\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\,0<t\leq t_{0}.
\end{equation}
\end{lemma}

We  deduce from the lemma  above the following result.

\begin{lemma} \label{lem2.2}
The semigroup $T(t)$ is continuously differentiable and compact on $Z$ 
for $t>0$; 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\,0<t\leq t_{0}.
\end{equation}
\end{lemma}

\begin{proof}
The continuous differentiability of $T(t)$ follows from the continuity of
 $\frac{\partial \Gamma}{\partial t}$ on $[0,l]\times[0,l]\times]0,\infty[$. 
Then, $T(t)$ maps  $Z$ into $D(A)$ for $t>0$, $AT(t)\in L(Z)$ 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 $0<t<t_{u}$  and $u(t)$ satisfies 
\eqref{eq3.1} (\text{resp. }\eqref{eq3.2}), 
where $[0,t_{u}[$ is the maximum interval of existence of $u(t)$.

\item[(ii)]  If $u(t)$ is  bounded  in $Z$ then, for any $t_{0}>0$ 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
$$
u(t)=T(t)u_{0}+\int_{0}^{t}T(t-s)f(s,u(s))ds,\quad 0<t<t_{u}.
$$
Since $T(t)$ is continuously differentiable, we have $T(t)u_{0}\in D(A)$, 
for $0<t<t_{u}$ and $AT(t)\in L(Z)$, for $t>0$. Let $\varepsilon$, $T_{0}$ 
and $T_{1}$ be such that $0<\varepsilon<T_{0}\leq T_{1}<t_{u}$ and rewrite 
the equality above  as follows
$$
u(t)=T(t-\varepsilon)u(\varepsilon)+\int_{\varepsilon}^{t}T(t-s)f(s,u(s))ds,
\quad \varepsilon\leq t\leq T_{1}.
$$
The map $t\to T(t-\varepsilon)u(\varepsilon)$ is continuously differentiable 
on $]\varepsilon,T_{1}]$ and $T(t-\varepsilon)u(\varepsilon)\in D(A)$
fo rall $t\in]\varepsilon,T_{1}]$. Let 
$$
v(t)=\int_{\varepsilon}^{t}T(t-s)f(s,u(s))ds,\quad \varepsilon\leq t\leq T_{1}.
$$ 
Since $f:[\varepsilon,T_{1}]\times Z\to Z$ is continuously differentiable,
 by \cite[Theorem 1.5]{Paz83}, $v$ is continuously differentiable on 
 $]\varepsilon,T_{1}]$ and   if $w(t)$ denotes the solution of the integral 
equation 
$$
w(t)=T(t-\varepsilon)f(\varepsilon,u(\varepsilon))+\int_{\varepsilon}^{t}\! T(t-s)
\frac{\partial}{\partial s}f(s,u(s))ds+\int_{\varepsilon}^{t}T(t-s)
\frac{\partial}{\partial u}f(s,u(s))w(s)ds
$$
on $[\varepsilon,T_{1}]$.  Then  
$$
\frac{d v}{d t}(t)=w(t)+\int_{\varepsilon}^{t}AT(t-\varepsilon)
\frac{\partial}{\partial u}f(s,u(s))u(\varepsilon)ds,\quad \forall\;t\in]
\varepsilon,T_{1}].
$$
Therefore, $v(t)\in D(A)$ for all $t\in ]\varepsilon,T_{1}]$. 
Hence, $u(t)=T(t-\varepsilon)u(\varepsilon)+v(t)\in D(A)$ for all
$t\in [T_{0},T_{1}]$ and $\frac{\partial u}{\partial t}\in C([T_{0},T_{1}];\;Z)$.
