\documentclass[reqno]{amsart}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 129, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2004/129\hfil A semilinear parabolic BVP]
{A semilinear parabolic boundary-value problem in bioreactors theory}
\author[Abdou Khadry Dram\'e\hfil EJDE-2004/129\hfilneg]
{Abdou Khadry Dram\'e}
\address{U.F.R. Sciences Appliqu\'ees et Technologie,
Universit\'e Gaston Berger de Saint-Louis, S\'en\'egal.
\hfill\break
INRA - U.M.R Analyse des Syst\`emes et Biom\'etrie, 2,
Place Viala, 34060 Montpellier, France}
\email{drame@ensam.inra.fr}
\date{}
\thanks{Submitted September 10, 2004. Published November 10, 2004.}
\subjclass[2000]{92B05, 35B40, 35K60}
\keywords{Bioreactors; semilinear equation; asymptotically autonomous;
\hfill\break\indent
omega limit sets}
\begin{abstract}
In this paper, we analyze a dynamical model describing the behavior
of bioreactors with diffusion. We obtain a convergence result for
solutions of asymptotically autonomous semilinear parabolic equations
to steady state solutions of the limiting equations.
This allows us to establish the convergence of solutions of the
initial value problem that describes the dynamics of the bioreactor.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\section{Introduction}
We consider a Plug Flow bioreactor with diffusion
in which occurs a simple growth reaction (one biomass/one
substrate). The dynamics of this bioreactor are described by the following
system of partial differential equations
\begin{equation}\label{eq1}
\begin{gathered}
\frac{\partial S}{\partial t}=-q\frac{\partial
S}{\partial x}+d\frac{\partial^2 S}{\partial x^2}-\mu(S)X,\quad
(t,x)\in ]0,\infty[\times ]0,l[\\
\frac{\partial X}{\partial t}=-q\frac{\partial
X}{\partial x}+d\frac{\partial^2 X}{\partial x^2}+\mu(S)X,\quad
(t,x)\in ]0,\infty[\times ]0,l[\\
S(0,x)=S_{0}(x),\quad X(0,x)=X_{0}(x),\quad x\in ]0,l[\,,
\end{gathered}
\end{equation}
with the boundary conditions
\begin{equation}\label{eq2}
\begin{gathered}
d\frac{\partial S}{\partial x}(t,0)-qS(t,0)=-qS_{\rm in}\,,
\quad \frac{\partial S}{\partial x}(t,l)=0,\quad t\in ]0,\infty[,\\
d\frac{\partial X}{\partial x}(t,0)-qX(t,0)=-qX_{\rm in}\,,
\quad \frac{\partial X}{\partial x}(t,l)=0,\quad t\in ]0,\infty[\,.
\end{gathered}
\end{equation}
In \eqref{eq1}-\eqref{eq2}, $S$, $X$, $S_{\rm in}$, $X_{\rm in}$, $q$, $d$, $l$ and
$\mu$ denote substrate and biomass concentrations in the bioreactor, feed
substrate and biomass concentrations, the flow rate, the
diffusion rate, the length of the bioreactor and the kinetic function,
respectively.
Basically the first equation of \eqref{eq1} contains a yield coefficient $Y$, but
it is convenient to rescale $X$ to $\frac{X}{Y}$ in order to reduce
the number of parameters. For further details on the modeling, refer
to \cite{Dra042} or \cite{Smith952}.
This paper is devoted to the analysis of \eqref{eq1}-\eqref{eq2}:
we aim at proving
uniform boundedness of the solutions and describing their omega-limit sets.
To ease the analysis, we will perform in Section $2$ a linear change of
state variables which transforms \eqref{eq1} into two equations; one
of them is nonlinear, but the other one is linear. Next, in the same section,
we will show that the operator associated to this linear equation is the
infinitesimal generator of a strongly continuous semigroup on $C[0,l]$
(the Banach space of the continuous real-valued functions on $[0,l]$)
which is exponentially stable. As a consequence of this, the unique steady
state solution of the linear equation is globally exponentially stable in
$C[0,l]$. Following this, we will rewrite
\eqref{eq1}-\eqref{eq2} as a nonautonomous semilinear parabolic equation
\begin{equation} \label{int3}
\begin{gathered}
\frac{d u}{d t}=Au(t)+f(t,u),\\
u(0)=u_{0},
\end{gathered}
\end{equation}
where $A$ is a linear operator in the Banach space $C[0,l]$ with domain
$D(A)$ and \eqref{int3} is asymptotically autonomous with limiting equation
\begin{equation}\label{int4}
\begin{gathered}
\frac{d u}{d t}=Au(t)+g(u),\\
u(0)=u_{0}
\end{gathered}
\end{equation}
in the sense that:
\begin{itemize}
\item[(i)] \eqref{int3} and \eqref{int4} have a unique mild solution in $C[0,l]$,
respectively,
\item[(ii)] $\lim_{t \to \infty}f(t,u)=g(u)$ uniformly in
$u$ on bounded subsets of $C[0,l]$.
