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May  2018, 38(5): 2629-2653. doi: 10.3934/dcds.2018111

Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous kirchhoff wave models

Zhijian Yang , and  Yanan Li

School of Mathematics and Statistics, Zhengzhou University, No.100, Science Road, Zhengzhou 450001, China

* Corresponding author: Zhijian Yang

Received  May 2017 Revised  November 2017 Published  March 2018

Fund Project: This work is supported by National Natural Science Foundation of China (No.11671367).

In this paper, we are concerned with the existence and stability of pullback exponential attractors for a non-autonomous dynamical system. (ⅰ) We propose two new criteria for the discrete dynamical system and continuous one, respectively. (ⅱ) By applying the criteria to the non-autonomous Kirchhoff wave models with structural damping and supercritical nonlinearity we construct a family of pullback exponential attractors which are stable with respect to perturbations.

Keywords: Pullback exponential attractor, stability with respect to perturbations, non-autonomous Kirchhoff wave equations, structural damping, supercritical nonlinearity.
Mathematics Subject Classification: Primary: 37L15, 37L30; Secondary: 35B40, 35B41.
Citation: Zhijian Yang, Yanan Li. Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous kirchhoff wave models. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2629-2653. doi: 10.3934/dcds.2018111
References:
[1]

S. Bosia and S. Gatti, Pullback exponential attractor for a Cahn-Hilliard-Navier-Stokes system in 2D, Dynamics of PDE, 11 (2014), 1-38.  doi: 10.4310/DPDE.2014.v11.n1.a1.

[2]

A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Suárez, Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system, J.Differential Equations, 236 (2007), 570-603.  doi: 10.1016/j.jde.2007.01.017.

[3]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: theoretical results, Commun. Pure Appl. Anal., 12 (2013), 3047-3071.  doi: 10.3934/cpaa.2013.12.3047.

[4]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: properties and applications, Commun. Pure Appl. Anal., 13 (2014), 1114-1165. 

[5]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional NonAutonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer, New York, 2013.

[6]

I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, AKTA, Kharkiv, 1999.

[7]

I. Chueshov, Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping, J. Abstr. Differ. Equ. Appl., 1 (2010), 86-106. 

[8]

I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262.  doi: 10.1016/j.jde.2011.08.022.

[9]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015.

[10]

R. Czaja and M. Efendiev, Pullback exponential attractors for nonautonomous equations Part I: Semilinear parabolic problems, J. Math. Anal. Appl., 381 (2011), 748-765.  doi: 10.1016/j.jmaa.2011.03.053.

[11]

R. Czaja, Pullback exponential attractors with admissible exponential growth in the past, Nonlinear Analysis, 104 (2014), 90-108.  doi: 10.1016/j.na.2014.03.020.

[12]

M. Efendiev, A. Miranville and S. Zelik, Exponential Attractors for a Nonlinear Reaction-Diffusion System in $ \mathbb{R}^3$, C. R. Acad. Sci. Paris Sr. I Math., 330 (2000), 713-718.  doi: 10.1016/S0764-4442(00)00259-7.

[13]

M. Efendiev, S. Zelik and A. Miranville, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems, Proceedings of the Royal Society of Edinburgh, 135 (2005), 703-730.  doi: 10.1017/S030821050000408X.

[14]

M. Efendiev, Y. Yamamoto and A. Yagi, Exponential attractors for non-autonomous dissipative systems, Journal of the Mathematical Society of Japan, 63 (2011), 647-673.  doi: 10.2969/jmsj/06320647.

[15]

G. Kirchhoff, Vorlesungen über Mechanik, (German) [Lectures on Mechanics], Teubner, Stuttgart, 1883.

[16]

P. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112.  doi: 10.1142/S0219493703000632.

[17]

K. Kuratowski, Sur les espaces complets, Fund. Math., 15 (1930), 301-309. 

[18]

J. A. Langa, A. Miranville and J. Real, Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357. 

[19]

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

[20]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl., 146 (1986), 65-96. 

[21]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988.

[22]

Y. H. Wang and C. K. Zhong, Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models, Discrete Cont. Dyn. Sys., 33 (2013), 3189-3209.  doi: 10.3934/dcds.2013.33.3189.

[23]

Z. J. Yang and Y. Q. Wang, Global attractor for the Kirchhoff type equation with a strong dissipation, J. Differential Equations, 249 (2010), 3258-3278.  doi: 10.1016/j.jde.2010.09.024.

