doi: 10.3934/dcdsb.2022046
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Attractor of the Kirchhoff type plate equation with memory and nonlinear damping on the whole time-dependent space

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

* Corresponding author

Received  August 2021 Revised  January 2022 Early access March 2022

Fund Project: Ma and Liu are supported by NSF grant(11961059, 12101502)

First of all, by virtue of the Faedo-Galerkin procedure, we obtain existence of solution for the Kirchhoff type plate equation with memory and nonlinear damping on $ \mathbb{R}^{n}. $ Secondly, using a new method of asymptotic contractive functions presented in [9] as well as the tail estimates we prove the asymptotic compactness of solution process. Finally, existence of the time-dependent attractor on $ \mathbb{R}^{n} $ is shown. The results are new and they are the extension and improvement of [9].

Citation: Tingting Liu, Qiaozhen Ma, Ling Xu. Attractor of the Kirchhoff type plate equation with memory and nonlinear damping on the whole time-dependent space. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022046
References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

[2]

H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472.  doi: 10.1115/1.4011138.

[3]

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

[4]

M. Conti and V. Pata, On the time-dependent Cattaneo law in space dimension one, Appl. Math. Compu., 259 (2015), 32-44.  doi: 10.1016/j.amc.2015.02.039.

[5]

M. ContiV. 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]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.

[7]

A. K. 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.

[8]

A. K. Khanmamedov, Global attractors for the plate equation with a localized damping and a critical exponent in an unbounded domain, J. Differential Equations, 255 (2006), 528-548.  doi: 10.1016/j.jde.2005.12.001.

[9]

T. T. Liu and Q. Z. Ma, Time-dependent attractor for plate equations on $\mathbb{R}^{n}$, J. Math. Anal. Appl., 479 (2019), 315-332.  doi: 10.1016/j.jmaa.2019.06.028.

[10]

T. T. Liu and Q. Z. Ma, Time-dependent asymptotic behavior of the solution for plate equations with linear memory, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4595-4616.  doi: 10.3934/dcdsb.2018178.

[11]

T. T. Liu and Q. Z. Ma, The existence of time-dependent strong pullback attractors for non-autonomous plate equations(Chinese), translation in Chinese J. Contemp. Math., 38 (2017), 101-118. 

[12]

Q. F. MaS. 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.

[13]

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.

[14]

F. J. MengJ. Wu and C. X. Zhao, Time-dependent global attractor for extensible Berger equation, J. Math. Anal. Appl., 469 (2019), 1045-1069.  doi: 10.1016/j.jmaa.2018.09.050.

[15]

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

[16]

F. Morillas and J. Valero, Attractors for reaction-diffusion equations in $\mathbb{R}^n$ with continuous nonlinearity, Asymptotic Anal., 44 (2005)1, 111–130.

[17]

V. Pata, Attractors for a damped wave equation on $\mathbb{R}^3$ with linear memory, Math. Methods Appl. Sci., 23 (2000), 633-653.  doi: 10.1002/(SICI)1099-1476(20000510)23:7<633::AID-MMA135>3.0.CO;2-C.

[18]

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

[19]

D. PlinioG. 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.

[20]

J. E. M. River and L. H. Fatori, Smoothing effect and propagations of singularities for viscoelastic plates, J. Math. Anal. Appl., 206 (1997), 397-427.  doi: 10.1006/jmaa.1997.5223.

[21] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001. 
[22]

M. A. J. Silva and T. F. Ma, On a viscoelastic plate equation with history setting and perturbation of p-Laplacian type, Ima J. Appl. Math., 78 (2013), 1130-1146.  doi: 10.1093/imamat/hxs011.

[23]

M. A. J. Silva and T. F. Ma, Long-time dynamics for a class of Kirchhoff models with memory, J. Math. Phys., 54 (2013), 021505, 15pp. doi: 10.1063/1.4792606.

