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The sharp lifespan for a system of multiple speed wave equations: Radial case

Marvin Koonce, Jason Metcalfe

TL;DR

This work proves almost global existence for a radial, multi-speed wave system with small initial data, showing the lifespan $T_\varepsilon$ can be as large as $\exp(\tilde{c}/\varepsilon^2)$. The authors adapt a restricted invariant-vector-field method (primarily the scaling vector field) and combine integrated local energy estimates, $r^p$-weighted estimates, Morawetz-type scaling, and ghost weights to obtain robust decay in space-time. A key feature is reducing the problem to a $(1+1)$-dimensional formulation and using space-time Klainerman–Sobolev inequalities to control nonlinear interactions across speeds. Hardy-type inequalities tailored to multiple speeds and careful dyadic decompositions near light cones drive the nonlinear estimates. The results advance the understanding of long-time behavior for quasilinear, multi-speed wave systems and suggest avenues toward removing radial symmetry and extending to background geometries.

Abstract

Ohta examined a system of multiple speed wave equations with small initial data and demonstrated a finite time blowup. We show, in the radial case, that the same system exists almost globally with the same lifespan as a lower bound. To do this, we use integrated local energy estimate, $r^p$ weighted local energy estimates, the Morawetz estimate that results from using the scaling vector field as a multiplier, and mixed speed ghost weights.

The sharp lifespan for a system of multiple speed wave equations: Radial case

TL;DR

This work proves almost global existence for a radial, multi-speed wave system with small initial data, showing the lifespan can be as large as . The authors adapt a restricted invariant-vector-field method (primarily the scaling vector field) and combine integrated local energy estimates, -weighted estimates, Morawetz-type scaling, and ghost weights to obtain robust decay in space-time. A key feature is reducing the problem to a -dimensional formulation and using space-time Klainerman–Sobolev inequalities to control nonlinear interactions across speeds. Hardy-type inequalities tailored to multiple speeds and careful dyadic decompositions near light cones drive the nonlinear estimates. The results advance the understanding of long-time behavior for quasilinear, multi-speed wave systems and suggest avenues toward removing radial symmetry and extending to background geometries.

Abstract

Ohta examined a system of multiple speed wave equations with small initial data and demonstrated a finite time blowup. We show, in the radial case, that the same system exists almost globally with the same lifespan as a lower bound. To do this, we use integrated local energy estimate, weighted local energy estimates, the Morawetz estimate that results from using the scaling vector field as a multiplier, and mixed speed ghost weights.

Paper Structure

This paper contains 10 sections, 10 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Notation and Decay Estimates
  3. Local Energy Estimates
  4. Proof of Theorem \ref{['main_theorem']}
  5. Boundedness
  6. Decay bounds
  7. Bound on terms $(I)_j,\dots, (V)_j$
  8. Bound on terms $(VI)_j,\dots, (VIII)_j$
  9. Bound on terms $(IX)_j,\dots, (XII)_j$
  10. Convergence

Key Result

Theorem 1.1

Suppose that $V_{(j)}, W_{(j)}\in C^\infty_c({\mathbb{R}})$ and $V_{(j)}(x)=-V_{(j)}(-x), W_{(j)}(x)=-W_{(j)}(-x)$ for $j=0,1$. Then there exist constants $\tilde{c}, \varepsilon_0 >0$ such that when $0<\varepsilon < \varepsilon_0$, ohta_1d_system_odd has a unique solution $(V,W)\in C^\infty([4,T_\v

Theorems & Definitions (21)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1: MTT
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 11 more