Life span of small solutions to a system of wave equations. (July 2016) - Hidano, Kunio; Yokoyama, Kazuyoshi
Abstract: We study the Cauchy problem with small initial data for a system of semilinear wave equations □ u = | v | q, □ v = | ∂ t u | 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 ) .