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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 47, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/47\hfil
 Well-posedness and asynchronous exponential growth]
{Well-posedness and asynchronous exponential growth
of solutions of a two-phase cell \\ division model}

\author[M. Bai, S. Cui\hfil EJDE-2010/47\hfilneg]
{Meng Bai, Shangbin Cui}  % in alphabetical order

\address{Meng Bai \newline
Department of Mathematics, Sun Yat-Sen University,
Guangzhou, Guangdong 510275, China}
\email{baimeng.clare@yahoo.com.cn}

\address{Shangbin Cui \newline
Department of Mathematics, Sun Yat-Sen University,
Guangzhou, Guangdong 510275, China}
\email{cuisb3@yahoo.com.cn}


\thanks{Submitted March 3, 2010. Published April 6, 2010.}
\thanks{Supported by grant 10771223 from the  National Natural Science
Foundation of China}
\subjclass[2000]{35L02, 35P99}
\keywords{Cell division model; two-phase; well-posedness;
\hfill\break\indent  asynchronous exponential growth}

\begin{abstract}
 In this article we study a two-phase cell division model.
 The cells of the two different phases have different growth rates.
 We mainly consider the model of equal mitosis. By using the
 semigroup theory, we prove that this model is well-posed in
 suitable function spaces and its solutions have the property
 of asynchronous exponential growth as time approaches infinity.
 The corresponding model of asymmetric mitosis is also studied
 and similar results are obtained.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

In the study of cell division, it has been recognized that the cell
cycle can be divided into two major phase: The interphase and the
\textbf{M} (mitosis) phase (cf. \cite{PE,JC}). In the
interphase cells only increase their size and replicate their DNA,
and do not undergo mitosis, whereas in the M phase the
fully grown cells segregate the replicated chromosomes to opposite
ends of the molecular scaffold (termed the spindle) and then cleave
between them in a process known as cytokinesis to produce two
daughter cells. The cells in the two phases are observably
different.

In this paper we study a mathematical model describing the
proliferation of cells which are divided into two different phases:
mitotic phase and non-mitotic phase. We refer these two phases as
$m$-phase and $n$-phase, respectively. We denote by $m(t,x)$ and
$n(t,x)$ the densities of $m$-phase cells and $n$-phase cells,
respectively, of size $x$ (with a maximal size normalized to $x=1$)
at time $t$. We assume that two daughter cells have equal sizes
(i.e. \emph{equal mitosis}). In particular, we assume that the two
phases have different growth rates $\gamma_1(x)$ and $\gamma_2(x)$,
respectively. Then the model reads as follows:
\begin{equation} \label{e1.1}
\begin{gathered}
\begin{aligned}
\frac{\partial m}{\partial t}+\frac{\partial
(\gamma_1(x)m)}{\partial x}
&=-B(x)m(t,x)-\nu(x)m(t,x)+\mu(x)n(t,x),\\
&\quad 0<x<1,\;t>0,\\
\frac{\partial n}{\partial t}+\frac{\partial
(\gamma_2(x)n)}{\partial x}
&=-\nu(x)n(t,x)-\mu(x)n(t,x)\\
&\quad +\begin{cases}
  4B(2x)m(t,2x),  &   0\leq x\leq\frac{1}{2},\; t>0 \\
  0, & \frac{1}{2}<x\leq 1, \; t>0,
\end{cases}
\end{aligned}\\
m(t,0)=0,\quad  n(t,0)=0,\quad t>0,\\
 m(0,x)=m_0(x),\quad  n(0,x)=n_0(x),\quad 0<x<1.
\end{gathered}
\end{equation}
Here $\mu(x)$ represents the transferring rate of cells from $n$-phase
to $m$-phase, $\nu(x)$ represents the death rate of the cells, and
$B(x)$ represents the mitosis rate of the cells in $m$-phase.

For the one-pase cell division model, it has been proved by Perthame
and Ryzhik in \cite{PR} (see also  \cite[Chapter 4]{P}) and Michel,
Mischler and Perthame in \cite{MMP} by using the generalized
relative entropy method that the problems are globally well-posed
and the solutions exhibit so called \emph{asynchronous exponential
growth} (cf. \cite{FH,SW,DVW,EN,AG}).
The purpose of this work is to extend these results to
 model \eqref{e1.1}, but using a different method -- the semigroup
method. We shall prove that under suitable assumptions on $\mu$,
$\nu$ and $B$, problem \eqref{e1.1} is globally well-posed, and its
solution possesses the properties of asynchronous exponential
growth.

The anonymous referee called our attention to a recent work
by  Perthame and Touaoula \cite{PT}, where a different multi-species
cell division model is studied. In that model it is assumed
that each cell can divide at
most $I$ times in its lifespan, so that all cells are divided into
$I$-generations. All cells grow at a same constant rate and each of
the cells in the $i$-th ($1\leq i\leq I-1$) generation divide into
two cells of equal size at a rate $B_{i}(x)$ when mitosis occurs.
One of the two cells, called the daughter cell, resumes a cycle at
the generation 1, while the other cell, called the mother cell,
enters the generation $i+1$. By establishing existence of
eigenvalues with positive eigenvectors of the eigenvalue problem and
its adjoint problem and using the general relative entropy method,
those authors proved that the solutions of their model have the
property of asynchronous exponential growth. Unlike that model, in
the model under this study we assume that cells consist of two
different phases: the mitotic phase and the non-mitotic phase. Cells
in the mitotic phase can divide into two cells of non-mitotic phase,
while cells in the non-mitotic phase do not undergo mitotic. This is
the main difference between this work and the reference \cite{PT}.

Throughout this paper, the transferring rate $\mu(x)$, the death
rate $\nu(x)$, the equal mitosis rate $B(x)$, and the growth rates
$\gamma_1(x)$ and $\gamma_2(x)$ are supposed to satisfy the
following conditions:
\begin{itemize}

\item[(H1)] $\mu$ and $\nu$ are nonnegative and continuous functions
defined in $[0,1]$. Moreover, $\mu(x)>0$ for almost all
$x\in(0,1)$;

\item[(H2)] $B$ is a nonnegative and continuous function defined in
$[0,1]$ with $B(x)>0$ for $x\in(0,1)$ and $B(x)=0$ for otherwise.

