Generically sharp decay for quasilinear wave equations with null condition

TL;DR

This paper analyzes the precise decay behavior of solutions to three-dimensional quasilinear wave equations with null condition, establishing optimal decay rates for generic initial data posed on a hyperboloid.

Contribution

It demonstrates the optimal decay rate $v^{-1}u^{-1}$ for solutions with initial data on a hyperboloid, refining previous global existence results.

Findings

Decay rate $v^{-1}u^{-1}$ is optimal for generic initial data.

Solutions exhibit precise pointwise asymptotic behavior.

Results apply to small smooth initial data on hyperboloids.

Abstract

We are interested in the three-dimensional quasilinear wave equations with null condition. Global existence and pointwise decay for this model have been proved in the celebrated works of Klainerman \cite{Klainerman86} and Christodoulou \cite{Christodoulou86} for small smooth initial data. In this work, we illustrate the precise pointwise asymptotic behavior of the solutions for initial data posed on a hyperboloid and show that the decay rate $v^{-1}u^{-1}$ is optimal for a generic set of initial data.