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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

Alexandre Nolasco de Carvalho 1, and  Stefanie Sonner 2,

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.
Keywords: non-autonomous damped wave equation., non-autonomous dynamical systems, Exponential attractors, non-autonomous Chafee-Infante equation, pullback and forwards attractors.
Mathematics Subject Classification: Primary: 37L25, 37B55; Secondary: 37L30, 35B4.
Citation: Alexandre Nolasco de Carvalho, Stefanie Sonner. Pullback exponential attractors for evolution processes in Banach spaces: Properties and applications. Communications on Pure and 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, Elsevier, Amsterdam, 2003.

[2]

J. Arrieta, A. N. Carvalho and J. K. Hale, A damped hyperbolic equation with critical exponent , Comm. Partial Differential Equations, 17 (1992), 841-866. doi: 10.1080/03605309208820866.

[3]

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.

[4]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Appl. Math. Sci., 182, Springer, 2012. doi: 10.1007/978-1-4614-4581-4.

[5]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: theoretical results , Commun. Pure and Appl. Anal., 12 (2013), 3047-3071. doi: 10.3934/cpaa.2013.12.3047.

[6]

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

[7]

H. Crauel, A. Debussche and F. Flandoli, Random attractors , J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225.

[8]

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.

[9]

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.

[10]

D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers and Differential Operators, Cambridge University Press, New York, 1996. doi: 10.1017/CBO9780511662201.

[11]

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.

[12]

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.

[13]

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.

[14]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, Rhode Island, 1988.

[15]

A. N. Kolmogorov and V. M. Tihomirov, $\varepsilon$-entropy and $\varepsilon$-capacity of sets in functional spaces , Amer. Math. Soc. Transl. Ser. 2, 17 (1961), 277-364.

[16]

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.

[17]

J. A. Langa, J. C. Robinson and A. Suárez, Stability, instability and bifurcation phenomena in non-autonomous differential equations , Nonlinearity, 15 (2002), 887-903. doi: 10.1088/0951-7715/15/3/322.

[18]

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.

[19]

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.

[20]

X. Mora, Semilinear parabolic problems define semiflows on $C^k$ spaces , Trans. Amer. Math. Soc., 278 (1983), 21-55. doi: 10.2307/1999300.

[21]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.

[22]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[23]

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

[24]

A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer-Verlag, Berlin-Heidelberg, 2010. doi: 10.1007/978-3-642-04631-5.

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Elsevier, Amsterdam, 2003.

[2]

J. Arrieta, A. N. Carvalho and J. K. Hale, A damped hyperbolic equation with critical exponent , Comm. Partial Differential Equations, 17 (1992), 841-866. doi: 10.1080/03605309208820866.

[3]

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.

[4]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Appl. Math. Sci., 182, Springer, 2012. doi: 10.1007/978-1-4614-4581-4.

[5]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: theoretical results , Commun. Pure and Appl. Anal., 12 (2013), 3047-3071. doi: 10.3934/cpaa.2013.12.3047.

[6]

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

[7]

H. Crauel, A. Debussche and F. Flandoli, Random attractors , J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225.

[8]

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.

[9]

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.

[10]

D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers and Differential Operators, Cambridge University Press, New York, 1996. doi: 10.1017/CBO9780511662201.

[11]

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.

[12]

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.

[13]

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.

[14]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, Rhode Island, 1988.

[15]

A. N. Kolmogorov and V. M. Tihomirov, $\varepsilon$-entropy and $\varepsilon$-capacity of sets in functional spaces , Amer. Math. Soc. Transl. Ser. 2, 17 (1961), 277-364.

[16]

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.

[17]

J. A. Langa, J. C. Robinson and A. Suárez, Stability, instability and bifurcation phenomena in non-autonomous differential equations , Nonlinearity, 15 (2002), 887-903. doi: 10.1088/0951-7715/15/3/322.

[18]

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.

[19]

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.

[20]

X. Mora, Semilinear parabolic problems define semiflows on $C^k$ spaces , Trans. Amer. Math. Soc., 278 (1983), 21-55. doi: 10.2307/1999300.

[21]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.

[22]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[23]

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

[24]

A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer-Verlag, Berlin-Heidelberg, 2010. doi: 10.1007/978-3-642-04631-5.

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