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

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.