Gen. Math. Notes, Vol. 16, No. 2, June, 2013, pp.32-47
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2013
www.i-csrs.org
Available free online at http://www.geman.in
On the Invariant Regions and Global Existence
of Solutions for Four Components Reaction-Diffusion
Systems With a Full Matrix of Diffusion Cofficients
and Nonhomogeneous Boundary Conditions
Nouri Boumaza1 and Abdelatif Toualbia2
1,2
LAMIS - Laboratory of Mathematics
Computer Science and Systems
University of Tebessa, Algeria
1 E-mail: nboumaza@gmail.com
2 E-mail:toualbiadz@yahoo.fr
(Received: 24-2-13 / Accepted: 31-3-13)
Abstract
In this work we consider a reaction-diffusion systems (four equations) with a
full matrix of diffusion and with nonhomogeneous boundary conditions, by using the
Invariant regions and Lyapunov functional method we established the global existence
of solution.
Keywords: Reaction-Diffusion systems, Invariant regions, Matrice of diffusion,
Global Existence.
33
On the Invariant Regions and Global Existence...
1
Introduction
We consider the folowing reaction-diffusion system
∂u
∂t − a∆u − b∆v − c∆w − d∆z = f1 (u, v, w, z)
∂v
∂t − c∆u − a∆v − b∆z = f2 (u, v, w, z)
∂w
∂t − b∆u − a∆w − c∆z = f3 (u, v, w, z)
∂z
∂t − d∆u − c∆v − b∆w − a∆z = f4 (u, v, w, z)
in
in
in
in
with the bondary conditions

∂u


= β1
λu + (1 − λ)


∂η







∂v


= β2
λv + (1 − λ)



∂η
R+ × Ω
R+ × Ω
R+ × Ω
R+ × Ω
(1.1)
(1.2)
(1.3)
(1.4)
in R+ × ∂Ω
in R+ × ∂Ω
(1.5)


∂w


λw + (1 − λ)
= β3


∂η








∂z

 λz + (1 − λ)
= β4
∂η
in R+ × ∂Ω
in R+ × ∂Ω
and the intial data
u(0, x) = u0 (x); v(0, x) = v0 (x); w(0, x) = w0 (x) ; z(0, x) = z0 (x), in Ω
(1.6)
i. For nonhomogeneous Robin bondary conditions, we use
0 < λ < 1, and β i ∈ R.i = 1, 2, 3, 4.
ii. For nonhomogeneous Neumann bondary conditions, we use
λ = β i = 0,
i = 1, 2, 3, 4.
iii. For nonhomogeneous Dirichlet bondary conditions, we use
1 − λ = β i = 0,
i = 1, 2, 3, 4.
Where Ω is an open bounded domain of class C 1 in Rn , with bondary ∂Ω and
∂
denotes the outward normal derivative on ∂Ω. a, b , c and d are positive constants
∂η
satisfying the condition a + 21 (d − µ) (a − d) which reflects the parabolicity of the
system and implies at the same time that the matrix of diffusion


a b c d
 c a 0 b 

A=
(1.7)
 b 0 a c 
d c b a
34
Nouri Boumaza et al.
is positive definite; that is the eingenvalues λ1 , λ2 , λ3 and λ4 (λ1 < λ2 < λ3 < λ4 )
of its transposed are positive.
The initial data (u0 , v0 , w0 , z0 ) ∈ R4 are assumed to be in the folowing region :


2(b − c)
2(b − c)






(v0 − w0 ) + z0
(β 2 − β 3 ) + β 4
β1 ≤ −
u0 ≤ −






d+µ
d+µ









2(b − c)
2(b − c)





(v0 − w0 ) + z0
(β 2 − β 3 ) + β 4
u0 ≤ −
β1 ≤ −







d−µ
d−µ
X 
2(b + c)
2(b + c)
=
if
u0 +
(v0 + w0 ) + z0 ≥ 0
(β 2 + β 3 ) + β 4 ≥ 0
β1 +






d
−
µ
d−µ








 and


and












2(b
+
c)
2(b + c)


 β1 +

 u0 +
(v0 + w0 ) + z0 ≥ 0
(β 2 + β 3 ) + β 4 ≥ 0
d+µ
d+µ
where
µ=
p
4b2 + 8bc + 4c2 + d2
We suppose that the reaction termes f1 , f2 , f3 and f4 are continuously differentiable, polynomially bounded on Σ satisfying:
2(b−c)
−f1 − 2(b−c)
(f
−
f
)
+
f
(1.8)
2
3
4 (− d+µ (v − w) + z, v, w, z) ≥ 0
d+µ
−f1 − 2(b−c)
(f
−
f
)
+
f
(− 2(b−c)
(1.9)
2
3
4
d−µ
d−µ (v − w) + z, v, w, z) ≥ 0
2(b+c)
(1.10)
f1 + 2(b+c)
d−µ (f2 + f3 ) + f4 (− d−µ (v + w) − z, v, w, z) ≥ 0
2(b+c)
f1 + 2(b+c)
(1.11)
d+µ (f2 + f3 ) + f4 (− d+µ (v + w) − z, v, w, z) ≥ 0
for all v, w, z ≥ 0. And for positive constants C11 , C12 , C13 ≤ 1.
(C11 f1 + C12 f2 + C13 f3 + f4 ) (u, v, w, z) ≤ C1 (u + v + w + z + 1);
(1.12)
for all u, v, w, z in Σ where C1 is a positive constant.
2
2.1
Existence
Local Existence
The usual norms in spaces Lp (Ω), L∞ (Ω) and C(Ω) are respectively denote by:
Z
1
p
kukp =
|u(x)|p dx
Ω
Ω
kuk∞
= max |u(x)| .
x∈Ω
Lp (Ω)
For any initial data in C(Ω) or
, p ∈ (1, +∞); local existence and uniqueness of solutions to the initial value problem (1.1)-(1.5) folow from the basic existence
theory for abstract semilinear differential equations ( see A. Friedman[3], D.Henry[4]
and A Pazzy[12]). The solutions are classical on [0, Tmax [ ; where Tmax denotes the
eventual blowing-up time in L∞ (Ω).

























