end{equation}
So we update \eqref{e3.76} as
\begin{equation} \label{e3.78}
 \frac{1}{2}\frac{d}{dt}\|\Phi_t\|^2
+\frac{3\alpha}{4}\|\Delta\Phi_t\|^2\leq c_{66}\|\Phi_t\|^2
+c_{68}\big(\|\nabla\mu\|^2+\|\Delta\Phi\|^2\big)
+\frac{\alpha}{2}\|\nabla\Phi_t\|^2.
\end{equation}
Combining \eqref{e3.74} and \eqref{e3.78} we obtain
\begin{equation} \label{e3.79}
\frac{d}{dt}\Big(\|\nabla\mu\|^2+\|\Phi_t\|^2\Big)
 +{\alpha}\Big(\|\nabla\Phi_t\|^2+\|\Delta\Phi_t\|^2\Big)
 \leq c_{69}\big(\|\Phi_t\|^2+\|\nabla\mu\|^2+\|\Delta\Phi\|^2\big).
\end{equation}
After integrating \eqref{e3.79} in time and using \eqref{e3.3}
and \eqref{e3.31} we have
\begin{equation} \label{e3.80}
\|\nabla\mu(\cdot,t)\|^2+\|\Phi_t(\cdot,t)\|^2
+\int_0^t\big(\|\nabla\Phi_t\|^2+\|\Delta\Phi_t\|^2\big)d\tau
 \leq c_{70},\quad \forall\,t\ge0.
\end{equation}
Substituting \eqref{e3.80} in \eqref{e3.70} we have
$\|U_t(\cdot,t)\|^2\leq c_{71}$. This completes the proof.
\end{proof}

As consequences of previous lemmas, we have the following result.


\begin{lemma} \label{lem3.7}
Under the assumptions of Theorem \ref{thm1.2}, it holds
\begin{equation} \label{e3.81}
\|\Phi(\cdot,t)\|^2_{H^4}+\|\mu(\cdot,t)\|^2_{H^2}\leq c_{72},
\quad \forall\,t\ge0.
\end{equation}
\end{lemma}

\begin{proof}
First, by \eqref{e3.45} we have
\begin{equation} \label{e3.82}
 \begin{aligned}
  \|\mu(\cdot,t)\|_{H^2}^2\leq c_{40}\big(\|\Phi_t(\cdot,t)\|^2
+\|\Phi(\cdot,t)\|_{H^2}^2\big)+\|\mu(\cdot,t)\|^2.
 \end{aligned}
\end{equation}
Then the uniform estimate of $ \|\mu(\cdot,t)\|_{H^2}^2$
follows from Lemma \ref{lem3.4} and Lemma \ref{lem3.6}.

Second, by Lemma \ref{lem2.4} and \eqref{e3.1}$_2$ we have
\begin{equation} \label{e3.83}
 \|\Phi(\cdot,t)\|_{H^4}^2\leq c_{73}\big(\|\mu(\cdot,t)\|_{H^2}^2
+\|F'(\phi)(\cdot,t)\|_{H^2}^2\big).
\end{equation}
Using the second condition $(H_2)$ on $F$, \eqref{e3.31} and
\eqref{e3.73}, it is straightforward to show that
$\|F'(\phi)(\cdot,t)\|_{H^2}^2\leq c_{74}$,
which together with the uniform bound of $\|\mu(\cdot,t)\|_{H^2}^2$
imply that $\|\Phi(\cdot,t)\|_{H^4}^2\leq c_{75}$.
This completes the proof.
\end{proof}

\section{Large time asymptotic behavior}

In this section we prove Theorem \ref{thm1.2}, based
on a sequence of accurate combinations of energy estimates. For
the convenience of the reader, we first collect some
uniform-in-time estimates. From \eqref{e3.31}, \eqref{e3.66}
and \eqref{e3.81} we have, for any $t\ge0$:
\begin{equation} \label{e4.1}
\big(\|\Phi\|^2_{H^4}+\|\mu\|^2_{H^2}+\|\Phi_t\|^2
+\|F^{(p-n)}(\phi)\|^2_{L^\infty}+
 \|U\|^2_{W^{1,4}}+\|U\|^2_{L^\infty}+\|U_t\|^2\big)(t)\leq c_{76}.
\end{equation}