\item[(H3)]  $\gamma_1,\gamma_2\in C^{1}[0,1]$;
$\gamma_1(x),\gamma_2(x)>0$ for almost all $x\in [0,1]$; Moreover,
$\gamma_1(x)\neq\gamma_2(x)$ for $x\in [0,1]$ and
$\gamma_2(2x)\neq2\gamma_1(x)$ for $x\in [0,1]$.

\end{itemize}

Our first main result considers well-posedness of  \eqref{e1.1}
and reads as follows.


\begin{theorem} \label{thm1.1}
 For any pair of functions $(m_0,n_0)\in W^{1,1}(0,1)\times
W^{1,1}(0,1)$ such that $(m_0(0),n_0(0))=(0,0)$, problem \eqref{e1.1}
has a unique solution
\[
(m,n)\in C([0,\infty),W^{1,1}(0,1)\times
W^{1,1}(0,1))\cap C^{1}([0,\infty),L^{1}[0,1]\times L^{1}[0,1]),
\]
and for any $T>0$, the mapping $(m_0,n_0)\mapsto(m,n)$ from the
space
$$
  \{(m_0,n_0)\in W^{1,1}(0,1)\times
  W^{1,1}(0,1):(m_0(0),n_0(0))=(0,0)\}
$$
to $C([0,T],W^{1,1}(0,1)\times W^{1,1}(0,1))\cap
C^{1}([0,T],L^{1}[0,1]\times L^{1}[0,1])$ is continuous.
\end{theorem}


The proof of this result will be given in Section 2. From the proof
of this theorem we shall see that for any $(m_0,n_0)\in
W^{1,1}(0,1)\times W^{1,1}(0,1)$ we have $(m(t),n(t))=
T(t)(m_0,n_0)$, for all $t\geq 0$, where $(T(t))_{t\geq 0}$ is a
strongly continuous semigroup in the space $X=L^{1}[0,1]\times
L^{1}[0,1]$. Thus, for any $(m_0,n_0)\in X=L^{1}[0,1]\times
L^{1}[0,1]$, $(m(t),n(t))= T(t)(m_0,n_0)$ is well-defined for all
$t\geq 0$, and $(m,n)\in C([0,\infty),X)$. As usual, for any
$(m_0,n_0)\in X$ we call the vector function $t\mapsto (m(t),n(t))=
T(t)(m_0,n_0)$ (for $t\geq 0$) a \emph{mild solution} of
\eqref{e1.1}.

Our second main result studies the asymptotic behavior of the
solution of  \eqref{e1.1}. Before stating this result, we
introduce the eigenvalue problem
\begin{equation} \label{e1.2}
\begin{gathered}
 (\gamma_1(x)\hat{m}(x))'+\lambda\hat{m}(x)
=-B(x)\hat{m}(x)-\nu(x)\hat{m}(x)+\mu(x)\hat{n}(x),\quad 0<x<1,\\
  (\gamma_2(x)\hat{n}(x))'+\lambda\hat{n}(x)=
-\nu(x)\hat{n}(x)-\mu(x)\hat{n}(x)+
\begin{cases}
  4B(2x)\hat{m}(2x),  &   0\leq x\leq\frac{1}{2}, \\
  0, & \frac{1}{2}<x\leq 1,
\end{cases}\\
 \hat{m}(0)=0, \quad   \hat{n}(0)=0,\\
 \int_0^1\hat{m}(x)dx+\int_0^1\hat{n}(x)dx=1.
\end{gathered}
\end{equation}
and its conjugate problem
\begin{equation} \label{e1.3}
\begin{gathered}
-\gamma_1(x)\varphi'(x)+\lambda\varphi(x)
=-B(x)\varphi(x)-\nu(x)\varphi(x)+2B(x)\psi(\frac{x}{2}),\quad 0<x<1,
\\
-\gamma_2(x)\psi'(x)+\lambda\psi(x)=
-\nu(x)\psi(x)-\mu(x)\psi(x)+\mu(x)\varphi(x),\quad0<x<1,
\\
 \varphi(1)=0, \quad   \psi(1)=0,\\
  \int_0^1[\hat{m}(x)\varphi(x)dx+\hat{n}(x)\psi(x)]dx =1.
\end{gathered}
\end{equation}
Then the second main result reads as follows.


\begin{theorem} \label{thm1.2}
  There exists a constant $\lambda$ and a
strongly positive vector $(\hat{m},\hat{n})\in L^{1}[0,1]\times
L^{1}[0,1]$ satisfying \eqref{e1.2} such that
$$
\lim_{t\to\infty}e^{-\lambda
t}(m(t,\cdot),n(t,\cdot))=\int_0^1[m_0(x)\varphi(x)+n_0(x)\psi(x)]dx
(\hat{m},\hat{n}).
$$
where $(\varphi(x),\psi(x))\in L^{\infty}[0,1]\times L^{\infty}[0,1]
$ is the strongly positive solution of \eqref{e1.3}.
\end{theorem}

The proof of this result will be given in Section 3. The parameter
$\lambda$ is called the \emph{intrinsic rate of natural increase} or
\emph{Malthusian parameter} (see \cite{T}).

The layout of the rest part is as follows. In Section 2 we reduce
model \eqref{e1.1} into an abstract Cauchy problem and establish the
well-posedness of it by means of strongly continuous semigroups. In
Section 3 we prove that the solution of  model \eqref{e1.1} has
asynchronous exponential growth. In section 4 we consider extensions
of the above results to the asymmetric counterpart of model
\eqref{e1.1}, and establish similar results as
Theorems \ref{thm1.1} and \ref{thm1.2}.

\section{Well-posedness}

In this section we use the semigroup theory to study well-posedness
of  \eqref{e1.1}. We introduce the following spaces:
\begin{gather*}
  X=L^1[0,1]\times L^1[0,1], \quad \text{with norm }
  \|(u,v)\|_{X}=\|u\|_1+\|v\|_1, \\
  E=\{(u,v)\in W^{1,1}(0,1)\times W^{1,1}(0,1):   u(0)=0, v(0)=0\},
\end{gather*}
with norm $  \|(u,v)\|_{E}=\|u\|_{W^{1,1}}+\|v\|_{W^{1,1}}$.

