NECESSARY CONDITIONS FOR LOCAL AND GLOBAL
EXISTENCE TO A REACTION-DIFFUSION SYSTEM WITH
FRACTIONAL DERIVATIVES
KAMEL HAOUAM AND MOURAD SFAXI
Received 6 August 2006; Revised 21 September 2006; Accepted 10 October 2006
We give some necessary conditions for local and global existence of a solution to reactiondiffusion system of type (FDS) with temporal and spacial fractional derivatives. As in the
case of single equation of type (STFE) studied by M. Kirane et al. (2005), we prove that
these conditions depend on the behavior of initial conditions for large |x|.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction
This paper deals with the following reaction-diffusion system of type (FDS):
(FDS)
⎧
⎨Dα u + (−Δ)β/2 u = |v | p
0|t
⎩Dδ v + (−Δ)γ/2 v = |u|q
0|t
in RN × R+ ,
in RN × R+ ,
(1.1)
where N ≥ 1, p and q are two positive reals.
For α ∈ (0,1) (resp., δ ∈ (0,1)), “Dα0|t ” (resp., “Dδ0|t ”) denotes the time derivative of
order α (resp., δ) in the sense of Caputo (see Definition 1.2, see also [9]). While, for β ∈
[1,2] (resp., γ ∈ [1,2]), “(−Δ)β/2 ” (resp., “(−Δ)γ/2 ”) stands for the β/2-fractional (resp.,
γ/2-fractional) power of the Laplacian with respect to x and defined by
(−Δ)β/2 v(x) = Ᏺ−1 |ξ |β Ᏺ(v)(ξ) (x),
(1.2)
where Ᏺ denotes the Fourier transform and Ᏺ−1 its inverse.
The system (FDS) is completed by the following initial conditions:
u(·,0) = u0 ,
v(·,0) = v0 ,
(1.3)
and we will assume that both u0 and v0 are nonnegative continuous functions.
The system (FDS) was considered in the case α = δ = 1, β = γ = 2, by many authors in
several contexts, see [2, 3, 5] (with ν = μ = 1). Moreover, concerning nonexistence result
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2006, Article ID 85203, Pages 1–8
DOI 10.1155/IJMMS/2006/85203
2
Necessary conditions for local and global existence
and basing one’s argument on [4], Escobedo and Herero proved in [2] that if pq ≥ 1, then
the only global solution of the system (FDS), reduced to the following reaction-diffusion
problem:
ut − Δu = v p ,
(1.4)
vt − Δv = uq ,
is the trivial one, that is, u ≡ v ≡ 0, while, in a recent paper (see [7]), the authors study the
system (FDS) and they find a bound on N leading to the absence of global nonnegative
solutions. More precisely, recovering the case studied in [2] (when α = δ = 1, β = γ = 2),
they proved the following.
Theorem 1.1. If p > 1 and q > 1 and supposing
δ/q + α − 1 − 1/ pq α/ p + δ − 1 − 1/ pq
,
N ≤ max
δ/γqp + α/βq
α/βpq + δ/γ p
,
(1.5)
then the system (FDS) does not admit nontrivial global weak nonnegative solution.
Therefore, they also establish some necessary conditions for the existence of local and
global solutions to the following problem:
⎧
⎨Dα u + (−Δ)β/2 u = h|u| p
0|t
⎩u(·,0) = u0 ≥ 0
(STFE)
in RN × R+ ,
in RN ,
(1.6)
and these conditions depend on the behavior of the initial data u0 and on the function h
for large |x|. Similar results can be found in [6] and [1].
Our results can be viewed as analogous to those obtained in [7], since the system (FDS)
can be considered as a system of two equations of type (STFE). Therefore, we can extend
these results to the more general system,
Dα0|t u + (−Δ)β/2 |u|m−1 u = h|v| p + g |u|r
in RN × R+ ,
Dδ0|t v + (−Δ)γ/2 |v|m−1 v = k|u|q + l|v|s
in RN × R+ ,
(1.7)
under some suitable conditions on h, g, k, and l. Of course, taking such form for reaction
terms, all results presented here will depend also on the functions h, g, k, and l.
For the convenience of the reader, we recall here some definitions and properties of
fractional derivatives in the sense of Caputo and of Riemann-Liouville.
Definition 1.2. The left-handed derivative and the right-handed derivative in the sense of
Caputo for ψ ∈ L1 (0,T) are defined, respectively, by
t
ψ (t)
dσ,
Γ(1 − α) 0 (t − σ)α
T
α ψ (t)
1
Dt|T ψ (t) = −
dσ,
Γ(1 − α) t (σ − t)α
Dα0|t ψ (t) =
1
where Γ denotes, as usual, the Euler gamma function.
