  We explore the inflationary dynamics leading to formation of closed domain
walls in course of evolution of an axion like particle (ALP) field whose
Peccei-Quinn-like phase transition occurred well before inflationary epoch.
Evolving after inflation, the domain walls may leave their imprint in
stochastic gravitational waves background, in the frequency range accessible
for the pulsar timing array measurements. We derive the characteristic strain
power spectrum produced by the distribution of the closed domain walls and
relate it with the recently reported NANOGrav signal excess. We found that the
slope of the frequency dependence of the characteristic strain spectrum
generated by the domain walls is very well centered inside the range of the
slopes in the signal reported by the NANOGrav. Analyzing the inflationary
dynamics of the ALP field, in consistency with the isocurvature constraint, we
revealed those combinations of the parameters where the signal from the
inflationary induced ALPs domain walls could saturate the amplitude of the
NANOGrav excess. The evolution of big enough closed domain walls may incur in
formation of wormholes with the walls escaping into baby universes. We studied
the conditions, when closed walls escaped into baby universes could leave a
detectable imprint in the stochastic gravitational waves background.
