# Almost global existence for the nonlinear Klein-Gordon equation in the nonrelativistic limit

## TL;DR

This paper proves that small solutions to the one-dimensional nonlinear Klein-Gordon equation with a convolution potential remain small over long times uniformly for large measure sets of the parameter c, in the nonrelativistic limit.

## Contribution

It establishes almost global existence results for the nonlinear Klein-Gordon equation in the nonrelativistic limit with uniform bounds across large measure parameter sets.

## Key findings

- Solutions with small initial data stay small for long times.
- The results are uniform in the parameter c for c ≥ 1.
- The set of c values for which the results hold has large measure.

## Abstract

We study the one-dimensional nonlinear Klein-Gordon (NLKG) equation with a convolution potential, and we prove that solutions with small $H^s$ norm remain small for long times. The result is uniform with respect to $c \geq 1$, which however has to belong to a set of large measure.