Asymptotic Analysis of a Parabolic Problem in a Thick Two-Level Junction - Durante, T.; Mel’‎nyk, T. A. (Журнал математичної фізики, аналізу, геометрії, 2007)
We consider an initial boundary value problem for the heat equation in a plane two-level junction $Ω_ε$ which is the union of a domain and a large number $2N$ of thin rods with the variable thickness of order $ε=\mathcal{O}(N^{-1})$. 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 $α\ge 1$ and $β\ge 1$, 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 $L^2(0,T;H^1(Ω_ε))$ is obtained and the convergence theorem is proved with minimal conditions for the right-hand sides. Mathematics Subject Classification: 35B27, 74K30, 35K20, 35B40, 35C20.