LONG TIME SOLUTIONS FOR QUASILINEAR HAMILTONIAN PERTURBATIONS OF SCHRODINGER AND KLEIN-GORDON EQUATIONS ON TORI

We consider quasilinear, Hamiltonian perturbations of the cubic Schrodinger and of the cubic (derivative) Klein-Gordon equations on the d-dimensional torus. If is an element of << 1 is the size of the initial datum, we prove that the lifespan of solutions is strictly larger than the local existence time c(-2). More precisely, concerning the Schrodinger equation we show that the lifespan is at least of order O(is an element of(-4)), and in the Klein-Gordon case we prove that the solutions exist at least for a time of order O(is an element of(-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(is an element of(-10/3-)), 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).