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March  2020, 40(3): 1937-1961. doi: 10.3934/dcds.2020100

Uniform attractors for non-autonomous plate equations with $ p $-Laplacian perturbation and critical nonlinearities

Xin-Guang Yang 1, , Marcelo J. D. Nascimento 2,, and  Maurício L. Pelicer 3,

1. 

College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China
2. 

Departamento de Matemática, Universidade Federal de São Carlos, 13565-905 São Carlos SP, Brazil
3. 

Departamento de Ciências, Campus Regional de Goioerê, Universidade Estadual de Maringá, 87360-000, Goioerê, PR, Brazil

* Corresponding author: Marcelo J. D. Nascimento

Received  July 2019 Revised  October 2019 Published  December 2019

Fund Project: The first author is partially supported by the Key Project of Science and Technology of Henan Province (Grant No. 182102410069) and the Fund of Young Backbone Teacher in Henan Province (No. 2018GGJS039)
The second author is partially supported by FAPESP grant # 2017/06582-2, Brazil.

This paper is concerned with the long-time behavior for a class of non-autonomous plate equations with perturbation and strong damping of
$ p $
-Laplacian type
$ u_{tt} + \Delta^2 u + a_{\epsilon}(t) u_t - \Delta_p u - \Delta u_t + f(u) = g(x,t), $
in bounded domain
$ \Omega\subset \mathbb{R}^N $
 with smooth boundary and critical nonlinear terms. The global existence of weak solution which generates a continuous process has been presented firstly, then the existence of strong and weak uniform attractors with non-compact external forces also derived. Moreover, the upper-semicontinuity of uniform attractors under small perturbations has also obtained by delicate estimate and contradiction argument.
Keywords: Non-autonomous problem, plate equation, $ p $-Laplacian, uniform attractors.
Mathematics Subject Classification: Primary: 35L75, 37B55; Secondary: 35B41, 74K20.
Citation: 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 and Continuous Dynamical Systems, 2020, 40 (3) : 1937-1961. doi: 10.3934/dcds.2020100
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second edition. Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003.

[2]

D. Andrade, M. A. Jorge Silva and T. F. Ma, Exponential stability for a plate equation with $p$-Laplacian and memory terms, Math. Meth. Appl. Sci., 35 (2012), 417-426.  doi: 10.1002/mma.1552.

[3]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.

[4]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium publications, 49. American Mathematical Society, Providence, 2002.

[5]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-Posedness and Long-Time Dynamics, Springer Monographs in Mathematics. Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.

[6]

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

[7]

I. Chueshov and I. Lasiecka, Existence, uniqueness of weakly solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models, Discrete Contin. Dyn. Syst., 15 (2006), 777-809.  doi: 10.3934/dcds.2006.15.777.

[8]

B. W. Feng, X.-G. Yang and Y. M. Qin, Uniform attractors for a nonautonomous extensible plate equation with a strong damping, Math. Meth. Appl. Sci., 40 (2017), 3479-3492.  doi: 10.1002/mma.4239.

[9]

A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, Recherches en Mathématiques Appliquées, 17. Masson, Paris, 1991.

[10]

M. A. Jorge 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.

[11]

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

[12]

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.

[13]

J. U. Kim, A boundary thin obstacle problem for a wave equation, Comm. Partial Differential Equations, 14 (1989), 1011-1026.  doi: 10.1080/03605308908820640.

[14]

S. S. Lu, Attractors for non-autonomous 2D Navier-Stokes equations with less regular normal forces, J. Differential Equations, 230 (2006), 196-212.  doi: 10.1016/j.jde.2006.07.009.

[15]

S. S. Lu, H. Q. Wu and C. K. Zhong, Attractors for non-autonomous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst., 13 (2005), 701-719.  doi: 10.3934/dcds.2005.13.701.

