# Existence of Global Solutions Via Invariant Regions for a Generalized
Reaction-Diffusion System with a Tri-diagonal Toeplitz Matrix of Diffusion
Coefficients

Salem ABDELMALEK  S. ABDELMALEK: Department of Mathematics, College of
Sciences, Yanbu, Taibah University, Saudi Arabia.
and,
Department of mathematics, University of Tebessa 12002 Algeria.
<EMAIL_ADDRESS>
###### Abstract.

The aim of this paper is to construct invariant regions of a generalized
$m$-component reaction-diffusion system with a tri-diagonal Toeplitz matrix of
diffusion coefficients and prove the global existence of solutions using
Lyapunov functional. The paper assumes nonhomogeneous boundary conditions and
polynomial growth for the non-linear reaction term.

###### Key words and phrases:

Reaction-Diffusion Systems, Invariant Regions, Global Existence

###### 2000 Mathematics Subject Classification:

Primary 35K45, 35K57

##  1\. Introduction

Reaction-diffusion systems arise in many applications ranging from chemistry
and biology to engineering. They have been the subject of countless studies in
the past few decades. One of the most important aspects of this broad field is
proving the global existence of solutions under certain assumptions and
restrictions. We quote the recent papers of Amann [4, 5] who studies the
problem in $W^{1,p}$ and $W^{2,p}$ spaces with $p>n.$ An excellent reference
for a dynamic theory of reaction-diffusion systems is the book of Henry [12].

In 2001, Kouachi [15] followed on previous work and showed the global
existence of solutions assuming the reaction terms of a $2\times 2$ diagonal
system exhibit a polynomial growth. This was later generalized by Kouachi for
an arbitrary $2\times 2$ Toeplitz matrix. In [2], the author of this work
studied the $3\times 3$ case under the same assumptions and restrictions.
Abdelmalek and Kouachi [3] also showed the global existence of solutions for
an $m$-component reaction–diffusion system ($m\geq 2$) with a diagonal
diffusion matrix and reaction terms of polynomial growth.

An important factor in the study of reaction diffusion systems is the
characteristics of the diffusion matrix. Although in some cases the matrix is
diagonal, in many cases cross diffusion terms exist. For instance, many
chemical and biological operations are described by reaction-diffusion systems
with a tri-diagonal matrix of diffusion coefficients, (see, e.g., Cussler [9]
and [10]). Other examples include the modelling of epidemics [7], ecology [16]
and biochemistry [6], where cross-diffusion appears to be a very relevant
problem to be analyzed. In this paper, tri-diagonal diffusion matrices have
been considered and sufficient conditions have been given for global existence
steady states.

The purpose of this paper is to prove the global existence of solutions with
nonhomogeneous Neumann, Dirichlet, or Robin conditions and a polynomial growth
of reaction terms. The polynomial growth is established through a mere single
inequality as we shall show. The main contribution of this paper is the fact
that we take a general Toeplitz matrix as opposed to the symmetry constraint
assumed in [1].

Throughout this paper, we consider an $m$-component system, with $m\geq 2$:

$\dfrac{\partial U}{\partial t}-D\Delta U=F\left(U\right)\text{\ in
}\Omega\times\left(0,+\infty\right),$ (1.1)

with the boundary conditions:

$\alpha U+\left(1-\alpha\right)\partial_{\eta}U=B\text{\ on
}\partial\Omega\times\left(0,+\infty\right),$ (1.2)

or

$\alpha U+\left(1-\alpha\right)D\partial_{\eta}U=B\text{\ on
}\partial\Omega\times\left(0,+\infty\right),$ (1.3)

in the case of non-diagonal boundary conditions, and the initial data:

$U\left(x,0\right)=U_{0}\left(x\right)\text{ on}\;\Omega.$ (1.4)

We consider three types of boundary conditions:

  1. (i)

Nonhomogeneous Robin boundary conditions, corresponding to

$0<\alpha<1,\text{ }B\in\mathbb{R}^{m};$

  2. (ii)

Homogeneous Neumann boundary conditions, corresponding to

$\alpha=0\text{ and }B\equiv 0;$

  3. (iii)

Homogeneous Dirichlet boundary conditions, corresponding to

$1-\alpha=0\text{ and }B\equiv 0.$

In the context of this work, $\Omega$ is an open bounded domain of class
$C^{1}$ in $\mathbb{\mathbb{R}}^{n}$ with boundary $\partial\Omega$,
$\dfrac{\partial}{\partial\eta}$ denotes the outward normal derivative on
$\partial\Omega$, and

$\displaystyle U$ $\displaystyle:=\left(u_{1},...,u_{m}\right)^{T},$
$\displaystyle F$ $\displaystyle:=\left(f_{1},...,f_{m}\right)^{T},$
$\displaystyle B$ $\displaystyle:=\left(\beta_{1},...,\beta_{m}\right)^{T}.$

The diffusion matrix is assumed to be a tri-diagonal Toeplitz one of the form

$D:=\left(\begin{array}[c]{ccccc}a&b&0&\cdots&0\\\ c&a&b&\ddots&\vdots\\\
0&c&a&\ddots&0\\\ \vdots&\ddots&\ddots&\ddots&b\\\
0&\cdots&0&c&a\end{array}\right)_{m\times m},$

where $a,b$ and $c$ are supposed to be strictly positive constants satisfying:

$\cos\frac{\pi}{m+1}<\frac{a}{b+c},$ (1.5)

which reflects the parabolicity of the system.

The initial data are assumed to be in the regions:

$\Sigma_{\mathfrak{L},\mathfrak{Z}}:=\left\\{U_{0}\in\mathbb{\mathbb{R}}^{m}:\left\langle
V_{z},U_{0}\right\rangle\leq 0\leq\left\langle
V_{\ell},U_{0}\right\rangle,\text{ }\ell\in\mathfrak{L},\text{
}z\in\mathfrak{Z}\right\\},$ (1.6)
$\mathfrak{L}\cap\mathfrak{Z}=\varnothing,\mathfrak{L}\cup\mathfrak{Z}=\left\\{1,2,...,m\right\\},$
(1.7)

subject to

$\left\langle V_{z},B\right\rangle\leq 0\leq\left\langle
V_{\ell},B\right\rangle,\text{ }\ell\in\mathfrak{L},z\in\mathfrak{Z}.$

The vector $V_{\ell}=\left(v_{1\ell},...,v_{m\ell}\right)^{T}$ are defined as

$v_{k\ell}=\sqrt{\mu^{k}}\sin\frac{k\left(m+1-\ell\right)\pi}{m+1},\text{
}k=1,...,m,$

with

$\mu:=\frac{b}{c}.$

The notation $\left\langle\cdot,\cdot\right\rangle$ denotes the inner product
in $\mathbb{R}^{m}.$

From (1.7) we can clearly see that there are in fact $2^{m}$ regions. One of
the main contributions of this paper is that unlike previous studies we cover
_all_ possible regions. Hence, the work carried out here is a generalization
of previous studies. The most important of these studies are discussed below.

In 2002, Kouachi [14] studied the case $m=2$, for which the parabolicity
condition we use here (1.5) reduces to the same condition employed in [14]:
$2a>\left(b+c\right)$. Although in this case $2^{2}=4$ regions exist, the
study of Kouachi considered only a couple of these regions. Setting $m=2$ in
(1.6) yields the following regions:

  * •

If $\mathfrak{L}=\left\\{1,2\right\\},\mathfrak{Z}=\emptyset$ then,

$\Sigma_{\mathfrak{L},\mathfrak{Z}}=\left\\{\left(u_{1}^{0},u_{2}^{0}\right)^{T}\in\mathbb{\mathbb{R}}^{2}:u_{1}^{0}\geq\sqrt{\mu}\left|u_{2}^{0}\right|\text{
if }\beta_{1}\geq\sqrt{\mu}\left|\beta_{2}\right|\right\\}.$

  * •

If $\mathfrak{L}=\left\\{2\right\\},\mathfrak{Z}=\left\\{1\right\\}$ then,

$\Sigma_{\mathfrak{L},\mathfrak{Z}}=\left\\{\left(u_{1}^{0},u_{2}^{0}\right)^{T}\in\mathbb{\mathbb{R}}^{2}:\sqrt{\mu}u_{2}^{0}\geq\left|u_{1}^{0}\right|\text{
if }\sqrt{\mu}\beta_{2}\geq\left|\beta_{1}\right|\right\\}.$

  * •

If $\mathfrak{L}=\emptyset,\mathfrak{Z}=\left\\{1,2\right\\}$ then,

$\Sigma_{\mathfrak{L},\mathfrak{Z}}=\left\\{\left(u_{1}^{0},u_{2}^{0}\right)^{T}\in\mathbb{\mathbb{R}}^{2}:-u_{1}^{0}\geq\sqrt{\mu}\left|u_{2}^{0}\right|\text{
if }-\beta_{1}\geq\sqrt{\mu}\left|\beta_{2}\right|\right\\}.$

  * •

If $\mathfrak{L}=\left\\{1\right\\},\mathfrak{Z}=\left\\{2\right\\}$ then,

$\Sigma_{\mathfrak{L},\mathfrak{Z}}=\left\\{\left(u_{1}^{0},u_{2}^{0}\right)^{T}\in\mathbb{\mathbb{R}}^{2}:-\sqrt{\mu}u_{2}^{0}\geq\left|u_{1}^{0}\right|\text{
if }-\sqrt{\mu}\beta_{2}\geq\left|\beta_{1}\right|\right\\}.$

In fact the last two of these regions were not considered in [14].

