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June  2022, 27(6): 3101-3129. doi: 10.3934/dcdsb.2021175

Strong attractors and their robustness for an extensible beam model with energy damping

School of Mathematics and Statistics, Zhengzhou University, , Zhengzhou 450001, China

* Corresponding author: Zhijian Yang

Received  August 2020 Published  June 2022 Early access  July 2021

Fund Project: The authors are supported by National Natural Science Foundation of China (Grant No. 11671367)

This paper investigates the existence of strong global and exponential attractors and their robustness on the perturbed parameter for an extensible beam equation with nonlocal energy damping in $ \Omega\subset{\mathbb R}^N $: $ u_{tt}+\Delta^2 u-\kappa\phi(\|\nabla u\|^2)\Delta u-M(\|\Delta u\|^2+\|u_t\|^2)\Delta u_t+f(u) = h $, where $ \kappa \in \Lambda $ (index set) is an extensibility parameter, and where the "strong" means that the compactness, the attractiveness and the finiteness of the fractal dimension of the attractors are all in the topology of the stronger space $ {\mathcal H}_2 $ where the attractors lie in. Under the assumptions that either the nonlinearity $ f(u) $ is of optimal subcritical growth or even $ f(u) $ is a true source term, we show that (ⅰ) the semi-flow originating from any point in the natural energy space $ {\mathcal H} $ lies in the stronger strong solution space $ {\mathcal H}_2 $ when $ t>0 $; (ⅱ) the related solution semigroup $ S^\kappa(t) $ has a strong $ ({\mathcal H},{\mathcal H}_2) $-global attractor $ {\mathscr A}^\kappa $ for each $ \kappa $ and the family of $ {\mathscr A}^\kappa, \kappa\in \Lambda $ is upper semicontinuous on $ \kappa $ in the topology of stronger space $ {\mathcal H}_2 $; (ⅲ) $ S^\kappa(t) $ has a strong $ ({\mathcal H},{\mathcal H}_2) $-exponential attractor $ \mathfrak {A}^\kappa_{exp} $ for each $ \kappa $ and it is Hölder continuous on $ \kappa $ in the topology of $ {\mathcal H}_2 $. These results break through long-standing existed restriction for the attractors of the extensible beam models in energy space and show the optimal topology properties of them in the stronger phase space.

Citation: Yue Sun, Zhijian Yang. Strong attractors and their robustness for an extensible beam model with energy damping. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3101-3129. doi: 10.3934/dcdsb.2021175
References:
[1]

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

[2]

A. V. Balakrishnan and L. W. Taylor, Distributed Parameter Nonlinear Damping Models for Flight Structures, Proceedings Damping 89, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989.

[3]

J. M. Ball, Stability theory for an extensible beam, J. Differential Equations, 14 (1973), 399-418.  doi: 10.1016/0022-0396(73)90056-9.

[4]

H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472.  doi: 10.1115/1.4011138.

[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.

[6] J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge university press, Cambridge, 2000.  doi: 10.1017/CBO9780511526404.
[7]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008) doi: 10.1090/memo/0912.

[8]

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

[9]

M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal., 4 (2005), 705-720.  doi: 10.3934/cpaa.2005.4.705.

[10]

R. W. Dickey, Dynamic stability of equilibrium states of the extensible beam, Proc. Amer. Math. Soc., 41 (1973), 94-102.  doi: 10.1090/S0002-9939-1973-0328290-8.

[11]

A. Eden and A. J. Milani, Exponential attractor for extensible beam equations, Nonlinearity, 6 (1993), 457-479.  doi: 10.1088/0951-7715/6/3/007.

[12]

P. G. Geredeli and J. T. Webster, Qualitative results on the dynamics of a Berger plate with nonlinear boundary damping, Nonlinear Analysis: Real World Applications, 31 (2016), 227-256.  doi: 10.1016/j.nonrwa.2016.02.002.

