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On the rigid-lid approximation of shallow water Bingham
Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China |
$ \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*} $ |
$ S(t) $ |
References:
[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-Gwaiz, V. 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. Google Scholar |
[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. Benaissa, A. 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. Berchio, A. 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. Caraballo, A. N. Carvalho, J. 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. Datko, J. 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. Messaoudi, S. 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. Nicaise, C. 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. Scott, In 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. Google Scholar |
show all references
References:
[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-Gwaiz, V. 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. Google Scholar |
[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. Benaissa, A. 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. Berchio, A. 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. Caraballo, A. N. Carvalho, J. 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. Datko, J. 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. Messaoudi, S. 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. Nicaise, C. 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. Scott, In 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. Google Scholar |
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