American Institute of Mathematical Sciences

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September  2021, 14(9): 3319-3336. doi: 10.3934/dcdss.2021076

Non-autonomous weakly damped plate model on time-dependent domains

Penghui Zhang 1, , Zhaosheng Feng 2, and  Lu Yang 1,

1. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China
2. 

School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg, Texas 78539, USA

Received  January 2020 Revised  October 2020 Published  September 2021 Early access  June 2021

Fund Project: This work was supported by the NSFC grants(12071192)

We are concerned with dynamics of the weakly damped plate equation on a time-dependent domain. Under the assumption that the domain is time-like and expanding, we obtain the existence of time-dependent attractors, where the nonlinear term has a critical growth.

Keywords: Plate equation, non-cylindrical domain, non-autonomous system, pullback attractor, critical exponent.
Mathematics Subject Classification: 37L05, 35B40, 35B41.
Citation: Penghui Zhang, Zhaosheng Feng, Lu Yang. Non-autonomous weakly damped plate model on time-dependent domains. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3319-3336. doi: 10.3934/dcdss.2021076
References:
[1]

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T. F. Ma, P. Marín-Rubio and C. M. Surco Chuño, Dynamics of wave equation with moving boundary, J. Differential Equations, 262 (2017), 3317-3342.  doi: 10.1016/j.jde.2016.11.030.  Google Scholar

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T. F. Ma and T. M. Souza, Pullback dynamics of non-autonomous wave equations with acoustic boundary condition, Differential Integral Equations, 30 (2017), 443-462.   Google Scholar

[22]

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

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[25]

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[26]

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Z. Wang and S. Zhou, Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise, Discrete Contin. Dyn. Syst., 38 (2018), 4767-4817.  doi: 10.3934/dcds.2018210.  Google Scholar

[29]

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

[30]

Z. Yang and Z. Liu, Longtime dynamics of the quasi-linear wave equations with structural damping and supercritical nonlinearities, Nonlinearity, 30 (2017), 1120-1145.  doi: 10.1088/1361-6544/aa599f.  Google Scholar

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Z. Yang and Z. Liu, Stability of exponential attractors for a family of semilinear wave equations with gentle dissipation, J. Differential Equations, 264 (2018), 3976-4005.  doi: 10.1016/j.jde.2017.11.035.  Google Scholar

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F. Zhou, C. Sun and X. Li, Dynamics for the damped wave equations on time-dependent domains, Discrete Contin. Dyn. Syst. B, 23 (2018), 1645-1674.  doi: 10.3934/dcdsb.2018068.  Google Scholar

show all references

References:
[1]

M. M. Al-Gharabli and S. A. Messaoudi, Existence and a general decay result for a plate equation with nonlinear damping and a logarithmic source term, J. Evol. Equ., 18 (2018), 105-125.  doi: 10.1007/s00028-017-0392-4.  Google Scholar

[2]

C. Bardos and G. Chen, Control and stabilization for the wave equation, Part Ⅲ: Domain with moving boundary, SIAM J. Control Optim., 19 (1981), 123-138.  doi: 10.1137/0319010.  Google Scholar

[3]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[4]

X. Chen and A. Friedman, A free boundary problem for an elliptic-hyperbolic system: An application to tumor growth, SIAM J. Math. Anal., 35 (2003), 974-986.  doi: 10.1137/S0036141002418388.  Google Scholar

[5]

I. Chueshov and I. Lasiecka, Attroctors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512.  doi: 10.1007/s10884-004-4289-x.  Google Scholar

[6]

I. Chueshov and I. Lasiecka, Global attractors for von Karman evolutions with a nonlinear boundary dissipation, J. Differential Equations, 198 (2004), 196-231.  doi: 10.1016/j.jde.2003.08.008.  Google Scholar

[7]

M. Conti, V. Pata and R. Temam, Attrators for processes on time-dependent space. Application to Wave equation, J. Differential Equations, 255 (2013), 1254-1277.  doi: 10.1016/j.jde.2013.05.013.  Google Scholar

[8]

D. R. da Costa, C. P. Dettmann and E. D. Leonel, Escape of particles in a time-dependent potential well, Phys. Rev. E, 83 (2011), 066211. doi: 10.1103/PhysRevE.83.066211.  Google Scholar

[9]

