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

Abstract

We study radial solutions in a ball of 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 8π. We show that a critical mass phenomenon also occurs for N ⩾ 3, but with a strongly different qualitative behaviour. More precisely, if the total mass of cells is smaller or equal to the critical mass , then the cell density converges to a regular steady state that is supported 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 , then all radial 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 that are supported strictly inside the ball.