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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 23, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/23\hfil Reaction-diffusion system]
{On a reaction-diffusion system associated with brain lactate kinetics}

\author[R. Guillevin, A. Miranville, A. Perrillat-Mercerot \hfil EJDE-2017/23\hfilneg]
{R\'emy Guillevin, Alain Miranville,  Ang\'elique Perrillat-Mercerot}

\address{R\'emy Guillevin \newline
Universit\'e de Poitiers,
Laboratoire de Math\'ematiques et Applications,
UMR CNRS 7348  SP2MI, Equipe DACTIM-MIS,
Boulevard Marie et Pierre Curie - T\'el\'eport 2,
F-86962 Chasseneuil Futuroscope Cedex, France. \newline
CHU de Poitiers, 2 Rue de la Mil\'etrie,
F-86021 Poitiers, France}
\email{remy.guillevin@chu-poitiers.fr}

\address{Alain Miranville \newline
Universit\'e de Poitiers,
Laboratoire de Math\'ematiques et Applications,
UMR CNRS 7348  SP2MI, Equipe DACTIM-MIS,
Boulevard Marie et Pierre Curie - T\'el\'eport 2,
F-86962 Chasseneuil Futuroscope Cedex, France}
\email{alain.miranville@math.univ-poitiers.fr}

\address{Ang\'elique Perrillat-Mercerot \newline
Universit\'e de Poitiers,
Laboratoire de Math\'ematiques et Applications,
UMR CNRS 7348  SP2MI, Equipe DACTIM-MIS,
Boulevard Marie et Pierre Curie - T\'el\'eport 2,
F-86962 Chasseneuil Futuroscope Cedex, France}
\email{Angelique.Perrillat@math.univ-poitiers.fr}


\dedicatory{Communicated by Marco Squassina}

\thanks{Submitted November 18, 2016. Published January 19, 2017.}
\subjclass[2010]{35K57, 35K67, 35B45}
\keywords{Brain lactate kinetics; spatial diffusion; reaction-diffusion system;
\hfill\break\indent  well-posedness; regularity; linear stability}

\begin{abstract}
 Our aim in this article is to study properties of a reaction-diffusion
 system which is related with brain lactate kinetics, when spatial diffusion
 is taken into account. In particular, we prove the existence and uniqueness
 of nonnegative solutions and obtain linear stability results. We also derive
 $L^\infty $-bounds on the solutions. These results give insights on the
 therapeutic management of glioma.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

Glioma (also called glial tumors) appear to be the most frequent primary brain tumors.
In particular, the so-called grade-II low-grade glioma
take a significant place, as they inevitably evolve to anaplastic
transformation, with a very poor prognosis. We can note that delay and kinetics
of this transformation are highly variable and the occurrence of
commutation into anaplastic glioma is a decisive factor. The prediction of this
moment constitutes a challenge and is a deciding factor for therapeutic management.
Though currently unpredictable by clinical data and morphological medical
imaging, some data concerning glioma metabolism are accessible noninvasively
by magnetic resonance imaging. Furthermore, we can obtain metabolite concentrations,
such as creatine and lactate, by means of magnetic resonance spectroscopy.
A major challenge is to explain the variations of magnetic resonance data observed
during the transformation of low grade glioma and suggest new therapies.
We refer the interested readers to \cite{lahutte} and the references therein
for more details.

In view of this, the system of ODE's
\begin{gather}\label{i1}
\frac{du}{dt}+\kappa ({u\over {k+u}}-{v\over {k'+v}})=J,\quad \kappa ,
\; k,\; k',\; J>0, \\
\label{i2}
\epsilon\frac{dv}{dt}+Fv+\kappa ({v\over {k'+v}}-{u\over {k+u}})=FL,\quad
 \epsilon ,\; F,\; L>0,
\end{gather}
where $\epsilon $ is a small parameter, was proposed and studied
as a model for brain lactate kinetics
 (see \cite{C,lahutte,LCFG,L}). In this context,
$u=u(t)$ and $v=v(t)$ correspond to the lactate concentrations in an interstitial
(i.e., extra-cellular) domain and in a capillary domain, respectively.
Furthermore, the nonlinear term
$\kappa ({u\over {k+u}}-{v\over {k'+v}})$ stands for a co-transport through the
brain-blood boundary
(see \cite{cotransport}). Finally, $J$ and $F$ are forcing and input terms,
 respectively, assumed frozen.
In particular, the model has a unique stationary point
which is asymptotically stable. This is consistent
with clinical observations which suggest that, within a short time scale
from minutes to days, metabolite concentrations within the tumor are nearly constant.
Furthermore, as discussed in \cite{lahutte}, a therapeutic perspective of such a
result is to have the stationary point outside the viability domain, where
cell necrosis occurs.

Now, it is reasonable to expect that the lactate concentrations vary according to
the position in the brain, meaning that, in view of more precise models,
one should consider a PDE's system. Furthermore, spatial diffusion, meaning that
the lactates spread (note that there is a flux into different compartments),
should also be taken into account. This suggests reaction-diffusion models
to describe the lactate kinetics.

The simplest possible corresponding PDE's (reaction-diffusion) system, accounting
 for spatial diffusion, reads  (see also \cite{new})
\begin{gather}\label{i3}
\frac{\partial u}{\partial t}-\alpha \Delta u+\kappa ({u\over {k+u}}-{v\over {k'+v}})
=J, \quad \alpha >0, \\
\label{i4}
\epsilon \frac{\partial v}{\partial t}-\beta \Delta v+Fv
+\kappa ({v\over {k'+v}}-{u\over {k+u}})=FL,\quad \beta >0,
\end{gather}
where $u=u(x,t)$ and $v=v(x,t)$,
which we consider in a bounded and regular domain $\Omega $ of
$\mathbb{R}^N$, $N=1$, $2$ or $3$,
together with Neumann boundary conditions,
$$
\frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}
=0\quad\text{on }\Gamma ,
$$
where $\Gamma =\partial \Omega $ and $\nu $ is the unit outer normal vector.

Actually, more precise models should also account for the geometry,
i.e., the different compartments (interstitial, capillary).
In the simplified model \eqref{i3}-\eqref{i4},
even if, initially, meaning in the initial conditions, the two compartments
are separated, $u$ and $v$ diffuse in each of them: the total lactate
concentration should thus be seen as
the sum $u+v$. Finally, although we kept here, at first approximation,
the same nonlinear terms as in \eqref{i1}-\eqref{i2},
this should be further investigated in view of more realistic PDE's
models; this paper can thus be seen as a first step to understand the mathematical
difficulties related with reaction-diffusion systems with such nonlinearities/such a
coupling. We will address more elaborate models elsewhere.

A first step, to validate  the model, is to show that it still satisfies
the important (in view of therapeutic management) properties of the ODE's system.
 More precisely, two major issues are the boundedness of the lactate concentrations
(related with the viability domain of the glial tumors) and the stability of
 the unique (spatially homogeneous in the case of the PDE's system)
steady state (related with therapeutic protocols).

The mathematical analysis of \eqref{i3}-\eqref{i4} (and, in particular,
the well-posedness) appears to be challenging, due to the coupling terms,
especially for negative initial data
(though biologically irrelevant, this makes sense from a mathematical point of view).

In \cite{miranville}, the well-posedness of the following singular
reaction-diffusion equation was studied:
\begin{equation}\label{i5}
\frac{\partial u}{\partial t}-\Delta u+Fu+\kappa {{u}\over {k+u}}=f(x,t),\quad F\ge 0,
\end{equation}
corresponding to the case where either $u$ or $v$ is known in \eqref{i3}
and \eqref{i4}; we can also think of \eqref{i5} as an equation in each compartment,
 assuming that the lactate concentration is known in the other one.

In this paper, we prove the existence and uniqueness of regular nonnegative
solutions to \eqref{i3}-\eqref{i4}, for nonnegative initial data.
We also derive $L^\infty $-bounds on these solutions.
We then study the linear stability of the unique spatially homogeneous steady state.
We can note that this spatially homogeneous equilibrium is the same as
the unique equilibirum for \eqref{i1}-\eqref{i2}, meaning that \eqref{i3}-\eqref{i4}
contains important and relevant features of the original ODE's model.

