Life span of small solutions to a system of wave equations

We study the Cauchy problem with small initial data for a system of semilinear wave equations □u=|v|q□u=|v|q, □v=|∂tu|p□v=|∂tu|p in nn-dimensional space. When n≥2n≥2, we prove that blow-up can occur for arbitrarily small data if (p,q)(p,q) lies below a curve in the pp–qq plane. On the other hand, we show a global existence result for n=3n=3 which asserts that a portion of the curve is in fact the borderline between global-in-time existence and finite time blow-up. We also estimate the maximal existence time and get its upper bound, which is sharp at least for (n,p,q)=(2,2,2)(n,p,q)=(2,2,2) and (3,2,2)(3,2,2).