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In the paper, for the 3D quasilinear Klein-Gordon equation with the smallinitial data posed on the product space $\mathbb{R}^{2}\times \mathbb{T}$, wefocus on the lower bound of the lifespan of the smooth solution. When the sizeof initial data is bounded by $\varepsilon_0>0$, by the space-time resonancemethod, 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.

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