Folia Mathematica Acta Universitatis Lodziensis Vol. 13, No. 1, pp. 3–19 c  2006 for University of  L´ od´ z Press MEAN FIELD MODELS FOR SELF-GRAVITATING PARTICLES PIOTR BILER ‡ , ROBERT STA ´ NCZY ‡‡,‡‡‡ Abstract. This paper describes generalizations of results (presented during SNA 2004 conference) on the behavior of solutions of particular systems de- scribing the interaction of gravitationally attracting particles that obey either Maxwell–Boltzmann or Fermi–Dirac statistics. 1. Introduction and density-pressure relations We consider parabolic-elliptic systems of the form n t = ∇· (D ∗ (∇p + n∇ϕ)), (1) ∆ϕ = n, (2) which appear in statistical mechanics as hydrodynamical (mean field) mod- els for self-interacting particles. Here n = n(x, t) ≥ 0 is the density function defined for (x, t) ∈ Ω × R + ,Ω ⊂ R d , ϕ = ϕ(x, t) is the Newtonian potential generated by the particles of density n, and the pressure p ≥ 0 is determined by the density-pressure relation with a sufficiently regular function p (3) p = p(n, ϑ). The parameter ϑ> 0 plays the role of the temperature, and D ∗ > 0 is a diffusion coefficient which may depend on n, ϑ, ϕ, x, . . . Such systems can be studied either in the canonical ensemble (i.e. the isothermal setting), ‡ University of Wroc law, Institute of Mathematics, Grunwaldzki 2/4 Sq., 50-384 Wroc law, Poland. E-mail: Piotr.Biler@math.uni.wroc.pl. ‡‡ University of  L´od´ z, Faculty of Mathematics, Banacha 22 St., 90-238  L´od´ z, Poland. E-mail: stanczr@math.uni.lodz.pl. ‡‡‡ University of Wroc law, Faculty of Mathematics, Grunwaldzki 2/4 Sq., 50-384 Wroc law, Poland. E-mail: stanczr@math.uni.wroc.pl. Key words and phrases: generalized Chavanis-Sommeria-Robert model, mean field equation, nonlinear nonlocal parabolic system, steady states, finite time blow up of solu- tions. AMS subject classifications: 35Q, 35K60, 35B40, 82C21.