\documentclass[reqno]{amsart}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2002(2002), No. 84, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
\thanks{\copyright 2002 Southwest Texas State University.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE--2002/84\hfil
Constructing universal pattern formation processes]
{Constructing universal pattern formation processes
governed by reaction-diffusion systems}
\author[Sen-Zhong Huang\hfil EJDE--2002/84\hfilneg]
{Sen-Zhong Huang}
\address{Sen-Zhong Huang\hfill\break
Fachbereich Mathematik, Universit\"at Rostock,\hfill\break
Universit\"atsplatz 1, D-18055 Rostock, F. R. Germany}
\email{sen-zhong.huang@mathematik.uni-rostock.de}
\date{}
\thanks{Submitted August 28, 2002. Published October 4, 2002.}
\subjclass[2000]{35B40, 70G60, 35Q99}
\keywords{Attractor, pattern formation}
\begin{abstract}
For a given connected compact subset $K$ in
$\mathbb{R}^n$ we construct a smooth map $F$ on
$\mathbb{R}^{1+n}$ in such a way that the corresponding
reaction-diffusion system
$u_t=D\Delta u+F(u)$ of $n+1$
components $u=(u_0,u_1,\dots ,u_n)$, accompanying with the
homogeneous Neumann boundary condition, has an attractor
which is isomorphic to $K$. This implies the following
universality: The make-up of a pattern with arbitrary complexity
(e.g., a fractal pattern) can be realized by a reaction-diffusion
system once the vector supply term $F$ has been previously
properly constructed.
\end{abstract}
\maketitle
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\section{Introduction}
Reaction-diffusion systems of the form
$$
u_t=D\Delta u+F(u),\quad x\in\Omega,t>0,
$$
model the diffusion through a domain $\Omega\subset\mathbb{R}^k$
of $m$ interacting species or chemicals, where the $i$-th
component $u_i$ of $u=(u_1,\dots ,u_m)$ represents the density or
concentration of the $i$-th reactants and $D=(d_1,\dots ,d_m)$ is
the matrix of the diffusion constants $d_i>0$
(cf. \cite{Mu77,Mu89,CH93,S83,T88}). One hard
mathematical task for the practical use of these models is to find
appropriate vector supply terms $F$ in such a way that the pattern
formation process governed by the corresponding reaction-diffusion system
coincides with the phenomenon observed in the laboratory
experiments or in nature. Famous examples in this direction
include the Kolmogorov-Fischer equations modeling the two-species
interactions, the Field-Noyes equations modeling the
Belousov-Zhabotinsky reactions in chemical kinetics, the
Hodgkin-Huxley equations or the FitzHugh-Nagumo equations modeling
the nerve impulse transmission
(cf. \cite{Mu77,Mu89,CH93,S83, T88}), and the Murray equations
(\cite{Mu81a}, \cite{Mu81b}) modeling the mammalian coat markings.
However, as commented by J.D. Murray \cite[p.238]{Mu89}, ``for
most practical models of real world situations it is premature, to
say the least, to spend too much time on sophistical
generalisations before the simpler versions have been shown to be
inadequate when compared with experiments," and, ``the
generalisations seem endless."
Murray's comment raises implicitly a very crucial theoretical problem,
namely, the following \textit{universality problem}:
Can the reaction-diffusion systems model each phenomenon of
\textit{arbitrary complexity}?
Because the pattern formation process is the main subject
in question, we can pose the above universality
problem in a more restrictive way, i.e.,
we ask if the complexity of patterns modeled by
the reaction-diffusion systems can be allowed to be arbitrary.
A pattern is the eventual result of a time evolution of a biological
or chemical or physical process and thus has the following two
main features: a) Long-time effect -- it describes the
long-time behavior of that process, and
b) Great randomness of the initial conditions -- the
initial conditions which lead to the same pattern
have a great randomness due to the circumstance fluctuation.
Based on this observation, we see that a pattern is a kind of
attractor. Here, by an attractor for a reaction-diffusion system we mean
the mathematical object which attracts an open set of initial
data in such a way that the trajectories starting from this initial
data set eventually end up on the attractor in question
(this is just the long-time effect of an attractor).
The openness of the set of initial data guarantees the required
great randomness of initial data which lead to the same pattern
(attractor). Moreover, this openness corresponds to the practical
need (e.g., for the computer simulation) that there is a positive
probability that computed trajectories will tend to the attractor.
Using the concept of attractors, the above universality problem
can be mathematically more exactly reformulated as follows:
{\it Can the complexity of attractors for the reaction-diffusion systems
be allowed to be arbitrary?}
In this article we will tackle the ``universality
problem'' mathematically posed in this way.
What we will do is to give a method of constructing
the vector supply term $F$ for the purpose that the corresponding
reaction-diffusion system has an attractor whose complexity is allowed to be
arbitrary in some sense.
In the following statement, $\Omega$ is any given smooth bounded
domain in $\mathbb{R}^k$. Moreover, for every fixed $n\in\mathbb{N}$ we
identify the Euclidean space $\mathbb{R}^n$ as a subspace of the
product Banach space $C(\bar{\Omega})^n$ by identifying a point
$(a_1,\dots ,a_n)\in\mathbb{R}^n$ with the vector of constant
functions $(a_1{\bf 1},\dots ,a_n{\bf 1})\in C(\bar{\Omega})^n$.