\subsection*{Notation}

We denote by $((\cdot ,\cdot ))$ the usual $L^2$-scalar product, with associated
norm $\| \cdot \| $. More generally, $\| \cdot \| _X$ denotes the norm
on the Banach space $X$.

Throughout the paper, the same letters $c$ and $c'$ denote (generally positive)
constants which may vary from line to line.

\section{Existence, uniqueness and regularity of nonnegative solutions}

We consider the initial and boundary value problem
\begin{gather}\label{1}
\frac{\partial u}{\partial t}-\alpha \Delta u
+\kappa ({{k'}\over {k'+v}}-{k\over {k+u}})=J, \\
\label{2}
\epsilon \frac{\partial v}{\partial t}-\beta \Delta v+Fv
 +\kappa ({k\over {k+u}}-{{k'}\over {k'+v}})=FL, \\
\label{3}
\frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=0\quad\text{on }
 \Gamma , \\
\label{4}
u|_{t=0}=u_0,\quad v|_{t=0}=v_0.
\end{gather}
Note that \eqref{1}-\eqref{2} are equivalent to \eqref{i3}-\eqref{i4}.
We assume that
\begin{equation}\label{5}
(u_0,v_0)\in (H^3(\Omega )\cap H^2_{\rm N}(\Omega ))^2,\quad
 u_0\ge 0,\; v_0\ge 0\text{  a.e. } x,
\end{equation}
where
$$
H^2_{\rm N}(\Omega )=\{w\in H^2(\Omega ): {{\partial w}\over {\partial \nu }}=0
\text{ on }\Gamma \}.
$$

\begin{theorem}\label{theorem1}
We assume that \eqref{5} holds. Then, \eqref{1}-\eqref{4} possesses a unique
solution $(u,v)$ such that
\begin{equation}\label{6}
u\ge 0,\quad v\ge 0\quad\text{a.e. } (x,t)
\end{equation}
and, for all $T>0$,
\begin{gather*}
(u,v)\in L^\infty (0,T;(H^3(\Omega )\cap H^2_{\rm N}(\Omega ))^2)
\cap L^2(0,T;H^4(\Omega )^2), \\
(\frac{\partial u}{\partial t},\frac{\partial v}{\partial t})\in
L^\infty (0,T;H^1(\Omega )^2)\cap L^2(0,T;H^2(\Omega )^2).
\end{gather*}
\end{theorem}

\begin{proof} (a) Uniqueness:
Let $(u_1,v_1)$ and
$(u_2,v_2)$ be two such solutions to \eqref{1}-\eqref{3} with initial data
$(u_{0,1},v_{0,1})$ and
$(u_{0,2},v_{0,2})$, respectively. We set $(u,v)=(u_1-u_2,v_1-v_2)$ and
$(u_0,v_0)= (u_{0,1}-u_{0,2},v_{0,1}-v_{0,2})$ and have
\begin{gather}\label{7}
\frac{\partial u}{\partial t}-\alpha \Delta u+\kappa ({{ku}\over {(k+u_1)(k+u_2)}}-
{{k'v}\over {(k'+v_1)(k'+v_2)}})=0, \\
\label{8}
\epsilon \frac{\partial v}{\partial t}-\beta \Delta v+Fv+\kappa (
{{k'v}\over {(k'+v_1)(k'+v_2)}}-{{ku}\over {(k+u_1)(k+u_2)}})=0, \\
\label{9}
\frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=0\quad\text{on }
 \Gamma , \\
\label{10}
u|_{t=0}=u_0,\quad  v|_{t=0}=v_0.
\end{gather}
We multiply \eqref{7} by $u$ and \eqref{8} by $v$ and integrate over $\Omega $
and by parts. Summing the two resulting equalities, we easily obtain,
noting that $u_i\ge 0$, $v_i\ge 0$ a.e. $(x,t)$, $i=1,2$, so that
$$
0\le {{k}\over {(k+u_1)(k+u_2)}} \le {1\over k},\quad
0\le {{k'}\over {(k'+v_1)(k'+v_2)}}\le {1\over {k'}}\quad\text{a.e. } (x,t),
$$
the differential inequality
$$
{1\over 2}{d\over {dt}}(\| u\| ^2+\epsilon \| v\| ^2)\le
\kappa ({1\over k}+{1\over {k'}})\vert ((u,v))\vert .
$$
This yields
\begin{equation}\label{11}
{d\over {dt}}(\| u\| ^2+\epsilon \| v\| ^2)
\le c(\| u\| ^2+\epsilon \| v\| ^2),
\end{equation}
whence, owing to Gronwall's lemma,
\begin{equation}\label{12}
\begin{aligned}
&\| u_1(t)-u_2(t)\| ^2+\epsilon \| v_1(t)-v_2(t)\| ^2\\
&\le e^{ct}(\| u_{0,1}-u_{0,2}\| ^2+\epsilon (\| v_{0,1}-v_{0,2}\| ^2),
\quad t\ge 0.
\end{aligned}
\end{equation}
This yields the uniqueness, as well as the continuous dependence with respect to the
initial data in the $L^2$-norm.
\smallskip

\noindent (b) Regularity estimates:
We assume that \eqref{1}-\eqref{4} possesses a solution $(u,v)$ such that
\eqref{6} is satisfied and the estimates below are justified.

We multiply \eqref{1} by $u$ and \eqref{2} by $v$ and find, summing the two
resulting equalities,
$$
{1\over 2}{d\over {dt}}(\| u\| ^2+\epsilon \| v\| ^2)
+\alpha \| \nabla u\| ^2+\beta \| \nabla v\| ^2
\le (J+\kappa )\int _\Omega u\, dx+(FL+\kappa )\int _\Omega v\, dx,
$$
which yields
\begin{equation}\label{13}
{d\over {dt}}(\| u\| ^2+\epsilon \| v\| ^2)
+c(\| \nabla u\| ^2+\| \nabla v\| ^2)
\le c'(\| u\| ^2+\epsilon \| v\| ^2)+c'',\ c>0,
\end{equation}
 whence estimates on $u$ and $v$ in $L^\infty (0,T;L^2(\Omega ))$ and
$L^2(0,T;H^1(\Omega ))$, for all $T>0$.

\begin{remark} \label{rmk2.2} \rm
Multiplying \eqref{2} by $v$ gives
$$
{\epsilon \over 2}{d\over {dt}}\| v\| ^2+\beta \| \nabla v\| ^2+
F\| v\| ^2\le c\| v\| ,
$$
which yields
$$
{\epsilon }{d\over {dt}}\| v\| ^2+F\| v\| ^2\le c,
$$
whence, owing to Gronwall's lemma,
$$
\| v(t)\| ^2\le e^{-{F\over \epsilon}t}\| v_0\| ^2+c,\ t\ge 0,
$$
so that the estimate on $v$ in $L^2(\Omega )$ is global in time and even dissipative
(in the sense that it is independent of time and
bounded sets of initial data, at least for large times). We can also note that
this estimate only depends on the initial datum for $v$.
\end{remark}

Next, we multiply \eqref{1} by $-\Delta u$ and \eqref{2} by $-\Delta v$ and have,
summing the two resulting equalities,
$$
{1\over 2}{d\over {dt}}(\| \nabla u\| ^2+\epsilon \| \nabla v\| ^2)
+\alpha \| \Delta u\| ^2+\beta \| \Delta v\| ^2
\le c(\| \Delta u\| +\| \Delta v\| ),
$$
owing again to \eqref{6}, which yields
\begin{equation}\label{14}
{d\over {dt}}(\| \nabla u\| ^2+\epsilon \| \nabla v\| ^2)+c
(\| \Delta u\| ^2+\| \Delta v\| ^2)\le c',\quad  c>0,
\end{equation}
and we obtain estimates on $u$ and $v$ in $L^\infty (0,T;H^1(\Omega ))$ and
$L^2(0,T;H^2(\Omega ))$, for all $T>0$.

