A Sufficient Condition for Blowup of the Nonlinear Klein-Gordon Equation with Positive Initial Energy in FLRW Spacetimes

TL;DR

This paper establishes a sufficient condition for the blowup of solutions to the nonlinear Klein-Gordon equation with positive initial energy in FLRW spacetimes, extending concavity methods from Minkowski space.

Contribution

It introduces a new sufficient condition for blowup in FLRW spacetimes and adapts the concavity method and energy inequality proof to this geometric setting.

Findings

Derived a blowup criterion for the nonlinear Klein-Gordon equation in FLRW spacetimes.

Extended the concavity method to curved spacetime geometries.

Proved an energy inequality using a geometric approach.

Abstract

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, $E(t_0)\geq E(t)$, also proved herein using a geometric method.