Asymptotic Analysis of a Parabolic Problem in a Thick Two-Level Junction

Abstract

We consider an initial boundary value problem for the heat equation in a plane two-level junction Ωε; which is the union of a domain and a large number 2N of thin rods with the variable thickness of order ε = O(N^-1). The thin rods are divided into two levels depending on boundary conditions given on their sides. In addition, the boundary conditions depend on the parameters α ≥ 1 and β ≥ 1, and the thin rods from each level are ε-periodically alternated. The asymptotic analysis of this problem for different values of α and β is made as ε → 0. The leading terms of the asymptotic expansion for the solution are constructed, the asymptotic estimate in the Sobolev space L² (0; T; H¹(Ωε)) is obtained and the convergence theorem is proved with minimal conditions for the right-hand sides.