Global existence of solutions for some coupled systems of reaction-diffusion equations

0.09 MB·English

by  Abdelmalek Salem

Global Existence of Solutions for Some
1
1 Coupled Systems of Reaction-Diffusion
0
2
Abdelmalek Salem
n
a
J
Department of mathematics
9
2 University of Tebessa, 12002 Algeria
[email protected]
]
P
A
Youkana Amar
.
h
t
a Department of Mathematics
m
University of Batna, 5000 Algeria
[ [email protected]

2
v
Abstract
7
8
6 The aim of this work is to study the global existence of solutions for
5 some coupled systems of reaction diffusion which describe the spread
.
9 within a population of infectious disease. We consider a triangular ma-
0
trixdiffusionandweshowthatwecanproveglobalexistenceofclassical
0
1 solutions for the nonlinearities of weakly exponential growth.
:
v
i Mathematics Subject Classification: 35K57, 35K57
X
r
a Keywords: Reaction-Diffusionsystems, GlobalExistence, LyapunovFunc-
tional
1 Introduction
Inthis workwe shall beconcerned witha reaction-diffusionsystem oftheform:
du a∆u = Λ λ(t)f(u,v) µu in R+ Ω
dt − − − × (1.1)
dv b∆u d∆v = λ(t)f(u,v) µv in R+ Ω
(cid:26) dt − − − ×
with boundary conditions
∂u ∂v
= = 0 on R+ ∂Ω (1.2)
∂η ∂η ×
426 S. Abdelmalek and A. Youkana
and the nonnegative continuous and bounded initial data
−
u(0,x) = u (x), v(0,x) = v (x) in Ω (1.3)
0 0
where Ω is a bounded domain of class C1 in Rn, with boundary ∂Ω, ∂ is the
∂η
outward normal derivative to ∂Ω.
The constants a,b,d, Λ, µ are such that
a > 0, b > 0, d a b, µ > 0, and Λ 0. (H.1)
− ≥ ≥
We assume that t λ(t) is a nonnegative and bounded function in C(R+)
→
with 0 λ(t) λ and nonlinearity f is a nonnegative continuously differen-
≤ ≤
tiable function on (0,+ ) (0,+ ) satisfying
∞ × ∞
b
log(1+f(.,η))
f (0,η) = 0 for all η in R+ and lim = 0. (H.2)
η→+∞ η
The reaction-diffusion system (1.1) (1.3) may be viewed as a diffusive
−
epidemic model where u and v represent the nondimensional population den-
sities of susceptibles and invectives, respectively. We can consider the system
(1.1) (1.3) as a model describing the spread of an infection disease (such as
−
AIDS for instance) within a population assumed to be divided into the suscep-
tible and infective classes as precised (for further motivation see for instance
[2, 3] and the references therein).
We note that in the case Λ = µ = 0, this problem corresponds to the
problem of R. H.Martin and much works have been done in the literature and
positive answers have been under different forms for subgrowth nonlinearity
(se for instance [1, 9, 5, 7, 8, 11]).
We mentioned that in the case Λ > 0, it is not obvious to obtain global
existence of solutions when the nonlinearity are at most polynomial growth
and the methods as above can not be applied.
Our mainpurpose is to study the global existence of solutions of thesystem
(1.1) (1.3) with nonlinearities of weakly exponential growth.
−
2 Preliminaries results
2.1 Local existence of solutions.
We denote by Wm,p(Ω), the Sobolev space of order m 0 for 1 p
≥ ≤ ≤
− −
+ and by C(Ω) the Banach space of continuous functions on Ω.
∞
We can convert equations (1.1) (1.3) to an abstract first order system in
−
− −
the Banach space X = C(Ω) C(Ω) of the form
×
dU(t) = A˘U(t)+F(U(t))
dt t > 0,
U(0) = U X
(cid:26) 0 ∈
Reaction-diffusion 427
where
A˜ : D (A) D (B) X,
∞ ∞
× →
with
− ∂z
D (A) = D (B) = z W2,p(Ω) for all p > n, ∆z C(Ω), = 0 ,
∞ ∞
∈ ∈ ∂η
(cid:26) (cid:27)
A˜U = (Au,Bv)
and
F(U) = (Λ λf (u,v);λf (u,v)).
−
Itisclearthatfromthegeneraltheoryofsemigroupwededucetheexistence
of an unique local classical solution in some interval ]0,T∗[, where T∗ is the
eventual blowing-up time in L∞(Ω) (see for example Henry [6] or Pazy [12]).
2.2 Positivity of solutions.
From the nonnegativity of the initial data u on Ω, one easily deduce from the
0
maximum applied to the first equation of (1.1) that the component u remains
nonnegative and bounded on (0,T∗) Ω so that
×
Λ
0 u(t,x) K := max( u , ).
≤ ≤ k 0k∞ µ
In order to obtain the positivity of v we assume that
Λ
u . (H.3)
k 0k∞ ≤ µ
Lemma 1 Let (u,v) be the solution of (1.1) (1.3).
−
If the initial data v satisfies the condition
0
b Λ
v ( u )
0 ≥ d a µ −k 0k∞
−
then for all (t,x) (0,T∗) Ω we have
∈ ×
b Λ
v(t,x) ( u(t,x)).
≥ d a µ −
−
Proof. Let us consider w = v b (Λ u) and in this way the system
− d−a µ −
(1.1) (1.3) may be equivalent to the system
−
du a∆u = Λ λf(u,v) µu
dt − − − in R+ Ω (2.1)
dw d∆w = (1 b )λf(u,v) µw ×
(cid:26) dt − − d−a −
428 S. Abdelmalek and A. Youkana
with
u(0,.) = u (.) 0
0 ≥ in Ω. (2.2)
w(0,.) = w (.) 0
0
(cid:26) ≥
Using (H.1) and maximum principle to the second equation of the system
(2.1) we obtain
w(t,x) 0 (t,x) (0,T∗) Ω
≥ ∀ ∈ ×
that means
b Λ
v(t,x) ( u(t,x)) 0 for all (t,x) (0,T∗) Ω. (2.3)
≥ d a µ − ≥ ∈ ×
−
Now we are able to state our main result.
3 Statement and proof of the main results
Using the idea of Haraux and Youkana [5], let us consider the functional
J(t) = 1+δ(1+u+u2 eεwdx
ZΩ
(cid:0) (cid:1)
where δ and ε are constants such that
2
µ 2√ab 1
0 < δ min ,2 (3.1)
≤ 2Λ(1+2K) a+b · 1+2K 
!
 