We first reduce  problem \eqref{e1.3} into an initial value problem of
an abstract differential equation in the space $X$. For this purpose
we introduce the linear operators $A$, $B$ and $C$ in $X$ as
follows:
\begin{gather*}
A(u,v)=(-(\gamma_1(x)u(x))',-(\gamma_2(x)v(x))'), \quad
 \text{with domain } D(A)=E,
\\
  B(u,v)=(B_1(u,v),B_2(u,v)), \quad \text{for }
  (u,v)\in X,
\\
C(u,v)=(C_1(u,v),C_2(u,v)), \quad \text{for } (u,v)\in X,
\end{gather*}
where
\begin{gather*}
  B_1(u,v)=-B(x)u(x)-\nu(x)u(x),\\
  B_2(u,v)=-\mu(x)v(x)-\nu(x)v(x) \\
  C_1(u,v)=-\mu(x)v(x),\\
  C_2(u,v)=\begin{cases}
  4B(2x)u(2x)\;\;\;  &
   \text{for } 0\leq x\leq\frac{1}{2}, \\
  0 & \text{for }\frac{1}{2}<x\leq 1\,.
\end{cases}
\end{gather*}
We now let
$L=A+B+C$ with domain $D(L)=D(A)=E$.
We note that $A\in\mathcal{L}(E,X)$,
$B\in\mathcal{L}(X)$,
$C\in\mathcal{L}(X)$, and $L\in\mathcal{L}(E,X)$.
Later on we shall
regard $A$ and $L$ as unbounded linear operators in $X$.

Using these notation, we see that  \eqref{e1.1} can be
rewritten as the following abstract initial value problem of an
ordinary differential equation in the Banach space $X$:
\begin{equation} \label{e2.1}
\begin{gathered}
U'(t)=LU(t)\quad \text{for }t>0,\\
U(0)=U_0,
\end{gathered}
\end{equation}
where $U(t)=(m(t),n(t))$ and $U_0=(m_0(x),n_0(x))$.

Thus, to prove that  \eqref{e1.1} is well-posed in $X$, we only
need to show that the operator $L$ generates a strongly continuous
semigroup in $X$.

\begin{lemma} \label{lem2.1}
The operator $A+B$ generates a strongly
continuous semigroup \\ $\{T_1(t)\}_{t\geq 0}$ in $X$.
\end{lemma}


\begin{proof} Let $F\in X$ and $U(t)=T_1(t)F$. We write $F=(f,g)$,
$U(t)=(u(t,\cdot),v(t,\cdot))$. Then $(u,v)$ is the solution of
the  problem
\begin{gather*}
\frac{\partial u}{\partial t}+\frac{\partial
(\gamma_1(x)u)}{\partial x} =-a_1(x)u(t,x), \quad 0\leq x\leq 1,\;t>0,
\\
\frac{\partial v}{\partial t}+\frac{\partial
(\gamma_2(x)v)}{\partial x}=-a_2(x)v(t,x),\quad 0\leq x\leq 1,\;t>0,
\\
  u(t,0)=0,\quad  v(t,0)=0,\quad t>0,\\
  u(0,x)=f(x),\quad  v(0,x)=g(x),\quad 0\leq x\leq 1,
\end{gather*}
where $a_1(x)=B(x)+\nu(x)$ and $a_2(x)=\mu(x)+\nu(x)$. Let
$S_1(t,x)$ and $S_2(t,x)$ be the solution of the following two
equations
\begin{equation} \label{e2.2}
\begin{gathered}
\frac{dS_1}{dt}(t,x)=\gamma_{1}(S_1(t,x)),\quad S_1(0,x)=x,\\
\frac{dS_2}{dt}(t,x)=\gamma_{2}(S_2(t,x)),\quad S_2(0,x)=x
\end{gathered}
\end{equation}
Then
\begin{equation} \label{e2.3}
S_1(t,x)=G_1^{-1}(t+G_1(x)),\quad
S_2(t,x)=G_2^{-1}(t+G_2(x))
\end{equation}
where
\begin{equation} \label{e2.4}
G_1(x)=\int^{x}_{0}\frac{d\xi}{\gamma_{1}(\xi)},\quad
G_2(x)=\int^{x}_{0}\frac{d\xi}{\gamma_{2}(\xi)}
\end{equation}
By using the standard characteristic method, we obtain
\begin{gather}
  u(t,x)=\begin{cases}
 \frac{E_1(x)}{\gamma_{1}(x)}
\frac{\gamma_{1}(S_1(-t,x))}{E_1(S_1(-t,x))}f(S_1(-t,x)),& 0<S_1(-t,x)\\
0,&\text{elsewhere},
\end{cases} \label{e2.5}\\
  v(t,x)=\begin{cases}
 \frac{E_2(x)}{\gamma_{2}(x)}
\frac{\gamma_{2}(S_2(-t,x))}{E_2(S_2(-t,x))}g(S_2(-t,x)),&
0<S_2(-t,x), \\
0,&\text{elsewhere},
\end{cases}\label{e2.6}
\end{gather}
where
$$
E_1(x)=\exp\Big(-\int^{x}_{0}\frac{a_1(s)}{\gamma_1(s)}ds\Big),\quad
E_2(x)=\exp\Big(-\int^{x}_{0}\frac{a_2(s)}{\gamma_2(s)}ds\Big)\,.
$$
Obviously, $\{T_1(t)\}_{t\geq 0}$ is a strongly continuous semigroup
in $X$.
\end{proof}

Since $L=A+B+C$ and $C\in\mathcal{L}(X)$, by using the above lemma
and a well-known perturbation theorem for generators of strongly
continuous semigroups in Banach spaces, we get the following result.


\begin{lemma} \label{lem2.2}
 The operator $L$ generates a strongly
continuous semigroup $\{T(t)\}_{t\geq 0}$ in $X$.
\end{lemma}

By this lemma and a well-known result in the theory of strongly
continuous semigroups, we have the following result.

\begin{theorem} \label{thm2.3}
 For any given initial data $U_{0}\in E$,
the initial value problem \eqref{e2.1} has a unique solution
$U\in C([0,+\infty);E)\cap C^{1}([0,+\infty);X)$, given by
$$
U(t)=T(t)U_0 \quad \text{for } t\geq 0.
$$
\end{theorem}

Since \eqref{e2.1} is an abstractly rewritten form of \eqref{e1.1},
by this theorem we see that Theorem \ref{thm1.1} follows.


