Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 94, pp. 1–11.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
BOUNDEDNESS AND LARGE-TIME BEHAVIOR OF
SOLUTIONS FOR A GIERER-MEINHARDT SYSTEM OF THREE
EQUATIONS
SAFIA HENINE, SALEM ABDELMALEK, AMAR YOUKANA
Abstract. The aim of this work is to prove the uniform boundedness and the
existence of global solutions for Gierer-Meinhardt model of three substance
described by reaction-diffusion equations with Neumann boundary conditions.
Based on a Lyapunov functional we establish the asymptotic behaviour of the
solutions.
1. Introduction
In this article, we consider the Gierer-Meinhardt type system of three equations
∂u
− a1 ∆u = −b1 u + f (u, v, w), in R+ × Ω,
∂t
∂v
− a2 ∆v = −b2 v + g(u, v, w), in R+ × Ω,
∂t
∂w
− a3 ∆w = −b3 w + h(u, v, w), in R+ × Ω,
∂t
(1.1)
where
up1
+ σ1 (x),
v q1 (wr1 + c)
up2
g(u, v, w) = ρ2 (x, u, v, w) q2 r2 + σ2 (x),
v w
up3
h(u, v, w) = ρ3 (x, u, v, w) q3 r3 + σ3 (x),
v w
with homogeneous Neumann boundary conditions
f (u, v, w) = ρ1 (x, u, v, w)
∂u
∂v
∂w
=
=
=0
∂η
∂η
∂η
on R+ × ∂Ω,
(1.2)
and initial data
u(0, x) = ϕ1 (x),
v(0, x) = ϕ2 (x),
w(0, x) = ϕ3 (x),
2000 Mathematics Subject Classification. 35K57.
Key words and phrases. Gierer-Meinhardt system; Lyapunov functional;
Uniform boundedness.
c 2015 Texas State University - San Marcos.
Submitted January 8, 2015. Published April 14, 2015.
1
in Ω.
(1.3)
2
S. HENINE, S. ABDELMALEK, A. YOUKANA
EJDE-2015/94
Here Ω is an open bounded domain in RN with smooth boundary ∂Ω and outer
normal η(x). The constants c, pi , qi , ri , ai and bi , i = 1, 2, 3 are real numbers such
that
c, pi , qi , ri ≥ 0, and ai , bi > 0,
and
q1
r1 r1
q1 0 < p1 − 1 < max p2 min
, , 1 , p3 min
, ,1 .
(1.4)
q2 + 1 r2
r3 + 1 q3
The initial data are assumed to be positive and continuous functions on Ω̄. For
i = 1, 2, 3, we assume that σi are positive functions in C(Ω̄), and ρi are positive
bounded functions in C 1 (Ω̄ × R3+ ).
In 1972, following the ingenious idea of Turing [15], Gierer and Meinhardt [2]
proposed a mathematical model for pattern formations of spatial tissue structure
of hydra in morphogenesis, a biological phenomenon discovered by Trembley in
1744 [14]. It can be expressed in the following system
up
∂u
= a1 ∆u − µ1 u + q + σ, in R+ × Ω,
∂t
v
(1.5)
ur
∂v
= a2 ∆v − µ2 v + s , in R+ × Ω,
∂t
v
N
on a bounded Ω ⊂ R , with the homogeneous Neumann boundary conditions and
positive initial data: a1 , a2 , µ1 , µ2 and σ are positive constants, and p, q, r, s are non
negative constants satisfying the relation
p−1
q
<
.
r
s+1
The existence of global solutions to the system (1.5) is proved by Rothe [11] with
special cases N = 3, p = 2, q = 1, r = 2 and s = 0. The Rothe’s method can not
be applied (at least directly) to general p, q, r, s. Wu and Li [16] obtained the same
results for the problem (1.5) so long as u, v −1 and σ are suitably small. Li, Chen
and Qin [7] showed that the solutions of this problem are bounded all the time for
each pair of initial values in L∞ (Ω) if
p−1
q
< min 1,
.