In 2007, the author of this work [2] studied the case $m=3$ for which the
parabolicity condition is $\sqrt{2}a>\left(b+c\right)$, resulting from the
direct substitution of $m=3$ in (1.5). The total number of regions in this
case is $2^{3}=8$ of which only 4 regions were however studied.

In 2014 the author [1] elaborated on the generalized $m$-component case with a
tri-diagonal matrix having equal upper and lower diagonal elements, i.e.
$\left(b=c\right)$. Substituting $b=c$ in (1.5) yields the same condition used
in [1]: $2b\cos\frac{\pi}{m+1}<a$.

The aim of this work is to prove the global existence of solutions. The
necessary proofs are similar for all the invariant regions. Hence we only
focus on one of the regions and present a generalization at the end of the
paper.

Consider the region with $\mathfrak{L}=\left\\{1,2,...,m\right\\}$ and
$\mathfrak{Z}=\emptyset$ yielding

$\Sigma_{\mathfrak{L},\emptyset}=\left\\{U_{0}\in\mathbb{\mathbb{R}}^{m}:\left\langle
V_{\ell},U_{0}\right\rangle\geq 0,\text{ }\ell\in\mathfrak{L},\right\\},$
(1.8)

subject to

$\left\langle V_{\ell},B\right\rangle\geq 0,\text{ }\ell\in\mathfrak{L.}$

In order to establish the global existence of solutions in this region we
diagonalize the diffusion matrix $D$. We define the reaction diffusion
functions as:

$\digamma\left(W\right):=\left(\digamma_{1},\digamma_{2},...,\digamma_{m}\right)^{T},\text{
}\digamma_{\ell}:=\left\langle V_{\ell},F\right\rangle,$ (1.9)

where the variable $W=\left(w_{1},w_{2},...,w_{m}\right)^{T}$ is given by

$W:=\left(w_{1},w_{2},...,w_{m}\right)^{T},w_{\ell}:=\left\langle
V_{\ell},U\right\rangle.$ (1.10)

The functions $\digamma_{\ell}$ must satisfy the following three conditions:

  1. (A1)

be continuously differentiable on $\mathbb{R}_{+}^{m}$ for all $\ell=1,...,m$,
satisfying $\digamma_{\ell}(w_{1},...,w_{\ell-1},0,w_{\ell+1},...,w_{m})\geq
0$, for all $w_{\ell}\geq 0;$ $\ell=1,...,m$.

  2. (A2)

be of polynomial growth (see the work of Hollis and Morgan [13]), which means
that for all $\ell=1,...,m$:

$\left|\digamma_{\ell}\left(W\right)\right|\leq C_{1}\left(1+\left\langle
W,1\right\rangle\right)^{N},n\in\mathbb{N}\text{,on
}\left(0,+\infty\right)^{m}.$ (1.11)

  3. (A3)

satisfy the inequality:

$\left\langle S,\digamma\left(W\right)\right\rangle\leq
C_{2}\left(1+\left\langle W,1\right\rangle\right),$ (1.12)

where

$S:=\left(d_{1},d_{2},...,d_{n-1},1\right)^{T},$

for all $w_{\ell}\geq 0,$ $\ell=1,...,m,$. All the constants $d_{\ell}$
satisfy $d_{\ell}\geq\overline{d_{\ell}},$ $\ell=1,...,m$ where
$\overline{d_{\ell}},$ $\ell=1,...,m,$ are sufficiently large positive
constants. Here $C_{1}$ and $C_{2}$ are uniformly bounded positive functions
defined on $\mathbb{R}_{+}^{m}$.

##  2\. Some Properties of the Diffusion Matrix and Parabolicity

######  Proposition 1.

A quadratic form $Q=\left\langle X,AX\right\rangle=X^{T}AX$, with $A$ being a
symmetric matrix, is positive definite for every non-zero column vector $X$ if
all the principal minors in the top-left corner of $A$ are positive. If $A$ is
non-symmetric, $Q$ is positive definite iff the principal minors in the top-
left corner of $\frac{1}{2}\left(A+A^{T}\right)$ are positive.

######  Lemma 1.

The reaction-diffusion system (1.1) satisfies the parabolicity condition if
(1.5) is satisfied.

###### Proof.

The system (1.1) satisfies the parabolicity condition if the matrix
$\left(D+D^{T}\right)$ is positive definite. The matrix $\left(D+D^{T}\right)$
is symmetric tri-diagonal with off-diagonal elements
$\frac{1}{2}\left(b+c\right)$. In [1] a similar matrix with off-diagonal
elements $b$ and the parabolicity condition

$2b\cos\frac{\pi}{m+1}<a,$

is considered. Substituting $b$ with $\frac{1}{2}\left(b+c\right)$ yields
(1.5). ∎

######  Lemma 2 (see [8] ).

The eigenvalues $\overline{\lambda}_{\ell}<\overline{\lambda}_{\ell-1};$
$\ell=2,...,m,$ of $D^{T}$ are positive and are given by

$\overline{\lambda}_{\ell}:=a+2\sqrt{bc}\cos\left(\frac{\ell\pi}{m+1}\right),$
(2.1)

with the corresponding eigenvectors being $\overline{V}_{\ell}=V_{m+1-\ell},$
for $\ell=1,...,m$. Therefore, $D^{T}$ is diagonalizable.

In the remainder of this work we require an ascending order of the
eigenvalues. In order to simplify the indices in the formulas to come we
define

$\lambda_{\ell}:=\bar{\lambda}_{m+1-\ell}=a+2\sqrt{bc}\cos\left(\frac{(m+1-\ell)\pi}{m+1}\right);\text{
}\ell=1,...,m,$ (2.2)

thus $\lambda_{\ell}<\lambda_{\ell+1};$ $\ell=2,...,m.$

###### Proof.

Recall that the diffusion matrix is positive definite, implying that its
eigenvalues are necessarily positive. For a given eigenpair
$\left(\overline{\lambda},X\right)$ the components of
$\left(D^{T}-\overline{\lambda}I\right)X=0$ are

$bx_{k-1}+\left(a-\overline{\lambda}\right)x_{k}+cx_{k+1}=0,\text{
}k=1,...,m,$

with $x_{0}=x_{m+1}=0,$ or equivalently,

$x_{k+2}+\left(\frac{a-\overline{\lambda}}{c}\right)x_{k+1}+\mu
x_{k}=0,k=0,...,m-1,$

whose solutions are

$x_{k}=\left\\{\begin{array}[c]{ll}\alpha r_{1}^{k}+\beta r_{2}^{k},&\text{if
}r_{1}\neq r_{2},\\\ \alpha\rho^{k}+\beta k\rho^{k},&\text{if
}r_{1}=r_{2}=\rho,\end{array}\right.$

where $\alpha$ and $\beta$ are arbitrary constants.

For the eigenvalue problem at hand, $r_{1}$ and $r_{2}$ must be distinct.
Putting $x_{k}=\alpha r_{1}^{k}+\beta r_{2}^{k}$, and $x_{0}=x_{m+1}=0$ yields

$\left\\{\begin{array}[c]{l}0=\alpha+\beta\\\ 0=\alpha r_{1}^{m+1}+\beta
r_{2}^{m+1}\end{array}\right.\Rightarrow\left(\frac{r_{1}}{r_{2}}\right)^{m+1}=\frac{-\beta}{\alpha}=1\Rightarrow\frac{r_{1}}{r_{2}}=e^{\frac{2i\pi\ell}{m+1}}.$

Therefore we see that $r_{1}=r_{2}e^{\frac{2i\pi\ell}{m+1}}$ for
$1\leq\ell\leq m$. This together with

$r^{2}+\left(\frac{a-\overline{\lambda}}{c}\right)r+\mu=\left(r-r_{1}\right)\left(r-r_{2}\right)\Rightarrow\left\\{\begin{array}[c]{l}r_{1}r_{2}=\mu\\\
r_{1}+r_{2}=-\frac{a-\overline{\lambda}}{c}\end{array}\right.,$

leads to $r_{1}=\sqrt{\mu}e^{\frac{i\pi\ell}{m+1}}$,
$r_{2}=\sqrt{\mu}e^{-\frac{i\pi\ell}{m+1}}$, and

$\overline{\lambda}=a+2\sqrt{cb}\left(e^{\frac{i\pi\ell}{m+1}}+e^{-\frac{i\pi\ell}{m+1}}\right)=a+2a+2\sqrt{cb}\cos\left(\frac{\ell\pi}{m+1}\right)\text{.}$

Thus the eigenvalues of $D^{T}$ are given by

$\overline{\lambda}_{\ell}=a+2\sqrt{cb}\cos\left(\frac{\ell\pi}{m+1}\right),$

for $\ell=1,...,m.$

Since the eigenvalues are all distinct (as $\cos\theta$ is strictly decreasing
on $\left(0,\pi\right),$ and $b\neq 0\neq c$), then $D$ is necessarily
diagonalizable.

