Long time behaviour of the nonlinear Klein-Gordon equation in the nonrelativistic limit, I

Stefano Pasquali

Published 2017-03-05Version 1

We study the the nonlinear Klein-Gordon (NLKG) equation on a manifold $M$ in the nonrelativistic limit, namely as the speed of light $c$ tends to infinity. In particular, we consider a higher-order normalized approximation of NLKG (which corresponds to the NLS at order $r=1$), and prove that when $M$ is a smooth compact manifold or $\mathbb{R}^d$, the solution of the approximating equation approximates the solution of the NLKG locally uniformly in time. When $M=\mathbb{R}^d$, $d \geq 3$, we prove that small radiation solutions of the order $r$ normalized equation approximate solutions of NLKG up to times of order $\mathcal{O}(c^{2(r-1)})$ for any $r>1$.