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On nonexistence of global solutions of a quasilinear riser equation

  • Jorge A. Esquivel-Avila EMAIL logo
Veröffentlicht/Copyright: 23. Dezember 2017
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Abstract

We consider a quasilinear equation of fourth order that models the mechanical vibrations of a marine riser. We study the nonexistence of global solutions for any real value of the initial energy. To this end, we construct a new invariant set and improve previous results.

MSC 2010: 35L70; 35B35; 35B40

Acknowledgements

I want to thank to the referee for the very useful suggestions to improve the final version of the work.

References

[1] S. Alinhac, Blowup for Nonlinear Hyperbolic Equations, Progr. Nonlinear Differential Equations Appl. 17, Birkhäuser, Boston, 1995. 10.1007/978-1-4612-2578-2Suche in Google Scholar

[2] J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser. (2) 28 (1977), no. 112, 473–486. 10.1093/qmath/28.4.473Suche in Google Scholar

[3] V. Bayrak and M. Can, Global nonexistence and numerical instabilities of the vibrations of a riser, Math. Comput. Appl. 2 (1997), 45–52. Suche in Google Scholar

[4] J. A. Esquivel-Avila, Blow up and boundedness for high energies of a quasilinear riser equation, Nonlinear Anal. 71 (2009), no. 5–6, 1925–1932. 10.1016/j.na.2009.01.028Suche in Google Scholar

[5] J. A. Esquivel-Avila, Decay and nonexistence of global solutions of a quasilinear riser equation, Math. Methods Appl. Sci. 33 (2010), no. 1, 71–79. 10.1002/mma.1151Suche in Google Scholar

[6] A. Gmira and M. Guedda, A Note on the global nonexistence of solutions to vibrations of a riser, Arab. J. Sci. Eng. 27 (2002), 197–206. Suche in Google Scholar

[7] J. Hao, S. Li and Y. Zhang, Blow up and global solutions for a quasilinear riser problem, Nonlinear Anal. 67 (2007), 974–980. 10.1016/j.na.2006.06.042Suche in Google Scholar

[8] 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, Z. Angew. Math. Mech. 77 (1997), 209–215. 10.1002/zamm.19970770310Suche in Google Scholar

[9] M. Köhl, An extended Liapunov approach to the stability assessment of marine risers, Z. Angew. Math. Mech. 73 (1993), no. 2, 85–92. 10.1002/zamm.19930730208Suche in Google Scholar

[10] N. Kutev, N. Kolkovska and M. Dimova, Sign-preserving functionals and blow up to Klein–Gordon equation with arbitrary high energy, Appl. Anal. 95 (2016), 860–873. 10.1080/00036811.2015.1038994Suche in Google Scholar

[11] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt=-Au+(u), Trans. Amer. Math. Soc. 192 (1974), 1–21. 10.1090/S0002-9947-1974-0344697-2Suche in Google Scholar

[12] S. Messaoudi and B. Said-Houari, A global nonexistence result for the nonlinearly damped multi-dimensional boussinesq equation, Arab. J. Sci. Eng. 31 (2006), 57–68. Suche in Google Scholar

[13] L. E. Payne and D. E. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22 (1975), 273–303. 10.1007/BF02761595Suche in Google Scholar

[14] M. G. Sun, The stress boundary layers of a slender riser in a steady flow, Adv. Hydrodyn. 4 (1986), 2–43. Suche in Google Scholar

[15] Y. Wang, A sufficient condition for finite time blow up of the nonlinear Klein–Gordon equations with arbitrarily positive initial energy, Proc. Amer. Math. Soc. 136 (2008), no. 10, 3477–3482. 10.1090/S0002-9939-08-09514-2Suche in Google Scholar

Received: 2017-04-22
Revised: 2017-11-05
Accepted: 2017-11-30
Published Online: 2017-12-23
Published in Print: 2018-07-01

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