May  2011, 10(3): 885-915. doi: 10.3934/cpaa.2011.10.885

Robust exponential attractors for non-autonomous equations with memory

1. 

Institut für Mathematik, Johann Wolfgang Goethe Universität, D-60054 Frankfurt am Main, Germany

2. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla

3. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

Received  October 2008 Revised  February 2009 Published  December 2010

The aim of this paper is to consider the robustness of exponential attractors for non-autonomous dynamical systems with line memory which is expressed through convolution integrals. Some properties useful for dealing with the memory term for non-autonomous case are presented. Then, we illustrate the abstract results by applying them to the non-autonomous strongly damped wave equations with linear memory and critical nonlinearity.
Citation: Peter E. Kloeden, José Real, Chunyou Sun. Robust exponential attractors for non-autonomous equations with memory. Communications on Pure and Applied Analysis, 2011, 10 (3) : 885-915. doi: 10.3934/cpaa.2011.10.885
References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," NorthHolland, Amsterdam, 1992.

[2]

S. Borini and V. Pata, Uniform attractors for a strongly damped wave equation with linear memory, Asymptot. Anal., 20 (1999), 263-277.

[3]

D. N. Cheban, "Global Attractors of Non-autonomous Dissipative Dynamical Systems," World Scientific Publishing, 2004. doi: doi:10.1142/9789812563088.

[4]

V. V. Chepyzhov and A. Miranville, Trajectory and global attractors of dissipative hyperbolic equations with memory, Commun. Pure Appl. Anal., 4 (2005), 115-142.

[5]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," Amer. Math. Soc. Colloq. Publ., Vol. 49, Amer. Math. Soc., Providence, RI, 2002.

[6]

M. Conti, V. Pata and M. Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J., 55 (2006), 169-215. doi: doi:10.1512/iumj.2006.55.2661.

[7]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. doi: doi:10.1007/BF00251609.

[8]

F. Di Plinio and V. Pata, Robust exponential attractors for the strongly damped wave equation with memory. I, Russian Journal of Mathematical Physics, 15 (2008), 301-315. doi: doi:10.1134/S1061920808030014.

[9]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, "Exponential Attractors for Dissipative Evolution Equations," Research in Applied Mathematics, 37, John-Wiley, New York, 1994.

[10]

M. Efendiev, A. Miranville and S. V. Zelik, Exponential attractors and finite-dimensional reduction of non-autonomous dynamical systems, Proc. Royal Soc. Edin., 135A (2005), 703-730. doi: doi:10.1017/S030821050000408X.

[11]

G. A. Francfort and P. M. Suquet, Homogenization and mechanical dissipation in thermo-viscoelasticity, Arch. Rational Mech. Anal., 96 (1986), 879-895. doi: doi:10.1007/BF00251909.

[12]

H. Gao and J. E. Muñoz Rivera, On the exponential stability of thermoelastic problem with memory, Appl. Anal., 78 (2001), 379-403. doi: doi:10.1080/00036810108840942.

[13]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, A construction of a robust family of exponential attractors, Proc. Amer. Math. Soc., 134 (2006), 117-127. doi: doi:10.1090/S0002-9939-05-08340-1.

[14]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, Memory relaxation of first order evolution equations, Nonlinearity, 18 (2005), 1859-1883. doi: doi:10.1088/0951-7715/18/4/023.

[15]

C. Gatti, A. Miranville, V. Pata and S. V. Zelik, Attractors for semilinear equations of viscoelasticity with very low dissipation, R. Mountain J. Math., 38 (2008), 1117-1138. doi: doi:10.1216/RMJ-2008-38-4-1117.

[16]

C. Giorgi, A. Marzocchi and V. Pata, Asymptotic behavior of a semilinear problem in heat conduction with memory, Nonl. Diff. Equa. Appl., 5 (1998), 333-354. doi: doi:10.1007/s000300050049.

[17]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, in "Evolution Equations, Semigroups and Functional Analysis" (A. Lorenzi and B. Ruf, Eds.), 155-178, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Boston, 2002.

[18]

M. Grasselli and V. Pata, Robust exponential attractors for a phase-field system with memory, J. Evol. Equ., 5 (2005), 465-483. doi: doi:10.1007/s00028-005-0199-6.

[19]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," AMS, Providence, RI, 1988.

[20]

F. J. Hickernell, Time-dependent critical layers in shear flows on the beta-plane, J. Fluid Mech., 142 (1984), 431-449. doi: doi:10.1017/S0022112084001178.

