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Logarithmic scaling of the collapse in the critical Keller-Segel equation. (English) Zbl 1278.35029

Summary: 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, \(t = t_c\), the critical collapse is characterized by the \(L \propto (t_c t)^{1/2}\) scaling law with logarithmic modification, where \(L\) is 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.

MSC:

35B44 Blow-up in context of PDEs
35A20 Analyticity in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
92C17 Cell movement (chemotaxis, etc.)
35C06 Self-similar solutions to PDEs
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