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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 161, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/161\hfil Almost periodic solutions]
{Almost periodic solutions of distributed parameter biochemical
systems with time delay and time varying input}

\author[A. K. Drame, M. R. Kothari, P. R. Wolenski \hfil EJDE-2013/161\hfilneg]
{Abdou K. Drame, Mangala R. Kothari, Peter R. Wolenski}  % in alphabetical order

\address{Abdou K. Drame \newline
Dept of Mathematics,  
La Guardia Community College,
The City University of New York, 
31-10 Thomson Ave., 
Long Island City, NY 11101, USA}
\email{adrame@lagcc.cuny.edu}

\address{Mangala R. Kothari \newline
Dept of Mathematics,  
La Guardia Community College, 
The City University of New York, 
31-10 Thomson Ave., 
Long Island City, NY 11101, USA}
\email{mkothari@lagcc.cuny.edu}

\address{Peter R. Wolenski \newline
 Dept of Mathematics, Louisiana State University,
 Baton Rouge, LA 70803, USA}
\email{wolenski@math.lsu.edu}

\thanks{Submitted October 18, 2012. Published July 12, 2013.}
\subjclass[2000]{92B05, 35B15, 35K60}
\keywords{Time delay; almost periodic solutions;
  biochemical system; \hfill\break\indent 
 partial functional differential equations}

\begin{abstract}
 In this article we study the existence of almost periodic solutions 
 for distributed parameters biochemical system, with time delay
 when the input $S_{\rm in}$ is time dependent. This study is motivated 
 by the input begin time dependent in many applications, and by
 the importance of almost periodically varying environments. 
 Using the semigroup method, we prove that if the input is almost periodic 
 then the system has an almost periodic solution.

\end{abstract}

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


\section{Introduction}

The article is concerned with the existence of almost periodic 
solutions of a distributed parameters biochemical system, with time delay 
in the growth response when the input is time dependent
($S_{\rm in}=S_{\rm in}(t)$) and is an Almost periodic function
of the time $t$. 
The study of the existence of periodic and almost periodic solutions 
is important in the theory of bioengineering systems because of 
the periodically and almost periodically time varying environments. 
In the previous decades, many authors have introduced various types of 
delay into the dynamical models of bioengineering systems to understand 
the oscillations (periodic solutions) observed experimentally on 
chemostat systems (see references in \cite{Dra12}). However, most of 
these studies were focused on time varying systems and few consider 
space varying cases. Recently, Drame et al \cite{Dra12} studied a distributed
parameters biochemical system with time delay in the growth response. 
They proved the existence of periodic solutions (oscillations) 
for large values of the delay parameter. In \cite{Dra12}, 
the authors assumed the inlet substrate concentration to be constant.
However, in applications, this quantity may be time dependent and periodic 
or almost periodic in time. In this paper, we consider a distributed 
parameter biochemical system with time delay and almost periodic time 
varying inlet substrate concentration. The basis of the model under study 
is derived from the work performed on anaerobic digestion in the pilot 
fixed bed reactor of the LBE-INRA in Narbonne (France) and is inspired 
from the dynamical models built and validated on the process 
(see \cite{Ber01,Sch04}). Using semigroup approach, we prove the existence 
of almost periodic solutions for the system. We consider the  
dynamical system
\begin{equation}\label{e1.1}
\begin{gathered}
\frac{\partial S(t)}{\partial t}=d\frac{\partial^2 S(t)}{\partial x^2}
-q\frac{\partial S(t)}{\partial x}-k\mu(S(t),X(t))X(t)\\
\frac{\partial X(t)}{\partial t}=-k_d X(t)+\mu(S(t-r),X(t-r))X(t-r),
\end{gathered}
\end{equation}
with the boundary conditions: for all $t\geq 0$,
\begin{equation}\label{e1.2}
d\frac{\partial S}{\partial x}(t,0)-q S(t,0)+q S_{\rm in}(t)=0\quad \text{and}\quad
\frac{\partial S}{\partial x}(t,L)=0,
\end{equation}
and initial condition: for all $-r\leq s\leq 0$, $0\leq x\leq L$,
\begin{equation}\label{e1.3}
S(s,x)=\Phi_1(s,x),\quad X(s,x)=\Phi_2(s,x)
\end{equation}
where the initial data $\Phi_1$ and $\Phi_2$ are continuous functions on 
$[-\tau, 0]\times [0, L]$. 

The parameters $d, q, k, k_d, S_{\rm in}, \mu, r, L$ are positive and represent
the diffusion coefficient, the superficial fluid velocity, the yield coefficient, 
the death rate of the biomass, the inlet limiting substrate concentration,
 the specific growth function or growth response, the delay parameter and 
the length of the reactor, respectively.
In the rest of the paper, we will assume that the length $L$ is $1$.

