\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 18, 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 (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2007/18\hfil Travelling wavefronts for a bio-reactor equation]
{Existence and uniqueness of travelling
wavefronts for a bio-reactor equations with distributed delays}
\author[Z. Zhao, Y. Xu, Y. Li \hfil EJDE-2007/18\hfilneg]
{Zhihong Zhao, Yuantong Xu, Yongjin Li} % in alphabetical order
\address{Zhihong Zhao \newline
Department of Mathematics, Sun Yat-sen University, Guangzhou
510275, China} \email{zhaozhihong01@yahoo.com.cn}
\address{Yuantong Xu \newline
Department of Mathematics, Sun Yat-sen University, Guangzhou,
510275, China} \email{xyt@mail.sysu.edu.cn}
\address{Yongjin Li \newline
Department of Mathematics, Sun Yat-sen University, Guangzhou,
510275, China} \email{stslyj@mail.sysu.edu.cn}
\thanks{Submitted November 24, 2006. Published January 25, 2007.}
\thanks{Supported by grants 10471155 from NNSF of China
and 031608 from NSF of Guangdong}
\subjclass[2000]{34B18, 39A10}
\keywords{Bio-reactor model; travelling wavefronts; distributed
delay; \hfill\break\indent
heteroclinic orbit; singular perturbation theory; center manifold theorem}
\begin{abstract}
We consider the diffusive single species growth in a plug flow
reactor model with distributed delay. For small delay, existence
and uniqueness of such wavefronts are proved when the convolution
kernel assumes the strong generic delay kernel. The approaches
used in this paper are geometric singular perturbation theory and
the center manifold theorem.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{proposition}[theorem]{Proposition}
\section{Introduction}
There has been considerable interest recently in the system of
reaction-diffusion equations
\begin{equation}
\begin{gathered}
S_t=\varepsilon S_{xx}- \alpha S_x - f(S)u \\
u_t=u_{xx}-\alpha u_x + (f(S)-k)u,
\end{gathered}\label{e1.1}
\end{equation}
as a mathematical model to study some problems in biology and
chemical reaction. Most recently \eqref{e1.1} has been derived in
\cite{b1} to study a single population microbial growth for a
limiting nutrient in a flow reactor, where $\alpha>0$ is the flow
velocity, $S(x,t)$ and $u(x,t)$ are the concentrations of nutrient
and microbial population at position $x$ and time $t$, respectively.
We refer readers to \cite{b1,h2,s1} and the references therein for
further details of model description. To best describe this
phenomenon \cite{s1}, we consider an infinitely long flow reactor.
Suppose that the amount $S^0$ of nutrient is input at a constant
velocity $\alpha$ at one end of the flow reactor, says at $x =
-\infty$. On the other hand, assume that the nutrient uptake
function $f$ satisfies $f(0)=0$, $f'>0$ and $f(S^0)>k$ (see
\cite{s1}), where $k>0$ is the cell death rate. We naturally expect
that the nutrient should be sufficient for growth upstream of the
pulse and be depleted below the level at which bacteria can grow
downstream of the pulse. Hence one many expect that a hump-shaped
bacteria population density $u(x, t)$ moves towards the other end of
reactor. That is, we expect that there are constants $c$, $S_0$
with $f(S_0)0$ are constants, and suppose that
$f$ satisfies $f(0)=0, \ f'>0$ and $f(S^*)=k$ for some positive
number $S^*$. Then, given $S^0>S^*$ and there is a unique $S_0 \in
(0, S^*)$ such that \eqref{e1.1} has a travelling wavefronts $S(x+ct),
u(x+ct)$ satisfying the boundary condition \eqref{e1.2} if and only if
$c+\alpha \geq C^*:=\sqrt{4(f(S^0)-k)}$. Moreover, $S(z)$ is
strictly decreasing and $u(z)$ is strictly positive for
$z \in \mathbb{R}$.
