- Research Article
- Open Access
Global Asymptotic Stability of Solutions to Nonlinear Marine Riser Equation
© Şevket Gür. 2010
Received: 28 May 2010
Accepted: 14 September 2010
Published: 15 September 2010
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.
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 .
2. Decay Estimate
From this inequality it follows that the zero solution (1.4)–(1.6) is globally asymptotically stable.
Special thanks to Prof. Dr. Varga Kalantarov.
- 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.19930730208View ArticleMathSciNetMATHGoogle Scholar
- Kalantarov 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.19970770310MathSciNetView ArticleMATHGoogle Scholar
- Nakao M: Remarks on the existence and uniqueness of global decaying solutions of the nonlinear dissipative wave equations. Mathematische Zeitschrift 1991, 206(2):265–276.MathSciNetView ArticleMATHGoogle Scholar
- 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/BF00282203MathSciNetView ArticleMATHGoogle Scholar
- Marcati P: Decay and stability for nonlinear hyperbolic equations. Journal of Differential Equations 1984, 55(1):30–58. 10.1016/0022-0396(84)90087-1MathSciNetView ArticleMATHGoogle Scholar
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