In the right hand side of the first equation of \eqref{e1.1}, the last term 
represents the growth response while the other terms represent the 
hydrodynamics (i.e. diffusion and convection terms). As the biomass is 
assumed to be fixed, there is no hydrodynamic term in the right hand side 
of the second equation of \eqref{e1.1}. The first term in right hand side 
of this equation represents the mortality of biomass and the last one 
represents the growth response with delay. The delay is considered in the 
second equation of \eqref{e1.1} only as it is assumed to be caused by 
memory effects of micro-organisms, on one hand. On the other hand, 
the substrate is apparently instantaneously consumed although the biomass 
growth takes place with some delay. This can be explained e.g. by the 
absorption of the (liquid) substrate into the (solid) biomass, 
a phenomenon that might be fast with respect to the conversion of substrate 
into biomass, at least fast enough to emphasize the disappearance of the 
substrate from the liquid medium before its conversion into biomass exhibits 
its effects. The resulting dynamical model is a system of partial functional 
differential equations  with almost periodic boundary conditions. 
One of the most attracting areas of qualitative theory of partial functional 
differential equations is the existence of periodic and almost periodic 
solutions due to the important roles of periodically and almost periodically 
varying environments play in many biological and ecological dynamical systems. 
Compared with periodic effects, almost periodic effects are more frequent 
\cite{Lon12}. The existence of almost periodic (or pseudo-, weighted almost 
periodic) solutions of partial differential equations has been increasingly 
studied by many authors (see e.g. \cite{Dam12,Dia11,Hen11} and references therein).
 In these studies, the semigroup or evolution family is compact and 
exponentially stable, and therefore cannot be applied to the system under 
consideration here. As mentioned above, the current work is motivated, 
on one hand, by a recent paper by Drame et al \cite{Dra12}, where the 
authors considered existence question of periodic solutions of biochemical 
distributed parameters system with time delay. On the other hand, in applications 
the inlet concentration of substrate in biological processes are time varying 
and may be periodic or almost periodic function of time.

The fundamental assertion we prove in this paper is that if the input 
$S_{\rm in}(t)$ is an almost periodic function of $t$ then the
system \eqref{e1.1}-\eqref{e1.3} has an almost periodic.
So, we make the following assumption:


\noindent\textbf{Assumption A1.} 
The input $S_{\rm in}(t)$ is an Almost Periodic function of $t$, 
that is: for any $\varepsilon>0$, there exists $l(\varepsilon)>0$ 
such that any interval of length $l(\varepsilon)$ contains a number 
$\tau$ such that
$$
|S_{\rm in}(t+\tau)-S_{\rm in}(t)|<\varepsilon\quad
 \text{ for all } t\in\mathbb{R}.
$$
To guaranty the existence and some regularity of solution of the 
system \eqref{e1.1}-\eqref{e1.3}, we make the assumption that:

\noindent\textbf{Assumption A2.} 
The function $\mu $ is smooth (to say of class $C^2$, for example) 
and bounded on $(0,1)\times (0,\infty)$.


In the next section, we give some preliminary properties of the system 
\eqref{e1.1}-\eqref{e1.3} and recall some definitions and properties 
of Almost periodic functions.

\section{Preliminaries}

Let us consider the following state spaces $Z_1=Z_2=C[0,1]$,
$Z=Z_1\times Z_2$, $\mathcal{C}=C([-r,0];Z)$ and the positive cones
\begin{gather*}
Z^+=\{v\in Z: v_i(z)\geq 0,\,\forall z\in [0,1],\,i=1,2\},\\
\mathcal{C}^+=\{\varphi\in\mathcal{C}:\varphi(s)\in Z^+,\,\forall -r\leq s\leq 0\}.
\end{gather*}
We also adopt the following notation: 
for a continuous function $w: [-r,\,b)\to Z,\;b>0$,
we define $w_t \in \mathcal{C}$, $t\in[0,\,b)$, by $w_t(s)=w(t+s)$ for 
all $-r\leq s\leq 0$. Therefore, it is not difficult to see that the 
map $t\to w_t$ is continuous from $[0,b)$ into $\mathcal{C}$.

