# 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

(1.1)
(1.2)

with the following boundary conditions:

(1.3)

and the initial data

(1.4)

where Ω is an open bounded domain in $R n$ with boundary Ω of class $C 1$, $∂ ∂ η$ 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, $Π≥0$. The eigenvalues $λ 1$ and $λ 2$ ($λ 1 < λ 2$) of the matrix are positive. We assume that

$λ 1

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

(1.5)

For more details, one may consult .

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

(1.6)

$(ξ,η)∈Σ⟹0≤f(ξ,η)≤φ(ξ) ( 1 + η ) β ,$
(1.7)

where $β≥1$ and φ is a nonnegative function of class $C(R)$ such that

$lim ξ → − ∞ φ ( ξ ) ξ =0.$
(1.8)

Melkemi et al.  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  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 . Here, we follow the same reasoning as in , 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 ).

Remark 1 If $a, then we have $λ 1 . 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 $∥ ∥ p$ the norm in $L p (Ω)$, and by $∥ ∥ ∞$ the norm in $L ∞ (Ω)$ or $C( Ω ¯ )$.

For any initial data in $C( Ω ¯ )$ or $L p (Ω)$, $p∈]1,+∞[$, 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  and Pazy ). The solutions are classical on $]0; T ∗ [$, where $T ∗$ denotes the eventual blowing-up time in $L ∞ (Ω)$.

Furthermore, if $T ∗ <+∞$, then

$lim t ↑ T ∗ ( ∥ u ( t ) ∥ ∞ + ∥ v ( t ) ∥ ∞ ) =+∞.$

Therefore, if there exists a positive constant C such that

$∥ u ( t ) ∥ ∞ + ∥ v ( t ) ∥ ∞ ≤C,∀t∈]0, T ∗ [,$

then $T ∗ =+∞$.

Multiplying (1.2) first through by $− a − λ 2 c$ and adding (1.1) and then by $a − λ 1 c$ and subtracting (1.1), we get

(2.1)
(2.2)

with the boundary conditions

(2.3)

and the initial data

(2.4)

where

$w ( t , x ) = u ( t , x ) − a − λ 2 c v ( t , x ) , z ( t , x ) = − u ( t , x ) + a − λ 1 c v ( t , x )$
(2.5)

for any $(t,x)$ in $]0, T ∗ [×Ω$ and

(2.6)

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

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

Now, to prove that the region $Σ 1$ is invariant for system (2.1)-(2.4), it suffices to show that $(Π−(1+ a − λ 2 c )F(0,z))≥0$ for $z≥0$, and $(−Π+(1+ a − λ 1 c )F(w,0))≥0$, for $w≥0$, see .

From (1.6), its clear that the region $Σ 1$ is invariant for system (2.1)-(2.4) and from (2.5) we have

$v ( t , x ) = c λ 2 − λ 1 ( w ( t , x ) + z ( t , x ) ) , u ( t , x ) = a − λ 1 λ 2 − λ 1 w ( t , x ) + a − λ 2 λ 2 − λ 1 z ( t , x ) .$
(2.7)

Remark 2 We note that if $(ξ,η)∈Σ$, then $ξ∈R$ and $η≥0$.

## 3 Existence of global solutions

A simple application of the comparison theorem [, Theorem 10.1] to system (2.1)-(2.4) implies that for any initial conditions $w 0 ≥0$ and $z 0 ≥0$, we have

$0≤w(t,x)≤max ( ∥ w 0 ∥ ∞ , Π σ ) =K.$
(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 $∥ − Π + ( 1 + a − λ 1 c ) F ( w , z ) − σ z ∥ p$ for some $p> n 2$, i.e.

$∥ − Π + ( 1 + a − λ 1 c ) F ( w , z ) − σ z ∥ p ≤C,$

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 $∥ z ∥ p$ on $]0, T ∗ [$ in order to get that of $∥ z ∥ ∞$ on $]0, T ∗ [$.

