A note on blow-up results for semilinear wave equations in de Sitter and anti-de Sitter spacetimes

TL;DR

This paper establishes finite-time blow-up results for semilinear wave equations in de Sitter and anti-de Sitter spacetimes by analyzing spatial averages and deriving lower bounds through iterative slicing techniques.

Contribution

It introduces new blow-up criteria for these equations under specific conditions on the nonlinear term's time dependence, expanding understanding of solution behavior in curved spacetimes.

Findings

Finite-time blow-up of solutions under certain conditions.

Development of lower bound estimates for spatial averages.

Application of slicing and iteration methods to curved spacetime wave equations.

Abstract

In this work we derive some blow-up results for semilinear wave equations both in de Sitter and anti-de Sitter spacetimes. By requiring suitable conditions on a time-dependent factor in the nonlinear term, we prove the blow-up in finite time of the spatial averages of local in time solutions. In particular, we derive a sequence of lower bound estimates for the spatial average by combining a suitable slicing procedure with an iteration frame for this time-dependent functional.