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

Abstract

For the 2-D quasilinear wave equation (∂t2-Δx)u+∑i,j=02gij(∂u)∂iju=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 (∂t2-Δx)u+∑i,j=02gij(u, ∂u)∂iju=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.