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Long-Time Behavior of Solution for Autonomous Suspension Bridge Equations with State-Dependent Delay
School of Mathematics and Information Engineering, Longdong University, Qingyang, China
School of Mathematics and Statistics, Northwest Normal University, Lanzhou, China
School of Mathematics and Information Engineering, Longdong University, Qingyang, China
- 1 School of Mathematics and Information Engineering, Longdong University, Qingyang, China
- 2 School of Mathematics and Statistics, Northwest Normal University, Lanzhou, China
- 3 School of Mathematics and Information Engineering, Longdong University, Qingyang, China
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Abstract
This work is devoted to the following suspension bridge with state-dependent delay: . The mai n goal of this paper is to investigate the long-time behavior of the system. Under suitable hypothesis, the quasi-stability estimates of the system are established, base d on which the existence of global attractor with finite fractal dimension is obtained. Furthermore, the existence of exponential attractor is proved.
KeywordsSuspension Bridge EquationState-Dependent DelayGlobal AttractorExponential AttractorQuasi-Stability
- Caraballo, T., Kloeden, P.E. and Real, J. (2004) Pullback and Forward Attractors for a Damped Wave Equation with Delays. Stochastics and Dynamics, 4, 405-423. https://doi.org/10.1142/S0219493704001139
- Marin-Rubio, P. and Real, J. (2010) Pullback Attractors for 2D-Navier-Stokes Equation with Delay in Continuous and Sub-linear Operators. Discrete and Continuous Dynamical Systems, 26, 989-1006. https://doi.org/10.3934/dcds.2010.26.989
- Wang, Y.J. (2015) Pullback Attractors for a Damped Wave Equation with Delays. Stochastics and Dynamics, 15, Article ID: 1550003. https://doi.org/10.1142/S0219493715500033
- Wang, S.P. and Ma, Q.Z. (2019) Uniform Attractors for the Non-autonomous Suspension Bridge Equation with Time Delay. Journal of Inequalities and Applications, 180, 1-17. https://doi.org/10.1186/s13660-019-2133-4
- Aouadi, M. (2020) Global and exponential Attractors for Extensible Thermoelastic Plate with Timevarying Delay. Journal of Differential Equations, 5, 4079-4115. https://doi.org/10.1016/j.jde.2020.03.026
- Wang, S.P. and Ma, Q.Z. (2020) Existence of Pullback Attractors for the Non- autonomous Suspension Bridge Equation with Time Delay. Discrete and Continuous Dynamical Systems, Series B, 4, 1299-1316. https://doi.org/10.3934/dcdsb.2019221
- Rezounenko, A.V. (2009) Differential Equations with Discrete State-dependent Delay: Uniqueness and Well-posedness in the Space of Continuous Functions. Nonlinear Analysis, Theory, Methods and Applications, 70, 3978-3986. https://doi.org/10.1016/j.na.2008.08.006
- Chueshov, I. and Rezounenko, A. (2015) Finite-dimensional Global Attractors for Parabolic Nonlinear Equations with State-dependent Delay. Communications on Pure and Applied Analysis, 14, 1685-1704. https://doi.org/10.3934/cpaa.2015.14.1685
- Chueshov, I. and Rezounenko, A. (2015) Dynamics of Second Order in Time Evolution Equations with State-dependent Delay. Nonlinear Analysis, Theory, Methods and Applications, 123-124, 126-149. https://doi.org/10.1016/j.na.2015.04.013
- Wang, S.P., Ma, Q.Z. and Shao, X.K. (2022) Dynamics of Suspension Bridge Equation with Delay. Journal of Dynamics and Differential Equations. https://doi.org/10.1007/s10884-022-10130-9
- Hartung, F., Krisztin, T., Walther H.-O. and Wu, J.H. (2006) Chapter 5 Functional Differential Equations with State-dependent Delays: Theory and Applications. Handbook of Differential Equations: Ordinary Differential Equations, 3, 435-545. https://doi.org/10.1016/S1874-5725(06)80009-X