  We consider the semilinear heat equation $$u_t-\Delta u=f(u) $$
  for a large class of non scale invariant nonlinearities of the form
  $f(u)=u^pL(u)$, where $p>1$ is Sobolev subcritical and $L$ is a slowly
varying function (which includes for instance logarithms and their powers and
iterates, as well as some strongly oscillating functions).
  For any positive radial decreasing blow-up solution, we obtain the sharp,
global blow-up profile in the scale of the original variables $(x, t)$, which
takes the form:
  $$u(x,t)=(1+o(1))\,G^{-1}\bigg(T-t+\frac{p-1}{8p}\frac{|x|^2}{|\log
|x||}\bigg),
  \ \hbox{as $(x,t)\to (0,T)$, \quad where } G(X)=\int_{X}^{\infty}\frac{
ds}{f(s)}.$$ This estimate in particular provides the sharp final space profile
and the refined space-time profile.
  As a remarkable fact and completely new observation, our results reveal a
{\it structural universality} of the global blow-up profile, being given by the
"resolvent" $G^{-1}$ of the ODE, composed with a universal, time-space building
block, which is the same as in the pure power case.
