On Glassey's conjecture for semilinear wave equations in Friedmann-Lema\^itre-Robertson-Walker spacetime

TL;DR

This paper investigates finite-time blow-up of solutions to nonlinear wave equations in flat FLRW spacetimes, extending Glassey's conjecture to cosmological models with new lifespan bounds and blow-up results.

Contribution

It establishes the FLRW spacetime version of Glassey's conjecture for time derivative nonlinearities and provides new blow-up results for space-time derivatives.

Findings

Solutions blow up in finite time in FLRW spacetimes.

Upper bounds on the lifespan of blow-up solutions are derived.

Blow-up results are extended to space-time derivative nonlinearities.

Abstract

Consider nonlinear wave equations in the spatially flat Friedmann-Lema\^itre-Robertson-Walker (FLRW) spacetimes. We show blow-up in finite time of solutions and upper bounds of the lifespan of blow-up solutions to give the FLRW spacetime version of Glassey's conjecture for the time derivative nonlinearity. We also show blow-up results for the space time derivative nonlinearity.