Forum Math. 3 (1991), 205-218 Forum Mathematicum © de Gruyter 1991 Hyperbolic Perturbation Problems Involving Time Derivatives on the Boundary* Pierluigi Colli and Jose-Francisco Rodrigues (Communicated by Pierre-Arnaud Raviart) Abstract. This paper is devoted to the analysis of the asymptotic behaviour of an hyperbolic transmission problem in a fixed domain surrounded by a layer whose thickness goes to zero and where the coefficients ρ and σ of the second and first time derivative, respectively, go to infinity. We characterize all the possible limit cases according to the limits α and β of the products δσ and δρ, respectively, and, when α = β = -f oo, also according to the limit γ of the quotient σ/ρ. In particular, if α = β = 0 we have a Neumann boundary condition, if 0 < α -f + β < -f oo we have a mixed boundary condition involving the time derivatives of the trace of the solution, otherwise we obtain Dirichlet boundary conditions depending on the initial data and on each value of 7, 0 < γ < + oo. 1980 Mathematics Subject Classification (1985 Revision): 35B25, 35B40, 35L20. 1. Introduction Perturbation problems related to thin layers arise naturally in mathematical-physics theories. They lead to interesting problems of asymptotic analysis in boundary value problems (see [L], [S], for instance), where the rate of the Singular parameters has a special role. Recently, the authors have considered in [CR] a diffusion-transmission problem with thin layers, where the limit problem exhibits the time derivative on the boundary in contrast to the cases where the perturbation acts only on the elliptic operator s in [HS] or [S]. In this work we extend the results of [CR] to the following transmission problem for the wave equation. Let Ω 0 be a bounded domain of IR n (n > 1) with smooth * This work has been done in the exchange framework between the Istituto di Analisi Numerica of C.N.R. in Pavia/Italy and the Centro de Matem tica e Aplicas es Fundamentais of I.N.I.C. in Lisboa/Portugal. Brought to you by | University of Arizona Authenticated Download Date | 5/27/15 5:13 AM