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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 S¸ evket G ¨ ur Department of Mathematics, Sakarya University, 54100 Sakarya, Turkey Correspondence should be addressed to S¸evket G ¨ ur, sgur@sakarya.edu.tr Received 28 May 2010; Revised 25 August 2010; Accepted 14 September 2010 Academic Editor: Michel C. Chipot Copyright q 2010 S¸evket G ¨ ur. 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 f ollowing initial boundary value problem describing the dynamics of marine riser: mu tt  EIu xxxx −  N eff u x  x  au x  bu t | u t |  0,x∈  0,l  ,t>0, 1.1 u  0,t   u xx  0,t   u  l, t   u xx  l, t   0,t>0, 1.2 where EI is the flexural rigidity of the riser, N eff 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, K ¨ ohl 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 mu tt  ku xxxx −  a  x  u x  x  γu tx  bu t | u t | p  0 1.3 2 Journal of Inequalities and Applications under boundary conditions 1.2.Herep, m, k,andb are given positive numbers, γ is given real number, ax is a C 1 0,l function, and ax ≥−c 0 > 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: u tt  kΔ 2 u − aΔu  n  i1 γ i u tx i  b | u t | p u t  0,x∈ Ω,t>0, 1.4 u  x, 0   u 0  x  ,u t  x, 0   u 1  x  ,x∈ Ω, 1.5 u Δu  0,x∈ ∂Ω,t >0, 1.6 where Ω ⊂ R n is a bounded domain with sufficiently smooth boundary ∂Ω. k, b,andp are given positive numbers, and a, γ i , i  1, ,nare 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 → ∞. I n this work, ·stands for the norm in L 2 Ω. 2. Decay Estimate Theorem 2.1. Suppose that k, b, and p are arbitrary positive numbers, and number a satisfies a  kλ 1  m 0 > 0, 2.1 where λ 1 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 ∈  0, 4 n − 2  when n ≥ 3. 2.2 Then the following estimate holds: 1 2  u t  2  m 0 2  ∇u  2 ≤ ⎧ ⎨ ⎩ At −p1/p2 ,p∈  0, 1  , At −2/p2 ,p≥ 1,t∈  1, ∞  , 2.3 where A depends only on the initial data and the numbers a, b, p, γ i , i  1, ,n, and λ 1 . Journal of Inequalities and Applications 3 Proof. We multiply 1.4 by u t and integrate over Ω: d dt  1 2  u t  2  k 2  Δu  2  a 2  ∇u  2   n  i1 γ i  Ω u tx i u t dx  b  Ω | u t | p2 dx  0. 2.4 Since n  i1 γ i  Ω u tx i u t dx  n  i1 γ i  Ω ∂ ∂x i  1 2 u 2 t  dx  0, 2.5 we obtain d dt  1 2  u t  2  k 2  Δu  2  a 2  ∇u  2   b  Ω | u t | p2 dx  0. 2.6 Let δ>0. Multiplying 1.4 by δu, integrating over Ω and adding to 2.6,weobtain d dt  1 2  u t  2  k 2  Δu  2  a 2  ∇u  2  δ  u, u t   − δ  u t  2  kδ  Δu  2  aδ  ∇u  2  δ n  i1 γ i  Ω u tx i udx  bδ  Ω | u t | p u t udx  b  Ω | u t | p2 dx  0. 