(1.6)
r
s+1
Masuda and Takahashi [8] considered the generalized Gierer-Meinhardt system
∂ui
= ai ∆ui − µi ui + gi (x, u1 , u2 ), in R+ × Ω (i = 1, 2),
∂t
where ai , µi , i = 1, 2 are positive constants, and
(1.7)
up1
+ σ1 (x),
uq2
ur
g2 (x, u1 , u2 ) = ρ2 (x, u1 , u2 ) 1s + σ2 (x),
u2
g1 (x, u1 , u2 ) = ρ1 (x, u1 , u2 )
with σ1 (·) (resp. σ2 (·)) is a positive (resp. non-negative) C 1 function on Ω̄, and ρ1
(resp. ρ2 ) is a non negative (resp. positive) bounded and C 1 function on Ω̄ × R2+ .
They extended the result of global existence of solutions for (1.7) of Li, Chen and
Qin [7] to
2
p−1
<
,
(1.8)
r
N +2
EJDE-2015/94
and
UNIFORM BOUNDEDNESS OF SOLUTIONS
3
ϕ1 , ϕ2 ∈ W 2,l (Ω),
l > max{N, 2},
(1.9)
∂ϕ1
∂ϕ2
=
= 0 on ∂Ω and ϕ1 ≥ 0, ϕ2 > 0 in Ω̄.
∂η
∂η
Jiang [6] obtained the same results as Masuda and Takahashi [8] by another
method such that (1.6) and (1.9) are satisfied.
Abdelmalek, et al [1] considered the following Gierer-Meinhardt system of three
equations
up1
∂u
+ σ, in R+ × Ω,
− a1 ∆u = −b1 u + q1 r1
∂t
v (w + c)
∂v
up2
(1.10)
− a2 ∆v = −b2 v + q2 r2 , in R+ × Ω,
∂t
v w
∂w
u p3
− a3 ∆w = −b3 w + q3 r3 , in R+ × Ω,
∂t
v w
with homogeneous Neumann boundary conditions
∂u
∂v
∂w
=
=
= 0 on R+ × ∂Ω,
(1.11)
∂η
∂η
∂η
and the initial data
u(0, x) = ϕ1 (x) > 0,
v(0, x) = ϕ2 (x) > 0,
(1.12)
w(0, x) = ϕ3 (x) > 0
in Ω, and ϕi ∈ C(Ω̄) for all i = 1, 2, 3. Under the condition (1.4) and by using a
suitable Lyapunov functional, they studied the global existence of solutions for the
system (1.10)–(1.12). Their method gave only the existence of global solutions, and
they did not obtain results about the uniform boundedness of solutions on (0, +∞).
For the asymptotic behavior of the solutions, Wu and Li [16] considered the
system
∂u1
up
= a1 ∆u1 − u1 + 1q + σ1 (x), in R+ × Ω,
∂t
u2
(1.13)
∂u2
ur
τ
= a2 ∆u2 − u2 + 1s + σ2 (x), in R+ × Ω,
∂t
u2
with the constant of relaxation time τ > 0, and they proved that if σ1 ≡ σ2 ≡ 0
q
and τ > p−1
, then (u(t, x), v(t, x)) → (0, 0) uniformly on Ω̄ as t → +∞.
Under suitable conditions on τ and on the initial data, Suzuki and Takagi [12,13]
also studied the behavior of the solutions for (1.13) with the constant of relaxation
time τ .
We first treat the uniform boundedness of the solutions for Gierer-Meinhardt
system of three equations by proving that the Lyapunov function argument proposed in [1] can be adapted to our situation. Interestingly, we show that the same
Lyapunov function satisfies a differential inequality from which the uniform boundedness of the solutions is deduced for any positive time. Then under reasonable
conditions on the coefficients b1 , b2 and b3 , and by using the uniform boundedness
of the solutions and the Lyapunov function which is non-increasing function, we
deal with the long-time behavior of solutions as the time goes to +∞. In particular
we are concerned with σ1 ≡ 0, σ2 and σ3 are non-negative constants to assure that
σ3
σ2
lim ku(t, .)k∞ = lim kv(t, .) − k∞ = lim kw(t, .) − k∞ = 0 .
