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

Анотація

We consider an initial boundary value problem for the heat equation in a plane two-level junction W_e which is the union of a domain and a large number 2N of thin rods with the variable thickness of order e = 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 a ≥ 1 and b ≥ 1, and the thin rods from each level are e-periodically alternated. The asymptotic analysis of this problem for different values of a and b is made as e → 0. The leading terms of the asymptotic expansion for the solution are constructed, the asymptotic estimate in the Sobolev space L²(0; T;H¹(W_e)) is obtained and the convergence theorem is proved with minimal conditions for the right-hand sides.

Mathematics Subject Classification: 35B27, 74K30, 35K20, 35B40, 35C20.