Almost global existence for some Hamiltonian PDEs on manifolds with globally integrable geodesic flow

In this paper we prove an abstract result of almost global existence for small and smooth solutions of some semilinear PDEs on Riemannian manifolds with globally integrable geodesic flow. Some examples of such manifolds are Lie groups (including flat tori), homogeneous spaces and rotational invariant surfaces. As applications of the abstract result we prove almost global existence for a nonlinear Schr & ouml;dinger equation with a convolution potential and for a nonlinear beam equation. We also prove Hs stability of the ground state in NLS equation. The proof is based on a normal form procedure and the combination of the arguments used in Bambusi and Langella (2022 arXiv:2202.04505) to bound the growth of Sobolev norms in linear systems and a generalization of the arguments in Bambusi et al (2024 Commun. Math. Phys. 405 253-85).