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

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 ${\epsilon }_0>0$, by the space-time resonance method, it is shown that smooth solution exists up to the time $e^{c_0\mathrm{/}{\epsilon }_0^2}$ with ${\epsilon }_0$ being sufficiently small and $c_0>0$ being some suitable constant.