In this paper we study the nonlinear parabolic system @tu = u+a|x| |v| p 1 v, @tv = v + b|x| ⇢ |u| q 1 u, t > 0, x 2 R N \{0}, N 1, a, b 2 R, 0  < min(N, 2), 0 < ⇢ < min(N, 2), p, q > 1. Under conditions on the parameters p, q, and ⇢ we show the existence and uniqueness of global solutions for initial values small with respect of some norms. In particular, we show the existence of self-similar solutions with initial value = ('1, '2), where '1, '2 are homogeneous initial data. We also prove that some global solutions are asymptotic for large time to self-similar solutions.