We recall the notion {\it $\omega$-limit set} $\omega(\psi,E)$
which is used to describe the asymptotic behavior of a map
$\psi:[0,\infty)\to E$ in a Banach space $E$ and is defined as the
set of all points $p\in E$ for which there exists a sequence
$(t_k)_{k\in\mathbb{N}}$ (probably depending on the points $p$) such that
$t_k\to\infty$ and $\Vert\psi(t_k)-p\Vert\to 0$ as $k\to\infty$.
For our present situation, the notion $\omega(u,C)$ for a map
$u:[0,\infty)\to C(\bar{\Omega})^{1+n}$ is used to denote the
$\omega-$limit set of $u$ which is calculated in the Banach space
$C(\bar{\Omega})^{1+n}$. Our main result reads as follows.
\begin{theorem} \label{thm1.1}
Let $n\in\mathbb{N}$ and $K\subset\mathbb{R}^n$ a given connected compact
subset of arbitrary complexity. Then there exists a vector supply
term $F_K$ of the form
$$
F_K(\phi,v)=(A(\phi),A'(\phi)v+f(\phi))
\quad for\quad\phi\in\mathbb{R},v\in\mathbb{R}^n,
\eqno(1.1a)
$$
where $A$ is a smooth function on $\mathbb{R}$ and $f$ a smooth map on
$\mathbb{R}^n$, such that the corresponding reaction-diffusion system
$$
\begin{gathered}
u_t=D\Delta u+F_K(u),\quad x\in\Omega,t>0,\\
\frac{du}{dn}=0\quad\mbox{on}\quad\partial\Omega
\end{gathered}
\eqno(1.1b)
$$
of $n+1$ components $u=(\phi,v)$, accompanying with the zero flux
boundary condition, generates a dynamical system in the state
space $C(\bar{\Omega})^{1+n}$ with the following properties
(i)-(ii):\\
(i) For each initial value $u_0\in C(\bar{\Omega})^{1+n}$ the
reaction-diffusion system (1.1a-b) has a global unique solution
$u$, $u(0)=u_0$, such that $u$ is continuous in
$\bar{\Omega}\times [0,\infty)$, $u_t,\Delta u$ as well as all partial
derivatives $\frac{\partial u}{\partial x_i}$ are continuous in
$\bar{\Omega}\times (0,\infty)$. \\
(ii) Each solution $u$ of the reaction-diffusion system (1.1a-b)
starting from an initial value $u_0=(\phi_0,v_0)\in
C(\bar{\Omega})^{1+n}$ such that either $\phi_0>0$ or $\phi_0<0$
is asymptotically stable and converges to the set
$\tilde{K}:=\{0\}\times K$ in the sense that
$\omega(u,C)=\tilde{K}$. As result, the connected compact set
$\tilde{K}$ is an attractor for the reaction-diffusion system (1.1a-b)
settled in the state space $C(\bar{\Omega})^{1+n}$.
\end{theorem}
Here we choose the zero flux boundary condition (i.e., the
homogeneous Neumann boundary condition) and the positive initial
condition, since they are probably the most interested boundary
and initial conditions in the biological or chemical situation.
Namely, the zero flux boundary condition reflects the
self-organization mechanism of pattern while the positive initial
condition restricts the pattern formation process to such a
beginning circumstance that each of the reactants has a positive
distribution all over the reaction domain. Thus, our Theorem 1.1
should be considered as a partial but affirmative answer to the
universality problem stated above and implies that any pattern
(here $\tilde{K}$) which is isomorphic to a connected compact
subset (here $K$) of the Euclidean space $\mathbb{R}^n$ can be
seen as the final result of the pattern formation process governed
by some appropriate reaction-diffusion system of $n+1$ components.
Moreover, it implies the following universality: The make-up of a pattern
$\tilde{K}\cong K\subset\mathbb{R}^n$ with arbitrary complexity
(e.g., a fractal pattern \cite{Ma83}) can be realized by a reaction-diffusion
system of the form (1.1a-b) once the vector supply term $F_K$ has
been previously properly constructed.
Although the above construction is the product of
theoretic thoughts, we are also interested in whether
it is possible to derive such reaction-diffusion systems from any
sequence of reasonable biochemical or physical situations.
To have a close look of the mechanism of the system (1.1a-b),
we write $D=\mbox{diag}(d_0,\tilde{D})$ and
then decompose the reaction-diffusion system (1.1a-b) into the
reaction-diffusion equation for the first component $\phi$
$$
\begin{gathered}
\phi_t=d_0\Delta\phi+A(\phi),\quad x\in\Omega,t>0,\\
\frac{d\phi}{dn}=0\quad\mbox{on}\quad\partial\Omega
\end{gathered} \eqno(1.2a)
$$
and the linear inhomogeneous reaction-diffusion system for the rest $n$
components $v$
$$
\begin{gathered}
v_t=\tilde{D}\Delta v+A'(\phi)v+f(\phi),
\quad x\in\Omega,t>0,\\
\frac{dv}{dn}=0\quad\mbox{on}\quad\partial\Omega.
\end{gathered} \eqno(1.2b)
$$
In this way, it becomes clear that the first component
$\phi$ of the vector $u=(\phi,v)$ works as the
\textit{activator} and the rest components $v$ as the
\textit{inhibitor} of the reaction-diffusion system (1.1a-b). It is
worthy noting that the activator system (1.2a) is
self-contained and the inhibitor system (1.2b) is
an inhomogeneous nonautonomous {\it linear} system
with an external force $f(\phi)$ and an internal
friction force $A'(\phi)v$ supplied by the activator.
