A sufficient condition for blowup of the nonlinear Klein–Gordon equation with positive initial energy in FLRW spacetimes

In this paper we demonstrate a sufficient condition for blowup of the nonlinear Klein-Gordon equation with arbitrarily positive initial energy in Friedmann-Lema\ⁱtre-Robertson-Walker spacetimes. This is accomplished using an established concavity method that has been employed for similar PDEs in Minkowski space. This proof relies on the energy inequality associated with this equation, E (t₀) E (t), also proved herein using a geometric method.