Robust attractors for a perturbed non-autonomous extensible beam equation with nonlinear nonlocal damping

Yanan Li; Zhijian Yang; Fang Da

doi:10.3934/dcds.2019261

Article Contents

2019, Volume 39, Issue 10: 5975-6000. Doi: 10.3934/dcds.2019261

This issue Previous Article Traveling waves of a full parabolic attraction-repulsion chemotaxis system with logistic source Next Article On the isomorphism problem for non-minimal transformations with discrete spectrum

Robust attractors for a perturbed non-autonomous extensible beam equation with nonlinear nonlocal damping

School of Mathematics and Statistics, Zhengzhou University, No.100, Science Road, Zhengzhou 450001, China

^* Corresponding author: Zhijian Yang
^* Corresponding author: Zhijian Yang

Received: November 30, 2018

Revised: March 31, 2019

Published: July 2019

The work is supported by National Nature Science Foundation of China (No.11671367)

Abstract Full Text(HTML) Related Papers Cited by

Abstract

The paper investigates the attractors and their robustness for a perturbed non-autonomous extensible beam equation with nonlinear nonlocal damping. We prove that the related evolution process has a finite-dimensional pullback attractor $ \mathscr{A}_\kappa $ and a pullback exponential attractor $ \mathscr{M}^\kappa_{exp} $ for each extensibility parameter $ \kappa\in [0,1] $, respectively, and both of them are stable on the perturbation $ \kappa $. In particular, these stability holds for the global and exponential attractors when the non-autonomous dynamical system degenerates to an autonomous one, so the results of the paper deepen and extend those in recent literatures [22,33].

Keywords:

Non-autonomous extensible beam equation,
nonlinear nonlocal damping,
pullback attractor,
pullback exponential attractor,
robustness of attractors.

