Abstract. In this paper, we study the asymptotic behaviour, the geodesic completeness and the energy condition for the coupled Einstein-Maxwell-Boltzmann-Massive scalar field system with pseudo-tensor of pressure in the sources which rules the dynamincs of kind of charged pure matter in the presence of a massive scalar field which is a continuity of the work we have done. We then prove that, at late times, the Hubble variable of the space time goes to strictly positive limit.

We consider in this work the Boltzmann equation in the presence of a Yang-Mills fields in temporal gauge, which generalizes to the non-Abelian case the electromagnetic field. We prove, using the method presented by N. Noutchegueme and R. D. Ayissi [2], a local in time existence and uniqueness theorem for the regular solutions.