The small data solutions of general 3-D quasilinear wave equations. II

This paper is a continuation of the work in [8], where the authors established the global existence of smooth small data solutions to the general 3-D quasilinear wave equation ∑i,j=03gij(u,∂u)∂ij2u=0 when the weak null condition holds. In the present paper, we show that the smooth small data solutions of equation ∑i,j=03gij(u,∂u)∂ij2u=0 will blow up in finite time when the weak null condition does not hold and a generic nondegenerate condition on the initial data is satisfied, moreover, a precise blowup time is completely determined. Therefore, collecting the main results in this paper and [8], we have given a basically complete study on the blowup or global existence of small data solutions to the 3-D quasilinear wave equation ∑i,j=03gij(u,∂u)∂ij2u=0.