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$ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values
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 |
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.
References:
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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. |
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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. |
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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. |
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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.
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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. |
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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