2.7 Using the method integrating by parts, we get δ n  i1 γ i  Ω u tx i udx  −δ n  i1 γ i  Ω u x i u t dx. 2.8 Hence we obtain d dt  1 2  u t  2  k 2  Δu  2  a 2  ∇u  2  δ  u, u t   − δ  u t  2  kδ  Δu  2  aδ  ∇u  2 − δ n  i1 γ i  Ω u x i u t dx  bδ  Ω | u t | p u t udx  b  Ω | u t | p2 dx  0. 2.9 Let E 1  t   1 2  u t  2  k 2  Δu  2  a 2  ∇u  2  δ  u, u t  . 2.10 4 Journal of Inequalities and Applications Then we have from 2.9 d dt E 1  t   δ  u t  2 − kδ  Δu  2 − aδ  ∇u  2  δ   γ    Ω | ∇u || u t | dx  bδ  Ω | u t | p u t udx − b  Ω | u t | p2 dx, 2.11 where |γ|   γ 2 1  γ 2 2  ··· γ 2 n . Using Cauchy-Schwarz and Young’s inequalities, we can get the following estimate: δ   γ    Ω | ∇u || u t | dx ≤ δ 2 2   γ   2  ∇u  2  1 2  u t  2 . 2.12 It is not difficult to see that  ∇u  ≤ λ −1/2 1  Δu  . 2.13 Using inequalities 2.12 and 2.13 in 2.11,weobtain d dt E 1  t    δ  1 2   u t  2 −  δk − δ 2 2λ 1   γ   2   Δu  2 − aδ  ∇u  2  bδ  Ω | u t | p u t udx − b  Ω | u t | p2 dx. 2.14 Let 0 <δ< 2λ 1 k   γ   2 , 2.15 then L  δk − δ 2 2λ 1   γ   2 > 0. 2.16 From 2.14,weget d dt E 1  t    δ  1   u t  2  bδ  Ω | u t | p1 udx − b  Ω | u t | p2 dx −  1 2  u t  2  aδ  ∇u  2  L  Δu  2  . 2.17 Journal of Inequalities and Applications 5 Let E  t   1 2  u t  2  k 2  Δu  2  a 2  ∇u  2 , 2.18 E  t  ≥ 1 2  u t  2  kλ 1 2  ∇u  2  a 2  ∇u  2 ≥ 1 2  u t  2  m 0 2  ∇u  2 ≥ min  1 2 , m 0 2    u t  2   ∇u  2  . 2.19 From 2.6, we have d dt E  t   −b  Ω | u t | p2 dx ≤ 0. 2.20 Therefore Et is a Lyapunov functional. From 2.20,wefindthat E  t  − E  0   −b  t 0  Ω | u t | p2 dx ds. 2.21 Since Et ≥ 0, we obtain  t 0  Ω | u t | p2 dx ds  E  0  b . 2.22 If a is nonnegative, then we have 1 2  u t  2  aδ  ∇u  2  L  Δu  2 ≥ min  1, 2δ, 2L k  E  t  ≥ D 1 E  t  , 2.23 where D 1  min{1, 2δ, 2L/k}. If a is negative, then, using 2.13, we have 1 2  u t  2  aδ  ∇u  2  L  Δu  2 ≥ 1 2  u t  2   aδ λ 1  L   Δu  2 ≥ min  1, 2 k  aδ λ 1  L  1 2  u t  2  k 2  Δu  2  ≥ D 2 E  t  , 2.24 where D 2  min{1, 2/kaδ/λ 1  L}. Therefore if a either nonnegative or negative then it is clear that 1 2  u t  2  aδ  ∇u  2  L  Δu  2 ≥ DE  t  , 2.25 6 Journal of Inequalities and Applications where D  min{1, 2δ, 2L/k, 2/kaδ/λ 1  L} and 0 <δ<2m 0 /|γ| 2 .Using2.25,weobtain from 2.17 d dt E 1  t    δ  1   u t  2 − DE  t   bδ  Ω | u t | p1 | u | dx − b  Ω | u t | p2 dx. 2.26 Integrating 2.26 with respect to t, we can get E 1  t  − E 1  0    δ  1   t 0  Ω | u t | 2 dx ds − D  t 0 E  s  ds  bδ  t 0  Ω | u t | p1 | u | dx ds − b  t 0  Ω | u t | p2 dx ds, DtE  t   D  t 0 E  s  ds   E 1  0  − E 1  t    δ  