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Pullback attractors of nonautonomous and stochastic multivalued dynamical systems. (English) Zbl 1018.37048

The existence of pullback global attractors for multivalued processes generated by differential inclusions is studied. Mainly one deals with nonautonomous multivalued dynamical systems in which the trajectories can be unbounded in time and with nonautonomous stochastic multivalued dynamical systems. Let X be a complete metric space, P(X) the set of all subsets of X. Denote d ={(t,s) z :ts}. The map U: d ×XP(X) is called a multivalued dynamical process (MDP) on X if 1) U(t,t,·) is the identical map, 2) U(t,s,x)U(t,τ,U(τ,s,x)), xX, s<τ<t.

At first the authors proved abstract results on the existence of ω-limit sets and global attractors and studied their topological properties (compactness, connectedness). Then they applied the abstract results to nonautonomous differential inclusions of the reaction-diffusion type in which the forcing term can grow polynomially in time and they gave applications to stochastic differential inclusions with additive and multiplicative noises. MDP is defined as a two-parameter family of multivalued maps. The attraction of any bounded set of the phase space to the global attractor is uniform with respect to the first parameter. The rate of attraction and the attractor itself can depend on the second parameter. The theory is extended to the cases with stochastic terms in the model.

MSC:
37L55Infinite-dimensional random dynamical systems; stochastic equations
35B41Attractors (PDE)
35K55Nonlinear parabolic equations
35K57Reaction-diffusion equations