Mathematics Subject Classification: Primary: 37L30, 37L15, 35B40; Secondary: 35B41, 35B20.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	A. V. Babin and M. I. Vishik, Attractors of Evolutionary Equations, North-Holland, Amsterdam, 1992.
[2]	M. Berger, A new approach to the large deflection of plate, J. Appl. Mech., 22 (1955), 465-472.
[3]	T. Caraballo, G. Łukaszewicz and J. Reala, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111.
[4]	M. M. Cavalcanti, V. N. D. Cavalcanti and T. F. Ma, Exponential decay of the viscoelastic Euler-Bernoulli equation with a nonlocal dissipation in general domains, Differential Integral Equations, 17 (2004), 495-510.
[5]	M. M. Cavalcanti, V. N. D. Cavalcanti and J. A. Soriano, Global existence and asymptotic stability for the nonlinear and generalized damped extensible plate equation, Commun. Contemp. Math., 6 (2004), 705-731. doi: 10.1142/S0219199704001483.
[6]	A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Surez, Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system, J. Differential Equations, 236 (2007), 570-603. doi: 10.1016/j.jde.2007.01.017.
[7]	A. N. Carvalho, J. A. Langa and J. C. Robinson, On the continuity of pullback attractors for evolution processes, Nonlinear Anal., 71 (2009), 1812-1824. doi: 10.1016/j.na.2009.01.016.
[8]	A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical System, Springer Science+Business, Media, LLC, 2013. doi: 10.1007/978-1-4614-4581-4.
[9]	I. Chueshov and I. Lasiecka, Attractors for second order evolution equations, J. Dynam. Diff. Eqs., 16 (2004), 469-512. doi: 10.1007/s10884-004-4289-x.
[10]	I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs of AMS, 195 (2008), viii+183 pp. doi: 10.1090/memo/0912.
[11]	I. Chueshov and I. Lasiecka, Von Karman Evolutions; Wellposedness and Long Time Behavior, Springer, 2010. doi: 10.1007/978-0-387-87712-9.
[12]	I. Chueshov and S. Kolbasin, Long-time dynamics in plate models with strong nonlinear damping, Commun. Pure Appl. Anal., 11 (2012), 659-674. doi: 10.3934/cpaa.2012.11.659.
[13]	I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015. doi: 10.1007/978-3-319-22903-4.
[14]	A. Eden and A. J. Milani, Exponential attractors for extensible beam equations, Nonlinearity, 6 (1993), 457-479. doi: 10.1088/0951-7715/6/3/007.
[15]	M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems, Proceedings of the Royal Society of Edinburgh, 135 (2005), 703-730. doi: 10.1017/S030821050000408X.
[16]	T. Fastovska, Upper semicontinuous attractor for 2D Mindlin-Timoshenko thermoelastic model with memory, Commun. Pure Appl. Anal., 6 (2007), 83-101. doi: 10.3934/cpaa.2007.6.83.
[17]	M. M. Freitas, P. Kalita and J. A. Langa, Continuity of non-autonomous attractors for hyperbolic perturbation of parabolic equations, J. Differential Equations, 264 (2018), 1886-1945. doi: 10.1016/j.jde.2017.10.007.
[18]	J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Differential Equations, 73 (1998), 197-214. doi: 10.1016/0022-0396(88)90104-0.
[19]	J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988.
[20]	L. T. Hoang, E. J. Olson and J. C. Robinson, Continuity of pullback and uniform attractors, J. Differential Equations, 264 (2018), 4067-4093. doi: 10.1016/j.jde.2017.12.002.
[21]	M. A. Jorge Silva and V. Narciso, Attractors and their properties for a class of nonlocal extensible beams, Discrete Contin. Dyn. Syst., 35 (2015), 985-1008. doi: 10.3934/dcds.2015.35.985.
[22]	M. A. Jorge Silva and V. Narciso, Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping, Evolution Equations and Control Theory, 6 (2017), 437-470. doi: 10.3934/eect.2017023.
[23]	P. E. Kloeden, Pullback attractors of non-autonomous semi-dynamical systems, Stoch. Dyn., 3 (2003), 101-112. doi: 10.1142/S0219493703000632.
[24]	H. Lange and G. Perla Menzala, Rates of decay of a nonlocal beam equation, Differential Integral Equations, 10 (1997), 1075-1092.
[25]	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.
[26]	M. Nakao, Global attractors for nonlinear wave equations with nonlinear dissipative terms, J. Differential Equations, 227 (2006), 204-229. doi: 10.1016/j.jde.2005.09.013.
[27]	S. K. Patcheu, On a global solution and asymptotic behaviour for the generalized damped extensible beam equation, J. Differential Equations, 135 (1997), 299-314. doi: 10.1006/jdeq.1996.3231.
[28]	J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1986), 65-96. doi: 10.1007/BF01762360.
[29]	C. Y. Sun, D. M. Cao and J. Q. Duan, Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity, Nonlinearity, 19 (2006), 2645-2665. doi: 10.1088/0951-7715/19/11/008.
[30]	Y. H. Wang, Pullback attractors for non-autonomous wave equations with critical exponent, Nonlinear Anal., 68 (2008), 365-376. doi: 10.1016/j.na.2006.11.002.
[31]	Y. H. Wang and C. K. Zhong, Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models, Discrete Contin. Dyn. Syst., 33 (2013), 3189-3209. doi: 10.3934/dcds.2013.33.3189.
[32]	S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36.
[33]	Z. J. Yang, On an extensible beam equation with nonlinear damping and source terms, J. Differential Equations, 254 (2013), 3903-3927. doi: 10.1016/j.jde.2013.02.008.
[34]	Z. J. Yang and Y. N. Li, Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous Kirchhoff wave models, Discrete Contin. Dyn. Syst., 38 (2018), 2629-2653. doi: 10.3934/dcds.2018111.
[35]	M. C. Zelati, Global and exponential attractors for the singularly perturbed extensible beam, Discrete Contin. Dyn. Syst., 25 (2009), 1041-1060. doi: 10.3934/dcds.2009.25.1041.