DOI: 10.1007/s00208-004-0565-7 Math. Ann. (2004) Mathematische Annalen Global regular and singular solutions for a model of gravitating particles Piotr Biler · Marco Cannone · Ignacio A. Guerra · Grzegorz Karch Received: 8 November 2002 / Published online: 13 July 2004 – © Springer-Verlag 2004 Abstract. The existence of solutions of a nonlinear parabolic equation describing the gravita- tional interaction of particles is studied for the initial data in spaces of (generalized) pseudo- measures. This approach permits us to relax regularity assumptions on the initial conditions and to prove asymptotic stability results for the above problem. Mathematics Subject Classification (2000): 35B40, 35K15, 82C21 1. Introduction We are concerned in this paper with a construction of solutions of the Cauchy problem for the nonlinear equation u t = u +∇· (u∇ ϕ), (1.1) ∇ ϕ =∇ E d ∗ u, where the drift coefficient ∇ ϕ is determined by u in a nonlocal way, E d (z) = -((d - 2)σ d ) -1 |z| 2-d being the fundamental solution of the Laplacian in R d , σ d = 2π d/2 /Ŵ(d/2) — the area of the unit sphere in R d , d ≥ 3. The initial condition u(0)(x) = u(x, 0) (1.2) supplements the above parabolic-elliptic system considered for x ∈ R d and t> 0. Physical interpretations of (1.1) are connected with astrophysical models of gravitational self-interaction of massive particles in a cloud or a nebula. Indeed, P. Biler, G. Karch Instytut Matematyczny, UniwersytetWroclawski, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland and Institute of Mathematics, Polish Academy of Sciences, Warsaw (2002–2003) (e-mail: {Piotr.Biler, Grzegorz.Karch}@math.uni.wroc.pl) M. Cannone Universit´ e de Marne-la-Vall´ ee, Laboratoire d’Analyse et de Math´ ematiques Appliqu´ ees, Cit´ e Descartes, 77454, Marne-la-Vall´ ee Cedex 2, France (e-mail: cannone@math.univ-mlv.fr) I. A. Guerra Centre for Mathematical Modeling, Universidad de Chile, FCFM, Casilla 170 Correo 3, Santiago, Chile (e-mail: iguerra@dim.uchile.cl)