Global Asymptotic Stability of Solutions to Nonlinear Marine Riser Equation
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Research Article Global Asymptotic Stability of Solutions to Nonlinear Marine Riser Equation ¨ S¸evket Gur Department of Mathematics, Sakarya University, 54100 Sakarya, Turkey Correspondence should be addressed to S¸evket Gur, ¨ [email protected] Received 28 May 2010; Revised 25 August 2010; Accepted 14 September 2010 Academic Editor: Michel C. Chipot Copyright q 2010 S¸evket Gur. ¨ 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, Kohl ¨ 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
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