Random pullback exponential attractors: general existence results for random dynamical systems in Banach spaces. (English) Zbl 1457.37096
Summary: 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.
MSC:
| 37L30 | Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems |
| 37L55 | Infinite-dimensional random dynamical systems; stochastic equations |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 35B41 | Attractors |
Keywords:
random dynamical system; exponential attractor; random pullback attractor; fractal dimension; \(\epsilon\)-entropyReferences:
| [1] | T. Caraballo, Random exponential attractors for stochastic damped wave equations,, in preparation. |
| [2] | A. N. Carvalho, Pullback exponential attractors for evolution processes in Banach spaces: Theoretical results,, Comm. Pure Appl. Anal., 12, 3047 (2013) · Zbl 1390.37126 · doi:10.3934/cpaa.2013.12.3047 |
| [3] | A. N. Carvalho, Pullback exponential attractors for evolution processes in Banach spaces: properties and applications,, Comm. Pure Appl. Anal., 13, 1141 (2014) · Zbl 1336.37059 · doi:10.3934/cpaa.2014.13.1141 |
| [4] | I. Chueshov, <em>Monotone Random Systems Theory and Applications</em>,, Lecture Notes in Math., 1779 (2002) · Zbl 1023.37030 · doi:10.1007/b83277 |
| [5] | H. Crauel, Nonautonomous and random attractors,, Jahresber. Dtsch. Math.- Ver., 117, 173 (2015) · Zbl 1358.37041 · doi:10.1365/s13291-015-0115-0 |
| [6] | R. Czaja, Pullback exponential attractors for nonautonomous equations part I: Semilinear parabolic equations,, J. Math. Anal. Appl., 381, 748 (2011) · Zbl 1233.35041 · doi:10.1016/j.jmaa.2011.03.053 |
| [7] | A. Eden, <em>Exponential Attractors for Dissipative Evolution Equations</em>,, Research in Applied Mathematics (1994) · Zbl 0842.58056 |
| [8] | D. E. Edmunds, <em>Function Spaces, Entropy Numbers and Differential Operators</em>,, Cambridge University Press (1996) · Zbl 0865.46020 · doi:10.1017/CBO9780511662201 |
| [9] | M. A. Efendiev, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems,, Proc. R. Soc. Edinburgh Sect. A, 135, 703 (2005) · Zbl 1088.37005 · doi:10.1017/S030821050000408X |
| [10] | A. N. Kolmogorov, \( \varepsilon \)-entropy and \(\varepsilon \)-capacity of sets in functional spaces,, Amer. Math. Soc. Transl. Ser. 2, 17, 277 (1961) |
| [11] | J. A. Langa, Pullback exponential attractors,, Discrete Contin. Dyn. Syst., 26, 1329 (2010) · Zbl 1303.37032 · doi:10.3934/dcds.2010.26.1329 |
| [12] | A. Shirikyan, Exponential attractors for random dynamical systems and applications,, Stoch. Partial Differ. Equ. Anal. Comput., 1, 241 (2013) · Zbl 1320.35089 · doi:10.1007/s40072-013-0007-1 |
| [13] | S. Zhou, Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative noise,, J. Differential Equations, 263, 2247 (2017) · Zbl 1364.37158 · doi:10.1016/j.jde.2017.03.044 |
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