Improved decay for quasilinear wave equations close to asymptotically flat spacetimes including black hole spacetimes

Abstract

We study the quasilinear wave equation $\Box_{g}\phi=0$ where the metric $g = g(\phi,t,x)$ is close to and asymptotically approaches $g(0,t,x)$, which equals the Schwarzschild metric or a Kerr metric with small angular momentum, as time tends to infinity. Under only weak assumptions on the metric coefficients, we prove an improved pointwise decay rate for the solution $\phi$. One consequence of this rate is that for bounded $|x|$, we have the integrable decay rate $|\phi(t,x)| \le Ct^{-1-\min(\delta,1)}$ where $\delta>0$ is a parameter governing the decay, near the light cone, of the coefficient of the slowest-decaying term in the quasilinearity. We also obtain the same aforementioned pointwise decay rates for the quasilinear wave equation $(\Box_{\tilde g} + B^{\alpha}(t,x)\partial_\alpha + V(t,x))\phi=0$ with a more general asymptotically flat metric $\tilde g = \tilde g(\phi,t,x)$ and with other time-dependent asymptotically flat lower order terms.