Invariant regions and global existence of solutions for reaction-diffusion systems with a general full matrix

Abstract

The aim of this study is to prove the global existence in time of solutions for reaction-diffusion systems. We make use of the appropriate techniques which are based on invariant regions and Lyapunov functional methods. We consider a full matrix of diffusion coefficients and we show the global existence of the solutions.

MSC:35K45, 35K57.

1 Introduction

We are mainly interested in the global existence in time of solutions to a reaction-diffusion system of the form

$$\frac{\partial u}{\partial t}-a\mathrm{\Delta }u-b\mathrm{\Delta }v=\mathrm{\Pi }-f(u,v)-\sigma u\text{in }]0,+\mathrm{\infty }[\times \mathrm{\Omega },$$

(1.1)

$$\frac{\partial v}{\partial t}-c\mathrm{\Delta }u-d\mathrm{\Delta }v=f(u,v)-\sigma v\text{in }]0,+\mathrm{\infty }[\times \mathrm{\Omega }$$

(1.2)

with the following boundary conditions:

$$\frac{\partial u}{\partial \eta }=\frac{\partial v}{\partial \eta }=0\text{in }]0,+\mathrm{\infty }[\times \partial \mathrm{\Omega }$$

(1.3)

and the initial data

$$u(0,x)=u_0,v(0,x)=v_0\text{in }\mathrm{\Omega },$$

(1.4)

where Ω is an open bounded domain in ${\mathbb{R}}^n$ with boundary ∂ Ω of class $C^1$, $\frac{\partial }{\partial \eta }$ denotes the outward normal derivative on ∂ Ω, Δ denotes the Laplacian operator with respect to the x variable, a, b, c, d, σ are positive constants satisfying the condition ${(b+c)}^2<4ad$, which reflects the parabolicity of the system and implies at the same time that the matrix of diffusion is positive definite, $\mathrm{\Pi }\ge 0$. The eigenvalues ${\lambda }_1$ and ${\lambda }_2$ (${\lambda }_1<{\lambda }_2$) of the matrix are positive. We assume that

$${\lambda }_1<a<d<{\lambda }_2<a+c,$$

and the initial data are assumed to be in the following region:

$$\mathrm{\Sigma }=\{(u_0,v_0)\in {\mathbb{R}}^2\text{ such that }\frac{a-{\lambda }_2}{c}{v}_0\le u_0\le \frac{a-{\lambda }_1}{c}{v}_0\}.$$

(1.5)

For more details, one may consult [1].

The function f is a nonnegative continuously differentiable function on Σ such that

$$f(\frac{a-{\lambda }_2}{c}\eta ,\eta )=0\text{and}f(\frac{a-{\lambda }_1}{c}\eta ,\eta )\ge \frac{\mathrm{\Pi }}{(1+\frac{a-{\lambda }_1}{c})}\text{for all }\eta \ge 0.$$

(1.6)

In addition we suppose that

$$(\xi ,\eta )\in \mathrm{\Sigma }⟹0\le f(\xi ,\eta )\le \phi (\xi ){(1+\eta )}^{\beta },$$

(1.7)

where $\beta \ge 1$ and φ is a nonnegative function of class $C(\mathbb{R})$ such that

$$\underset{\xi \to -\mathrm{\infty }}{lim}\frac{\phi (\xi )}{\xi }=0.$$

(1.8)

Melkemi et al. [2] established the existence of global solutions (eventually uniformly bounded in time) using a novel approach that involved the use of a Lyapunov function for system (1.1)-(1.4) when $c=b=0$. Along the same lines, Rebai [3] has proved the global existence of solutions for system (1.1)-(1.4), in the case $b=0$, $c>0$ (triangular matrix). The present investigation is a continuation of results obtained in [3]. Here, we follow the same reasoning as in [2], in the study of system (1.1)-(1.4), when $b>0$, $c>0$, that is, for a model that involves a general full matrix.

The components $u(t,x)$ and $v(t,x)$ represent either chemical concentrations or biological population densities and system (1.1)-(1.2) is a mathematical model describing various chemical and biological phenomena (see, e.g., Cussler [4]).

Remark 1 If $a<d$, then we have ${\lambda }_1<a<d<{\lambda }_2$. We note that the condition of parabolicity implies that $det(A)=ad-bc>0$, where A is the matrix of diffusion.

