Partial Normal Form for the Semilinear Klein–Gordon Equation with Quadratic Potentials and Algebraic Non-resonant Masses

2021 
In this paper, we study small solutions of the non-linear Klein–Gordon equation on $$\mathbb {R}^{d}$$ with quadratic potential. The goal is to understand for which masses, we can apply a normal form procedure. We prove two mains theorems. Our first contribution gives explicit for which the procedure of partial normal form works due to algebraic assumptions. The second one shows that this strategy works for almost all m in the sense of Lebesgue measure.
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