The dissipative wave equation with a critical quintic nonlinearity in smooth bounded three dimensional domain is considered. Based on the recent extension of the Strichartz estimates to the case of bounded domains, the existence of a compact global attractor for the solution semigroup of this equation is established. Moreover, the smoothness of the obtained attractor is also shown.

In this paper, we establish the existence and boundedness of solutions of a doubly nonlinear parabolic system. We also obtain the existence of a global attractor and the regularity property for this attractor in [L(Ω)] and 2 ∏ i=1 B 1+σi,pi ∞ (Ω).

The entropy of a dynamical system S' on a hyperbolic attractor with respect to the Bowen-Ruelle-Sina) measure is obtained as a limit of entropy characteristics of small random perturbations x' of S'. Both the case of perturbations only in some neighborhood of an attractor and global perturbations of a flow with hyperbolic attracting sets are considered.