NECESSARY CONDITIONS FOR THE EXISTENCE OF
GLOBAL SOLUTIONS TO SYSTEMS OF
FRACTIONAL DIFFERENTIAL EQUATIONS
K.M. Furati
1
and M. Kirane
2
Dedicated to Professor Paul L. Butzer
on the occasion of his 80th birthday
Abstract
Necessary conditions for the existence of global solutions to certain 2×2systems of fractional differential equations are presented. It is demonstrated
that fractional derivatives of lower order have a strong influence on the
character of the solutions. Our method of proof relies on a suitable choice
of the test function in the weak formulation of the solutions to the system
under study.
2000 Mathematics Subject Classification: 26A33
Key Words & Phrases: fractional differential systems, blow-up, blowingup solutions
1. Introduction
Recent studies, see for example [5, 6, 7] and their extensive list of references (articles and books), demonstrate the role played by fractional derivatives in the mathematical modeling of various practical situations in mechanics, physical chemistry, biology and finance.
In this article, we want to present exponents threshold for the existence
of global solutions to the system of equations containing fractional derivatives
282
K.M. Furati, M. Kirane
½
α (u − u ) = f (t) |v|q + F (t),
a u0 (t) + b D0+
0
β
c v 0 (t) + d D0+
(v − v0 ) = g(t) |u|p + G(t),
t>0
t>0
(1)
subject to the initial conditions
u(0) = u0 ,
v(0) = v0 ,
(2)
where 1 < p, q, 0 < α, β < 2, a, b, c, d are constants, the functions f (t), g(t)
are positive functions for t > 0, while F (t) and G(t) are given functions with
nonnegative averages. There is a number of interesting articles dealing with
local existence, existence of global solutions and their behavior, blowing-up
solutions, estimates of the interval of existence, and lower-bounds of solutions for various equations of Abel or Volterra type that are connected to
fractional differential equations, see [1, 2, 3, 4, 5, 8, 9]. For systems of fractional differential equations we are concerning, the interesting dissertation
[8] deals with local and global existence under natural conditions (sub-linear
growth of the nonlinearities).
The type of the nonlinearities in system (1) appears in systems describing processes of heat diffusion and combustion in two component continua with nonlinear heat conductance and volumetric release (they read
ut = ∆u + v q , vt = ∆v + up , the sub-script t stands for the time derivative, while ∆ is the Laplace operator). So, the chosen nonlinearities may
be viewed as a prototype of nonlinearities. Other choice could be ur v q and
up v s or ev and eu .
By choosing a = c = f (t) = g(t) = 1, F (t) = G(t) = 0, and b = d = 0
in (1) we obtain the particular system
½ 0
u (t) = |v|q , t > 0
.
(3)
v 0 (t) = |u|p , t > 0
This system has, for pq > 1, the solution
u(t) = C1 (T max − t)−(q+1)/(pq−1) ,
v(t) = C2 (T max − t)−(p+1)/(pq−1) ,
for 0 < t < T max < ∞, where
£
¤1/(pq−1)
C1 = (p + 1)q (q + 1)/(pq − 1)q+1
,
£
¤
1/(pq−1)
C2 = (p + 1)(q + 1)p /(pq − 1)p+1
.
For pq > 1 the solution blows-up in the finite time
NECESSARY CONDITIONS FOR THE EXISTENCE OF . . .
µ
T
max
=
C2 u0
C1 v0
283
¶ q−p
pq−1
.
However, the particular system containing only fractional derivatives,
½ α
D0+ (u − u0 ) = |v|q , t > 0,
β
D0+
(v − v0 ) = |u|p , t > 0,
(4)
has, for pq > 1 the blowing-up solution (due to Trujillo [11] in case of a
single equation)
max
u(t) = C1,α,β (Tα,β
− t)−(α+qβ)/(pq−1) ,
max
v(t) = C2,α,β (Tα,β
− t)−(β+pα)/(pq−1)
max < ∞, where
for 0 < t < Tα,β
¶ q µ
¶ 1
µ
pq−1
pq−1
Γ(δ + 1)
Γ(% + 1)
C1,α,β =
,
Γ(δ + 1 − β)
Γ(% + 1 − α)
and
µ
¶ 1 µ
¶ p
pq−1
pq−1
Γ(δ + 1)
Γ(% + 1)
,
C2,α,β =
Γ(δ + 1 − β)
Γ(% + 1 − α)
with
qβ + α
pα + β
,
%=−
.
