Global Existence And Growth Of H~s Norms For Solutions Of Some Dispersive Equations On Manifolds

First we consider long time existence for solutions to semi-linear Klein-Gordon equations with a quadratic potential on the whole space. Using a normal form method to control the Sobolev energy of the solution, we prove that, given weakly dacaying small data, the solution exists over a longer interval than the one given by local existence theory, for almost every value of the positive mass. The difficulty in comparison with some similar results on the sphere comes from the fact that two successive eigenvaluesλ,λ'of (?) on Rd may be separated by a distance as small as 1/λ.Then we consider semi-linear Klein-Gordon equations on the torus. Using the same method, we prove that, given smooth small data, the solution exists over a longer interval than the one given by local existence theory, for almost all the positive mass. The gain on the exponent is independent of the space dimension. In high space dimensions, the result is better than that of Delort [15].At last we improve the method of Delort [16] to show that, for the linear Schrodinger equation with a time dependent potential on the torus, if the potential is Gevrey function of orderμwithμ> 1, then the solution has at most logarithmically growing Hs-norm with respect to time. Our result contains that of Wang [47].