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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 76, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/76\hfil Existence of solutions]
{Existence of solutions for a quasi-linear phase separation of multi-component system}

\author[D. Zhang, R. Liu \hfil EJDE-2018/76\hfilneg]
{Dongpei Zhang, Ruikuan Liu}

\address{Dongpei Zhang \newline
Department of Mathematics,
 Sichuan University,
 Chengdu, Sichuan 610064, China}
\email{zhangdpscu@163.com}

\address{Ruikuan Liu \newline
Department of Mathematics,
 Sichuan University,
 Chengdu, Sichuan 610064, China}
\email{liuruikuan2008@163.com}

\dedicatory{Communicated by Jerome A Goldstein}

\thanks{Submitted May 5, 2017. Published March 18, 2018.}
\subjclass[2010]{35J58, 35J62, 35K52, 35K59}
\keywords{Multi-component system; the acute angle principle;
\hfill\break\indent $T$-weakly continuous operators; global weak solution}

\begin{abstract}
 This article formulates a new model of the phase separation of multi-component
 system,  which is a fourth-order quasi-linear evolution  partial
 differential equation. By using the acute angle principle, we obtain
 a weak solution of the corresponding steady-state equations.
 In addition,  we show that the quasi-linear dynamic equations have
 at least one global weak solution, based on the $T$-weakly continuous
 operators theory.
\end{abstract}

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


\section{Introduction and statement of main results}

Phase separation of multi-component, which consists of $N$ $(N\geq 2)$ different 
kinds of components, is a fundamental physical phenomenon. When the temperature
of the system $T>T_c$ ($T_c$ is the critical temperature), 
the concentration of $N$ different kinds of components is homogeneous 
distribution. However, the temperature $T<T_c$, the multi-component system 
may lead to phase separation, i.e., the concentration which is homogeneous 
distribution undergoes changes leading to heterogeneous spatial distribution. 
In the case that  $N=2$,  it is the binary mixture system described by the
 well-known Cahn-Hilliard equations \cite{CH}.
There have been many mathematical studies  on the dynamics of the 
Cahn-Hilliard equations, see 
\cite{AM,AWS,ggg,fff,ttt,ooo,H,eee,LSW,LSWZ,aaa,M,ccc,hhh,ddd}
 and the references therein.

 Note that the existence, uniqueness, regularity and numerical approximate 
solution of the version of stochastic Cahn-Hilliard equation have attracted 
much attentions \cite{DG,G,ZLZ}. As we known, there are few mathematical 
researches for the phase separation of multi-component systems. 
 For  the phase separation of a multi-component alloy by the finite element method, 
we refer the readers to \cite{iii,jjj,sss,nnn}. For the phase separation of 
multi-component mixture with interfacial free energy,
Elliott and Luckhaus\cite{uuu} studied a nonlinear multi-component diffusion 
equation incorporating uphill diffusion and capillarity effects. 
 Moreover, Elliott and Garcke\cite{ttt} derived a model of fourth-order 
degenerate parabolic partial differential equations for the phase separation 
in multi-component systems by  considering the possibility of a concentration 
dependence of the mobility matrix. It is worth pointing out that they also 
showed some properties of the model and proved a global existence result 
for the degenerate system.

  Based on the equilibrium phase transition dynamics theory established by
 Ma and Wang \cite{lll,bbb}, we derive a fourth-order quasi-linear dynamic
 model for phase separation of multi-component system with Ginzburg-Landau 
free energy.  The fourth-order quasi-linear dynamic equations can be expressed 
as follows
\begin{equation}\label{a}
 \frac{\partial u_k}{\partial t}=D_i[a_{ij}^{kl}(x,\mathbf{u},
 \nabla \mathbf{u},D^2\mathbf{u})D_j\Delta u_{l}]-f^{k}
 (x,\mathbf{u},\nabla \mathbf{u},\Delta \mathbf{u}),
 \end{equation}
with the  initial-boundary value conditions
 \begin{gather}\label{aa}
 \mathbf{u}(x,0)=\mathbf{\varphi}(x), \\
\label{ab}
 \mathbf{u}|_{\partial\Omega}=0,~\Delta \mathbf{u}|_{\partial\Omega}=0,
  \end{gather}
and the physical condition
 \begin{equation}\label{ac}
  \int_{\Omega}\mathbf{u}dx=0,
  \end{equation}
where $\Omega\subset \mathbb{R}^{n}$ is a bounded open set,
$\mathbf{u}=(u_1,u_2,\dots,u_{m})~(m\geq2)$ is the unknown function,
$1\leq k,l\leq m$, $1\leq i,j\leq n$.
The boundary conditions \eqref{ab} show that there is no component on
the boundary. And the physical condition \eqref{ac} indicates that
the system satisfies the certain physical conservation laws.

When $ \mathbf{u}$ is in equilibrium state, 
i.e., $\frac{\partial \mathbf{u}}{\partial t}=0$,  the corresponding  
stationary equations of \eqref{a}--\eqref{ac}  can be expressed as
\begin{equation}\label{stationary}
\begin{gathered}
 D_i[a_{ij}^{kl}(x,\mathbf{u},\nabla \mathbf{u},D^2
\mathbf{u})D_j\Delta u_{l}]-f^{k}(x,\mathbf{u},\nabla \mathbf{u},
\Delta \mathbf{u})=0, \\ 
 \mathbf{u}|_{\partial\Omega}=0,\quad \Delta \mathbf{u}|_{\partial\Omega}=0, \\ 
\int_{\Omega}\mathbf{u}dx=0,
  \end{gathered}
 \end{equation}
where $x\in\Omega \subset \mathbb{R}^{n}$,
$\mathbf{u}=(u_1,\dots,u_{m})$, $1\leq k$, $l\leq m$, $1\leq i,j\leq n$.

The main aim of this article is to study the existence of global weak solution 
for the dynamic system \eqref{a}--\eqref{ac} and the existence of weak solution 
for the corresponding  stationary equations \eqref{stationary}. 
The main techniques  are the $T$-weakly continuous operators theory for the
evolution partial differential equations established by Ma et al \cite{lll,bbb,kkk} 
and the acute angle principle for weakly continuous operators proposed by Ma et al
 \cite{qqq,bbb,kkk}, respectively.

