Abstract
We prove that small smooth solutions of weakly semi-linear Klein-Gordon equations on the torus \( \mathbb{T}^d \) (d ≥ 2) exist over a larger time interval than the one given by local existence theory, for almost every value of the mass. We use a normal form method for the Sobolev energy of the solution. The difficulty, in comparison with previous results obtained on the sphere, comes from the fact that the set of differences of eigenvalues of \( \sqrt { - \Delta } \) on \( \mathbb{T}^d \) (d ≥ 2) is dense in ℝ.
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This work was partially supported by the ANR project Equa-disp.
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Delort, JM. On long time existence for small solutions of semi-linear Klein-Gordon equations on the torus. J Anal Math 107, 161–194 (2009). https://doi.org/10.1007/s11854-009-0007-2
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DOI: https://doi.org/10.1007/s11854-009-0007-2