L^2-based stability of blowup with log correction for semilinear heat equation
arxiv(2024)
摘要
We propose an alternative proof of the classical result of type-I blowup with
log correction for the semilinear equation. Compared with previous proofs, we
use a novel idea of enforcing stable normalizations for perturbation around the
approximate profile and establish a weighted H^k stability, thereby avoiding
the use of a topological argument and the analysis of a linearized spectrum.
Therefore, 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 argument to blowups
that are not exactly self-similar 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.
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