Almost global smooth solutions of the 3D quasilinear Klein-Gordon equations on the product space $\mathbb{R}^{2}\times \mathbb{T}$

TL;DR

This paper proves that small initial data for the 3D quasilinear Klein-Gordon equation on a product space leads to solutions that exist for an exponentially long time, using the space-time resonance method.

Contribution

It establishes almost global existence of smooth solutions for the 3D quasilinear Klein-Gordon equation on  space, extending previous results to this setting.

Findings

Solutions exist up to time e^{c_0/\u03b5_0^2} for small initial data 0

The space-time resonance method effectively estimates lifespan

Small initial data ensures long-term smooth solutions

Abstract

In the paper, for the 3D quasilinear Klein-Gordon equation with the small initial data posed on the product space $\mathbb{R}^{2}\times \mathbb{T}$, we focus on the lower bound of the lifespan of the smooth solution. When the size of initial data is bounded by $\varepsilon_0>0$, by the space-time resonance method, it is shown that smooth solution exists up to the time $e^{c_{0}/\varepsilon_{0}^2}$ with $\varepsilon_0$ being sufficiently small and $c_0>0$ being some suitable constant.