For $p≥2$, we put

$α= ( λ 2 − λ 1 ) 2 4 λ 1 λ 2 ,α(p)= p α + 1 p − 1 , M p =K+ Π σ α ( p ) .$
(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

$d d t ∫ Ω wdx+ ( 1 + a − λ 2 c ) ∫ Ω F(w,z)dx+σ ∫ Ω wdx=Π|Ω|.$
(3.3)

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

$d d t ∫ Ω wdx=Π|Ω|− ( 1 + a − λ 2 c ) ∫ Ω F(w,z)dx−σ ∫ Ω wdx.$

□

Lemma 2 Assume that $p≥2$ and let

$G q (t)= ∫ Ω [ q w + exp ( − p − 1 p α + 1 ln ( α ( p ) ( M p − w ) ) ) z p ] dt,$
(3.4)

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

$d d t G q (t)≤−(p−1)σ G q +s.$
(3.5)

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

Let

$h(w)=− p − 1 p α + 1 ln ( α ( p ) ( M p − w ) ) ,$
(3.6)

then

$G q (t)=q ∫ Ω wdx+N(t),$
(3.7)

where

$N(t)= ∫ Ω e h ( w ) z p dx.$
(3.8)

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

$d d t N=H+S,$
(3.9)

where

$H = − λ 2 ∫ Ω ( ( h ′ ( w ) ) 2 + h ″ ( w ) ) e h ( w ) z p ( ∇ w ) 2 d x − p ( λ 2 + λ 1 ) ∫ Ω h ′ ( w ) e h ( w ) z p − 1 ∇ w ∇ z d x − λ 1 ∫ Ω p ( p − 1 ) e h ( w ) z p − 2 ( ∇ z ) 2 d x$
(3.10)

and

$S = Π ∫ Ω h ′ ( w ) e h ( w ) z p d x + ∫ Ω [ p z p − 1 ( 1 + a − λ 1 c ) F ( w , z ) − ( 1 + a − λ 2 c ) h ′ ( w ) z p F ( w , z ) ] e h ( w ) d x − σ ∫ Ω h ′ ( w ) w e h ( w ) z p d x − p σ ∫ Ω e h ( w ) z p d x − p Π ∫ Ω e h ( w ) z p − 1 d x .$
(3.11)

We observe that H is given by

$H=− ∫ Ω Q e h ( w ) dx,$

where

$Q = λ 2 ( ( h ′ ( w ) ) 2 + h ″ ( w ) ) z p ( ∇ w ) 2 + p ( λ 2 + λ 1 ) h ′ ( w ) z p − 1 ∇ w ∇ z + λ 1 p ( p − 1 ) z p − 2 ( ∇ z ) 2$
(3.12)

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

$( p ( λ 2 + λ 1 ) h ′ ( w ) z p − 1 ) 2 −4 λ 1 λ 2 p(p−1) ( ( h ′ ( w ) ) 2 + h ″ ( w ) ) z 2 p − 2 ≤0,$
(3.13)

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

$h ′ (w)= 1 α ( p ) ( M p − w ) , h ″ (w)= α ( p ) [ α ( p ) ( M p − w ) ] 2 .$
(3.14)

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

$4 λ 1 λ 2 p z 2 p − 2 { p [ α 1 ( α ( p ) ( M p − w ) ) 2 − α ( p ) ( α ( p ) ( M p − w ) ) 2 ] + 1 + α ( p ) ( α ( p ) ( M p − w ) ) 2 } = 0 ,$
(3.15)

since

$pα−pα(p)+1+α(p)=0,$

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

$H=− ∫ Ω Q e h ( w ) dx≤0,$
(3.16)

and the second term S can be estimated as

$S ≤ ∫ Ω ( Π h ′ ( w ) − σ p ) e h ( w ) z p d x + ∫ Ω [ p z p − 1 ( 1 + a − λ 1 c ) F ( w , z ) − h ′ ( w ) z p ( 1 + a − λ 2 c ) F ( w , z ) ] e h ( w ) d x ≤ − ( p − 1 ) σ ∫ Ω e h ( w ) z p d x + ∫ Ω [ ( 1 + a − λ 1 c ) p z p − 1 F ( w , z ) − ( 1 + a − λ 2 c ) h ′ ( w ) z p F ( w , z ) ] e h ( w ) d x ,$
(3.17)

since

$h ′ (w)= 1 α ( p ) ( M p − w ) ≤ 1 α ( p ) ( M p − K ) = σ Π .$
(3.18)

On the other hand

$− h ′ ( w ) = − 1 α ( p ) ( M p − w ) ≤ − 1 α ( p ) M p , h ( w ) ≤ − 1 α ( p ) ln Π σ .$
(3.19)