First, we define the following two spaces, which are crucial to our theorems 
and the proofs.
\begin{gather*}
H_2=\Big\{\mathbf{u}\in H^2(\Omega,\mathbb{R}^{m}):
 \int_{\Omega}\mathbf{u}dx=0,\mathbf{u}|_{\partial\Omega}=0\Big\},\\
X_2=\Big\{\mathbf{u}\in W^{3,2}(\Omega,\mathbb{R}^{m})
 \cap W^{2,p_2}(\Omega,R^{m}): \int_{\Omega}\mathbf{u}dx=0,
\mathbf{u}|_{\partial\Omega}=0,\Delta \mathbf{u}|_{\partial\Omega}=0\Big\},
\end{gather*}
where $p_2> 2$.

We make the following assumptions:
\begin{itemize}
\item[(A1)] $a_{ij}^{kl}(x,z,\xi,\eta)$ and $f^{k}(x,z,\xi,\eta)$, 
$1\leq k,l\leq m$, $1\leq i,j\leq n$, satisfy the Carath\'eodory conditions.

\item[(A2)] There exists a $\lambda> 0$, such that
 $$
a_{ij}^{kl}\zeta_i^{k}\zeta_j^{l}\geq \lambda |\zeta|^2,
\quad\text{for any } \zeta\in \mathbb{R}^{nm}\backslash \{0\}.
$$

\item[(A3)] $f^{k}(x,z,\xi,\eta)(1\leq k\leq m)$ satisfy the 
structural conditions
\begin{gather*}
D_{\eta}f^{k}(x,z,\xi,\eta)\geq \delta> 0,\\
f^{k}(x,z,\xi,\eta)\eta_k\geq C_1|\eta|^{p_2}-C_2,
\end{gather*}
where $\delta> 0$, $C_1$, $C_2\geq 0$ are constants, $p_2> 2$. 

\item[(A4)] $a_{ij}^{kl}(x,z,\xi,\eta)$ and $f^{k}(x,z,\xi,\eta)$ satisfy the 
 increasing conditions
\begin{gather*}
|a_{ij}^{kl}(x,z,\xi,\eta)|
\leq \begin{cases}
C(|\eta|^{\frac{q_3}{2}}+|\xi|^{\frac{q_2}{2}}+|z|^{\frac{q_1}{2}}+1),
 &n>\max\{6,2p_2\},\\
\mu_3(|z|)(|\eta|^{\frac{q_3}{2}}+|\xi|^{\frac{q_2}{2}}+1),
 &\max\{4,p_2\}<n<\max\{6,2p_2\},\\
\mu_{4}(|\xi|,|z|)(|\eta|^{\frac{q_3}{2}}+1), &p_2< n<\max\{4,p_2\}.
\end{cases}
\\
|f^{k}(x,z,\xi,\eta)|
\leq  \begin{cases}
C(|\eta|^{\frac{q_3}{p_2}}+|\xi|^{\frac{q_2}{p_2}}+|z|^{\frac{q_1}{p_2}}+1),
 & n>\max\{6,2p_2\},\\
\mu_1(|z|)(|\eta|^{\frac{q_3}{p_2}}+|\xi|^{\frac{q_2}{p_2}}+1),
 & \max\{4,p_2\}<n<\max\{6,2p_2\},\\
\mu_2(|\xi|,|z|)(|\eta|^{{\frac{q_3}{p_2}}}+1),
 & p_2< n<\max\{4,p_2\}.
\end{cases}
\end{gather*}
where $C>0$ is a constant, $\mu_i(i=1,2,3,4)$ are monotonically increasing
and continuous functions.
$q_1<\max\{\frac{2n}{n-6},\frac{np_2}{n-2p_2}\}$,
$q_2<\max\{\frac{np_2}{n-p_2},\frac{2n}{n-4}\}$,
$q_3<\max\{p_2,\frac{2n}{n-2}\}$.
\end{itemize}

For the stationary equations \eqref{stationary}, we have the following 
existence result.

\begin{theorem} \label{thm1.1}
Assume that {\rm (A1)--(A4)} hold, then  \eqref{stationary} have at least one 
weak solution $\mathbf{u}\in X_2$.
\end{theorem}

For the evolution equations \eqref{a}--\eqref{ac}, the structural condition 
(A3)  can be replaced by the following condition:
\begin{itemize}
\item[(A3')]
$f^{k}(x,z,\xi,\eta)(1\leq k\leq m)$ satisfy the  structural condition
\[
f^{k}(x,z,\xi,\eta)\eta_k\geq C_1|\eta|^{p_2}
-C_2(|\eta|^2+|\xi|^2+|z|^2)-g_1(x),
\]
where $C_1,~C_2\geq 0$ are constants, $p_2> 2$, $g_1(x)\in L^{1}(\Omega)$.
\end{itemize}
Now, we give the existence of global weak solution for 
 system \eqref{a}--\eqref{ac}.

\begin{theorem} \label{thm1.2}
 Let $\mathbf{\varphi}\in H_2$, and {\rm (A1), (A2), (A3') (A4)} hold. 
Then  \eqref{a}--\eqref{ac} have at least one global weak solution
$$
\mathbf{u}\in L^p_{\rm loc}((0,\infty),X_2)\cap L^{\infty}_{\rm loc}((0,\infty),H_2).
$$
\end{theorem}


\begin{remark} \label{rmk1.3} \rm
Here we need to introduce the space mentioned in  Theorem \ref{thm1.2}.
For a Banach space $X$, we let
$$
L^p((0,T),X)=\Big\{u:(0,T)\to X : \Big(\int_0^{T}\|u\|^pdt\Big)^{1/p}
 < \infty\Big\},
$$
where $p=(p_1,p_2,\dots,p_{m})$, $p_i\geq1~(1\leq i\leq m)$,
 $\|u\|^p=\sum_{i=1}^{m}|u|_i^{p_i}$, $|\cdot|_i$
is the semi-norm in $X$ and $\|\cdot\|_{X}=\sum_{i=1}^{m}|\cdot|_i$.

