Long time classical solutions of quasilinear Klein-Gordon equations with small weakly decaying initial data

It is well known that for the quasilinear Klein-Gordon equation with quadratic nonlinearity and sufficiently decaying small initial data, there exists a global smooth solution if the space dimensions $d\ge 2$. When the initial data are of size $\epsilon >0$ in the Sobolev space, for the semilinear Klein-Gordon equation satisfying the null condition, the authors in the article (Delort and Fang, 2000 [11]) prove that the solution exists in time $[0,T_{\epsilon })$ with $T_{\epsilon }\ge C{e}^{C{\epsilon }^{-\mu }}$ ($\mu =1$ if $d\ge 3$, $\mu =2/3$ if $d=2$). In the present paper, we will focus on the general quasilinear Klein-Gordon equation without the null condition and further show that the existence time of the solution can be improved to $T_{\epsilon }=+\infty$ if $d\ge 3$ and $T_{\epsilon }\ge e^{C{\epsilon }^{-2}}$ if $d=2$. In addition, for $d=2$ and any fixed number $\alpha >0$, if the weighted $L^2$ norm of the initial data with the weight ${(1+|x\left|\right)}^{\alpha }$ is small, then the solution exists globally and scatters to a free solution. Our arguments are based on the introduction of a new good unknown, the Strichartz estimate, the weighted $L^2$-norm estimate and the resonance analysis.