[21]

V. A. Marchenko and E. Y. Khruslov, "Homogenization of Partial Differential Equations," Birkhäuser, Boston, 2006.

[22]

A. Miranville, V. Pata and S. V. Zelik, Exponential attractors for singularly perturbed damped wave equations: a simple construction, Asymptot. Anal., 53 (2007), 1-12.

[23]

A. Miranville and S. V. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations, Evolutionary Equations," Volume 4, C. M. Dafermos and M. Pokorny, eds., Elsevier, Amsterdam 2008, p.103.

[24]

V. Pata and M. Squassina, On the strongly damped wave equation, Comm. Math. Phys., 253 (2005), 511-533. doi: doi:10.1007/s00220-004-1233-1.

[25]

V. Pata, and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.

[26]

G. R. Sell, "Topological Dynamics and Differential Equations," Van Nostrand-Reinbold, London, 1971.

[27]

C. Sun, D. Cao and J. Duan, Non-autonomous wave dynamics with memory - Asymptotic regularity and uniform attractor, Discrete Contin. Dyn. Syst., Series B, 9 (2008), 743-761.

[28]

C. Sun and M. Yang, Dynamics of the nonclassical diffusion equations, Asymptot. Anal., 59 (2008), 51-81.

[29]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer-Verlag New York, 1997.

[30]

M. Yang and C. Sun, Dynamics of strongly damped wave equations in locally uniform spaces: attractors and asymptotic regularity, Trans. Amer. Math. Soc., 361 (2009), 1069-1101. doi: doi:10.1090/S0002-9947-08-04680-1.

[31]

S. V. Zelik, Asymptotic regularity of solutions of a non-autonomous damped wave equation with a critical growth exponent, Comm. Pure Appl. Anal., 3 (2004), 921-934. doi: doi:10.3934/cpaa.2004.3.921.

show all references

References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," NorthHolland, Amsterdam, 1992.

[2]

S. Borini and V. Pata, Uniform attractors for a strongly damped wave equation with linear memory, Asymptot. Anal., 20 (1999), 263-277.

[3]

D. N. Cheban, "Global Attractors of Non-autonomous Dissipative Dynamical Systems," World Scientific Publishing, 2004. doi: doi:10.1142/9789812563088.

[4]

V. V. Chepyzhov and A. Miranville, Trajectory and global attractors of dissipative hyperbolic equations with memory, Commun. Pure Appl. Anal., 4 (2005), 115-142.

[5]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," Amer. Math. Soc. Colloq. Publ., Vol. 49, Amer. Math. Soc., Providence, RI, 2002.

[6]

M. Conti, V. Pata and M. Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J., 55 (2006), 169-215. doi: doi:10.1512/iumj.2006.55.2661.

[7]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. doi: doi:10.1007/BF00251609.

[8]

F. Di Plinio and V. Pata, Robust exponential attractors for the strongly damped wave equation with memory. I, Russian Journal of Mathematical Physics, 15 (2008), 301-315. doi: doi:10.1134/S1061920808030014.

[9]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, "Exponential Attractors for Dissipative Evolution Equations," Research in Applied Mathematics, 37, John-Wiley, New York, 1994.

[10]

M. Efendiev, A. Miranville and S. V. Zelik, Exponential attractors and finite-dimensional reduction of non-autonomous dynamical systems, Proc. Royal Soc. Edin., 135A (2005), 703-730. doi: doi:10.1017/S030821050000408X.

[11]

G. A. Francfort and P. M. Suquet, Homogenization and mechanical dissipation in thermo-viscoelasticity, Arch. Rational Mech. Anal., 96 (1986), 879-895. doi: doi:10.1007/BF00251909.

[12]

H. Gao and J. E. Muñoz Rivera, On the exponential stability of thermoelastic problem with memory, Appl. Anal., 78 (2001), 379-403. doi: doi:10.1080/00036810108840942.

[13]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, A construction of a robust family of exponential attractors, Proc. Amer. Math. Soc., 134 (2006), 117-127. doi: doi:10.1090/S0002-9939-05-08340-1.

[14]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, Memory relaxation of first order evolution equations, Nonlinearity, 18 (2005), 1859-1883. doi: doi:10.1088/0951-7715/18/4/023.

[15]

C. Gatti, A. Miranville, V. Pata and S. V. Zelik, Attractors for semilinear equations of viscoelasticity with very low dissipation, R. Mountain J. Math., 38 (2008), 1117-1138. doi: doi:10.1216/RMJ-2008-38-4-1117.

[16]

C. Giorgi, A. Marzocchi and V. Pata, Asymptotic behavior of a semilinear problem in heat conduction with memory, Nonl. Diff. Equa. Appl., 5 (1998), 333-354. doi: doi:10.1007/s000300050049.