[13]

J. HowellI. Lasiecka and J. T. Webster, Quasi-stability and exponential attractors for a non-gradient system-applications to Piston-Theoretic plates with internal damping, Evolution Equations and Control Theory, 5 (2016), 567-603.  doi: 10.3934/eect.2016020.

[14]

J. HowellD. Toundykov and J. T. Webster, A cantilevered extensible beam in axial flow: Semigroup well-posedness and post-flutter regimes, SIAM J. Math. Anal., 50 (2018), 2048-2085.  doi: 10.1137/17M1140261.

[15]

M. A. Jorge Silva and V. Narciso, Long-time behavior for a plate equation with nonlocal weak damping, Differ. Integral Equations, 27 (2014), 931-948. 

[16]

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.

[17]

M. A. Jorge Silva and V. Narciso, Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping, Evol. Equ. Control Theory, 6 (2017), 437-470.  doi: 10.3934/eect.2017023.

[18]

M. A. Jorge SilvaV. Narciso and A. Vicente, On a beam model related to flight structures with nonlocal energy damping, Discrete Contin. Dyn. Syst., 24 (2019), 3281-3298.  doi: 10.3934/dcdsb.2018320.

[19]

J. R. Kang, Global attractor for an extensible beam equation with localized nonlinear damping and linear memory, Math. Methods Appl. Sci., 34 (2011), 1430-1439.  doi: 10.1002/mma.1450.

[20]

H. Lange and G. P. Menzala, Rates of decay of a nonlocal beam equation, Differ. Integral Equations, 10 (1997), 1075-1092. 

[21]

Y. N. LiZ. J. Yang and F. Da, Robust attractor for a perturbed non-autonomous extensible beam equation with nonlinear nonlocal damping, Discrete Contin. Dyn. Syst., 39 (2019), 5975-6000.  doi: 10.3934/dcds.2019261.

[22]

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.

[23]

T. F. MaV. Narciso and M. L. Pelicer, Long-time behavior of a model of extensible beams with nonlinear boundary dissipations, J. Math. Anal. Appl., 396 (2012), 694-703.  doi: 10.1016/j.jmaa.2012.07.004.

[24]

F. J. Meng, J. Wu abd C. X. Zhao, Time-dependent global attractor for extensible Berger equation, J. Math. Anal. Appl., 469 (2019), 1045–1069. doi: 10.1016/j.jmaa.2018.09.050.

[25]

T. Niimura, Attractors and their stability with respect to rotational inertia for nonlinear extensible beam equations, Discrete Contin. Dyn. Syst., 40 (2020), 2561-2591.  doi: 10.3934/dcds.2020141.

[26]

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.

[27]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[28]

S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36.  doi: 10.1115/1.4010053.

[29]

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.

[30]

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.

[31]

Z. J. Yang and F. Da, Stability of attractors for the Kirchhoff wave equation with strong damping and critical nonlinearities, J. Math. Anal. Appl., 469 (2019), 298-320.  doi: 10.1016/j.jmaa.2018.09.012.

[32]

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.

[33]

C. X. Zhao, S. Ma and C. K. Zhong, Long-time behavior for a class of extensible beams with nonlocal weak damping and critical nonlinearity, J. Math. Phys., 61 (2020), 032701. doi: 10.1063/1.5128686.

show all references

References:
[1]

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

[2]

A. V. Balakrishnan and L. W. Taylor, Distributed Parameter Nonlinear Damping Models for Flight Structures, Proceedings Damping 89, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989.

[3]

J. M. Ball, Stability theory for an extensible beam, J. Differential Equations, 14 (1973), 399-418.  doi: 10.1016/0022-0396(73)90056-9.

[4]

H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472.  doi: 10.1115/1.4011138.

[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.

[6] J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge university press, Cambridge, 2000.  doi: 10.1017/CBO9780511526404.
[7]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008) doi: 10.1090/memo/0912.

[8]

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

[9]

M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal., 4 (2005), 705-720.  doi: 10.3934/cpaa.2005.4.705.

