Late-time tail for a scalar quasilinear wave equation satisfying the weak null condition

TL;DR

This paper derives the late-time asymptotic behavior of solutions to a class of scalar quasilinear wave equations satisfying the weak null condition, revealing a superposition of decay rates and connecting it to linear wave solutions.

Contribution

It provides the first detailed asymptotic formula for solutions to weak null condition equations, linking nonlinear decay to linear wave behavior and establishing rigidity results.

Findings

Late-time asymptotics described by linear wave solutions

Superposition of decay rates differs from null condition equations

Solutions with faster decay than expected must be trivial

Abstract

We consider a class of scalar quasilinear wave equations in three spatial dimensions satisfying the weak null condition. For solutions arising from small, localized, smooth data, we give an asymptotic formula describing the global asymptotics towards the future. We prove that the late-time asymptotics is given by a continuous superposition of decay rates, in stark contrast to equations satisfying a null condition. The asymptotic formula we obtain is given in terms of a solution to the linear wave equation. Combining this with analysis on the linear wave equation, we strengthen some rigidity results of the third author, showing in particular that any solution with a faster time decay than expected away from the wave zone must vanish identically.