2 Local existence and invariant regions

Throughout the text we shall denote by ${\parallel \parallel }_p$ the norm in $L^p(\mathrm{\Omega })$, and by ${\parallel \parallel }_{\mathrm{\infty }}$ the norm in $L^{\mathrm{\infty }}(\mathrm{\Omega })$ or $C(\overline{\mathrm{\Omega }})$.

For any initial data in $C(\overline{\mathrm{\Omega }})$ or $L^p(\mathrm{\Omega })$, $p\in ]1,+\mathrm{\infty }[$, local existence and uniqueness of solutions to the initial value problem (1.1)-(1.4) follow from the basic existence theory for abstract semilinear differential equations (see Henry [5] and Pazy [6]). The solutions are classical on $]0;T^{\ast }[$, where $T^{\ast }$ denotes the eventual blowing-up time in $L^{\mathrm{\infty }}(\mathrm{\Omega })$.

Furthermore, if $T^{\ast }<+\mathrm{\infty }$, then

$$\underset{t\uparrow T^{\ast }}{lim}({\parallel u(t)\parallel }_{\mathrm{\infty }}+{\parallel v(t)\parallel }_{\mathrm{\infty }})=+\mathrm{\infty }.$$

Therefore, if there exists a positive constant C such that

$${\parallel u(t)\parallel }_{\mathrm{\infty }}+{\parallel v(t)\parallel }_{\mathrm{\infty }}\le C,\mathrm{\forall }t\in ]0,T^{\ast }[,$$

then $T^{\ast }=+\mathrm{\infty }$.

Multiplying (1.2) first through by $-\frac{a-{\lambda }_2}{c}$ and adding (1.1) and then by $\frac{a-{\lambda }_1}{c}$ and subtracting (1.1), we get

$$\frac{\partial w}{\partial t}-{\lambda }_2\mathrm{\Delta }w=\mathrm{\Pi }-(1+\frac{a-{\lambda }_2}{c})F(w,z)-\sigma w\text{in }]0,T^{\ast }[\times \mathrm{\Omega },$$

(2.1)

$$\frac{\partial z}{\partial t}-{\lambda }_1\mathrm{\Delta }z=-\mathrm{\Pi }+(1+\frac{a-{\lambda }_1}{c})F(w,z)-\sigma z\text{in }]0,T^{\ast }[\times \mathrm{\Omega },$$

(2.2)

with the boundary conditions

$$\frac{\partial w}{\partial \eta }=\frac{\partial z}{\partial \eta }=0\text{in }]0,T^{\ast }[\times \partial \mathrm{\Omega },$$

(2.3)

and the initial data

$$w(0,x)=w_0(x),z(0,x)=z_0(x)\text{in }\mathrm{\Omega },$$

(2.4)

where

$$\begin{array}{r}w(t,x)=u(t,x)-\frac{a-{\lambda }_2}{c}v(t,x),\\ z(t,x)=-u(t,x)+\frac{a-{\lambda }_1}{c}v(t,x)\end{array}$$

(2.5)

for any $(t,x)$ in $]0,T^{\ast }[\times \mathrm{\Omega }$ and

$$F(w,z)=f(u,v)\text{for all }(u,v)\text{ in }\mathrm{\Sigma }.$$

(2.6)

To prove that Σ is an invariant region for system (1.1)-(1.4) it suffices to prove that the region

$${\mathrm{\Sigma }}_1=\{(w_0,z_0)\in {\mathbb{R}}^2\text{ such that }{w}_0\ge 0,z_0\ge 0\}$$

is invariant for system (2.1)-(2.4).

Now, to prove that the region ${\mathrm{\Sigma }}_1$ is invariant for system (2.1)-(2.4), it suffices to show that $(\mathrm{\Pi }-(1+\frac{a-{\lambda }_2}{c})F(0,z))\ge 0$ for $z\ge 0$, and $(-\mathrm{\Pi }+(1+\frac{a-{\lambda }_1}{c})F(w,0))\ge 0$, for $w\ge 0$, see [7].

From (1.6), its clear that the region ${\mathrm{\Sigma }}_1$ is invariant for system (2.1)-(2.4) and from (2.5) we have

$$\begin{array}{r}v(t,x)=\frac{c}{{\lambda }_2-{\lambda }_1}(w(t,x)+z(t,x)),\\ u(t,x)=\frac{a-{\lambda }_1}{{\lambda }_2-{\lambda }_1}w(t,x)+\frac{a-{\lambda }_2}{{\lambda }_2-{\lambda }_1}z(t,x).\end{array}$$

(2.7)

Remark 2 We note that if $(\xi ,\eta )\in \mathrm{\Sigma }$, then $\xi \in \mathbb{R}$ and $\eta \ge 0$.

