# Homogenization of a parabolic equation in perforated domain with Neumann boundary condition

• A. K. Nandakumaran
• M. Rajesh
Article

## Abstract

In this article, we study the homogenization of the family of parabolic equations over periodically perforated domains
$$\begin{gathered} \partial _t b(\tfrac{x}{\varepsilon },u_\varepsilon ) - diva(\tfrac{x}{\varepsilon },u_\varepsilon ,\nabla u_\varepsilon ) = f(x,t) in \Omega _\varepsilon \times (0,T), \hfill \\ a(\tfrac{x}{\varepsilon },u_\varepsilon ,\nabla u_\varepsilon ) \cdot v_\varepsilon = 0 on \partial S_\varepsilon \times (0,T), \hfill \\ u_\varepsilon = 0 on \partial \Omega \times (0,T), \hfill \\ u_\varepsilon (x,0) = u_0 (x) in \Omega _\varepsilon \hfill \\ \end{gathered}$$
. Here, ΩɛS ɛ 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 [15]. The homogenization for a fixed domain and$$b(\tfrac{x}{\varepsilon },u_\varepsilon ) \equiv b(u_\varepsilon )$$ has been done by Jian [11].

## Keywords

Homogenization perforated domain two-scale convergence correctors

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