Small data solutions of 3-D quasilinear wave equations under null condition

For the 3-D quasilinear wave equation ∑i,j=03gij(u)∂iju=0, a global existence result has been shown by H. Lindblad. In this paper, we consider this 3-D quasilinear wave equation ∑i,j=03gij(u)∂iju=∑i,j=03fij∂iu∂ju with small initial data and fij satisfying null condition. Through deriving decay estimates of low derivatives and energy estimates for high derivatives, combined with the properties of null condition, a global existence solution is also established by continuous induction.