  Fast radio bursts (FRBs) are mysterious transient sources. If extragalactic,
as suggested by their relative large dispersion measures, their brightness
temperatures must be extremely high. Some FRB models (e.g. young pulsar model,
magnetar giant flare model, or supra-massive neutron star collapse model)
suggest that they may be associated with a synchrotron nebula. Here we study a
synchrotron-heating process by an FRB in a self-absorbed synchrotron nebula. If
the FRB frequency is below the synchrotron self-absorption frequency of the
nebula, electrons in the nebula would absorb FRB photons, leading to a harder
electron spectrum and enhanced self-absorbed synchrotron emission. In the
meantime, the FRB flux is absorbed by the nebula electrons. We calculate the
spectra of FRB-heated synchrotron nebulae, and show that the nebula spectra
would show a significant hump in several decades near the self-absorption
frequency. Identifying such a spectral feature would reveal an embedded FRB in
a synchrotron nebula.