We then differentiate \eqref{1} and \eqref{2} with respect to time to have
\begin{gather}\label{16}
{\partial \over {\partial t}}\frac{\partial u}{\partial t}
-\alpha \Delta \frac{\partial u}{\partial t}+\kappa ({k\over {(k+u)^2}}
\frac{\partial u}{\partial t}
-{{k'}\over {(k'+v)^2}}\frac{\partial v}{\partial t})=0,\\
\label{17}
\epsilon {\partial \over {\partial t}}\frac{\partial v}{\partial t}
-\beta \Delta \frac{\partial v}{\partial t}+F\frac{\partial v}{\partial t}
+\kappa ({{k'}\over {(k'+v)^2}}\frac{\partial v}{\partial t}
-{{k}\over {(k+u)^2}}\frac{\partial u}{\partial t})=0, \\
\label{18}
{\partial \over {\partial \nu }}\frac{\partial u}{\partial t}=
{\partial \over {\partial \nu }}\frac{\partial v}{\partial t}=0\quad\text{on }
 \Gamma ,
\end{gather}
where the initial data
\begin{gather}\label{19}
\frac{\partial u}{\partial t}(0)=J+\alpha \Delta u_0+\kappa ({k\over {k+u_0}}-
{{k'}\over {k'+v_0}}), \\
\label{20}
\frac{\partial v}{\partial t}(0)={1\over \epsilon }
(FL+\beta \Delta v_0-Fv_0+\kappa ({{k'}\over {k'+v_0}}-{k\over {k+u_0}}))
\end{gather}
belong to $H^1(\Omega )$.

We multiply \eqref{16} by $\frac{\partial u}{\partial t}$ and \eqref{17} by
$\frac{\partial v}{\partial t}$ and obtain, summing the two resulting
 equalities and owing once more to \eqref{6},
\begin{equation}\label{21}
{d\over {dt}}(\| \frac{\partial u}{\partial t}\| ^2+\epsilon
\| \frac{\partial v}{\partial t}\| ^2)+c(\| \nabla \frac{\partial u}{\partial t}\| ^2+
\| \nabla \frac{\partial v}{\partial t}\| ^2)\le
c'(\| \frac{\partial u}{\partial t}\| ^2+\epsilon
\| \frac{\partial v}{\partial t}\| ^2),\ c>0,
\end{equation}
whence estimates on $\frac{\partial u}{\partial t}$ and
$\frac{\partial v}{\partial t}$ in $L^\infty (0,T;L^2(\Omega ))$ and
$L^2(0,T;H^1(\Omega ))$, for all $T>0$.

We also multiply \eqref{16} by $-\Delta \frac{\partial u}{\partial t}$ and \eqref{17}
by $-\Delta \frac{\partial v}{\partial t}$ and find
\begin{align*}
&{d\over {dt}}(\| \nabla \frac{\partial u}{\partial t}\| ^2+\epsilon
\| \nabla \frac{\partial v}{\partial t}\| ^2)
 +c(\| \Delta \frac{\partial u}{\partial t}\| ^2+
\| \Delta \frac{\partial v}{\partial t}\| ^2) \\
&\le c'(\| \frac{\partial u}{\partial t}\| +
\| \frac{\partial v}{\partial t}\| )(\| \Delta \frac{\partial u}{\partial t}\| +
\| \Delta \frac{\partial v}{\partial t}\| ),
\end{align*}
which yields
\begin{equation}\label{22}
{d\over {dt}}(\| \nabla \frac{\partial u}{\partial t}\| ^2
 +\epsilon \| \nabla \frac{\partial v}{\partial t}\| ^2)
 +c(\| \Delta \frac{\partial u}{\partial t}\| ^2
 +\| \Delta \frac{\partial v}{\partial t}\| ^2)
\le c'(\| \frac{\partial u}{\partial t}\| ^2+
\| \frac{\partial v}{\partial t}\| ^2),
\end{equation}
with $c>0$,
whence estimates on $\frac{\partial u}{\partial t}$ and
$\frac{\partial v}{\partial t}$ in $L^\infty (0,T;H^1(\Omega ))$ and
$L^2(0,T;H^2(\Omega ))$, for all $T>0$.
We now rewrite \eqref{1}-\eqref{2} as an elliptic system, for $t>0$ fixed,
\begin{gather}\label{23}
-\alpha \Delta u=g(x,t),\quad \frac{\partial u}{\partial \nu}=0\quad\text{on }
 \Gamma , \\
\label{24}
-\beta \Delta v=h(x,t),\ \frac{\partial v}{\partial \nu}=0\quad\text{on } \Gamma ,
\end{gather}
where
\begin{gather}\label{25}
g=J-\frac{\partial u}{\partial t}-\kappa ({{k'}\over {k'+v}}-{k\over {k+u}}), \\
\label{26}
h=FL-\epsilon \frac{\partial v}{\partial t}-Fv
-\kappa ({{k}\over {k+u}}-{{k'}\over {k'+v}})
\end{gather}
belong to $L^\infty (0,T;L^2(\Omega ))$ and
$L^2(0,T;H^1(\Omega ))$, $\forall T>0$. Indeed, note that
$$
{\partial \over {\partial x_i}}{1\over {k+u}}
=-{1\over {(k+u)^2}}{{\partial u}\over {\partial x_i}},\quad
{\partial \over {\partial x_i}}{1\over {k'+v}}
=-{1\over {(k'+v)^2}}{{\partial v}\over {\partial x_i}},\quad
i=1,\dots, n.
$$
It thus follows from standard elliptic regularity results
(see, e.g., \cite{ADN1} and \cite{ADN2}) that
$(u,v)\in L^\infty (0,T;H^2(\Omega )^2)\cap L^2(0,T;H^3(\Omega )^2)$,
for all $T>0$.
This, in turn, yields that $g$ and $h$ belong to $L^\infty (0,T;H^1(\Omega ))$
and $L^2(0,T;H^2(\Omega ))$, for all $T>0$. Indeed, note that
$$
{{\partial ^2}\over {\partial x_i\partial x_j}}{1\over {k+u}}
= {2\over {(k+u)}^3}{{\partial u}\over {\partial x_i}}
 {{\partial u}\over {\partial x_j}}
-{1\over {(k+u)^2}}{{\partial ^2u}\over {\partial x_i\partial x_j}},\
i,\; j=1,\dots, n,
$$
and
\begin{align*}
\| {{\partial ^2}\over {\partial x_i\partial x_j}}{1\over {k+u}}\|
&\le c(\| {{\partial u}\over {\partial x_i}}{{\partial u}\over {\partial x_j}}\|
+\| {{\partial ^2u}\over {\partial x_i\partial x_j}}\| ) \\
&\le c(\| {{\partial u}\over {\partial x_i}}\| _{L^4(\Omega )}
\| {{\partial u}\over {\partial x_j}}\| _{L^4(\Omega )}
+\| {{\partial ^2u}\over {\partial x_i\partial x_j}}\| ) \\
&\le c(\| u\| ^2_{H^2(\Omega )}+1),
\end{align*}
where $i,j=1, \dots, n$, owing to the continuous embedding
$H^1(\Omega )\subset L^4(\Omega )$. We proceed in a similar way for $v$.
Again, standard elliptic regularity results
yield that $(u,v)\in L^\infty (0,T;H^3(\Omega )^2)\cap L^2(0,T;H^4(\Omega )^2)$,
for all $T>0$.
\smallskip

(c) Existence of nonnegative solutions:
We consider the  initial and boundary value problem
\begin{gather}\label{a1}
\frac{\partial u}{\partial t}-\alpha \Delta u+\kappa ({{u}\over {k+\vert u\vert }}-
{{v}\over {k'+\vert v\vert }})=J, \\
\label{a2}
\epsilon \frac{\partial v}{\partial t}-\beta \Delta v+Fv
+\kappa ({{v}\over {k'+\vert v\vert }}- {{u}\over {k+\vert u\vert }})=FL, \\
\label{a3}
\frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=0\quad
\text{on } \Gamma , \\
\label{a4}
u|_{t=0}=u_0,\quad  v|_{t=0}=v_0.
\end{gather}

Noting that $f(s)={s\over {c+\vert s\vert }}$, $c>0$ given, is of class
 $\mathcal C^1$, where
$f'(s)={c\over {(c+\vert s\vert )^2}}$ is bounded on $\mathbb{R}$, so that
$f$ is also Lipschitz continuous,
we can prove the existence and uniqueness
of the weak (i.e., variational) solution to \eqref{a1}-\eqref{a4}
 (we refer the interested reader to, e.g.,
\cite{marion}, \cite{smoller}
and \cite{temam} for developments on reaction-diffusion equations and systems).
Furthermore, this solution satisfies regularity estimates which are similar
to those derived above and is thus strong  (i.e., it satisfies
\eqref{a1}-\eqref{a4} a.e. $(x,t)$). This, together
with the existence of a solution, can be done by considering a standard
Galerkin scheme, taking a spectral basis associated with the spectrum of the
minus Laplace operator, with Neumann boundary conditions, as Galerkin basis.

