# American Institute of Mathematical Sciences

September  2017, 6(3): 437-470. doi: 10.3934/eect.2017023

## Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping*

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

Corresponding author

* Dedicated to the memory of Professor Igor Chueshov

Received  January 2017 Revised  May 2017 Published  July 2017

Fund Project: The first author was supported by supported by CNPq grant 441414/2014-1.
The second author was partially supported by CAPES grant 20132268.

In this paper we consider new results on well-posedness and long-time dynamics for a class of extensible beam/plate models whose dissipative effect is given by the product of two nonlinear terms. The addressed model contains a nonlocal nonlinear damping term which generalizes some classes of dissipations usually given in the literature, namely, the linear, the nonlinear and the nonlocal frictional ones. A first mathematical analysis of such damping term is presented and represents the main novelty in our approach.

Citation: Marcio Antonio Jorge da Silva, Vando Narciso. Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping*. Evolution Equations & Control Theory, 2017, 6 (3) : 437-470. doi: 10.3934/eect.2017023
 [1] A. V. Balakrishnan and L. W. Taylor, Distributed Parameter Nonlinear Damping Models for Flight Structures in: Proceedings ''Daming 89", Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989. Google Scholar [2] 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 [3] M. M. Cavalcanti, V. N. Domingos 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 [4] 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 [5] R. C. Charão, E. Bisognin, V. Bisognin and A. F. 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Anal., 11 (2012), 659-674.  doi: 10.3934/cpaa.2012.11.659.  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), viii+183 pp. doi: 10.1090/memo/0912.  Google Scholar [11] 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.  Google Scholar [12] I. Chueshov, I. Lasiecka and D. Toundykov, Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent, Discrete Contin. Dyn. Syst., 20 (2008), 459-509.  doi: 10.3934/dcds.2008.20.459.  Google Scholar [13] 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 [14] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar [15] 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 [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] 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 [18] A. Kh. Khanmamedov, Global attractors for the plate equation with a localized damping and critical exponent in an unbounded domain, J. Differential Equation, 225 (2006), 528-548.  doi: 10.1016/j.jde.2005.12.001.  Google Scholar [19] A. Kh. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation, J. Math. Anal. Appl., 318 (2006), 92-101.  doi: 10.1016/j.jmaa.2005.05.031.  Google Scholar [20] A. Kh. Khanmamedov, A global attractor for the plate equation with displacement-dependent damping, Nonlinear Anal., 74 (2011), 1607-1615.  doi: 10.1016/j.na.2010.10.031.  Google Scholar [21] S. Kolbasin, Attractors for Kirchhoff's equation with a nonlinear damping coefficient, Nonlinear Anal., 71 (2009), 2361-2371.  doi: 10.1016/j.na.2009.01.187.  Google Scholar [22] 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 [23] H. Lange and G. Perla Menzala, Rates of decay of a nonlocal beam equation, Differential Integral Equations, 10 (1997), 1075-1092.   Google Scholar [24] P. Lazo, Global solutions for a nonlinear wave equation, Appl. Math. Comput., 200 (2008), 596-601.  doi: 10.1016/j.amc.2007.11.056.  Google Scholar [25] J. -L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires Dunod, Paris, 1969.  Google Scholar [26] J. -L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar [27] 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 [28] 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 [29] L. A. Medeiros and M. Milla Miranda, On a nonlinear wave equation with damping, Rev. Mat. Univ. Complut. Madrid, 3 (1990), 213-231.   Google Scholar [30] M. Nakao, Convergence of solutions of the wave equation with a nonlinear dissipative term to the steady state, Mem. Fac. Sci. Kyushu Univ. Ser. A, 30 (1976), 257-265.  doi: 10.2206/kyushumfs.30.257.  Google Scholar [31] M. Nakao, On the decay of solutions of some nonlinear dissipative wave equations in higher dimensions, Math. Z., 193 (1986), 227-234.  doi: 10.1007/BF01174332.  Google Scholar [32] M. Nakao, Global attractors for wave equations with nonlinear dissipative terms, J. Differential Equations, 227 (2006), 204-229.  doi: 10.1016/j.jde.2005.09.013.  Google Scholar [33] M. Potomkin, Asymptotic behavior of thermoviscoelastic Berger plate, Commun. Pure Appl. Anal., 9 (2010), 161-192.  doi: 10.3934/cpaa.2010.9.161.  Google Scholar [34] J. Simon, Compact sets in the space $$L^{p}(0,T;B) , Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. 