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

$x_{k}=\alpha\mu^{\frac{k}{2}}\left(e^{\frac{2i\pi k}{m+1}}-e^{-\frac{2i\pi
k}{m+1}}\right)=2i\alpha\mu^{\frac{k}{2}}\sin\left(\frac{k}{m+1}\pi\right).$

Setting $\alpha=\frac{1}{2i}$ yields a particular eigenvector associated to
$\overline{\lambda}_{\ell}$ given by

$\overline{V}_{\ell}=\left(\mu^{\frac{1}{2}}\sin\left(\frac{1\ell\pi}{m+1}\right),\mu^{\frac{2}{2}}\sin\left(\frac{2\ell\pi}{m+1}\right),...,\mu^{\frac{m}{2}}\sin\left(\frac{m\ell\pi}{m+1}\right)\right)^{t}.$

Since the eigenvectors are all distinct then
$\left\\{\overline{V}_{1},\overline{V}_{2},...,\overline{V}_{m}\right\\}$ is a
complete linearly independent set, hence
$\left(\overline{V}_{1}\shortmid\overline{V}_{2}\shortmid...\shortmid\overline{V}_{m}\right)$
_diagonalizes_ $D$.

Now let us prove that

$\overline{\lambda}_{\ell}<\overline{\lambda}_{\ell-1};\text{ }\ell=2,...,m.$

We have

$\ell>\ell-1\Rightarrow\frac{\ell\pi}{m+1}>\frac{\left(\ell-1\right)\pi}{m+1}.$

Once again using the fact that $\cos\theta$ is strictly decreasing on
$\left(0,\pi\right),$ we deduce that

$\cos\left(\frac{\ell\pi}{m+1}\right)<\cos\left(\frac{\left(\ell-1\right)\pi}{m+1}\right),$

whereupon

$\overline{\lambda}_{\ell}=a+2\sqrt{cb}\cos\left(\frac{\ell\pi}{m+1}\right)<a+2\sqrt{cb}\cos\left(\frac{\left(\ell-1\right)\pi}{m+1}\right)=\overline{\lambda}_{\ell-1}.$

∎

######  Lemma 3.

The eigenvalues of the matrix $D$ are positive, i.e. $\lambda_{\ell}>0$ and
$\det D>0.$

###### Proof.

Recall that $\lambda_{\ell}<\lambda_{\ell+1};$ $\ell=1,...,m-1$, i.e.

$\lambda_{1}<\lambda_{2}<...<\lambda_{m}.$

We want to show that $\lambda_{1}>0$. First, we have

$\lambda_{1}=a+2\sqrt{cb}\cos\left(\frac{m}{m+1}\pi\right)>0,$ (2.3)

which implies

$a>2\sqrt{bc}\left[-\cos\left(\frac{m}{m+1}\pi\right)\right].$

From condition (1.5), we obtain

$a>\left(c+b\right)\left(\cos\frac{\pi}{m+1}\right).$ (2.4)

Note that

$\frac{m}{m+1}>\frac{1}{2}\Rightarrow\cos\left(\frac{m}{m+1}\pi\right)<\cos\frac{\pi}{2}=0;$
(2.5)

furthermore

$\left(c+b\right)\geq 2\sqrt{bc}.$ (2.6)

We also have

$\cos\left(\frac{\pi}{m+1}\right)+\cos\left(\frac{m}{m+1}\pi\right)=0$ (2.7)

since

$\displaystyle\cos\left(\frac{\pi}{m+1}\right)+\cos\left(\frac{m}{m+1}\pi\right)$
$\displaystyle=2\cos\left(\frac{\frac{\pi}{m+1}+\frac{m}{m+1}\pi}{2}\right)\cos\left(\frac{\frac{\pi}{m+1}-\frac{m}{m+1}\pi}{2}\right)$
$\displaystyle=2\cos\left(\frac{\pi}{2}\right)\cos\left(\frac{m-1}{m+1}\frac{\pi}{2}\right)=0.$

Now, from (2.5), (2.6), and (2.7), we obtain

$\left(c+b\right)\left(\cos\frac{\pi}{m+1}\right)\geq
2\sqrt{bc}\left[-\cos\left(\frac{m}{m+1}\pi\right)\right],$ (2.8)

and from (2.4) and (2.8), we get

$a>2\sqrt{bc}\left[-\cos\left(\frac{m}{m+1}\pi\right)\right],$

which concludes the proof of (2.3) and guarantees that all eigenvalues of
$D^{T}$ are positive. Furthermore since the eigenvalues of $D$ are the same as
those of $D^{T}$ we conclude that $\det D>0$. ∎

##  3\. Local Existence and Invariant Regions

The usual norms in spaces $L^{p}(\Omega)$, $L^{\infty}(\Omega)$ and
$C(\overline{\Omega})$ are denoted respectively by:

$\displaystyle\left\|u\right\|_{p}^{p}$
$\displaystyle=\frac{1}{\left|\Omega\right|}\int_{\Omega}\left|u(x)\right|^{p}dx;$
$\displaystyle\left\|u\right\|_{\infty}$
$\displaystyle=ess\underset{x\in\Omega}{\sup}\left|u(x)\right|,$

and

$\left\|u\right\|_{C(\overline{\Omega})}=\underset{x\in\overline{\Omega}}{\max}\left|u(x)\right|.$

It is well-known that in order to prove the global existence of solutions to a
reaction-diffusion system (see Henry [12]) it suffices to derive a uniform
estimate of the associated reaction term on $\left[0,T_{\max}\right)$ in the
space $L^{p}(\Omega)$ for some $p>n/2$. Our aim is to construct Lyapunov
polynomial functionals allowing us to obtain $L^{p}$-bounds on the components,
which leads to global existence. Since the reaction terms are continuously
differentiable on $\mathbb{R}_{+}^{m}$, then for any initial data in
$C(\overline{\Omega})$ it is straightforward to directly check their Lipschitz
continuity on bounded subsets of the domain of a fractional power of the
operator

$\mathfrak{D}=-\operatorname*{diag}\left(\lambda_{1}\Delta,\lambda_{2}\Delta,...,\lambda_{m}\Delta\right).$
(3.1)

The assumption (1.5) implies that $D\Delta$ is a strongly elliptic operator in
the sense of Petrowski, see Friedman [11].

######  Proposition 2.

Diagonalizing system (1.1) yields:

$W_{t}-\operatorname*{diag}\left(\lambda_{1},\lambda_{2},...,\lambda_{m}\right)\Delta
W=\digamma\left(W\right)\text{ in }\Omega\times\left(0,+\infty\right),$ (3.2)

with the boundary condition

$\alpha W+\left(1-\alpha\right)\partial_{\eta}W=\Lambda\text{\ \ \ \ \ on
}\partial\Omega\times\left(0,+\infty\right),$ (3.3)

or

$\alpha
W+\left(1-\alpha\right)\operatorname*{diag}\left(\lambda_{1},\lambda_{2},...,\lambda_{m}\right)\partial_{\eta}W=\Lambda\text{\
on }\partial\Omega\times\left(0,+\infty\right),$ (3.4)

and the initial data

$W\left(x,0\right)=W_{0}\text{ }\ \ \ \ \text{on }\Omega$

###### Proof.

The eigenvectors of the diffusion matrix associated with the eigenvalues
$\lambda_{\ell}$ are defined as $V_{\ell}=\left(v_{\ell 1},v_{\ell
2},...,v_{\ell m}\right)^{T}$. Let us consider the diagonalizing matrix of
eigenvectors $P=\left(V_{1}\shortmid V_{2}\shortmid...\shortmid V_{m}\right)$
and define the solution vector $U$ and the reaction terms vector $F$. Pre-
multiplying the system by $P^{T}$ yields

$\displaystyle U_{t}-D\Delta U$ $\displaystyle=F$ $\displaystyle
P^{T}U_{t}-\Delta P^{T}DU$ $\displaystyle=P^{T}F$ $\displaystyle
P^{T}U_{t}-\Delta P^{T}D\left(P^{T}\right)^{-1}P^{T}U$ $\displaystyle=P^{T}F.$
(3.5)