We then multiply \eqref{a1} by $-u^-$, where $u^-=\min (0,-u)$, and have
\begin{equation}\label{a5}
{1\over 2}{d\over {dt}}\| u^-\| ^2+\alpha \| \nabla u^-\| ^2+
\kappa \int _\Omega {{\vert u^-\vert ^2}\over {k+\vert u\vert }}\, dx+
\kappa \int _\Omega {{vu^-}\over {k'+\vert v\vert }}\, dx\le 0.
\end{equation}
Noting that $v=v^+-v^-$, where $v^+=\max (0,v)$, and $u^-$ and $v^+$
are nonnegative, it follows that
\begin{equation}\label{a6}
{d\over {dt}}\| u^-\| ^2\le 2\kappa \int _\Omega {{v^-u^-}\over {k'+\vert v\vert }}\,
 dx,
\end{equation}
whence
\begin{equation}\label{a7}
{d\over {dt}}\| u^-\| ^2\le c(\| u^-\| ^2+\epsilon \| v^-\| ^2).
\end{equation}
Similarly, multiplying \eqref{a2} by $-v^-$, we obtain
\begin{equation}\label{a8}
\epsilon {d\over {dt}}\| v^-\| ^2\le c(\| u^-\| ^2+\epsilon \| v^-\| ^2).
\end{equation}
Summing \eqref{a7} and \eqref{a8}, we find
\begin{equation}\label{a9}
{d\over {dt}}(\| u^-\| ^2+\epsilon \| v^-\| ^2)
\le c(\| u^-\| ^2+\epsilon \| v^-\| ^2).
\end{equation}
Gronwall's lemma finally yields
\begin{equation}\label{a10}
\| u^-(t)\| ^2+\epsilon \| v^-(t)\| ^2\le e^{ct}(\| u^-_0\| ^2
+\epsilon \| v^-_0\| ^2),
\end{equation}
whence $u^-=v^-=0$ (recall that $u_0\ge 0$ and $v_0\ge 0$
a.e. $x$) and $u\ge 0$ and $v\ge 0$ a.e. $(x,t)$. This means that
$(u,v)$ is solution to \eqref{1}-\eqref{4} and the regularity estimates
derived above are now fully justified.
\end{proof}

\begin{remark}\label{nid} \rm
(i) Actually, the biologically relevant quadrant $\{u\ge 0,\ v\ge 0\}$ is an
invariant region (see \cite{smoller}) for both systems \eqref{1}-\eqref{2}
and \eqref{a1}-\eqref{a2}, meaning that solutions starting from this region
cannot leave it. We chose however to give a proof of the nonnegativity
of the solutions, also having in mind more elaborate models
which we will study in forthcoming papers.
\smallskip

\noindent (ii) We now assume that
\begin{equation}\label{n1}
u_0\ge -\delta _1\quad\text{and}\quad  v_0\ge -\delta _2\quad\text{a.e. }x,
\end{equation}
where $\delta _1$ and $\delta _2$ are positive (and are intended to be small).
We then consider the  modified initial and boundary value problem
\begin{gather}\label{n10}
\frac{\partial u}{\partial t}-\alpha \Delta u
+\kappa ({u\over {k-\delta _1+\vert u+\delta _1\vert }}-
{v\over {k'-\delta _2+\vert v+\delta _2\vert }})=J, \\
\label{n11}
\epsilon \frac{\partial v}{\partial t}-\beta \Delta v
+Fv+\kappa ({v\over {k'-\delta _2+\vert v+\delta _2\vert }}-
{u\over {k-\delta _1+\vert u+\delta _1\vert }})=FL, \\
\label{n11bis}
\frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=0
\quad\text{on } \Gamma , \\
\label{n12}
u|_{t=0}=u_0,\quad  v|_{t=0}=v_0,
\end{gather}
 where $\delta _1$ and $\delta _2$ are chosen such that $k-\delta _1>0$
and $k'-\delta _2>0$. The existence and uniqueness of the solution to
\eqref{n10}-\eqref{n12}
is straightforward. Next, we set $\tilde u=u+\delta _1$ and $\tilde v=v+\delta _2$.
These functions are solutions to
\begin{gather}\label{n10'}
{{\partial \tilde u}\over {\partial t}}-\alpha \Delta \tilde u
+\kappa ({{\tilde u}\over {k-\delta _1+\vert \tilde u\vert }}-
{{\tilde v}\over {k'-\delta _2+\vert \tilde v\vert }})=\tilde J, \\
\label{n11'}
\epsilon {{\partial \tilde v}\over {\partial t}}-\beta \Delta \tilde v
+F\tilde v+\kappa ({{\tilde v}\over {k'-\delta _2+\vert \tilde v\vert }}-
{{\tilde u}\over {k-\delta _1+\vert \tilde u\vert }})=\tilde F, \\
\label{n11ter}
{{\partial \tilde u}\over {\partial \nu }}
={{\partial \tilde v}\over {\partial \nu }}=0\quad\text{on } \Gamma , \\
\label{n12'}
\tilde u|_{t=0}=u_0+\delta _1,\quad  \tilde v|_{t=0}=v_0+\delta _2,
\end{gather}
 where
\begin{gather*}
\tilde J=J+\kappa ({{\delta _1}\over {k-\delta _1+\vert \tilde u\vert }}-
{{\delta _2}\over {k'-\delta _2+\vert \tilde v\vert }}), \\
\tilde F=F(L+\delta _2)-\kappa ({{\delta _1}\over {k-\delta _1+\vert \tilde u\vert }}-
{{\delta _2}\over {k'-\delta _2+\vert \tilde v\vert }}).
\end{gather*}
Choosing $\delta _1$ and $\delta _2$ such that $\tilde J\ge 0$ and $\tilde F\ge 0$
(in particular, these hold when $\delta _1$ and $\delta _2$ are small enough)
and noting that $\tilde u(0)\ge 0$ and $\tilde v(0)\ge 0$ a.e. $x$,
 we can prove, as in the proof of Theorem
\ref{theorem1}, that $\tilde u(x,t)\ge 0$ and $\tilde v(x,t)\ge 0$ a.e. $(x,t)$,
so that $(u,v)$ is solution to \eqref{1}-\eqref{3} (here, the quadrant
$\{u\ge -\delta _1,\ v\ge -\delta _2\}$
is an invariant region). For general negative initial data however,
the existence of a global
in time solution is much more involved.
\end{remark}


\begin{theorem}
Under the assumptions of Theorem \ref{theorem1}, the solution $(u,v)$
to \eqref{1}-\eqref{4} such that \eqref{6} holds satisfies
\begin{gather*}
\| u(t)\| _{L^\infty (\Omega )}\le \| u_0\| _{L^\infty (\Omega )}
+(J+\kappa )t,\ t\ge 0, \\
\| v(t)\| _{L^\infty (\Omega )}\le e^{-{F\over \epsilon}t}\| v_0\| _{L^\infty (\Omega )}
+{{FL+\kappa }\over F},\ t\ge 0.
\end{gather*}
\end{theorem}