\begin{definition}[{\cite[p. 55]{Cor89}}] \label{d2.1}
A function $F(t,x)$ is called almost periodic in $t$, uniformly with 
respect to $x\in Z$, if for any $\varepsilon>0$ there exists a number 
$l(\varepsilon)$ such that any interval of the real line of length
 $l(\varepsilon)$ contains at least one number $\tau$ such that
$$
|F(t+\tau,x)-F(t,x)|<\varepsilon\quad \text{ for all } t\in\mathbb{R}, \, x\in Z.
$$
\end{definition}

\begin{theorem}[{\cite[Theorem 2.6]{Cor89}}] \label{th2.1}
A necessary and sufficient condition for a function $F(t,x)$ to be almost
 periodic in $t$, uniformly with respect to $x\in \Omega$, where $\Omega$ 
is a bounded and closed set, is that for any sequence $\{F(t+\tau_n,x)\}$, 
we can extract a subsequence $\{F(t+r_n,x)\}$  that satisfies the 
Cauchy uniform convergence.
\end{theorem}


Let us consider the following auxiliary problem from \eqref{e1.1}-\eqref{e1.2} 
(recall $L=1$):
 \begin{equation}\label{e2.1}
\begin{gathered}
\frac{\partial w(t)}{\partial t} 
= d\frac{\partial^2 w(t)}{\partial x^2}-q\frac{\partial w(t)}{\partial x},
\\
d\frac{\partial w}{\partial x}(t,0) = q w(t,0)-q S_{\rm in}(t),\quad 
\frac{\partial w}{\partial x}(t,1)=0.
\end{gathered}
\end{equation}
In section 3, we will show that under assumption A1, the auxiliary
 problem \eqref{e2.1} has an {almost periodic} solution.  
Observe that if $w$ is an almost periodic solution of \eqref{e2.1} 
then the function $f$, defined by:
for any $u\in\mathcal{C}$,
\begin{gather*}
(f_1(t,u))(x)=-k\mu(u_1(0,x)+w(t,x),u_2(0,x))u_2(0,x)\\
(f_2(t,u))(x)=\mu(u_1(-r,x)+w(t,x),u_2(-r,x))u_2(-r,x)
\end{gather*}
and $f=(f_1, f_2)$, is almost periodic in $t$ uniformly with respect to $u$.
Now, let $w(t,x)$ be a solution of  \eqref{e2.1} and let us introduce
$$
u_1(t,x)=S(t,x)-w(t,x) \text{ and } u_2(t,x)=X(t,x)\quad \text{for all } 
t\geq 0 \text{ and } 0\leq x\leq 1,
$$ 
where $S$ and $X$ are as in \eqref{e1.1}-\eqref{e1.3}.
Then $u=(u_1,u_2)$ satisfies the equations
\begin{equation}\label{e2.2}
\begin{gathered}
\frac{\partial u_1(t)}{\partial t}
=d\frac{\partial^2 u_1(t)}{\partial x^2}-q\frac{\partial u_1(t)}{\partial x}
+f_1(t,u_{1t}, u_{2t})\\
\frac{\partial u_2(t)}{\partial t}=-k_d u_2(t)+f_2(t,u_{1t},u_{2t}),\\
d\frac{\partial u_1}{\partial x}(t,0)=q u_1(t,0) , \quad 
\frac{\partial u_1}{\partial x}(t,1)=0.
\end{gathered}
\end{equation}
Let us define the operators
\begin{gather*}
D(A_1)=\{u_1\in C^2[0,1]: d\frac{\partial u_1}{\partial x}(0)
=qu_1(0),\;\frac{\partial u_1}{\partial x}(1)=0\};
\\
A_1u_1=d\frac{\partial^2 u_1}{\partial x^2}-q\frac{\partial u_1}{\partial x};
\\
D(A_2)=C[0,1], \quad A_2=-k_d I;
\\
D(A)=D(A_1)\otimes D(A_2),\quad A=\operatorname{diag}(A_1,A_2).
\end{gather*}
By the same arguments as in \cite{Dra08}, we can show that the operator $A$ 
is the infinitesimal generator of a $C_0$-semigroup of bounded linear 
operators $T(t)$ on $Z$, given by $T(t)=\operatorname{diag}(T_1(t),T_2(t))$, 
where $T_1(t)$ and $T_2(t)$ are the $C_0$-semigroups generated by $A_1$ 
and $A_2$, respectively. Moreover, the semigroup $T_1(t)$ is compact in 
$C^1[0,1]$ and $T(t)$ is analytic.
The system \eqref{e2.2} can be written as the following abstract
 Cauchy problem
\begin{equation}\label{e2.3}
\begin{gathered}
\frac{du(t)}{dt}=Au(t)+f(t,u_t),\\
u(s)=u_0(s), \quad \text{for } -r\leq s\leq 0,
\end{gathered}
\end{equation}
where, as in \eqref{e1.3}, the initial data $u_0$ is in $C([-\tau,0], Z)$.

\begin{definition}\label{d2.2}\rm \quad
\begin{itemize}
\item A mild solution of \eqref{e2.3} (equivalently of \eqref{e1.1}-\eqref{e1.3}) 
is a continuous function $u:[-\tau,t_u)\to Z$, with $t_u>0$, satisfying
\begin{gather*}
u(t)=T(t)\theta(0)+\int_0^t T(t-s)f(s,u_s)ds,\quad 0\leq t\leq t_u\\
u_0=\theta\in\mathcal{C},
\end{gather*}
where $u_s\in\mathcal{C}$ is given by $u_s(\sigma)=u(s+\sigma)$, 
$-\tau\leq \sigma\leq 0$.