The term $P^{T}U$ can be simplified as follows

$\displaystyle P^{T}U$ $\displaystyle=\left(V_{1}\shortmid
V_{2}\shortmid...\shortmid V_{m}\right)^{T}U$
$\displaystyle=\left(\left\langle V_{1},U\right\rangle,\left\langle
V_{2},U\right\rangle,...,\left\langle V_{m},U\right\rangle\right)^{T}$
$\displaystyle=\left(w_{1},w_{2},...,w_{m}\right)^{T}=W.$ (3.6)

Hence, $P^{T}U_{t}=W_{t}$. Similarly,

$\displaystyle P^{T}F$ $\displaystyle=\left(V_{1}\shortmid
V_{2}\shortmid...\shortmid V_{m}\right)^{T}F$
$\displaystyle=\left(\left\langle V_{1},F\right\rangle,\left\langle
V_{2},F\right\rangle,...,\left\langle V_{m},F\right\rangle\right)^{T}$
$\displaystyle=\left(\digamma_{1},\digamma_{2},...,\digamma_{m}\right)^{T}=\digamma.$
(3.7)

Furthermore we have the similarity transformation

$\displaystyle P^{T}D\left(P^{T}\right)^{-1}$
$\displaystyle=P^{T}\left(D^{T}\right)^{T}\left(P^{-1}\right)^{T}$
$\displaystyle=\left(D^{T}P\right)^{T}\left(P^{-1}\right)^{T}$
$\displaystyle=\left(P^{-1}D^{T}P\right)^{T}$
$\displaystyle=\left(\operatorname*{diag}\left(\lambda_{1},\lambda_{2},...,\lambda_{m}\right)\right)^{T}$
$\displaystyle=\operatorname*{diag}\left(\lambda_{1},\lambda_{2},...,\lambda_{m}\right).$
(3.8)

Substituting (3.6), (3.7), and (3.8) in (3.5) results in the equivalent system
(3.2). The boundary condition (3.3) can be obtained by pre-multiplying (1.2)
by $P^{T}$:

$\displaystyle\alpha U+\left(1-\alpha\right)\partial_{\eta}U$
$\displaystyle=B,$ $\displaystyle\alpha
P^{T}U+\left(1-\alpha\right)\partial_{\eta}P^{T}U$ $\displaystyle=P^{T}B.$
(3.9)

Simplifying the term $P^{T}B$ yields

$\displaystyle P^{T}B$ $\displaystyle=\left(V_{1}\shortmid
V_{2}\shortmid...\shortmid V_{m}\right)^{T}B$
$\displaystyle=\left(\left\langle V_{1},B\right\rangle,\left\langle
V_{2},B\right\rangle,...,\left\langle V_{m},B\right\rangle\right)^{T}$
$\displaystyle=\left(\rho_{1}^{0},\rho_{2}^{0},...,\rho_{m}^{0}\right)^{T}:=\Lambda.$
(3.10)

Substituting (3.6) and (3.9) in (3.10) gives the boundary condition for the
equivalent system (3.3).

Pre-multiplying (1.3) by $P^{T}$ yields

$\displaystyle\alpha U+\left(1-\alpha\right)D\partial_{\eta}U$
$\displaystyle=B,$ $\displaystyle\alpha
P^{T}U+\left(1-\alpha\right)\partial_{\eta}P^{T}DU$ $\displaystyle=P^{T}B,$
$\displaystyle\alpha
P^{T}U+\left(1-\alpha\right)\partial_{\eta}P^{T}D\left(P^{T}\right)^{-1}P^{T}U$
$\displaystyle=P^{T}B.$ (3.11)

Substituting (3.7), (3.8), and (3.10) results in the equivalent boundary
condition in (3.4).

We note that condition (1.5) guarantees the parabolicity of the system (1.1),
which implies that this system is equivalent to that described by (3.2) in the
region:

$\displaystyle\Sigma_{\mathfrak{L},\mathfrak{Z}}$
$\displaystyle=\left\\{U_{0}\in\mathbb{\mathbb{R}}^{m}:\left\langle
V_{\ell},U_{0}\right\rangle\geq 0,\text{ }\ell\in\mathfrak{L}\right\\}$
$\displaystyle=\left\\{U_{0}\in\mathbb{\mathbb{R}}^{m}:w_{\ell}^{0}=\left\langle
V_{\ell},U_{0}\right\rangle\geq 0,\text{ }\ell\in\mathfrak{L}\right\\}$

with

$\rho_{\ell}^{0}=\left\langle V_{\ell},B\right\rangle\geq
0,\ell\in\mathfrak{L}.$

This implies that the components $w_{\ell}$ are necessarily positive. ∎

The local existence and uniqueness of solutions to the initial system (1.1),
with initial data in $C(\overline{\Omega})$ or $L^{p}(\Omega)$,
$p\in\left(1,+\infty\right)$, follows from the basic existence theory for
abstract semi-linear differential equations (Henry [12] ). The solutions are
classical on $\left(0,T_{\max}\right)$, where $T_{\max}$ denotes the eventual
blow up time in $L^{\infty}(\Omega)$. The local solution is continued globally
by apriori estimates. Once the invariant regions are constructed, one can
apply the Lyapunov technique and establish the global existence of a unique
solution for (1.1).

######  Proposition 3.

The system (3.2) admits a unique classical solution $W$ on$\
\Omega\times(0,T_{\max})$; moreover we have the alternative

$\text{If }T_{\max}<\infty\text{ then }\underset{t\nearrow
T_{\max}}{\lim}\overset{m}{\underset{\ell=1}{\sum}}\left\|w_{\ell}\left(t,.\right)\right\|_{\infty}=\infty\text{,}$
(3.12)

where $T_{\max}$
$\left(\left\|w_{1}^{0}\right\|_{\infty},\left\|w_{2}^{0}\right\|_{\infty},...,\left\|w_{m}^{0}\right\|_{\infty}\right)$
denotes the eventual blow-up time.

##  4\. Main Result

Before we present the main result of this paper, let us define

$K_{l}^{r}=K_{r-1}^{r-1}K_{l}^{r-1}-\left[H_{l}^{r-1}\right]^{2},\text{
}r=3,...,l\text{,}$ (4.1)

where

$H_{l}^{r}=\underset{1\leq\ell,\kappa\leq
l}{\det}\left(\left(a_{\ell,\kappa}\right)_{\begin{subarray}{c}\ell\neq
l,...r+1\\\ \kappa\neq
l-1,..r\end{subarray}}\right)\overset{k=r-2}{\underset{k=1}{\prod}}\left(\det\left[k\right]\right)^{2^{\left(r-k-2\right)}},\text{
}r=3,...,l-1\text{,}$ $K_{l}^{2}=\underset{\text{positive
value}}{\underbrace{\bar{\lambda}_{1}\bar{\lambda}_{l}\overset{l-1}{\underset{k=1}{\prod}}\theta_{k}^{2\left(p_{k}+1\right)^{2}}\overset{m-1}{\underset{k=l}{\prod}}\theta_{k}^{2\left(p_{k}+2\right)^{2}}}}\left[\overset{l-1}{\underset{k=1}{\prod}}\theta_{k}^{2}-A_{1l}^{2}\right],$

and

$H_{l}^{2}=\underset{\text{positive
value}}{\underbrace{\bar{\lambda}_{1}\sqrt{\bar{\lambda}_{2}\bar{\lambda}_{l}}\theta_{1}^{2\left(p_{1}+1\right)^{2}}\overset{l-1}{\underset{k=2}{\prod}}\theta_{k}^{\left(p_{k}+2\right)^{2}+\left(p_{k}+1\right)^{2}}\overset{m-1}{\underset{k=l}{\prod}}\theta_{k}^{2\left(p_{k}+2\right)^{2}}}}\left[\theta_{1}^{2}A_{2l}-A_{12}A_{1l}\right].$

Here $\underset{1\leq\ell,\kappa\leq
l}{\det}\left(\left(a_{\ell,\kappa}\right)_{\begin{subarray}{c}\ell\neq
l,...r+1\\\ \kappa\neq l-1,..r\end{subarray}}\right)$ denotes the determinant
of the $r$-square symmetric matrix obtained from
$\left(a_{\ell,\kappa}\right)_{1\leq\ell,\kappa\leq m}$ by removing the
$\left(r+1\right)^{\text{th}}$, $\left(r+2\right)^{\text{th}}$, …,
$l^{\text{th}}$ rows and the
$r^{\text{th}},\left(r+1\right)^{\text{th}},...,\left(l-1\right)^{\text{th}}$
columns, and $\det\left[1\right],...,\det\left[m\right]$ are the minors of the
matrix $\left(a_{\ell,\kappa}\right)_{1\leq\ell,\kappa\leq m}.$ The elements
of the matrix are:

$a_{\ell\kappa}=\frac{\lambda_{\ell}+\lambda_{\kappa}}{2}\theta_{1}^{p_{1}^{2}}...\theta_{\left(\ell-1\right)}^{p_{\left(\ell-1\right)}^{2}}\theta_{\ell}^{\left(p_{\ell}+1\right)^{2}}...\theta_{\kappa-1}^{\left(p_{\left(\kappa-1\right)}+1\right)^{2}}\theta_{\kappa}^{\left(p_{\kappa}+2\right)^{2}}...\theta_{\left(m-1\right)}^{\left(p_{\left(m-1\right)}+2\right)^{2}},$
(4.2)

where $\lambda_{\ell}$ is defined in (2.1)-(2.2). Note that
$A_{\ell\kappa}=\dfrac{\lambda_{\ell}+\lambda_{\kappa}}{2\sqrt{\lambda_{\ell}\lambda_{\kappa}}}$
for all $\ell,\kappa=1,...,m$, and $\theta_{\ell},$
$\ell=1,...,\left(m-1\right)$ are positive constants.