\end{itemize}
Many works available in the literature are devoted to the study of the
asymptotic behavior of solutions of equations of type \eqref{int3} and/or
\eqref{int4} (see \cite{Cha75,Chen89,Mat78,Mat79,Pol91,Pol92,Pol96,Pol021,
Pol022,Pol03,Smith952}, etc.). In the earlier works of N.
Chafee \cite{Cha75} and H. Matano \cite{Mat78,Mat79}, the authors dealt
with equations of type \eqref{int4} with Neumann and Robin boundary conditions.
In \cite{Cha75}, one-dimensional equation was considered and the author used
the energy function as a Lyapunov function of \eqref{int4} to prove that
the omega-limit sets of solutions consist of steady state solutions of
\eqref{int4}. Observe that this result is proved under the strong assumption
that the initial value is continuously differentiable. In \cite{Mat79},
Matano proved a more general result. He considered \eqref{int4} in $C(D)$,
where $D$ is a bounded domain of $\mathbb{R}^{N}$, $N\geq 1$.
He established that omega-limit sets of bounded solutions of \eqref{int4}
consist of its steady state solutions. In \cite{Mat78}, he considered
one-dimensional equation and proved that the omega-limit sets contain at
most one element, that is, each solution of \eqref{int4} either blows up
or converges to steady state solution. More recently, Pol\`a$\check{c}$ik et al.
investigated the asymptotic behavior of solutions of \eqref{int4} with
Dirichlet, Neumann and Robin conditions (see
\cite{Pol91,Pol92,Pol96,Pol021,Pol022,Pol03}). They established that the
omega-limit set of bounded solutions of \eqref{int4} can be a set of
continuum of steady state solutions
(\cite{Pol91,Pol96,Pol021,Pol022}).
However, the knowledge of the behavior of solutions of \eqref{int4} does not
give any a priori information on the structure of the omega-limit sets of
solutions of \eqref{int3}. In \cite{Chen89} the one-dimensional case was
considered. It is proved therein that if $f$ is periodic then any
bounded solution of \eqref{int3} converges to a periodic solution of \eqref{int3}.
In \cite{Smith952}, the system of type \eqref{eq1}-\eqref{eq2} has been studied
by Smith for a class of monotonic kinetic functions. In this case,
the limiting equation \eqref{int4} generates a monotone dynamical system.
However, the author does not establish any result on the behavior of
solutions of the nonautonomous equation (equivalently \eqref{eq1}-\eqref{eq2}),
as it is mentioned in his remarks section. His result on the asymptotic
behavior of the solutions of the limiting equation are valid only for
monotonic kinetic functions.
In this paper, we extend the earlier result of \cite{Mat79} to
asymptotically autonomous nonlinear equations. In Theorem \ref{thm3.4},
we prove that the $\omega$-limit set of any bounded solution of the
nonautonomous equation \eqref{int3} is nonempty and it is contained in a
set of steady state solutions of \eqref{int4}. This result relies neither
on a particular form of $f$ deduced from the reduction of \eqref{eq1}
nor on the one-dimensional aspect of the equations.
It is also established for equations in abstract
Banach spaces with more general properties on $f$
(see remarks following the proof of Theorem \ref{thm3.4}). On the other hand,
Theorem \ref{thm3.4} can be applied to many models in practical applications
since we do not consider a particular class of kinetic functions.
Based on Theorem \ref{thm3.4} and \cite[Theorem A]{Mat78},
in Theorem \ref{thm3.5} we show
that every solution of \eqref{int3} that starts in a certain given set,
is bounded and converges to a unique steady state solution of \eqref{int4}.
We finally apply Theorem \ref{thm3.4} to the limiting equation although it is
autonomous.
We introduce the following assumptions. Observe that they are often
fullfiled by kinetic models in practical applications.
\begin{itemize}
\item[A1] $\mu(s)>0$ for $s>0$, $\mu(s)=0$ for $s\leq 0$,
$\mu$ is bounded as $s\to +\infty$.
\item[A2] The function $s\to\mu(s)$ is twice continuously
differentiable. Moreover, $\mu$ and $\mu'$ are Holder continuous
in $\mathbb{R}$ (of exponent $\gamma$).
\end{itemize}
\section{Preliminaries}
Let us consider the new function $U(t,x)=S(t,x)+X(t,x)$ and let us
introduce the notation $M=S_{\rm in}+X_{\rm in}$. Then $U(t,x)$ satisfies:
\begin{equation}\label{lin1}
\begin{gathered}
\frac{\partial U}{\partial t}=d\frac{\partial^2 U}{\partial x^2}-q\frac{\partial
U}{\partial x},\quad (t,x)\in ]0,\infty[\times ]0,l[,\\
U(0,x)=U_{0}(x), \quad x\in ]0,l[,\\
d\frac{\partial U}{\partial x}(t,0)=q(U(t,0)-M)\,,
\quad \frac{\partial U}{\partial x}(t,l)=0,\quad t\in ]0,\infty[,
\end{gathered}
\end{equation}
with $U_{0}(x)=S_{0}(x)+X_{0}(x)$.