35
On the Invariant Regions and Global Existence...
2.2
Invariant Regions
P
Proposition
1 Soppose that the functions
P
P f1 , f2 , f3 and f4 points into the region
on ∂
then for any (u0 , v0 , w0 , z0 ) inP , the solution (u(t, .), v(t; .), w(t; .), z(t, .))
of the problem (1.1)-(1.6) remains in
for any time.
Proof. The proof folows the same way to that in S.Kouachi[5]
2(b − c)
2(b − c)
• Multiplying successively (1.1) by −1, (1.2) by −
, (1.3) by
and
d+µ
d+µ
(1.4) by 1.
Ading the first result, the second, the third and the forth results we get (2.1).
2(b − c)
2(b − c)
, (1.3) by
and
• Multiplying successively (1.1) by −1, (1.2) by −
d−µ
d−µ
last one by 1.
Ading the first result, the second result, the third and the forth results we get
(2.2).
2(b + c)
• Multiplying successively (1.1) by 1, (1.2) and (1.3) by
, and (1.4) by
d+µ
1.
Ading the first result, the second result, the third and the forth results we get
(2.3).
2(b + c)
• Multiplying successively (1,1) by 1, the second and the third by
and
d−µ
last one by 1.
Ading the first result, the second result, the third and the forth results we get
(2.4).
Finaly we obtain the system
∂U
1
− a − (d + µ) ∆U = F1 (U, V, W, Z)
(2.1)
∂t
2
1
∂V
− a − (d − µ) ∆V = F2 (U, V, W, Z)
(2.2)
∂t
2
∂W
1
− a + (d − µ) ∆W = F3 (U, V, W, Z)
(2.3)
∂t
2
∂Z
1
− a + (d + µ) ∆Z = F4 (U, V, W, Z)
(2.4)
∂t
2
with the boundary conditions

∂U


λU + (1 − λ)
= ρ1


∂η



∂V


 λV + (1 − λ)
= ρ2
∂η
∂W


λW + (1 − λ)
= ρ3


∂η



∂Z


 λZ + (1 − λ)
= ρ4
∂η
in ]0, T ∗ [ × ∂Ω
in ]0, T ∗ [ × ∂Ω
(2.5)
in ]0, T ∗ [ × ∂Ω
in ]0, T ∗ [ × ∂Ω
36
Nouri Boumaza et al.
and the initial data
where
U (0, x)=U0 (x); V (0, x)=V0 (x); W (0, x)=W0 (x); Z(0, x)=Z0 (x)
(2.6)

2(b − c)


(w(t, x) − v(t, x)) + z(t, x)
U (t, x) = −u(t, x) +



d+µ




2(b − c)

 V (t, x) = −u(t, x) +
(w(t, x) − v(t, x)) + z(t, x)
d−µ
2(b + c)


 W (t, x) = u(t, x) +
(v(t, x) + w(t, x)) + z(t, x)


d−µ



2(b + c)


(v(t, x) + w(t, x)) + z(t, x)
 Z(t, x) = u(t, x) +
d+µ
(2.7)
for any (t, x) in ]0, T ∗ [ × Ω,and

2(b − c)


(f3 − f2 ) + f4 (u, v, w, z)
F1 (U, V, W, Z) = −f1 +


d+µ




2(b − c)


 F2 (U, V, W, Z) = −f1 +
(f3 − f2 ) + f4 (u, v, w, z)
d−µ
2(b + c)



(f2 + f3 ) + f4 (u, v, w, z)
F3 (U, V, W, Z) = f1 +


d−µ



2(b + c)


 F4 (U, V, W, Z) = f1 +
(f2 + f3 ) + f4 (u, v, w, z)
d+µ
P
for all (u, v, w, z) in
,

λ1 = a − 21 (d + µ)



λ2 = a − 12 (d − µ)
λ = a + 12 (d − µ)


 3
λ4 = a + 12 (d + µ)
and
(2.8)

2(b − c)


ρ1 = −β 1 +
(β 3 − β 2 ) + β 4



d+µ




2(b − c)

 ρ2 = −β 1 +
(β 3 − β 2 ) + β 4
d−µ
2(b + c)



ρ3 = β 1 +
(β 2 + β 3 ) + β 4


d
−
µ



2(b + c)


