Large mass self-similar solutions of the parabolic-parabolic Keller--Segel model of chemotaxis

TL;DR

This paper investigates self-similar solutions of the parabolic-parabolic Keller--Segel model, revealing conditions under which solutions with mass exceeding the known threshold can exist globally, unlike in the parabolic-elliptic case.

Contribution

It demonstrates the existence of large mass self-similar solutions in the parabolic-parabolic Keller--Segel system, extending understanding beyond the classical mass threshold.

Findings

Self-similar solutions can exist with mass above 8? in the parabolic-parabolic case.

Such solutions can be global even when mass exceeds the known blow-up threshold.

The results highlight differences between parabolic-parabolic and parabolic-elliptic Keller--Segel models.

Abstract

In two space dimensions, the parabolic-parabolic Keller--Segel system shares many properties with the parabolic-elliptic Keller--Segel system. In particular, solutions globally exist in both cases as long as their mass is less than 8?. However, this threshold is not as clear in the parabolic-parabolic case as it is in the parabolic-elliptic case, in which solutions with mass above 8? always blow up. Here we study forward self-similar solutions of the parabolic-parabolic Keller--Segel system and prove that, in some cases, such solutions globally exist even if their total mass is above 8?, which is forbidden in the parabolic-elliptic case.