\end{proposition}
The objective of the present paper is to address the question of the
existence and uniqueness of travelling wavefronts solution of the
following more general version of the system \eqref{e1.1} with
$\varepsilon = 0$,
\begin{equation}
\begin{gathered}
S_t=- \alpha S_x - f(S)(g*u) \\
u_t=u_{xx}-\alpha u_x + (f(S)-k)(g*u),
\end{gathered}\label{e1.5}
\end{equation}
where the convolution $g*u$ is defined by
\begin{equation}
(g*u)(x,t)=\int_{-\infty}^t g(t-s)u(x,s)ds \label{e1.6}
\end{equation}
and the kernel $g:[0,\infty) \to [0,\infty)$ satisfies
\begin{equation}
g(t)\geq 0,\quad\forall t\geq 0\quad\text{and}\quad
\int_0^{\infty}g(t)dt=1. \label{e1.7}
\end{equation}
The delay kernel $g$ are frequently of the form
\begin{equation}
g(t)=\delta(t-\tau),\quad
g(t)=\frac{1}{\tau}e^{-t/\tau},\quad
g(t)=\frac{t}{\tau^2}e^{-t/\tau}. \label{e1.8}
\end{equation}
In each of these cases, the parameter $\tau > 0$ measures the
delay. The first of these kernels gives rise to a model having a
discrete time-delay, where $\delta$ denotes Dirac's delta
function. The other two kernels in \eqref{e1.8} are called weak and
strong generic delay kernels. The ``weak'' case
$g(t)=\frac{1}{\tau}e^{-t/\tau}$ reflects the idea that the
importance of the past decreases exponentially the further one
looks into the past. The ``strong'' case
$g(t)=\frac{t}{\tau^2}e^{-t/\tau}$ can be regarded as a smoothed
out version of the discrete delay case $g(t)=\delta(t-\tau)$. This
strong kernel implies that a particular time in the past, namely
$\tau$ time units ago, is more important than any other since
kernel achieves its unique maximum at $t=\tau$.
The remaining part of this paper is organized as follows. Section
2 is devote to some preliminary discussion mainly focus on the
particular case of the kernel. In Section 3, we establish the
existence and uniqueness of travelling wavefronts solutions when
$\tau$ is small. Geometrical singular perturbation theory and the
center manifold theory play a major role in the proofs.
\section{preliminaries}
The purpose of this section is to establish propositions that will
serve main proof of the existence and uniqueness of the travelling
wavefront.
At first, the results on travelling fronts for the non-delay
equation is needed. \eqref{e1.3} with $\varepsilon=0$ can read as a
system of first-order equations
\begin{equation}
\begin{gathered}
u''= (c+\alpha)u' -(f(S)-k)u \\
S'= - f(S)u/(c+\alpha)
\end{gathered}\label{e2.1}
\end{equation}
which is equivalent to (see \cite{s1})
\begin{equation}
\begin{gathered}
u'= (c+\alpha)[-G(S^0)+u+G(S)] \\
S'= - f(S)u/(c+\alpha)
\end{gathered}\label{e2.2}
\end{equation}
where
\begin{equation}
\begin{gathered}
G(S)=S - k \int_{S^*}^S \frac{1}{f(s)} ds,\quad S>0.
\end{gathered}\label{e2.3}
\end{equation}
The function $G$ satisfies
\begin{equation}
G(0_+)=G(+\infty)=+\infty,\quad
G(S^*)=S^*,\quad
G'(S)=\frac{f(S)-k}{f(S)},\quad
G''(S)>0. \label{e2.4}
\end{equation}
The following lemma yields the existence of a travelling wavefront
solution of the non-delay equation \eqref{e2.2}.
\begin{lemma} [\cite{s1}]\label{lem2.1}
If $c+\alpha \geq C^*:=\sqrt{4(f(S^0)-k)}$, then in the $(S,u)$
phase plane, a heteroclinic connection exists between the critical
points $(S,u)=(S^0,0)$ and $(S_0,0)$ for $S^0>S^*$, $G(S_0)=G(S^0)$
and $S(\cdot)$ is strictly decreasing and $u(\cdot)$ is positive and
unimodal.
\end{lemma}
We return now to the delay equation \eqref{e1.5}. The travelling wavefronts
is a solution of the form $S(x,t)=S(z)$, $u(x,t)=u(z)$, where
$z=x+ct$ and $c>0$ is called wave speed, satisfies
\begin{equation}
\begin{gathered}
0=- (\alpha+c)S' - f(S)(g*u) \\
0=u''-(\alpha+c) u' + (f(S)-k)(g*u)
\end{gathered}\label{e2.5}
\end{equation}
with
\begin{equation}
(g*u)(z)=\int_0^{\infty} g(w)u(z-cw)dw. \label{e2.6}
\end{equation}
We shall seek leftward-moving waves, thus we take
\begin{equation}
S(-\infty)=S^0,\quad S(+\infty)=S_0,\quad
u(-\infty)=0,\quad u(+\infty)=0. \label{e2.7}
\end{equation}
Next, we shall analyze \eqref{e2.5} for travelling wavefronts in
the particular case when the kernel $g$ is the third of \eqref{e1.8}, the
strong generic delay case. The corresponding calculations for the
weak kernel are similar and will be omitted. Recall that the
parameter $\tau$ measures the delay. It is useful to reference
\eqref{e2.5} as
\begin{equation} \label{e2.8}
\begin{gathered}
u''= (c+\alpha)u' -(f(S)-k)(g*u) \\
S'= - \frac{f(S)}{(c+\alpha)}(g*u).
