## 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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