\subsection{Decay of $(\Phi,\mu)$}

\noindent{\bf Step 1.} First, by \eqref{e3.1}$_2$ we have
\begin{equation} \label{e4.2}
\begin{aligned}
\|\mu-F'(\bar{\phi})\|^2_{H^3}
&=\|-\alpha\Delta\Phi+F'(\phi)-F'(\bar{\phi})\|^2+\|\nabla\mu\|_{H^2}^2\\
&\leq 2\alpha^2\|\Delta\Phi\|^2+2\|F'(\phi)-F'(\bar{\phi})\|^2
+\|\nabla\mu\|_{H^2}^2.
\end{aligned}
\end{equation}
Using \eqref{e4.1} and Lemma \ref{lem2.3} we estimate the last two terms
on the RHS of \eqref{e4.2} as follows:
\begin{equation} \label{e4.3}
\begin{aligned}
&2\|F'(\phi)-F'(\bar{\phi})\|^2+\|\nabla\mu\|_{H^2}^2\\
&\leq 2\|F''(\zeta)\|^2_{L^\infty}\|\Phi\|^2+c_{77}\|\nabla\Delta\mu\|^2\\
&\leq c_{78}\|\Phi\|^2+c_{79}\big(\|\nabla\Phi_t\|^2+\|\nabla (U\cdot\nabla\Phi)\|^2\big)\\
&\leq c_{78}\|\Phi\|^2+c_{80}\big(\|\nabla\Phi_t\|^2+\|\nabla U\|^2\|\nabla\Phi\|^2_{\infty}+\|U\|_\infty^2\|\nabla^2\Phi\|^2\big)\\
&\leq c_{81}\big(\|\nabla\Phi_t\|^2+\|\Phi\|^2_{H^3}\big).
\end{aligned}
\end{equation}
Combining \eqref{e4.2} and \eqref{e4.3} we have
\begin{equation} \label{e4.4}
\begin{aligned}
\|\mu-F'(\bar{\phi})\|^2_{H^3}\leq c_{82}\big(\|\nabla\Phi_t\|^2+\|\Phi\|^2_{H^3}\big).
\end{aligned}
\end{equation}
Combining \eqref{e3.25} and \eqref{e4.4} we then have
\begin{equation} \label{e4.5}
\|\mu-F'(\bar{\phi})\|^2_{H^3}\leq c_{83}\big(\|\Phi_t\|_{H^1}^2+\|\Phi\|^2_{H^2}+\|\nabla\mu\|^2\big).
\end{equation}

Second, by Lemma \ref{lem2.4} and \eqref{e3.1}$_2$ we have
\begin{equation} \label{e4.6}
\begin{aligned}
\|\Phi\|^2_{H^5}\leq c_{8}\|\nabla\Delta^2\Phi\|^2
\leq c_{84}\|\nabla\Delta\mu\|^2+c_{85}\|\nabla\Delta F'(\phi)\|^2.
\end{aligned}
\end{equation}
By direct calculations and Sobolev embeddings we can show that
\begin{equation} \label{e4.7}
\begin{aligned}
\|\nabla F'(\phi)\|_{H^2}^2
&\leq c_{86}\|F'''(\phi)\|_{L^\infty}^2\big(\|\nabla\Phi\|^4_{H^1}
+2\|\nabla\Phi\|^2_{H^1}\|\nabla\Phi\|^2_{H^2}\big)\\
&\quad+c_{87}\|F''''(\phi)\|_{L^\infty}^2\|\nabla\Phi\|_{H^2}^4
\|\nabla\Phi\|^2.
\end{aligned}
\end{equation}
Using \eqref{e4.1} we obtain from \eqref{e4.7} that
$\|\nabla F'(\phi)\|_{H^2}^2\leq c_{88}\|\nabla\Phi\|^2_{H^2}$, which,
together with \eqref{e4.6}, implies that
\begin{equation} \label{e4.8}
\|\Phi\|^2_{H^5}\leq c_{89}\big(\|\nabla\Phi_t\|^2
+\|\Phi\|_{H^3}^2\big).
\end{equation}
Combining \eqref{e4.5}, \eqref{e4.8} and \eqref{e3.25} we have
\begin{equation} \label{e4.9}
\|\mu-F'(\bar{\phi})\|^2_{H^3}+\|\Phi\|^2_{H^5}
\leq c_{90}\big(\|\Phi_t\|_{H^1}^2+\|\Phi\|^2_{H^2}+\|\nabla\mu\|^2\big).
\end{equation}
Therefore, it suffices to show the decay of RHS of \eqref{e4.9}
in order to prove the decay of $\Phi$ and $\mu$.