\end{gathered}
\end{equation}
Thus
$$
g(t)=\frac{t}{\tau^2}e^{-t/\tau},\quad \tau>0,
$$
and we define
$$
p(z)=(g*u)(z)=\int_0^\infty \frac{t}{\tau^2}e^{-t/\tau}u(z-ct)dt.
$$
Differentiating with respect to $z$, we obtain
$$
\frac{dp}{dz}=\frac{1}{c\tau}(p-q),
$$
where
$$
q(z)=\int_0^\infty \frac{1}{\tau}e^{-t/\tau}u(z-ct)dt.
$$
Similarly,
$$
\frac{dq}{dz}=\frac{1}{c\tau}(q-u).
$$
If we further denote $u'=v$, then \eqref{e2.8} with the kernel given
above can be replaced by the system
\begin{equation} \label{e2.9}
\begin{gathered}
u'= v \\
v'= (c+\alpha)v -(f(S)-k)p \\
S'= - \frac{f(S)}{(c+\alpha)}p\\
c \tau p'= p-q \\
c \tau q'= q-u,
\end{gathered}
\end{equation}
Note that if $\tau=0$, then \eqref{e2.9} reduces to
\begin{equation}
\begin{gathered}
u'= v \\
v'= (c+\alpha)v -(f(S)-k)u \\
S'= - f(S)u/(c+\alpha),
\end{gathered}\label{e2.10}
\end{equation}
the autonomous ordinary differential system for travelling wavefronts
solutions of \eqref{e1.5} in the non-delay case.
For $\tau>0$, \eqref{e2.9} defines a system of ODEs whose solutions
evolve in the five-dimensional $(u, v, S, p, q)$ phase space. In
this phase space, $E=\{(0,0,S,0,0)\}$ is the one-dimensional
manifold of critical for \eqref{e2.9}. A travelling wavefronts solution of
the \eqref{e2.8} will exist if among the solutions of \eqref{e2.9}, there
exists a heteroclinic connection between the two critical points in
$E$.
Then, we will show that \eqref{e2.9} has travelling wavefronts for
sufficiently small $\tau >0$ by the geometric singular perturbation
theory and the center manifold theorem. Note that when $\tau=0$,
system \eqref{e2.9} does not define a dynamical system in $\mathbb{R}^5$.
This problem may be overcome by the transformation $z=\tau \eta$,
under which the system becomes
\begin{equation} \label{e2.11}
\begin{gathered}
\dot u= \tau v \\
\dot v= \tau[(c+\alpha)v -(f(S)-k)p] \\
\dot S= \tau [-\frac{f(S)}{(c+\alpha)}p]\\
c \dot p= p-q \\
c \dot q= q-u, \\
\end{gathered}
\end{equation}
where a dot on top of a variable denotes differentiation with respect
to $\eta$. We refer to \eqref{e2.9} as the slow system and
\eqref{e2.11} as the fast system. The two
are equivalent when $\tau > 0$.
Consider the fast system \eqref{e2.11}, for $\tau =0$, then the flow
of that system is confined to the set
\begin{equation}
M_0=\{(u,v,S,p,q) \in \mathbb{R}^5 : p=u,\ q=u \},
\label{e2.12}
\end{equation}
which is, therefore, a three-dimensional invariant manifold for
\eqref{e2.9}. $E \subset M_0$ and an easy calculation shows that the
eigenvalues of the Jacobian, on setting $\tau = 0$, has 3 zero
eigenvalues corresponding the tangent space of $M_0$ and two same
positive eigenvalues, namely $\frac{1}{c}$. Thus, $M_0$ is normally
hyperbolic manifold.