Then we can define
\[
L^p_{\rm loc}((0,\infty),X)=\{u(t)\in X : u\in L^p((0,T),X),\text{ for any }T>0\}.
\]
\end{remark}

\begin{remark} \label{rmk1.4} \rm
According to the definition of the space $L^p((0,T),X)$,
 it is easy to see that  $p=(2,p_2)$ in  Theorem \ref{thm1.2}.
\end{remark}

The rest of this paper is organized as follows. 
The preliminaries, the acute angle principle for weakly continuous operators
 and the $T$-weakly continuous operators theory for parabolic equations are 
given in Section 2. In Section 3, we first introduce some basic physical
 quantities and then derive the fourth-order quasi-linear dynamic equations 
of phase separation of multi-component system. Section 4 is devoted to 
proving the main results.

\section{Preliminaries}

In this section, we  introduce the acute principle for the weakly continuous
 operators and the $T$-weakly continuous operators theory for the evolution 
equations respectively.

\subsection{Acute angle principle for weakly continuous operators}

Weakly continuous operators theory is a useful tool to solve the existence 
of elliptic equations \cite{ppp}. Here, we mainly introduce the definition 
and the acute angle principle for weakly continuous operators proposed
 by Ma in \cite{bbb,kkk}.

Let $X$ be a linear space and $X_1,~X_2$ be the completion of $X$ with 
the norm $\|\cdot\|_1,~\|\cdot\|_2$, respectively. Let $X_1$ be a separable 
Banach space and $X_2$ be a reflexive Banach space. $X_1^{*}$ is the dual 
space of $X_1$ and $X\subset X_2$. There is a linear operator $L$ satisfying
$$
L:X\to X_1 \text{ is a one-to-one and dense linear operator}.
$$

\begin{definition} \label{def2.1} \rm
A mapping $G:X_2\to X_1^{*}$ is called weakly continuous. If for any 
$\{u_n\}\subset X_2,~u_n\rightharpoonup u_0$ in $X_2$, we have
$$
\lim_{n\to \infty}\langle G(u_n),v\rangle=\langle G(u_0),v\rangle,
\quad\text{for any } v\in X_1.
$$
\end{definition}

The following lemma for weakly continuous operator is crucial to our proof.

\begin{lemma}[Acute angle principle] \label{lem2.2} 
Suppose that $G:X_2\to X_1^{*}$ is weakly continuous. 
Let $U\subset X_2$ be a bounded open set and $0\in U$. If
$$
\langle G(u),Lu\rangle\geq 0,~\quad\text{for any } u\in\partial U\cap X,
$$
then the equation $G(u)=0$ has a solution in $X_2$.
\end{lemma}


\subsection{$T$-weakly continuous operators theory for parabolic equations}
The $T$-weakly continuous operators theory was established by Ma \cite{bbb},
 which  can effectively solve the global weak solutions for many  nonlinear
 problems \cite{lll,bbb,kkk,WWL}.

Assume that the nonlinear evolution  equations  can be expressed as the 
abstract form
 \begin{equation}\label{i} 
\begin{gathered}
 \frac{du}{dt}=\widetilde{G}u,~~0<t<\infty,\\
 u(0)=\varphi,
  \end{gathered}
\end{equation}
where $\varphi\in H$, $H$ is a Hilbert space. $u:[0,\infty)\to H$ is the 
unknown function.

Let $Y_1$ and $Y_2$ be Banach spaces, $Y_1,Y_2\subset H$ and 
$Y_1^{*}$ be the dual space of $Y_1$.


\subsection*{Basic definitions and lemmas}
First, we introduce the definition of global weak solution for the 
equations \eqref{i}.

\begin{definition} \label{def2.3} \rm
Let $\varphi\in H$.~$u\in L^p_{\rm loc}((0,\infty),Y_2)
\cap L^{\infty}_{\rm loc}((0,\infty),H)$ is called a global weak solution 
of \eqref{i}, if $u$ satisfies the following equality:
$$
\langle u(t),v\rangle_{H}=\int_0^{t}\langle \widetilde{G}u,v\rangle d\tau
+\langle \varphi,v\rangle_{H}.
$$
for any $v\in Y_1\subset H$.
\end{definition}

Next we give the definitions of uniformly weak convergence and $T$-weak continuity.

\begin{definition} \label{def2.4} \rm
Let $\{u_n\}\subset L^p((0,T),Y_2),~u_0\in L^p((0,T),Y_2)$.
We say that $u_n\rightharpoonup u_0$ in $L^p((0,T),Y_2)$ is uniformly
 weakly convergent, if $\{u_n\}\subset L^{\infty}((0,T),H)$ is bounded 
and satisfies
\begin{gather*}
u_n\rightharpoonup u_0 ~\text{in}~L^p((0,T),Y_2),\\
\lim_{n\to \infty}\int_0^{T}|\langle u_n-u_0,v\rangle_{H}|^2dt=0,
 \quad\text{for any } v\in H.
\end{gather*}
\end{definition}

\begin{definition} \label{def2.5} \rm
A mapping $\widetilde{G}:Y_2\times (0,\infty)\to Y_1^{*}$ is called 
$T$-weakly continuous. If for any $p=(p_1,p_2,\dots,p_{m})$, $0<T<\infty$ and 
$u_n\rightharpoonup u_0$ is uniformly weakly convergent in $L^p((0,T),Y_2)$,
 we have
$$
\lim_{n\to \infty}\int_0^{T}\langle \widetilde{G}u_n,v\rangle dt
=\int_0^{T}\langle \widetilde{G}u_0,v\rangle dt,\quad\text{for any } v\in Y_1.
$$
\end{definition}

The following two elementary lemmas will be used later. 
Their proofs can be found in  \cite{bbb}
.
\begin{lemma} \label{lem2.6}
Let $\Omega\subset \mathbb{R}^{n}$ be a bounded set, 
$\{u_n\}\subset L^p((0,T),W^{s,p}(\Omega))(s\geq 1, p\geq2)$ 
be a bounded sequence and $\{u_n\}$ is uniformly weakly convergent to
 $u_0\in L^p((0,T),W^{s,p}(\Omega))$. Then for any $|\alpha|\leq s-1$, we have
\[
D^{\alpha}u_n\to D^{\alpha}u_0 \quad\text{in } L^2((0,T)\times\Omega).
\]
\end{lemma}

\begin{lemma} \label{lem2.7}
Let $\Omega\subset\mathbb{ R}^{n}$ be an open set, the function
$f:\Omega\times \mathbb{R}^{N}\to \mathbb{R}^{1}$ satisfy the 
Carath\'eodory conditions and
\[
|f(x,\xi)|\leq C\sum_{i=1}^{N}|\xi_i|^{p_i/p}+b(x),
\]
where $C>0$ is a constant and $p_i,p> 1$, $b(x)\in L^p(\Omega)$.

