Dynamic singular perturbation problems for multi-structures (Q2722279)

The main objective of the present paper is to describe some recent developments in asymptotic analysis of boundary value problems in multi-structures, i.e. domains dependent on a small parameter \(\varepsilon\) in such a way that the limit region, as \(\varepsilon\to 0,\) consists of subsets of different space dimensions. It is demonstrated on simple examples the efficiency of the method of compound asymptotic expansions applied to mathematical models of multi-structures. Three types of problems are considered: 1) an initial boundary value problem in a domain defined as a union of several two-dimensional thin bodies; it involves an asymptotic algorithm developed to construct an asymptotic approximation of the solution in the domain with junction; 2) vibration of a layered structure with an imperfect interface represented by a thin and ``soft'' layer is analysed; 3) a dynamic problem for a 1D--3D multi-structure defined as a union of a three-dimensional body and a set of thin cylinders is considered. The asymptotic results presented in this paper are based on the method of compound asymptotic expansions.