Weighted Space Part Of The Dissipative System Is The Existence Of Random Attractors

Lattice dynamical systems (LDSs) are infinite systems of ordinary differential equations or difference equations indexed by points in a lattice, Through the coupling, lattice dynamical systems show a complex nature of space-time dynamics. Lattice dynamical systems study the asymptotic behavior of the field of modern mathematical physics, one of the most important problems, and consider the existence of the global attractor is an effective way to deal with this problem, that is to find a compact invariant set, which attracts all the orbits.Random attraction is an important dynamical behavior that has received the increasing research. Recently, authors in [1 0 ] introduce a weighted space, and prove the existence of global attractor in such space. Motivated by this idea, we will prove (1) has a random attractor in weighted space. We will introduce a new weigh, define a semi-group on lμ2×lμ2. Prove the existence of the attractor of the following system:Firstly, we establish space lμ2×lμ2 and prove the uniqueness and existence of the solutions of partly dissipative stochastic lattice dynamical systems in weighted spaces lμ2×lμ2. Furthermore, by stochastic analysis, we show that the solutions of the equation are dependent continuously on initial conditions. Then we estimate the ends of the solutions of the system by stochastic analysis and nonlinear analysis.Finally, we establish the existence of the global random attractor with the set of tempered bounded sets by proving the asymptotic compactness of the random dynamical systems.In section one, the background and significance of stochastic dynamical systems and the global random attractor is introduced .In section two, we introduce some basic concepts related to stochastic dynamical systems and the global random attractor. Meanwhile, we present some notations and give a simple description of the system.In section three, some definitions and related lemmas are present. Received the uniqueness and existence of the solutions of partly dissipative stochastic lattice dynamical systems in weighted and then receive existence random dynamical systems.In section four, we firstly prove an important lemma 4.2 for the main theory. And then prove the main lemma 4.3 received the existence of the systems .