A first order asymptotic expansion of the solution of a singularly perturbed problem for the telegraph equations

The boundary value problem (S), (IC), (BC) below is considered, where ∊ is a positive small parameter. This problem is nonlinear because f0 in (BC) is assumed to be a nonlinear function We indicate a first order expansion of the solution (see (1.1) below), including some correctora (boundary layer functions). In Section 2 of this paper we formally compute the correctors Vo, Vl and indicate the problems verified by U 1 and R (the remainder of order 1). Section 3 is devoted to the existence and higher order regularity of the solutions of the problems involved in our treatement. Finally, in Section 4, estimates of the remainder components are performed in order to validate our asymptotic expansion.