δ=−
pq − 1
pq − 1
The blow-up time for this system is
¶ α(1−p)−β(1−q)
µ
pq−1
C2,α,β u0
max
Tα,β =
,
C1,α,β v0
which tends, when α → 1, β → 1, to T max .
For the system (1), no explicit solutions are available. In this paper we
present a result concerning necessary conditions of existence.
The proof is based on a suitable choice of the test function in the weak
formulation of the problem.
2. Preliminaries
We start by introducing some definitions and preliminary results needed
for the principle result. For 0 < α < 1, we define the left-handed and righthanded fractional derivatives in the Riemann-Liouville sense by
284
K.M. Furati, M. Kirane
α
D0+
f (t)
and
1
d
=
Γ(1 − α) dt
DTα − f (t) =
Z
Z
d
−1
Γ(1 − α) dt
t
0
T
t
f (σ)
dσ,
(t − σ)α
f (σ)
dσ,
(σ − t)α
respectively.
It is shown in [10], Lemma 2.2, that for f ∈ AC([0, T ]),
·
¸
Z t
1
f (0)
f 0 (σ)
α
+
dσ .
D0+ f (t) =
α
Γ(1 − α) tα
0 (t − σ)
See also [7]. Note that
α
D0+
(f
1
− f (0))(t) =
Γ(1 − α)
Z
t
(5)
(6)
f 0 (σ)
α
dσ =: c D0+
f,
(t − σ)α
0
which is the Caputo derivative of f . For the right-hand derivative, we have
·
¸
Z T
1
f (T )
f 0 (σ)
α
DT − f (t) =
−
dσ .
(7)
α
Γ(1 − α) (T − t)α
t (σ − t)
It is shown in ([10], Corollary 2, p.46) that for functions f, g ∈ C([0, t]) with
α g, D α f exist at every point in [0, T ] and are continuous, we have
D0+
T−
Z T
Z T
α
f (t) D0+
g(t) dt =
DTα − f (t) g(t) dt,
0
0
which is a formula for integration by parts.
To announce the preparatory lemmas, let
½
(1 − t/T )λ , 0 ≤ t ≤ T,
ϕ(t) =
0,
t > T.
λ ≥ 2.
(8)
Then we have the following
Lemma 1. Let ϕ(t) be as in (8). Then
Z
0
T
DTα − ϕ(t) dt = Kα,λ T 1−α ,
Kα,λ =
λΓ(λ − α)
.
(λ − α + 1) Γ(λ − 2α + 1)
P r o o f. Using (5), we compute
Z T
λ
(T − σ)λ−1
α
−λ
DT − ϕ(t) =
T
dσ.
Γ(1 − α)
(σ − t)α
t
Using the Euler change of variable
y=
σ−t
,
T −t
(9)
NECESSARY CONDITIONS FOR THE EXISTENCE OF . . .
we can write
DTα − ϕ(t)
λ
=
T −λ (T − t)λ−α
Γ(1 − α)
Z
0
1
285
(1 − y)λ−α−1
dy.
yα
The last integral is the Beta function of Euler:
Z 1
(1 − y)λ−α−1
Γ(λ − α)Γ(1 − α)
dy =
.
α
y
Γ(λ − 2α + 1)
0
Thus,
λΓ(λ − α)
DTα − ϕ(t) =
T −λ (T − t)λ−α .
Γ(λ − 2α + 1)
(10)
By integration we obtain the result.
Lemma 2. Let ϕ be as in (8) and p > 1. Then for λ > p − 1,
Z T
ϕ1−p (t) |ϕ0 (t)|p dt = Λp T 1−p ,
(11)
0
where
Λp =
For λ > αp − 1,
Z
0
T
λp
.
λ+1−p
ϕ1−p (t) |DTα − ϕ(t)|p = Λp,α T 1−αp ,
h
ip
Γ(λ−α)
λp
Λp,α = λ+1−pα
.