(β 2 + β 3 ) + β 4
 ρ4 = β 1 +
d+µ
First, let’s notice that the condition of parabolicity of
the system (1.1)-(1.4)1 implies
1
the one of the (2.1)-(2.4)
system;
since
a
+
(d
−
µ)
(a − d) > 0 =⇒ a + 2 (d − µ) >
2
1
1
0; a − 2 (d + µ) > 0 and a − 2 (d − µ) > 0.
Now, it suffices to prove that the region
Σ = {U ≥ 0, V ≥ 0, W ≥ 0, Z ≥ 0} = R+ × R+ × R+ × R+ ,
is invariant region for the system (2.1)-(2.4).
37
On the Invariant Regions and Global Existence...
Since , from (1.8) − (1.9) − (1.10) − (1.11) we have F1 (0, V, W, Z) ≥ 0 it suffices
to be
2(b−c)
2(b−c)
−f1 (− 2(b−c)
(v
−
w)
+
z,
v,
w,
z)
−
(f
−
f
)
(−
(v
−
w)
+
z,
v,
w,
z
2
3
d+µ
d+µ
d+µ
+f4 (− 2(b−c)
d+µ (v − w) + z, v, w, z) ≥ 0, for all V, W, Z ≥ 0 and all v, w, z ≥ 0,
then
U (t, x) ≥ 0 for all (t, x) in ]0, T ∗ [ × Ω,
thanks to the invariant region method (see Smoller[13 ]), and because F2 (U, 0, W, Z) ≥
0 it suffices to be
2(b−c)
2(b−c)
−f1 (− 2(b−c)
(v
−
w)
+
z,
v,
w,
z)
−
(f
−
f
)
(−
(v
−
w)
+
z,
v,
w,
z
2
3
d−µ
d−µ
d−µ
+f4 (− 2(b−c)
d−µ (v − w) + z, v, w, z) ≥ 0, for all U, W, Z ≥ 0 and all v, w, z ≥ 0,
then
V (t, x) ≥ 0 for all (t, x)in ]0, T ∗ [ × Ω, and F3 (U, V, 0, Z) ≥ 0
it suffices to be
f1 (− 2(b+c)
d−µ
2(b+c)
d−µ
(− 2(b+c)
d−µ
(v + w) − z, v, w, z) +
(f2 + f3 )
(v + w) − z, v, w, z
2(b + c)
+f4 (−
(v + w) − z, v, w, z) ≥ 0, for all U, V, Z ≥ 0 and all v, w, z ≥ 0,
d−µ
then
W (t, x) ≥ 0 for all (t, x)in ]0, T ∗ [ × Ω, and F4 (U, V, W, 0) ≥ 0
it suffices to be
f1 (− 2(b+c)
d+µ (v + w) − z, v, w, z) +
2(b+c)
d+µ
(f2 + f3 ) (− 2(b+c)
(v
+
w)
−
z,
v,
w,
z
d+µ
+f4 (− 2(b+c)
d+µ (v + w) − z, v, w, z) ≥ 0, for all U, V, W ≥ 0 and all v, w, z ≥ 0,
then
Z(t, x) ≥ 0 for all (t, x) in ]0, T ∗ [ × Ω,
P
then
is an invariant region for the system (1.1) − (1.4).
P Then the system (1.1) − (1.4)with boundary condition (1.5) and initial data in
is equivalent to system (2.1) − (2.4) with bondary conditions (2.5) and positive
initiale data (2.6).
Once invariant regions are constructed, one can apply Lyapunov technique and
estabish global existence of unique solutions for (1.1)-(1.6)
2.3
Global Existence
As the determinant of the linear algebraic system (2.7) with regard to variables
u, v, w and z, is different from zero, then to prove global existence of solutions of
problem (1.1) − (1.6), one needs to prove it for problem (2.1) − (2.6). To this subject
; it is well-known that (see Henry[4]) it sufficies to derive an uniform estimate of
38
Nouri Boumaza et al.
kF1 (U, V, W, Z)kp , kF2 (U, V, W, Z)kp ,kF3 (U, V, W, Z)kp and kF4 (U, V, W, Z)kp on
N
[0, T ∗ [ for some p >
(N = dim Ω).
2
λ1 + λ2
λ1 + λ3
λ1 + λ4
λ2 + λ3
Let’s put A12 = √
, A13 = √
, A14 = √
, A23 = √
,
2 λ1 λ2
2 λ1 λ3
2 λ1 λ4
2 λ2 λ3
λ2 + λ4
A24 = √
; θ1 , θ2 and θ3 are three positive constants sufficiently large such that
2 λ2 λ4
θ1 > A12 ,
(θ21 − A212 ) θ22 − A223 > (A13 − A12 A23 )2
(θ21 − A212 ) θ22 − A223 θ23 − A234 > (A13 − A12 A23 )2 + (A14 − A13 A24 )2
(2.9)
(2.10)
(2.11)
The main result of the paper is as follows:
Theorem 1. Let (U (t, .), V (t, .), W (t, .), Z(t, .)) be any positive solution of
(2.1) − (2.6) and let the functional
Z
L(t) = Hn (U (t, x), V (t, x), W (t, x), Z(t, .))dx,
