Grow-up rate and refined asymptotics for a two-dimensional Patlak-Keller-Segel model in a disk

TL;DR

This paper analyzes the long-term behavior of solutions to a specific two-dimensional Patlak-Keller-Segel chemotaxis model in a disk, revealing a unique grow-up rate of density behaving like e^{√(2t)} and providing refined asymptotic estimates.

Contribution

It establishes the precise grow-up rate and refined asymptotics for solutions at critical mass, confirming a conjectured behavior with rigorous proofs.

Findings

Density grows like e^{√(2t)} for large time

Solutions exhibit mass concentration phenomena in infinite time

Provides rigorous asymptotic estimates matching formal predictions

Abstract

We consider a special case of the Patlak-Keller-Segel system in a disc, which arises in the modelling of chemotaxis phenomena. For a critical value of the total mass, the solutions are known to be global in time but with density becoming unbounded, leading to a phenomenon of mass-concentration in infinite time. We establish the precise grow-up rate and obtain refined asymptotic estimates of the solutions. Unlike in most of the similar, recently studied, grow-up problems, the rate is neither polynomial nor exponential. In fact, the maximum of the density behaves like $e^{\sqrt{2t}}$ for large time. In particular, our study provides a rigorous proof of a behaviour suggested by Sire and Chavanis [Phys. Rev. E, 2002] on the basis of formal arguments.