Existence of the global attractor for the plate equation with nonlocal nonlinearity in \mathbb{R} ^{n}

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January 2016, 21(1): 151-172. doi: 10.3934/dcdsb.2016.21.151

Existence of the global attractor for the plate equation with nonlocal nonlinearity in $ \mathbb{R} ^{n}$

Azer Khanmamedov ^1, and Sema Simsek ^1,

1.	Department of Mathematics, Faculty of Science, Hacettepe University, Beytepe 06800, Ankara, Turkey

Received March 2015 Revised August 2015 Published November 2015

Abstract
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We consider Cauchy problem for the semilinear plate equation with nonlocal nonlinearity. Under mild conditions on the damping coefficient, we prove that the semigroup generated by this problem possesses a global attractor.

Keywords: global attractor., Plate equation.

Mathematics Subject Classification: 35B41, 35G20, 74K2.

Citation: Azer Khanmamedov, Sema Simsek. Existence of the global attractor for the plate equation with nonlocal nonlinearity in $ \mathbb{R} ^{n}$. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 151-172. doi: 10.3934/dcdsb.2016.21.151

References:

[1]	Z. Arat, A. Khanmamedov and S. Simsek, Global attractors for the plate equation with nonlocal nonlinearity in unbounded domains, Dynamics of PDE, 11 (2014), 361-379. doi: 10.4310/DPDE.2014.v11.n4.a4.
[2]	J. Ball, Global attractors for semilinear wave equations, Discr. Cont. Dyn. Sys., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31.
[3]	F. Bucci and I. Chueshov, Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations, Discrete Contin. Dyn. Syst., 22 (2008), 557-586. doi: 10.3934/dcds.2008.22.557.
[4]	T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford University Press, New York, 1998.
[5]	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.
[6]	I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer, Berlin, 2010. doi: 10.1007/978-0-387-87712-9.
[7]	E. Dowell, Aeroelasticity of Plates and Shells, Nordhoff, Leyden, 1975.
[8]	E. Dowell, A Modern Course in Aeroelasticity, Springer, 2015. doi: 10.1007/978-3-319-09453-3.
[9]	A. Kh. Khanmamedov, Existence of a global attractor for the plate equation with a critical exponent in an unbounded domain, Applied Mathematics Letters, 18 (2005), 827-832. doi: 10.1016/j.aml.2004.08.013.
[10]	A. Kh. Khanmamedov, Global attractors for the plate equation with a localized damping and a critical exponent in an unbounded domain, J. Differential Equations, 225 (2006), 528-548. doi: 10.1016/j.jde.2005.12.001.
[11]	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.
[12]	A. Kh. Khanmamedov, Global attractors for 2-D wave equations with displacement dependent damping, Math. Methods Appl. Sci., 33 (2010), 177-187. doi: 10.1002/mma.1161.
[13]	A. Kh. Khanmamedov, A global attractors for plate equation with displacement-dependent damping, Nonlinear Analysis, 74 (2011), 1607-1615. doi: 10.1016/j.na.2010.10.031.
[14]	S. Kolbasin, Attractors for Kirchoff's equation with a nonlinear damping coefficient, Nonlinear Analysis, 71 (2009), 2361-2371. doi: 10.1016/j.na.2009.01.187.
[15]	W. Krolikowski and O. Bang, {Solitons in nonlocal nonlnear media: Exact solutions, Physical Review E, 63 (2000), 016610.
[16]	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.
[17]	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.
[18]	M. Potomkin, {On transmission problem for Berger plates on an elastic base, Journal of Mathematical Physics, Analysis, Geometry, 7 (2011), 96-102.
[19]	M. Potomkin, A nonlinear transmission problem for acompound plate with thermoelastic part, Math. Methods Appl. Sci., 35 (2012), 530-546. doi: 10.1002/mma.1589.
[20]	J. Simon, Compact sets in the space $L_p(0,T;B)$, Annali Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.
[21]	A. Snyder and J. Mitchell, Accessible Solitons, Science, 276 (1997), 1538-1541. doi: 10.1126/science.276.5318.1538.
[22]	L. Yang, Uniform attractor for non-autonomous plate equation with a localized damping and a critical nonlinearity, J. Math. Anal. Appl., 338 (2008), 1243-1254. doi: 10.1016/j.jmaa.2007.06.011.
[23]	G. Yue and C. Zhong, Global attractors for plate equations with critical exponent in locally uniform spaces, Nonlinear Analysis, 71 (2009), 4105-4114. doi: 10.1016/j.na.2009.02.089.

