  In this paper we prove a result of almost global existence for some abstract
nonlinear PDEs on flat tori and apply it to some concrete equations, namely a
nonlinear Schr\"odinger equation with a convolution potential, a beam equation
and a quantum hydrodinamical equation. We also apply it to the stability of
plane waves in NLS. The main point is that the abstract result is based on a
nonresonance condition much weaker than the usual ones, which rely on the
celebrated Bourgain's Lemma which provides a partition of the "resonant sites"
of the Laplace operator on irrational tori.
