Timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null condition

TL;DR

This paper derives the asymptotic behavior of solutions to a scalar quasilinear wave equation satisfying the weak null condition and shows conditions under which solutions must vanish, extending understanding of wave decay and scattering.

Contribution

It provides an explicit asymptotic formula for solutions inside the light cone and links decay conditions to solution vanishing, based on scattering data and previous asymptotic completeness results.

Findings

Derived asymptotic formula for solutions inside the light cone.

Proved solutions vanish under certain decay assumptions.

Connected decay conditions to solution nullity using scattering data.

Abstract

We study the timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null condition. Given a global solution $u$ to the scalar wave equation with sufficiently small $C_c^\infty$ initial data, we derive an asymptotic formula for this global solution inside the light cone (i.e. for $|x|<t$). It involves the scattering data obtained in the author's asymptotic completeness result in arXiv:2105.11573. Using this asymptotic formula, we prove that $u$ must vanish under some decaying assumptions on $u$ or its scattering data, provided that the wave equation violates the null condition.