\noindent{\bf Step 2.} We recall \eqref{e3.17},
\begin{equation} \label{e4.10}
\frac{d}{dt}\Big(\frac{\alpha}{2}\|\nabla\Phi\|^2
+\int_{\Omega}F(\phi)d\textbf{x}\Big)+\|\nabla\mu\|^2
=-\int_\Omega\mu(\nabla\Phi\cdot U)d\mathbf{x}.
\end{equation}
Due to the structure of the function $F(\cdot)$, there may
be a constant term in the integral
${\int_{\Omega}F(\phi)d\textbf{x}}$ in general, which is
impossible to decay. In order to resolve this issue, we observe,
since $\int_\Omega(\phi-\overline{\phi}) d\mathbf{x}=0$,
it holds that
\begin{align*}
\int_\Omega F(\phi)-F(\overline{\phi})d\mathbf{x}
&=\int_\Omega
F'(\overline{\phi})(\phi-\overline{\phi})d\mathbf{x}+\frac{1}{2}\int_\Omega
F''(\xi)(\phi-\overline{\phi})^2d\mathbf{x}\\
&=\frac{1}{2}\int_\Omega
F''(\xi)\Phi^2d\mathbf{x},\ \
\mathrm{for\ some\ }\xi\ \mathrm{between}\ \phi\ \mathrm{and}\ \bar{\phi}.
\end{align*}
Then we update \eqref{e4.10} as
\begin{equation} \label{e4.11}
\frac{d}{dt}\Big(\frac{\alpha}{2}\|\nabla\Phi\|^2+
\frac{1}{2}\int_{\Omega}F''(\xi)\Phi^2d\textbf{x}\Big)+\|\nabla\mu\|^2=-\int_\Omega\mu(\nabla\Phi\cdot
U)d\mathbf{x}.
\end{equation}
Using \eqref{e4.1} and Lemma \ref{lem2.4} we estimate the RHS
of \eqref{e4.11} as
\begin{equation} \label{e4.12}
\begin{aligned}
\Big|-\int_{\Omega}\mu(\nabla\Phi\cdot U)d\textbf{x}\Big|
&=\Big|\int_\Omega\Phi U\cdot \nabla\mu d\textbf{x}\Big|\\
&\leq \frac{1}{2}\|\nabla\mu\|^2+\frac{1}{2}\|U\|_{L^4}^2\|\Phi\|_{L^4}^2\\
&\leq \frac{1}{2}\|\nabla\mu\|^2+c_{91}\|\Phi\|^2_{H^1}\\
&\leq \frac{1}{2}\|\nabla\mu\|^2+c_{92}\|\Delta\Phi\|^2.
\end{aligned}
\end{equation}
Substituting \eqref{e4.12} in \eqref{e4.11}, we have
\begin{equation} \label{e4.13}
\frac{d}{dt}\Big(\alpha\|\nabla\Phi\|^2+\int_\Omega
F''(\xi)\Phi^2d\mathbf{x}\Big)+\|\nabla\mu\|^2\leq
c_{93}\|\Delta\Phi\|^2.
\end{equation}

\noindent{\bf Step 3.} Recalling \eqref{e3.34} and using \eqref{e4.1}
and Lemma \ref{lem2.4} we have
\begin{align*}
&\frac{d}{dt}\Big(\frac{\alpha}{2}\|\Delta\Phi\|^2
+\frac{1}{2}\int_{\Omega}F''(\phi)|\nabla\Phi|^2 d\textbf{x}\Big)
+\|\Phi_t\|^2\\
&=\frac{1}{2}\int_{\Omega}F'''(\phi)\Phi_t|\nabla\Phi|^2d\textbf{x}-
\int_{\Omega}\Phi_t(U\cdot\nabla\Phi)d\textbf{x}\\
&\leq \frac{1}{2}\|\Phi_t\|^2+c_{94}(\|\nabla\Phi\|^4_{H^1}
+\|\nabla\Phi\|^2_{H^1})\\
&\leq  \frac{1}{2}\|\Phi_t\|^2+c_{95}\|\Delta\Phi\|^2,
\end{align*}
which yields
\begin{equation} \label{e4.14}
\frac{d}{dt}\Big(\alpha\|\Delta\Phi\|^2
+\int_{\Omega}F''(\phi)|\nabla\Phi|^2
d\textbf{x}\Big)+\|\Phi_t\|^2\leq c_{96}\|\Delta\Phi\|^2.
\end{equation}

\noindent{\bf Step 4.} From \eqref{e3.77} and \eqref{e4.1} we have
\begin{equation} \label{e4.15}
\frac{1}{2}\frac{d}{dt}\|\Phi_t\|^2+\alpha\|\Delta\Phi_t\|^2
\leq \frac{\alpha}{4}\|\Delta\Phi_t\|^2
+\frac{\alpha}{2}\|\nabla\Phi_t\|^2+c_{97}(\|\Phi_t\|^2
+\|\Delta\Phi\|^2).
\end{equation}
Combining \eqref{e3.74} and \eqref{e4.15}, we have
\begin{equation} \label{e4.16}
\frac{d}{dt}\big(\|\nabla\mu\|^2+\|\Phi_t\|^2\big)
+\alpha\big(\|\nabla\Phi_t\|^2+\|\Delta\Phi_t\|^2\big)\leq
c_{98}\big(\|\Phi_t\|^2+\|\Delta\Phi\|^2\big).
\end{equation}