######  Theorem 1.

Suppose that the functions $\digamma_{\ell},$ $\ell=1,...,m$, are of
polynomial growth and satisfy the condition (1.12) for some sufficiently large
positive constants $D_{\ell},$ $\ell=1,...,m$. Let
$\left(w_{1}\left(t,.\right),w_{2}\left(t,.\right),...,w_{m}\left(t,.\right)\right)$
be a solution of (3.2)-(3.3) and

$L(t)=\int_{\Omega}H_{p_{m}}\left(w_{1}\left(t,x\right),w_{2}\left(t,x\right),...,w_{m}\left(t,x\right)\right)dx,$
(4.3)

where

$H_{p_{m}}\left(w_{1},...,w_{m}\right)=\overset{p_{m}}{\underset{p_{m-1}=0}{\sum}}...\overset{p_{2}}{\underset{p_{1}=0}{\sum}}C_{p_{m}}^{p_{m-1}}...C_{p_{2}}^{p_{1}}\theta_{1}^{p_{1}^{2}}...\theta_{\left(m-1\right)}^{p_{\left(m-1\right)}^{2}}w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}...w_{m}^{p_{m}-p_{m-1}},$

with $p_{m}$ a positive integer and
$C_{p_{\kappa}}^{p_{\ell}}=\frac{p_{\kappa}!}{p_{\ell}!\left(p_{\kappa}-p_{\ell}\right)!}$.
Furthermore suppose that the following condition is satisfied

$K_{l}^{l}>0,\text{ }l=2,...,m\text{.}$ (4.4)

where $K_{l}^{l}$ was defined in (4.1).Then it follows that the functional $L$
is uniformly bounded on the interval $\left[0,T^{\ast}\right],$
$T^{\ast}<T_{\max}$.

######  Corollary 1.

Under the assumptions of Theorem 1, all solutions of (3.2)-(3.3) with positive
initial data in $L^{\infty}\left(\Omega\right)$ are in
$L^{\infty}\left(0,T^{\ast};L^{p}\left(\Omega\right)\right)$, for some $p\geq
1.$

######  Proposition 4.

Under the assumptions of theorem 1 and given that the condition (1.5) is
satisfied, all solutions of (3.2)-(3.3) with positive initial data in
$L^{\infty}\left(\Omega\right)$ are global for some $p>\dfrac{Nn}{2}$.

For the proof of Theorem 1, we first need to define some preparatory Lemmas.

######  Lemma 4 (see [3]).

let $H_{p_{m}}$ be the homogeneous polynomial defined in (4.3), we have

$\displaystyle\partial_{w_{1}}H_{p_{m}}$
$\displaystyle=p_{m}\overset{p_{m}-1}{\underset{p_{m-1}=0}{\sum}}...\overset{p_{2}}{\underset{p_{1}=0}{\sum}}C_{p_{m}-1}^{p_{m-1}}...C_{p_{2}}^{p_{1}}\theta_{1}^{\left(p_{1}+1\right)^{2}}...\theta_{\left(m-1\right)}^{\left(p_{\left(m-1\right)}+1\right)^{2}}$
$\displaystyle
w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}w_{3}^{p_{3}-p_{2}}...w_{m}^{\left(p_{m}-1\right)-p_{m-1}}.$
(4.5) $\displaystyle\partial_{w_{\ell}}H_{p_{m}}$
$\displaystyle=p_{m}\overset{p_{m}-1}{\underset{p_{m-1}=0}{\sum}}...\overset{p_{2}}{\underset{p_{1}=0}{\sum}}C_{p_{m}-1}^{p_{m-1}}...C_{p_{2}}^{p_{1}}\theta_{1}^{p_{1}^{2}}...\theta_{\ell-1}^{p_{\left(\ell-1\right)}^{2}}\theta_{\ell}^{\left(p_{\ell}+1\right)2}...\theta_{\left(m-1\right)}^{\left(p_{\left(m-1\right)}+1\right)^{2}}$
$\displaystyle
w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}w_{3}^{p_{3}-p_{2}}...w_{m}^{\left(p_{m}-1\right)-p_{m-1}},\text{
}\ell=2,...,m-1,$ (4.6) $\displaystyle\partial_{w_{m}}H_{p_{m}}$
$\displaystyle=p_{m}\overset{p_{m}-1}{\underset{p_{m-1}=0}{\sum}}...\overset{p_{2}}{\underset{p_{1}=0}{\sum}}C_{p_{m}-1}^{p_{m-1}}...C_{p_{3}}^{p_{2}}C_{p_{2}}^{p_{1}}\theta_{1}^{p_{1}^{2}}\theta_{2}^{p_{2}^{2}}...\theta_{\left(m-1\right)}^{p_{\left(m-1\right)}^{2}}$
$\displaystyle
w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}w_{3}^{p_{3}-p_{2}}...w_{m}^{\left(p_{m}-1\right)-p_{m-1}}.$
(4.7)

######  Lemma 5 (see [3]).

We have

$\displaystyle\partial_{w_{1}^{2}}H_{n}$
$\displaystyle=p_{m}\left(p_{m}-1\right)\overset{p_{m}-2}{\underset{p_{m-1}=0}{\sum}}...\overset{p_{3}}{\underset{p_{2}=0}{\sum}}\overset{p_{2}}{\underset{p_{1}=0}{\sum}}C_{p_{m}-2}^{p_{m-1}}...C_{p_{2}}^{p_{1}}$
$\displaystyle\theta_{1}^{\left(p_{1}+2\right)^{2}}...\theta_{\left(m-1\right)}^{\left(p_{\left(m-1\right)}+2\right)^{2}}w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}...w_{m}^{\left(p_{m}-2\right)-p_{m-1}},$
(4.8) $\displaystyle\partial_{w_{\ell}^{2}}H_{n}$
$\displaystyle=p_{m}\left(p_{m}-1\right)\overset{p_{m}-2}{\underset{p_{m-1}=0}{\sum}}...\overset{p_{2}}{\underset{p_{1}=0}{\sum}}C_{p_{m}-2}^{p_{m-1}}...C_{p_{2}}^{p_{1}}$
$\displaystyle\theta_{1}^{p_{1}^{2}}\theta_{2}^{p_{2}^{2}}...\theta_{\ell-1}^{p_{\ell-1}^{2}}\theta_{\ell}^{\left(p_{\ell}+2\right)^{2}}...\theta_{\left(m-1\right)}^{\left(p_{\left(m-1\right)}+2\right)^{2}}$
$\displaystyle
w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}...w_{m}^{\left(p_{m}-2\right)-p_{m-1}},$
(4.9)

for all $\ell=2,...,m-1$, and

$\displaystyle\partial_{w_{\ell}w_{\kappa}}H_{n}$
$\displaystyle=p_{m}\left(p_{m}-1\right)\overset{p_{m}-2}{\underset{p_{m-1}=0}{\sum}}...\overset{p_{2}}{\underset{p_{1}=0}{\sum}}C_{p_{m}-2}^{p_{m-1}}...C_{p_{2}}^{p_{1}}$
$\displaystyle\theta_{1}^{p_{1}^{2}}...\theta_{\ell-1}^{p_{\ell-1}^{2}}\theta_{\ell}^{\left(p_{\ell}+1\right)^{2}}...\theta_{\kappa-1}^{\left(p_{\kappa-1}+1\right)^{2}}\theta_{\kappa}^{\left(p_{\kappa}+2\right)^{2}}...\theta_{\left(m-1\right)}^{\left(p_{\left(m-1\right)}+2\right)^{2}}$
$\displaystyle
w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}...w_{m}^{\left(p_{m}-2\right)-p_{m-1}}$
(4.10)

for all $1\leq\ell<\kappa\leq m$,

$\displaystyle\partial_{w_{m}^{2}}H_{n}$
$\displaystyle=p_{m}\left(p_{m}-1\right)\overset{p_{m}-2}{\underset{p_{m-1}=0}{\sum}}...\overset{p_{2}}{\underset{p_{1}=0}{\sum}}C_{p_{m}-2}^{p_{m-1}}...C_{p_{2}}^{p_{1}}\theta_{1}^{p_{1}^{2}}...\theta_{\left(m-1\right)}^{p_{\left(m-1\right)}^{2}}$
$\displaystyle
w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}...w_{m}^{\left(p_{m}-2\right)-p_{m-1}}.$
(4.11)

