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

We study the 2D coupled wave–Klein-Gordon systems with semilinear null nonlinearities Q0 and Qαβ. The main result states that solution to system exists globally provided initial data is small in some weighted Sobolev space, which does not necessarily have compact support, we also asymptotic behavior of solution. In particular, show optimal time decay scattering major difficulties lie slow nature wave Klein-Gordon components two space dimensions, addition, extra arise due presence form divergence compatible equations. To overcome difficulties, a new observation for structure required.