\noindent{\bf Step 5.}
Taking $L^2$ inner product of \eqref{e3.75} with $\mu_t$ we have
\begin{equation} \label{e4.17}
\frac{\alpha}{2}\frac{d}{dt}\|\nabla\Phi_t\|^2
+\|\nabla\mu_t\|^2
=\int_\Omega(U_t\Phi+U\Phi_t)\cdot\nabla\mu_td\mathbf{x}
-\int_\Omega F''(\phi)\Phi_t\Phi_{tt}d\mathbf{x}.
\end{equation}
For the last term on the RHS of \eqref{e4.17}, we have
\begin{equation*}
 -\int_\Omega F''(\phi)\Phi_t\Phi_{tt}d\mathbf{x}=-\frac{1}{2}\frac{d}{dt}\int_\Omega F''(\phi)\Phi_t^2d\mathbf{x}+\frac{1}{2}\int_\Omega F'''(\phi)\Phi_t^3d\mathbf{x}.
\end{equation*}
So we update \eqref{e4.17} as
\begin{equation} \label{e4.18}
\begin{aligned}
&\frac{d}{dt}\Big(\alpha\|\nabla\Phi_t\|^2+\int_\Omega F''(\phi)\Phi_t^2d\mathbf{x}\Big)+2\|\nabla\mu_t\|^2\\
&=2\int_\Omega(U_t\Phi+U\Phi_t)\cdot\nabla\mu_td\mathbf{x}+\int_\Omega F'''(\phi)\Phi_t^3d\mathbf{x}.
\end{aligned}
\end{equation}
Using \eqref{e4.1} and Lemma \ref{lem3.5} we
estimate the first two terms on the RHS of \eqref{e4.18} as
\begin{equation} \label{e4.19}
 \begin{aligned}
\Big|2\int_\Omega(U_t\Phi+U\Phi_t)\cdot\nabla\mu_td\mathbf{x}\Big|
&\leq \|\nabla\mu_t\|^2+\|U_t\|^2\|\Phi\|^2_{L^\infty}
 +\|U\|^2_{L^\infty}\|\Phi_t\|^2\\
&\leq \|\nabla\mu_t\|^2+c_{99}\|\Phi\|^2_{H^2}+c_{76}\|\Phi_t\|^2\\
&\leq \|\nabla\mu_t\|^2+c_{100}\|\Delta\Phi\|^2+c_{101}\|\Delta\Phi_t\|^2.
 \end{aligned}
\end{equation}
Similarly, for the term involving $\Phi_t^3$, we have
\begin{equation} \label{e4.20}
\begin{aligned}
\Big|\int_\Omega F'''(\phi)\Phi_t^3d\mathbf{x}\Big|
&\leq \|F'''(\phi)\|_\infty\|\Phi_t\|_{L^3}^3\\
&\leq c_{76}\|\Phi_t\|^2_\infty\|\Phi_t\|_{L^1}\\
&\leq c_{102}\|\Delta\Phi_t\|^2\|\Phi_t\|
\leq c_{103}\|\Delta\Phi_t\|^2.
\end{aligned}
\end{equation}
Substituting \eqref{e4.19} and \eqref{e4.20} in \eqref{e4.18}, we have
\begin{equation} \label{e4.21}
\frac{d}{dt}\Big(\alpha\|\nabla\Phi_t\|^2+\int_\Omega
F''(\phi)\Phi_t^2d\mathbf{x}\Big)+\|\nabla\mu_t\|^2\leq
c_{104}\big(\|\Delta\Phi_t\|^2+\|\Delta\Phi\|^2\big).
\end{equation}

\noindent{\bf Step 6.} In this step, we make combinations
of energy estimates, which will be used to prove the exponential
decay of $\Phi$ and $\mu$.
First, we collect energy inequalities from {\bf Steps 2--5}:


