Long-time existence for the semilinear Klein-Gordon equation on a compact boundary-less Riemannian manifold. (English) Zbl 1365.37056
Authors’ abstract: We investigate the long-time existence of small and smooth solutions for the semilinear Klein-Gordon equation on a compact boundary-less Riemannian manifold. Without any spectral or geometric assumption, our first result improves the lifespan obtained by the local theory. The previous result is proved under a generic condition of the mass. As a by-product of the method, we examine the particular case, where the manifold is a multidimensional torus, and we give explicit examples of algebraic masses for which we can improve the local existence time. The analytic part of the proof relies on multilinear estimates of eigenfunctions and estimates of small divisors proved by J. M. Delort and J. Szeftel [Am. J. Math. 128, No. 5, 1187–1218 (2006; Zbl 1108.58023)]. The algebraic part of the proof relies on a multilinear version of the Roth theorem proved by W. M. Schmidt [Diophantine approximation. Lect. Notes Math. 785. Berlin, etc.: Springer-Verlag. (1980; Zbl 0421.10019)].
Reviewer: Jesús Hernández (Madrid)
MSC:
| 37K40 | Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems |
| 35P10 | Completeness of eigenfunctions and eigenfunction expansions in context of PDEs |
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
| 37K25 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
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