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

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 $N\ge 3$, 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.

35 pages

References in corpus (4)

• Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations
• Local and Global Well-Posedness for Aggregation Equations and Patlak-Keller-Segel Models with Degenerate Diffusion
• Large mass self-similar solutions of the parabolic-parabolic Keller--Segel model of chemotaxis
• Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion

Cited by in corpus (2)

• How strong a logistic damping can prevent blow-up for the minimal Keller-Segel chemotaxis system?
• Exponential speed of uniform convergence of the cell density toward equilibrium for subcritical mass in a Patlak-Keller-Segel model