(1.8)
K. Haouam and M. Sfaxi 3
Up to replace ψ by ψ and to keep the derivative operator before the integral in expressions (1.8), we obtain the definitions of the left-handed derivative and the right-handed
derivative in the sense of Riemann-Liouville denoted, respectively, by D0α|t and Dtα|T . See
[8] for more details.
Recall that the Caputo derivative is related to the Riemann-Liouville one by the following formula:
Dα0|t ψ(t) = D0α|t ψ(t) − ψ(0) .
(1.9)
Finally, taking into account the following integration by parts formula:
T
0
T
D0α|t f (t)g(t)dt =
0
f (t) Dtα|T g (t)dt,
(1.10)
we adopt the following.
Definition 1.3. For 0 < T ≤ ∞, it is said that (u,v) is a local weak solution to (FDS) defined
on QT (QT := RN × (0,T)) if
∩ Lq QT ,dx dt ,
v ∈ C [0,T];L1loc RN ∩ L p QT ,dx dt ,
u ∈ C [0,T];L1loc RN
(1.11)
and satisfies
p
QT
|v | ϕ +
QT
u0 Dtα|T ϕ =
QT
|u| ψ +
QT
QT
q
uDtα|T ϕ +
v0 Dtδ|T ψ
=
QT
QT
u(−Δ)β/2 ϕ,
vDtδ|T ψ
+
(1.12)
v(−Δ) ψ
γ/2
QT
2,1
(QT ) satisfying ϕ(·,T) = ψ(·,T) = 0. If T = +∞, it is said
for all test functions ϕ,ψ ∈ Cx,t
that (u,v) is a global weak solution.
2. Statement of the results
Our main result is the following.
Theorem 2.1. Assume that p, q > 1 and let (u,v) be a local solution (T < +∞) of problem
(FDS). Then, the following estimates hold:
liminf u0 (x) ≤ CT −(α+pδ)/(pq−1) ,
(2.1)
liminf v0 (x) ≤ C T −(δ+qα)/(pq−1) ,
(2.2)
|x|→∞
|x|→∞
where C and C are some positive constants.
4
Necessary conditions for local and global existence
Proof. Thanks to variational formulation (1.12), we have
QT
u0 Dtα|T ϕ ≤
QT
QT
uDtα|T ϕ +
QT
u(−Δ)β/2 ϕ,
v0 Dtδ|T ψ ≤
QT
vDtδ|T ψ +
QT
(2.3)
v(−Δ)γ/2 ψ
2,1
(QT ) satisfying ϕ(·,T) = ψ(·,T) = 0.
for all nonnegative test functions ϕ,ψ ∈ Cx,t
Using Hölder’s inequality, we get
QT
QT
uDtα|T ϕ ≤
u(−Δ)β/2 ϕ ≤
1/q QT
|u|q ψ
QT
1/q q
QT
|u| ψ
QT
α q −q /q
D ϕ ψ
t |T
1/q
,
(−Δ)β/2 ϕq ψ −q /q
(2.4)
1/q
.
And thus
QT
u0 Dtα|T ϕ ≤
1/q
QT
|u|q ψ
Ꮽ,
(2.5)
where
Ꮽ=
QT
α q −q /q
D ϕ ψ
t |T
1/q
+
QT
(−Δ)β/2 ϕq ψ −q /q
1/q
.
(2.6)
As above, using once again Hölder’s inequality, we obtain
QT
v0 Dtδ|T ψ ≤
QT
1/ p
|v | p ϕ
Ꮾ,
(2.7)
where
Ꮾ=
QT
δ p − p / p
D ψ ϕ
t |T
1/ p
+
QT
(−Δ)γ/2 ψ p ϕ− p / p
1/ p
.
(2.8)
Furthermore, keeping the first terms in the left-hand sides of (1.12) and recalling that
both u0 and v0 are nonnegative functions, we obtain as above
QT
|v | p ϕ ≤
1/q
QT
q
QT
|u|q ψ
|u| ψ ≤
1/ p
p
QT
|v | ϕ
Ꮽ,
(2.9)
Ꮾ.
K. Haouam and M. Sfaxi 5
Consequently,
QT
1−1/ pq
|v | p ϕ
≤ Ꮾ1/q Ꮽ,
QT
|u|q ψ
1−1/ pq
(2.10)
≤ ᏮᏭ1/ p .
