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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 247, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/247\hfil Invariant regions and global solutions]
{Invariant regions and global solutions for reaction-diffusion systems 
with a tridiagonal symmetric Toeplitz matrix of diffusion coefficients}

\author[Salem Abdelmalek \hfil EJDE-2014/247\hfilneg]
{Salem Abdelmalek}  % in alphabetical order

\address{Salem Abdelmalek \newline
Department of Mathematics, College of Sciences, Yanbu, 
Taibah University, Saudi Arabia}
\email{sallllm@gmail.com}

\thanks{Submitted June 20, 2014. Published November 21, 2014.}
\subjclass[2000]{35K45, 35K57}
\keywords{Reaction-diffusion systems; invariant regions; global solution}

\begin{abstract}
 In this article we construct the invariant regions for
 $m$-component reaction-diffusion systems with a tridiagonal symmetric
 Toeplitz matrix of diffusion coefficients and with nonhomogeneous boundary
 conditions. We establish the  existence of global solutions, and use Lyapunov
 functional methods. The nonlinear reaction term is assumed to be of
 polynomial growth.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

In recent years, the  existence of global solutions for nonlinear parabolic
systems has received considerable attention. Among valuable works is the one
by Morgan \cite{Morgan1}, where where all the components
satisfy the same boundary conditions (Neumann or Dirichlet), and the reaction
terms are polynomially bounded and satisfy certain inequalities. Hollis,
later, completed the work of Morgan and established global existence in the
presence of mixed boundary conditions subject to certain structure
requirements of the system. In 2007, Abdelmalek and Kouachi \cite{Abdelmalek3}
show that solutions of $m$-component reaction-diffusion systems with a
diagonal diffusion matrix exist globally (for $m\geq2$) and reaction terms of
polynomial growth. In the case of $2\times2$-systems, Haraux and Youkana \cite{Haraux}
using a judicious Lyapunov functional, succeeded in considering sub-exponential
non-linearities. Kouachi and Youkana \cite{Kouachi5} generalized the
results of Haraux and Youkana \cite{Haraux} to the triangular case.
Then, Kanel and Kirane \cite{kanel1,kanel2} proved the global existence
for a full matrix of diffusion coefficients  under certain  restrictions.

The results obtained in this work prove the  existence of global solutions
with nonhomogeneous Neumann, Dirichlet, or Robin conditions. The reaction
terms are again assumed to be of polynomial growth and satisfy a mere single
inequality. The diffusion matrix is a tri-diagonal symmetric Toeplitz matrix.

In this article, we  use the following notation and assumptions: we
denote by $m\geq2$ the number of equations of the system (i.e. an
$m$-component system):
\begin{equation} \label{1.1}
\begin{gathered}
\frac{\partial u_{1}}{\partial t}-a\Delta u_{1}-b\Delta u_{2}=f_{1}(U)  , \\
\frac{\partial u_{\ell}}{\partial t}-b\Delta u_{\ell-1}-a\Delta u_{\ell
}-b\Delta u_{\ell+1}=f_{\ell}(  U),\quad \ell=2,\dots,m-1,\\
\frac{\partial u_{m}}{\partial t}-b\Delta u_{m-1}-a\Delta u_{m}=f_{m}(U)  ,
\end{gathered}
\end{equation}
with the boundary conditions
\begin{equation}
\alpha u_{\ell}+(  1-\alpha)  \partial_{\eta}u_{\ell}=\beta_{\ell
},\quad \ell=1,\dots,m,\text{ on }\partial\Omega\times\{  t>0\},\label{1.2}
\end{equation}
and the initial data
\begin{equation}
u_{\ell}(0,x)=u_{\ell}^{0}(x),\quad \ell=1,\dots,m,\text{ on } \Omega,\label{1.3}
\end{equation}
where
\begin{itemize}
\item[(i)] for nonhomogeneous Robin boundary conditions, we use
$0<\alpha<1$, $\beta_{\ell}\in\mathbb{R}$, $\ell=1,\dots,m$;

\item[(ii)] for homogeneous Neumann boundary conditions, we use
$\alpha=\beta_{\ell}=0$, $\ell=1,\dots,m$;

\item[(iii)] for homogeneous Dirichlet boundary conditions, we use
$1-\alpha=\beta_{\ell}=0$, $\ell=1,\dots,m$.
\end{itemize}
Here $\Omega$ is an open bounded domain of class $C^{1}$ in
$\mathbb{\mathbb{R}}^{n}$ with boundary $\partial\Omega$,
$\frac{\partial}{\partial\eta}$
denotes the outward normal derivative on $\partial\Omega$, and
 $U=(u_{\ell})  _{\ell=1}^{m}$. The constants $a$ and $b$ are supposed to
be strictly positive and satisfy the condition
\begin{equation}
2b\cos\frac{\pi}{m+1}<a.\label{1.4}
\end{equation}
The initial data are assumed to be\ in the regions:
\begin{equation}
\begin{aligned}
\Sigma_{\mathfrak{L},\mathfrak{Z}}
=\Big\{&  ( u_{1}^{0},\dots,u_{m}^{0})  \in\mathbb{R}^{m}:
\sum_{k=1}^{m}u_{k}^{0}\sin\frac{(  m+1-\ell)  k\pi}{m+1}\geq 0,\\
&\ell\in\mathfrak{L},\;
\sum_{k=1}^{m}u_{k}^{0}\sin \frac{(m+1-z) k\pi}{m+1}
 \leq0,\;z\in\mathfrak{Z}\Big\},
\end{aligned} \label{1.5}
\end{equation}
with
\begin{gather*}
\sum_{k=1}^{m}\beta_{k}\sin\frac{(  m+1-\ell)  k\pi}{m+1}\geq 0,\quad
\ell\in\mathfrak{L},\\
\sum_{k=1}^{m}\beta_{k}\sin \frac{(m+1-z) k\pi}{m+1}\leq 0,\quad
z\in\mathfrak{Z},
\end{gather*}
where
\[
\mathfrak{L}\cap\mathfrak{Z}=\emptyset,\quad
\mathfrak{L} \cup\mathfrak{Z}=\{1,2,\dots,m\} .
\]
Hence, we can see that there are $2^{m}$ regions. The subsequent work is
similar for all of these regions as will be shown at the end of the paper. Let
us now examine the first region and then comment on the remaining cases. The
chosen region is the case where $L=\{ 1,2,\dots,m\}  $ and
$\mathfrak{Z}=\emptyset$: we have
\begin{equation}
\Sigma_{\mathfrak{L},\emptyset}
=\big\{  (  u_{1}^{0},\dots,u_{m}^{0})  \in\mathbb{R}^{m}:
\sum_{k=1}^{m}u_{k}^{0}\sin\frac{(  m+1-\ell)  k\pi
}{m+1}\geq0,\;\ell\in\mathfrak{L}\big\}  \label{1.6}
\end{equation}
with
\[
\sum_{k=1}^{m}\beta_{k}\sin\frac{(  m+1-\ell)  k\pi}{m+1}
\geq0,\quad \ell\in\mathfrak{L}.
\]
The aim is now to study the existence of global solutions for the
reaction-diffusion system \eqref{1.1} in this region. To achieve this
aim, we need to diagonalize the diffusion matrix, see formula \eqref{5.1}.
First, let us define the reaction diffusion functions as
\begin{equation}
F_{\ell}(  w_{1},w_{2},\dots,w_{m})  =\sum_{k=1}^{m}f_{k}(
U)  \sin\frac{(  m+1-\ell)  k\pi}{m+1},\label{1.8}
\end{equation}
where
\begin{equation}
w_{\ell}=\sum_{k=1}^{m}u_{k}\sin\frac{(  m+1-\ell)  k\pi}
{m+1}.\label{1.7}
\end{equation}
The defined functions that satisfy the following three conditions:
\begin{itemize}
\item[(A1)] The functions $F_{\ell}$ are continuously differentiable on
$\mathbb{R}_{+}^{m}$ for all $\ell=1,\dots,m$, and satisfy
$F_{\ell}(w_{1},\dots,w_{\ell -1},0,w_{\ell+1},\dots,w_{m})\geq0$,
for all $w_{\ell} \geq0$, $\ell=1,\dots,m$;

