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

In this paper, for the 3D quadratic nonlinear Klein-Gordon equation on the product space \(\mathbb{R}^{2}\times \mathbb{T}\) $${\mathbb{R}}^2\times \mathbb{T}$$ , we focus on the lower bound of the lifespan of the smooth solution with slowly decaying initial data. When the size of initial data is bounded by ϵ₀ > 0, it is shown that a smooth solution exists up to the time \(\mathrm{e}^{c_{0}}/\epsilon_{0}^{2}\) $${\mathrm{e}}^{c_0}/{ϵ}_0^2$$ with ϵ₀ being sufficiently small and c₀ > 0 being some suitable constant. Note that the solution of the corresponding 3D linear homogeneous Klein-Gordon equation on \(\mathbb{R}^{2}\times \mathbb{T}\) $${\mathbb{R}}^2\times \mathbb{T}$$ only admits the optimal time-decay rate (1 + t)⁻¹, from which we generally derive the lifespan of the nonlinear Klein-Gordon equation up to \(\mathrm{e}^{c_{0}/\epsilon_{0}}\) $${\mathrm{e}}^{c_0/{ϵ}_0}$$ rather than the more precise \(\mathrm{e}^{c_{0}/\epsilon_{0}^{2}}\) $${\mathrm{e}}^{c_0/{ϵ}_0^2}$$ here.