\section{Asynchronous exponential growth}

In this section we study the asymptotic behavior of the solution of
 \eqref{e1.1}. We shall prove that the semigroup $(T(t))_{t\geq
0}$ has the property of asynchronous exponential growth on $X$. For
this purpose, we shall prove that the semigroup $(T(t))_{t\geq 0}$
is \emph{positive, eventually norm continuous, eventually compact}
and \emph{irreducible}. Recall (see \cite{EN} and \cite{AG}) that a
strongly continuous semigroup $(T(t))_{t\geq 0}$ in a Banach lattice
$X$ is said to be \emph{positive} if $0\leq f \in X$ implies
$T(t)f\geq 0$ for all $t\geq0$; it is said to be \emph{eventually
continuous} if there exists $t_{0}\geq 0$ such that the mapping
$t\mapsto T(t)$ is continuous from $[t_{0},\infty)$ to
$\mathcal{L}(X)$; it is said to be \emph{eventually compact} if there
exists $t_{0}\geq 0$ such that the operator $T(t)$ is compact for
all $t\geq t_0$. Moreover, $(T(t))_{t\geq 0}$ is said to be \emph{
irreducible} if $\forall$ $\varphi \in X$, $\psi \in X^{\ast}$ (the
linear and topological dual of $X$), $\varphi>0$, $\psi>0$, we have
that $\langle T(t_{0})\varphi,\psi\rangle> 0$ for some $t_{0}> 0$,
where $\langle\cdot,\cdot\rangle$ denotes the dual product between
$X$ and $X^{\ast}$.

We denote by $s(L)$ the \emph{spectral bound} of $L$; i.e.,
\begin{equation} \label{e3.1}
  s(L)=\sup\{\mathop{\rm Re}\lambda:\lambda\in\sigma(L)\}.
\end{equation}
If the above assertions on the semigroup $(T(t))_{t\geq 0}$ are
proved, then by a well-known result in the theory of semigroups we
see that $s(L)$ is a dominant eigenvalue of $L$ (i.e.,
$s(L)\in\sigma(L)$ and $\mathop{\rm Re}\lambda<s(L)$ for all
$\lambda\in\sigma(L)\backslash\{s(L)\}$), and it is a first-order
pole of $R(\lambda,L)$ with an one-dimensional residue $P$ (see
Corollary V.3.2, Theorem VI.1.12 and Corollary VI.1.13 in
\cite{EN}). By \cite[Corollary V.3.3]{EN}, we then obtain the
assertion in Theorem \ref{thm1.2}. Thus, in the sequel we step by step prove
the above assertions about the semigroup $(T(t))_{t\geq 0}$.

\begin{lemma} \label{lem3.1}
The semigroup $(T(t))_{t\geq 0}$ is
positive.
\end{lemma}

\begin{proof}
 From \eqref{e2.5} and \eqref{e2.6}, we  see that
$\{T_1(t)\}_{t\geq 0}$ is positive. Since $C$ is a positive operator
on $X$, then the claim follows from \cite[Corollary 1.11]{EN}.
\end{proof}

\begin{lemma} \label{lem3.2}
 The semigroup $(T(t))_{t\geq 0}$ is
eventually norm continuous.
\end{lemma}

\begin{proof}
 From \eqref{e2.5} and \eqref{e2.6} we see that $T_1(t)=0$ for
$t>\max\{G_1(1),G_2(1)\}$. This particularly implies that
$(T_1(t))_{t\geq 0}$ is norm continuous for
$t>\max\{G_1(1),G_2(1)\}$. Thus, by \cite[Theorem III.1.16]{EN},
the desired assertion follows if we prove that the mapping $t\mapsto
K(t)\in\mathcal{L}(X)$ is continuous for $t>0$, where $K(t)$ is the
operator in $X$ defined by
$$
K(t)F=\int^{t}_{0}T_1(t-r)CT_1(r)Fdr \quad \text{for } F\in X.
$$
Using the representations of $T_1(t)$
(given by \eqref{e2.5} and \eqref{e2.6}) and
$C$  we see that for $F=(f,g)\in X$,
\begin{align*}
& T_1(t-r)CT_1(r)F\\
&=  \Big(c_{1}(x,t,r)g(S_1(-t+r,S_2(-r,x))),
 c_{2}(x,t,r)f(S_2(-t+r,2S_1(-r,x)))\Big),
\end{align*}
where $c_{i}(x,t,r)$ ($i=1,2$) are continuous functions. Hence
\begin{align*}
  K(t)F=\Big(&\int_{0}^{t}
   c_{1}(x,t,r)g(S_1(-t+r,S_2(-r,x)))dr,\\
&\int_{0}^{t}  c_{2}(x,t,r)f(S_2(-t+r,2S_1(-r,x)))dr\Big),
\end{align*}
We substitute $\xi_{1}=S_1(-t+r,S_2(-r,x))$ for $r$ in the first
term of $K(t)F$ and $\xi_{2}=S_2(-t+r,2S_1(-r,x))$ for $r$ in the
second term of $K(t)F$. Because of the assumption (H3), We can find
that
\begin{gather*}
\frac{d\xi_{1}}{dr}= \gamma_1(\xi_{1})
\Big(1-\frac{\gamma_2(S_2(-r,x))}{\gamma_1(S_2(-r,x))}\Big)\neq 0,\\
\frac{d\xi_{2}}{dr}= \frac{\gamma_2(\xi_2)}{\gamma_2(2S_1(-r,x))}
(\gamma_2(2S_1(-r,x))-2\gamma_1(S_1(-r,x)))\neq 0.
\end{gather*}
Then  we can easily verify that the mapping $t\mapsto K(t)$ from
$(0,+\infty)$ to $\mathcal{L}(X)$ is continuous. Hence the desired
assertion follows. This proves lemma 3.2.
\end{proof}


\begin{lemma} \label{lem3.3}
 The semigroup $(T(t))_{t\geq 0}$ is
eventually compact.
\end{lemma}

\begin{proof} Since $R(\lambda,A)$ is compact and $B+C$ is the
bounded operator, we conclude that $R(\lambda,L)=R(\lambda,A+B+C)$
is compact. Consequently, $R(\lambda,L)T(t)$ is compact for all
$t>0$. Since $(T(t))_{t\geq 0}$ is eventually norm continuous,
 by \cite[Lemma II.4.28]{EN}, it follows that $(T(t))_{t\geq 0}$ is
eventually compact. This completes the proof.
\end{proof}


