Global attractor of a contact parabolic problem in a thin two-layer domain

A semilinear parabolic equation is considered in the union of two bounded thin cylindrical domains $\Omega_{1,\varepsilon}=\Gamma\times(0,\varepsilon)$ and $\Omega_{2,\varepsilon}=\Gamma\times(-\varepsilon,0)$ adjoining along their bases, where $\Gamma$ is a domain in $\mathbb R^d$, $d\leqslant3$. The unknown functions are related by means of an interface condition on the common base $\Gamma$. This problem can serve as a reaction-diffusion model describing the behaviour of a system of two components interacting at the boundary. The intensity of the reaction is assumed to depend on $\varepsilon$ and the thickness of the domains, and to be of order $\varepsilon^\alpha$.
Under investigation are the limiting properties of the evolution semigroup $S_{\alpha,\varepsilon}(t)$, generated by the original problem as $\varepsilon\to0$ (that is, as the domain becomes ever thinner). These properties are shown to depend essentially on the exponent $\alpha$. Depending on whether $\alpha$ is equal to, greater than, or smaller than 1, the original system can have three distinct systems of equations on $\Gamma$ as its asymptotic limit. The continuity properties of the global attractor of the semigroup $S_{\alpha,\varepsilon}(t)$ as $\varepsilon\to0$ are established under natural assumptions.