t→+∞
t→+∞
t→+∞
b2
b3
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S. HENINE, S. ABDELMALEK, A. YOUKANA
EJDE-2015/94
2. Notation and preliminary results
2.1. Existence of local solutions. For i = 1, 2, 3 we set
ϕi = minϕi (x), ϕ̄i = maxϕi (x),
x∈Ω̄
x∈Ω̄
¯
ρi = min ρi (x, ξ), ρ¯i = max ρi (x, ξ),
x∈Ω̄,ξ∈R3+
x∈Ω̄,ξ∈R3+
¯
σi = minσi (x), σ̄i = maxσi (x).
x∈Ω̄
x∈Ω̄
¯
The basic existence theory for abstract semi linear differential equations directly
leads to a local existence result to system (1.1)–(1.3) (see, Henry [5]). All solutions
are classical on (0, T ) × Ω, T < Tmax , where Tmax (ku0 k∞ , kw0 k∞ ) denotes the
eventual blowing-up time in L∞ (Ω).
2.2. Positivity of solutions.
Lemma 2.1. If (u, v, w) is a solution of the problem (1.1)–(1.3), then for all (t, x) ∈
(0, Tmmax ) × Ω, we have
(1)
u(t, x) ≥ e−b1 t ϕ1 > 0,
v(t, x) ≥ e−b2 t ϕ2 > 0,
w(t, x) ≥ e−b3 t ϕ3 > 0.
(2)
u(t, x) ≥ min σ1 /b1 , ϕ1 = m1 ,
v(t, x) ≥ min σ2 /b2 , ϕ2 = m2 ,
w(t, x) ≥ min σ3 /b3 , ϕ3 = m3 .
The proof of the above lemma follows immediate from the maximum principle.
3. Boundedness of solutions
For proving the existence of global solutions for (1.1)–(1.3), it suffices to prove
that the solutions remains bounded in (0, T ) × Ω̄. One of the main results of this
paper reads as follows.
Theorem 3.1. Assume that (1.4) holds. Let (u, v, w) be a solution to (1.1)–(1.3),
and let
Z
uα (t, x)
L(t) =
dx, for all t ∈ (0, T ),
(3.1)
β
γ
Ω v (t, x)w (t, x)
where α, β and γ are positive constants satisfying the following conditions:
3b + b 1
(a1 + a2 )2
2
3
α > 2 max 1,
,
>
,
(3.2)
b1
β
2a1 a2
and
1
2β
−
(a1 + a2 )2 1 (a1 + a3 )2 (α − 1)(a2 + a3 ) (a1 + a2 )(a1 + a3 ) 2
p
−
>
−
.
√
4a1 a2
2γ
4a1 a3
2α a2 a3
4 a21 a2 a3
(3.3)
EJDE-2015/94
UNIFORM BOUNDEDNESS OF SOLUTIONS
5
Then there exists a positive constant C such that for all t ∈ (0, T ),
d
L(t) ≤ −(αb1 − 3b2 β − γb3 )L(t) + C.
dt
(3.4)
Corollary 3.2. Under the assumptions of Theorem 3.1, all solutions of (1.1)–(1.3)
with positive initial data in C(Ω̄) are global and uniformly bounded on (0, +∞) × Ω̄.
Before proving the above theorem we first need the following technical lemma.
Lemma 3.3. Suppose that x > 0, y > 0 and z > 0, then for each group of indices
r, p, q, δ, θ, λ and ξ satisfies λ < p < δ (not necessarily positive), and any constant
Λ > 0, we have
xδ
xλ
xp
− p−λ
δ−p
,
≤
Λ
+
Λ
yq zr
yθ zξ
y η1 z η2
where
η1 = [q(δ − λ) − θ(p − λ)](δ − p)−1 ,
η2 = [r(δ − λ) − ξ(p − λ)](δ − p)−1 .
Proof. We can write
δ(p−λ) θ(p−λ) ξ(p−λ) λ(δ−p) θ(p−λ)
ξ(p−λ)
xp
x δ−λ y δ−λ −q z δ−λ −r .
= x δ−λ y − δ−λ z − δ−λ
q
r
y z
By using Young’s inequality we obtain
p−λ
xδ
xλ
xp
≤ ε θ ξ + ε− δ−p η1 η2 ,
q
r
y z
y z
y z
where
η1 = [q(δ − λ) − θ(p − λ)](δ − p)−1 ,
η2 = [r(δ − λ) − ξ(p − λ)](δ − p)−1 .