\item A function $u\in C([-\tau, t_u), Z)\cap C^1((0,t_u), Z)$ satisfying 
$u(t)\in D(A)$, for $0<t<t_u$, and $u(t)$ satisfies \eqref{e2.3} 
is called a classical solution.
\end{itemize}
\end{definition}

\section{The auxiliary problem and existence of solutions for \eqref{e2.3}}

As we mentioned earlier, in this section we prove that under assumption 
A1 the auxiliary problem \eqref{e2.1} has an  almost periodic solution,
 by using the method of sub-super solutions. Later, we will also discuss 
the existence of solutions of \eqref{e2.3}.

Since $S_{\rm in}(t)$ is  almost periodic,  by \cite[Theorem 1.2]{Cor89}, 
$S_{\rm in}(t)$ is continuous and bounded on the real line. 
Let $\underline{S}\geq 0$ and $\bar S\geq 0$ be such that
$$ 
\underline{S}\leq S_{\rm in}(t)\leq \frac{1}{2}\bar S, \quad \text{for all } 
t\in\mathbb{R},
$$
and let $S_*=\frac{q}{2d}\underline{S}$,  and $S^*=\frac{q}{2d}\bar S$.
To prove the following remark, we assume that:

\noindent\textbf{Assumption A3:} $1+q\leq \frac{d}{2}$.

\begin{remark}\label{r3.1} \rm
Assume that {\rm A3} holds. Then there exists a function $g$ such that 
for all $t\geq 0$
$$
0\leq g(t)\leq A=\min(S_*,\frac{2q^2 \underline{S}}{4d+qd}), \quad
 0\leq g'(t)\leq 8d g(t)-4qS_*, \quad t\geq 0.
$$
\end{remark}

\begin{proof}
Let $y(t)=\frac{At}{1+t}$,  for $t\geq 0$. We have $y(t)\leq A$,  for $t\geq 0$.
 In addition, for $t\geq 1$, $y(t)\geq \frac{A}{2}$. On the other hand, under 
assumption \textbf{A3}, we have $A\geq \frac{2q}{d}S_*$. Therefore, for 
$t\geq 1$, we obtain
$$
8dy(t)-4qS_*\geq 8dy(t)-2dA\geq 8dy(t)-4dy(t)=4dy(t)\geq 2dA.
$$
Also, $y'(t)=\frac{A}{(1+t)^2}\leq 2dA$ for all
$t\geq -1+\frac{1}{\sqrt{2d}}$. Taking $t_0=\max(1, -1+\frac{1}{\sqrt{2d}})$, 
we obtain $0\leq y'(t)\leq 8dy(t)-4qS_*$ for all $t\geq t_0$.
Finally, define the function $g$ to be $g(t)=y(t+t_0)$ for all $t\geq 0$.  
\end{proof}


\begin{lemma}\label{l3.1}
Assume {\rm A1} and {\rm A3} hold. Then  \eqref{e2.1} has an almost periodic
solution,   $w(t,x)$,  in the sense of definition \ref{d2.1}.
\end{lemma}

\begin{proof}
 By  \cite[Theorem 22.3]{Hes91},  equation \eqref{e2.1} will have a
 stable periodic solution, which in turn will be almost periodic, 
if it has a properly ordered pair of strict sub- and super solutions.
Let
$$
\varphi(t,x)=g(t)(\frac{1}{2}-x)^2+S_*(-1-x),\quad 
\psi(t,x)=-g(t)(\frac{1}{2}-x)^2+S_*(1+x),
$$
for $t\geq 0$ , $0\leq x\leq 1$,
where $g$ is as given in Remark \ref{r3.1}. We will show that $\varphi$ 
and $\psi$ are properly ordered pair of strict sub- and super solutions of 
the equation \eqref{e2.1}.