[10]

R. W. Dickey, Dynamic stability of equilibrium states of the extensible beam, Proc. Amer. Math. Soc., 41 (1973), 94-102.  doi: 10.1090/S0002-9939-1973-0328290-8.

[11]

A. Eden and A. J. Milani, Exponential attractor for extensible beam equations, Nonlinearity, 6 (1993), 457-479.  doi: 10.1088/0951-7715/6/3/007.

[12]

P. G. Geredeli and J. T. Webster, Qualitative results on the dynamics of a Berger plate with nonlinear boundary damping, Nonlinear Analysis: Real World Applications, 31 (2016), 227-256.  doi: 10.1016/j.nonrwa.2016.02.002.

[13]

J. HowellI. Lasiecka and J. T. Webster, Quasi-stability and exponential attractors for a non-gradient system-applications to Piston-Theoretic plates with internal damping, Evolution Equations and Control Theory, 5 (2016), 567-603.  doi: 10.3934/eect.2016020.

[14]

J. HowellD. Toundykov and J. T. Webster, A cantilevered extensible beam in axial flow: Semigroup well-posedness and post-flutter regimes, SIAM J. Math. Anal., 50 (2018), 2048-2085.  doi: 10.1137/17M1140261.

[15]

M. A. Jorge Silva and V. Narciso, Long-time behavior for a plate equation with nonlocal weak damping, Differ. Integral Equations, 27 (2014), 931-948. 

[16]

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.

[17]

M. A. Jorge Silva and V. Narciso, Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping, Evol. Equ. Control Theory, 6 (2017), 437-470.  doi: 10.3934/eect.2017023.

[18]

M. A. Jorge SilvaV. Narciso and A. Vicente, On a beam model related to flight structures with nonlocal energy damping, Discrete Contin. Dyn. Syst., 24 (2019), 3281-3298.  doi: 10.3934/dcdsb.2018320.

[19]

J. R. Kang, Global attractor for an extensible beam equation with localized nonlinear damping and linear memory, Math. Methods Appl. Sci., 34 (2011), 1430-1439.  doi: 10.1002/mma.1450.

[20]

H. Lange and G. P. Menzala, Rates of decay of a nonlocal beam equation, Differ. Integral Equations, 10 (1997), 1075-1092. 

[21]

Y. N. LiZ. J. Yang and F. Da, Robust attractor for a perturbed non-autonomous extensible beam equation with nonlinear nonlocal damping, Discrete Contin. Dyn. Syst., 39 (2019), 5975-6000.  doi: 10.3934/dcds.2019261.

[22]

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.

[23]

T. F. MaV. Narciso and M. L. Pelicer, Long-time behavior of a model of extensible beams with nonlinear boundary dissipations, J. Math. Anal. Appl., 396 (2012), 694-703.  doi: 10.1016/j.jmaa.2012.07.004.

[24]

F. J. Meng, J. Wu abd C. X. Zhao, Time-dependent global attractor for extensible Berger equation, J. Math. Anal. Appl., 469 (2019), 1045–1069. doi: 10.1016/j.jmaa.2018.09.050.

[25]

T. Niimura, Attractors and their stability with respect to rotational inertia for nonlinear extensible beam equations, Discrete Contin. Dyn. Syst., 40 (2020), 2561-2591.  doi: 10.3934/dcds.2020141.

[26]

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.

[27]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[28]

S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36.  doi: 10.1115/1.4010053.

[29]

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.

[30]

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.

[31]

Z. J. Yang and F. Da, Stability of attractors for the Kirchhoff wave equation with strong damping and critical nonlinearities, J. Math. Anal. Appl., 469 (2019), 298-320.  doi: 10.1016/j.jmaa.2018.09.012.

[32]

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.

[33]

C. X. Zhao, S. Ma and C. K. Zhong, Long-time behavior for a class of extensible beams with nonlocal weak damping and critical nonlinearity, J. Math. Phys., 61 (2020), 032701. doi: 10.1063/1.5128686.

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