Available online at www.tjnsa.com
J. Nonlinear Sci. Appl. 9 (2016), 1600–1608
Research Article
Oscillation properties for solutions of a kind of
partial fractional differential equations with damping
term
Wei Nian Li∗, Weihong Sheng
Department of Mathematics, Binzhou University, Shandong 256603, PR China.
Communicated by R. Saadati
Abstract
The aim of the present paper is to obtain sufficient conditions for oscillation of solutions of partial
fractional differential equations with the damping term of the form
1+α
α
D+,t
u(x, t) + p(t)D+,t
u(x, t) = a(t)∆u(x, t) +
m
X
ai (t)∆u(x, t − τi )
i=1
Z
− q(x, t)
t
(t − ξ)−α u(x, ξ)dξ, (x, t) ∈ Ω × R+ ≡ G.
0
c
Two examples are given to illustrate the main results. 2016
All rights reserved.
Keywords: Oscillation, fractional partial differential equation, Riemann–Liouville derivative, damping
term.
2010 MSC: 35B05, 35R11, 26A33.
1. Introduction
Fractional differential equations are generalizations of classical differential equations to an arbitrary order
and have been widely applied in various areas of engineering, physics, mechanics, chemistry, aerodynamics,
finance, applied mathematics, bioengineering, nonlinear control and so on. There has been a significant
∗
Corresponding author
Email addresses: wnli@263.net (Wei Nian Li), wh-sheng@163.com (Weihong Sheng)
Received 2015-09-18
W. N. Li, W. Sheng, J. Nonlinear Sci. Appl. 9 (2016), 1600–1608
1601
development in fractional differential equations in recent years. For example, see the monographs [1, 4, 7,
9, 14].
Recently, the research on the theory of partial fractional differential equations is a very interesting topic
and some results are established. Jafari et al.[6] presented the iterative Laplace transform method to derive
exact and approximate analytical solutions of fractional partial differential equations. Chen and Jiang[2]
exploited the techniques of Lie group analysis to solve n–order linear fractional partial differential equation
0
and nonlinear fractional reaction diffusion convection equation. Zheng [12] applied the (G /G) method to
seek exact solutions for several fractional partial differential equations including the space–time fractional
(2 + 1) dimensional dispersive long wave equations, the (2 + 1)–dimensional space time fractional Nizhnik–
Novikov–Veselov system and the time fractional fifth–order Sawada–Kotera equation. Based on a fractional
complex transformation, Zheng and Feng[13] obtained Jacobi elliptic function solutions to fractional partial
differential equations.
At the same time, a few papers [5, 8, 10, 11] studied the oscillation of fractional partial differential equations which involve the Riemann–Liouville fractional partial derivative. In [11], Prakash et al. considered
the oscillation of the fractional partial differential equation
Z t
∂
α
(t − ν)−α u(x, ν)dν = a(t)∆u(x, t),
(1.1)
(r(t)D+,t u(x, t)) + q(x, t)f
∂t
0
where (x, t) ∈ Ω × R+ ≡ G.
Harikrishnan et al.[5] studied the oscillatory behavior of the fractional partial differential equation of the
form
α
α
D+,t
r(t)D+,t
u(x, t) + q(x, t)f (u(x, t)) = a(t)∆u(x, t) + g(x, t), (x, t) ∈ G.
(1.2)
Prakash et al.[10] established the oscillation of a nonlinear fractional partial differential equation with
damping and forced term of the form
α (r(t)D α u(x, t)) + p(t)D α u(x, t) + q(x, t)f (u(x, t)) = a(t)∆u(x, t) + g(x, t), (x, t) ∈ G.
D+,t
+,t
+,t
(1.3)
In [8], Li investigated the forced oscillation of fractional partial differential equations with the damping
term of the form
∂ α
α
(1.4)
D+,t u(x, t) + p(t)D+,t
u(x, t) = a(t)∆u(x, t) − q(x, t)u(x, t) + f (x, t), (x, t) ∈ G.
∂t
2. Formulation of the Problems
In this paper, we are concerned with establishing criteria for oscillation of solutions of partial fractional
differential equations with the damping term of the form
1+α
D+,t
u(x, t)
+
α
p(t)D+,t
u(x, t)
= a(t)∆u(x, t) +
m
X
ai (t)∆u(x, t − τi )
i=1
Z
− q(x, t)
t
(2.1)
−α
(t − ξ)
u(x, ξ)dξ, (x, t) ∈ Ω × R+ ≡ G,
0
where Ω is a bounded domain in Rn with a piecewise smooth boundary ∂Ω, α ∈ (0, 1) is a constant and
α u(x, t) is the Riemann–Liouville fractional derivative of order α of u with respect to t, ∆ is the Laplacian
D+,t
in Rn .
