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 $$u_t-\mathrm{\Delta }u=f(u)$$ for a large class of non scale invariant nonlinearities of the form $f(u)=u^{p}L(u)$, 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: $$u(x,t)=(1+o(1))G^{-1}(T-t+\frac{p-1}{8p}\frac{\mathrm{\mid }x{\mathrm{\mid }}^2}{\mathrm{\mid }\log \mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }}),\text{as }(x,t)\to (0,T)\text{, }\text{where }G(X)={\int }_X^{\mathrm{\infty }}\frac{ds}{f(s)}\mathrm{.}$$ 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.