Spatial-skin effect for a spectral problem with “slightly heavy” concentrated masses in a thick cascade junction

Abstract

The asymptotic behavior (as ε→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=(ε−1) of ε−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 O(ε). The density of the junction is of order O(ε−α) on the rods from the second class and O(1) outside of them. There exist five qualitatively different cases in the asymptotic behavior of eigenvibrations as ε→0, namely the cases of “light” concentrated masses (α∈(0,1)), “middle” concentrated masses (α=1), “slightly heavy” concentrated masses (α∈(1,2)), “intermediate heavy” concentrated masses (α=2), and “very heavy” concentrated masses (α>2). In the paper we study the influence of the concentrated masses on the asymptotic behavior of the eigen-magnitudes if α∈(1,2).