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

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February 2021, 26(2): 907-942. doi: 10.3934/dcdsb.2020147

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

Wenjun Liu ^, and Hefeng Zhuang

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

^* Corresponding author: Wenjun Liu

Received April 2019 Revised November 2019 Published February 2021 Early access May 2020

Fund Project: 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

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) $

.

Keywords: Suspension bridge, Delay, Global attractor, Absorbing set, Asymptotic smoothness.

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

Citation: Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147

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.
[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.

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.
[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.

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