\begin{proof}
We note that, owing to \eqref{6},
\begin{gather}\label{27}
\frac{\partial u}{\partial t}-\alpha \Delta u\le J+\kappa , \\
\label{28}
\epsilon \frac{\partial v}{\partial t}-\beta \Delta v+Fv\le FL+\kappa .
\end{gather}
We multiply \eqref{27} by $u^{m+1}$, $m\in \mathbb{N} $, and have
$$
{1\over {m+2}}{d\over {dt}}\| u\| ^{m+2}_{L^{m+2}(\Omega )}+\alpha (m+1)
\int _\Omega u^{m}\vert \nabla u\vert ^2\, dx
\le (J+\kappa )\int _\Omega u^{m+1}\, dx,
$$
 which yields
\begin{equation}\label{29}
\| u\| ^{m+1}_{L^{m+2}(\Omega )}{d\over {dt}}\| u\| _{L^{m+2}(\Omega )}\le
(J+\kappa ){\rm Vol}(\Omega )^{1\over {m+2}}
\| u\| ^{m+1}_{L^{m+2}(\Omega )}.
\end{equation}
Therefore, $\| u(t)\| _{L^{m+2}(\Omega )}=0$ or $\| u(t)\| _{L^{m+2}(\Omega )}>0$
and
\begin{equation}\label{30}
{d\over {dt}}\| u\| _{L^{m+2}(\Omega )}\le
(J+\kappa ){\rm Vol}(\Omega )^{1\over {m+2}}.
\end{equation}
In the later case (note that it follows from the
regularity given in Theorem \ref{theorem1} that $u$ is continuous, both in space
and time), for $t>0$ given, either $\| u(s)\| _{L^{m+2}(\Omega )}>0$, for all
$s\in (0,t]$, in which case
\begin{equation}\label{31}
\| u(t)\| _{L^{m+2}(\Omega )}\le \| u_0\| _{L^{m+2}(\Omega )}
+(J+\kappa ){\rm Vol}(\Omega )^{1\over {m+2}}t,
\end{equation}
or there exists $t_0\in (0,t]$ such that $\| u(t_0)\| _{L^{m+2}(\Omega )}=0$
and $\| u(s)\| _{L^{m+2}(\Omega )}>0$, for all $s\in (t_0,t]$, in which
case
$$
\| u(t)\| _{L^{m+2}(\Omega )}\le
(J+\kappa ){\rm Vol}(\Omega )^{1\over {m+2}}(t-t_0),
$$
so that \eqref{31} again holds. Noting that \eqref{31}
 also holds when $\| u(t)\| _{L^{m+2}(\Omega )}=0$, we obtain, passing
to the limit $m\to +\infty $ (see, e.g., \cite{alikakos}),
\begin{equation}\label{32}
\| u(t)\| _{L^{\infty }(\Omega )}\le \| u_0\| _{L^{\infty }(\Omega )}
+(J+\kappa )t,\quad  t\ge 0.
\end{equation}

Proceeding in a similar way for \eqref{28}, we find
\begin{equation}\label{33}
\epsilon {d\over {dt}}\| v\| _{L^{m+2}(\Omega )}+
F\| v\| _{L^{m+2}(\Omega )}\le
(FL+\kappa ){\rm Vol}(\Omega )^{1\over {m+2}}.
\end{equation}
Employing Gronwall's lemma, we deduce from \eqref{33} that
\begin{equation}\label{34}
\| v(t)\| _{L^{m+2}(\Omega )}\le e^{-{F\over \epsilon}t}\| v_0\| _{L^{m+2}(\Omega )}
+{{FL+\kappa }\over F}{\rm Vol}(\Omega )^{1\over {m+2}},\quad  t\ge 0.
\end{equation}
Passing to the limit $m\to +\infty $, we finally have
\begin{equation}\label{35}
\| v(t)\| _{L^{\infty }(\Omega )}\le e^{-{F\over \epsilon }t}
\| v_0\| _{L^{\infty }(\Omega )}
+{{FL+\kappa }\over F},\ t\ge 0,
\end{equation}
which completes the proof.
\end{proof}

\begin{remark} \label{rmk2.5} \rm
 (i) In particular,  from \eqref{35}
(which is a dissipative estimate) it follows that
if $\| v_0\| _{L^\infty (\Omega)}\le R$
and $\delta >0$ is given,
then there exists $t_0=t_0(R,\delta )>0$ such that
\begin{equation}\label{r2}
\| v(t)\| _{L^\infty (\Omega)}\le {{FL+\kappa }\over F}+\delta ,\quad t\ge t_0.
\end{equation}
Let now $M$ be such that $F(L-M)+\kappa \le 0$, i.e., $M\ge  {{FL+\kappa }\over F}$,
and $v_0$ be such that $0\le v_0\le M$ a.e. $x$. Setting
$\tilde v=v-M$, we obtain
\begin{gather}\label{36}
\epsilon {{\partial \tilde v}\over {\partial t}}-\beta \Delta \tilde v
+F\tilde v\le 0, \\
\label{n11ter''}
{{\partial \tilde v}\over {\partial \nu }}=0\quad\text{on }\Gamma ,
\end{gather}
where $\tilde v(0)\le 0$. Multiplying \eqref{36} by ${\tilde v}^+$, we easily find
$$
{d\over {dt}}\| {\tilde v}^+\| ^2\le 0,
$$
whence ${\tilde v}^+=0$ and $0\le v\le M$ a.e. $(x,t)$ (compare with \eqref{r2}),
meaning that the capillary lactate concentration is uniformly bounded
(or ultimately uniformly bounded in \eqref{r2}).
Now, we have not been able to derive
a similar upper bound on the interstitial lactate concentration $u$.
We can note that, in the biological model, outside a bounded viability domain,
cell necrosis occurs (see \cite{lahutte}), meaning that one expects viable
trajectories to be uniformly bounded.
\smallskip

\noindent (ii) Multiplying \eqref{i3} by $u+k$, integrating over $\Omega  $
and by parts, we obtain
$$
{{dE}\over {dt}}+\alpha \| \nabla u\| ^2+\kappa \| u\| _{L^1(\Omega )}
=((J+{{\kappa v}\over {k'+v}},u+k)),
$$
where
$$
E={1\over 2}\| u\| ^2+k\| u\| _{L^1(\Omega )}.
$$
Noting that $v$ is uniformly bounded (we assume that, say,
$0\le v_0\le {{FL+\kappa }\over F}$), we take, for
$\kappa $, $J$, $F$ and $L$ given small enough and $k'$ large enough such that
$$
J+{{\kappa v}\over {k'+v}}<\kappa .
$$
We thus deduce that
$$
{{dE}\over {dt}}+\alpha \| \nabla u\| ^2+c\| u\| _{L^1(\Omega )}
\le c',\quad c>0,
$$
which yields, noting that
\begin{align*}
\alpha \| \nabla u\| ^2+c\| u\| _{L^1(\Omega )}
&\ge c'(\| \nabla u\| +\| u\| _{L^1(\Omega )})-c'' \\
&\ge c'(\| u\| +\| u\| _{L^1(\Omega )})-c'',
\end{align*}
the differential inequality
\begin{equation}\label{f1}
{{dE}\over {dt}}+c\sqrt E\le c',\quad c>0.
\end{equation}
Set $E^*=(c'/ c)^2$, where $c$ and $c'$ are the same constants as
in \eqref{f1}, so that
$$
{{dE^*}\over {dt}}+c\sqrt {E^*}=c'.
$$
It then follows from comparison arguments that
\begin{equation}\label{f2}
E(t)\le \max (E(0),E^*),\quad t\ge 0,
\end{equation}
and we finally deduce that the $L^2$-norm of $u$ is uniformly bounded.
\end{remark}

