Advanced Search
Article Contents
Article Contents

Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback

  • * Corresponding author: Wenjun Liu

    * Corresponding author: Wenjun Liu 

The first author is supported by the National Natural Science Foundation of China [grant number 11771216], the Key Research and Development Program of Jiangsu Province (Social Development) [grant number BE2019725], the Six Talent Peaks Project in Jiangsu Province [grant number 2015-XCL-020] and the Qing Lan Project of Jiangsu Province

Abstract Full Text(HTML) Related Papers Cited by
  • In the present paper, we consider a suspension bridge problem with a nonlinear delay term in the internal feedback. Namely, we investigate the following equation:

    $ \begin{equation*} u_{tt}+ \Delta^2 u + \delta_1 g_1 (u_t (x,y,t))+ \delta_2 g_2 (u_t (x,y, t-\tau))+ h(u(x,y,t)) = f(x,y), \end{equation*} $

    together with some suitable initial data and boundary conditions. We prove the global existence of solutions by means of the energy method combined with the Faedo-Galerkin procedure under a certain relation between the weight of the delay term in the feedback and the weight of the nonlinear frictional damping term without delay. Moreover, we establish the existence of a global attractor for the above-mentioned system by proving the existence of an absorbing set and the asymptotic smoothness of the semigroup $ S(t) $.

