December  2017, 37(12): 6383-6403. doi: 10.3934/dcds.2017277

Random pullback exponential attractors: General existence results for random dynamical systems in Banach spaces

1. 

Departamento Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, C/ Tarfia s/n, 41012 Sevilla, Spain

2. 

Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstr. 36, 8010 Graz, Austria

* Corresponding author: Stefanie Sonner

Received  March 2017 Revised  July 2017 Published  August 2017

Fund Project: The first author was partially supported by FEDER (EU) and Ministerio de Economía y Competitividad (Spain) grant MTM2015-63723-P and by the Junta de Andalucía under the Proyecto de Excelencia P12-FQM-1492

We derive general existence theorems for random pullback exponential attractors and deduce explicit bounds for their fractal dimension. The results are formulated for asymptotically compact random dynamical systems in Banach spaces.

Citation: Tomás Caraballo, Stefanie Sonner. Random pullback exponential attractors: General existence results for random dynamical systems in Banach spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6383-6403. doi: 10.3934/dcds.2017277
References:
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T. Caraballo and S. Sonner, Random exponential attractors for stochastic damped wave equations, in preparation.Google Scholar

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

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A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: properties and applications, Comm. Pure Appl. Anal., 13 (2014), 1141-1165. doi: 10.3934/cpaa.2014.13.1141. Google Scholar

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

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A. Shirikyan and S. Zelik, Exponential attractors for random dynamical systems and applications, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 241-281. doi: 10.1007/s40072-013-0007-1. Google Scholar

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show all references

References:
[1]

T. Caraballo and S. Sonner, Random exponential attractors for stochastic damped wave equations, in preparation.Google Scholar

[2]

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

[3]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: properties and applications, Comm. Pure Appl. Anal., 13 (2014), 1141-1165. doi: 10.3934/cpaa.2014.13.1141. Google Scholar

[4]

I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Math., 1779, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277. Google Scholar

[5]

H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber. Dtsch. Math.-Ver., 117 (2015), 173-206. doi: 10.1365/s13291-015-0115-0. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

A. Shirikyan and S. Zelik, Exponential attractors for random dynamical systems and applications, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 241-281. doi: 10.1007/s40072-013-0007-1. Google Scholar

[13]

S. Zhou, Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative noise, J. Differential Equations, 263 (2017), 2247-2279. doi: 10.1016/j.jde.2017.03.044. Google Scholar

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