We finally have

\begin{theorem}\label{theorem3}
We further assume that $J\ge \kappa $, $FL\ge \kappa $ and $u_0>0$ and $v_0>0$ a.e.
$x$. Let $(u,v)$ be the solution to \eqref{1}-\eqref{4} such that \eqref{6} holds.
Then, $u>0$ and $v>0$ a.e. $(x,t)$ and
$$
u(x,t)\ge {1\over {\| {1\over {u_0}}\| _{L^{\infty }(\Omega )}}},\quad
v(x,t)\ge {{e^{-{F\over \epsilon }t}}\over {\| {1\over {v_0}}
\| _{L^{\infty }(\Omega )}}}\quad\text{a.e. }(x,t).
$$
\end{theorem}

\begin{proof}
We first note that from \eqref{1}-\eqref{2} it follows that
\begin{gather}\label{37}
\frac{\partial u}{\partial t}-\alpha \Delta u\ge J-\kappa , \\
\label{38}
\epsilon \frac{\partial v}{\partial t}-\beta \Delta v+Fv\ge FL-\kappa .
\end{gather}
Multiplying \eqref{37} by $-{1\over u}$, we have
$$
{d\over {dt}}\int _\Omega \ln {1\over u}\, dx\le 0,
$$
whence
$$
\int _\Omega \ln {1\over {u(t)}}\, dx
\le \int _\Omega \ln {1\over {u_0}}\, dx,\quad  t\ge 0,
$$
and $u(x,t)>0$ a.e. $(x,t)$. We proceed in a similar way to prove that
$v(x,t)>0$ a.e. $(x,t)$.

Next, we multiply \eqref{37} by $-{1\over {u^{m+1}}}$, $m\in \mathbb{N} $, and find that
$$
{1\over {m}}{d\over {dt}}\| {1\over u}\| ^{m}_{L^{m}(\Omega )}+\alpha (m+1)
\int _\Omega {{\vert \nabla u\vert ^2}\over {u^{m+2}}}\, dx
\le 0,
$$
whence
\begin{equation}\label{39}
\| {1\over {u(t)}}\| _{L^{m}(\Omega )}\le \| {1\over {u_0}}\| _{L^{m}(\Omega )},\quad
 t\ge 0.
\end{equation}
Passing to the limit $m\to +\infty $, we deduce that
\begin{equation}\label{40}
\| {1\over {u(t)}}\| _{L^{\infty }(\Omega )}
\le \| {1\over {u_0}}\| _{L^{\infty }(\Omega )},\quad t\ge 0.
\end{equation}
Multiplying  \eqref{38} by $-{1\over {v^{m+1}}}$, $m\in \mathbb{N} $, we have
$$
{\epsilon \over {m}}{d\over {dt}}\| {1\over v}\| ^{m}_{L^{m}(\Omega )}
\le F\| {1\over v}\| ^{m}_{L^{m}(\Omega )},
$$
which yields
\begin{equation}\label{29'}
\epsilon {d\over {dt}}\| {1\over v}\| _{L^{m}(\Omega )}
\le F\| {1\over v}\| _{L^{m}(\Omega )},
\end{equation}
whence, employing Gronwall's lemma,
\begin{equation}\label{30'}
\| {1\over {v(t)}}\| _{L^{m}(\Omega )}
\le \| {1\over {v_0}}\| _{L^{m}(\Omega )}e^{{F\over
\epsilon }t}, \quad t\ge 0.
\end{equation}
Passing to the limit $m\to +\infty $, we deduce that
\begin{equation}\label{31'}
\| {1\over {v(t)}}\| _{L^{\infty }(\Omega )}
\le \| {1\over {v_0}}\| _{L^{\infty }(\Omega )}e^{{F\over \epsilon }t},\quad t\ge 0,
\end{equation}
which completes the proof.
\end{proof}

\begin{remark} \label{rmk2.7} \rm
Proceeding as in the proof of Theorem \ref{theorem1} (see also
Remark \ref{nid}, (ii)), we can prove that, if
$$
u_0\ge \delta _1\quad\text{and}\quad  v_0\ge \delta _2\quad\text{a.e. } x,
$$
where $\delta _1$ and $\delta _2$ are positive and small enough, then
$$
u(x,t)\ge \delta _1\quad\text{and}\quad
 v(x,t)\ge \delta _2\quad\text{a.e. }(x,t).
$$
\end{remark}

\begin{remark} \label{rmk2.8} \rm
It is interesting to note that, as far as the $L^\infty $-estimates are concerned,
the system behaves as if it were uncoupled.
\end{remark}

\section{A stability result}

As shown in \cite{C,lahutte,LCFG,L}, \eqref{1}-\eqref{3} possesses a
unique spatially homogeneous
stationary solution $(u,v)=({\overline u},{\overline v})$ given by
\begin{gather}\label{s1}
{\overline v}=L+{J\over F}>0, \\
\label{s2}
{\overline u}={{k({J\over \kappa }+{{{\overline v}}\over {k'+{\overline v}}})}\over
{1-({J\over \kappa }+{{{\overline v}}\over {k'+{\overline v}}})}}.
\end{gather}
Note that ${\overline u}$ is not necessarily positive. We thus assume in what
follows that
\begin{equation}\label{s3}
{\overline u}>0.
\end{equation}

The linearized (around $({\overline u},{\overline v})$) system reads
\begin{gather}\label{s4}
{{\partial U}\over {\partial t}}-\alpha \Delta U+\kappa
({k\over {(k+{\overline u})^2}}U-{{k'}\over {(k'+{\overline v})^2}}V)=0, \\
\label{s5}
\epsilon {{\partial V}\over {\partial t}}-\beta \Delta V+FV+\kappa
({{k'}\over {(k'+{\overline v})^2}}V-{k\over {(k+{\overline u})^2}}U)=0, \\
\label{s6}
{{\partial U}\over {\partial \nu }}={{\partial V}\over {\partial \nu }}=0\quad\text{on }
\Gamma , \\
\label{s7}
U|_{t=0}=U_0,\quad  V|_{t=0}=V_0.
\end{gather}
Noting that \eqref{s4}-\eqref{s5} is a linear system, it is not difficult
to prove the following result.

\begin{theorem}\label{theorem4}
We assume that $(U_0,V_0)\in L^2(\Omega )^2$. Then \eqref{s4}-\eqref{s7}
possesses a unique weak solution $(U,V)$ such that, for all $T>0$,
$$
(U,V)\in L^\infty (0,T;L^2(\Omega )^2)\cap L^2(0,T;H^1(\Omega )^2).
$$
If we further assume that $(U_0,V_0)\in H^1(\Omega )^2$, then, for all $T>0$,
$$
(U,V)\in L^\infty (0,T;H^1(\Omega )^2)\cap L^2(0,T;H^2(\Omega )^2).
$$
Finally, if $(U_0,V_0)\in H^2_{\rm N}(\Omega )^2$, then,  for all $T>0$,
$$
(U,V)\in L^\infty (0,T;H^2_{\rm N}(\Omega )^2)\cap L^2(0,T;H^3(\Omega )^2)
$$
and the solution is strong.
\end{theorem}

\begin{remark} \label{rmk3.2} \rm
It is easy to prove that, if $U_0\ge 0$ and $V_0\ge 0$ a.e. $x$,
then $U\ge 0$ and $V\ge 0$ a.e. $(x,t)$.
\end{remark}

\begin{theorem}\label{theorem6}
The stationary solution $({\overline u},{\overline v})$ is linearly stable
in $L^2(\Omega )^2$.
\end{theorem}

