Boundedness of solutions to the critical fully parabolic quasilinear one-dimensional Keller-Segel system

TL;DR

This paper proves that solutions to a one-dimensional critical Keller-Segel system remain uniformly bounded over time, demonstrating the absence of a critical mass phenomenon in this setting.

Contribution

It establishes the uniform boundedness of solutions for the one-dimensional critical Keller-Segel system using Lyapunov functionals, contrasting with higher-dimensional cases.

Findings

Solutions are uniformly bounded in time

No critical mass phenomenon occurs in 1D with critical diffusion

Utilizes Lyapunov functionals for analysis

Abstract

In this paper we consider a one-dimensional fully parabolic quasilinear Keller-Segel system with critical nonlinear diffusion. We show uniform-in-time boundedness of solutions, which means, that unlike in higher dimensions, there is no critical mass phenomenon in the case of critical diffusion. To this end we utilize estimates from a well-known Lyapunov functional and a recently introduced new Lyapunov-like functional in [3].