If $\{u_{i_k}\}\subset L^{p_i}(\Omega)~(1\leq i\leq N)$ is bounded and
$\{u_{i_k}\}$ converges to $\{u_i\}$ by measure in $\Omega_0$ for any
bounded subregion $\Omega_0\subset \Omega$, then for any $v\in L^{p'}(\Omega)$,
we have
\[
\lim_{k\to\infty}\int_{\Omega}f(x,u_{1_k},\dots,u_{N_k})v\,dx
=\int_{\Omega}f(x,u_1,\dots,u_{N})v\,dx,
\]
where $p'$ satisfies $\frac{1}{p'}+\frac{1}{p}=1$.
\end{lemma}


\subsection*{Existence  of a global weak solution for nonlinear parabolic equations}

First, we introduce the following function spaces
\begin{gather*}
Y\subset Y_2\subset Y_1\subset H,\\
Y_2\subset H_2\subset H_1\subset H,
\end{gather*}
where $Y$ is a linear space, $Y_1,Y_2$ are Banach spaces, $H,H_1$ and 
$H_2$ are Hilbert spaces. We remark that all inclusion relations are dense 
embedding.

Moreover, suppose that there exists an operator $\overline{L}$ satisfying 
the following conditions
\begin{equation}\label{j} 
\begin{gathered}
\overline{L}: Y\to Y_1 \text{ is a one-to-one and dense linear operator},\\
\langle \overline{L}u,v\rangle_{H}=\langle u,v\rangle_{H_2},\quad\text{for any }
 u,v\in Y. 
  \end{gathered}
\end{equation}
In addition,  there exists a sequence $\{e_k\}_{k=1}^{\infty}\subset Y$ such that
\begin{equation}\label{k}
\overline{L}e_k=\rho_ke_k,\quad k=1,2,\dots,
\end{equation}
where $\rho_k\neq 0$, $\{e_k\}_{k=1}^{\infty}$ is the common orthogonal
basis of $H$.

Here we also assume that $\widetilde{G}:Y_2\times (0,\infty)\to Y_1^{*}$ 
satisfies the following inequality,
\begin{equation}\label{l}
\langle \widetilde{G}u,\overline{L}u\rangle
\leq -C_1\|u\|_{Y_2}^p+C_2\|u\|_{H_2}^2+f(t),
\end{equation}
where $p=(p_1,p_2,\dots,p_{m})$, $p_i>1$ ($1\leq i\leq m$),~
$\|u\|_{Y_2}^p=\sum_{i=1}^{m}|u|_i^{p_i}$, $|\cdot|_i$ is the semi-norm in 
$Y_2$, $\|u\|_{Y_2}=\sum_{i=1}^{m}|u|_i$, $C_1,C_2>0$ are constants, 
$f\in L_{\rm loc}^{1}(0,\infty)$.

Then we give the following existence result of global weak solutions for the 
nonlinear parabolic equations \eqref{i}.

\begin{lemma} \label{lem2.8}
 Assume that \eqref{j}--\eqref{l} hold. 
If $\widetilde{G}:Y_2\times (0,\infty)\to Y_1^{*}$ is $T$-weakly continuous, 
then   problem \eqref{i} has a global weak solution
$$
u\in  L^p_{\rm loc}((0,\infty),Y_2)\cap L^{\infty}_{\rm loc}((0,\infty),H_2)
$$
for any $\varphi\in H_2$.
\end{lemma}

\section{Dynamic equations of phase separation of multi-component system}
In this section, we devote to deriving the new dynamic model 
\eqref{a}--\eqref{ac} of phase separation of multi-component system by 
using the equilibrium phase transition dynamics theory founded by Ma 
and Wang\cite{lll}.


\subsection{Basic physical quantities}

Let $\Sigma$ be a multi-component system mixed by $m+1$ different kinds 
of components $A_1,\dots,A_{m+1}~(m\geq 2)$. $u_k~(1\leq k\leq m+1)$
is the molar density of $A_k$, i.e.,
$$
u_k(x)=\text{the molar number of $A_k$ in unit volume at }x\in\Omega.
 $$
Note that $u_1,\dots,u_{m+1}$ satisfy the relation
$$
u_1+u_2+\dots +u_{m+1}=\text{constant}.
$$
It is worth noticing that the order parameter $\mathbf{u}$ contains only 
$m$ independent variables, i.e., $\mathbf{u}=(u_1,u_2,\dots,u_{m})$. 
In fact $\mathbf{u}=(u_1,u_2,\dots,u_{m})$ is the unknown function.

Based on the physical experiments,  this system is also related to the 
temperature $T$ and the container volume $|\Omega|$.
 Hence, we regard  $T$ and $|\Omega|$ as the control parameters. More generally, 
the control parameter can be expressed as
$$
\kappa=(T,|\Omega|,\omega_1,\dots,\omega_{m}),
$$
where $\omega_k$ is the proportion of $A_k$ in the multi-component system.

\subsection{A new dynamic model}

In this subsection, we are focused on obtaining the dynamic equations \eqref{a}
 for the order parameter $\mathbf{u}$.