(2.12)
Ω
where
Hn (U, V, W, Z) =
j
n X
k X
X
2
2
2
Cnk Ckj Cji θi1 θj2 θk3 U i V j−i W k−j Z n−k .
k=0 j=0 i=0
Thene, the functional L is uniformly bounded on the interval [0, T ∗ ] , T ∗ < Tmax .
Corollary 1.PSuppose that the
P functions f1 , f2 , f3 and f4 are countinuously
differentiable on
, point into ∂
and satisfy
condition (1.12). Then all solutions
P
of (1.1) − (1.6) with the initial data in
and uniformly bounded on Ω are in
L∞ (0, T ∗ ; Lp (Ω)) for all p ≥ 1
Proposition 2. Under the hypothesis of corolly 1, if the reactions f1 , f2 , f3
and f4 are
P polynomially bounded, then all solutions of (1.1) − (1.6) with the initial
data in
and uniformly bounded on Ω are global time.
2.4
Proofs
For the proof of theorem 1, we need some preparatory lemmas.
Lemma 1. Let Hn be the homogeneous polynomial defined by (2.11). Then
n−1
k
P P
∂Hn
∂U
= n
∂Hn
∂V
= n
n−1
k
P P
∂Hn
∂W
= n
n−1
k
P P
∂Hn
∂Z
= n
j
P
k=0 j=0 i=0
j
P
k=0 j=0 i=0
j
P
k=0 j=0 i=0
n−1
k
P P
j
P
k=0 j=0 i=0
(i+1)2
k
Cn−1
Ckj Cji θ1
2
(j+1)2
θ2
(j+1)2
k
Cn−1
Ckj Cji θi1 θ2
2
2
2
2
(k+1)2
θ3
(k+1)2
k
Cn−1
Ckj Cji θi1 θj2 θ3
2
(k+1)2
θ3
U i V j−i W k−j Z (n−1)−k ,
U i V j−i W k−j Z (n−1)−k ,
U i V j−i W k−j Z (n−1)−k ,
k
Cn−1
Ckj Cji θi1 θj2 θk3 U i V j−i W k−j Z (n−1)−k ,
39
On the Invariant Regions and Global Existence...
Proof of lemma 1. see Said Kouachi[6].
Lemma 2. The second partial derivatives of Hn are given by
n−2
k
P P
j
P
∂ 2 Hn
∂U 2
= n(n-1)
∂ 2 Hn
∂U V
= n(n − 1)
n−2
k
P P
∂ 2 Hn
∂U W
= n(n − 1)
n−2
k
P P
∂ 2 Hn
∂U Z
= n(n − 1)
n−2
k
P P
= n(n − 1)
n−2
k
P P
∂ 2 Hn
∂V W
= n(n − 1)
n−2
k
P P
∂ 2 Hn
∂V Z
= n(n − 1)
n−2
k
P P
∂ 2 Hn
∂W 2
= n(n − 1)
k
n−2
P P
∂ 2 Hn
∂W Z
= n(n − 1)
n−2
k
P P
= n(n − 1)
n−2
k
P P
∂ 2 Hn
∂V 2
∂ 2 Hn
∂Z 2
k=0 j=0 i=0
(i+2)2 (j+2)2 (k+2)2 i j−i
θ2
θ3
U V W k−j Z (n−2)−k ,
k
Cn−2
Ckj Cji θ1
j
P
k=0 j=0 i=0
j
P
k=0 j=0 i=0
j
P
k=0 j=0 i=0
j
P
k=0 j=0 i=0
j
P
k=0 j=0 i=0
j
P
k=0 j=0 i=0
j
P
k=0 j=0 i=0
j
P
k=0 j=0 i=0
j
P
k=0 j=0 i=0
(i+1)2 (j+2)2 (k+2)2 i j-i
θ2
θ3
U V W k-j Z (n-2)-k ,
k
Cn−2
Ckj Cji θ1
(i+1)2
k
Cn−2
Ckj Cji θ1
(i+1)2
k
Cn−2
Ckj Cji θ1
(j+1)2
θ2
(j+1)2
θ2
2
(j+2)2
2
(j+1)2
2
(j+1)2
k
Cn−2
Ckj Cji θi1 θ2
k
Cn−2
Ckj Cji θi1 θ2
k
Cn−2
Ckj Cji θi1 θ2
(k+2)2
U i V j−i W k−j Z (n−2)−k ,
(k+1)2
U i V j−i W k−j Z (n−2)−k
θ3
θ3
(k+2)2
U i V j−i W k−j Z (n−2)−k ,
(k+2)2
U i V j−i W k−j Z (n−2)−k ,
(k+1)2
U i V j−i W k−j Z (n−2)−k ,
θ3
θ3
θ3
2
2
(k+2)2
U i V j−i W k−j Z (n−2)−k ,
2
2
(k+1)2
U i V j−i W k−j Z (n−2)−k ,
2
2
k
Cn−2
Ckj Cji θi1 θj2 θ3
k
Cn−2
Ckj Cji θi1 θj2 θ3
2
k
Cn−2
Ckj Cji θi1 θj2 θk3 U i V j−i W k−j Z (n−2)−k ,
Proof of lemma 2. see Said Kouachi[6].
Proof of Theorem 1. Differentiating L with respect to t yields
Z ∂U
∂V
∂W
∂Z
0
L (t) =
∂U Hn
+ ∂V Hn
+ ∂W Hn
+ ∂Z Hn
dx
∂t
∂t
∂t
∂t
ZΩ
=
(λ1 ∂U Hn ∆U + λ2 ∂V Hn ∆V + λ3 ∂W Hn ∆W + λ4 ∂Z Hn ∆Z) dx
ΩZ
+ (F1 ∂U Hn + F2 ∂V Hn + F3 ∂W Hn + F4 ∂Z Hn ) dx
Ω
= I + J.
Using Green’s formula and applying lemma 1 we get I = I1 + I2 , where
R ∂V
∂W
∂Z
I1 =
λ1 ∂U Hn ∂U
+λ
∂
H
+λ
∂
H
+λ
∂
H
2
n
3
n
4
n
V
W
Z
∂η
∂η
∂η
∂η dx
(2.13)
∂Ω
Z n−2 k j
P P P k
I2 = −n(n − 1)
Cn−2 Ckj Cji (T Bijk )T t dx,
Ω k=0 j=0 i=0
((2.14))
40
Nouri Boumaza et al.
where