It is easy to see that \eqref{lin1} has a unique steady state solution
$\bar{U}$ and $\bar U(x)= M$, for all $x\in[0,l]$.
Let $Z=C[0,l]$. We define the linear operator
\begin{gather*}
D(A)=\{v\in C^{2}[0,l] : d\frac{\partial v}{\partial
x}(0)-\frac{q}{2}v(0)=0,\;d\frac{\partial v}{\partial
x}(l)+\frac{q}{2}v(l)=0\}, \\
Av=d\frac{\partial^{2}
v}{\partial x^{2}}-\frac{q^{2}}{4d}v,\quad \forall\, v\in D(A).
\end{gather*}
Note that if $u(t,x)=e^{-\frac{q}{2d}x}(U(t,x)-M)$, where $U(t,x)$ is a
solution of \eqref{lin1}, then we have $u(t)\in D(A)$ as long as
$U(t,x)$ is defined and $t>0$. Moreover,
\begin{equation}\label{lin2}
\begin{gathered}
\frac{d u}{d t}=Au(t),\\
u(0)=u_{0}.
\end{gathered}
\end{equation}
The linear operator $A$ is closed, densely defined and
$A+\delta I$ is dissipative in $Z$, where $\delta=\frac{q^{2}}{4d}$.
Moreover, for any $\lambda>0$ and $f\in Z$, the ordinary differential
equation $\lambda u-Au=f$ has a unique solution $u\in D(A)$.
Then, $\lambda -A$ is surjective for $\lambda >0$. It follows that $A$ is the
infinitesimal generator of a $C_{0}$-semigroup of contractions $T(t)$ on
$Z$ (see \cite[Theorem 3.15]{Eng00} or \cite[Theorem 4.3]{Paz83})
and
\begin{equation}\label{ine1}
\| T(t)\|_{L(Z)}\leq e^{-\delta t}, \quad \forall\;t\geq
0.
\end{equation}
Further, if $\Gamma(x,y,t)$ denotes the fundamental solution of
\[
\frac{\partial v}{\partial
t}=d\frac{\partial^{2} v}{\partial x^{2}}-\delta v, \quad
(t,x)\in ]0,\infty[\times]0,l[
\]
and
\[
d\frac{\partial v}{\partial
x}(t,0)=\frac{q}{2} v(t,0); \; d\frac{\partial
v}{\partial x}(t,l)=-\frac{q}{2}v(t,l),\quad t>0,
\]
then the semigroup $T(t)$ is given by
\begin{equation}\label{ine2}
(T(t)v)(x)=\int_{0}^{l}\Gamma(x,y,t)v(y)dy, \quad\forall\,t> 0,\quad \forall\,v\in Z.
\end{equation}
(see \cite{Mat79}).
Let us recall \cite[Lemma 2.2]{Mat79}.
\begin{lemma} \label{lem2.1}
The functions $\Gamma$ and $\frac{\partial \Gamma}{\partial t}$ are
continuous in $[0,l]\times[0,l]\times]0,\infty[$. Moreover, given any
$t_{0}>0$, there exists a constant $C_{0}>0$ such that
\begin{equation}\label{ine3}
\sup_{0\leq x\leq l}\int_{0}^{l}\vert\frac{\partial \Gamma}{\partial t}(x,y,t)
\vert dy\leq \frac{C_{0}}{t},\quad \forall\,00$; i.e: $T(t):Z\to Z$ is compact and for any $v\in Z$,
the map $t\to T(t)v$ is continuously differentiable for $t>0$.
Moreover, for any given $t_{0}>0$, there exists $C_{0}>0$ such that
\begin{equation}\label{ine4}
\| AT(t)\|_{L(Z)}\leq \frac{C_{0}}{t},\quad \forall\,00$, $AT(t)\in L(Z)$ for $t>0$ and
$AT(t)v=\frac{d}{dt}T(t)v$ for all $t>0$ and all $v\in Z$. Hence,
\eqref{ine4} follows from \eqref{ine2} and \eqref{ine3}. Since $\Gamma$ is
continuous on the compact $[0,l]\times[0,l]$ for any fixed $t>0$,
the compactness of $T(t)$ follows from Ascoli-Arzel\`a's Theorem
(see \cite[P. 85]{Yos68}).
\end{proof}
\noindent \textbf{Remarks:}
Indeed, $T(t)$ defines an analytic semigroup (see \cite[P. 121]{Smith952}.
However, it is more interesting to consider the properties stated in
Lemma \ref{lem2.2} since the condition of continuous differentiability and \eqref{ine4}
is weaker than analyticity condition. Moreover, the condition in Lemma
\ref{lem2.2}
is sufficient to establish the main result in this paper and it is satisfied
in much more situations if one thinks of generalization
(see remarks in Section 3).