$\bullet$ For $\varphi$: The scripts $\varphi_t$, $\varphi_x$, and 
$\varphi_{xx}$ represent the first and second partial derivatives of 
$\varphi$ with respect to $t$ and $x$, respectively. We have
\begin{gather*}
 \varphi_t(t,x)=g'(t)(\frac{1}{2}-x)^2\leq \frac{1}{4}g'(t),\\
 \varphi_x(t,x)=-2g(t)(\frac{1}{2}-x)-S_*, \\
 \varphi_{xx}(t,x)=2g(t).
\end{gather*}

(i) If $g'(t)\leq 8dg(t)-4qS_*$ as in Remark \ref{r3.1}, then 
$\frac{1}{4}g'(t)\leq 2dg(t)+2qg(t)(\frac{1}{2}-x)+qS_*$
and therefore, 
$$
\varphi_t(t,x)\leq d\varphi_{xx}-q\varphi_x.
$$

(ii) $d\varphi_x(t,0)-q\varphi(t,0)=-dg(t)-dS_*-\frac{q}{4}g(t)+qS_*
=-(d+\frac{q}{4})g(t)-\frac{q}{2}\underline{S}+\frac{q^2}{2d}\underline{S}$. 
Therefore, if $0\leq g(t)\leq A$ as in Remark \ref{r3.1}, then
$$
d\varphi_x(t,0)-q\varphi(t,0)\geq -\frac{q}{2}\underline{S}\geq -q S_{\rm in}(t).
$$

(iii) $\varphi_x(t,1)=g(t)-S_*<0$ since $g$ satisfies the conditions in 
Remark \ref{r3.1}. 

Combining (i), (ii), and (iii), it follows that $\varphi$ is a strict 
sub-solution of \eqref{e2.1}.

$\bullet$ For $\psi$. Similarly, the scripts $\psi_t$, $\psi_x$, and $\psi_{xx}$
 represent the first and second partial derivatives of $\psi$ with respect
 to $t$ and $x$, respectively. We have
\begin{gather*}
 \psi_t(t,x)=-g'(t)(\frac{1}{2}-x)^2\geq -\frac{1}{4}g'(t),\\
 \psi_x(t,x)=2g(t)(\frac{1}{2}-x)+S^*,\\
 \psi_{xx}(t,x)=-2g(t).
\end{gather*}

(i) If $g'(t)\leq 8dg(t)-4qS_*$ as in Remark \ref{r3.1}, then 
$-\frac{1}{4}g'(t)\geq -2dg(t)-2qg(t)(\frac{1}{2}-x)+qS^*$
and therefore, 
$$
\psi_t(t,x)\geq d\psi_{xx}-q\psi_x.
$$

(ii) $d\psi_x(t,0)-q\psi(t,0)=dg(t)-dS^*+\frac{q}{4}g(t)-qS^*
=(d+\frac{q}{4})g(t)-\frac{q}{2}\bar S-\frac{q^2}{2d}\bar S$. 
Therefore, if $0\leq g(t)\leq A$ as in Remark \ref{r3.1}, then
$$
d\psi_x(t,0)-q\psi(t,0)\leq -\frac{q}{2}\bar S\leq -q S_{\rm in}(t).
$$

(iii) $\psi_x(t,1)=-g(t)+S^*>0$ since $g$ satisfies the conditions in Remark 
\ref{r3.1}. 

Combining (i), (ii), and (iii), it follows that $\psi$ is a strict 
super-solution of  \eqref{e2.1}.

Now, let us show that $\varphi(t,x)\leq \psi(t,x)$  for all 
$t\geq 0$ and $0\leq x\leq 1$. Using Remark \ref{r3.1},
$$
\psi(t,x)-\varphi(t,x)=-2g(t)(\frac{1}{2}-x)^2 +(S^* + S_*)(1+x)
\geq (S^* + S_*)(1+x- (\frac{1}{2}-x)^2) \geq 0,
$$
for all  $t\geq 0$ and  $0\leq x\leq 1$. 

Then, $\varphi$ and $\psi$ are properly ordered pair of strict sub- and 
super solutions of \eqref{e2.1}. Therefore, by \cite[Theorem 22.3]{Hes91},
 the equation \eqref{e2.1} has a stable periodic solution, $w(t,x)$, 
which in turn is  almost periodic. 
\end{proof}

From assumption {\rm A2}, the function $f: \mathcal{C}\to Z$ is continuously 
differentiable. Also, the semigroup $T(t)$ is a $C_0$-semigroup on $Z$. 
Then, by the usual existence and regularity theorem
 (see \cite[Theorem 3,1]{Dra12}, \cite[Theorem 1]{Mar76}, 
\cite[Theorem 1.5, p. 187]{Paz83}), we have the following theorem.

\begin{theorem}\label{th3.2}
Assume {\rm A2} holds. For any $\theta\in\mathcal{C}$,  system \eqref{e2.3} 
(and equivalently \eqref{e1.1}-\eqref{e1.3}) has a unique mild solution 
$u(t)$ with initial condition $\theta$. Moreover, $u(t)$ is a classical 
solution of \eqref{e2.3} for all $t>0$. Also, if we denote 
by $\mathbb{T}(t)\theta=u(t,\theta)$ the solution of \eqref{e2.3},
 then $\mathbb{T}(t)$ is a nonlinear $C_0$-semigroup on $Z$.
\end{theorem}

