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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 & 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.  doi: 10.4310/DPDE.2014.v11.n4.a4.  Google Scholar

[2]

J. Ball, Global attractors for semilinear wave equations,, Discr. Cont. Dyn. Sys., 10 (2004), 31.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[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.  doi: 10.3934/dcds.2008.22.557.  Google Scholar

[4]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford University Press, (1998).   Google Scholar

[5]

I. Chueshov and S. Kolbasin, Long-time dynamics in plate models with strong nonlinear damping,, Commun. Pure Appl. Anal., 11 (2012), 659.  doi: 10.3934/cpaa.2012.11.659.  Google Scholar

[6]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations,, Springer, (2010).  doi: 10.1007/978-0-387-87712-9.  Google Scholar

[7]

E. Dowell, Aeroelasticity of Plates and Shells,, Nordhoff, (1975).   Google Scholar

[8]

E. Dowell, A Modern Course in Aeroelasticity,, Springer, (2015).  doi: 10.1007/978-3-319-09453-3.  Google Scholar

[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.  doi: 10.1016/j.aml.2004.08.013.  Google Scholar

[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.  doi: 10.1016/j.jde.2005.12.001.  Google Scholar

[11]

A. Kh. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation,, J. Math. Anal. Appl., 318 (2006), 92.  doi: 10.1016/j.jmaa.2005.05.031.  Google Scholar

[12]

A. Kh. Khanmamedov, Global attractors for 2-D wave equations with displacement dependent damping,, Math. Methods Appl. Sci., 33 (2010), 177.  doi: 10.1002/mma.1161.  Google Scholar

[13]

A. Kh. Khanmamedov, A global attractors for plate equation with displacement-dependent damping,, Nonlinear Analysis, 74 (2011), 1607.  doi: 10.1016/j.na.2010.10.031.  Google Scholar

[14]

S. Kolbasin, Attractors for Kirchoff's equation with a nonlinear damping coefficient,, Nonlinear Analysis, 71 (2009), 2361.  doi: 10.1016/j.na.2009.01.187.  Google Scholar

[15]

W. Krolikowski and O. Bang, {Solitons in nonlocal nonlnear media: Exact solutions,, Physical Review E, 63 (2000).   Google Scholar

[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.  doi: 10.1016/j.na.2010.07.023.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2012.07.004.  Google Scholar

[18]

M. Potomkin, {On transmission problem for Berger plates on an elastic base,, Journal of Mathematical Physics, 7 (2011), 96.   Google Scholar

[19]

M. Potomkin, A nonlinear transmission problem for acompound plate with thermoelastic part,, Math. Methods Appl. Sci., 35 (2012), 530.  doi: 10.1002/mma.1589.  Google Scholar

[20]

J. Simon, Compact sets in the space $L_p(0,T;B)$,, Annali Mat. Pura Appl., 146 (1987), 65.  doi: 10.1007/BF01762360.  Google Scholar

[21]

A. Snyder and J. Mitchell, Accessible Solitons,, Science, 276 (1997), 1538.  doi: 10.1126/science.276.5318.1538.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2007.06.011.  Google Scholar

[23]

G. Yue and C. Zhong, Global attractors for plate equations with critical exponent in locally uniform spaces,, Nonlinear Analysis, 71 (2009), 4105.  doi: 10.1016/j.na.2009.02.089.  Google Scholar

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.  doi: 10.4310/DPDE.2014.v11.n4.a4.  Google Scholar

[2]

J. Ball, Global attractors for semilinear wave equations,, Discr. Cont. Dyn. Sys., 10 (2004), 31.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[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.  doi: 10.3934/dcds.2008.22.557.  Google Scholar

[4]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford University Press, (1998).   Google Scholar

[5]

I. Chueshov and S. Kolbasin, Long-time dynamics in plate models with strong nonlinear damping,, Commun. Pure Appl. Anal., 11 (2012), 659.  doi: 10.3934/cpaa.2012.11.659.  Google Scholar

[6]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations,, Springer, (2010).  doi: 10.1007/978-0-387-87712-9.  Google Scholar

[7]

E. Dowell, Aeroelasticity of Plates and Shells,, Nordhoff, (1975).   Google Scholar

[8]

E. Dowell, A Modern Course in Aeroelasticity,, Springer, (2015).  doi: 10.1007/978-3-319-09453-3.  Google Scholar

[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.  doi: 10.1016/j.aml.2004.08.013.  Google Scholar

[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.  doi: 10.1016/j.jde.2005.12.001.  Google Scholar

[11]

A. Kh. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation,, J. Math. Anal. Appl., 318 (2006), 92.  doi: 10.1016/j.jmaa.2005.05.031.  Google Scholar

[12]

A. Kh. Khanmamedov, Global attractors for 2-D wave equations with displacement dependent damping,, Math. Methods Appl. Sci., 33 (2010), 177.  doi: 10.1002/mma.1161.  Google Scholar

[13]

A. Kh. Khanmamedov, A global attractors for plate equation with displacement-dependent damping,, Nonlinear Analysis, 74 (2011), 1607.  doi: 10.1016/j.na.2010.10.031.  Google Scholar

[14]

S. Kolbasin, Attractors for Kirchoff's equation with a nonlinear damping coefficient,, Nonlinear Analysis, 71 (2009), 2361.  doi: 10.1016/j.na.2009.01.187.  Google Scholar

[15]

W. Krolikowski and O. Bang, {Solitons in nonlocal nonlnear media: Exact solutions,, Physical Review E, 63 (2000).   Google Scholar

[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.  doi: 10.1016/j.na.2010.07.023.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2012.07.004.  Google Scholar

[18]

M. Potomkin, {On transmission problem for Berger plates on an elastic base,, Journal of Mathematical Physics, 7 (2011), 96.   Google Scholar

[19]

M. Potomkin, A nonlinear transmission problem for acompound plate with thermoelastic part,, Math. Methods Appl. Sci., 35 (2012), 530.  doi: 10.1002/mma.1589.  Google Scholar

[20]

J. Simon, Compact sets in the space $L_p(0,T;B)$,, Annali Mat. Pura Appl., 146 (1987), 65.  doi: 10.1007/BF01762360.  Google Scholar

[21]

A. Snyder and J. Mitchell, Accessible Solitons,, Science, 276 (1997), 1538.  doi: 10.1126/science.276.5318.1538.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2007.06.011.  Google Scholar

[23]

G. Yue and C. Zhong, Global attractors for plate equations with critical exponent in locally uniform spaces,, Nonlinear Analysis, 71 (2009), 4105.  doi: 10.1016/j.na.2009.02.089.  Google Scholar

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