Averaging of trajectory attractors of evolution equations with rapidly oscillating terms

Evolution equations containing rapidly oscillating terms with respect to th e spatial variables or the time variable are considered. The trajectory att ractors of these equations are proved to approach the trajectory attractors of the equations whose terms are the averages of the corresponding terms o f the original equations. The corresponding Cauchy problems are not assumed here to be uniquely soluble. At the same time if the Cauchy problems for t he equations under consideration are uniquely soluble, then they generate s emigroups having global attractors. These global attractors also converge t o the global attractors of the averaged equations in the corresponding spac es. These results are applied to the following equations and systems of mathema tical physics: the 3D and 2D Navier-Stokes systems with rapidly oscillating external forces, reaction-diffusion systems, the complex Ginzburg-Landau e quation, the generalized Chafee-Infante equation, and dissipative hyperboli c equations with rapidly oscillating terms and coefficients.