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A note on asymptotic expansion for a periodic boundary condition. (English) Zbl 0911.35014

The author presents an asymptotic expansion of solutions to the heat equation \( \partial _t u_\varepsilon =\Delta u_\varepsilon +f_\varepsilon \) for \(\varepsilon \to 0\). The space variable belongs to a domain \(\Omega =\{x\in \mathbb{R}^3\); \((x_1,x_2)\in (0,l)^2,\;\theta (x_1,x_2)<x_3<d\}\) and periodic Dirichlet-Neuman boundary conditions with the period of order \(\varepsilon \) are imposed on the set \(\{x\in \partial \Omega\); \(x_3=\theta (x_1,x_2)\}\times (0,T)\).

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35K05 Heat equation
35B10 Periodic solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
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