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

$p ( 1 + a − λ 1 c ) z p − 1 F ( w , z ) − ( 1 + a − λ 2 c ) h ′ ( w ) z p F ( w , z ) ≤ ( p ( 1 + a − λ 1 c ) z p − 1 − 1 α ( p ) M p ( 1 + a − λ 2 c ) z p ) F ( w , z ) .$

Then for $η 0 =p(1+ a − λ 1 c )( 1 ( 1 + a − λ 2 c ) +1)α(p) M p >0$, and for $0≤ξ≤K$, $η≥ η 0$, we have

$( p ( 1 + a − λ 1 c ) η p − 1 − 1 α ( p ) M p ( 1 + a − λ 2 c ) η p ) F ( ξ , η ) = [ p ( 1 + a − λ 1 c ) η − ( 1 + a − λ 2 c ) α ( p ) M p ] η p F ( ξ , η ) ≤ 0 .$

On the other hand, we deduce that the function

$(ξ,η)→p ( 1 + a − λ 1 c ) η p − 1 − 1 α ( p ) M p ( 1 + a − λ 2 c ) η p$

is bounded on the compact interval $[0, η 0 ]$; then there exists $c 1 >0$ such that

$p ( 1 + a − λ 1 c ) z p − 1 F(w,z)− ( 1 + a − λ 2 c ) h ′ (w) z p F(w,z)≤ c 1 F(w,z).$
(3.20)

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

$S≤−(p−1)σN+ c 1 ∫ Ω F(w,z) e h ( w ) dx≤−(p−1)σN+ c 1 e − 1 α ( p ) ln Π σ ∫ Ω F(w,z)dx,$

and putting

$q= c 1 ( 1 + a − λ 2 c ) e − 1 α ( p ) ln Π σ ,$
(3.21)

by (3.3), we have

$S≤−(p−1)σN+qΠ|Ω|−q d d t ∫ Ω w(t,x)dx,$
(3.22)

and from (3.7), it follows that

$S≤−(p−1)σ G q +q ( ( p − 1 ) σ K + Π ) |Ω|−q d d t ∫ Ω w(t,x)dx,$
(3.23)

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

$d d t G q ≤−(p−1)σ G q +s,$
(3.24)

where

$s=q ( ( p − 1 ) σ K + Π ) |Ω|.$
(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 $[0,+∞[×Ω$.

Proof Multiplying the inequality (3.24) by $e ( p − 1 ) σ t$ and then integrating, we deduce that there exists a positive constant $C>0$ independent of t, such that

$G q (t)≤C.$
(3.26)

From (3.6), we observe that

$e h ( w ) ≥ e − 1 α ( p ) ln α ( p ) M p ,$
(3.27)

and it follows, for all $p≥2$, that

$∫ Ω z p dx≤ e 1 α ( p ) ln ( K α ( p ) + Π σ ) G q (t)≤ C 1 (p),$
(3.28)

where

$C 1 (p)=C e 1 α ( p ) ln ( K α ( p ) + Π σ ) ,$
(3.29)

and as we select $p> n 2$ we can proceed to bound $∥ − Π + ( 1 + a − λ 1 c ) F ( w , z ) − σ z ∥ p$.

Let

$A= max ξ 0 ≤ ξ ≤ K 1 φ(ξ),$
(3.30)

where

$K 1 = a − λ 1 λ 2 − λ 1 K,$

and $ξ 0$ is such that

$ξ≤ ξ 0 ⟹φ(ξ)<|ξ|,$
(3.31)

using (1.7), we deduce

$F(w,z)=f(u,v)≤φ(u) ( 1 + v ) β ,$

which implies

$∫ Ω F p ( w , z ) d x ≤ ∫ Ω ( φ ( u ) ) p ( 1 + v ) β p d x = ∫ u ≤ ξ 0 ( φ ( u ) ) p ( 1 + v ) β p d x + ∫ ξ 0 ≤ u ( φ ( u ) ) p ( 1 + v ) β p d x ≤ ∫ u ≤ ξ 0 | u | p ( 1 + v ) β p d x + A p ∫ ξ 0 ≤ u ( 1 + v ) β p d x .$

