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## The global attractor for a class of extensible beams with nonlocal weak damping

 Department of Mathematics, Nanjing University, Nanjing, 210093, China

* Corresponding author: Chengkui Zhong

Received  January 2019 Published  September 2019

Fund Project: This work is partly supported by the NSFC (11731005, 11801228) and Postgraduate Research and Practice Innovation Program of Jiangsu Province(KYCX19-0027)

The goal of this paper is to study the long-time behavior of a class of extensible beams equation with the nonlocal weak damping
 $\begin{eqnarray*} u_{tt}+\Delta^2 u-m(\|\nabla u\|^2)\Delta u +\| u_t\|^{p}u_t+f(u) = h, \rm{in}\; \Omega\times\mathbb{R^{+}}, p\geq0 \end{eqnarray*}$
on a bounded smooth domain
 $\Omega\subset\mathbb{R}^{n}$
with hinged (clamped) boundary condition. Under some suitable conditions on the Kirchhoff coefficient
 $m(\|\nabla u\|^2)$
and the nonlinear term
 $f(u)$
, the well-posedness is established by means of the monotone operator theory and the existence of a global attractor is obtained in the subcritical case, where the asymptotic smooothness of the semigroup is verified by the energy reconstruction method.
Citation: Chunxiang Zhao, Chunyan Zhao, Chengkui Zhong. The global attractor for a class of extensible beams with nonlocal weak damping. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019197
##### References:
 [1] A. C. Biazutti and H. R. Crippa, Global attractor and inertial set for the beam equation, Appl. Anal., 55 (1994), 61-78. doi: 10.1080/00036819408840290. Google Scholar [2] E. H. de Brito, The damped elastic stretched string equation generalized: Existence, uniqueness, regularity and stability, Applicable Anal., 13 (1982), 219-233. doi: 10.1080/00036818208839392. Google Scholar [3] J. M. Ball, Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90. doi: 10.1016/0022-247X(73)90121-2. Google Scholar [4] 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 [5] J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31. Google Scholar [6] P. Biler, Remark on the decay for damped string and beam equations, Nonlinear Anal., 10 (1986), 839-842. doi: 10.1016/0362-546X(86)90071-4. Google Scholar [7] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces., Noordhoff International Publishing, Leiden, 1976,352 pp. Google Scholar [8] M. M. Cavalcanti, V. N. Domingos 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 [9] I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, AKTA, Kharkiv, 1999. 436 pp. Google Scholar [10] 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 [11] 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 [12] M. Coti 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 [13] R. W. Dickey, Free vibrations and dynamic buckling of the extensible beam, J.Math. Anal.Appl., 29 (1970), 443-454. doi: 10.1016/0022-247X(70)90094-6. Google Scholar [14] 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 [15] J. G. Eisley, Nonlinear vibration of beams and rectangular plates, Z. Angew. Math. Phys., 15 (1964), 167-175. doi: 10.1007/BF01602658. Google Scholar [16] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. AMS, Providence, RI, 1988. Google Scholar [17] H. Lange and G. Perla Menzala, Rates of decay of a nonlocal beam equation, Differential Integral Equations, 10 (1997), 1075-1092. Google Scholar [18] J.-L. Lions, On some questions in boundary value problems in mathematical physics, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Rio de Janeiro, North-Holland, Amsterdam-New York, 30 (1978), 284–346. Google Scholar [19] H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472. Google Scholar [20] 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 [21] L. A. Medeiros, On a new class of nonlinear wave equations, J. Math. Anal. Appl., 69 (1979), 252-262. doi: 10.1016/0022-247X(79)90192-6. Google Scholar [22] F. J. Meng, J. Wu and 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 [23] F. J. Meng, M. H. Yang and C. K. Zhong, Attractors for wave equations with nonlinear damping on time-dependent space, Discrete. Contin. Dyn. Syst. Ser. B, 21 (2016), 205-225. doi: 10.3934/dcdsb.2016.21.205. Google Scholar [24] S. Kouémou 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 [25] V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506. doi: 10.1088/0951-7715/19/7/001. Google Scholar [26] I. Perai, Multiplicity of Solutions for the $p$-Laplacian, 1997.Google Scholar [27] M. A. Jorge Silva and V. Narciso, Long-time behavior for a plate equation with nonlocal weak damping, Differential Integral Equations, 27 (2014), 931-948. Google Scholar [28] 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 [29] M. A. J. da 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 [30] J. Simon, Régularité de la solution d'une équation non linéaire dans ${{\mathbf{R}}^{N}}$, Journées d'Analyse Non Linéaire, Lecture Notes in Math., Springer, Berlin, 665 (1978), 205–227. Google Scholar [31] J. Simon, Compact sets in the space $L^{p}(0, T;B)$, Ann. Mat. Pure Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar [32] R. E. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, 49. AMS, Providence, RI, 1997. Google Scholar [33] 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. Google Scholar [34] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. Google Scholar [35] C. F. Vasconcellos and L. M. Teixeira, Existence, uniqueness and stabilization for a nonlinear plate system with nonlinear damping, Ann. Fac. Sci. Toulouse Math., 8 (1999), 173-193. doi: 10.5802/afst.928. Google Scholar [36] S. Woinowsky-Krieger, The efect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36. Google Scholar [37] L. Yang and X. Wang, Existence of attractors for the non-autonomous Berger equation with nonlinear damping, Electron. J. Differential Equations, (2017), 14 pp. Google Scholar [38] L. Yang, Uniform attractor for non-autonomous plate equation with a localized damping and a critical nonlinearity, J. Math. Anal. Appl., 338 (2008), 1243-1254. doi: 10.1016/j.jmaa.2007.06.011. Google Scholar [39] Z. J. Yang, On an extensible beam equation with nonlinear damping and source terms, J.Dierential Equations, 254 (2013), 3903-3927. doi: 10.1016/j.jde.2013.02.008. Google Scholar

