Unboundedness of solutions to a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 1 and to a critical one in higher dimensions

TL;DR

This paper proves that solutions to certain supercritical chemotaxis systems become unbounded over time, regardless of initial conditions, in both one and higher dimensions, using a contradiction approach with virial inequalities.

Contribution

It introduces a novel contradiction method employing virial inequalities and Liapunov functionals to establish unboundedness in supercritical Keller-Segel systems across dimensions.

Findings

Unbounded solutions occur in 1D for any initial mass.

Unboundedness also occurs in higher dimensions for large mass.

Method applies to both one-dimensional and higher-dimensional cases.

Abstract

Unboundedness of solutions is shown to occur in a one-dimensional quasilinear parabolicparabolic chemotaxis system for any initial mass. Our result is also independent of the relation between the speeds of the diffusion of cells and chemoattractant. The proof is achieved by contradiction. It uses a virial type inequality together with a boundedness from below of the Liapunov functional associated to the system. Moreover, the same method is applied also in higher dimensions and an unboundedness result is shown in the case of critical quasilinear fully parabolic Keller-Segel system for mass large enough in dimensions n\geq 3. Key words: chemotaxis, infinite-time blowup.