According to the Ginzburg-Landau mean field theory, the free energy of a 
$m+1$-components system(see\cite{lll}) can be expressed as
\begin{equation}\label{b}
H(\mathbf{u},\kappa)=\int_{\Omega}
\Big[\frac{1}{2}\sum_{k=1}^{m}\mu_k|\nabla u_k|^2+g(\mathbf{u},\kappa)\Big]dx,
\end{equation}
where $\mu_k=\mu_k(\kappa)\geq 0$ is the physical parameter.
$g(\mathbf{u},\kappa)$ is a polynomial on $\mathbf{u}$, which can be given by
\begin{equation}\label{c}
g(\mathbf{u},\kappa)=\sum_{1\leq |\gamma|\leq 2r}
a_{\gamma}u_1^{\gamma_1}u_2^{\gamma_2}\dots u_{m}^{\gamma_{m}},
\quad \gamma=(\gamma_1,\gamma_2,\dots \gamma_{m}).
\end{equation}

Based on the equilibrium phase transition dynamics theory (see\cite{lll}), 
the following dynamic equations can be deduced from \eqref{b}--\eqref{c}:
\begin{equation}\label{d}
\begin{aligned}
\frac{\partial u_k}{\partial t}
&=-\beta_k\nabla\cdot\Big[\sum_{l=1}^{m}L_{kl}\nabla(\mu_{l}\Delta u_{l}
 -g_{l}(\mathbf{u},\kappa))\Big]\\
&\quad +\nabla\cdot\Big(\sum_{l=1}^{m}L_{kl}\nabla\phi_{l}
(\mathbf{u},\kappa)\Big),
 \end{aligned}
\end{equation}
where $\beta_k>0$, $L_{kl}=L_{kl}(\mathbf{u},D\mathbf{u})~(1\leq k,l\leq m)$
is positive and symmetric, and 
$g_{l}(\mathbf{u},\kappa)=\frac{\partial}{\partial u_{l}}g(\mathbf{u},\kappa)$. 
$\phi_{l}$ is independent of $u_{l}$ and satisfies
\begin{equation}\label{e}
\int_{\Omega}\sum_{k,l=1}^{m}L_{kl}\nabla(\mu_k\Delta u_k-g_k)
\cdot\nabla\phi_{l}dx=0,
\end{equation}
where $g_k(\mathbf{u},\kappa)
=\frac{\partial}{\partial u_k}g(\mathbf{u},\kappa)$.

In this paper, we consider the more general case that  the equations \eqref{d} are
quasi-linear. Meanwhile,  we take  $\phi_{l}(\mathbf{u},\kappa)=0$ in 
\eqref{d} and \eqref{e}, which has no material impact to the main characteristics 
of this physical system. Furthermore, we supplement with the initial-boundary 
conditions \eqref{aa}--\eqref{ab} and the physical conservation laws
condition \eqref{ac}. Therefore, we obtain the modified dynamic model 
\eqref{a}--\eqref{ac}, which is a fourth-order quasi-linear
evolution partial differential equations.


\section{Proofs of main results}

\subsection{Proof of Theorem \ref{thm1.1}}
Now we will apply Lemma \ref{lem2.2} and  Lemma \ref{lem2.7}
 to prove the existence of a 
weak solution for
the steady state equations \eqref{stationary}.
We will prove  Theorem \ref{thm1.1} in three steps.
\smallskip

\noindent\textbf{Step 1.} Define the operator $G$. Let
\begin{gather*}
  X=\Big\{\mathbf{u}\in C^{\infty}(\Omega,\mathbb{R}^{m}): 
\int_{\Omega}\mathbf{u}dx=0, \mathbf{u}|_{\partial\Omega}=0, 
 \Delta \mathbf{u}|_{\partial\Omega}=0\Big\}, \\
  X_1=\{\mathbf{u}\in C^{\infty}(\Omega,\mathbb{R}^{m}): \mathbf{u}|_{\partial\Omega}=0\},\\
  X_2=\Big\{\mathbf{u}\in W^{3,2}(\Omega,\mathbb{R}^{m})\cap W^{2,p_2}
(\Omega,\mathbb{R}^{m}): \int_{\Omega}\mathbf{u}dx=0, \mathbf{u}|_{\partial\Omega}=0, 
\Delta \mathbf{u}|_{\partial\Omega}=0\Big\}.
\end{gather*}
According to the general definition of weak solution, we define the operator 
$G:X_2\to X_1^{*}$ by the inner product from
\[
\langle G\mathbf{u},\mathbf{v}\rangle=\int_{\Omega}[a_{ij}^{kl}(x,\mathbf{u},
\nabla \mathbf{u},D^2\mathbf{u})D_j\Delta u_{l}D_iv_k+f^{k}(x,\mathbf{u},
\nabla \mathbf{u},\Delta \mathbf{u})v_k]dx,
\]
where $\mathbf{v}=(v_1,v_2,\dots,v_{m})\in X_1$, $X_1^{*}$ is the dual space of 
$X_1$.
From (A4), it is easy to show that the operator $G$ is a bounded operator.
\smallskip

\noindent\textbf{Step 2.} Check  the conditions for the acute angle principle.
Let $L=\Delta:X\to X_1$. The conditions (A2) and (A3) imply that
\begin{equation} \label{AB}
\begin{aligned}
\langle G\mathbf{u},\Delta\mathbf{u}\rangle
&=\int_{\Omega}[a_{ij}^{kl}(x,\mathbf{u},\nabla\mathbf{u},D^2\mathbf{u})
  D_j\Delta u_{l}D_i\Delta u_k+f^{k}(x,\mathbf{u},\nabla \mathbf{u},
 \Delta \mathbf{u})\Delta u_k]dx\\
&\geq \lambda\int_{\Omega}|\nabla(\Delta \mathbf{u})|^2dx
 +C_1\int_{\Omega}|\Delta \mathbf{u}|^{p_2}dx-C_2.
\end{aligned}
\end{equation}
By \eqref{AB},  it is clear that
$$
\langle G\mathbf{u},\Delta\mathbf{u}\rangle\geq 0,\quad
\text{for any $\mathbf{u}\in X_2$ and $\|\mathbf{u}\|_{X_2}$ is large enough},
$$
which implies that the operator $G:X_2\to X_1^{*}$ satisfies the condition
of Lemma \ref{lem2.2}.
\smallskip

