Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 504670, 8 pages
doi:10.1155/2010/504670
Research Article
Global Asymptotic Stability of Solutions to
Nonlinear Marine Riser Equation
S¸ evket G
¨
ur
Department of Mathematics, Sakarya University, 54100 Sakarya, Turkey
Correspondence should be addressed to S¸evket G
¨
ur,
Received 28 May 2010; Revised 25 August 2010; Accepted 14 September 2010
Academic Editor: Michel C. Chipot
Copyright q 2010 S¸evket G
¨
ur. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
This paper studies initial boundary value problem of fourth-order nonlinear marine riser equation.
By using multiplier method, it is proven that the zero solution of the problem is globally
asymptotically stable.
1. Introduction
The straight-line vertical position of marine risers has been investigated with respect to
dynamic stability 1. It studies the f ollowing initial boundary value problem describing the
dynamics of marine riser:
mu
tt
EIu
u
l, t
u
xx
l, t
0,t>0, 1.2
where EI is the flexural rigidity of the riser, N
eff
is the “effective tension”, a is the coefficient
of the Coriolis force, b is the coefficient of the nonlinear drag force, and m is the mass line
density. u represents the riser deflection.
By using the Lyapunov function technique, K
¨
ohl has shown that the zero solution of
the problem is stable.
In 2, Kalantarov and Kurt have studied the initial boundary value problem for the
equation
mu
tt
ku
xxxx
−
a
obtained.
Similar results for the higher-order nonlinear wave equations are obtained in 5.
In this study, we consider the following initial boundary value problem for the
multidimensional version of 1.1:
u
tt
kΔ
2
u − aΔu
n
i1
γ
i
u
tx
i
b
|
u
t
|
p
u
t
0,x∈ Ω,t>0,
1.4
u
x, 0
a kλ
1
m
0
> 0, 2.1
where λ
1
is the first eigenvalue of the operator −Δ with the homogeneous Dirichlet boundary condition.
p is an arbitrary positive number when n ≤ 2 and
p ∈
0,
4
n − 2
when n ≥ 3. 2.2
Then the following estimate holds:
1
2
u
t
2
m
0
2
∇u
d
dt
1
2
u
t
2
k
2
Δu
2
a
2
∇u
2
n
i1
γ
t
dx
n
i1
γ
i
Ω
∂
∂x
i
1
2
u
2
t
dx 0,
2.5
we obtain
d
dt
1
2
u
t
1
2
u
t
2
k
2
Δu
2
a
2
∇u
2
δ
u, u
t
− δ
t
|
p
u
t
udx b
Ω
|
u
t
|
p2
dx 0.
2.7
Using the method integrating by parts, we get
δ
n
i1
γ
i
Ω
u
tx
i
udx −δ
n
a
2
∇u
2
δ
u, u
t
− δ
u
t
2
kδ
Δu
2
aδ
∇u
2
− δ
dx 0.
2.9
Let
E
1
t
1
2
u
t
2
k
2
Δu
2
a
2
∇u
2
2
δ
γ
Ω
|
∇u
||
u
t
|
dx bδ
Ω
|
u
t
|
p
u
t
udx − b
Ω
|
u
t
|
dx ≤
δ
2
2
γ
2
∇u
2
1
2
u
t
2
.
2.12
It is not difficult to see that
∇u
δk −
δ
2
2λ
1
γ
2
Δu
2
− aδ
∇u
2
bδ
Ω
|
u
t
|
p
u
γ
2
> 0.
2.16
From 2.14,weget
d
dt
E
1
t
δ 1
u
t
2
bδ
Ω
|
u
.
2.17
Journal of Inequalities and Applications 5
Let
E
t
1
2
u
t
2
k
2
Δu
2
a
2
∇u
1
2
u
t
2
m
0
2
∇u
2
≥ min
1
2
,
m
0
2
u
t
2
− E
0
−b
t
0
Ω
|
u
t
|
p2
dx ds.
2.21
Since Et ≥ 0, we obtain
t
0
Ω
|
u
t
|
p2
dx ds
E
E
t
≥ D
1
E
t
, 2.23
where D
1
min{1, 2δ, 2L/k}.
If a is negative, then, using 2.13, we have
1
2
u
t
2
aδ
∇u
2
L
Δu
L
1
2
u
t
2
k
2
Δu
2
≥ D
2
E
t
,
2.24
where D
2
min{1, 2/kaδ/λ
1
L}.
2
.Using2.25,weobtain
from 2.17
d
dt
E
1
t
δ 1
u
t
2
− DE
t
bδ
Ω
|
u
t
|
t
0
Ω
|
u
t
|
2
dx ds − D
t
0
E
s
ds
bδ
t
0
Ω
|
u
t
|
p1
|
0
− E
1
t
δ 1
t
0
Ω
|
u
t
|
2
dx ds
bδ
t
0
Ω
|
− E
1
t
δ 1
t
0
Ω
|
u
t
|
2
dx ds bδ
t
0
Ω
|
u
t
|
2
∇u
2
δ
u, u
t
1
2
u
t
2
k
2
Δu
2
a
2
∇u
2
−
δ
2
u
t
2
1
2
u
t
2
m
0
2
∇u
2
−
δ
2λ
1
δ
λ
1
∇u
2
,
2.29
thus for
0 <δ<min
2λ
1
k
γ
2
, 1,m
0
λ
1
,
2m
0
d
1
1
2
min
1 − δ, m
0
−
δ
λ
1
.
2.32
Therefore,
E
1
0
− E
1
t
E
1
Ω
|
u
t
|
p1
|
u
|
dx ds.
2.34
Now we can estimate the right-hand side of 2.34 from below. Due to Holder inequality and
2.22,weobtain
δ 1
t
0
Ω
|
u
t
|
2
dx ds
δ 1
0
b
2/p2
t
0
Ω
dx ds
p/p2
C
1
t
p/p2
,
2.35
where C
1
is a positive constant depending on the initial data and the parameters of 1.4.
Using the Holder inequality and the Sobolev imbedding H
1
⊂ L
p2
,weobtain
bδ
t
0
Ω
|
u
|
p2
dx ds
1/p2
bδ
t
0
Ω
|
u
t
|
p2
dx ds
p1/p2
t
0
∇u
p2
ds
1/p2
,
2.36
where C
2
is a positive constant depending on Ω.Dueto2.22 and
∇u
2
2E
0
a kλ
1
,
2.37
we obtain
bδ
t
0
E
0
b
p1/p2
.
2.39
Therefore
DtE
t
E
1
0
C
1
t
p/p2
C
3
t
1/p2
,
E
⎧
⎨
⎩
At
−p1/p2
,p∈
0, 1
,
At
−2/p2
,p≥ 1,
2.41
where A D
−1
E
1
0C
1
C
3
. Hence we have from 2.19
1
2
u
t
2
Special thanks to Prof. Dr. Varga Kalantarov.
References
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¨
ohl, “An extended Liapunov approach to the stability assessment of marine risers,” Zeitschrift f
¨
ur
Angewandte Mathematik und Mechanik, vol. 73, no. 2, pp. 85–92, 1993.
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wave equation, describing the dynamics of marine risers,” Zeitschrift f
¨
ur Angewandte Mathematik und
Mechanik, vol. 77, no. 3, pp. 209–215, 1997.
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dissipative wave equations,” Mathematische Zeitschrift, vol. 206, no. 2, pp. 265–276, 1991.
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Archive for Rational Mechanics and Analysis, vol. 100, no. 2, pp. 191–206, 1988.
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vol. 55, no. 1, pp. 30–58, 1984.
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