Global existence and scattering of small data smooth solutions to quasilinear wave systems on R 2 × T $\mathbb {R}^2\times \mathbb {T}$ , II

In our previous paper [Fei Hou, Fei Tao, Huicheng Yin, Global existence and scattering of small data smooth solutions to a class of quasilinear wave systems on $R^2\times T$, Preprint (2024), arXiv:2405.03242], for the $Q_0$-type quadratic nonlinearities, we have shown the global well-posedness and scattering properties of small data smooth solutions to the quasilinear wave systems on $R^2\times T$. In this paper, we start to solve the global existence problem for the remaining $Q_{\alpha \beta }$-type nonlinearities. By combining these results, we have established the global well-posedness of small solutions on $R^2\times T$ for the general 3-D quadratically quasilinear wave systems when the related 2-D null conditions are fulfilled.