In this article we revisit the global existence result of the wave-Klein-Gordon model of the system of the self-gravitating massive field. Our new observation is that, by applying the conformal energy estimates on hyperboloids, we obtain mildly increasing energy estimate up to the top order for the Klein-Gordon component, which clarify the question on the hierarchy of the energy bounds of the Klein-Gordon component in our previous work. Furthermore, a uniform-in-time energy estimate is established for the wave component up to the top order, as well as a scattering result. These improvements indicate that the partial conformal symmetry of the Einstein-massive scalar system will play an important role in the global analysis.

We are interested in the Klein–Gordon–Zakharov system in ${\mathbb{R}}^{1+2}$, which is an important model in plasma physics with extensive mathematical studies. The system can be regarded as semilinear coupled wave and Klein–Gordon equations with nonlinearities violating the null conditions. Without the compactness assumptions on the initial data, we aim to establish the existence of small global solutions, and in addition, we want to illustrate the optimal pointwise decay of the solutions. Furthermore, we show that the Klein–Gordon part of the system enjoys linear scattering, while the wave part has uniformly bounded low-order energy. None of these goals is easy because of the slow pointwise decay nature of the linear wave and Klein–Gordon components in ${\mathbb{R}}^{1+2}$. We tackle the difficulties by carefully exploiting the properties of the wave and the Klein–Gordon components, and by relying on the ghost weight energy estimates to close higher order energy estimates. This appears to be the first pointwise decay result and the first scattering result for the Klein–Gordon–Zakharov system in ${\mathbb{R}}^{1+2}$ without compactness assumptions.