Since  $T_{0}$ and $T_{1}$ are any given numbers in $]0,t_{u}[$, we have 
$u\in C([0,t_{u}[;\;Z)\cap C^{1}(]0,t_{u}[;\;Z)$ and 
$u(t)\in D(A)$  for all $0<t<t_{u}$. Moreover, $u(t)$ satisfies 
\eqref{eq3.1} on $[0,t_{u}[$.\\
$(ii)$ Let $0<a<t_{0}$ and $\| u(t)\|_{Z}\leq N_{0} $, 
$\| f(t,u(t))\|_{Z}\leq N_{1}$, for all $t\geq 0$.  We have 
\begin{align*}
Au(t_{0}+t)
&=AT(t_{0})u(t)+\int_{0}^{t_{0}-a}AT(t_{0}-s)f(s+t,u(s+t))ds\\
&+\int_{t_{0}-a}^{t_{0}}AT(t_{0}-s)f(s+t,u(s+t))ds.
\end{align*}
By \eqref{ine4}, we have
\begin{equation}\label{e3.3}
\begin{aligned}
&\| AT(t_{0})u(t)\|_{Z}+\int_{0}^{t_{0}-a}\| AT(t_{0}-s)\|\,
\| f(s+t,u(s+t))\|_{Z}ds\\
&\leq \frac{C_{0}N_{0}}{t_{0}}+C_{0}N_{1}
\ln(\frac{t_{0}}{a}),
\end{aligned}
\end{equation}
where $\| AT(t)\|$ denotes the norm of $AT(t)$ in $L(Z)$. Moreover, 
one can check readily that 
\begin{align*}
&\int_{t_{0}-a}^{t_{0}}AT(t_{0}-s)f(s+t,u(s+t))ds\\
&=\int_{t_{0}-a}^{t_{0}}AT(t_{0}-s)\left(f(s+t,u(s+t))-f(t_{0}+t, u(s+t))\right)ds\\
&\quad +\int_{t_{0}-a}^{t_{0}}AT(t_{0}-s)\left(f(t_{0}+t,u(s+t))-f(t_{0}+t, u(t_{0}+t))
\right)ds\\
&\quad +\int_{t_{0}-a}^{t_{0}}AT(t_{0}-s)f(t_{0}+t, u(t_{0}+t))ds.
\end{align*}
Under Hypotheses (A1) and (A2), $\mu(r)$ is bounded as 
$r\to \infty$ and $f$ is locally Lipschitz continuous in $u$, uniformly in $t$. 
Then, let $\mu_{0}$ be a constant such that 
$\vert \mu(r)\vert\leq \mu_{0}$ for all $r\in\mathbb{R}$ and let  $L_{0}$ be 
the  (local) Lipschitz constant of $f$ with respect to the second variable
 since $u(t)$ is bounded. Using \eqref{ine1} and \eqref{ine4} we have, for all 
$s\in [t_{0}-a,t_{0}]$,
\begin{align*}
\| f(s+t,u(s+t))-f(t_{0}+t,u(s+t))\|_{Z}
&\leq \mu_{0}\| T(s+t)V_{0}-T(t_{0}+t)V_{0}\|_{Z}\\
&\leq \mu_{0}(t_{0}-s) \| T(t)\|\,\| AT(s)\|\,\| V_{0}\|_{Z}\\
&\leq \mu_{0}C_{0}\frac{(t_{0}-s)}{t_{0}-a} \| V_{0}\|_{Z},
\end{align*}
where $V_{0}(x)=e^{-\frac{q}{2d} x}(U_{0}(x)-M)$ for all $x\in [0,l]$ 
(and $T(\tau)V_{0}$ is a solution of \eqref{lin2}). 
Moreover,
\begin{align*}
&\|\int_{t_{0}-a}^{t_{0}}AT(t_{0}-s)f(t_{0}+t, u(t_{0}+t))ds\|_{Z}\\
&=\| (I-T(a))f(t_{0}+t,u(t_{0}+t))\|_{Z}\leq 2N_{1}.
\end{align*}
It follows that 
\begin{align*}
&\| \int_{t_{0}-a}^{t_{0}}AT(t_{0}-s)f(s+t,u(s+t))ds\|_{Z}\\
&\leq \frac{aC_{0}^{2}\mu_{0} \| V_{0}\|_{Z}}{t_{0}-a}+2N_{1}
+ L_{0}\int_{t_{0}-a}^{t_{0}}\| AT(t_{0}-s)\|\,\| u(s+t))-u(t_{0}+t)\|_{Z}.
\end{align*}
