  In this paper we prove global and almost global existence theorems for
nonlinear wave equations with quadratic nonlinearities in infinite homogeneous
waveguides. We can handle both the case of Dirichlet boundary conditions and
Neumann boundary conditions. In the case of Neumann boundary conditions we need
to assume a natural nonlinear Neumann condition on the quasilinear terms. The
results that we obtain are sharp in terms of the assumptions on the dimensions
for the global existence results and in terms of the lifespan for the almost
global results. For nonlinear wave equations, in the case where the infinite
part of the waveguide has spatial dimension three, the hypotheses in the
theorem concern whether or not the Laplacian for the compact base of the
waveguide has a zero mode or not.
