LONG TIME DYNAMICS OF THE KLEIN-GORDON EQUATION IN THE NON-RELATIVISTIC LIMIT

In this thesis I study the non-relativistic limit ($c \to \infty$) of the nonlinear Klein-Gordon (NLKG) equation on a manifold $M$, namely \[ \frac{1}{c^2} u_{tt} - \Delta u + c^2 u + \lambda |u|^{2(l-1)}u = 0, \; \; t \in \R, x \in M \] where $\lambda = \pm 1$, $l \geq 2$. The aim of the present work is to discuss the convergence of solutions of the NLKG to solutions of a suitable nonlinear Schr\"odinger (NLS) equation, and to study whether such convergence may hold for large (namely, of size$O(c^r)$ with $r \geq 1$) timescales. In particular I obtain the following results: (1) when $M$ is a general manifold, I show that the solution of NLS describes well the solution of the original equation up to times of order $\cO(1)$; (2) when $M=\mathbb{R}^d$,$d \geq 3$, I consider higher order approximations of NLKG and prove that small radiation solutions of the approximating equation describe well solutions of NLKG up to times of order $O(c^{2r})$ for any $r \geq 1$; (3) when $M=[0,\pi] \subset \mathbb{R}$ I consider the NLKG equation with aconvolution potential and prove existence for long times of solutionsin $H^s$ uniformly in $c$, which however has to belong to a set of large measure. I also get some new dispersive estimates for a Klein-Gordon type equation with a potential.