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

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 tt 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 −λu3−λu3 (λ>0λ>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)F(u). We also prove the global existence of a solution using the repulsiveness of FF. In the particular case F(u)=−λu3F(u)=−λu3, the problem is a relativistic model for a quantum particle with repulsive self-interaction and tunnel effect at a semi-infinite potential step.