Late-time tails of wave maps coupled to gravity

TL;DR

This paper investigates the late-time decay behavior of wave map solutions coupled with gravity, showing specific power-law decay rates at different infinities for small initial data.

Contribution

It provides the first detailed analysis of decay rates for wave maps coupled to gravity with small initial data, extending understanding of their asymptotic behavior.

Findings

Solutions decay as t^{-(2 ext{l}+2)} at future timelike infinity

Solutions decay as u^{-( ext{l}+1)} at future null infinity

Decay rates depend on the equivariance index ext{l}

Abstract

We consider the late-time asymptotic behavior for solutions of Einstein's equations with the wave map matter. Solutions starting from small compactly supported $\ell$-equivariant initial data with $\ell\geq 1$ are shown to decay as $t^{-(2\ell+2)}$ at future timelike infinity and as $u^{-(\ell+1)}$ at future null infinity.