Bijk
(i+2)2 (j+2)2 (k+2)2
θ2
θ3
λ1 +λ2
2 a12
λ1 θ 1





=





λ1 +λ2
2 a12
2
(j+2)2
λ2 θi1 θ2
λ1 +λ3
2 a13
(k+2)2
λ2 +λ3
2 a23
θ3
λ1 +λ3
2 a13
λ2 +λ3
2 a23
λ1 +λ4
2 a14
λ2 +λ4
2 a24
2
2
(k+2)2
λ3 θi1 θj2 θ3
λ3 +λ4
2 a34

λ1 +λ4
2 a14
λ2 +λ4
2 a24
λ3 +λ4
2 a34
2
2
λ4 θi1 θj2 θk3
2











where
(i+1)2 (j+2)2 (k+2)2
(i+1)2 (j+1)2 (k+2)2
(i+1)2 (j+1)2 (k+1)2
θ2
θ3
; a13 = θ1
θ2
θ3
; a14 = θ1
θ2
θ3
;
2
2
2
2
2
i2 (j+1) (k+2)
i2 (j+1) (k+1)
i2 j 2 (k+1)
,
θ1 θ2
θ3
; a24 = θ1 θ2
θ3
; a34 = θ1 θ2 θ3
a12 = θ1
a23 =
for i = 0, .., j; j = 0, .., k; k = 0, ..., n − 2; and T = (∇U, ∇V, ∇W, ∇Z)
We prove that there exists a positive constant C2 independent of t ∈ [0, Tmax [
such that
I1 ≤ C2 for all t ∈ [0, Tmax [
(2.15)
and that
I2 ≤ 0
(2.16)
for several boundary conditions.
i) If 0 < λ < 1, using the boundary conditions (2.5) we get
I1 =
R
(λ1 ∂U Hn (γ 1 − αU ) + λ2 ∂V Hn (γ 2 − αV ) + λ3 ∂W Hn (γ 3 − αW ) + λ4 ∂Z Hn (γ 4 − αZ)) dx,
∂Ω
where α =
Since
λ
ρi
and γ i =
, i = 1, 2, 3, 4.
1−λ
1−λ
K(U, V, W, Z) = λ1 ∂U Hn (γ 1 − αU ) + λ2 ∂V Hn (γ 2 − αV ) + λ3 ∂W Hn (γ 3 − αW )
+λ4 ∂Z Hn (γ 4 − αZ) = Pn−1 (U, V, W, Z) − Qn (U, V, W, Z)
where Pn−1 and Qn are polynomials with positive coefficients and respective degrees
(n − 1) and n and since the solution is positive, then,
lim sup K(U, V, W, Z) = −∞
(2.17)
(|U |+|V |+|W |)−→∞
which proves that K is uniformly bounded on R4+ and consequently (2.15).
(ii) If λ = 0, then I1 = 0 on [0, Tmax [.
(iii) The case of homogeneous Dirichlet conditions is trivial, since in this case
the positivity of the solution on [0, Tmax [×Ω implies ∂η U ≤ 0, ∂η V ≤ 0, ∂η W ≤ 0
,and ∂η Z ≤ 0 on [0, Tmax [×∂Ω. Consequently one gets again (2.15) with C2 = 0.
41
On the Invariant Regions and Global Existence...
Now we prove (2.16). The quadratic forms ( with respect to ∇U, ∇V, ∇W and
∇Z ) associated with the matrixes Bijk , i = 0, ..., j; j = 0, ..., k and k = 0, ..., n − 2
are positive since their main determinants ∆1 , ∆2 ,∆3 and ∆4 are positive , too .
To see this, we have :
(i+2)2 (j+2)2 (k+2)2
θ2
θ3
1. ∆1 = λ1 θ1
0, ..., n − 2
> 0 for i = 0, ..., j ; j = 0, ..., k ; and k =
2.
∆2 = (i+2)2 (j+2)2 (k+2)2
θ2
θ3
2
2
2
λ1 +λ2 (i+1) (j+2) (k+2)
θ2
θ3
2 θ1
λ1 θ 1
2
2
2
λ1 +λ2 (i+1) (j+2) (k+2)
θ2
θ3
2 θ1
2(i+1)2 2(j+2)2 2(k+2)2 2
= λ1 λ2 θ1
θ2
θ3
(θ1
2
(j+2)2 (k+2)2
θ3
λ2 θi1 θ2
− A212 ),
for i = 0, ..., j ; j = 0, ..., k and k = 0, ..., n − 2. Using (2.9), we get ∆2 > 0.
3.
(i+2)2 (j+2)2 (k+2)2
λ1 θ 1
θ2
θ3
λ1 +λ2
2 a12
∆3 = λ1 +λ3
2 a13
λ1 +λ3
2 a13
λ1 +λ2
2 a12
2
(j+2)2 (k+2)2
θ3
λ2 +λ3
2 a23
λ2 θi1 θ2
λ2 +λ3
2 a23
2
2
(k+2)2
λ3 θi1 θj2 θ3
2(i+2)2 2(i+1)2 2(j+2)2 2(j+1)2 2(k+1)2 k2
θ1
θ2
θ2
θ3
θ3
= λ1 λ2 λ3 θ1
× (θ21 − A212 ) θ22 − A223 − (A13 − A12 A23 )2
for i = 0, ..., j ; j = 0, ..., k and k = 0, ..., n − 2.Using (2.10), we get ∆3 > 0.
4.
2(i+2)2 2(i+1)2 2(j+1)2 j 2 2(k+1)2 k2
∆4 = |Bijk | = λ1 λ2 λ3 λ43 θ1
θ1
θ2
θ2 θ3
θ3
× (θ21 − A212 ) θ22 − A223 θ23 − A234 − (A13 − A12 A23 )2 − (A14 − A13 A24 )2
for i = 0, ..., j ; j = 0, ..., k and k = 0, ..., n − 2.Using (2.11) , we get ∆4 > 0.
Substituting the expressions of the partial derivatives given by lemma 1 in the
42
Nouri Boumaza et al.
second integral, yields
Z
j
n−1
k X
XX
2
2
2
2
2
2
2
k C j C i θ (i+1) θ (j+1) θ (k+1) F + θ i2 θ (j+1) θ (k+1) F + θ i2 θ j θ (k+1) F
Cn−1
1
2
3
1 2
1 2 3
1
2
3
3
k j
Ω k=0 j=0 i=0
!
Z
j
n−1
k P
P P
i2 j 2 k 2
k C j C i U i V j−i W k−j Z (n−1)−k
i
j−i
k−j
(n−1)−k
Cn−1
n
+θ1 θ2 θ3 F4 U V W
Z
dx =
k j
n
J=
(i+1)2 (j+1)2 (k+1)2
θ3
θ2
θ1
2
2 2
θi1 θj2 θk3
×
Z
n
=
k=0 j=0 i=0
Ω
(k+1)2
+
θ3
θ3k
j
P
n−1
k
P P
2
F1 +
2 (j+1)2 (k+1)2
θ3
θi1 θ2
2
2 2
θi1 θj2 θk3
k=0 j=0 i=0
Ω
2 (k+1)2
2
F2 +
θi1 θj2 θ3
2
2
θi1 θj2!θk3
2
k C j C i U i V j−i W k−j Z (n−1)−k
Cn−1
k j
2
2
2
F3 + F4 θi1 θj2 θk3 dx.
(i+1)2 (j+1)2 (k+1)2
θ3
θ2
i2 j 2 k2
θ1 θ2 θ3
θ1
F1 +
(j+1)2 (k+1)2
θ3
j 2 k2
θ2 θ3
θ2
2
2
2
F3 + F4 θi1 θj2 θk3 dx.
Using the expressions (2.8), we obtain
n−1
k
P P
Z
J=
j
P
!
k C j C i U i V j−i W k−j Z (n−1)−k
Cn−1
k j
n
j=0 i=0
k=0
(i+1)2 (j+1)2 (k+1)2
(j+1)2 (k+1)2
(k+1)2
θ2
θ3
θ3
θ1
θ2
θ3
× −
−
+ k2 + 1 f1
2
2 2
2
2
θ3
θi1 θj2 θk3
θj2 θk3
2
2
2
(i+1) (j+1) (k+1)
(j+1)2 (k+1)2
(k+1)2
θ1
θ2
θ3
θ3
2(b−c) θ2
2(b+c) θ3
+ − 2(b−c)
−
+
+
j 2 k2
k2
d+µ
d−µ
d−µ
i2 j 2 k2
Ω
θ1 θ2 θ3
+
(i+1)2 (j+1)2 (k+1)2
(j+1)2 (k+1)2
θ2
θ3
θ3
2(b−c) θ2
2
2
2
j 2 k2
d−µ
i j
k
θ1 θ2 θ3
θ2 θ3
(i+1)2 (j+1)2 (k+1)2
(j+1)2 (k+1)2
(k+1)2
θ1
θ2
θ3
θ2
θ3
θ3
2
2
2 2
2
2
θk3
θi1 θj2 θk3
θj2 θk3
2(b−c) θ1
d+µ
+
θ3
θ2 θ3
+
+
(k+1)2
2(b+c) θ3
2
d−µ
θk
2(b+c)
d+µ
2(b+c)
d+µ
+
+
3
2
2
2
+ 1 f4 θi1 θj2 θk3 dx
+
f2
f3
And so we get the following inequality:
n−1
k
P P
Z
J≤
!
j
P
k C j C i U i V j−i W k−j Z (n−1)−k
n
Cn−1
k j
k=0 j=0 i=0
 (i+1)2 (j+1)2 (k+1)2