\begin{lemma} \label{lem3.4}
 The semigroup $(T(t))_{t\geq 0}$ is irreducible.
\end{lemma}

\begin{proof}
 Since $R(\lambda,L)=\int^{+\infty}_{0} e^{-\lambda t}T(t)dt$,
for all $Re\lambda >s(L)$ (see [\cite{EN},Lemma VI.1.9]),
we have that for all $F=(f(x),g(x))\in X$,
$\Psi=(\psi_1,\psi_2)\in X^{\ast}$, $F> 0$, $\Psi>0$,
$$
\langle\Psi,R(\lambda,L)F\rangle=\int^{+\infty}_{0} e^{-\lambda
t}\langle\Psi,T(t)F\rangle dt\,.
$$
If we prove that $\langle\Psi,R(\lambda,L)F\rangle>0$ for some
$\lambda>0$, then from the above equation it follows that there
exists a $t_{0}>0$ such that $\langle\Psi,T(t)F\rangle>0$, and the
desired assertion then follows. Let $\pi_1$ and $\pi_2$ be the
projections onto the first and second coordinates, respectively. We
will prove that $\pi_1(R(\lambda,L)F)(x)>0$ and
$\pi_2(R(\lambda,L)F)(x)>0$ for almost all $x\in[0,1]$. In the
sequel we find the expression of $R(\lambda,L)$. For
$F=(f(x),g(x))\in X$, we solve the equation
\begin{equation} \label{e3.2}
(\lambda I-L)U=F.
\end{equation}
By writing $U=(u(x),v(x))$ and $F=(f(x),g(x))$, we see that
 \eqref{e3.2} can be rewritten as
\begin{gather*}
(\gamma_1(x)u(x))'+\lambda u(x)+a_{1}(x)u(x)=f(x)+\mu(x)v(x)\quad
\text{for } 0<x<1, \\
(\gamma_2(x)u(x))'+\lambda v(x)+a_{2}(x)v(x)
=g(x)+\begin{cases}
  4B(2x)u(2x) &  \text{for }0< x\leq\frac{1}{2}, \\
  0  & \text{for } \frac{1}{2}<x<1,
\end{cases}\\
 u(0)=0,\quad  v(0)=0
\end{gather*}
where $a_1(x)=B(x)+\nu(x)$, $a_2(x)=\mu(x)+\nu(x)$. Then, we have
\begin{gather}
 u(a)=\int^{x}_{0}\frac{\varepsilon_{1\lambda}(x)f(s)}
{\varepsilon_{1\lambda}(s)\gamma_{1}(s)}ds+
\int^{x}_{0}\frac{\varepsilon_{1\lambda}(x)\mu(s)v(s)}
{\varepsilon_{1\lambda}(s)\gamma_{1}(s)}ds \label{e3.3}
\\
v(a)=\begin{cases}
  \int^{x}_{0}\frac{\varepsilon_{2\lambda}(x)g(s)}
  {\varepsilon_{2\lambda}(s)\gamma_{2}(s)}ds+
4\int^{x}_{0}\frac{\varepsilon_{2\lambda}(x)B(2s)u(2s)}
{\varepsilon_{2\lambda}(s)\gamma_{2}(s)}ds  &
   \text{for } 0< x\leq\frac{1}{2}, \\[4pt]
 \int^{x}_{0}\frac{\varepsilon_{2\lambda}(x)g(s)}
  {\varepsilon_{2\lambda}(s)\gamma_{2}(s)}ds+
4\int^{\frac{1}{2}}_{0}\frac{\varepsilon_{2\lambda}(x)B(2s)u(2s)}
{\varepsilon_{2\lambda}(s)\gamma_{2}(s)}ds
  & \text{for } \frac{1}{2}<x<1,
\end{cases} \label{e3.4}
\end{gather}
where
\begin{gather*}
\varepsilon_{1\lambda}(x)=\exp\big\{-
\int^{x}_{0}\frac{\lambda+a_1(y)+\gamma_1'(y)}{\gamma_1(y)}dy\big\},
\\
\varepsilon_{2\lambda}(x)=\exp\big\{-
\int^{x}_{0}\frac{\lambda+a_2(y)+\gamma_2'(y)}{\gamma_2(y)}dy\big\}.
\end{gather*}
For each $\lambda\in \mathbb{C}$, we define the following operators, on
$X$,
\begin{equation} \label{e3.5}
\begin{aligned}
&H_{\lambda}(f_1(x),f_2(x))\\
&=\Bigg(
\int^{x}_{0}\frac{\varepsilon_{1\lambda}(x)\mu(s)f_2(s)}
{\varepsilon_{1\lambda}(s)\gamma_{1}(s)}ds,
\begin{cases}
4\int^{x}_{0}\frac{\varepsilon_{2\lambda}(x)B(2s)f_1(2s)}
{\varepsilon_{2\lambda}(s)\gamma_{2}(s)}ds &
   \text{for } 0< x\leq\frac{1}{2}, \\
4\int^{\frac{1}{2}}_{0}\frac{\varepsilon_{2\lambda}(x)B(2s)f_1(2s)}
{\varepsilon_{2\lambda}(s)\gamma_{2}(s)}ds
  & \text{for }\frac{1}{2}<x<1,
\end{cases}
\Bigg)
\end{aligned}
\end{equation}
\begin{equation}\label{e3.6}
S_{\lambda}(f_1(x),f_2(x))=\Big(
\int^{x}_{0}\frac{\varepsilon_{1\lambda}(x)f_1(s)}
{\varepsilon_{1\lambda}(s)\gamma_{1}(s)}ds,
 \int^{x}_{0}\frac{\varepsilon_{2\lambda}(x)f_2(s)}
 {\varepsilon_{2\lambda}(s)\gamma_{2}(s)}ds\Big)\,.
\end{equation}
Since
$$
\|H_{\lambda}(f_1(x),f_2(x))\| \to 0(\lambda\to +\infty)
$$
there exists $\lambda^{*}>0$ such that $\|H_{\lambda}\|<1$ for
$\lambda\geq \lambda^{*}$. This implies $(I-H_{\lambda})^{-1}$
exists for $\lambda\geq\lambda^{*}$. Then the resolvent of $L$ is
\begin{equation} \label{e3.7}
R(\lambda,L)F=(I-H_{\lambda})^{-1}S_{\lambda}F=
\sum^{\infty}_{n=0}(H_{\lambda})^{n}S_{\lambda}F,
\quad \text{for }\lambda>\lambda^{*}
\end{equation}
For $0\leq(f(x),g(x))\in \mathbf{X}$ and $(f(x),g(x))\neq 0$,
without loss of generality, we can assume that
$0\leq f\in L^{1}[0,1]$ and $f(x)>0$ for almost all
$x\in [x_{0},x_{1}]$. Then
\begin{gather*}
\pi_1(S_{\lambda}(f,g))(x)>0, \quad \text{for } x\in[x_{0},1] \\
\pi_2(H_{\lambda}S_{\lambda}(f,g))(x)>0, \quad
 \text{for } x\in[\frac{x_{0}}{2},1] \\
\pi_1(H_{\lambda}H_{\lambda}S_{\lambda}(f,g))(x)>0, \quad
 \text{for } x\in[\frac{x_{0}}{2},1] \\
\pi_2(H_{\lambda}H_{\lambda}H_{\lambda}S_{\lambda}(f,g))(x)>0,
\quad \text{for } x\in[\frac{x_{0}}{4},1]
\end{gather*}
Continuing in this way, we obtain $\pi_1(R(\lambda,L)F)(x)>0$ and
$\pi_2(R(\lambda,L)F)(x)>0$ for almost all $x\in[0,1]$. If we assume
that $g(x)>0$ for almost all $x\in [x_{0},x_{1}]$, the result is the
same. This completes the proof.
\end{proof}


