\documentclass[reqno]{amsart}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 99, 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-2004/99\hfil On the role of the equal-area]
{On the role of the equal-area condition in internal layer
stationary solutions to a class of reaction-diffusion systems}
\author[J. Crema, A. S. do Nascimento\hfil EJDE-2004/99\hfilneg]
{Janete Crema, Arnaldo Simal do Nascimento} % in alphabetical order
\address{Janete Crema \hfill\break
Instituto de Ci\^encias Matem\'aticas e de Computa\c c\~ao-USP,
S. Carlos, S. P., Brasil}
\email{janete@icmc.sc.usp.br}
\address{Arnaldo Simal do Nascimento \hfill\break
Departamento de Matem\'atica, Universidade Federal de S\~ao Carlos,
S. Carlos, S.P., Brasil}
\email{arnaldon@dm.ufscar.br}
\date{}
\thanks{Submitted January 13, 2003. Published August 12, 2004.}
\thanks{A. S. do Nascimento was supported by CNPq, Brazil}
\subjclass[2000]{35B25, 35B35, 35K57, 35R35}
\keywords{Reaction-diffusion system; internal transition layer;
\hfill\break\indent
equal-area condition}
\begin{abstract}
We present necessary conditions for the formation of internal
transition layers in stationary solutions to some singularly
perturbed reaction-diffusion systems. In particular we prove that
the well-known equal-area condition which is always assumed in a
typical set of sufficient conditions for existence of such
solutions is actually a necessary hypothesis. Examples of
existence and nonexistence of these solutions are given.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{defi}{Definition}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{theorem}{Theorem}[section]
\newtheorem{coro}{Corollary}[section]
\section{Introduction}
The prime concern in this paper is to present a necessary condition
for the formation of internal transition layers for stationary
solutions in $N$-dimensional domains of a singularly perturbed
reaction-diffusion system. This system often appears in the literature
and has the general form
\begin{gather*}
u_t=\epsilon\mathop{\rm div}\left(h_1(x)\nabla u\right)+f(x,u ,{\bf v}),
\quad\mbox{in } \mathbb{R}^+ \times \Omega \\
{\bf v}_t=\mathop{\rm div}\left({\bf h_2}(x)\nabla {\bf v}\right)+ {\bf
g}(x,u ,{\bf v}),\quad\mbox{in } \mathbb{R}^+ \times \Omega \\
\frac{\partial {\bf v}}{\partial \hat{n}} = {\bf 0}\quad
(\mbox{or }{\bf v}=(0,0,\dots ,0),\quad \mbox{on } \mathbb{R}^+ \times \partial\Omega,
\end{gather*}
where $\Omega$ is a smooth domain in $\mathbb{R}^N$ and the bold
letters stand for vector-valued functions. However in order to put
our work into perspective let us consider a simpler system of
reaction-diffusion equations of activator-inhibitor type:
\begin{equation} \label{e1.1}
\begin{gathered}
u_t={\epsilon} \Delta u+f(u,v),\quad (t,x)\in \mathbb{R}^+ \times \Omega \\
v_t=\Delta v+g(u ,v ),\quad (t,x)\in \mathbb{R}^+ \times \Omega\\
\frac{\partial u}{\partial \hat{n}}=\frac{\partial v}{\partial \hat{n}}
= 0,\quad (t,x) \in R^+ \times \partial {\Omega},
\end{gathered}
\end{equation}
where $\epsilon$ is a small positive parameter and $\Omega$
a smooth domain in $\mathbb{R}^N$, $N \geq 1$.
Part of the available literature on this problem is devoted to the
study of \eqref{e1.1} in the
context of spatial pattern formation as it appears in many different fields such as
mathematical biology, chemical reactions, morphogenesis, combustion, etc..
See \cite{FNH}, for instance, for a survey on this issue.
Roughly speaking we will say that a uniformly bounded family
$\Phi^{\epsilon}=(u_{\epsilon},v_{\epsilon})$ of stationary
(meaning that $u_t=v_t=0$) solutions to \eqref{e1.1} develops
internal transition layer as $\epsilon \to 0$ if the component
$u_{\epsilon}$ exhibits a sharp spatial transition between two
different states. These internal transition layer stationary
solutions will be referred to as ITLS solutions, and a rigorous
definition will be provided.
Typically the setting in which the issue of spatial pattern
formation in reaction-diffusion systems is considered involves
the choice of a specific parameter region (the rates of diffusion
and/or reaction) and the geometry of the zero-level set
(nullcline) of the reaction terms $f$ and $g$.
For one-dimensional domains, and using different techniques, existence
(sometimes stability too) of ITLS solutions to \eqref{e1.1}
has been established in \cite{FN1,FN2,saka,MTH,FNH,Ni1}, for
instance. There is a vast literature on the subject but the
references above best suit our purposes.
However, regardless of the particular technique used, whenever proving the existence of
ITLS solutions to \eqref{e1.1}, the following hypotheses are tacitly assumed.
The zero set of $f$,
$$ Z=\{(u,v)\in \mathbb{R}^2 : f(u,v)=0\} $$
has at least two different solutions $u=h_{-}(v)$ and
$u=h_{+}(v)$,
$h_{-}(v)< h_{+}(v)$, in a suitable domain which contains a real number $v=v^*$
satisfying
$$\int_{h_{-}(v^*)}^{h_{+}(v^*)}f(s,v^*)\,ds=0.$$
Usually $Z$ is supposed to take a sigmoidal form in the
$(u,v)$-plane. This assumption is known as the equal-area condition
(or rule) and it is always assumed as a sufficient condition for
existence of ITLS solutions to \eqref{e1.1}. Sometimes it does not
appear explicitly but in an equivalent form, as a Melnikov
integral, for instance. See \cite{lin2}, for this matter.
It might seems at first sight that this hypothesis is not
necessary for proving existence of such solutions.
We give herein a rigorous mathematical proof
that this is not the case. Rather it is a necessary
condition.
Of course for each phenomenon the mathematical system
models there is a physical mechanism underlying the equal-area
condition whenever internal transition layer is created.
