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

Abstract

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-8/3javax.xml.bind.JAXBElement@f17399) 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-10/3javax.xml.bind.JAXBElement@3095a7a6), 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).