  Axion-like particles (ALPs) are a compelling candidate for dark matter (DM).
Their production is associated with the formation of a string-wall network.
This system must annihilate, producing gravitational waves and non-relativistic
ALPs. We show that these gravitational waves, if produced at temperatures below
100 eV, could be detected by future cosmological probes for ALPs with mass from
$10^{-16}$ to $10^{6}$ eV that could constitute the entirety of the DM.
