Wave equations with concentrated nonlinearities

In this paper, we address the problem of wave dynamics in the presence of concentrated nonlinearities. Given a vector field V on an open subset of Cn and a discrete set Y is part of R3 with n elements, we define a nonlinear operator ΔV,Y on L2(R3) which coincides with the free Laplacian when restricted to regular functions vanishing at Y, and which reduces to the usual Laplacian with point interactions placed at Y when V is linear and represented by a Hermitian matrix. We then consider the nonlinear wave equation Φ-doubledot = ΔV,YΦ and study the corresponding Cauchy problem, giving an existence and uniqueness result when V is Lipschitz. The solution of such a problem is explicitly expressed in terms of the solutions of two Cauchy problems: one relative to a free wave equation and the other relative to an inhomogeneous ordinary differential equation with delay and principal part ζ-dot + V(ζ). The main properties of the solution are given and, when Y is a singleton, the mechanism and details of blow-up are studied