Γ(λ+1−2α)
(12)
P r o o f. Using (8), we have
Z T
Z Th
¯p
i1−p ¯
¯
¯
1−p
0
p
ϕ (t) |ϕ (t)| dt =
T −λ (T − t)λ
¯−λT −λ (T − t)λ−1 ¯ dt
0
0
Z T
= λp T −λ(1−p) T −λp
(T − t)λ(1−p)+(λ−1)p dt
0
Z T
λp
T λ+1−p
=
T 1−p .
= λp T −λ
(T − t)λ−p dt = λp T −λ
λ
+
1
−
p
λ
+
1
−
p
0
Using (8) and (10), we compute
Z T
ϕ1−p (t) |DTα − ϕ(t)|p dt
0
Z
=
0
T
h
i1−p
−λ
λ
T (T − t)
¯
¯p
¯ λΓ(λ − α)
¯
−λ
λ−α ¯
¯
¯ Γ(λ − 2α + 1) T (T − t)
¯ dt
286
K.M. Furati, M. Kirane
·
¸p Z T
λΓ(λ − α)
−λ
(T − t)λ(1−p)+p(λ−α) dt
=T
T
Γ(λ − 2α + 1)
0
·
¸
·
¸
Z
λΓ(λ − α) p −λ T
λΓ(λ − α) p −λ T λ+1−pα
λ−pα
=
T
(T − t)
dt
T
.
Γ(λ − 2α + 1)
Γ(λ − 2α + 1)
λ + 1 − pα
0
−λ(1−p)
Remark 1. Note that the bound in (11) is a limiting case of the bound
in (12) as α → 1. Moreover, since Γ(ζ) is increasing for ζ ≥ 2, for λ ≥ 3 we
have
Λp,α < Λp .
For the rest of this paper we assume
λ > max{3, p − 1, q − 1}.
(13)
For clarity, we consider first the case a = b = c = d = 1 (this is possible
by a scaling argument), f (t) = g(t) = 1, F (t) = G(t) = 0 in (1). We will
comment on this choice later. The reduced system we have, is:
½ 0
α (u − u ) = |v|q ,
u (t) + D0+
t > 0,
0
(14)
β
0
p
v (t) + D0+ (v − v0 ) = |u| , t > 0.
It is classical to show (see [8]) that system (14) subject to conditions (2)
has a solution
¡
¢2
(u, v) ∈ C 1 (0, Tmax ) ∩ C([0, Tmax )
with the life span Tmax such that Tmax + lim {|u(t)| + |v(t)|} = +∞.
t→Tmax
In what follows, we treat separately the two cases 0 < α, β < 1 and
1 < α, β < 2.
3. The case 0 < α, β < 1
3.1. The main result
The main result is given by the following theorem.
Theorem 1. Let 1 < p, q and u0 > 0, v0 > 0, then if
1
β
1
α
1−
≤ α + , or 1 −
≤β+ ,
pq
p
pq
q
the system (14) subject to (2) admits no global solutions.
(15)
NECESSARY CONDITIONS FOR THE EXISTENCE OF . . .
287
P r o o f. Our proof is by contradiction. Let (u, v) be a global solution
and ϕ as in (8) with the restriction (13) on λ. It is easy to see that (u, v)
satisfies
Z
T
0
Z
µ
Z
|v| ϕ + u0 1 +
T
µ
Z
|u| ϕ + v0 1 +
T
q
T
0
¶
DTα − ϕ
0
p
0
Z
Z
T
=−
¶
DTβ − ϕ
T
0
u DTα − ϕ,
(16)
v DTβ − ϕ.
(17)
uϕ +
Z
0
Z
T
=−
0
0
T
vϕ +
0
0
Using the Hölder inequality, we have
Z
Z
T
0
uϕ =
1/p
uϕ
0
ϕ
−1/p
0
T
0
Z
0
T
p
0
T
µZ
T
v ϕ0 ≤
0
Z
¶1/p µZ
|u| ϕ
T
ϕ ≤
0
Z
and
µZ
T
T
≤
¶1/q µZ
q
|v| ϕ
¶1/p µZ
|u| ϕ
p
0
µZ
v DTβ − ϕ
T
0
ϕ−q /q |ϕ0 |q
¶1/q µZ
|v| ϕ
q
≤
T
where p + p0 = pp0 and q + q 0 = qq 0 .
