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December  2018, 23(10): 4595-4616. doi: 10.3934/dcdsb.2018178

Time-dependent asymptotic behavior of the solution for plate equations with linear memory

Tingting Liu and  Qiaozhen Ma ,

School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

* Corresponding author

Received  September 2017 Revised  January 2018 Published  December 2018 Early access  June 2018

Fund Project: Ma is supported by NSF grant(11561064, 11361053), and partly supported by NWNU-LKQN-14-6.

In this article, we consider the long-time behavior of solutions for the plate equation with linear memory. Within the theory of process on time-dependent spaces, we investigate the existence of the time-dependent attractor by using the operator decomposition technique and compactness of translation theorem and more detailed estimates. Furthermore, the asymptotic structure of time-dependent attractor, which converges to the attractor of fourth order parabolic equation with memory, is proved. Besides, we obtain a further regular result.

Keywords: Plate equation, linear memory, time-dependent attractor, regularity, compactness transitivity theorem, asymptotic structure.
Mathematics Subject Classification: Primary: 35B40; 35K30; Secondary: 37B55; 35B25.
Citation: Tingting Liu, Qiaozhen Ma. Time-dependent asymptotic behavior of the solution for plate equations with linear memory. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4595-4616. doi: 10.3934/dcdsb.2018178
References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[2]

J. M. Ball, Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90.  doi: 10.1016/0022-247X(73)90121-2.  Google Scholar

[3]

J. M. Ball, Stability theory for an extensible beam, J. Differential Equations, 14 (1973), 399-418.  doi: 10.1016/0022-0396(73)90056-9.  Google Scholar

[4]

S. Borini and V. Pata, Uniform attractors for a strongly damped wave equations with linear memory, Asymptot. Anal., 20 (1999), 263-277.   Google Scholar

[5]

M. Conti, V. Pata and R. Temam, Attractors for processes on time-dependent spaces. Applications to wave equations, J. Differential Equations, 255 (2013), 1254-1277.  doi: 10.1016/j.jde.2013.05.013.  Google Scholar

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M. Conti and V. Pata, Asymptotic structure of the attractor for processes on time-dependent spaces, Nonlinear Analysis RWA, 19 (2014), 1-10.  doi: 10.1016/j.nonrwa.2014.02.002.  Google Scholar

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M. Conti and V. Pata, On the time-dependent Cattaneo law in space dimension one, Applied Mathematic and Computation, 259 (2015), 32-44.  doi: 10.1016/j.amc.2015.02.039.  Google Scholar

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F. Di Plinio, G. S. Duane and R. Temam, Time-Dependent attractor for the oscillon equation, Discrete Contin. Dyn. Syst., 29 (2011), 141-167.  doi: 10.3934/dcds.2011.29.141.  Google Scholar

[9]

A. Kh. Khanmamedov, Existence of a global attractor for the plate equation with a critical exponent in an unbounded domain, Appl. Math. Lett., 18 (2005), 827-832.  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), 528-548.  doi: 10.1016/j.jde.2005.12.001.  Google Scholar

[11]

T. T. Liu and Q. Z. Ma, Existence of time-dependent global attractors for plate equation, J. East China Normal University(Chinese), 2 (2016), 35-44.   Google Scholar

[12]

T. T. Liu and Q. Z. Ma, The existence of time-dependent strong pullback attractors for nonautonomous plate equations, Chinese Annals of Mathematics(Chinese), 38 (2017), 125-144; Chinese Journal of Contemporary Mathematics(English), 2 (2017), 101-118.  Google Scholar

[13]

Q. Z. Ma, Y. Yang and X. L. Zhang, Existence of exponential attractors for the plate equations with strong damping, Elec. J. Differential Equations, 114 (2013), 1-10.   Google Scholar

[14]

W. J. Ma and Q. Z. Ma, Attractors for stochastic strongly damped plate equation with additive noise, Elec. J. Differential Equations, 111 (2013), 1-12.   Google Scholar

[15]

Q. F. Ma, S. H. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractor for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559.  doi: 10.1512/iumj.2002.51.2255.  Google Scholar

[16]

F. J. Meng, M. H. Yang and C. K. Zhong, Attractors for wave equations with nonlinear damping on time-dependent space, Discrete. Contin. Dyn. Syst. B., 21 (2016), 205-225.  doi: 10.3934/dcdsb.2016.21.205.  Google Scholar

[17]

F. J. Meng and C. C. Liu, Necessary and sufficient conditions for the existence of time-dependent global attractor and application, J. Math. Phys., 58(2017), 032702, 9 pp. doi: 10.1063/1.4978329.  Google Scholar

[18]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.   Google Scholar

[19]

G. S. Sell and Y. You, Dynamics of Evolution Equations, Springer-Verlag, New York, 2002. Google Scholar

[20]

J. Simon, Compact sets in the space LP (0; T; B), Ann. Math. Pura. Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[21]

S. Woinowsky, The effect of axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36.   Google Scholar

[22]

