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

作者：Montaru Alexandre^*

来源：Nonlinearity, 2013, 26(9): 2669-2701.

DOI：10.1088/0951-7715/26/9/2669

We study radial solutions in a ball of RN 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 >= 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 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 (M) over bar, 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.

• 出版日期2013-9