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

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

* Corresponding author

Received  April 2019 Revised  June 2019 Published  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.

Citation: Suping Wang, Qiaozhen Ma. Existence of pullback attractors for the non-autonomous suspension bridge equation with time delay. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1299-1316. doi: 10.3934/dcdsb.2019221
References:
[1]

T. CaraballoG. 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.  Google Scholar

[2]

T. CaraballoP. 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.  Google Scholar

[3]

J. García-LuengoP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[10]

Q. Z. MaS. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

C. Y. Sun and K. X. Zhu, Pullback attractors for nonclassical diffusion equations with delays, J. Math. Phys, 56 (2015), 092703, 20 pp.  Google Scholar

[19]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics abd Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[20]

C. K. ZhongQ. 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.  Google Scholar

show all references

References:
[1]

T. CaraballoG. 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.  Google Scholar

[2]

T. CaraballoP. 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.  Google Scholar

[3]

J. García-LuengoP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[10]

Q. Z. MaS. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

C. Y. Sun and K. X. Zhu, Pullback attractors for nonclassical diffusion equations with delays, J. Math. Phys, 56 (2015), 092703, 20 pp.  Google Scholar

[19]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics abd Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[20]

C. K. ZhongQ. 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.  Google Scholar

[1]

Theodore Tachim Medjo. Pullback $ \mathbb{V}-$attractor of a three dimensional globally modified two-phase flow model. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2141-2169. doi: 10.3934/dcds.2018088

[2]

Fuzhi Li, Dongmei Xu, Jiali Yu. Regular measurable backward compact random attractor for $ g $-Navier-Stokes equation. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3137-3157. doi: 10.3934/cpaa.2020136

[3]

Anhui Gu. Weak pullback mean random attractors for non-autonomous $ p $-Laplacian equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3863-3878. doi: 10.3934/dcdsb.2020266

[4]

Mustapha Yebdri. Existence of $ \mathcal{D}- $pullback attractor for an infinite dimensional dynamical system. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021036

[5]

Gongwei Liu, Baowei Feng, Xinguang Yang. Longtime dynamics for a type of suspension bridge equation with past history and time delay. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4995-5013. doi: 10.3934/cpaa.2020224

[6]

Siqi Chen, Yong-Kui Chang, Yanyan Wei. Pseudo $ S $-asymptotically Bloch type periodic solutions to a damped evolution equation. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021017

[7]

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

[8]

Jiahao Qiu, Jianjie Zhao. Maximal factors of order $ d $ of dynamical cubespaces. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 601-620. doi: 10.3934/dcds.2020278

[9]

Lin Du, Yun Zhang. $\mathcal{H}_∞$ filtering for switched nonlinear systems: A state projection method. Journal of Industrial & Management Optimization, 2018, 14 (1) : 19-33. doi: 10.3934/jimo.2017035

[10]

Monica Motta, Caterina Sartori. On ${\mathcal L}^1$ limit solutions in impulsive control. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1201-1218. doi: 10.3934/dcdss.2018068

[11]

Yu-Zhao Wang. $ \mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116

[12]

Rakesh Nandi, Sujit Kumar Samanta, Chesoong Kim. Analysis of $ D $-$ BMAP/G/1 $ queueing system under $ N $-policy and its cost optimization. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020135

[13]

Abdeladim El Akri, Lahcen Maniar. Uniform indirect boundary controllability of semi-discrete $ 1 $-$ d $ coupled wave equations. Mathematical Control & Related Fields, 2020, 10 (4) : 669-698. doi: 10.3934/mcrf.2020015

[14]

Xue-Li Song, Yan-Ren Hou. Pullback $\mathcal{D}$-attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 991-1009. doi: 10.3934/dcds.2012.32.991

[15]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195

[16]

Edcarlos D. Silva, José Carlos de Albuquerque, Uberlandio Severo. On a class of linearly coupled systems on $ \mathbb{R}^N $ involving asymptotically linear terms. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3089-3101. doi: 10.3934/cpaa.2019138

[17]

Abdelwahab Bensouilah, Van Duong Dinh, Mohamed Majdoub. Scattering in the weighted $ L^2 $-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2735-2755. doi: 10.3934/cpaa.2019122

[18]

Debora Amadori, Fatima Al-Zahrà Aqel. On the decay in $ W^{1,\infty} $ for the 1D semilinear damped wave equation on a bounded domain. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021080

[19]

María Anguiano, Alain Haraux. The $\varepsilon$-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors. Evolution Equations & Control Theory, 2017, 6 (3) : 345-356. doi: 10.3934/eect.2017018

[20]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2635-3652. doi: 10.3934/dcds.2020378

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (211)
  • HTML views (282)
  • Cited by (0)

Other articles
by authors

[Back to Top]