\item[(A2)] The functions $F_{\ell}$ are of polynomial growth (see Hollis and
Morgan \cite{Hollis3}), which means that for all $\ell=1,\dots,m$ with integer
$N\geq1$,
\begin{equation}
| F_{\ell}(  W)  | \leq C_{1}\Big(
1+\sum_{\ell=1}^m w_{\ell}\Big)  ^{N}\quad \text{on }(
0,+\infty)  ^{m}; \label{1.10}
\end{equation}

\item[(A3)] The inequality
\begin{equation}
\overset{m-1}{\underset{\ell=1}{\sum}}D_{\ell}F_{\ell}(  W)
+F_{m}(  W)  \leq C_{2}\Big(  1+\overset{m}{\underset{\ell=1}
{\sum}}w_{\ell}\Big)  , \label{1.11}
\end{equation}
holds for all $w_{\ell}$ $\geq0$, $\ell=1,\dots,m$, and all constants
$D_{\ell }\geq\overline{D_{\ell}}$, $\ell=1,\dots,m$ where $\overline{D_{\ell}}$,
$\ell=1,\dots,m$ are positive constants sufficiently large. Note that $C_{1}$
and $C_{2}$ are positive and uniformly bounded functions defined on
$\mathbb{R}_{+}^{m}$.
\end{itemize}

\section{Preliminary observations and notation}

The usual norms in spaces $L^{p}(\Omega)$, $L^{\infty}(\Omega)$ and
$C(\overline{\Omega})$ are denoted respectively by
\begin{equation} \label{3.1}
\| u\| _{p}^{p}
=\frac{1}{| \Omega| }\int_{\Omega}| u(x)| ^{p}dx,\quad
\| u\| _{\infty} =\operatorname{ess,sup}_{x\in\Omega} | u(x)|,\quad
\| u\| _{C(\overline{\Omega})}  =\max_{x\in \overline{\Omega}}| u(x)| .
\end{equation}

It is well-known that to prove the existence of global solutions to a
reaction-diffusion system (see Henry \cite{Henry}), it suffices to derive a
uniform estimate of the associated reaction term on $[  0;T_{\rm max}) $
in the space $L^{p}(\Omega)$ for some $p>n/2$. Our aim is to
construct polynomial Lyapunov functionals allowing us to obtain
$L^{p}$-bounds on the components, which leads to global existence.
Since the reaction terms are continuously differentiable on
$\mathbb{R}_{+}^{m}$, it follows that for any initial data in
$C(\overline{\Omega})$, it
is easy to check directly their Lipschitz continuity on bounded subsets of the
domain of a fractional power of the operator
\begin{equation}
\mathfrak{D}=-\begin{pmatrix}
\lambda_{1}\Delta & 0 & \dots & 0\\
0 & \lambda_{2}\Delta & \dots & 0\\
\vdots & \vdots & \ddots & \vdots\\
0 & 0 & \dots & \lambda_{m}\Delta
\end{pmatrix} . \label{3.2}
\end{equation}
Under these assumptions, the local existence result is well known (see Friedman
\cite{Friedman} and Pazy \cite{Pazy}).

Assumption (A1) contains smoothness and quasi-positivity conditions that
guarantee local existence and nonnegativity of solutions as long as they
exist, via the maximum principle (see Smoller \cite{Smoller}).
 Assumption (A3) is the usual polynomial growth condition necessary to
obtain uniform bounds from $p$-dependent $L^{P}$ estimates.
(see\ Abdelmalek and Kouachi
\cite{Abdelmalek3}, and Hollis and Morgan \cite{Hollis4}).

\section{Some properties of the diffusion matrix}

\begin{lemma} \label{lem1}
Considering the reaction-diffusion system in \eqref{1.1}, the resulting
$m\times m$ diffusion matrix is given by
\begin{equation}
A=\begin{pmatrix}
a & b & 0 & \cdots & 0 & 0\\
b & a & b & \ddots & 0 & 0\\
0 & b & a & \ddots & \vdots & \vdots\\
\vdots & \ddots & \ddots & \ddots & b & 0\\
0 & \cdots & 0 & b & a & b\\
0 & \cdots & 0 & 0 & b & a
\end{pmatrix}. \label{2.1}
\end{equation}
This matrix is said to be positive definite if the condition in \eqref{1.4}
is satisfied.
\end{lemma}

\begin{proof}
The proof of this lemma can be found in \cite{Johnson}. Note that if the
matrix is positive definite, it follows that $\det A>0$.
\end{proof}

\begin{lemma} \label{Lemma0}
The eigenvalues $( \lambda_{\ell}<\lambda_{\ell-1};\text{
}\ell=2,\dots,m)  $ of $A$ are positive and are given by
\begin{equation}
\lambda_{\ell}=a+2b\cos(  \frac{\ell\pi}{m+1})  ,\label{2.2}
\end{equation}
with the corresponding eigenvectors
\[
v_{\ell}=\big(  \sin\frac{\ell\pi}{m+1},\sin\frac{2\ell\pi}{m+1}
,\dots,\sin\frac{m\ell\pi}{m+1}\Big)  ^{t},
\]
for $\ell=1,\dots,m$. Hence, we conclude that $A$ is diagonalizable.
 For simplicity, we write
\begin{equation}
\bar{\lambda}_{\ell}=\lambda_{m+1-\ell}=a+2b\cos(  \frac{(m+1-\ell)\pi
}{m+1}),\quad \ell=1,\dots,m;\label{2.3}
\end{equation}
thus $\bar{\lambda}_{\ell}<\bar{\lambda}_{\ell+1}$, $\ell=2,\dots,m$.
\end{lemma}

\begin{proof}
Recall that the diffusion matrix is positive definite, hence its eigenvalues
are necessarily positive. For an eigenpair $(\lambda,X)  $, the
components in $( A-\lambda I) X=0$ are
\[
bx_{k-1}+(  a-\lambda)  x_{k}+bx_{k+1}=0,\quad k=1,\dots,m,
\]
with $x_{0}=x_{m+1}=0$, or equivalently,
\[
x_{k+2}+(  \frac{a-\lambda}{b})  x_{k+1}+x_{k}=0,\quad k=0,\dots,m-1.
\]

We seek solutions in the form $x_{k}=\xi r^{k}$ for constants $\xi$ and $r$.
This leads to the quadratic equation
\[
r^{2}+\big(  \frac{a-\lambda}{b}\big)  r+1=0,
\]
with roots $r_{1}$ and $r_{2}$. The general solution of $x_{k+2}+(
\frac{a-\lambda}{b})  x_{k+1}+x_{k}=0$ is
\[
x_{k}=\begin{cases}
\alpha r_{1}^{k}+\beta r^{k}, & \text{if }r_{1}\neq r_{2},\\
\alpha\rho^{k}+\beta k\rho^{k}, & \text{if  }r_{1}=r_{2}=\rho,
\end{cases}
\]
where $\alpha$ and $\beta$ are arbitrary constants.

