Exploding solutions for a nonlocal quadratic evolution problem

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].