An immediate conclusion of our results is that if the above
equal-area condition on $f$ does not hold then, as long
as concentration phenomenon is concerned, we can only expect
formation of spikes and/or boundary layer for stationary solutions
of the system considered, just to mention the simplest geometric
configurations that can occur (see example A.4 in Applications).
See \cite{N1,N6,ko}, for instance, for cases of
a single scalar equation where the equal-area condition does not
hold and spike and boundary layer solutions are obtained.
The present work is an extension to a class of systems of results
obtained in \cite{N4} for a single scalar elliptic equation and at
the same time an improvement of the approach used therein. In
order to be more specific let us briefly describe one particular
case of the main result in \cite{N4}.
Let the constants $\alpha$ and $\beta$, $\alpha <\beta$, satisfy
$f(x,\alpha)=f(x,\beta)=0,\, \forall x \in \Omega \subset \mathbb{R}^N, N
\geq 1$ and consider a smooth $(N-1)$-dimensional compact manifold
without boundary $\Gamma$ such that $\Gamma \subset \Omega$.
Suppose that the boundary-value problem
\begin{equation} \label{escalar}
\begin{gathered}
\epsilon\, \mathop{\rm div}\left(h_1(x)\nabla u\right)+f(x,u)=0,\quad x\in
\Omega \\
\frac{\partial u}{\partial \hat{n}} = 0 \quad \mbox{on } \partial\Omega,
\end{gathered}
\end{equation}
has a family $\{u_{\epsilon}\}$ of solutions which develops inner
transition layer with interface $\Gamma $ connecting the states
$\alpha$ to $\beta$. Then necessarily
\begin{equation} \label{e1.3}
\int_{\Gamma}\{\int_{\alpha}^{\beta}f(x,s)ds\}
x \cdot \hat{\eta}(x)\,dS=0\,.
\end{equation}
where $\hat {\eta}$ stands for
the outward unit normal vector on $\Gamma$. In particular if
$f$ does not depend on $x$ then
\begin{equation}
\int_{\alpha}^{\beta}f(s)ds\,=0.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
If in \cite{N4} we had allowed the interface $\Gamma$ to intersect
$\partial \Omega$ in a proper way (as we do herein) then still for
the case $\alpha$ and $\beta$ constants and $f$ independent of
$x$, \eqref{e1.3} would become $$ \big(\int_{\alpha}^{\beta}f(s)ds
\big) \int_{\Gamma} x \cdot \hat{\eta}(x)\,dS=0. $$ Then we would
only recover the equal-area condition at the price of requiring
the interface $\Gamma$ to be a subset of the boundary of a
star-shaped set, as the above equality shows. This is so because
in \cite{N4} we used the vector-field $\widetilde{X}(x)=x$.
Resorting to a vector-field $X(x)$ which when restricted to the
interface $\Gamma$ coincides with the normal vector-field
$\hat{\eta}(x)$ on $\Gamma$ (used in the present work) would
allows us to recover the equal-area condition without the
star-shape condition since then $X(x) \cdot \hat{\eta}(x)=1$ on
$\Gamma$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Herein we let $\alpha$ and $\beta$ be
functions of the space-variable $x$ and in order to obtain any
meaningful conclusion we generalize the Pohozaev procedure by
working with the vector field $X(x)$ described above.
As an illustration we describe the corresponding version
of the above equal-area condition for \eqref{e1.1} which is obtained in
the present work. Let
$\Phi^{\epsilon}=(u_{\epsilon},v_{\epsilon})$ be a family of
ITLS solutions to \eqref{e1.1} with interface $\Gamma \subset
\overline { \Omega}$ in the sense that $\Phi^{\epsilon}
\stackrel{\epsilon\to 0}{\to} (u_0, v_0)$, uniformly on
compact sets of $\overline{\Omega}\backslash \Gamma$, where
$$
u_{0}(x)={{\alpha}} (x)\chi _{\Omega _{\alpha}}(x)+{\beta}
(x)\chi_{\Omega _{\beta}}(x),\quad x \in
\Omega= \Omega _{\alpha}\cup \Gamma \cup \Omega _{\beta}
$$
and
$\chi_{A}$ stands for the characteristic function of the set $A$.
Then we show that necessarily $f(\alpha (x),\,v_0(x))=0=f(\beta
(x),v_0(x))$, for all $x\in \Omega \backslash \Gamma$ and
there must exist constants
$\bar{\alpha},\bar{\beta}\,(\bar{\alpha}< \bar{\beta}) $ and $\bar{v}$
such that
\begin{equation} \label{equalarea}
\int_{\bar{\alpha}}^{\bar{\beta}}f(s,\bar{v})ds=0,
\end{equation}
where $\bar{\alpha}= \alpha(\bar{x})$ and $\bar{\beta}=
\beta(\bar{x})$, for some $\bar{x} \in \Gamma$.
\section{Necessity for the formation of internal layers}
The main theorem is stated in a more general framework than those considered in the
references supplied. Although existence of ITLS solutions to the full system
considered below seems to be difficulty our results imply that
whenever trying to do so the appropriate equal-area condition must
be assumed.
Henceforth the following system will be considered:
\begin{equation}\label{eprincipal}
\begin{gathered}
u_t=\epsilon \mathop{\rm div}\left(h_1(x)\nabla u\right)+f(x,u ,{\bf v}),\quad
x\in \Omega \\
{\bf v}_t=\mathop{\rm div}\left({\bf h_2}(x)\nabla {\bf v}\right)
+ {\bf g}(x,u ,{\bf v}),\quad x\in \Omega \\ \frac{\partial
{\bf v}}{\partial \hat{n}} ={\bf 0}\quad
(\mbox{or }{\bf v}=(0,0,\dots ,0) \quad \mbox{on } \partial\Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a smooth domain in $\mathbb{R}^N$, $N \geq 1$,
$0<\epsilon \leq \epsilon_{0} $
for some small $\epsilon_0$; ${\bf g}=(g_1, g_2,\dots ,g_n)$ and $f$,
$g_i$ are functions in
$C^1(\Omega \times \mathbb{R} \times \mathbb{R}^n)$,
${\bf h_2}=(h_{2,1}, h_{2,2},\dots h_{2,n})$ with ${{\bf h_2 \nabla v}}=(h_{2,1}\nabla v_1, \dots , h_{2,n}\nabla v_n)$,
$h_1, h_{2,i} \in C^{1,\nu}(\Omega)$,
$0<\nu<1$, satisfying $0 < m < h_1,h_{2,i} < M$, for $i=1,\dots n$,
and some constants $m$ and $M$. Since our definition of internal
transition layer will be local in space we do not need any
boundary condition on $u$. On the other hand the boundary
condition on ${\bf v}$ is used just once for technical reasons.
