Taiwanese Journal of Mathematics

Upper Semicontinuity of Random Attractor for a Kirchhoff Type Suspension Bridge Equation with Strong Damping and White Noise

Ling Xu and Qiaozhen Ma

Advance publication

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Abstract

This paper is devoted to the well-posedness and long-time behavior of a stochastic Kirchhoff type suspension bridge equation with strong damping. The existence of the random attractor for a Kirchhoff type suspension bridge equation with white noise is established. Moreover, the upper semicontinuity of random attractors is also provided when the coefficient of random term approaches zero.

Article information

Source
Taiwanese J. Math., Advance publication (2019), 25 pages.

Dates
First available in Project Euclid: 9 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1565337620

Digital Object Identifier
doi:10.11650/tjm/190708

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 35Q35: PDEs in connection with fluid mechanics 35B40: Asymptotic behavior of solutions

Keywords
a Kirchhoff type suspension bridge equation random dynamical system random attractor white noise upper semicontinuity

Citation

Xu, Ling; Ma, Qiaozhen. Upper Semicontinuity of Random Attractor for a Kirchhoff Type Suspension Bridge Equation with Strong Damping and White Noise. Taiwanese J. Math., advance publication, 9 August 2019. doi:10.11650/tjm/190708. https://projecteuclid.org/euclid.twjm/1565337620


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References

  • N. U. Ahmed and H. Harbi, Mathematical analysis of dynamic models of suspension bridges, SIAM J. Appl. Math. 58 (1998), no. 3, 853–874.
  • L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Spring-verlag, New York, 1998.
  • I. Bochicchio, C. Giorgi and E. Vuk, Long-term damped dynamics of the extensible suspension bridge, Int. J. Differ. Equ. 2010 (2010), Art. ID 383420, 19 pp.
  • I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Mathematics 1779, Springer-Verlag, Berlin, 2002.
  • H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields 100 (1994), no. 3, 365–393.
  • X. Fan, Random attractor for a damped sine-Gordon equation with white noise, Pacific J. Math. 216 (2004), no. 1, 63–76.
  • X. Fan and Y. Wang, Fractal dimension of attractors for a stochastic wave equation with nonlinear damping and white noise, Stoch. Anal. Appl. 25 (2007), no. 2, 381–396.
  • L. D. Humphreys, Numerical mountain pass solutions of a suspension bridge equation, Nonlinear Anal. 28 (1997), no. 11, 1811–1826.
  • A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis, SIAM Rev. 32 (1990), no. 4, 537–578.
  • Q. Ma, S. Wang and X. Chen, Uniform compact attractors for the coupled suspension bridge equations, Appl. Math. Comput. 217 (2011), no. 4, 6604–6615.
  • Q. Ma and C. Zhong, Existence of global attractors for the coupled system of suspension bridge equations, J. Math. Anal. Appl. 308 (2005), no. 1, 365–379.
  • ––––, Existence of strong solutions and global attractors for the coupled suspension bridge equations, J. Differential Equations 246 (2009), no. 10, 3755–3775.
  • W. Ma and Q. Ma, Attractors for stochastic strongly damped plate equations with additive noise, Electron. J. Differential Equations 2013 (2013), no. 111, 12 pp.
  • P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge, Arch. Rational Mech. Anal. 98 (1987), no. 2, 167–177.
  • J. Y. Park and J. R. Kang, Pullback $\mathscr{D}$ -attractors for non-autonomous suspension bridge equations, Nonlinear Anal. 71 (2009), no. 10, 4618–4623.
  • ––––, Global attractors for the suspension bridge equations with nonlinear damping, Quart. Appl. Math. 69 (2011), no. 3, 465–475.
  • ––––, Uniform attractor for non-autonomous suspension bridge equations with localized damping, Math. Methods Appl. Sci. 34 (2011), no. 4, 487–496.
  • A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44, Springer-Verlag, New York, 1983.
  • B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electron. J. Differential Equations 2009 (2009), no. 139, 18 pp.
  • ––––, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations 253 (2012), no. 5, 1544–1583.
  • S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech. 17 (1950), 35–36.
  • M. Yang, J. Duan and P. Kloeden, Asymptotic behavior of solutions for random wave equations with nonlinear damping and white noise, Nonlinear Anal. Real World Appl. 12 (2011), no. 1, 464–478.
  • M. Yang and P. E. Kloeden, Random attractors for stochastic semi-linear degenerate parabolic equations, Nonlinear Anal. Real World Appl. 12 (2011), no. 5, 2811–2821.
  • C. Zhong, Q. Ma and C. Sun, Existence of strong solutions and global attractors for the suspension bridge equations, Nonlinear Anal. 67 (2007), no. 2, 442–454.