Almost global solutions of 1D nonlinear Klein-Gordon equations with small weakly decaying initial data

It has been known that if the initial data decay sufficiently fast at space infinity, then 1D Klein-Gordon equations with quadratic nonlinearity admit classical solutions up to time $e^{C/\epsilon^2}$ while $e^{C/\epsilon^2}$ is also the upper bound of the lifespan, where $C>0$ is some suitable constant and $\epsilon>0$ is the size of the initial data. In this paper, we will focus on the 1D nonlinear Klein-Gordon equations with weakly decaying initial data. It is shown that if the $H^s$-Sobolev norm with $(1+|x|)^{1/2+}$ weight of the initial data is small, then the almost global solutions exist; if the initial $H^s$-Sobolev norm with $(1+|x|)^{1/2}$ weight is small, then for any $M>0$, the solutions exist on $[0,\epsilon^{-M}]$. Our proof is based on the dispersive estimate with a suitable $Z$-norm and a delicate analysis on the phase function.