We consider positive radial decreasing blow-up solutions of the semilinear heat equation \(\begin{aligned} u_t-\Delta u=f(u):=e^{u}L(e^{u}),\quad x\in \Omega ,\ t>0, \end{aligned}\) where \(\Omega =\mathbb {R}^n\) or \(\Omega =B_R\) and L is a slowly varying function (which includes for instance logarithms and their powers and iterates, as well as some strongly oscillating unbounded functions). We characterize the asymptotic blow-up behavior and obtain the sharp, global blow-up profile in the scale of the original variables (x, t). Namely, assuming for instance \(u_t\ge 0\) , we have \(\begin{aligned} u(x,t)\!=\!G^{-1}\bigg (T\!-\!t\!+\!\frac{1}{8}\frac{|x|^2}{|\log |x||}\bigg )\!+\!o(1)\ \ \hbox { as } (x,t)\!\rightarrow \! (0,T),\quad \hbox { where } G(X)\!=\!\int _{X}^{\infty } \frac{ds}{f(s)}. \end{aligned}\) This estimate in particular provides the sharp final space profile and the refined space-time profile. For exponentially growing nonlinearities, such results were up to now available only in the scale invariant case \(f(u)=e^u\) . Moreover, this displays a universal structure of the global blow-up profile, given by the resolvent \(G^{-1}\) of the ODE composed with a fixed time-space building block, which is robust with respect to the factor \(L(e^u)\) .