This mechanism simplicity throws light on the
possibility of deriving reaction-diffusion systems of the form (1.1a-b)
from real world situations.
To close this introduction, we give a remark echoing with the
referee's comments on the previous version of the present article.
Starting from the pioneering work of A.M. Turing \cite{Tu52},
there is a continuous interest in the mathematical modelings by
the so-called ``parameter space method'', i.e., one looks for a
vector supply term $F$ of the form $F(u)=G(u,\alpha)$ which
depends on a set of parameters $\alpha$, and the choice of the
form of such vector supply terms is in general based on the
following philosophy: the form should be as simple as possible.
Therefore, the usual candidate for such forms is the class of {\it
elementary functions} like polynomials or rational functions. It
is just very surprising that with such simple choice one has had a
very deep look at the secrets of the developmental ground plan of
the nature. A lot of such examples are prepared in the book
\cite{Mu89} of J.D. Murray and in the article \cite{CH93} of M.C.
Cross and P.C. Hohenberg. One of them is the very fascinating ones
posed by J.D. Murray \cite{Mu81b} for modeling the mammalian coat
markings. Murray's model is a reaction-diffusion system of the form
\[
v_t=\Delta v+\gamma [a-v-g(v,w)],\quad
w_t=\beta\Delta w+\gamma [\alpha (b-w)-g(v,w)],
\]
with $g(v,w)=\rho vw/(1+v+\kappa v^2)$ and seven parameters
$a,b,\alpha,\beta,\gamma,\rho$ and $\kappa$. Murray's work is
based on a previous model experimentally posed by D. Thomas
\cite{Th75}. However, the fascinating aspects of Thomas' model, as
shown by Murray, is certainly unexpected. In connecting to our
work presented in this article, one would like to know if the
elementary functions choice of the vector supply terms $F$ in the
reaction-diffusion systems is adequate for the common purposes.
The author is very indebted to the anonymous referee for
her/his valuable discussions.
\section{Constructive proof of Theorem 1.1}
This is divided into two main steps: the construction of the
smooth function $A$ as well as the smooth map $f$ in
the nonlinearity $F_K(\phi,v)=(A(\phi),A'(\phi)v+f(\phi))$
and the analysis of the resulted reaction-diffusion system (1.1b). \smallskip
\noindent{\bf 2A. Tools}\quad First we recall two construction results from
\cite{Hu00}. Starting with a Banach space $E$, we define the {\it
$\omega-$limit set} $\omega(\psi,E)$ (resp., {\it $\alpha-$limit
set} $\alpha(\psi,E)$) of a continuous path $\psi:(a,b)\to E$ to
be the set consisting of each point $p\in E$ for which there
exists a sequence $(s_k)_{k\geq 1}\subset (a,b)$ such that
$\Vert\psi(s_k)-p\Vert\to 0$ as $k\to\infty$ and $s_k\to b$ as
$k\to\infty$ (resp., $s_k\to a$ as $k\to\infty$). Equivalently, we
have the following more compact representations
\[
\omega(\psi,E)=\bigcap_{k=2}^\infty\overline{\psi([b-(b-a)/k,b))},
\quad
\alpha(\psi,E)=\bigcap_{k=2}^\infty\overline{\psi(a,a+(b-a)/k]}
\]
Here the closure is taken in $E$. These two limit sets describe
the behavior of the path $\psi$ near the endpoints $a$ and $b$. In
general, we do not assume that $\psi(s)$ has a limit as $s$
approaching to endpoints; namely, a path $\psi$ which serves to
satisfy our following requirements should be very irregular near
endpoints.
It is not difficult to establish the following
two useful properties of limit sets:
Each $\omega-$limit (resp., $\alpha-$limit) set
is closed and connected once it is not empty. Recall that a closed subset
in a Banach space is said {\it connected} if this set can not be
decomposed into the union of two disjoint closed subsets. Thus,
connectedness is sometimes refered to as ``indecomposibility''.
\begin{lemma} \label{lm2.1}
Let $(E,\Vert\cdot\Vert)$ be a given Banach space.\\
{\bf (i)} For each connected compact subset $K$ in $E$ there
exists a path $\psi: (0,1]\to E$ of class $C^\infty$ such that
$\alpha(\psi,E)=K$.\\
{\bf (ii)} For every $C^\infty$ path
$\varphi:(0,1]\to E$ there exists a $C^\infty$ function
$A:\mathbb{R}\to\mathbb{R}$ satisfying the conditions (c1)-(c2) below:
\\
{\bf (c1)}\quad $A$ is an odd function which is strictly negative
and concave in the interval $(0,\infty)$, i.e., there holds
$-A(-s)=A(s)<0$ and $A''(s)<0$ for all $s>0$. Moreover,
$A^{(m)}(0)=0$ for all $m\in\{0\}\cup\mathbb{N}$.
\\
{\bf (c2)}\quad The function $A$ smooths the given path $\varphi$
in such a way that
\[
\lim_{s\to 0} |A(s)^{(m)}|\cdot\Vert\varphi(s)^{(l)}\Vert=0
\quad\forall m,l\in\{0\}\cup\mathbb{N}.
