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 and a discrete set with n elements, we define a nonlinear operator Δ_V,Y on 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 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 . The main properties of the solution are given and, when Y is a singleton, the mechanism and details of blow-up are studied. © 2005 IOP Publishing Ltd.