Existence of pullback attractors for the non-autonomous suspension bridge equation with time delay

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April 2020, 25(4): 1299-1316. doi: 10.3934/dcdsb.2019221

Existence of pullback attractors for the non-autonomous suspension bridge equation with time delay

Suping Wang and Qiaozhen Ma ^,

School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

^* Corresponding author

Received April 2019 Revised June 2019 Published April 2020 Early access September 2019

Fund Project: Ma is supported by NSF grant(11561064, 11761062), and partly supported by NWNU-LKQN-14-6.

We investigate the long-time behavior of solutions for the suspension bridge equation when the forcing term containing some hereditary characteristic. Existence of pullback attractor is shown by using the contractive function methods.

Keywords: Suspension bridge equation, time delay, pullback attractor, pullback $ \mathcal{D}- $asymptotically compact, contractive function.

Mathematics Subject Classification: Primary: 35B40, 37B55; Secondary: 35B41, 35K30.

Citation: Suping Wang, Qiaozhen Ma. Existence of pullback attractors for the non-autonomous suspension bridge equation with time delay. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1299-1316. doi: 10.3934/dcdsb.2019221

References:

[1]	T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal, 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111.
[2]	T. Caraballo, P. E. Kloeden and J. Real, Pullback and forward attractors for a damped wave equation with delays, Stoch. Dyn, 4 (2004), 405-423. doi: 10.1142/S0219493704001139.
[3]	J. García-Luengo, P. Marín-Rubio and J. Real, Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays, Commun. Pure Appl. Anal, 14 (2015), 1603-1621. doi: 10.3934/cpaa.2015.14.1603.
[4]	J. García-Luegngo and P. Marín-Rubio, Reaction-diffusion equations with non-autonomous force in $H^{-1}$ and delays under measurability conditions on the driving delay term, J. Math. Anal. Appl, 417 (2014), 80-95. doi: 10.1016/j.jmaa.2014.03.026.
[5]	A. Kh. Khanmamedov, Global attractors for a non-autonomous von Karman equations with nonlinear interior dissipation, Math. Anal. Appl, 318 (2006), 92-101. doi: 10.1016/j.jmaa.2005.05.031.
[6]	A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridge: Some new connections with nonlinear analysis, SIAM Rev, 32 (1990), 537-578. doi: 10.1137/1032120.
[7]	Q. Z. Ma and C. K. Zhong, Existence of strong solutions and global attractors for the coupled suspension bridge equations, J. Differential Equations, 246 (2009), 3755-3775. doi: 10.1016/j.jde.2009.02.022.
[8]	Q. Z. Ma and C. K. Zhong, Existence of global attractors for the coupled system of suspension bridge equations, J. Math. Anal. Appl, 308 (2005), 365-379. doi: 10.1016/j.jmaa.2005.01.036.
[9]	Q. Z. Ma and C. K. Zhong, Existence of global attractors for the suspension bridge equations, J. Sichuan University (Natural Science Bridge Edition), 43 (2006), 271-276.
[10]	Q. Z. Ma, S. P. Wang and X. B. Chen, Uniform compact attractors for the coupled suspension bridge equations, Appl. Math. Comput, 217 (2011), 6604-6615. doi: 10.1016/j.amc.2011.01.045.
[11]	Q. Z. Ma and B. L. Wang, Existence of pullback attractors for the coupled suspension bridge equation, Electronic. J. Differential Equations, 2011 (2011), 1-10.
[12]	Q. Z. Ma and L. Xu, Random attractors for the coupled suspension bridge equations with white noises, Appl. Math. Comput, 306 (2017), 38-48. doi: 10.1016/j.amc.2017.02.019.
[13]	Q. Z. Ma and L. Xu, Random attractors for the extensible suspension bridge equation with white noise, Comput. Appl. Math., 70 (2015), 2895-2903. doi: 10.1016/j.camwa.2015.09.029.
[14]	P. J. McKenna and W. Walter, Nonlinear oscillation in a suspension bridge, Arch. Ration. Mech. Appl. Sci, 98 (1987), 167-177. doi: 10.1007/BF00251232.
[15]	J. Y. Park and J. R. Kang, Pullback D-attractors for non-autonomous suspension bridge equations, Nonlinear Anal, 71 (2009), 4618-4623. doi: 10.1016/j.na.2009.03.025.
[16]	S. H. Park, Long-time behavior for suspension bridge equations with time delay, Z. Angew. Math. Phys., 69 (2018), 1-12. doi: 10.1007/s00033-018-0934-9.
[17]	S. H. Park, Long-time dynamics of a von Karman equation with time delay, Appl. Math. Lett., 75 (2018), 128-134. doi: 10.1016/j.aml.2017.07.004.
[18]	C. Y. Sun and K. X. Zhu, Pullback attractors for nonclassical diffusion equations with delays, J. Math. Phys, 56 (2015), 092703, 20 pp.
[19]	R. Temam, Infinite Dimensional Dynamical Systems in Mechanics abd Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
[20]	C. K. Zhong, Q. Z. Ma and C. Y. Sun, Existence of strong solutions and global attractors for the suspension bridge equations, Nonlinear Anal., 67 (2007), 442-454. doi: 10.1016/j.na.2006.05.018.

