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

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

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.

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]

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. ContiV. 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. FatoriM. A. Jorge SilvaT. 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. KloedenP. 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. KloedenJ. 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. LasieckaT. 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. LimacoL. 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. MaP. 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. MengM. 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 PlinioG. 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. ZhouC. 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. ContiV. 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. FatoriM. A. Jorge SilvaT. 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. KloedenP. 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. KloedenJ. 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. LasieckaT. 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. LimacoL. 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. MaP. 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. MengM. 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 PlinioG. 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. ZhouC. 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

[1]

Zhaojuan Wang, Shengfan Zhou. Random attractor for stochastic non-autonomous damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 545-573. doi: 10.3934/dcds.2017022

[2]

Flank D. M. Bezerra, Vera L. Carbone, Marcelo J. D. Nascimento, Karina Schiabel. Pullback attractors for a class of non-autonomous thermoelastic plate systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3553-3571. doi: 10.3934/dcdsb.2017214

[3]

Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195

[4]

T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037

[5]

Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194

[6]

Xue-Li Song, Yan-Ren Hou. Pullback $\mathcal{D}$-attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 991-1009. doi: 10.3934/dcds.2012.32.991

[7]

Shuang Yang, Yangrong Li. Forward controllability of a random attractor for the non-autonomous stochastic sine-Gordon equation on an unbounded domain. Evolution Equations & Control Theory, 2020, 9 (3) : 581-604. doi: 10.3934/eect.2020025

[8]

Shengfan Zhou, Linshan Wang. Kernel sections for damped non-autonomous wave equations with critical exponent. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 399-412. doi: 10.3934/dcds.2003.9.399

[9]

Olivier Goubet, Wided Kechiche. Uniform attractor for non-autonomous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 639-651. doi: 10.3934/cpaa.2011.10.639

[10]

Xin-Guang Yang, Marcelo J. D. Nascimento, Maurício L. Pelicer. Uniform attractors for non-autonomous plate equations with $ p $-Laplacian perturbation and critical nonlinearities. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1937-1961. doi: 10.3934/dcds.2020100

[11]

Feng Zhou, Chunyou Sun. Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3767-3792. doi: 10.3934/dcdsb.2016120

[12]

Mirelson M. Freitas, Anderson J. A. Ramos, Manoel J. Dos Santos, Eraldo R. N. Fonseca. Attractors and pullback dynamics for non-autonomous piezoelectric system with magnetic and thermal effects. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3745-3765. doi: 10.3934/cpaa.2021129

[13]

Suping Wang, Qiaozhen Ma. Existence of pullback attractors for the non-autonomous suspension bridge equation with time delay. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1299-1316. doi: 10.3934/dcdsb.2019221

[14]

Bo You, Yanren Hou, Fang Li, Jinping Jiang. Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1801-1814. doi: 10.3934/dcdsb.2014.19.1801

[15]

Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4899-4912. doi: 10.3934/dcdsb.2019036

[16]

Peter E. Kloeden, Jacson Simsen. Pullback attractors for non-autonomous evolution equations with spatially variable exponents. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2543-2557. doi: 10.3934/cpaa.2014.13.2543

[17]

Xueli Song, Jianhua Wu. Non-autonomous 2D Newton-Boussinesq equation with oscillating external forces and its uniform attractor. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020102

[18]

Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887

[19]

Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021004

[20]

Zhaojuan Wang, Shengfan Zhou. Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (127)
  • HTML views (115)
  • Cited by (0)

Other articles
by authors

[Back to Top]