Let
Z T
I :=
|u|p ϕ,
0
Z T
0
0
A :=
ϕ−p /p |ϕ0 |p ,
Z 0T
0
0
C :=
ϕ−p /p |DTα − ϕ|p ,
0
0 p0
¶1/p0
|ϕ |
(18)
0
0
|DTα − ϕ|p
−q 0 /q
0
|DTβ − ϕ|q
ϕ
T
ϕ
Z
T
J :=
Z 0T
B :=
Z0 T
D :=
0
,
(19)
¶1/p0
,
(20)
,
(21)
¶1/q0
|v|q ϕ,
0
0
ϕ−q /q |ϕ0 |q ,
0
0
β
ϕ−q /q |Dt|T
ϕ|q .
We use (18), (19), (20) and (21), to write (16) and (17) in the forms
µ
¶
Z T
³
´
0
0
α
J + u0 1 +
DT − ϕ ≤ I 1/p A1/p + C 1/p ,
and
,
¶1/q0
−p0 /p
0
0
ϕ
0
0
T
−p0 /p
0
0
µZ
u DTα − ϕ
T
(22)
0
µ
Z
I + v0 1 +
0
T
¶
DTβ − ϕ
³
´
0
0
≤ J 1/q B 1/q + D1/q .
Using inequalities (22) and (23), we obtain the inequalities
(23)
288
K.M. Furati, M. Kirane
and
from which we have
J
(24)
³
´
0
0
I ≤ J 1/q B1/q + D1/q ,
(25)
1
1− pq
³
´³
´1/p
0
0
0
0
≤ A1/p + C 1/p
B 1/q + D1/q
,
(26)
1
1− pq
³
´
³
´
0
0 1/q
0
0
≤ A1/p + C 1/p
B1/q + D1/q .
(27)
and
I
³
´
0
0
J ≤ I 1/p A1/p + C 1/p ,
Using the elementary inequality
(x + y)1/r ≤ x1/r + y 1/r ,
r ≥ 1, x > 0, y > 0,
we estimate the inequalities (26) and (27) as
³ 1
1 ´ ³
1
1 ´
1− 1
J pq ≤ A p0 + C p0
B pq0 + D pq0 ,
and
I
1
1− pq
³ 1
1 ´ ³
1
1 ´
≤ A qp0 + C qp0
B q0 + D q0 .
(28)
(29)
From Lemma 2, with λ ≥ max{p0 , q 0 } − 1, we have
0
0
A = Λp0 T 1−p ,
0
C = Λp0 ,α T 1−αp ,
B = Λq0 T 1−q ,
0
D = Λq0 ,β T 1−βq ,
(30)
and thus (28) takes the form
J
1
1− pq
≤ K1 (T s1 + T s2 + T s3 + T s4 ) ,
(31)
where
½ 1 1
¾
1
1
1
1
1
1
1
1
0
0
p0
pq 0
pq 0
p0
p0
pq 0
p0
pq 0
K1 = max Λp0 Λq0 , Λq0 Λp0 ,α , Λp0 Λq0 ,β , Λp0 ,α Λq0 ,β = Λpp0 Λqpq0 ,
and
³
s1 = −
s3 = −
1
³ pq
1
pq
´
+
+
1
p´,
β
p ,
³
s2
1
³ pq
1
− pq
=−
s4 =
´
+
+
1
p´+ 1 − α
β
p +1−α
= s1 + 1 − α,
= s3 + 1 − α.
Note that s1 < s3 < 0 and s2 < s4 . Thus J is bounded for s4 ≤ 0. This is
equivalent to
1
β
1−
≤α+ .
(32)
pq
p
NECESSARY CONDITIONS FOR THE EXISTENCE OF . . .
Now, from (23),
Z
v0
289
³
´
0
0
DTβ − ϕ ≤ J 1/q B1/q + D1/q .
T
0
From Lemma 1 and (30),
³
´
0
0
v0 ≤ Kβ,λ T β−1 J 1/q B1/q + D1/q
≤ K J 1/q (T z1 + T z2 ) ,
(33)
where
z1 = −2 + β +
1
1
= −1 + β − < 0,
0
q
q
and
z2 = −1 +
1
1
= − < 0,
0
q
q
(34)
1
0
−1
K = Λqq0 Kβ,λ
.