Then Lemma 3.3 is proved.
Proof of Theorem 3.1. Let (u, v, w) be the solution of system (1.1)–(1.3) in (0, T ).
Differentiating L(t) respect to t, we obtain L0 (t) = I + J, where
Z α−1
Z
Z
u
uα
uα
I = a1 α
∆u
dx
−
a
β
∆v
dx
−
a
γ
∆w dx,
2
3
β γ
β+1 w γ
β γ+1
Ωv w
Ω v
Ω v w
Z
uα−1+p1
J = (−αb1 + βb2 + γb3 )L(t) + α ρ1 (x, u, v, w) β+q1 γ+r1 dx
v
w
Ω
Z
Z
α+p2
u
uα+p3
− β ρ2 (x, u, v, w) β+1+q2 γ+r2 dx − γ ρ3 (x, u, v, w) β+q3 γ+1+r3 dx
v
v
w
w
Ω
Ω
Z
Z
Z
α−1
α
α
u
u
u
+ α σ1 (x) β γ dx − β σ2 (x) β+1 γ dx − γ σ3 (x) β γ+1 dx.
v
w
v
w
v
w
Ω
Ω
Ω
Using Green’s formula, for all t ∈ (0, T ), we obtain (see [1])
I ≤ 0.
(3.5)
6
S. HENINE, S. ABDELMALEK, A. YOUKANA
EJDE-2015/94
Now let us estimate the term J. For all t ∈ (0, T ) we have
Z
Z
uα+p2
uα−1+p1
dx
−
βρ
dx
J ≤ − αb1 + βb2 + γb3 L(t) + αρ¯1
2
β+q1 w γ+r1
β+1+q2 w γ+r2
Ωv
Ωv
Z
Z
uα−1
uα+p3
− ρ3 γ
dx
+
α
σ
¯
dx.
1
β+q3 w γ+1+r3
β γ
Ωv w
Ωv
(3.6)
Applying Lemma 3.3 with p = α − 1, q = θ = β, r = γ, δ = α, ξ = γ and λ = 0,
one gets
Z
Z
Z α−1
uα
1
u
dx
≤
βb
dx
+
C
dx,
(3.7)
ασ¯1
2
1
β
γ
β
γ
β
γ
Ωv w
Ωv w
Ωv w
where C1 = ασ¯1 ( αβbσ¯21 )1−α .
Now, we choose 1 ∈ (0, α) such that
q1 p2 − (p1 − 1)(1 + q2 )
q1 − 1 − q2
+α
≥ 0,
1 (p2 + 1 − p1 )
p2 + 1 − p1
r1 − r2
r1 p2 − r2 (p1 − 1)
+α
≥ 0.
γ+α
1 (p2 − p1 + 1)
p2 − p1 + 1
β+α
Again, applying Lemma 3.3 for p = α − 1 + p1 , q = β + q1 , r = γ + r1 , δ = α + p2 ,
θ = β + 1 + q2 , ξ = γ + r2 and λ = α − 1 , we obtain
Z
Z
Z
uα−1+p1
up2 +α
uα−1
αρ¯1
dx
≤
βρ
dx
+
C
dx,
(3.8)
2
2
q1 +β w r1 +γ
q2 +β+1 w r2 +γ
η1 η2
Ωv
Ωv
Ωv w
where
η1 = β + [q1 p2 − (q2 + 1)(p1 − 1) + 1 (q1 − q2 − 1)](p2 − p1 + 1)−1 ,
η2 = γ + [r1 p2 − r2 (p1 − 1) + 1 (r1 − r2 )](p2 − p1 + 1)−1 ,
βρ2
p1 −1+1
and C2 = αρ¯1 ( αρ¯1 )− p2 −p1 +1 .
