# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations. (English) Zbl 1188.37076

The authors introduce the notion of quasi-continuity for a dynamical system $\varphi$ on a Banach space $B$ to mean that for any sequence $\left({t}_{n},{x}_{n}\right)$ converging to $\left(t,x\right)$ in $\left[0,\infty \right)×B$, for which ${\left(\varphi \left({t}_{n}\right){x}_{n}\right)}_{n\in ℕ}$ is bounded, $\varphi \left({t}_{n}\right){x}_{n}$ converges weakly to $\varphi \left(t\right)x$. This property is weaker than norm-to-weak continuity. The authors invoke this notion for random dynamical systems on a separable $B$ in order to derive a sufficient condition for the existence of a random attractor, using asymptotic compactness properties formulated in terms of the Kuratowski measure of non-compactness.

The result is applied to stochastic reaction-diffusion equations on $D\subset {ℝ}^{d}$ bounded, with a finite-dimensional additive Wiener process taking values in ${L}^{\infty }\left(D\right)$. The nonlinearity of the drift being polynomially bounded, one has to consider solutions in ${L}^{p}\left(D\right)$ for suitable $p>2$. The induced random dynamical system is quasi-continuous, but it fails to be norm-to-weak continuous. It is shown to have a finite-dimensional random attractor $A$ for bounded subsets of ${L}^{2}\left(D\right)$, where attraction holds with respect to the ${L}^{p}\left(D\right)$ norm for every $p\ge 2$. Finally, a comparison estimate for the fractal dimensions of $A$ with respect to ${L}^{p}\left(D\right)$ norms, $p\ge 2$, is given.

##### MSC:
 37L55 Infinite-dimensional random dynamical systems; stochastic equations 35B41 Attractors (PDE) 35R60 PDEs with randomness, stochastic PDE 37L30 Attractors and their dimensions, Lyapunov exponents 60H15 Stochastic partial differential equations
##### References:
 [1] Arnold, L.: Random dynamical system, Springer monogr. Math. (1998) [2] Arnold, L.; Schmalfuss, B.: Lyapunov’s second method for random dynamical systems, J. differential equations 177, 235-265 (2001) · Zbl 1040.37035 · doi:doi:10.1006/jdeq.2000.3991 [3] Brzezniak, Z.; Li, Y. H.: Asymptotic compactness and absorbing sets for 2D stochastic Navier – Stokes equations on some unbounded domains, Trans. amer. Math. soc. 358, 5587-5629 (2006) · Zbl 1113.60062 · doi:doi:10.1090/S0002-9947-06-03923-7 [4] Caraballo, T.; Langa, J. A.: Stability and random attractors for a reaction – diffusion equation with multiplicative noise, Discrete contin. Dyn. syst. 6, 875-892 (2000) · Zbl 1011.37031 · doi:doi:10.3934/dcds.2000.6.875 [5] Caraballo, T.; Langa, J. A.; Robinson, J. C.: Upper semi-continuity of attractors for random perturbations of dynamical systems, Comm. partial differential equations 23, 1557-1581 (1998) · Zbl 0917.35169 · doi:doi:10.1080/03605309808821394 [6] Crauel, H.: Random point attractors versus random set attractors, J. London math. Soc. 63, 413-427 (2001) · Zbl 1011.37032 · doi:doi:10.1017/S0024610700001915 [7] Crauel, H.; Debussche, A.; Flandoli, F.: Random attractors, J. dynam. Differential equations 9, 307-341 (1997) · Zbl 0884.58064 · doi:doi:10.1007/BF02219225 [8] Crauel, H.; Flandoli, F.: Attractors for random dynamical systems, Probab. theory related fields 100, 365-393 (1994) · Zbl 0819.58023 · doi:doi:10.1007/BF01193705 [9] Debussche, A.: Hausdorff dimension of a random invariant set, J. math. Pures appl. 77, 967-988 (1998) · Zbl 0919.58044 · doi:doi:10.1016/S0021-7824(99)80001-4 [10] Deimling, K.: Nonlinear functional analysis, (1985) · Zbl 0559.47040 [11] Kloeden, P. E.; Langa, J. A.: Flattening, squeezing and the existence of random attractors, Proc. R. Soc. lond. Ser. A 463, 163-181 (2007) · Zbl 1133.37323 · doi:doi:10.1098/rspa.2006.1753 [12] Langa, J. A.; Robinson, J. C.: Fractal dimension of a random invariant set, J. math. Pures appl. 85, 269-294 (2006) · Zbl 1134.37364 · doi:doi:10.1016/j.matpur.2005.08.001 [13] Ma, Q. F.; Wang, S. H.; Zhong, C. K.: Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana univ. Math. J. 51, 1541-1559 (2002) · Zbl 1028.37047 · doi:doi:10.1512/iumj.2002.51.2255 [14] Marion, M.: Attractors for reaction – diffusion equations: existence and estimate of their dimension, Appl. anal. 25, 101-147 (1987) · Zbl 0609.35009 · doi:doi:10.1080/00036818708839678 [15] Pazy, A.: Semigroups of linear operators and applications to partial differential equations, (1983) [16] Da Prato, G.; Zabczyk, J.: Stochastic equations in infinite dimensions, Encyclopedia math. Appl. (1992) · Zbl 0761.60052 [17] Robinson, J. C.: Stability of random attractors under perturbation and approximation, J. differential equations 186, 652-669 (2002) · Zbl 1020.37033 · doi:doi:10.1016/S0022-0396(02)00038-4 [18] Robinson, J. C.: Infinite-dimensional dynamical systems: an introduction to dissipative parabolic pdes and the theory of global attractors, (2001) [19] Schmalfuss, B.: Backward cocycle and attractors of stochastic differential equations, International seminar on applied mathematics — nonlinear dynamics: attractor approximation and global behavior, 185-192 (1992) [20] Sun, C. Y.; Zhong, C. K.: Attractors for the semi-linear reaction – diffusion equation with distribution derivatives in unbounded domains, Nonlinear anal. 63, 49-65 (2005) · Zbl 1082.35036 · doi:doi:10.1016/j.na.2005.04.034 [21] Sun, C. Y.; Yang, M. H.; Zhong, C. K.: Global attractors for the wave equation with nonlinear damping, J. differential equations 227, 427-443 (2006) · Zbl 1101.35021 · doi:doi:10.1016/j.jde.2005.09.010 [22] Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics, (1997) [23] Wang, Y. J.; Zhong, C. K.; Zhou, S. F.: Pullback attractors of non-autonomous dynamical systems, Discrete contin. Dyn. syst. 16, 587-614 (2006) · Zbl 1179.37105 · doi:doi:10.3934/dcds.2006.16.587 [24] Wu, D. L.; Zhong, C. K.: The attractors for the non-homogeneous non-autonomous Navier – Stokes equations, J. math. Anal. appl. 321, 426-444 (2006) · Zbl 1111.35042 · doi:doi:10.1016/j.jmaa.2005.08.044 [25] Zhong, C. K.; Yang, M. H.; Sun, C. Y.: The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction – diffusion equations, J. differential equations 223, 367-399 (2006) · Zbl 1101.35022 · doi:doi:10.1016/j.jde.2005.06.008