Stability of a wave and Klein-Gordon system with mixed coupling

We are interested in establishing stability results for a system of semilinear wave and Klein-Gordon equations with mixed coupling nonlinearities, that is, we consider all of the possible quadratic nonlinear terms of the type of wave and Klein-Gordon interactions. The main difficulties are due to the absence of derivatives on the wave component in the nonlinearities. By doing a transformation on the wave equation, we reveal a hidden null structure. Next by using the scaling vector field on the wave component only, which was generally avoided, we are able to get very good $L^2$--type estimates on the wave component. Then we distinguish high order and low order energies of both wave and Klein-Gordon components, which allows us to close the bootstrap argument.