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Homogenization of a parabolic equation in perforated domain with Neumann boundary condition. (English) Zbl 1199.35016

Summary: In this article, we study the homogenization of the family of parabolic equations over periodically perforated domains \[ \partial_tb(\tfrac{x}{\varepsilon}, u_\varepsilon)-\text{div}\,a(\tfrac{x}{\varepsilon}, u_\varepsilon,\nabla u_\varepsilon)=f(x,t)\quad\text{in}\;\Omega_\varepsilon\times(0,T), \]
\[ a(\tfrac{x}{\varepsilon}, u_\varepsilon,\nabla u_\varepsilon)\cdot \nu_\varepsilon=0\quad\text{on}\;\partial S_\varepsilon\times(0,T), \]
\[ u_\varepsilon=0 \quad\text{on}\;\partial\Omega_\varepsilon\times(0,T), \]
\[ u_\varepsilon(x,0)=u_0(x)\quad\text{in}\;\Omega_\varepsilon. \]
Here, \(\Omega_\varepsilon = \Omega\setminus S_\varepsilon\) is a periodically perforated domain. We obtain the homogenized equation and corrector results. The homogenization of the equations on a fixed domain was studied by the authors [Electron. J. Differ. Equ. 2001, Paper No. 17, 19 p., electronic only (2001; Zbl 1052.35023)]. The homogenization for a fixed domain and \(b(\tfrac{x}{\varepsilon },u_\varepsilon ) \equiv b(u_\varepsilon )\) has been done by H. Jian [Acta Math. Appl. Sin., Engl. Ser. 16, No. 1, 100–110 (2000; Zbl 0957.35076)].

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35K20 Initial-boundary value problems for second-order parabolic equations
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