For the eigenvalue problem at hand, $r_{1}$ and $r_{2}$ must be distinct
-otherwise $x_{k}=\alpha\rho^{k}+\beta k\rho^{k}$, and $x_{0}=x_{m+1}=0$
implies that each $x_{k}=0$, which is impossible because $X$ is an
eigenvector. Hence, $x_{k}=\alpha r_{1}^{k}+\beta r^{k}$, and
$x_{0} =x_{m+1}=0$ yields
\[
\left\{\begin{array}[c]{l}
0=\alpha+\beta\\
0=\alpha r_{1}^{m+1}+\beta r_{2}^{m+1}
\end{array} \right\}
  \Rightarrow(  \frac{r_{1}}{r_{2}})  ^{m+1}=\frac{-\beta
}{\alpha}=1\Rightarrow\frac{r_{1}}{r_{2}}=e^{\frac{2i\pi\ell}{m+1}};
\]
therefore, $r_{1}=r_{2}e^{\frac{2i\pi\ell}{m+1}}$ for some $1\leq\ell\leq m$.
This coupled with
\[
r^{2}+(  \frac{a-\lambda}{b})  r+1=(  r-r_{1})  (
r-r_{2})  \Rightarrow\{
\begin{array}[c]{l}
r_{1}r_{2}=1,\\
r_{1}+r_{2}=-(  \frac{a-\lambda}{b})  ,
\end{array}
\]
leads to $r_{1}=e^{\frac{i\pi\ell}{m+1}}$, $r_{2}=e^{-\frac{i\pi\ell}{m+1}}$,
and
\[
\lambda=a+b\big(  e^{\frac{i\pi\ell}{m+1}}+e^{-\frac{i\pi\ell}{m+1}}\big)
=a+2b\cos(  \frac{\ell\pi}{m+1})  \text{.}
\]
The eigenvalues of $A$ can, therefore, be given by
\[
\lambda_{\ell}=a+2b\cos(  \frac{\ell\pi}{m+1})  ,\text{ for }
\ell=1,\dots,m.
\]


Since these $\lambda_{\ell}$'s are all distinct ($\cos\theta$ is a strictly
decreasing function of $\theta$ on $(  0,\pi)  $, and $b\neq0$),
$A$ is necessarily diagonalizable.

Finally, the $\ell^{th}$ component of any eigenvector associated with
$\lambda_{\ell}$ satisfies $x_{k}=\alpha r_{1}^{k}+\beta r_{2}^{k}$ with
$\alpha+\beta=0$, thus
\[
x_{k}=\alpha\big(  e^{\frac{2i\pi k}{m+1}}-e^{-\frac{2i\pi k}{m+1}}\big)
=2i\alpha\sin\big(  \frac{k}{m+1}\pi\big)  .
\]
Setting $\alpha=1/(2i)$ yields a particular eigenvector associated with
$\lambda_{\ell}$ given by
\[
v_{\ell}=\Big( \sin(  \frac{1\ell\pi}{m+1})  ,\sin(
\frac{2\ell\pi}{m+1})  ,\dots,\sin(  \frac{m\ell\pi}{m+1})
\Big)  ^{t}.
\]
Because the $\lambda_{\ell}$'s are distinct, $\{  v_{1},v_{2}
,\dots,v_{m}\}  $, is a complete linearly independent set, so
$(v_{1} | v_{2}|\dots| v_{m})  $ is the diagonal form
of $A$.

Now, let us prove that
\[
\lambda_{\ell}<\lambda_{\ell-1},\text{ }\ell=2,\dots,m.
\]
We have
$\ell>\ell-1$, whereupon
\[
\frac{\ell\pi}{m+1}>\frac{(  \ell-1)  \pi}{m+1},
\]
The function $\cos\theta$ is strictly decreasing in $\theta$ on
 $( 0,\pi)  $, thus we have
\[
\cos\big(  \frac{\ell\pi}{m+1}\big)  <\cos\big(  \frac{(
\ell-1)  \pi}{m+1}\big)  .
\]
Finally, multiplying both sides of the inequality by $2b$ and adding $a$
yields
\[
\lambda_{\ell}=a+2b\cos\big(  \frac{\ell\pi}{m+1}\big)
<a+2b\cos\big(
\frac{(  \ell-1)  \pi}{m+1}\big)  =\lambda_{\ell-1}.
\]
\end{proof}

\section{Main results}

\begin{proposition}\label{prop1}
The eigenvectors of the diffusion matrix associated with
the eigenvalues $\overline{\lambda}_{\ell}$ are defined as $\overline{v}
_{\ell}=(  \overline{v}_{\ell1},\overline{v}_{\ell2},\dots,\overline
{v}_{\ell m})  ^{T}$. They satisfy the equations:
\begin{gather}
\frac{\partial w_{\ell}}{\partial t}-\bar{\lambda}_{\ell}\Delta w_{\ell
}=F_{\ell}(  w_{1},w_{2},\dots,w_{m})  , \label{5.1}
\\
\alpha w_{\ell}+(  1-\alpha)  \partial_{\eta}w_{\ell}=\rho_{\ell}\quad
\text{on }\partial\Omega\times\{ t>0\}  , \label{5.2}
\end{gather}
where the reaction term $F_{\ell}$, and $w_{\ell}$ are given in \eqref{1.7}
and \eqref{1.8}, respectively.
\end{proposition}

Note that condition \eqref{1.4} guarantees the parabolicity of the proposed
reaction-diffusion system in \eqref{1.1}--\eqref{1.3}, which implies it is
equivalent to \eqref{5.1}--\eqref{5.2} in the region
\[
\Sigma_{\mathfrak{L},\emptyset}=\big\{ (  u_{1}^{0},\dots,u_{m}
^{0})  \in\mathbb{R}^{m}:w_{\ell}^{0}=\sum_{k=1}^{m}u_{k}^{0}\sin\frac{(
m+1-\ell)  k\pi}{m+1}\geq0,\text{ }\ell\in\mathfrak{L}\}
\]
with
\[
\rho_{\ell}^{0}=\sum_{k=1}^{m}\beta_{k}\sin\frac{(  m+1-\ell)
k\pi}{m+1}\geq0,\quad \ell\in\mathfrak{L}.
\]
This implies that the components $w_{\ell}$ are necessarily positive.

\begin{proposition} \label{prop2}
System \eqref{5.1}--\eqref{5.2} admits a unique classical solution
$(w_{1},w_{2},\\ \dots,w_{m})$ on $(0,T_{\rm max})\times\Omega$.
\begin{equation}
\text{If $T_{\rm max}<\infty$  then }\lim_{t\nearrow T_{\rm max}}
\sum_{\ell=1}^m \| w_{\ell}(  t,.)
\| _{\infty}=\infty, \label{3.3}
\end{equation}
where $T_{\rm max}$ $(  \| w_{1}^{0}\| _{\infty
},\| w_{2}^{0}\| _{\infty},\dots,\| w_{m}
^{0}\| _{\infty})  $ denotes the eventual blow-up time.
\end{proposition}

