The wave equation near flat Friedmann-Lema\^itre-Robertson-Walker and Kasner Big Bang singularities

TL;DR

This paper studies the behavior of solutions to the wave equation near Big Bang singularities in flat FLRW and Kasner spacetimes, showing generic blow-up results for certain initial data configurations.

Contribution

It characterizes initial data sets leading to blow-up of wave solutions near singularities without assuming symmetries, using weighted energy estimates.

Findings

Solutions blow up near the Big Bang hypersurface for generic initial data.

Blow-up rates are inverse polynomial for FLRW and logarithmic for Kasner backgrounds.

Initial data with dominant Neumann conditions lead to solution blow-up.

Abstract

We consider the wave equation, $\square_g\psi=0$, in fixed flat Friedmann-Lema\^itre-Robertson-Walker and Kasner spacetimes with topology $\mathbb{R}_+\times\mathbb{T}^3$. We obtain generic blow up results for solutions to the wave equation towards the Big Bang singularity in both backgrounds. In particular, we characterize open sets of initial data prescribed at a spacelike hypersurface close to the singularity, which give rise to solutions that blow up in an open set of the Big Bang hypersurface $\{t=0\}$. The initial data sets are characterized by the condition that the Neumann data should dominate, in an appropriate $L^2$-sense, up to two spatial derivatives of the Dirichlet data. For these initial configurations, the $L^2(\mathbb{T}^3)$ norms of the solutions blow up towards the Big Bang hypersurfaces of FLRW and Kasner with inverse polynomial and logarithmic rates respectively. Our method is based on deriving suitably weighted energy estimates in physical space. No symmetries of solutions are assumed.