# American Institute of Mathematical Sciences

March  2012, 11(2): 659-674. doi: 10.3934/cpaa.2012.11.659

## Long-time dynamics in plate models with strong nonlinear damping

 1 Department of Mechanics and Mathematics, Kharkov National University, Kharkov, 61077, Ukraine

Received  September 2010 Revised  January 2011 Published  October 2011

We study long-time dynamics of a class of abstract second order in time evolution equations in a Hilbert space with the damping term depending both on displacement and velocity. This damping represents the nonlinear strong dissipation phenomenon perturbed with relatively compact terms. Our main result states the existence of a compact finite dimensional attractor. We study properties of this attractor. We also establish the existence of a fractal exponential attractor and give the conditions that guarantee the existence of a finite number of determining functionals. In the case when the set of equilibria is finite and hyperbolic we show that every trajectory is attracted by some equilibrium with exponential rate. Our arguments involve a recently developed method based on the "compensated" compactness and quasi-stability estimates. As an application we consider the nonlinear Kirchhoff, Karman and Berger plate models with different types of boundary conditions and strong damping terms. Our results can be also applied to the nonlinear wave equations.
Citation: Igor Chueshov, Stanislav Kolbasin. Long-time dynamics in plate models with strong nonlinear damping. Communications on Pure & Applied Analysis, 2012, 11 (2) : 659-674. doi: 10.3934/cpaa.2012.11.659
##### References:
 [1] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North-Holland, (1992). Google Scholar [2] V. Barbu, I. Lasiecka and M. A. Rammaha, On nonlinear wave equations with degenerate damping and source terms,, Trans. Amer. Math. Soc., 357 (2005), 2571. doi: 10.1090/S0002-9947-05-03880-8. Google Scholar [3] A. Carvalho and J. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities,, Pacific J. Math., 207 (2002), 287. doi: 10.2140/pjm.2002.207.287. Google Scholar [4] J. W. Cholewa and T. Dlotko, Strongly damped wave equation in uniform spaces,, Nonlinear Anal., 64 (2006), 174. doi: 10.1016/j.na.2005.06.021. Google Scholar [5] I. D Chueshov, The theory of functionals that uniquely determine the asymptotic dynamics of infinite-dimensional dissipative systems,, Russian Math. Surveys, 53 (1998), 731. doi: 10.1070/RM1998v053n04ABEH000057. Google Scholar [6] I. D. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems,", (Russian) Acta, (1999). Google Scholar [7] I. Chueshov and V. Kalantarov, Determining functionals for nonlinear damped wave equations,, Mat. Fiz. Anal. Geom., 8 (2001), 215. 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. doi: 10.1016/j.na.2010.04.072. Google Scholar [9] I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping,, J. Dynam. Differential Equations, 16 (2004), 469. doi: 10.1007/s10884-004-4289-x. 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). Google Scholar [11] I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations,", Sprin\-ger, (2010). doi: 10.1007/978-0-387-87712-9. Google Scholar [12] B. Cockburn, D. A. Jones and E. Titi, Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems,, Math. Comp., 66 (1997), 1073. doi: 10.1090/S0025-5718-97-00850-8. Google Scholar [13] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential attractors for dissipative evolution equations,, RAM: Research in Applied Mathematics, 37 (1994). Google Scholar [14] C. Foias and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension deux,, (French), 39 (1967), 1. Google Scholar [15] S. Gatti and V. Pata, A one-dimensional wave equation with nonlinear damping,, Glasg. Math. J., 48 (2006), 419. doi: 10.1017/S0017089506003156. Google Scholar [16] J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", American Mathematical Society, (1988). Google Scholar [17] V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation,, J. Differential Equations, 247 (2009), 1120. doi: 10.1016/j.jde.2009.04.010. Google Scholar [18] A. Khanmamedov, A strong global attractor for the 3D wave equation with displacement dependent damping,, Appl. Math. Lett., 23 (2010), 928. doi: 10.1016/j.aml.2010.04.013. Google Scholar [19] S. Kolbasin, Attractors for Kirchhoff's equation with a nonlinear damping coefficient,, Nonlin. Anal., 71 (2009), 2361. doi: 10.1016/j.na.2009.01.187. Google Scholar [20] O. Ladyzhenskaya, A dynamical system generated by the Navier-Stokes equations,, J. Soviet Math., 3 (1975), 458. doi: 10.1007/BF01084684. Google Scholar [21] I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations,", Cambridge University Press, (2000). Google Scholar [22] J. L. Lions, "Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires,", (French), (1969). Google Scholar [23] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in, (2008), 103. Google Scholar [24] V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations,, Nonlinearity, 19 (2006), 1495. doi: 10.1088/0951-7715/19/7/001. Google Scholar [25] V. Pata and S. Zelik, Attractors and their regularity for 2-D wave equations with nonlinear damping,, Adv. Math. Sci. Appl., 17 (2007), 225. Google Scholar [26] V. Pata and S. Zelik, Global and exponential attractors for 3-D wave equations with displacement dependent damping,, Math. Methods Appl. Sci., 29 (2006), 1291. doi: 10.1002/mma.726. Google Scholar [27] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl., 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar [28] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Springer, (1988). Google Scholar [29] G. Raugel, Global attractors in partial differential equations,, in, 2 (2002), 885. Google Scholar [30] M. Yang and C. Sun, Dynamics of strongly damped wave equations in locally uniform spaces: attractors and asymptotic regularity,, Trans. Amer. Math. Soc., 361 (2009), 1069. doi: 10.1090/S0002-9947-08-04680-1. Google Scholar