Before we present the main result of this paper, let us define
\begin{equation}
K_{l}^{r}=K_{r-1}^{r-1}\times K_{l}^{r-1}-[  H_{l}^{r-1}]
^{2},\quad r=3,\dots,l, \label{mycond}
\end{equation}
where
\begin{gather*}
H_{l}^{r}=\underset{1\leq\ell,\kappa\leq l}{\det}\Big(  (
a_{\ell,\kappa})  _{\substack{\ell\neq l,\dots r+1\\
\kappa\neq l-1,\dots,r}}\Big)  \prod_{k=1}^{k=r-2}
\big( \det[ k] \big)  ^{2^{(  r-k-2)  }},\quad r=3,\dots,l-1,
\\
K_{l}^{2}=\underset{\text{positive value}}{\underbrace{\bar{\lambda}_{1}
\bar{\lambda}_{l}
\prod^{l-1}_{k=1} \theta_{k}^{2(p_{k}+1)  ^{2}}
\prod^{m-1}_{k=l} \theta_{k}^{2( p_{k}+2)  ^{2}}}}
\Big[  \prod^{l-1}_{k=1} \theta_{k}^{2}-A_{1l}^{2}\Big]  ,
\\
H_{l}^{2}=\underset{\text{positive value}}{\underbrace{\bar{\lambda}_{1}
\sqrt{\bar{\lambda}_{2}\bar{\lambda}_{l}}\theta_{1}^{2(  p_{1}+1)
^{2}}
\prod^{l-1}_{k=2} \theta_{k}^{(  p_{k}+2)
^{2}+(  p_{k}+1)  ^{2}}\prod^{m-1}_{k=l} \theta
_{k}^{2(  p_{k}+2)  ^{2}}}}\big[  \theta_{1}^{2}A_{2l}
-A_{12}A_{1l}\big]  .
\end{gather*}
The expression $\underset{1\leq\ell,\kappa\leq l}{\det}(  (  a_{\ell,\kappa
})  _{\substack{\ell\neq l,\dots r+1\\ \kappa\neq l-1,\dots r}})  $
denotes the determinant of $r$ square symmetric matrix obtained from
$(a_{\ell,\kappa})  _{1\leq\ell,\kappa\leq m}$ by removing the
$(r+1)  ^{\text{th}},(r+2)  ^{\text{th}},\dots,l^{\text{th}}$
rows and the $r^{\text{th}},(  r+1)  ^{\text{th}},\dots,(
l-1)  ^{\text{th}}$ columns. where $\det[1]  ,\dots,\det
[  m]  $ \ are the minors of the matrix $(  a_{\ell,\kappa
})  _{1\leq\ell,\kappa\leq m}$. The elements of the matrix are:
\begin{equation}
a_{\ell\kappa}=\frac{\bar{\lambda}_{\ell}+\bar{\lambda}_{\kappa}}{2}\theta
_{1}^{p_{1}^{2}}\dots\theta_{(  \ell-1)  }^{p_{(  \ell-1)
}^{2}}\theta_{\ell}^{(  p_{\ell}+1)  ^{2}}\dots\theta_{\kappa
-1}^{(  p_{(  \kappa-1)  }+1)  ^{2}}\theta_{\kappa
}^{(  p_{\kappa}+2)  ^{2}}\dots\theta_{(  m-1)  }^{(
p_{(  m-1)  }+2)  ^{2}}. \label{1.13}
\end{equation}
where $\bar{\lambda}_{\ell}$ in \eqref{2.2}--\eqref{2.3}. Note that
$A_{\ell\kappa}=\frac{\bar{\lambda}_{\ell}+\bar{\lambda}_{\kappa}}
{2\sqrt{\bar{\lambda}_{\ell}\bar{\lambda}_{\kappa}}}$ for all $\ell
,\kappa=1,\dots,m$. and $\theta_{\ell};\ell=1,\dots,(  m-1)  $ are
positive constants.

\begin{theorem}\label{thm1}
Suppose that the functions $F_{\ell}$, $\ell=1,\dots,m$, are of
polynomial growth and satisfy condition \eqref{1.11} for some positive
constants $D_{\ell}$, $\ell=1,\dots,m$, sufficiently large.
Let $(w_{1}(t,\cdot),w_{2}(t,\cdot) ,\dots,w_{m}(t,\cdot))$ be the
solution of \eqref{5.1}--\eqref{5.2} and
\begin{equation}
L(t)=\int_{\Omega}H_{p_{m}}(  w_{1}(  t,x)  ,w_{2}(
t,x)  ,\dots,w_{m}(  t,x)  )  dx, \label{5.3}
\end{equation}
where
\begin{align*}
&H_{p_{m}}(  w_{1},\dots,w_{m})\\
&=\sum^{p_{m}}_{p_{m-1}=0}  \dots \sum^{p_{2}}_{p_{1}=0}
C_{p_{m}}^{p_{m-1} }\dots C_{p_{2}}^{p_{1}}\theta_{1}^{p_{1}^{2}}\dots\theta_{(  m-1)
}^{p_{(  m-1)  }^{2}}w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}
\dots w_{m}^{p_{m}-p_{m-1}},
\end{align*}
with $p_{m}$ a positive integer and
$C_{p_{\kappa}}^{p_{\ell}}=\frac{p_{\kappa}!}{p_{\ell}!(  p_{\kappa}-p_{\ell})!}$.
Also suppose that the following condition is satisfied
\begin{equation}
K_{l}^{l}>0;\quad l=2,\dots,m,  \label{1.12}
\end{equation}
where $K_{l}^{l}$ was defined in \eqref{mycond}. Then, the functional $L$ is
uniformly bounded on the interval $[  0,T^{\ast}]  ,T^{\ast}<T_{\rm max}$.
\end{theorem}

\begin{corollary}\label{coro1}
Under the assumptions of Theorem \ref{thm1}, all
solutions of \eqref{5.1}--\eqref{5.2} with positive initial data in
$L^{\infty}(  \Omega)  $ are in $L^{\infty}(  0,T^{\ast};L^{p}(\Omega)  )  $
for some $p\geq1$.
\end{corollary}

\begin{proposition} \label{prop3}
Under the assumptions of Theorem \ref{thm1}, and
assuming the condition \eqref{1.4} is satisfied, all solutions of
\eqref{5.1}--\eqref{5.2} with positive initial data in
$L^{\infty}(\Omega)  $ are global for some $p>\frac{Nn}{2}$.
\end{proposition}

\section{Proofs of main results}
For the proof of Theorem \ref{thm1}, we first need to define some
preparatory Lemmas.