As a consequence of \eqref{ine1}, the steady state solution $\bar{U}\equiv M$
of \eqref{lin1} is globally exponentially stable in $Z$.
Following this, it can be seen that
\eqref{eq1}-\eqref{eq2} is equivalent to the following semilinear parabolic
equation
\begin{equation}\label{eq2.5}
\begin{gathered}
\frac{\partial u}{\partial t}=d\frac{\partial^{2} u}{\partial
x^{2}}-q\frac{\partial u}{\partial x}+\tilde{f}(t,u), \quad (t,x)\in
]0,\infty[\times ]0,l[\,,\\
u(0,x)=u_{0}(x),\quad x\in ]0,l[\,,\\
d\frac{\partial u}{\partial x}(t,0)=q(u(t,0)-S_{\rm in});
\quad \frac{\partial u}{\partial x}(t,l)=0, \quad t\in ]0,\infty[\,,
\end{gathered}
\end{equation}
where $\tilde{f}(t,u)=-\mu(u)(U(t)-u)$ and
$U(t)$ is the solution of the linear equation \eqref{lin1}.
We have that $\tilde{f}$ is continuous in $t$ and locally Lipschitz
continuous in $u$, uniformly in $t$ and
$\displaystyle\lim_{t \to \infty}\tilde{f}(t,u)=\tilde{g}(u)=-\mu(u)(M-u)$ uniformly in
$u$ on bounded subsets of $Z$ under assumptions (A1)-(A2).
Equation \eqref{eq2.5} is then asymptotically autonomous according to the
previous definition and its limiting equation is
\begin{equation}\label{eq4}
\begin{gathered}
\frac{\partial u}{\partial t}=d\frac{\partial^2 u}{\partial x^2}-q\frac{\partial
u}{\partial x}-\mu(u)(M-u),\quad
(t,x)\in ]0,\infty[\times ]0,l[\,,\\
u(0,x)=u_{0}(x),\quad x\in ]0,l[\,,\\
d\frac{\partial u}{\partial x}(t,0)=q(u(t,0)-S_{\rm in});
\quad \frac{\partial u}{\partial x}(t,l)=0, \quad t\in ]0,\infty[\,.
\end{gathered}
\end{equation}
\section{Main results}
We give here our main result on the asymptotic behavior of solutions of
the nonautonomous equation \eqref{eq2.5} (and equivalently the system
\eqref{eq1}-\eqref{eq2}). Equation \eqref{eq4} is also analyzed.
\subsection{The nonautonomous equation}
Instead of \eqref{eq2.5} and \eqref{eq4}, we consider the following equations
\begin{equation}\label{eq3.1}
\begin{gathered}
\frac{\partial u}{\partial t}=d\frac{\partial^{2}
u}{\partial x^{2}}-\frac{q^{2}}{4d}u+f(t,u), \quad (t,x)\in ]0,\infty[\times
]0,l[\,,\\
u(0,x)=u_{0}(x),\quad x\in ]0,l[\,,\\
d\frac{\partial u}{\partial x}(t,0)=\frac{q}{2}u(t,0);\quad
d\frac{\partial u}{\partial x}(t,l)=-\frac{q}{2}u(t,l),\quad t\in ]0,\infty[
\end{gathered}
\end{equation}
and
\begin{equation}\label{eq3.2}
\begin{gathered}
\frac{\partial u}{\partial t}=d\frac{\partial^{2}
u}{\partial x^{2}}-\frac{q^{2}}{4d}u+g(u),\quad
(t,x)\in ]0,\infty[\times ]0,l[\,,\\
u(0,x)=u_{0}(x),\quad x\in ]0,l[, \\
d\frac{\partial u}{\partial x}(t,0)=\frac{q}{2}u(t,0),\quad
d\frac{\partial u}{\partial
x}(t,l)=-\frac{q}{2}u(t,l),\quad t\in ]0,\infty[\,,
\end{gathered}
\end{equation}
where $f:[0,\infty[\times Z\to Z$ is continuous and
$f:]0,\infty[\times Z\to Z$, $g:Z\to Z$ are continuously differentiable and
$\lim_{t \to \infty}f(t,u)=g(u)$ uniformly in $u$ on bounded subsets
of $Z$. These
equations are deduced from \eqref{eq2.5} and \eqref{eq4} respectively by
introducing $u(t,x)=e^{-\frac{q}{2d}x}(v(t,x)-S_{\rm in})$ for any solution
$v$ of \eqref{eq2.5} (respectively \eqref{eq4}) as in Section 2. So, it
is equivalent to study \eqref{eq3.1} in order to understand the behavior of
solutions of \eqref{eq2.5}. Note that for any $u_{0}\in Z$, \eqref{eq3.1}
(resp. \eqref{eq3.2}) has a unique mild solution on some interval
$[0,t_{u}[$, that is: $u\in C([0,_,t_{u}[;Z)$ and is solution of the
integral equation $u(t)=T(t)u_{0}+\int_{0}^{t}T(t-s)f(s,u(s))ds$
(resp. $u(t)=T(t)u_{0}+\int_{0}^{t}T(t-s)g(u(s))ds$) on $[0,t_{u}[$.
