  We investigate the relation between the detection of the $11.3\,\mu$m PAH
feature in the nuclear ($\sim 24-230\,$pc) regions of 22 nearby Seyfert
galaxies and the properties of the cold molecular gas. For the former we use
ground-based (0.3-0.6" resolution) mid-infrared (mid-IR) spectroscopy. The cold
molecular gas is traced by ALMA and NOEMA high (0.2-1.1") angular resolution
observations of the CO(2-1) transition. Galaxies with a nuclear detection of
the $11.3\,\mu$m PAH feature contain more cold molecular gas (median $1.6\times
10^7\,M_\odot$) and have higher column densities ($N({\rm H}_2) = 2 \times
10^{23}\,{\rm cm}^{-2}$) over the regions sampled by the mid-IR slits than
those without a detection. This suggests that molecular gas plays a role in
shielding the PAH molecules in the harsh environments of Seyfert nuclei.
Choosing the PAH molecule naphthalene as an illustration, we compute its
half-life in the nuclear regions of our sample when exposed to 2.5keV hard
X-ray photons. We estimate shorter half-lives for naphthalene in nuclei without
a $11.3\,\mu$m PAH detection than in those with a detection. The Spitzer/IRS
PAH ratios on circumnuclear scales ($\sim$ 4" $\sim$ 0.25-1.3kpc) are in
between model predictions for neutral and partly ionized PAHs. However, Seyfert
galaxies in our sample with the highest nuclear H$_2$ column densities are not
generally closer to the neutral PAH tracks. This is because in the majority of
our sample galaxies, the CO(2-1) emission in the inner $\sim$ 4" is not
centrally peaked and in some galaxies traces circumnuclear sites of strong star
formation activity. Spatially resolved observations with the MIRI
medium-resolution spectrograph (MRS) on the James Webb Space Telescope will be
able to distinguish the effects of an active galactic nucleus (AGN) and star
formation on the PAH emission in nearby AGN.