3 Existence of global solutions

A simple application of the comparison theorem [[7], Theorem 10.1] to system (2.1)-(2.4) implies that for any initial conditions $w_0\ge 0$ and $z_0\ge 0$, we have

$$0\le w(t,x)\le max({\parallel w_0\parallel }_{\mathrm{\infty }},\frac{\mathrm{\Pi }}{\sigma })=K.$$

(3.1)

To prove the global existence of the solutions of problem (1.1)-(1.4), one needs to prove it for problem (2.1)-(2.4). As regards this subject, it is well known that it suffices to derive a uniform estimate of ${\parallel -\mathrm{\Pi }+(1+\frac{a-{\lambda }_1}{c})F(w,z)-\sigma z\parallel }_p$ for some $p>\frac{n}{2}$, i.e.

$${\parallel -\mathrm{\Pi }+(1+\frac{a-{\lambda }_1}{c})F(w,z)-\sigma z\parallel }_p\le C,$$

where C is a nonnegative constant independent of t.

From the assumptions (1.7) and (1.8), we are led to establish the uniform boundedness of the ${\parallel z\parallel }_p$ on $]0,T^{\ast }[$ in order to get that of ${\parallel z\parallel }_{\mathrm{\infty }}$ on $]0,T^{\ast }[$.

For $p\ge 2$, we put

$$\alpha =\frac{{({\lambda }_2-{\lambda }_1)}^2}{4{\lambda }_1{\lambda }_2},\alpha (p)=\frac{p\alpha +1}{p-1},M_p=K+\frac{\mathrm{\Pi }}{\sigma \alpha (p)}.$$

(3.2)

We firstly introduce the following lemmas, which are useful in our main results.

Lemma 1 Let $(w,z)$ be a solution of (2.1)-(2.4). Then

$$\frac{d}{dt}{\int }_{\mathrm{\Omega }}wdx+(1+\frac{a-{\lambda }_2}{c}){\int }_{\mathrm{\Omega }}F(w,z)dx+\sigma {\int }_{\mathrm{\Omega }}wdx=\mathrm{\Pi }|\mathrm{\Omega }|.$$

(3.3)

Proof We integrate both sides of (2.1), satisfied by w, which is positive and then we obtain

$$\frac{d}{dt}{\int }_{\mathrm{\Omega }}wdx=\mathrm{\Pi }|\mathrm{\Omega }|-(1+\frac{a-{\lambda }_2}{c}){\int }_{\mathrm{\Omega }}F(w,z)dx-\sigma {\int }_{\mathrm{\Omega }}wdx.$$

 □

Lemma 2 Assume that $p\ge 2$ and let

$$G_q(t)={\int }_{\mathrm{\Omega }}[qw+exp(-\frac{p-1}{p\alpha +1}ln(\alpha (p)(M_p-w)))z^p]dt,$$

(3.4)

where $(w,z)$ is the solution of (2.1)-(2.4) on $]0,T^{\ast }[$. Then under the assumptions (1.7)-(1.8) there exist two positive constants $q>0$ and $s>0$ such that

$$\frac{d}{dt}{G}_q(t)\le -(p-1)\sigma G_q+s.$$

(3.5)

Proof The proof is similar to that in Melkemi et al. [2].

Let

$$h(w)=-\frac{p-1}{p\alpha +1}ln(\alpha (p)(M_p-w)),$$

(3.6)

then

$$G_q(t)=q{\int }_{\mathrm{\Omega }}wdx+N(t),$$

(3.7)

where

$$N(t)={\int }_{\mathrm{\Omega }}{e}^{h(w)}{z}^{p}dx.$$

(3.8)

Differentiating $N(t)$ with respect to t and using the Green formula one obtains

$$\frac{d}{dt}N=H+S,$$

(3.9)

where

$$\begin{array}{rcl}H& =& -{\lambda }_2{\int }_{\mathrm{\Omega }}({(h^{\prime }(w))}^2+h^{″}(w))e^{h(w)}{z}^p{(\mathrm{\nabla }w)}^{2}dx\\ -p({\lambda }_2+{\lambda }_1){\int }_{\mathrm{\Omega }}{h}^{\prime }(w)e^{h(w)}{z}^{p-1}\mathrm{\nabla }w\mathrm{\nabla }zdx\\ -{\lambda }_1{\int }_{\mathrm{\Omega }}p(p-1)e^{h(w)}{z}^{p-2}{(\mathrm{\nabla }z)}^{2}dx\end{array}$$