[17]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, in "Evolution Equations, Semigroups and Functional Analysis" (A. Lorenzi and B. Ruf, Eds.), 155-178, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Boston, 2002.

[18]

M. Grasselli and V. Pata, Robust exponential attractors for a phase-field system with memory, J. Evol. Equ., 5 (2005), 465-483. doi: doi:10.1007/s00028-005-0199-6.

[19]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," AMS, Providence, RI, 1988.

[20]

F. J. Hickernell, Time-dependent critical layers in shear flows on the beta-plane, J. Fluid Mech., 142 (1984), 431-449. doi: doi:10.1017/S0022112084001178.

[21]

V. A. Marchenko and E. Y. Khruslov, "Homogenization of Partial Differential Equations," Birkhäuser, Boston, 2006.

[22]

A. Miranville, V. Pata and S. V. Zelik, Exponential attractors for singularly perturbed damped wave equations: a simple construction, Asymptot. Anal., 53 (2007), 1-12.

[23]

A. Miranville and S. V. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations, Evolutionary Equations," Volume 4, C. M. Dafermos and M. Pokorny, eds., Elsevier, Amsterdam 2008, p.103.

[24]

V. Pata and M. Squassina, On the strongly damped wave equation, Comm. Math. Phys., 253 (2005), 511-533. doi: doi:10.1007/s00220-004-1233-1.

[25]

V. Pata, and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.

[26]

G. R. Sell, "Topological Dynamics and Differential Equations," Van Nostrand-Reinbold, London, 1971.

[27]

C. Sun, D. Cao and J. Duan, Non-autonomous wave dynamics with memory - Asymptotic regularity and uniform attractor, Discrete Contin. Dyn. Syst., Series B, 9 (2008), 743-761.

[28]

C. Sun and M. Yang, Dynamics of the nonclassical diffusion equations, Asymptot. Anal., 59 (2008), 51-81.

[29]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer-Verlag New York, 1997.

[30]

M. Yang and C. Sun, Dynamics of strongly damped wave equations in locally uniform spaces: attractors and asymptotic regularity, Trans. Amer. Math. Soc., 361 (2009), 1069-1101. doi: doi:10.1090/S0002-9947-08-04680-1.

[31]

S. V. Zelik, Asymptotic regularity of solutions of a non-autonomous damped wave equation with a critical growth exponent, Comm. Pure Appl. Anal., 3 (2004), 921-934. doi: doi:10.3934/cpaa.2004.3.921.

[1]

Chunyou Sun, Daomin Cao, Jinqiao Duan. Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 743-761. doi: 10.3934/dcdsb.2008.9.743

[2]

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

[3]

Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure and Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809

[4]

Sylvia Novo, Rafael Obaya, Ana M. Sanz. Exponential stability in non-autonomous delayed equations with applications to neural networks. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 517-536. doi: 10.3934/dcds.2007.18.517

[5]

Bixiang Wang. Multivalued non-autonomous random dynamical systems for wave equations without uniqueness. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 2011-2051. doi: 10.3934/dcdsb.2017119

[6]

Ling Xu, Jianhua Huang, Qiaozhen Ma. Random exponential attractor for stochastic non-autonomous suspension bridge equation with additive white noise. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2021318

[7]

Bixiang Wang. Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 269-300. doi: 10.3934/dcds.2014.34.269

[8]

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

[9]

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

[10]

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

[11]

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

[12]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703

[13]

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

[14]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Forwards dynamics of non-autonomous dynamical systems: Driving semigroups without backwards uniqueness and structure of the attractor. Communications on Pure and Applied Analysis, 2020, 19 (4) : 1997-2013. doi: 10.3934/cpaa.2020088

[15]

Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure and Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281

[16]

José A. Langa, James C. Robinson, Aníbal Rodríguez-Bernal, A. Suárez, A. Vidal-López. Existence and nonexistence of unbounded forwards attractor for a class of non-autonomous reaction diffusion equations. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 483-497. doi: 10.3934/dcds.2007.18.483

[17]

V. V. Chepyzhov, M. I. Vishik, W. L. Wendland. On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 27-38. doi: 10.3934/dcds.2005.12.27

[18]

T. Tachim Medjo. Non-autonomous 3D primitive equations with oscillating external force and its global attractor. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 265-291. doi: 10.3934/dcds.2012.32.265

[19]

Rafael Obaya, Víctor M. Villarragut. Direct exponential ordering for neutral compartmental systems with non-autonomous $\mathbf{D}$-operator. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 185-207. doi: 10.3934/dcdsb.2013.18.185

[20]

Fang Li, Bo You. Pullback exponential attractors for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 55-80. doi: 10.3934/dcdsb.2019172

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (62)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]