[16]

S. S. Lu, Attractors for non-autonomous reactive-diffusion systems with symbols without strong translation compactness, Asymptot. Anal., 54 (2007), 197-210. 

[17]

S. Ma, C. K. Zhong and H. T. Song, Attractors for non-autonomous 2D Navier-Stokes equations with less regular symbols, Nonlinear Anal., 71 (2009), 4215-4222.  doi: 10.1016/j.na.2009.02.107.

[18]

S. Ma, X. Y. Cheng and H. T. Li, Attractors for non-autonomous wave equations with a new class of external forces, J. Math. Anal. Appl., 337 (2008), 808-820.  doi: 10.1016/j.jmaa.2007.03.108.

[19]

S. Ma and C. K. Zhong, The attractors for weakly damped non-autonomous hyperbolic equations with a new class of external forces, Discrete Contin. Dyn. Syst., 18 (2007), 53-70.  doi: 10.3934/dcds.2007.18.53.

[20]

T. F. Ma and M. L. Pelicer, Attractors for weakly damped beam equations with $p$-Laplacian, Discrete Contin. Dyn. Syst. Suppl., (2013), 525–534. doi: 10.3934/proc.2013.2013.525.

[21]

C. Y. Sun, D. M. Cao and J. Q. Duan, Uniform attractors for non-autonomous wave equations with nonlinear damping, SIAM J. Appl. Dyn. Syst., 6 (2007), 293-318.  doi: 10.1137/060663805.

[22]

C. Y. Sun, D. M. Cao and J. Q. Duan, Non-autonomous wave dynamics with memory-asymptotic regularity and uniform attractor, Discrete Contin. Dyn. Syst. B, 9 (2008), 743-761.  doi: 10.3934/dcdsb.2008.9.743.

[23]

X. G. Yang, Z. H. Fan and K. Li, Uniform attractor for non-autonomous Boussinesq-type equation with critical nonlinearity, Math. Meth. Appl. Sci., 39 (2016), 3075-3087.  doi: 10.1002/mma.3753.

[24]

Z. J. Yang, Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term, J. Differential Equations, 187 (2003), 520-540.  doi: 10.1016/S0022-0396(02)00042-6.

[25]

Z. J. Yang, Longtime behavior for a nonlinear wave equation arising in elasto-plastic flow, Math. Meth. Appl. Sci., 32 (2009), 1082-1104.  doi: 10.1002/mma.1080.

[26]

Z. J. Yang, Finite-dimensional attractors for the Kirchhoff models with critical exponents, J. Math. Phys., 53 (2012), 032702, 15 pp. doi: 10.1063/1.3694730.

[27]

Z. J. Yang and B. Jin, Global attractor for a class of Kirchhoff models, J. Math. Phys., 50 (2009), 032701, 29 pp.

[28]

S. Zelik, Strong uniform attractors for non-autonomous dissipative PDEs with non translation-compact external force, Discrete Contin. Dyn. Syst., 20 (2015), 781-810.  doi: 10.3934/dcdsb.2015.20.781.

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second edition. Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003.

[2]

D. Andrade, M. A. Jorge Silva and T. F. Ma, Exponential stability for a plate equation with $p$-Laplacian and memory terms, Math. Meth. Appl. Sci., 35 (2012), 417-426.  doi: 10.1002/mma.1552.

[3]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.

[4]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium publications, 49. American Mathematical Society, Providence, 2002.

[5]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-Posedness and Long-Time Dynamics, Springer Monographs in Mathematics. Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.

[6]

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

[7]

I. Chueshov and I. Lasiecka, Existence, uniqueness of weakly solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models, Discrete Contin. Dyn. Syst., 15 (2006), 777-809.  doi: 10.3934/dcds.2006.15.777.

[8]

B. W. Feng, X.-G. Yang and Y. M. Qin, Uniform attractors for a nonautonomous extensible plate equation with a strong damping, Math. Meth. Appl. Sci., 40 (2017), 3479-3492.  doi: 10.1002/mma.4239.

