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 and 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, Amsterdam, 1992.

[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), 2571-2611. doi: 10.1090/S0002-9947-05-03880-8.

[3]

A. Carvalho and J. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities, Pacific J. Math., 207 (2002), 287-310. doi: 10.2140/pjm.2002.207.287.

[4]

J. W. Cholewa and T. Dlotko, Strongly damped wave equation in uniform spaces, Nonlinear Anal., 64 (2006), 174-187. doi: 10.1016/j.na.2005.06.021.

[5]

I. D Chueshov, The theory of functionals that uniquely determine the asymptotic dynamics of infinite-dimensional dissipative systems, Russian Math. Surveys, 53 (1998), 731-776. doi: 10.1070/RM1998v053n04ABEH000057.

[6]

I. D. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems," (Russian) Acta, Kharkov, 1999; English translation: Acta, Kharkov, 2002; see also http://www.emis.de/monographs/Chueshov/.

[7]

I. Chueshov and V. Kalantarov, Determining functionals for nonlinear damped wave equations, Mat. Fiz. Anal. Geom., 8 (2001), 215-227.

[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.

[9]

I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512. doi: 10.1007/s10884-004-4289-x.

[10]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), no. 912.

[11]

I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations," Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.

[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), 1073-1087. doi: 10.1090/S0025-5718-97-00850-8.

[13]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential attractors for dissipative evolution equations, RAM: Research in Applied Mathematics, 37, Masson, Paris, 1994.

[14]

C. Foias and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension deux, (French), Rend. Sem. Mat. Univ. Padova, 39 (1967), 1-34.

[15]

S. Gatti and V. Pata, A one-dimensional wave equation with nonlinear damping, Glasg. Math. J., 48 (2006), 419-430. doi: 10.1017/S0017089506003156.

[16]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," American Mathematical Society, Providence, RI, 1988.

[17]

V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations, 247 (2009), 1120-1155. doi: 10.1016/j.jde.2009.04.010.

[18]

A. Khanmamedov, A strong global attractor for the 3D wave equation with displacement dependent damping, Appl. Math. Lett., 23 (2010), 928-934. doi: 10.1016/j.aml.2010.04.013.

[19]

S. Kolbasin, Attractors for Kirchhoff's equation with a nonlinear damping coefficient, Nonlin. Anal., 71 (2009), 2361-2371. doi: 10.1016/j.na.2009.01.187.

[20]

O. Ladyzhenskaya, A dynamical system generated by the Navier-Stokes equations, J. Soviet Math., 3 (1975), 458-479. doi: 10.1007/BF01084684.

[21]

I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations," Cambridge University Press, Cambridge, 2000.

[22]

J. L. Lions, "Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires," (French), Dunod, Paris, 1969.

[23]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations: Evolutionary Equations" (eds. C.M. Dafermos and M. Pokorny), North-Holland, Amsterdam (2008), 103-200.

[24]

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.

[25]

V. Pata and S. Zelik, Attractors and their regularity for 2-D wave equations with nonlinear damping, Adv. Math. Sci. Appl., 17 (2007), 225-237.

[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), 1291-1306. doi: 10.1002/mma.726.

[27]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.

[28]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer, New York, 1988.

[29]

G. Raugel, Global attractors in partial differential equations, in "Handbook of Dynamical Systems," 2, Elsevier Sciences, Amsterdam (2002), 885-992.

[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), 1069-1101. doi: 10.1090/S0002-9947-08-04680-1.

show all references

References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992.

[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), 2571-2611. doi: 10.1090/S0002-9947-05-03880-8.

[3]

A. Carvalho and J. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities, Pacific J. Math., 207 (2002), 287-310. doi: 10.2140/pjm.2002.207.287.

[4]

J. W. Cholewa and T. Dlotko, Strongly damped wave equation in uniform spaces, Nonlinear Anal., 64 (2006), 174-187. doi: 10.1016/j.na.2005.06.021.

[5]

I. D Chueshov, The theory of functionals that uniquely determine the asymptotic dynamics of infinite-dimensional dissipative systems, Russian Math. Surveys, 53 (1998), 731-776. doi: 10.1070/RM1998v053n04ABEH000057.

[6]

I. D. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems," (Russian) Acta, Kharkov, 1999; English translation: Acta, Kharkov, 2002; see also http://www.emis.de/monographs/Chueshov/.

[7]

I. Chueshov and V. Kalantarov, Determining functionals for nonlinear damped wave equations, Mat. Fiz. Anal. Geom., 8 (2001), 215-227.

[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.

[9]

I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512. doi: 10.1007/s10884-004-4289-x.

[10]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), no. 912.

[11]

I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations," Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.

[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), 1073-1087. doi: 10.1090/S0025-5718-97-00850-8.

[13]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential attractors for dissipative evolution equations, RAM: Research in Applied Mathematics, 37, Masson, Paris, 1994.

[14]

C. Foias and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension deux, (French), Rend. Sem. Mat. Univ. Padova, 39 (1967), 1-34.

[15]

S. Gatti and V. Pata, A one-dimensional wave equation with nonlinear damping, Glasg. Math. J., 48 (2006), 419-430. doi: 10.1017/S0017089506003156.

[16]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," American Mathematical Society, Providence, RI, 1988.

[17]

V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations, 247 (2009), 1120-1155. doi: 10.1016/j.jde.2009.04.010.

[18]

A. Khanmamedov, A strong global attractor for the 3D wave equation with displacement dependent damping, Appl. Math. Lett., 23 (2010), 928-934. doi: 10.1016/j.aml.2010.04.013.

[19]

S. Kolbasin, Attractors for Kirchhoff's equation with a nonlinear damping coefficient, Nonlin. Anal., 71 (2009), 2361-2371. doi: 10.1016/j.na.2009.01.187.

[20]

O. Ladyzhenskaya, A dynamical system generated by the Navier-Stokes equations, J. Soviet Math., 3 (1975), 458-479. doi: 10.1007/BF01084684.

[21]

I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations," Cambridge University Press, Cambridge, 2000.

[22]

J. L. Lions, "Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires," (French), Dunod, Paris, 1969.

[23]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations: Evolutionary Equations" (eds. C.M. Dafermos and M. Pokorny), North-Holland, Amsterdam (2008), 103-200.

[24]

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.

[25]

V. Pata and S. Zelik, Attractors and their regularity for 2-D wave equations with nonlinear damping, Adv. Math. Sci. Appl., 17 (2007), 225-237.

[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), 1291-1306. doi: 10.1002/mma.726.

[27]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.

[28]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer, New York, 1988.

[29]

G. Raugel, Global attractors in partial differential equations, in "Handbook of Dynamical Systems," 2, Elsevier Sciences, Amsterdam (2002), 885-992.

[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), 1069-1101. doi: 10.1090/S0002-9947-08-04680-1.

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