A critical blow-up exponent in a chemotaxis system with nonlinear signal production

This paper is concerned with radially symmetric solutions of the Keller-Segel system with nonlinear signal production, as given by {u(t) = Delta(u) - del . (u del v), 0 = Delta v - mu(t) + f (u), mu(t) := 1/vertical bar Omega vertical bar integral(Omega)integral(u(., t)), in the ball Omega = B-R(0) subset of R-n for n >= 1 and R > 0, where f is a suitably regular function generalizing the prototype determined by the choice f (u) = u(kappa), u >= 0, with kappa > 0. The main results assert that if in this setting the number kappa satisfies kappa > 2/n, (star) then for any prescribed mass level m > 0, there exist initial data u0 with integral(Omega) u(0) = m, for which the solution of the corresponding Neumann initial-boundary value problem blows up in finite time. The condition in (star) is essentially optimal and is indicated by a complementary result according to which in the case kappa < 2/n, for widely arbitrary initial data, a global bounded classical solution can always be found.