Long time smooth solutions of 3D cubic quasilinear wave systems with small weakly decaying initial data

For the 3D cubic quasilinear wave system ${\mathrm{\square }}_{c_i}{u}^i=G^i(u,\mathrm{\partial }u,{\mathrm{\partial }}^{2}u)={\displaystyle \sum _{\begin{array}{c}0\le \mathrm{\mid }\alpha \mathrm{\mid },\mathrm{\mid }\beta \mathrm{\mid },\mathrm{\mid }\gamma \mathrm{\mid }\le 1\\ 1\le j,k,l\le m\end{array}}{g}_{\alpha \beta \gamma }^{ijkl}{\mathrm{\partial }}^{\alpha }{u}^j{\mathrm{\partial }}^{\beta }{u}^k{\mathrm{\partial }}^{\gamma }{u}^l}$, it is well known that global solution $u$ exists when the small smooth initial data $(u,{\mathrm{\partial }}_{t}u){\mathrm{\mid }}_{t=0}$ $=(u_0(x),u_1(x))$ are compactly supported or decay rapidly at spatial infinity. However, when $(u_0,u_1)\in (H^{s+1},H^s)$ with $s>\frac{5}{2}$ are small, it remains unknown whether $u$ exists globally or not. In this paper, we show that if $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{H^{N+1}}+\mathrm{\parallel }{u}_1{\mathrm{\parallel }}_{H^N}\le \epsilon$ ($N\ge 6$) is small, then the almost global solution $u$ exists in $[0,T_{\epsilon }]$ with $T_{\epsilon }\ge e^{C{\epsilon }^{-1}}$ for the general $G(u,\mathrm{\partial }u,{\mathrm{\partial }}^{2}u)$ depending on $u$ and $T_{\epsilon }\ge e^{C{\epsilon }^{-2}}$ for the nonlinearity $G(\mathrm{\partial }u,{\mathrm{\partial }}^{2}u)$ independent of $u$, respectively. In addition, if ${\displaystyle \sum _{\mathrm{\mid }a\mathrm{\mid }\le 5}\mathrm{\parallel }⟨x{⟩}^{\mu }{\mathrm{\partial }}_x^a(u_0,u_1){\mathrm{\parallel }}_{L^2}\le \epsilon }$ holds for any fixed constant $\mu \in (0,1)$, then the solution $u$ exists globally and meanwhile the scattering property of $u$ is derived. Our main ingredients consist in establishing a series of new weighted $L^{\mathrm{\infty }}-L^2$ estimates and Strichartz estimates based on the strong Huygens’ principle for 3D linear wave equations.