This boundary condition could have been suppressed as well at the
price of adding another (not restrictive) hypothesis in the
definition of boundary layer. We will comment on this in the
appropriate place.
We now state and justify our definition.
\begin{defi} \label{desenvolvepadrao} \rm
Let $\mathcal{U} $ be an open connect set
in $\Omega$ and let $\Gamma \subset \overline{\mathcal{U}} $ be
an $(N-1)$-dimensional smooth (at least $C^2$) compact connected
orientable manifold whose boundary $\partial \Gamma $ is such
that $\partial \Gamma \cap \partial \Omega$ is a smooth
$(N-2)$-dimensional submanifold of $\partial \Omega$.
We will say that an $\epsilon$-family of stationary solutions to
\eqref{eprincipal},
$$\Phi^{\epsilon}=\left\{(u_{\epsilon},{\bf v}_{\epsilon}\right )\in
[C^1(\overline{\mathcal{U}})\cap C^2(\mathcal{U})]^{N+1},\quad
0<\epsilon < \epsilon _0\},
$$
develops internal transition layer, as $\epsilon \to 0$, in $\mathcal{U}$
with interface $\Gamma$ if:
\begin{itemize}
\item The family $\Phi^{\epsilon}$ is bounded in $\overline{\Omega}$ uniformly
for $0< \epsilon< \epsilon_{0}$.
\item
$u_{\epsilon} \stackrel{\epsilon\to 0}{\to}u_{0},$
uniformly on compact sets of
$\overline{\mathcal{U} }\backslash \Gamma$,
where $u_{0}$ is given by
\[ %2.2
u_{0}(x)={\alpha} (x)\chi _{\mathcal{U}_{\alpha}}(x)+\beta
(x)\chi_{\mathcal{U}_{\beta}}(x)
\]
for some functions $\alpha$ and $\beta$ in $C^0({\mathcal{U}})$,
$\alpha(x)< \beta(x)$ for $x \in \Gamma$ and ${\mathcal
U}=\mathcal{U}_{\alpha} \cup \Gamma \cup \mathcal{U}_{\beta}$, where
$\mathcal{U}_{\alpha}$ and $\mathcal{U}_{\beta }$ are disjoint open
connect sets.
\item ${\bf v}_{\epsilon} \stackrel{\epsilon\to 0}{\to}{\bf v}_{0}$
uniformly in $\overline {\mathcal{U}}. $
\end{itemize}
In this case we will refer to $\Phi ^{\epsilon}$, as a
family of ITLS solutions to \eqref{eprincipal} in $\mathcal{U}$ with
interface $\Gamma $.
\end{defi}
This definition is consistent with known existence results for the
one-dimensional case, when ${\bf v}$ is a scalar function. Indeed
consider \eqref{e1.1} with $\Omega=(0\,,1)=I$ and $0< \epsilon< \epsilon _0$.
Then the existence of a family $\Phi^{\epsilon}$ of ITLS
solutions (as defined above) is proved, for instance, in
\cite{FN1}. See also \cite{saka} and \cite{MTH} for related
results. Actually $\Phi^{\epsilon}$ is a $C^2_{\epsilon}(I) \times
C^1(\bar{I})$ bounded family where $C^p_{\epsilon}(\bar{I})$ is the
space of $p$-times continuous differentiable functions on
$\bar{I}$ with the norm
$\|u\|_{C^p_{\epsilon}}=\sum_{j=0}^{p}|\epsilon^j\frac{d^j}{dx^j}u(x)|$.
The above definition, which is local in space,
suffices for our purposes and it allows for the existence of more
than one transition-layer surface (interface) in $\Omega$.
\begin{remark} \label{rmk2.1} \rm
As in \cite{N4} we could equally well have considered the case in
which the interface $\Gamma$ does not intersect $\partial \Omega$.
Actually this case is somehow easier and will be omitted.
\end{remark}
\begin{remark} \label{rmk2.2} \rm
The question of how the interface $\Gamma$ of an eventual
family of ITLS solutions to \eqref{eprincipal} intersects the boundary of
$\Omega$ is in general very difficult. It is known that in some
simple scalar equations the intersection is orthogonal. However
since this is not the issue herein only restriction on the
smoothness of the intersection will be assumed.
\end{remark}
The next theorem states what is the main result of the present work.
\begin{theorem} \label{thmprincipal}
Let $\mathcal{U}\subset \Omega \subset \mathbb{R} ^N$ be an
smooth open bounded connect set and $\Phi
^{\epsilon}=\{(u_{\epsilon},{\bf v}_{\epsilon})\}_{0< \epsilon < \epsilon _0 }$ a
family of ITLS solutions to problem \eqref{eprincipal}, in
$\mathcal{U}$ with interface $\Gamma $.
Then $f(x,u_{0} (x),{\bf v}_{0}(x))=0$ on $\mathcal{U}\backslash \Gamma$ and
\begin{equation} \label{condicaogeral}
\int_{\Gamma }\int_{\alpha (x)}^{\beta (x)} f(x,s,{\bf
v}_{0}(x))ds\;dS \end{equation} where $dS$ stands for the element
of $(N-1)$-dimensional surface measure.
\end{theorem}
The following lemma will play an important role in
the proof of Theorem \ref{thmprincipal}.
\begin{lemma}\label{lm2.1}
Under the conditions of Theorem \ref{thmprincipal},
we have
$$
\lim_{\epsilon\to 0} \epsilon\int_{\partial {\mathcal{U}}} |\nabla
u_{\epsilon}(x) |^2 dS = 0.
$$
\end{lemma}
\begin{proof} It suffices to show that
\begin{itemize}
\item[(a)] $\lim_{\epsilon\to 0} |\epsilon^{1/2}\nabla u_{\epsilon}(x)| = 0$,
a.e. in $\partial \mathcal{U}\backslash \partial \Gamma$; and
\item[(b)] There exists $M>0$ such that $\left|\epsilon^{1/2}\,\nabla
u_{\epsilon}(x)\right|\leq M$, a.e. in $ x\in \partial \mathcal
{U}\backslash \partial \Gamma,$ uniformly for $ 0 < \epsilon <\epsilon _0$.