\begin{lemma}\label{Lemma1}
With\ $H_{p_{m}}$ being the homogeneous polynomial defined by
\eqref{5.3}, differentiating in $w_{1}$ yields
\begin{equation}
\begin{aligned}
\partial_{w_{1}}H_{p_{m}}
& =p_{m}\sum^{p_{m}-1}_{p_{m-1}=0} \dots
\sum^{p_{2}}_{p_{1}=0} C_{p_{m}-1}^{p_{m-1}}\dots
C_{p_{2}}^{p_{1}}\theta_{1}^{(  p_{1}+1)  ^{2}}\dots\theta
_{(  m-1)  }^{(  p_{(  m-1)  }+1)  ^{2}} \\
&\quad\times  w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}w_{3}^{p_{3}-p_{2}}\dots
w_{m}^{( p_{m}-1)  -p_{m-1}}\,.
\end{aligned}\label{5.4}
\end{equation}
Similarly for $\ell=2,\dots,m-1$, we have
\begin{equation}
\begin{aligned}
\partial_{w_{\ell}}H_{p_{m}}
&  =p_{m}\sum^{p_{m}-1}_{p_{m-1}=0} \dots
\sum^{p_{2}}_{p_{1}=0} C_{p_{m}-1}^{p_{m-1}}\dots
C_{p_{2}}^{p_{1}}\theta_{1}^{p_{1}^{2}}\dots\theta_{\ell-1}^{p_{(
\ell-1)  }^{2}}\theta_{\ell}^{(  p_{\ell}+1)  2}
\dots\theta_{(  m-1)  }^{(  p_{(  m-1)  }+1)^{2}} \\
&\quad\times  w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}w_{3}^{p_{3}-p_{2}}\dots w_{m}^{(
p_{m}-1)  -p_{m-1}}. \label{5.5}
\end{aligned}
\end{equation}
Finally, differentiating in $w_{m}$ yields
\begin{equation}
\begin{aligned}
\partial_{w_{m}}H_{p_{m}}
&  =p_{m}\sum^{p_{m}-1}_{p_{m-1}=0} \dots
\sum^{p_{2}}_{p_{1}=0} C_{p_{m}-1}^{p_{m-1}}\dots
C_{p_{3}}^{p_{2}}C_{p_{2}}^{p_{1}}\theta_{1}^{p_{1}^{2}}\theta_{2}
^{p_{2}^{2}}\dots\theta_{(  m-1)  }^{p_{(  m-1)  }^{2}} \\
&\quad\times  w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}w_{3}^{p_{3}-p_{2}}\dots w_{m}^{(
p_{m}-1)  -p_{m-1}}.
\end{aligned} \label{5.6}
\end{equation}
\end{lemma}

\begin{lemma} \label{Lemma2}
The second partial derivative of $H_{p_{m}}$ in $w_{1}$ is
\begin{equation}
\begin{aligned}
\partial_{w_{1}^{2}}H_{n}
&  =p_{m}(  p_{m}-1)  \sum^{p_{m}-2}_{p_{m-1}=0} \dots
\sum^{p_{3}}_{p_{2}=0} \sum^{p_{2}}_{p_{1}=0} C_{p_{m}-2}^{p_{m-1}}\dots C_{p_{2}
}^{p_{1}} \\
&\times  \theta_{1}^{(  p_{1}+2)  ^{2}}\dots\theta_{(  m-1)
}^{(  p_{(  m-1)  }+2)  ^{2}}w_{1}^{p_{1}}w_{2}
^{p_{2}-p_{1}}\dots w_{m}^{(  p_{m}-2)  -p_{m-1}}.
\end{aligned} \label{5.7}
\end{equation}
Similarly, we obtain
\begin{equation}
\begin{aligned}
\partial_{w_{\ell}^{2}}H_{n}
&  =p_{m}(  p_{m}-1)  \sum^{p_{m}-2}_{p_{m-1}=0} \dots
\sum^{p_{2}}_{p_{1}=0} C_{p_{m}-2}^{p_{m-1}}\dots C_{p_{2}}^{p_{1}} \\
&\quad \times  \theta_{1}^{p_{1}^{2}}\theta_{2}^{p_{2}^{2}}\dots\theta_{\ell-1}^{p_{\ell
-1}^{2}}\theta_{\ell}^{(  p_{\ell}+2)  ^{2}}\dots\theta_{(
m-1)  }^{(  p_{(  m-1)  }+2)  ^{2}}
  w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}\dots w_{m}^{(  p_{m}-2)  -p_{m-1}}.
\end{aligned} \label{5.8}
\end{equation}
for all\ $\ell=2,\dots,m-1$,
\begin{equation}
\begin{aligned}
\partial_{w_{\ell}w_{\kappa}}H_{n}
&  =p_{m}(  p_{m}-1)
\sum^{p_{m}-2}_{p_{m-1}=0} \dots
\sum^{p_{2}}_{p_{1}=0} C_{p_{m}-2}^{p_{m-1}}\dots C_{p_{2}}^{p_{1}} \\
&\quad\times  \theta_{1}^{p_{1}^{2}}\dots\theta_{\ell-1}^{p_{\ell-1}^{2}}\theta_{\ell
}^{(  p_{\ell}+1)  ^{2}}\dots\theta_{\kappa-1}^{(  p_{\kappa
-1}+1)  ^{2}}\theta_{\kappa}^{(  p_{\kappa}+2)  ^{2}
}\dots\theta_{(  m-1)  }^{(  p_{(  m-1)  }+2)
^{2}} \\
&\quad\times  w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}\dots w_{m}^{(  p_{m}-2)  -p_{m-1}}
\end{aligned}\label{5.9}
\end{equation}
for all $1\leq\ell<\kappa\leq m$. Finally, the second derivative in $w_{m}$
is
\begin{equation}
\begin{aligned}
\partial_{w_{m}^{2}}H_{n}
&  =p_{m}(  p_{m}-1)  \sum^{p_{m}-2}_{p_{m-1}=0} \dots
\sum^{p_{2}}_{p_{1}=0} C_{p_{m}-2}^{p_{m-1}}\dots
C_{p_{2}}^{p_{1}}\theta_{1}^{p_{1}^{2}}
\dots\theta_{(  m-1)  }^{p_{(  m-1)  }^{2}} \\
&  w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}\dots w_{m}^{(  p_{m}-2)  -p_{m-1}}.
\end{aligned}\label{5.10}
\end{equation}
\end{lemma}

\begin{lemma}[\cite{Abdelmalek3}] \label{Lemma3}
Let $A$ be the $m$-square symmetric
matrix defined by $A=(  a_{\ell\kappa})$, with $1\leq\ell,\kappa\leq m$,
then
\begin{equation}
\begin{gathered}
K_{m}^{m}=\det[m]  \prod^{k=m-2}_{k=1} (\det[  k]  )  ^{2^{(  m-k-2)  }},\quad
 m>2,\\
K_{2}^{2}=\det[2]  ,
\end{gathered}  \label{5.11}
\end{equation}
where
\begin{gather*}
K_{m}^{l} =K_{l-1}^{l-1}K_{m}^{l-1}-(  H_{m}^{l-1})  ^{2},\quad l=3,\dots,m,\\
H_{m}^{l}   =\underset{1\leq\ell,\kappa\leq m}{\det}
\Big(  (a_{\ell,\kappa})  _{\substack{\ell\neq m,\dots l+1\\\kappa\neq
m-1,\dots l}}\Big)  \prod^{k=l-2}_{k=1} \Big(  \det[
k] \Big)  ^{2^{(  l-k-2)  }},\quad l=3,\dots,m-1,\\
K_{m}^{2}   =a_{11}a_{mm}-(  a_{1m})  ^{2},\text{ }H_{m}
^{2}=a_{11}a_{2m}-a_{12}a_{1m}.
\end{gather*}
\end{lemma}

