  We investigate the cosmological constraints on axion models where the domain
wall number is greater than one. In these models, multiple domain walls
attached to strings are formed, and they survive for a long time. Their
annihilation occurs due to the effects of explicit symmetry breaking term which
might be raised by Planck-scale physics. We perform three-dimensional lattice
simulations and compute the spectra of axions and gravitational waves produced
by long-lived domain walls. Using the numerical results, we estimated relic
density of axions and gravitational waves. We find that the existence of
long-lived domain walls leads to the overproduction of cold dark matter axions,
while the density of gravitational waves is too small to observe at the present
time. Combining the results with other observational constraints, we find that
the whole parameter region of models are excluded unless an unacceptable
fine-tuning exists.
