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Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity. (English) Zbl 1223.39010

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.

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)
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