\noindent\textbf{Step 3.} Verify the weak continuity of the operator $G$.
Let $\{\mathbf{u}_n\}\subset X_2,~\mathbf{u}_n\rightharpoonup \mathbf{u}_0$ in
$X_2$. Based on the  Definition \ref{def2.1}, we only need to prove that the following 
limit holds
\begin{equation} \label{AC}
\begin{aligned}
&\lim_{n\to\infty}\int_{\Omega}[a_{ij}^{kl}(x,\mathbf{u}_n,\nabla \mathbf{u}_n,D^2
 \mathbf{u}_n)D_j\Delta u_{nl}D_iv_k+f^{k}(x,\mathbf{u}_n,
 \nabla \mathbf{u}_n,\Delta \mathbf{u}_n)v_k]dx\\
&=\int_{\Omega}[a_{ij}^{kl}(x,\mathbf{u}_0,
 \nabla \mathbf{u}_0,D^2\mathbf{u}_0)D_j\Delta u_{0l}D_iv_k+f^{k}
 (x,\mathbf{u}_0,\nabla \mathbf{u}_0,\Delta \mathbf{u}_0)v_k]dx.
\end{aligned}
\end{equation}
for any $\mathbf{v}\in X_1$.

We should divide  \eqref{AC} into the following two parts.
\begin{gather}\label{p}
\lim_{n\to\infty}\int_{\Omega}f^{k}(x,\mathbf{u}_n,\nabla \mathbf{u}_n,
\Delta \mathbf{u}_n)v_kdx=\int_{\Omega}f^{k}(x,\mathbf{u}_0,
\nabla \mathbf{u}_0,\Delta \mathbf{u}_0)v_kdx, \\
\label{AD}
\begin{aligned}
&\lim_{n\to\infty}\int_{\Omega}a_{ij}^{kl}(x,\mathbf{u}_n,
 \nabla \mathbf{u}_n,D^2\mathbf{u}_n)D_j\Delta u_{nl}D_iv_kdx\\
&=\int_{\Omega}a_{ij}^{kl}(x,\mathbf{u}_0,\nabla \mathbf{u}_0,
D^2\mathbf{u}_0)D_j\Delta u_{0l}D_iv_kdx.
\end{aligned}
\end{gather}
By the compact embedding theorem, it is easy to check the following relations
\begin{equation}\label{o}
(\mathbf{u}_n,D\mathbf{u}_n,D^2\mathbf{u}_n)
\to (\mathbf{u}_0,D\mathbf{u}_0,D^2\mathbf{u}_0) \quad\text{in }
\begin{cases}
L^{q_1}\times L^{q_2}\times L^{q_3},\\
C^0\times L^{q_2}\times L^{q_3},\\
C^0\times C^0\times L^{q_3},
\end{cases}
\end{equation}
where $q_1<\max\{\frac{2n}{n-6},\frac{np_2}{n-2p_2}\}$,
$q_2<\max\{\frac{np_2}{n-p_2},\frac{2n}{n-4}\}$,
$q_3<\max\{p_2,\frac{2n}{n-2}\}$.
Combining (A4), \eqref{o} and Lemma \ref{lem2.7}, it is easy to see that
\eqref{p} is  valid.

Notice that \eqref{AD} is equivalent to
\begin{equation} \label{AE}
\begin{aligned}
&\lim_{n\to\infty}\int_{\Omega}[a_{ij}^{kl}(x,\mathbf{u}_n,
 \nabla\mathbf{u}_n,D^2\mathbf{u}_n)D_j\Delta u_{nl}\\
&-a_{ij}^{kl}(x,\mathbf{u}_0,\nabla \mathbf{u}_0,
 D^2\mathbf{u}_0)D_j\Delta u_{0l}]D_iv_kdx=0.
\end{aligned}
\end{equation}
Moreover, the left part of \eqref{AE} can be rewritten as
\begin{equation}\label{AF}
\begin{aligned}
   &\lim_{n\to\infty}\int_{\Omega}[a_{ij}^{kl}(x,\mathbf{u}_n,
\nabla \mathbf{u}_n,D^2\mathbf{u}_n)D_j\Delta u_{nl}\\
   &-a_{ij}^{kl}(x,\mathbf{u}_0,\nabla \mathbf{u}_0,D^2\mathbf{u}_0)D_j
\Delta u_{0l}]D_iv_kdx\\
   &=\lim_{n\to\infty}\Big\{\int_{\Omega}[a_{ij}^{kl}(x,\mathbf{u}_n,
\nabla \mathbf{u}_n,D^2\mathbf{u}_n)-a_{ij}^{kl}(x,\mathbf{u}_0,
\nabla \mathbf{u}_0,D^2\mathbf{u}_0)]D_j\Delta u_{nl}D_iv_kdx\\
   &\quad +
   \int_{\Omega}a_{ij}^{kl}(x,\mathbf{u}_0,\nabla \mathbf{u}_0,D^2
\mathbf{u}_0)[D_j\Delta u_{nl}-D_j\Delta u_{0l}]D_iv_kdx\Big\}.
\end{aligned}
\end{equation}
Analogously, under the assumption (A4),  we get following  
equality basing on \eqref{o} and  Lemma \ref{lem2.7},
\begin{equation}\label{q}
\begin{aligned}
&\lim_{n\to\infty}\int_{\Omega}[a_{ij}^{kl}(x,\mathbf{u}_n,
 \nabla \mathbf{u}_n,D^2\mathbf{u}_n) \\
&-a_{ij}^{kl}(x,\mathbf{u}_0,\nabla \mathbf{u}_0,D^2\mathbf{u}_0)]
D_j\Delta u_{nl}D_iv_kdx=0.
\end{aligned}
\end{equation}

For the second term on the right hand of  \eqref{AF}, it is not 
difficult to derive the following result from 
$\mathbf{u}_n\rightharpoonup \mathbf{u}_0$ in $X_2$,
\begin{equation}\label{r}
\lim_{n\to\infty}\int_{\Omega}a_{ij}^{kl}(x,\mathbf{u}_0,
\nabla \mathbf{u}_0,D^2\mathbf{u}_0)[D_j\Delta u_{nl}-D_j
\Delta u_{0l}]D_iv_kdx=0.
\end{equation}
Obviously, \eqref{q} and \eqref{r} infer that \eqref{AD} holds true.
Then the weak continuity of the operator $G:X_2\to X_1^{*}$ is obtained.

