Long Times Existence Of Nonlinear Wave Equation On Compact Manifold

This thesis is devoted to the study of the long time and the global existence of nonlinear wave equations on some special manifolds, such as sphere, torus and waveguides.For nonlinear Klein-Gordon equation on sphere, Delort have proved, by using the technique of normal form, that the estimates of the existence time of the solution given by the local existence theory can be improved when the nonlinearity satisfy some property. Based on the work of Delort, we give the detailed expressions of the normal form of some special nonlinearity, and proved that the result of Delort can be improved for some special cases.For the case of waveduide R~2×M, where M be Zoll manifolds, if the initial data is small enough, we can use the special distribution of eigenvalues of Laplace-Beltrami operator on Zoll manifold, the estimate of the energy and the technique of normal form to prove the existence of global solutions for nonlinear Klein-Gordon equations.For the wave equation on the torus, we look for some special global solutions-periodic solutions. This can be consider as the problem of periodic boundary. The difficulty of this problem is the small denominator. To cope this difficult, using some special numbers as period and the method of variational, we can prove the existence of periodic solutions.In the end, we use the method of Galerkin to prove the existence of global weak solutions for the Schr(o|¨)dinger-Klein-Gordon equations on the sphere or torus.