\begin{proof}
We multiply \eqref{s4} by ${k\over {(k+{\overline u})^2}}U$ and \eqref{s5}
by ${{k'}\over {(k'+{\overline v})^2}}V$
and obtain, summing the two resulting equalities,
\begin{equation}\label{s14}
\begin{aligned}
&{1\over 2}{d\over {dt}}({k\over {(k+{\overline u})^2}}\| U\| ^2
 +\epsilon {{k'}\over {(k'+{\overline v})^2}}\| V\| ^2) \\
&+{{\alpha k}\over {(k+{\overline u})^2}}\| \nabla U\| ^2+
{{\beta k'}\over {(k'+{\overline v})^2}}\| \nabla V\| ^2
 +{{Fk'}\over {(k'+{\overline v})^2}}\| V\| ^2 \\
&+\kappa ({{k^2}\over {(k+{\overline u})^4}}\| U\| ^2
 +{{{k'}^2}\over {(k'+{\overline v})^4}}
\| V\| ^2-{{2kk'}\over {(k+{\overline u})^2(k'+{\overline v})^2}}
 \int _\Omega UV\, dx)=0.
\end{aligned}
\end{equation}
Noting that
\begin{align*}
&{{k^2}\over {(k+{\overline u})^4}}\| U\| ^2+{{{k'}^2}\over {(k'+{\overline v})^4}}
\| V\| ^2-{{2kk'}\over {(k+{\overline u})^2(k'+{\overline v})^2}}\int _\Omega UV\, dx\\
&=\int _\Omega \big({k\over {(k+{\overline u})^2}}U-{{k'}\over {(k'+{\overline v})^2}}
V\big )^2\, dx\ge 0,
\end{align*}
we deduce from \eqref{s14} that
\begin{equation}\label{s16}
{d\over {dt}}({k\over {(k+{\overline u})^2}}\| U\| ^2
+\epsilon {{k'}\over {(k'+{\overline v})^2}}\| V\| ^2)\le 0,
\end{equation}
whence
\begin{equation}\label{s17}
{k\over {(k+{\overline u})^2}}\| U(t)\| ^2
+\epsilon {{k'}\over {(k'+{\overline v})^2}}\| V(t)\| ^2
\le {k\over {(k+{\overline u})^2}}\| U_0\| ^2
 +\epsilon {{k'}\over {(k'+{\overline v})^2}} \| V_0\| ^2
\end{equation}
and the result follows.
\end{proof}

\begin{remark} \label{rmk3.4} \rm
Similarly, multiplying \eqref{s4} by $-{k\over {(k+{\overline u})^2}}\Delta U$ and
\eqref{s5} by $-{{k'}\over {(k'+{\overline v})^2}}\Delta V$,
we obtain the linear stability of the stationary solution
$({\overline u},{\overline v})$ in $H^1(\Omega )^2$.
\end{remark}

Actually, we can do better and prove the following result.

\begin{theorem}\label{theorem7}
The stationary solution $({\overline u},{\overline v})$ is linearly exponentially
stable, in the sense that all eigenvalues $s\in \mathbb{C} $ associated with the linear
system \eqref{s4}-\eqref{s6} satisfy $\mathcal R e(s)\le -\xi $, for a given
$\xi >0$, $\mathcal R e$ denoting the real part.
\end{theorem}

\begin{proof}
We look for solutions of the form
\begin{equation}\label{R1}
U(x,t)=\hat U(x)e^{st},\quad  V(x,t)=\hat V(x)e^{{s\over \epsilon }t},
\end{equation}
for $s\in \mathbb{C} $. Inserting this into \eqref{s4}-\eqref{s6}, we have
\begin{gather}\label{R2}
-\alpha \Delta \hat U+s\hat U+\kappa ({k\over {(k+{\overline u})}^2}
\hat U-{{k'}\over {(k'+{\overline v})}^2}\hat V)=0, \\
\label{R3}
-\beta \Delta \hat V+(s+F)\hat V+\kappa ({{k'}\over {(k'+{\overline v})}^2}
\hat V-{{k}\over {(k+{\overline u})}^2}\hat U)=0, \\
\label{R4}
{{\partial \hat U}\over {\partial \nu }}={{\partial \hat V}\over {\partial \nu }}=0
\quad\text{on }\Gamma .
\end{gather}
Summing \eqref{R2} and \eqref{R3}, we obtain
\begin{gather}\label{R5}
-\Delta (\alpha \hat U+\beta \hat V)+{{s+F}\over \beta }(\alpha \hat U+\beta \hat V)
+(s-{{\alpha (s+F)}\over \beta })\hat U=0, \\
\label{R6}
{\partial \over {\partial \nu }}(\alpha \hat U+\beta \hat V)=0
\quad\text{on }\Gamma ,
\end{gather}
so that
\begin{equation}\label{R7}
\hat V={{\alpha F+(\alpha -\beta )s}\over {\beta ^2}}
(-\Delta +{{s+F}\over \beta }I)^{-1}\hat U
-{\alpha \over \beta }\hat U.
\end{equation}
 Inserting this into \eqref{R2}, we find
\begin{equation}\label{R8}
-\alpha \Delta \hat U+\delta \hat U-\gamma
(-\Delta +{{s+F}\over \beta }I)^{-1}\hat U=0,
\end{equation}
where
\begin{gather*}
\delta =s+{{\kappa k}\over {(k+{\overline u})}^2}
+{{\kappa k'\alpha }\over {\beta (k'+{\overline v})}^2}, \\
\gamma ={{\kappa k'(\alpha F+(\alpha -\beta )s)}\over {\beta ^2(k'+{\overline v})}^2}.
\end{gather*}
This yields
\begin{equation}\label{R11}
\alpha \Delta ^2\hat U-({{\alpha (s+F)}\over \beta }+\delta )\Delta \hat U+
({{\delta (s+F)}\over \beta }-\gamma )\hat U=0,
\end{equation}
 where, in view of \eqref{R4} and \eqref{R8},
\begin{equation}\label{R12}
{{\partial \hat U}\over {\partial \nu }}
={{\partial \Delta \hat U}\over {\partial \nu }}=0 \quad\text{on }\Gamma .
\end{equation}
 We further note that, setting
$$
k_1={{\kappa k}\over {(k+{\overline u})}^2},\quad
k_2={{\kappa k'}\over {(k'+{\overline v})}^2},
$$
we have
\begin{gather}\label{13'}
{{\alpha (s+F)}\over \beta }+\delta =(1+{\alpha \over \beta })s+
k_1+{\alpha {k_2}\over \beta }
+{\alpha F\over \beta }, \\
\label{R13}
{{\delta (s+F)}\over \beta }-\gamma ={1\over \beta }(s^2+(k_1
+k_2+F)s+k_1F).
\end{gather}
Thus to study the stability of $(\overline u,\overline v)$, we need 
to study the eigenvalues/eigenvectors of problem \eqref{R11}-\eqref{R12}.

We first assume that $s\in \mathbb{R}$. Then, when $s\ge 0$, noting that
${{\alpha (s+F)}\over \beta }+\delta >0$
and ${{\delta (s+F)}\over \beta }-\gamma >0$, we easily prove that the only
solution to \eqref{R11}-\eqref{R12} is the trivial one, $\hat U\equiv 0$.
Furthermore, \eqref{R11}-\eqref{R12} can have nontrivial solutions only when
$$
s\in \big[-{{b+\sqrt \theta }\over 2}, -{{b-\sqrt \theta }\over 2}\big],\quad
-{{b-\sqrt \theta }\over 2}<0,
$$
 where
$$
\theta =(k_1
+k_2+F)^2-4k_1F=(k_1-F)^2+k_2^2+2k_1k_2+2k_2F>0
$$
and
$b=k_1+k_2+F$, or
$$
{{\alpha (s+F)}\over \beta }+\delta \le 0,
$$
i.e.,
$$
s\le -{1\over {1+{\alpha \over \beta }}}({{\alpha F}\over \beta }+
k_1+{{\alpha k_2}\over \beta })<0.
$$
Therefore, necessarily,
\begin{equation}\label{R18}
s\le \max \big(-{{b-\sqrt \theta }\over 2},
-{1\over {1+{\alpha \over \beta }}}({{\alpha F}\over \beta }+
k_1+{{\alpha k_2}\over \beta })\big)<0.
\end{equation}

We now assume that $s\in \mathbb{C} \backslash \mathbb{R}$. Setting $s=\zeta +i\eta $,
$\eta \in \mathbb{R}\backslash \{0\}$, we obtain,
multiplying \eqref{R11} by the conjugate of $\hat U$, integrating over $\Omega $
and by parts and taking the imaginary part,
\begin{equation}\label{R19}
\eta (1+{\alpha \over \beta })\| \vert \nabla \hat U\vert \| ^2
+{{\eta(k_1+k_2+F)+2\zeta \eta }\over \beta }\| \vert \hat U\vert \| ^2=0.
\end{equation}
Therefore, when $\zeta \ge 0$, then, necessarily, $\hat U\equiv 0$. Furthermore,
\eqref{R19} can have nontrivial solutions only when
\begin{equation}\label{R20}
\zeta \le -{{k_1 +k_2+F}\over 2}<0,
\end{equation}
which completes the proof.
\end{proof}

\begin{remark} \label{rmk3.6} \rm
 (i) In \cite{C,lahutte,LCFG,L}, it was proved that $({\overline u},{\overline v})$
is a node for the linearized
system associated with \eqref{i1}-\eqref{i2}, meaning that
it is linearly exponentially stable.

