Almost global smooth solutions of the 3D quasilinear Klein-Gordon equations on the product space R2 x T

In this paper, for the 3D quadratic nonlinear Klein-Gordon equation on the product space R-2 x 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 epsilon(0) > 0, it is shown that a smooth solution exists up to the time e(epsilon 0/epsilon 02) with epsilon(0) being sufficiently small and c(0) > 0 being some suitable constant. Note that the solution of the corresponding 3D linear homogeneous Klein-Gordon equation on R-2 x 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 e(c0/epsilon 0) rather than the more precise e(epsilon 0/epsilon 02) here.