\begin{document}$ Z_{\mathcal N} $\end{document} QCD axion and axion-like particle. A general mixing case in which the heavy and light mass eigenvalues do not necessarily have to coincide with the axion masses is considered. To form the domain walls, the axions should start to oscillate slightly before the level crossing. The domain walls must annihilate before dominating the Universe to avoid cosmological catastrophe. Then, we focus our attention on the GW emission from the domain wall annihilation and the PBH formation from the domain wall collapse. We show the predicted GW spectra with a peak frequency \begin{document}$\sim 0.2~ \rm nHz$\end{document} and peak amplitude \begin{document}$ \sim 5\times 10^{-9} $\end{document}, which can be tested by future pulsar timing array projects. In addition, during the domain wall annihilation, the closed walls could shrink to the Schwarzschild radius and collapse into the PBH. We find that PBHs in the mass range of \begin{document}$ \mathcal{O}(10^5-10^8) M_\odot $\end{document} could potentially form in this scenario and account for a small fraction (\begin{document}$ \sim 10^{-5} $\end{document}) of the cold dark matter."> Gravitational waves and primordial black holes from axion domain walls in level crossing -
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