We consider a nonlinear parabolic equation with fractional
diffusion which arises from modelling chemotaxis in bacteria. We
prove the wellposedness, continuation criteria and smoothness of
local solutions. In the repulsive case we prove global wellposedness
in Sobolev spaces. Finally in the attractive case, we prove that for
a class of smooth initial data the $L_x^\infty$-norm of the
corresponding solution blows up in finite time. This solves a
problem left open by Biler and Woyczy\'nski [Biler, P. and Woyczy\'Nski, W.A.:
Global and exploding solutions for nonlocal quadratic evolution problems.
SIAM J. Appl. Math. {\bf 59} (1999), no. 3, 845-869].