Boundary layers and glancing blow-up in nonlinear geometric optics

We construct rigorous geometric optics expansions of high order for semilin ear hyperbolic boundary problems with oscillatory data. The errors approach zero in L-infinity as the wavelength epsilon --> 0. To achieve such errors it is necessary to incorporate profiles of the glancing, elliptic, and hyp erbolic boundary layers into the expansions. The analysis of the glancing b oundary layer forces the introduction of a third scale 1/root epsilon in ad dition to the usual oscillatory (1/epsilon) and spatial (1) scales. The evo lution of the leading part of the glancing profile is governed by a semilin ear Schrodinger-type equation with nonhomogeneous boundary conditions. The description of the elliptic boundary layer involves complex phases and comp lex transport equations. We also construct examples showing that when glancing modes of order at lea st 3 are present, the maximal time of existence T-epsilon of the exact solu tion u(epsilon) can approach 0 as epsilon --> 0. The blow-up mechanism is d ifferent from the types of focusing known to occur in free space. (C) 2000 Editions scientifiques et medicales Elsevier SAS.