\begin{proof}[Proof of Theorem \ref{thm1}]
The aim is to prove that $L(t)$ is uniformly
bounded on the interval $[  0,T^{\ast}]  ,T^{\ast}<T_{\rm max}$. Let
us start by differentiating $L$ with respect to $t$:
\begin{align*}
L'(t)  &  =\int_{\Omega}\partial_{t}H_{p_{m}}dx=\int_{\Omega}
 \sum_{\ell=1}^m \partial_{w_{\ell}}H_{p_{m}}\frac{\partial
 w_{\ell}}{\partial t}dx\\
&  =\int_{\Omega}\sum_{\ell=1}^m \partial_{w_{\ell}}H_{p_{m}}
(\bar{\lambda}_{\ell}\Delta w_{\ell}+F_{\ell})  dx\\
&  =\int_{\Omega}\sum_{\ell=1}^m \bar{\lambda}_{\ell}\partial_{w_{\ell}}H_{p_{m}}
 \Delta w_{\ell}dx
+\int_{\Omega}\sum^{m}_{\ell=1} \partial_{w_{\ell}}H_{p_{m}}F_{\ell}dx\\
&  =I+J,
\end{align*}
where
\begin{gather}
I=\int_{\Omega}\sum_{\ell=1}^m \bar{\lambda}_{\ell
}\partial_{w_{\ell}}H_{p_{m}}\Delta w_{\ell}dx, \label{5.11a}
\\
J=\int_{\Omega}\sum^{m}_{\ell=1}
\partial_{w_{\ell}}H_{p_{m}}F_{\ell}dx. \label{5.12}
\end{gather}
Using Green's formula, we can divide $I$ into two parts $I_{1}$ and $I_{2}$
where
\begin{gather}
I_{1}=\int_{\partial\Omega}\sum_{\ell=1}^m \bar{\lambda
}_{\ell}\partial_{w_{\ell}}H_{p_{m}}\partial_{\eta}w_{\ell}dx, \label{5.13}
\\
I_{2}=-\int_{\Omega}\Big[  \Big(  \Big(  \frac{\overline{\lambda}_{\ell
}+\overline{\lambda}_{\kappa}}{2}\partial_{w_{\kappa}w_{\ell}}H_{p_{m}
}\Big)  _{1\leq\ell,\kappa\leq m}\Big)  T\Big]   T\,dx, \label{5.14}
\end{gather}
for $p_{1}=0,\dots,p_{2}$, $p_{2}=0,\dots,p_{3}$ $\dots p_{m-1}=0,\dots,p_{m}-2$ and
\[
T=(  \nabla w_{1},\nabla w_{2},\dots,\nabla w_{m})  ^{t}.
\]
Applying Lemmas \ref{Lemma1} and \ref{Lemma2} yields
\begin{equation}
\begin{aligned}
&\Big(  \frac{\bar{\lambda}_{\ell}+\bar{\lambda}_{\kappa}}{2}\partial
_{w_{\kappa}w_{\ell}}H_{p_{m}}\Big)  _{1\leq\ell,\kappa\leq m}\\
&= p_{m}(  p_{m}-1)  \sum^{p_{m}-2}_{p_{m-1}=0} \dots
\sum^{p_{2}}_{p_{1}=0} C_{p_{m}-2}^{p_{m-1}}\dots C_{p_{2}}^{p_{1}}
\Big( (  a_{\ell\kappa})  _{1\leq\ell,\kappa\leq
m}\Big)  w_{1}^{p_{1}}\dots w_{m}^{(  p_{m}-2)  -p_{m-1}},
\end{aligned} \label{5.15}
\end{equation}
where $(  a_{\ell\kappa})  _{1\leq\ell,\kappa\leq m}$ is the matrix
defined in formula \eqref{1.13}.

Now, the proof of positivity for $I$ is reduced to proving that there exists a
positive constant $C_{4}$ independent of $t\in[  0,T_{\rm max})  $
such that
\begin{equation}
I_{1}\leq C_{4}\quad \text{for all }t\in[  0,T_{\rm max})  , \label{5.16}
\end{equation}
and that
\begin{equation}
I_{2}\leq0, \label{5.17}
\end{equation}
for several boundary conditions. First, let us prove the formula in
\eqref{5.16}:

(i) If $0<\alpha<1$ , then using the boundary conditions \eqref{1.2} we obtain
\[
I_{1}=\int_{\partial\Omega}\sum_{\ell=1}^m \bar{\lambda
}_{\ell}\partial_{w_{\ell}}H_{p_{m}}(  \gamma_{\ell}-\sigma w_{\ell
})  dx,
\]
where $\sigma=\frac{\alpha}{1-\alpha}$ and $\gamma_{\ell}=\frac{\beta_{\ell
}}{1-\alpha}$, for\ $\ell=1,\dots m$.
Since
\[
H(  W)  =\sum_{\ell=1}^m \overline
{\lambda}_{\ell}\partial_{w_{\ell}}H_{p_{m}}(  \gamma_{\ell}-\sigma
w_{\ell})  =P_{n-1}(  W)  -Q_{n}(  W) ,
\]
where $P_{n-1}$ and $Q_{n}$, are polynomials with positive coefficients
and degrees $n-1$ and $n$, respectively, and since the solution is positive,
it follows that
\begin{equation}
\underset{\sum_{\ell=1}^m | w_{\ell}|
\to +\infty}{\lim\sup}H(  W)  =-\infty,\label{5.18}
\end{equation}
which proves that $H$ is uniformly bounded on $\mathbb{R}_{+}^{m}$
 and consequently \eqref{5.16}.

(ii) If $\alpha=0$, then $I_{1}=0$ on $[  0,T_{\rm max})  $.

(iii) The case of homogeneous Dirichlet conditions is trivial, since in this
case the positivity of the solution on
$[  0,T_{\rm max}) \times\Omega$ implies
$\partial_{\eta}w_{\ell}\leq0,\forall\ell=1,\dots m$ on
$[  0,T_{\rm max})  \times\partial\Omega$. Consequently, one obtains
the same result in \eqref{5.16} with $C_{4}=0$.
Hence, the proof of \eqref{5.16} is complete.

Now, we pass to the proof of \eqref{5.17}:
Recall the matrix $(  a_{\ell\kappa})  _{1\leq\ell,\kappa\leq m}$
which was defined in formula \eqref{1.13}. The quadratic forms (with respect
to $\nabla w_{\ell}$, $\ell=1,\dots,m$) associated with the matrix $(
a_{\ell\kappa})  _{1\leq\ell,\kappa\leq m}$, with $p_{1}=0,\dots,p_{2}$,
$p_{2}=0,\dots,p_{3}$ \dots $p_{m-1}=0,\dots,p_{m}-2$, is positive definite since its
minors $\det[  1] $, $\det[  2] ,\dots \det[m]  $ are all positive.
Let us examine these minors and prove their
positivity by induction.

The first minor
\[
\det[  1]  =\lambda_{1}\theta_{1}^{(  p_{1}+2)  ^{2}
}\theta_{2}^{(  p_{2}+2)  ^{2}}\dots \theta_{(  m-1)
}^{(  p_{(  m-1)  }+2)  ^{2}}>0
\]
is trivial for $p_{1}=0,\dots,p_{2}$, $p_{2}=0,\dots,p_{3}$ \dots $p_{m-1}
=0,\dots,p_{m}-2$.

For the second minor $\det[  2]  $,\ according to Lemma
\ref{Lemma3}, we obtain
\[
\det[  2]  =K_{2}^{2}=\lambda_{1}\lambda_{2}\theta_{1}^{2(
p_{1}+1)  ^{2}}\overset{m-1}{\underset{k=2}{\Pi}}\theta_{k}^{2(
p_{k}+2)  ^{2}}[  \theta_{1}^{2}-A_{12}^{2}]  .
\]
Using \eqref{1.12}\ for $l=2$\ we get $\det[  2]  >0$.