We assume throughout this paper that
(A1) a ∈ C([0, ∞); [0, ∞)), p ∈ C([0, ∞); [0, ∞)), ai ∈ C([0, ∞); [0, ∞)) and τi ≥ 0 is a constant, i =
1, 2, · · · , m;
(A2) q ∈ C(G; R+ ) and q(t) = minx∈Ω q(x, t).
Consider one of the two following boundary conditions
W. N. Li, W. Sheng, J. Nonlinear Sci. Appl. 9 (2016), 1600–1608
1602
∂u(x, t)
+ g(x, t)u(x, t) = 0, (x, t) ∈ ∂Ω × R+ ,
∂N
(2.2)
u(x, t) = 0, (x, t) ∈ ∂Ω × R+ ,
(2.3)
or
where N is the unit exterior normal vector to ∂Ω and g(x, t) is a nonnegative continuous function on
∂Ω × R+ .
By a solution of the problem (2.1)–(2.2)(or (2.1)–(2.3)), we mean a function u(x, t) which satisfies (2.1)
on G and the boundary condition (2.2)(or (2.3)).
A solution u(x, t) of the problem (2.1)–(2.2)(or (2.1)–(2.3)) is said to be oscillatory in G if it is neither
eventually positive nor eventually negative, otherwise it is non-oscillatory.
Definition 2.1 ([7]). The Riemann–Liouville fractional integral of order α > 0 of a function y : R+ → R
on the half–axis R+ is given by
Z t
1
α
I+ y(t) :=
(t − ξ)α−1 y(ξ)dξ for t > 0
(2.4)
Γ(α) 0
provided the right hand side is pointwise defined on R+ , where Γ is the gamma function.
Definition 2.2 ([7]). The Riemann–Liouville fractional derivative of order α > 0 of a function y : R+ → R
on the half–axis R+ is given by
ddαe
dαe−α
α y(t) :=
D+
I
y
(t)
dtdαe +
Z t
(2.5)
1
ddαe
dαe−α−1
=
(t
−
ξ)
y(ξ)dξ
for
t
>
0
Γ(dαe − α) dtdαe 0
provided the right hand side is pointwise defined on R+ , where dαe is the ceiling function of α.
Definition 2.3 ([7]). The Riemann–Liouville partial fractional derivative of order 0 < α < 1 with respect
to t of a function u(x, t) is given by
α
D+,t
u(x, t)
1
∂
:=
Γ(1 − α) ∂t
Z
t
(t − ξ)−α u(x, ξ)dξ
(2.6)
0
provided the right hand side is pointwise defined on R+ .
The following lemmas are very useful in the proof of our main results.
d
α y(t) and D α+m y(t) exist,
Lemma 2.4 ([7]). Let α ≥ 0, m ∈ N and D = dt
. If the fractional derivatives D+
+
then
α+m
α
Dm (D+
y(t)) = D+
y(t).
(2.7)
Lemma 2.5 ([11]). Let
Z
e =:
E(t)
0
e 0 (t) = Γ(1 − α)Dα y(t).
Then E
+
t
(t − ξ)−α y(ξ)dξ
for
α ∈ (0, 1)
and
t > 0.
(2.8)
W. N. Li, W. Sheng, J. Nonlinear Sci. Appl. 9 (2016), 1600–1608
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3. Oscillation of the Problem (2.1)–(2.2)
Theorem 3.1. Assume that for some t0 > 0,
Z ∞
Z t
p(ξ)dξ dt = ∞
exp −
and
(3.1)
t0
t0
Z
∞
q(s)v(s)ds = ∞,
(3.2)
t0
where
t
Z
v(t) = exp
p(s)ds , t ≥ t0 .
(3.3)
t0
Then every solution of the problem (2.1)–(2.2) is oscillatory in G.
Proof. Suppose to the contrary that there is a nonoscillatory solution u(x, t) to the problem (2.1)–(2.2)
which has no zero in Ω × [t0 , ∞) for some t0 > 0. Without loss of generality, we may assume that u(x, t) >
0 and u(x, t − τi ) > 0 in Ω × [t1 , ∞) for t1 ≥ t0 , i = 1, 2, · · · , m.
