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Oscillations semi-linéaires multiphasées compatibles en dimension 2 ou 3 d’espace. (French) Zbl 0736.35001
The author studies the solutions for \(\varepsilon\to0\) of the problem: \[ Pu_ \varepsilon+Q(x,u_ \varepsilon,du_ \varepsilon)=\sum_{k\in K}f_ k^ \varepsilon(x,\varphi_ k(x)/\varepsilon), \qquad u_{\varepsilon| t<0}=0, \] where \(P\) is a strictly hyperbolic second order operator, in space dimension 2 or 3, \(Q\) is a suitable quadratic form, and \(\varphi_ k(x)\) are phase functions of \(P\), for \(k\) in a finite set of indices \(K\). The author proves a result of global existence and gives a precise asymptotic expansion of \(u_ \varepsilon\) for \(\varepsilon\to 0\). Similar results in the one dimensional case were proved by J. L. Joly and J. Rauch [Journ. Equations Dériv. Partielles, St. Jean-De-Monts 1986, Conf. No. 11 (1986; Zbl 0614.35002)].
Reviewer: L.Rodino (Torino)

35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
35G25 Initial value problems for nonlinear higher-order PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
Zbl 0614.35002
Full Text: DOI
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