Abstract:
The Ginzburg-Landau equation with rapidly oscillating terms in the equation and boundary conditions in a
perforated domain was considered. Proof was given that the trajectory attractors of this equation converge
weakly to the trajectory attractors of the homogenized Ginzburg-Landau equation. To do this, we use the
approach from the articles and monographs of V.V. Chepyzhov and M.I. Vishik about trajectory attractors
of evolutionary equations, and we also use homogenization methods that appeared at the end of the 20th
century. First, we use asymptotic methods to construct asymptotics formally, and then we justify the form
of the main terms of the asymptotic series using functional analysis and integral estimates. By defining the
corresponding 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 theorems and
prove them by using auxiliary lemmas. We prove that the trajectory attractors of this equation tend in a
weak sense to the trajectory attractors of the homogenized Ginzburg-Landau equation in the subcritical
case, and they disappear in the supercritical case.