[9]

A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, Recherches en Mathématiques Appliquées, 17. Masson, Paris, 1991.

[10]

M. A. Jorge 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.

[11]

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

[12]

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.

[13]

J. U. Kim, A boundary thin obstacle problem for a wave equation, Comm. Partial Differential Equations, 14 (1989), 1011-1026.  doi: 10.1080/03605308908820640.

[14]

S. S. Lu, Attractors for non-autonomous 2D Navier-Stokes equations with less regular normal forces, J. Differential Equations, 230 (2006), 196-212.  doi: 10.1016/j.jde.2006.07.009.

[15]

S. S. Lu, H. Q. Wu and C. K. Zhong, Attractors for non-autonomous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst., 13 (2005), 701-719.  doi: 10.3934/dcds.2005.13.701.

[16]

S. S. Lu, Attractors for non-autonomous reactive-diffusion systems with symbols without strong translation compactness, Asymptot. Anal., 54 (2007), 197-210. 

[17]

S. Ma, C. K. Zhong and H. T. Song, Attractors for non-autonomous 2D Navier-Stokes equations with less regular symbols, Nonlinear Anal., 71 (2009), 4215-4222.  doi: 10.1016/j.na.2009.02.107.

[18]

S. Ma, X. Y. Cheng and H. T. Li, Attractors for non-autonomous wave equations with a new class of external forces, J. Math. Anal. Appl., 337 (2008), 808-820.  doi: 10.1016/j.jmaa.2007.03.108.

[19]

S. Ma and C. K. Zhong, The attractors for weakly damped non-autonomous hyperbolic equations with a new class of external forces, Discrete Contin. Dyn. Syst., 18 (2007), 53-70.  doi: 10.3934/dcds.2007.18.53.

[20]

T. F. Ma and M. L. Pelicer, Attractors for weakly damped beam equations with $p$-Laplacian, Discrete Contin. Dyn. Syst. Suppl., (2013), 525–534. doi: 10.3934/proc.2013.2013.525.

[21]

C. Y. Sun, D. M. Cao and J. Q. Duan, Uniform attractors for non-autonomous wave equations with nonlinear damping, SIAM J. Appl. Dyn. Syst., 6 (2007), 293-318.  doi: 10.1137/060663805.

[22]

C. Y. Sun, D. M. Cao and J. Q. Duan, Non-autonomous wave dynamics with memory-asymptotic regularity and uniform attractor, Discrete Contin. Dyn. Syst. B, 9 (2008), 743-761.  doi: 10.3934/dcdsb.2008.9.743.

[23]

X. G. Yang, Z. H. Fan and K. Li, Uniform attractor for non-autonomous Boussinesq-type equation with critical nonlinearity, Math. Meth. Appl. Sci., 39 (2016), 3075-3087.  doi: 10.1002/mma.3753.

[24]

Z. J. Yang, Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term, J. Differential Equations, 187 (2003), 520-540.  doi: 10.1016/S0022-0396(02)00042-6.

[25]

Z. J. Yang, Longtime behavior for a nonlinear wave equation arising in elasto-plastic flow, Math. Meth. Appl. Sci., 32 (2009), 1082-1104.  doi: 10.1002/mma.1080.

[26]

Z. J. Yang, Finite-dimensional attractors for the Kirchhoff models with critical exponents, J. Math. Phys., 53 (2012), 032702, 15 pp. doi: 10.1063/1.3694730.

[27]

Z. J. Yang and B. Jin, Global attractor for a class of Kirchhoff models, J. Math. Phys., 50 (2009), 032701, 29 pp.

[28]

S. Zelik, Strong uniform attractors for non-autonomous dissipative PDEs with non translation-compact external force, Discrete Contin. Dyn. Syst., 20 (2015), 781-810.  doi: 10.3934/dcdsb.2015.20.781.

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