\begin{lemma} \label{lem3.1}
Assume that (A1)-(A2) hold. Then
\begin{itemize}
\item[(i)] For any $u_{0}\in Z$, the mild solution $u(t)$ of \eqref{eq3.1}
(resp. of \eqref{eq3.2}) is a classical solution; i.e.,
$u\in C([0,t_{u}[;Z)\cap C^{1}(]0,t_{u}[;Z)$,
$u(t)\in D(A)$, for all $00$ the
subsets $\{Au(t), \,t\geq t_{0}\}$ and
$\{\frac{\partial u(t)}{\partial t}, t\geq t_{0}\}$ are bounded in $Z$.
\end{itemize}
\end{lemma}
\begin{proof}
We give the proof only for solutions of \eqref{eq3.1} since the other case
is similar.
(i) The mild solution $u(t)$ of \eqref{eq3.1} is given by
$$
u(t)=T(t)u_{0}+\int_{0}^{t}T(t-s)f(s,u(s))ds,\quad 00$. Let $\varepsilon$, $T_{0}$
and $T_{1}$ be such that $0<\varepsilon0$. As $u(t)$ is bounded in $Z$,
we have $\| f(t,u(t))\|_{Z}\leq N$, for $t\geq 0$ where $N>0$.
The compactness of $K$ follows from \cite[Lemma 2.4]{Paz83}.
\end{proof}
Let us define the functional
$$
J(t,v)=\int_{0}^{l}\Big(\frac{d}{2}\big(\frac{\partial v}{\partial x}\big)^{2}
-\int_{0}^{v}F(t,x,w)dw\Big)dx +
\frac{q}{4}(v^{2}(0)+v^{2}(l)),
$$
where $F(t,x,w)=-\left[\frac{q^{2}}{4d}w+e^{-\alpha
x}\mu(e^{\alpha x}w+S_{\rm in})(U(t,x)-e^{\alpha x} w-S_{\rm in})\right]$,
$\alpha=\frac{q}{2d}$.
For any solution $u(t)$ of \eqref{eq3.1}, $J(t,u(t))$ is defined
and the following statement holds.
\begin{lemma} \label{lem3.3}
If $u(t)$ is a solution of \eqref{eq3.1}, then
$$
\frac{d }{dt}\left(J(t,u(t))\right)
=\int_{0}^{l}-\big(\frac{\partial u}{\partial t}\big)^{2}dx
-\int_{0}^{l}\Big(\int_{0}^{u(t,x)}
\frac{\partial F}{\partial t}(t,x,w)\,dw\Big) dx
$$
for $00$ there exists $C>0$ such that
\begin{equation}\label{eq3.4}
\sup_{0\leq t\leq t_{0}}\| v_{m}(t)-v_{n}(t)\|_{Z}\leq C\left(\| u_{m}-u_{n}\|_{Z}
+ \mu_{0}\| T(t_{m})V_{0}-T(t_{n})V_{0}\|_{Z}\right).
\end{equation}
It follows from \eqref{eq3.4} that there exists a continuous function
$h:[0,\infty[\to Z$ such that
$$\lim_{n \to \infty}\sup_{0\leq t\leq t_{0}}\| v_{n}(t)-h(t)\|_{Z}=0
\quad \text{for any given} \;t_{0}>0\,.
$$
On the other hand, for all $t>0$,
\begin{equation}\label{eq3.5}
\lim_{n \to \infty}\| f(t+t_{n}, v_{n}(t))-g(v_{n}(t))\|_{Z}
\leq \lim_{n \to \infty}\sup_{w\in B}\| f(t+t_{n},w)-g(w)\|_{Z}=0.
\end{equation}
So, rewriting \eqref{eq3.3} as
$$
v_{n}(t)=T(t)u_{n}+\int_{0}^{t}T(t-s)(f(s+t_{n},v_{n}(s))-g(v_{n}(s)))+\int_{0}^{t}T(t-s)g(v_{n}(s))ds
$$
and passing to the limit when $n\to +\infty$, we have
\begin{equation}\label{eq3.6}
h(t)=T(t)\varphi+\int_{0}^{t}T(t-s)g(h(s))ds,\quad t\geq 0.
\end{equation}
It follows from \eqref{eq3.6} that $h(t)$ is a mild solution of \eqref{eq3.2}
and by Lemma \ref{lem3.1} (i), $h(t)$ is a classical solution of \eqref{eq3.2}.
By Lemma \ref{lem3.1} (i), we have $v_{n}(t)\in D(A)$ for $n\geq 0$ and $t>0$.
Moreover,
\begin{align*}
Av_{n}(t)&=AT(t)u_{n}+\int_{0}^{t}AT(t-s)(f(s+t_{n},v_{n}(s))-g(v_{n}(s)))ds\\
&\quad +\int_{0}^{t}AT(t-s)g(v_{n}(s))ds.
