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
Şevket Gür
Department of Mathematics, Sakarya University, 54100 Sakarya, Turkey
Correspondence should be addressed to Şevket Gür, sgur@sakarya.edu.tr
Received 28 May 2010; Revised 25 August 2010; Accepted 14 September 2010
Academic Editor: Michel C. Chipot
Copyright q 2010 Şevket Gür. 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 following initial boundary value problem describing the
dynamics of marine riser:
mutt EIuxxxx − Neff ux x aux but |ut | 0,
u0, t uxx 0, t ul, t uxx l, t 0,
x ∈ 0, l, t > 0,
t > 0,
1.1
1.2
where EI is the flexural rigidity of the riser, Neff 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, Köhl 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
mutt kuxxxx − axux x γutx but |ut |p 0
1.3
2
Journal of Inequalities and Applications
under boundary conditions 1.2. Here p, m, k, and b are given positive numbers, γ is given
real number, ax is a C1 0, l function, and ax ≥ −c0 > 0for all x ∈ 0, l. It is shown that
the zero solution of the problem 1.3-1.2 is globally asymptotically stable, that is, the zero
solution is stable and all solutions of this problem are tending to zero when t → ∞. Moreover
the polynomial decay rate for solutions is established.
There are many articles devoted to the investigation of the asymptotic behavior of
solutions of nonlinear wave equations with nonlinear dissipative terms see, e.g. 3, 4,
where theorems on asymptotic stability of the zero solution and estimates of the zero solution
and the estimates of the rate of decay of solutions to second order wave equations are
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:
n
utt kΔ2 u − aΔu γi utxi b|ut |p ut 0,
x ∈ Ω, t > 0,
1.4
i1
ux, 0 u0 x,
ut x, 0 u1 x,
u Δu 0,
x ∈ Ω,
x ∈ ∂Ω, t > 0,
1.5
1.6
where Ω ⊂ Rn is a bounded domain with sufficiently smooth boundary ∂Ω. k, b, and p are
given positive numbers, and a, γi , i 1, . . . , n are given real numbers.
Following 2, 5, we prove that all solutions of the problem 1.4–1.6 are tending to
zero with a polynomial rate as t → ∞. In this work, · stands for the norm in L2 Ω.
2. Decay Estimate
Theorem 2.1. Suppose that k, b, and p are arbitrary positive numbers, and number a satisfies
a kλ1 m0 > 0,
2.1
where λ1 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∈
4
0,
n−2
when n ≥ 3.
2.2
Then the following estimate holds:
⎧
⎨At−p1/p2 , p ∈ 0, 1,
1
m
0
2
2
ut ∇u ≤
⎩At−2/p2 ,
2
2
p ≥ 1, t ∈ 1, ∞,
where A depends only on the initial data and the numbers a, b, p, γi , i 1, . . . , n, and λ1 .
2.3
Journal of Inequalities and Applications
3
Proof. We multiply 1.4 by ut and integrate over Ω:
n
d 1
k
a
2
2
2
utxi ut dx b
|ut |p2 dx 0.
ut Δu ∇u γi
dt 2
2
2
Ω
Ω
i1
2.4
Since
n
n
γi
utxi ut dx γi
i1
Ω
i1
Ω
∂
∂xi
1 2
u dx 0,
2 t
2.5
we obtain
d 1
k
a
|ut |p2 dx 0.
ut 2 Δu2 ∇u2 b
dt 2
2
2
Ω
2.6
Let δ > 0. Multiplying 1.4 by δu, integrating over Ω and adding to 2.6, we obtain
d 1
k
a
ut 2 Δu2 ∇u2 δu, ut − δut 2 kδΔu2
dt 2
2
2
n
aδ∇u2 δ γi
utxi udx bδ
|ut |p ut udx b
|ut |p2 dx 0.
i1
Ω
Ω
2.7
Ω
Using the method integrating by parts, we get
n
n
δ γi
utxi udx −δ γi
uxi ut dx.
i1
Ω
i1
Ω
2.8
Hence we obtain
d 1
k
a
ut 2 Δu2 ∇u2 δu, ut − δut 2 kδΔu2 aδ∇u2
dt 2
2
2
n
p
− δ γi
uxi ut dx bδ
|ut | ut udx b
|ut |p2 dx 0.
i1
Ω
Ω
2.9
Ω
Let
E1 t k
a
1
ut 2 Δu2 ∇u2 δu, ut .
