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Critical chemotactic collapse. (English) Zbl 1236.92014

Summary: A Keller-Segel model describes macroscopic dynamics of bacterial colonies and biological cells as well as dynamics of a gas of self-gravitating Brownian particles. Bacteria secret chemicals which attract other bacteria so that they move towards a chemical gradient creating nonlocal attraction between bacteria. If bacterial (or Brownian particle) densities exceed a critical value then the densities collapse (blow up) in a finite time which corresponds to bacterial aggregation or gravitational collapse. Collapse in the Keller-Segel model has striking qualitative similarities with a nonlinear Schrödinger equation including critical collapse in two dimensions and supercritical collapse in three dimensions. A self-similar solution near a blow up point is studied in the critical two-dimensional case and has the form of a rescaled steady state solution which contains a critical number of bacteria. Time dependence of scaling of that solution has a square root scaling law with logarithmic modification.

MSC:

92C17 Cell movement (chemotaxis, etc.)
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
35Q41 Time-dependent Schrödinger equations and Dirac equations
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