  We extend the semilinear framework developed by the two authors and the
non-trapping quasilinear theory developed by the first author to solve
quasilinear wave equations with normally hyperbolic trapping. The most
well-known example that fits into our general framework is wave-type equations
on Kerr-de Sitter space. The key advance is an adaptation of the Nash-Moser
iteration to our framework.
