Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system

We study the Neumann initial boundary value problem for the fully parabolic Keller Segel system [GRAPHICS] , where Omega is a ball in R-n with n >= 3. It is proved that for any prescribed m > 0 there exist radially symmetric positive initial data (u(0), v(0)) is an element of C-0((Omega) over bar) x W-l,W- (infinity)(Omega) with integral(Omega) u(o) = m such that the corresponding solution blows up in finite time. Moreover, by providing an essentially explicit blow-up criterion it is shown that within the space of all radial functions, the set of such blow-up enforcing initial data indeed is laige in an appropriate sense; in particular, this set is dense with respect to the topology of L-P (Omega) x W-1,W- 2n(Omega) for any p is an element of (1, 2n/n+2). (C) 2013 Elsevier Masson SAS. All rights reserved.