(3.10)

and

$$\begin{array}{rcl}S& =& \mathrm{\Pi }{\int }_{\mathrm{\Omega }}{h}^{\prime }(w)e^{h(w)}{z}^{p}dx\\ +{\int }_{\mathrm{\Omega }}[p{z}^{p-1}(1+\frac{a-{\lambda }_1}{c})F(w,z)-(1+\frac{a-{\lambda }_2}{c})h^{\prime }(w)z^{p}F(w,z)]e^{h(w)}dx\\ -\sigma {\int }_{\mathrm{\Omega }}{h}^{\prime }(w)w{e}^{h(w)}{z}^{p}dx-p\sigma {\int }_{\mathrm{\Omega }}{e}^{h(w)}{z}^{p}dx-p\mathrm{\Pi }{\int }_{\mathrm{\Omega }}{e}^{h(w)}{z}^{p-1}dx.\end{array}$$

(3.11)

We observe that H is given by

$$H=-{\int }_{\mathrm{\Omega }}Q{e}^{h(w)}dx,$$

where

$$\begin{array}{rcl}Q& =& {\lambda }_2({(h^{\prime }(w))}^2+h^{″}(w))z^p{(\mathrm{\nabla }w)}^2+p({\lambda }_2+{\lambda }_1)h^{\prime }(w)z^{p-1}\mathrm{\nabla }w\mathrm{\nabla }z\\ +{\lambda }_{1}p(p-1)z^{p-2}{(\mathrm{\nabla }z)}^2\end{array}$$

(3.12)

is a quadratic form with respect to ∇w and ∇z, which is nonnegative if

$${(p({\lambda }_2+{\lambda }_1)h^{\prime }(w)z^{p-1})}^2-4{\lambda }_1{\lambda }_{2}p(p-1)({(h^{\prime }(w))}^2+h^{″}(w))z^{2p-2}\le 0,$$

(3.13)

and we have chosen $h(w)$ such that

$$h^{\prime }(w)=\frac{1}{\alpha (p)(M_p-w)},h^{″}(w)=\frac{\alpha (p)}{{[\alpha (p)(M_p-w)]}^2}.$$

(3.14)

It is easy to see that the left-hand side of (3.13) can be written as

$$\begin{array}{c}4{\lambda }_1{\lambda }_{2}p{z}^{2p-2}\{p[\alpha \frac{1}{{(\alpha (p)(M_p-w))}^2}-\frac{\alpha (p)}{{(\alpha (p)(M_p-w))}^2}]\hfill \\ +\frac{1+\alpha (p)}{{(\alpha (p)(M_p-w))}^2}\}=0,\hfill \end{array}$$

(3.15)

since

$$p\alpha -p\alpha (p)+1+\alpha (p)=0,$$

the inequality (3.13) holds, $Q\ge 0$, and consequently

$$H=-{\int }_{\mathrm{\Omega }}Q{e}^{h(w)}dx\le 0,$$

(3.16)

and the second term S can be estimated as

$$\begin{array}{rcl}S& \le & {\int }_{\mathrm{\Omega }}(\mathrm{\Pi }{h}^{\prime }(w)-\sigma p)e^{h(w)}{z}^{p}dx\\ +{\int }_{\mathrm{\Omega }}[p{z}^{p-1}(1+\frac{a-{\lambda }_1}{c})F(w,z)-h^{\prime }(w)z^p(1+\frac{a-{\lambda }_2}{c})F(w,z)]e^{h(w)}dx\\ \le & -(p-1)\sigma {\int }_{\mathrm{\Omega }}{e}^{h(w)}{z}^{p}dx\\ +{\int }_{\mathrm{\Omega }}[(1+\frac{a-{\lambda }_1}{c})p{z}^{p-1}F(w,z)-(1+\frac{a-{\lambda }_2}{c})h^{\prime }(w)z^{p}F(w,z)]e^{h(w)}dx,\end{array}$$

(3.17)

since

$$h^{\prime }(w)=\frac{1}{\alpha (p)(M_p-w)}\le \frac{1}{\alpha (p)(M_p-K)}=\frac{\sigma }{\mathrm{\Pi }}.$$

