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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 |
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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