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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}$

1. 

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

Received  March 2015 Revised  August 2015 Published  November 2015

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.
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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