Abstract

We report on recent progress on the description of blow-up asymptotics for the semilinear heat equation , where is a genuinely non scale invariant nonlinearity, of the form , where and is a slowly varying function (which includes for instance logarithms and their powers and iterates, as well as some strongly oscillating functions). Also the power function can in some cases be replaced by the exponential.

 

We obtain three types of results (under suitable assumptions):

(a) Type I blow-up estimates (in connection with point (b))

(b) Liouville type theorems for entire solutions

(c) Sharp global blow-up profile in the scale of the original variables (which in particular implies the sharp final and refined space-time profiles).

 

As a remarkable fact, the result (c) reveals a structural universality of the global blow-up profile, being given by the ``resolvent'' of the ODE, composed with a universal, time-space building block, which is the same as in the pure power case. (Joint works with L. Chabi or P. Quittner) 

 

Organizer: Van Tien Nguyen (NTU)