\end{align*}
Since $T(t)$ is continuously differentiable, $AT(t)\in L(Z)$ for $t>0$.
Then, using $\eqref{eq3.5}$ and $\eqref{eq3.6}$, we have
$$
\lim_{n \to \infty}Av_{n}(t)=Ah(t)\quad \text{in }Z \quad\text{for}\; t>0.
$$
Hence,
$$
\lim_{n \to \infty}\frac{\partial v_{n}(t)}{\partial t}
=\frac{\partial h(t)}{\partial t}\quad \text{in } Z\quad\text{for } t>0.
$$
Now we aim to prove that $\frac{\partial h}{\partial t}=0$ in $]0,\;\infty[$.
Let $t_{0}>0$, by Lemma \ref{lem3.3} we have
$$
\int_{t_{0}}^{t}\int_{0}^{l}\big(\frac{\partial u}{\partial s}\big)^{2}\,dx\,ds
=J(t_{0},u(t_{0}))-J(t,u(t))-\int_{t_{0}}^{t}\int_{0}^{l}\int_{0}^{u(s,x)}
\frac{\partial F}{\partial s}(s,x,w)\,dw\,dx\,ds
$$
for $t\geq t_{0}$.
Since $u(t)$ is bounded in $Z$, it follows from Lemma \ref{lem3.1} (ii) that
$J(t,u(t))$ remains bounded for $t\geq t_{0}$. Let
\begin{align*}
\xi(t)&=\int_{t_{0}}^{t}\int_{0}^{l}\int_{0}^{u(s,x)}
\frac{\partial F}{\partial s}(s,x,w)\,dw\,dx\,ds \\
&=\int_{t_{0}}^{t}\int_{0}^{l}\int_{0}^{u(s,x)}e^{-\alpha x}
\mu(e^{\alpha x}w+S_{\rm in})\frac{\partial U}{\partial s}(s,x)dw\,dx\,ds\\
&=\int_{t_{0}}^{t}\int_{0}^{l}\frac{\partial }{\partial s}
\left(e^{-\alpha x}(U(s,x)-M)\right)k(s,x)\,dx\,ds,
\end{align*}
where $k(t,x)=\int_{0}^{u(t,x)}\mu(e^{\alpha x}w+S_{\rm in})dw$,
$\alpha=\frac{q}{2d}$ and $U(t,x)$ is the solution of the linear
equation \eqref{lin1}. Then,
\begin{align*}
\xi(t)&=-\int_{0}^{l}\int_{t_{0}}^{t}\left(e^{-\alpha x}(U(s,x)-M)\right)
\frac{\partial k}{\partial s}(s,x)ds dx \\
&\quad + \int_{0}^{l}e^{-\alpha x}[(U(t,x)-M)k(t,x)-(U(t_{0},x)-M)k(t_{0},x)]dx
\end{align*}
and $\frac{\partial k}{\partial t}(t,x)=\mu(e^{\alpha x} u(t,x)
+S_{\rm in})\frac{\partial u}{\partial t}(t,x)$.
By Lemma \ref{lem3.1} (ii), $\frac{\partial u}{\partial t}(t)$ remains bounded in
$Z$ for $t\geq t_{0}$ and therefore
$\vert\frac{\partial k}{\partial t}(t,x) \vert$ remains also bounded for
$t\geq t_{0}$ and $x\in [0,l]$. Furthermore, by \eqref{ine1} we have
$$
\sup_{0\leq x\leq l}\vert e^{-\alpha x} (U(t,x)-M)\vert
\leq \sup_{0\leq x\leq l}\vert e^{-\alpha x}(U_{0}(x)-M)\vert e^{-\delta t},
\quad \forall\,t\geq 0.
$$
Since $u(t)$ is bounded in $Z$, it follows that $\xi(t)$ is bounded for
$t\geq t_{0}$. Hence,
\begin{equation}\label{eq3.7}
\int_{t_{0}}^{\infty}\int_{0}^{l}\big(\frac{\partial u}{\partial t}\big)^{2}\,dx\,dt
<\infty,\quad\forall\;t_{0}>0.
\end{equation}
Let $00$ such that
$\| f(t,u)-f(t',v)\|_{Z}\leq C(\vert t-t'\vert+\| u-v\|_{Z})$ for
$t,t'\in \mathbb{R}_{+}$, $u,v\in B$.
Let $u(t)$ be a precompact, classical solution of \eqref{int3} satisfying
$$
\int_{t_{0}}^{\infty}\|\frac{\partial u}{\partial t}(t)\|_{Z}dt<\infty,\quad
\text{for some}\;t_{0}>0.
$$
Then, the omega-limit set $\omega(u_{0})$ of $u(t)$ is nonempty, it is
contained in $D(A)$ and it consists of steady state solutions of \eqref{int4}.
