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

Abstract

We study the 2D coupled wave–Klein-Gordon systems with semilinear null nonlinearities Q0 and Qαβ. The main result states that the solution to the 2D coupled system exists globally provided that the initial data is small in some weighted Sobolev space, which does not necessarily have compact support, and we also study the asymptotic behavior of the solution. In particular, we show the optimal time decay and scattering 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 Q0 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 Q0 is required.