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Generically sharp decay and blowing up at infinity for a weak null wave system

Shijie Dong, Siyuan Ma, Yue Ma, Xu Yuan

Abstract

We study a system of semilinear wave equations satisfying the weak null condition, which can be regarded as a simplified model for the Einstein vacuum equations. The main objective is to establish precise pointwise decay estimates, as both lower and upper bounds of decay, for small data solutions. Specifically, we show that the difference between the solution and its leading-order term is dominated by lower-order terms that decay faster in the retarded time variable $u=t-r$. Moreover, we prove that these pointwise decay estimates are sharp for a generic class of small initial data decaying sufficiently fast. As applications of these estimates, we demonstrate that the energy of one component of the solution admits a lower bound that generically grows to infinity as $t\to +\infty$, which can be interpreted as ``blowing up at infinity." Furthermore, we verify that this component generically exhibits an energy cascade from high to low frequencies.

Generically sharp decay and blowing up at infinity for a weak null wave system

Abstract

We study a system of semilinear wave equations satisfying the weak null condition, which can be regarded as a simplified model for the Einstein vacuum equations. The main objective is to establish precise pointwise decay estimates, as both lower and upper bounds of decay, for small data solutions. Specifically, we show that the difference between the solution and its leading-order term is dominated by lower-order terms that decay faster in the retarded time variable . Moreover, we prove that these pointwise decay estimates are sharp for a generic class of small initial data decaying sufficiently fast. As applications of these estimates, we demonstrate that the energy of one component of the solution admits a lower bound that generically grows to infinity as , which can be interpreted as ``blowing up at infinity." Furthermore, we verify that this component generically exhibits an energy cascade from high to low frequencies.
Paper Structure (25 sections, 22 theorems, 265 equations)

This paper contains 25 sections, 22 theorems, 265 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Application to generic energy cascade and blow-up at infinity
  4. Relation to the Einstein equations in the wave gauge
  5. Relevant literature
  6. Major difficulties and key ideas
  7. Organization
  8. Preliminaries and almost sharp decay estimates
  9. Notation and conventions
  10. Elementary analytic estimates
  11. Global existence and almost sharp decay
  12. Precise decay of the solution
  13. Alternative forms of the wave system
  14. Estimates in Region $\mathrm{II}$
  15. Precise decay of $\phi$
  16. ...and 10 more sections

Key Result

Theorem 1.2

Let $N\in \mathbb{N}^{+}$ with $N\ge 6$. There exists an $\epsilon_{0}>0$ such that for all $\epsilon\in (0,\epsilon_{0})$ and all initial data $(\phi_{0},\phi_1, \psi_{0}, \psi_1)$ satisfying the following smallness condition the Cauchy problem equ:main--eq:ID admits a global-in-time solution $(\phi,\psi)$. Moreover, the following estimates of asymptotic behavior for $(\phi,\psi)$ are valid for

Theorems & Definitions (48)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5: Generic energy cascade and blow-up at infinity
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Lemma 2.1: Klainerman-Sobolev inequality
  • Lemma 2.2: Estimate for the null form $Q_0$
  • ...and 38 more