Longtime dynamics for a type of suspension bridge equation with past history and time delay. (English) Zbl 1460.35039
Summary: In this paper, we investigate a suspension bridge equation with past history and time delay effects, defined in a bounded domain \(\Omega\) of \(\mathbb{R}^N\). Many researchers have considered the well-posedness, energy decay of solution and existence of global attractors for suspension bridge equation without memory or delay. But as far as we know, there are no results on the suspension bridge equation with both memory and time delay. The purpose of this paper is to show the existence of a global attractor which has finite fractal dimension by using the methods developed by Chueshov and Lasiecka. Result on exponential attractors is also proved. We also establish the exponential stability under some conditions. These results are extension and improvement of earlier results.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B41 | Attractors |
| 35L35 | Initial-boundary value problems for higher-order hyperbolic equations |
| 35L76 | Higher-order semilinear hyperbolic equations |
| 35L90 | Abstract hyperbolic equations |
| 35R09 | Integro-partial differential equations |
| 74D10 | Nonlinear constitutive equations for materials with memory |
| 93D20 | Asymptotic stability in control theory |
References:
| [1] | Y. An, On the suspension bridge equations and the relevant problems, Ph.D thesis, 2001. |
| [2] | R. O. Araújo; T. F. Ma; Y. Qin, Long-time behavior of a quasilinear viscoelastic equation with past history, J. Differ. Equ., 254, 4066-4087 (2013) · Zbl 1282.35078 · doi:10.1016/j.jde.2013.02.010 |
| [3] | A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Amsterdam, North-Holland, 1992. · Zbl 0778.58002 |
| [4] | A. R. A. Barbosa; T. F. Ma, long-time dynamics of an extensible plate equation with thermal memory, J. Math. Anal. Appl., 416, 143-165 (2014) · Zbl 1297.35031 · doi:10.1016/j.jmaa.2014.02.042 |
| [5] | I. D. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer Monogr. Math., Springer, New York, 2010. · Zbl 1298.35001 |
| [6] | C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37, 297-308 (1970) · Zbl 0214.24503 · doi:10.1007/BF00251609 |
| [7] | Q. Dai; Z. Yang, Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 65, 885-903 (2014) · Zbl 1312.35021 · doi:10.1007/s00033-013-0365-6 |
| [8] | B. Feng; X. Yang, Long-term dynamics for a nonlinear Timoshenko system with delay, Appl. Anal., 96, 606-625 (2017) · Zbl 1362.37149 · doi:10.1080/00036811.2016.1148139 |
| [9] | B. Feng, On a semilinear Timoshenko-Coleman-Gurtin system: Quasi-stability and attractors, Discrete Contin. Dyn. Syst., 37, 4729-4751 (2017) · Zbl 1364.35359 · doi:10.3934/dcds.2017203 |
| [10] | J. R. Kang, Long-time behavior of a suspension bridge equations with past history, Appl. Math. Comput., 265, 509-519 (2015) · Zbl 1410.35084 · doi:10.1016/j.amc.2015.04.116 |
| [11] | J. R. Kang, Global attractor for suspension bridge equations with memeory, Math. Meth. Appl. Sci., 39, 762-775 (2016) · Zbl 1338.35060 · doi:10.1002/mma.3520 |
| [12] | M. Kirane; B. Said Houari, Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62, 1065-1082 (2011) · Zbl 1242.35163 · doi:10.1007/s00033-011-0145-0 |
| [13] | A. C. Lazer; P. J. McKenna, Large-amplitude periodic oscillations in suspension bridge: some new connections with nonlinear analysis, SIAM Rev., 32, 537-578 (1990) · Zbl 0725.73057 · doi:10.1137/1032120 |
| [14] | G. Liu; H. Zhang, Well-posedness for a class of wave equation with past history and a delay, Z. Angew. Math. Phys., 67, 1-14 (2016) · Zbl 1347.35144 · doi:10.1007/s00033-015-0593-z |
| [15] | G. Liu; H. Yue; H. Zhang, Long time behavior for a wave equation with time delay, Taiwan. J. Math., 27, 2017-129 (2017) · Zbl 1361.35105 · doi:10.11650/tjm.21.2017.7246 |
| [16] | Q. Z. Ma; C. K. Zhong, Existence of global attractors for the suspension bridge equations, J. Sichuan Univ., 43, 271-276 (2006) · Zbl 1111.35089 |
| [17] | Q. Z. Ma; C. K. Zhong, Existence of strong solutions and global attractors for the coupled suspension bridge equations, J. Differ. Equ., 246, 3755-3775 (2009) · Zbl 1176.35036 · doi:10.1016/j.jde.2009.02.022 |
| [18] | P. J. McKenna; W. Walter, Nonlinear oscillation in a suspension bridge, Nonlinear Anal., 39, 731-743 (2000) |
| [19] | S. Nicaise; C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differ. Equ., 41, 1-20 (2011) · Zbl 1215.35098 |
| [20] | S. Nicaise; 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, 1561-1585 (2006) · Zbl 1180.35095 · doi:10.1137/060648891 |
| [21] | J. Y. Park; J. R. Kang, Pullback D-attractors for non-autonomous suspension bridge equations, Nonlinear Anal., 71, 4618-4623 (2009) · Zbl 1173.35606 · doi:10.1016/j.na.2009.03.025 |
| [22] | J. Y. Park; J. R. Kang, Global attractors for the suspension bridge equations with nonlinear damping, Q. Appl. Math., 69, 465-475 (2011) · Zbl 1227.35081 · doi:10.1090/S0033-569X-2011-01259-1 |
| [23] | S. H. Park, Long-time behavior for suspension bridge equations with time delay, Z. Angew. Math. Phys., 69 (2018), Art. 45. · Zbl 1400.35035 |
| [24] | G. Q. Xu; S. P. Yung; L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12, 770-785 (2006) · Zbl 1105.35016 · doi:10.1051/cocv:2006021 |
| [25] | Z. Yang, Existence and energy decay of solutions for the Euler-Bernoulli viscoelastic equation with a delay, Z. Angew. Math. Phys., 66, 727-745 (2015) · Zbl 1326.35188 · doi:10.1007/s00033-014-0429-2 |
| [26] | C. K. Zhong; Q. Z. Ma; C. Y. Sun, Existence of strong solutions and global attractors for the suspension bridge equations, Nonlinear Anal., 67, 442-454 (2007) · Zbl 1125.34019 · doi:10.1016/j.na.2006.05.018 |
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