According to Fenichel's Invariant Manifold Theorem
(see \cite{f1}, \cite {j1}),
there exist a locally invariant three-dimensional
manifold $M_{\tau}$ with $\tau$ is sufficiently small. It can be
written in the form
\begin{equation}
M_\tau = \big\{(u,v,S,p,q) \in \mathbb{R}^5 :
p=u+\widetilde{h}_1(u,v,S,\tau),\ q=u+\widetilde{h}_2(u,v,S,\tau)
\big\}, \label{e2.13}
\end{equation}
where the functions $\widetilde{h}_1$, $\widetilde{h}_2$ are smooth
functions defined on a compact domain, and satisfies
$\widetilde{h}_1(u,v,S,0)=\widetilde{h}_2(u,v,S,0)=0$ and thus that
\begin{equation}
\widetilde{h}_1(u,v,S,\tau)=\tau \overline{h}_1(u,v,S,\tau),\
\widetilde{h}_2(u,v,S,\tau)=\tau \overline{h}_2(u,v,S,\tau).
\label{e2.14}
\end{equation}
Since $\tau$ is small, $\widetilde{h}_1,\widetilde{h}_2$ can be
expanded into the form of Taylor series about $\tau$, and
$\overline{h}_1,\overline{h}_2$ can express as
\begin{equation}
\begin{gathered}
\overline{h}_1(u,v,S,\tau)=\overline{h}_1^1(u,v,S)+\tau
\overline{h}_1^2(u,v,S)+\dots,\\
\overline{h}_2(u,v,S,\tau)=\overline{h}_2^1(u,v,S)+\tau
\overline{h}_2^2(u,v,S)+\dots.
\end{gathered}\label{e2.15}
\end{equation}
By substituting \eqref{e2.14} into \eqref{e2.9}, we see that
$\overline{h}_1,\overline{h}_2$ must satisfy
$$
c(v+\tau(\frac{\partial \overline{h}_1}{\partial u}v
+\frac{\partial \overline{h}_1}
{\partial v}((c+\alpha)v -(f(S)-k)(u+\tau
\overline{h}_1))-\frac{\partial \overline{h}_1}{\partial
S}\frac{f(S)}{(c+\alpha)}(u+\tau \overline{h}_1)))=
\overline{h}_1-\overline{h}_2
$$
and
$$
c(v+\tau(\frac{\partial
\overline{h}_2}{\partial u}v+\frac{\partial \overline{h}_2}
{\partial v}((c+\alpha)v -(f(S)-k)(u+\tau
\overline{h}_1))-\frac{\partial \overline{h}_2}{\partial
S}\frac{f(S)}{(c+\alpha)}(u+\tau
\overline{h}_1)))=\overline{h}_2
$$
Substitute \eqref{e2.15} into the above
two equations and comparing powers of $\tau$ yields, we obtain
\begin{equation}
\begin{gathered}
\overline{h}_1(u,v,S,\tau)=2cv+3\tau c^2((c+\alpha)v
-(f(S)-k)u)+\dots\,,\\
\overline{h}_2(u,v,S,\tau)=cv+\tau c^2((c+\alpha)v
-(f(S)-k)u)+\dots \,.
\end{gathered}\label{e2.16}
\end{equation}
We study the flow of \eqref{e2.9} restricted to $M_\tau$ and show that
it has a travelling front solution.
The slow system \eqref{e2.9} restricted to $M_\tau$ is given by
\begin{equation}
\begin{gathered}
u'= v \\
v'= (c+\alpha)v -(f(S)-k)(u+\tau \overline{h}_1(u,v,S,\tau)) \\
S'= - \frac{f(S)}{c+\alpha}(u+\tau
\overline{h}_1(u,v,S,\tau)).
\end{gathered}\label{e2.17}
\end{equation}
which is equal to
\begin{equation}
\begin{gathered}
u'= v \\
v'= (c+\alpha)v -(f(S)-k)u+\tau h_1(u,v,S,\tau) \\
S'= - \frac{f(S)}{c+\alpha}u+\tau h_2(u,v,S,\tau),
\end{gathered}\label{e2.18}
\end{equation}
where $h_1(u,v,S,\tau)=-(f(S)-k)\overline{h}_1(u,v,S,\tau)$,
$h_2(u,v,S,\tau)=- \frac{f(S)}{c+\alpha}\overline{h}_1(u,v,S,\tau)$.
Note that when $\tau=0$, this system reduces to the corresponding
system for the non-delay \eqref{e2.10}. It is easily verified that for
$\tau>0$, system \eqref{e2.18} still has one-dimensional manifold of
critical $E=(0,0,S)$.
\section{main results}
In this section, we discuss the existence and uniqueness of
travelling wavefronts solutions of \eqref{e1.5}
The ideas of the following proof are similar to those of Smith and
Zhao \cite{s1} who were considering the question of persistence of
travelling wavefronts solutions in an equation with a fourth-order
spatial derivative but no time delay.