(2.11)
Applying (2.10) and (2.11), respectively, in (2.5) and (2.7), we get
1−1/ pq
u0 Dtα|T ϕ
QT
≤ Ꮾ1/q Ꮽ,
(2.12)
≤ ᏮᏭ1/ p .
(2.13)
1−1/ pq
QT
v0 Dtδ|T ψ
Now, we consider some test functions in (2.12) and (2.13), introduced in [7], of the form
⎧
⎪
⎪
⎨
t
x
1−
ϕ(x,t) = ψ(x,t) = Φ
T
R ⎪
⎪
⎩0,
l
,
0 < t ≤ T,
(2.14)
t > T,
where Φ ∈ W 1,∞ (RN ) is nonnegative, with support in {R < |x| < 2R} and satisfies
(−Δ)β/2 Φ
(−Δ)γ/2 Φ
+
≤ kΦ
for some constant k > 0,
+
≤ hΦ
for some constant h > 0.
(2.15)
The exponent l, introduced in (2.14), is any positive real number if
min p −
1
1−δ
,q −
1
1−α
≥ 0,
(2.16)
< 0,
(2.17)
and l > max(αq − 1,δ p − 1) if
min p −
1
1−δ
,q −
1
1−α
where p and q are, respectively, the conjugate exponents of p and q.
Moreover, note that
Dtα|T ϕ(x,t) = ΛT −α Φ
x
R
t
1−
T
l−α
,
(2.18)
,
(2.19)
where Λ := Γ(1 + l)/Γ(1 + l − α).
Similarly,
Dtδ|T ψ(x,t) = ΥT −δ Φ
where Υ := Γ(1 + l)/Γ(1 + l − δ).
x
R
1−
t
T
l−δ
6
Necessary conditions for local and global existence
Next, as in [7], consider the change of variables
t = Tτ,
x = Ry.
(2.20)
We get
QT
u0 Dtα|T ϕdx dt =
ΛT 1−α RN
l−α+1
RN
u0 (Ry)Φ(y)d y.
(2.21)
Taking into account (2.15), we obtain
Ꮽ≤
Λq T 1−αq RN
(l − α)q − l(q /q) + 1
RN
Φ(y)d y
1/q
+
TR−βq +N k q
l+1
RN
Φ(y)d y
1/q
(2.22)
or
Ꮽ≤R
N/q
⎧
⎨
⎫
T 1/q R−β k ⎬
ΛT 1/q −α
+
⎩ (l − α)q − l(q /q) + 11/q
(l + 1)1/q ⎭
Φ(y)d y
RN
1/q
.
(2.23)
Analogously, using once again (2.15), we get
Ꮾ≤R
N/ p
⎧
⎨
⎫
T 1/ p R−γ h ⎬
ΥT 1/ p −δ
+
⎩ (l − δ)p − l(p / p) + 1 1/ p
(l + 1)1/ p ⎭
RN
Φ(y)d y
1/ p
.
(2.24)
Hence, (2.21), (2.23), and (2.24) with inequality (2.12) lead to
T (1−α)(1−1/ pq)
RN
1−1/ pq
u0 (Ry)Φ(y)d y
≤ C1 T 1/q −α + C2 T 1/q R−β
C3 T 1/ p −δ + C4 T 1/ p R−γ
1/q
RN
Φ(y)d y
1−1/ pq
,
(2.25)
where C1 , C2 , C3 , and C4 are positive constants independent on R and T. Consequently,
T
(1−α)(1−1/ pq)
1−1/ pq
inf u0 (Ry)
| y |>1
1/q
≤ C1 T 1/q −α + C2 T 1/q R−β C3 T 1/ p −δ +C4 T 1/ p R−γ
.
(2.26)
Finally, letting R → +∞ in (2.26), we conclude that
T (1−α)(1−1/ pq) liminf u0 (x)
|x|→∞
1−1/ pq
≤ CT 1/q −α T 1/ p q−δ/q .
(2.27)
Estimate (2.1) is then proved.
Using estimate (2.13) and applying the change of variables t = Tτ and x = Ry in the
expressions of Ꮽ and Ꮾ, we obtain estimate (2.2).
K. Haouam and M. Sfaxi 7
In the sequel, we also assume that p, q > 1.
Concerning existence of global and local solution, we give the following necessary condition results.
Corollary 2.2. Suppose that the system (FDS) admits a nontrivial global nonnegative weak
solution. Then
liminf u0 (x) = liminf v0 (x) = 0.
|x|→∞
(2.28)
|x|→∞
Corollary 2.3. If liminf |x|→∞ u0 (x) = +∞ or if liminf |x|→∞ v0 (x) = +∞, then the system
(FDS) cannot have nontrivial local nonnegative weak solution.
