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

Authors:

A K Nandakumaran M Rajesh

Institution:

(1) Department of Mathematics, Indian Institute of Science, 560 012 Bangalore, India

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, Ω_ɛ=ΩS _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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