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Estimate of blow-up and relaxation time for self-gravitating Brownian particles and bacterial populations
Pierre-Henri Chavanis and Clément Sire
par - 11 avril 2006
We determine an asymptotic expression of the blow-up time 
for self-gravitating Brownian particles or bacterial populations (chemotaxis) close to the critical point. 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 asymptotic expression of the relaxation time close but above the critical temperature and derive a large time asymptotic expansion for the density profile exactly at the critical point.