  We study positive blowing-up solutions of the system: $$u_{t}-\delta\Delta
u=v^p,\,\,\, v_{t}-\Delta v=u^{q},$$ as well as of some more general systems.
For any $p,\,q>1$, we prove single-point blow-up for any radially decreasing,
positive and classical solution in a ball. This improves on previously known
results in 3 directions:
  (i) no type I blow-up assumption is made (and it is known that this property
may fail);
  (ii) no equidiffusivity is assumed, i.e. any $\delta>0$ is allowed;
  (iii) a large class of nonlinearities $F(u,v)$, $G(u,v)$ can be handled,
which need not follow a precise power behavior.
  As side result, we also obtain lower pointwise estimates for the final
blow-up profiles.
