Abstract

In this Talk we will discuss the system of Einstein equation coupled with a positive mass scalar field. The problem to be solved is the stability of the Minkowski vacuum or, in another word, the global-in-time existence of the regular solution associated to a small, regular initial data. Contrary to the existent result which concentrated mainly on the vacuum case or coupling with mass-zero filed, our case leads to a system of coupled wave-Klein-Gordon system. For this, we apply the hyperboloidal foliation method that we have recently developed specifically on this kind of system. But due to the complexity of the problem, we also developed new techniques including the $L^{\infty – L}\infty$ estimates of Klein-Gordon equation in curved space-time, the $L^{\infty – L}\infty$ estimate on wave equations based on spherical-mean expression. We also re-state the existent techniques in the hyperboloidal foliation framework such as the integration along characteristics, the decomposition of quasi-null terms and the estimates based on wave coordinate conditions.