Asymptotic behavior of 2D Wave-Klein-Gordon coupled system under null condition

We study the 2D coupled wave-Klein-Gordon systems with semi-linear null nonlinearities $Q_0$ and $Q_{\alpha\beta}$. The main result states that the solution to the 2D coupled systems exists globally provided that the initial data are small in some weighted Sobolev space, which do not necessarily have compact support, and we also show the optimal time decay of the solution. The major difficulties lie in the slow decay nature of the wave and the Klein-Gordon components in two space dimensions, in addition, extra difficulties arise due to the presence of the null form $Q_0$ which is not of divergence form and is not compatible with the Klein-Gordon equations. To overcome the difficulties, a new observation for the structure of the null form $Q_0$ is required.