\begin{corollary} \label{coro3.5}
 $\sigma(L)\neq\emptyset$.
\end{corollary}

\begin{proof} This follows from \cite[Theorem C-III.3.7]{AG},
which states that if a semigroup is irreducible, positive and
eventually compact, then the spectrum of its generator is not
empty.
\end{proof}


\begin{corollary} \label{coro3.6}
 $s(L)>-\infty$ and $s(L)\in \sigma(L)$.
\end{corollary}

\begin{proof}
 The first assertion is an immediately consequence of
Corollary \ref{coro3.5}. The second assertion follows from the positivity of
the semigroup $(T(t))_{t\geq 0}$ and the fact $s(L)>-\infty$;
see \cite[Theorem VI.1.10]{EN}.
\end{proof}

By Lemmas \ref{lem3.1}--\ref{lem3.4}, Corollary \ref{coro3.6}
and \cite[Corollary V.3.3]{EN},
we conclude that there exists an eigenvalue $\lambda$ of $L$
associated with a strictly positive eigenvector $(\hat{m},\hat{n})$
such that
\begin{equation} \label{e3.8}
\lim_{t\to+\infty}e^{-\lambda
t}(m(t,x),n(t,x))=C(\hat{m}(x),\hat{n}(x))
\end{equation}
where $\lambda=s(L)$. In the sequel we find the constant $C$. we
know that $\lambda=s(L)$ is the dominant eigenvalue of the
eigenvalue problem
\begin{equation} \label{e3.9}
\begin{gathered}
 (\gamma_1(x)\hat{m}(x))'+\lambda\hat{m}(x)
=-B(x)\hat{m}(x)-\nu(x)\hat{m}(x)+\mu(x)\hat{n}(x),\quad 0<x<1,\\
  (\gamma_2(x)\hat{n}(x))'+\lambda\hat{n}(x)=
-\nu(x)\hat{n}(x)-\mu(x)\hat{n}(x)+
\begin{cases}
  4B(2x)\hat{m}(2x),  &    0\leq x\leq\frac{1}{2}, \\
  0, & \frac{1}{2}<x\leq 1,
\end{cases}
\\
 \hat{m}(0)=0, \quad   \hat{n}(0)=0,
\end{gathered}
\end{equation}
and the corresponding eigenvector $(\hat{m},\hat{n})$ is strongly
positive in $(0,1)$; i.e., $\hat{m}(x)>0$ and $\hat{n}(x)>0$ for all
$0<x<1$. We normalize $(\hat{m},\hat{n})$ such that
$$
  \int_0^1\hat{m}(x)dx+\int_0^1\hat{n}(x)dx=1.
$$
Let $(\varphi,\psi)$ be the eigenvector of the conjugate problem of
\eqref{e3.9}; i.e.,
\begin{equation} \label{e3.10}
\begin{gathered}
-\gamma_1(x)\varphi'(x)+\lambda\varphi(x)
=-B(x)\varphi(x)-\nu(x)\varphi(x)+2B(x)\psi(\frac{x}{2}),\;\;0<x<1,
\\
-\gamma_2(x)\psi'(x)+\lambda\psi(x)=
-\nu(x)\psi(x)-\mu(x)\psi(x)+\mu(x)\varphi(x),\;\;0<x<1,
\\
 \varphi(1)=0, \quad   \psi(1)=0.
\end{gathered}
\end{equation}
We normalize $(\varphi,\psi)$ such that
$$
  \int_0^1\hat{m}(x)\varphi(x)dx+\int_0^1\hat{n}(x)\psi(x)dx=1.
$$
Then $\varphi$ and $\psi$ are also strictly positive in $(0,1)$, due
to a similar reason as that for $\hat{m}$ and $\hat{n}$. Now we
consider the function
$\int_0^1[m(t,x)\varphi(x)+n(t,x)\psi(x)]e^{-\lambda t}dx$. From
\eqref{e1.1} and \eqref{e3.10} we easily obtain
$$
  \frac{d}{dt}\int_0^1[m(t,x)\varphi(x)+n(t,x)\psi(x)]e^{-\lambda t}dx=0.
$$
Hence
$$
  \int_0^1[m(t,x)\varphi(x)+n(t,x)\psi(x)]e^{-\lambda t}dx=
  \int_0^1[m_0(x)\varphi(x)+n_0(x)\psi(x)]dx
$$
for all $t\geq 0$. Letting $t\to\infty$ and using \eqref{e3.8}, we get
$$
  C\int_0^1[\hat{m}(x)\varphi(x)dx+\hat{n}(x)\psi(x)]dx=
  \int_0^1[m_0(x)\varphi(x)+n_0(x)\psi(x)]dx.
$$
Since $\int_0^1[\hat{m}(x)\varphi(x)dx+\hat{n}(x)\psi(x)]dx =1$, we
obtain
$$
  C=\int_0^1[m_0(x)\varphi(x)+n_0(x)\psi(x)]dx.
$$
This completes the proof of Theorem \ref{thm1.2}.