Similarly, for the third minor $\det[3]  $, and again using
Lemma \ref{Lemma3}, we have
\[
K_{3}^{3}=\det[  3]  \det[  1]  .
\]
Since $\det[  1]  >0$, we conclude that
\[
\operatorname*{sign}(K_{3}^{3})=\operatorname*{sign}(\det[  3]  ).
\]
Again, using \eqref{1.12}\ for $l=3$, we obtain $\det[3]  >0$.

To conclude the proof, let us suppose $\det[ k]  >0$ for
$k=1,2,\dots,l-1$ and show that $\det[l]$ is necessarily positive. We have
\begin{equation}
\det[ k] >0,\;k=1,\dots,(  l-1)  ,\Rightarrow
\prod^{k=l-2}_{k=1} (  \det[  k]  )
^{2^{(  l-k-2)  }}>0. \label{5.20}
\end{equation}
From Lemma \ref{Lemma3}, we obtain
\[
K_{l}^{l}=\det[  l] \prod^{k=l-2}_{k=1} (  \det[  k]  )
^{2^{(  l-k-2)  }},
\]
 and from \eqref{5.20}, we obtain
$\operatorname*{sign}(K_{l}^{l})=\operatorname*{sign}(  \det[
l]  )$. Since $K_{l}^{l}>0$ according to \eqref{1.12} then
$\det[  l]  >0$ and the proof of \eqref{5.17} is finished. It
follows from \eqref{5.16} and \eqref{5.17} that $I$ is bounded. Now, let us
prove that $J$ in \eqref{5.12} is bounded. Substituting the expressions of the
partial derivatives given by \ref{Lemma1} in the second integral of
\eqref{5.12} yields
\begin{align*}
J &  =\int_{\Omega} \Big[  p_{m}\sum^{p_{m}-1}_{p_{m-1}=0}
 \dots\sum^{p_{2}}_{p_{1}=0} C_{p_{m}-1}^{p_{m-1}}\dots C_{p_{2}
  }^{p_{1}}w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}\dots w_{m}^{p_{m}-1-p_{m-1}}\Big] \\
&\quad \times \Big( \prod^{m-1}_{\ell=1}\theta_{\ell}^{(
p_{\ell}+1)  ^{2}}F_{1}+\sum^{m-1}_{\kappa=2} 
\prod^{\kappa-1}_{k=1}\theta_{k}^{p_{k}^{2}}
\prod^{m-1}_{\ell=\kappa} \theta_{\ell}^{(  p_{\ell}+1)
^{2}}F_{\kappa}+\prod^{m-1}_{\ell=1}\theta_{\ell}^{p_{\ell
}^{2}}F_{m}\Big)  dx\\
&  =\int_{\Omega}\Big[  p_{m}\sum^{p_{m}-1}_{p_{m-1}=0} 
\dots\sum^{p_{2}}_{p_{1}=0} C_{p_{m}-1}^{p_{m-1}}\dots C_{p_{2}
}^{p_{1}}w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}\dots w_{m}^{p_{m}-1-p_{m-1}}\Big] \\
& \quad\times \Big(  \frac{\prod^{m-1}_{\ell=1}\theta_{\ell}^{(
p_{\ell}+1)  ^{2}}}{\prod^{m-1}_{\ell=1}\theta_{\ell
}^{p_{\ell}^{2}}}F_{1}+\sum^{m-1}_{\kappa=2} \frac
{\prod^{\kappa-1}_{k=1}\theta_{k}^{p_{k}^{2}}
\prod^{m-1}_{\ell=\kappa} \theta_{\ell}^{(  p_{\ell}+1)
^{2}}}{\prod^{m-1}_{\ell=1}\theta_{\ell}^{p_{\ell}^{2}}
}F_{\kappa}+F_{m}\Big)  \prod^{m-1}_{\ell=1}\theta_{\ell}^{p_{\ell}^{2}}dx\\
&  =\int_{\Omega}\Big[  p_{m}\overset{p_{m}-1}{\underset{p_{m-1}=0}{\sum}
}\dots\sum^{p_{2}}_{p_{1}=0} C_{p_{m}-1}^{p_{m-1}}\dots C_{p_{2}
}^{p_{1}}w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}\dots w_{m}^{p_{m}-1-p_{m-1}}\Big] 
\\
&\quad\times\Big(  \prod^{m-1}_{\ell=1}\frac{\theta_{\ell}^{(
p_{\ell}+1)  ^{2}}}{\theta_{\ell}^{p_{\ell}^{2}}}F_{1}
+\sum^{m-1}_{\kappa=2}   \prod^{\kappa-1}_{k=1}\theta_{k}^{p_{k}^{2}} 
\prod^{m-1}_{\ell=\kappa} 
\frac{\theta_{\ell}^{(  p_{\ell}+1)  ^{2}}}{\theta_{\ell}
^{p_{\ell}^{2}}}F_{\kappa}+F_{m}\Big)  
\prod^{m-1}_{\ell=1} \theta_{\ell}^{p_{\ell}^{2}}dx.
\end{align*}
Hence, using condition \eqref{1.11}, we deduce that
\[
J\leq C_{5}\int_{\Omega}\Big[  \sum^{p_{m}-1}_{p_{m-1}=0} \dots
\sum^{p_{2}}_{p_{1}=0} C_{p_{2}}^{p_{1}}\dots C_{p_{m}
-1}^{p_{m-1}}w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}\dots w_{m}^{p_{m}-1-p_{m-1}}(
1+\sum_{\ell=1}^m w_{\ell}) \Big]  dx.
\]
To prove that the functional $L$ is uniformly bounded on the interval 
$[0,T^{\ast}]  $, let us first write
\begin{align*}
&  \sum^{p_{m}-1}_{p_{m-1}=0}\dots
\sum^{p_{2}}_{p_{1}=0} C_{p_{2}}^{p_{1}}\dots C_{p_{m}-1}^{p_{m-1}}
w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}\dots w_{m}^{p_{m}-1-p_{m-1}}
\Big(1+\sum_{\ell=1}^m w_{\ell}\Big) \\
&  =R_{p_{m}}(  W)  +S_{p_{m}-1}(  W)  ,
\end{align*}
where $R_{p_{m}}(  W)  $ and $S_{p_{m}-1}(  W)  $ are
two homogeneous polynomials of degrees $p_{m}$ and $p_{m}-1$, respectively.
Since all of the polynomials $H_{p_{m}}$ and $R_{p_{m}}$ are of degree 
$p_{m}$, there exists a positive constant $C_{6}$ such that
\begin{equation}
\int_{\Omega}R_{p_{m}}(  W)  dx\leq C_{6}\int_{\Omega}H_{p_{m}}(  W)  dx.
 \label{5.21}
\end{equation}
Applying H\"{o}lder's inequality to the following integral
one obtains
\[
\int_{\Omega}S_{p_{m}-1}(  W)  dx\leq( \operatorname{meas}\Omega)
^{\frac{1}{p_{m}}}\Big( \int_{\Omega}(  S_{p_{m}-1}(  W)
)  ^{\frac{p_{m}}{p_{m}-1}}dx\Big)  ^{\frac{p_{m}-1}{p_{m}}}.
\]
Since for all $w_{1},w_{2,\dots},w_{m-1}\geq0$ and $w_{m}>0$,
\[
\frac{(  S_{p_{m}-1}(  W)  )  ^{\frac{p_{m}}{p_{m}-1}}
}{H_{p_{m}}(  W)  }=\frac{(  S_{p_{m}-1}(  x_{1}
,x_{2},\dots,x_{m-1},1)  )  ^{\frac{p_{m}}{p_{m}-1}}}{H_{p_{m}
}(  x_{1},x_{2},\dots,x_{m-1},1)  },
\]
where for all $\ell\in\{1,2,\dots,m-1\} $, 
$x_{\ell}=\frac{w_{\ell }}{w_{\ell+1}}$ 
and
\[
\lim_{x_{\ell}\to+\infty} \frac{(  S_{p_{m}-1}(
x_{1},x_{2},\dots,x_{m-1},1)  )  ^{\frac{p_{m}}{p_{m}-1}}}{H_{p_{m}
}(  x_{1},x_{2},\dots,x_{m-1},1)  }<+\infty,
\]
one asserts that there exists a positive constant $C_{7}$ such that
\begin{equation}
\frac{(  S_{p_{m}-1}(  W)  )  ^{\frac{p_{m}}{p_{m}-1}}
}{H_{p_{m}}(  W)  }\leq C_{7},\quad \text{for all }w_{1},w_{2}
,\dots,w_{m}\geq0. \label{5.22}
\end{equation}
Hence, the functional $L$ satisfies the differential inequality
\[
L'(  t)  \leq C_{6}L(  t)  +C_{8}L^{\frac{p_{m}-1}{p_{m}}}(  t)  ,
\]
which for $Z=L^{\frac{1}{p_{m}}}$ can be written as
\begin{equation}
p_{m}Z'\leq C_{6}Z+C_{8}. \label{5.23}
\end{equation}
A simple integration gives the uniform bound of the functional $L$ on the
interval $[ 0,T^{\ast}]$. This completes the proof.
\end{proof}