Integrating (2.1) with respect to x over the domain Ω, we have
Z
Ω
1+α
D+,t
u(x, t)dx
Z
Z
α
D+,t
u(x, t)dx
+ p(t)
= a(t)
Ω
∆u(x, t)dx +
Ω
Z
−
i=1
Ω
∂Ω
ai (t)∆u(x, t − τi )dx
Ω
Z t
q(x, t)
(t − ξ)−α u(x, ξ)dξ dx, t ≥ t1 .
Ω
Green’s formula and (2.2) yield
Z
Z
∆u(x, t)dx =
m Z
X
(3.4)
0
∂u(x, t)
dS = −
∂N
Z
g(x, t)u(x, t)dS ≤ 0, t ≥ t1
(3.5)
∂Ω
and
Z
Z
∆u(x, t − τi )dx =
Ω
∂Ω
Z
∂u(x, t − τi )
dS
∂N
=−
(3.6)
g(x, t − τi )u(x, t − τi )dS ≤ 0, t ≥ t1 , i = 1, 2, · · · , m,
∂Ω
where dS is the surface element on ∂Ω. Noting the assumpition (A2), we have
Z
Z Z t
Z t
q(x, t)
(t − ξ)−α u(x, ξ)dξ dx ≥ q(t)
(t − ξ)−α u(x, ξ)dξ dx
Ω
0
Ω
0
Z t
Z
−α
= q(t) (t − ξ)
u(x, ξ)dx dξ, t ≥ t1 .
0
Let
(3.7)
Ω
Z
V (t) =
u(x, t)dx.
Ω
Combining (3.4)–(3.7), we obtain
1+α
α
D+
V (t) + p(t)D+
V (t) ≤ −q(t)E(t), t ≥ t1 ,
where
Z
E(t) =
0
t
(t − ξ)−α V (ξ)dξ > 0.
(3.8)
(3.9)
W. N. Li, W. Sheng, J. Nonlinear Sci. Appl. 9 (2016), 1600–1608
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Using Lemma 2.4, it follows from (3.3), (3.8) and (3.9) that
α
(D+
V (t))v(t)
0
1+α
α
= (D+
V (t))v(t) + (D+
V (t))p(t)v(t)
(3.10)
≤ −q(t)E(t)v(t) < 0, t ≥ t1 .
α V (t))v(t) is strictly decreasing in [t , ∞) and we can claim that D α V (t) > 0 for t ≥ t . In fact,
Hence, (D+
1
1
+
α
if D+ V (t) ≤ 0 for t ≥ t1 , then there exists a T ≥ t1 such that
α
α
(D+
V (t))v(t) < (D+
V (T ))v(T ) = C1 < 0, t > T.
(3.11)
By Lemma 2.5, from (3.11) we have
0
E (t)
C1
α
= D+
V (t) <
= C1 exp −
Γ(1 − α)
v(t)
Z
t
p(s)ds , t ≥ T.
(3.12)
Z t
p(ξ)dξ ds, t > T.
exp −
(3.13)
t0
Integrating (3.12) from T to t, we obtain
Z
t
E(t) < E(T ) + Γ(1 − α)C1
t0
T
Letting t → ∞ in (3.13), we have limt→∞ E(t) = −∞, which contradicts with the fact E(t) > 0.
Define the function
α V (t))v(t)
(D+
w(t) =
, t ≥ t1 .
E(t)
(3.14)
Then w(t) > 0 and
0
0
α V (t))v(t)]
α V (t))v(t)E (t)
[(D+
(D+
w (t) =
−
E(t)
E 2 (t)
1+α
α
α V (t))2 v(t)
(D+ V (t))v(t) + (D+ V (t))p(t)v(t) Γ(1 − α)(D+
−
=
E(t)
E 2 (t)
≤ −q(t)v(t), t ≥ t1 .
0
(3.15)
Integrating (3.15) from t1 to t, we obtain
Z
t
w(t) ≤ w(t1 ) −
q(s)v(s)ds.
(3.16)
t1
Letting t → ∞ in (3.16) and noting the assumption (3.2), we can obtain a contradiction with w(t) > 0. The
proof of Theorem 3.1 is complete.
Theorem 3.2. Assume that (3.1) holds and
Z ∞
q(s) −
t0
p2 (s)
ds = ∞, t0 > 0.
4Γ(1 − α)
(3.17)
Then every solution of the problem (2.1)–(2.2) is oscillatory in G.