The proof is almost the same one as above. However, the existence of $h$ is
proved by application of Ascoli-Arzela's Theorem to the subset
$\{v_{n},n\geq 0\}$ of $C(]0,\infty[;\;Z)$ and the equicontinuity is
established in the same manner as the estimates of
$\| u(s+t)-u(t_{0}+t)\|_{Z}$ in the proof of Lemma \ref{lem3.1}(ii). \smallskip
Now we can apply Theorem \ref{thm3.4} to prove the convergence of solutions of
\eqref{eq2.5}.
Let
$$
\mathcal{K}_{0}=\{u\in Z, \; 0\leq u(x)\leq U_{0}(x)\}.
$$
\begin{theorem} \label{thm3.5}
Assume that (A1) and (A2) hold. Then, for any $u_{0}\in \mathcal{K}_{0}$,
there exists a unique steady state solution $\bar u$ of \eqref{eq4} such that
the solution $u(t)$ of \eqref{eq2.5} converges to $\bar u$ in $Z$.
\end{theorem}
\begin{proof}
Let $u_{0}\in \mathcal{K}_{0}$. $U(t,x)$ is then an upper-solution of
\eqref{eq2.5} and by the standard comparison Theorem, we have
$0\leq u(t,x)\leq U(t,x)$, for $t\geq 0$, and $x\in [0,l]$
(see \cite[Chap 3 Theorem 8]{Prot67}. As $U(t,x)$ is bounded then $u(t,x)$ is also
bounded and by Theorem \ref{thm3.4} we have that $\omega (u_{0})$ is nonempty and
consists of steady state solutions of \eqref{eq4}. Then, it follows from
\cite[Theorem A]{Mat78} that $\omega(u_{0})$ contains exactly one steady
state solution (the proof in \cite{Mat78} can be easily extended to the
nonautonomous case since as in the autonomous case $\omega (u_{0})$
consists of solutions of autonomous ordinary differential equations).
\end{proof}
\subsection{The limiting equation}
Let
$$
\mathcal{K}_{M}=\left\{u\in Z,\; 0\leq u(x)\leq M\right\}\,.
$$
\begin{proposition} \label{prop3.6}
Assume that(A1) and (A2) hold. For any
$u_{0}\in\mathcal{K}_{M}$, the solution $u(t)$ of \eqref{eq4} remains in
$\mathcal{K}_{M}$ (i.e. for all $t\geq 0$, $u(t)\in\mathcal{K}_{M}$)
and there exists a unique steady state solution $\bar u$ of \eqref{eq4}
such that $u(t)$ converges to $\bar u$ in $Z$.
\end{proposition}
\begin{proof}
Let $h(w)=\mu(w)\vert M-w\vert$, for $w\in\mathbb{R}$ and
$w_{0}=\max(S_{\rm in},\;\| u_{0}\|_{Z}$). Assumption (A1) implies
$$
-\mu(w)(M-w)\leq h(w),\quad\forall\;w\in\mathbb{R}.
$$
Consider the solution $w(t)$ of the
ordinary differential equation
\begin{gather*}
\frac{dw}{dt}=h(w),\\
w(0)=w_{0}.
\end{gather*}
We deduce from the standard comparison theorem that
$$
0\leq u(t,x)\leq w(t)\leq M,\quad\text{for }t\geq 0 \;\text{and all }x\in[0,l].
$$
The convergence of $u(t)$ to steady state solution of \eqref{eq4} follows
from Theorem \ref{thm3.4} above and \cite[Theorem A]{Mat78}.
To apply Theorem \ref{thm3.4} to \eqref{eq4}, we have to consider the functional
$$
J_{1}(u)=\int_{0}^{l}\Big(\frac{d}{2}\big(\frac{\partial u}{\partial x}\big)^{2}
-\int_{0}^{u}F(x,w)dw\Big)dx +
\frac{q}{4}\left(u^{2}(0)+u^{2}(l)\right),
$$
where $F(x,w)=-(\frac{q^{2}}{4d}w+e^{-\alpha x}\mu(e^{\alpha x}w
+S_{\rm in})(X_{\rm in}-e^{\alpha x}w))$ for
$x\in [0,l]$ and $w\in\mathbb{R}$, instead of $J(t,u(t))$. Therefore,
$\frac{d}{dt}J_{1}(u(t))=-\int_{0}^{l}
\left(\frac{\partial u}{\partial t}\right)^{2}dx$
for solutions of the corresponding transformed equation \eqref{eq3.2}.
\end{proof}
\subsection*{Acknowledgments}
The author would like to express his gratitude to Professors C. Lobry,
M. T. Niane, A. Rapaport and F. Mazenc for their helpfull remarks
and suggestions.
\begin{thebibliography}{99}
\bibitem{Cha75} N. Chafee; Asymptotic behavior for solutions of a
one-dimensional parabolic equation with homogeneous Neumann boundary
conditions, {\em J. Differential Equations}, 18 (1975), 111-134.