\end{itemize}
Once this has been accomplished an application of Lebesgue
Bounded Convergence Theorem will conclude the proof.
Due to smoothness of $\partial \Omega$ we can take without loss of
generality $\mathcal{U}$ smooth and such that $\partial
\mathcal{U} \cap \partial \Omega \subset \Gamma$. This will
prevent us from taking on Schauder estimates on portions of
$\partial \Omega$, which is a delicate and very technical matter,
and at same time this type of neighborhood of $\Gamma$ will
suffice for our purposes.
A standard procedure will be used and therefore we only sketch the proof. It is based on a
blow-up technique and Schauder estimates that have been used in
the scalar case. Therefore only the points in which the proof
differs from the scalar case will be stressed. See \cite{N4}, for
more details.
Firstly, for $\overline {x}\in \partial \mathcal{U} \backslash \partial
\Gamma$ we define a $C^2$ local change of variables $\Sigma$, with
$\Sigma (\overline{x})=0$, which straightens $\partial\mathcal{U}$ near
$\overline{x}$\, and then set
\[ \tilde{u}_\epsilon(y) = u_\epsilon({\Sigma}^{-1}(y)), \quad \mbox{for } y\in
\overline{B^+_\rho},
\]
where $\overline{B^+_\rho}$ stands for the positive hemisphere
of the ball of radius $\rho$ and center at the origin. Let us
consider $\{\epsilon _k\}$ any positive sequence converging to $0$.
Now define scaled functions $\omega_k(z)$ and $\theta_k(z)$ by
$\omega_k(z)= \tilde{u}_{\epsilon_k}\big( \epsilon^{1/2}_kz\big)$,
${\bf \theta}_k(z)= {\bf \tilde{v}}_{\epsilon_k}\big(
\epsilon^{1/2}_kz\big)$
for $z\in \overline{B^+_{\rho /\epsilon^{1/2}_k}}$.
All the coefficients in the new differential equation for
$\omega_k$ are $C^{\nu}$ bounded, uniformly in $k$. For a fixed
$\rho$, we set
\[ \rho_k :=\big(\rho /\epsilon^{1/2}_k\big)
\overset{k\to\infty}{\to} \infty\ .
\]
Let $R_m$ be a monotone increasing sequence of positive numbers
such that $R_m\to +\infty$, as $m\to \infty$. For
each $m$, there is $k_m$ such that $2R_m < \rho_k$, for
$k\geq k_m$.
Since $\{ u_\epsilon\}_{0\leq \epsilon \leq \epsilon _0}$ and
$\{ {\bf v}_\epsilon\}_{_{0\leq \epsilon \leq \epsilon _0}}$ are bounded in
$\overline{\mathcal{U}}$, uniformly on $\epsilon$, it follows that
$\|{\bf \theta}_k\|_{C(\overline{B^+_{2R_m}})}$,
$\|\omega_k\|_{C(\overline{B^+_{2R_m}})}\leq K_1 $, for
some constant $K_1$ which is independent of $k$.
Thus by \cite[Theorem 8.24]{GT}, we conclude that ${\bf \theta}_{k}$ and
$\omega_{k}$ are locally $C^{\nu}$ bounded in
$\overline{B^+_{2R_m}}$, uniformly in $k$. Interior Schauder
estimates in $B_{R_m}$ (here the fact that $\partial \mathcal{U}
\cap \partial \Omega \subset \Gamma$ comes into play)
yield that $\omega_k$ is $C^{2,\nu}$ bounded in $\overline{B^+_{2R_m}}$,
uniformly for
$k \geq k_m$. Then by a diagonal process we can extract a
subsequence, still labelled $\{ \omega_k\}$, such that $\omega_k
\to\omega_o$ in $C^{2}_{\rm loc}(\mathbb{R}^N_+)$ where
$\mathbb{R}^N_+ = \{ z\in \mathbb{R}^N : z_N\geq 0\}$.
Consequently, for $B^+_1=\{z\in \mathbb{R}^N : z_N\geq 0 \mbox{ and }
|z|\leq 1\}$ we have $|\omega _k - \omega _0|_{C^2(B_1^+)}\to 0 $.
But from the definition of $\omega _k$ we conclude that
$\omega_0\equiv \beta (\overline x)$ or $\omega_0\equiv
\alpha(\overline x)$, and so $\omega _0(z)$ is a constant function
in $B^+_1$.
In particular $\lim_{k\to\infty} |\nabla \omega_k(0)|=0$ and then
if $ \epsilon _k \to 0$, $\epsilon _k\in (0,\epsilon _0)$,
\[
\lim_{k\to \infty}\left|\epsilon^{1/2}_k\,\nabla u_{\epsilon_k}(\overline {x})\right|
= 0
\]
for any $\overline{x} \in \partial \mathcal{U} \backslash \partial
\Gamma$. Since $ \partial \mathcal{U} \cap \partial \Gamma$ has zero
$(N-1)$-dimension surface measure, (a) follows.
Finally standard Schauder estimates may be evoked to obtain
(b). Thus our claim is proved.
\end{proof}
\vspace{1cm} But note that if $\Phi_{\epsilon}=(u_{\epsilon},v_{\epsilon})$
is a family of ITLS solutions on $\mathcal{U}$ then it still will be on
any smooth open $\hat{\mathcal{U}}\subset \mathcal{U}$ containing $\Gamma$.
