Asymptotics of the elements of attractors corresponding to singularly perturbed parabolic equations

In a domain $\Omega^n\Subset\mathbf R^n$ we consider the first boundary value problem for a quasilinear parabolic fourth-order equation with a small parameter in the highest derivatives, which degenerates for $\varepsilon=0$ into a second order equation. It is well known that the semigroup corresponding to this problem has an attractor, that is, an invariant attracting set in the phase space. In this paper we investigate the structure of this attractor by means of an asymptotic expansion in $\varepsilon$.
The dominant term of the asymptotics is the solution of a second-order equation. The asymptotic expansion also contains boundary layer functions, which are responsible for the deterioration of the differential properties of the elements of the attractor near the boundary. The asymptotics constructed in this way (with an estimate of the remainder) enable us to study the differential properties of attractors and their behavior as $\varepsilon\to0$ in any interior subdomain $\Omega'$, $\overline\Omega'\subset\Omega$.
For simplicity, the investigation is carried out in the case when $\Omega$ is a bounded cylindrical domain. The generalization to $\Omega\Subset\mathbf R^n$ does not present any difficulties.