In an analoguous way, we have
Z
Z
Z α−1
uα
1
u
dx
≤
b
β
dx
+
C
dx,
C2
2
3
η
η
β
γ
η
1
2
3 η4
Ωv w
Ωv
Ω v
(3.9)
where
−1
η3 = β + α[−1
≥ 0,
1 (q1 p2 − (q2 + 1)(p1 − 1)) + q1 − q2 − 1](p2 − p1 + 1)
−1
η4 = γ + α[−1
≥ 0,
1 (r1 p2 − r2 (p1 − 1)) + r1 − r2 ](p2 − p1 + 1)
α−1
and C3 = C2 ( bC22β )− 1 .
Or, we choose 2 ∈ (0, α) such that
q1 p3 − q3 (p1 − 1)
q1 − q3
+α
≥ 0,
2 (p3 − p1 + 1)
p3 − p1 + 1
r1 − r2 − 1
r1 p3 − (r3 + 1)(p1 − 1)
γ+α
+α
≥ 0.
2 (p3 − p1 + 1)
p3 − p1 + 1
β+α
Now, applying Lemma 3.3 with p = p1 + α − 1, q = q1 + β, r = r1 + γ, δ = p3 + α,
θ = q3 + β, ξ = r3 + γ + 1 and λ = α − 2 , we find that
Z
Z
Z
uα−1+p1
uα+p3
uα−2
αρ¯1
dx
≤
γρ
dx
+
C
dx,
(3.10)
3
4
β+q1 w γ+r1
β+q3 w γ+1+r3
η5 η6
Ωv
Ωv w
Ωv
EJDE-2015/94
UNIFORM BOUNDEDNESS OF SOLUTIONS
7
where
η5 = β + [q1 p3 − q3 (p1 − 1) + 2 (q1 − q3 )](p3 − p1 + 1)−1 ,
η6 = γ + [r1 p3 − (r3 + 1)(p1 − 1) + 2 (r1 − r3 − 1)](p3 − p1 + 1)−1 ,
γρ3
p1 −1+2
and C4 = αρ¯1 ( αρ¯1 )− p3 −p1 +1 . In the same way, we obtain
Z
Z
Z
uα
1
uα−2
dx
≤
b
β
dx,
dx
+
C
C4
2
5
η
η8
η
β
γ
η
6
5
7
Ωv w
Ωv w
Ωv w
where
(3.11)
−1
η7 = β + α[−1
≥ 0,
2 (q1 p3 − q3 (p1 − 1)) + q1 − q3 ](p3 − p1 + 1)
−1
η8 = γ + α[−1
≥ 0,
2 (r1 p3 − (r3 + 1)(p1 − 1)) + r1 − r3 − 1](p3 − p1 + 1)
α−2
and C5 = C4 ( bC24β )− 2 .
From (3.6)–(3.11) there exists a positive constant C such that
L0 (t) ≤ −(b1 α − 3βb2 − γb3 )L(t) + C,
∀t ∈ (0, T ).
Then the proof is complete.
Proof of Corollary 3.2. Since
C
for all t ∈ (0, T ),
αb1 − 3b2 β − γb3
then there exist non-negative constants C6 , C7 and C8 independent of t such that
L(t) ≤ L(0) +
kf (u, v, w) − b1 ukN ≤ C6 ,
kg(u, v; w) − b2 vkN ≤ C7 ,
kh(u, v, w) − b3 wkN ≤ C8 .
Since (ϕ1 , ϕ2 , ϕ3 ) ∈ (C(Ω̄))3 , we conclude from the Lp -Lq -estimate (see Henry
[5], Haraux and Kirane [4]) that
u ∈ L∞ ((0, T ), L∞ (Ω)),
v ∈ L∞ ((0, T ), L∞ (Ω)),
w ∈ L∞ ((0, T ), L∞ (Ω)).
Finally, we deduce that the solutions of the system (1.1)–(1.3) are global and uniformly bounded on (0, +∞) × Ω̄.
Remark 3.4. It is clear that the results of this section are valid when σ1 ≡ σ2 ≡
σ3 ≡ 0.