1   t 0  Ω | u t | 2 dx ds  bδ  t 0  Ω | u t | p1 | u | dx ds − b  t 0  Ω | u t | p2 dx ds, 2.27 DtE  t    E 1  0  − E 1  t    δ  1   t 0  Ω | u t | 2 dx ds  bδ  t 0  Ω | u t | p1 | u | dx ds. 2.28 Using Poincare’s and Cauchy-Schwarz inequalities, we can estimate E 1 t from below: E 1  t   1 2  u t  2  k 2  Δu  2  a 2  ∇u  2  δ  u, u t   1 2  u t  2  k 2  Δu  2  a 2  ∇u  2 − δ |  u, u t  |  1 2  u t  2  m 0 2  ∇u  2 − δ 2  u  2 − δ 2  u t  2  1 2  u t  2  m 0 2  ∇u  2 − δ 2λ 1  ∇u  2 − δ 2  u t  2  1 2  1 − δ   u t  2  1 2  m 0 − δ λ 1   ∇u  2 , 2.29 thus for 0 <δ<min  2λ 1 k   γ   2 , 1,m 0 λ 1 , 2m 0   γ   2  2.30 the following estimate holds: E 1  t   d 1   u t  2   ∇u  2  , 2.31 Journal of Inequalities and Applications 7 where d 1  1 2 min  1 − δ, m 0 − δ λ 1  . 2.32 Therefore, E 1  0  − E 1  t   E 1  0  , 2.33 DtE  t   E 1  0    δ  1   t 0  Ω | u t | 2 dx ds  bδ  t 0  Ω | u t | p1 | u | dx ds. 2.34 Now we can estimate the right-hand side of 2.34 from below. Due to Holder inequality and 2.22,weobtain  δ  1   t 0  Ω | u t | 2 dx ds   δ  1    t 0  Ω | u t | p2 dx ds  2/p2   t 0  Ω dx ds  p/p2   δ  1   E  0  b  2/p2   t 0  Ω dx ds  p/p2  C 1 t p/p2 , 2.35 where C 1 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 ⊂ L p2 ,weobtain bδ  t 0  Ω | u t | p1 | u | dx ds  bδ   t 0  Ω | u t | p2 dx ds  p1/p2   t 0  Ω | u | p2 dx ds  1/p2  bδ   t 0  Ω | u t | p2 dx ds  p1/p2   t 0  u  p2 p2 ds  1/p2  C 2 bδ   t 0  Ω | u t | p2 dx ds  p1/p2   t 0  ∇u  p2 ds  1/p2 , 2.36 where C 2 is a positive constant depending on Ω.Dueto2.22 and  ∇u  2  2E  0  a  kλ 1 , 2.37 we obtain bδ  t 0  Ω | u t | p1 | u | dx ds  C 3 t 1/p2 , 2.38 8 Journal of Inequalities and Applications where C 3  bδC 2  2E  0  a  kλ 1  1/2  E  0  b  p1/p2 . 2.39 Therefore DtE  t   E 1  0   C 1 t p/p2  C 3 t 1/p2 , E  t   D −1  E 1  0  t −1  C 1 t −2/p2  C 3 t −p1/p2  . 2.40 It follows then that for large values of t, t ≥ 1, the following estimate is valid: E  t  ≤ ⎧ ⎨ ⎩ At −p1/p2 ,p∈  0, 1  , At −2/p2 ,p≥ 1, 2.41 where A D −1 E 1 0C 1  C 3 . Hence we have from 2.19 1 2  u t  2  m 0 2  ∇u  2 ≤ ⎧ ⎨ ⎩ At −p1/p2 ,p∈  0, 1  , At −2/p2 ,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. K ¨ ohl, “An extended Liapunov approach to the stability assessment of marine risers,” Zeitschrift f ¨ ur 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 f ¨ ur 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. . 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 S¸. 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. 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