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

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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, Ωɛ=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].

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

    A. K. Nandakumaran & M. Rajesh

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  1. A. K. Nandakumaran
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  2. M. Rajesh
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Correspondence to M. Rajesh.

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Nandakumaran, A.K., Rajesh, M. Homogenization of a parabolic equation in perforated domain with Neumann boundary condition. Proc. Indian Acad. Sci. (Math. Sci.) 112, 195–207 (2002). https://doi.org/10.1007/BF02829651

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