Vol. 52 (2003) REPORTS ON MATHEMATICAL PHYSICS No. 2 GLOBAL AND EXPLODING SOLUTIONS IN A MODEL OF SELF-GRAVITATING SYSTEMS PIOTR BILER Instytut Matematyki, Uniwersytet Wroclawski, pl. Grunwaldzki 2/4, 5&384 Wroclaw, Poland (e-mail: Piotr.Biler@math.uni.wroc.pl) and TADEUSZ NADZIEJA Instytut Matematyki, Uniwersytet Zielonogbrski, ul. Podg6rna 50, 65-246 Zielona G&a, Poland (e-mail: T.Nadzieja@im.uz.zgora.pl) (Received November 25, 2002) We study asymptotic properties of solutions to an extension to arbitrary dimensions of the astrophysical model proposed by Chavanis et al. to explain phenomena of gravi- tational collapse in clouds of self-gravitating particles. In particular, we show that in the two-dimensional case the solutions can be continued to global ones, while in three space dimensions large data of negative energy blow up in a finite time. Relations between isother- mal, Streater’s energy-transport and the present models are also studied. Keywords: Chavanis-Sommeria-Robert model, mean field equations, drift-diffusion and en- ergy-transport systems, finite time blow-up of solutions, asymptotics of solutions. 2000 Mathematics Subject Classification: 35Q, 35K60, 35B40, 82C21. 1. Introduction and equations Our aim in this paper is to give some properties of solutions of a system proposed by Chavanis, Sommeria and Robert in [17], and studied recently in [16, 18, 25, 261. The model consists of a nonlinear parabolic equation of drift-diffusion type for the density u = u(x, t) of a cloud of particles, coupled with a Poisson equation for the gravitational potential generated by themselves, and supplemented with the energy relation which determines the temperature 0 = z?(t) uniform in the domain Q c R”, see Eqs. (l)-(3) below. These equations have been derived from a maximum entropy production princi- ple in the context of statistical mechanics of stellar systems, see [17]. The resulting Eq. (1) can be viewed as a generalization of the classical Smoluchowski equation introduced in the seminal paper [27], describing self-gravitating particles that are sub- 12051