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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 and Continuous Dynamical Systems - S, 2021, 14 (9) : 3319-3336. doi: 10.3934/dcdss.2021076
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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

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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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[19]

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

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[19]

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

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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