Limits of the Solutions to the Initial-Boundary Dirichlet Problem for the Semilinear Klein-Gordon Equation with Two Small Parameters

Abstract

We study the behavior of solutions uεδ to the problem (Pεδ) in two different cases: (i) when ε → 0 and δ ≥ δ0 > 0; (ii) when ε → 0 and δ → 0. We obtain some a priori estimates of solutions to the perturbed problem, which are uniform with respect to parameters, and a relationship between solutions to both problems. We establish that the solution to the unperturbed problem has a singular behavior, relative to the parameters, in the neighbourhood of t = 0. We show the boundary layer and boundary layer function in both cases.