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

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 estimates, $ 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.