Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres

Consider a nonlinear Klein-Gordon equation on a compact manifold M with smooth Cauchy data of size ε → 0. Denote by Tε the maximal time of existence of a smooth solution. One always has Tε ≥ c ε^−k+1 if the nonlinearity vanishes at order k at 0. We prove that when $M=\text{T}^d$ , k=2, and the equation is quasilinear, Tε ≥ cε⁻². When $M=\text{S}^{d-1}$ , the equation is semilinear, and k = 2p, we show that Tε ≥ cε^−2p for almost every mass. The proof of these results relies on a systematic use of normal forms methods.