\section{Main Result}

In this section, we prove our main result on the existence of almost periodic
 solutions of the system \eqref{e2.2} which is equivalent to the distributed 
parameters biochemical system \eqref{e1.1}-\eqref{e1.3}. Let us first 
prove the following lemma.

\begin{lemma}\label{l4.1}
Under assumption {\rm A2}, the solutions of  \eqref{e2.2} are bounded.
\end{lemma}

\begin{proof} 
Let $u(t)=(u_1(t), u_2(t))$ be a solution of \eqref{e2.2}. Let us start
 with the second component $u_2(t)$. In integral form, we have
$$
u_{2t}(.)=T_2(t)\theta_2(.)+\int_0^t T_2(t-s)f_2(s,u_s)ds.
$$
Therefore,
$$
|u_{2t}(.)|\leq |\theta_2(.)|+N\int_0^t e^{-\gamma(t-s)}|u_{2s}|ds,
$$
where $N$ is an upper bound of the function $\mu$. Recall that 
$$
(f_2(t,u))(x)=\mu(u_1(-r,x)+w(t,x),u_2(-r,x))u_2(-r,x).
$$
Applying the Gronwall inequality, we obtain
$$
|u_{2t}(.)|\leq |\theta_2(.)|\exp\Big(N\int_0^t e^{-\gamma(t-s)}ds\Big)
\leq |\theta_2(.)|\exp\big(\frac{N}{\gamma}\big).
$$
Therefore, $u_2$ is bounded in $Z_2$. Now, observe that 
$(f_1(t,u))(x)=-k\mu(u_1(0,x)+w(t,x),u_2(0,x))u_2(0,x)$. 
Since the function $\mu$ is bounded and we just proved that $u_2$ 
is also bounded, then $f_1(s,u_s)$ is also bounded. Using the integral form
 $$
u_1(t)=T_1(t)\theta_1(0)+\int_0^tT_1(t-s)f_1(s,u_s)ds
$$
of $u_1$, we deduce from \cite[p.236, Lemma 2.4]{Paz83} that $u_1$ has a 
compact closure in $Z_1$. Therefore, $u_1$ is bounded in $Z_1$. 
\end{proof}

Now, we can state and prove our main result.

\begin{theorem}\label{th4.1}
Assume {\rm A1--A3} hold. Then,  system \eqref{e2.2} has an almost periodic
 solution, $u=(u_1,u_2)$, in the sense of definition \ref{d2.1}.
\end{theorem}

\begin{proof} 
Observe that the function $f$ in the system \eqref{e2.2} is almost periodic 
in $t$, uniformly with respect to $u$, since the function $w(t,x)$ in 
the expression of $f$ is an almost periodic solution of \eqref{e2.1}.

Let $u(t)=(u_1(t), u_2(t))$ be a bounded solution of \eqref{e2.2} 
corresponding to the almost periodic solution $w(t,x)$ of \eqref{e2.1}, 
with initial condition $u(s)=(\theta_1(s), \theta_2(s))$, $r\leq s\leq 0$. 
In integral form, we have
\begin{gather*}
u_1(t)=T_1(t)\theta_1(0)+\int_0^tT_1(t-s)f_1(s,u_s)ds,\\
u_2(t)=T_2(t)\theta_2(0)+\int_0^tT_2(t-s)f_2(s,u_s)ds.
\end{gather*}
Since the semigroup $T_1(t)$ is compact and $f_1$ is bounded, then we can 
deduce from \cite[p. 236, Lemma 2.4]{Paz83} that $u_1(t)$ has a compact 
closure in $Z_1$. Therefore, for any sequence $u_1(t+\tau_n)$, we can 
extract a subsequence $u_1(t+r_n)$ that satisfies the Cauchy uniform 
convergence in $Z_1$.

