\(L^2\)-Based Stability of Blowup with Log Correction for Semilinear Heat Equation

Abstract

We propose an alternative proof of the classical result of Type-I blowup with log correction for the semilinear heat equation. Compared with previous proofs, we use a novel idea of enforcing stable normalizations for perturbations around the approximate profile and we establish a weighted \(H^k\) stability, thereby avoiding the use of a topological argument and the analysis of a linearized spectrum. Consequently, this approach can be adopted even if we only have a numerical profile and do not have explicit information on the spectrum of its linearized operator. This result generalizes the \(L^2\)-based stability framework beyond exactly self-similar blowup and can be adapted to higher dimensions. Numerical results corroborate the effectiveness of our normalization, even in the large perturbation regime beyond our theoretical setting.