Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system

TL;DR

This paper proves that solutions to a one-dimensional quasilinear chemotaxis system can blow up in finite time if the initial mean value exceeds a certain threshold, using a novel virial identity and Liapunov functional analysis.

Contribution

It introduces a new virial-type identity and leverages the boundedness of a Liapunov functional to establish finite time blow-up in a one-dimensional chemotaxis model.

Findings

Finite time blow-up occurs when initial mean exceeds threshold

Novel virial identity developed for analysis

Liapunov functional boundedness is crucial for proof

Abstract

Finite time blow-up is shown to occur for solutions to a one-dimensional quasilinear parabolic-parabolic chemotaxis system as soon as the mean value of the initial condition exceeds some threshold value. The proof combines a novel identity of virial type with the boundedness from below of the Liapunov functional associated to the system, the latter being peculiar to the one-dimensional setting.