Abstract:
This paper is devoted to studying the reaction–diffusion systems with rapidly
oscillating coefficients in the equations and in boundary conditions in domains with locally
periodic oscillating boundary; on this boundary a Robin boundary condition is imposed.
We consider the supercritical case, when the homogenization changes the Robin boundary
condition on the oscillating boundary is to the homogeneous Dirichlet boundary condition
in the limit as the small parameter, which characterizes oscillations of the boundary, tends
to zero. In this case, we prove that the trajectory attractors of these systems converge in a
weak sense to the trajectory attractors of the limit (homogenized) reaction–diffusion systems
in the domain independent of the small parameter. For this aim we use the homogenization
theory, asymptotic analysis and the approach of V.V. Chepyzhov and M.I. Vishik concerning
trajectory attractors of dissipative evolution equations. The homogenization method and
asymptotic analysis are used to derive the homogenized reaction–diffusion system and to
prove the convergence of solutions. First we define the appropriate auxiliary functional
spaces with weak topology, then, we prove the existence of trajectory attractors for these
systems and formulate the main Theorem. Finally, we prove the main convergence result
with the help of auxiliary lemmas.