\]
\end{lemma}
The topological assertion in (i) has just been proved in
\cite[Lemma 1]{Hu00} (a sketch proof can be also found in the
Appendix of \cite{Hu01}). It is worthy noting that this assertion
implies the following fact: A compact subset $K$ of a Banach space
$E$ is connected if and only if it is the $\alpha-$limit set of
some smooth path in $E$. The assertion in Lemma 2.1-(ii) is
very similar to the ones in \cite[Lemma 2]{Hu00}. However, the
concavity and the convexity of the function $A$ here is new. For
the sake of completeness, below we give a sketch of its proof.
\noindent{\bf Sketch proof of Lemma 2.1-(ii):}\quad
We begin with $k\in\{0\}\cup\mathbb{N}$ and put
\[
\rho_k(s):=\max_{s \leq r\leq 1}\Vert\varphi^{(k)}(r)\Vert
\quad (s\in (0,1]).
\]
Each $\rho_k$ is nonincreasing and continuous. Note that for each
$s\in (0,1]$ there are at most finitely many $k\in\{0\}\cup\mathbb{N}$
such that $k<1/s$. Therefore, the following
\[
\tilde{\rho}(s):=\sum_{\{k<1/s\}}\rho_k(s)
\quad (0~~1$,
we obtain a nondecreasing continuous function $\rho$. Define
\[
g(s):=\int_0^s \beta(s-r)\rho(r)\, dr
\quad (s\geq 0)
\]
where the function $\beta:[0,\infty)\to\mathbb{R}$ is given by
$\beta(r):=r\exp(-1/r)$ for $r>0$ and $\beta(0):=0$. An elementary
computation yields that $\beta^{(m)}(0)=0$ for all $m\in\mathbb{N}$,
$\beta''(r)= r^{-4}\beta(r)$ for all $r>0$. This implies that the
function $g$ is an increasing and convex function of class
$C^\infty$ and
$$
g^{(m)}(s)=\int_0^s\beta^{(m)}(s-r)\rho(r)\, dr
\eqno(2.1)
$$
for all $m\geq 1$ and all $s\geq 0$. Finally, the odd function $A$
is defined by $A(s):=-g(s)$ for $s\leq 0$ and $A(s):=-g(-s)$ for
$s<0$. The desired properties in (c1) follow directly from
the related ones of the function $g$. To see the smoothing
property in (c2), we fix $l,m\in\mathbb{N}\cup\{0\}$ and consider
$s\in (0,1)$ such that $s<1/(1+l)$. On one hand, we have by
definition that
$\tilde{\rho}(s)\geq\rho_l(s)\geq\Vert\varphi^{(l)}(s)\Vert$ and
thus
$$
\rho(s)=s/(1+\tilde{\rho}(s))\leq s/(1+\Vert\varphi^{(l)}(s)\Vert).
\eqno(2.2a)
$$
On the other hand, we use (2.1) combining with
the monotonicity of the function $\rho$ to find that
$$
|A^{(m)}(s)|=|g^{(m)}(s)|\leq c_m\cdot\rho(s)
\eqno(2.2b)
$$
with $c_m:=\int_0^1|\beta^{(m)}(r)|\, dr$. Taking into account the
estimates in (2.2a-b), we see that
\[
|A^{(m)}(s)|\cdot\Vert\varphi^{(l)}(s)\Vert\leq c_m\cdot s
\]
for all $s\in (0,1)$ being such that $s<1/(1+l);$ yielding the
convergence $|A^{(m)}(s)|\cdot\Vert\varphi^{(l)}(s)\Vert\to 0$ as
$s\to 0$. \smallskip
\noindent {\bf 2B. Construction of the vector
supply term}\quad From now on $K\subset\mathbb{R}^n$ is a given
connected compact subset of arbitrary complexity.
As the first step of the construction we will apply Lemma
2.1 under the choice $E=\mathbb{R}^n\subset C(\bar{\Omega})^n$ to
find the function $A$ and the map $f$. Here, as before, we have
identified $\mathbb{R}^n$ with the subspace of constant functions
in the product Banach space $C(\bar{\Omega})^n$. Starting with the
set $K$ we first apply Lemma 2.1-(i) to obtain a $C^\infty$
path $\tilde{\psi}:(0,1]\to\mathbb{R}^n$ such that
$\alpha(\tilde{\psi},\mathbb{R}^n)=K$. We then apply Lemma
2.1-(ii) under the choice of the path $\varphi=\tilde{\psi}$ to
obtain an odd $C^\infty$ function $A:\mathbb{R}\to\mathbb{R}$ satisfying the
conditions (c1)-(c2) there.
To construct the smooth map $f$, we need extend the path
$\tilde{\psi}$ to a smooth path $\psi$ defined on $(0,\infty)$. To
this end, we choose a $C^\infty$ function $\gamma$ on $\mathbb{R}$ such
that $\gamma(s)=1$ for all $s<1/4$ and $\gamma(s)=0$ for all
$s>1/2$. The method of constructing such functions is well-known,
see e.g. \cite[p.42]{Hi91}. We set
$\psi(s):=\gamma(s)\tilde{\psi}(s)$ for $s\in (0,1]$ and
$\psi(s)={\bf 0}$ for all $s>1$. Then
$\psi:(0,\infty)\to\mathbb{R}^n$ is a smooth path such that
$$
\alpha(\psi,\mathbb{R}^n)=\alpha(\tilde{\psi},\mathbb{R}^n)=K.
\eqno(2.3a)
$$
We define the map $f:\mathbb{R}\to\mathbb{R}^n$ by
$$
f(\phi):=\begin{cases}
A(\phi)\psi'(\phi)-A'(\phi)\psi(\phi)
&\mbox{for } \phi>0,\\
0 \quad &\mbox{for } \phi=0,\\
A(-\phi)\psi'(-\phi)-A'(-\phi)\psi(-\phi)
&\mbox{for } \phi<0.
