Global existence and causality for a hyperbolic transmission problem with a repulsive nonlinearity

Abstract

It is well known that the solution of the classical linear wave equation with an initial condition with compact support and vanishing initial velocity also has a compact support included in a set depending on time: the support of the solution at time t is causally related to that of the initial condition. Reed and Simon have shown that for a real-valued Klein–Gordon equation with (nonlinear) right-hand side − λ u 3 ( λ > 0 ), causality still holds. We show the same property for a one-dimensional Klein–Gordon problem but with transmission and with a more general repulsive nonlinear right-hand side F ( u ) . We also prove the global existence of a solution using the repulsiveness of F . In the particular case F ( u ) = − λ u 3 , the problem is a relativistic model for a quantum particle with repulsive self-interaction and tunnel effect at a semi-infinite potential step.