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May  2014, 13(3): 1141-1165. doi: 10.3934/cpaa.2014.13.1141

## Pullback exponential attractors for evolution processes in Banach spaces: Properties and applications

 1 Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Caixa postal 668, 13560-970 São Carlos, São Paulo, Brazil 2 BCAM Basque Center for Applied Mathematics, Mazarredo 14, E-48009 Bilbao, Basque Country

Received  April 2013 Revised  September 2013 Published  December 2013

This article is a continuation of our previous work [5], where we formulated general existence theorems for pullback exponential attractors for asymptotically compact evolution processes in Banach spaces and discussed its implications in the autonomous case. We now study properties of the attractors and use our theoretical results to prove the existence of pullback exponential attractors in two examples, where previous results do not apply.
Citation: Alexandre Nolasco de Carvalho, Stefanie Sonner. Pullback exponential attractors for evolution processes in Banach spaces: Properties and applications. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1141-1165. doi: 10.3934/cpaa.2014.13.1141
##### References:
 [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces,, 2nd edition, (2003). Google Scholar [2] J. Arrieta, A. N. Carvalho and J. K. Hale, A damped hyperbolic equation with critical exponent ,, \emph{Comm. Partial Differential Equations}, 17 (1992), 841. doi: 10.1080/03605309208820866. Google Scholar [3] T. Caraballo, A. N. Carvalho, J. A. Langa and L. F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes ,, \emph{Nonlinear Anal.}, 72 (2010), 1967. doi: 10.1016/j.na.2009.09.037. Google Scholar [4] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems,, \emph{Appl. Math. Sci.}, 182 (2012). doi: 10.1007/978-1-4614-4581-4. Google Scholar [5] A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: theoretical results ,, \emph{Commun. Pure and Appl. Anal.}, 12 (2013), 3047. doi: 10.3934/cpaa.2013.12.3047. Google Scholar [6] V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics,, Amer. Math. Soc., (2002). Google Scholar [7] H. Crauel, A. Debussche and F. Flandoli, Random attractors ,, \emph{J. Dynam. Differential Equations}, 9 (1997), 307. doi: 10.1007/BF02219225. Google Scholar [8] R. Czaja and M. A. Efendiev, Pullback exponential attractors for nonautonomous equations part I: Semilinear parabolic equations ,, \emph{J. Math. Anal. Appl.}, 381 (2011), 748. doi: 10.1016/j.jmaa.2011.03.053. 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] D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers and Differential Operators,, Cambridge University Press, (1996). doi: 10.1017/CBO9780511662201. Google Scholar [11] M. A. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\R ^3$ ,, \emph{C. R. Acad. Sci. Paris Sr. I Math.}, 330 (2000), 713. doi: 10.1016/S0764-4442(00)00259-7. Google Scholar [12] M. A. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems ,, \emph{Proc. R. Soc. Edinburgh Sect. A}, 135A (2005), 703. doi: 10.1017/S030821050000408X. Google Scholar [13] M. A. Efendiev, Y. Yamamoto and A. Yagi, Exponential attractors non-autonomous dissipative systems ,, \emph{J. Math. Soc. Japan}, 63 (2011), 647. doi: 10.2969/jmsj/06320647. Google Scholar [14] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, American Mathematical Society, (1988). Google Scholar [15] A. N. Kolmogorov and V. M. Tihomirov, $\varepsilon$-entropy and $\varepsilon$-capacity of sets in functional spaces ,, \emph{Amer. Math. Soc. Transl. Ser. 2}, 17 (1961), 277. Google Scholar [16] J. A. Langa, A. Miranville and J. Real, Pullback exponential attractors ,, \emph{Discrete Contin. Dyn. Syst.}, 26 (2010), 1329. doi: 10.3934/dcds.2010.26.1329. Google Scholar [17] J. A. Langa, J. C. Robinson and A. Suárez, Stability, instability and bifurcation phenomena in non-autonomous differential equations ,, \emph{Nonlinearity}, 15 (2002), 887. doi: 10.1088/0951-7715/15/3/322. Google Scholar [18] J. A. Langa and B. Schmalfuss, Finite dimensionality of attractors for non-autonomous dynamical systems given by partial differential equations ,, \emph{Stoch. Dyn.}, 4 (2004), 385. doi: 10.1142/S0219493704001127. Google Scholar [19] P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems ,, \emph{Nonlinear Anal.}, 71 (2009), 3956. doi: 10.1016/j.na.2009.02.065. Google Scholar [20] X. Mora, Semilinear parabolic problems define semiflows on $C^k$ spaces ,, \emph{Trans. Amer. Math. Soc.}, 278 (1983), 21. doi: 10.2307/1999300. Google Scholar [21] C. V. Pao, Nonlinear Parabolic and Elliptic Equations,, Plenum Press, (1992). Google Scholar [22] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [23] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, 2nd edition, (1997). Google Scholar [24] A. Yagi, Abstract Parabolic Evolution Equations and Their Applications,, Springer-Verlag, (2010). doi: 10.1007/978-3-642-04631-5. Google Scholar