\bibitem{Chen89} Xu-Yan Chen and H. Matano;
Convergence, Asymptotic Periodicity and Finite-Point Blow-Up in
One-Dimensional Semilinear Heat Equations,
{\em Journal of Differential Equations} 78, (1989), 160-190.
\bibitem{Dra041} A. K. Dram\'e, J. Harmand, A. Rapaport and C. Lobry;
Multiple Steady State Profiles in Interconnected Biological Systems,
to appear in {\em Math. and Computer Modeling of Dynamical Systems}.
\bibitem{Dra042} A. K. Dram\'e, C. Lobry, A. Rapaport and J. Harmand;
Multiple positive solutions for a two point boundary value problem in
bioreactors theory, in preparation.
\bibitem{Eng00} K. J. Engel and R. Nagel;
One-Parameter Semigroups for Linear Evolution Equations,
Graduate Texts in Mathematics $194$, {\em Springer}, 2000.
\bibitem{Fri64} A. Friedman; Partial differential equations of parabolic type,
{\em Prentice-Hall}, 1964.
\bibitem{Har90} A. Haraux, Syst\`emes dynamiques dissipatifs et applications;
{\em Masson}, 1990.
\bibitem{Kon95} M. Konstantin, Hal Smith and Horst R. Thieme;
Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions,
{\em Transactions of the American Math. Society}, 347-5 (1995), 1669-1685.
\bibitem{Laa00} M. Laabissi, M. E. Achhab, J.J. Winkin and D. Dochain;
Trajectory analysis of nonisothermal tubular reactor nonlinear models, {\em
Systems and Control Letters}, 42-3 (2000), 169-184.
\bibitem{Mat78}H. Matano; Convergence of solutions of one-dimensional
semilinear parabolic equations, {\em J. Math. Kyoto Univ.} 18-2 (1978),
221-227.
\bibitem{Mat79} H. Matano; Asymptotic behavior and stability of solutions of
semilinear diffusion equations, {\em Publ. RIMS, Kyoto Univ.}, 15 (1979),
401-454.
\bibitem{Paz83} A. Pazy; Semigroups of Linear Operators and Applications to
Partial Differential Equations, {\em Springer-Verlag}, 1983.
\bibitem{Prot67} M. H. Protter and H. F. Weinberger; Maximum principles in
differential equations, {\em Prentice-Hall}, 1967.
\bibitem{Pol91} P. Pol\'a$\check{c}$ik, P. Brunovsk\'y, X. Mora and J. Sola-Morales;
Asymptotic behavior of solutions of semilinear elliptic equations on an
unbounded strip, {\em Acta Math. Univ. Comenianae} LX 2 (1991), 163-183.
\bibitem{Pol92} P. Pol\'a$\check{c}$ik and A. Haraux;
Convergence to a positive equilibrium for some nonlinear evolution equations,
{\em Acta Math. Univ. Comenianae}, LXI (1992), 129-141.
\bibitem{Pol96} P. Pol\'a$\check{c}$ik and K. P. Rybakowski;
Nonconvergent bounded trajectories in semilinear heat equations,
{\em J. Differential Equations}, 124 (1996), 472-494.
\bibitem{Pol021} P. Pol\'a$\check{c}$ik and H. Matano;
Existence of $L^{1}$-connections between equilibria of a semilinear
parabolic equation, {\em J. Dynam. Differential Equations} 14 (2002), 463-491.
\bibitem{Pol022} P. Pol\'a$\check{c}$ik and F. Simondon;
Nonconvergent bounded solutions of semilinear heat equations on arbitrary
domains, {\em J. Differential Equations} 186 (2002), 586-610.
\bibitem{Pol03}P. Pol\'a$\check{c}$ik; Asymptotic symmetry of positive solutions
of parabolic equations, {\em International conference of Differential Equations.
Hasselt-Belgium}, July 22-26, 2003.
\bibitem{Rob76} H. Robert and J. R. Martin;
Nonlinear operators and differential equations in Banach
spaces, {\em Wiley-Interscience Publication}, 1967.
\bibitem{Sch01} R. Schnaubelt;
Asymptotically autonomous parabolic evolution equations, {\em J. Evol. Equ.}
1 (2001) 19-37.
\bibitem{Sho97} R. E. Showalter; Monotone Operators in Banach Sapce and
Nonlinear Partial Differential Equations, {\em American Mathematical Society},
1997.
\bibitem{Smith951} Hal L. Smith and P. Waltman; The Theory of the
Chemostat. Dynamics of Microbial Competition, {\em Cambridge University
Press}, 1995.
\bibitem{Smith952} Hal L. Smith; Monotone dynamical systems:
An introduction to the theory of competitive and cooperative systems,
{\em Math. Surveys Monogr. 41, AMS, Providence, RI}, 1995.
\bibitem{Yos68} K. Yosida; Functional Analysis, {\em Springer-Verlag Berlin
Heidelberg New-York}, 1968.
\end{thebibliography}
\end{document}