[24]

Z. J. Yang, P. Y. Ding and L. Li, Longtime dynamics of the Kirchhoff equation with fractional damping and supercritical nonlinearity, J. Math. Anal. Appl., 442 (2016), 485-510.  doi: 10.1016/j.jmaa.2016.04.079.

[25]

S. F. Zhou and X. Y. Han, Pullback exponential attractors for non-autonomous lattice systems, J Dyn. Diff. Equat., 24 (2012), 601-631.  doi: 10.1007/s10884-012-9260-7.

show all references

References:
[1]

S. Bosia and S. Gatti, Pullback exponential attractor for a Cahn-Hilliard-Navier-Stokes system in 2D, Dynamics of PDE, 11 (2014), 1-38.  doi: 10.4310/DPDE.2014.v11.n1.a1.

[2]

A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Suárez, Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system, J.Differential Equations, 236 (2007), 570-603.  doi: 10.1016/j.jde.2007.01.017.

[3]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: theoretical results, Commun. Pure Appl. Anal., 12 (2013), 3047-3071.  doi: 10.3934/cpaa.2013.12.3047.

[4]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: properties and applications, Commun. Pure Appl. Anal., 13 (2014), 1114-1165. 

[5]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional NonAutonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer, New York, 2013.

[6]

I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, AKTA, Kharkiv, 1999.

[7]

I. Chueshov, Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping, J. Abstr. Differ. Equ. Appl., 1 (2010), 86-106. 

[8]

I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262.  doi: 10.1016/j.jde.2011.08.022.

[9]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015.

[10]

R. Czaja and M. Efendiev, Pullback exponential attractors for nonautonomous equations Part I: Semilinear parabolic problems, J. Math. Anal. Appl., 381 (2011), 748-765.  doi: 10.1016/j.jmaa.2011.03.053.

[11]

R. Czaja, Pullback exponential attractors with admissible exponential growth in the past, Nonlinear Analysis, 104 (2014), 90-108.  doi: 10.1016/j.na.2014.03.020.

[12]

M. Efendiev, A. Miranville and S. Zelik, Exponential Attractors for a Nonlinear Reaction-Diffusion System in $ \mathbb{R}^3$, C. R. Acad. Sci. Paris Sr. I Math., 330 (2000), 713-718.  doi: 10.1016/S0764-4442(00)00259-7.

[13]

M. Efendiev, S. Zelik and A. Miranville, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems, Proceedings of the Royal Society of Edinburgh, 135 (2005), 703-730.  doi: 10.1017/S030821050000408X.

[14]

M. Efendiev, Y. Yamamoto and A. Yagi, Exponential attractors for non-autonomous dissipative systems, Journal of the Mathematical Society of Japan, 63 (2011), 647-673.  doi: 10.2969/jmsj/06320647.

[15]

G. Kirchhoff, Vorlesungen über Mechanik, (German) [Lectures on Mechanics], Teubner, Stuttgart, 1883.

[16]

P. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112.  doi: 10.1142/S0219493703000632.

[17]

K. Kuratowski, Sur les espaces complets, Fund. Math., 15 (1930), 301-309. 

[18]

J. A. Langa, A. Miranville and J. Real, Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357. 

[19]

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

[20]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl., 146 (1986), 65-96. 

[21]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988.

[22]

Y. H. Wang and C. K. Zhong, Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models, Discrete Cont. Dyn. Sys., 33 (2013), 3189-3209.  doi: 10.3934/dcds.2013.33.3189.

[23]

Z. J. Yang and Y. Q. Wang, Global attractor for the Kirchhoff type equation with a strong dissipation, J. Differential Equations, 249 (2010), 3258-3278.  doi: 10.1016/j.jde.2010.09.024.

[24]

Z. J. Yang, P. Y. Ding and L. Li, Longtime dynamics of the Kirchhoff equation with fractional damping and supercritical nonlinearity, J. Math. Anal. Appl., 442 (2016), 485-510.  doi: 10.1016/j.jmaa.2016.04.079.

[25]

S. F. Zhou and X. Y. Han, Pullback exponential attractors for non-autonomous lattice systems, J Dyn. Diff. Equat., 24 (2012), 601-631.  doi: 10.1007/s10884-012-9260-7.

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