    Mathematics Subject Classification: Primary: 35B40; Secondary: 35B41, 35L57.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] V. I. Arnold, Mathematical Methods of Classical Mecanics, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.
    [2] M. Al-GwaizV. Benci and F. Gazzola, Bending and stretching energies in a rectangular plate modeling suspension bridges, Nonlinear Anal., 106 (2014), 18-34.  doi: 10.1016/j.na.2014.04.011.
    [3] O. H. Ammann, T. von Karman and G. B. Woodruff, The Failure of the Tacoma Narrows Bridge, Federal Works Agency, 1941.
    [4] J. M. W. Brownjohn, Observations on non-linear dynamic characteristics of suspension bridges, Earthquake Engng. Struct. Dyn., 23 (1994), 1351-1367.  doi: 10.1002/eqe.4290231206.
    [5] A. Benaissa and M. Bahlil, Global existence and energy decay of solutions to a nonlinear Timoshenko beam system with a delay term, Taiwanese J. Math., 18 (2014), 1411-1437.  doi: 10.11650/tjm.18.2014.3586.
    [6] A. Benaissa, A. Benaissa and S. A. Messaoudi, Global existence and energy decay of solutions for the wave equation with a time varying delay term in the weakly nonlinear internal feedbacks, J. Math. Phys., 53 (2012), 123514, 19 pp. doi: 10.1063/1.4765046.
    [7] A. BenaissaA. Benguessoum and S. A. Messaoudi, Global existence and energy decay of solutions to a viscoelastic wave equation with a delay term in the non-linear internal feedback, Int. J. Dyn. Syst. Differ. Equ., 5 (2014), 1-26.  doi: 10.1504/IJDSDE.2014.067080.
    [8] E. BerchioA. Ferrero and F. Gazzola, Structural instability of nonlinear plates modelling suspension bridges: Mathematical answers to some long-standing questions, Nonlinear Anal. Real World Appl., 28 (2016), 91-125.  doi: 10.1016/j.nonrwa.2015.09.005.
    [9] T. CaraballoA. N. CarvalhoJ. A. Langa and F. Rivero, A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor, Nonlinear Anal., 74 (2011), 2272-2283.  doi: 10.1016/j.na.2010.11.032.
    [10] M. M. Cavalcanti et al., Stabilization of a suspension bridge with locally distributed damping, Math. Control Signals Systems, 30 (2018), Art. 20, 39 pp. doi: 10.1007/s00498-018-0226-0.
    [11] I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512.  doi: 10.1007/s10884-004-4289-x.
    [12] I. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with non-linear boundary/interior damping, J. Differential Equations, 233 (2007), 42-86.  doi: 10.1016/j.jde.2006.09.019.
    [13] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. doi: 10.1090/memo/0912.
    [14] R. DatkoJ. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.  doi: 10.1137/0324007.
    [15] L. H. Fatori et al., Long-time behavior of a class of thermoelastic plates with nonlinear strain, J. Differential Equations, 259 (2015), 4831-4862. doi: 10.1016/j.jde.2015.06.026.
    [16] A. Ferrero and F. Gazzola, A partially hinged rectangular plate as a model for suspension bridges, Disc. Cont. Dynam. Syst., 35 (2015), 5879-5908.  doi: 10.3934/dcds.2015.35.5879.
    [17] F. Gazzola, Nonlinearity in oscillating bridges, Electron. J. Differential Equations, 2013 (2013), 47 pp.
    [18] F. Gazzola and Y. Wang, Modeling suspension bridges through the von Kármán quasilinear plate equations, in Contributions to Nonlinear Elliptic Equations and Systems, Progr. Nonlinear Differential Equations Appl., Birkhäuser/Springer, Cham., 86 (2015), 269–297 doi: 10.1007/978-3-319-19902-3_18.
    [19] Z.-J. Han and G.-Q. Xu, Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks, ESAIM Control Optim. Calc. Var., 17 (2011), 552-574.  doi: 10.1051/cocv/2010009.
    [20] A. Kh. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation, J. Math. Anal. Appl., 318 (2006), 92-101.  doi: 10.1016/j.jmaa.2005.05.031.
    [21] A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents, J. Differential Equations, 230 (2006), 702-719.  doi: 10.1016/j.jde.2006.06.001.
    [22] W. Lacarbonara, Nonlinear Structural Mechanics, Springer, New York, 2013. doi: 10.1007/978-1-4419-1276-3.
    [23] W. J. Liu and H. F. Zhuang, Global existence, asymptotic behavior and blow-up of solutions for a suspension bridge equation with nonlinear damping and source terms, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 67, 35 pp. doi: 10.1007/s00030-017-0491-5.
    [24] S. A. Messaoudi et al., The global attractor for a suspension bridge with memory and partially hinged boundary conditions, ZAMM Z. Angew. Math. Mech., 97 (2017), 159-172. doi: 10.1002/zamm.201600034.
    [25] N. Mezouar, M. Abdelli and A. Rachah, Existence of global solutions and decay estimates for a viscoelastic Petrovsky equation with a delay term in the non-linear internal feedback, Electron. J. Differential Equations, 2017 (2017), Paper No. 58, 25 pp.
    [26] S. A. MessaoudiS. E. Mukiawa and E. D. Cyril, Finite dimensional global attractor for a suspension bridge problem with delay, C. R. Math. Acad. Sci. Paris, 354 (2016), 808-824.  doi: 10.1016/j.crma.2016.05.014.
    [27] T. F. Ma and V. Narciso, Global attractor for a model of extensible beam with nonlinear damping and source terms, Nonlinear Anal., 73 (2010), 3402-3412.  doi: 10.1016/j.na.2010.07.023.
    [28] P. J. McKenna and C. O. Tuama, Large torsional oscillations in suspension bridges visited again: Vertical forcing creates torsional response, Amer. Math. Monthly, 108 (2001), 738-745.  doi: 10.1080/00029890.2001.11919805.
    [29] P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge, Arch. Rational Mech. Anal., 98 (1987), 167-177.  doi: 10.1007/BF00251232.
    [30] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.  doi: 10.1137/060648891.
    [31] S. NicaiseC. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 693-722.  doi: 10.3934/dcdss.2011.4.693.
    [32] J. Y. Park and J. R. Kang, Global attractor for hyperbolic equation with nonlinear damping and linear memory, Sci. China Math., 53 (2010), 1531-1539.  doi: 10.1007/s11425-010-3110-z.
    [33] J.-Y. Park and J.-R. Kang, Global attractors for the suspension bridge equations with nonlinear damping, Quart. Appl. Math., 69 (2011), 465-475.  doi: 10.1090/S0033-569X-2011-01259-1.
    [34] S.-H. Park, Long-time behavior for suspension bridge equations with time delay, Z. Angew. Math. Phys., 69 (2018), Art. 45, 12 pp. doi: 10.1007/s00033-018-0934-9.
    [35] R. H. Plaut and F. M. Davis, Sudden lateral asymmetry and torsional oscillations of section models of suspension bridges, J. Sound Vib., 307 (2007), 894-905.  doi: 10.1016/j.jsv.2007.07.036.
    [36] R. ScottIn the Wake of Tacoma: Suspension Bridges and the Quest for Aerodynamic Stability, ASCE Press, 2001.  doi: 10.1061/9780784405420.
    [37] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci., Vol. 68, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-1-4684-0313-8.
    [38] X. Wang, L. Yang and Q. Ma, Uniform attractors for non-autonomous suspension bridge-type equations, Bound. Value Probl., 2014 (2014), 14 pp. doi: 10.1186/1687-2770-2014-75.
    [39] Y. Wang, Finite time blow-up and global solutions for fourth order damped wave equations, J. Math. Anal. Appl., 418 (2014), 713-733.  doi: 10.1016/j.jmaa.2014.04.015.
    [40] L. Yang and C.-K. Zhong, Global attractor for plate equation with nonlinear damping, Nonlinear Anal., 69 (2008), 3802-3810.  doi: 10.1016/j.na.2007.10.016.
    [41] Z. Yang, Global attractor for a nonlinear wave equation arising in elastic waveguide model, Nonlinear Anal., 70 (2009), 2132-2142.  doi: 10.1016/j.na.2008.02.114.
    [42] Z. Yang and X. Li, Finite-dimensional attractors for the Kirchhoff equation with a strong dissipation, J. Math. Anal. Appl., 375 (2011), 579-593.  doi: 10.1016/j.jmaa.2010.09.051.
    [43] Z. Yang and Y. Wang, Global attractor for the Kirchhoff type equation with a strong dissipation, J. Differential Equations, 249 (2010), 3258-3278. 
  • 加载中

Article Metrics

HTML views(518) PDF downloads(332) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint