Homogenization of Attractors to the Chafee–Infante Equations in a Perforated Domain with Locally Periodic Obstacles: Subcritical and Supercritical Cases

Abstract

In this paper, we investigate the Chafee–Infante equation in a locally periodic perforated domain with a small parameter \(\varepsilon>0\) characterizing the diameter of the cavities and the distance between them. The asymptotic behavior of the problem depends on the scaling of the boundary effects, characterized by the parameter \(\varepsilon^\theta\) in Fourier condition on the boundary of the cavities, leading to two distinct regimes. In contrast to the critical case (\(\theta=1\)), in which the condition at the boundary of the cavities gives a potential term and new coefficients of the elliptic operator to the homogenized equation, in the subcritical and supercritical cases the situation changes radically.

The subcritical case (\( \theta > 1\)): The influence of boundary conditions on perforations does not contribute to the potential term but still changes the coefficients of the homogenized equation. We prove that both trajectory and global attractors of the original problem converge to those of the homogenized problem.

The supercritical case (\(\theta < 1\)): In this case, the solutions degenerate as the small parameter tends to zero. It has been proven that the attractors also tend to zero in the corresponding topology.

In the subcritical case, the homogenized limit is rigorously derived via auxiliary cell problems and asymptotic expansions. Appropriate functional spaces with weak topology are introduced, and the existence of trajectory attractors is established. In the supercritical case, we use integral estimates to prove the degeneration of solutions and attractors.