# American Institute of Mathematical Sciences

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.

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##### 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.
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