\end{cases} \eqno(2.3b)
$$
Clearly, $f$ vanishes outside the compact interval $[-1/4,1/4]$.
Moreover, $f$ is an even map, i.e., $f(-\phi)=f(\phi)$ for all
$\phi\in\mathbb{R}$. With $A,f$ constructed as above, we take the smooth
map
$$
F_K(u):=(A(\phi),A'(\phi)v+f(\phi))
\quad \mbox{for}\quad
u=(\phi,v)\in\mathbb{R}\times\mathbb{R}^n
\eqno(2.4a)
$$
as the vector supply term, and thus have a reaction-diffusion system
$$
\begin{gathered}
u_t=D\Delta u+F_K(u)\quad x\in\Omega,t>0\\
\frac{du}{dn}=0\quad\mbox{on}\quad\partial\Omega
\end{gathered} \eqno(2.4b)
$$
accompanying with the zero flux boundary condition.
As observed in the Introduction, with
$D=\mbox{diag}(d_0,\tilde{D})$ the reaction-diffusion system (2.4a-b) decomposes
into the reaction-diffusion equation for the {\it activator} consisting of
the first component $\phi:$
$$
\begin{gathered}
\phi_t=d_0\Delta\phi+A(\phi),\quad x\in\Omega,t>0,\\
\frac{d\phi}{dn}=0\quad\mbox{on}\quad\partial\Omega
\end{gathered} \eqno(2.5a)
$$
and the linear inhomogeneous reaction-diffusion system for the {\it inhibitor}
consisting of the rest $n$ components $v:$
$$
\begin{gathered}
v_t=\tilde{D}\Delta v+A'(\phi)v+f(\phi),
\quad x\in\Omega,t>0,\\
\frac{dv}{dn}=0\quad\mbox{on}\quad\partial\Omega.
\end{gathered} \eqno(2.5b)
$$
Since the function $A$ is odd so that $A'$ is even and
the map $f$ is even, it follows from the equivalent form (2.5)
that $u=(\phi,v)$ solves (2.4) if and only if
$\tilde{u}:=(-\phi,v)$ solves (2.4). \smallskip
\noindent{\bf 2C. Analysis of the activator system (2.5a)}
\quad We study first its kinetic reduction $\xi'=A(\xi)$. Since
the smooth function $A$ is negative in the interval $(0,\infty)$
(cf. condition (c1)), the following initial value problem
$$
\xi'(t)=A(\xi(t)), t\geq 0;\quad
\xi(0)=s\in\mathbb{R}.
\eqno(2.6a)
$$
admits a unique smooth solution $\xi(t,s)$ which exists for all
$t\geq 0$. As a consequence of the oddness of $A$ and the
uniqueness of the solutions, there holds $\xi(t,-s)=-\xi(t,s)$ for
all $(t,s)$ and $\xi(t,s)>0$ if $s>0$. Moreover, the negativity
$A(s)<0$ ($s>0$) implies that
$$
\lim_{t\to\infty}\xi(t,s)=0=\lim_{t\to\infty}\xi(t,-s)
\quad\mbox{for all}\quad s\geq 0.
\eqno(2.6b)
$$
Let $\sigma,\tau$, $\sigma>\tau>0$, be fixed. Set
$\xi_1(t):=\xi(t,\sigma)$, $\xi_2(t):=\xi(t,\tau)$ and
$\eta(t):=\xi_1(t)-\xi_2(t)$ for $t\geq 0$. Since $\sigma>\tau$,
an elementary comparison result yields that $\xi_1(t)>\xi_2(t)$
and thus $\eta(t)>0$ for all $t\geq 0$. Using the concavity of the
function $A(s)$ for $s>0$, we have
\[
A(\xi_1(t))-A(\xi_2(t))\leq A'(\xi_2(t))(\xi_1(t)-\xi_2(t)).
\]
Therefore,
\[
\eta'(t)=A(\xi_1(t))-A(\xi_2(t))\leq A'(\xi_2(t))\eta(t).
\]
Integrating this inequality yields
\[
\eta(t)\leq\eta(0)\cdot\exp(\int_0^t A'(\xi_2(s))\, ds).
\]
To compute the integral, we differentiate the equation
$\xi_2'(t)=A(\xi_2(t))$ and obtain
\[
A'(\xi_2(t))=\xi_2''(t)/\xi_2'(t)=\frac{d}{dt}\ln (-\xi_2'(t)).
\]
It follows that
\[
\int_0^t A'(\xi_2(s))\, ds=\ln(\xi_2'(t)/\xi_2'(0))=
\ln (A(\xi_2(t))/A(\tau))
\]
and thus $\eta(t)\leq\eta(0)A(\xi_2(t))/A(\tau)$. In terms of the
solutions $\xi(\cdot,\sigma)$ and $\xi(\cdot,\tau)$, this implies
that
$$
0<\xi(t,\sigma)-\xi(t,\tau)
\leq\sigma\cdot\frac{A(\xi(t,\tau))}{A(\tau)}
\quad (t\geq 0);\quad\sigma>\tau>0.
\eqno(2.6c)
$$
Next we turn to the study of solutions for the
activator system (2.5a). Consider an initial
value $\phi_0\in C(\bar{\Omega})$ and set
$$
\tau_0:=\min_{x\in\bar{\Omega}}\phi_0(x),\quad
\sigma_0:=\max_{x\in\bar{\Omega}}\phi_0(x).
