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

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.