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A note on the stability of self-similar blow-up solutions for superconformal semilinear wave equations

Jie Liu

TL;DR

This work addresses the stability of the full family of self-similar blow-up solutions for the semilinear wave equation in the superconformal regime by exploiting a Lorentz transformation in self-similar variables. The authors introduce a spectral framework and prove a spectral equivalence between linearized operators around any Lorentz-transformed profile $\kappa_d$ and the ODE blow-up profile $\kappa_0$, thereby reducing mode stability to the known $\kappa_0$ case. Building on this, they establish linear stability and, via a nonlinear fixed-point/perturbative argument with symmetry-projected corrections, prove asymptotic stability of the entire $(T, x_0, d)$-family of self-similar blow-up solutions in a backward light cone, extending Merle–Zaag’s results to the superconformal setting and broadening Ostermann’s recent work to the full ODE blow-up family. The results provide a robust spectral-and-nonlinear stability paradigm for non-radial, Lorentz-boosted self-similar blow-up in higher dimensions, with potential applicability to related problems where symmetry-induced instabilities arise. Key technical ingredients include the Lorentz-invariant reformulation in similarity variables, discrete-spectrum equivalence, resolvent and semigroup bounds for the linearized flow, and nonlinear stability via projection onto the stable subspace and controlled evolution of the unstable directions.

Abstract

In this note, we investigate the stability of self-similar blow-up solutions for superconformal semilinear wave equations in all dimensions. A central aspect of our analysis is the spectral equivalence of the linearized operators under Lorentz transformations in self-similar variables. This observation serves as a useful tool in proving mode stability and provides insights that may aid the study of self-similar solutions in related problems. As a direct consequence, we establish the asymptotic stability of the ODE blow-up family, extending the classical results of Merle and Zaag [Merle-Zaag, 2007, 2016] to the superconformal case and generalizing the recent findings of Ostermann [Ostermann, 2024] to include the entire ODE blow-up family.

A note on the stability of self-similar blow-up solutions for superconformal semilinear wave equations

TL;DR

This work addresses the stability of the full family of self-similar blow-up solutions for the semilinear wave equation in the superconformal regime by exploiting a Lorentz transformation in self-similar variables. The authors introduce a spectral framework and prove a spectral equivalence between linearized operators around any Lorentz-transformed profile and the ODE blow-up profile , thereby reducing mode stability to the known case. Building on this, they establish linear stability and, via a nonlinear fixed-point/perturbative argument with symmetry-projected corrections, prove asymptotic stability of the entire -family of self-similar blow-up solutions in a backward light cone, extending Merle–Zaag’s results to the superconformal setting and broadening Ostermann’s recent work to the full ODE blow-up family. The results provide a robust spectral-and-nonlinear stability paradigm for non-radial, Lorentz-boosted self-similar blow-up in higher dimensions, with potential applicability to related problems where symmetry-induced instabilities arise. Key technical ingredients include the Lorentz-invariant reformulation in similarity variables, discrete-spectrum equivalence, resolvent and semigroup bounds for the linearized flow, and nonlinear stability via projection onto the stable subspace and controlled evolution of the unstable directions.

Abstract

In this note, we investigate the stability of self-similar blow-up solutions for superconformal semilinear wave equations in all dimensions. A central aspect of our analysis is the spectral equivalence of the linearized operators under Lorentz transformations in self-similar variables. This observation serves as a useful tool in proving mode stability and provides insights that may aid the study of self-similar solutions in related problems. As a direct consequence, we establish the asymptotic stability of the ODE blow-up family, extending the classical results of Merle and Zaag [Merle-Zaag, 2007, 2016] to the superconformal case and generalizing the recent findings of Ostermann [Ostermann, 2024] to include the entire ODE blow-up family.

Paper Structure

This paper contains 18 sections, 20 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main result
  4. Idea of the proof
  5. Notation
  6. Equivalence of Discrete Spectra and Mode Stability
  7. The Lorentz transformation in self-similar variables
  8. Unstable eigenvalues induced by symmetries
  9. Mode stability
  10. Spectral equivalence and linear stability
  11. Functional setup
  12. Spectral analysis of the linearized operator
  13. Uniform growth bounds
  14. Proof of \ref{['MainTHM']} and \ref{['thm-trapping']}
  15. Nonlinear stability
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $d_0\in \mathbb{B}^N$. Let $(N,p,k,R)\in\mathbb{N}\times\mathbb{R}_{>1}\times\mathbb{N}\times\mathbb{R}_{\ge1}$ with $k>\frac{N}{2}$ and $R<|d_0|^{-1}$. Put For all $0 < \varepsilon < \omega_{p}$ there are constants $0<\delta_{\varepsilon}<1-|d_0|$ and $C_{\varepsilon} > 1$ such that for all $0<\delta\leq\delta_{\varepsilon}$, $C\geq C_{\varepsilon}$ and all real-valued $(f,g)\in C^{\infty}(\

Theorems & Definitions (46)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3: Behavior of solutions of equation \ref{['nlw-ss-s2']} near $\kappa(d, y)$
  • Remark 1.4
  • Lemma 2.1: The Lorentz transform in self-similar variables
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4: ostermann2024stable
  • Definition 2.5: Mode stability
  • Proposition 2.6: Transformation of the linearized equation
  • ...and 36 more