show all references

##### References:
 [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces,, 2nd edition, (2003). Google Scholar [2] J. Arrieta, A. N. Carvalho and J. K. Hale, A damped hyperbolic equation with critical exponent ,, \emph{Comm. Partial Differential Equations}, 17 (1992), 841. doi: 10.1080/03605309208820866. Google Scholar [3] T. Caraballo, A. N. Carvalho, J. A. Langa and L. F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes ,, \emph{Nonlinear Anal.}, 72 (2010), 1967. doi: 10.1016/j.na.2009.09.037. Google Scholar [4] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems,, \emph{Appl. Math. Sci.}, 182 (2012). doi: 10.1007/978-1-4614-4581-4. Google Scholar [5] A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: theoretical results ,, \emph{Commun. Pure and Appl. Anal.}, 12 (2013), 3047. doi: 10.3934/cpaa.2013.12.3047. Google Scholar [6] V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics,, Amer. Math. Soc., (2002). Google Scholar [7] H. Crauel, A. Debussche and F. Flandoli, Random attractors ,, \emph{J. Dynam. Differential Equations}, 9 (1997), 307. doi: 10.1007/BF02219225. Google Scholar [8] R. Czaja and M. A. Efendiev, Pullback exponential attractors for nonautonomous equations part I: Semilinear parabolic equations ,, \emph{J. Math. Anal. Appl.}, 381 (2011), 748. doi: 10.1016/j.jmaa.2011.03.053. 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] D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers and Differential Operators,, Cambridge University Press, (1996). doi: 10.1017/CBO9780511662201. Google Scholar [11] M. A. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\R ^3$ ,, \emph{C. R. Acad. Sci. Paris Sr. I Math.}, 330 (2000), 713. doi: 10.1016/S0764-4442(00)00259-7. Google Scholar [12] M. A. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems ,, \emph{Proc. R. Soc. Edinburgh Sect. A}, 135A (2005), 703. doi: 10.1017/S030821050000408X. Google Scholar [13] M. A. Efendiev, Y. Yamamoto and A. Yagi, Exponential attractors non-autonomous dissipative systems ,, \emph{J. Math. Soc. Japan}, 63 (2011), 647. doi: 10.2969/jmsj/06320647. Google Scholar [14] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, American Mathematical Society, (1988). Google Scholar [15] A. N. Kolmogorov and V. M. Tihomirov, $\varepsilon$-entropy and $\varepsilon$-capacity of sets in functional spaces ,, \emph{Amer. Math. Soc. Transl. Ser. 2}, 17 (1961), 277. Google Scholar [16] J. A. Langa, A. Miranville and J. Real, Pullback exponential attractors ,, \emph{Discrete Contin. Dyn. Syst.}, 26 (2010), 1329. doi: 10.3934/dcds.2010.26.1329. Google Scholar [17] J. A. Langa, J. C. Robinson and A. Suárez, Stability, instability and bifurcation phenomena in non-autonomous differential equations ,, \emph{Nonlinearity}, 15 (2002), 887. doi: 10.1088/0951-7715/15/3/322. Google Scholar [18] J. A. Langa and B. Schmalfuss, Finite dimensionality of attractors for non-autonomous dynamical systems given by partial differential equations ,, \emph{Stoch. Dyn.}, 4 (2004), 385. doi: 10.1142/S0219493704001127. Google Scholar [19] P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems ,, \emph{Nonlinear Anal.}, 71 (2009), 3956. doi: 10.1016/j.na.2009.02.065. Google Scholar [20] X. Mora, Semilinear parabolic problems define semiflows on $C^k$ spaces ,, \emph{Trans. Amer. Math. Soc.}, 278 (1983), 21. doi: 10.2307/1999300. Google Scholar [21] C. V. Pao, Nonlinear Parabolic and Elliptic Equations,, Plenum Press, (1992). Google Scholar [22] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [23] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, 2nd edition, (1997). Google Scholar [24] A. Yagi, Abstract Parabolic Evolution Equations and Their Applications,, Springer-Verlag, (2010). doi: 10.1007/978-3-642-04631-5. Google Scholar
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