The Null Condition and Global Existence for Nonlinear Wave Equations on Slowly Rotating Kerr Spacetimes

TL;DR

This paper proves global existence and decay of solutions to a nonlinear wave equation satisfying a null condition on slowly rotating Kerr spacetimes, using the vector field method and decay properties of linear waves.

Contribution

It establishes global stability results for nonlinear wave equations on Kerr backgrounds under small initial data, leveraging improved decay estimates.

Findings

Solutions exist globally for small initial data

Solutions decay at a quantitative rate

Utilizes decay properties of linear wave equations on Kerr

Abstract

We study a semilinear equation with derivatives satisfying a null condition on slowly rotating Kerr spacetimes. We prove that given sufficiently small initial data, the solution exists globally in time and decays with a quantitative rate to the trivial solution. The proof uses the robust vector field method. It makes use of the decay properties of the linear wave equation on Kerr spacetime, in particular the improved decay rates in the region $\{r\leq \frac{t}{4}\}$.