Asymptotic behavior of the Cahn-Hilliard-Oono equation

Our aim in this article is to study the asymptotic behavior, in terms of finite-dimensional attractors, of the Cahn-Hilliard-Oono equation. This equation differs from the usual Cahn-Hilliard equation by the presence of a term of the form \( \epsilon u,\ \epsilon >0\), which takes into account long-ranged interactions. In particular, we prove the existence of a robust family of exponential attractors as \(\epsilon\) goes to \(0\).