Note that system \eqref{e2.10} is equivalent to \eqref{e2.1}.
According to Lemma \ref{lem2.1} and \cite {s1}, for $00$, the
positive branch of the one-dimensional stable manifold of
$(0,0,S_0)$ for system \eqref{e2.10}, $W_0^s(S_0)$, connect to
$(0,0,S^0)$, where $G(S_0)=G(S^0)>S^*$. We want to show that for
$\tau>0$ but very small, the positive branch of one-dimensional
stable manifold of $(0,0,S_0)$ for system \eqref{e2.18},
$W_\tau ^s(S_0)$, also connects to $(0,0,S^0)$.
We may describe the local
stable manifold as the forward orbit $\{x_\tau (z): z \geq 0\}$ of
\eqref{e2.18} through a point $x_\tau :=x_\tau (0)$ on the local stable
manifold, which depends continuously on $\tau$, and by a compact
piece of the global stable manifold we mean
$\{x_\tau (z): z \geq -Z\}\ (Z \gg 1)$, with endpoint
$x_\tau (-Z)$. We expect that such
a compact piece of $W_\tau ^S(S_0)$ has endpoint nearby
$(0,0,S^0)$ for small $\tau > 0$. The next result indicates what
happens to the backward orbit through this endpoint.
\begin{lemma} \label{lem3.1}
Given $S^0>S^*$ and $\delta_0>0$, there exists $\tau_0$, $\delta >0$
such that if $\xi=(u,v,S)$ satisfies $|\xi - (0,0,S^0)|<\delta$ and
$0 \leq \tau <\tau_0$, then the solution of starting at $\xi$,
$x^\tau(z)=(u^\tau(z),v^\tau(z),S^\tau(z))$, satisfies
$|x^\tau(z)-(0,0,S^0)|<\delta_0$ for all $z<0$, and there exist
$\beta^\tau=(0,0,S^\tau)$ such that $x^\tau(z)\to
\beta^\tau$ as $z\to -\infty$.
\end{lemma}
\begin{proof}
Appending an equation for $\tau$ to \eqref{e2.18}, we shall
argument the system \eqref{e2.18} with equation for $\tau$.
\begin{equation}
\begin{gathered}
u'= v \\
v'= (c+\alpha)v -(f(S)-k)u+\tau h_1(u,v,S,\tau) \\
S'= - \frac{f(S)}{c+\alpha}u+\tau h_2(u,v,S,\tau)\\
\tau '= 0
\end{gathered}\label{e3.1}
\end{equation}
We apply the center manifold theory in \cite {c1} to the time
reversed system \eqref{e3.1}. Note that this four-dimensional system has
the two-dimensional manifold of critical given by $N=\{u=v=0\}$.
Focus on one of steady states $N^0=(0,0,S^0,0)$. A change of
variables $S\to S_1$ given by
\begin{equation}
S_1=S-S^0+\frac{f(S^0)}{r}(u-\frac{v}{c+\alpha}),\quad r=f(S^0)-k.
\label{e3.2}
\end{equation}
Translates $N^0$ to the origin and de-couples the linear parts of
the time reversed system \eqref{e3.1}. Then resulting system is
\begin{equation}
\begin{gathered}
u'= -v \\
v'= ru-(c+\alpha)v +(f(S)-f(S^0))u-\tau h_1(u,v,S,\tau) \\
S_1'=\frac{1}{(c+\alpha)r}(f(S)-f(S^0))u-\tau h_2(u,v,S,\tau)\\
\tau '= 0
\end{gathered}\label{e3.3}
\end{equation}
where $S$ is determined by \eqref{e3.2}. We let $x=(S_1, \tau)$ and
$y=(u,v)$, then \eqref{e3.3} has the form
\begin{equation}
\begin{gathered}
x'=Ax+f(x,y) \\
y'=By+g(x,y),
\end{gathered}
\end{equation}
where $A$ is the zero matrix and
$B = \begin{pmatrix}
0 & -1\\
r & -C \end{pmatrix}$
where $C=c+\alpha$, all the eigenvalues of $B$ have negative real
parts,
$$
f(x,y)=f(u,v,S_1,\tau)= \begin{pmatrix}
\frac{f(S)-f(S^0)}{Cr}u-\tau h_2(u,v,S,\tau)\\
0\end{pmatrix}
$$
and
$$
g(x,y)=g(u,v,S_1,\tau)=\begin{pmatrix}
0 \\
(f(S)-f(S^0))u-\tau h_1(u,v,S,\tau)) \end{pmatrix}.