When T → ∞, we obtain the contradiction 0 < v0 ≤ 0.
A similar analysis could be performed by showing that I is bounded
when
1
α
1−
≤β+ ,
(35)
pq
q
which leads via (22) to the contradiction 0 < u0 ≤ 0.
3.2. Remarks
Remark 2. We can obtain a bound on the blow-up time as follows.
Since s4 = maxi {si }, we have from (31)
ps4
p
J 1/q ≤ (4K1 ) pq−1 T pq−1 .
Moreover, from (34), we have z1 < z2 . It follows from (33) that
v0 ≤ Cv T s ,
where
s=
pα + β
< 0,
1 − pq
2p
1+ pq−1
Cv = 2
p
K1pq−1 K.
So, a bound on the blow-up time is given by
µ ¶1
v0 s
Tv =
.
Cv
A similar bound can be obtained in terms of u0 given by
290
K.M. Furati, M. Kirane
µ
Tu =
u0
Cu
¶1
s
e
.
The bound on the blow-up time is then given by the min {Tu , Tv }.
Remark 3.
condition
We can obtain the same result with the less restrictive
u0 + v0 > 0.
Remark 4. When system (1) is considered, the analysis above shows
that the proof goes through, if the conditions on the terms F and G
Z ∞
Z ∞
F (t) dt ≥ 0,
G(t) dt ≥ 0
0
0
are imposed. Of course, the threshold exponents will depend on the behavior
of the functions f (t) and g(t) at infinity.
To explain this point, let us recall the weak formulation of solutions to
the system (1)
µ Z T
¶
Z T
Z T
Z T
Z T
q
α
0
F (t) ϕ +
f (t)|v| ϕ + u0 1+
DT − ϕ = −
uϕ +
u DTα − ϕ,
0
Z
0
Z
T
G(t) ϕ +
0
0
µ Z
T
p
g(t)|u| ϕ + v0 1+
0
0
¶
Z
T
β
DT − ϕ = −
0
0
T
Z
v ϕ0 +
0
0
T
v DTβ − ϕ.
If, for example,
f (t) ≥ tl1 ,
tÀ1
for
l1 > 0,
f (t) ≥ tl2 ,
tÀ1
for
l2 > 0,
then one has to use the estimates
µZ T
¶1/p µZ
Z T
0
|u| |ϕ | ≤
|u|p ϕg(t)
0
Z
T
0
|u| |DTα − ϕ|
Z
T
0
µZ
T
≤
µZ
0
T
0
|v| |ϕ | ≤
Z
0
0
T
|v| |DTβ − ϕ|
µZ
0
≤
0
T
¶1/p
0
(g(t)ϕ)−p /p ϕ
,
0
¶1/p µZ
p
|u| ϕg(t)
¶1/q µZ
|v| f (t)ϕ
T
T
−p0 /p
(g(t)ϕ)
0
T
q
−q 0 /q
(f (t) ϕ)
0
|DTα − ϕ|p
0 q0
¶1/q µZ
|v| ϕ f (t)
0
T
−q 0 /q
(f (t)ϕ)
,
¶1/q0
|ϕ |
0
q
¶1/p0
0
|DTβ − ϕ|q
,
¶1/q0
.
NECESSARY CONDITIONS FOR THE EXISTENCE OF . . .
291
When the scaling argument is used as before to conclude, it appears that
the behavior of f (t) and g(t) will have an influence on the exponents.
Let us pass now to the second case.
4. The case 0 < p, q ≤ 1
In this case because the nonlinearity is sub-linear, as it is expected the
solutions are global.
Theorem 2. Let u0 > 0, v0 > 0, and 0 < p, q ≤ 1. Then any solution
to system (14) emerging from (u0 , v0 ) is global.
P r o o f. Integrating the first equation of system (14) from ε to t, we
obtain
Z t
Z t
α
u(t) +
D0+ (u − u0 ) = u(ε) +
vp
ε
ε
or
Z t
Z t
Z ε
u(τ )
u0 1−α
u(τ )
u0 1−α
u(t) +
t
−
ε
+
vp,
−
= u(ε) +
α
α
1−α
1−α
0 (t − τ )
ε
0 (ε − τ )
which leads to the estimate
Z ε
Z t
u(τ )
u0 1−α
u(t) ≤
t
+
vp.
