The results on the local existence and continuation criteria obtained by G. Rein in [4] are extended to the case with a non-zero cosmological constant. It is also shown that for the spherically symmetric case and a positive cosmological constant there is a large class of initial data with global existence and inflationary asymptotics in the future as in the case of plane or hyperbolic symmetry treated in [7]. Furthermore we analyze the behaviour of the energy-momentum tensor at late times.

We prove, for the relativistic Boltzmann equation on a Bianchi Type I space-time, a global existence and uniqueness theorem, for arbitrarily large initial data.

Abstract. We prove a global in time existence theorem for the initial value problem for the Einstein-Boltzmann system, with positive cosmological constant and arbitrarily large initial data, in the spatially homogeneous case, in a Robertson-Walker space-time.Key words. Global existence in time, Einstein-Boltzmann, Cosmological constant AMS subject classifications. 83xx IntroductionIn the mathematical study of General Relativity, one of the main problems is to establish the existence and to give the properties of global solutions of the Einstein equations coupled to various field equations. The knowledge of the global dynamics of the relativistic kinetic matter is based on such results. In the case of Collisionless matter, the phenomena are governed by the Einstein-Vlasov system in the pure gravitational case, and by this system coupled to other fields equations, if other fields than the gravitational field are involved. [22] for the Einstein-Vlasov system with a cosmological constant. Now in the case of Collisional matter, the Einstein-Vlasov system is replaced by the Einstein-Boltzmann system, that seems to be the best approximation available and that describes the case of instantaneous, binary and elastic collisions. In contrast with the abundance of works in the collisionless case, the literature is very poor in the collisional case. If, due to its importance in collisional kinetic theory, several authors studied and proved global results for the single Boltzmann equation, see [5], [4] , [11] for the non-relativistic case, and [7], for the full relativistic case, very few authors studied the Einstein-Boltzmann system, see [2] for a local existence theorem. It then seems interesting for us, to extend to the collisional case, some global results obtained in the collisionless case. This was certainly the objective of the author in [13] and [14], in which he studied the existence of global solutions of the Einstein-Boltzmann system. Unfortunately, several points of the work are far from clear; such as, the use of a formulation which is valid only for the non-relativistic Boltzmann equation, or, concerning the Einstein equations, to abandon the evolution equations which are really relevant, and to concentrate only on the constraint equations, which, in the homogeneous case studied, reduce as we will see to a question of choice for the initial data.In this paper, we study the collisional evolution of a kind of uncharged massive particles, under the only influence of their own gravitational field, which is a function of the position of the particles.

We prove in the cases of spherical, plane and hyperbolic symmetry a local in time existence theorem and continuation criteria for cosmological solutions of the Einstein-Vlasov-scalar field system, with the sources generated by a distribution function and a scalar field, subject to the Vlasov and wave equations respectively. This system describes the evolution of self-gravitating collisionless matter and scalar waves within the context of general relativity. In the case where the only source is a scalar field it is shown that a global existence result can be deduced from the general theorem.

We prove a global in time existence theorem, for the initial value problem for the Einstein-Boltzmann system, with arbitrarily large initial data, in the homogeneous case, in a Bianchi Type I space-time.