Modulation theory for the flat blow-up solutions of nonlinear heat equation

Abstract

In this paper, we revisit the proof of the existence of a solution to the semilinear heat equation in one space dimension with a flat blow-up profile, already proved by Bricmont and Kupainen together with Herrero and Velázquez. Though our approach relies on the well-celebrated method, based on the reduction of the problem to a finite-dimensional one, then the use of a topological 'shooting method' to solve the latter, the novelty of our approach lays in the use of a modulation technique to control the projection of the zero eigenmode arising in the problem. Up to our knowledge, this is the first time where modulation is used with this kind of profiles. We do hope that this simplifies the argument.