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<p>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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(H^k\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mi>k</mi> </msup> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-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.</p>

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

  • 期刊论文

  • 发表日期：

    2026年4月10日

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  • Thomas Y. Hou,
  • Van Tien Nguyen,
  • Yixuan Wang

摘要

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 \(H^k\) 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 \(L^2\) 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.