\begin{center}
\begin{tabular}{|r|l|}
 \hline
 \eqref{e4.13} & ${\frac{d}{dt}\Big(\alpha\|\nabla\Phi\|^2+\int_\Omega
F''(\xi)\Phi^2d\mathbf{x}\Big)+\|\nabla\mu\|^2\leq
c_{93}\|\Delta\Phi\|^2}$\\
\hline
 \eqref{e4.14} & ${\frac{d}{dt}\Big(\alpha\|\Delta\Phi\|^2+\int_{\Omega}F''(\phi)|\nabla\Phi|^2
d\textbf{x}\Big)+\|\Phi_t\|^2\leq c_{96}\|\Delta\Phi\|^2}$\\
\hline
 \eqref{e4.16} & ${\frac{d}{dt}\big(\|\nabla\mu\|^2+\|\Phi_t\|^2\big)+\alpha\big(\|\nabla\Phi_t\|^2+\|\Delta\Phi_t\|^2\big)\leq
c_{98}\big(\|\Phi_t\|^2+\|\Delta\Phi\|^2\big)}$\\
\hline
 \eqref{e4.21} & ${\frac{d}{dt}\Big(\alpha\|\nabla\Phi_t\|^2+\int_\Omega
F''(\phi)\Phi_t^2d\mathbf{x}\Big)+\|\nabla\mu_t\|^2\leq
c_{104}\big(\|\Delta\Phi_t\|^2+\|\Delta\Phi\|^2\big)}$\\
\hline
\end{tabular}
\end{center}

First, multiply \eqref{e4.16} by $\frac{2c_{104}}{\alpha}$
and then add \eqref{e4.21} to obtain
\begin{equation} \label{e4.22}
\frac{d}{dt}\big[J_0(t)\big]+K_0(t)\leq
c_{105}\big(\|\Phi_t\|^2+\|\Delta\Phi\|^2\big),
\end{equation}
where
\begin{gather*}
J_0(t)\equiv
\frac{2c_{104}}{\alpha}\big(\|\nabla\mu\|^2+\|\Phi_t\|^2\big)+\alpha\|\nabla\Phi_t\|^2+\int_{\Omega}F''(\phi)\Phi_t^2
d\textbf{x},\\
K_0(t)\equiv
c_{104}\big(2\|\nabla\Phi_t\|^2+\|\Delta\Phi_t\|^2\big)+\|\nabla\mu_t\|^2.
\end{gather*}

Second, multiply \eqref{e4.14} by $2c_{105}$ then add \eqref{e4.22} to
obtain
\begin{equation} \label{e4.23}
\frac{d}{dt}\big[J_1(t)\big]+K_1(t)\leq c_{106}\|\Delta\Phi\|^2,
\end{equation}
where
\begin{gather*}
J_1(t) \equiv J_0(t)+
2c_{105}\Big(\alpha\|\Delta\Phi\|^2+\int_{\Omega}F''(\phi)|\nabla\Phi|^2
d\textbf{x}\Big),\\
K_1(t) \equiv c_{105}\|\Phi_t\|^2+ K_0(t).
\end{gather*}

Third, by coupling \eqref{e4.23} and \eqref{e4.13}, we have
\begin{equation} \label{e4.24}
\frac{d}{dt}\big[J_2(t)\big]+K_2(t)\leq c_{107}\|\Delta\Phi\|^2,
\end{equation}
where
\[
J_2(t) \equiv J_1(t)+\Big(\alpha\|\nabla\Phi\|^2+\int_\Omega
F''(\xi)\Phi^2d\mathbf{x}\Big),\quad
K_2(t) \equiv K_1(t)+\|\nabla\mu\|^2.
\]
Recalling \eqref{e3.7}, we have
\begin{equation} \label{e4.25}
\frac{d}{dt}\|\Phi\|^2+2\alpha_1\|\Delta\Phi\|^2\le0.
\end{equation}
Then, multiply \eqref{e4.25} by $\frac{c_{107}}{\alpha_1}$ then add
\eqref{e4.24} to obtain
\begin{equation} \label{e4.26}
\frac{d}{dt}\big[J_3(t)\big]+K_3(t)\leq 0,
\end{equation}
where
$$
J_3(t)\equiv J_2(t)+\frac{c_{107}}{\alpha_1}\|\Phi\|^2,\quad
K_3(t)\equiv K_2(t)+c_{107}\|\Delta\Phi\|^2.
$$

\noindent{\bf Step 7.} In this step we  apply the condition
$\alpha_1=\alpha-F_3c_0>0$ to uncover the secret hidden
in \eqref{e4.26},
which will give the desired decay estimates of $\Phi$ and $\mu$.
First, by \eqref{e4.1} and Poincar\'e inequality we have
\begin{align*}
&\frac{2c_{104}}{\alpha}\big(\|\nabla\mu\|^2+\|\Phi_t\|^2\big)
 +\alpha_1\|\nabla\Phi_t\|^2\\
&\leq J_0(t)\\
&\leq \frac{2c_{104}}{\alpha}\big(\|\nabla\mu\|^2+\|\Phi_t\|^2\big)
+\alpha\|\nabla\Phi_t\|^2+c_{76}\|\Phi_t\|^2,
\end{align*}
which implies
$$
J_0(t)\cong \|\nabla\mu\|^2+\|\Phi_t\|^2_{H^1}.
$$
where the symbol $\cong$ denotes the equivalence of quantities.
Also, by Poincar\'e inequality, we have
\begin{align*}
& c_{104}\big(c_0\|\Phi_t\|^2+\|\nabla\Phi_t\|^2
 +\|\Delta\Phi_t\|^2\big)+\|\nabla\mu_t\|^2\\
&\leq K_0(t)\\
&\leq c_{104}\big(\|\Phi_t\|^2+2\|\nabla\Phi_t\|^2
+\|\Delta\Phi_t\|^2\big)+\|\nabla\mu_t\|^2.
\end{align*}
Therefore,
$K_0(t)\cong \|\Phi_t\|^2_{H^2}+\|\nabla\mu_t\|^2$.