Corollary 2.4. If A := liminf |x|→∞ u0 (x) > 0 and B := liminf |x|→∞ v0 (x) > 0, then
C
,
A
T (α+pδ)/(pq−1) ≤
T (δ+pα)/(pq−1) ≤
C
,
B
(2.29)
where C and C denote the constants introduced in (2.1) and (2.2).
Our second main result is the following.
Theorem 2.5. Supposing that problem (FDS) admits a nontrivial global nonnegative weak
solution, then there exist positive constants H and K such that
liminf u0 (x)|x|(α+pδ)/(pq−1) ≤ H,
(2.30)
liminf v0 (x)|x|(δ+qα)/(pq−1) ≤ K.
(2.31)
|x|→∞
|x|→∞
Proof. Let us return to expression (2.25) and multiply by |x|(α+pδ)/(pq−1) |x|−(α+pδ)/(pq−1)
inside the integral of both members. Taking into account the fact that suppΦ ⊂ {1 < | y | <
2}, expression (2.25) becomes
T (1−α)(1−1/ pq) inf u0 (x)|x|(α+pδ)/(pq−1)
1−1/ pq
|x|>R
≤ C1 T
1/q −α
+ C2 T
1/q −β
R
C3 T
1/ p −δ
+ C4 T
1/ p −γ 1/q
R
(2.32)
(2R)
(α+pδ)/ pq
.
Taking T = R, it follows
inf u0 (x)|x|(α+pδ)/(pq−1)
|x|>R
1−1/ pq
1/q
≤ 2(α+pδ)/ pq C1 + C2 Rα−β C3 + C4 Rδ −γ
.
(2.33)
Since α < β and δ < γ, we pass to the limit with respect to R and we obtain
liminf u0 (x)|x|(α+pδ)/(pq−1)
|x|→∞
1−1/ pq
1/q
≤ 2(α+pδ)/ pq C1 C3 .
1/q
(2.34)
Hence, inequality (2.30) holds for H := 2(α+pδ)/(pq−1) {C1 C3 } pq/(pq−1) . As before, we ob1/ p
tain inequality (2.31) with K := 2(δ+qα)/(pq−1) {C1 C3 } pq/(pq−1) .
8
Necessary conditions for local and global existence
References
[1] P. Baras and R. Kersner, Local and global solvability of a class of semilinear parabolic equations,
Journal of Differential Equations 68 (1987), no. 2, 238–252.
[2] M. Escobedo and M. A. Herrero, Boundedness and blow up for a semilinear reaction-diffusion
system, Journal of Differential Equations 89 (1991), no. 1, 176–202.
[3] M. Fila, H. A. Levine, and Y. Uda, A Fujita-type global existence—global non-existence theorem
for a system of reaction diffusion equations with differing diffusivities, Mathematical Methods in
the Applied Sciences 17 (1994), no. 10, 807–835.
[4] H. Fujita, On the blowing up of solutions of the Cauchy problem for ut = Δu + u1+α , Journal of the
Faculty of Science. University of Tokyo. Section IA 13 (1966), 109–124 (1966).
[5] V. A. Galaktionov, S. P. Kurdyumov, and A. A. Samarskiı̆, A parabolic system of quasilinear equations. I, Differentsial’nye Uravneniya 19 (1983), no. 12, 2123–2140 (Russian).
[6] A. S. Kalashnikov, The heat equation in a medium with nonuniformly distributed nonlinear
heat sources or absorbers, Vestnik Moskovskogo Universiteta. Seriya I. Matematika, Mekhanika
(1983), no. 3, 20–24.
[7] M. Kirane, Y. Laskri, and N.-E. Tatar, Critical exponents of Fujita type for certain evolution equations and systems with spatio-temporal fractional derivatives, Journal of Mathematical Analysis
and Applications 312 (2005), no. 2, 488–501.
[8] J. D. Munkhammar, Riemann-Liouville fractional derivatives and the Taylor-Riemann series,
Project Report, Department of Mathematics, Uppsala University, Uppsala, 2004.
[9] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198,
Academic Press, California, 1999.
Kamel Haouam: Latp, Centre de Mathématiques et Informatiques, Université de Provence,
39 rue F., 13453 Marseille cedex 13, France
E-mail address: haouam@cmi.univ-mrs.fr
Mourad Sfaxi: Latp, Centre de Mathématiques et Informatiques, Université de Provence,
39 rue F., 13453 Marseille cedex 13, France
E-mail address: sfaxi@cmi.univ-mrs.fr