Wave Equations with Concentrated Nonlinearities

In this paper we address the problem of wave dynamics in presence of concentrated nonlinearities. Given a vector field V on an open subset of C and a discrete set Y ⊂ R with n elements, we define a nonlinear operator ∆V,Y on L (R) 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 is represented by an Hermitean matrix. We then consider the nonlinear wave equation φ̈ = ∆V,Y φ and study the corresponding Cauchy problem, giving an existence and uniqueness result in the case V is Lipschitz. The solution of such a problem is explicitly expressed in terms of the solutions of two Cauchy problem: one relative to a free wave equation and the other relative to an inhomogeneous ordinary differential equation with delay and principal part ζ̇ +V (ζ). Main properties of the solution are given and, when Y is a singleton, the mechanism and details of blow-up are studied.