ON A DESCRIPTION OF LONG-TIME BEHAVIOR OF DISSIPATIVE PERTURBATIONS OF INFINITE-DIMENSIONAL HAMILTONIAN-SYSTEMS

We present some recent results on long-time behaviour and limit regime s for a class of nonlinear partial differential equations which are di ssipative perturbations of Hamiltonian systems. Contrasting with unper turbed case there exist finite dimensional global attractors for the c onsidered equations. In a rather unified framework we construct infini te families of approximate inertial manifolds (AIMs) which are finite dimensional smooth surfaces in a phase space of the system whose small vicinities attract all solutions and contain the global attractor. Us ing the properties of AIMs we can establish localization theorems for the attractor and suggest a new approximate method for investigation o f the long-time dynamics. A similar method for parabolic equations is known as a nonlinear Galerkin method. As examples we consider both dis sipative perturbations of well-known integrable systems and some model s of elastic solids subjected to nonconservative loads.