Two dimensional wave-Klein-Gordon equations with a below-critical nonlinearity

In this paper we investigate the small data global existence and pointwise decay of solutions to two systems of coupled wave-Klein-Gordon equations in two spatial dimensions. In particular, we consider critical (in the sense of time decay) semilinear nonlinearities for the wave equation and below-critical semilinear nonlinearities for the Klein-Gordon equation, a situation that has not been studied before in the context of coupled wave and Klein-Gordon equations. An interesting feature of our two systems is that the below-critical nonlinearity causes the Klein-Gordon field to lose its linear behaviour close to the light cone, even though it enjoys optimal time decay.