(j+1)2 (k+1)2
(k+1)2
θ2
θ3
θ1
θ2
θ3
θ3
−
−
+
+
1
2
2
2 2
2
2


θk3
θi1 θj2 θk3
θj2 θk3

 f1
×
 θ(i+1)2 θ(j+1)2 θ(k+1)2

(j+1)2 (k+1)2
(k+1)2
θ
θ
θ
1
2
3
2
3
3
+
1
+
+
2
2
2
2
2
2
θk3
θi1 θj2 θk3
θj2 θk3

(i+1)2 (j+1)2 (k+1)2
(j+1)2 (k+1)2
(k+1)2
θ2
θ3
θ3
2(b−c) θ1
2(b−c) θ2
2(b+c) θ3
−
−
+
+
2
2
2 2
2
2
d−µ
d−µ
 d+µ
θk3
θi1 θj2 θk3
θj2 θk3

+
(i+1)2 (j+1)2 (k+1)2
(j+1)2 (k+1)2
(k+1)2
θ3
θ1
θ2
θ2
θ3
θ3
+
+
+1
2
2
2
2
2
j
k2
i j
k
k
Ω
θ1 θ2 θ3

2
j2
2
×θi1 θ2 θk3 dx,


 f2

θ3
(i+1)2 (j+1)2 (k+1)2
(j+1)2 (k+1)2
θ2
θ3
θ3
2(b−c) θ2
2(b+c)
+
+ d−µ
2
j 2 k2
d−µ
i2 j
k2
θ1 θ2 θ3
θ2 θ3
(i+1)2 (j+1)2 (k+1)2
(j+1)2 (k+1)2
(k+1)2
θ1
θ2
θ3
θ2
θ3
θ3
+
+
2
2
2
2
2
2
θk3
θi1 θj2 θk3
θj2 θk3
2(b−c) θ1
 d+µ
+