and
δ d a
0 < ε min 1, − . (3.2)
≤ 1+δ(K +K2) b
(cid:18) (cid:19)
Our main results are as follows:
Theorem 2 Let (u,v)be a solution of (1.1) (1.3) on (0,T∗),then there exist
−
a positive constant γsuch that for all t (0,T∗) the functional
∈
J(t) = 1+δ(1+u+u2 eεvdx (3.3)
ZΩ
(cid:0) (cid:1)
satisfies the relation
d µ
J(t) J(t)+γ. (3.4)
dt ≤ −2
Corollary 3 Under the hypothesis (H.1) (H.3), the solutions of the parabolic
−
system (1.1) (1.3) with nonnegative and bounded initial data u ,v are global
0 0
−
and uniformly bounded in (0,+ ) Ω.
∞ ×
Reaction-diffusion 429
Proof. [Proof of the theorem] By simple use of Green’s formula and from
equations (1.1) (1.3) we have for all t (0,T∗)
− ∈
d
J(t) = G+H +H +H
1 2 3
dt
where
G = δ [2a+bε(1+2u)]eεv u 2dx
− Ω |∇ |
ε [(a+d)δ(1+2u)+bε(1+δ(u+u2))]eεv u vdx
−R Ω ∇ ∇
dε2 [1+δ(u+u2)]eεv v 2dx
− R Ω |∇ |
and R
H = (Λ δ(1+2u) µu δ(1+2u) µ)(1+δ(u+u2)eεvdx
1 Ω 1+δ(u+u2) − 1+δ(u+u2) −
H2 = RΩ ε− 1+δ(δ1(+u+2uu)2) λf (u,v)(1+δ(u+u2))eεvdx
H = µ (cid:16)(1 εv)eεv(1(cid:17)+δ(u+u2)dx.
3 R Ω −
We observe thRat G involves a quadratic form with respect to u and v
∇ ∇
Q = δ(2a+bε(1+2u))eεv u 2
|∇ |
+ε[(a+d)δ(1+2u)+bε(1+δ(u+u2))]eεv u v
∇ ∇
+dε2(1+δ(u+u2))eεv v 2
|∇ |
and the discriminant
D = [ε[(a+d)δ(1+2u)+bε(1+δ(u+u2))]]2
−
4[δ(2a+bε(1+2u))][dε2(1+δ(u+u2))],
may be non-positive since the constants δ and ε satisfies (3.1) and (3.2).
Consequently
G 0 for all t in (0,T∗).
≤
Concerning the terms H , i = 1,2,3 we have again from (3.1) (3.2) where
i
−
now
µ δ
0 < δ , 0 < ε , (3.5)
≤ 2Λ(1+2K) ≤ 1+δ(K +K2)
one checks that
δ(1+2u) δ(1+2u) µ
Λ µu µ 2Λδ(1+2K) µ , (3.6)
1+δ(u+u2) − 1+δ(u+u2) − ≤ − ≤ −2
and
δ(1+2u) δ
ε ε 0, (3.7)
− 1+δ(u+u2) ≤ − 1+δ(K +K2) ≤
from we deduce
µ
H J(t)
1
≤ −2
430 S. Abdelmalek and A. Youkana
and
H 0.
2
≤
Now for the term H ,one observe that the function
3
π : η (1 εη)eεη,
→ −
is bounded on R+. Indeed, one has
dπ
(η) = ε2ηeεη 0,
dη − ≤
so that π is nonincreasing in [0,+ [ and
∞
max(1 εη)eεη = 1.
η≥0 −
In this way one can deduce that there a positive constant γ
γ = µ 1+δ K +K2 Ω
| |
such that (cid:0) (cid:0) (cid:1)(cid:1)
H γ. (3.8)
3
≤
Aditioning G, H , H and H we get
1 2 3
d µ
J(t) = G+H +H +H J (t)+γ. (3.9)
dt 1 2 3 ≤ −2
Thus, the proof of theorem is complete.
Now we are able to prove the global existence for the solutions of (1.1)
−
(1.3).
Proof. [Proof of the corollary] From (H.2) we see that there exist a positive
consant C such that
v
1+f(.,v) Cen, v 0. (3.10)
≤ ∀ ≥
where ε is chosen as in (3.2). By virtue of (3.9), it is seen that
J(t) C, t (0,T∗)
≤ ∀ ∈
and it follows in particular that
f(.,v) L∞((0,T∗);Ln(Ω)).
∈
By the regularizing effect of the heat equation [6, 4] we conclude that
u L∞((0,T∗); L∞(Ω)).
∈
Finally the solutions of the system (1.1) (1.3) are global and uniformly
−
bounded on (0,+ ) Ω and the corollary is completely proved.
∞ ×
ACKNOWLEDGEMENTS. We would like to thank a lot Prof. M.
KIRANE (Universit´e de la Rochelle, FRANCE)for his help and guidance..
Reaction-diffusion 431
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