Let $\Delta s=t_{0}-s$, for all $s\in [t_{0}-a,t_{0}]$. 
We have $\Delta s\geq 0$ and 
\begin{align*}
&\| u(s+t)-u(t_{0}+t)\|_{Z}\\\
&\leq \| (T(t_{0})-T(s))u(t)\|_{Z}
+\int_{\max(s-\Delta s,0)}^{t_{0}}\| T(t_{0}-\tau)f(\tau+t,u(\tau+t))\|_{Z}d\tau
\\
&\quad +\int_{\max(s-\Delta s,0)}^{s}\| T(s-\tau)f(\tau+t,u(\tau+t))\|_{Z}d\tau
\\
&\quad +\int_{0}^{\max(s-\Delta s,0)}\| \left(T(t_{0}-\tau)-T(s-\tau)\right)
f(\tau+t,u(\tau+t))\|_{Z}d\tau.
\end{align*}
Then,
\begin{align*}
&\| u(s+t)-u(t_{0}+t)\|_{Z}\\
&\leq \frac{C_{0}N_{0}}{t_{0}-a}\Delta s+3N_{1}\Delta s\\
&\quad +\int_{\min(s,\Delta s)}^{s}\| \left(T(\tau+\Delta s)-T(\tau)\right)f
(s-\tau+t,u(s-\tau+t))\|_{Z}d\tau\\
&\leq \frac{C_{0}N_{0}}{t_{0}-a}\Delta s+3N_{1}\Delta s\\
&\quad + \int_{\min(s,\Delta s)}^{s}\int_{0}^{\Delta s}\| T(\sigma)\|\,\| AT(\tau)\|\,
\| f(s-\tau+t,u(s-\tau+t))\|_{Z}d\sigma d\tau.
\end{align*}
Using \eqref{ine1} and \eqref{ine4} again and the estimate on $f$, we have 
\begin{align*}
\| u(s+t)-u(t_{0}+t)\|_{Z} 
&\leq \frac{C_{0}N_{0}}{t_{0}-a}\Delta s+3N_{1}\Delta s
+\int_{\min(s,\Delta s)}^{s}\frac{C_{0}N_{1}}{\tau}\Delta s d\tau\\
&\leq \frac{C_{0}N_{0}}{t_{0}-a}\Delta s+3N_{1}\Delta s+C_{0}N_{1} 
\ln(\frac{s}{\min(s,\Delta s)})\Delta s \\
&\leq \left(\frac{C_{0}N_{0}}{t_{0}-a}+3N_{1}+C_{0}N_{1}
\max\big(\ln(\frac{s}{t_{0}-a}),\;\ln(\frac{s}{a})\big)\right)\Delta s \\
&\leq \left(\frac{C_{0}N_{0}}{t_{0}-a}+3N_{1}+C_{0}N_{1} N_{2}\right)(t_{0}-s), 
\end{align*}
where $N_{2}=\max \left(\ln( \frac{t_{0}}{t_{0}-a}),\; 
\ln(\frac{t_{0}}{a})\right)$.
Then, using \eqref{ine4} once again, we have
\begin{equation}\label{e3.4}
\begin{aligned}
&\| \int_{t_{0}-a}^{t_{0}}AT(t_{0}-s)f(s+t,u(s+t))ds\|_{Z}\\
&\leq \frac{aC_{0}^{2}\mu_{0} \| V_{0}\|_{Z}}{t_{0}-a}+2N_{1}+aL_{0}C_{0}
\big(\frac{C_{0}N_{0}}{t_{0}-a}+3N_{1}+C_{0}N_{1} N_{2}\big).
\end{aligned}
\end{equation} 
It follows from \eqref{e3.3} and \eqref{e3.4} that 
\begin{align*}
\| Au(t_{0}+t)\|_{Z}
&\leq \frac{C_{0}N_{0}}{t_{0}}+C_{0}N_{1}\ln(\frac{t_{0}}{a})
+\frac{aC_{0}^{2}\mu_{0} \| V_{0}\|_{Z}}{t_{0}-a}+2N_{1}\\
&\quad +aL_{0}C_{0}\big(\frac{C_{0}N_{0}}{t_{0}-a}+3N_{1}+C_{0}N_{1} N_{2}\big),
\end{align*}
for any $t\geq 0$.  
Hence, $Au(t_{0}+t)$ remains bounded in $Z$ for  $t\geq 0$. 
Since $\| f(t,u(t))\|_{Z}\leq N_{1}$ and $u(t)$ is a classical solution 
of \eqref{eq3.1} then, $\frac{\partial u}{\partial t}(t_{0}+t)$ also remains 
bounded for $t\geq 0$ and  Lemma \ref{lem3.1} is proved. 
\end{proof} 