4. Asymptotic behavior of the solutions
In this section, we study the asymptotic behavior of the solutions for the system
∂u
− a1 ∆u = −b1 u + f (u, v, w), in R+ × Ω,
∂t
∂v
(4.1)
− a2 ∆v = −b2 v + g(u, v, w), in R+ × Ω,
∂t
∂w
− a3 ∆w = −b3 w + h(u, v, w), in R+ × Ω,
∂t
where
up1
+ σ1 ,
f (u, v, w) = ρ1 (x, u, v, w) q1 r1
v (w + c)
8
S. HENINE, S. ABDELMALEK, A. YOUKANA
EJDE-2015/94
up2
+ σ2 ,
v q 2 w r2
p3
u
h(u, v, w) = ρ3 (x, u, v, w) q3 r3 + σ3 ,
v w
with homogeneous Neumann boundary conditions
g(u, v, w) = ρ2 (x, u, v, w)
∂u
∂v
∂w
=
=
=0
∂η
∂η
∂η
on R+ × ∂Ω,
(4.2)
and initial data
u(0, x) = ϕ1 (x),
v(0, x) = ϕ2 (x),
w(0, x) = ϕ3 (x)
in Ω.
(4.3)
Here σ1 , σ2 and σ3 are non negative constants.
Before stating the results, let us expose some simple facts concluded from the
result of the previous section. From Theorem 3.1, and by using classical method
of a semi group and a power fractional (see [5]) we can find the positive constants
M1 , M2 and M3 explicitly (see [9]) such that
ku(t, .)k∞ M1 ,
kv(t, .)k∞ ≤ M2 ,
kw(t, .)k∞ ≤ M3 .
Let us consider the same function as in Theorem 3.1,
Z
uα (t, x)
dx, ∀t ∈ (0, +∞),
L(t) =
β
γ
Ω v (t, x)w (t, x)
where α, β and γ are positive constants satisfying the following conditions
α > 2 max(1,
and
1
2β
−
(a1 + a2 )2
1
>
,
β
2a1 a2
3b2 + b3
),
b1
(a1 + a2 )2 1 (a1 + a3 )2 (α − 1)(a2 + a3 ) (a1 + a2 )(a1 + a3 ) 2
p
−
>
−
.
√
4a1 a2
2γ
4a1 a3
2α a2 a3
4 a21 a2 a3
The main result in this section reads as follows.
Theorem 4.1. Assume (1.4) holds. Let (u, v, w) be the solution of (4.1)–(4.3) in
(0, +∞). Suppose that σ1 = 0, and
b1 >
βb2 + γb3 + K
,
2
(4.4)
where
p1 −1
βρ2
K=
αρ¯1 ( αρ¯1 )− p2 −p1 +1
[q p2 −(q2 +1)(p1 −1)](p2 −p1 +1)−1
m2 1
[r p2 −r2 (p1 −1)](p2 −p1 +1)−1
m3 1
,
or
p1 −1
γρ3
K=
αρ¯1 ( αρ¯1 )− p3 −p1 +1
[q p −q (p −1)](p −p1 +1)−1
3
m2 1 3 3 1
Then for all t ∈ (0, +∞) we have
ϕα
1 (x)
dx.
β
γ
Ω ϕ2 (x)ϕ3 (x)
Z
L(t) ≤
[r p3 −(r3 +1)(p1 −1)](p3 −p1 +1)−1
m3 1
.
EJDE-2015/94
UNIFORM BOUNDEDNESS OF SOLUTIONS
9
Corollary 4.2. Under the assumptions of Theorem 4.1, for all positive initial data
in C(Ω̄) we have
ku(t, .)k∞ → 0 as t → +∞,
σ2
kv(t, .) − k∞ → 0 as t → +∞,
b2
σ3
kw(t, .) − k∞ → 0 as t → +∞.
b3
Proof of Theorem 4.1. From (3.5) and (3.6), we obtain for all t ∈ (0, +∞)
Z
uα−1+p1
dx
L0 (t) ≤ −(αb1 − βb2 − γb2 )L(t) + αρ¯1
β+q
1 w γ+r1
Ωv
Z
Z
uα+p2
uα+p3
− βρ2
dx
−
γρ
dx.
