Long-time existence for semi-linear Klein-Gordon equations with quadratic potential

We prove that small smooth solutions of semi-linear Klein-Gordon equations with quadratic potential exist over a longer interval than the one given by local existence theory, for almost every value of mass. We use normal form for the Sobolev energy. The difficulty in comparison with some similar results on the sphere comes from the fact that two successive eigenvalues λ, λ′ of √ −∆+ |x|2 may be separated by a distance as small as 1 √ λ . 0 Introduction Let −∆+ |x|2 be the harmonic oscillator on Rd. This paper is devoted to the proof of lower bounds for the existence time of solutions of non-linear Klein-Gordon equations of type (∂ t −∆+ |x| +m)v = v v|t=0 = ǫv0 ∂tv|t=0 = ǫv1 where m ∈ R+, xα∂ xvj ∈ L2 when |α|+ |β| ≤ s+ 1− j (j = 0, 1) for a large enough integer s, and where ǫ > 0 is small enough. The similar equation without the quadratic potential |x|2, and with data small, smooth and compactly supported, has global solutions when d ≥ 2 (see Klainerman [18] and Shatah [23] for dimensions d ≥ 3, Ozawa, Tsutaya and Tsutsumi [22] when d = 2). The situation is drastically different when we replace −∆ by −∆ + |x|2, since the latter operator has pure point spectrum. This prevents any time decay for solutions of the linear equation. Because of that, the question of long time existence for Klein-Gordon equations associated to the harmonic oscillator is similar to the corresponding problem on compact manifolds. For the equation (∂2 t −∆ +m2)v = vκ+1 on the circle S1, it has been proved by Bourgain [6] and Bambusi [1], that for almost every m > 0, the above equation has solutions defined on intervals The author is supported by NSFC 10871175. email address: [email protected]