Journal
NONLINEARITY
Volume 26, Issue 11, Pages 3011-3041Publisher
IOP PUBLISHING LTD
DOI: 10.1088/0951-7715/26/11/3011
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Funding
- NSF [DMS 0719895, DMS 0807131]
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A reduced Keller-Segel equation (RKSE) is a parabolic-elliptic system of partial differential equations which describes bacterial aggregation and the collapse of a self-gravitating gas of Brownian particles. We consider RKSE in two dimensions, where solution has a critical collapse (blow-up) if the total number of bacteria exceeds a critical value. We study the self-similar solutions of RKSE near the blow-up point. Near the collapse time, t = tc, the critical collapse is characterized by the L. (tc -t) 1/2 scaling law with logarithmic modification, whereLis the spatial width of the collapsing solution. We develop an asymptotic perturbation theory for these modifications and show that the resulting scaling agrees well with numerical simulations. The quantitative comparison of the theory and simulations requires several terms of the perturbation series to be taken into account.
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