  Recent observations in X-rays and gamma-rays of nearby FRI radio galaxies
have raised the question of the origin of the emission detected in the
termination structures of their jets. The study of these structures can give
information on the conditions for particle acceleration and radiation at the
front shocks. In addition, an evolutionary scenario can help to disentangle the
origin of the detected X-ray emission in young FRI sources, like some Gigahertz
Peaked Spectrum AGNs. This work focuses on the nature and detectability of the
radiation seen from the termination regions of evolving FRI jets. We use the
results of a relativistic, two-dimensional numerical simulation of the
propagation of an FRI jet, coupled with a radiation model, to make predictions
for the spectra and lightcurves of the thermal and non-thermal emission at
different stages of the FRI evolution. Our results show that under moderate
magnetic fields, the synchrotron radiation would be the dominant non-thermal
channel, appearing extended in radio and more compact in X-rays, with
relatively small flux variations with time. The shocked jet synchrotron
emission would dominate the X-ray band, although the shocked ISM/ICM thermal
component alone may be significant in old sources. Inverse Compton scattering
of CMB photons could yield significant fluxes in the GeV and TeV bands, with a
non-negligible X-ray contribution. The IC radiation would present a bigger
angular size in X-rays and GeV than in TeV, with fluxes increasing with time.
We conclude that the thermal and non-thermal broadband emission from the
termination regions of FRI jets could be detectable for sources located up to
distances of a few 100 Mpc.
