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

We study the quasilinear wave equation $\Box_{g}ϕ=0$ where the metric $g = g(ϕ,t,x)$ 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 $ϕ$. One consequence of this rate is that for bounded $|x|$, we have the integrable decay rate $|ϕ(t,x)| \le Ct^{-1-\min(δ,1)}$ where $δ>0$ 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 $(\Box_{\tilde g} + B^α(t,x)\partial_α+ V(t,x))ϕ=0$ with a more general asymptotically flat metric $\tilde g = \tilde g(ϕ,t,x)$ and with other time-dependent asymptotically flat lower order terms.