Abstract
We determine an exact asymptotic expression of the blow-up time for self-gravitating Brownian particles or bacterial populations (chemotaxis) close to the critical point in . We show that with , where represents the inverse temperature (for Brownian particles) or the mass (for bacterial colonies), and is the critical value of above which the system blows up. This result is in perfect agreement with the numerical solution of the Smoluchowski-Poisson system. We also determine the exact asymptotic expression of the relaxation time close to but above the critical temperature and derive a large time asymptotic expansion for the density profile exactly at the critical point.
- Received 13 May 2004
DOI:https://doi.org/10.1103/PhysRevE.70.026115
©2004 American Physical Society