Critical mass on the Keller-Segel system with signal-dependent motility

TL;DR

This paper investigates the Keller-Segel system with density-dependent motility, demonstrating a critical mass phenomenon where solutions are globally bounded below a threshold and may blow up above it, depending on initial cell mass.

Contribution

It establishes the existence of a critical mass threshold for the Keller-Segel system with exponential decay motility in a bounded domain.

Findings

Existence of a critical mass $m_*$ for global boundedness.

Solutions blow up if initial mass exceeds $m_*$.

Solutions are globally bounded if initial mass is below $m_*$.

Abstract

This paper is concerned with the global boundedness and blowup of solutions to the Keller-Segel system with density-dependent motility in a two-dimensional bounded smooth domain with Neumman boundary conditions. We show that if the motility function decays exponentially, then a critical mass phenomenon similar to the minimal Keller-Segel model will arise. That is there is a number $m_*>0$, such that the solution will globally exist with uniform-in-time bound if the initial cell mass (i.e. $L^1$-norm of the initial value of cell density) is less than $m_*$, while the solution may blow up if the initial cell mass is greater than $m_*$.