Similarly, we have
\begin{center}
\begin{tabular}{|r|l|}
 \hline
$J_1(t)\cong \|\nabla\mu\|^2+\|\Phi_t\|^2_{H^1}+\|\Phi\|^2_{H^2}$
 & $K_1(t)\cong \|\Phi_t\|^2_{H^2}+\|\nabla\mu_t\|^2$\\
\hline \hline
$J_2(t)\cong \|\nabla\mu\|^2+\|\Phi_t\|^2_{H^1}+\|\Phi\|^2_{H^2}$
 & $K_2(t)\cong \|\Phi_t\|^2_{H^2}+\|\nabla\mu_t\|^2+\|\nabla\mu\|^2$\\
\hline  \hline
$J_3(t)\cong \|\nabla\mu\|^2+\|\Phi_t\|^2_{H^1}+\|\Phi\|^2_{H^2}$
 & $K_3(t)\cong \|\Phi_t\|^2_{H^2}+\|\nabla\mu_t\|^2+\|\nabla\mu\|^2+\|\Phi\|^2_{H^2}$\\
\hline
\end{tabular}
\end{center}

Then it is clear that there exists a constant $c_{108}>0$
independent of $t$ such that
$c_{108}J_3(t)\leq K_3(t)$,
which, together with \eqref{e4.26}, implies
\begin{equation*}
\frac{d}{dt}\big(J_3(t)\big)+c_{108}J_3(t)\leq 0,
\end{equation*}
which gives
$$
J_3(t)\leq J_3(0)e^{-c_{108}t},\quad \forall\,t\ge0.
$$
We then have
\begin{equation*}
\|\nabla\mu(\cdot,t)\|^2+\|\Phi_t(\cdot,t)\|^2_{H^1}
+\|\Phi(\cdot,t)\|^2_{H^2}\leq
c_{109}e^{-c_{108}t},\quad \forall\,t\ge0,
\end{equation*}
which, together with \eqref{e4.9}, implies
\begin{equation} \label{e4.27}
\|\Phi(\cdot,t)\|^2_{H^5}+\|(\mu-F'(\bar{\phi}))(\cdot,t)\|^2_{H^3}\leq
c_{110}e^{-c_{108}t},\quad \forall\,t\ge0,
\end{equation}

\subsection{Decay of $\Theta$}

In this section, we show the exponential decay of $\|\Theta\|_{H^3}$.
Again, this will be done by combining uniform estimates of the
solution.

\noindent{\bf Step 1.} Since
$\Theta|_{\partial\Omega}=0$, by Lemma \ref{lem2.2} and \eqref{e4.1} we have
\begin{equation} \label{e4.28}
\begin{aligned}
 \|\Theta\|_{H^3}^2
&\leq c_{111}\big(\|\Theta_t\|_{H^1}^2+\|U\cdot\nabla\Theta\|_{H^1}^2\big)\\
 &\leq c_{112}\big(\|\Theta_t\|_{H^1}^2+\|U\|_{L^\infty}^2
 \|\nabla\Theta\|^2+ \|\nabla U\|^2_{L^4}\|\nabla\Theta\|^2_{L^4}
 +\|U\|_{L^\infty}^2\|\nabla^2\Theta\|^2\big)\\
&\leq c_{113}\big(\|\Theta_t\|_{H^1}^2+\|\Theta\|^2_{H^2}\big).
\end{aligned}
\end{equation}