\begin{proof}[Proof of Corollary \ref{coro1}]
The proof of is an
immediate consequence of Theorem \ref{thm1} and the inequality
\begin{equation}
\int_{\Omega}\Big(  \sum_{\ell=1}^m w_{\ell}(
t,x)  \Big)  ^{p}dx\leq C_{9}L(  t)  \quad \text{on }[0,T^{\ast}]  . \label{5.24}
\end{equation}
for some $p\geq1$.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop2}]
From Corollary \ref{coro1}, there exists a positive constant $C_{10}$ 
such that
\begin{equation}
\int_{\Omega}\Big( \sum_{\ell=1}^m w_{\ell}(
t,x)  +1\Big)  ^{p}dx\leq C_{10}\quad \text{on }[  0,T_{\rm max}). \label{5.25}
\end{equation}
From \eqref{1.10}, we have that for all $\ell   \in\{  1,2,\dots,m\}$, 
 \begin{equation}
| F_{\ell}(  W)  | ^{\frac{p}{N}}    \leq
C_{11}(  W)  \Big(  \sum_{\ell=1}^m W_{\ell
}(  t,x)  \Big)  ^{p}\quad \text{on }[  0,T_{\rm max})\times\Omega. \label{5.26}
\end{equation}
Since $w_{1},w_{2},\dots,w_{m}$ are in $L^{\infty}(  0,T^{\ast}
;L^{p}(  \Omega)  )  $ and $\frac{p}{N}>\frac{n}{2}$, 
the solution is global.
\end{proof}

\section{Final remarks}

Recall that the eigenvectors of the diffusion matrix associated with the
eigenvalue $\overline{\lambda}_{\ell}$ is defined as 
$\overline{v}_{\ell}=(  \overline{v}_{\ell 1},\overline{v}_{\ell2},
\dots,\overline{v}_{\ell m})  ^{t}$. 
It is important to note that if $\overline{v}_{\ell}$ is an
eigenvector then so is $(-1)\overline{v}_{\ell}$. 
In the region considered in
previous sections, we only used the positive $\overline{v}_{\ell}$. 
The remainder of the $2^{m}$ regions can be formed using negative versions of the
eigenvectors. In each region, the reaction-diffusion system with a
diagonalized diffusion matrix is formed by multiplying each of the $m$
equations in \eqref{1.1} by the corresponding element of either 
$\overline {v}_{\ell}$ or $(-1)\overline{v}_{\ell}$ and then adding the 
$m$ equations together. The equations multiplied by elements of 
$\overline{v}_{\ell}$ form a set $\mathfrak{L}$, whereas the equations 
multiplied by elements of $(-1)\overline{v}_{\ell}$ form a set $\mathfrak{Z}$. 
Hence, we can define the region in the form
\begin{align*}
\Sigma_{\mathfrak{L},\mathfrak{Z}}
=\Big\{& (  u_{1}^{0},u_{2} ^{0},\dots,u_{m}^{0})  \in\mathbb{R}^{m}:
w_{\ell}^{0}=\sum_{k=1}^{m}u_{k}^{0}\overline{v}_{\ell k}\geq0,\\
&\ell \in\mathfrak{L}, \;
w_{z}^{0}=(  -1)  \sum_{k=1}^{m}u_{k}^{0}\overline{v}_{zk}
\geq0,\; z\in\mathfrak{Z}
\Big\}
\end{align*}
with
\begin{gather*}
\rho_{\ell}^{0}=\sum_{k=1}^{m}\beta_{k}v_{(  m+1-\ell)  k}\geq0, \quad
\ell\in\mathfrak{L,}\\
\rho_{\ell}^{0}=(  -1)  \sum_{k=1}^{m}\beta_{k}v_{(m+1-z)  k}\geq0, \quad 
z\in\mathfrak{Z}.
\end{gather*}
Using Lemma \ref{Lemma0} we obtain
\begin{align*}
\Sigma_{\mathfrak{L},\mathfrak{Z}}
=\Big\{& (  u_{1}^{0},u_{2}^{0},\dots,u_{m}^{0})  \in \mathbb{R}^{m}:
w_{\ell}^{0}=\sum_{k=1}^{m}u_{k}^{0}\sin\frac{(  m+1-\ell)  k\pi
}{m+1}\geq0,\\
&\ell\in\mathfrak{L}, \; 
w_{z}^{0}=(-1) \sum_{k=1}^{m}u_{k}^{0}\sin\frac{(m+1-z)  k\pi}{m+1}\geq0,\;
z\in\mathfrak{Z}\Big\},
\end{align*}
with
\begin{gather*}
\rho_{\ell}^{0}=\sum_{k=1}^{m}\beta_{k}\sin\frac{(  m+1-\ell)
k\pi}{m+1}\geq0, \quad \ell\in\mathfrak{L}\\
\rho_{z}^{0}=(-1)  \sum_{k=1}^{m}\beta_{k}\sin\frac{(
m+1-z)  k\pi}{m+1}\geq0, \quad z\in\mathfrak{Z}, \\
\mathfrak{L}\cap\mathfrak{Z}=\emptyset,\quad
\mathfrak{L}\cup \mathfrak{Z}=\{ 1,2,\dots,m\}.
\end{gather*}

\subsection*{Acknowledgements}
The author wants to thank Prof. M. Kirane and the anonymous referee for their
suggestions that improved the quality of this article.

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\end{document}