######  Lemma 6 (see [3]).

Let $A$ be the $m$-square symmetric matrix defined by
$A=\left(a_{\ell\kappa}\right)_{1\leq\ell,\kappa\leq m}$. Then the following
property holds:

$\left\\{\begin{array}[c]{l}K_{m}^{m}=\det\left[m\right]\overset{k=m-2}{\underset{k=1}{\prod}}\left(\det\left[k\right]\right)^{2^{\left(m-k-2\right)}},\text{
\ \ }m>2,\\\ K_{2}^{2}=\det\left[2\right],\end{array}\right.$ (4.12)

where

$\displaystyle K_{m}^{l}$
$\displaystyle=K_{l-1}^{l-1}K_{m}^{l-1}-\left(H_{m}^{l-1}\right)^{2},l=3,...,m,$
$\displaystyle H_{m}^{l}$ $\displaystyle=\underset{1\leq\ell,\kappa\leq
m}{\det}\left(\left(a_{\ell,\kappa}\right)_{\begin{subarray}{c}\ell\neq
m,...l+1\\\ \kappa\neq
m-1,..l\end{subarray}}\right)\overset{k=l-2}{\underset{k=1}{\prod}}\left(\det\left[k\right]\right)^{2^{\left(l-k-2\right)}},\text{
}l=3,...,m-1,$ $\displaystyle K_{m}^{2}$
$\displaystyle=a_{11}a_{mm}-\left(a_{1m}\right)^{2},\text{
}H_{m}^{2}=a_{11}a_{2m}-a_{12}a_{1m}.$

###### Proof of Theorem 1.

we prove that $L(t)$ is uniformly bounded on the interval
$\left[0,T^{\ast}\right],T^{\ast}<T_{\max}$. we have:

$\displaystyle L^{\prime}(t)$
$\displaystyle=\int_{\Omega}\partial_{t}H_{p_{m}}dx=\int_{\Omega}\overset{m}{\underset{\ell=1}{\sum}}\partial_{w_{\ell}}H_{p_{m}}\frac{\partial
w_{\ell}}{\partial t}dx$
$\displaystyle=\int_{\Omega}\overset{m}{\underset{\ell=1}{\sum}}\partial_{w_{\ell}}H_{p_{m}}\left(\lambda_{\ell}\Delta
w_{\ell}+\digamma_{\ell}\right)dx$
$\displaystyle=\int_{\Omega}\overset{m}{\underset{\ell=1}{\sum}}\lambda_{\ell}\partial_{w_{\ell}}H_{p_{m}}\Delta
w_{\ell}dx+\int_{\Omega}\overset{m}{\underset{\ell=1}{\sum}}\partial_{w_{\ell}}H_{p_{m}}\digamma_{\ell}dx=I+J,$

where

$I=\int_{\Omega}\overset{m}{\underset{\ell=1}{\sum}}\lambda_{\ell}\partial_{w_{\ell}}H_{p_{m}}\Delta
w_{\ell}dx,$ (4.13)

and

$J=\int_{\Omega}\overset{m}{\underset{\ell=1}{{\displaystyle\sum}}}\partial_{w_{\ell}}H_{p_{m}}\digamma_{\ell}dx.$
(4.14)

Using Green’s formula we can divide $I$ into two parts: $I_{1}$ and $I_{2}$,
where

$I_{1}=\int_{\partial\Omega}\overset{m}{\underset{\ell=1}{\sum}}\lambda_{\ell}\partial_{w_{\ell}}H_{p_{m}}\partial_{\eta}w_{\ell}dx,$
(4.15)

and

$I_{2}=-\int_{\Omega}\left\langle
T,\left(\left(\frac{\lambda_{\ell}+\lambda_{\kappa}}{2}\partial_{w_{\kappa}w_{\ell}}H_{p_{m}}\right)_{1\leq\ell,\kappa\leq
m}\right)T\right\rangle dx,$ (4.16)

for $p_{1}=0,...,p_{2},$ $p_{2}=0,...,p_{3}$ $...p_{m-1}=0,...,p_{m}-2$ and
$T=\left(\nabla w_{1},\nabla w_{2},...,\nabla w_{m}\right)^{T}.$ Applying
Lemmas 4 and 5 yields

$\left.\begin{array}[c]{l}\left(\frac{\lambda_{\ell}+\lambda_{\kappa}}{2}\partial_{w_{\kappa}w_{\ell}}H_{p_{m}}\right)_{1\leq\ell,\kappa\leq
m}=\\\
p_{m}\left(p_{m}-1\right)\overset{p_{m}-2}{\underset{p_{m-1}=0}{\sum}}...\overset{p_{2}}{\underset{p_{1}=0}{\sum}}C_{p_{m}-2}^{p_{m-1}}...C_{p_{2}}^{p_{1}}\left(\left(a_{\ell\kappa}\right)_{1\leq\ell,\kappa\leq
m}\right)w_{1}^{p_{1}}...w_{m}^{\left(p_{m}-2\right)-p_{m-1}},\end{array}\right.$
(4.17)

where $\left(a_{\ell\kappa}\right)_{1\leq\ell,\kappa\leq m}$ is the matrix
defined in (4.2).

Now, in order to prove that $I$ is bounded, we will show that there exists a
positive constant $C_{4}$ independent of $t\in\left[0,T_{\max}\right)$ such
that

$I_{1}\leq C_{4}\text{ for all }t\in\left[0,T_{\max}\right),$ (4.18)

and that

$I_{2}\leq 0,$ (4.19)

for several boundary conditions. First let us prove (4.18):

(i) If $0<\alpha<1$ , then using the boundary conditions (1.2) we get

$I_{1}=\int_{\partial\Omega}\overset{m}{\underset{\ell=1}{\sum}}\lambda_{\ell}\partial_{w_{\ell}}H_{p_{m}}\left(\gamma_{\ell}-\sigma_{\ell}w_{\ell}\right)dx,$

where $\sigma_{\ell}=\dfrac{\alpha}{1-\alpha}$ and
$\gamma_{\ell}=\dfrac{\beta_{\ell}}{1-\alpha_{\ell}}$, for $\ell=1,...m$. For
the second type of boundary condition (3.4),
$\sigma_{\ell}=\dfrac{\alpha}{\lambda_{\ell}\left(1-\alpha\right)}$ and
$\gamma_{\ell}=\dfrac{\beta_{\ell}}{\lambda_{\ell}\left(1-\alpha\right)}$.

Since
$H\left(W\right)=\overset{m}{\underset{\ell=1}{\sum}}\lambda_{\ell}\partial_{w_{\ell}}H_{p_{m}}\left(\gamma_{\ell}-\sigma_{\ell}w_{\ell}\right)=P_{n-1}\left(W\right)-Q_{n}\left(W\right)$,
where $P_{n-1}$ and $Q_{n}$ are polynomials with positive coefficients and
respective degrees $n-1$ and $n$, and since the solution is positive it
follows that

$\underset{\overset{m}{\underset{\ell=1}{\sum}}\left|w_{\ell}\right|\rightarrow+\infty}{\lim\sup}H\left(W\right)=-\infty,$
(4.20)

which proves that $H$ is uniformly bounded on $\mathbb{R}_{+}^{m}$ and
consequently proves (4.18).

(ii) If for all $\ell=1,...m:\alpha=0$, then $I_{1}=0$ on
$\left[0,T_{\max}\right)$.

(iii) The case of homogeneous Dirichlet conditions is trivial since in this
case the positivity of the solution on $\left[0,T_{\max}\right)\times\Omega$
implies $\partial_{\eta}w_{\ell}\leq 0,\forall\ell=1,...m$ on
$\left[0,T_{\max}\right)\times\partial\Omega$. Consequently one obtains the
same result in (4.18) with $C_{4}=0$.

Hence the proof of (4.18) is complete.

Now we move to the proof of (4.19).

Consider the matrix $\left(a_{\ell\kappa}\right)_{1\leq\ell,\kappa\leq m}$
which we defined in (4.2). The quadratic form (with respect to $\nabla
w_{\ell},$ $\ell=1,...,m$) associated with the matrix
$\left(a_{\ell\kappa}\right)_{1\leq\ell,\kappa\leq m}$, with
$p_{1}=0,...,p_{2},$ $p_{2}=0,..,p_{3}$ … $p_{m-1}=0,...,p_{m}-2$, is positive
definite since its minors $\det\left[1\right]$, $\det\left[2\right]$,$...$
$\det\left[m\right]$ are all positive. Let us prove their positivity by
induction.