\noindent {\bf Step 2.} By taking temporal derivative of
$\eqref{e3.1}_4$ we obtain
\begin{equation} \label{e4.29}
 \Theta_{tt}+U_t\cdot\nabla\Theta+U\cdot\nabla\Theta_t
=\kappa\Delta\Theta_t.
\end{equation}
Taking $L^2$ inner product of \eqref{e4.29} with $\Theta_t$ and using
\eqref{e4.1} we have
 \begin{align*}
  \frac{1}{2}\frac{d}{dt}\|\Theta_t\|^2+\kappa\|\nabla\Theta_t\|^2
&=\int_\Omega \Theta(U_t\cdot\nabla\Theta_t)d\mathbf{x}\\
&\leq \frac{\kappa}{2}\|\nabla\Theta_t\|^2
 +\frac{1}{2\kappa}\|U_t\|^2\|\Theta\|^2_{L^\infty}\\
&\leq \frac{\kappa}{2}\|\nabla\Theta_t\|^2+c_{114}\|\Theta\|^2_{H^2},
 \end{align*}
which gives
\begin{equation} \label{e4.30}
\frac{d}{dt}\|\Theta_t\|^2+{\kappa}\|\nabla\Theta_t\|^2\leq
c_{115}\|\Theta\|^2_{H^2}.
\end{equation}
Substituting \eqref{e3.55} in \eqref{e4.30}, we have
\begin{equation} \label{e4.31}
\frac{d}{dt}\|\Theta_t\|^2+\kappa\|\nabla\Theta_t\|^2\leq
c_{116}\big(\|\nabla\Theta\|^2+\|\Theta_t\|^2\big).
\end{equation}
Now, we recall \eqref{e3.58},
\begin{equation} \label{e4.32}
\frac{d}{dt}\Big(\frac{c_{52}}{\kappa}\|\Theta\|^2
+\frac{\kappa}{2}\|\nabla\Theta\|^2\Big)
+c_{52}\|\nabla\Theta\|^2+\frac{1}{2}\|\Theta_t\|^2\leq
0.
\end{equation}

By absorbing the RHS of \eqref{e4.31} into the LHS of \eqref{e4.32}
we have
\begin{equation} \label{e4.33}
\frac{d}{dt}\big(J_4(t)\big) +K_4(t)\le0,
\end{equation}
where
\begin{equation} \label{e4.34}
\begin{gathered}
J_4(t)\cong\|\Theta\|^2+\|\nabla\Theta\|^2+\|\Theta_t\|^2
 \cong\|\Theta\|^2_{H^2}+\|\Theta_t\|^2,\\
K_4(t)\cong\|\nabla\Theta\|^2+\|\Theta_t\|^2+\|\nabla\Theta_t\|^2
 \cong\|\Theta\|^2_{H^2}+\|\Theta_t\|^2_{H^1}.
\end{gathered}
\end{equation}

\noindent{\bf Step 3.} By taking $L^2$ inner product of \eqref{e4.29}
with $\Theta_{tt}$ and using Cauchy-Schwarz inequality, we have
\begin{equation} \label{e4.35}
\frac{\kappa}{2}\frac{d}{dt}\|\nabla\Theta_t\|^2
 +\frac{1}{2}\|\Theta_{tt}\|^2\leq
\|U_t\|^2\|\nabla\Theta\|^2_{L^\infty}
 +\|U\|_{L^\infty}^2\|\nabla\Theta_t\|^2.
\end{equation}
For the RHS of \eqref{e4.35}, using \eqref{e4.1}, Lemma \ref{lem2.1}
and Lemma \ref{lem2.2} with $m=0,p=3$, we have
\begin{equation} \label{e4.36}
\begin{aligned}
\|U_t\|^2\|\nabla\Theta\|^2_{L^\infty}
+\|U\|_{L^\infty}^2\|\nabla\Theta_t\|^2
&\leq c_{117}\big(\|\Theta\|^2_{W^{2,3}}+\|\nabla\Theta_t\|^2\big)\\&\leq c_{118}\big(\|\Theta_t\|^2_{L^3}+\|U\cdot\nabla\Theta\|^2_{L^3}+\|\nabla\Theta_t\|^2\big)\\
&\leq c_{119}\big(\|\Theta_t\|^2_{H^1}+\|U\|_{L^\infty}^2\|\nabla\Theta\|^2_{H^1}+\|\nabla\Theta_t\|^2\big)\\
&\leq c_{120}\big(\|\Theta_t\|^2_{H^1}+\|\nabla\Theta\|^2\big),
\end{aligned}
\end{equation}
where we have used \eqref{e3.55} for $\|\nabla\Theta\|^2_{H^1}$.
So we update \eqref{e4.35} as
\begin{equation} \label{e4.37}
\kappa\frac{d}{dt}\|\nabla\Theta_t\|^2+\|\Theta_{tt}\|^2\leq
c_{121}\Big(\|\Theta_t\|^2_{H^1}+\|\nabla\Theta\|^2\Big).
\end{equation}
Then it is clear that, by absorbing the RHS of \eqref{e4.37} into
the LHS of \eqref{e4.33}, it holds that
\begin{equation} \label{e4.38}
\frac{d}{dt}\big(J_5(t)\big)+K_5(t)\le0,
\end{equation}
where
\begin{equation} \label{e4.39}
J_5(t)\cong\|\Theta\|^2_{H^2}+\|\Theta_t\|^2_{H^1},\quad
K_5(t)\cong\|\Theta\|^2_{H^2}+\|\Theta_t\|^2_{H^1}+\|\Theta_{tt}\|^2.
\end{equation}
Therefore,
\begin{equation} \label{e4.40}
\|\Theta(\cdot,t)\|^2_{H^2}+\|\Theta_t(\cdot,t)\|^2_{H^1}\leq
c_{122} e^{-c_{123} t},\quad \forall\,t\ge0.
\end{equation}
which, together with \eqref{e4.28}, implies that
\begin{equation} \label{e4.41}
\|\Theta(\cdot,t)\|^2_{H^3}\leq c_{124} e^{-c_{123} t},\quad \forall\,
t\ge0.
\end{equation}

