October  2019, 39(10): 5975-6000. doi: 10.3934/dcds.2019261

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

Received  December 2018 Revised  April 2019 Published  July 2019

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

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].

Citation: Yanan Li, Zhijian Yang, Fang Da. Robust attractors for a perturbed non-autonomous extensible beam equation with nonlinear nonlocal damping. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5975-6000. doi: 10.3934/dcds.2019261
References:
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A. V. Babin and M. I. Vishik, Attractors of Evolutionary Equations, North-Holland, Amsterdam, 1992. Google Scholar

[2]

M. Berger, A new approach to the large deflection of plate, J. Appl. Mech., 22 (1955), 465-472. Google Scholar

[3]

T. CaraballoG. 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. Google Scholar

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M. M. CavalcantiV. 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. Google Scholar

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M. M. CavalcantiV. 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. Google Scholar

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A. N. CarvalhoJ. A. LangaJ. 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. Google Scholar

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A. N. CarvalhoJ. 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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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I. Chueshov and I. Lasiecka, Von Karman Evolutions; Wellposedness and Long Time Behavior, Springer, 2010. doi: 10.1007/978-0-387-87712-9. Google Scholar

[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. Google Scholar

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I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015. doi: 10.1007/978-3-319-22903-4. Google Scholar

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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. Google Scholar

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M. EfendievA. 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. Google Scholar

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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. Google Scholar

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M. M. FreitasP. 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. Google Scholar

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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. Google Scholar

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J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988. Google Scholar

[20]

L. T. HoangE. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

P. E. Kloeden, Pullback attractors of non-autonomous semi-dynamical systems, Stoch. Dyn., 3 (2003), 101-112. doi: 10.1142/S0219493703000632. Google Scholar

[24]

H. Lange and G. Perla Menzala, Rates of decay of a nonlocal beam equation, Differential Integral Equations, 10 (1997), 1075-1092. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[29]

C. Y. SunD. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolutionary Equations, North-Holland, Amsterdam, 1992. Google Scholar

[2]

M. Berger, A new approach to the large deflection of plate, J. Appl. Mech., 22 (1955), 465-472. Google Scholar

[3]

T. CaraballoG. 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. Google Scholar

[4]

M. M. CavalcantiV. 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. Google Scholar

[5]

M. M. CavalcantiV. 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. Google Scholar

[6]

A. N. CarvalhoJ. A. LangaJ. 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. Google Scholar

[7]

A. N. CarvalhoJ. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

I. Chueshov and I. Lasiecka, Von Karman Evolutions; Wellposedness and Long Time Behavior, Springer, 2010. doi: 10.1007/978-0-387-87712-9. Google Scholar

[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. Google Scholar

[13]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015. doi: 10.1007/978-3-319-22903-4. Google Scholar

[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. Google Scholar

[15]

M. EfendievA. 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. Google Scholar

[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. Google Scholar

[17]

M. M. FreitasP. 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. Google Scholar

[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. Google Scholar

[19]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988. Google Scholar

[20]

L. T. HoangE. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

P. E. Kloeden, Pullback attractors of non-autonomous semi-dynamical systems, Stoch. Dyn., 3 (2003), 101-112. doi: 10.1142/S0219493703000632. Google Scholar

[24]

H. Lange and G. Perla Menzala, Rates of decay of a nonlocal beam equation, Differential Integral Equations, 10 (1997), 1075-1092. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[29]

C. Y. SunD. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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