Logarithmic-type Scaling of the Collapse of Keller-Segel Equation

Keller‐Segel equation (KS) is a parabolic‐elliptic system of partial differential equations with applications to bacterial aggregation and collapse of self‐gravitating gas of brownian particles. KS has striking qualitative similarities with nonlinear Schrodinger equation (NLS) including critical collapse (finite time point‐wise singularity) in two dimensions. The self‐similar solutions near blow up point are studied for KS in two dimensions together with time dependence of these solutions. We found logarithmic‐type modifications to (t0−t)1/2 scaling law of self‐similar solution in qualitative analogy with log‐log modification for NLS. We found very good agreement between the direct numerical simulations of KS and the analytical results obtained by developing a perturbation theory for logarithmic‐type modifications. It suggests that log‐log modification in NLS also could be verified in a similar way.