On blowup dynamics in the Keller-Segel model of chemotaxis

TL;DR

This paper studies the blowup behavior in the Keller-Segel model of chemotaxis, providing formal derivations, partial rigorous results, and numerical confirmation of aggregation dynamics.

Contribution

It offers a formal derivation and partial rigorous analysis of blowup dynamics in the Keller-Segel equations, aligning with known solutions.

Findings

Numerical simulations confirm the theoretical blowup behavior.

Derived formula matches known solutions for special cases.

Partial rigorous results support the formal derivation.

Abstract

We investigate the (reduced) Keller-Segel equations modeling chemotaxis of bio-organisms. We present a formal derivation and partial rigorous results of the blowup dynamics of solution of these equations describing the chemotactic aggregation of the organisms. Our results are confirmed by numerical simulations and the formula we derive coincides with the formula of Herrero and Vel\'{a}zquez for specially constructed solutions.