The first minor

$\det\left[1\right]=\lambda_{1}\theta_{1}^{\left(p_{1}+2\right)^{2}}\theta_{2}^{\left(p_{2}+2\right)^{2}}..\theta_{\left(m-1\right)}^{\left(p_{\left(m-1\right)}+2\right)^{2}}>0$

for $p_{1}=0,...,p_{2},$ $p_{2}=0,...,p_{3}$ … $p_{m-1}=0,...,p_{m}-2$.

For the second minor $\det\left[2\right]$, and according to Lemma 6, we have:

$\det\left[2\right]=K_{2}^{2}=\lambda_{1}\lambda_{2}\theta_{1}^{2\left(p_{1}+1\right)^{2}}\overset{m-1}{\underset{k=2}{\Pi}}\theta_{k}^{2\left(p_{k}+2\right)^{2}}\left[\theta_{1}^{2}-A_{12}^{2}\right],$

using (4.4) for $l=2$ we get $\det\left[2\right]>0.$

Similarly for the third minor $\det\left[3\right]$, and again using Lemma 6,
we have:

$K_{3}^{3}=\det\left[3\right]\det\left[1\right].$

Since $\det\left[1\right]>0$, we conclude that

$\operatorname*{sign}(K_{3}^{3})=\operatorname*{sign}(\det\left[3\right]).$

Again, using (4.4) for $l=3$ yields $\det\left[3\right]>0$.

To finish the proof let us suppose $\det\left[k\right]>0$ for $k=1,2,...,l-1$
and show that $\det[l]$ is necessarily positive. We have

$\det\left[k\right]>0,k=1,...,\left(l-1\right)\Rightarrow\overset{k=l-2}{\underset{k=1}{\prod}}\left(\det\left[k\right]\right)^{2^{\left(l-k-2\right)}}>0.$
(4.21)

From Lemma 6 we obtain
$K_{l}^{l}=\det\left[l\right]\overset{k=l-2}{\underset{k=1}{\prod}}\left(\det\left[k\right]\right)^{2^{\left(l-k-2\right)}}$,
and from (4.21) we get
$\operatorname*{sign}(K_{l}^{l})=\operatorname*{sign}\left(\det\left[l\right]\right)$.
Since $K_{l}^{l}>0$ according to (4.4) then $\det\left[l\right]>0$ and the
proof of (4.19) is concluded. It then follows from (4.18) and (4.19) that $I$
is finished.

Now let us prove that $J$ in (4.14) is bounded. Substituting the expressions
of the partial derivatives given by Lemma 4 in the second integral of (4.14)
yields

$\displaystyle J$
$\displaystyle=\int_{\Omega}\left[p_{m}\overset{p_{m}-1}{\underset{p_{m-1}=0}{\sum}}...\overset{p_{2}}{\underset{p_{1}=0}{\sum}}C_{p_{m}-1}^{p_{m-1}}...C_{p_{2}}^{p_{1}}w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}...w_{m}^{p_{m}-1-p_{m-1}}\right]$
$\displaystyle\left(\overset{m-1}{\underset{\ell=1}{\Pi}}\theta_{\ell}^{\left(p_{\ell}+1\right)^{2}}\digamma_{1}+\overset{m-1}{\underset{\kappa=2}{\sum}}\overset{\kappa-1}{\underset{k=1}{\prod}}\theta_{k}^{p_{k}^{2}}\overset{m-1}{\underset{\ell=\kappa}{\prod}}\theta_{\ell}^{\left(p_{\ell}+1\right)^{2}}\digamma_{\kappa}+\overset{m-1}{\underset{\ell=1}{\prod}}\theta_{\ell}^{p_{\ell}^{2}}\digamma_{m}\right)dx$
$\displaystyle=\int_{\Omega}\left[p_{m}\overset{p_{m}-1}{\underset{p_{m-1}=0}{\sum}}...\overset{p_{2}}{\underset{p_{1}=0}{\sum}}C_{p_{m}-1}^{p_{m-1}}...C_{p_{2}}^{p_{1}}w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}...w_{m}^{p_{m}-1-p_{m-1}}\right]$
$\displaystyle\left(\overset{m-1}{\underset{\ell=1}{\Pi}}\frac{\theta_{\ell}^{\left(p_{\ell}+1\right)^{2}}}{\theta_{\ell}^{p_{\ell}^{2}}}\digamma_{1}+\overset{m-1}{\underset{\kappa=2}{\sum}}\overset{\kappa-1}{\underset{k=1}{\prod}}\theta_{k}^{p_{k}^{2}}\overset{m-1}{\underset{\ell=\kappa}{\prod}}\frac{\theta_{\ell}^{\left(p_{\ell}+1\right)^{2}}}{\theta_{\ell}^{p_{\ell}^{2}}}\digamma_{\kappa}+\digamma_{m}\right)\overset{m-1}{\underset{\ell=1}{\prod}}\theta_{\ell}^{p_{\ell}^{2}}dx$
$\displaystyle=\int_{\Omega}\left[p_{m}\overset{p_{m}-1}{\underset{p_{m-1}=0}{\sum}}...\overset{p_{2}}{\underset{p_{1}=0}{\sum}}C_{p_{m}-1}^{p_{m-1}}...C_{p_{2}}^{p_{1}}w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}...w_{m}^{p_{m}-1-p_{m-1}}\right]$
$\displaystyle\left\langle\left(\overset{m-1}{\underset{\ell=1}{\prod}}\frac{\theta_{\ell}^{\left(p_{\ell}+1\right)^{2}}}{\theta_{\ell}^{p_{\ell}^{2}}},\theta_{1}^{p_{1}^{2}}\overset{m-1}{\underset{\ell=2}{\prod}}\frac{\theta_{\ell}^{\left(p_{\ell}+1\right)^{2}}}{\theta_{\ell}^{p_{\ell}^{2}}},....,\overset{m-2}{\underset{k=1}{\prod}}\theta_{k}^{p_{k}^{2}}\frac{\theta_{m-1}^{\left(p_{m-1}+1\right)^{2}}}{\theta_{m-1}^{p_{m-1}^{2}}},1\right),\digamma\right\rangle\overset{m-1}{\underset{\ell=1}{\prod}}\theta_{\ell}^{p_{\ell}^{2}}dx.$

Hence using the condition (1.12) we deduce that

$J\leq
C_{5}\int_{\Omega}\left[\overset{p_{m}-1}{\underset{p_{m-1}=0}{\sum}}...\overset{p_{2}}{\underset{p_{1}=0}{\sum}}C_{p_{2}}^{p_{1}}...C_{p_{m}-1}^{p_{m-1}}w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}...w_{m}^{p_{m}-1-p_{m-1}}\left(1+\left\langle
W,1\right\rangle\right)\right]dx.$

To prove that the functional $L$ is uniformly bounded on the interval
$\left[0,T^{\ast}\right]$ we write

$\displaystyle\overset{p_{m}-1}{\underset{p_{m-1}=0}{\sum}}...\overset{p_{2}}{\underset{p_{1}=0}{\sum}}C_{p_{2}}^{p_{1}}...C_{p_{m}-1}^{p_{m-1}}w_{1}^{p_{1}}w_{2}^{p_{2}-p_{1}}...w_{m}^{p_{m}-1-p_{m-1}}\left(1+\left\langle
W,1\right\rangle\right)$
$\displaystyle=R_{p_{m}}\left(W\right)+S_{p_{m}-1}\left(W\right),$

where $R_{p_{m}}\left(W\right)$ and $S_{p_{m}-1}\left(W\right)$ are two
homogeneous polynomials of degrees $p_{m}$ and $p_{m}-1,$ respectively. Since
all the polynomials $H_{p_{m}}$ and $R_{p_{m}}$ are of degree $p_{m}$ then
there exists a positive constant $C_{6}$ such that

$\int_{\Omega}R_{p_{m}}\left(W\right)dx\leq
C_{6}\int_{\Omega}H_{p_{m}}\left(W\right)dx.$ (4.22)

Applying Hölder’s inequality to the integral
$\int_{\Omega}S_{p_{m}-1}\left(W\right)dx,$ one obtains

$\int_{\Omega}S_{p_{m}-1}\left(W\right)dx\leq\left(meas\Omega\right)^{\frac{1}{p_{m}}}\left(\int_{\Omega}\left(S_{p_{m}-1}\left(W\right)\right)^{\frac{p_{m}}{p_{m}-1}}dx\right)^{\frac{p_{m}-1}{p_{m}}}.$

Using the fact that for all $w_{1},w_{2,...},w_{m-1}\geq 0$ and $w_{m}>0,$

$\dfrac{\left(S_{p_{m}-1}\left(W\right)\right)^{\frac{p_{m}}{p_{m}-1}}}{H_{p_{m}}\left(W\right)}=\dfrac{\left(S_{p_{m}-1}\left(x_{1},x_{2},...,x_{m-1},1\right)\right)^{\frac{p_{m}}{p_{m}-1}}}{H_{p_{m}}\left(x_{1},x_{2},...,x_{m-1},1\right)},$

where we have
$\forall\ell\in\left\\{1,2,...,m-1\right\\}:x_{\ell}=\frac{w_{\ell}}{w_{\ell+1}}$,
and

