Caraballo, T.; Morillas, F.; Valero, J. Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity. (English) Zbl 1223.39010 J. Difference Equ. Appl. 17, No. 1-2, 161-184 (2011). The authors study the asymptotic behavior of solutions of the stochastic first order lattice dynamical system with additive noise of the form\[ \frac{du_i}{dt} = v(u_{i-1}-2u_i+u_{i+1})-h_i(u_i)-f_i(u_i)+a_i\frac{dw_i(t)}{dt},\quad u_i(0)=u_i^0,\;i\in\mathbb Z, \]where \(w_i(t)\) are real independent two-sided Brownian motions. The nonlinear terms \(f\) and \(h\) are assumed to satisfy certain continuity conditions together with growth and dissipative conditions, but no Lipschitz continuity, so that uniqueness of the solution fails to be true. Using the theory of multi-valued random dynamical systems, existence of a random global attractor is proven. Reviewer: Dora Selesi (Novi Sad) Cited in 33 Documents MSC: 39A50 Stochastic difference equations 39A30 Stability theory for difference equations 37H10 Generation, random and stochastic difference and differential equations 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems 37B25 Stability of topological dynamical systems 37L55 Infinite-dimensional random dynamical systems; stochastic equations 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) Keywords:stochastic lattice dynamical systems; set-valued dynamical system; random attractor; asymptotic behavior; additive noise; independent two-sided Brownian motions; random global attractor PDFBibTeX XMLCite \textit{T. Caraballo} et al., J. Difference Equ. Appl. 17, No. 1--2, 161--184 (2011; Zbl 1223.39010) Full Text: DOI Link References: [1] DOI: 10.1016/j.jmaa.2007.06.054 · Zbl 1127.37051 [2] Arnold L., Random Dynamical Systems (1998) · Zbl 0906.34001 [3] DOI: 10.1142/S0218127401002031 · Zbl 1091.37515 [4] DOI: 10.1142/S0219493706001621 · Zbl 1105.60041 [5] DOI: 10.1023/B:JODY.0000009745.41889.30 · Zbl 1041.37040 [6] DOI: 10.1007/s11464-008-0028-7 · Zbl 1155.60324 [7] DOI: 10.1006/jmaa.2001.7497 · Zbl 0987.60074 [8] DOI: 10.1016/S0362-546X(00)00216-9 · Zbl 1004.37035 [9] DOI: 10.1007/s00245-004-0802-1 · Zbl 1066.60058 [10] DOI: 10.3934/dcds.2008.21.415 · Zbl 1155.60025 [11] T. Caraballo, M.J. Garrido-Atienza, B. Schmalfuß, and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B 14 (2010), pp. 439–455 · Zbl 1201.60063 [12] Castaing C., Lecture Notes in Mathematics 580, in: Convex Analysis and Measurable Multifunctions (1977) · Zbl 0346.46038 [13] Hu S., Handbook of Multivalued Analysis. Volume 1: Theory (1997) · Zbl 0887.47001 [14] DOI: 10.1016/j.jde.2005.06.002 · Zbl 1084.35092 [15] DOI: 10.3934/dcds.2007.19.711 · Zbl 1149.37039 [16] Kato S., Funkcialaj Ekvacioj. 19 pp 239– (1976) [17] DOI: 10.1142/S0218127409023196 · Zbl 1170.37337 [18] DOI: 10.1080/10236190701859211 · Zbl 1171.35339 [19] Schmalfuß B., International Conference on Differential Equations, Vols. 1, 2 (Berlin, 1999) pp 684– (2000) · Zbl 0944.60071 [20] DOI: 10.1016/j.jmaa.2006.08.070 · Zbl 1112.37076 [21] DOI: 10.1080/10236190701859542 · Zbl 1143.37050 [22] DOI: 10.1088/0951-7715/20/8/010 · Zbl 1130.34053 [23] DOI: 10.1016/j.jde.2004.02.005 · Zbl 1173.37331 [24] DOI: 10.1016/j.jde.2005.06.024 · Zbl 1091.37023 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.