# Critical chemotactic collapse

## TL;DR

This paper analyzes the critical collapse phenomena in a Keller-Segel model of bacterial chemotaxis, revealing self-similar solutions and scaling laws near blow-up points, with parallels to nonlinear Schrödinger equations.

## Contribution

It provides a detailed study of self-similar solutions and scaling laws at the critical collapse in the Keller-Segel model, highlighting mathematical similarities to nonlinear Schrödinger equations.

## Key findings

- Collapse occurs when bacterial density exceeds a critical value.
- Self-similar solutions describe the blow-up behavior near collapse.
- Scaling laws involve a square root dependence with logarithmic corrections.

## Abstract

A Keller-Segel model describes macroscopic dynamics of bacterial colonies and biological cells. Bacteria secret chemical which attracts other bacteria so that they move towards chemical gradient creating nonlocal attraction between bacteria. If bacterial density exceeds a critical value then the density collapses (blows up) in a finite time which corresponds to bacterial aggregation. Collapse in the Keller-Segel model has striking qualitative similarities with a nonlinear Schrodinger equation including critical collapse in two dimensions and supercritical in three dimensions. A self-similar solution near blow up point is studied in the critical case and it has a form of a rescaled steady state solution which contains a critical number of bacteria. Time dependence of scaling of that solution has square root scaling law with logarithmic corrections.