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. Google Scholar [36] D. Wang and J. Zhang, Global attractor for a nonlinear plate equation with supported boundary conditions, J. Math. Anal. Appl., 363 (2010), 468-480. doi: 10.1016/j.jmaa.2009.09.020. Google Scholar [37] Y. Zhijian, 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 show all references ##### References:  [1] A. V. Balakrishnan and L. W. Taylor, Distributed Parameter Nonlinear Damping Models for Flight Structures in: Proceedings ''Daming 89", Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989. Google Scholar [2] 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 [3] M. M. Cavalcanti, V. N. Domingos 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 [4] 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 [5] R. C. Charão, E. Bisognin, V. Bisognin and A. F. Pazoto, Asymptotic behavior of a Bernoulli-Euler type equation with nonlinear localized damping, Contributions to Nonlinear Analysis -Progress in nonlinear partial differential equations and their applications, 66 (2005), 67-91. doi: 10.1007/3-7643-7401-2_5. Google Scholar [6] I. Chueshov, Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping, J. Abstr. Differ. Equ. Appl., 1 (2010), 86-106. Google Scholar [7] I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262. doi: 10.1016/j.jde.2011.08.022. Google Scholar [8] I. Chueshov and S. Kolbasin, Plate models with state-dependent damping coefficient and their quasi-static limits, Nonlinear Anal., 73 (2010), 1626-1644. doi: 10.1016/j.na.2010.04.072. Google Scholar [9] 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 [10] 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. Google Scholar [11] 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. Google Scholar [12] I. Chueshov, I. Lasiecka and D. Toundykov, Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent, Discrete Contin. Dyn. Syst., 20 (2008), 459-509. doi: 10.3934/dcds.2008.20.459. Google Scholar [13] 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 [14] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025. Google Scholar [15] 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 [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] 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 [18] A. Kh. Khanmamedov, Global attractors for the plate equation with a localized damping and critical exponent in an unbounded domain, J. Differential Equation, 225 (2006), 528-548. doi: 10.1016/j.jde.2005.12.001. Google Scholar [19] A. Kh. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation, J. Math. Anal. Appl., 318 (2006), 92-101. doi: 10.1016/j.jmaa.2005.05.031. Google Scholar [20] A. Kh. Khanmamedov, A global attractor for the plate equation with displacement-dependent damping, Nonlinear Anal., 74 (2011), 1607-1615. doi: 10.1016/j.na.2010.10.031. Google Scholar [21] S. Kolbasin, Attractors for Kirchhoff's equation with a nonlinear damping coefficient, Nonlinear Anal., 71 (2009), 2361-2371. doi: 10.1016/j.na.2009.01.187. Google Scholar [22] 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 [23] H. Lange and G. Perla Menzala, Rates of decay of a nonlocal beam equation, Differential Integral Equations, 10 (1997), 1075-1092. Google Scholar [24] P. Lazo, Global solutions for a nonlinear wave equation, Appl. Math. Comput., 200 (2008), 596-601. doi: 10.1016/j.amc.2007.11.056. Google Scholar [25] J. -L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires Dunod, Paris, 1969. Google Scholar [26] J. -L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, 1972. Google Scholar [27] 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 [28] 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 [29] L. A. Medeiros and M. Milla Miranda, On a nonlinear wave equation with damping, Rev. Mat. Univ. Complut. Madrid, 3 (1990), 213-231. Google Scholar [30] M. Nakao, Convergence of solutions of the wave equation with a nonlinear dissipative term to the steady state, Mem. Fac. Sci. Kyushu Univ. Ser. A, 30 (1976), 257-265. doi: 10.2206/kyushumfs.30.257. Google Scholar [31] M. Nakao, On the decay of solutions of some nonlinear dissipative wave equations in higher dimensions, Math. Z., 193 (1986), 227-234. doi: 10.1007/BF01174332. Google Scholar [32] M. Nakao, Global attractors for wave equations with nonlinear dissipative terms, J. Differential Equations, 227 (2006), 204-229. doi: 10.1016/j.jde.2005.09.013. Google Scholar [33] M. Potomkin, Asymptotic behavior of thermoviscoelastic Berger plate, Commun. Pure Appl. Anal., 9 (2010), 161-192. doi: 10.3934/cpaa.2010.9.161. Google Scholar [34] J. Simon, Compact sets in the space$$L^{p}(0,T;B), Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  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.   Google Scholar [36] D. Wang and J. Zhang, Global attractor for a nonlinear plate equation with supported boundary conditions, J. Math. Anal. Appl., 363 (2010), 468-480.  doi: 10.1016/j.jmaa.2009.09.020.  Google Scholar [37] Y. Zhijian, 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
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