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$L^2$-based stability of blowup with log correction for semilinear heat equation

Thomas Y. Hou, Van Tien Nguyen, Yixuan Wang

TL;DR

The paper develops an $L^2$-based stability framework for Type-I blowup with log correction in the semilinear heat equation by employing a dynamic rescaling formulation with carefully chosen normalization constraints. It introduces dimension-specific scaling in $1$D and $n$D, derives ODEs for the scaling parameters, and proves stability of perturbations in a singular weighted $L^2$ space and a higher-order energy, leading to finite-time blowup with the logarithmic correction. Crucially, the approach avoids spectral analysis and remains applicable when only numerical or implicit approximate profiles are available, enabling higher-dimensional generalizations. Numerical experiments in 1D and 2D corroborate the theoretical predictions, showing convergence to the approximate steady state and correct log-scaling in the rescaled variables.

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.

$L^2$-based stability of blowup with log correction for semilinear heat equation

TL;DR

The paper develops an -based stability framework for Type-I blowup with log correction in the semilinear heat equation by employing a dynamic rescaling formulation with carefully chosen normalization constraints. It introduces dimension-specific scaling in D and D, derives ODEs for the scaling parameters, and proves stability of perturbations in a singular weighted space and a higher-order energy, leading to finite-time blowup with the logarithmic correction. Crucially, the approach avoids spectral analysis and remains applicable when only numerical or implicit approximate profiles are available, enabling higher-dimensional generalizations. Numerical experiments in 1D and 2D corroborate the theoretical predictions, showing convergence to the approximate steady state and correct log-scaling in the rescaled variables.

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 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 -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.
Paper Structure (19 sections, 2 theorems, 114 equations, 2 figures)

This paper contains 19 sections, 2 theorems, 114 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Literature review and main contributions
  3. Notations
  4. Dynamic rescaling formulation and normalization conditions
  5. 1D case
  6. nD case
  7. Stability of perturbation and blowup
  8. $L^2$ stability analysis
  9. Higher order stability analysis
  10. Finite time blowup
  11. Numerical experiments
  12. 1D case
  13. 2D case
  14. Local well-posedness
  15. Boundedness of the heat kernel:
  16. ...and 4 more sections

Key Result

Theorem 1

Let $k=2n+10$. There exist positive constants $C_0$ and $\lambda_0$ such that the following holds. For any $\lambda\in(0,\lambda_0)$, if $g$ is evenWe call a multivariate function $a$ even if it is even in each coordinate, i.e. $a(\xi_1 x_1,\dots,\xi_n x_n)=a(x_1,\dots,x_n)$ for all $\xi_i\in\{-1,1\ blows up in finite time $T<\infty$. Moreover, as $t\to T$ we have convergence in $\mathcal{E}_k$:

Figures (2)

  • Figure 1: Left: Comparison of the profile to the approximate steady state. Right: Plot of the residue multiplied by the rescaled time.
  • Figure 2: Left: Fitting the law of the normalization constants in 1D. Right: 2D

Theorems & Definitions (9)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 1
  • proof
  • proof : Proof of the bootstrap bound \ref{['bootstrapbound']}