$$
We have $f(0,0)=0$, $f'(0,0)=0$, and $g(0,0)=0$,
$g'(0,0)=0$. \cite[Theorem 1]{c1} asserts there exists a center
manifold for \eqref{e3.3}, but we already know that the center manifold
which is unique in our case, is just the manifold of critical $N$
(see \cite{s1}). The dynamical on $N$ is trivial:
\begin{equation} \label{e3.5}
\begin{gathered}
S_1'= 0 \\
\tau '= 0,
\end{gathered}
\end{equation}
Since critical point $(S_1, \tau)=(0,0)$ is stable for the dynamics
on $N$, from \cite[throem 2]{c1}, we get that the origin is stable for
\eqref{e3.3}. Furthermore, by the second assertion of \cite[Theorem 2]{c1},
a solution $(u(z),v(z),S_1(z),\tau)$ of \eqref{e3.3} which start $(0,0,S_1^0,
\tau)$ near the origin, such that as $z\to +\infty$,
$$
u(z)=O(e^{-\gamma z}),\quad v(z)=O(e^{-\gamma z}),\quad
S_1(z)=S_1^0+O(e^{-\gamma z}),
$$
where $\gamma>0$. Thus, we get $S(z)=S^0+S_1^0+O(e^{-\gamma z})$.
This is exactly what we assert above.
\end{proof}
Now we prove the main results in this section.
\begin{theorem}\label{thm3.2}
Let $S_0$ satisfy $0S^*$ satisfy
$G(S^0)=G(S_0)$. If $\tau>0$ is sufficiently small and $c+\alpha>0$
the system \eqref{e1.5} has a unique travelling wavefronts solution
$(S(z),u(z))(z=x+ct)$ connecting $(S^0,0)$ and $(S_0,0)$ with
$u(z)>0$ for $z\approx +\infty$.
\end{theorem}
\begin{proof}
For $00$ is as in
Lemma 3.1, then there exists $\tau_1>0$, such that for all $\tau
\in[0, \tau_1)$, a compact piece of $W_\tau ^S(S_0)$ has end point
with in distance $\delta$ of $(0,0,S^0)$. We can assume that
$\tau_1<\tau_0$ of Lemma \ref{lem3.1} and so, according to Lemma
\ref{lem3.1}, the backward continuation of the compact piece of
$W_\tau ^S(S_0)$ is asymptotic to a point $\beta^\tau=(0,0,S^\tau)$
satisfying $|\beta^\tau-(0,0,S^0)|<\delta_0$. Thus, we have shown
the existence of a heteroclinc orbit for \eqref{e2.18} connecting
$(0,0,S_0)$ to $\beta^\tau$. That is there exists a heteroclinc
orbit of \eqref{e2.9} connecting $(0,0,\widehat{S},0,0)$ to
$(0,0,S_0,0,0)$ where $\beta^\tau=(0,0,\widehat{S})$.
Next, we prove that $\widehat{S}=S^0$. As in \eqref{e1.4}, we have
\begin{equation}
(\alpha + c)(\widehat{S}-S_0)=k \int_{-\infty}^{+\infty}(g*
u)(z)dz=k \int_{-\infty}^{+\infty}p(z)dz.
\label{e3.6}
\end{equation}
From then third equation of \eqref{e2.9} we find that
$$
p=-\frac{(c+\alpha)S'}{f(S)},
$$
which, substituting into \eqref{e3.6} and integrating, lead to
\begin{equation}
G(\widehat{S})=G(S_0), \label{e3.7}
\end{equation}
where $G$ is defined by \eqref{e2.3}, by \eqref{e2.4} we get that
$\widehat{S}(\tau)=S^0$. Consequently, the heteroclinc orbit of
\eqref{e2.9} connecting $(0,0,S^0,0,0)$ to $(0,0,S_0,0,0)$
\end{proof}
\begin{remark} \label{rmk3.1} \rm
The travelling wave solution described in the Theorem \ref{thm3.2} depends on
$\tau$ and $c+\alpha$.
\end{remark}
Note that we make no assertions about the signs of $u$ and $S'$. In
the next theory, we take up these issues.
\begin{theorem}\label{thm3.3}
Let $S_0$ satisfies $0
C^*:=\sqrt{4(f(S^0)-k)}$. If $\tau>0$ is sufficiently small, then
the travelling wavefronts solution described in Theorem \ref{thm3.2},
$(S(x+ct),u(x+ct))$, has the property that $S(\cdot)$ is strictly
decreasing and $u(\cdot)$ is positive and unimodal.