(3.18)

On the other hand

$$\begin{array}{r}-h^{\prime }(w)=\frac{-1}{\alpha (p)(M_p-w)}\le \frac{-1}{\alpha (p)M_p},\\ h(w)\le \frac{-1}{\alpha (p)}ln\frac{\mathrm{\Pi }}{\sigma }.\end{array}$$

(3.19)

Taking into account the fact that $z\ge 0$, and using (3.19), we observe that

$$\begin{array}{c}p(1+\frac{a-{\lambda }_1}{c})z^{p-1}F(w,z)-(1+\frac{a-{\lambda }_2}{c})h^{\prime }(w)z^{p}F(w,z)\hfill \\ \le (p(1+\frac{a-{\lambda }_1}{c})z^{p-1}-\frac{1}{\alpha (p)M_p}(1+\frac{a-{\lambda }_2}{c})z^p)F(w,z).\hfill \end{array}$$

Then for ${\eta }_0=p(1+\frac{a-{\lambda }_1}{c})(\frac{1}{(1+\frac{a-{\lambda }_2}{c})}+1)\alpha (p)M_p>0$, and for $0\le \xi \le K$, $\eta \ge {\eta }_0$, we have

$$\begin{array}{c}(p(1+\frac{a-{\lambda }_1}{c}){\eta }^{p-1}-\frac{1}{\alpha (p)M_p}(1+\frac{a-{\lambda }_2}{c}){\eta }^p)F(\xi ,\eta )\hfill \\ =[\frac{p(1+\frac{a-{\lambda }_1}{c})}{\eta }-\frac{(1+\frac{a-{\lambda }_2}{c})}{\alpha (p)M_p}]{\eta }^{p}F(\xi ,\eta )\le 0.\hfill \end{array}$$

On the other hand, we deduce that the function

$$(\xi ,\eta )\to p(1+\frac{a-{\lambda }_1}{c}){\eta }^{p-1}-\frac{1}{\alpha (p)M_p}(1+\frac{a-{\lambda }_2}{c}){\eta }^p$$

is bounded on the compact interval $[0,{\eta }_0]$; then there exists $c_1>0$ such that

$$p(1+\frac{a-{\lambda }_1}{c})z^{p-1}F(w,z)-(1+\frac{a-{\lambda }_2}{c})h^{\prime }(w)z^{p}F(w,z)\le c_{1}F(w,z).$$

(3.20)

From (3.17) and (3.20), we deduce immediately the following inequality:

$$S\le -(p-1)\sigma N+c_1{\int }_{\mathrm{\Omega }}F(w,z)e^{h(w)}dx\le -(p-1)\sigma N+c_1{e}^{\frac{-1}{\alpha (p)}ln\frac{\mathrm{\Pi }}{\sigma }}{\int }_{\mathrm{\Omega }}F(w,z)dx,$$

and putting

$$q=\frac{c_1}{(1+\frac{a-{\lambda }_2}{c})}{e}^{\frac{-1}{\alpha (p)}ln\frac{\mathrm{\Pi }}{\sigma }},$$

(3.21)

by (3.3), we have

$$S\le -(p-1)\sigma N+q\mathrm{\Pi }|\mathrm{\Omega }|-q\frac{d}{dt}{\int }_{\mathrm{\Omega }}w(t,x)dx,$$

(3.22)

and from (3.7), it follows that

$$S\le -(p-1)\sigma G_q+q((p-1)\sigma K+\mathrm{\Pi })|\mathrm{\Omega }|-q\frac{d}{dt}{\int }_{\mathrm{\Omega }}w(t,x)dx,$$

(3.23)

and from (3.7) and (3.9), we conclude that

$$\frac{d}{dt}{G}_q\le -(p-1)\sigma G_q+s,$$

(3.24)

where

$$s=q((p-1)\sigma K+\mathrm{\Pi })|\mathrm{\Omega }|.$$

(3.25)

 □

Now we can establish the global existence and uniform boundedness of the solutions of (2.1)-(2.4).

Theorem 1 Under the assumptions (1.7) and (1.8), the solutions of (2.1)-(2.4) are global and uniformly bounded on $[0,+\mathrm{\infty }[\times \mathrm{\Omega }$.