\section{Two-phase Asymmetric Cell Division Model}

 In this section we study the
 two-phase cell division model
\begin{equation} \label{e4.1}
\begin{gathered}
\frac{\partial m}{\partial
t}+\frac{(\gamma_1(x)\partial m)}{\partial x}
=-\nu(x)m(t,x)-B(x)m(t,x)+\mu(x)n(t,x),\\
\quad 0<x<1,\;t>0,\\
\frac{\partial n}{\partial t}+\frac{(\gamma_2(x)\partial n)}{\partial x}
=-\nu(x)n(t,x)-\mu(x)n(t,x)+\int^{1}_{0}b(x,y)m(t,y)dy,\\
\quad 0<x< 1,\;t>0, \\
m(t,0)=0,\quad  n(t,0)=0,\\
m(0,x)=m_0(x),\quad  n(0,x)=n_0(x),\quad 0<x< 1,
\end{gathered}
\end{equation}
This model describes asymmetric division of cells; i.e., the $m$-phase
cell of size $y$ is divided into one $n$-phase cell of size $x$ and
another $n$-phase cell of size $y-x$.  The notations $\gamma_{1}(x)$,
$\gamma_2(x)$, $\mu(x)$, $\nu(x)$, and $B(x)$ have the same meaning
as the corresponding notation in  \eqref{e1.1}. For consistency
with the modelling we have to impose
\begin{gather}
b(x,y)\geq0 \quad \text{for } y\geq x \quad \text{and}\quad
b(x,y)=0 \quad \text{for }y<x, \label{e4.2}\\
 \int^{y}_{0}b(x,y)dx=2B(y), \label{e4.3}\\
 \int^{y}_{0}x b(x,y)dx=yB(y), \label{e4.4}\\
b(x,y)=b(y-x,y). \label{e4.5}
\end{gather}
We still assume that $\mu(x)$, $\nu(x)$ and $B(x)$ satisfy the
assumptions (H1) and (H2). We only assume that
$\gamma_1,\gamma_2\in C^{1}[0,1]$ and $\gamma_1(x),\gamma_2(x)>0$
for almost all $x\in [0,1]$ in this section. Besides, we assume that
$b(\cdot,y)\in C[0,1]$ for any fixed $y\in [0,1]$.

To establish well-posedness of  \eqref{e4.1}, we redefine the
operator $C_2$ in Section 2 as
$$
C_2(u,v)=\int^{1}_{0}b(x,y)u(y)dy\quad \text{for }  (u,v)\in X,
$$
and let $C_1(u,v)$ be as before. We note that the redefined operator
$C(u,v)=(C_1(u,v),C_2(u,v))$ is bounded on $X$. Similar arguments as
that in Section 2 yield that the redefined operator $L=A+B+C$
generates a strongly continuous semigroup $(T_{2}(t))_{t\geq 0}$ on
$X$. Then we can obtain the same assertion as Theorem \ref{thm1.1} about
model \eqref{e4.1}.

Second, we will obtain the asynchronous exponential growth for
 \eqref{e4.1}. Note that the redefined operator $C$ is still positive
on $X$. Then a similar argument as in the proof of Lemma \ref{lem3.1} shows
that the semigroup $(T_{2}(t))_{t\geq 0}$ is positive. The proofs of
the eventual compactness and the irreducibility of this semigroup
have some differences from those given in Lemmas \ref{lem3.3}
and \ref{lem3.4}. We
thus give them in the following two lemmas.


\begin{lemma} \label{lem4.1}
The semigroup $(T_{2}(t))_{t\geq 0}$ is
eventually norm continuous and eventually compact.
\end{lemma}

\begin{proof}
 In view of the Fr\'{e}chet-Kolmogorov compactness
criterion in $L^{1}$ we conclude from
\begin{equation} \label{e4.6}
\begin{aligned}
\Big|\int^{1}_{0}b(x,y)u(y)dy-\int^{1}_{0}b(x',y)u(y)dy\Big|
&\leq \int^{1}_{0}|b(x,y)-b(x',y)||u(y)|dy\\
&\leq \|b(x,y)-b(x',y)\|_{\infty}\|u(y)\|_{L^{1}}
\end{aligned}
\end{equation}
and the continuity of $b$ that the operator $C$ is compact. From
\eqref{e2.5} and \eqref{e2.6}, we can easily see that the semigroup
$(T_{1}(t))_{t\geq 0}$ generated by the operator $A+B$ is compact
for $t>\max\{G_{1}(1),G_{2}(1)\}$. Hence, the semigroup
$(T_{2}(t))_{t\geq 0}$ is compact for $t>\max\{G_{1}(1),G_{2}(1)\}$;
see \cite[Lemma III.1.14]{EN}. By \cite[Lemma II.4.22]{EN}, the
semigroup $(T_{2}(t))_{t\geq 0}$ is norm continuous for
$t>\max\{G_{1}(1),G_{2}(1)\}$. That completes the proof.
 \end{proof}


\begin{lemma} \label{lem4.2}
The semigroup $(T_2(t))_{t\geq 0}$ is
irreducible.
\end{lemma}