 (ii) As mentioned in the introduction, a therapeutic perspective of such a
result is to have the (spatially homogeneous) steady state outside the viability
domain, where cell necrosis occurs (see \cite{lahutte}).

 (iii) An important question is whether there are other (not spatially homogeneous)
equilibria. This will be addressed elsewhere.
\end{remark}

\section{Concluding remarks}

Possible extensions of our results are the following ones.

 (i) We can consider a time dependent electrical stimulus $F=F(t)$, where
$F$ is continuous and satisfies
$$
0<F_1\le  F(t)\le F_2,\ t\ge 0.
$$
In particular, these assumptions are satisfied by the continuous piecewise linear
stimulus considered in experiments (see \cite{HW}), where $F(0)=F_0>0$,
$F(t)=F_1t$, $t\in [t_0,t_1]$, $F_1>0$, and $F(t)=F_0$, $t\ge t_f$.
In that case, the well-posedness
and some regularity results (here, we cannot differentiate the equations
with respect to time) obtained in this
paper still hold, with minor modifications.


 (ii) We can also consider a forcing term $J=J(x,t,u)$ (such a forcing
term accounts for the interactions with a third
intracellular compartment (which includes both neurons and astrocytes))
such that $J$
is continuous on $\mathbb{R}\times \mathbb{R}^+\times \mathbb{R}$, of class
$\mathcal C^1$ with respect to $t$, $0\le J(x,t,s)\le J_1$,
$\vert {{\partial J}\over {\partial t}} (x,t,s)\vert \le J_2$,
$(x,t,s)\in \mathbb{R}\times \mathbb{R}^+\times \mathbb{R}$, and $J$ is
Lipschitz continuous with respect to $s$, uniformly in $x$ and $t$,
$$
\vert J(x,t,s_1)-J(x,t,s_2)\vert \le c\vert s_1-s_2\vert ,\quad
 x\in \mathbb{R},\; t\in \mathbb{R}^+,\; s_1, s_2 \in \mathbb{R}.
$$
In that case, the well-posedness and regularity results obtained in this
paper still hold, with minor modifications.

 (iii) An interesting problem is to study the limit as $\epsilon $ goes to $0$
in \eqref{i4}. This will be addressed elsewhere.


 (iv) A more general ODE's model for brain lactate kinetics reads
\begin{gather*}
\frac{du}{dt}+\kappa _1({u\over {k+u}}-{p\over {k_n+p}})
+\kappa _2({u\over {k+u}}-{q\over {k_a+q}})
+\kappa ({u\over {k+u}}-{v\over {k'+v}})=J_0, \\
{{dp}\over {dt}}+\kappa _1({p\over {k_n+p}}-{u\over {k+u}})=J_1, \\
{{dq}\over {dt}}+\kappa _2({q\over {k_a+q}}-{u\over {k+u}})
+\kappa _a({q\over {k_a+q}}-{v\over {k'+v}})=J_2, \\
\epsilon\frac{dv}{dt}+Fv+\kappa ({v\over {k'+v}}-{u\over {k+u}})
+\kappa _a({v\over {k'+v}}-{q\over {k_a+q}})=FL,
\end{gather*}
where all the constants are nonnegative. In this model,
the intracellular compartment splits into neurons and astrocytes. It also
includes transports through cell membranes and
a direct transport from capillary to intracellular astrocytes.
We refer the reader to \cite{AC} for more details. It would also be interesting
to construct and study corresponding PDE's models.

\begin{thebibliography}{99}

\bibitem{ADN1} S. Agmon, A. Douglis, L. Nirenberg;
\emph{Estimates near the boundary for solutions of elliptic partial differential
equations, I}, {Commun. Pure Appl. Math.}, \textbf{12} (1959), 623--727.

\bibitem{ADN2} S. Agmon, A. Douglis, L. Nirenberg;
\emph{Estimates near the boundary for solutions of elliptic partial differential
equations, II}, {Commun. Pure Appl. Math.}, \textbf{17} (1964), 35--92.

\bibitem{alikakos} N. D. Alikakos;
\emph{$L^p$ bounds of solutions to reaction-diffusion equations},
{Commun. Partial Differ. Eqns.},  \textbf{4} (1979), 827--868.

\bibitem{AC} A. Aubert and R. Costalat;
\emph{Interaction between astrocytes and neurons studied using a mathematical
 model of compartmentalized energy metabolism},
{J. Cereb. Blood Flow Metab.}, \textbf{25} (2005), 1476--1490.

\bibitem{C} R. Costalat, J.-P. Fran\c{c}oise, C. Menuel,
M. Lahutte, J.-N. Vall\'ee, G. de Marco, J. Chiras, R. Guillevin;
\emph{Mathematical modeling of metabolism and hemodynamics}, {Acta Biotheor.},
 \textbf{60} (2012), 99--107.

\bibitem{HW} Y. Hu, G. S. Wilson;
\emph{A temporary local energy pool coupled to neuronal activity: fluctuations
of extracellular lactate levels in rat brain monitored with rapid-response
enzyme-based sensor}, {J. Neurochem.}, \textbf{69} (1997), 1484--1490.

\bibitem{lahutte} M. Lahutte-Auboin;
\emph{Mod\'elisation biomath\'ematique du m\'etabolisme
\'energ\'etique c\'er\'ebral : r\'eduction de mod\`ele et approche
multi-\'echelle, application \`a l'aide \`a la d\'ecision pour la
pathologie des gliomes}, PhD thesis, Universit\'e Pierre et Marie Curie,
2015.

\bibitem{LCFG} M. Lahutte-Auboin, R. Costalat, J.-P. Fran\c{c}oise, R. Guillevin;
\emph{Dip and buffering in a fast-slow system associated to brain lactate
kinetics}, Preprint, 2013.

\bibitem{L} M. Lahutte-Auboin, R. Guillevin,
J.-P. Fran\c coise, J.-N. Vall\'ee, R. Costalat;
\emph{On a minimal model for hemodynamics
and metabolism of lactate: application to low grade
glioma and therapeutic strategies}, {Acta Biotheor.}, \textbf{61} (2013), 79--89.

\bibitem{cotransport} J. Keener, J. Sneyd;
\emph{Mathematical physiology},
Second edition, Interdisciplinary Applied Mathematics, Vol. 8,
  Springer-Verlag, New York, 2009.

\bibitem{marion} M. Marion;
\emph{Attractors for reaction-diffusion equations; Existence and estimate of
their dimension}, {Appl. Anal.}, \textbf{25} (1987), 101--147.

\bibitem{new} B. Mendoza-Juez, A. Mart\'inez-Gonz\'alez, G. F. Calvo, 
V. M. Per\'ez-Garc\'ia;
\emph{A mathematical model for the glucose-lactate metabolism of in vitro 
cancer cells},
 Bull. Math. Biol., \textbf{74} (2012), 1125--1142.

\bibitem{miranville} A. Miranville;
\emph{A singular reaction-diffusion equation
associated with brain lactate kinetics}, Math. Models Appl. Sci., to appear.

\bibitem{smoller} J. Smoller;
\emph{Shock waves and reaction-diffusion equations},
Springer-Verlag, New York, 1983.

\bibitem{temam} R. Temam;
\emph{Infinite-dimensional dynamical systems in mechanics and  physics},
Second edition, Applied Mathematical Sciences, Vol. 68,
  Springer-Verlag, New York, 1997.

\end{thebibliography}

\end{document}