θ2 θ3
2(b+c)
d+µ
(k+1)2
θ3
θk3
+1
2

2(b+c)
+ d+µ



 f3 + f4 


F2
43
On the Invariant Regions and Global Existence...
using condition (1.10) , and relation (2.6) successively, we get
Z Z
J ≤ C4
n
Ω Ω
p
n−1
XX
p
Cn−1
Cpq U q V p−q W (n−1)−p (U + V + W + 1) dx.
p=0 q=0
Following the same reasoning as in S. Kouachi[6], a straight forward calculation
shows that
n−1
0
L (t) ≤ C5 L(t) + C6 L n (t) on [0, T ∗ ],
1
which for Z = L n can be written as
0
nZ ≤ C5 Z + C6 .
A simple integration gives the uniform bound of the functional L on the interval
[0, T ∗ [ ; this ends the proof of the theorem.
Proof of corollary 1. The proof of this corolarly is an immediate consequence
of the theorem and the inequality
Z
(U + V + W + Z)P dx ≤ C8 L(t) on [0, T ∗ [,
Ω
for some P ≥ 1, taking into consideration expressions (2.7).
Proof of proposition 2. As it has been mentioned above; it suffices to derive
a uniforme estimate of kF1 (U, V, W, Z)kp , kF2 (U, V, W, Z)kp , kF3 (U, V, W, Z)kp and
N
kF4 (U, V, W, Z)kp on [0, T ∗ [ for some p >
. Since the functions f1 , f2 , f3 and f4
2
P
are polynomially bounded on , then using relations (2.5), (2.7) and (2.8), we get
that F1 , F2 , F3 and F4 are polynomially bounded, too and the proof becomes an
immediate consequence of corollary 1.
3
Final Remarks
1. In the case when the system (1.1) − (1.4) rewriten as follows:
∂u
− a∆u − b∆v − c∆w = f1 (u, v, w, z)
∂t
∂v
− c∆u − a∆v − d∆w − b∆z = f2 (u, v, w, z)
∂t
∂w
− b∆u − d∆v − a∆w − c∆z = f3 (u, v, w, z)
∂t
∂z
c∆v − b∆w − a∆z = f4 (u, v, w, z)
∂t
in R+ × Ω
(3.1)
in R+ × Ω
(3.2)
in R+ × Ω
(3.3)
in R+ × Ω
(3.4)
with the same boundary conditions (1.5) and initial data (1.6), and the diffusion matrix

a
 c

 0
d
b
a
b
c
c
b
a
b

d
0 
,
c 
a
44
Nouri Boumaza et al.
All the previous results remain valid in the region
P
2(b − c)
= (u0 , v0 , w0 , z0 ) ∈ R4 such that:u0 ≤ −
(v0 − w0 ) + z0
d−µ
2(b − c)
2(b + c)
, u0 ≤ −
(v0 − w0 ) + z0 , , u0 +
(v0 + w0 ) + z0 ≥ 0
d+µ
d − µ
2(b + c)
(v0 + w0 ) + z0 ≥ 0,
and u0 +
d+µ
(3.5)
and system (2.1) − (2.4) becomes







where
∂U
∂t
∂V
∂t
∂W
∂t
∂Z
∂t
−
−
−
−
a − 12 (d − µ) ∆U = F11 (U, V, W, Z)
a − 12 (d + µ) ∆V = F21 (U, V, W, Z)
a + 12 (d − µ) ∆W = F31 (U, V, W, Z)
a + 12 (d + µ) ∆Z = F41 (U, V, W, Z)
(3.6)

2(b − c)


(w(t, x) − v(t, x)) + z(t, x)
U (t, x) = −u(t, x) +



d+µ




2(b − c)

 V (t, x) = −u(t, x) +
(w(t, x) − v(t, x)) + z(t, x)
d−µ
2(b + c)



W (t, x) = u(t, x) +
(v(t, x) + w(t, x)) + z(t, x)


d
−
µ



2(b + c)


(v(t, x) + w(t, x)) + z(t, x)
 Z(t, x) = u(t, x) +
d+µ
for all (t, x) in ]0, T ∗ [×Ω

2(b − c)


(f
−
f
)
+
f
F
(U,
V,
W,
Z)
=
−f
+
3
2
4 (u, v, w, z)
11
1


d−µ




2(b − c)


 F12 (U, V, W, Z) = −f1 +
(f3 − f2 ) + f4 (u, v, w, z)
d+µ
2(b + c)



F13 (U, V, W, Z) = f1 +
(f2 + f3 ) + f4 (u, v, w, z)


d−µ



2(b + c)


 F14 (U, V, W, Z) = f1 +
(f2 + f3 ) + f4 (u, v, w, z)
d+µ
(3.7)
P
for all (u, v, w, z) in
With the boundary conditions

∂U


λU + (1 − λ)
= ρ1


∂η




∂V

 λV + (1 − λ)
= ρ2
∂η
∂W



λW + (1 − λ)
= ρ3


∂η



∂Z

 λZ + (1 − λ)
= ρ4
∂η
in ]0, T ∗ [ × ∂Ω
in ]0, T ∗ [ × ∂Ω
(3.8)
in
]0, T ∗ [
× ∂Ω
in ]0, T ∗ [ × ∂Ω
45
On the Invariant Regions and Global Existence...
where

2(b − c)


(β 3 − β 2 ) + β 4
ρ1 = −β 1 +



d−µ




2(b − c)

