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doi: 10.3934/eect.2022013
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## Attractors for a class of extensible beams with strong nonlinear damping

 1 Department of Mathematics, State University of Londrina, Londrina, 86057-970, Brazil 2 Nucleus of Exact and Earth Sciences, State University of Mato Grosso do Sul, Dourados, 79804-970, Brazil

* Corresponding author. Partially supported by FUNDECT Grant 219/2016. Email: vnarciso@uems.br

Received  May 2021 Revised  January 2022 Early access March 2022

We concern to stablish the existence and qualitative properties of the compact global attractor associate to solutions of a class of extensible beam equations with strong nonlinear damping arising from the wave model proposed by Prestel [18].

Citation: Eduardo Henrique Gomes Tavares, Vando Narciso. Attractors for a class of extensible beams with strong nonlinear damping. Evolution Equations and Control Theory, doi: 10.3934/eect.2022013
##### References:
 [1] F. Aloui, I. B. Hassen and A. Haraux, Compactness of trajectories to some nonlinear second order evolution equation and applications, Math. Pures Appl., 100 (2013), 295-326.  doi: 10.1016/j.matpur.2013.01.002. [2] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste România, Bucharest, Noordhoff International Publishing, Leiden, 1976,352 pp. [3] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5. [4] H. Brézis, Équations et inéquations non linéaires dans les spaces vectoriels en dualité, Ann. Ins. Fourier (Grenoble), 18 (1968), 115-175.  doi: 10.5802/aif.280. [5] H. Brézis and F. E. Browder, Some properties of higher-order Sobolev spaces, Journal de Mathematiques Pures et Appliquees, 61 (1982), 245-259. [6] M. M. Cavalcanti, V. N. Domingos Cavalcanti, M. A. Jorge Silva and V. Narciso, Stability for extensible beams with a single degenerate nonlocal damping of Balakrishnan-Taylor type, Journal of Differential Equations, 290 (2021), 197-222.  doi: 10.1016/j.jde.2021.04.028. [7] I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Communication in Partial Differential Equations, 27 (2002), 1901-1951.  doi: 10.1081/PDE-120016132. [8] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. doi: 10.1090/memo/0912. [9] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations: Well-Posedness and Long-Time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9. [10] M. A. J. da Silva and V. Narciso, Attractors and their properties for a class of nonlocal extensible beams, Discrete and Continuous Dynamical Systems, 35 (2015), 985-1008.  doi: 10.3934/dcds.2015.35.985. [11] M. A. J. da 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. [12] P. Ding and Z. Yang, Longtime behavior for an extensible beam equation with rotational inertia and structural nonlinear damping, Journal of Mathematical Analysis and Applications, 496 (2021), 25 pp. doi: 10.1016/j.jmaa.2020.124785. [13] A. Haraux, Damping out of transient states for some semilinear, quasi-autonomous systems of hyperbolic type, Rend. Accad. Naz. Sci. XL Mem. Mat., 7 (1983), 89-136. [14] A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents, Journal of Differential Equations, 230 (2006), 702-719.  doi: 10.1016/j.jde.2006.06.001. [15] I. Lasiecka, T. F. Ma and R. N. Monteiro, Global smooth attractors for dynamics of thermal shallow shells without vertical dissipation, Trans. Amer. Math. Soc., 371 (2019), 8051-8096.  doi: 10.1090/tran/7756. [16] I. Lasiecka, T. F. Ma and R. N. Monteiro, Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1037-1072.  doi: 10.3934/dcdsb.2018141. [17] 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. [18] M.-A. Prestel, Forced oscillations for the solutions of a nonlinear hyperbolic equation, J. Nonlinear Anal., 6 (1982), 209-216.  doi: 10.1016/0362-546X(82)90089-X. [19] R. E. Showalter, Monotone Operator in Banach Spaces and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, 49. American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/049. [20] Z. 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.

show all references

##### References:
 [1] F. Aloui, I. B. Hassen and A. Haraux, Compactness of trajectories to some nonlinear second order evolution equation and applications, Math. Pures Appl., 100 (2013), 295-326.  doi: 10.1016/j.matpur.2013.01.002. [2] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste România, Bucharest, Noordhoff International Publishing, Leiden, 1976,352 pp. [3] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5. [4] H. Brézis, Équations et inéquations non linéaires dans les spaces vectoriels en dualité, Ann. Ins. Fourier (Grenoble), 18 (1968), 115-175.  doi: 10.5802/aif.280. [5] H. Brézis and F. E. Browder, Some properties of higher-order Sobolev spaces, Journal de Mathematiques Pures et Appliquees, 61 (1982), 245-259. [6] M. M. Cavalcanti, V. N. Domingos Cavalcanti, M. A. Jorge Silva and V. Narciso, Stability for extensible beams with a single degenerate nonlocal damping of Balakrishnan-Taylor type, Journal of Differential Equations, 290 (2021), 197-222.  doi: 10.1016/j.jde.2021.04.028. [7] I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Communication in Partial Differential Equations, 27 (2002), 1901-1951.  doi: 10.1081/PDE-120016132. [8] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. doi: 10.1090/memo/0912. [9] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations: Well-Posedness and Long-Time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9. [10] M. A. J. da Silva and V. Narciso, Attractors and their properties for a class of nonlocal extensible beams, Discrete and Continuous Dynamical Systems, 35 (2015), 985-1008.  doi: 10.3934/dcds.2015.35.985. [11] M. A. J. da 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. [12] P. Ding and Z. Yang, Longtime behavior for an extensible beam equation with rotational inertia and structural nonlinear damping, Journal of Mathematical Analysis and Applications, 496 (2021), 25 pp. doi: 10.1016/j.jmaa.2020.124785. [13] A. Haraux, Damping out of transient states for some semilinear, quasi-autonomous systems of hyperbolic type, Rend. Accad. Naz. Sci. XL Mem. Mat., 7 (1983), 89-136. [14] A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents, Journal of Differential Equations, 230 (2006), 702-719.  doi: 10.1016/j.jde.2006.06.001. [15] I. Lasiecka, T. F. Ma and R. N. Monteiro, Global smooth attractors for dynamics of thermal shallow shells without vertical dissipation, Trans. Amer. Math. Soc., 371 (2019), 8051-8096.  doi: 10.1090/tran/7756. [16] I. Lasiecka, T. F. Ma and R. N. Monteiro, Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1037-1072.  doi: 10.3934/dcdsb.2018141. [17] 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. [18] M.-A. Prestel, Forced oscillations for the solutions of a nonlinear hyperbolic equation, J. Nonlinear Anal., 6 (1982), 209-216.  doi: 10.1016/0362-546X(82)90089-X. [19] R. E. Showalter, Monotone Operator in Banach Spaces and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, 49. American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/049. [20] Z. 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.
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