The non-local parabolic equation v(t) = Delta v + lambda e(v)/ integral(Omega)e(v) in Omega x (0, T) associated with Dirichlet boundary and initial conditions is considered here. This equation is a simplified version of the full chemotaxis system. Let lambda* be such that the corresponding steady-state problem has no solutions for lambda > lambda*, then it is expected that blow-up should occur in this case. In fact, for lambda > lambda* and any bounded domain Omega subset of R-2 it is proven, using Trudinger-Moser's inequality, that integral(Omega)e(v(x,t)) dx -> infinity oo as t -> T-max <= infinity. Moreover, in this case, some properties of the blow-up set are provided. For the two-dimensional radially symmetric problem, i.e. when Omega = B(0, 1), where it is known that lambda* = 8 pi, we prove that v blows up in finite time T* < co for lambda > 8 pi and this blow-up occurs only at the origin r = 0 (single-point blow-up, mass concentration at the origin).