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 $\Omega_\varepsilon$ which is the union of a domain and a large number $2N$ of thin rods with the variable thickness of order $\varepsilon=\mathcal{O}(N^{-1})$. 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 $\alpha\ge 1$ and $\beta\ge 1$, and the thin rods from each level are $\varepsilon$-periodically alternated. The asymptotic analysis of this problem for different values of $\alpha$ and $\beta$ is made as $\varepsilon\to 0$. The leading terms of the asymptotic expansion for the solution are constructed, the asymptotic estimate in the Sobolev space $L^2(0,T;H^1(\Omega_\varepsilon))$ is obtained and the convergence theorem is proved with minimal conditions for the right-hand sides.

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