Refined Blow-up Behavior for Reaction-Diffusion Equations with Non Scale Invariant Exponential Nonlinearities

We consider positive radial decreasing blow-up solutions of the semilinear heat equation u(t)-Delta u=f(u):=e(u )L(e(u)), x is an element of Omega ,t>0, where Omega =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 )>= 0, we have u(x, t)=G(-1 )(T-t+1/8 |x|(2)/|log |x||) +o(1) as (x, t)->(0,T), where } G(X)=integral(infinity )(x)ds/f(s). 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)).