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Long-time dynamics in plate models with strong nonlinear damping
1. | Department of Mechanics and Mathematics, Kharkov National University, Kharkov, 61077, Ukraine |
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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