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Prevalence of stable periodic solutions in the forced relativistic pendulum equation
Time-dependent asymptotic behavior of the solution for plate equations with linear memory
School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China |
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.
References:
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A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1994.
doi: 10.1007/978-1-4612-0873-0. |
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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. |
[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. |
[4] |
S. Borini and V. Pata,
Uniform attractors for a strongly damped wave equations with linear memory, Asymptot. Anal., 20 (1999), 263-277.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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] |
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.
|
[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. |
[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.
|
[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.
|
[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. |
[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. |
[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. |
[18] |
V. Pata and A. Zucchi,
Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.
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[19] |
G. S. Sell and Y. You,
Dynamics of Evolution Equations, Springer-Verlag, New York, 2002. |
[20] |
J. Simon,
Compact sets in the space LP (0; T; B), Ann. Math. Pura. Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[21] |
S. Woinowsky,
The effect of axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36.
|
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[4] |
S. Borini and V. Pata,
Uniform attractors for a strongly damped wave equations with linear memory, Asymptot. Anal., 20 (1999), 263-277.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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] |
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.
|
[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. |
[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.
|
[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.
|
[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. |
[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. |
[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. |
[18] |
V. Pata and A. Zucchi,
Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.
|
[19] |
G. S. Sell and Y. You,
Dynamics of Evolution Equations, Springer-Verlag, New York, 2002. |
[20] |
J. Simon,
Compact sets in the space LP (0; T; B), Ann. Math. Pura. Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[21] |
S. Woinowsky,
The effect of axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36.
|
[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. |
[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. |
[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. |
[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. |
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