Refined behavior and structural universality of the blow-up profile for the semilinear heat equation with non scale invariant nonlinearity

We consider the semilinear heat equation \(\begin{aligned} u_t-\Delta u=f(u) \end{aligned}\) $$\begin{array}{c}\hfill u_t-\mathrm{\Delta }u=f(u)\end{array}$$ for a large class of non scale invariant nonlinearities of the form \(f(u)=u^pL(u)\) $f(u)=u^{p}L(u)$ , where \(p>1\) $p>1$ 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: \(\begin{aligned} u(x,t)= & (1+o(1))\,G^{-1}\bigg (T-t+\frac{p-1}{8p}\frac{|x|^2}{|\log |x||}\bigg ), \ \hbox {as }(x,t){\rightarrow } (0,T)\hbox {,}\\ \hbox {where } G(X)= & \int _{X}^{\infty }\frac{ ds}{f(s)}. \end{aligned}\) $$\begin{array}{cc}\hfill u(x,t)=& (1+o(1))G^{-1}(T-t+\frac{p-1}{8p}\frac{{|x|}^2}{|log|x||}),\text{as}(x,t)\to (0,T)\text{,}\hfill \\ \hfill \text{where}G(X)=& {\int }_X^{\infty }\frac{\mathit{ds}}{f(s)}.\hfill \end{array}$$ 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 structural universality of the global blow-up profile, being given by the “resolvent” \(G^{-1}\) $G^{-1}$ of the ODE, composed with a universal, time-space building block, which is the same as in the pure power case.