Spatial-Skin Effect for a Spectral Problem with "Slightly Heavy" Concentrated Masses in a Thick Cascade Junction

Abstract

The asymptotic behavior (as $\varepsilon \to 0$) of eigenvalues and eigenfunctions of a boundary- value problem for the Laplace operator in a thick cascade junction with concentrated masses is studied. This cascade junction consists of the junction’s body and a great number $5N = \mathcal(\varepsilon^{−1})$ of $\varepsilon$−alternating thin rods belonging to two classes. One class consists of rods of finite length and the second one consists of rods of small length of order $\mathcal{O}(\varepsilon)$. The density of the junction is of order $\mathcal{O}(\varepsilon^{−\alpha})$ on the rods from the second class and $\mathcal{O}(1)$ outside of them. There exist five qualitatively different cases in the asymptotic behavior of eigenvibrations as $\varepsilon \to 0$, namely the cases of “light” concentrated masses ($\alpha \in (0, 1)$), “middle” concentrated masses ($\alpha = 1$), “slightly heavy” concentrated masses ($\alpha \in (1, 2)$), “intermediate heavy” concentrated masses ($\alpha = 2$), and “very heavy” concentrated masses ($\alpha > 2$). In the paper we study the influence of the concentrated masses on the asymptotic behavior of the eigen-magnitudes if $\alpha \in (1, 2)$.