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# Global Asymptotic Stability of Solutions to Nonlinear Marine Riser Equation

*Journal of Inequalities and Applications*
**volume 2010**, Article number: 504670 (2010)

## Abstract

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:

where is the flexural rigidity of the riser, is the "effective tension", is the coefficient of the Coriolis force, is the coefficient of the nonlinear drag force, and is the mass line density. 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

under boundary conditions (1.2). Here , and are given positive numbers, is given real number, is a function, and . 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 . 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):

where is a bounded domain with sufficiently smooth boundary . , and are given positive numbers, and , 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 . In this work, stands for the norm in .

## 2. Decay Estimate

Theorem 2.1.

Suppose that , and are arbitrary positive numbers, and number satisfies

where is the first eigenvalue of the operator with the homogeneous Dirichlet boundary condition. is an arbitrary positive number when and

Then the following estimate holds:

where A depends only on the initial data and the numbers , , , , and .

Proof.

We multiply (1.4) by and integrate over :

Since

we obtain

Let . Multiplying (1.4) by , integrating over and adding to (2.6), we obtain

Using the method integrating by parts, we get

Hence we obtain

Let

Then we have from (2.9)

where . Using Cauchy-Schwarz and Young's inequalities, we can get the following estimate:

It is not difficult to see that

Using inequalities (2.12) and (2.13) in (2.11), we obtain

Let

then

From (2.14), we get

Let

From (2.6), we have

Therefore is a Lyapunov functional. From (2.20), we find that

Since , we obtain

If is nonnegative, then we have

where .

If is negative, then, using (2.13), we have

where .

Therefore if either nonnegative or negative then it is clear that

where and . Using (2.25), we obtain from (2.17)

Integrating (2.26) with respect to , we can get

Using Poincare's and Cauchy-Schwarz inequalities, we can estimate from below:

thus for

the following estimate holds:

where

Therefore,

Now we can estimate the right-hand side of (2.34) from below. Due to Holder inequality and (2.22), we obtain

where is a positive constant depending on the initial data and the parameters of (1.4).

Using the Holder inequality and the Sobolev imbedding , we obtain

where is a positive constant depending on . Due to (2.22) and

we obtain

where

Therefore

It follows then that for large values of , , the following estimate is valid:

where . Hence we have from (2.19)

From this inequality it follows that the zero solution (1.4)–(1.6) is globally asymptotically stable.

## References

Köhl M: An extended Liapunov approach to the stability assessment of marine risers.

*Zeitschrift für Angewandte Mathematik und Mechanik*1993, 73(2):85–92. 10.1002/zamm.19930730208Kalantarov VK, Kurt A: 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*1997, 77(3):209–215. 10.1002/zamm.19970770310Nakao M: Remarks on the existence and uniqueness of global decaying solutions of the nonlinear dissipative wave equations.

*Mathematische Zeitschrift*1991, 206(2):265–276.Haraux A, Zuazua E: Decay estimates for some semilinear damped hyperbolic problems.

*Archive for Rational Mechanics and Analysis*1988, 100(2):191–206. 10.1007/BF00282203Marcati P: Decay and stability for nonlinear hyperbolic equations.

*Journal of Differential Equations*1984, 55(1):30–58. 10.1016/0022-0396(84)90087-1

## Aknowledgment

Special thanks to Prof. Dr. Varga Kalantarov.

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**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Gür, Ş. Global Asymptotic Stability of Solutions to Nonlinear Marine Riser Equation.
*J Inequal Appl* **2010**, 504670 (2010). https://doi.org/10.1155/2010/504670

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DOI: https://doi.org/10.1155/2010/504670