2
2
2
2.10
4
Journal of Inequalities and Applications
Then we have from 2.9
d
E1 t δut 2 − kδΔu2 − aδ∇u2
dt
δγ |∇u||ut |dx bδ
|ut |p ut udx − b
|ut |p2 dx,
Ω
Ω
2.11
Ω
where |γ| γ12 γ22 · · · γn2 . Using Cauchy-Schwarz and Young’s inequalities, we can get
the following estimate:
δγ Ω
1
δ2 2
γ ∇u2 ut 2 .
2
2
|∇u||ut |dx ≤
2.12
It is not difficult to see that
∇u ≤ λ−1/2
Δu.
1
2.13
Using inequalities 2.12 and 2.13 in 2.11, we obtain
d
E1 t dt
δ2 2
1
2
γ
δ
ut − δk −
Δu2
2
2λ1
2
p
− aδ∇u bδ
|ut | ut udx − b
|ut |p2 dx.
Ω
2.14
Ω
Let
2λ1 k
0 < δ < 2 ,
γ 2.15
then
L δk −
δ2 2
γ > 0.
2λ1
2.16
From 2.14, we get
d
E1 t δ 1ut 2 bδ
dt
Ω
|ut |p1 udx − b
Ω
|ut |p2 dx −
1
ut 2 aδ∇u2 LΔu2 .
2
2.17
Journal of Inequalities and Applications
5
Let
Et k
a
1
ut 2 Δu2 ∇u2 ,
2
2
2
2.18
kλ1
a
1
m0
1
ut 2 ∇u2 ∇u2 ≥ ut 2 ∇u2
2
2
2
2
2
1 m0 ≥ min ,
ut 2 ∇u2 .
2 2
Et ≥
2.19
From 2.6, we have
d
Et −b
dt
Ω
|ut |p2 dx ≤ 0.
2.20
Therefore Et is a Lyapunov functional. From 2.20, we find that
Et − E0 −b
t 0
Ω
|ut |p2 dx ds.
2.21
Since Et ≥ 0, we obtain
t 0
Ω
|ut |p2 dx ds E0
.
b
2.22
If a is nonnegative, then we have
1
2L
Et ≥ D1 Et,
ut 2 aδ∇u2 LΔu2 ≥ min 1, 2δ,
2
k
2.23
where D1 min{1, 2δ, 2L/k}.
If a is negative, then, using 2.13, we have
1
1
aδ
L Δu2
ut 2 aδ∇u2 LΔu2 ≥ ut 2 2
2
λ1
2 aδ
1
k
≥ min 1,
L
ut 2 Δu2 ≥ D2 Et,
k λ1
2
2
2.24
where D2 min{1, 2/kaδ/λ1 L}.
Therefore if a either nonnegative or negative then it is clear that
1
ut 2 aδ∇u2 LΔu2 ≥ DEt,
2
2.25
6
Journal of Inequalities and Applications
where D min{1, 2δ, 2L/k, 2/kaδ/λ1 L} and 0 < δ < 2m0 /|γ|2 . Using 2.25, we obtain
from 2.17
d
E1 t δ 1ut 2 − DEt bδ
dt
Ω
|ut |
p1
|u|dx − b
Ω
|ut |p2 dx.
2.26
Integrating 2.26 with respect to t, we can get
E1 t − E1 0 δ 1
t 0
bδ
DtEt D
t t
0
Ω
Ω
|ut |2 dx ds − D
|ut |
Esds
0
p1
|u|dx ds − b
t 0
Ω
|ut |p2 dx ds,
Esds E1 0 − E1 t δ 1
0
bδ
t
t Ω
0
|ut |
p1
|u|dx ds − b
DtEt E1 0 − E1 t δ 1
t 0
Ω
2.27
2
|ut | dx ds
|ut |p2 dx ds,
2
Ω
Ω
0
t 0
t |ut | dx ds bδ
t 0
Ω
|ut |p1 |u|dx ds.
