Long time solutions of semilinear Klein-Gordon equations on the torus

It is well established that the quadratic nonlinear Klein-Gordon equation admits global solutions for sufficiently small and rapidly decaying initial data in spatial dimensions $ d\geq 2 $. If the initial periodic data is merely in the unweighted Sobolev space of the size $ \varepsilon\geq 0 $ on the torus (side length $ 2\pi $), for semilinear Klein-Gordon equations, Fang and Zhang [9] proved that the long-time existence of the solutions has a lower bound $ T_{\varepsilon}\geq c\varepsilon^{-(3/2-\rho)} $ with some constants $ c, \rho>0 $. In this paper, we will show that, without the null condition, the lifespan can be improved to $ T_{R,\varepsilon}\geq cR\left(\varepsilon^2\ln R\right)^{-1} $ for $ \varepsilon\leq c\left(\sqrt{\ln R}\right)^{-1 } $ on the square torus $ \mathbb{T}^{2}_{R} $ (side length $ R\geq10000 $), where $ c $ is a positive constant. For $ d\geq3, $ we further show that the lifespan of solution can be improved to $ T_{R,\varepsilon}\geq cR\varepsilon^{-2} $ ($ T_{R,\varepsilon}\rightarrow \infty $ as $ R\rightarrow \infty $). The main ingredients are the bootstrap estimates and Strichartz estimates.