\begin{lemma} \label{lem3.2}
Assume that (A1) and (A2)  hold.
Let $u(t)$ be a bounded solution of \eqref{eq3.1} (resp. of \eqref{eq3.2}) then, 
$K=\overline{\{u(t),\;t\geq 0\}}$ is compact in $Z$,
where $\overline{E}$ denotes the closure of $E$.
\end{lemma}

\begin{proof} 
By Lemma \ref{lem2.2}, $T(t)$ is compact for $t>0$. 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 $0<t<t_{u}$.
\end{lemma}

\begin{proof} 
First, we deduce from Lemma \ref{lem3.1} (i) and \cite[Chap 3 Theorem 10]{Fri64} that for 
any $u_{0}\in Z$ the solution $u(t)$ of \eqref{eq3.1} has continuous partial 
derivatives $\frac{\partial^{3} u}{\partial x^{3}}$ and  
$\frac{\partial^{2} u}{\partial t\partial x}$ on 
$]0,t_{u}[\times ]0,l[$. Then the following calculation is well founded. 
By deriving and integrating by parts, we have 
\begin{align*}
&\frac{\partial}{\partial t}\int_{0}^{l}\Big(\frac{d}{2}\Big(\frac{\partial
u}{\partial x}\Big)^{2}-\int_{0}^{u}F(t,x,w)dw\Big)dx\\
&=\int_{0}^{l}\left(d\frac{\partial^{2}u}{\partial
t\partial x}\frac{\partial u}{\partial x}-F(t,x,u)
\frac{\partial u}{\partial t}\right)dx
-\int_{0}^{l}\int_{0}^{u(t,x)}\frac{\partial F}{\partial t}(t,x,w)dw dx \\
&=\int_{0}^{l}-\left(\frac{\partial u}{\partial t}\right)^{2} dx +
d\frac{\partial u}{\partial t}\frac{\partial u}{\partial
x}\vert_{x=l}-d\frac{\partial u}{\partial t}\frac{\partial u}{\partial
x}\vert_{x=0}
-\int_{0}^{l}\int_{0}^{u(t,x)}\frac{\partial F}{\partial t}(t,x,w)dw dx\,.
\end{align*}
Since $u(t)$ satisfies the boundary conditions in \eqref{eq3.1},
$$
d\frac{\partial u}{\partial t}\frac{\partial u}{\partial x}\vert_{x=l}-d\frac{\partial u}{\partial t}\frac{\partial u}{\partial
x}\vert_{x=0}=-\frac{q}{4}\frac{\partial}{\partial t}
\Big(v^{2}(t,0)+v^{2}(t,l)\Big).
$$
Hence,
$$ 
\frac{d }{dt}\left(J(t,u(t))\right)
=\int_{0}^{l}-\Big(\frac{\partial u}{\partial t}\Big)^{2}dx
-\int_{0}^{l}\int_{0}^{u(t,x)}\frac{\partial F}{\partial t}(t,x,w)dw dx,
$$
for $0<t<t_{u}$ and for any solution $u(t)$ of \eqref{eq3.1}. 
\end{proof}

 Now we can state the main  result dealing with the asymptotic behavior 
 of solutions of \eqref{eq3.1}.
 