+
u(ε)
+
α
(ε
−
τ
)
1
−
α
0
ε
If |u|∞,T := max0≤t≤T u(t), for any T > 0, then we have the estimate
ε1−α
u0 1−α
|u|∞,T ≤
|u|∞,T + u(ε) +
t
+ t|v|q∞,T .
(36)
1−α
1−α
Similarly, we obtain, via the second equation of system (14), the estimate
ε1−β
v0 1−β
|v|∞,T ≤
|v|∞,T + v(ε) +
t
+ t|u|p∞,T .
(37)
1−α
1−β
Inequalities (36) and (37) can be arranged in the form
 ³
´
 1 − ε1−α |u|∞,T ≤ u(ε) + u0 t1−α + t|v|q
∞,T
1−α ´
1−α
³
 1 − ε1−β |v|∞,T ≤ v(ε) + v0 t1−β + t|u|p .
∞,T
1−β
1−β
(38)
As 0 ≤ p ≤ 1 and 0 ≤ q ≤ 1, the system (38) allows to obtain the a priori
estimates
|u|∞,T ≤ A(t), |v|∞,T ≤ B(t)
with two continuous functions A(t) and B(t) that are easy to compute.
Whereupon, any solution emerging from the initial data (u0 , v0 ) is global.
292
K.M. Furati, M. Kirane
However, in the case where 0 < p < 1, 0 < q < 1, one may expect
extinction of any solution, i.e., there exists a time t∗ > 0 such that u(t) ≡ 0
and v(t) ≡ 0 for any time t ≥ t∗ .
5. The case 1 < α, β < 2
5.1. The main result
We set α = 1 + α
e and β = 1 + βe with 0 < α
e, βe < 1.
The main result of this section is the following
Theorem 3. Let 1 < p, q, 1 < α, β < 2 and u0 > 0, v0 > 0, then the
system (14) subject to (2) and u0 (0) = u1,0 ≥ 0, v 0 (0) = v1,0 ≥ 0 admits no
global solutions.
P r o o f. The weak formulation in this case reads:
½
¾
Z T
Z T
Z T
1+α
e
q
t · DT1+−αe ϕ(t)
|v| ϕ + u0 1 +
DT − ϕ(t) + u1,0
0
0Z
0 Z
T
T
0
1+α
e
=
u(t) DT − ϕ(t) −
uϕ ,
Z
0
T
0Z
½
Z
|u|p ϕ + v0 1 +
T
=
0
(39)
0
¾
Z T
e
e
DT1+−β ϕ(t) + v1,0
t · DT1+−β ϕ(t)
0 Z
0
T
0
1+βe
v(t) DT − ϕ(t) −
vϕ .
T
(40)
0
With the choice of the test function ϕ as in (8), we clearly have
Z T
Z T
´
¯T ³
1+α
e
DT − ϕ(t) = −
D · DTαe − ϕ(t) = −DTαe − ϕ(t)¯0 = DTαe − ϕ (0).
0
0
Recall that
DTαe − ϕ(t) =
λΓ(λ − α
e)
T −λ (T − t)λ−αe .
Γ(1 + λ − 2e
α)
¡
¢
So, in particular, DTαe − ϕ (0) =
On the other side,
Z T
Z
t · DT1+−αe ϕ(t) = −
0
0
T
λΓ(λ−α
e)
T −αe .
Γ(1+λ−2α
e)
Z
t · D · DTαe − ϕ(t) =
0
Now, for example, (39) can be rewritten in the form
T
DTαe − ϕ(t) ≥ 0.
NECESSARY CONDITIONS FOR THE EXISTENCE OF . . .
Z
T
0Z
³
|v|q ϕ + u0 1 +
=
0
T
´
λΓ(λ − α
e)
T −αe + u1,0 Kαe,λ T 1−αe
Γ(1Z+ λ − 2e
α)
u(t) DT1+−αe ϕ(t)
T
−
0
293
(41)
uϕ .
