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Dispersive limits in the homogenization of the wave equation. (English) Zbl 1070.35006

The author studies a scalar wave equation (the Klein-Gordon equation) with a large potential in a periodic medium. The period is supposed of order \(\varepsilon\) and the potential is of order \(\varepsilon^{-2}\). The homogenized limit depends on the sign of the first cell eingenvalue \(\lambda_1\). If \(\lambda_1=0\), then the homogenized problem gives a standard wave equation. If \(\lambda_1\neq 0\), then, upon changing the time scale to focus on large times of order \(\varepsilon^{-1}\), the author obtains dispersive homogenized problems, i.e. equations which are not of second order in time. If \(\lambda_1 < 0\), the homogenized equation becomes parabolic, while, for \(\lambda_1 > 0\), the homogenized equation is of Schrödinger type.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35L20 Initial-boundary value problems for second-order hyperbolic equations
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References:

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