3
β+1+q
γ+r
β+q
2
2
3 w γ+1+r3
w
Ωv
Ωv
(4.5)
Now, we apply Lemma 3.3 for p = α − 1 + p1 , q = β + q1 , r = γ + r1 , δ = α + p2 ,
θ = β + 1 + q2 , ξ = γ + r2 and λ = α we obtain
Z
Z
Z
uα−1+p1
uα+p2
uα
αρ¯1
dx
≤
βρ
dx
+
A
dx,
(4.6)
2
1
β+q1 w γ+r1
β+1+q2 w γ+r2
η9 η10
Ωv
Ωv
Ωv w
where
η9 = β + [q1 p2 − (q2 + 1)(p1 − 1)](p2 − p1 + 1)−1 > 0,
η10 = γ + [r1 p2 − r2 (p1 − 1)](p2 − p1 + 1)−1 > 0,
βρ2
p1 −1
and A1 = αρ¯1 ( αρ¯1 )− p2 −p1 +1 .
Or, applying Lemma 3.3 for p = α − 1 + p1 , q = β + q1 , r = γ + r1 , δ = α + p3 ,
θ = β + q3 , ξ = γ + 1 + r3 and λ = α, we obtain
Z
Z
Z
uα−1+p1
uα+p3
uα
αρ¯1
dx
≤
γρ
dx
+
A
dx,
(4.7)
3
2
β+q1 w γ+r1
β+q3 w γ+1+r3
η11 w η12
Ωv
Ωv
Ωv
where
η11 = β + [q1 p3 − q3 (p1 − 1)](p3 − p1 + 1)−1 > 0,
η12 = γ + [r1 p3 − (r3 + 1)(p1 − 1)](p3 − p1 + 1)−1 > 0,
γρ3
p1 −1
and A2 = αρ¯1 ( αρ¯1 )− p3 −p1 +1 . By combining (4.5) with (4.6) and (4.7) we obtain
L0 (t) ≤ −(αb1 − βb2 − γb3 − K)L(t),
∀t ∈ (0, +∞),
(4.8)
where
p1 −1
βρ2
K=
αρ¯1 ( αρ¯1 )− p2 −p1 +1
[q p2 −(q2 +1)(p1 −1)](p2 −p1 +1)−1
m2 1
[r p2 −r2 (p1 −1)](p2 −p1 +1)−1
m3 1
,
or
p1 −1
γρ3
K=
αρ¯1 ( αρ¯1 )− p3 −p1 +1
[q p3 −q3 (p1 −1)](p3 −p1 +1)−1
m2 1
[r p3 −(r3 +1)(p1 −1)](p3 −p1 +1)−1
m3 1
.
Using (4.4) we deduce that the function t 7−→ L(t) is a non-increasing function.
This completes the proof of Theorem 4.1.
10
S. HENINE, S. ABDELMALEK, A. YOUKANA
EJDE-2015/94
Proof of Corollary 4.2. Setting for all (t, x) ∈ (0, +∞) × Ω:
h1 (t, x) = u(t, x),
σ2
h2 (t, x) = v(t, x) − ,
b2
σ3
h3 (t, x) = w(t, x) − .
b3
For i = 1, 2, 3 we have
upi
dhi
(4.9)
− ai ∆hi = −bi hi + ρi (x, u, v, w) qi ri .
dt
v w
Multiplying (4.9) by hi (t, x), i = 1, 2, 3 and integrating over [0, t] × Ω we obtain
Z
Z tZ
Z tZ
1
2
2
h dx + ai
|∇hi | dx ds + bi
h2i dx ds
2 Ω i
0 Ω
0 Ω
Z
Z tZ
1
upi
2
hi (0)dx +
=
hi ρi (x, u, v) qi ri dx ds.
2 Ω
v w
0 Ω
From (4.8), for all t ∈ (0, +∞), and for i = 1, 2, 3 we obtain
Z tZ
Z Z
uα
upi
M pi M β M γ t
dx ds < +∞.
hi ρi (x, u, v) qi ri dx ds ≤ ρ¯i Mi 1qi α2 r3i
β
v w
m2 m1 m3 0 Ω v wγ
0 Ω
One obviously deduces that for i = 1, 2, 3,
Z +∞ Z
hi (t, .) ∈ L2 (Ω),
|∇hi |2 dxds < +∞,
0
Ω
Z +∞ Z
2
hi dx ds < +∞,
0
Ω
so that Barbalate’s lemma [3, Lemma 1.2.2] permits to conclude that
lim khi (t, .)k2 = 0,
t→+∞
i = 1, 2, 3.
