Late-time tails of wave maps coupled to gravity

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.