Scattering towards the singularity for the wave equation and the linearized Einstein-scalar field system in Kasner spacetimes

Li, Warren

doi:10.2140/apde.2026.19.1459

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Scattering towards the singularity for the wave equation and the linearized Einstein-scalar field system in Kasner spacetimes

Warren Li

Vol. 19 (2026), No. 8, 1459–1546

DOI: 10.2140/apde.2026.19.1459

Abstract

We consider the scalar wave equation $□_{g} ϕ = 0$ and the linearized Einstein-scalar field system around generalized Kasner spacetimes with spatial topology $𝕋^{D}$ . In suitable regimes for the Kasner exponents, it is known that solutions to such equations arising from regular Cauchy data (e.g., at $t = 1$ ) have certain quantitative blow-up asymptotics near the initial time (i.e., $t = 0$ ) singularity of Kasner. For instance, solutions to the wave equation behave as $ϕ (t, x) \approx ψ_{\infty} (x) \log t + φ_{\infty} (x)$ near $t = 0$ .

This article provides a description, and proof, of a scattering theory for the above equations, linking Cauchy data at $t = 1$ and suitable asymptotic data at $t = 0$ in Kasner. For the scalar wave equation, this means a Hilbert space isomorphism between $(ϕ, \partial_{t} ϕ)$ at $t = 1$ and the functions $(ψ_{\infty}, φ_{\infty})$ . A curious detail is that certain quantities, e.g., $ψ_{\infty}$ , feature a gain of half a derivative when compared to $\partial_{t} ϕ$ at $t = 1$ .

The study of the linearized Einstein-scalar field system reveals further interesting phenomena, including differences between diagonal and off-diagonal components of certain tensors in the scattering theory, and that the losses of derivatives feature a sensitive dependence on the anisotropy of the background Kasner spacetime. In fact, though our result holds for the entire subcritical regime of background Kasner exponents, the number of derivatives lost and gained in the scattering theory can become unbounded as one nears the boundary of this regime.

Keywords

general relativity, Kasner, big bang, scattering, regularity, nonlinear wave equations

Mathematical Subject Classification

Primary: 35Q76, 83C75

Secondary: 35L10, 35P25, 83F05

Milestones

Received: 6 February 2024

Revised: 28 July 2025

Accepted: 2 November 2025

Published: 8 October 2026

Authors

Warren Li
Department of Mathematics Princeton University Princeton, NJ United States	Stanford University Stanford, CA United States

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Vol. 19, No. 8, 2026