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- 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-18p|x|²| |x||), \ as (x, t) (0, T), where G (X) =ₗ^ dsf (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.