Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2

TL;DR

This paper proves that radially symmetric solutions to a 2D Keller-Segel system with nonlinear diffusion can blow up in finite time for any positive mass, highlighting the importance of the non-decay condition for solution behavior.

Contribution

It establishes finite-time blowup results for a supercritical Keller-Segel system in 2D with nonlinear diffusion and cross-diffusion, emphasizing the role of the non-decay assumption.

Findings

Finite-time blowup occurs for any positive mass in the system.

The non-decay assumption is crucial for finite-time blowup to occur.

Without non-decay, solutions may only exhibit infinite-time blowup or boundedness.

Abstract

In this paper we prove finite-time blowup of radially symmetric solutions to the quasilinear parabolic-parabolic two-dimensional Keller-Segel system for any positive mass. This is done in case of nonlinear diffusion and also in the case of nonlinear cross-diffusion provided the nonlinear chemosensitivity term is assumed not to decay. Moreover, it is shown that the above-mentioned lack of non-decay assumption is essential with respect to keeping the dichotomy finite-time blowup against boundedness of solutions. Namely, we prove that without the non-decay assumption possible asymptotic behaviour of solutions includes also infinite-time blowup.