show all references

References:

[1]	T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal, 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111.
[2]	T. Caraballo, P. E. Kloeden and J. Real, Pullback and forward attractors for a damped wave equation with delays, Stoch. Dyn, 4 (2004), 405-423. doi: 10.1142/S0219493704001139.
[3]	J. García-Luengo, P. Marín-Rubio and J. Real, Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays, Commun. Pure Appl. Anal, 14 (2015), 1603-1621. doi: 10.3934/cpaa.2015.14.1603.
[4]	J. García-Luegngo and P. Marín-Rubio, Reaction-diffusion equations with non-autonomous force in $H^{-1}$ and delays under measurability conditions on the driving delay term, J. Math. Anal. Appl, 417 (2014), 80-95. doi: 10.1016/j.jmaa.2014.03.026.
[5]	A. Kh. Khanmamedov, Global attractors for a non-autonomous von Karman equations with nonlinear interior dissipation, Math. Anal. Appl, 318 (2006), 92-101. doi: 10.1016/j.jmaa.2005.05.031.
[6]	A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridge: Some new connections with nonlinear analysis, SIAM Rev, 32 (1990), 537-578. doi: 10.1137/1032120.
[7]	Q. Z. Ma and C. K. Zhong, Existence of strong solutions and global attractors for the coupled suspension bridge equations, J. Differential Equations, 246 (2009), 3755-3775. doi: 10.1016/j.jde.2009.02.022.
[8]	Q. Z. Ma and C. K. Zhong, Existence of global attractors for the coupled system of suspension bridge equations, J. Math. Anal. Appl, 308 (2005), 365-379. doi: 10.1016/j.jmaa.2005.01.036.
[9]	Q. Z. Ma and C. K. Zhong, Existence of global attractors for the suspension bridge equations, J. Sichuan University (Natural Science Bridge Edition), 43 (2006), 271-276.
[10]	Q. Z. Ma, S. P. Wang and X. B. Chen, Uniform compact attractors for the coupled suspension bridge equations, Appl. Math. Comput, 217 (2011), 6604-6615. doi: 10.1016/j.amc.2011.01.045.
[11]	Q. Z. Ma and B. L. Wang, Existence of pullback attractors for the coupled suspension bridge equation, Electronic. J. Differential Equations, 2011 (2011), 1-10.
[12]	Q. Z. Ma and L. Xu, Random attractors for the coupled suspension bridge equations with white noises, Appl. Math. Comput, 306 (2017), 38-48. doi: 10.1016/j.amc.2017.02.019.
[13]	Q. Z. Ma and L. Xu, Random attractors for the extensible suspension bridge equation with white noise, Comput. Appl. Math., 70 (2015), 2895-2903. doi: 10.1016/j.camwa.2015.09.029.
[14]	P. J. McKenna and W. Walter, Nonlinear oscillation in a suspension bridge, Arch. Ration. Mech. Appl. Sci, 98 (1987), 167-177. doi: 10.1007/BF00251232.
[15]	J. Y. Park and J. R. Kang, Pullback D-attractors for non-autonomous suspension bridge equations, Nonlinear Anal, 71 (2009), 4618-4623. doi: 10.1016/j.na.2009.03.025.
[16]	S. H. Park, Long-time behavior for suspension bridge equations with time delay, Z. Angew. Math. Phys., 69 (2018), 1-12. doi: 10.1007/s00033-018-0934-9.
[17]	S. H. Park, Long-time dynamics of a von Karman equation with time delay, Appl. Math. Lett., 75 (2018), 128-134. doi: 10.1016/j.aml.2017.07.004.
[18]	C. Y. Sun and K. X. Zhu, Pullback attractors for nonclassical diffusion equations with delays, J. Math. Phys, 56 (2015), 092703, 20 pp.
[19]	R. Temam, Infinite Dimensional Dynamical Systems in Mechanics abd Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
[20]	C. K. Zhong, Q. Z. Ma and C. Y. Sun, Existence of strong solutions and global attractors for the suspension bridge equations, Nonlinear Anal., 67 (2007), 442-454. doi: 10.1016/j.na.2006.05.018.

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