Stability of perturbed wave equations on Kerr black hole spacetimes

I will discuss a recent work with Gustav Holzegel, in which we prove integrated decay bounds for solutions of the geometric wave equation with small linear perturbations on Kerr black hole spacetimes. Our proof adapts the framework introduced by Dafermos, Rodnianski, and Shlapentokh-Rothman for the homogeneous wave equation on Kerr spacetimes. When adding the perturbative term one must also compensate for obstructions caused by the necessary degeneration of Morawetz-type estimates in these spacetimes, which is due to the presence of trapped null geodesics. Mathematically, the key mechanism to our approach is the construction of a pseudodifferential commutator W, such that for the commuted equation one may obtain a nondegenerate Morawetz-type estimate.