Proof Multiplying the inequality (3.24) by $e^{(p-1)\sigma t}$ and then integrating, we deduce that there exists a positive constant $C>0$ independent of t, such that

$$G_q(t)\le C.$$

(3.26)

From (3.6), we observe that

$$e^{h(w)}\ge e^{\frac{-1}{\alpha (p)}ln\alpha (p)M_p},$$

(3.27)

and it follows, for all $p\ge 2$, that

$${\int }_{\mathrm{\Omega }}{z}^{p}dx\le e^{\frac{1}{\alpha (p)}ln(K\alpha (p)+\frac{\mathrm{\Pi }}{\sigma })}{G}_q(t)\le C_1(p),$$

(3.28)

where

$$C_1(p)=C{e}^{\frac{1}{\alpha (p)}ln(K\alpha (p)+\frac{\mathrm{\Pi }}{\sigma })},$$

(3.29)

and as we select $p>\frac{n}{2}$ we can proceed to bound ${\parallel -\mathrm{\Pi }+(1+\frac{a-{\lambda }_1}{c})F(w,z)-\sigma z\parallel }_p$.

Let

$$A=\underset{{\xi }_0\le \xi \le K_1}{max}\phi (\xi ),$$

(3.30)

where

$$K_1=\frac{a-{\lambda }_1}{{\lambda }_2-{\lambda }_1}K,$$

and ${\xi }_0$ is such that

$$\xi \le {\xi }_0⟹\phi (\xi )<|\xi |,$$

(3.31)

using (1.7), we deduce

$$F(w,z)=f(u,v)\le \phi (u){(1+v)}^{\beta },$$

which implies

$$\begin{array}{rcl}{\int }_{\mathrm{\Omega }}{F}^p(w,z)dx& \le & {\int }_{\mathrm{\Omega }}{(\phi (u))}^p{(1+v)}^{\beta p}dx\\ =& {\int }_{u\le {\xi }_0}{(\phi (u))}^p{(1+v)}^{\beta p}dx+{\int }_{{\xi }_0\le u}{(\phi (u))}^p{(1+v)}^{\beta p}dx\\ \le & {\int }_{u\le {\xi }_0}|u{|}^p{(1+v)}^{\beta p}dx+A^p{\int }_{{\xi }_0\le u}{(1+v)}^{\beta p}dx.\end{array}$$

From (2.7), we have

$$\begin{array}{rcl}|u{|}^p& =& |\frac{a-{\lambda }_1}{{\lambda }_2-{\lambda }_1}w(t,x)+\frac{a-{\lambda }_2}{{\lambda }_2-{\lambda }_1}z(t,x){|}^p\le {(\frac{a-{\lambda }_1}{{\lambda }_2-{\lambda }_1}w(t,x)+\frac{{\lambda }_2-a}{{\lambda }_2-{\lambda }_1}z(t,x))}^p\\ \le & {\left(\frac{{\lambda }_2-a}{{\lambda }_2-{\lambda }_1}\right)}^p{(w(t,x)+z(t,x))}^p,\end{array}$$