0
Following the same lines as in the first part, we obtain the estimate
µ
¶
λΓ(λ − α
e)
|v|q ϕ + u0 1 +
T −αe + u1,0 Kαe,λ T 1−αe
Γ(1 + λ − 2e
α)
0
µZ T
¶1/p ³
´
0
0
≤
|u|p ϕ
C1 T 1−p + C2 T 1−(1+αe)p ≤ CIB,
Z
T
(42)
0
0
0
where B = T 1−p (1 + T −αep ), C1 , C2 , are constants and C := max {C1 , C2 }.
If we set
³
´
0
e 0
D = T 1−q 1 + T −βq ,
we may then write the inequality (C is a constant that may change from
line to line)
J q + A ≤ C BI.
with a clear meaning of A.
We can obtain a similar inequality via (40)
I p + C ≤ C DJ .
with a clear meaning of C.
So,
½
J q + A ≤ C BI
I p + C ≤ C DJ
from which we can easily obtain the estimate
´³
´
0 ³
0
e 0 1/q
q− 1
1−p0 + 1−q
q
J p ≤CT
1 + T −αep
1 + T −βq
where C > 0 os a constant. The requirement p > 1, q > 1 leads to a
contradiction when we let T → +∞.
5.2. Remarks
Remark 5. The intermediate case 1 < α < 2 and 0 < β < 1 can be
handled in the same manner as the cases treated here. We refrain to give
the details.
294
K.M. Furati, M. Kirane
Remark 6. Let us mention that our analysis may be used to tackle
the more general system
( Pi=N αi m (t)
Pi=M δi li (t)
i
i
− um
− v0li ) = f1 (t, |u|, |v|),
0 )+
i=1 D0+ (u
i=1 D0+ (v
Pi=K βi ni (t)
P
σi
ki (t) − uki )
− v0ni ) + i=L
= f2 (t, |u|, |v|),
0
i=1 D0+ (v
i=1 D0+ (u
t > 0, where 0 < αi , βi , δi , σi , 0 < m1 ≤ mi (t) ≤ m2 < ∞, 0 < n1 ≤ ni (t) ≤
n2 < ∞, 0 < l1 ≤ li (t) ≤ l2 < ∞, 0 < k1 ≤ ki (t) ≤ k2 < ∞, and with either
(
f1 (t, |u|, |v|) ≥ c1 tϑ1 |u|p1 + c2 tϑ2 |v|p2 ,
f2 (t, |u|, |v|) ≥ c3 tϑ3 |u|p3 + c4 tϑ4 |v|p4 ,
for t > 0, where 0 < c1 , c2 , c3 , c4 , 0 < ϑ1 , ϑ2 , ϑ3 , ϑ4 , 1 < p1 , p2 , p3 , p4 , or
(
f1 (t, |u|, |v|) ≥ d1 tϑ1 |u|p1 |v|p2 ,
f2 (t, |u|, |v|) ≥ d2 tϑ2 |u|p3 |v|p4 .
with 0 < d1 , d2 , ϑ1 , ϑ2 , 1 < p1 , p2 , p3 , p4 .
Remark 7. Let us stress again that the study of systems containing
fractional derivatives can be drastically different from that of systems of
ordinary differential equations; here is an example: it is easy to see that the
system
( 0
u (t) = up1 v q2 t > 0,
v 0 (t) = up2 v q1
can be decoupled





t > 0,
u(0) = u0 > 0, v(0) = u0 > 0,
into
³
´q2 /r2
u0 (t) = up1 rr21 ur1 − r2 C0
t > 0,
³
´q1 /r1
v 0 (t) = v p2 rr12 v r2 + r1 C0
t > 0,
where we have set
ur1
v r2
ur1
v r2
−
= C0 = 0 − 0 , t > 0,
r1
r2
r1
r2
that is easy to handle. However, we are unaware of a method that can
decouple systems of nonlinear fractional differential equations.
NECESSARY CONDITIONS FOR THE EXISTENCE OF . . .
