Long time solutions for quasilinear Hamiltonian perturbations of Schrödinger and Klein-Gordon equations on tori

We consider quasilinear, Hamiltonian perturbations of the cubic Schrödinger and of the cubic (derivative) Klein–Gordon equations on the d -dimensional torus. If ϵ ≪ 1 is the size of the initial datum, we prove that the lifespan of solutions is strictly larger than the local existence time ϵ − 2 . More precisely, concerning the Schrödinger equation we show that the lifespan is at least of order O ( ϵ − 4 ) , and in the Klein–Gordon case we prove that the solutions exist at least for a time of order O ( ϵ − 8 ∕ 3 − ) as soon as d ≥ 3 . Regarding the Klein–Gordon equation, our result presents novelties also in the case of semilinear perturbations: we show that the lifespan is at least of order O ( ϵ − 1 0 ∕ 3 − ) , improving, for cubic nonlinearities and d ≥ 4 , the general results of Delort (J. Anal. Math. 107 (2009), 161–194) and Fang and Zhang (J. Differential Equations 249:1 (2010), 151–179).