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 & 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, (1992).   Google Scholar

[2]

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

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

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

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V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics,", Amer. Math. Soc. Colloq. Publ., (2002).   Google Scholar

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

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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.  doi: doi:10.1134/S1061920808030014.  Google Scholar

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A. Eden, C. Foias, B. Nicolaenko and R. Temam, "Exponential Attractors for Dissipative Evolution Equations,", Research in Applied Mathematics, (1994).   Google Scholar

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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.  doi: doi:10.1017/S030821050000408X.  Google Scholar

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

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

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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.  doi: doi:10.1090/S0002-9939-05-08340-1.  Google Scholar

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

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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.  doi: doi:10.1216/RMJ-2008-38-4-1117.  Google Scholar

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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.  doi: doi:10.1007/s000300050049.  Google Scholar

[17]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory,, in, (2002), 155.   Google Scholar

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

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J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", AMS, (1988).   Google Scholar

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

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V. A. Marchenko and E. Y. Khruslov, "Homogenization of Partial Differential Equations,", Birkh\, (2006).   Google Scholar

[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.   Google Scholar

[23]

A. Miranville and S. V. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in, (2008).   Google Scholar

[24]

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

[25]

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

[26]

G. R. Sell, "Topological Dynamics and Differential Equations,", Van Nostrand-Reinbold, (1971).   Google Scholar

[27]

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

[28]

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

[29]

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

[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.  doi: doi:10.1090/S0002-9947-08-04680-1.  Google Scholar

[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.  doi: doi:10.3934/cpaa.2004.3.921.  Google Scholar

show all references

References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", NorthHolland, (1992).   Google Scholar

[2]

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

[3]

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

[4]

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

[5]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics,", Amer. Math. Soc. Colloq. Publ., (2002).   Google Scholar

[6]

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

[7]

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

[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.  doi: doi:10.1134/S1061920808030014.  Google Scholar

[9]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, "Exponential Attractors for Dissipative Evolution Equations,", Research in Applied Mathematics, (1994).   Google Scholar

[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.  doi: doi:10.1017/S030821050000408X.  Google Scholar

[11]

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

[12]

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

[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.  doi: doi:10.1090/S0002-9939-05-08340-1.  Google Scholar

[14]

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

[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.  doi: doi:10.1216/RMJ-2008-38-4-1117.  Google Scholar

[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.  doi: doi:10.1007/s000300050049.  Google Scholar

[17]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory,, in, (2002), 155.   Google Scholar

[18]

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

[19]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", AMS, (1988).   Google Scholar

[20]

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

[21]

V. A. Marchenko and E. Y. Khruslov, "Homogenization of Partial Differential Equations,", Birkh\, (2006).   Google Scholar

[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.   Google Scholar

[23]

A. Miranville and S. V. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in, (2008).   Google Scholar

[24]

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

[25]

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

[26]

G. R. Sell, "Topological Dynamics and Differential Equations,", Van Nostrand-Reinbold, (1971).   Google Scholar

[27]

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

[28]

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

[29]

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

[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.  doi: doi:10.1090/S0002-9947-08-04680-1.  Google Scholar

[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.  doi: doi:10.3934/cpaa.2004.3.921.  Google Scholar

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