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##### References:
 [1] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North-Holland, (1992). Google Scholar [2] V. Barbu, I. Lasiecka and M. A. Rammaha, On nonlinear wave equations with degenerate damping and source terms,, Trans. Amer. Math. Soc., 357 (2005), 2571. doi: 10.1090/S0002-9947-05-03880-8. Google Scholar [3] A. Carvalho and J. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities,, Pacific J. Math., 207 (2002), 287. doi: 10.2140/pjm.2002.207.287. Google Scholar [4] J. W. Cholewa and T. Dlotko, Strongly damped wave equation in uniform spaces,, Nonlinear Anal., 64 (2006), 174. doi: 10.1016/j.na.2005.06.021. Google Scholar [5] I. D Chueshov, The theory of functionals that uniquely determine the asymptotic dynamics of infinite-dimensional dissipative systems,, Russian Math. Surveys, 53 (1998), 731. doi: 10.1070/RM1998v053n04ABEH000057. Google Scholar [6] I. D. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems,", (Russian) Acta, (1999). Google Scholar [7] I. Chueshov and V. Kalantarov, Determining functionals for nonlinear damped wave equations,, Mat. Fiz. Anal. Geom., 8 (2001), 215. 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. doi: 10.1016/j.na.2010.04.072. Google Scholar [9] I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping,, J. Dynam. Differential Equations, 16 (2004), 469. doi: 10.1007/s10884-004-4289-x. 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). Google Scholar [11] I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations,", Sprin\-ger, (2010). doi: 10.1007/978-0-387-87712-9. Google Scholar [12] B. Cockburn, D. A. Jones and E. Titi, Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems,, Math. Comp., 66 (1997), 1073. doi: 10.1090/S0025-5718-97-00850-8. Google Scholar [13] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential attractors for dissipative evolution equations,, RAM: Research in Applied Mathematics, 37 (1994). Google Scholar [14] C. Foias and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension deux,, (French), 39 (1967), 1. Google Scholar [15] S. Gatti and V. Pata, A one-dimensional wave equation with nonlinear damping,, Glasg. Math. J., 48 (2006), 419. doi: 10.1017/S0017089506003156. Google Scholar [16] J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", American Mathematical Society, (1988). Google Scholar [17] V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation,, J. Differential Equations, 247 (2009), 1120. doi: 10.1016/j.jde.2009.04.010. Google Scholar [18] A. Khanmamedov, A strong global attractor for the 3D wave equation with displacement dependent damping,, Appl. Math. Lett., 23 (2010), 928. doi: 10.1016/j.aml.2010.04.013. Google Scholar [19] S. Kolbasin, Attractors for Kirchhoff's equation with a nonlinear damping coefficient,, Nonlin. Anal., 71 (2009), 2361. doi: 10.1016/j.na.2009.01.187. Google Scholar [20] O. Ladyzhenskaya, A dynamical system generated by the Navier-Stokes equations,, J. Soviet Math., 3 (1975), 458. doi: 10.1007/BF01084684. Google Scholar [21] I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations,", Cambridge University Press, (2000). Google Scholar [22] J. L. Lions, "Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires,", (French), (1969). Google Scholar [23] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in, (2008), 103. Google Scholar [24] V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations,, Nonlinearity, 19 (2006), 1495. doi: 10.1088/0951-7715/19/7/001. Google Scholar [25] V. Pata and S. Zelik, Attractors and their regularity for 2-D wave equations with nonlinear damping,, Adv. Math. Sci. Appl., 17 (2007), 225. Google Scholar [26] V. Pata and S. Zelik, Global and exponential attractors for 3-D wave equations with displacement dependent damping,, Math. Methods Appl. Sci., 29 (2006), 1291. doi: 10.1002/mma.726. Google Scholar [27] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl., 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar [28] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Springer, (1988). Google Scholar [29] G. Raugel, Global attractors in partial differential equations,, in, 2 (2002), 885. Google Scholar [30] M. Yang and C. Sun, Dynamics of strongly damped wave equations in locally uniform spaces: attractors and asymptotic regularity,, Trans. Amer. Math. Soc., 361 (2009), 1069. doi: 10.1090/S0002-9947-08-04680-1. Google Scholar
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