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 uₜ-Δ u=f(u) for a large class of non scale invariant nonlinearities of the form f(u)=uᵖL(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+p-1/8p|x|²/|log |x||), \ as(x,t)→ (0,T), where G(X)=∫X∞ 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 { 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.