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Homogenization of a degenerate parabolic problem in a highly heterogeneous medium with highly anisotropic fibers. (English) Zbl 1165.80302

Summary: We consider the homogenization of a heat transfer problem in a periodic medium, consisting of a set of highly anisotropic fibers surrounded by insulating layers, the whole being embedded in a third material having a conductivity of order 1. The conductivity along the fibers is of order 1, but the conductivities in the transverse direction and in the insulating layers are very small, and related to the scales \(\mu \) and \(\lambda \) respectively. We assume that \(\mu \) (resp. \(\lambda \)) tends to zero with a rate \(\mu =\mu (\epsilon )\) (resp. \(\lambda =\lambda (\epsilon ))\), where \(\epsilon \) is the size of the basic periodicity cell. The heat capacities \(c_i\) of the \(i\)-th component are positive, but may vanish at some subsets, such that the problem can be degenerate (parabolic-elliptic). We show that the critical values of the problem are \(\gamma=\text{lim}_{\epsilon\to 0}\frac{\epsilon^2}{\mu}\) and \(\delta=\text{lim}_{\epsilon\to 0}\frac{\epsilon^2}{\lambda}\), and we identify the homogenized limit depending on whether \(\gamma \) and \(\delta \) are zero, strictly positive, finite or infinite.

MSC:

80A20 Heat and mass transfer, heat flow (MSC2010)
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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