November  2013, 12(6): 3047-3071. doi: 10.3934/cpaa.2013.12.3047

Pullback exponential attractors for evolution processes in Banach spaces: Theoretical results

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, Spain

Received  October 2012 Revised  February 2013 Published  May 2013

We construct exponential pullback attractors for time continuous asymptotically compact evolution processes in Banach spaces and derive estimates on the fractal dimension of the attractors. We also discuss the corresponding results for autonomous processes.
Citation: Alexandre Nolasco de Carvalho, Stefanie Sonner. Pullback exponential attractors for evolution processes in Banach spaces: Theoretical results. Communications on Pure and Applied Analysis, 2013, 12 (6) : 3047-3071. doi: 10.3934/cpaa.2013.12.3047
References:
[1]

T. Caraballo, A. N. Carvalho, J. A. Langa and L. F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Anal., 72 (2010), 1967-1976. doi: 10.1016/j.na.2009.09.037.

[2]

A. N. Carvalho, J. A. Langa and J. C. Robinson, "Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems," Appl. Math. Sci., 182, Springer, 2012.

[3]

J. W. Cholewa, R. Czaja and G. Mola, Remarks on the fractal dimension of bi-space global and exponential attractors, Boll. Unione Mat. Ital., 1 (2008), 121-146.

[4]

D. N. Cheban, P. E. Kloeden and B. Schmalfuss, The relationship between pullback, forward and global attractors of nonautonomous dynamical systems, Nonlinear Dyn. and Syst. Theory, 2 (2002), 9-28.

[5]

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

[6]

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

[7]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, "Exponential Attractors for Dissipative Evolution Equations," Research in Applied Mathematics, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.

[8]

M. A. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\R ^3$, C. R. Acad. Sci. Paris Sr. I Math., 330 (2000), 713-718. doi: 10.1016/S0764-4442(00)00259-7.

[9]

M. A. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems, Proc. R. Soc. Edinburgh Sect.A, 135A (2005), 703-730. doi: 10.1017/S030821050000408X.

[10]

M. A. Efendiev, Y. Yamamoto and A. Yagi, Exponential attractors non-autonomous dissipative systems, J. Math. Soc. Japan, 63 (2011), 647-673. doi: 10.2969/jmsj/06320647.

[11]

J. A. Langa, A. Miranville and J. Real, Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357. doi: 10.3934/dcds.2010.26.1329.

[12]

J. A. Langa and B. Schmalfuss, Finite dimensionality of attractors for non-autonomous dynamical systems given by partial differential equations, Stoch. Dyn., 4 (2004), 385-404. doi: 10.1142/S0219493704001127.

[13]

P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Anal., 71 (2009), 3956-3963. doi: 10.1016/j.na.2009.02.065.

[14]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Appl. Math. Sci. 68, Springer Verlag, New York, 1997.

show all references

References:
[1]

T. Caraballo, A. N. Carvalho, J. A. Langa and L. F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Anal., 72 (2010), 1967-1976. doi: 10.1016/j.na.2009.09.037.

[2]

A. N. Carvalho, J. A. Langa and J. C. Robinson, "Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems," Appl. Math. Sci., 182, Springer, 2012.

[3]

J. W. Cholewa, R. Czaja and G. Mola, Remarks on the fractal dimension of bi-space global and exponential attractors, Boll. Unione Mat. Ital., 1 (2008), 121-146.

[4]

D. N. Cheban, P. E. Kloeden and B. Schmalfuss, The relationship between pullback, forward and global attractors of nonautonomous dynamical systems, Nonlinear Dyn. and Syst. Theory, 2 (2002), 9-28.

[5]

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

[6]

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

[7]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, "Exponential Attractors for Dissipative Evolution Equations," Research in Applied Mathematics, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.

[8]

M. A. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\R ^3$, C. R. Acad. Sci. Paris Sr. I Math., 330 (2000), 713-718. doi: 10.1016/S0764-4442(00)00259-7.

[9]

M. A. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems, Proc. R. Soc. Edinburgh Sect.A, 135A (2005), 703-730. doi: 10.1017/S030821050000408X.

[10]

M. A. Efendiev, Y. Yamamoto and A. Yagi, Exponential attractors non-autonomous dissipative systems, J. Math. Soc. Japan, 63 (2011), 647-673. doi: 10.2969/jmsj/06320647.

[11]

J. A. Langa, A. Miranville and J. Real, Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357. doi: 10.3934/dcds.2010.26.1329.

[12]

J. A. Langa and B. Schmalfuss, Finite dimensionality of attractors for non-autonomous dynamical systems given by partial differential equations, Stoch. Dyn., 4 (2004), 385-404. doi: 10.1142/S0219493704001127.

[13]

P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Anal., 71 (2009), 3956-3963. doi: 10.1016/j.na.2009.02.065.

[14]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Appl. Math. Sci. 68, Springer Verlag, New York, 1997.

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