show all references

##### References:
 [1] A. C. Biazutti and H. R. Crippa, Global attractor and inertial set for the beam equation, Appl. Anal., 55 (1994), 61-78. doi: 10.1080/00036819408840290. Google Scholar [2] E. H. de Brito, The damped elastic stretched string equation generalized: Existence, uniqueness, regularity and stability, Applicable Anal., 13 (1982), 219-233. doi: 10.1080/00036818208839392. Google Scholar [3] J. M. Ball, Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90. doi: 10.1016/0022-247X(73)90121-2. Google Scholar [4] 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 [5] J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31. Google Scholar [6] P. Biler, Remark on the decay for damped string and beam equations, Nonlinear Anal., 10 (1986), 839-842. doi: 10.1016/0362-546X(86)90071-4. Google Scholar [7] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces., Noordhoff International Publishing, Leiden, 1976,352 pp. Google Scholar [8] M. M. Cavalcanti, V. N. Domingos 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 [9] I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, AKTA, Kharkiv, 1999. 436 pp. Google Scholar [10] 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 [11] 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 [12] M. Coti 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 [13] R. W. Dickey, Free vibrations and dynamic buckling of the extensible beam, J.Math. Anal.Appl., 29 (1970), 443-454. doi: 10.1016/0022-247X(70)90094-6. Google Scholar [14] 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 [15] J. G. Eisley, Nonlinear vibration of beams and rectangular plates, Z. Angew. Math. Phys., 15 (1964), 167-175. doi: 10.1007/BF01602658. Google Scholar [16] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. AMS, Providence, RI, 1988. Google Scholar [17] H. Lange and G. Perla Menzala, Rates of decay of a nonlocal beam equation, Differential Integral Equations, 10 (1997), 1075-1092. Google Scholar [18] J.-L. Lions, On some questions in boundary value problems in mathematical physics, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Rio de Janeiro, North-Holland, Amsterdam-New York, 30 (1978), 284–346. Google Scholar [19] H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472. Google Scholar [20] 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 [21] L. A. Medeiros, On a new class of nonlinear wave equations, J. Math. Anal. Appl., 69 (1979), 252-262. doi: 10.1016/0022-247X(79)90192-6. Google Scholar [22] F. J. Meng, J. Wu and 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 [23] F. J. Meng, M. H. Yang and C. K. Zhong, Attractors for wave equations with nonlinear damping on time-dependent space, Discrete. Contin. Dyn. Syst. Ser. B, 21 (2016), 205-225. doi: 10.3934/dcdsb.2016.21.205. Google Scholar [24] S. Kouémou 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 [25] V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506. doi: 10.1088/0951-7715/19/7/001. Google Scholar [26] I. Perai, Multiplicity of Solutions for the $p$-Laplacian, 1997.Google Scholar [27] M. A. Jorge Silva and V. Narciso, Long-time behavior for a plate equation with nonlocal weak damping, Differential Integral Equations, 27 (2014), 931-948. Google Scholar [28] 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 [29] M. A. J. da 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 [30] J. Simon, Régularité de la solution d'une équation non linéaire dans ${{\mathbf{R}}^{N}}$, Journées d'Analyse Non Linéaire, Lecture Notes in Math., Springer, Berlin, 665 (1978), 205–227. Google Scholar [31] J. Simon, Compact sets in the space $L^{p}(0, T;B)$, Ann. Mat. Pure Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar [32] R. E. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, 49. AMS, Providence, RI, 1997. Google Scholar [33] 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. Google Scholar [34] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. Google Scholar [35] C. F. Vasconcellos and L. M. Teixeira, Existence, uniqueness and stabilization for a nonlinear plate system with nonlinear damping, Ann. Fac. Sci. Toulouse Math., 8 (1999), 173-193. doi: 10.5802/afst.928. Google Scholar [36] S. Woinowsky-Krieger, The efect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36. Google Scholar [37] L. Yang and X. Wang, Existence of attractors for the non-autonomous Berger equation with nonlinear damping, Electron. J. Differential Equations, (2017), 14 pp. Google Scholar [38] L. Yang, Uniform attractor for non-autonomous plate equation with a localized damping and a critical nonlinearity, J. Math. Anal. Appl., 338 (2008), 1243-1254. doi: 10.1016/j.jmaa.2007.06.011. Google Scholar [39] Z. J. Yang, On an extensible beam equation with nonlinear damping and source terms, J.Dierential Equations, 254 (2013), 3903-3927. doi: 10.1016/j.jde.2013.02.008. Google Scholar
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