Now, let us consider the component $u_2(t)$ of the solution. 
Let $u_2(t+\tau_n)$ be a sequence. Since $f_2(t,u)$ is almost periodic 
in $t$, uniformly in $u$, by Theorem \ref{th2.1}, we can extract
 a subsequence $f(t+r_n,u)$ which satisfies the Cauchy uniform convergence. 
In integral form, we have
$$
u_{2t}(.)=T_2(t)\theta_2(.)+\int_0^t T_2(t-s)f_2(s,u_s)ds.
$$
Let $i, j$ be positive integers. We have
\begin{align*}
|u_{2(t+i)}-u_{2(t+j)}|
&\leq |T_2(t+i)-T_2(t+j)||\theta_2(.)|+|\int_0^{t+i}T_2(t+i-s)f_2(s,u_s)ds \\
&\quad - \int_0^{t+j}T_2(t+j-s)f_2(s,u_s)ds|
\end{align*}
Introducing the change of variables in the integrals, we obtain:
\begin{align*}
&|u_{2(t+i)}-u_{2(t+j)}|\\
&\leq |e^{-\gamma(t+i)}-e^{-\gamma(t+j)}||\theta_2(.)|
 +\int_0^t T_2(t-s)|f_2(s+i, u_{s+i})-f_2(s+j,u_{s+j})|ds\\
&\quad+|-\int_{-i}^0 T_2(t-s)f_2(s+i, u_{s+i})ds 
 +\int_{-j}^0 T_2(t-s)f_2(s+j,u_{s+j})ds|\\
&\leq  |e^{-\gamma(t+i)}-e^{-\gamma(t+j)}||\theta_2(.)|
 +\int_0^t T_2(t-s)|f_2(s+i, u_{s+i})-f_2(s+i,u_{s+j})|ds\\
&\quad +\int_0^t T_2(t-s)|f_2(s+i, u_{s+j})-f_2(s+j,u_{s+j})|ds\\
&\quad +|-\int_{-i}^0 T_2(t-s)f_2(s+i, u_{s+i})ds 
 +\int_{-j}^0 T_2(t-s)f_2(s+j,u_{s+j})ds|\\
&\leq  |e^{-\gamma(t+i)}-e^{-\gamma(t+j)}||\theta_2(.)|
 +L\int_0^t e^{-\gamma(t-s)}|u_{1(s+i)})-u_{1(s+j)})|ds \\
&\quad + L\int_0^t e^{-\gamma(t-s)}|u_{2(s+i)})-u_{2(s+j)})|ds\\
&\quad +\int_0^t e^{-\gamma(t-s)}|f_2(s+i, u_{s+j})-f_2(s+j,u_{s+j})|ds\\
&\quad +|-\int_{-i}^0 T_2(t-s)f_2(s+i, u_{s+i})ds 
 +\int_{-j}^0 T_2(t-s)f_2(s+j,u_{s+j})ds|,
\end{align*}
where $L$ is the Lipschitz constant of of the function $f_2$.
Observe that
\begin{itemize}
\item[(i)] For any $\varepsilon>0$, there exists $l_1(\varepsilon)>0$, 
such that if $i,\,j\geq l_1(\varepsilon)$ then 
$|e^{-\gamma(t+i)}-e^{-\gamma(t+j)}|<\varepsilon$.

\item[(ii)] Since $u_1(t)$ has a compact closure, then: 
for any $\varepsilon>0$, there exists $l_2(\varepsilon)>0$, such that 
if $i,j\geq l_2(\varepsilon)$, then 
$|u_{1(t+i)}-u_{1(t+j)}|<\varepsilon$. Therefore,
    $$
    L\int_0^t e^{-\gamma(t-s)}|u_{1(s+i)}-u_{1(s+j)}|ds
\leq L\varepsilon \int_0^t e^{-\gamma(t-s)}ds
=\frac{L\varepsilon }{\gamma}(1-e^{-\gamma t})<\frac{L\varepsilon}{\gamma}.
    $$

\item[(iii)] By Theorem \ref{th2.1}:
for any $\varepsilon>0$, there exists $l_3(\varepsilon)>0$, such that if 
$i,j\geq l_3(\varepsilon)$ then
$$
|f_2(s+i,u_{s+j})-f_2(s+j, u_{s+j})|<\varepsilon.
$$ 
Therefore,
$$
\int_0^t e^{-\gamma(t-s)}|f_2(s+i,u_{s+j})-f_2(s+j,u_{s+j})|ds
\leq \varepsilon \int_0^t e^{-\gamma(t-s)}ds\leq \frac{\varepsilon}{\gamma}.
$$

