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 $\square u = |v|^p$, $\square v = |\partial_t u|^p$ in $n$-dimensional space. When $n \geq 2$, 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 $(3, 2, 2)$.

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