then

$$\begin{array}{c}{\int }_{\mathrm{\Omega }}{F}^p(w,z)dx\le {\int }_{u\le {\xi }_0}{\left(\frac{{\lambda }_2-a}{{\lambda }_2-{\lambda }_1}\right)}^p{(w+z)}^p{(1+\frac{c}{{\lambda }_2-{\lambda }_1}(w+z))}^{\beta p}dx\hfill \\ \phantom{{\int }_{\mathrm{\Omega }}{F}^p(w,z)dx\le }+A^p{\int }_{{\xi }_0\le u}{(1+\frac{c}{{\lambda }_2-{\lambda }_1}(w+z))}^{\beta p}dx\hfill \\ \phantom{{\int }_{\mathrm{\Omega }}{F}^p(w,z)dx}\le max(A^p,{\left(\frac{{\lambda }_2-a}{{\lambda }_2-{\lambda }_1}\right)}^p)({\int }_{u\le {\xi }_0}{(w+z)}^p{(1+\frac{c}{{\lambda }_2-{\lambda }_1}(w+z))}^{\beta p}dx\hfill \\ \phantom{{\int }_{\mathrm{\Omega }}{F}^p(w,z)dx\le }+{\int }_{{\xi }_0\le u}{(1+\frac{c}{{\lambda }_2-{\lambda }_1}(w+z))}^{\beta p}dx)\hfill \\ \phantom{{\int }_{\mathrm{\Omega }}{F}^p(w,z)dx}\le max(A^p,{\left(\frac{{\lambda }_2-a}{{\lambda }_2-{\lambda }_1}\right)}^p)({\int }_{\mathrm{\Omega }}{(w+z)}^p{(1+\frac{c}{{\lambda }_2-{\lambda }_1}(w+z))}^{\beta p}dx\hfill \\ \phantom{{\int }_{\mathrm{\Omega }}{F}^p(w,z)dx\le }+{\int }_{\mathrm{\Omega }}{(1+\frac{c}{{\lambda }_2-{\lambda }_1}(w+z))}^{\beta p}dx),\hfill \\ {\int }_{\mathrm{\Omega }}{(w+z)}^p{(1+\frac{c}{{\lambda }_2-{\lambda }_1}(w+z))}^{\beta p}dx\hfill \\ \le 2^{\beta p-1}({\int }_{\mathrm{\Omega }}{(w+z)}^p+{\left(\frac{c}{{\lambda }_2-{\lambda }_1}\right)}^{\beta p}{(w+z)}^{(\beta +1)p}dx)\hfill \\ \le 2^{(\beta +1)p-2}(K^P|\mathrm{\Omega }|+C_1(p))+2^{(2\beta +1)p-2}{\left(\frac{c}{{\lambda }_2-{\lambda }_1}\right)}^{\beta p}(K^{(\beta +1)p}|\mathrm{\Omega }|+C_1((\beta +1)p))\hfill \\ =C_2(\beta ,p,K,\mathrm{\Omega }),\hfill \\ {\int }_{\mathrm{\Omega }}{(1+\frac{c}{{\lambda }_2-{\lambda }_1}(w+z))}^{\beta p}dx\hfill \\ \le 2^{\beta p-1}(|\mathrm{\Omega }|+{\left(\frac{c}{{\lambda }_2-{\lambda }_1}\right)}^{\beta p}\times 2^{\beta p-1}(K^{\beta p}|\mathrm{\Omega }|+C_1(\beta p)))\hfill \\ =C_3(\beta ,p,K,\mathrm{\Omega }).\hfill \end{array}$$

Consequently

$${\int }_{\mathrm{\Omega }}{F}^p(w,z)dx\le C_4(A,\beta ,p,K,\mathrm{\Omega }).$$

Finally

$$\begin{array}{rcl}{\parallel -\mathrm{\Pi }+\frac{a-{\lambda }_1}{c}F(w,z)-\sigma z\parallel }_p& =& {\parallel \frac{a-{\lambda }_1}{c}F(w,z)-(\sigma z+\mathrm{\Pi })\parallel }_p\\ \le & \frac{a-{\lambda }_1}{c}{\parallel F(w,z)\parallel }_p+\sigma {\parallel z\parallel }_p+\mathrm{\Pi }|\mathrm{\Omega }|\\ \le & \frac{a-{\lambda }_1}{c}\sqrt[p]{C_4(A,\beta ,p,K)}+\sigma \sqrt[p]{C_1(p)}+\mathrm{\Pi }|\mathrm{\Omega }|\\ =& C_5(A,\beta ,p,K,\mathrm{\Omega },\sigma ).\end{array}$$

(3.32)

Using the regularity results for the solutions of the parabolic equations in [5], we conclude that the solutions of problem (2.1)-(2.4) are uniformly bounded on $[0,+\mathrm{\infty }[\times \mathrm{\Omega }$. □

By (2.7), it is easy to see that the solutions of problem (1.1)-(1.4) are also uniformly bounded on $[0,+\mathrm{\infty }[\times \mathrm{\Omega }$.

Remark 3 Because $0\le w(t,x)\le K$ and $z(t,x)\ge 0$, we deduce that

$$-\mathrm{\infty }\le u(t,x)\le \frac{a-{\lambda }_1}{{\lambda }_2-{\lambda }_1}K=K_1.$$

Remark 4 We note that $\frac{a-{\lambda }_2}{{\lambda }_2-{\lambda }_1}<0$ and $\frac{{\lambda }_2-a}{{\lambda }_2-{\lambda }_1}\ge \frac{a-{\lambda }_1}{{\lambda }_2-{\lambda }_1}$, because ${\lambda }_2+{\lambda }_1=a+d$ and $d>a$.

We conclude by noting that the study of the global existence of strongly coupled systems has been a major development, and several articles are devoted to this subject. In our opinion, many other systems with non-constant diffusion matrix which are in the actual scope of the results previously given, should be taken in consideration and studied with more interest.