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

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It is well-known that the solution of the classical linear wave equation with compactly supported initial condition and vanishing initial velocity is also compactly supported in a set depending on time: the support of the solution at time t is causally related to that of the initially given condition. Reed and Simon have shown that for a real-valued Klein-Gordon equation with (nonlinear) right-hand side - λu³, 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. We also prove the global existence of a solution using the repulsiveness of F. In the particular case F(u) = - λu³, the problem is a physical model for a quantum particle submitted to self-interaction and to a potential step.