H. B. Xiao, Asymptotic dynamtics of plate equation with a critical exponent on unbounded domain, Nonlinear Analysis, 70 (2009), 1288-1301.  doi: 10.1016/j.na.2008.02.012.  Google Scholar

[23]

H. B. Xiao, Compact attractors of fourth order parabolic equation on Rn, Applied Mathematic and Computation, 219 (2013), 9827-9837.  doi: 10.1016/j.amc.2013.03.121.  Google Scholar

[24]

L. Yang and C. K. Zhong, Global attractor for plate equation with nonlinear damping, Nonlinear Analysis, 69 (2008), 3802-3810.  doi: 10.1016/j.na.2007.10.016.  Google Scholar

[25]

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.  Google Scholar

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[2]

J. M. Ball, Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90.  doi: 10.1016/0022-247X(73)90121-2.  Google Scholar

[3]

J. M. Ball, Stability theory for an extensible beam, J. Differential Equations, 14 (1973), 399-418.  doi: 10.1016/0022-0396(73)90056-9.  Google Scholar

[4]

S. Borini and V. Pata, Uniform attractors for a strongly damped wave equations with linear memory, Asymptot. Anal., 20 (1999), 263-277.   Google Scholar

[5]

M. Conti, V. Pata and R. Temam, Attractors for processes on time-dependent spaces. Applications to wave equations, J. Differential Equations, 255 (2013), 1254-1277.  doi: 10.1016/j.jde.2013.05.013.  Google Scholar

[6]

M. Conti and V. Pata, Asymptotic structure of the attractor for processes on time-dependent spaces, Nonlinear Analysis RWA, 19 (2014), 1-10.  doi: 10.1016/j.nonrwa.2014.02.002.  Google Scholar

[7]

M. Conti and V. Pata, On the time-dependent Cattaneo law in space dimension one, Applied Mathematic and Computation, 259 (2015), 32-44.  doi: 10.1016/j.amc.2015.02.039.  Google Scholar

[8]

F. Di Plinio, G. S. Duane and R. Temam, Time-Dependent attractor for the oscillon equation, Discrete Contin. Dyn. Syst., 29 (2011), 141-167.  doi: 10.3934/dcds.2011.29.141.  Google Scholar

[9]

A. Kh. Khanmamedov, Existence of a global attractor for the plate equation with a critical exponent in an unbounded domain, Appl. Math. Lett., 18 (2005), 827-832.  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), 528-548.  doi: 10.1016/j.jde.2005.12.001.  Google Scholar

[11]

T. T. Liu and Q. Z. Ma, Existence of time-dependent global attractors for plate equation, J. East China Normal University(Chinese), 2 (2016), 35-44.   Google Scholar

[12]

T. T. Liu and Q. Z. Ma, The existence of time-dependent strong pullback attractors for nonautonomous plate equations, Chinese Annals of Mathematics(Chinese), 38 (2017), 125-144; Chinese Journal of Contemporary Mathematics(English), 2 (2017), 101-118.  Google Scholar

[13]

Q. Z. Ma, Y. Yang and X. L. Zhang, Existence of exponential attractors for the plate equations with strong damping, Elec. J. Differential Equations, 114 (2013), 1-10.   Google Scholar

[14]

W. J. Ma and Q. Z. Ma, Attractors for stochastic strongly damped plate equation with additive noise, Elec. J. Differential Equations, 111 (2013), 1-12.   Google Scholar

[15]

Q. F. Ma, S. H. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractor for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559.  doi: 10.1512/iumj.2002.51.2255.  Google Scholar

[16]

F. J. Meng, M. H. Yang and C. K. Zhong, Attractors for wave equations with nonlinear damping on time-dependent space, Discrete. Contin. Dyn. Syst. B., 21 (2016), 205-225.  doi: 10.3934/dcdsb.2016.21.205.  Google Scholar

[17]

F. J. Meng and C. C. Liu, Necessary and sufficient conditions for the existence of time-dependent global attractor and application, J. Math. Phys., 58(2017), 032702, 9 pp. doi: 10.1063/1.4978329.  Google Scholar

[18]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.   Google Scholar

[19]

G. S. Sell and Y. You, Dynamics of Evolution Equations, Springer-Verlag, New York, 2002. Google Scholar

[20]

J. Simon, Compact sets in the space LP (0; T; B), Ann. Math. Pura. Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[21]

S. Woinowsky, The effect of axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36.   Google Scholar

[22]

H. B. Xiao, Asymptotic dynamtics of plate equation with a critical exponent on unbounded domain, Nonlinear Analysis, 70 (2009), 1288-1301.  doi: 10.1016/j.na.2008.02.012.  Google Scholar

[23]

H. B. Xiao, Compact attractors of fourth order parabolic equation on Rn, Applied Mathematic and Computation, 219 (2013), 9827-9837.  doi: 10.1016/j.amc.2013.03.121.  Google Scholar

[24]

L. Yang and C. K. Zhong, Global attractor for plate equation with nonlinear damping, Nonlinear Analysis, 69 (2008), 3802-3810.  doi: 10.1016/j.na.2007.10.016.  Google Scholar

[25]

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.  Google Scholar

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