\begin{proof}
 The proof of this lemma is similar as that in Lemma \ref{lem3.4}
except for the definition of the operators $H_{\lambda}$. Here for
each $\lambda\in \mathbb{C}$, we define
\begin{align*}
H_{\lambda}(f_1(x),f_2(x))
&=\Big(
\int^{x}_{0}\frac{\varepsilon_{1\lambda}(x)\mu(s)f_2(s)}
{\varepsilon_{1\lambda}(s)\gamma_{1}(s)}ds,
\int^{x}_{0}\frac{\varepsilon_{2\lambda}(x)}
{\varepsilon_{2\lambda}(s)\gamma_{2}(s)}\int^{1}_{s}b(s,y)f_1(y)\,dy\,ds\Big)
\\
&=\Big(\int^{x}_{0}\frac{\varepsilon_{1\lambda}(x)\mu(s)f_2(s)}
{\varepsilon_{1\lambda}(s)\gamma_{1}(s)}ds,
\int^{x}_{0}f_1(y)\int^{y}_{0}\frac{\varepsilon_{2\lambda}(x)b(s,y)}
  {\varepsilon_{2\lambda}(s)\gamma_{2}(s)}\,ds\,dy\\
&\quad +\int^{1}_{x}f_1(y)\int^{x}_{0}\frac{\varepsilon_{2\lambda}(x)b(s,y)}
  {\varepsilon_{2\lambda}(s)\gamma_{2}(s)}\,ds\,dy\Big),
\end{align*}
where $\varepsilon_{1\lambda}(x)$ and $\varepsilon_{2\lambda}(x)$
are correspondingly the same with those appearing in the proof of
Lemma \ref{lem3.4}.  We also note that \eqref{e4.2}, \eqref{e4.3} and the assumption on
$B(x)$ play a important role to obtain that
$\pi_1(R(\lambda,L)F)(x)>0$ and $\pi_2(R(\lambda,L)F)(x)>0$ for
almost all $x\in[0,1]$.
\end{proof}

We also have that there exist an eigenvalue $\lambda$ of the
redefined operator $L$ (which is also the spectral bound of the
redefined operator $L$) and the strictly positive associated
eigenvector $(\hat{m}(x),\hat{n}(x))$; i.e.,
\begin{equation} \label{e4.7}
\begin{gathered}
(\gamma_1(x)\hat{m}(x))'+\lambda\hat{m}(x)
=-\nu(x)\hat{m}(x)-B(x)\hat{m}(x)+\mu(x)\hat{n}(x),\quad 0<x<1,\\
  (\gamma_2(x)\hat{n}(x))'+\lambda\hat{n}(x)
=-\nu(x)\hat{n}(x)-\mu(x)\hat{n}(x)+\int^{1}_{0}b(x,y)\hat{m}(y)dy,
\quad 0<x< 1,\\
 \hat{m}(0)=0,\quad  \hat{n}(0)=0,
\end{gathered}
\end{equation}
The conjugate problem of \eqref{e4.8} is as follows
\begin{equation} \label{e4.8}
\begin{gathered}
-\gamma_1(x)\varphi'(x)+\lambda\varphi(x)
=-\nu(x)\varphi(x)-B(x)\varphi(x)+\int^{1}_{0}b(y,x)\psi(y)dy,\quad
0<x<1,\\
  -\gamma_2(x)\psi'(x)+\lambda\psi(x)
=-\nu(x)\psi(x)-\mu(x)\psi(x)+\mu(x)\varphi(x),\quad 0<x< 1,\\
\varphi(1)=0,\quad  \psi(1)=0,
\end{gathered}
\end{equation}
The rest argument is similar to that in Section 3 and is therefore
omitted. Hence, we can obtain the same assertion as Theorem \ref{thm1.2}
about  model \eqref{e4.1}.


\subsection*{Acknowledgements}
The authors are greatly in debt to the
anonymous referee for his/her valuable comments and suggestions on
modifying this manuscript.


\begin{thebibliography}{00}

\bibitem{PR} B. Perthame and L. Ryzhik;
Exponential decay for the fragmentation or cell-division equation,
\emph{J. Diff. Equ.}, 210(2005), 155-177.

\bibitem{MMP} P. Michel, S. Mischler and B. Perthame;
 General entropy inequality: an
illustration on growth models, \emph{J. Math. Pures et Appl.},
84(2005), 1235--1260.

\bibitem{M} P. Michel;
 Existence of a solution to the cell division eigenproblem,
\emph{Model. Math. Meth. Appl. Sci.}, vol.16, suppl. issue 1(2006),
1125--1153.

\bibitem{PT} B. Perthame and T. M. Touaoula;
Analysis of a cell system with finite divisions,
\emph{Bol. Soc. Esp. Mat. Apl.}, no. 44 (2008), 53-77.

\bibitem{FH} J. Z. Farkas and P. Hinow;
 On a size-structured two-phase population model with infinite
states-at-birth, arXiv:0903.1649, March 2009.

\bibitem{F} J. Z. Farkas;
 Note on asychronous exponential
growth for structured population models,
\emph{Nonlinear Analysis: TMA}, 67(2007), 618-622.

\bibitem{T} H. R. Thieme;
 Balanced exponential growth of operator semigroups,
\emph{J. Math. Anal. Appl.}, 223(1998), 30-49.

\bibitem{SW}O. Arina, E. S\'{a}nchez and G. F. Webb;
Necessary and sufficient conditions
for asynchronous exponential growth in age structured cell
populations with Quiescence, \emph{J. Math. Anal. Appl.}, 215(1997),
499-513.

\bibitem{DVW} J. Dyson, R. Villella-Bressan and G. F. Webb;
Asynchronous exponential growth in
an age structured population of proliferating and quienscent cells,
\emph{Math. Biosci.}, 177-178(2002), 73-83.

\bibitem{P} B. Perthame;
 \emph{Transport Equations in Biology},
Birkh\"{a}user Verlag, Basel, 2007.

\bibitem{EN} K-J. Engel and R. Nagel;
One-Parameter Semigroups for Linear Evolution Equations,
Springer, New York, 2000.

\bibitem{CH} Ph. Cl\'{e}ment, H. Heijmans, S. Angenent,
C. van Duijin, and B. de Pagter; One-Parameter Semigroups,
North-holland, Amsterdam, 1987.

\bibitem[13]{AG} W. Arendt, A. Grabosch, G. Greiner, U. Groh,
H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck;
One-Parameter Semigroups of Positive Operators, Springer-Verlag,
Berlin, (1986), North-holland, Amsterdam, 1987.

\bibitem{P2} A. Pazy;
 Semigroups of Linear Operators and Applications to
Partial Differential Equations, Springer, New York, 1983.

\bibitem{PE} T. D. Pollard and W. C. Earnshaw;
\emph{Cell Biology}, Elsevier Science, New York, 2002.

\bibitem{JC} J. Celis;
\emph{Cell Biology}, third edition, Elsevier Academic Press, 2006.

\end{thebibliography}

\end{document}