## 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

### Keywords:

homogenized problem; Klein-Gordon equation; periodic medium
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### References:

 [1] Allaire, G., Homogenization and two-scale convergence , SIAM J. Math. Anal.23(6), p. 1482-1518 (1992). · Zbl 0770.35005 [2] Allaire, G., Capdeboscq, Y., Homogenization of a spectral problem in neutronic multigroup diffusion, Comput. Methods Appl. Mech. Engrg.187, p. 91-117 (2000). · Zbl 1126.82346 [3] Allaire, G., Malige, F., Analyse asymptotique spectrale d’un problème de diffusion neutronique, C. R. Acad. Sci. Paris Série I 324, p. 939-944 (1997). · Zbl 0879.35153 [4] Allaire, G., Piatnitski, A., Uniform Spectral Asymptotics for Singularly Perturbed Locally Periodic Operators, Com. in PDE27, p. 705-725 (2002). · Zbl 1026.35012 [5] Bensoussan, A., Lions, J.-L., Papanicolaou, G., Asymptotic analysis for periodic structures, North-Holland, Amsterdam, 1978. · Zbl 0404.35001 [6] Brahim-Otsmane, S., Francfort, G., Murat, F., Correctors for the homogenization of the wave and heat equations, J. Math. Pures Appl. (9) 71, p. 197-231 (1992). · Zbl 0837.35016 [7] Castro, C. , Zuazua, E., Low frequency asymptotic analysis of a string with rapidly oscillating density, SIAM J. Appl. Math.60(4), p. 1205-1233 (2000 ). · Zbl 0967.34074 [8] Francfort, G., Murat, F., Oscillations and energy densities in the wave equation, Comm. Partial Differential Equations17, p. 1785-1865 (1992). · Zbl 0803.35010 [9] Gérard, P., Microlocal defect measures, Comm. Partial Diff. Equations16, p. 1761-1794 (1991 ). · Zbl 0770.35001 [10] Kozlov, S., Reducibility of quasiperiodic differential operators and averaging, Transc. Moscow Math. Soc., 2, p. 101-126 (1984). · Zbl 0566.35036 [11] Lions, J.-L. , Contrôlabilité exacte, perturbations et stabilisation des systèmes distribués , Masson, Paris ( 1988). · Zbl 0653.93002 [12] Nguetseng, G., A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal.20(3), p. 608-623 (1989). · Zbl 0688.35007 [13] Orive, R., Zuazua, E., Pazoto, A., Asymptotic expansion for damped wave equations with periodic coefficients, Math. Mod. Meth. Appl. Sci.11, p. 1285-1310 (2001 ). · Zbl 1013.35014 [14] Sanchez-Palencia, E., Non homogeneous media and vibration theory , 127, Springer Verlag (1980 ). · Zbl 0432.70002 [15] Tartar, L., H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh115A, p. 193-230 (1990). · Zbl 0774.35008 [16] Vanninathan, M., Homogenization of eigenvalue problems in perforated domains, Proc. Indian Acad. Sci. Math. Sci.90, p. 239-271 (1981). · Zbl 0486.35063
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