Consequently
making $u_t=0$ in the first equation of \eqref{eprincipal},
integrating and using the Divergence Theorem and Lemma \ref{lm2.1} we
conclude that for any $\hat {\mathcal{U}}$ such that $ \Gamma
\subset \hat {\mathcal{U}} \subset \mathcal{U}$
$$\lim_{\epsilon\to 0}\int_{\hat {\mathcal{U}}}f(x,u_{\epsilon}(x),{\bf
v}_{\epsilon}(x))=0, $$
For $\epsilon \to 0$, the family
$(u_{\epsilon}, {\bf v}_{\epsilon}) \to (u_0,{\bf v}_0)$ uniformly in
any compact set $K\subset \hat{\mathcal{U}}\backslash \Gamma$. So
by boundness and regularity of $f$ we conclude that for any open
set $\hat{\mathcal{U}}$ such that $\hat{\mathcal{U}}\subset
\mathcal{U}$,
$$
\int_{\hat{\mathcal U}}f(x,u_{0}(x),{\bf v}_{0}(x))=0
$$
We have thus proved the following lemma.
\begin{lemma} \label{lm2.2}
Under the conditions of Theorem \ref{thmprincipal},
$f(x,u_{0} (x), {\bf v}_{0}(x))=0$ for all
$x\in \mathcal{U} \backslash \Gamma$.
\end{lemma}
\begin{proof}[Proof of Theorem \ref{thmprincipal}]
Let $\hat{\eta}$ stands for the normal vector field to $\Gamma$ (which by
hypothesis is $C^2$) and let us take a $C^1$ vector field
$X:\overline{\mathcal{U}}\rightarrow \mathbb{R} ^N$
so that when restrict to $\Gamma$ it coincides with $\hat{\eta}$.
As in the Pohozaev procedure, making $u_t=0$ in the first equation
of \eqref{eprincipal}, multiplying it by $X(x).\nabla u_{\epsilon}$ and
integrating over ${\mathcal{U}} $ we obtain
\begin{equation}
\int_{\mathcal{U}}\{ \epsilon \mathop{\rm div} (h_1(x)\nabla u_{\epsilon})(X(x) \cdot \nabla u_{\epsilon})+
f(x,u_{\epsilon},{\bf v}_{\epsilon})\,X(x) \cdot \nabla u_{\epsilon}
\}\,dx=0
\end{equation}
Working with the first term of this equality and using that
\begin{align*}
&\mathop{\rm div}[X(x) \cdot \nabla u h_1 \nabla u]\\
&= X(x) \cdot \nabla u \mathop{\rm div}(h_1 \nabla u)+
h_1\Big[\sum _{i,k=1}^N \frac{\partial X_k}{\partial x_i}u_{x_k} u_{x_i}+
X(x) \cdot \nabla (\frac{|\nabla u|^2}{2})\Big]
\end{align*}
along with the Divergence Theorem, it follows that
\begin{equation} \label{pohozaev}
\begin{aligned}
&- \epsilon \int_{\partial {\mathcal{U}} }h_1(x) X(x)
\cdot\nabla u_{\epsilon} \frac{\partial u_{\epsilon}}{\partial \hat{n}}\,dS
+ \frac{\epsilon }{2} \int_{\partial {\mathcal{U}}
} h_1(x)|\nabla u_\epsilon |^2 X(x)\cdot \hat{n}\,dS \\
&- \frac{\epsilon }{2} \int_{\mathcal{U}} |\nabla u_{\epsilon }|^2
X(x)\cdot \nabla h_1\,dx
- \frac{\epsilon}{2} \int_{\mathcal{U}} h_1(x)|\nabla u_{\epsilon }|^2
\mathop{\rm div}X(x)\,dx\\
&+ \epsilon \int _{\mathcal{U}} \sum_{i,k=1}^N h_1 \frac{\partial
X_k}{\partial x_i} u_{x_k}u_{x_i} \\
&= \int_{\mathcal{U}}
f(x,u_{\epsilon},v_{\epsilon}) X(x) \cdot\nabla u_{\epsilon} \,dx.
\end{aligned}
\end{equation}
We claim that the left hand side of this equality goes to $0$, as
$\epsilon \to 0$. In fact, this holds for the first and second terms
by virtue of Lemma \ref{lm2.1}.
By utilizing energy estimates on $\mathcal{U}$ applied to the
first equation of \eqref{eprincipal} (with $u_t=0$) and Lemma \ref{lm2.2}
we obtain
\begin{equation}\label{gradu}
\lim_{\epsilon \to 0} {\epsilon } \int_{ \mathcal{U}} |\nabla
u_{\epsilon }(x)|^2 \, dx =0\,.
\end{equation}
So the third, fourth, and fifth terms of
(\ref{pohozaev}) approach zero, as $\epsilon \to 0$. Hence
\begin{equation} \label{condicaof}
\lim_{\epsilon \to 0}\, \int_{\mathcal
U}f(x,u_{\epsilon },{\bf v}_{\epsilon }) X(x)\cdot \nabla
u_{\epsilon}\,dx = 0.
\end{equation}
Note that if $F(x,u)=\int_{\theta}^{u}f(x,s,{\bf v})\,ds$,
\begin{equation}\label{F(x,u)}
\mathop{\rm div} [\,X(x) F(x,u)] =f(x,u,{\bf v})\nabla u
\cdot X(x) + \int _ {\theta }^{u} X(x) \cdot \nabla
_xf(x,s,{\bf v})ds + \mathop{\rm div}X(x) F(x,u),
\end{equation}
where
\[
\nabla_xf(x,s,{\bf v}(x))=\partial_xf(x,s,{\bf v}(x))+\sum
\partial_{3,i}f(x,s,{\bf v}(x))\nabla { v_i}(x)\partial _{x}f(x,s,t_1, t_2,\dots
t_n)
\]
is the gradient of $f$ with respect to $x$ and
$\partial_{3,i}(x,s,t_1, t_2,\dots t_n)$ is the partial derivative of $f$ with
respect to $t_i$, $i=1,2,\dots n$. The Divergence Theorem yields
\begin{equation} \label{expressaof}
\begin{aligned}
& \int_{\mathcal{U}}f(x,u_{\epsilon }, {\bf v}_{\epsilon })\, X(x)
\cdot \nabla u_{\epsilon} \, dx \\
&= \int_{\partial {\mathcal{U}}} X(x)\cdot \hat{n} \int_{\theta }^{u_{\epsilon }}
f(x,s,{\bf v}_{\epsilon })\, ds\, dS \\
&-\int _{\mathcal{U}}\{ \int _{\theta }^{u_{\epsilon }} X(x)\cdot \nabla _x f(x,s,{\bf v}_{\epsilon })ds
+ \mathop{\rm div}X(x) \int _{\theta } ^{u_{\epsilon }}
f(x,s,{\bf v}_{\epsilon })ds\} dx
\end{aligned}
\end{equation}
Now each element of the right-hand term of (\ref{expressaof})
will be analyzed.