Proof. Suppose to the contrary that there is a nonoscillatory solution u(x, t) to the problem (2.1)–(2.2)
which has no zero in Ω × [t0 , ∞) for some t0 > 0. Without loss of generality, we may assume that u(x, t) >
0 and u(x, t − τi ) > 0 in Ω × [t1 , ∞) for t1 ≥ t0 , i = 1, 2, · · · , m.
α V (t) > 0 for t ≥ t hold. Defined the
As in the proof of Theorem 3.1, we obtain that (3.7) and D+
1
function
Dα V (t)
, t ≥ t1 .
(3.18)
w(t)
e = +
E(t)
W. N. Li, W. Sheng, J. Nonlinear Sci. Appl. 9 (2016), 1600–1608
1605
Then w(t)
e > 0 and
0
0
w
e (t) =
≤
=
=
≤
1+α
α V (t)
D+
V (t) E (t)D+
−
E(t)
E 2 (t)
α
α V (t))2
−p(t)D+ V (t) − q(t)E(t) Γ(1 − α)(D+
−
E(t)
E 2 (t)
2
−q(t) − p(t)w(t)
e − Γ(1 − α)w
e (t)
2
p
p(t)
p2 (t)
−q(t) −
Γ(1 − α)w(t)
e + p
+
4Γ(1 − α)
2 Γ(1 − α)
p2 (t)
, t ≥ t1 .
− q(t) −
4Γ(1 − α)
(3.19)
Integrating (3.19) from t1 to t, we obtain
Z t
w(t)
e ≤ w(t
e 1) −
t1
p2 (s)
q(s) −
ds.
4Γ(1 − α)
(3.20)
Letting t → ∞ in (3.20), we can obtain a contradiction with w(t)
e > 0. This completes the proof of Theorem
3.2.
4. Oscillation of the Problem (2.1)–(2.3)
The following fact [3] will be used.
The smallest eigenvalue β0 of the Dirichlet problem
∆ω(x) + βω(x) = 0, in Ω,
ω(x) = 0, on ∂Ω,
is positive and the corresponding eigenfunction ϕ(x) is positive in Ω.
Theorem 4.1. Assume that the conditions of Theorem 3.1 hold. Then every solution of the problem (2.1)–
(2.3) is oscillatory in G.
Proof. Suppose to the contrary that there is a nonoscillatory solution u(x, t) to the problem (2.1)–(2.3)
which has no zero in Ω × [t0 , ∞) for some t0 > 0. Without loss of generality, we may assume that u(x, t) >
0 and u(x, t − τi ) > 0 in Ω × [t1 , ∞) for t1 ≥ t0 , i = 1, 2, · · · , m.
Multiplying both sides of (2.1) by ϕ(x) and integrating with respect to x over the domain Ω, we have
Z
Z
1+α
α
D+,t u(x, t)ϕ(x)dx + p(t) D+,t
u(x, t)ϕ(x)dx
Ω
Ω
Z
= a(t)
∆u(x, t)ϕ(x)dx +
Ω
i=1
Z
−
m Z
X
q(x, t)
Ω
Z
t
−α
(t − ξ)
ai (t)∆u(x, t − τi )ϕ(x)dx
u(x, ξ)dξ ϕ(x)dx, t ≥ t1 .
0
Green’s formula and (2.3) yield
Z
Z
Z
∆u(x, t)ϕ(x)dx =
u(x, t)∆ϕ(x)dx = −β0
u(x, t)ϕ(x)dx ≤ 0, t ≥ t1 ,
Ω
(4.1)
Ω
Ω
(4.2)
Ω
and
Z
Z
∆u(x, t − τi )ϕ(x)dx =
Ω
u(x, t − τi )∆ϕ(x)dx
Z
= −β0
u(x, t − τi )ϕ(x)dx ≤ 0, t ≥ t1 , i = 1, 2, · · · , m.
Ω
Ω
(4.3)
W. N. Li, W. Sheng, J. Nonlinear Sci. Appl. 9 (2016), 1600–1608
1606
It follows from the assumpition (A2) that
Z Z t
Z
Z t
(t − ξ)−α u(x, ξ)dξ ϕ(x)dx ≥ q(t)
(t − ξ)−α u(x, ξ)dξ ϕ(x)dx
q(x, t)
0
Ω
0
Ω
Z t
Z
−α
u(x, ξ)ϕ(x)dx dξ, t ≥ t1 .
