  The purpose of this paper is to show how local energy decay estimates for
certain linear wave equations involving compact perturbations of the standard
Laplacian lead to optimal global existence theorems for the corresponding small
amplitude nonlinear wave equations with power nonlinearities. To achieve this
goal, at least for spatial dimensions $n=3$ and 4, we shall show how the
aforementioned linear decay estimates can be combined with "abstract
Strichartz" estimates for the free wave equation to prove corresponding
estimates for the perturbed wave equation when $n\ge3$. As we shall see, we are
only partially successful in the latter endeavor when the dimension is equal to
two, and therefore, at present, our applications to nonlinear wave equations in
this case are limited.
