A critical blow-up exponent in a chemotaxis system with nonlinear signal production
Abstract This paper is concerned with radially symmetric solutions of the Keller–Segel system with nonlinear signal production, as given by in the ball for and R > 0, where f is a suitably regular function generalizing the prototype determined by the choice , , with . The main results assert that if in this setting the number κ satisfies then for any prescribed mass level m > 0, there exist initial data u 0 with , for which the solution of the corresponding Neumann initial-boundary value problem blows up in finite time. The condition in ( ) is essentially optimal and is indicated by a complementary result according to which in the case , for widely arbitrary initial data, a global bounded classical solution can always be found.