On the Asymptotics of Attractors of the Ginzburg–Landau Complex Equation in a Perforated Domain With an Oscillating Boundary: Critical Case

ABSTRACT We analyze the asymptotic behavior of trajectory attractors associated with the Ginzburg–Landau complex equation in a perforated domain. This domain features a rapidly oscillating outer boundary. The analysis centers on the scenario in which the parameters defining the pore sizes, the distance between them, as well as the amplitude and frequency of the boundary oscillation, simultaneously approach zero. The asymptotic behavior of attractors related to an initial boundary value problem for complex Ginzburg–Landau equations in perforated domains, specifically in the critical case (which involves the emergence of additional potential in the homogenized equation), is examined by Bekmaganbetov K. A., Chechkin G. A., Chepyzhov V. V., and Tolemis A. A. This study is presented in their pape titled “Homogenization of Attractors to Ginzburg–Landau Equations in Media with Locally Periodic Obstacles:Critical Case”, Mathematics Series 3 no. 111 (2023): pp. 11–27 (published in the Bulletin of the Karaganda University). By defining the auxiliary function spaces with a weak topology, we derive a limit (homogenized) equation and prove the existence of a trajectory attractor for this equation. Subsequently, we formulate the main theorem regarding the weak convergence of attractors and prove it based on auxiliary lemmas.