[24]

S. Woinowsky, The effect of axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36.  doi: 10.1115/1.4010053.

[25]

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

[26]

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

[27]

Z. J. Yang and P. Y. Ding, Longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on $\mathbb{R}^{N}$, J. Math. Anal. Appl., 434 (2016), 1826-1851.  doi: 10.1016/j.jmaa.2015.10.013.

[28]

X. B. Yao and Q. Z. Ma, Global attractors of the extensible plate equations with nonlinear damping and memory, J. Funct. Spaces, 2017 (2017), Art. ID 4896161, 10 pp. doi: 10.1155/2017/4896161.

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

[2]

H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472.  doi: 10.1115/1.4011138.

[3]

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

[4]

M. Conti and V. Pata, On the time-dependent Cattaneo law in space dimension one, Appl. Math. Compu., 259 (2015), 32-44.  doi: 10.1016/j.amc.2015.02.039.

[5]

M. ContiV. 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]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.

[7]

A. K. 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.

[8]

A. K. Khanmamedov, Global attractors for the plate equation with a localized damping and a critical exponent in an unbounded domain, J. Differential Equations, 255 (2006), 528-548.  doi: 10.1016/j.jde.2005.12.001.

[9]

T. T. Liu and Q. Z. Ma, Time-dependent attractor for plate equations on $\mathbb{R}^{n}$, J. Math. Anal. Appl., 479 (2019), 315-332.  doi: 10.1016/j.jmaa.2019.06.028.

[10]

T. T. Liu and Q. Z. Ma, Time-dependent asymptotic behavior of the solution for plate equations with linear memory, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4595-4616.  doi: 10.3934/dcdsb.2018178.

[11]

T. T. Liu and Q. Z. Ma, The existence of time-dependent strong pullback attractors for non-autonomous plate equations(Chinese), translation in Chinese J. Contemp. Math., 38 (2017), 101-118. 

[12]

Q. F. MaS. 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.

[13]

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.

[14]

F. J. MengJ. Wu and C. X. Zhao, Time-dependent global attractor for extensible Berger equation, J. Math. Anal. Appl., 469 (2019), 1045-1069.  doi: 10.1016/j.jmaa.2018.09.050.

[15]

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

[16]

F. Morillas and J. Valero, Attractors for reaction-diffusion equations in $\mathbb{R}^n$ with continuous nonlinearity, Asymptotic Anal., 44 (2005)1, 111–130.

[17]

V. Pata, Attractors for a damped wave equation on $\mathbb{R}^3$ with linear memory, Math. Methods Appl. Sci., 23 (2000), 633-653.  doi: 10.1002/(SICI)1099-1476(20000510)23:7<633::AID-MMA135>3.0.CO;2-C.

[18]

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

[19]

D. PlinioG. 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.

[20]

J. E. M. River and L. H. Fatori, Smoothing effect and propagations of singularities for viscoelastic plates, J. Math. Anal. Appl., 206 (1997), 397-427.  doi: 10.1006/jmaa.1997.5223.

[21] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001. 
[22]

M. A. J. Silva and T. F. Ma, On a viscoelastic plate equation with history setting and perturbation of p-Laplacian type, Ima J. Appl. Math., 78 (2013), 1130-1146.  doi: 10.1093/imamat/hxs011.

[23]

M. A. J. Silva and T. F. Ma, Long-time dynamics for a class of Kirchhoff models with memory, J. Math. Phys., 54 (2013), 021505, 15pp. doi: 10.1063/1.4792606.

[24]

S. Woinowsky, The effect of axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36.  doi: 10.1115/1.4010053.

[25]

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

[26]

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

[27]

Z. J. Yang and P. Y. Ding, Longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on $\mathbb{R}^{N}$, J. Math. Anal. Appl., 434 (2016), 1826-1851.  doi: 10.1016/j.jmaa.2015.10.013.

[28]

X. B. Yao and Q. Z. Ma, Global attractors of the extensible plate equations with nonlinear damping and memory, J. Funct. Spaces, 2017 (2017), Art. ID 4896161, 10 pp. doi: 10.1155/2017/4896161.

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