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 square u = vertical bar v vertical bar(q), square v = vertical bar partial derivative(t) u vertical bar(p) in n-dimensional space. When n >= 2, we prove that blow-up can occur for arbitrarily small data if (p, q) lies below a curve in the p-q plane. On the other hand, we show a global existence result for n = 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) and (3, 2, 2). (C) 2016 Elsevier Ltd. All rights reserved.