\eqno(2.7a)
$$
By the classical existence and uniqueness theorem (cf. e.g.
\cite{S83}) there exists a $T>0$ such that the equation (2.5a) has
a unique solution $\phi$ in the time interval $[0,T)$,
$\phi(0)=\phi_0$, with the property that $\phi$ is continuous in
$\bar{\Omega}\times [0,T)$, $\phi_t,\Delta\phi$ as well as all
partial derivatives $\frac{\partial\phi}{\partial x_i}$ are
continuous in $\bar{\Omega}\times (0,T)$. Moreover, if $T<\infty$,
then there should hold $\lim_{t\to
T}\Vert\phi(t)\Vert_\infty=\infty$. We claim that $T=\infty$,
i.e., the solution $\phi$ to (2.5a) with initial value
$\phi(0)=\phi_0\in C(\bar{\Omega})$ exists for all times $t\geq
0$. To see this, we note that both $\xi(\cdot,\sigma_0)$ and
$\xi(\cdot,\tau_0)$ are also solutions of (2.5a) such that
\[
\xi(0,\tau_0)=\tau_0\leq\phi_0(x)\leq
\sigma_0=\xi(0,\sigma_0)\quad\forall x\in\bar{\Omega}.
\]
Hence, the classical comparison theorem
(see e.g. \cite[Theorem 10.1, p.94]{S83})
yields that
$$
\xi(t,\tau_0)\leq\phi(x,t)\leq\xi(t,\sigma_0)
\quad\forall x\in\bar{\Omega},t\in [0,T).
\eqno(2.7b)
$$
This implies in particular the uniform boundedness of the solution
$\phi(t)$ in the time interval $[0,T)$ and thus we have
$T=\infty$. Moreover, since both solutions $\xi(t,\tau_0)$ and
$\xi(t,\sigma_0)$ converge to $0$ as $t\to\infty$, we have that
$\Vert\phi(t)\Vert_\infty\to 0$ as $t\to\infty$. To find one more
result, we assume $\phi_0>0$. It follows that $\tau_0>0$. Hence,
we are in the position to apply (2.6c) under the choice of the
positive constants $\tau,\sigma$ such that $\tau\leq\tau_0$ and
$\sigma\geq\sigma_0=\Vert\phi_0\Vert_\infty$, and obtain that
$$
\begin{gathered}
\Vert\phi(t)-\xi(t,\tau)\Vert_\infty\leq
\sigma\cdot\frac{A(\xi(t,\tau))}{A(\tau)}
\quad (t\geq 0);\\
\forall \tau,\sigma: 0<\tau\leq\tau_0\leq\sigma_0\leq\sigma.
\end{gathered} \eqno(2.7c)
$$
\noindent{\bf 2D. Final step of the proof of Theorem 1.1}\quad We prove
first the assertion in (i). For this purpose, we consider an
initial value $u_0=(\phi_0,v_0)\in C(\bar{\Omega})\times
C(\bar{\Omega})^n$. By the result in 2C, the equation (2.5a)
has a unique global solution $\phi$, $\phi(0)=\phi_0$, with the
property that $\phi$ is continuous in $\bar{\Omega}\times
[0,\infty)$, $\phi_t,\Delta\phi$ as well as all partial
derivatives $\frac{\partial\phi}{\partial x_i}$ are continuous in
$\bar{\Omega}\times (0,\infty)$. Moreover, there holds the
estimate in (2.7b) for the general case and the estimate in (2.7c)
for the particular case $\phi_0>0$. Substituting this global
solution $\phi$ into (2.5b), we have a linear system. It follows
from the basic linear theory (see e.g., \cite{P83}, \cite{EN00})
that the corresponding linear system (2.5b) has a unique global
solution $v$, $v(0)=v_0$, with the property that $v$ is
continuous in $\bar{\Omega}\times [0,\infty)$, $v_t,\Delta v$ as
well as all partial derivatives $\frac{\partial v}{\partial x_i}$
are continuous in $\bar{\Omega}\times (0,\infty)$. Thus, we have
established the global existence and uniqueness assertion in (i).
To prove the convergence assertion in (ii), we note that if
$u(t)=(\phi(t),v(t))\in C(\bar{\Omega})^{1+n}$ is a global
solution of (2.5a-b) then $\tilde{u}(t):=(-\phi(t),v(t))$ is also
a solution of (2.5a-b). Hence, it suffices to prove the assertion
in (ii) for the case $\phi_0>0$. To this end, we fix two positive
constants $\sigma,\tau$, $\tau<\sigma$. We consider a global
solution $u=(\phi,v)$ with initial value $u_0=(\phi_0,v_0)$ such
that $\phi_0\in C(\bar{\Omega})$ satisfies
$$
\tau\leq\phi_0(x)\leq\sigma\quad
\forall x\in\bar{\Omega}.
\eqno(2.8a)
$$
Then, we have, using (2.7b-c), that
$$
\begin{gathered}
\Vert\phi(t)-\zeta(t)\Vert_\infty\leq
c_1\cdot (-A(\zeta(t)))\\
\zeta(t)\leq\phi(x,t)\quad (\forall x\in\bar{\Omega})
\end{gathered} \quad (t\geq 0)
\eqno(2.8b)
$$
with the positive constant $c_1:=\sigma/(-A(\tau))>0$ and positive
function $\zeta(t):=\xi(t,\tau)$ ($t\geq 0$). Since $A$ is
strictly concave in the interval $(0,\infty)$, the function $A'$
is decreasing there and thus
$$
A'(\phi(x,t))\leq A'(\zeta(t))\quad
\forall x\in\bar{\Omega},t\geq 0.