Therefore, we can immediately get that  problem \eqref{stationary} 
has a weak solution by using Lemma \ref{lem2.2}.

\subsection{Proof of Theorem \ref{thm1.2}}
We now apply Lemma \ref{lem2.8} to  prove the system \eqref{a}--\eqref{ac} has a
 global weak solution.
The proof is divided into three steps.
\smallskip

\noindent\textbf{Step 1.} Define the operator $\widetilde{G}$. Let
\begin{gather*}
 X=\Big\{\mathbf{u}\in C^{\infty}(\Omega,\mathbb{R}^{m}):
  \int_{\Omega}\mathbf{u}dx=0,\mathbf{u}|_{\partial\Omega}=0,
 \Delta \mathbf{u}|_{\partial\Omega}=0\Big\}, \\
 X_1=\{\mathbf{u}\in C^{\infty}(\Omega,\mathbb{R}^{m}):
  \mathbf{u}|_{\partial\Omega}=0\},\\
 X_2=\Big\{\mathbf{u}\in W^{3,2}(\Omega,\mathbb{R}^{m})\cap W^{2,p_2}
 (\Omega,R^{m}): \int_{\Omega}\mathbf{u}dx=0,\mathbf{u}|_{\partial\Omega}=0,
 \Delta \mathbf{u}|_{\partial\Omega}=0\Big\},\\
 H=\Big\{\mathbf{u}\in L^2(\Omega,\mathbb{R}^{m}):
 \int_{\Omega}\mathbf{u}dx=0\Big\},\\
 H_1=\Big\{\mathbf{u}\in H^{1}(\Omega,\mathbb{R}^{m}):
 \int_{\Omega}\mathbf{u}dx=0,\mathbf{u}|_{\partial\Omega}=0\Big\},\\
 H_2=\Big\{\mathbf{u}\in H^2(\Omega,\mathbb{R}^{m}):
 \int_{\Omega}\mathbf{u}dx=0,\mathbf{u}|_{\partial\Omega}=0\Big\}.
\end{gather*}
According to the  Definition \ref{def2.3}, we define the operator 
$\widetilde{G}:X_2\times (0,\infty)\to X_1^{*}$ by the  inner product form
\[
\langle \widetilde{G}\mathbf{u},\mathbf{v}\rangle
=\int_{\Omega}[-a_{ij}^{kl}(x,\mathbf{u},\nabla \mathbf{u},
D^2\mathbf{u})D_j\Delta u_{l}D_iv_k-f^{k}(x,\mathbf{u},\nabla \mathbf{u},
\Delta \mathbf{u})v_k]dx,
\]
where $\mathbf{v}\in X_1$.
By  assumption (A4), it is easy to check that the $\widetilde{G}$ is
a bounded operator.
\smallskip

\noindent\textbf{Step 2.} Check  conditions \eqref{j}--\eqref{l}.
Let $\overline{L}=\Delta:X\to X_1$. It is obvious that \eqref{j} and \eqref{k}
 are valid.
It follows from  assumptions (A2) and (A3')  that
\begin{equation}\label{AH}
\begin{aligned}
&\langle \widetilde{G}\mathbf{u},\Delta\mathbf{u}\rangle\\
&=\int_{\Omega}[-a_{ij}^{kl}(x,\mathbf{u},\nabla\mathbf{u},
 D^2\mathbf{u}) D_j\Delta u_{l}D_i\Delta u_k-f^{k}(x,\mathbf{u},
 \nabla \mathbf{u},\Delta \mathbf{u})\Delta u_k]dx\\
&\leq -\lambda\int_{\Omega}|\nabla(\Delta \mathbf{u})|^2dx
 -C_1\int_{\Omega}|\Delta \mathbf{u}|^{p_2}dx\\
&\quad +C_2\int_{\Omega}(|\Delta \mathbf{u}|^2+|\nabla \mathbf{u}|^2
 +|\mathbf{u}|^2)dx+\int_{\Omega}g_1(x)dx,
\end{aligned}
\end{equation}
which implies that \eqref{l} holds true.
\smallskip

