  We study the localization and decay properties as well as the thermal
conductance of one-dimensional plasmons. Our model contains a Luttinger-liquid
part with spatially random plasmon velocity and interaction parameter as well
as a nonlinearity that is cubic in density. The scaling of the decay rate of
plasmons is obtained in several regimes. At sufficiently high frequencies, it
describes the inelastic life time of localized plasmon excitations that crosses
over to the clean result with lowering frequencies. For higher frequencies, we
analyze implications of many-body-localization effects that lead to a
suppression of the decay rate. We find that the thermal conductance depends in
a non-trivial fashion on the system size $L$. Specifically, it scales as
$L^{-1/2}$ for sufficiently short wires and crosses over to $L^{-2/3}$ scaling
for longer wires.