\eqno(2.8c)
$$
We want to show that there exist positive constants $c_2,c_3$,
depending only on the given constants $\sigma,\tau$, such that
$$
\begin{gathered}
\Vert w(t)\Vert_\infty\leq
c_2\cdot\Vert w(0)\Vert_\infty\cdot (-A(\zeta(t)))+
c_3\cdot\zeta(t/2)\\
\mbox{with}\quad w(t):=v(t)-\psi(\xi(t))
\end{gathered}
\quad (\forall t\geq 0).
\eqno(2.9)
$$
Once (2.9) was proved, we derive the asymptotic stability of the
solution $u$ and the equality $\omega(u,C)=\tilde{K}$ as follows.
On the one hand, we use (2.9) combining with the fact that
$\zeta(t)\to 0$ and $A(\zeta(t))\to 0$ as $t\to\infty$ (see
(2.6b)) as well as the equality $\alpha(\psi,\mathbb{R}^n)=K$ (see
(2.3a)) to find that
\[
\omega(v,C)=\alpha(\psi,C)=\alpha(\psi,\mathbb{R}^n)=K,
\]
giving the desired equality $\omega(u,C)=\{0\}\times K=\tilde{K}$.
On the other hand, we let $\hat{u}=(\hat{v},\hat{\phi})$ be
another global solution of (2.5a-b) with an initial value
$\hat{u}_0=(\hat{v}_0,\hat{\phi}_0)$ such that
\[
\tau\leq\hat{\phi}_0(x)\leq\sigma\quad \forall x\in\bar{\Omega}.
\]
Then we can apply (2.9) to the new solution $\hat{u}$
and obtain that
\[
\Vert \hat{w}(t)\Vert_\infty\leq
c_2\cdot\Vert\hat{w}(0)\Vert_\infty\cdot (-A(\zeta(t)))+
c_3\cdot\zeta(t/2)
\quad\forall t\geq 0
\]
with $\hat{w}(t):=\hat{v}(t)-\psi(\xi(t))$ ($t\geq 0$).
It yields that
\[
\Vert v(t)-\hat{v}(t)\Vert_\infty\leq
c_2(\Vert w(0)\Vert_\infty+\Vert\hat{w}(0)\Vert_\infty)\cdot
(-A(\zeta(t)))+2c_3\cdot\zeta(t/2)
\]
for all $t\geq 0$. Obviously, this implies the asymptotic
stability of the solution $u$. Now we prove (2.9). To this end, we
note that
\begin{eqnarray*}
w_t &=& v_t-\zeta'\psi'(\zeta)=v_t-A(\zeta)\psi'(\zeta)\\
&=&\tilde{D}\Delta (w+\psi(\zeta))+A'(\phi)(w+\psi(\zeta))+
f(\phi)-A(\zeta)\psi'(\zeta)\\
&=&\tilde{D}\Delta w+A'(\phi)w+ [A'(\phi)-A'(\zeta)]\psi(\zeta)+
[f(\phi)-f(\zeta)]\\
& &+[f(\zeta)+A'(\zeta)\psi(\zeta)-A(\zeta)\psi'(\zeta)].
\end{eqnarray*}
By the definition of the map $f$ (see (2.3b)), the term in
the last bracket vanishes. Therefore,
$$
w_t=\tilde{D}\Delta w+A'(\phi)w+g(t)
\eqno(2.10a)
$$
with
$$
g(t):=[A'(\phi(t))-A'(\zeta(t))]\psi(\zeta(t))+
[f(\phi(t))-f(\zeta(t))]\quad (t\geq 0).
\eqno(2.10b)
$$
To estimate the term $g(t)$, we note that the spatially uniform
solution $\zeta$ is bounded in time, and (2.8a) implies that the
solution $\phi$ is uniformly bounded in space-time. Moreover, the
smooth map $f$ has a compact support, and the smooth path $\psi$
is bounded by the compactness of the given set $K$. Hence, there
exists a positive constant $c_4$ (essentially depending only on
the given constants $\sigma,\tau$ as well as on the map $\psi$)
such that
$$
\Vert g(t)\Vert_\infty\leq c_4\cdot\Vert\phi(t)-\zeta(t)\Vert_\infty
$$
for all $t\geq 0$. This implies by (2.8b) that
$$
\Vert g(t)\Vert_\infty\leq c_5\cdot (-A(\zeta(t)))\quad (t\geq 0)
\eqno(2.10c)
$$
with $c_5:=c_4\sigma/(-A(\tau))>0$. To treat the solution $w$ of
the inhomogeneous linear evolution equation (2.10a), we recall
from the basic linear theory (see e.g., \cite{P83}, \cite{EN00})
the following representation
$$
w(t)=U(t,0)w(0)+\int_0^t U(t,s)g(s)\, ds\quad (t\geq 0),
\eqno(2.11)
$$
where $\{U(t,s):t\geq s\geq 0\}$ is the evolution family
associated with the nonautonomous linear system
$$
\begin{gathered}
W_t=\tilde{D}\Delta W+A'(\phi)W,
\quad x\in\Omega,t>0,\\
\frac{dW}{dn}=0\quad\mbox{on}\quad\partial\Omega
\end{gathered} \eqno(2.12a)
$$
settled in the state space $C(\bar{\Omega})^n$. Since solutions to
(2.12a) with positive initial data keep positive, each operator
$U(t,s)$ is positive, i.e.,
$$
U(t,s)h\geq 0\quad\mbox{whenever}\quad h\geq 0.