\end{theorem}
\begin{proof}
For $\tau=0$, we can get \eqref{e2.10} is the same as \eqref{e2.1} and from
the second equation of \eqref{e2.1}, $u(z),u'(z)$ satisfies
$$
\frac{u'(z)}{u(z)}=-\frac{u'(z)}{S'(z)}\frac{f(S(z))}{c+\alpha}.
$$
Letting $z\to -\infty$, from \cite[Corollary 2.1]{c1}, the
ratio approaches
$$
2(\frac{r^0}{c+\alpha})[\frac{1}{1+(1-\chi)^\frac{1}{2}}]
<2\frac{r^0}{c+\alpha}<\frac{c+\alpha}{2},
$$
where $\chi = \frac {4(f(S^0) - k)} {(c + \alpha )^2}$ and we use
$\frac{(c+\alpha)^2}{4}>r^0=f(S^0)-k$ in the last inequality. If
$Z$ is sufficiently large, then $u(-Z),u'(-Z)>0$ and
$$
\frac{u'(-Z)}{u(-Z)}<\frac{c+\alpha}{2}.
$$
By continuity, for $\tau>0$ sufficiently small, we have that $u>0$
along the part of the heteroclinic orbit which lies outside the
small $\delta-$neighborhood of $(0,0,S^0)$ identified in
Lemma \ref{lem3.1}.
By choosing $Z$ larger if necessary, we can assume that
$u(z),v(z),S(z)$ belongs to the $\delta-$neighborhood of $(0,0,S^0)$
for $z<-Z$, that $u(z)>0$ for $-Z\leq z < \infty$, $v(-Z)>0$ and
that
\begin{equation}
\frac{v(-Z)}{u(-Z)}<\frac{c+\alpha}{2}. \label{e3.8}
\end{equation}
We wish to show that $u(z)>0$ for all $z$. Therefore, it is only
necessary to consider $(u(z),v(z),S(z))$ for $z \leq -Z$.
It is useful to reverse "time" by setting $z\to -z$, then we
consider the heteroclinic orbit for $(u(z),v(z),S(z))$ for $z \geq
Z$, which belongs to the $\delta-$neighborhood of $(0,0,S^0)$. Now
we replacing $(u,v)$ in \eqref{e2.18} by polar coordinates $(\rho,
\theta)$, then we get
\begin{equation}
\begin{gathered}
\rho^2 \theta '= -(c+\alpha)uv +ru^2 + v^2-\tau u h_1\\
\rho \rho'= -(1-r)uv-(c+\alpha)v^2 -\tau v h_1\\
S'=\frac{f(S)}{c+\alpha}u-\tau h_2,\\
\end{gathered}\label{e3.9}
\end{equation}
where $r=f(S)-k$ depend on $S(z)$. We are interested in \eqref{e3.9}
for $z \geq Z$ where $S(z)-S^0$ is so small that
$\frac{(c+\alpha)^2}{4}-r>0$. By \eqref{e3.8}, we see that $(u(Z),v(Z))$
belong to the open first quadrant and that
$$
0<\theta(Z)=\tan^{-1}(\frac{v(Z)}{u(Z)})<\theta_0
:=\tan^{-1}(\frac{c+\alpha}{2}).
$$
If $\theta(z)=0$(i.e., $v=0$), the first equation of \eqref{e3.9} become
$\rho^2 \theta '= ru^2 -\tau u h_1$ substituting \eqref{e3.8} into it and
we get $\rho^2 \theta '= [r-3\tau^2c^2r^2]u^2 +O(\tau^3)$, for
$\tau$ is sufficiently small, the sign of $\theta'(z)$ depend on
$r$, thus $\theta'(z)>0$ whenever $\theta(z)=0$. If
$\theta(z)=\theta_0$(i.e., $v=\frac{c+\alpha}{2}u$), the first
equation of \eqref{e3.9} become $\rho^2 \theta '=
-(\frac{(c+\alpha)^2}{4}-r)u^2 -\tau u h_1$ the same way we get the
sign of $\theta'(z)$ depend on $-(\frac{(c+\alpha)^2}{4}-r)$, thus
$\theta'(z)<0$ whenever $\theta(z)=\theta_0$. Thus, $0\leq
\theta(z)\leq \theta_0$ for all $z\geq Z$ and, in particular,
$u(z)>0$ for $z\geq Z$. Thus, $u(z)>0$ for all $z$.