By hypothesis, $\{u_{\epsilon}\}$ and $\{{\bf v}_{\epsilon }\}$ converge
uniformly on compact sets $K\subset \overline{\mathcal{U}}
\backslash \Gamma$ and $\overline {\mathcal{U}} $, respectively. So
by regularity of $f$ we obtain for any $x \in \overline {\mathcal
U} \backslash \Gamma$ that
$$\int_{\theta }^{u_{\epsilon
}(x)}f(x,s,{\bf v}_{\epsilon }(x))\, ds \stackrel{\epsilon \to
0}{\to} \int_{\theta }^{u_0 (x)} f(x,s,{\bf
v}_0(x))\, ds,$$
with the same result when we take $\partial _xf$ or $
\partial _{3,i}f$, $i=1,2,\dots n$, instead of $f$. Then applying Lebesgue Convergence Theorem
we conclude that
\begin{equation}
\label{fronteira} \int_{\partial {\mathcal{U}} } X(x)\cdot \hat{n}
\int_{\theta }^{u_{\epsilon }}f(x,s,{\bf v}_{\epsilon })\, ds\,dS
\stackrel{\epsilon \to 0}{\to} \int_{\partial{\mathcal
U}} X(x)\cdot \hat{n} \int_{\theta }^{u_0} f(x,s,{\bf v}_0)\, ds\, dS
\end{equation}
and
\begin{equation}
\label{derivf}
\begin{array}{l}
\int _{{\mathcal{U}}}\int _{\theta }^{u_{\epsilon }}
X(x)\,\partial _xf(x,s,{\bf v}_{\epsilon })ds dx
\stackrel{\epsilon \to 0}{\to}
\int _{{\mathcal{U}}}\int _{\theta }^{u_0 }
X(x)\,\partial _x f(x,s,{\bf v}_0)dsdx.
\end{array}
\end{equation}
Recalling that $\mathcal{U}\backslash \Gamma=\mathcal
U_{\alpha}\cup \mathcal{U} _{\beta} $, for $\sigma \in
\{\alpha,\beta\}$ we obtain
\begin{equation}
\label{Nf}
\begin{array}{l}
\int _{{\mathcal{U}}_{\sigma}} \mathop{\rm div}X(x) \int
_{\theta }^{u_{\epsilon }} f(x,s,{\bf v}_{\epsilon })ds dx
\stackrel{\epsilon \to 0}{\to}
\int _{{\mathcal{U}}_{\sigma}}\mathop{\rm div}X(x)\int _{\theta
}^{\sigma } f(x,s,{\bf v}_0)dsdx.
\end{array}
\end{equation}
We also have that for $i=1,2,\dots n$,
\begin{equation}
\label{pa3f}
X(x) \int_{\theta }^{u_{\epsilon }}\partial_{3,i}f(x,s,{\bf v}_{\epsilon}(x))\, ds
\to X(x) \int_{\theta }^{u_0} \partial_{3,i}f(x,s,{\bf v}_0(x))\,
ds\end{equation}
strongly in $L^2(\mathcal{U})$. Then if $\nabla {\bf v}_{\epsilon}$
to converge weakly in $L^2(\mathcal{U})$ we will have
\begin{equation} \label{pa2f}
\int_{\mathcal{U}} \int_{\theta}^{u_{\epsilon}}X(x)\cdot \nabla _xf(x,s,{\bf v}_{\epsilon
}(x))ds\,dx \to \int_{\mathcal{U}} \int_{\theta}^{u_{0}}X(x)\cdot \nabla _xf(x,s,{\bf
v}_{0}(x))ds\,dx.
\end{equation}
To establish the weak convergence of $\{{\bf v}_{\epsilon
}\}$ in $H^1(\mathcal{U})$ note that energy estimates on the second
equation of \eqref{eprincipal} (with ${\bf v}_t=0$) and boundness
of ${\bf g}$, $u_{\epsilon}$, and ${\bf v}_{\epsilon}$ give us the boundness
of $\nabla {\bf v}_{\epsilon}$ in $L^2(\Omega)$, uniformly on $\epsilon$.
Moreover ${\bf v}_{\epsilon }\to {\bf v}_0$ uniformly in
$\overline{\mathcal{U}}$. So ${\bf v}_{\epsilon }$ is bounded in
$H^1(\mathcal{U})$ and $\nabla {\bf v}_{\epsilon}$ converges weakly to
$\nabla {\bf v}_0$ in $L^2(\Omega)$.
Passing to the limit in (\ref{expressaof}), as $\epsilon \to 0$, and
using (\ref{condicaof}), (\ref{fronteira}) to (\ref{Nf}) and
({\ref{pa2f}}) we obtain
\begin{align*}
\label{provisoria}
0 & = \int _{\partial {\mathcal{U}}}X(x)\cdot \hat{n}\int_{\theta }^{u_0(x)}f(x,s,{\bf v}_0(x))ds\,dS \\
&\quad - \int_{{\mathcal{U}}_{\alpha } }\{ \int _{\theta}^{\alpha (x)}
X(x)\cdot \nabla _{x}f(x,s,{\bf v}_0(x))+\mathop{\rm div} X(x)f(x,s,{\bf v}_0(x))\,ds\}\,dx \\
& \quad- \int_{{\mathcal{U}}_{\beta} }\{ \int _{\theta}^{\beta (x)}X(x)\cdot
\nabla _{x}f(x,s,{\bf v}_0(x))+\mathop{\rm div} X(x) f(x,s,{\bf
v}_0(x))\,ds\}\,dx\,.
