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

TL;DR

This paper investigates the lifespan of solutions to a system of multiple speed wave equations in the radial case, showing almost global existence with a lifespan comparable to the blowup time for small initial data.

Contribution

It extends previous blowup results by establishing almost global existence for the radial case using advanced energy estimates and multiplier techniques.

Findings

Solutions exist almost globally in time for small initial data.

The lifespan matches the lower bound established by blowup results.

Advanced energy estimates are effective for analyzing multi-speed wave systems.

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.