 ρ2 = −β 1 +
(β 3 − β 2 ) + β 4
d+µ
2(b + c)



(β 2 + β 3 ) + β 4
ρ3 = β 1 +


d−µ



2(b + c)


(β 2 + β 3 ) + β 4
 ρ4 = β 1 +
d+µ
and initial data (1.5).
The conditions (1.8) − (1.12) become respectively:
2(b+c)
(f
+
f
)
+
f
f1 + 2(b+c)
2
3
4 (− d−µ (v + w) − z, v, w, z) ≥ 0
d−µ
2(b−c)
−f1 − 2(b−c)
d+µ (f2 − f3 ) + f4 (− d+µ (v − w) + z, v, w, z) ≥ 0
2(b+c)
f1 + 2(b+c)
d−µ (f2 + f3 ) + f4 (− d−µ (v + w) − z, v, w, z) ≥ 0
2(b+c)
f1 + 2(b+c)
d+µ (f2 + f3 ) + f4 (− d+µ (v + w) − z, v, w, z) ≥ 0
2. When the system (1.1) − (1.4) rewritten as follws:
 ∂u

∂t − a∆u − b∆v − c∆w − d∆z = f1 (u, v, w, z)

 ∂v
∂t − c∆u − a∆v − b∆w = f2 (u, v, w, z)
∂w


∂t − b∆v − a∆w − c∆z = f3 (u, v, w, z)
 ∂z
∂t − d∆u − c∆v − b∆w − a∆z = f4 (u, v, w, z)
or
 ∂u

∂t − a∆u − b∆v − c∆w − d∆z = f1 (u, v, w, z)

 ∂v
∂t − a∆v − c∆w − b∆z = f2 (u, v, w, z)
,
∂w

− b∆u − c∆v − a∆w = f3 (u, v, w, z)

∂t
 ∂z
∂t − d∆u − c∆v − b∆w − a∆z = f4 (u, v, w, z)
all the previous results remain valid in the region
P
2(b − c)
= (u0 , v0 , w0 , z0 ) ∈ R4 such that:u0 ≤ −
(v0 − w0 ) + z0 ,
d−µ
2(b − c)
2(b + c)
u0 ≤ −
(v0 − w0 ) + z0 , u0 +
(v0 + w0 ) + z0 ≥ 0
d+µ
d−µ
2(b + c)
and u0 +
(v0 + w0 ) + z0 ≥ 0 ,
d+µ
for the first system.
P
=
(u0 , v0 , w0 , z0 ) ∈ R4 such that:u0 ≤ −
2(b − c)
(v0 − w0 ) + z0 ,
d−µ
2(b + c)
2(b − c)
(v0 − w0 ) + z0 , u0 +
(v0 + w0 ) + z0 ≥ 0
d+µ
d−µ
2(b + c)
and u0 +
(v0 + w0 ) + z0 ≥ 0 ,
d+µ
u0 ≤ −
for the second system
(3.9)
46
Nouri Boumaza et al.
Acknowledgements
The authors are grateful to the referees for their valuable suggestions to improve
the quality of paper.
References
[1] S. Abdelmalek, Invariant regions and global existence of solution for reactiondiffusion systems with a tridiagonal matrix of diffusion coefficients and nonhomogeneous boundary conditions, Journal of Applied Mathematics, 2007(Article
ID 12375) (2007), 15 pages.
[2] E.L. Cusler, Multicomponent Diciffusion, Chemical Engineering Monographs
(Vol.3), Elsevier Publishing Scientific Company, Amsterdam, (1976).
[3] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall,
Inc, Englewood Cliffs, N.J, (1964).
[4] D. Henry, Geometric theory of semi-linear parabolic equations, Lecture Notes
in Mathematics 840, Springer -Verlag, New-York, (1984).
[5] S. Kouachi, Existence of global solutions to reaction-diffusion systems via a
Lyapunov functional, Electron. J. Diff. Equ, 2001(68) (2001), 1-10.
[6] S. Kouachi, Existence of global solutions to reaction-diffusion systems with
nonhomogeneous boundary conditions via a Lypunov functional, Electron. J.
Diff. Equ, 2002(88) (2002), 1-13.
[7] S. Kouachi, Global existence of solutions for reaction-diffusion systems with a
full matrix of diffusion coefficients and nonhomogeneous boundary condition,
Electron. J. Diff. Equ, 2002(2) (2002), 1-10.
[8] S. Kouachi, Invariant regions and global existence of solution for reactiondiffusion systems with a full matrix of diffusion coefficient and nonhomogeneous
boundary conditions, Georgian Mathematical Journal, 11(2) (2004), 349-359.
[9] S. Kouachi and A. Youkana, Global existence for a class of reaction-diffusion
systems, Bulletin of the Polish Academy of Science, 49(3) (2001), 303-308.
[10] S. Kouachi and B. Rebai, Ivariant regions and the global existence for reactiondiffusion systems with a tridiagonal matrix of diffusion coefficients, Memoirs
on Differential Equations and Mathematical Physics, 51(2010), 93-108.
[11] K. Masuda, On the global existence and asymptotic behaviour of solutions of
reaction-diffusion equations, Hokkaido Math. J., 12(3) (1983), 303-308.
[12] A. Pazzy, Semigroups of linear operators and applications to partial differential
equations, Applied Math. Sciences, Springer -Verlag, New-York, (1983).
On the Invariant Regions and Global Existence...
47
[13] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag,
New-York, (1983).
[14] F. Webb, A reaction diffusion model for a deterministic diffusive epidemic, J.
Math. Anal. Appl., 84(1981), 150-161.