L. C. Evans, Partial Differential Equations, 2nd ed., vol. 19, American Mathmatical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[10]

L. H. Fatori, M. A. Jorge Silva, T. F. Ma and Z. Yang, Long-time behavior of a class of thermoelastic plates with nonlinear strain, J. Differential Equations, 259 (2015), 4831-4862.  doi: 10.1016/j.jde.2015.06.026.  Google Scholar

[11]

Z. Feng, Duffing-van der Pol-type oscillator systems, Discrete Contin. Dyn. Syst. S, 7 (2014), 1231-1257.  doi: 10.3934/dcdss.2014.7.1231.  Google Scholar

[12]

A. Kh. Khanmamedov, Existence of a global attractor for the plate equation with a critical exponent in unbounded domain, Appl. Math. Lett., 18 (2005), 827-832.  doi: 10.1016/j.aml.2004.08.013.  Google Scholar

[13]

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

[14]

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.  Google Scholar

[15]

P. E. Kloeden, P. Marín-Rubio and J. Real, Pullback attractors for a semilinear heat equation in non-cylindrical domain, J. Differential Equations, 244 (2008), 2062-2090.  doi: 10.1016/j.jde.2007.10.031.  Google Scholar

[16]

P. E. Kloeden, J. Real and C. Sun, Pullback attractors for a semilinear heat equation on time-varying domains, J. Differential Equations, 246 (2009), 4702-4730.  doi: 10.1016/j.jde.2008.11.017.  Google Scholar

[17]

I. Lasiecka, T. F. Ma and R. N. Monteiro, Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1037-1072.  doi: 10.3934/dcdsb.2018141.  Google Scholar

[18]

J. Limaco, L. A. Mederios and E. Zuazua, Existence, uniqueness and contrallability for parabolic equations in non-cylindrical domain, Mat. Contemp., 23 (2002), 49-70.   Google Scholar

[19]

J.-L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaries, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[20]

T. F. Ma, P. Marín-Rubio and C. M. Surco Chuño, Dynamics of wave equation with moving boundary, J. Differential Equations, 262 (2017), 3317-3342.  doi: 10.1016/j.jde.2016.11.030.  Google Scholar

[21]

T. F. Ma and T. M. Souza, Pullback dynamics of non-autonomous wave equations with acoustic boundary condition, Differential Integral Equations, 30 (2017), 443-462.   Google Scholar

[22]

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

[23]

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

[24]

H. M. Soner and S. E. Shreve, A free boundary problem related to singular stochastic control: Parabolic case, Comm. Partial Differential Equations, 16 (1991), 373-424.  doi: 10.1080/03605309108820763.  Google Scholar

[25]

J. Stefan, $\ddot{U}$ber die Theorie der Eisbildung, insbesondere $\ddot{u}$ber die Eisbildung im Polarmeere, Ann. Phys., 278 (1891), 269-286.  doi: 10.1002/andp.18912780206.  Google Scholar

[26]

C. Sun and Y. Yuan, $L^p$-type pullback attractors for a semilinear heat equation on time- varying domains, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1029-1052.  doi: 10.1017/S0308210515000177.  Google Scholar

[27]

Z. Wang and S. Zhou, Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise, Discrete Contin. Dyn. Syst., 37 (2017), 2787-2812.  doi: 10.3934/dcds.2017120.  Google Scholar

[28]

Z. Wang and S. Zhou, Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise, Discrete Contin. Dyn. Syst., 38 (2018), 4767-4817.  doi: 10.3934/dcds.2018210.  Google Scholar

[29]

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

[30]

Z. Yang and Z. Liu, Longtime dynamics of the quasi-linear wave equations with structural damping and supercritical nonlinearities, Nonlinearity, 30 (2017), 1120-1145.  doi: 10.1088/1361-6544/aa599f.  Google Scholar

[31]

Z. Yang and Z. Liu, Stability of exponential attractors for a family of semilinear wave equations with gentle dissipation, J. Differential Equations, 264 (2018), 3976-4005.  doi: 10.1016/j.jde.2017.11.035.  Google Scholar

[32]

F. Zhou, C. Sun and X. Li, Dynamics for the damped wave equations on time-dependent domains, Discrete Contin. Dyn. Syst. B, 23 (2018), 1645-1674.  doi: 10.3934/dcdsb.2018068.  Google Scholar

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