# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021175
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## 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 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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021175
##### References:
 [1] A. V. Babin and M. I. Vishik, Attractors of Evolutionary Equations, North-Holland, Amsterdam, 1992.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge university press, Cambridge, 2000.  doi: 10.1017/CBO9780511526404.  Google Scholar [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.  Google Scholar [8] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer International Publishing Switzerland, 2015. doi: 10.1007/978-3-319-22903-4.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] J. Howell, I. 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.  Google Scholar [14] J. Howell, D. 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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [18] M. A. Jorge Silva, V. 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.  Google Scholar [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.  Google Scholar [20] H. Lange and G. P. Menzala, Rates of decay of a nonlocal beam equation, Differ. Integral Equations, 10 (1997), 1075-1092.   Google Scholar [21] Y. N. Li, Z. 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.  Google Scholar [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.  Google Scholar [23] T. F. Ma, V. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### References:
 [1] A. V. Babin and M. I. Vishik, Attractors of Evolutionary Equations, North-Holland, Amsterdam, 1992.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge university press, Cambridge, 2000.  doi: 10.1017/CBO9780511526404.  Google Scholar [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.  Google Scholar [8] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer International Publishing Switzerland, 2015. doi: 10.1007/978-3-319-22903-4.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] J. Howell, I. 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.  Google Scholar [14] J. Howell, D. 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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [18] M. A. Jorge Silva, V. 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.  Google Scholar [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.  Google Scholar [20] H. Lange and G. P. Menzala, Rates of decay of a nonlocal beam equation, Differ. Integral Equations, 10 (1997), 1075-1092.   Google Scholar [21] Y. N. Li, Z. 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.  Google Scholar [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.  Google Scholar [23] T. F. Ma, V. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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