  We give an elementary new argument for global existence and exponential decay
of solutions of quasilinear wave equations on Schwarzschild-de Sitter black
hole backgrounds, for appropriately small initial data. The core of the
argument is entirely local, based on time translation invariant energy
estimates in spacetime slabs of fixed time length. Global existence then
follows simply by iterating this local result in consecutive spacetime slabs.
We infer that an appropriate future energy flux decays exponentially with
respect to the energy flux of the initial data.
