A semilinear parabolic-elliptic chemotaxis system with critical mass in any space dimension

TL;DR

This paper investigates a chemotaxis model in higher dimensions, revealing a critical mass phenomenon where solutions either stabilize or blow up, contrasting with the well-known two-dimensional case.

Contribution

It extends the understanding of the critical mass phenomenon in chemotaxis systems to higher dimensions with nonlinear sensitivity, identifying new behaviors and solution types.

Findings

Critical mass phenomenon occurs for all dimensions N ≥ 2.

Solutions with mass ≤ critical mass converge to steady states.

Solutions with mass > critical mass blow up in finite time.

Abstract

We study radial solutions in a ball of $\mathbb{R}^N$ of a semilinear, parabolic-elliptic Patlak-Keller-Segel system with a nonlinear sensitivity involving a critical power. For $N = 2$, the latter reduces to the classical linear model, well-known for its critical mass $8\pi$. We show that a critical mass phenomenon also occurs for $N \geq 3$, but with a strongly different qualitative behaviour. More precisely, if the total mass of cells is smaller or equal to the critical mass M, then the cell density converges to a regular steady state with support strictly inside the ball as time goes to infinity. In the case of the critical mass, this result is nontrivial since there exists a continuum of stationary solutions and is moreover in sharp contrast with the case $N = 2$ where infinite time blow-up occurs. If the total mass of cells is larger than M, then all solutions blow up in finite time. This actually follows from the existence (unlike for $N = 2$) of a family of self-similar, blowing up solutions with support strictly inside the ball.