\noindent\textbf{Step 3.}
 Verify the condition for the $T$-weak continuity of the operator 
$\widetilde{G}$.
Let $\{\mathbf{u}_n\}\subset L^p((0,T),X_2)\cap L^{\infty}((0,T),H_2)$, 
$\mathbf{u}_n\rightharpoonup \mathbf{u}_0$ in $L^p((0,T),X_2)$ be uniformly
 weakly convergent.
By  definition \ref{def2.5}, we only need to show the following limit holds,
\begin{equation}\label{AJ}
\begin{aligned}
&\lim_{n\to\infty}\int_0^{t}\int_{\Omega}[-a_{ij}^{kl}
 (x,\mathbf{u}_n,\nabla \mathbf{u}_n,D^2\mathbf{u}_n)D_j\Delta u_{nl}D_iv_k\\
&-f^{k}(x,\mathbf{u}_n,\nabla \mathbf{u}_n,\Delta \mathbf{u}_n)v_k]\,dx\,d\tau\\
&=\int_0^{t}\int_{\Omega}[-a_{ij}^{kl}(x,\mathbf{u}_0,\nabla \mathbf{u}_0,D^2
 \mathbf{u}_0)D_j\Delta u_{0l}D_iv_k\\
&\quad -f^{k}(x,\mathbf{u}_0,\nabla \mathbf{u}_0,\Delta \mathbf{u}_0)v_k]\,dx\,d\tau.
\end{aligned}
\end{equation}
Obviously, \eqref{AJ} can be divided into the following two parts.
\begin{gather}\label{v}
\begin{aligned}
&\lim_{n\to\infty}\int_0^{t}\int_{\Omega}f^{k}(x,\mathbf{u}_n,\nabla \mathbf{u}_n,
 \Delta \mathbf{u}_n)v_k\,dx\,d\tau\\
&=\int_0^{t}\int_{\Omega}f^{k}(x,\mathbf{u}_0,\nabla \mathbf{u}_0,
 \Delta \mathbf{u}_0)v_k\,dx\,d\tau.
\end{aligned} \\ 
\label{AK}
\begin{aligned}
&\lim_{n\to\infty}\int_0^{t}\int_{\Omega}a_{ij}^{kl}(x,\mathbf{u}_n,
 \nabla \mathbf{u}_n,D^2\mathbf{u}_n)D_j\Delta u_{nl}D_iv_k\,dx\,d\tau\\
&=\int_0^{t}\int_{\Omega}a_{ij}^{kl}(x,\mathbf{u}_0,\nabla \mathbf{u}_0,
D^2\mathbf{u}_0)D_j\Delta u_{0l}D_iv_k\,dx\,d\tau.
\end{aligned}
\end{gather}
Owing to $\{\mathbf{u}_n\}\subset L^p((0,T),X_2)\cap L^{\infty}((0,T),H_2)$, 
$\mathbf{u}_n\rightharpoonup \mathbf{u}_0$ in $L^p((0,T),X_2)$ is uniformly 
weakly convergent, we can derive the following convergence properties by 
using the  Lemma \ref{lem2.6},
\begin{equation}\label{t}
\begin{gathered}
\mathbf{u}_n\to \mathbf{u}_0 ~~\text{in}~~ L^2((0,T)\times\Omega), \\
D\mathbf{u}_n\to D\mathbf{u}_0 ~~\text{in}~~ L^2((0,T)\times\Omega),\\
D^2\mathbf{u}_n\to D^2\mathbf{u}_0 ~~\text{in}~~ L^2((0,T)\times\Omega),
\end{gathered}
\end{equation}
which infer that $\{\mathbf{u}_n\}$, $\{D\mathbf{u}_n\}$ and $\{D^2\mathbf{u}_n\}$
converge to $\mathbf{u}_0$, $D\mathbf{u}_0$ and $D^2\mathbf{u}_0$ by measure 
in $\Omega\times (0,T)$, respectively. Then, together the assumption 
(A4) with Lemma \ref{lem2.7},
we see that \eqref{v} holds.


Note that \eqref{AK}  is equivalent to
\begin{equation} \label{AL}
\begin{aligned}
&\lim_{n\to\infty}\int_0^{t}\int_{\Omega}[a_{ij}^{kl}(x,\mathbf{u}_n,
 \nabla \mathbf{u}_n,D^2\mathbf{u}_n)D_j\Delta u_{nl}\\
&-a_{ij}^{kl}(x,\mathbf{u}_0,\nabla \mathbf{u}_0,D^2\mathbf{u}_0)
D_j\Delta u_{0l}]D_iv_k\,dx\,d\tau=0.
\end{aligned}
\end{equation}
Furthermore, the left part of  \eqref{AL} can be rewritten as
\begin{equation}\label{AO}
\begin{aligned}
&\lim_{n\to\infty}\int_0^{t}\int_{\Omega}[a_{ij}^{kl}(x,\mathbf{u}_n,
 \nabla \mathbf{u}_n,D^2\mathbf{u}_n)D_j\Delta u_{nl} \\
&-a_{ij}^{kl}(x,\mathbf{u}_0,\nabla \mathbf{u}_0,D^2\mathbf{u}_0)
 D_j\Delta u_{0l}]D_iv_k\,dx\,d\tau\\
&=\lim_{n\to\infty}\Big\{\int_0^{t}\int_{\Omega}[a_{ij}^{kl}(x,\mathbf{u}_n,
 \nabla \mathbf{u}_n,D^2\mathbf{u}_n)\\
&\quad -a_{ij}^{kl}(x,\mathbf{u}_0,\nabla \mathbf{u}_0,D^2\mathbf{u}_0)]
D_j\Delta u_{nl}D_iv_k\,dx\,d\tau\\
&\quad +\int_0^{t}\int_{\Omega}a_{ij}^{kl}(x,\mathbf{u}_0,
 \nabla \mathbf{u}_0,D^2\mathbf{u}_0)[D_j\Delta u_{nl}
 -D_j\Delta u_{0l}]D_iv_k\,dx\,d\tau\Big\}.
\end{aligned}
\end{equation}
Combining  assumption (A4), \eqref{t} and  Lemma \ref{lem2.7}, it is clear that
\begin{equation}\label{w}
\begin{aligned}
&\lim_{n\to\infty}\int_0^{t}\int_{\Omega}[a_{ij}^{kl}(x,\mathbf{u}_n,\nabla \mathbf{u}_n,D^2\mathbf{u}_n)\\
&-a_{ij}^{kl}(x,\mathbf{u}_0,\nabla \mathbf{u}_0,D^2\mathbf{u}_0)]
D_j\Delta u_{nl}D_iv_k\,dx\,d\tau=0.
\end{aligned}
\end{equation}

Because $\mathbf{u}_n\rightharpoonup \mathbf{u}_0$ in $L^p((0,T),X_2)$ 
which is uniformly weakly convergent,  it is easy to see that the following
 limit holds
\begin{equation}\label{x}
\lim_{n\to\infty}\int_0^{t}\int_{\Omega}a_{ij}^{kl}(x,\mathbf{u}_0,
\nabla \mathbf{u}_0,D^2\mathbf{u}_0)[D_j\Delta u_{nl}-D_j\Delta u_{0l}]
D_iv_k\,dx\,d\tau=0.
\end{equation}

 Note that \eqref{w} and \eqref{x} imply  that \eqref{AK} holds. Hence, 
$G:X_2\times(0,\infty)\to X_1^{*}$ is $T$-weakly continuous.

Consequently, from Lemma \ref{lem2.8}, we can easily obtain  that
 problem \eqref{a}--\eqref{ac} has one global weak solution
$$
\mathbf{u}\in L^p_{\rm loc}((0,\infty),X_2)\cap L^{\infty}_{\rm loc}((0,\infty),H_2).
$$


\subsection*{Acknowledgments} We thank Professor Tian Ma
for his valuable discussions during the preparation of this work. 
Ruikuan Liu was supported by the NSFC (No. 11771306). 
The authors are very grateful to the editor and the
anonymous referees for their valuable suggestions.

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\end{document}