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Stable Type I blow-up for the one-dimensional wave equation with time-derivative nonlinearity

Oliver Gough

TL;DR

We address finite-time blow-up for the 1D wave equation with a quadratic time-derivative nonlinearity, proving the nonexistence of nontrivial exact self-similar profiles and constructing a five-parameter family of generalized self-similar blow-up solutions that are smooth inside the past light cone. A detailed spectral/semigroup framework is developed to analyse the linearised operator, including a subcoercivity-based decomposition, resolvent estimates, and hypergeometric analysis to establish mode stability. Nonlinear stability is achieved via a Lyapunov–Perron scheme with modulation, yielding exponential decay of perturbations in similarity variables and convergence to a modulated generalized self-similar profile. The results provide a rigorous, explicit mechanism for stable Type I blow-up in a derivative-nonlinear wave equation, extending the 1D theory beyond ODE blow-up profiles.

Abstract

We study finite-time blow-up for the one-dimensional nonlinear wave equation with a quadratic time-derivative nonlinearity, \[ u_{tt}-u_{xx}=(u_t)^2,\qquad (x,t)\in\mathbb R\times[0,T). \] Building on the work of Ghoul, Liu, and Masmoudi \cite{ghoul2025blow} on the spatial-derivative analogue, we establish the non-existence of smooth, exact self-similar blow-up profiles. Instead we construct an explicit family of \emph{generalised self-similar} solutions, bifurcating from the ODE blow-up, that are smooth within the past light cone and exhibit type-I blow-up at a prescribed point \((x_0,T)\). We further prove asymptotic stability of these profiles under small perturbations in the energy topology.

Stable Type I blow-up for the one-dimensional wave equation with time-derivative nonlinearity

TL;DR

We address finite-time blow-up for the 1D wave equation with a quadratic time-derivative nonlinearity, proving the nonexistence of nontrivial exact self-similar profiles and constructing a five-parameter family of generalized self-similar blow-up solutions that are smooth inside the past light cone. A detailed spectral/semigroup framework is developed to analyse the linearised operator, including a subcoercivity-based decomposition, resolvent estimates, and hypergeometric analysis to establish mode stability. Nonlinear stability is achieved via a Lyapunov–Perron scheme with modulation, yielding exponential decay of perturbations in similarity variables and convergence to a modulated generalized self-similar profile. The results provide a rigorous, explicit mechanism for stable Type I blow-up in a derivative-nonlinear wave equation, extending the 1D theory beyond ODE blow-up profiles.

Abstract

We study finite-time blow-up for the one-dimensional nonlinear wave equation with a quadratic time-derivative nonlinearity, Building on the work of Ghoul, Liu, and Masmoudi \cite{ghoul2025blow} on the spatial-derivative analogue, we establish the non-existence of smooth, exact self-similar blow-up profiles. Instead we construct an explicit family of \emph{generalised self-similar} solutions, bifurcating from the ODE blow-up, that are smooth within the past light cone and exhibit type-I blow-up at a prescribed point \((x_0,T)\). We further prove asymptotic stability of these profiles under small perturbations in the energy topology.
Paper Structure (30 sections, 25 theorems, 252 equations)

This paper contains 30 sections, 25 theorems, 252 equations.

Table of Contents

  1. Introduction
  2. Related work
  3. Outline of the stability proof
  4. Notation
  5. Construction of smooth blow-up solutions
  6. Non-existence of smooth self-similar blow-up
  7. Construction of generalised self-similar solutions
  8. The case $p=1$:
  9. The case $p<0$.
  10. General solution for $p\in(0,1)$
  11. Solutions obtained from the Lorentz transformation
  12. Mode stability
  13. Symmetry–generated eigenmodes
  14. A generalised neutral mode ($\lambda=0$).
  15. Frobenius Theory and Hypergeometric functions
  16. ...and 15 more sections

Key Result

Theorem 1.1

1) For any $T>0$ and $x_0\in\mathbb{R}$, there are no nontrivial smooth exact self-similar blow-up solutions to PDE in the past light cone Equivalently, apart from the symmetry-induced constant profiles $\widetilde{U} \equiv\kappa$, no smooth profile of the form $u(x,t)=\widetilde{U} (y)$ exists in $\{|y|\le 1\}$. 2) There exists a five-parameter family of smooth generalised self-similar blow-up

Theorems & Definitions (58)

  • Theorem 1.1: Non-existence of exact smooth self-similar blow-up solutions
  • Remark 1.1: Lorentz boost viewpoint
  • Remark 1.2: Values of $p$
  • Theorem 1.2: Asymptotic stability of generalised self-similar solutions
  • Remark 1.3: Symmetry $q=\pm1$
  • Remark 1.4: Similarity normalisation
  • Remark 2.1
  • Proposition 3.1
  • proof
  • Definition 3.2
  • ...and 48 more