For the third equation of \eqref{e2.18} substituting (2.16) into it and
we get $$S'= - f(S)u/(c+\alpha)-\tau vf(S) +O(\tau^2)=-
f(S)(\frac{u}{c+\alpha}+\tau v)+O(\tau^2)$$ Since $u>0$, $\tau$ is
sufficiently small, we have $S'<0$ for all $z \in \mathbb{R}$.
From \cite{c1}, we can get \eqref{e2.9} is equals to
\begin{equation}
\begin{gathered}
u'= (c+\alpha)[-G(S^0)+u+G(S)] \\
S'= -\frac{f(S)}{c+\alpha} p\\
c \tau p'= p-q \\
c \tau q'= q-u
\end{gathered}\label{e3.10}
\end{equation}
where $G$ is defined by \eqref{e2.3}. It has two critical points
$(0,S^0,0,0)$ and $(0,S_0,0,0)$. The linearized matrix $J$ of
system \eqref{e3.10} is
\begin{equation}
J(u,S,p,q)=\begin{pmatrix}
c+\alpha & -(c+\alpha)G'(S) & 0 & 0 \\
0 & \frac{f'(S)}{c+\alpha}p & -\frac{f(S)}{c+\alpha} & 0 \\
0 & 0 & \frac{1}{c\tau} & -\frac{1}{c\tau} \\
-\frac{1}{c\tau} & 0 & 0 & \frac{1}{c\tau}
\end{pmatrix}\label{e3.11}
\end{equation}
The eigenvalues $\lambda$ of this matrix at critical points satisfy
$$
(c\tau)^2\lambda^4-((c\tau)^2(c+\alpha)+2c\tau)\lambda^3
+(1+2c\tau(c+\alpha))\lambda^2-(c+\alpha)\lambda-(f(S)-k)=0
$$
At critical point $(0,S^0,0,0)$, for $S^0>S^*$, sufficiently small
$\tau$ and $c>0$, this equation has four positive real part. At
critical point $(0,S_0,0,0)$, for $S_0~~0
$$
Since $S'<0$, then $v=u'$ is negative when $z$ is very close to
$+\infty$. As $u(\pm\infty)=0$, $v=u'$ admits at least one zero. By
the first equation of \eqref{e3.10} and \eqref{e2.17}, it follows that
\begin{equation}
\begin{aligned}
v'&=u''\\
&=(c+\alpha)G'(S)S'=(k-f(S))p\\
&=(k-f(S))u+O(\tau^2)\quad \text{whenever } v(z)=0.
\end{aligned}\label{e3.12}
\end{equation}
Let $z_0$ be the largest zero of $v$. Then $v'(z_0)\leq 0$ and
$v<0$, hold for any $z>z_0$. Suppose $v'(z_0)=0$, for $\tau$ is
sufficiently small, then \eqref{e3.12} implies that
$v''(z_0)=-f'(S)S'(z_0)u(z_0)+O(\tau^2)>0$, hence $v(z_0)=0$ is the
local minimum of $v(z)$ around $z_0$, which contradicts the choice
of $z_0$. Hence, we get $v'(z_0)< 0$ i.e., $S(z_0)> S^*$. Since
$S'(z)<0$, for all $z \in \mathbb{R}$, we get $S(z)>S^*$ hold for
any $z\in (-\infty, z_0)$, hence $v'<0$ hold for any $z\in (-\infty,
z_0)$. Thus, $v(z)$ admits no zero in $(-\infty, z_0)$ and $v>0$. So
$v=u'$ has precisely one zero $z_0$ and $v(z)>0$ for all
$z \in (-\infty, z_0)$ and $v(z)<0$ for all $z \in (z_0, \infty)$. Hence
$u(z)$ is positive and unimodal.
\end{proof}
\begin{remark} \label{rmk3.2} \rm
We have considered travelling wavefronts of a plug flow reactor model
\eqref{e1.1} with $\varepsilon=0$, and with distributed delay in the form
of an integral convolution in time, mainly using strong generic
kernel. It should certainly be applicable in principle to \eqref{e1.1} with
$\varepsilon>0$ involving time delay
\begin{equation}
\begin{gathered}
S_t=\varepsilon S_{xx}- \alpha S_x - f(S)(g*u) \\
u_t=u_{xx}-\alpha u_x + (f(S)-k)(g*u)
\end{gathered}\label{e3.13}
\end{equation}
and other coupled system.
\end{remark}
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