Abstract
We study the asymptotic behavior of trajectory attractors of the Ginzburg–Landau complex equation in a perforated domain with a rapidly oscillating outer boundary when the parameter characterizing the pore sizes and the distance between them, as well as the amplitude and frequency of the boundary oscillation, tends to zero. The asymptotic behaviour of attractors to an initial boundary value problem for complex Ginzburg–Landau equations in perforated domains for the critical case (appearance of additional potential in the homogenized equation) is studied by Bekmaganbetov K.A., Chechkin G.A., Chepyzhov V.V., and Tolemis, A.A. in “Homogenization of Attractors to Ginzburg–Landau Equations in Media with Locally Periodic Obstacles: Critical Case” (Bulletin of the Karaganda university. Mathematics series.- 2023.- v. 3 (111).- p. 11–27). By defining the auxiliary function spaces with weak topology, we derive a limit (homogenized) equation and prove the existence of a trajectory attractor for this equation. Then, we formulate the main Theorem of weak convergence of attractors and prove it on the base of auxiliary lemmas.