From (2.7), we have

$| u | p = | a − λ 1 λ 2 − λ 1 w ( t , x ) + a − λ 2 λ 2 − λ 1 z ( t , x ) | p ≤ ( a − λ 1 λ 2 − λ 1 w ( t , x ) + λ 2 − a λ 2 − λ 1 z ( t , x ) ) p ≤ ( λ 2 − a λ 2 − λ 1 ) p ( w ( t , x ) + z ( t , x ) ) p ,$

then

$∫ Ω F p ( w , z ) d x ≤ ∫ u ≤ ξ 0 ( λ 2 − a λ 2 − λ 1 ) p ( w + z ) p ( 1 + c λ 2 − λ 1 ( w + z ) ) β p d x ∫ Ω F p ( w , z ) d x ≤ + A p ∫ ξ 0 ≤ u ( 1 + c λ 2 − λ 1 ( w + z ) ) β p d x ∫ Ω F p ( w , z ) d x ≤ max ( A p , ( λ 2 − a λ 2 − λ 1 ) p ) ( ∫ u ≤ ξ 0 ( w + z ) p ( 1 + c λ 2 − λ 1 ( w + z ) ) β p d x ∫ Ω F p ( w , z ) d x ≤ + ∫ ξ 0 ≤ u ( 1 + c λ 2 − λ 1 ( w + z ) ) β p d x ) ∫ Ω F p ( w , z ) d x ≤ max ( A p , ( λ 2 − a λ 2 − λ 1 ) p ) ( ∫ Ω ( w + z ) p ( 1 + c λ 2 − λ 1 ( w + z ) ) β p d x ∫ Ω F p ( w , z ) d x ≤ + ∫ Ω ( 1 + c λ 2 − λ 1 ( w + z ) ) β p d x ) , ∫ Ω ( w + z ) p ( 1 + c λ 2 − λ 1 ( w + z ) ) β p d x ≤ 2 β p − 1 ( ∫ Ω ( w + z ) p + ( c λ 2 − λ 1 ) β p ( w + z ) ( β + 1 ) p d x ) ≤ 2 ( β + 1 ) p − 2 ( K P | Ω | + C 1 ( p ) ) + 2 ( 2 β + 1 ) p − 2 ( c λ 2 − λ 1 ) β p ( K ( β + 1 ) p | Ω | + C 1 ( ( β + 1 ) p ) ) = C 2 ( β , p , K , Ω ) , ∫ Ω ( 1 + c λ 2 − λ 1 ( w + z ) ) β p d x ≤ 2 β p − 1 ( | Ω | + ( c λ 2 − λ 1 ) β p × 2 β p − 1 ( K β p | Ω | + C 1 ( β p ) ) ) = C 3 ( β , p , K , Ω ) .$

Consequently

$∫ Ω F p (w,z)dx≤ C 4 (A,β,p,K,Ω).$

Finally

$∥ − Π + a − λ 1 c F ( w , z ) − σ z ∥ p = ∥ a − λ 1 c F ( w , z ) − ( σ z + Π ) ∥ p ≤ a − λ 1 c ∥ F ( w , z ) ∥ p + σ ∥ z ∥ p + Π | Ω | ≤ a − λ 1 c C 4 ( A , β , p , K ) p + σ C 1 ( p ) p + Π | Ω | = C 5 ( A , β , p , K , Ω , σ ) .$
(3.32)

Using the regularity results for the solutions of the parabolic equations in , we conclude that the solutions of problem (2.1)-(2.4) are uniformly bounded on $[0,+∞[×Ω$. □

By (2.7), it is easy to see that the solutions of problem (1.1)-(1.4) are also uniformly bounded on $[0,+∞[×Ω$.

Remark 3 Because $0≤w(t,x)≤K$ and $z(t,x)≥0$, we deduce that

$−∞≤u(t,x)≤ a − λ 1 λ 2 − λ 1 K= K 1 .$

Remark 4 We note that $a − λ 2 λ 2 − λ 1 <0$ and $λ 2 − a λ 2 − λ 1 ≥ a − λ 1 λ 2 − λ 1$, because $λ 2 + λ 1 =a+d$ 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.

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## Acknowledgements

The author would like to thank the anonymous referee for his/her valuable suggestions.

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Correspondence to Khaled Boukerrioua.

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The author declares that they have no competing interests.

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Boukerrioua, K. Invariant regions and global existence of solutions for reaction-diffusion systems with a general full matrix. J Inequal Appl 2014, 24 (2014). https://doi.org/10.1186/1029-242X-2014-24 