Profile decomposition for the wave equation outside a convex obstacle

We study the nonlinear wave equation: (1) squareu + \u\(4) u = 0 in R-t x Ohm. with Dirichlet boundary conditions, where Ohm is the exterior of a strictly convex domain of R-3. We first prove a structure theorem for bounded energ y sequences of solutions to the linear wave equation in Ohm, following a me thod introduced by H. Bahouri and the second author. The proof requires a n on-concentration theorem for such sequences, the proof of which involves se mi-classical measures. We then infer, using Strichartz estimates proved by H. Smith and C. Sogge, the description of bounded energy sequences of solut ions of (1), up to remainder terms, small both in energy and in Strichartz norms. As corollaries, we obtain an a priori bound for the Strichartz norms in terms of the energy, as well as a Lipschitz estimate for the nonlinear evolution group. (C) 2001 Editions scientifiques et medicales Elsevier SAS. AMS classification: 35L40, 35L25.