Abstract
This paper is concerned with the random attractors for a class of second-order stochastic lattice dynamical systems. We first prove the uniqueness and existence of the solutions of second-order stochastic lattice dynamical systems in the spaceF=lλ2×l2. Then, by proving the asymptotic compactness of the random dynamical systems, we establish the existence of the global random attractor. The system under consideration is quite general, and many existing results can be regarded as the special case of our results.
Highlights
IntroductionWe consider the following second-order stochastic lattice dynamical system:. l2, u = (ui)i∈Zn ∈ l2 are real-value (ξi)i∈Zn and η = (ηi)i∈Zn are given vectors satisfying bounded conditions; g = (gi)i∈Zn ∈ l2; f(u) = (fi(ui))i∈Zn and h(u) = (hi(ui))i∈Zn are nonlinear terms satisfying some growth assumptions to be given later;
We consider the following second-order stochastic lattice dynamical system:ü + ξAu + h (u) + Au + ηu + f (u) = g + Ẇ (t), t > 0, (1)u (0) =i∈Zn = u0,u (0) = (u1i,0)i∈Zn = u10, where u functions =on(uRi)+i∈;Zξn ∈ =l2, u =i∈Zn ∈ l2 are real-valuei∈Zn and η =i∈Zn are given vectors satisfying bounded conditions; g =i∈Zn ∈ l2; f(u) = (fi(ui))i∈Zn and h(u) = (hi(ui))i∈Zn are nonlinear terms satisfying some growth assumptions to be given later;A is the linear operator on l2
For the second-order SLDS with stochastic noises on the lattice Z or Zk, the existence of the random attractor is receiving the attention from research community [22, 23]
Summary
We consider the following second-order stochastic lattice dynamical system:. l2, u = (ui)i∈Zn ∈ l2 are real-value (ξi)i∈Zn and η = (ηi)i∈Zn are given vectors satisfying bounded conditions; g = (gi)i∈Zn ∈ l2; f(u) = (fi(ui))i∈Zn and h(u) = (hi(ui))i∈Zn are nonlinear terms satisfying some growth assumptions to be given later;. Bates et al [19] first studied the existence of global random attractor for a class of first-order dynamical systems driven by white noises on lattice Z. For the second-order SLDS with stochastic noises on the lattice Z or Zk, the existence of the random attractor is receiving the attention from research community [22, 23]. [22] investigated the asymptotic behavior for a class of second-order stochastic lattice dynamical systems and proved the existence of the random attractor for the concerned second-order SLDS. Paper [23] addressed the asymptotic behavior of solutions to second-order SLDS with random coupled coefficients and multiplicative white noises in weighted spaces of infinite sequences and discussed the existence of a tempered random bounded absorbing set and a random attractor for the SLDS.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
