Life span of small solutions to a system of wave equations

Abstract

We study the Cauchy problem with small initial data for a system of semilinear wave equations $\square u=|v{|}^p$, $\square v=|{\partial }_{t}u{|}^p$ in $n$-dimensional space. When $n\ge 2$, we prove that blow-up can occur for arbitrarily small data if $(p,q)$ lies below a curve in $p$-$q$ 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 an upper bound, which is sharp at least for $(n,p,q)=(2,2,2)$ and $(3,2,2)$.

26 pages