Small data solutions of 2-D quasilinear wave equations under null conditions

For the 2-D quasilinear wave equation $\left({?}_t^2-{\Delta }_x\right)u+\sum _{i,j=0}^2{g}^{ij}\left(?u\right){?}_{ij}u=0$ satisfying null condition or both null conditions, a blowup or global existence result has been shown by Alinhac. In this paper, we consider a more general 2-D quasilinear wave equation $\left({?}_t^2-{\Delta }_x\right)u+\sum _{i,j=0}^2{g}^{ij}\left(u,?u\right){?}_{ij}u=0$ satisfying null conditions with small initial data and the coefficients depending simultaneously on u and ?u. Through construction of an approximate solution, combined with weighted energy integral method, a quasi-global or global existence solution are established by continuous induction.