2.28
Using Poincare’s and Cauchy-Schwarz inequalities, we can estimate E1 t from below:
k
a
1
ut 2 Δu2 ∇u2 δu, ut 2
2
2
E1 t k
a
1
ut 2 Δu2 ∇u2 − δ|u, ut |
2
2
2
m0
δ
δ
1
ut 2 ∇u2 − u2 − ut 2
2
2
2
2
2.29
1
m0
δ
δ
∇u2 − ut 2
ut 2 ∇u2 −
2
2
2λ1
2
1
1
δ
m0 −
1 − δut 2 ∇u2 ,
2
2
λ1
thus for
2λ1 k
2m0
0 < δ < min 2 , 1, m0 λ1 , 2
γ γ 2.30
the following estimate holds:
E1 t d1 ut 2 ∇u2 ,
2.31
Journal of Inequalities and Applications
7
where
1
δ
.
d1 min 1 − δ, m0 −
2
λ1
2.32
Therefore,
E1 0 − E1 t E1 0,
t t DtEt E1 0 δ 1
|ut |2 dx ds bδ
|ut |p1 |u|dx ds.
Ω
0
2.33
2.34
Ω
0
Now we can estimate the right-hand side of 2.34 from below. Due to Holder inequality and
2.22, we obtain
δ 1
t 0
2
Ω
|ut | dx ds δ 1
t |ut |
Ω
0
E0
δ 1
b
p2
2/p2 t dx ds
Ω
0
2/p2 t dx ds
2.35
p/p2
Ω
0
p/p2
C1 tp/p2 ,
dx ds
where C1 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 ⊂ Lp2 , we obtain
t bδ
0
Ω
|ut |
p1
|u|dx ds bδ
t Ω
0
bδ
p2
t 0
C2 bδ
|ut |
|ut |
p1/p2 t
dx ds
0
t 0
dx ds
0
p2
Ω
p1/p2 t Ω
|ut |
p2
1/p2
Ω
|u|
p2
dx ds
1/p2
p2
up2 ds
p1/p2 t
dx ds
1/p2
∇u
p2
ds
,
0
2.36
where C2 is a positive constant depending on Ω. Due to 2.22 and
2E0
,
a kλ1
2.37
|ut |p1 |u|dx ds C3 t1/p2 ,
2.38
∇u2 we obtain
t bδ
0
Ω
8
Journal of Inequalities and Applications
where
C3 bδC2
2E0
a kλ1
1/2 E0
b
p1/p2
.
2.39
Therefore
DtEt E1 0 C1 tp/p2 C3 t1/p2 ,
Et D−1 E1 0t−1 C1 t−2/p2 C3 t−p1/p2 .
2.40
It follows then that for large values of t, t ≥ 1, the following estimate is valid:
Et ≤
⎧
⎨At−p1/p2 ,
⎩At−2/p2 ,
p ∈ 0, 1,
p ≥ 1,
2.41
where A D−1 E1 0 C1 C3 . Hence we have from 2.19
⎧
⎨At−p1/p2 ,
1
m0
ut 2 ∇u2 ≤
⎩At−2/p2 ,
2
2
p ∈ 0, 1,
p ≥ 1, t ∈ 1, ∞.
2.42
From this inequality it follows that the zero solution 1.4–1.6 is globally asymptotically
stable.
Aknowledgment
Special thanks to Prof. Dr. Varga Kalantarov.
References
1 M. Köhl, “An extended Liapunov approach to the stability assessment of marine risers,” Zeitschrift für
Angewandte Mathematik und Mechanik, vol. 73, no. 2, pp. 85–92, 1993.
2 V. K. Kalantarov and A. Kurt, “The long-time behavior of solutions of a nonlinear fourth order
wave equation, describing the dynamics of marine risers,” Zeitschrift für Angewandte Mathematik und
Mechanik, vol. 77, no. 3, pp. 209–215, 1997.
3 M. Nakao, “Remarks on the existence and uniqueness of global decaying solutions of the nonlinear
dissipative wave equations,” Mathematische Zeitschrift, vol. 206, no. 2, pp. 265–276, 1991.
4 A. Haraux and E. Zuazua, “Decay estimates for some semilinear damped hyperbolic problems,”
Archive for Rational Mechanics and Analysis, vol. 100, no. 2, pp. 191–206, 1988.
5 P. Marcati, “Decay and stability for nonlinear hyperbolic equations,” Journal of Differential Equations,
vol. 55, no. 1, pp. 30–58, 1984.