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Global existence for a Fritz John equation in expanding FLRW spacetimes

João L. Costa, Jesús Oliver, Flavio Rossetti

Abstract

We study the family of semilinear wave equations $\square_{\mathbf{g}_p}φ=(\partial_tφ)^2$, on fixed expanding FLRW spacetimes, having $\mathbb{R}^3$ spatial slices and undergoing a power law expansion, with scale factor $a(t)=t^p$, $0< p \le 1$. This is a natural generalization to a non-stationary background of a famous Fritz John ''blow-up'' equation in $\mathbb{R}^{1+3}$ (corresponding to $p=0$, i.e. the case in which $\mathbf{g}_0$ is the Minkowski metric). While, in Minkowski spacetime ($p=0$), non-trivial solutions to this equation are known to diverge in finite time, here we prove that, on the referred FLRW backgrounds ($0<p\leq 1$), sufficiently small, smooth, and compactly supported initial data yield global-in-time solutions to the future. Previous work, co-authored by the first two authors, considered accelerated expanding spacetimes ($p>1$) and relied on the integrability of the inverse of the scale factor to establish future global well-posedness. In the current work, where such an integrability condition is lacking, we rely on a vector field method that captures and combines dispersive estimates with the spacetime expansion to control the solution and suppress the nonlinear blow-up mechanism. To achieve this, we commute the Laplace-Beltrami operator with a boosts-free subset of the Poincaré algebra and employ Klainerman-Sideris types of inequalities. Our strategy is general and is developed to handle the non-stationary nature of FLRW spacetimes. While we focus solely on this Fritz John type of equation, which serves as a prototype to study blow-up of non-linear waves, our approach provides a rigorous proof of the regularizing effects of spacetime expansion and can be exploited for a wider range of applications and nonlinearities.

Global existence for a Fritz John equation in expanding FLRW spacetimes

Abstract

We study the family of semilinear wave equations , on fixed expanding FLRW spacetimes, having spatial slices and undergoing a power law expansion, with scale factor , . This is a natural generalization to a non-stationary background of a famous Fritz John ''blow-up'' equation in (corresponding to , i.e. the case in which is the Minkowski metric). While, in Minkowski spacetime (), non-trivial solutions to this equation are known to diverge in finite time, here we prove that, on the referred FLRW backgrounds (), sufficiently small, smooth, and compactly supported initial data yield global-in-time solutions to the future. Previous work, co-authored by the first two authors, considered accelerated expanding spacetimes () and relied on the integrability of the inverse of the scale factor to establish future global well-posedness. In the current work, where such an integrability condition is lacking, we rely on a vector field method that captures and combines dispersive estimates with the spacetime expansion to control the solution and suppress the nonlinear blow-up mechanism. To achieve this, we commute the Laplace-Beltrami operator with a boosts-free subset of the Poincaré algebra and employ Klainerman-Sideris types of inequalities. Our strategy is general and is developed to handle the non-stationary nature of FLRW spacetimes. While we focus solely on this Fritz John type of equation, which serves as a prototype to study blow-up of non-linear waves, our approach provides a rigorous proof of the regularizing effects of spacetime expansion and can be exploited for a wider range of applications and nonlinearities.
Paper Structure (15 sections, 19 theorems, 155 equations, 1 table)

This paper contains 15 sections, 19 theorems, 155 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary version of the main results
  3. Main techniques of the proof
  4. Related works and outlook
  5. Outline of the paper
  6. Setup and Main Results
  7. Vector fields and notation
  8. Main Results I: Linear Estimates
  9. Main Results II: Nonlinear Estimates
  10. Energy Formalism
  11. Commutators
  12. Pointwise Decay Estimates
  13. Proof of Theorem 2.3
  14. Proof of Theorem 2.4, case 0 < p < 1
  15. Proof of Theorem 2.4, case p=1

Key Result

Theorem 1.1

Let $0 < p \le 1$. Given $\phi \colon [t_0, t_1) \times \mathbb{R}^3 \to \mathbb{R}$ sufficiently regular, then:

Theorems & Definitions (34)

  • Theorem 1.1: Improved energy estimate
  • Theorem 1.2: Global pointwise estimate
  • Theorem 1.3: Small data global existence of solutions to the Fritz John equation in non-accelerated expanding spacetimes
  • Corollary 1.4: Small data global existence of solutions to the Fritz John equation in expanding spacetimes
  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5: On regularity
  • Remark 2.6
  • ...and 24 more