Life span of small solutions to a system of wave equations

TL;DR

This paper investigates the lifespan of small solutions to a coupled wave system, demonstrating conditions for blow-up and global existence, and providing sharp estimates for the maximal existence time in certain dimensions.

Contribution

It establishes blow-up results for small data below a certain curve and proves global existence for specific cases, identifying the borderline between these behaviors.

Findings

Blow-up can occur for arbitrarily small data below a certain curve in the p-q plane.

Global existence is proven for the case n=3, with the curve acting as a borderline.

Sharp upper bounds for the maximal existence time are obtained for specific parameter sets.

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 $n \geq 2$, 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 $(3, 2, 2)$.