On the other hand, since the orbits {hi (t, .)/t ≥ 0, i = 1, 2, 3} are relatively compact
in C(Ω̄) (see [4]), it follows readily that
lim khi (t, .)k∞ = 0,
t→+∞
Then proof of Corollary 4.2 is complete.
i = 1, 2, 3.
Acknowledgments. The authors want to thank Prof. M. Kirane and the anonymous referee for their suggestions that improved the quality of this article.
References
[1] S. Abdelmalek, H. Louafi, A. Youkana; Global Existence of Solutions for Gierer-Meinhardt
System with Three Equations. Electronic Journal of Differential Equations, Vol. 2012 (2012),
No, 55. pp. 1-8.
[2] A. Gierer, H. Meinhardt; A Theory of Biological Pattern Formation. Kybernetik, 1972, 12:3039.
[3] K. Gopalsamy; Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74 of Mathematics and its Applications, Kliwer Academic Publishers, Dordrecht,
the Netherlands, 1992.
[4] A. Haraux, M. Kirane; Estimations C 1 pour des problèmes paraboliques semi-linéaires. Ann.
Fac. Sci. Toulouse 5(1983), 265-280.
[5] D. Henry; Geometric Theory of Semi-linear Parabolic Equations. Lecture Notes in Mathematics 840, Springer-Verlag,New-York, 1984.
EJDE-2015/94
UNIFORM BOUNDEDNESS OF SOLUTIONS
11
[6] H. Jiang; Global existence of Solutions of an Activator-Inhibitor System, Discrete and continuous Dynamical Systems. V14,N4 April 2006.p 737-751.
[7] M. Li, S. Chen, Y. Qin; Boundedness and Blow Up for the general Activator-Inhibitor Model,
Acta Mathematicae Applicarae Sinica, vol. 11 No.1. Jan, 1995.
[8] K. Masuda, K. Takahashi; Reaction-Diffusion Systems in the Gierer-Meinhardt theory of
biological pattern formation. Japan J. Appl. Math., 4(1):47-58, 1987.
[9] L. Melkemi, A. Z. Mokrane, A. Youkana; Boundedness and Large-Time Behavior Results
for a Diffusive Epidemic Model, Journal of Applied Mathematics, Volume 2007, Article ID
17930, 15 pages.
[10] P. Quitner, Ph. Souplet; Superlinear Parabolic Problems. Blow-up, Global Existence and
Steady States. Birkhäuser Verlag AG, Basel. Boston. Berlin (2007).
[11] F. Rothe; Global Solutions of Reaction-Diffusion Equations. Lecture Notes in Mathematics,
1072, Springer-Verlag, Berlin, (1984).
[12] K. Suzuki, I. Takagi; Behavior of Solutions to an Activator-Inhibitor System with Basic
Production Terms, Proceeding of the Czech-Japanese Seminar in Applied Mathematics 2008.
[13] K. Suzuki and I. Takagi; On the Role of Basic Production Terms in an Activator-Inhibitor
System Modeling Biological Pattern Formation, Funkcialaj Ekvacioj, 54 (2011) 237-274.
[14] A. Trembley; Mémoires pour servir à l’histoire d’un genre de polypes d’eau douce, abras en
forme de cornes. 1744.
[15] A. M. Turing; The chemical basis of morphogenesis. Philosophical Transactions of the Royal
Society (B), 237: 37-72, 1952.
[16] J. Wu and Y. Li; Global Classical Solution for the Activator-Inhibitor Model. Acta Mathematicae Applicatae Sinica (in Chinese), 1990, 13: 501-505.
Safia Henine
Department of Mathematics, University of Batna 05000, Algeria
E-mail address: henine.safia@yahoo.fr
Salem Abdelmalek
Department of Mathematics, College of Sciences, Yanbu Taibah University, Saudi Arabia.
Department of Mathematics, University of Tebessa 12002, Algeria
E-mail address: sabdelmalek@taibahu.edu.sa
Amar Youkana
Department of Mathematics, University of Batna 05000, Algeria
E-mail address: youkana amar@yahoo.fr