An uncoupled limit model for a high-contrast problem in a thin multi-structure

We investigate a degenerating elliptic problem in a multi-structure $\Omega_\varepsilon$ of $ \mathbb{R}^3$, in the framework of the thermal stationary conduction with highly contrasting diffusivity. Precisely, $\Omega_\varepsilon$ consists of a fixed base $\Omega^-$ surmounted by a thin cylinder $\Omega_\varepsilon^+$ with height $1$ and cross-section with a small diameter of order $\varepsilon$. Moreover, $\Omega^+_\varepsilon$ contains a cylindrical core, always with height $1$ and cross-section with diameter of order $\varepsilon$, with conductivity of order $1$, surrounded by a ring with conductivity of order $\varepsilon^2$. Also $\Omega^-$ has conductivity of order $\varepsilon^2$. By assuming that the temperature is zero on the top and on the bottom of the boundary of $\Omega_\varepsilon$, while the flux is zero on the remaining part of the boundary, under a suitable choice of the source term we prove that the limit problem, as $\varepsilon$ vanishes, boils down to two uncoupled problems: one in $\Omega^-$ and one in $\Omega^+_1$. Moreover, the problem in $\Omega^+_1$ is nonlocal.