295
6. Numerical analysis
For the numerical treatment of the system, we write (14) in the form
( 0
α (u − u ) + |v|q , t > 0,
u (t) = −D0+
0
(43)
β
v 0 (t) = −D0+ (v − v0 ) + |u|p , t > 0
α (u − u ) and D β (v − v ) as perturbations.
which views D0+
0
0
0+
Next we write

Z t
©
ª

α

−D0+
(u − u0 )(σ) + |v|q (σ) dσ,
 u(t) = u0 +
Z 0t n
o

β

 v(t) = v0 +
−D0+
(v − v0 )(σ) + |u|p (σ) dσ,
t > 0,
t > 0,
0
and use the iterative scheme

Z tn
o

α
(n)

−D0+
(u(n−1) − u0 )(σ) + |v (n−1) (σ)|q dσ,
 u (t) = u0 +
Z0 t n
o

β

 v (n) (t) = v0 +
−D0+
(v (n−1) − v0 )(σ) + |u(n−1) (σ)|p dσ,
0
for n = 1, 2, . . ., starting with u0 = u0 , v 0 = v0 .
The figures show the blowing-up character of the solutions for various
values of the parameters.
1,2E8
1,6E7
1E8
1,2E7
8E7
6E7
8E6
4E7
4E6
2E7
0
-10
-5
0
0
5
t
10
-10
-5
0
5
10
t
Figure 1:
Curve u(3) at left, and curve v (3) at right, when
p = q = 2, α = β = 21 .
296
K.M. Furati, M. Kirane
8E15
1,6E21
6E15
1,2E21
4E15
8E20
2E15
4E20
0
-10
-5
0
0
5
10
-10
-5
0
5
t
10
t
Figure 2:
Curve u(4) at left, and curve v (4) at right, when
p = q = 2, α = β = 21 .
3500
1E8
1,6E21
3000
8E7
2500
1,2E21
6E7
2000
8E20
1500
4E7
1000
4E20
2E7
500
0
-10
-5
0
0
5
t
10
-10
-5
0
0
5
10
-10
-5
0
t
Figure 3: Curve u(n) , n = 2, 3, 4 when p = 2, q = 4, α =
5
10
t
1
2
and β = 34 .
NECESSARY CONDITIONS FOR THE EXISTENCE OF . . .
1,6E7
297
6E15
400
5E15
1,2E7
300
4E15
8E6
3E15
200
2E15
4E6
100
1E15
0
0
-10
-5
0
5
t
10
-10
-5
0
0
5
10
-10
-5
0
t
Figure 4: Curve v (n) , n = 2, 3, 4 when p = 2, q = 4, α =
5
10
t
1
2
and β = 34 .
References
[1] M.R. Arias and J.M.F. Castillo, Lower estimates for nonlinear Volterra
equations. Nonlinear Analysis, TMA 38 (1999), 351-360.
[2] M.R. Arias and R. Benitez, Aspects of the behaviour of solutions of
nonlinear Abel equations. Nonlinear Analysis, TMA 54 (2003), 12411249.
[3] P.J. Bushell, On a class of Volterra and Fredholm non-linear integral
equations, Math. Proc. Camb. Phil. Soc. 79 (1976), 329-335.
[4] P.J. Bushell and W. Okrasinski, On the maximum interval of existence for solutions to some non-linear Volterra integral equations with
convolution kernel. Bull. London Math. Soc. 28 (1996), 59-65.
[5] R. Gorenflo and S. Vessella, Abel integral Equations. Lecture Notes in
Mathematics, No 1461, Springer-Verlag (1991).
[6] R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order. In: A. Carpinteri and F. Mainardi
(Ed-s), Fractals and Fractional Calculus in Continuum Mechanics,
Springer Verlag, Wien (1997), 223-276.
[7] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier (2006).
[8] M.W. Michalski, Derivatives of noninteger order and their applications. Dissertations Mathematicae, CCCXXVIII (1993).
[9] W. Mydlarczyk, W. Okrasiński and C.A. Roberts, Blow-up solutions
to a system of nonlinear Volterra equations. J. Math. Anal. Appl.
301(2005), 208-218.
298
K.M. Furati, M. Kirane
[10] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and
Derivatives: Theory and Applications. Gordon and Beach, Yverdon
and New York (1993).
[11] J.J. Trujillo, Private communication to M. Kirane.
Received: May 3, 2008
1
King Fahd University of Petroleum & Minerals
Department of Mathematics & Statistics
Dhahran 31261, SAUDI ARABIA
e-mail: [email protected]
2
Laboratoire de Mathématiques Pôle Sciences et Technologie
Université de La Rochelle
Avenue M. Crépeau, 17042 La Rochelle Cedex, FRANCE
e-mail: [email protected]