\eqno(2.12b)
$$
We compare the linear system (2.12a) with the following
nonautonomous linear system
$$
\begin{gathered}
\tilde{W}_t=\tilde{D}\Delta\tilde{W}+A'(\zeta(t))\tilde{W},
\quad x\in\Omega,t>0,\\
\frac{d\tilde{W}}{dn}=0\quad\mbox{on}\quad\partial\Omega
\end{gathered} \eqno(2.13a)
$$
settled in the same state space $C(\bar{\Omega})^n$. It is easily
verified, using the equality $\zeta'=A(\zeta)$, that the
evolution family $\{V(t,s):t\geq s\geq 0\}$ associated with
(2.13a) is given by
$$
V(t,s)=\frac{A(\zeta(t))}{A(\zeta(s))} T(t-s)
\quad (t\geq s\geq 0),
\eqno(2.13b)
$$
where $\{T(t):t\geq 0\}$ is the semigroup generated by the
operator $\tilde{D}\Delta$ for which each operator $T(t)$ is a
positive contraction on the state space $C(\bar{\Omega})^n$. We
claim that
$$
U(t,s)\leq V(t,s)\quad\forall t\geq s\geq 0.
\eqno(2.14a)
$$
To see this, we fix some $s\geq 0$ and consider a solution $W$ of
(2.12a) in the interval $[s,\infty)$ with a positive initial value
$W(s)=h\geq 0$. It follows from the definition of evolution
families that this solution $W$ is given by $W(t)=U(t,s)h$ for all
$t\geq s$. We have $W(t)\geq 0$ for all $t\geq s$. Hence, by
(2.8c) we find that
\[
W_t=\tilde{D}\Delta W+A'(\phi)W\leq
\tilde{D}\Delta W+A'(\zeta)W.
\]
Applying the classical comparison theorem (see e.g., \cite[Theorem
10.1, p.94]{S83}) to each component of $W$, we conclude that
$W(t)\leq\tilde{W}(t)$ ($\forall t\geq s$), where $\tilde{W}$ is
the positive solution of (2.13a) with the same initial value
$\tilde{W}(s)=h$ and thus is given by $\tilde{W}(t)=V(t,s)h$ for
all $t\geq s$, again by the definition of evolution families.
This establishes (2.14a). As consequence of (2.14a), we find
particularly that
$$
\Vert U(t,s)\Vert_\infty\leq \Vert V(t,s)\Vert_\infty
\leq \frac{A(\zeta(t))}{A(\zeta(s))}\quad (t\geq s\geq 0).
\eqno(2.14b)
$$
We are now in the position to finish the proof of (2.9). Using the
representation (2.11) of $w$, we have
\begin{eqnarray*}
\Vert w(t)\Vert_\infty &\leq &
\Vert U(t,0)\Vert_\infty\cdot\Vert w(0)\Vert_\infty+
\int_0^t\Vert U(t,s)\Vert_\infty\cdot\Vert g(s)\Vert_\infty\, ds\\
&\leq &\frac{A(\zeta(t))}{A(\tau)}\cdot\Vert w(0)\Vert_\infty
+\int_0^t \frac{A(\zeta(t))}{A(\zeta(s))}\cdot\Vert g(s)\Vert_\infty
\, ds\quad (\mbox{by (2.14b)})\\
&\leq &\frac{A(\zeta(t))}{A(\tau)}\cdot\Vert w(0)\Vert_\infty+
\int_0^t\frac{A(\zeta(t))}{A(\zeta(s))}\cdot c_5\cdot(-A(\zeta(s))
\, ds\quad (\mbox{by (2.10c)}).
\end{eqnarray*}
Consequently,
$$
\Vert w(t)\Vert_\infty\leq
\frac{A(\zeta(t))}{A(\tau)}\cdot\Vert w(0)\Vert_\infty+
c_5\cdot (-tA(\zeta(t))
\quad\forall t\geq 0.
\eqno(2.15a)
$$
To estimate the term $(-tA(\zeta(t))$, we note that the relation
$\zeta'=A(\zeta)<0$ combining with the monotonicity of the
function $A$ implies that the function $t\mapsto A(\zeta(t))$ is
increasing. Hence,
\[
\frac{t}{2}A(\zeta(t))\geq\int_{t/2}^t A(\zeta(s))\, ds
=\int_{t/2}^t \zeta'(s)\, ds=\zeta(t)-\zeta(t/2)\geq -\zeta(t/2)
\]
and thus
$$
(-tA(\zeta(t))\leq 2\zeta(t/2)\quad\forall t\geq 0.
\eqno(2.15b)
$$
Taking (2.15a-b) together, we find finally that
$$
\Vert w(t)\Vert_\infty\leq
\frac{A(\zeta(t))}{A(\tau)}\cdot\Vert w(0)\Vert_\infty+
2c_5\cdot\zeta(t/2)\quad
\mbox{for all}\quad t\geq 0.
\eqno(2.16)
$$
Therefore, the estimate (2.9) holds true
under the choice of the positive constants $c_2:=1/(-A(\tau))$
and $c_3:=2c_5$. \hfill $\Box$
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