\item[(iv)] Finally, for the last integral we have:
\begin{align*}
I&=|-\int_{-i}^0 T_2(t-s)f_2(s+i, u_{s+i})ds 
 +\int_{-j}^0 T_2(t-s)f_2(s+j,u_{s+j})ds| \\
&\leq \int_{-i}^0 T_2(t-s)|f_2(s+j, u_{s+j})-f_2(s+i,u_{s+i})|ds\\
&\quad  +\int_{-j}^{-i} T_2(t-s)|f_2(s+j,u_{s+j})|ds\\
&\leq \int_{-i}^0 T_2(t-s)|f_2(s+j, u_{s+j})-f_2(s+j,u_{s+i})|ds\\
&\quad + \int_{-i}^0 T_2(t-s)|f_2(s+j, u_{s+i})-f_2(s+i,u_{s+i})|ds 
 +N_2\int_{-j}^{-i} e^{-\gamma(t-s)}ds,
\end{align*}
where $N_2$ is an upper bound of the function $f_2$.
If we proceed as in $(i)$, $(ii)$ and $(iii)$, we obtain:
  for any $\varepsilon>0$, there exists $l_4(\varepsilon)>0$, such that 
if $i,j\geq l_4(\varepsilon)$ then
\begin{align*}
I&\leq \varepsilon L\int_{-i}^0 e^{-\gamma(t-s)}ds
 +\varepsilon \int_{-i}^0 e^{-\gamma(t-s)}ds 
 + \frac{N_2}{\gamma}(e^{-\gamma(t+i)}-e^{-\gamma(t+j)})\\
&\leq \frac{\varepsilon L}{\gamma} + \frac{\varepsilon}{\gamma} 
+\frac{\varepsilon N_2}{\gamma} 
=\varepsilon\Big(\frac{L+N_2 +1}{\gamma}\Big).
\end{align*}
\end{itemize}
Combining, (i), (ii), (iii) and (iv),  for 
$i,j\geq l(\varepsilon)=\min\big(l_1(\varepsilon),\,l_2(\varepsilon),
\,l_3(\varepsilon),\,l_4(\varepsilon)\big)$, we obtain
\begin{align*}
&|u_{2(t+i)}-u_{2(t+j)}|\\
&\leq \varepsilon+\frac{L\varepsilon}{\gamma}+\frac{\varepsilon}{\gamma}
 +\varepsilon\big(\frac{L+N_2 +1}{\gamma}\big)
 +L\int_0^t e^{-\gamma(t-s)}|u_{2(s+i)}-u_{2(s+j)}|ds\\
&\leq  \varepsilon\big(1+\frac{L}{\gamma}+\frac{1}{\gamma}
 +\frac{L+N_2 +1}{\gamma}\big)
 +L\int_0^t e^{-\gamma(t-s)}|u_{2(s+i)}-u_{2(s+j)}|ds.
\end{align*}
Applying the Gronwall inequality, we obtain
\begin{align*}
|u_{2(t+i)}-u_{2(t+j)}|
&\leq \varepsilon\big(\frac{2+\gamma + 2L+N_2}{\gamma}\big)
 \exp\Big(L\int_0^t e^{-\gamma(t-s)}ds\Big)\\
&\leq \varepsilon\big(\frac{2+\gamma + 2L+N_2}{\gamma}\big)
 \exp\Big(\frac{L}{\gamma}(1-e^{-\gamma t})\Big)\\
&\leq \varepsilon\big(\frac{2+\gamma + 2L+N_2}{\gamma}\big)
\exp\big(\frac{L}{\gamma}\big).
\end{align*}
Therefore, taking 
$\varepsilon'=\varepsilon\big(\frac{2+\gamma + 2L+N_2}{\gamma}\big)
e^{L/\gamma}$ and interchanging the roles of $\varepsilon$ and 
$\varepsilon'$, we obtain:
for any $\varepsilon>0$, there exists $l(\varepsilon)>0$, such that if 
$i,j\geq l(\varepsilon)$ then $|u_{2(t+i)}-u_{2(t+j)}|<\varepsilon$.
That is the component $u_2(t)$ of the solution is also  almost periodic. 
Hence, the system \eqref{e2.2} has an  almost periodic solution. 
\end{proof}

\subsection*{Final remarks}
This paper was devoted to the qualitative analysis of a distributed 
parameter biochemical systems with time delay and time varying input. 
The basis for the system studied here is derived from the work performed 
on anaerobic digestion in the pilot fixed bed reactor of 
LBE-INRA in Narbonne (France) and is mainly inspired from the dynamical 
models built and validated on the process (see \cite{Ber01,Sch04}). 
The growth function in \cite{Ber01}, \cite{Sch04} and subsequently in 
\cite{Dra08}, is expressed via the law:
\begin{equation}\label{e5.1}
\mu(S,X)=\mu_0\frac{S}{K_S X+S+\frac{1}{K_i}S^2}
\end{equation}
which clearly satisfies the assumption {\rm A2} of our present paper. 
In \eqref{e5.1}, the reaction is considered autocatalytic; i.e.,
the biomass(microorganism) is not only a product of the reaction, but 
also a catalyst of that reaction.
Therefore, although our present work is inspired from dynamical models
 built and validated on the process, our main result can be applied 
to many different situations with different models of the growth 
function (or reaction term).

\subsection*{Acknowledgments}
We are thankful to the anonymous referees for their thorough examination
 of the paper, making comments that substantially improved the manuscript.


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\end{document}