= q(t) (t − ξ)
Ω
0
Let
(4.4)
Z
u(x, t)ϕ(x)dx, t ≥ t1 .
U (t) =
Ω
Combining (4.1)–(4.4), we have
1+α
α
D+
U (t) + p(t)D+
U (t) ≤ −q(t)E(t), t ≥ t1 ,
where
Z
E(t) =
t
(4.5)
(t − ξ)−α U (ξ)dξ > 0.
(4.6)
0
Using Lemma 2.4, it follows from (4.5) and (4.6) that
α
(D+
U (t))v(t)
0
1+α
α
= (D+
U (t))v(t) + (D+
U (t))p(t)v(t)
(4.7)
≤ −q(t)E(t)v(t) < 0, t ≥ t1 .
α U (t))v(t) is strictly decreasing in [t , ∞) and D α U (t) > 0 for t ≥ t .
Hence, (D+
1
1
+
The remainder of the proof is similar to that of Theorem 3.1 and we omit it here. The proof of Theorem
4.1 is complete.
It is not difficult to prove the following theorem.
Theorem 4.2. Assume that the conditions of Theorem 3.2 hold. Then every solution of the problem (2.1)–
(2.3) is oscillatory in G.
5. Examples
In this section, we give two examples to illustrate our main results.
Example 5.1. Consider the following partial fractional differential equation
5
1 14
1
D+4 u(x, t) + D+,t
u(x, t) = et ∆u(x, t) + 2t∆u(x, t − 1)
t
8
Z
1
1 t
3
− x + 2
(t − ξ)− 4 u(x, ξ)dξ, (x, t) ∈ (0, π) × R+ ,
t
0
(5.1)
with the boundary condition
ux (0, t) = ux (π, t) = 0, t > 0.
(5.2)
Here α = 41 , Ω = (0, π), n = 1, m = 1, a1 (t) = 2t, τ1 = 1, p(t) = 1t , a(t) = 18 et and q(x, t) = x3 +
q(t) = t12 . It is easy to see that
Z
∞
Z
t
exp −
t0
Z
∞
v(t) = exp
t0
Z
t
t0
Z
t
1 dξ dt =
ξ
exp −
p(ξ)dξ dt =
t0
t0
Z
p(ξ)dξ = exp
t
t0
Z
∞
t0
1 t
dξ = ,
ξ
t0
t0
dt = ∞,
t
1
.
t2
Hence
(5.3)
(5.4)
W. N. Li, W. Sheng, J. Nonlinear Sci. Appl. 9 (2016), 1600–1608
and
Z
∞
Z
∞
q(s)v(s) =
t0
t0
1 s
ds =
s2 t0
Z
∞
t0
1
ds = ∞, t0 > 0.
t0 s
1607
(5.5)
Therefore, the conditions of Theorem 3.1 are fulfilled. Then every solution of the problem (5.1)–(5.2)
oscillates in (0, π) × R+ .
Example 5.2. Consider the following partial fractional differential equation
6
1 15
t
1
D+5 u(x, t) + D+,t
u(x, t) = 3e−t ∆u(x, t) + ∆(x, t − )
t
3
3
Z t
1
1
(t − ξ)− 5 u(x, ξ)dξ, (x, t) ∈ (0, π) × R+ ,
− 2x2 +
t 0
(5.6)
with the boundary condition
u(0, t) = u(π, t) = 0, t > 0.
(5.7)
Here α = 51 , Ω = (0, π), n = 1, m = 1, p(t) = 1t , a(t) = 3e−t , a1 (t) = 3t , τ1 = 31 and q(x, t) = 2x2 + 1t .
Hence q(t) = 1t .
We easily see that
Z ∞
Z ∞
Z ∞
Z t
Z t1 t0
dξ dt =
dt = ∞
(5.8)
exp −
p(ξ)dξ dt =
exp −
t
t0
t0
t0
t0
t0 ξ
and
Z
∞
t0
Z ∞
1
1
p2 (s)
ds =
−
ds = ∞, t0 > 0.
q(s) −
s 4Γ( 45 )s2
4Γ( 45 )
t0
(5.9)
Therefore, the conditions of Theorem 4.2 are satisfied. Then every solution of the problem (5.6)–(5.7)
oscillates in (0, π) × R+ .
Acknowledgements
This work is supported by the National Natural Science Foundation of China (10971018). The authors
thank the referee very much for his valuable comments and suggestions on this paper.
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