\subsection{Uniform estimates of $U$ and $\omega$}

In this section, we show  uniform estimates for $U$ and $\omega$
as indicated in Theorem \ref{thm1.2}. For this purpose, we
observe, for any $p\ge2$, it holds that
\begin{equation} \label{e4.42}
\begin{aligned}
\frac{d}{dt}\|\omega\|_{L^p}
&\leq
2\|\nabla\mu\|_{L^{2p}}\|\nabla\Phi\|_{L^{2p}}+\|\nabla\Theta\|_{L^{p}}\\
&\leq 2\|\nabla\mu\|_{L^\infty}|\Omega|^{\frac{1}{2p}}\|\nabla\Phi\|_{L^\infty}|\Omega|^{\frac{1}{2p}}
+\|\nabla\Theta\|_{L^\infty}|\Omega|^{1/p}\\
&\leq c_{127}\max\{1,|\Omega|\}\big(\|\nabla\mu\|_{H^2}
 \|\nabla\Phi\|_{H^2}+\|\nabla\Theta\|_{H^2}\big).
\end{aligned}
\end{equation}

Using the decay estimates for $\Phi,\mu$ and $\Theta$, we update
\eqref{e4.42} as
\begin{equation} \label{e4.43}
\frac{d}{dt}\|\omega\|_{L^p}\leq c_{128} e^{-c_{129}t},\quad \forall\,
t\ge0,
\end{equation}
where the constants $c_{128}$ and $c_{129}$ are independent of $p\ge2$.
Upon integrating \eqref{e4.43} in time, we have
\begin{equation} \label{e4.44}
\|\omega(\cdot,t)\|_{L^p}\leq
c_{130}+\|\omega_0\|_{L^\infty}\max\{1,|\Omega|\},\quad \forall\,
t\ge0.
\end{equation}
Letting $p\to\infty$ in \eqref{e4.44} we have
\begin{equation} \label{e4.45}
\|\omega(\cdot,t)\|_{L^\infty}\leq c_{131},\quad \forall\,t\ge0.
\end{equation}
Moreover, by Lemma \ref{lem2.3},
\begin{align*}
\|U(\cdot,t)\|_{W^{1,p}}
&\leq c_7(p)\big(\|\omega(\cdot,t)\|_{L^p}+\|U(\cdot,t)\|_{L^p}\big)\\
&\leq c_7(p)\big(\|\omega(\cdot,t)\|_{L^\infty}
 +\|U(\cdot,t)\|_{L^\infty}\big)\max\{1,|\Omega|\}.
\end{align*}
By \eqref{e4.1} and \eqref{e4.45}, we have
$$
\|U(\cdot,t)\|_{W^{1,p}}\leq c_{132}(p),\quad \forall\,t\ge0,\;
\forall\,1\leq p<\infty.
$$
This estimate, together with \eqref{e4.27}, \eqref{e4.41}
and \eqref{e4.45}, completes
the proof of Theorem \ref{thm1.2}.

We complete this section with the following remark.

\begin{remark} \label{rmk4.1} \rm
Using the arguments in this paper one can show that Theorem
\ref{thm1.2} still holds if the Dirichlet boundary condition for
$\theta$ is replaced by the Neumann boundary condition
$\nabla\theta\cdot \mathbf{n}|_{\partial\Omega}=0$. In this case,
the asymptotic state of $\theta$ is
$\tilde{\theta}=\frac{1}{|\Omega|}\int_\Omega\theta_0(\mathbf{x})
d\mathbf{x}$, which is a constant.
\end{remark}


\subsection*{Acknowledgements}
The author was partially supported by NSF through grant DMS 0807406
and under Agreement No. 0635561. The author would also likes to
thank  Professor Ronghua Pan, for his support
and encouragement.

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\end{document}