Existence of solutions of semilinear wave equations with time-dependent propagation speed and time derivative nonlinearity

TL;DR

This paper investigates the existence, lifespan, and blow-up behavior of solutions to semilinear wave equations with time-dependent speeds and derivatives, relevant to cosmological spacetimes like FLRW, de Sitter, and anti-de Sitter.

Contribution

It provides new lifespan estimates and conditions for global existence or blow-up for these wave equations in various cosmological models.

Findings

Lower bounds on solution lifespan in expanding spacetimes.

Finite-time blow-up results in contracting universes.

Applicability of integrability conditions to specific spacetime models.

Abstract

Consider wave equations with time derivative nonlinearity and time-dependent propagation speed which are generalized versions of the wave equations in the Friedmann-Lema\^itre-Robertson-Walker (FLRW) spacetime, the de Sitter spacetime and the anti-de Sitter space time. We show lower bounds of the lifespan of solutions as well as the global existence by providing an integrability condition on the propagation speed function, which is applicable to the nonlinear wave equation in the expanding FLRW spacetime including the de Sitter spacetime. We also prove that blow-up in a finite time occurs for the generalized form of the equation in contracting universes such as the anti-de Sitter spacetime, as well as upper bounds of the lifespan of blow-up solutions.