\end{align*}
By (\ref{F(x,u)}) and Lemma \ref{lm2.2} it follows that
\begin{align*}
0 =& \int_{\partial
{\mathcal{U}}} X(x)\cdot\hat{n} \quad F(x,u_0(x))\,dS -
\int_{{\mathcal{U}}_{\alpha }}\mathop{\rm div}\{X(x)F(x,\alpha (x))\}dx \\
&- \int_{{\mathcal{U}}_{\beta }}\mathop{\rm div}\{X(x)F(x,\beta (x))\}dx.
\end{align*}
The Divergence Theorem implies
\begin{equation}
\label{condicaoareageral} \int_{\Gamma} \{
\int_{\alpha (x)}^{\beta (x)}f(x,s,{\bf v}_0(x))ds\,\} \, X(x)
\cdot \hat {\eta}\, dS=0.
\end{equation}
Due to the way $X$ was taken we obtain
\begin{equation}
\label{pcondicao}
\int_{\Gamma} \{ \int_{\alpha (x)}^{\beta (x)}f(x,s,{\bf
v}_0(x))ds\,\} \, \hat{\eta}\cdot \hat {\eta} \,dS=
\int_{\Gamma} \{ \int_{\alpha (x)}^{\beta (x)}f(x,s,{\bf v}_0(x))ds\,\} \,dS=0,
\end{equation}
thus proving \eqref{condicaogeral}.
\end{proof}
\begin{remark} \label{rmk2.3} \rm
It is worthwhile to note that boundary condition ${\bf
v}_{\epsilon}=0$ or $\frac{\partial {\bf v}_{\epsilon}}{\partial
\hat{\eta}}=0$ in $\mathbb{R}^+\times
\partial \Omega$ was need in order to obtaining (\ref{pa2f}). Without the
boundary condition on ${\bf v}_{\epsilon}$, the same conclusion could
have been obtained had we required boundness of $\{{\bf
v}_{\epsilon}\}$ in $C^1(\overline{\Omega})$, uniformly in $\epsilon$, in
Definition \ref{desenvolvepadrao}.
This additional hypothesis is not restrictive since the existence
of a family $\Phi^{\epsilon}=(u_{\epsilon},{\bf v}_{\epsilon})$ which
satisfies also this condition is proved, for instance, in
\cite{saka}, for the case $\Omega=[0,1]$.
\end{remark}
In the next results our goal is to recover the equal-area
condition on $f$.
\begin{coro} \label{coro0}
If $\Omega =(0,1)$ then the interface $\Gamma $
is a point $ \overline{x} \in (0,1)$ and condition
(\ref{condicaogeral}) becomes
\begin{equation} \label{quase}
\int_{\alpha (\overline{x})}^{\beta (\overline{x})}
f(\overline{x},s,{\bf v}_{0}(\overline{x}))\;ds=0
\end{equation}
\noindent
which is the known equal-area
condition for $f$.
\end{coro}
\begin{coro} \label{coro1}
If $\Phi ^{\epsilon}$ is a family of ITLS solutions to
\eqref{eprincipal} as in Theorem \ref{thmprincipal}, then there
must exist $\overline{x}\in \Gamma$ such that $(\overline{x},
\alpha(\overline{x}) ,{\bf v}_{0}(\overline{x}))$ and
$(\overline{x}, \beta(\overline{x}) ,{\bf v}_{0}(\overline{x}))$
are roots of $f$ and
\begin{equation}\label{1equalarea}
\int_{\alpha(\overline{x})}^{\beta(\overline{x})}f(\overline{x},s,{\bf
v}_{0}(\overline{x}) )\, ds=0.
\end{equation}
\end{coro}
This corollary follows from the continuity of $\alpha$, $\beta $, ${\bf v}_{0}$,
and $f$.
Next we provide sufficient conditions so that $\alpha(x)$,
$\beta(x)$ and ${\bf v}_0(x)$ be constant functions, thus
recovering the equal-area formula for $f$.
First of all we observe that if ${\bf g}$ is allowed to depend on
$\epsilon$, i.e., ${\bf g}={\bf g}(\epsilon,x,u,{\bf v})$ is a
continuous function in any point $(\epsilon,x,u,{\bf v})$ then the
conclusions of Theorem \ref{thmprincipal} still remains true. In particular we
have
\begin{coro} \label{coro2}
With the notation of Theorem \ref{thmprincipal} let us
take $f=f(u,{\bf v})$ and ${\bf g}={\bf g}(\epsilon,x,u,{\bf v})$.
If $\{(u_{\epsilon}, \bf{v}_{\epsilon})\}$ is a family of ITLS
solutions in $\Omega$ with interface $\Gamma$ and if ${\bf
g}(\epsilon,x,u_{\epsilon}(x),{\bf v}_{\epsilon}(x))\stackrel{\epsilon \to
0}{\to}{\bf 0}$ a.e. in $\Omega$ then ${\bf v}_{0}$ is a
constant vector-valued function. Moreover if for any constant
vector ${\bf c}=(c_1,c_2,\dots ,c_n)$ there holds that the set
$\{s: f(s,{\bf c})=0\}$ is discrete, then $\alpha _0$ and $\beta_0$
are also constant functions. In this case (\ref{condicaogeral})
simplifies to
\begin{equation}\label{2equalarea}
\int_{\alpha _0}^{\beta _0}f(s,{\bf v}_{0})ds=0
\end{equation}
\end{coro}
\begin{proof} Since $(u_{\epsilon},{\bf v}_{\epsilon})$ satisfies
\eqref{eprincipal} (with time derivatives vanishing), by multiplying the second
equation by ${\bf v}_{\epsilon}$, integrating on $\Omega$ and passing
to the limit as $\epsilon \to 0$, we conclude that ${\bf v} _0$ is a
constant vector-valued function.
