### Abstract

We consider the evolution of the density and temperature of
11 three-dimensional cloud of self-interacting particles. This phenomenon
is modeled by a parabolic equation for the density distribution combining
temperature-dependent diffusion and convection driven by the gradient
of the gravitational potential. This equation is coupled with Poisson's
equation for the potential generated by the density distribution. The
system preserves mass by imposing a zero-flux boundary condition. Finally
the temperature is fixed by energy conservation; that is, the sum of
kinetic energy (temperature) and gravitational energy remains constant
in time. This model is thermodynamically consistent, obeying the first
and the second laws of thermodynamics. We prove local existence and
uniqueness of weak solutions for the system, using a Schauder fixed-point
theorem. In addition, we give sufficient conditions for global-in-time existence
and blow-up for radially symmetric solutions. We do this using
a comparison principle for an equation for the accumulated radial mass.

Original language | English |
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Pages (from-to) | 133-158 |

Number of pages | 26 |

Journal | Advances in Differential Equations |

Volume | 9 |

Issue number | 1-2 |

Publication status | Published - 2004 |

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## Cite this

Duijn, van, C. J., Guerra Benavente, I. A., & Peletier, M. A. (2004). Global existence conditions for a nonlocal problem arising in statistical mechanics.

*Advances in Differential Equations*,*9*(1-2), 133-158.