\begin{theorem} \label{thm3.4}
Assume that (A1) and (A2) hold and  
let $u_{0}\in Z$ be such that $u(t)$ is a bounded solution of
\eqref{eq3.1}. Then, the omega limit set $\omega(u_{0})$ of  $u(t)$ is nonempty, 
it is contained in $C^{2}[0,l]$ and it consists of
steady state solutions of  \eqref{eq3.2}. 
\end{theorem}

\begin{proof} Let $K=\overline{\{u(t),t\geq 0\}}$. By Lemma \ref{lem3.2}, $K$ is compact 
in $Z$. Then, $\omega(u_{0})$ is nonempty. Let $\varphi\in\omega(u_{0})$, 
there exists a sequence $(t_{n})_{n\geq 0}$ such that $t_{n}\to +\infty$ and  
$u(t_{n})\to \varphi$ in $Z$ as $n\to +\infty$. 
Let $u_{n}=u(t_{n})$ and $v_{n}(t)=u(t+t_{n})$ for $n\geq 0$ and $t\geq 0$. 
We have
\begin{equation}\label{eq3.3}
\begin{aligned}
v_{n}(t)&=T(t)u_{n}+\int_{t_{n}}^{t+t_{n}}T(t+t_{n}-s)f(s,u(s))ds\\
&=T(t)u_{n}+\int_{0}^{t}T(t-s)f(s+t_{n},v_{n}(s))ds.
\end{aligned}
\end{equation}
The set $B=\{v_{n}(t), n\geq 0, t\geq 0\}$ is bounded in $Z$ and $f$ 
is locally Lipschitz continuous in $u$, uniformly in $t$.  Moreover, 
$$
\| f(s+t_{n},v_{n}(s))-f(s+t_{m},v_{n}(s))\|_{Z}
\leq \mu_{0}\| T(t_{n})V_{0}-T(t_{m})V_{0}\|_{Z}, \quad \text{for all } s\geq 0,
$$
where $\mu_{0}$ is a constant such that $\vert \mu(r)\vert\leq \mu_{0}$
for all $r\in\mathbb{R}$  and $V_{0}(x)=e^{-\alpha x}(U_{0}(x)-M)$
for all $x\in [0,l]$. Then, by Gronwall's inequality, we have:
For all $t_{0}>0$ 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 $0<t_{0}<t_{1}<\infty$. From \eqref{eq3.7}, we have
$$ 
\lim_{n \to \infty}\int_{t_{0}}^{t_{1}}\int_{0}^{l}
\big(\frac{\partial v_{n}}{\partial t}(t)\big)^{2}\,dx\,dt
=\lim_{n \to \infty}\int_{t_{0}+t_{n}}^{t_{1}+t_{n}}\int_{0}^{l}
\big(\frac{\partial u}{\partial t}(t)\big)^{2}\,dx\,dt=0.
$$
Then, regarding $h$ as a function of $(t,x)$, we have 
$$
\int_{t_{0}}^{t_{1}}\int_{0}^{l}\big(\frac{\partial h}{\partial t}\big)^{2}\,dx\,dt=0.
$$
It follows that $\frac{\partial h}{\partial t}=0$ on any compact set 
$[t_{0},t_{1}]\times [0,l]$. Then, $\frac{\partial h}{\partial t}=0$ 
in $]0,\infty[\times [0,l]$ and therefore $h(t)=\varphi$ in $Z$ for $t\geq 0$. 
Hence, $\varphi\in D(A)$ and $A\varphi+g(\varphi)=0$.
This proves that $\omega(u_{0})\subset C^{2}[0,l]$ and for any 
$\varphi\in\omega(u_{0})$, we have 
\begin{gather*}
d\frac{\partial^{2}\varphi}{\partial x^{2}}-\frac{q^{2}}{4d}\varphi+g(\varphi)=0,
\quad x\in ]0,l[,\\
d\frac{\partial \varphi}{\partial x}(0)-\frac{q}{2}\varphi(0)=0,\quad 
d\frac{\partial \varphi}{\partial x}(l)+\frac{q}{2}\varphi(l)=0.
\end{gather*}
\end{proof}

\noindent\textbf{Remarks:}
Theorem \ref{thm3.4} can be stated in a more general form: 
Consider an asymptotically autonomous nonlinear equation of type
 \eqref{int3} with limiting equation \eqref{int4} on a Banach space $Z$. 
Assume that the linear operator $A$ is the infinitesimal generator of a  
$C_{0}$-semigroup of contractions on $Z$ which is continuously differentiable 
and satisfies \eqref{ine4} and that $f$ is Lipschitz continuous 
(locally with respect to $u$)  in the sense that for any bounded subset $B$ 
of $Z$, there is a constant $C>0$  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. 

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\end{thebibliography}





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