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On blowup for the supercritical quadratic wave equation

Elek Csobo, Irfan Glogić, Birgit Schörkhuber

Abstract

We study singularity formation for the focusing quadratic wave equation in the energy supercritical case, i.e., for $d \geq 7$. We find in closed form a new, non-trivial, radial, self-similar blowup solution $u^*$ which exists for all $d \geq 7$. For $d=9$, we study the stability of $u^*$ without any symmetry assumptions on the initial data and show that there is a family of perturbations which lead to blowup via $u^*$. In similarity coordinates, this family represents a co-dimension one Lipschitz manifold modulo translation symmetries. In addition, in $d=7$ and $d=9$, we prove non-radial stability of the well-known ODE blowup solution. Also, for the first time we establish persistence of regularity for the wave equation in similarity coordinates.

On blowup for the supercritical quadratic wave equation

Abstract

We study singularity formation for the focusing quadratic wave equation in the energy supercritical case, i.e., for . We find in closed form a new, non-trivial, radial, self-similar blowup solution which exists for all . For , we study the stability of without any symmetry assumptions on the initial data and show that there is a family of perturbations which lead to blowup via . In similarity coordinates, this family represents a co-dimension one Lipschitz manifold modulo translation symmetries. In addition, in and , we prove non-radial stability of the well-known ODE blowup solution. Also, for the first time we establish persistence of regularity for the wave equation in similarity coordinates.

Paper Structure

This paper contains 28 sections, 31 theorems, 448 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Conditional stability of blowup via $u^*$
  5. Stable ODE blowup without symmetry
  6. Related results
  7. Notation
  8. The stability problem in similarity coordinates
  9. Stability of $\mathbf U_a$
  10. Time-evolution for small perturbations - Modulation ansatz
  11. The free wave evolution in similarity variables
  12. Linearization around a self-similar solution - Preliminaries on the structure of the spectrum
  13. Spectral analysis for perturbations around $\mathbf U_a$ - The case $d=9$
  14. Analysis of the spectral ODE
  15. Proof of Proposition \ref{['prop:SpectralODE']} for $\ell\in \{0,1\}$
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $d=9$. Define functions $h_j: \mathbb{R}^9 \to \mathbb{R}$, $j=1,2$ by There exist constants $M>0$, $\delta>0$, and $\omega>0$, such that for all real valued $(f,g)\in C^\infty(\overline{\mathbb B^{9}_2})\times C^\infty(\overline{\mathbb B^{9}_2})$ satisfying the following holds: There are parameters $a\in \overline{\mathbb B^{9}_{M\delta/\omega}}$, $x_0\in \overline{\mathbb B^{9}_\delta

Theorems & Definitions (68)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.2
  • Remark 1.5
  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.1
  • ...and 58 more