$\underset{x_{\ell}\rightarrow+\infty}{\lim}\dfrac{\left(S_{p_{m}-1}\left(x_{1},x_{2},...,x_{m-1},1\right)\right)^{\frac{p_{m}}{p_{m}-1}}}{H_{p_{m}}\left(x_{1},x_{2},...,x_{m-1},1\right)}<+\infty,$

one asserts that there exists a positive constant $C_{7}$ such that

$\dfrac{\left(S_{p_{m}-1}\left(W\right)\right)^{\frac{p_{m}}{p_{m}-1}}}{H_{p_{m}}\left(W\right)}\leq
C_{7},\text{ for all }w_{1},w_{2},...,w_{m}\geq 0.$ (4.23)

Hence the functional $L$ satisfies the differential inequality

$L^{\prime}\left(t\right)\leq
C_{6}L\left(t\right)+C_{8}L^{\frac{p_{m}-1}{p_{m}}}\left(t\right),$

which for $Z=L^{\frac{1}{p_{m}}}$ can be written as

$p_{m}Z^{\prime}\leq C_{6}Z+C_{8}.$ (4.24)

A simple integration gives the uniform bound of the functional $L$ on the
interval $\left[0,T^{\ast}\right]$. This ends the proof of the theorem. ∎

###### Proof of Corollary 1.

It is an immediate consequence of Theorem 1 and the inequality

$\int_{\Omega}\left\langle W,1\right\rangle^{p}dx\leq
C_{9}L\left(t\right)\text{ on }\left[0,T^{\ast}\right].$ (4.25)

for some $p\geq 1.$ ∎

###### Proof of Proposition 4.

From Corollary 1, it follows that there exists a positive constant $C_{10}$
such that

$\int_{\Omega}\left(\left\langle W,1\right\rangle+1\right)^{p}dx\leq
C_{10}\text{ on }\left[0,T_{\max}\right).$ (4.26)

From (1.11), we have

$\displaystyle\text{for any }\ell$
$\displaystyle\in\left\\{1,2,...,m\right\\}:$
$\displaystyle\left|\digamma_{\ell}\left(W\right)\right|^{\frac{p}{N}}$
$\displaystyle\leq C_{11}\left(W\right)\left\langle W,1\right\rangle^{p}\text{
on }\left[0,T_{\max}\right)\times\Omega.$ (4.27)

Since $w_{1},w_{2},...,w_{m}$ are in
$L^{\infty}\left(0,T^{\ast};L^{p}\left(\Omega\right)\right)$ and
$\dfrac{p}{N}>\dfrac{n}{2},$ then as discussed in section 2, the solution is
global. ∎

##  5\. Construction of Invariant Regions

Recall that the eigenvector of the diffusion matrix associated with the
eigenvalue $\lambda_{\ell}$ is defined as $V_{\ell}=\left(v_{\ell 1},v_{\ell
2},...,v_{\ell m}\right)^{T}$. In the region that we considered in previous
sections, we used the diagonalizing matrix $P=\left(V_{1}\shortmid
V_{2}\shortmid...\shortmid V_{m}\right)$. In general the diagonalizing matrix
can be written as

$P=\left(\left(-1\right)^{i_{1}}V_{1}\shortmid\left(-1\right)^{i_{2}}V_{2}\shortmid...\shortmid\left(-1\right)^{i_{m}}V_{m}\right),$

with the powers $i_{\ell}$

$i_{\ell}=1\text{ or }2,\text{ for }\ell=1,...,m.$

Now one can subdivide the indices $\ell$ into two disjoint sets $\mathfrak{Z}$
and $\mathfrak{L}$, such that

$\left\\{\begin{array}[c]{c}i_{\ell}=1\Rightarrow\ell\in\mathfrak{Z}\\\
i_{\ell}=2\Rightarrow\ell\in\mathfrak{L.}\end{array}\right.$

It is then straightforward to notice that

$\mathfrak{L}\cap\mathfrak{Z}=\phi,\text{
}\mathfrak{L}\cup\mathfrak{Z}=\left\\{1,2,...,m\right\\}.$

Hence the number of possible permutations for $\mathfrak{Z}$ and
$\mathfrak{L}$ is $2^{m}$.

Recall that

$W_{0}=P^{T}U_{0}=\left(w_{1}^{0},w_{2}^{0},...,w_{m}^{0}\right)^{T}.$

Since we have $2^{m}$ different diagonalizing matrices $P^{T},$ we can write

$W_{0}=\left\\{\begin{array}[c]{l}w_{\ell}^{0}=\left\langle
V_{\ell},U_{0}\right\rangle,\text{ }\ell\in\mathfrak{L,}\\\
w_{\ell}^{0}=\left\langle\left(-1\right)V_{\ell},U_{0}\right\rangle,\text{
}\ell\in\mathfrak{Z.}\end{array}\right.$

This along with (1.8) guarantees that the elements of $W_{0}$ are positive,
i.e.

$\left.\begin{array}[c]{l}\Sigma_{\mathfrak{L},\mathfrak{Z}}=\left\\{U_{0}\in\mathbb{\mathbb{R}}^{m}:w_{\ell}^{0}=\left\langle
V_{\ell},U_{0}\right\rangle\geq 0,\text{ }\right.\\\ \text{ \ \ \ \ \ \ \ \ \
\ \ \ \ \
}\left.\ell\in\mathfrak{L,}w_{\ell}^{0}=\left\langle\left(-1\right)V_{\ell},U_{0}\right\rangle\geq
0,\text{ }\ell\in\mathfrak{Z}\right\\}.\end{array}\right.$

## References

  * [1] S. Abdelmalek, Invariant regions and global solutions for reaction-diffusion systems with a tridiagonal symmetric Toeplitz matrix of diffusion coefficients, Electron. J. Diff. Equ., Vol. 2014 (2014), No. 247, pp. 1-14.
  * [2] S. Abdelmalek, Invariant regions and global existence of solutions for reaction-diffusion systems with a tri-diagonal matrix of diffusion coefficients and nonhomogeneous boundary conditions, Journal of Applied Mathematics, Vol (2007), 1-15 .
  * [3] S. Abdelmalek and S. Kouachi, Proof of existence of global solutions for $m$-component reaction-diffusion systems with mixed boundary conditions via the Lyapunov functional method, Journal of Physics A, (2007) vol. 40, 12335-12350.
  * [4] H. Amann, Dynamic theory of quasi-linear parabolic systems II. Reaction-diffusion systems. Differential and Integral Equations 3(1) (1990) 13-75.
  * [5] H. Amann, Dynamic theory of quasi-linear parabolic systems III . Global existence. Math. Z. vol 202(2). (1989) 219-254.
  * [6] R. Aris, The mathematical theory of diffusion and reaction in permeable catalysts, Vol. I and II, Oxford University Press, New York, 1975.
  * [7] V. Capasso and A Di Liddo, Global attractivity for reaction-diffusion systems.The case of nondiagonal diffusion matrices J.Math.Anal.Appl. 177 (1993),510-529.
  * [8] D. Carl, Matrix analysis applied linear algebra, SIAM: Society for Industrial and Applied Mathematics, 200, Meyer.
  * [9] E. L. Cussler, Multicomponent diffusion. Chemical Engineering Monographs, Vol.3, Elsevier Publishing Scientific Company, Amesterdam, 1976.
  * [10] E. L. Cussler, Diffusion, mass transfer in fiuid system. Second Edition, Cambridge University Press, Cambridge, 1997.
  * [11] A. Friedman, Partial differential equations of parabolic type. Krieger Pub Co (1983).
  * [12] D. Henry, Geometric theory of semi-linear parabolic equations. Lecture Notes in Mathematics 840, Springer-Verlag, New-York, 1984.
  * [13] S. L. Hollis and J. J. Morgan, On the blow-up of solutions to some semilinear and quasilinear reaction-diffusion systems, Rocky Mountain J. Math. vol 14. (1994), 1447-1465.
  * [14] S. Kouachi, Global existence of solutions for reaction-diffusion systems with a full matrix of diffusion coefficients and nonhomogeneous boundary conditions, Electronic Journal of Qualitative Theory of Differential Equations, no. 2, pp. 1–10, 2002.
  * [15] S. Kouachi, Existence of global solutions to reaction-di usion systems via a lyapunov functional. Electron. J. Diff. Eqns Vol. 2001(2001), No. 68, pp. 1-10.
  * [16] A. Okubo, and S. A. Levin, Diffusion and ecological problems: Modern perspectives, Second Edition Interdisciplinary Applied Mathematics, V14, (2001) Springer Science+Business Media New York.