show all references

References:

[1]	Z. Arat, A. Khanmamedov and S. Simsek, Global attractors for the plate equation with nonlocal nonlinearity in unbounded domains, Dynamics of PDE, 11 (2014), 361-379. doi: 10.4310/DPDE.2014.v11.n4.a4.
[2]	J. Ball, Global attractors for semilinear wave equations, Discr. Cont. Dyn. Sys., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31.
[3]	F. Bucci and I. Chueshov, Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations, Discrete Contin. Dyn. Syst., 22 (2008), 557-586. doi: 10.3934/dcds.2008.22.557.
[4]	T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford University Press, New York, 1998.
[5]	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.
[6]	I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer, Berlin, 2010. doi: 10.1007/978-0-387-87712-9.
[7]	E. Dowell, Aeroelasticity of Plates and Shells, Nordhoff, Leyden, 1975.
[8]	E. Dowell, A Modern Course in Aeroelasticity, Springer, 2015. doi: 10.1007/978-3-319-09453-3.
[9]	A. Kh. Khanmamedov, Existence of a global attractor for the plate equation with a critical exponent in an unbounded domain, Applied Mathematics Letters, 18 (2005), 827-832. doi: 10.1016/j.aml.2004.08.013.
[10]	A. Kh. Khanmamedov, Global attractors for the plate equation with a localized damping and a critical exponent in an unbounded domain, J. Differential Equations, 225 (2006), 528-548. doi: 10.1016/j.jde.2005.12.001.
[11]	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.
[12]	A. Kh. Khanmamedov, Global attractors for 2-D wave equations with displacement dependent damping, Math. Methods Appl. Sci., 33 (2010), 177-187. doi: 10.1002/mma.1161.
[13]	A. Kh. Khanmamedov, A global attractors for plate equation with displacement-dependent damping, Nonlinear Analysis, 74 (2011), 1607-1615. doi: 10.1016/j.na.2010.10.031.
[14]	S. Kolbasin, Attractors for Kirchoff's equation with a nonlinear damping coefficient, Nonlinear Analysis, 71 (2009), 2361-2371. doi: 10.1016/j.na.2009.01.187.
[15]	W. Krolikowski and O. Bang, {Solitons in nonlocal nonlnear media: Exact solutions, Physical Review E, 63 (2000), 016610.
[16]	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.
[17]	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.
[18]	M. Potomkin, {On transmission problem for Berger plates on an elastic base, Journal of Mathematical Physics, Analysis, Geometry, 7 (2011), 96-102.
[19]	M. Potomkin, A nonlinear transmission problem for acompound plate with thermoelastic part, Math. Methods Appl. Sci., 35 (2012), 530-546. doi: 10.1002/mma.1589.
[20]	J. Simon, Compact sets in the space $L_p(0,T;B)$, Annali Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.
[21]	A. Snyder and J. Mitchell, Accessible Solitons, Science, 276 (1997), 1538-1541. doi: 10.1126/science.276.5318.1538.
[22]	L. Yang, Uniform attractor for non-autonomous plate equation with a localized damping and a critical nonlinearity, J. Math. Anal. Appl., 338 (2008), 1243-1254. doi: 10.1016/j.jmaa.2007.06.011.
[23]	G. Yue and C. Zhong, Global attractors for plate equations with critical exponent in locally uniform spaces, Nonlinear Analysis, 71 (2009), 4105-4114. doi: 10.1016/j.na.2009.02.089.

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