# Global Asymptotic Stability of Solutions to Nonlinear Marine Riser Equation

- Şevket Gür
^{1}Email author

**2010**:504670

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

© Şevket Gür. 2010

**Received: **28 May 2010

**Accepted: **14 September 2010

**Published: **15 September 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

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.

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].

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.

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

Proof.

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

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

## Declarations

### Aknowledgment

Special thanks to Prof. Dr. Varga Kalantarov.

## Authors’ Affiliations

## References

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## Copyright

This article is published under license to BioMed Central Ltd. 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.