Global existence conditions for a non-local problem arising in statistical mechanics

We consider the evolution of the density and temperature of athree-dimensional cloud of self-interacting particles. This phenomenon ismodelled by a parabolic equation for the density distributioncombining temperature-dependent diffusion and convection drivenby the gradient of the gravitational potential. This equation iscoupled with Poisson's equation for the potential generated by thedensity distribution. The system preserves mass by imposing azero-flux boundary condition. Finally the temperature is fixed byenergy 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 thesecond law of thermodynamics. We prove local existence anduniqueness of weak solutions for the system, using a Schauderfixed-point theorem. In addition, we give sufficient conditions forglobal in time existence and blow-up for radially symmetricsolutions. We do this using a comparison principle for anequation for the accumulated radial mass.