By our hypotheses, $f(u_{\epsilon },
{\bf v}_{\epsilon }) \stackrel{\epsilon \to 0} \to f(u_{0},{\bf v}_{0})$,
uniformly in compact sets
$K\subset \Omega \backslash \Gamma$. Thus Lemma \ref{lm2.2} implies
$f(u_0(x),{\bf v}_{0})=0 $ and therefore $u_{0}(x)\in
\{s;f(s,{\bf v}_{0})=0\}$ for any $x \in \Omega \backslash
\Gamma$. But this is a discrete set and
$u_{\epsilon} \stackrel{\epsilon \to 0} \to u_{0}$ uniformly in compact
sets $K\subset\Omega\backslash \Gamma$. Therefore
$u_{0}=\alpha \chi_{\Omega_{\alpha }}+\beta \chi_{\Omega_{\beta }}$ is a
constant function on each connect component of $\Omega\backslash
\Gamma$, i.e., $\alpha $ and $\beta$ are constant functions on
$\Omega _{\alpha}$ and $\Omega _{\beta} $, respectively. But in
particular $(u_{\epsilon }, {\bf v}_{\epsilon })$ is a family of
ITLS solutions in a open set $\mathcal{U} \subset \Omega $ with
$\mathcal{U}$ as in Theorem \ref{thmprincipal}. Consequently
(\ref{condicaogeral}) holds and by above considerations it
simplifies to \eqref{2equalarea}.
\end{proof}
\begin{remark} \label{rmk2.4} \rm
Under the hypothesis of Corollary \ref{coro2} the nodal curve of
$f$ must intersect the set $\{s: f(s,{\bf v}_{0})=0\}$ at least
three times in order that \eqref{2equalarea} holds.
\end{remark}
\section{Applications}
We single out four examples from the extensive existing
bibliography concerning existence of internal transition layers
for such systems and conclude that the different forms of the
equal-area assumed therein are in fact necessary conditions.
\subsection*{A.1 Spatial dependent reactions terms}
The following model problem is considered, for instance, in \cite{lin}:
\begin{equation} \label{e3.1}
\begin{gathered}
u_t={\epsilon}^2 u_{xx}+(1-u^2)(u-a)-v-\frac{k}{\omega}\sin(\omega
x+b) \\ v_t= \frac{1}{\sigma}v_{xx}+(\delta u-v),\quad x\in \
[0,1]\\ u_{x}=v_{x}=0\quad \mbox{for}\; x \in \{0,1\}.
\end{gathered}
\end{equation}
where $a \in (-1,0)$, $\epsilon$ and $\sigma$ are positive and small,
$\delta>0$, $k>0$, $\omega>0$, and $b \in \mathbb{R}$.
In particular the existence of a family of stationary solution to
\eqref{e3.1} which develops internal transition layer, as
$\epsilon \to 0$, is proved.
Corollary \ref{coro0} implies that a necessary condition for
existence of such a family is that for some $x_0 \in [0,1]$ the
following holds
$$
\int_{h_{-}(v(x_0))}^{h_{+}(v(x_0))}[(1-\xi^2)(\xi-a)-v(x_0)-\frac{k}{\omega}\sin(\omega
x_0+b)]d\xi=0.
$$
But this is just hypothesis A.2 (p. 372) in
\cite{lin}, assumed therein as a sufficient condition.
We remark that in the notation of Corollary \ref{coro0}, the above equation reads
$$
\tilde{v}=\frac{1} {\beta (x_0)-\alpha (x_0)}
\int_{\alpha (x_0)}^{\beta(x_0)}[(1-\xi^2)(\xi-a)] \,d\xi
$$
where $\tilde{v}=v(x_0)+ \frac{k}{\omega}\sin(\omega x_0+b)$.
\subsection*{A.2 A rescaled system regarding instability of patterns}
Consider the reaction - diffusion system
\begin{equation} \label{suzukiparabolic}
\begin{gathered}
u_t={\epsilon}^2 \Delta u+f(u,v), \\
v_t=D\Delta v+g(u,v),\quad (x,t)\in \Omega \times (0,\infty)\\
\frac{\partial u}{\partial \hat{n}} = \frac{\partial v}{\partial
\hat{n}} = 0, \quad (x,t)\in \partial\Omega \times (0,\infty)
\end{gathered}
\end{equation}
where $u$ is the activator, $v$ is the inhibitor, $\Omega$ is
a smooth domain in $\mathbb{R}^N$, $D>0$ and $\epsilon$ a small positive
parameter. The nullcline of $f$ is sigmoidal and consists of three
smooth curves $u=h_{-}(v)$, $u=h_{0}(v)$ and $u=h_{+}(v)$ defined
on the intervals $I_-$, $I_0$ and $I_+$, respectively. Also if
min$I_-=\underline v$ and max$I_+=\overline v$ then the inequality
$h_{-}(v) \alpha)$ and $\bar{v}$ such that
$h(\alpha)=h(\beta)=\bar{v}$ and
$$\frac{1}{(\beta-\alpha)}\int_{\alpha}^{\beta}h(s) ds=\bar{v}.
$$
Hence if this condition is violated one can only expect existence
of pulses for the F-N system in the real line.
\subsection*{A.4 A model from morphogenesis}
As another application let us consider the system introduced in
\cite{GM} in the context of morphogenesis and which inspired many
related works:
\begin{equation} \label{e3.6}
\begin{gathered}
u_t={d_1} \Delta u -u+(u^p/v^q),\quad (t,x)\in \mathbb{R}^+ \times \Omega \\
\tau v_t=d_2 \Delta v -v+(u^r/v^s),\quad (t,x)\in \mathbb{R}^+ \times\Omega\\
\frac{\partial u}{\partial \hat{n}}=\frac{\partial v}{\partial \hat{n}}
= 0,\quad (t,x) \in R^+ \times \partial {\Omega,}
\end{gathered}
\end{equation}
where $d_1, d_2, p, q, r, \tau$ are positive constant, $s \geq 0$
and $$0< \frac{p-1}{q}< \frac{r}{s+1}. $$ Our results give a
rigorous proof to the heuristic fact that stationary solutions to
\eqref{e3.6} do not develop internal transition layers as $d_1 \to
0$. This will follow from Corollary \ref{coro2} along with the
fact that for each fixed $v$, say $\bar{v},$ the line
$(u,\bar{v})$ intersects the graph of the function $v=u^{p-1/q}$
at most twice thus making it impossible for \eqref{1equalarea} to
hold.
Therefore, as $d_1 \to 0$, among the simplest geometric configuration
possible, stationary solutions to (3.16) can only develop formation
of spikes and/or boundary layer.
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\end{document}